Année 2013 Thèse n° 2056
THÈSE pour le
DOCTORAT DE L’UNIVERSITÉ BORDEAUX Ecole doctorale : Sociétés, Politique et Santé Publique (SP2) Mention : Santé publique et Epidémiologie Option : Biostatistique
Présentée et soutenue publiquement Le 15 novembre 2013
Par Mélanie PRAGUE Née le 8 juin 1987 à Bordeaux
Utilisation des modèles dynamiques pour l'optimisation des traitements des patients infectés par le VIH Use of dynamical models for treatment optimization in HIV infected patients
Membres du Jury Daniela DE ANGELIS, Programme Leader, MRC Biostatistics, Cambridge ... Rapporteur France MENTRE, PU-PH Inserm U738, Université Paris Diderot ................... Rapporteur Robin CALLARD, Professor, University College London ............................... Examinateur Geneviève CHENE, PU-PH, Inserm U897, Bordeaux ...................................... Examinateur Daniel COMMENGES, DR, Inserm U897, Bordeaux....................................... Directeur de thèse Rodolphe THIEBAUT, PU-PH, Inserm U897, Bordeaux ................................. Membre invité Victor DE GRUTTOLA, Professor, Harvard School Public Health, Boston .... Membre invité
Remerciements
3
Remerciements Mes premi`eres pens´ees vont `a mon directeur de th`ese Daniel Commenges sans qui aucune de ces lignes n’aurait pu ˆetre ´ecrite. Merci de m’avoir guid´ee et de m’avoir consacr´e autant de temps. J’affectionne particuli`erement la visite impr´evue (pas si impr´evue) de 9h/10h du matin o` u nous pouvions ´echanger `a bˆatons rompus sur les dossiers en cours. Merci d’avoir partag´e avec moi vos connaissances, vos points de vue sur les m´ethodes statistiques et vos id´ees de recherche ; merci de m’avoir oblig´e a` savoir reconnaˆıtre et prononcer les lettres grecques ; merci pour vos nombreuses relectures d’articles et de ce manuscrit o` u mˆeme moi j’avais parfois l’impression de moins les avoir lus que vous. Finalement, merci pour vos conseils sur la r´edaction. Vous m’avez appris qu’un mot important peut ˆetre r´ep´et´e dans un texte scientifique, je n’ai donc pas peur de vous dire encore une fois merci. J’exprime tout aussi vivement la plus grande reconnaissance a` mon codirecteur de th`ese Rodolphe Thi´ ebaut. Malgr´e tes journ´ees de 42h et la pile de mail quotidienne que tu dois traiter, tu as toujours r´epondu efficacement et dans les d´elais `a la moindre de mes questions. Merci de m’avoir ´eveill´ee aux probl´ematiques m´edicales pour lesquelles je partais du niveau z´ero (peut-ˆetre mˆeme -10). J’ai particuli`erement appr´eci´e les d´ejeuners impr´evus sur le pouce o` u l’expression parler de choses importantes entre la poire et le dessert prend tout son sens. Merci de m’avoir donn´e de nombreux conseils avis´es et ouvert des opportunit´es. I would like to thank my two reviewers Daniela de Angelis et France Mentr´ e who gave their time to read this thesis. Daniela, I am very admirative of your knowledge in biostatistics. France, j’ai lu de nombreux de vos travaux qui m’ont beaucoup appris durant cette th`ese, merci. I sincerely thank Robin Callard and Genevi` eve Chˆ ene who also accepted to be part of the jury and to review my works. Robin, I am very glad that you can bring some immunological lights to this works. Genevi`eve, merci pour votre aide dans l’obtention des autorisations d’usage des jeux de donn´ees durant ces 3 ans. All of you, I am very grateful of your presence. My thoughts also go to Victor de Gruttola who, from the other side of the Atlantic, will attend my defense on Skype. Thank you for hiring me as a postdoc, I am very excited about beginning and I hope to meet your expectations. Je tiens aussi a` remercier les directeurs pass´es et actuels de l’ISPED et de l’unit´e Inserm U897 : Roger Salamon, Rachid Salmi et Christophe Tzourio. Je pense aussi a` l’ensemble du personnel du laboratoire pour l’esprit familial qui r`egne dans ces locaux, qui s’illustre bien dans la “traditionnelle journ´ee `a Carcan” avec huˆıtres et grillades. J’aimerais avoir la chance de pouvoir retravailler un jour dans cette structure. Je voudrais remercier particuli`erement R´ ejane pour toute son aide administrative. C’est toujours un plaisir de passer dans ton bureau, que ce soit pour organiser des d´eplacements ou pour prendre des fournitures. D´esormais, le bureau 45 pourra se targuer d’avoir une grosse agrafeuse comme Daniel. Merci a` Christian Weller, Coralie Thore et Evelyne Mouillet qui trouvent les articles plus vite que leur ombre et m’ont fait gagner un temps pr´ecieux. J’adresse une pens´ee aux informaticiens du CREDIM.
Remerciements
4
Je remercie donc toute l’´ equipe p´ edagogique qui a su me faire une place. Mes pens´ees vont en particulier a` Nancy pour sa bonne humeur mˆeme lorsque l’on veut changer un horaire dans le planning, Pierre qui sait (tr`es bien) conseiller tout en rˆalant (beaucoup), Alioum et Marthe-Aline pour leurs conseils. J’esp`ere avoir ´et´e une charg´ee de TD guillerette pour Marie Chavent, Marta, Fran¸cois Caron et Robin. Aussi ´etrange que cela puisse paraˆıtre, j’ai particuli`erement aim´e faire des croix, des traits et des ronds avec Fleur et Val´ erie. J’en profite aussi pour remercier mes anciens professeurs de pr´epa et de l’ENSAI pour m’avoir montr´e ce qu’il fallait faire (et ne pas faire). Merci aux ´ etudiants de m’avoir ´ecout´e d’une oreille bienveillante. Merci `a tous les permanents, doctorants, ing´enieurs et stagiaires de l’´ equipe biostatistique et de l’ISPED en g´en´eral pour les bons moments pass´es ensemble. J’aime la bonne humeur des RLP (r´eunion d’´equipe dinatoire trimestrielle) qui sont toujours l’occasion d’´echanger et de s’´echanger la derni`ere recette culinaire `a la mode. D´esol´ee pour les nombreuses “briques” au chocolat que j’ai malencontreusement pu apporter. Merci a` C´ ecile pour ses conseils jeux de soci´et´e. Un remerciement particulier aux colocataires successifs du Bureau 45 : Linda pour avoir partag´e ton exp´erience de jeune phD, Yassin pour nous avoir montr´e l’exemple l’an dernier, Paul pour l’achat d’une machine `a caf´e rutilante qui fait la jalousie de tout le couloir violet, Julie pour nos moments fille-fille a` mettre Ycare a` tue-tˆete dans le bureau, Audrey pour les potins du net venus de nul part qui arrivent toujours au bon moment, Lingling pour tes anecdotes chinoises et Loic pour ta bonne humeur et ton accent. J’ai aussi une pens´ee pour Alexandre chez qui je prends plaisir a` taper quand le bureau 1 est ferm´e. Je pense aussi aux filles que Paul appelait les paquidettes qui se reconnaitront et en particulier Fanny qui en plus de me sauver lorsque j’oublie mes clefs de bureau (soit 4 jours sur 5) est une bonne copine de bavardage. Loin des yeux, mais pas loin du cœur, dans les hauteurs du 3i`eme , je remercie aussi Boris sans qui les pauses caf´e(s) ne seraient pas si drˆoles. J’esp`ere que tu resteras un ami. Et puis les autres : J´ er´ emie×2, C´ elia, Mb´ ery, Ana, Sophie, M´ elanie, Matthieu et les autres (pas dans l’ordre). . . Bordelaise d’origine, j’ai pris un plaisir immense `a cˆotoyer de nouveau mes amis. Merci d’avoir su me changer les id´ees, me divertir et me d´ebaucher quand cette th`ese me sortait par les yeux. Vous ˆetes tous de belles personnes que je veux revoir a` Boston puis o` u que je sois. Ma Nono, tu es une fille extraordinaire, j’aime nos soir´ees `a dur´ees ind´efinies quand tu viens a` la maison. Grˆace `a toi et a` Momo j’ai pu booster ma positive attitude (pieds sur le bureau, mains sur les hanches !). Jb ma famille choisie, nous serons toujours cousins lorsque cela nous arrange. Talex et Tib mes deux compagnons de trappes, que ferions-nous sans Appolo ? Shin (Cendrillon) toujours prˆet a` tout, enfin, avant le coup de pompe de minuit. Les amis d’amis qui sont maintenant des amis Noel. Les +1 Remi et Marion×2 bout-en-trains en soir´ee. Florence la voisine d’enfance, bientˆot voisine de ville. Les week-ends langues de vip`ere comp´et’ de patin a` rigoler avec Morgan et Amande. La liste des personnes qui ont compt´e pour moi est ensuite longue mˆeme si certains font parti des intermittents du spectacle : merci a` Vathana, Ian, Mathias, Elsa, Morina, Bastien, Pauline, Romain, Benjamin et les autres (pas dans l’ordre). . . J’ai une pens´ee tendre pour ma famille qui, je le sais, est tr`es heureuse pour moi. Les premi`eres ann´ees de ce cette th`ese pass´ees en voisins avec mon papa nous ont permis de profiter l’un de l’autre, je n’oublierai jamais ces bons moments. Je ne dirai jamais assez
Remerciements
5
merci a` ma maman pour son amour et son soutien inconditionnel. Tu es toujours l`a pour me conseiller avec tendresse et vision ´eclair´ee. J’aime que l’on soit aussi proche et pouvoir tout te dire autour d’un caf´e ou d’un verre de vin. Merci a` Mamimi fournisseuse officielle de Guerlain et cuisini`ere hors pair qui m’a toujours r´ep´et´e que tant que je travaillais a` l’´ecole elle s’occupait de tout. J’ai de la chance de t’avoir, mˆeme si des-fois, enfin parfois j’oublie d’appeler. Je voulais vous dire que je vous aime tr`es fort. Je pense aussi `a mamie Marie, papi Jean et Papipi qui auraient ´et´e si fiers de voir ce manuscrit. Je pense aussi a` ma voisine Jacqueline qui m’a vu grandir. Merci a` Lisette qui reste notre petit ange gardien du jeudi et permet de mieux me concentrer sur le travail. Pour finir, je pense a` ma belle famille, en particulier H´ el` ene, Krister et Norman qui m’ont toujours accueillie comme leur propre enfant et m’ont beaucoup aid´e `a progresser en anglais. Pour finir, on laisse souvent le meilleur pour la fin, j’ai une pens´ee amoureuse pour Vincent. Tu sais me supporter apr`es des journ´ees de boulot trop longues o` u d´esormais tu m’envoies faire du sport ou me coucher pour ne pas avoir a` subir ma mauvaise humeur. Tu es mon ange depuis presque 9 ans de bonheur et avec qui je m’apprˆete `a partir a` l’aventure. J’esp`ere que tout le chemin que nous avons parcouru nous a rendu plus forts. Merci de m’aimer comme tu le fais tous les jours. L’aventure de la th`ese s’arrˆete ici. Ces derniers mots ferment un chapitre de trois ans de ma vie pass´ee a` ´etudier, travailler, mais aussi m’amuser, profiter de la vie et rire aupr`es de gens formidables professionnellement et personnellement. D`es demain, une nouvelle aventure commencera, mais avant cela je profite de ces derni`eres lignes pour remercier ceux que j’ai oubli´ e et qui n’en sont pas moins importants. Cette th`ese commence ici pour les lecteurs courageux que je tiens, pour finir, `a remercier.
Productions scientifiques li´ees a` la th`ese
7
Productions scientifiques li´ ees ` a la th` ese Articles li´ es ` a la th` ese Accept´ es – Treatment monitoring of HIV infected patients based on mechanistic models. Prague M., Commenges D., Drylewicz J. et Thi´ebaut R. Biometrics 68(3) :902-911 (sept. 2012) – Dynamical models of biomarkers and clinical progression for personalized medicine : the HIV context. Prague M., Commenges D. et Thi´ebaut R. Advanced Drug Delivery Reviews 65(7) :954-965 (Juin 2013) – NIMROD : A Program for Inference via Normal Approximation of the Posterior in Models with Random effects based on Ordinary Differential Equations. Prague M., Commenges D., Guedj J., Drylewicz J. et Thi´ebaut R. Computer Methods and Programs in Biomedicine 111(2) :447-458 (Aoˆ ut 2013) Soumis – From descriptive causal models to mechanistic models : effect of HAART on CD4 counts and viral load. Commenges D., Prague M., G´egout-Petit A. et Thi´ebaut R. – Quantifying and Predicting the Effect of exogenous Interleukin-7 on CD4+T cells in HIV-1 Infection. Thi´ebaut R., Drylewicz J., Prague M., Lacabaratz C., Beq S., Crough T., Sekaly R.P., Lederman M.M., Sereti I., Commenges D. et Levy Y.
Communications internationales dans un congr` es avec comit´ e de lecture (∗ orateur) Communications orales – From in vivo to in vitro quantification of antiretroviral drugs effects based on dynamical models of HIV. Prague∗ M., Commenges D. et Thi´ebaut R. HIV dynamics and evolution Utrecht - Pays-bas (Mai 2013) – Toward information synthesis with mechanistic models of HIV dynamics. Prague∗ M., Commenges D. et Thi´ebaut R. International society for Clinical Biostatistics Bergen - Norv`ege (Aoˆ ut 2012) – Bayesian MAP Estimation in Models with Random effects based on Ordinary Differential Equations applied to Treatment Monitoring in HIV. Prague∗ M. et Commenges D. Eurandom Workshop PEDSII Eindhoven - Pays-bas (juin 2012) – Treatment monitoring of HIV infected patients : optimal drug dose control. Prague∗ M., Commenges D., Drylewicz J. et Thi´ebaut R. International Biometric Society Channel Network Bordeaux - France (Avril 2011) – The stochastic system approach for causal modeling. Commenges∗ D., Prague M., G´egout-Petit A. et Thi´ebaut R. International Biometric Society Channel Network Glasgow - UK (Juillet 2013) – Using mechanistic models to design and analyze immune intervention trials : the example of phase I/II trials of IL-7 therapy in HIV-infected individuals. Thi´ebaut∗
Productions scientifiques li´ees a` la th`ese
8
R., Prague M., Drylewicz J., Jarne A., Croughs T., Levy Y. et Commenges D. International society for Clinical Biostatistics Munich - Allemagne (Aoˆ ut 2013) Communications affich´ ees – Modeling the effect of Interleukin-7 on CD4+ T cells. Jarne A., Prague M., Commenges D. et Thi´ebaut R. HIV dynamics and evolution Utrecht - Pays-bas (Mai 2013)
Communications nationales dans un congr` es avec comit´ e de lecture Communications orales – Inf´erence par Approximation Normale de l’a posteriori dans les mod`eles dynamiques `a effets mixtes. Prague∗ M., Commenges D., Guedj J., Drylewicz J. et Thi´ebaut R. Journ´ ees de la SFDS Toulouse - France (Juin 2013) – Illustration of information synthesis of clinical trials with mechanistic models of HIV dynamics. Prague∗ M., Commenges D. et Thi´ebaut R. GDR Statistiques et Sant´ e Rennes - France (Sept. 2012) – R Package “marqLevAlg” : the Marcquardt-Levenberg algorithm an alternative to “optimx” in minimization problems. Prague∗ M., Diakit´e A. et Commenges D. Rencontres R Bordeaux - France (Juill. 2012) – Estimation in Differential Equations and prediction of treatment response in HIV e Paris infected patients. Prague∗ M. et Commenges D. GDR Statistiques et Sant´ - France (Mai 2011) Communications affich´ ees – HIV and immune system dynamics modeling and drug dose reduction. Prague∗ M., Commenges D. et Thi´ebaut R. Universit´ e des jeunes chercheurs sur le VIH Sidaction Carry-le-rouet - France (Sept 2011)
S´ eminaires / Communications orales invit´ ees – Overview on dynamical models : treatment optimization in HIV infected patients. Prague M., Commenges D. et Thi´ebaut R. John Hopkins University lab meeting. Baltimore - US (Oct. 2013) – Individual predictions using mechanistic models Prague M., Commenges D. et Thi´ebaut R. Workshop Dynamic predictions for repeated markers and repeated events : models and validation. Bordeaux - France (Oct. 2013) – Estimation and treatment optimization in HIV infected patients. Prague M., et Commenges D. Department of methodological statistics Universit´e de Li`ege - Belgique (Dec. 2012)
Table des mati` eres Productions scientifiques Introduction
7 14
PARTIE I : Contexte biologique et th´ erapeutique de l’infection ` a VIH 17 1 L’infection ` a VIH
19
1.1
Donn´ees ´epid´emiologiques actuelles . . . . . . . . . . . . . . . . . . . . . . 20
1.2
Dynamique VIH-syst`eme immunitaire . . . . . . . . . . . . . . . . . . . . . 20
1.3
1.2.1
Structure du VIH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.2
Une organisation en quasi-esp`eces . . . . . . . . . . . . . . . . . . . 21
1.2.3
Cycle de r´eplication du VIH . . . . . . . . . . . . . . . . . . . . . . 21
1.2.4
R´eponse du syst`eme immunitaire au VIH . . . . . . . . . . . . . . . 22
Biomarqueurs et crit`eres de jugement cliniques de l’avanc´ee de l’infection . 24 1.3.1
Charge virale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.2
Num´eration des CD4 . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3.3
Histoire naturelle de la maladie . . . . . . . . . . . . . . . . . . . . 25
1.3.4
Les populations cellulaires d’int´erˆet dans cette th`ese . . . . . . . . . 27
2 Les strat´ egies de traitement dans les pays du Nord 2.1
29
Les th´erapies antir´etrovirales hautement actives . . . . . . . . . . . . . . . 30
Table des mati`eres
2.2
10
2.1.1
Les mol´ecules antir´etrovirales . . . . . . . . . . . . . . . . . . . . . 30
2.1.2
Les th´erapies par combinaisons d’antir´etroviraux (cART)
2.1.3
Les ´echappements th´erapeutiques . . . . . . . . . . . . . . . . . . . 34
2.1.4
Les effets secondaires . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.5
L’adh´erence aux traitements . . . . . . . . . . . . . . . . . . . . . . 35
. . . . . 32
M´edecine personnalis´ee : vers une individualisation des traitements . . . . 36 2.2.1
Individualisation du choix du traitement . . . . . . . . . . . . . . . 37
2.2.2
Surveillance des r´eponses aux traitements et adaptations . . . . . . 40
PARTIE II : Aspects math´ ematiques, statistiques et num´ eriques des mod` eles dynamiques 43 3 Les mod` eles d’analyse de la dynamique des biomarqueurs 3.1
3.2
45
Les mod`eles descriptifs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1.1
Mod`eles multivari´es . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.2
Les mod`eles causaux . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1.2.1
Concept de causalit´e . . . . . . . . . . . . . . . . . . . . . 47
3.1.2.2
G-calculs . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.2.3
Mod`eles structuraux marginaux . . . . . . . . . . . . . . . 49
Les mod`eles dynamiques m´ecanistes explicatifs . . . . . . . . . . . . . . . . 52 3.2.1
3.2.2
3.2.3
Vocabulaire et Notations
. . . . . . . . . . . . . . . . . . . . . . . 52
3.2.1.1
Diff´erents types de mod`eles dynamiques . . . . . . . . . . 52
3.2.1.2
Les syst`emes d’´equations diff´erentielles ordinaires (ODE) . 54
Algorithmes d’estimation en mod`eles NLME-ODE . . . . . . . . . . 56 3.2.2.1
M´ethodes d’inf´erence disponibles . . . . . . . . . . . . . . 56
3.2.2.2
Validit´e de l’approximation normale de l’a posteriori . . . 58
3.2.2.3
Inf´erence par approximation normale de l’a posteriori . . . 59
Les mod`eles dynamiques m´ecanistes pour le VIH . . . . . . . . . . 64 3.2.3.1
Mod`ele simple . . . . . . . . . . . . . . . . . . . . . . . . 64
Table des mati`eres
11
3.2.3.2
Complexification des m´ecanismes . . . . . . . . . . . . . . 65
3.2.3.3
Mod`eles simplifi´es : parcimonie et identifiabilit´e. . . . . . . 68
3.2.3.4
Pistes de mod´elisation des effets traitement . . . . . . . . 69
4 NIMROD pour l’inf´ erence en mod` ele dynamique 4.1
71
Le programme NIMROD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.1.1
Am´elioration de la convergence . . . . . . . . . . . . . . . . . . . . 72
4.1.2
Am´elioration des temps de calcul . . . . . . . . . . . . . . . . . . . 73
4.1.3
Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.1.4
Description et application `a la pharmacocin´etique : « NIMROD : A program for inference via a normal approximation of the posterior in models with random effects based on ordinary differential equations » dans Computer methods and Programs in Biomedecine , Volume 111, Issue 2, aoˆ ut 2013, Pages 447-458 . . . . . . . . . . . . . . . . 77
4.2
Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5 Causalit´ e et comparaison d’approches 5.1
93
Comparaison des mod`eles descriptifs et dynamiques . . . . . . . . . . . . . 94 5.1.1
Probl´ematique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.1.2
S´election des donn´ees . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.1.3
Aspects techniques et limites . . . . . . . . . . . . . . . . . . . . . . 96
5.1.4
Un continuum entre mod`eles descriptifs et explicatifs : « From the descriptive to mechanistic dynamical models : effect of HAART on CD4 count » . . . . . . . . . . . . . . . . 97
5.2
Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Table des mati`eres
12
PARTIE III : Vers une m´ edecine personnalis´ ee dans le domaine du VIH 117 6 Optimisation de la dose de traitement 6.1
Les probl´ematiques li´ees `a l’optimisation des traitements . . . . . . . . . . 120 6.1.1
6.1.2 6.2
119
´ Evaluation des qualit´es des mod`eles . . . . . . . . . . . . . . . . . . 120 6.1.1.1
Qualit´e d’ajustement . . . . . . . . . . . . . . . . . . . . . 120
6.1.1.2
Param`etres individuels pour la pr´ediction . . . . . . . . . 121
6.1.1.3
Qualit´e de pr´ediction . . . . . . . . . . . . . . . . . . . . . 123
Optimisation et th´eorie de la commande . . . . . . . . . . . . . . . 127
Adaptation de la dose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.2.1
D´efinition de la fonction objectif . . . . . . . . . . . . . . . . . . . . 127
6.2.2
Exploitation des propri´et´es dynamiques . . . . . . . . . . . . . . . . 128
6.2.3
Strat´egie adaptative d’optimisation de la dose de traitement : « Treatment monitoring of HIV infected patients based on mechanistic models » dans Biometrics , Volume 68, Issue 3, septembre 2012, Pages 902-911 . . . . . . . . . . . . . . . 129
6.3
Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7 Mod´ elisation des effets des cART, vers une optimisation 7.1
145
Les facteurs d’int´erˆet dans les mod`eles dynamiques appliqu´es au VIH : « Dynamical models fo biomarkers and clinical progression for personalized medicine : The HIV context » dans Advanced Drug Delivery Reviews , Volume 65, Juin 2013, Pages 954-965 . . . . 146
7.2
De l’in vitro a` l’in vivo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.2.1
L’information in vitro
. . . . . . . . . . . . . . . . . . . . . . . . . 159
7.2.2
De l’in vitro a` l’estimation d’effets traitement in vivo : « From in vitro to in vivo quantification of antiretroviral drugs effects based on dynamical models of HIV »
. . . . . 160
Table des mati`eres
Conclusion et Perspectives G´ en´ erales
13
173
Bibliographie
177
Annexes
201
A Mat´ eriel suppl´ ementaire Comp. Meth. and Prog. in Biomedecine
203
B Mat´ eriel suppl´ ementaire Biometrics
209
Introduction Dans ces travaux de th`ese, nous proposons des approches statistiques pour mettre en place des strat´egies d’optimisation de traitements qui pourraient par la suite ˆetre reprises dans des essais cliniques. Pour cela nous nous int´eressons a` la mod´elisation de l’interaction entre le syst`eme immunitaire et le VIH. Nous consid´erons et comparons diff´erentes techniques propos´ees dans la litt´erature pour finalement pr´ef´erer les mod`eles m´ecanistes dynamiques explicatifs. Cela demande de r´esoudre des probl`emes d’inf´erence dans ces mod`eles bas´es sur des ´equations diff´erentielles ordinaires avec des effets mixtes sur les param`etres. Nous d´eveloppons et utilisons des m´ethodes et des outils statistiques d’estimation. La mise en œuvre de ces m´ethodes permet ensuite une recherche de la meilleure mod´elisation des interactions entre le VIH et les diff´erentes populations de cellules immunitaires avec et sans traitement. Il s’agit de retenir un mod`ele coh´erent qui soit a` la fois capable d’ajuster et de pr´edire des donn´ees, puis de l’utiliser afin de proposer et de valider des strat´egies d’optimisation des traitements. Nous travaillons en particulier sur le concept de pr´edictions bay´esiennes dans les mod`eles dynamiques et consid´erons l’´evaluation de leur cr´edibilit´e et de leur incertitude. Nous profitons de la pr´esence d’effets al´eatoires pour a` la fois utiliser les donn´ees de l’ensemble de la population et effectuer des pr´edictions individuelles rendant possible l’individualisation des traitements. Par ailleurs, l’information sur le patient qui augmente au cours du temps, permet par une approche bay´esienne une mise `a jour adaptative des traitements. Comme application et illustration de l’usage des outils statistiques et des mod´elisations d´evelopp´ees dans cette th`ese, nous proposons donc des strat´egies d’individualisation adaptatives du traitement chez les patients infect´es par le VIH.
Introduction
16
La premi`ere partie pr´esente les enjeux et le contexte de l’infection a` VIH. Le chapitre 1 introduit quelques donn´ees ´epid´emiologiques de l’infection afin de situer la probl´ematique dans un cadre de sant´e publique puis ´enonce les bases biologiques et virologiques de l’infection sans traitement. Le chapitre 2 s’int´eresse aux traitements antir´etroviraux, pour contrˆoler l’infection `a VIH qui dans l’´etat actuel des connaissances ne peut pas ˆetre ´eradiqu´ee. Une perspective individuelle est adopt´ee, nous mettons en valeur pourquoi et comment la m´edecine personnalis´ee utilis´ee pour de nombreuses autres pathologies pourrait ˆetre appliqu´ee au VIH. La deuxi`eme partie met en avant les solutions math´ematiques, statistiques et num´eriques qui existent pour mod´eliser l’interaction entre VIH et syst`eme immunitaire. Le chapitre 3 pr´esente les diff´erents types de mod`eles pour l’analyse des biomarqueurs. Le chapitre 4 traite de l’inf´erence dans les mod`eles dynamiques bas´es sur des ´equations diff´erentielles ordinaires avec effets mixtes sur les param`etres et pr´esente un programme d’estimation appel´e NIMROD. Le chapitre 5 montre la sup´eriorit´e d’une approche dynamique compar´ee a` une analyse longitudinale descriptive des biomarqueurs pour analyser l’effet d’un traitement lorsque la mise sous traitement est conditionn´ee par les variables observ´ees et les variables d’int´erˆet. La troisi`eme partie s’int´eresse a` l’utilisation de ce type d’approches pour une prescription individualis´ee des traitements antir´etroviraux chez les patients infect´es par le VIH. Le chapitre 6 introduit des m´ethodes de pr´edictions bay´esiennes dans les mod`eles dynamiques et propose des solutions pour ´evaluer leurs qualit´es. Par la suite, ce chapitre pr´esente une strat´egie adaptative d’individualisation et d’optimisation de la dose de traitement bas´ee sur l’utilisation d’un indicateur d’´equilibre en mod`ele ODE qui identifie les ´etats de contrˆole de l’infection. Le chapitre 7 fait une revue de litt´erature des facteurs d’int´erˆet pour la m´edecine personnalis´ee dans le domaine du VIH. Pour conclure, ce chapitre pr´esente nos travaux actuels sur la prise en compte de r´esultats pharmacologiques in vitro, la mod´elisation des r´eservoirs latents, l’inclusion des mutations, l’´etude de l’adh´erence et des propositions de strat´egies d’individualisation du choix du traitement.
PARTIE I : Contexte biologique et th´ erapeutique de l’infection ` a VIH
Chapitre 1 L’infection ` a VIH R´ esum´ e : Dans ce chapitre, nous donnons le contexte li´e a` l’application biom´edicale de cette th`ese. Apr`es avoir expos´e en quoi l’infection `a VIH reste encore un enjeu de sant´e publique majeur dans le monde entier en ce d´ebut de XXIi`eme si`ecle, nous pr´esentons les aspects biologiques et immunologiques de cette infection. Au travers de ces explications, nous mettons en avant les biomarqueurs sp´ecifiques du VIH qui vont ensuite constituer des variables a` expliquer et pr´edire dans des mod`eles de dynamique de populations de cellules. Mots-cl´ es : biomarqueur ; charge virale ; cycle de r´eplication ; incidence ; lymphocyte T-CD4 ; mutation ; primo-infection ; quasi-esp`ece ; SIDA ; syst`eme immunitaire ; VIH.
Abstract : In this chapter, we present the context of the biomedical application of this PhD work. We show why HIV is still, in the 21th century, an important public health problem in the world. We present biological and immunological aspects of this infection. We explicit specific biomarkers for HIV that will afterward be variables to explain and to predict in populations of cells dynamics models. Key words : AIDS ; acute-infection ; biomarker ; HIV ; immune system ; incidence ; lymphocyte T-CD4 ; mutation ; quasispecies ; replication cycle ; viral load.
Chapitre 1 : L’infection a` VIH
20
Le virus de l’immunod´eficience humaine (VIH) a ´et´e d´ecouvert au d´ebut des ann´ees 80 [Barr´e-Sinoussi et al., 1983, Gallo et al., 1983] deux ans apr`es le premier cas clinique de syndrome d’immunod´eficience acquise (SIDA) [Stahl et al., 1982]. L’´epid´emie est class´ee en pand´emie en 1987 et est devenue un enjeu majeur de sant´e publique [Mann et al., 1988].
1.1
Donn´ ees ´ epid´ emiologiques actuelles
En 1995, le programme ONUSIDA de l’ONU destin´e a` coordonner l’action des agences sp´ecialis´ees pour lutter contre la pand´emie de VIH/SIDA est cr´e´e. En 2013, la pr´evalence [UNAIDS, 2013] ´etablie sur 186 pays (parmi les 193 pays membres de l’ONU) est de 34 millions [31.4 - 35.9 millions]. Malgr´e un nombre de personnes infect´ees environ 20% inf´erieur `a celui de 2001, l’incidence annuelle reste de 2.5 millions [2.2 - 2.8 millions]. Sur cette mˆeme p´eriode, l’intensification de l’usage et l’am´elioration des traitements ont permis de r´eduire de plus de 24% le nombre de d´ec`es dus au SIDA, amenant ce nombre `a 1.7 millions [1.5 - 1.9 millions] par an.
1.2 1.2.1
Dynamique VIH-syst` eme immunitaire Structure du VIH
Le VIH est un r´etrovirus appartenant a` la famille des lentivirus dont la pathologie est caract´eris´ee par une longue p´eriode entre l’infection et les manifestations cliniques. Deux types viraux et de nombreux sous-types, bas´es sur des diff´erences g´en´etiques, ont ´et´e identifi´es : le VIH-1 pr´esent dans le monde entier et le VIH-2 r´epandu principalement en Afrique de l’ouest. Le VIH est d’un aspect sph´erique (voir Figure 1.1), pour un diam`etre moyen de 145 nanom`etres. Une capside, comprenant un brin simple d’ARN en double exemplaires et des enzymes agissant dans le cycle de r´eplication du virus (voir Section 1.2.3), est contenue dans l’enveloppe virale jalonn´ee de glycoprot´eines : la gp41 et la gp120.
1.2 : Dynamique VIH-syst`eme immunitaire
1.2.2
21
Une organisation en quasi-esp` eces
Des variabilit´es du mat´eriel g´en´etique peuvent ˆetre transmises ou g´en´er´ees au sein d’un hˆote [Wittkop et al., 2011]. En effet, une grande diversit´e g´en´etique du virus existe a` l’´etat naturel, car le VIH, comme beaucoup de lentivirus, conserve ses capacit´es de r´eplication virale avec une grande tol´erance aux variations de l’ARN [Ndung’u and Weiss, 2012, Taylor et al., 2008]. Au regard du mat´eriel viral, le virus est pr´esent sous forme de quasiesp`ece au sein d’un hˆote, il exprime plusieurs g´enotypes au sein d’un mˆeme type et soustype viral [Goodenow et al., 1989]. Par pression de s´election, lors par exemple de prise de traitements, certains g´enotypes peuvent ˆetre amen´es a` devenir presque inexistants au profit d’autres g´enotypes plus r´esistants : on parle alors de mutations de r´esistance. Ce nouveau g´enotype viral peut alors croˆıtre aux d´epens des pr´ec´edents et r´eg´en´erer de nouveau, par mutations, une quasi-esp`ece de virus.
1.2.3
Cycle de r´ eplication du VIH
Le syst`eme immunitaire constitue une ligne de d´efense contre le VIH, mais il est aussi une cible. En effet, le virus ne disposant pas de tous les ´el´ements n´ecessaires a` sa r´eplication il profite de la machinerie cellulaire pour se multiplier. De plus, la structure de l’enveloppe conf`ere au VIH des propri´et´es de r´eplication, car elle lui permet d’infecter facilement des cellules immunitaires pr´esentant a` leur surface des r´ecepteurs CD4 (principalement les lymphocytes T-CD4+, cette question sera abord´ee dans la Section 1.2.4). L’attachement du virus (Figure 1.1.1) `a une cellule cible se fait par la fusion de la couche lipidique du virus a` la membrane plasmatique de la cellule, men´ee par les glycoprot´eines. Il y a ensuite relˆachement du mat´eriel viral dans le cytoplasme de la cellule [Clapham and McKnight, 2002] (Figure 1.1.2). Une ´etape de d´ecapsidation permet ensuite de verser le mat´eriel viral a` l’int´erieur de la cellule puis la transcriptase inverse (Figure 1.1.3) produit de l’ADN viral grˆace a` l’ARN viral. Le g´enome viral peut alors ˆetre achemin´e vers le noyau, quel que soit l’´etat d’activation de la cellule [Nisole and Sa¨ıb, 2004]. L’ADN viral est ensuite int´egr´e a` l’ADN chromosomique de la cellule infect´ee grˆace `a l’int´egrase (Figure 1.1.4). L’ARN
Chapitre 1 : L’infection a` VIH
22
messager (ARNm) est ensuite produit et export´e du noyau. Une fois sorti par les pores nucl´eaires, l’ARNm est lu par les ribosomes de la cellule qui assemblent des acides amin´es afin de constituer les prot´eines virales (Figure 1.1.5). C’est en particulier lors de ces deux ´etapes que se cr´ee la diversit´e g´en´etique par des erreurs de lecture de codons et de recopies. Ce ph´enom`ene est tr`es fr´equent, car la dynamique de renouvellement du virus est tr`es rapide, avec en moyenne plus de 0.68 milliard [0.1 - 2.1 milliards] de virions journaliers cr´e´es [Ho et al., 1995]. Cet ordre de grandeur a ´et´e retrouv´e par d’autres auteurs [Coffin et al., 1995, Herz et al., 1996, Nowak et al., 1995, Perelson et al., 1996]. Apr`es cet assemblage, le nouveau mat´eriel viral sort de la cellule infect´ee en arrachant une partie de la membrane cellulaire, ce qui lui conf`ere les propri´et´es d’enveloppe permettant sa r´eplication et donne naissance `a un nouveau virus infectieux apr`es maturation (Figure 1.1.6 et Figure 1.1.7).
1.2.4
R´ eponse du syst` eme immunitaire au VIH
De nombreuses populations de cellules participent `a la r´eponse immunitaire contre le VIH mais seulement une partie de ces cellules immunitaires permettent au virus de se r´epliquer et de persister au sein du corps de l’hˆote. Les lymphocytes T-CD4+, que nous appellerons par souci de simplicit´e CD4 par la suite, prolif`erent et activent d’autres cellules immunitaires qui vont agir directement sur le virus. Les lymphocytes B produisent des anticorps permettant la lutte anti-VIH par neutralisation du virus circulant puis son ´elimination par les monocytes macrophages. Les lymphocytes T-CD8+, quant `a eux, s’attaquent directement aux cellules infect´ees. Ainsi, un sch´ema proie-pr´edateur s’instaure entre les populations cellulaires. Les CD4 constituent le plus important groupe de cellules cibles. En effet, elles pr´esentent des r´ecepteurs capables de reconnaitre sp´ecifiquement les glycoprot´eines a` la surface du virus favorisant le sch´ema de r´eplication d´ecrit Section 1.2.3. Les CD4 activ´es sont d’autant plus susceptibles d’ˆetre infect´ees. Lors de leur d´esactivation, ces cellules migrent vers des compartiments tissulaires (organes lympho¨ıdes, cerveau . . .) formant des r´eservoirs a` VIH. Certains de ces CD4 sont infect´es de fa¸con latente, avec une
1.2 : Dynamique VIH-syst`eme immunitaire
23
Figure 1.1 : Cycle de r´eplication du VIH (adapt´e de [NIAID, 2013]) demi-vie longue [Finzi et al., 1999], et persistent dans ces r´eservoirs [Siliciano et al., 2003, Chun et al., 1998, Ribeiro et al., 2002, Grossman et al., 2002]. C’est entre autres l`a o` u d’autres types de cellules (macrophages, cellules dendritiques . . .) peuvent ˆetre infect´ees par ces CD4 infect´es latents. Ces cellules ne peuvent pas ˆetre cibl´ees par le syst`eme immunitaire, car le VIH est int´egr´e au noyau et donc ind´etectable. C’est le concept de r´eservoir latent du VIH qui constitue une probl´ematique de recherche actuelle (voir Section 7.2.2). Cependant, les m´ecanismes sont plus complexes et dans les organes lympho¨ıdes par exemple, les CD4 infect´es, les monocytes macrophages et les cellules dendritiques
Chapitre 1 : L’infection a` VIH
24
peuvent s’infecter par contacts synaptiques ce qui pourrait avoir un impact significatif sur la persistance du virus [Felts et al., 2010]. Nous n’utiliserons dans ces travaux de th`ese que les donn´ees mesur´ees dans le compartiment sanguin en faisant l’hypoth`ese qu’il s’agit d’un bon reflet de la dynamique globale.
1.3
Biomarqueurs et crit` eres de jugement cliniques de l’avanc´ ee de l’infection
Les infections opportunistes ´etant de plus en plus rares chez les patients sous traitement, le suivi des patients se fait essentiellement par des mesures biologiques. Il existe une large palette de biomarqueurs dont une revue de litt´erature peut ˆetre trouv´ee dans [Lange et al., 1989, Taylor et al., 1989, Vajpayee and Mohan, 2011]. Nous nous concentrons sur la charge virale et le nombre de CD4 qui constituent la dynamique principale du VIH.
1.3.1
Charge virale
La mesure du nombre de virus circulant, appel´ee « charge virale », correspond a` la concentration de copies d’ARN viral dans le sang : la quantification est donn´ee en copies par mL. On d´enombre entre 1 000 et 80 000 virions par millilitres (mL) en phase asymptomatique. Sa mesure est compliqu´ee par des seuils de d´etection. Le seuil de d´etection pour les grandes valeurs de charge virale (censure a` droite) est rarement atteint. De plus, il ne constitue pas une probl´ematique lors de l’analyse puisque, quelque soit le niveau exact atteint, il est porteur d’information : le patient est en ´echappement virologique. Concernant le seuil de d´etection pour les valeurs faibles et donc la censure `a gauche, initialement calibr´es pour d´etecter tout niveau sup´erieur a` 500 copies/mL, les kits de mesures biologiques sont d´esormais en routine a` 50 copies/mL. Plus coˆ uteuses, des mesures avec des seuils de d´etection inf´erieurs peuvent aussi ˆetre r´ealis´ees sur demande du praticien. Ce seuil de d´etection a` gauche masque l’information sur le patient (le pronostic peut ˆetre diff´erent
1.3 : Biomarqueurs et crit`eres de jugement cliniques de l’avanc´ee de l’infection
25
en fonction du nombre exact de copies pr´esentes) et doit ˆetre pris en compte statistiquement. Les m´ethodes d’imputation des valeurs en rempla¸cant les valeurs censur´ees par z´ero, le seuil de d´etection ou un quantile du seuil de d´etection conduisent a` des r´esultats instables vis-`a-vis d’une analyse de sensibilit´e. Ainsi, une alternative est de quantifier la contribution des donn´ees censur´ees a` la vraisemblance, ce qui conduit `a des estimations non biais´ees [Thi´ebaut et al., 2006]. Ce point statistique sera plus particuli`erement trait´e Section 3.2.2.3. Une charge virale faible, voire ind´etectable, est corr´el´ee avec un meilleur pronostic clinique [Murray et al., 1999]. De plus, cet indicateur permet la surveillance du traitement par le clinicien : un pic de charge virale peut indiquer un ´echappement th´erapeutique et le besoin de faire des tests de r´esistance aux traitements pour potentiellement changer le traitement du patient.
1.3.2
Num´ eration des CD4
La num´eration des CD4 se fait par une technique de cytom´etrie de flux et s’exprime en nombre de cellules par mm3 . Cependant, le nombre de CD4 est a` prendre avec pr´ecaution puisqu’il est connu pour avoir une grande variabilit´e avec des fluctuations selon les caract´eristiques du patient (ˆage, sexe . . .), mais aussi diurnes [Malone et al., 1990], avec un plus haut niveau de CD4 la journ´ee que la nuit. Des grandeurs usuelles sont pour les sujets sains entre 700 et 1 200 cellules/mm3 avec une moyenne estim´ee a` 830 cellules/mm3 [Bofill et al., 1992]. Un grand nombre de CD4 est associ´e a` de meilleures d´efenses immunitaires. De plus, dans le cadre du VIH, il a ´et´e montr´e que c’est un bon pr´edicteur de la progression de l’infection et de la survie [Mellors et al., 1997, Egger et al., 2002].
1.3.3
Histoire naturelle de la maladie
La dynamique VIH-syst`eme immunitaire est tr`es corr´el´ee avec le d´eroulement clinique de l’infection. L’´evolution naturelle passe par trois phases (Figure 1.2) : une phase de primo-infection entre 6 et 12 semaines apr`es l’infection, une phase asymptomatique de dur´ee variable et la phase SIDA, derni`ere phase de l’infection.
Chapitre 1 : L’infection a` VIH
26
Figure 1.2 : Graphe de la relation entre la charge virale VIH et le nombre de CD4 en moyenne au cours du temps chez un patient non trait´e. (adapt´e de [Wikipedia, 2013]). En phase de primo-infection, les d´efenses immunitaires s’affaissent et la charge virale pr´esente un pic. Cette phase est importante, car elle est tr`es corr´el´ee au pronostic immunologique, virologique et clinique du patient. Le pic viral et le nadir de CD4 (nombre minimum de CD4 mesur´e) durant cette p´eriode sont largement corr´el´es avec les niveaux d’´equilibre atteint ult´erieurement par le patient [Lindb¨ack et al., 2000], d´eterminant ainsi l’´evolution clinique du patient [Mellors et al., 1995]. Une explication possible est que des r´eservoirs latents de VIH sont remplis pendant la phase de primo-infection et responsable dans la suite de l’infection de rebonds virologiques [Sedaghat et al., 2007]. De nombreuses revues de litt´erature sur les probl´ematiques de la primo-infection peuvent ˆetre trouv´ees dans [Kaufmann et al., 1998, Kahn and Walker, 1998, Sedaghat et al., 2007, Drylewicz, 2009, Cohen et al., 2011]. La phase asymptomatique est une phase de latence clinique pouvant durer plusieurs ann´ees pendant laquelle le patient ne pr´esente aucun symptˆome li´e au VIH. Sans traitement, le niveau de CD4 est souvent faible, mais acceptable (autour de 500-600 cellules/mm3 ) et la charge virale est stable autour de 1000-10000 copies/mL. Cet ´etat ne signifie pas pour autant un arrˆet de la dynamique entre VIH et syst`eme immunitaire. En effet, c’est une p´eriode o` u les taux de cr´eation et de d´ec`es des CD4 et du VIH sont tr`es importants,
1.3 : Biomarqueurs et crit`eres de jugement cliniques de l’avanc´ee de l’infection
27
mais o` u les populations sont en ´equilibre [Ho et al., 1995, Coffin et al., 1995]. Cependant, apr`es une p´eriode de 8 ans en moyenne, la quasi-totalit´e des patients non trait´es infect´es par le VIH ´evolue vers un SIDA. En p´eriode de SIDA, les d´efenses immunitaires c`edent, avec un nombre de CD4 inf´erieur a` 200 cellules/mm3 , et la charge virale explose a` plus de 100 000 virions par mL. Les manifestations cliniques se caract´erisent par l’apparition de symptˆomes constitutionnels et de maladies opportunistes graduellement li´es au nombre de CD4 et pouvant entraˆıner la mort du patient. Cependant, la progression vers cette phase n’est pas irr´eversible et de nos jours un patient pour lequel le traitement est efficace peut sortir de cette phase.
1.3.4
Les populations cellulaires d’int´ erˆ et dans cette th` ese
Dans ces travaux de th`ese, les populations de cellules d’int´erˆet seront les CD4 pouvant ˆetre activ´es (T ) ou quiescents (Q). Les populations de cellules infect´ees sont not´ees avec un ast´erisque (∗). De plus, nous mod´eliserons la charge virale du VIH (V ), voir Figure 1.3 les pictogrammes associ´es. Ces aspects compartimentaux seront repris Section 3.2.3.
V
Q
T
Charge virale CD4 CD4 du VIH non infectés non infectés quiescents activés
T*
Q*
CD4 infectés activés
CD4 infectés latents
Figure 1.3 : Population de cellules d’int´erˆet et virus VIH dans la mod´elisation in vivo de la dynamique entre VIH et syst`eme immunitaire.
Chapitre 2 Les strat´ egies de traitement dans les pays du Nord R´ esum´ e : Dans ce chapitre, nous introduisons les diff´erents traitements disponibles pour les patients infect´es par le VIH et, plus particuli`erement, le mode d’action des antir´etroviraux et de leurs combinaisons (cART). Puis nous montrons comment les sp´ecificit´es g´en´etiques du VIH et les probl´ematiques d’effets ind´esirables des traitements font de cette infection une application potentielle `a la construction de strat´egies d’optimisation de traitements. Nous pr´esentons diff´erentes approches utilis´ees pour d’autres pathologies et ´enon¸cons les premi`eres avanc´ees en mati`ere de m´edecine personnalis´ee dans le domaine du VIH. Mots-cl´ es : antir´etroviraux ; combinaison de traitements antir´etroviraux (cART) ; ´echappement th´erapeutique ; effet secondaire ; inhibiteur de prot´ease (PI) ; inhibiteur de fusion (IF) ; inhibiteur d’int´egrase (II) ; inhibiteur non nucl´eosidique de la transcriptase inverse (NNRTI) ; inhibiteur nucl´eosidique de la transcriptase inverse (NRTI) ; ligne de conduite de traitement anti-VIH ; m´edecine personnalis´ee ; m´edecine stratifi´ee ; r´esistance ; surveillance des traitements ; test g´enotypique ; test ph´enotypique ; th´erapie cibl´ee ; VIH. Abstract : In this chapter, we list the treatments available for HIV-infected patients : we describe the mechanism of action of antiretroviral drugs and the therapies based on combinations of antiretroviral drugs (cART). Then, we show how genetic specificities of HIV and high levels of drugs adverse effects make this infection a good application for treatment monitoring and optimization. We review approaches used in other pathologies and highlight first steps toward personalized medicine in the HIV context. Key words : adverse effect ; anti-HIV treatment guidelines ; antiretroviral drugs ; combinaison of antiretroviral therapies (cART) ; fusion inhibitor (IF) ; genotypic test ; HIV ; integrase inhibitor (II) ; monitoring ; non nucleosidic inverse transcriptase inhibitor (NNRTI) ; nucleosidic inverse transcriptase inhibitor (NRTI) ; personalized medicine ; phenotypic test ; protease inhibitor (PI) ; resistance ; stratified medicine ; targeted therapy ; therapeuthic escape.
Chapitre 2 : Les strat´egies de traitement dans les pays du Nord
2.1
30
Les th´ erapies antir´ etrovirales hautement actives
Mˆeme si de nouvelles techniques, comme celles reposant sur le cas clinique du patient de Berlin [H¨ utter et al., 2009], laissent toujours esp´erer une possibilit´e de gu´erison pour les patients infect´es par le VIH, l’´eradication de l’infection par le VIH ne peut en g´en´eral pas ˆetre atteinte avec les traitements actuels, mˆeme apr`es une intensification [Dinoso et al., 2009]. Cela repose sur de multiples m´ecanismes de long terme : la pr´esence de r´eservoirs de cellules infect´ees CD4 latentes [Chun and Fauci, 1999], la r´eplication continuelle du virus ou encore l’´echec du syst`eme immunitaire a` reconnaˆıtre et ´eliminer les cellules infect´ees [Deeks et al., 2012]. Par cons´equent, les principaux objectifs de l’instauration du traitement antir´etroviral (ART) sont de r´eduire la morbidit´e associ´ee au VIH, de prolonger la dur´ee et la qualit´e de la survie, de restaurer ou au moins pr´eserver la fonction immunologique, de durablement supprimer la charge virale plasmatique du VIH et de pr´evenir la transmission du VIH. En d’autres termes, les antir´etroviraux permettent de ralentir la progression naturelle de la maladie et ainsi de r´eduire le nombre de d´ec`es li´e au VIH/SIDA. Ainsi, dans les cas les plus favorables, nous parlons de gu´erison fonctionnelle o` u l’organisme peut contrˆoler naturellement le VIH et le maintenir `a des niveaux tr`es faibles. Des ´etudes conduites au Danemark sugg`erent qu’une personne diagnostiqu´ee s´eropositive a` l’ˆage de 25 ans aurait une esp´erance de vie de plus de 64 ans en 2011 alors qu’elle n’´etait que de 33 ans en 1995. Ce chiffre est a` comparer avec 76 ans en population g´en´erale. L’usage de traitements antir´etroviraux multiplie donc par pr`es de 5 l’esp´erance de vie des personnes vivant avec le VIH [Lohse et al., 2007]. Les b´en´efices associ´es `a l’usage d’antir´etroviraux ne sont donc plus `a d´emontrer.
2.1.1
Les mol´ ecules antir´ etrovirales
Les mol´ecules antir´etrovirales sont classifi´ees en fonction de l’´etape de r´eplication du virus qu’elles inhibent. Elles ont ´et´e mises sur le march´e de mani`ere successive fournissant au clinicien une palette de plus en plus ´etoff´ee de traitements, voir Figure 2.1 pour la frise temporelle des autorisations de mise sur le march´e. C’est en 1987 que le premier
2.1 : Les th´erapies antir´etrovirales hautement actives
31
´ traitement pour le VIH est introduit et approuv´e par la FDA aux Etats-Unis : il s’agit de la Zidovudine (AZT) [Fischl et al., 1987]. C’est un inhibiteur nucl´eosidique de transcriptase inverse (NRTI) qui alt`ere la production d’ADN viral (voir Figure 1.1.3) en introduisant des codons illisibles. Depuis, sept autres NRTI ont ´et´e mis sur le march´e : abacavir (ABC), didanosine (DDI), emtricitabine (FTC), lamivudine (d4T), stavudine (3TC), tenofovir (TFV) et zalcitabine (ddC). L´eg`erement diff´erents par leur mode d’action, mais agissant au mˆeme niveau du cycle de r´eplication, les inhibiteurs de transcriptase inverse non nucl´eosidique (NNRTI) sont autoris´es en 1996 et bloquent l’´etape de production d’ADN viral (voir Figure 1.1.3). Il existe 5 NRTI : delavirdine (DLV), efavirenz (EFV), etravirine (ETV), nevirapine (NVP) et rilpivirine (RPV). En 1995, une nouvelle classe de mol´ecules appel´ees inhibiteurs de prot´ease (PI) est introduite. Ces mol´ecules empˆechent l’enzyme prot´ease du VIH (voir Figure 1.1.5) de r´eassembler une prot´eine virale fonctionnelle. Il existe 10 PI autoris´es : amprenavir (APV), atazanavir (ATZ), darunavir (DRV), fosamprenavir (FAPV), indinavir (IDV), lopinavirritonavir (LPV/r), nelfinavir (NFV), ritonavir (RTV), saquinavir (SQV) et tipranavir (TPV). La plupart des PI sont utilis´es en combinaison avec du RTV `a faible dose (not´e /r) afin d’am´eliorer les propri´et´es pharmacocin´etiques [Cooper et al., 2003]. Les inhibiteurs de fusion (IF) apparus en 2003 qui inhibent l’entr´ee du virus (voir Figure 1.1.1) sont de deux types : soit ils bloquent le co-r´ecepteur des cellules cibles et empˆechent la prot´eine gp120 de s’accrocher, tels que le marivaroc (MVC) ; soit ils cassent les liens ´etablis entre la cellule et le virus comme l’enfuvirtide (T20). Encore plus r´ecent, depuis 2007, les inhibiteurs d’int´egrase (II) agissent en empˆechant l’ADN viral de s’int´egrer `a l’ADN de la cellule hˆote (voir Figure 1.1.4) . Ces derni`eres mol´ecules sont particuli`erement prometteuses, car elles pr´esentent une dynamique d’action consid´erablement diff´erente de celles des autres mol´ecules antir´etrovirales. En effet [Sedaghat et al., 2008] ont montr´e une diminution de la charge virale plus rapide avec les II, ph´enom`ene souvent pronostic d’une meilleure r´eponse au traitement a` long terme.
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Mono – therapy
Dual – therapy
32
cART
87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07
NRTI NNRTI
AZT
ddI
ddC
d4T 3TC
ABC
FI
FTC
NVP DLV EFV SQV RTV NFV IDV
PI
TDF
APV LPV/r
ATV FPV
T20
II
08 09
TPV DRV
MVC RAL
Figure 2.1 : Dates d’autorisation de mise sur le march´e d’antir´etroviraux pour le VIH par la FDA par classes antir´etrovirales.
2.1.2
Les th´ erapies par combinaisons d’antir´ etroviraux (cART)
En 1994, l’essai clinique CONCORDE [Seligmann et al., 1994] met en ´evidence une dur´ee limit´ee d’efficacit´e de l’AZT en mono th´erapie a` cause d’une apparition rapide de r´esistances virales. Sur la base de ces premiers r´esultats, un int´erˆet grandissant a ´et´e port´e aux combinaisons d’antir´etroviraux (cART), voir [Caliendo and Hirsch, 1994] pour une revue des probl´ematiques de passage des mono `a des multi th´erapies. Par la suite, les premiers essais cliniques ont d´emontr´e qu’une cART est souvent plus efficace qu’un antir´etroviral administr´e en mono th´erapie [Hammer et al., 1996, Detels et al., 1998, Bierman et al., 2009, Katlama et al., 2010, Delfraissy et al., 2008]. Cependant, ce r´esultat ne s’applique qu’aux antir´etroviraux les plus anciens et peut ˆetre nuanc´e pour certains nouveaux antir´etroviraux. Par exemple le DRV/r dans l’essai clinique MONET est montr´e comme non inf´erieur par rapport `a des strat´egies de traitement par trith´erapies classiques [Arribas et al., 2010]. En d´epit du grand nombre d’antir´etroviraux existant, de nombreuses questions sont encore d´ebattues : quel est le meilleur moment pour commencer un traitement ? Quelle combinaison de traitement prescrire ? Faut-il interrompre les traitements ou les modifier sans apparition d’´echecs th´erapeutiques ? Pour r´epondre a` ces questions, des lignes de conduite publi´ees annuellement par les diff´erentes autorit´es de sant´e (WHO, NIH, HAS . . .) per-
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mettent de faire ´evoluer les pratiques cliniques ([Yeni et al., 2002] , [Hammer et al., 2008], [Thompson et al., 2010]).
La premi`ere interrogation consiste a` savoir quand commencer le traitement. Les seuils de mise sous cART ont ´evolu´e depuis 1996 et sont maintenant de traiter en dessous de 500 cellules CD4/mm3 [Sterne et al., 2009, Kitahata et al., 2009, Severe et al., 2010]. Des strat´egies de traitement pr´ecoce sont aussi `a l’´etude, cette probl´ematique est un sujet de recherche a` part enti`ere, et ne sera pas trait´e dans ces travaux, voir [Kilby et al., 2008, S´aez-Ciri´on et al., 2013, Syed et al., 2013] pour une introduction.
Une autre question est quels traitements administrer pour quels patients. Il existe des lignes de conduite sur les cART a` choisir en fonction de crit`eres cliniques et de l’historique du patient. Le rationnel de ces choix s’appuie sur des crit`eres d’efficacit´e, de s´ecurit´e, de tol´erance, de toxicit´e, de facilit´e d’usage des antir´etroviraux et sur leurs caract´eristiques intrins`eques dans la cART. L´eg`erement modifi´ee en fonction des avanc´ees, la composition des cART en terme de classes de traitement n’a que peu vari´e depuis les ann´ees 2000. En revanche, les mol´ecules incluses dans les cART sont de plus en plus puissantes. Pour les patients na¨ıfs de traitements, la cART consiste en 2 NRTI et un antir´etroviral suppl´ementaire PI, NNRTI ou II. Ce sont des traitements dits de premi`ere ligne. De mani`ere pr´ef´erentielle, en 2013, les recommandations du NIH sont de choisir entre quatre r´egimes comme premier traitement : TDF+FTC suppl´ement de EFV (NNRTI), ATV/r (PI), DRV/r (PI) ou RAL(II) selon l’avis du clinicien au regard du patient. Des essais ont montr´e que les cART bas´ees sur des NNRTI ou II sont d’une efficacit´e durable d’un point de vue virologique, mais beaucoup de patients ne les tol`erent pas. Ce probl`eme est moins pr´esent pour les cART bas´ees sur des PI. En revanche, la grande variabilit´e de r´eponse pharmacocin´etique a` cette classe de traitement [Acosta et al., 2000] peut entraˆıner des toxicit´es chez certains patients et peuvent justifier la n´ecessit´e d’un suivi personnalis´e. Par ailleurs, certaines cART sont fortement d´econseill´ees sauf exception cliniquement justifi´ees. Les doses de traitements en routine sont pour la plupart standardis´ees et ne
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d´ependent que du poids du patient. Cette contrainte est en partie due aux probl`emes de gal´enique et d’industrialisation des mol´ecules.
2.1.3
Les ´ echappements th´ erapeutiques
Apr`es l’initiation d’une premi`ere ligne de traitement, le patient peut avoir un ´echec virologique a` la cART : il est d´efini comme l’incapacit´e a` maintenir une r´eplication virale faible amenant a` une charge virale inf´erieure a` 200 copies/mL. Il est donc n´ecessaire de passer aussi vite que possible sur une seconde ligne de traitement afin d’´eviter l’accumulation de mutations de r´esistance [Hosseinipour et al., 2009] ; son choix est beaucoup plus qualitatif et repose sur l’avis d’expert du clinicien. La d´ecision est majoritairement orient´ee par des crit`eres objectifs comme les trajectoires des biomarqueurs, les ant´ec´edents m´edicaux, les tests de r´esistance, les habitudes de vie et les comorbidit´es du patient. Cependant, le clinicien explore en profondeur les raisons qualitatives de l’´echec virologique qui est le plus souvent li´e au manque d’adh´erence, a` l’intol´erance de la cART et aux interactions pharmacologiques avec la nourriture, les drogues ou les autres m´edications du patient. Cependant, il n’existe pas de consensus sur le choix de nouvelles cART. Le patient peut aussi exp´erimenter un ´echec immunologique, d´efini comme l’incapacit´e de maintenir un niveau ad´equat de CD4. Ce niveau d´epend de l’´etat du patient a` l’initiation du traitement, mais correspond en moyenne `a une augmentation du nombre de CD4 de 150 cellules/mm3 apr`es environ 5 mois de cART [Bartlett et al., 2001]. Dans ces cas, la meilleure strat´egie en terme d’am´elioration immunologique et virologique n’est pas de changer le cART ou d’ajouter des antir´etroviraux, mais de rechercher de potentielles coinfections [Hammer et al., 2010]. Par ailleurs, une possibilit´e pas encore recommand´ee ni valid´ee par les autorit´es de sant´e serait d’utiliser des traitements de restauration immune tels que les th´erapies g´eniques, les hormones de croissance ou encore les immunoth´erapies (comme l’interleukine 7, voir Section 6.1.1.1).
2.1 : Les th´erapies antir´etrovirales hautement actives
2.1.4
35
Les effets secondaires
Des effets secondaires ont ´et´e rapport´es avec l’usage de tous les antir´etroviraux et sont la raison principale de changements de cART, d’interruptions de traitement et de nonadh´erence [O’Brien et al., 2003]. Les plus fr´equents sont des maux de tˆete, des naus´ees, des vomissements, de la fatigue, une perte d’app´etit, des acc`es de fi`evre, des picotements, des diarrh´ees, des probl`emes de peaux et des troubles de r´epartition des graisses. D’autres effets peuvent engager le pronostic vital et toucher le syst`eme cardiovasculaire et le syst`eme nerveux central. Pour finir, l’apparition de comorbidit´es dues a` un vieillissement pr´ecoce de 10 a` 15 ans [Guaraldi et al., 2011] est un nouveau probl`eme dans les pratiques cliniques. Bien que pr´esents dans plus de 10% des patients, le coˆ ut des effets secondaires est en g´en´eral inf´erieur `a celui qu’aurait le coˆ ut d’une infection VIH non prise en charge [Lichtenstein et al., 2008]. Cependant, ces effets secondaires sont importants, car ils modifient l’apparence et la vie du patient ce qui favorise la non-adh´erence du traitement [Catz et al., 2000, Weiser et al., 2003] qui est un enjeu majeur de sa r´eussite.
2.1.5
L’adh´ erence aux traitements
L’adh´erence est le respect par le patient des prescriptions de son m´edecin concernant la dose et l’heure de la prise m´edicamenteuse. Elle varie de mani`ere g´en´erale pour un traitement entre 28% et 82% [Backes and Schentag, 1991]. La quantification de l’adh´erence est difficile et aucune m´ethode n’est optimale [Paterson et al., 2000, Arnsten et al., 2001, Fletcher et al., 2005]. Nous pr´esentons une liste non exhaustive des m´ethodes disponibles en commen¸cant par les plus consistantes et comparables entre essais, une revue de litt´erature est disponible dans [Costagliola and Barberousse, 2001, van Dulmen et al., 2007]. – Le dosage de la concentration plasmatique en traitement permet de v´erifier la prise du m´edicament. C’est la mesure la plus significativement associ´ee `a la r´eponse virologique [Fletcher et al., 2005] mais ell ne peut ˆetre effectu´ee que ponctuellement. – La surveillance ´electronique des capuchons des piluliers (ou MEMS Medication Event Monitoring System en anglais) [Averbuch et al., 1990] : A chaque fois qu’un
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pilulier est ouvert, la prise du traitement est suppos´ee. Cette technique est couteˆ use et encombrante [Samet et al., 2001]. Cependant elle demeure le seul moyen d’avoir une ´evaluation en continue de l’adh´erence. – Le d´ecompte non annonc´e de pilules [Kalichman et al., 2010] permet de calculer le nombre d’oublis mais pas le sch´ema d’adh´erence. – L’auto-report par auto-questionnaires (questionnaires ACTG [Chesney et al., 2000], PMAQ [Duong et al., 2001] . . .), carnet de route ou entretiens d´ependent en grande partir de la qualit´e des questions et de la fr´equence du recueil. Quel que soit la m´ethode de recueil, les recommandations en mati`ere de traitement antiVIH sont en faveur d’une adh´erence parfaite puisque l’adh´erence est un d´eterminant majeur de la r´eussite du traitement, en terme d’apparition du SIDA, de d´ec`es et de restauration immune [Garc´ıa de Olalla et al., 2002, Parienti et al., 2013]. Par exemple, il est montr´e qu’une diminution de 10% du taux d’adh´erence augmente de 90% les risques d’´echecs virologiques [Gardner et al., 2009]. De plus, une faible adh´erence est parfois associ´ee au retour de VIH sous forme de virus r´esistant aux drogues prescrites [Paterson et al., 2000] bien que ce dernier point reste discut´e en fonction de la classe de traitement utilis´ee [Bangsberg et al., 2003]. Les donn´ees d’adh´erence utilis´ees dans ces travaux de th`ese, lorsqu’elles sont disponibles, sont recueillies par auto-questionnaires ACTG. Nous supposons l’adh´erence bien report´ee. Cependant, nous discutons en Section 7.2.2 une potentielle surestimation de l’adh´erence [Liu et al., 2001] amenant `a une sous-estimation probable de l’efficacit´e in vivo des antir´etroviraux.
2.2
M´ edecine personnalis´ ee : vers une individualisation des traitements
L’am´elioration de l’efficacit´e et de la qualit´e de vie du patient est un enjeu applicatif des m´ethodes et des mod`eles statistiques d´evelopp´es, une voie d’exploration est l’individualisation des traitements. Dans cette section, nous d´efinissons le concept et les diff´erents
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types de m´edecine personnalis´ee. Ces techniques sont tr`es utilis´ees en oncologie et, depuis quelques ann´ees, apparaissent aussi dans le traitement de l’infection par VIH. Les diff´erences entre les individus peuvent expliquer pourquoi un traitement peut provoquer des effets ind´esirables ou ˆetre efficace uniquement chez certains patients. Ces diff´erences peuvent ˆetre li´ees aux caract´eristiques de l’individu, mais aussi a` celle des bases mol´eculaires de l’infection et du traitement. La m´edecine personnalis´ee permet d’identifier quels patients pr´esenteront la meilleure r´eponse th´erapeutique `a quels traitements, afin de prescrire le traitement le plus efficace et le plus sˆ ur. Comme soulign´e par Janet Woodcock [Woodcock, 2007], “le concept de m´edecine personnalis´ee n’est rien de plus que la pratique m´edicale elle-mˆeme en int´egrant scrupuleusement l’´etat de sant´e du patient en prenant la meilleure information individuelle disponible”. De nos jours, la majorit´e des cliniciens essaye naturellement de s’en rapprocher. Cependant, ceci n’est possible que s’il existe une compr´ehension m´ecaniste de la maladie et du traitement.
2.2.1
Individualisation du choix du traitement
Le premier pas vers la “m´edecine personnalis´ee” consiste en la stratification des patients en groupes. Pour cela il est n´ecessaire de cibler des biomarqueurs repr´esentatifs de l’´evolution de la maladie et/ou pr´edictifs d’une r´eponse positive. Puis, il faut savoir comment les utiliser pour ´etablir un diagnostic pr´ecis, pour choisir un traitement cibl´e quand il est disponible, pour suivre l’efficacit´e du traitement et pour ´etablir un pronostic sur l’´evolution du patient. Des lignes de conduite bas´ees sur des essais cliniques pass´es permettent de d´efinir des seuils pour de nombreux biomarqueurs et ainsi de d´efinir des sous-groupes de patients pour lequel un certain traitement est pr´econis´e. Cette m´edecine stratifi´ee est facilit´ee par l’apparition de tests ”compagnons”. La m´edecine stratifi´ee constitue un continuum vers la m´edecine individualis´ee o` u chaque patient constituerait une strate. Par exemple, en oncologie, les deux techniques sont pratiqu´ees s´equentiellement. Les patients sont d’abord stratifi´es en deux sous groupes : les bons r´epondeurs et les mauvais r´epondeurs par une analyse g´enomique. Les patients
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ayant une bonne signature tumorale re¸coivent une th´erapie standard alors que les mauvais r´epondeurs re¸coivent une th´erapie cibl´ee en fonction de la mutation pr´esente. Ce choix sp´ecifique peut s’effectuer `a la fois sur le type et la dose de traitement. L’un des premiers exemples associant une maladie et un traitement sp´ecifique est l’Herceptest. Il est apparu en 1998 et permettait d’identifier les patients avec un cancer du poumon m´etastatique de type HER-2, meilleurs r´epondeurs a` l’Herceptin [Perez et al., 1999]. Par ailleurs, la pr´esence de la mutation ALK entraˆıne un choix particulier de la mol´ecule de traitement [Choi et al., 2010]. Pour les cancers du sein, les patients exprimant le g`ene erbB2 b´en´eficient d’une plus forte dose de chimioth´erapie [Muss et al., 1994]. [Arias, 2011] propose un revue de la litt´erature concernant les th´erapies cibl´ees en fonction de la g´enomique, de la prot´eomique ou encore de l’imagerie des tumeurs. Cependant, le plus grand obstacle reste celui des connaissances ; en effet, le succ`es de cette m´edecine personnalis´ee d´epend en grande partie de la disponibilit´e de tests diagnostiques [Papadopoulos et al., 2006] pour identifier les patients qui peuvent b´en´eficier d’un traitement particulier.
Dans le domaine du VIH, la m´edecine personnalis´ee (au sens de la connaissance de l’hˆote et du pathog`ene [Law et al., 2013]) en est encore a` ses pr´emices, mais, le choix des mol´ecules de traitement peut en partie ˆetre orient´e par l’analyse des virus du patient. Avant ou en d´ebut de traitement, le virus pr´esent dans l’hˆote est majoritairement sauvage (chez 84% a` 94% des patients [Ross et al., 2007]). Cependant, des mutations sur au moins un codon de l’ARN viral arrivent 104 a` 105 fois par jour chez les patients infect´es par le VIH [Coffin et al., 1995]. Il est donc important de connaitre la quasi-esp`ece du patient pour orienter le choix des mol´ecules de traitement. Concernant les tests de r´esistance, le test g´enotypique s´equence l’ARN et donc les g`enes viraux afin de d´etecter certaines mutations qui conf`erent des r´esistances aux traitements. Il permet de mettre en valeurs des mutations identifi´ees et classifi´ees (bases de donn´ees de type [ANRS-AC11-database, 2013, Stanford-database, 2013]) en terme de r´esistance a` certaines classes de traitements, voir [Shafer, 2002] pour une revue de litt´erature. Un score de susceptibilit´e g´enotypique (GSS) peut ˆetre calcul´e comme le nombre
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total de traitements actifs, au vu des tests g´enotypiques, compar´e au nombre de traitements donn´es au patient. Par ailleurs, les tests ph´enotypiques consistent a` regarder la capacit´e du virus de l’hˆote a` se reproduire dans diff´erentes concentrations de traitements en comparaison avec un virus de r´ef´erence issu d’une source sauvage (WT). Longs et coˆ uteux, ils permettent de mani`ere analogue au GSS de construire un Score de susceptibilit´e ph´enotypique (PSS). Ces tests de r´esistance sont associ´es a` la r´eponse virologique [DeGruttola et al., 2000, Wittkop et al., 2008] et l’essai clinique HAVANA [Tural et al., 2002] a d´emontr´e que l’usage des tests g´enotypiques du VIH am´eliore les r´esultats virologiques. En effet, le choix de th´erapies cibl´ees est utilis´e en cas de r´esistances connues pour choisir un traitement auquel le patient est sensible [Meynard et al., 2002]. Ces tests ponctuels, souvent effectu´es a` l’initiation d’un nouveau traitement, ne constituent que des recommandations stratifi´ees [Mallal et al., 2008, F¨atkenheuer et al., 2005]. Cependant, l’identification incompl`ete des mutations virales qui conf`erent des r´esistances constitue une limitation a` leur usage, en particulier pour les inhibiteurs de prot´eases [Rabi et al., 2013]. Les caract´eristiques g´enomiques du patient infect´e par le VIH peuvent aussi jouer un rˆole sur la progression du VIH et la r´eponse au traitement. Par exemple, certaines versions de g`enes pr´esentent des interactions pharmacog´enomiques [Rodriguez-Novoa et al., 2006, Haas et al., 2011] qui conf`erent une hypersensibilit´e a` certains traitements. Ainsi, pour certaines mol´ecules telles que le MVC ou l’ABC, un d´epistage des polymorphismes individuels est r´ealis´e. D’autres all`eles indiqueraient des progresseurs lents ou rapides vers le SIDA [den Uyl et al., 2004, Itescu et al., 1992]. La connaissance encore tr`es parcellaire des g`enes impliqu´es est une limite `a leur utilisation.
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2.2.2
40
Surveillance des r´ eponses aux traitements et adaptations
L’approche g´en´erale de surveillance des r´eponses de traitements (Treatment Drug monitoring (TDM) en anglais) est la mˆeme pour toutes les maladies. Il s’agit de surveiller les doses de traitement par des mesures pharmacocin´etiques et les r´eponses du patient par des biomarqueurs ou des observations cliniques. Cela permet de d´eterminer les raisons d’effets secondaires, de toxicit´e ou d’´echecs de traitement. En effet, des polymorphismes enzymatiques peuvent expliquer que certaines mol´ecules ne se m´etabolisent pas de mani`ere analogue entre les individus. En d’autres mots, les m´etaboliseurs rapides pourraient ˆetre sur-dos´es et les m´etaboliseurs lents sous-dos´es expliquant les variabilit´es de r´eponses. Or, malgr´e les recherches pharmacog´enomiques, il n’existe pas de technique syst´ematique pour le d´efinir `a l’avance. Ainsi, pour la tuberculose, les TDM permettent de varier la dose de traitement administr´ee au patient pour la maintenir dans un intervalle cible afin d’obtenir la meilleure efficacit´e [Peloquin, 2002]. Pour des maladies o` u les traitements `a hautes doses sont tr`es agressifs, comme les antid´epresseurs tricycliques, les TDM permettent de d´etecter pr´ecocement une toxicit´e [Peveler et al., 1999]. Dans le domaine du VIH, pendant le suivi du traitement, un TDM est de plus en plus fr´equemment ´etabli. En 2002, un essai randomis´e sur 40 patients [Fletcher et al., 2002] a permis de montrer qu’un traitement de mˆeme dose chez chaque patient est sousoptimal par rapport `a des doses choisies par TDM. Cependant, un grand nombre de mol´ecules interviennent dans les cART et ne sont pas toutes aussi ad´equates pour le TDM. Par exemple, les RTI pr´esentent une ´etape de phosphorylisation qui rend difficile l’association de leurs concentrations sanguines et intracellulaires avec leurs effets [Moore et al., 1999, Bazzoli et al., 2010], d’o` u toute la difficult´e d’extrapoler et de comparer les effets in vitro et in vivo. Ce ph´enom`ene peut expliquer le fait que certains essais, tels que GENOPHAR [Bossi et al., 2004] ou encore PharmAdapt [Clevenbergh et al., 2002], ne trouvent pas de sup´eriorit´e significative des TDM par rapport aux strat´egies standard de traitement. Globalement, les TDM bas´es uniquement sur la pharmacocin´etique ne seraient pas suffisants, mais, en incluant d’autres facteurs telle qu’une analyse conjointe avec
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41
les biomarqueurs sp´ecifiques du VIH, la piste des TDM reste envisag´ee. Une revue de leurs diff´erents avantages et inconv´enients peut ˆetre trouv´ee dans [Gerber and Acosta, 2003, Rakhmanina and la Porte, 2012]. Ainsi, les TDM restent avantageux dans les situations o` u ils permettent au patient de continuer un traitement qu’ils auraient autrement arrˆet´e ou substitu´e par un autre traitement plus complexe [Schoenenberger et al., 2013]. Pour finir, nous pouvons noter que l’´evaluation par TDM des r´eponses aux traitements repose en grande partie sur l’adh´erence du patient et la possibilit´e de son observation. En effet, si par exemple l’objectif est de contrˆoler la concentration plasmatique d’´equilibre de traitement chez le patient, son recueil ne sera possible que si le patient a correctement pris son traitement les jours pr´ec´edents. Que l’on s’int´eresse `a des TDM ou au suivi simple de biomarqueurs, l’adh´erence est donc un facteur cl´e de la r´eussite de l’adaptation des traitements d’o` u leur analyse conjointe de plus en plus fr´equente [Mentr´e et al., 2005, Savic et al., 2012]. Finalement, une question int´eressante concernant l’adaptation des traitement reste la simplification des traitements via l’analyse du sch´ema de l’adh´erence : une adh´erence al´eatoire a-t-elle moins d’impact qu’une adh´erence de type traitement intermittents en cycles courts. Les connaissances actuelles ne permettent pas de trancher, certains essais tendent a` montrer un contrˆole acceptable de l’infection avec, par exemple, des traitements de type « une semaine sur deux » [Dybul et al., 2001], d’autres essais sont stopp´es pr´ecocement [Ananworanich et al., 2003]. Nous introduirons en perspectives comment ces travaux de th`ese associ´es `a un TDM pourraient permettre d’´evaluer l’impact des sch´emas d’adh´erence. Le passage en revue de ces techniques de choix et d’adaptation du traitement montre que l’avenir des m´ethodes d’adaptation des traitements dans le domaine du VIH semble ˆetre de faire du cas par cas, c’est pourquoi notre champ d’application sera l’optimisation des traitements par individualisation. Dans le contexte de cette th`ese, nous nous attacherons a` des travaux m´ethodologiques dans lesquelles nous tˆacherons de d´evelopper des outils statistiques pour am´eliorer l’ajustement, la s´election et la pr´ediction de mod`eles d´ecrivant la dynamique du VIH-1 pour rendre cela possible.
PARTIE II : Aspects math´ ematiques, statistiques et num´ eriques des mod` eles dynamiques
Chapitre 3 Les mod` eles d’analyse de la dynamique des biomarqueurs R´ esum´ e : L’objectif est d’analyser les donn´ees longitudinales de la charge virale et du nombre de CD4 des patients avec et sans cART. Nous pr´esentons dans ce chapitre deux approches : les mod`eles descriptifs pour lesquels nous d´ecrivons les mod`eles multivari´es et les mod`eles causaux, et les mod`eles m´ecanistes dynamiques bas´es sur des syst`emes d’´equations diff´erentielles. Nous pr´esentons chacun des mod`eles et les techniques d’inf´erence associ´ees dans le cadre de la mod´elisation VIH-syst`eme immunitaire. Mots-cl´ es : effets mixtes ; ´equations d’estimation g´en´eralis´ees (GEE) ; maximisation a posteriori (MAP) ; mod`ele causal ; mod`ele multivari´e ; mod`eles structuraux marginaux (MSM) ; mod`ele `a cellules cibles ; mod`ele m´ecaniste dynamique ; optimisation ; syst`emes d’´equations diff´erentielles ordinaires (ODE) ; fonction de vraisemblance.
Abstract : The objective is to analyze longitudinal data of viral load and CD4 count of patients with and without cART. This chapter presents two different approaches : descriptive models for which we introduce multivariate modeling and causal models, and dynamic mechanistic models based on differential equations systems. Each model is introduced in the specific context of modeling the interaction between HIV and the immune system, and we present the associated inference techniques. Key words : maximization a posteriori (MAP) ; causal models ; generalized estimating equations (GEE) ; likelihood ; marginal structural models (MSM) ; mechanistic dynamic models ; mixed effets ; multivariate modeling ; optimization ; ordinary differential equations systems (ODE) ; target cells model.
Chapitre 3 : Les mod`eles d’analyse de la dynamique des biomarqueurs
3.1 3.1.1
46
Les mod` eles descriptifs Mod` eles multivari´ es
Les mod`eles lin´eaires mixtes classiques [Laird and Ware, 1982] permettent de d´ecrire les trajectoires pour la charge virale et les CD4, voir [Boscardin et al., 1998] pour une revue. Ils ont ensuite ´et´e ´etendus pour prendre en compte la censure `a gauche de certains biomarqueurs [Jacqmin-Gadda et al., 2000], la sortie d’´etude informative [Lyles et al., 2000], les al´eas stochastiques [Taylor et al., 1994], ou encore pour mod´eliser conjointement des ´ev´enements de progression et de survie [De Gruttola and Tu, 1994]. Cependant, les CD4 et la charge virale sont des biomarqueurs intrins`equement corr´el´es, il existe une dynamique entre ces populations. Ainsi, un mod`ele conjoint bivari´e est montr´e comme plus adapt´e que deux mod`eles s´epar´es [Thi´ebaut et al., 2003, Thi´ebaut et al., 2005]. Soit Yi = (Yi1 , Yi2 )T les biomarqueurs observ´es pour un individu i = 1, . . . , n. Yi1 (resp. Yi2 ) est un vecteur de ni observations de la charge virale (resp. nombre de CD4). Un mod`ele conjoint `a effets mixtes s’´ecrit :
Yi = Xi β + Z i γ i + i , i ∼ N (0, Σ) γi ∼ N (0, G)
o` u β = (β 1 , β 2 )T est le vecteur des effets fixes associ´e a` Xi = (X1i , X2i )T o` u Xki est la matrice de taille ni × nkp , avec nkp le nombre de covariables explicatives pour les effets fixes pour le biomarqueur k = 1, 2. De plus, γi = (γi 1 , γi 2 )T est le vecteur des effets al´eatoires associ´e `a Zi = (Z1i , Z2i )T o` u Zki la matrice de taille ni × nkq , avec nkq le nombre de covariables associ´ees aux effets al´eatoires pour le biomarqueur k = 1, 2. La matrice de variance-covariance Σ est suppos´ee diagonale. Les ´el´ements diagonaux σ1 et σ2 sont les erreurs de mesure pour chaque biomarqueur. La matrice de variance-covariance des effets al´eatoires G, suppos´ee non structur´ee, permet de prendre en compte la corr´elation entre les deux biomarqueurs. Par le biais de l’hypoth`ese de normalit´e des erreurs de distribution telle qu’elle est faite ici, la solution de cette ´equation est le maximum de
3.1 : Les mod`eles descriptifs
47
vraisemblance qui peut ˆetre trouv´e par des algorithmes de maximisation de type NewtonRaphson ou encore l’algorithme Esp´erance-Maximisation (EM) [Dempster et al., 1977, Lindstrom and Bates, 1988]. Cependant, dans le cadre de cette th`ese, nous sommes plus particuli`erement int´eress´es par l’´evaluation des effets traitement ; ainsi les mod`eles pr´esent´es ci-dessus ne sont pas valides pour tous les types de donn´ees. En effet, si nous travaillons sur des essais randomis´es, alors les patients sont r´epartis de mani`ere al´eatoire entre le groupe traitement et le groupe contrˆole. Ainsi, une interpr´etation directe des coefficients des mod`eles de r´egression sera permise. En revanche, dans des donn´ees observationnelles, l’attribution du traitement n’est pas due au hasard, mais d´epend des marqueurs qui servent aussi a` ´evaluer l’efficacit´e du traitement. En d’autres termes, certaines des covariables explicatives (Xi , Zi ) sont fortement corr´el´ees avec les variables d’int´erˆet Yi . Ainsi, l’estimation de l’effet causal du traitement requiert une mod´elisation particuli`ere.
3.1.2
Les mod` eles causaux
3.1.2.1
Concept de causalit´ e
Dans la th´eorie traditionnelle de la causalit´e, des variables al´eatoires sont manipul´ees et repr´esentent la r´ealisation d’´ev´enements. Nous cherchons `a distinguer une association et une relation causale entre A (la cause) et Y (la cons´equence). Par exemple, A peut repr´esenter la mise sous traitement et Y l’impact sur un biomarqueur ou le biomarqueur lui-mˆeme. Il s’agit d’´etablir une relation entre A et Y en pr´esence de facteurs de confusion mesur´es L et non mesur´es U . Pour avoir une bonne compr´ehension de la situation, [Robins, 1987] a propos´e les graphes dirig´es acycliques (DAG) comme m´ethode de visualisation des probl`emes causaux (Figure 3.1). Plusieurs th´eories ont ´et´e d´evelopp´ees pour appr´ehender la causalit´e. Nous pouvons classer les principales graduellement en fonction de leur degr´e d’implication requis entre A et Y . Les th´eories probabilistes [Suppes, 1970] regardent si l’occurrence de A augmente la probabilit´e de Y toutes choses ´etant ´egales par ailleurs. Une association significative
Chapitre 3 : Les mod`eles d’analyse de la dynamique des biomarqueurs
48
Facteurs de confusion
L U
Traitement
Facteurs de confusion non mesurés
A Y
Biomarqueurs
Figure 3.1 : Mod`eles causaux : d´efinition de l’effet entre A et Y en fonction des facteurs de confusion L et U .
suffisamment forte doit ˆetre mise en ´evidence pour parler de causalit´e, cependant la magnitude requise et l’interpr´etation restent subjectives. Ainsi, le courant interventionniste ´etudie la causalit´e entre A et Y en regardant s’il est possible de changer Y en modifiant A [Eberhardt and Scheines, 2007] mais le choix de l’intervention peut ˆetre crucial quant a` la conclusion de l’analyse. Une approche plus radicale, bas´ee sur des th´eories contrefactuelles [Rubin, 1974, H¨ofler, 2005], d´efinit la causalit´e comme le fait que Y n’aurait pas pris les valeurs qu’il a prises si A ne s’´etait pas produit. Nous pr´esenterons ici des mod`eles dans le cadre de cette derni`ere approche bien qu’elle soit controvers´ee. En effet, le r´esultat de l’inf´erence est bas´e sur des hypoth`eses inobservables puisqu’il s’agit de l’´evaluation des cons´equences d’alternatives pass´ees non r´ealis´ees ; voir [Dawid, 2000] pour une revue. Par ailleurs, [Commenges and G´egout-Petit, 2009] soulignent que les DAG ne font pas intervenir le temps de mani`ere continu et ne permettent donc pas de prendre en compte des relations cycliques (comme le fait que les biomarqueurs peuvent eux aussi avoir un impact sur la mise sous traitement, voir Figure 5.1 Section 5). En effet, pour rester dans le cadre acyclique, la situation classique des facteurs de confusion d´ependant du temps se pr´esente en temps discret comme l’impact de A au temps t sur Y au temps t + 1. Dans la Section 3.2 et la Section 5 nous ´etendons ces concepts `a des mod`eles dynamiques o` u A et Y sont des processus.
3.1 : Les mod`eles descriptifs 3.1.2.2
49
G-calculs
Pour ces m´ethodes descriptives, l’id´ee est de corriger la provenance des donn´ees pour simuler le fait qu’elles viennent d’un essai randomis´e. Initialement, [Robins, 1987] s’est int´eress´e a` l’esp´erance d’observation de Y sachant le traitement en A en s’affranchissant par int´egration des facteurs de confusion L : E [Y|A = a] =
Z
`
E [Y|A = a, L = `] dP` .
ˆ [Y|A = a] utilisant un r´etrocalIl propose des algorithmes de calcul de l’estimateur E cul `a l’aide de m´ethodes it´eratives appel´ees G-calculs. Initialement, [Greenland, 2004] traite l’effet du traitement en binaire (oui/non) en ´evaluant par la quantit´e C1/0 = ˆ [Y|A = 1] − E ˆ [Y|A = 0], cependant les calculs sont tr`es lourds. Plus tard, des m´ethodes E semi-param´etriques de G-calcul ont ´et´e d´evelopp´ees [Robins et al., 1992] et permettent d’all´eger les calculs. Cela a permis de faire des hypoth`eses sur les diff´erences de niveaux de traitement (N1 , . . . , NK ) et de quantifier CN1 /0 , . . . , CNK /0 . Bien qu’ils permettent un bon ajustement sur les facteurs de confusion observ´es L, les G-calculs restent lourds en terme de calculs en particulier en cas de donn´ees censur´ees et ne seront pas utilis´es dans la suite de ces travaux.
3.1.2.3
Mod` eles structuraux marginaux
Une alternative est les mod`eles de r´egression ajustant sur les variables de choix du traitement qui sont des facteurs de confusion observ´es L [Greenland and Robins, 1986]. Cependant, les th´eories d’ajustement classiques ne suffisent pas. Ni l’introduction des L en tant que facteurs de confusion ni la stratification par L ne parvient a` estimer de mani`ere consistante l’effet total du traitement [Robins, 1989]. Ainsi, [Robins, 1998, Robins et al., 2000] ont d´evelopp´e une nouvelle classe de mod`eles appel´es : mod`eles structuraux marginaux (MSM) qui pond`erent l’inf´erence par des poids correspondant aux probabilit´es inverses de traitement et de censure pour chaque temps et pour chaque patient.
Chapitre 3 : Les mod`eles d’analyse de la dynamique des biomarqueurs
50
Si nous notons Aij+1 la mise sous traitement du patient i au temps j +1, cette quantit´e ¯ ij = (Ai0 , Ai1 , . . . , Aij ). De plus, d´epend des valeurs pass´ees des diff´erents traitements : A ¯ ij = (Li0 , Li1 , . . . , Lij ), o` elle d´epend des covariables d’int´erˆet jusqu’au temps j : L u Lij est le vecteur des p facteurs de confusion observ´es pour le patient i au temps j. De plus, les valeurs pass´ees des biomarqueurs Y sont incluses dans les facteurs de confusion, car elles influencent le choix du traitement, ainsi Yij ∈ Lij . Il est alors possible de construire un mod`ele marginal causal : ¯ ij , Li0 = µij . E Yij |A
(3.1)
¯ ij du G´en´eralement, ce type de mod`ele causal fait intervenir un mod`ele de r´egression en A type g(β0 + β1 Aij ) [Cole et al., 2003]. Le r´egresseur β1 repr´esente alors l’effet traitement. ¯ ij , L ¯ ij ), les variables ayant permis Pour estimer les coefficients β0 et β1 en ajustant sur (X le choix de mise sous traitement, nous calculons les probabilit´es inverses de traitement. Nous notons f (.|.) la valeur de la fonction de vraisemblance conditionnelle aux covariables des observations, les poids de probabilit´es inverses de traitement sont d´efinis par :
WijA
j Y ¯ ik−1 , Li0 , C ¯ ik = 0) f (Aik |A = ¯ ik−1 , L ¯ ik , C ¯ ik = 0) , f (Aik |A k=0
o` u ces poids WijA peuvent ˆetre vus comme la probabilit´e qu’un patient i re¸coive un traitement au temps j sachant les traitements pr´ec´edents re¸cus sur la probabilit´e que ce mˆeme patient re¸coive le traitement qui lui a ´et´e attribu´e sachant son historique de traitement et de r´eponse aux traitements. De mani`ere analogue, il faut aussi ajuster sur la sortie d’´etude. En d´efinissant un indicateur de censure Cij valant 1 si le patient i est censur´e a` la visite j et 0 sinon, il est possible de d´efinir des poids de probabilit´es inverses de censures : WijC
j+1 ¯ ik−1 = 0, A ¯ ik−1 , Li0 ) Y P Cik = 0|C = ¯ ik−1 = 0, A ¯ ik−1 , L ¯ ik−1 . P Cik = 0|C k=0
Ainsi, ajuster sur la mise sous traitement et la sortie d’´etude dans ces mod`eles est possible grˆace aux poids : Wij = WijA WijC .
3.1 : Les mod`eles descriptifs
51
En appelant σ 2 la variance de l’erreur r´esiduelle gaussienne du mod`ele d´ecrit par l’´equation 3.1, la contribution marginale `a la vraisemblance pour un patient i au temps j n’est pas la mˆeme pour tous les patients a` tous les temps. Elle est pond´er´ee par les probabilit´es d’attribution de traitement wij et s’´ecrit : ¯ ij , L ¯ ij ) = p(Yij |A
(
" 2 #)wij 1 1 yij − µij √ exp − . 2 σ 2πσ
(3.2)
Elle est aussi d´efinie pour les donn´ees censur´ees a` gauche [Cole et al., 2007]. L’estimation peut se faire ensuite par GEE (´equation d’estimation g´en´eralis´ee [Liang and Zeger, 1986, Diggle et al., 1994]) qui diff`ere du maximum de vraisemblance, car elle ne fait une hypoth`ese que sur les moments d’ordre un de Yij et le caract`ere born´e du moment d’ordre deux. L’approche GEE permet d’estimer les param`etres dans des mod`eles multivari´es o` u les variables d’int´erˆet sont corr´el´ees, car elle prend en compte une matrice de corr´elation de travail (not´ee par la suite Ri ). Il s’agit d’´etendre l’´equation du score d’une vraisemblance classique a` une vraisemblance pond´er´ee comme d´efinie ´equation 3.2. Nous notons Yi = (Yi1 , . . . , Yini )T et Wi = (Wi1 , . . . , Wini )T des vecteurs de taille ni . Nous reprenons les notations de la Section 3.1.1 : A . . . A ini i1 L1 . . . L 1 ini i1 Xi = , ... ... ... Lpi1 . . . Lpini
est la matrice de taille ni par le nombre de facteurs de confusion observ´es du mod`ele. Le mod`ele de r´egression marginal s’´ecrit donc : µi = g(Xi β). De plus, Ri la matrice de corr´elation de travail n´ecessaire pour les GEE sera choisie diagonale dans ces probl´ematiques pour ´eviter un biais r´esiduel quand certaines variables explicatives sont d´ependantes du temps [Sullivan Pepe and Anderson, 1994]. Ainsi, obtenir un estimateur de β, c’est trou-
Chapitre 3 : Les mod`eles d’analyse de la dynamique des biomarqueurs ver la solution de :
n X i=1
52
XiT Ri−1 Wi (Yi − Xi β) = 0.
Cette approche ne s’applique pas uniquement aux donn´ees longitudinales observationnelles. Dans le cadre d’essais randomis´es, cette approche est inutile, mais valide. En effet, lorsque l’attribution du traitement ne d´epend que des covariables au temps de base, ¯ ik−1 , Li0 , C ¯ ik = 0) = f (Aik |A ¯ ik−1 , L ¯ ik , C ¯ ik = 0) et donc ∀i, j, W A = 1 ce qui ref (Aik |A ij vient a` faire une GEE non pond´er´ee. Par ailleurs, cette approche peut ˆetre ´etendue en une GEE augment´ee qui permet de prendre en compte des probl´ematiques particuli`eres telles que des essais en sous-groupes ou des variables d’int´erˆet corr´el´ees [Stephens et al., 2012]. Ces mod`eles, qui peuvent ˆetre qualifi´es de descriptifs, conduisent `a des estimateurs biais´es s’il existe des facteurs de confusion non mesur´es pour la mise sous traitement ou la trajectoire des biomarqueurs [Cole and Hern´an, 2002]. Or, si cela est possible, il n’existe pas de m´ethode pour tester cette hypoth`ese sur les donn´ees.
3.2
Les mod` eles dynamiques m´ ecanistes explicatifs
Par opposition, les mod`eles m´ecanistes explicatifs ne reposent pas sur l’analyse de donn´ees, mais en premier lieu sur la transcription de connaissances physiopathologiques. En d’autres termes, par m´ecaniste nous voulons dire que nous essayons de traduire par des ´equations diff´erentielles les lois expliquant les ph´enom`enes observ´es. Dans notre cas, il s’agit de prendre en compte les interactions complexes existant entre le syst`eme immunitaire et le VIH par l’interm´ediaire de syst`emes d’´equations diff´erentielles.
3.2.1
Vocabulaire et Notations
3.2.1.1
Diff´ erents types de mod` eles dynamiques
Les syst`emes d’´equations diff´erentielles, autrement appel´es syst`emes dynamiques, sont utilis´es en physique, ing´enierie et ´epid´emiologie. Ils permettent d’exprimer comment un
3.2 : Les mod`eles dynamiques m´ecanistes explicatifs
53
syst`eme X(t) ´evolue `a l’instant t en fonction de lois physiques. Il existe diff´erents types de syst`emes dynamiques d´ecrits ci-dessous. – Les syst`emes d’´equations diff´erentielles ordinaires (ODE) relient les valeurs d’une fonction a` celles de ses d´eriv´ees. Il s’agit d’´ecrire une relation entre une quantit´e continue et ses taux de changements dans le temps. C’est, sauf rares digressions, cette classe de syst`emes dynamiques que nous utiliserons dans ces travaux de th`ese. Cependant, il existe des extensions d´ecrites ici. – Les syst`emes d’´equations diff´erentielles partielles (PDE) contiennent des fonctions multivari´ees (f (x, y, z)) et leurs d´eriv´ees partielles ( ∂f (x,y,z) ) par rapport `a certains ∂x param`etres. Elles sont particuli`erement utilis´ees pour repr´esenter des ph´enom`enes spatiaux multidimensionnels ou lorsqu’il existe des ph´enom`enes multi ´echelles a` mod´eliser [Jacquez et al., 1988]. Par exemple, lorsqu’il s’agit de mesurer l’effet des PI sur le d´eclin du virus de l’h´epatite humaine (HCV), les PDE sont utilis´ees, car il est important de dissocier une dynamique au niveau intra et extra cellulaire [Rong et al., 2013]. Ces m´ethodes ne sont `a ce jour que peu utilis´ees dans le domaine de la mod´elisation des biomarqueurs du VIH, car elles alourdissent les calculs. Cependant, le mode d’action de nouveaux antir´etroviraux, par exemple les II dont l’action se situe au niveau de l’int´egration de l’ADN viral au noyau [Sedaghat et al., 2008], pourra pousser a` prendre en compte ce type de mod´elisations, au prix de mesures beaucoup plus pr´ecises des activit´es intracellulaires. – Les syst`emes d’´equations diff´erentielles stochastiques (SDE) permettent de mod´eliser des trajectoires al´eatoires. Il s’agit d’ajouter un ´el´ement diff´erentiel Brownien ou un bruit au syst`eme lui-mˆeme. Majoritairement utilis´es dans des ´etudes de simulation, ces approches permettent de capturer la variabilit´e non expliqu´ee par les parties d´eterministes des ´equations diff´erentielles. Elles sont en particulier utilis´ees pour mod´eliser les mutations du virus [Ribeiro and Bonhoeffer, 2000, Savkovic et al., 2012, Wang et al., 2013], pour d´ecrire la dynamique du virus en primo-infection o` u les nombres de cellules sont petits et les temps de progression
Chapitre 3 : Les mod`eles d’analyse de la dynamique des biomarqueurs
54
inconnus et variables [Tan and Wu, 1998, Banks et al., 2003] ou durant les pics d’infection virale [Conway and Coombs, 2011]. 3.2.1.2
Les syst` emes d’´ equations diff´ erentielles ordinaires (ODE)
Dans le cadre des mod`eles non lin´eaires `a effets mixtes bas´es sur des ´equations diff´erentielles ordinaires (NLME-ODE), nous consid´erons un ´echantillon de n patients in` chaque instant t ∈ R+ , l’espace d’´etats est compos´e de K composantes d´ependants. A i Xi (t) = (X1i (t), . . . , XK (t)). Les trajectoires de ces composantes d´ependent de param`etres intrins`eques au mod`ele et aux individus ξ i = ξ1i , . . . , ξni p et des fonctions f et h suppo-
s´ees bijectives deux fois diff´erentiables :
dXi (t) dt
= f [Xi (t), ξ i (t)],
Xi (0) = h[ξ i (0)].
(3.3)
Afin d’ˆetre capables de prendre en compte des donn´ees longitudinales issues de plusieurs individus, nous adoptons une approche populationnelle. Ainsi, les param`etres ξ i sont mod´elis´es en fonction : – d’interceptes fixes (φ de taille np ) qui sont des effets fixes constants dans la population, et repr´esentent les valeurs moyennes des param`etres biologiques, – d’effets fixes de covariables (nomm´es β qui est une matrice de taille np × ne ) qui repr´esentent l’effet moyen des covariables sur les param`etres biologiques ; le vecteur de taille ne (le nombre de covariables) compos´e des covariables potentiellement d´ependantes du temps se note zi (t), – d’effets al´eatoires (ui de taille nq ≤ np ) repr´esentant une variabilit´e interindividuelle inexpliqu´ee et permettant une d´eviation de certains param`etres des individus par rapport aux valeurs fixes ; nous supposerons ui ∼ N (0, Σ) avec Σ une matrice diagonale pour simplifier les calculs mˆeme si [Dumont et al., 2013] montre l’avantage de consid´erer les corr´elations entre les effets al´eatoires ; nous posons Σ = (ωj2 )i=j , 0i6=j , i = 1, . . . , nq , j = 1, . . . , nq ].
3.2 : Les mod`eles dynamiques m´ecanistes explicatifs
55
i Une mod´elisation lin´eaire est propos´ee pour une transformation ξ˜ des param`etres ξ i .
En g´en´eral, les quantit´es repr´esent´ees par les param`etres sont des quantit´es positives et i nous adoptons une transformation logarithmique : ξ˜ = log(ξ i ). Cependant, d’autres types
de transformations peuvent ˆetre adopt´ees, par exemple, les transformations logistiques pour des param`etres a` valeur dans [0; 1]. Nous ´ecrivons le mod`ele statistique d´efini par un mod`ele `a effets mixtes : i ξ˜ = φ + β T z(t)i + ui .
(3.4)
Il n’est souvent pas possible d’observer un syst`eme en temps continu (∀t ∈ R) : en pratique chaque patient i a donc un certain nombre de mesures ni r´ealis´ees aux temps tij avec j = 1, . . . ni . De plus, les K composantes du syst`eme dynamique ne sont pas 1 i toujours observ´ees et seules K 0 ≤ K combinaisons (g11 [Xi (tij )] , . . . , gK 0 [X (tij )]) le sont.
Pour finir, pour rendre possible les proc´edures d’estimation, nous supposons qu’il existe 2 e et l’houne fonction de lien bijective g 2 (.) = [g12 (.), . . . , gK 0 (.)] assurant la normalit´
mosc´edasticit´e des erreurs de mesures [Box and Cox, 1964]. Nous notons g = g 1 og 2 la fonction multivari´ee qui relie les observations discr`etes et bruit´ees du syst`eme not´ees Yi (tij ) = [Y1i (tij ), . . . , YKi 0 (tij )] aux Xi (t). Le mod`ele d’observation s’´ecrit pour tout m = 1, . . . K 0 : Yijm = gm [Xi (tij )] + ijm , . ij ∼ N (0, V), V = (σ 2 )i=j , 0i6=j , i = 1, . . . , K 0 , j = 1, . . . , K 0 ) . j
(3.5)
Les mod`eles NLME-ODE sont donc d´efinis avec un mod`ele math´ematique (de la forme de l’´equation 3.3), un mod`ele statistique (de la forme de l’´equation 3.4) et un mod`ele d’observation (de la forme de l’´equation 3.5). Dans ce cadre, il existe deux grands types d’applications d´ecrits ci-dessous. – L’approche de simulation math´ematique, autrement appel´ee descendante, permet de simuler les trajectoires Xi (t, ξ i ) des composantes Xi (t) pour une fonction f
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et des param`etres ξ i donn´es. Lorsqu’il n’existe pas de solution analytique, cela est rendu possible par des m´ethodes de simulations num´eriques [Lambert, 1974, Gupta et al., 1985, Butcher, 2008]. [Hull et al., 1972] ont compar´e de nombreuses m´ethodes et montr´e que les m´ethodes de diff´erenciation (BDF - backward differentiation methods en anglais) [Gear, 1971] sont plus efficaces que les approches de Runge-Kutta ou d’extrapolations trap´ezo¨ıdales. Ainsi, nous avons choisi d’utiliser la m´ethode BDF du solveur LSODE (Livermore Solver for Ordinary Differential Equations) impl´ement´ee dans de nombreux langages, dont Fortran [Hindmarsh, 1983, Radhakrishnan and Hindmarsh, 1993]. – L’approche d’inf´erence statistique, appel´ee parfois ascendante r´esout le probl`eme inverse qui est d’inf´erer la valeur des param`etres du syst`eme en fonction des donn´ees Yi (t). C’est cet aspect qui nous int´eressera dans ces travaux de th`ese o` u nous proposons d’estimer le vecteur de param`etres : h i θ = (φk )k=1,...,np , (βk )k=1,...,ne , (ω)k=1,...,nq , (σk )k=1,...,K 0 . Pour r´esoudre ce probl`eme, nous avons besoin de r´esoudre r´ep´etitivement les syst`emes ODE pour diff´erentes valeurs de param`etres et donc d’utiliser l’approche descendante.
3.2.2
Algorithmes d’estimation en mod` eles NLME-ODE
3.2.2.1
M´ ethodes d’inf´ erence disponibles
Pour l’estimation des param`etres dans ces mod`eles, les m´ethodes non param´etriques par analyse fonctionnelle [Wang et al., 2012] sont encore tr`es confidentielles sans r´eelle interface utilisateurs ; cependant, des m´ethodes param´etriques supposant la normalit´e des observations Y existent. Les premi`eres approches consistent `a estimer les param`etres patient par patient puis `a agr´eger les donn´ees en utilisant des indicateurs de dispersion empiriques [Ribeiro et al., 2002, Dixit and Perelson, 2005] ou des moyennes modifi´ees
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en tenant compte de contraintes de dispersions sur les param`etres [Steimer et al., 1984, Davidian, 1995]. Cependant, ces approches ne permettent pas de tirer parti de l’information populationnelle contenue dans les donn´ees. Les m´ethodes des mod`eles `a effets mixtes non lin´eaires classiques [Lindstrom and Bates, 1990] bas´ees sur la formule analytique exacte de la vraisemblance ne peuvent pas ˆetre appliqu´ees, car la plupart du temps, les ´equations diff´erentielles utilis´ees n’ont pas de solution analytique (en particulier dˆ u a` leur non-lin´earit´e). La vraisemblance peut en revanche ˆetre calcul´ee en utilisant les solveurs num´eriques issus des m´ethodes descendantes, puis optimis´ee. Cependant, la pr´esence d’effets al´eatoires implique un mod`ele hi´erarchique o` u la vraisemblance conditionnelle aux effets al´eatoires peut ˆetre facilement calcul´ee, mais doit ˆetre int´egr´ee pour obtenir la vraisemblance marginale. Plusieurs approches bas´ees sur l’approximation de la vraisemblance ont ´et´e propos´ees [Pinheiro and Bates, 1995], mais les approches peuvent conduire `a des estimations contradictoires lorsque l’approximation choisie est trop grossi`ere au regard de la structure de la vraisemblance [Ding and Wu, 2001]. Une alternative est l’algorithme SAEM (Approximation stochastique de l’esp´erance Maximisation) impl´ement´e dans le logiciel MONOLIX [Kuhn and Lavielle, 2005] et dans NONMEM [Beal et al., 1992] qui est une version particuli`ere stochastique de l’algorithme EM [Dempster et al., 1977]. Cependant, en temps fini, la stabilisation a` des maxima locaux `a cause de diminution m´ecanique du pas de d´eplacement est l’un des principaux inconv´enients. Par ailleurs, l’usage de la vraisemblance hi´erarchique [Lee and Nelder, 1996] sans utilisation de corrections sp´ecifiques telles que le bootstrap est l´eg`erement biais´e [Commenges et al., 2011]. Ainsi, le calcul num´erique des int´egrales semble la meilleure des solutions, en particulier par les m´ethodes de quadratures adaptatives gaussiennes [Plan et al., 2012]. Bien qu’en pratique les int´egrales num´eriques multiples entraˆınent une pr´ecision de calcul limit´ee, cette m´ethode permet le calcul num´erique de la vraisemblance exacte. Ainsi, [Guedj et al., 2007a] ont propos´e une approche d’estimation directe par maximisation de la vraisemblance que nous proposons d’´etendre et d’utiliser.
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Par ailleurs, en compl´ement des approches fr´equentistes pr´esent´ees ci-dessus, des techniques bay´esiennes ont permis de grandes am´eliorations dans l’estimation des param`etres. En effet, il y a souvent des probl`emes d’identifiabilit´e [Guedj et al., 2007b, Miao et al., 2011]. L’inf´erence bay´esienne est souvent r´ealis´ee grˆace aux m´ethodes de Monte-Carlo par chaˆınes de Markov (MCMC) [Putter et al., 2002, Huang et al., 2006, Gilks et al., 1996] impl´e´ ment´ees par exemple dans WinBUGS [Lunn et al., 2000]. Evitant les int´egrations num´eriques, [Drylewicz et al., 2010] montre que ces m´ethodes sont tr`es coˆ uteuses en terme de temps de calcul et ne permettent donc pas parfois l’estimation. Plus rapides, les m´ethodes bas´ees sur des approximations de Laplace de la loi a posteriori, telles que la m´ethode INLA [Rue et al., 2009], ne sont pas utilisables quand les ODE utilis´ees ne poss`edent pas de solution analytique. Ainsi, l’algorithme de [Guedj et al., 2007a] a ´et´e ´etendu dans un cadre bay´esien approxim´e et est d´ecrit dans les Sections suivantes [Drylewicz et al., 2012].
3.2.2.2
Validit´ e de l’approximation normale de l’a posteriori
Le rationnel de cette approche repose sur le fait que la distribution a posteriori des param`etres p(θ|Y1 , . . . , Yn ) peut ˆetre approxim´ee. [Tierney and Kadane, 1986] ont propos´e une approximation bas´ee sur l’approximation de Laplace des int´egrales. Au vu de la complexit´e de notre probl´ematique, nous nous int´eressons a` l’approximation normale de l’a posteriori. Cette derni`ere peut ˆetre justifi´ee par la convergence de la distribution a posteriori vers une loi normale ´enonc´ee dans le th´eor`eme de Bernstein von-Mises ˆ n un estimateur asymptotiquement efficace de θ (tel [van der Vaart, 2000]. Nous notons θ que le maximum de vraisemblance) et Iθ−1 l’information de Fischer individuelle prise en n ce point. Alors, en distance de variation totale not´ee ||.||, dans le cas fr´equentiste param´etrique, nous avons la convergence en probabilit´e de la distribution a posteriori vers une ˆ n et de variance I −1 /n : loi normale d’esp´erance θ θn p(θ|Y1 , . . . , Yn ) − N
Iθ−1 ˆ θn , n n
! P −→ 0. n→+∞
(3.6)
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Il existe aussi d’autres versions de ce th´eor`eme, par exemple, la convergence peut ˆetre montr´ee en terme de risque de Kullback-Leiber [Clarke, 1999]. Par ailleurs, si nous notons θ ∗ la vraie valeur des param`etres dans la population, le th´eor`eme de consistance de Doob [van der Vaart, 2000] garantit la convergence en loi de la distribution a posteriori vers un dirac en θ ∗ : L
p(θ|Y1 , . . . , Yn ) −→ δθ∗ . n→+∞
3.2.2.3
Inf´ erence par approximation normale de l’a posteriori
Calcul de la vraisemblance p´ enalis´ ee En reprenant les notations d´efinies dans la Section 3.2.1.2, nous notons yijm les trai jectoires observ´ees des biomarqueurs et X(tijm , ξ˜ ) les trajectoires pr´edites pour les pai ram`etres ξ˜ pour le patient i au temps j du compartiment m. Il est possible d’´ecrire la
vraisemblance de l’individu i conditionnellement aux effets al´eatoires ui : 0
p∗ (yi |ui , zi , θ) =
ni Y K Y
1 √
σ 2π j=1 m=1 m
exp −
1 2
i yijm − gm (X(tijm , ξ˜ )) σm
!2
,
i o` u les trajectoires X(tijm , ξ˜ ) sont ´evalu´ees par une approche descendante a` l’aide d’un
solveur d’´equations diff´erentielles. Par ailleurs, il est n´ecessaire de prendre en compte les probl`emes m´ethodologiques pos´es par la censure a` gauche (notamment de la charge virale) [Jacqmin-Gadda et al., 2000]. En notant, ζm le seuil de censure a` gauche du biomarqueur m, δijm = Iyijm >ζm est une indicatrice de censure pour l’individu i au temps j, nous avons : ni Y 1 Y 1 √ exp − p(yi |ui , zi , θ) = σm 2π 2 K0
j=1 m=1
(
Ψ
i yijm − gm (X(tijm , ξ˜ )) σm
i ζm − gm (X(tijm , ξ˜ ) σm
!)1−δijm ,
!2 δijm
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o` u Ψ est la fonction de r´epartition d’une loi normale centr´ee r´eduite. Int´egrer par rapport aux effets al´eatoires, par exemple avec des m´ethodes d’int´egration num´erique, permet d’obtenir la contribution marginale de l’individu i a` la vraisemblance observ´ee. En notant Φ la loi normale des effets al´eatoire, nous avons :
i
p(yi |z , θ) =
Z
u∈Rq
p(yi |ui , zi , θ)Φ(u)du.
(3.7)
La vraisemblance observ´ee pour tous les patients, suppos´es ind´ependants et identiquement distribu´es, est obtenue de mani`ere multiplicative et s’´ecrit donc :
p(y|z, θ) =
n Y i=1
p(yi |zi , θ).
Afin de travailler dans un cadre bay´esien, un terme de p´enalit´e est construit. Il se base sur deux grands types d’a priori d´ecrits ci-dessous.
– Les a priori informatifs permettent d’introduire dans le mod`ele une information issue d’analyses pr´ec´edentes. C’est ce type d’a priori qui est choisi pour les effets fixes du mod`ele pour lesquels il existe des ´etudes in vitro et in vivo pr´ec´edentes les quantifiant : ((φk )k=1,...,np , (βk )k=1,...,ne )T ∼ N (E0 , V0 ). – Les a priori non informatifs qui permettent de rester dans un cadre bay´esien, alors mˆeme que l’on ne dispose pas d’information a priori. Pour les erreurs de mesure, σk2 k=1,...,K 0 , nous pourrons utiliser une loi de type Jeffreys (π(σ) = 1/σ). Pour
les variances des effets al´eatoires, (ω 2 )k=1,...,nq , nous pourrons choisir des a priori
demi-Cauchy de m´ediane s2 plutˆot que des lois gamma classiquement utilis´ees ; en effet, elles mettent une masse trop forte en z´ero, restreignant la magnitude des effets al´eatoires [Gelman, 2006, Polson and Scott, 2012]. Ainsi, nous prenons pour a priori π(ω) = (ω 2 + s2 )−1 . Un autre choix est de ne pas p´enaliser sur les variances des erreurs de mesure et les variances des effets al´eatoires et ainsi d’adopter comme a priori une loi uniforme.
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61
Ainsi, la fonction de p´enalisation pour la log-vraisemblance s’´ecrit :
J(θ) = J1 (θ) + J2 (θ),
avec, np ne X [φj − E(φj )]2 X [βj − E(βj )]2 J1 (θ) = + , 2V(φj ) 2V(βj ) j=1 j=1
et J2 (θ) = 0 si un a priori uniforme est choisi sur les termes de variances, sinon :
J2 (θ) = +
nq X
0
log(ωj 2 + sj 2 ) +
j=1
K X
log(σj ).
j=1
D’apr`es le th´eor`eme de Bayes nous avons p(θ|z, y) ∝ π(θ)p(y|z, θ). Le logarithme de la densit´e a posteriori est donc ´egal (`a une constante pr`es) a` la log-vraisemblance p´enalis´ee : LP (y, θ) = ln [p(y|z, θ)] −J(θ). {z } | L(y,θ)
Le maximum a posteriori (MAP) not´e θM AP = argmax(LP (y, θ)) est l’estimateur de θ :
θM AP =
ˆk φ
k=1,...,np
, βˆk
k=1,...,ne
, (ˆ ω )k=1,...,nq , (ˆ σk )k=1,...,K 0 .
[Diaconis and Freedman, 1986] ont prouv´e que le MAP est asymptotiquement ´equivalent au maximum de vraisemblance. En effet, le MAP peut ˆetre vu comme un barycentre pond´er´e entre les a priori et la valeur de l’estimateur de maximum de vraisemblance en th´eorie classique. Ainsi, la vraisemblance p´enalis´ee peut ˆetre approxim´ee par une loi normale (voir th´eor`eme de Bernstein von-Mises Section 3.2.2.2 ´equation 3.6) d’esp´erance θM AP et de variance H −1 (θM AP ), qui est l’inverse de la d´eriv´ee seconde par rapport aux param`etres de LP (y, θ) prise au mode θM AP . En pratique, on obtient une meilleure approximation de l’a posteriori par N (θM AP , H −1 (θM AP )) qu’en utilisant le maximum de vraisemblance N (θM V , H −1 (θM V )).
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Optimisation et d´ eplacements
L’algorithme de type Newton-Raphson peut ˆetre utilis´e pour trouver θ M AP . Le sch´ema it´eratif de cet algorithme permet de se d´eplacer sur la surface `a np +ne +nq +K 0 dimensions en fonction des pentes (U ) et de la courbure (H) : θ (k+1) = θ (k) − H −1 (θ (k) )U (θ (k) ). Dans un algorithme de Newton-Raphson classique, le d´eplacement requiert que H soit inversible. Si ce n’est pas le cas, on peut utiliser l’algorithme de Marquardt bas´e sur l’augmentation de la diagonale de la matrice H [Marquardt, 1963]. Nous prenons H ∗ = H + νI o` u I est la matrice identit´e et ν est un coefficient d’adaptation qui tend vers 0 si H devient inversible.
La fa¸con classique de calculer num´eriquement U et H consiste `a faire des diff´erenciations finies de la fonction objectif LP (y, θ). En notant dθ et dθ 0 deux petits d´eplacements, nous avons : P P ˜ (θ) = L (y, θ + dθ) − L (y, θ − dθ) , U 2dθ
et, LP (y, θ + dθ + dθ 0 ) − LP (y, θ + dθ) − LP (y, θ + dθ 0 ) + LP (y, θ) ˜ H(θ) = . dθdθ 0 La pr´ecision de l’approximation repose sur le choix de dθ, ainsi, [Guedj et al., 2007a] ont propos´e une alternative qui repose sur le calcul analytique des d´eriv´ees appel´ees scores. Afin d’illustrer cette approche, nous ne pr´esenterons ici les formules analytiques de calcul des scores que pour les effets fixes φ, pour les autres types de param`etres le lecteur peut se r´ef´erer `a [Guedj et al., 2007a]. Comme pour la vraisemblance, le score est calcul´e
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63
conditionnellement aux effets al´eatoires pour chaque individu, pour tout l = 1, . . . , np : U φl (yi |ui , θ) =
∂ln(p(yi |ui , zi , θ)) , i ∂ ξ˜ l
i ni X K0 i X 1 ∂gm (X(tijm , ξ˜ )) h i ˜ y − g (X(t , ξ )) , = ijm m ijm i 2 σ ˜ m ∂ ξ j=1 m=1 l
i
˜ ) (t,ξ ) i ˜ ∂ ξl
o` u, les ´equations de sensibilit´e 1 du syst`eme d’´equations diff´erentielles ( ∂X
k
peuvent
en g´en´eral ˆetre d´efinies analytiquement et permettent de calculer : 0
i i i K X ∂gm (X(t, ξ˜ )) ∂X k (t, ξ˜ ) ∂gm (X(t, ξ˜ )) = . i i i ∂ ξ˜ ∂X k (t, ξ˜ ) ∂ ξ˜ k=1 l
l
La contribution marginale individuelle au score est obtenue par int´egration num´erique : i
i
−1
U (θ) = (p(yi |z , θ))
Z
Rq
p(yi |ui , zi , θ)U (yi |ui , θ)Φ(u)du.
Le score global est obtenu en sommant pour tous les individus puis en le p´enalisant avec la d´eriv´ee de J(θ) :
P
U (θ) =
n X i=1
U i (θ) −
∂J(θ) . ∂θ
D´eriver une seconde fois le syst`eme d’´equations diff´erentielles est possible, mais am`enerait `a un syst`eme d’´equations de sensibilit´e secondes de grande, difficile a` manipuler. 1. Exemple Soit le syst`eme d’´equations diff´erentielles suivant, avec {a, b, c, e, f } ∈ R5 : dX = aX + bY + c dt dY = eX + f Y dt Les ´equations de sensibilit´e li´ees au param`etre a sont les suivantes : dX = aX + bY + c dt dY = eX + f Y dt dXa = X + aXa + bYa dt dY a = eX a + f Ya dt
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De plus, le calcul par diff´erentiation finie est coˆ uteux en terme de temps. Ainsi, certains auteurs tels que [Hedeker and Gibbons, 1994] ont propos´e de faire une approximation de H en l’´ecrivant comme la somme pour tous les individus de l’´echantillon du proP T duit scalaire des U : ni=1 U i U i , voir aussi l’algorithme BHHH [Berndt et al., 1974].
[Commenges et al., 2006] propose comme extension l’algorithme RVS (robust variance
scoring en anglais) permettant le calcul de H par son approximation G qui est la variance des scores. Un terme de p´enalit´e qui est la d´eriv´ee seconde J(θ) est ajout´e :
P
G (θ) =
n X i=1
T
U i (θ)U i (θ) −
1 ∂ 2 J(θ) . U (θ)U T (θ) + n ∂θ 2
Finalement, une recherche lin´eaire [Potra and Shi, 1995] du meilleur d´eplacement, fraction du d´eplacement initial calcul´e par U et G, permet d’optimiser la vraisemblance p´enalis´ee pour trouver θ M AP .
3.2.3
Les mod` eles dynamiques m´ ecanistes pour le VIH
3.2.3.1
Mod` ele simple
Les premiers travaux se sont int´eress´es a` la dynamique virale apr`es mise sous traitement. Ils partaient de la constatation biologique que « le virus V est produit par des cellules infect´ees `a un taux not´e π et meurt a` un taux µV suppos´e constant », l’´equation diff´erentielle s’´ecrit
dV dt
= π − µV V [Ho et al., 1995]. En faisant des hypoth`eses sur
les conditions initiales, nous pouvons r´esoudre de mani`ere analytique cette ´equation. Un mod`ele comparable peut ˆetre appliqu´e aux CD4 [Wei et al., 1995]. Le premier avantage des mod`eles dynamiques est ici clairement identifi´e puisqu’il permet aux cliniciens et aux biologistes une interpr´etation directe des param`etres de ces mod`eles comme des grandeurs physiques (essentiellement des taux de cr´eation et de d´ec`es). Mod´eliser conjointement les deux biomarqueurs conduit `a un mod`ele analogue a` un mod`ele proie-pr´edateur [Volterra, 1928]. En 1996, [Perelson et al., 1996] affinent le mod`ele en combinant la dynamique des CD4 (infect´es et non infect´es) `a celle du virus. Ils pr´e-
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65
sentent alors un mod`ele dynamique `a trois composantes (Figure 3.2). La r`egle biologique observ´ee est la suivante : « les cellules CD4 sont produites a` un taux λ, elles s’infectent a` un taux γV et meurent avec un taux µT . Le nombre de nouvelles cellules infect´ees est le terme γV T provenant d’une loi d’action de masse. Chaque cellule infect´ee T ∗ produit π virions. Les cellules T ∗ meurent a` un taux µT ∗ et les virions `a un taux µV » :
Figure 3.2 : Mod`ele de Perelson de la dynamique du VIH
3.2.3.2
dT dt dT ∗ dt dV dt
= λ − µT T − γV T = γV T − µT ∗ T ∗ = πT ∗ − µV V
Complexification des m´ ecanismes
Nous ne cherchons pas `a faire un catalogue des diff´erents mod`eles existants, de bonnes revues peuvent ˆetre trouv´ees dans [Perelson and Nelson, 1999, Le Corfec et al., 2000] et plus r´ecemment dans [Xiao et al., 2013]. Cependant, l’´evolution des connaissances sur l’infection par VIH a permis l’´evolution des mod´elisations dynamiques. Nous listons ici quelques exemples. – Il est int´eressant de distinguer des cellules CD4 quiescentes non infect´ees Q ou infect´ees Q∗ pour mod´eliser les r´eservoirs latents. Ces cellules s’activent en cellules T (resp. T ∗ ) au taux α (resp. α∗ ) et se d´esactivent au taux ρ (resp. ρ∗ ). – En plus des CD4, d’autres types de cellules sont susceptibles d’ˆetre infect´ees par le VIH (comme par exemple les macrophages). Ces populations de cellules ont g´en´eralement des dur´ees de vie plus longues ce qui peut jouer un rˆole sur la dynamique
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66
du VIH [Perelson et al., 1997]. Ainsi, elles peuvent ˆetre ajout´ees au mod`ele `a la fois en tant que cellules infect´ees (M ∗ ) ou non infect´ees (M ) dans le mod`ele. – Par ailleurs, les cellules infect´ees (∗) disparaissent du syst`eme soit par mort naturelle soit par l’action de cellules effectrices tueuses not´ees E [Bonhoeffer et al., 2000, Nowak and Bangham, 1996]. – Concernant le virus, une quasi-esp`ece de virus pr´eexiste dans l’organisme de l’hˆote [Goodenow et al., 1989]. En pr´esence de cART, les g´enotypes r´esistants ne sont plus en comp´etition avec d’autres g´enotypes plus robustes, ainsi, il y a ´emergence d’un virus r´esistant par s´election (voir Section 1.2.2). Une m´ethode pour repr´esenter cela consisterait a` supposer qu’il existe R composantes virales diff´erentes (V 1 , . . . , V R ), chacun repr´esentant un g´enotype [Nowak et al., 1990]. Chaque g´enotype i repr´esente une fraction F (i) de la population de virus. – L’apparition des PI qui agissent sur les propri´et´es infectieuses du VIH a permis d’´etendre le mod`ele pour inclure deux types de virus : une fraction fI de virus infectieux VI et une fraction (1 − fI ) de virus non infectieux VN I .
TE* Q
Q
Q
T T
M M
M
T *
T
T*
M
* *
E Q* Q*
Q*
E
ME *
E
E
M*
M * V1I
V RI
1
VI
1
VI
V1NI
VRNI
1
VNI
R
VNI
Figure 3.3 : Proposition de mod`ele dynamique complexe de l’interaction VIH-syst`eme immunitaire a` 7 + 2R composantes. Ce mod`ele est donn´e a` titre d’exemple pour illustrer la potentielle complexit´e des mod`eles NLME-ODE.
3.2 : Les mod`eles dynamiques m´ecanistes explicatifs
67
Nous pouvons inclure ces remarques aux mod´elisations existantes et proposer une nouvelle repr´esentation plus complexe (Figure 3.3). Pour cela, nous faisons l’hypoth`ese forte que tous les taux de d´ec`es et de cr´eation sont des constantes. Par exemple, pour la prolif´eration des CD4, nous n´egligeons qu’elle peut varier en particulier en fonction des ´etats du syst`eme par des termes de r´etrocontrˆole [Bains et al., 2009, Yates et al., 2007, Callard and Hodgkin, 2007]. Nous d´efinissons un mod`ele a` 7 + 2R composantes et au moins 23 + 2R param`etres ressemblant `a celui utilis´e par [Adams et al., 2005] :
dQ dt
= λQ − µQ Q − αQ + ρT
dT dt
= αQ − ρT − µT T − γT
dM dt
= λM − µM M − γM
dE dt
= λE − µE E PR
PR
PR
i=1
i=1
V iT
V iM
dT ∗ dt
= γT
dQ∗ dt
∗ = ρ∗ T ∗ − α∗ Q∗ − µQ∗ Q∗ − µE Q∗ Q E
dM ∗ dt
= γM
dVI1 dt ... dVIR dt dVN1 I dt ... dVNRI dt
i=1
PR
∗ ∗ ∗ ∗ ∗ V i T − µT ∗ T ∗ − µE T∗T E − ρ T + α Q
i=1
∗ V i M − µM ∗ M ∗ − µE M∗M E
I ∗ 1 = πT ∗ fTI ∗ F (1)T ∗ + πM ∗ fM ∗ F (1)M − µV 1 VI I
I ∗ R = πT ∗ fTI ∗ F (R)T ∗ + πM ∗ fM ∗ F (R)M − µV R VI I I ∗ 1 = πT ∗ (1 − fTI ∗ )F (1)T ∗ + πM ∗ (1 − fM ∗ )F (1)M − µV 1 VN I NI
NI ∗ R = πT ∗ (1 − fTN∗I )F (R)T ∗ + πM ∗ (1 − fM ∗ )F (R)M − µV R VN I NI
(3.8)
Chapitre 3 : Les mod`eles d’analyse de la dynamique des biomarqueurs 3.2.3.3
68
Mod` eles simplifi´ es : parcimonie et identifiabilit´ e.
Des mod`eles complexes, tel que celui pr´esent´e ´equation 3.8, sont difficiles `a manier et pr´esentent des limites quant a` leur utilisation. En effet, ils peuvent ˆetre utilis´es de mani`ere descendante, par simulations, en fixant des valeurs de param`etres [Kubiak et al., 2001, Callaway and Perelson, 2002, Smith and Wahl, 2005, Zhou et al., 2013]. Cependant, cela est peu informatif quant a` la quantification des aspects biologiques des ph´enom`enes. Par ailleurs, des probl`emes d’identifiabilit´e th´eorique et pratique [Petersen et al., 2001] rendent difficile les m´ethodes ascendantes et l’estimation des param`etres par les techniques pr´esent´ees Section 3.2.2. L’identifiabilit´e th´eorique d´epend de la structure du mod`ele en ad´equation avec le type de donn´ees disponibles. L’identifiabilit´e pratique fait rentrer en ligne de compte les ph´enom`enes statistiques li´es aux conditions exp´erimentales. Des indicateurs, majoritairement bas´es sur la matrice d’information de Fisher ou des approximations [Mentre et al., 1997, Retout and Mentr´e, 2003], permettent de donner une indication sur la faisabilit´e des estimations [Rothenberg, 1971, Walter and Pronzato, 1997, Guedj et al., 2007b]. D’autres m´ethodes, plus fastidieuses, reposent sur des simulations issues du monde de l’ing´enierie (multiple time point technique) et ´etudient les d´eriv´ees successives du syst`eme jusqu’`a l’ordre 3 [Xia and Moog, 2003, Wu et al., 2008]. Plus r´epandues, les approches bas´ees sur l’analyse de la sensibilit´e aux changements des param`etres par profil de vraisemblance [Raue et al., 2009] sont peu utilisables en grande dimension. En reprenant les ´equations 3.8, nous pouvons voir que certains ph´enom`enes sont concurrents. Par exemple, pour les cellules infect´ees, les d´ec`es par apoptose et les d´ec`es par r´eponses immunitaires de cellules effectrices ne peuvent pas ˆetre diff´erenci´es avec le niveau d’information recueillie dans la majorit´e des ´etudes non sp´ecifiques `a cette probl´ematique. Une premi`ere solution est de faire le choix de fixer certains param`etres `a des valeurs issues de la litt´erature [Guedj et al., 2007a, Drylewicz et al., 2010, Huang et al., 2006]. Cependant, cela impacte la qualit´e du mod`ele lorsque la valeur d’un param`etre est mal choisie [Banks et al., 2003, Xia and Moog, 2003]. Ainsi, une solution alternative est de d´efinir
3.2 : Les mod`eles dynamiques m´ecanistes explicatifs
69
des mod`eles simplifi´es en ad´equation avec l’information recueillie dans les ´etudes `a notre disposition et les ph´enom`enes biologiques les plus importants au regard de nos probl´ematiques. Ainsi, dans la suite de ces travaux nous utiliserons le mod`ele a` cellules cibles (voir Figure 3.4) :
dQ dt
= λ + ρT − αQ − µQ Q,
dT dt
= αQ − γT V − ρT − µT T,
dT ∗ dt
= γT V − µT ∗ T ,
dV dt
= πT ∗ − µV V.
(3.9)
∗
Figure 3.4 : Mod`ele `a cellules cibles de la dynamique du VIH
3.2.3.4
Pistes de mod´ elisation des effets traitement
Certains des param`etres du mod`ele d´efinit ´equation 3.9 peuvent ´evoluer au cours du temps, en particulier, lors de l’introduction d’un traitement. Pour pouvoir envisager une m´edecine personnalis´ee, il est n´ecessaire de d´efinir une fonction de lien entre le traitement (type/dose) et son impact sur le syst`eme dynamique : quels sont les param`etres impliqu´es ? Quelle est la forme de la r´eponse ? Les cART n’agissent pas directement sur le virus V , il existe un ph´enom`ene de m´ediation. Les cART inhibent certaines ´etapes de r´eplication du VIH au sein des cellules cibles T les empˆechant de devenir infect´ees T ∗ , d’o` u la baisse de charge virale. Ainsi, dans la suite de ces travaux nous supposerons que les cART jouent uniquement sur le param`etre d’infectivit´e du mod`ele a` cellules cibles γ. Cette mod´elisation est classique pour toutes
Chapitre 3 : Les mod`eles d’analyse de la dynamique des biomarqueurs
70
les classes d’antir´etroviraux sauf les PI. Cependant, il sera montr´e section 7.2.2 que cette approximation est acceptable dans le cadre du mod`ele `a cellules cibles. Concernant, la forme de la fonction pharmacodynamique ψ(.) liant la cART a` l’effet, nous avons compar´e plusieurs approches. En notant dij (t) la dose de traitement j prise par le patient i au temps t et βj un coefficient de r´egression estim´e, nous avons consid´er´e entre autres : – les fonctions en escaliers : ψ(dij (t)) = βj 1dij (t)>0 , – les fonctions puissance : ψ(dij (t)) = βj (dij (t))κj , o` u κj est une puissance estim´ee par profil de vraisemblance, – les fonctions sigmo¨ıdales :
ψ(dij (t))
bj aj = βj log 1 − 1/ 1 + di (t) , o` u aj et bj j
sont des coefficients estim´es identifiables avec des indicateurs pharmacodynamiques. – des fonctions pharmacodynamiques ´etablies in vitro : potentiel d’inhibition instantan´e (IIP) ou encore facteur d’inhibition (IF) (voir Section 7.2.2). Aucun des choix n’est uniform´ement meilleur. La s´election de la fonction ψ(.) se fera pour chaque probl´ematique par une approche de s´election de mod`eles (voir Section 6.1.1.1).
Chapitre 4 NIMROD pour l’inf´ erence en mod` ele dynamique R´ esum´ e : Dans ce chapitre, nous pr´esentons un logiciel d´edi´e a` l’estimation bay´esienne de param`etres dans des mod`eles dynamiques a` effets mixtes par maximisation a posteriori (MAP) bas´ee sur l’optimisation de la vraisemblance p´enalis´ee. Cependant, il permet aussi l’inf´erence par maximum de vraisemblance simple. Ce programme parall´elis´e, diffus´e et gratuitement t´el´echargeable s’appelle NIMROD (normal approximation inference in models with random effects based on ordinary differential equations). Nous comparons ses performances avec celles de logiciels et m´ethodes concurrents. L’exemple fil rouge est la pharmacocin´etique, car c’est l’un des syst`emes ODE le plus utilis´e. Mots-cl´ es : approche bay´esienne ; calcul parall`ele ; crit`eres de convergence ; distance au maximum (RDM) ; estimation conditionnelle de premier ordre (FOCE) ; maximum a posteriori (MAP) ; MCMC ; NLME-ODE ; optimisation num´erique.
Abstract : In this chapter, we present a software for Bayesian estimation of parameters in dynamical models with mixed effects. The algorithm is based on a posteriori maximization (MAP) based on the optimization of the penalized likelihood. However, it also allows classical maximum likelihood. This parallelized, disseminated and free for download program is called NIMROD (normal approximation inference in models with random effects based on ordinary differential equations). We compare the performances of this algorithm compared to existing methods. We present pharmacokinetics as leading example because it is one of the most used ODE system. Key words : a posteriori maximization (MAP) ; bayesian approach ; convergence criteria ; distance to maximum (RDM) ; first order conditional estimation (FOCE) ; MCMC ; NLME-ODE ; numerical optimization ; parallel computing.
Chapitre 4 : NIMROD pour l’inf´erence en mod`ele dynamique
4.1
72
Le programme NIMROD
L’algorithme pour l’estimation des param`etres en NLME-ODE pr´esent´e en Section 3.2.2.3 mis au point par [Guedj et al., 2007a] puis [Drylewicz et al., 2012] pr´esentait des ´echecs de convergence en particulier lorsque le nombre de param`etres ´etait important. De plus, il ´etait coˆ uteux en terme de temps de calcul et confidentiel au niveau de son utilisation. Nous avons effectu´e une refonte du programme avec des am´eliorations de l’algorithme amenant a` une meilleure convergence, une restructuration de l’architecture logicielle, la parall´elisation des calculs et la mise en place d’outils de diffusion. Nous pr´esentons l’outil NIMROD (normal approximation inference in models with random effects based on ordinary differential equations).
4.1.1
Am´ elioration de la convergence
Nous avons impl´ement´e un crit`ere de convergence, la distance relative au maximum (RDM), d´evelopp´e par [Commenges et al., 2006]. En reprenant les notations de la Section 3.2.1.2, le RDM en θ s’´ecrit : −1
U P (θ)T GP (θ) U P (θ) . RDM(θ) = np Le RDM permet d’´evaluer simultan´ement le degr´e de courbure de la surface et d’´evaluer le ratio entre l’erreur statistique et l’erreur num´erique. Ainsi, le RDM doit ˆetre inf´erieur a` 1 et aussi proche de 0 que possible. Dans des ´etudes de simulation, nous avons v´erifi´e les propri´et´es du RDM vis-`a-vis de la pr´ecision finale sur la valeur de la vraisemblance. Par ailleurs, nous avons mis en valeur que plus le RDM est faible plus la pr´ecision est grande sur la valeur des param`etres estim´es et donc plus il y a de chiffres significatifs. Cette premi`ere m´ethode a permis de mettre en avant des probl`emes sp´ecifiques de convergence lorsque les points de d´epart de l’algorithme sont loin du maximum (en simulation et sur des donn´ees r´eelles). Nous avons donc am´elior´e l’algorithme RVS (voir Section 3.2.2.3). En effet, loin du maximum la matrice G peut ˆetre une mauvaise approxi-
4.1 : Le programme NIMROD
73
mation de la matrice H. Ainsi, l’algorithme RVS peut avoir des difficult´es `a maximiser la fonction objectif c’est pourquoi nous avons introduit une possibilit´e de commutation (“switch” en anglais) vers un algorithme de Marquardt-Levenberg classique o` u la Hessienne est calcul´ee par diff´erentiations finies lorsque l’algorithme RVS ne progresse pas pendant plusieurs it´erations. Ce travail a par ailleurs donn´e lieu au d´eveloppement d’un package R de l’algorithme de Marquardt-Levenberg [Prague et al., 2012b] 1 . Pour finir, une attention particuli`ere a ´et´e port´ee a` la d´efinition des a priori. Nous avons retravaill´e le choix des a priori non informatifs et d´efini une m´ethodologie de choix d’esp´erance et de variance pour les a priori informatifs en fonction des donn´ees de litt´erature.
4.1.2
Am´ elioration des temps de calcul
Nous avons mis en ´evidence par profilage les sous-routines et fonctions les plus coˆ uteuses en terme de temps de calcul. Nous avons identifi´e que le calcul de la vraisemblance et le calcul des scores ´etaient les deux fonctions les plus consommatrices de ressources. Or, dans ces deux fonctions, les calculs sont hautement parall`eles puisque les individus sont suppos´es ind´ependants. Ainsi, leurs vraisemblances individuelles sont calcul´ees sans faire appel les unes aux autres. Nous avons donc pris le parti de parall´eliser le programme sur les individus : le nombre maximum de processeurs (nproc ) ou cœurs pour la parall´elisation active du programme est donc inf´erieur ou ´egal au nombre d’individus dans l’´etude (nproc ≤ n). Comme [Krawezik and Cappello, 2006], nous avons ensuite compar´e deux techniques de parall´elisation : OpenMP [Chapman et al., 2008] sur architectures a` m´emoire partag´ee et MPI [Snir et al., 1995] sur ordinateurs distants. La premi`ere est l´eg`erement plus rapide 1. Nous avons propos´e le package “marqLevAlg” impl´ementant l’algorithme de Levenberg-Marquardt sans contrainte [Marquardt, 1963]. L’impl´ementation du package “marqLevAlg” est classique au niveau de l’optimisation, mais comporte comme crit`ere de convergence particulier le RDM. Nous avons compar´e les r´esultats obtenus avec “marqLevAlg” avec ceux du package “optimx” [Nash and Varadhan, 2011] (Nelder-mead, BFGS ...) en particulier sur des exemples disponibles dans le manuel d’“optimx”. Avec des temps de calcul comparables, les estimations s’av`erent parfois meilleures en particulier pour des points de d´epart loin du minimum ou des surfaces non strictement convexes ou ´eloign´ees d’une forme quadratique. Concernant l’utilisation de ce package, des statistiques sont disponibles via un des serveurs CRAN depuis le 01/10/2012. Ainsi, nous pouvons voir qu’en environ 10 mois nous comptabilisons 726 t´el´echargements.
Chapitre 4 : NIMROD pour l’inf´erence en mod`ele dynamique
74
en terme de performances brutes (voir Table 4.1) en revanche, nous ne pouvons exploiter que des cœurs situ´es sur les mˆemes ordinateurs ce qui limite les ressources. C’est pourquoi nous avons choisi la technologie MPI. Au mˆeme moment, un calculateur haute performance de plus de 3000 cœurs 2 a ´et´e mis en service au m´esocentre aquitain (MCIA) nous permettant une gestion des jobs plus facile que sur les grilles de calculs. Avec ces techniques, nous obtenons par r´egression polynomiale que les temps de calcul ont ´et´e diminu´es d’un facteur n−0.85 proc environ.
nproc 1 2 8 32 128
Technique de parall´elisation : MPI Technique de parall´elisation : OpenMP Temps Facteur d’acc´el´eration Temps Facteur d’acc´el´eration 6 h 42 min 1.00 6 h 42 min 1.00 4 h 01 min 1.66 3 h 51 min 1.87 1 h 18 min 5.06 56 min 18 sec 7.14 22 min 8 sec 17.62 impossible 7 min 12 sec 55.64 impossible
Table 4.1 : Comparaison des temps de calcul apr`es parall´elisation OpenMP et MPI en fonction du nombre de coeurs utilis´es sur le cluster de calcul Avakas du MCIA. Les r´esultats sont obtenus sur le mod`ele VIH utilis´e dans l’article [Prague et al., 2012a] voir Section 6.2.3 avec 16 param`etres estim´es, 149 sujets et 2 compartiments observ´es.
4.1.3
Diffusion
L’´elaboration d’un package R de ce programme est complexe, car les fonctions devant ˆetre d´efinies par l’utilisateur pour changer le mod`ele sont intriqu´ees et leur d´efinition fait appel `a des calculs symboliques `a effectuer en Maple pour d´efinir les ´equations de sensibilit´e des syst`emes ODE. Nous avons donc d´ecid´e de diffuser au format Fortran le programme NIMROD (Normal approximation Inference in Models with Random effects based on Ordinary Differential equations). NIMROD actuellement est un programme libre de droits qui peut ˆetre modifi´e et redistribu´e sous les termes de la licence publique g´en´erale GNU version 2.0 ou ult´erieure publi´ee par la fondation du logiciel libre. Nous avons travaill´e sur la structure de NIMROD afin d’avoir une architecture modulaire et ergonoR Xeon R x5675 @ 3.06 GHz avec 3168 cœurs, 4 2. Avakas : 264 serveurs de calcul biprocesseur Intel GB RAM/cœur et une puissance de crˆete de 38,8 Tflop/s
4.1 : Le programme NIMROD
75
mique. Ainsi, elle se pr´esente en trois sous dossiers ce qui permet d’adapter au niveau de l’utilisateur le niveau de compr´ehension requise sur la programmation, voir Figure 4.1 :
Figure 4.1 : Architecture modulaire de NIMROD
– Pour les nouveaux utilisateurs, le dossier “users” contient toutes les sous-routines permettant la d´efinition d’un mod`ele du point de vue math´ematique, statistique et observationnel. Il peut ˆetre utilis´e comme une librairie et ˆetre remplac´e par un autre dossier correspondant a` un autre mod`ele ou modifi´e par l’utilisateur lui mˆeme. Ces dossiers-mod`eles-librairies sont disponibles en t´el´echargement sur le site de NIMROD. – Pour les utilisateurs interm´ediaires, le dossier “options” permet de modifier/r´egler tous les param`etres de convergence pour les algorithmes d’optimisation, les d´erivations, le solveur ODE, les quadratures adaptatives gaussiennes . . .. Les param`etres initialis´es par d´efaut sont choisis comme optimis´es pour des probl`emes g´en´eraux. Par exemple, augmenter le nombre de points dans les calculs des quadratures adaptatives gaussiennes peut dans certains cas allonger consid´erablement les temps de calcul sans pour autant am´eliorer la pr´ecision. De mˆeme, mettre des seuils de pr´ecision ODE trop grands par rapport `a la variabilit´e des trajectoires peut entraˆıner des erreurs dans les calculs des vraisemblances empˆechant la convergence. Ainsi, un mauvais r´eglage de ces param`etres peut avoir des cons´equences catastrophiques sur les propri´et´es de l’inf´erence.
Chapitre 4 : NIMROD pour l’inf´erence en mod`ele dynamique
76
– Pour les utilisateurs avanc´es, le dossier “blackbox” contient toutes les sous-routines et fonctions permettant l’inf´erence. Website Frequentation (03/2013 – 09/2013)
Bounce rate (03/2013 – 09/2013)
Visit duration (03/2013 – 09/2013)
Percentage of returns (03/2013 – 09/2013)
QR Code – Link to Website
Figure 4.2 : Statistiques de fr´equentation du site Internet NIMROD et QR code de r´ef´erencement du site Internet. Nous avons diffus´e NIMROD sur un site internet (QR code Figure 4.2) qui a ´et´e g´en´er´e par Doxygen [Laramee, 2011]. Ce site a pour but de fournir une interface de t´el´echargement au programme NIMROD et une solide documentation. Il est, a` ce jour, fr´equemment mis a` jour avec les nouvelles avanc´ees logicielles et la mise a` disposition de nouvelles librairies de programme pour traiter de nouveaux probl`emes dynamiques (pharmacocin´etique, mod`eles VIH, mod`ele de restauration immunitaire . . .). Les statistiques de fr´equentation des 6 derniers mois (avril-septembre 2013) produites par Google analytics sont pr´esent´ee Figure 4.2. La fr´equentation est mondiale et augmente au fil des mois avec un total de 302 visites, dont 143 visiteurs diff´erents, sur 6 mois. Parmi ces visites, 47% sont en fait des rebonds o` u les utilisateurs ne consultent pas le contenu des pages. Pour les autres, 26% cliquent sur le lien de t´el´echargement (soit environ 42 t´el´echargements en 6 mois) et la dur´ee moyenne de visite est de 2 minutes 50 secondes.
4.1 : Le programme NIMROD
4.1.4
77
Description et application ` a la pharmacocin´ etique
Le programme NIMROD permet d’analyser tous les types de mod`eles dynamiques bas´es sur des NLME-ODE avec pour seule restriction technique connue les limites de tailles de tableaux support´es par Fortran 95. La Table 4.2 pr´esente une comparaison des fonctionnalit´es entre NIMROD et les logiciels classiques utilis´es dans l’industrie et la recherche, tels que R [R Core Team, 2013], NONMEM [Beal et al., 1992] WinBUGS [Lunn et al., 2000] ou MONOLIX [Lavielle, 2005], Charact´eristiques R NONMEM Logiciel Utilisation payante X Interface clique bouton X Langage de mod´elisation X X Parall´elisme X X Analyse statistique Fr´equentiste param´etrique X X Fr´equentiste non param´etrique X X Bay´esienne param´etrique X X Bay´esienne non param´etrique X
MONOLIX X X X
X
WinBUGS NIMROD
X X
X
X
X
X
X
Table 4.2 : Fonctionnalit´es de R, NONMEM, MONOLIX, WinBUGS et NIMROD. Par souci de large diffusion, nous avons propos´e des exemples pour l’analyse de donn´ees pharmacocin´etiques d’absorption orale de traitements avec mesures sanguines en utilisant des mod`eles a` un compartiment (sang) et deux compartiments (sang et tissus) [Rosenbaum, 2011]. En effet, ce mod`ele NLME-ODE est l’un des plus utilis´e dans le domaine biom´edical. Nous avons compar´e les performances de NIMROD avec des algorithmes plus classiques (FOCE et MCMC) sur des donn´ees simul´ees et sur des donn´ees issues de l’essai clinique PUZZLE [Raguin et al., 2004] concernant la dynamique de concentration plasmatique de l’APV chez des patients infect´es par le VIH. Ce travail fait l’objet d’un article publi´e dans Computer methods and Programs in Biomedecine , Volume 111, Issue 2, aoˆ ut 2013, Pages 447-458 [Prague et al., 2013a]. Le mat´eriel suppl´ementaire ´electronique de cet article se trouve en annexe A.
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 1 1 ( 2 0 1 3 ) 447–458
journal homepage: www.intl.elsevierhealth.com/journals/cmpb
NIMROD: A program for inference via a normal approximation of the posterior in models with random effects based on ordinary differential equations Mélanie Prague a,b,∗ , Daniel Commenges a,b , Jérémie Guedj c , Julia Drylewicz d,e , Rodolphe Thiébaut a,b a
Univ. Bordeaux, ISPED, Centre INSERM U897-Epidemiologie-Biostatistique, F-33000 Bordeaux, France INSERM, ISPED, Centre INSERM U897-Epidemiologie-Biostatistique, F-33000, Bordeaux, France c Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, NM 87545, USA d Laboratory for Translational Immunology, University Medical Center Utrecht, 3508 AB, Utrecht, The Netherlands e Theoretical Biology & Bioinformatics, Department of Biology, Utrecht University, 3584 CH, Utrecht, The Netherlands b
a r t i c l e
i n f o
a b s t r a c t
Article history:
Models based on ordinary differential equations (ODE) are widespread tools for describing
Received 7 August 2012
dynamical systems. In biomedical sciences, data from each subject can be sparse making
Received in revised form
difficult to precisely estimate individual parameters by standard non-linear regression but
4 March 2013
information can often be gained from between-subjects variability. This makes natural the
Accepted 23 April 2013
use of mixed-effects models to estimate population parameters. Although the maximum
Keywords:
which can incorporate prior knowledge in a flexible way. However, the combination of dif-
Maximum likelihood
ficulties coming from the ODE system and from the presence of random effects raises a
Maximum a posteriori
major numerical challenge. Computations can be simplified by making a normal approxi-
likelihood approach is a valuable option, identifiability issues favour Bayesian approaches
HIV
mation of the posterior to find the maximum of the posterior distribution (MAP). Here we
Non-linear mixed-effects model
present the NIMROD program (normal approximation inference in models with random
Ordinary differential equations
effects based on ordinary differential equations) devoted to the MAP estimation in ODE
Pharmacokinetics
models. We describe the specific implemented features such as convergence criteria and an approximation of the leave-one-out cross-validation to assess the model quality of fit. In pharmacokinetics models, first, we evaluate the properties of this algorithm and compare it with FOCE and MCMC algorithms in simulations. Then, we illustrate NIMROD use on Amprenavir pharmacokinetics data from the PUZZLE clinical trial in HIV infected patients. © 2013 Elsevier Ireland Ltd. All rights reserved.
1.
Introduction
Mechanistic or semi-mechanistic models based on ordinary differential equations (ODE) can naturally capture dynamic interactions in time between biomarkers and help understanding biological mechanisms. Thus, their use is
particularly popular in pharmacometrics to describe the drug pharmacokinetics (PK) in plasma [43] or to analyse viral kinetics during treatment [27]. In the clinical setting, studies involve a relatively large number of patients, but technical and ethical reasons often limit the number of sampling measurements that can be done within each patient. In this context, non-linear mixed-effects
Corresponding author. Tel.: +33 557571136. E-mail address:
[email protected] (M. Prague). 0169-2607/$ – see front matter © 2013 Elsevier Ireland Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cmpb.2013.04.014 ∗
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c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 1 1 ( 2 0 1 3 ) 447–458
models are a better way than subject-by-subject analysis to handle the available information [42]. Inference for these models can be done by either frequentist or Bayesian approaches. Many algorithms have been suggested for frequentist approaches. Semi-parametric methods based on functional analysis with splines smoothing have been proposed by Ramsay et al. [36] and extended for mixed-effects by Wang et al. [44]. The main drawback of this approach relies in the choice of the splines basis, the order of the splines and knots [46]. Moreover, the parameter tuning the trade-off between fitting the data and satisfying the ODE system has to be chosen by cross-validation, which is computationally demanding. Alternatively, a large body of literature focuses on parametric approaches based on the likelihood computation using ODE numerical solvers. Because of random effects, likelihood computation also implies the computation of integrals with, most of the time, no closed-forms if the biological model is nonlinear. A method based on the approximation of the likelihood, namely the Laplacian methods (FO and FOCE), has been proposed by Pinheiro and Bates [28] and Lindstrom and Bates [21]; see Wang [45] for a detailed introduction. However, optimization based on an “exact” likelihood provides better results than methods based on approximation of the likelihood [8]. A solution for maximizing the likelihood is to use the stochastic approximation expectation maximisation (SAEM) algorithm proposed by Kuhn and Lavielle [20] implemented in MONOLIX. Of note, all these methods are also implemented in NONMEM. However, Plan et al. [30] suggested that among methods based on an exact likelihood, those based on Gaussian quadratures might be more precise. We proposed an algorithm based on Gaussian quadratures specially adapted for models defined by ODE [15], but its use was limited by the lack of an interface user. Lack of identifiability [16] is another issue arising in these mechanistic models and motivates a Bayesian approach frequently encountered in non-linear models with a large number of parameters. By incorporating information found in the literature, one may increase the information available and thus improve parameter identifiability. However, priors can also be introduced to take advantage of previous data analysis when raw data are not available. In this second context, the priors arise from the posteriors of previous data analysis in an information synthesis framework, see Drylewicz et al. [10] for an example in HIV context. In ODE-based models, the conventional MCMC algorithm (such as implemented in WinBUGS) for computing the marginal posterior distributions of the parameters is time consuming and convergence of chains is often difficult to assess. Other Bayesian methods for approximation of the posterior distributions, such as the integrated nested Laplace approximation (INLA) method [38] cannot be applied here because ODE do not have an analytical solution. Computationally less demanding, a way to use Bayesian approaches is to assume a normal approximation of the posterior (NAP). This is justified asymptotically by the Bernstein–Von Mises theorem [41]. Numerically, this amounts to compute the maximum of the posterior distribution (MAP) while the variance matrix is approximated by the inverse of the Hessian of minus the logarithm of the posterior. This approach was shown to work well in situations where MCMC may fail [10].
Here we detail the NAP approach and its implementation in a program called NIMROD (normal approximation inference in models with random effects based on ordinary differential equations). NIMROD can also be used for classical maximum likelihood inference. We introduce some new statistical and numerical features of the algorithm, such as the properties of the stopping criterion and the optimization of the algorithm for parallel computing. The paper is organized as follows. In Section 2 a statistical model based on an ODE system in a general form is described. Section 3 presents the algorithm and its statistical properties. In Section 4, the NIMROD program is introduced in a simplified user-manual style. Section 5 presents the analysis of simulated data and of data coming from a clinical trial. Section 6 concludes.
2.
Model and inference
2.1.
General model for the system
Let us consider an ODE model for a population of n subjects. For subject i, with i = 1, . . ., n, this can be written:
i dX (t) = f [Xi (t), !i (t)] dt i i X (0) = h[! (0)]
where ! i (t) = [!1i (t), . . . , !ni b (t)] is a vector of nb biological parameters which appear naturally in the ODE system and have a biological interpretation; Xi (t) = [X1i (t), . . . , XKi (t)] is the vector of the K state variables (or compartments). We let X[t, ! i (t)] = Xi (t) to underline that ! i (t) completely determines the trajectories Xi (t). We assume that f is a possibly non-linear function, twice differentiable with respect to ! i (t). To determine the initial value of the ODE system we suppose we are in one of those two cases: either Xi (0) comes from initial conditions in the experiment (see pharmacokinetics illustration Section 5.1) or we assume that the system is in a stable state before the experiment (see pharmacokinetics illustration Section 5.2). In this case, h can be computed by solving the equation dXi (t)/dt = 0 analytically or with a numerical ODE solver. Reparametrization of the system allows us to take constraints into account: we introduce one-to-one functions l (.), l = 1, . . ., nb and define transformed parameters !˜ li (t) = l [!li (t)]. For instance, biological parameters such as rates can be parametrized using a logarithmic transformation to ensure positivity, or parameters between 0 and 1 (such as probabilities or bioavailability) can be parametrized using a logistic transformation. A mixed-effects model for the !˜ li (t) allows introducing covariates and taking into account the betweensubject variations. In this approach, random effects give each subject a different value for a subset q ≤ nb of the biological parameters: i !˜l (t) = "l + ˇl zil (t) + uil ,
uil ∼N(0, ωl2 ).
where "l is the intercept and zil is a vector of ne (possibly timedependent) explanatory variables associated with the fixed effects of the lth biological parameter; ˇl is a vector of regression coefficients; uil is the individual random effect with ωl2 = 0 for l > q.
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In practice, we do not observe Xi (t) directly, but we have discrete-time observations Yi (tij ) of some functions of Xi (t). We assume that there are known link functions gm (.), m = 1, . . ., M, leading to an additive measurement error model. For i = 1, . . ., n, m = 1, . . ., M and j = 1, . . . ni (the number of observation times for subject i), we observe: i Ym (tij ) = gm [Xi (tij )] + $ijm ,
2 $ijm ∼N(0, %m ).
This observation scheme may be complicated by leftcensoring due to detection limits [18]. We denote by & the parameter vector to be estimated: & = [("l )l=1,...,n , (ˇl )l=1,...,ne , (ωl )l=1,...,q , (%l )l=1,...,M ] ∈ Rp . b
2.2.
Likelihood and NAP
The likelihood formula for the model and observations described in Section 2.1 can be found in Guedj et al. [15]. The individual likelihood given the random effects (L&F |u ) is easgm [Xi (tij )]
i
i (t ) Ym ij
log[P(&|Y)] = L(&) + log['(&)] + C, where C is the normalization constant, P(&|Y) the posterior distribution, L(&) the log-likelihood and '(&) the prior distribution. The normal approximation of the posterior is obtained by maximizing the penalized log-likelihood LP (&) = L(&) − J(&) (which does not involve the difficult normalization constant C) and computing its Hessian at the maximum value; here, J(&) is a penalization term equal to − log ['(&)] up to a constant. If we assume normal independent priors for the fixed effects, halfCauchy priors for the standard deviation of the random effects, as recommended by Gelman [12] with median parameter sj 2 and conventional one-dimension Jeffreys-type improper priors for the standard deviation of the measurement errors, the penalization function J(&) can be written:
p $ ["j − E0 ("j )]2 j=1
+
2var0 ("j )
q $ j=1
+
ne $ [ˇj − E0 (ˇj )]2 j=1
log(ωj 2 + sj 2 ) +
2var0 (ˇj )
m $
Credible sets and Bayesian p-values
In a frequentist approach, parameter significance is assessed by testing “ˇ = 0&& or “ˇ ≤ 0&& (resp. “ˇ ≥ 0&& ). In a Bayesian approach, once we have the marginal posterior distributions of the parameters, it is straightforward to compute (1 − ˛)credible sets of highest probability density (HPD-sets, Berger [1], part 4.3.2). We implemented a Bayesian p-value (Bp-value) as an analogue to the frequentist one (see Section 2.8 Box and Tiao [3]). For the one-sided test “ˇ ≤ 0&& , it is simply P(ˇ ≤ 0|Y). For the two-sided Bayesian p-value testing “ˇ = 0&& , it is such that the largest HPD credible set not containing zero has probability 1 − Bp-value. This can be computed as 2min[P(ˇ ≤ 0|Y), P(ˇ ≥ 0|Y)]. If all priors are flat, this Bp-value is equal to the frequentist p-value obtained from a two-sided Wald test. Moreover, as credible sets and confidence intervals of the same coverage are asymptotically equivalent, the Bp-value is also asymptotically consistent with the frequentist p-value for any prior.
2.4.
Model choice
i
ily computed as a function of since the are independent Gaussian variables. However, the computation of the gm [Xi (tij )] requires to solve the ODE system. This is done by using the Livermore solver DLSODE [17], specially adapted for stiff systems using backward difference formula and gear type method BDF from Radhakrishnan et al. [34]. Then, the likelihood (L&i ) is computed by integrating over the random effects via the adaptive Gaussian quadrature (implemented as in Genz and Keister [13]) as proposed by Pinheiro and Bates [29]. Bayes theorem gives:
J(&) =
2.3.
449
log(%j ),
j=1
where E0 and var0 are the expectation and the variance under the priors.
Commenges et al. [5,6] proposed an approximate crossvalidation criterion that can be used to choose estimators based on penalized likelihood. Taking advantage of the fact that we maximize a penalized likelihood, we propose to use LCVa for model choice. This criterion is given by: ˆ ˆ LCVa = −n−1 [L(&ˆ ) − Tr(HL−1 P ( & )HL ( & ))], where HL and HLP are the Hessians of minus the log-likelihood and the penalized log-likelihood respectively, computed in &ˆ . The criterion is implemented in NIMROD with HL and HLP approximated as indicated in Section 3.1. The lower the criterion, the better the model. Differences of criteria between two models can be interpreted: differences larger that 0.1 are considered as “large” while differences lower than 0.001 are considered as negligible [7].
3.
Algorithm
3.1.
Optimization procedure
Under the conditions of the Bernstein–Von Mises theorem, the posterior tends to a normal, thus, when n is large enough the penalized log-likelihood is close to a quadratic form in &. The use of the Newton–Raphson method is justified as this method is based on a quadratic approximation of the surface to maximize. However, with n not very large, the surface may be close to a quadric only in a region near the maximum and may not be convex everywhere. More robust methods such as the Levenberg–Marquardt algorithm [23] are called for. Moreover, in ODE mixed-effects models the likelihood is difficult to compute: we need to solve the ODE system and numerically compute multiple integrals; on top of that, both Newton–Raphson and Levenberg–Marquardt algorithms require first and second derivatives of the objective function which also have to be numerically computed. To decrease the computation time we use an approximation of the second derivative of the log-likelihood leading to the robust variance
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scoring (RVS) algorithm [4], an improved version of the BHHH algorithm [2]. The individual score for a given value of the parameter &, Ui (&), is calculated using Louis’ formula [22] and the systems of sensitivity equations (given by {df [Xi (t), ! i (t)]/d!li (t)}l=1,...,p ) as described in Guedj et al. [15]. Then, the penalized scores, UiP (&), are derived as UiP (&) = Ui (&) − (1/n)(∂J(&)/∂&). The observed log-likelihood and scores are calculated as the sum over %n P %n all the subjects: LP (&) = L (&), U(&) = U (&), UP (&) = i=1 i i=1 i % n P (&). The Hessian of −LP (&), denoted H (&), is approxiU LP i=1 i mated by an estimator of the variance of U plus the second derivative of the penalization J: G(&) =
n $
Ui (&)UiT (&) −
i=1
1 ∂2 J(&) U(&)UT (&) + . n ∂& 2
%n
For & close to the true value and n large, U (&)UiT (&) − i= i T (1/n)U(&)U (&) approximates HL (&) and G(&) approximates HLP (&), the Hessians of −L and −LP , respectively. The RVS algorithm with the calculations described before is implemented in NIMROD with an optional automatic switch to a classical Levenberg–Marquardt algorithm when the algorithm gets stuck, i.e. between two iterations there is neither log-likelihood change nor parameters displacement whereas the main convergence criterion based on RDM (see Section 3.2) is not met yet. Finally, a line search algorithm, such as in Potra and Shi [31], is used to adjust the displacement when the log-likelihood is not improved at the first try. Despite of all these tricks, it may happen that the log-likelihood cannot be improved, possibly due to too large errors in the computation of the scores. In that case, the displacement is done toward the highest proposed log-likelihood.
3.2.
Convergence criteria
Two commonly used criteria assess the stability of the algorithm concerning the displacement in the parameter space )& (k+1) = ||& (k+1) − & (k) || and the variation of the objective func(k+1) tion, here: )L& = LP (& (k+1) ) − LP (& (k) ). If both of these criteria take very low values, this means that the algorithm does not move any more. This is the case when & (k) is very close to &ˆ but this is not sufficient. Stopping rules may be defined by (k+1) )& (k+1) < *1 and )L& < *2 but this is not really satisfactory. In the first place, it is not easy to fix good thresholds (at least for *1 which may depend on the parameters order of magnitude). More importantly, these stopping rules are not directly linked to the properties of the surface to maximize. If the algorithm does not move, this is not a proof that it has reached the maximum: it may just get stuck and be unable to find a good direction to improve the objective function. A maximum is characterized by UP (&ˆ ) = 0. Thus, a norm of the gradient will be a good candidate as convergence criterion. Commenges et al. [4] proposed a criterion based on the metric G−1 that we call the relative distance to maximum (RDM): RDM(& (k) ) =
T
UP (& (k) ) G−1 (& (k) )UP (& (k) ) p
The name “RDM” can be justified by noting that T near the maximum we have UP (& (k) ) G−1 (& (k) )UP (& (k) ) ≈ T (& (k) − &ˆ ) HLP (& (k) )(& (k) − &ˆ ), that is a distance between & (k)
and &ˆ . Using the same metric, by neglecting the effect of T penalization, we have E∗ [(&ˆ − & ∗ ) I(& ∗ )(&ˆ − & ∗ )] ≈ p, where I is the information matrix, &* is the true value of parameters and E* is the expectation under the true probability. Thus, RDM is approximately the ratio of the distance of & (k) from &ˆ over the expectation of the distance between &ˆ and &*. It can also be interpreted as the ratio of the numerical error over the statistical error. This criterion is asymptotically invariant near the maximum to any one-to-one transformation of the parameters. Thanks to its interpretation, the threshold (called *3 ) can be chosen independently from the problem or from the data: it must be lower than 1 and as close as possible to 0. RDM is an indicator of the number of significant digits that are obtained. See Section 5.1.2 for the analysis of the impact of this threshold; 0.1 is a good default value. In all cases, we recommend to run the algorithm from several starting points to check reproducibility and assess the number of significant digits that have been obtained. As a secure stopping rule, one may combine the three criteria, although RDM is clearly the best criterion. In the case where G is not invertible, RDM cannot be computed and the program will stop only when the maximum number of iterations is reached; this case is a failure of convergence.
4.
The NIMROD program
4.1.
Packaging
The program is written in Fortran 90 and comes with a Maple routine which allows the user to calculate sensitivity equations for the ODE system. An open source code is available and can be modified with external accessible subroutines. The code is organized in two levels: a user level that has to be modified to specify a new model, and an optimization part that can be used as a black box for new users. Nevertheless, it is freely modifiable if necessary.
4.2.
Parallel computing
Depending on the number of random effects, number of subjects and number of parameters, the optimization can take from few minutes to several days. Parallel computing offers the possibility of speeding the program. Indeed, calculations presented in Section 3.1 are highly parallel: likelihood and scores are first calculated subject-by-subject, then aggregated. We implemented parallel computing over the subjects of the study. Thus, the program can efficiently use np ≤ n processors. Our experience shows that the speed-up approximately follows the power law of n−0.85 . As an example, treating a nonp linear model of HIV dynamics similar to the one used in Prague et al. [32] with 3 random effects, 149 subjects and 17 parameters takes about 7h30. Using 149 processors for this problem reduces the computation time in theory to 6.4 min (approximated by the power law formula to be 7.5 × 60 × 149−0.85 ); in practice, we observe similar order of magnitude (see Section 5.2 for an illustration in Pharmacokinetics models on real data).
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Table 1 – Comparison of FOCE, MCMC and NIMROD estimations over 100 simulation replicates in one-compartment PK model with absorption: percentage of failure, time to convergence, empirical standard deviation, overall parameters absolute bias and overall parameters RMSE. Simulation scenario {n, ni , %CP } n = 100
ni = 10
%CP = 1.0 %CP = 0.3
ni = 4
%CP = 1.0 %CP = 0.3
n = 20
ni = 10
%CP = 1.0 %CP = 0.3
ni = 4
%CP = 1.0 %CP = 0.3
Failure (%)
Empirical s.d. Overall abs. bias Overall RMSE
FOCE MCMC NIMROD FOCE MCMC NIMROD
51 0 0 8 0 0
0.8 354 100 0.5 345 99
0.147 0.155 0.049 0.051 0.031 0.026
0.204 0.052 0.039 0.056 0.006 0.010
0.266 0.167 0.067 0.076 0.033 0.029
FOCE MCMC NIMROD FOCE MCMC NIMROD
90 0 0 60 0 0
1 245 103 0.7 244 120
0.201 0.241 0.065 0.122 0.105 0.040
0.225 0.181 0.120 0.225 0.037 0.041
0.337 0.320 0.146 0.257 0.119 0.060
FOCE MCMC NIMROD FOCE MCMC NIMROD
51 0 5 19 0 1
0.2 179 96 0.1 179 125
0.261 0.215 0.079 0.113 0.114 0.051
0.252 0.028 0.052 0.074 0.029 0.024
0.362 0.220 0.099 0.135 0.118 0.058
FOCE MCMC NIMROD FOCE MCMC NIMROD
90 0 4 51 0 1
0.2 159 103 0.2 159 129
0.350 0.324 0.108 0.218 0.203 0.063
0.394 0.093 0.139 0.187 0.054 0.055
0.526 0.363 0.187 0.288 0.217 0.090
Technically, MPI (message passing interface) is a library of functions that exploit the existence of multiple processors [26]. It is a freely available internationally recognized standard. It can work on existing networks of workstations on processors with or without shared memory. Even if a super-computer is not available, one can still efficiently use the twelve cores that are available on standard workstations to obtain a speedup higher than 7.
4.3.
Time (s)
Requirements and abilities
The user must specify the number of parameters, the ODE system, the ODE sensitivity systems, the parameters transformations, the statistical model and the observational model. Structurally, there is no limitation in the number of equations and random effects. However, if the model complexity is too high the program can either be time consuming or fail to converge. In our experience, 5 to 6 equations in the system can be handled without problem but the number of random effects must be limited to l ≤ 5. Drylewicz et al. [9] proposed a forward selection of the random effects. Moreover, the program can handle censored data. Penalization as described in Section 2.2 can be set on all parameters or biological parameters only. The user can also easily implement his own penalizations. We advise to carefully choose the priors for each parameter by making a synthesis of the literature. For instance, defining a normal prior taking the mean between maximum and minimum found values (resp. the range divided by two) seems adequate to indicate the mean (resp. the standard deviation). In all cases, we recommend to perform a sensitivity analysis to priors.
NIMROD can operate two ways: inference or predictions. Concerning inference there are several algorithm options: foremost, the user must set the optimization algorithm by choosing either the RVS only, or Marquardt only or a combination of both. Maximum number of iterations (default is 200), starting points (default are priors means) and convergence thresholds (default are *1 = *2 = *3 = 0.1) have to be set. Predictions (see Web-Appendix E for details) with parametric empirical Bayes [19] are automatically done after the inference with the stopping points as input values. Plots are automatically generated.
5.
Illustration in pharmacokinetics
NIMROD can be used in complex non-linear ODE systems without analytic solution. However, pharmacokinetics (PK) models are maybe the most studied ODE systems. Thus, we based illustrations on a one-compartment and a twocompartments PK model with absorption with three random effects.
5.1. Simulation: one-compartment pharmacokinetics model with oral absorption We consider a one-compartment PK model with oral absorption (more details in Web-Appendix A). The amount of drug in the gastrointestinal compartment (AGI ) is described by a firstorder drug absorption equation (Eq. (1a)); the concentration of
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Fig. 1 – Posterior distributions for all the parameters in a one-compartment PK model with absorption obtained with MCMC method with WinBUGS. Burn-in length was 100,000 iterations, then we ran a 50,000-sampling phase with a 100-thinning interval. Convergence diagnostics were acceptable, mixing is displayed. Plain lines on posterior distribution is the density of a normal law for adequate mean and standard deviation.
drug in the plasma (AP ) at any time depends on the relative rate of drug absorption and elimination (Eq. (1b)):
dA GI = −ka AGI , dt
dAP dt
= ka AGI − ke AP .
(a) (1) (b)
The bio-availability cannot be estimated, thus we fixed it equal to one. Initial values are known and given by AGI (0) = dose and AP (0) = 0. Random effects were associated to all parameters but no covariate was introduced. We simulated data for 40 subjects with this one-compartment PK model for a unique initial dose of 100 mg by oral absorption. Parameters values for simulations are presented Table 2 (Rosenbaum [37], chapter 9). Only the plasma concentration CP = AP /V0 was supposed observed at 10 observation times (baseline, 0.5, 1, 1.5, 2, 3, 4, 6, 8 and 10 h). We assume that we observe, with %CP = 0.3: Y1i (tij ) = [CiP (tij )]0.25 + $ij1 ,
because in a one-compartment PK model with absorption, ka and ke are mathematically exchangeable without impacting the trajectories, this is called the flip-flop paradox [14].
$ij1 ∼N(0, %C2P ).
The choice of the penalization for Bayesian methods is presented in Web-Appendix B. However, we took rather tight priors (that can be found in Table 1 in Web-Appendix B)
5.1.1.
NAP and convergence criteria evaluation
To evaluate the quality of the normal approximation in this problem, we estimated the posteriors by the MCMC algorithm using WinBUGS. Fig. 1 shows that the NAP is very good, except for ωV˜ . For comparison of posteriors means, results obtained 0 with NIMROD are presented Table 1 in Web-Appendix B. The behaviour of the main convergence criterion (RDM) during the optimization with NIMROD is monitored in Fig. 2. The RDM decreases when the log-likelihood improves. A zoom of Fig. 2 for the last iterations with a better scale and parameters trajectories over iterations is displayed in Web-Appendix C. To evaluate the link between the RDM and reproducibility, we randomly generated 1000 starting points at 2 standard deviations from the prior mean. We ran NIMROD with a unique convergence criterion: *3 < 1.0. Convergence was reached for 97% of the starting points, otherwise the program was stopped after 1000 iterations. The final RDM value is necessarily smaller than 1.0, but takes randomly different values. The final
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453
Fig. 2 – Trajectories of the RDM and the log-likelihood along iterations during optimization with NIMROD from a random starting point to convergence.
RDM value correlates with the final log-likelihood (see Fig. 3a). At the stopping value, the variability of the log-likelihood values is smaller when the RDM is close to zero. We ran NIMROD again with *3 < 0.01 as a convergence criterion from the same 1000 starting points sample. This led to 96.3% of convergence. Compared to the previous convergence criterion, we observed less variability in the final log-likelihood values (Fig. 3b). When RDM is small enough, variability in &ˆ is reduced.
5.1.2.
Comparison with existing algorithms
We compared the NIMROD program results with two estimation techniques from the R software: the Laplacian first-order conditional method (FOCE) from the PKTools package using the nlme function [21] and the R2WinBUGS package for the MCMC approach. We did a gradient of simulations with different numbers of patients (n = 100 and n = 20), numbers of repeated measurements (ni = 10 with observation times at baseline, 0.5, 1, 1.5, 2, 3, 4, 6, 8 and 10 hours and ni = 4 with observation times at baseline, 1, 2 and 6 hours) and standard deviations of the measurement error (%CP = 0.3 and %CP = 1.0). We generated 100 simulation replicates for each scenario. For NIMROD and MCMC, we selected the same tight
priors as in previous simulations. For FOCE, convergence criteria were internal to PKTools; two main causes of failure were detected: maximum number of iterations reached before convergence and non-invertibility of the Hessian. For NIMROD, convergence criteria were *1 = 10−3 and *2 = *3 = 0.1. For MCMC, we used standard autocorrelation diagnostics, Geweke and Gelman and Rubin statistics tests to ensure the convergence of the chains. We ran 2 chains and monitored a 10,000-sample with a thinning interval of 2 after a burn-in phase of 10,000 iterations. Table 1 presents the percentage of failures (i.e. when the algorithm did not converge), the computation times, the empirical standard deviation in estimates over simulations, the mean absolute bias over all parameters (|E(&ˆ ) − & ∗ |) and the root mean square error (RMSE: 2 1/2
E[(&ˆ − & ∗ ) ] ). Of note, to compute these indicators for MCMC, &ˆ is approximated by the posterior mode of the parameters distributions. In this model, FOCE failed in more than 50% of simulations maybe mainly due to identifiability issues. Thus, NIMROD or MCMC (i.e. Bayesian techniques) should be preferred to FOCE. In this problem, NIMROD is generally faster than the MCMC method. More generally, it has been previously
Fig. 3 – Link between the final RDM and the final log-likelihood at convergence (L( !ˆ )) in NIMROD for 1000 starting points with two values of convergence threshold "3 .
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Table 2 – Comparison of FOCE, MCMC and NIMROD estimations for each parameter in one-compartment PK model with absorption (n = 20, ni = 10 and # CP = 0.3) over 100 simulation replicates: mean posteriors mean and standard deviation, empirical standard deviation, bias, RMSE and 95% coverage probabilities for each parameter in the model. Parameter values
Mean posterior mean
Empirical s.d.
Bias
0.11 0.14 0.10
0.21 0.19 0.08
−4.31 −4.24 −4.16
0.09 0.12 0.07
FOCE MCMC NIMROD
−4.73 −4.66 −4.55
ωk˜ e = 0.25
FOCE MCMC NIMROD
0.21 0.25 0.24
ωV˜ = 0.10
FOCE
0.06
MCMC NIMROD
0.08 0.10
0.05 0.10
0.06 0.05
ωk˜ a = 0.20
FOCE MCMC NIMROD
0.15 0.20 0.19
4.04 0.07 0.09
%CP = 0.3
FOCE MCMC NIMROD
0.28 0.28 0.28
0.06 0.02 0.02
k˜ e = −5.56
FOCE MCMC NIMROD
−5.44 −5.52 −5.60
˜ 0 = −4.19 V
FOCE MCMC NIMROD
k˜ a = −4.60
0
Mean posterior s.d.
RMSE
95% CP
0.119 0.043 −0.041
0.236 0.191 0.082
91.4 93.0 95.5
0.19 0.19 0.05
−0.115 −0.049 0.030
0.228 0.202 0.062
85.2 87.0 94.3
0.11 0.15 0.08
0.22 0.23 0.05
−0.129 −0.061 0.050
0.253 0.237 0.071
88.9 90.0 95.5
0.21 0.06 0.07
0.05 0.05 0.05
−0.043 −0.005 −0.011
0.065 0.047 0.053
100.0 98.0 97.7
0.05
−0.042
0.061
100.0
−0.020 −0.003
0.062 0.046
99.0 98.9
0.06 0.06 0.06
−0.046 −0.002 −0.010
0.076 0.058 0.060
100.0 95.0 100.0
0.02 0.02 0.02
−0.023 −0.022 −0.023
0.029 0.028 0.029
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19.6
shown that WinBUGS computation times were significantly greater for complex non-linear ODE model without analytical solution [10]. We ran again simulations where NIMROD failed with an allowance for a greater number of iterations: convergence was reached in all cases and results remained as good. The MCMC method passed all convergence tests. For all simulations, NIMROD has better empirical standard deviations of estimations (reduced by 11.3%) and RMSE (reduced by 10.3%) than MCMC, with no significant difference in absolute biases. Finally, for the simulation scenario (N = 20, ni = 10 and %CP = 0.3), which is a plausible observation scheme in PK, we computed the posterior mean estimated value, the posterior mean standard deviations, the empirical standard deviation of estimations in simulations, bias,
RMSE and 95% Coverage Probabilities (CP) for each parameters, results are presented Table 2. Some results for other simulation scenarios are presented in Web-Appendix D. For biological parameters, MCMC tends to underestimate the posterior standard deviation whereas NIMROD very slightly overestimates it. This may explain the better CP for NIMROD. Concerning standard deviation of the measurement error, CP are small compared to the expected 95% but this problem is shared by all methods and is probably due to underestimation of the variance of this quantity (see Olofsen et al. [25] for details). However, for most parameters estimated with NIMROD, empirical standard deviations and RMSE are smaller than those estimated with MCMC, with similar biases.
Table 3 – Inference in a one and two-compartments PK model with absorption for APV concentrations in HIV-infected patients: priors and posteriors means and standard deviations. Convergence criteria were "1 = "2 = 0.1 and "3 = 0.3. Parameters (in log) LCVa k˜ e ˜0 V k˜ a k˜ PT k˜ TP ˜T V
Priors mean (sd.) – −5.0 (3.0) −3.0 (3.0) −3.0 (3.0) −5.0 (2.0) −7.0 (2.0) −1.0 (0.5)
One-compartment w/abs. mean (sd.) 11.78
Two-compartments w/abs. mean (sd.) 9.65
−6.25 (0.01) −1.41 (0.04) −1.91 (0.13) – – –
−5.45 (0.01) −2.05 (0.10) −3.39 (0.10) −4.94 (0.01) −5.32 (0.01) −0.58 (0.41)
ωk˜ e ωV˜
0.5 0.5
– 2.49 (0.10)
0.19 (0.01) 0.48 (0.05)
ωk˜ a
1.0
0.45 (0.02)
0.68 (0.05)
%CP
1/% CP
0.66 (0.03)
0.52 (0.02)
0
455
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Fig. 4 – Examples of fits for APV PK dynamics in 4 HIV infected patients from PUZZLE study. Left side presents one-compartment PK model with absorption; Right side is two-compartments PK model with absorption. Triangles are observations. Plain lines are the predicted plasma concentration. Dashed lines represent the “95% measurement error ˆ ± 1.96# C . predictive interval” given by Y P
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90
Perspectives
Ce logiciel reste encore difficile d’utilisation en particulier parce que son param´etrage se fait par lignes de codes et non par des menus d´eroulants ou une interface graphique. Par ailleurs, il ne poss`ede pas de langage de description de mod`ele propre comme par exemple MLXTRAN pour NONMEM et MONOLiX, mais doit ˆetre modifi´e en Fortran. Ainsi, la cr´eation d’un package R permettrait une meilleure diffusion de NIMROD et constitue une priorit´e dans les ´etapes d’´evolution de ce logiciel. Concernant les performances de l’optimisation, nous avons remarqu´e que la convergence est parfois difficile a` atteindre dans des probl`emes de grandes dimensions. Bien qu’il soit difficile de faire la part entre des probl`emes d’identifiabilit´e et des probl`emes de pr´ecision, nous pourrions impl´ementer un passage de NIMROD en quadruple pr´ecision pour gagner en terme de chiffres significatifs dans les calculs (cod´e sur 128 bits au lieu de 64 bits en double pr´ecision). Par exemple, lorsque la surface de la log-vraisemblance p´enalis´ee est particuli`erement plate, une grande pr´ecision dans le calcul du score est n´ecessaire pour se d´eplacer vers le maximum. Par ailleurs, lors de l’int´egration num´erique, la quadrature adaptative gaussienne (QAG) est la troisi`eme fonction la plus coˆ uteuse en temps apr`es le calcul des vraisemblances et des scores non parall´elis´es. Son remplacement par d’autres m´ethodes ne semble pas ad´equat, car des ´etudes de simulations nous ont permis de confirmer sa sup´eriorit´e en terme de pr´ecision et de temps de calcul par rapport a` des m´ethodes trap´ezo¨ıdales et de quasi Monte-Carlo, ainsi, nous avons cherch´e a` optimiser la QAG. Le caract`ere adaptatif de la m´ethode rend les calculs plus longs, mais les r´esultats sont significativement moins biais´es lorsque la fonction a` int´egrer est ´eloign´ee d’une loi normale. La seule optimisation ` l’heure possible reste donc un meilleur choix du nombre de nœuds utilis´es par la QAG. A actuelle, l’utilisateur choisit un nombre minimal (nmin ) et maximal de nœuds (nmax ), la QAG r´ealise ensuite l’´evaluation de l’int´egrale pour tous les nombres de nœuds, en partant de nmin jusqu’`a ce que la pr´ecision soit suffisante. Ceci est long et pourrait ˆetre ´evit´e en
4.2 : Perspectives
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fixant le nombre de nœuds utilis´e. Nous cherchons actuellement `a modifier la fonction de QAG pour que cela soit possible. Par la suite, lors de la premi`ere it´eration de NIMROD, le nombre de nœuds pourrait ˆetre choisi par choix optimal par la QAG puis tenu fixe pour toutes les it´erations suivantes. De mani`ere plus g´en´erale, il serait int´eressant de d´efinir des m´ethodes pour choisir de mani`ere automatique et optimiser les param´etrages (seuil de pr´ecision solveur, dθ, . . .) de NIMROD lors de la premi`ere it´eration puis de les garder fixes. Finalement, certains probl`emes sp´ecifiques de nos collaborateurs sont d´ecrits ci-dessous et restent a` impl´ementer pour prendre en compte une plus large gamme de mod`eles. – Prise en compte d’un mod`ele d’erreur proportionnelle : c’est le cas lorsque l’erreur est suppos´ee proportionnelle a` la grandeur observ´ee. Ceci est possible, mais demande une r´e´ecriture compl`ete de la vraisemblance et des scores. En reprenant les ´equations 3.5 Section 3.2.1.2 le nouveau mod`ele d’observation serait : Yi (tij ) = gm [Xi (tij )](1 + ijm ) m . ∼ N (0, V) ij
– Estimation des points initiaux des syst`emes ODE : cela pourrait se faire en fonction des donn´ees pour prendre en compte des mod`eles o` u les points de d´epart ne sont pas les points d’´equilibre des syst`emes dynamiques. En reprenant l’´equation 3.3 Section 3.2.1.2, la fonction h ne serait plus d´efinie, mais inf´er´ee avec potentiellement des effets al´eatoires. Une solution pr´eliminaire consiste `a fixer les points initiaux du mod`ele math´ematique aux valeurs observ´ees pour chaque patient. Un assouplissement de cette m´ethode serait de fixer les points initiaux du mod`ele math´ematique a` la valeur moyenne observ´ee pour tous les patients et de rajouter des effets al´eatoires. – Estimation des param`etres n’intervenant pas dans les syst`emes ODE : par exemple, si l’effet traitement n’est actif qu’`a partir d’un certain temps η apr`es le d´ebut du traitement Ttrt nous aurions γ˜ (t) = γ0 + βδtrt(t)=1 δt>Ttrt +η . Le param`etre η sera
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a` estimer, mais n’interviendra pas dans le syst`eme ODE. Il en est de mˆeme pour des param`etres qui interviendraient dans l’´equation d’observation, par exemple si l’on suppose qu’on observe qu’une fraction des CD4 et que l’on veut estimer cette fraction. L’estimation est d’ores et d´ej`a possible par des m´ethodes de profil de vraisemblance. Cependant, elles sont longues et fastidieuses `a mettre en œuvre. Une alternative serait l’inf´erence directe, mais celle-ci passe par le calcul des scores qui utilise les ´equations de sensibilit´e. Il n’est donc pas possible d’inf´erer des param`etres ˜ et H ˜ par diff´erenn’y apparaissant pas. Une solution pourrait ˆetre de calculer les U ciation finie pour ces param`etres. Pour finir, dans certains cas NIMROD ´echoue a` estimer les param`etres en mod`eles NLMEODE. Dans ce cas, en g´en´eral, les m´ethodes MCMC classiques ´echouent. Lorsqu’il ne s’agit pas de probl`eme d’identifiabilit´e l’´echec vient pour MCMC de temps de calcul trop longs pour ´echantillonner la distribution a posteriori. Ainsi, nous pourrions penser `a utiliser des filtres particulaires qui reviennent a` faire du MCMC parall´elis´e [Del Moral et al., 2006]. Il s’agit alors d’approximer la distribution a posteriori par un nuage de r´ealisations pond´er´ees al´eatoirement que l’on propage et qui sont s´electionn´ees par des algorithmes d’´echantillonnage d’importance (“importance sampling”) o` u seules les particules de plus gros poids sont conserv´ees et r´epliqu´ees. Cependant, le d´eveloppement d’un tel algorithme d´epasse le cadre de cette th`ese.
Chapitre 5 Causalit´ e et comparaison d’approches R´ esum´ e : Dans ce chapitre, nous mettons en valeur les avantages et les inconv´enients des mod`eles descriptifs compar´es aux mod`eles m´ecanistes dynamiques explicatifs pour quantifier l’effet d’un traitement. Dans le cadre de donn´ees longitudinales observationnelles, nous r´ealisons une ´etude de comparaison de ces mod`eles sur la quantification d’un effet traitement par cART (cod´e de mani`ere binaire oui/non) sur le nombre de CD4 chez les patients infect´es par le VIH. Nous montrons que les r´esultats obtenus par les mod`eles dynamiques sont plus r´ealistes a` long terme vis-`a-vis des connaissances et des donn´ees observ´ees. Mots-cl´ es : causalit´e ; CD4 ; effet des cART ; facteur de confusion ; HIV ; m´ediation ; mod`eles descriptifs ; mod`eles structuraux marginaux (MSM) ; NLME-ODE ; processus stochastiques.
Abstract : The objective is to compare pros and cons of descriptive and mechanistic dynamic explicative models to quantify a treatment effect. In the settings of observationnal longitudinal data, we do a comparison study to investigate the suitability of these models to quantify the effect of cART (binary outcome yes/no) on CD4 in HIV-infected patients. We show that the dynamical model is more realistic for long-term trends regarding knowledge and observed data. Key words : causality ; cART effect ; CD4 ; confusion factor ; descriptive models ; HIV ; marginal structural models (MSM) ; mediation ; NLME-ODE ; stochastic process .
Chapitre 5 : Causalit´e et comparaison d’approches
5.1 5.1.1
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Comparaison des mod` eles descriptifs et dynamiques Probl´ ematique
Il s’agit d’estimer l’effet de cART `a l’aide de donn´ees issues d’une cohorte observationnelle. La d´etermination de l’effet des cART sur l’´evolution des biomarqueurs du VIH est difficile, car le traitement est initi´e et modifi´e en fonction des niveaux de charge virale et du nombre de cellules CD4 [Thompson et al., 2010, Dybul et al., 2002] qui sont aussi pr´edicteurs de leurs num´erations futures [Tarwater et al., 2001]. Comme nous l’avons vu dans la Section 3.1.2, cette probl´ematique peut ˆetre trait´ee a` l’aide de mod`eles structuraux marginaux. Nous nous proposons de faire une comparaison formelle des mod`eles descriptifs (Section 3.1) et explicatifs (Section 3.2) dans le contexte de cette probl´ematique. Ce travail fait suite `a des travaux plus th´eoriques sur la causalit´e r´ealis´es par [Commenges and G´egout-Petit, 2009, G´egout-Petit and Commenges, 2010] pr´esent´es et ´etendus dans la premi`ere partie de l’article situ´e en Section 5.1.4. Il s’agit d’´etendre les notions de causalit´e sur des variables al´eatoires a` des processus. Ainsi, A et Y seront des processus d´ependants et pourrons avoir un effet dynamique de l’un sur l’autre au cours du temps. Le ph´enom`ene causal pourra donc ˆetre repr´esent´e par des graphes dirig´es non acycliques (Figure 5.1). L’application a` laquelle j’ai plus particuli`erement particip´e applique l’approche m´ethodologique de [Cole et al., 2003, Cole et al., 2005, Cole et al., 2007] sur les donn´ees de la Multicentric AIDS Cohort Study (MACS) [kaslow et al., 1987] aux donn´ees de la cohorte GECSA Anrs C03.
5.1.2
S´ election des donn´ ees
Afin d’obtenir des r´esultats comparables sur les jeux de donn´ees de la cohorte GECSA ´ emiologie clinique du SIDA en Aquitaine) `a ceux obtenus sur Anrs C03 (Groupe d’Epid´ la MACS, nous avons s´electionn´e des patients sur le mˆeme sch´ema d’´etude (Figure 5.2). Les crit`eres sont d´efinis ci-dessous.
5.1 : Comparaison des mod`eles descriptifs et dynamiques
95
Facteurs de confusion
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Figure 5.1 : Mod`eles causaux pour des processus : A a un effet sur Y qui a un effet sur A.
– Les patients doivent avoir ´et´e inclus dans la cohorte entre avril 1996 et avril 2005. En effet, 1996 correspond aux premi`eres apparitions de cART. Apr`es 2005 de nombreux nouveaux traitements sont mis sur le march´e rendant l’effet des cART trop h´et´erog`ene pour ˆetre mod´elis´e de mani`ere binaire. – Les patients sous antir´etroviraux simples non cART sont exclus, car l’effet mesur´e est alors plus faible puisque ces traitements sont connus comme sous-optimaux. – Les patients avec des donn´ees manquantes sont exclus, car la GEE ne g`ere pas ce type de donn´ees sans imputation. – Les patients sous traitement a` l’entr´ee de l’´etude sont exclus, car l’effet de la cART est alors non identifiable. – Les patients n’ayant pas une visite tous les 6 mois [±2.5 mois] sont exclus, car pour la GEE les donn´ees doivent ˆetre ´equireparties. Nous nous sommes ensuite assur´e que la s´election n’avait pas cr´e´e de biais cons´equent sur les covariables d’int´erˆet et que les populations d’´etude avaient une description similaire avec la population recrut´ee dans la MACS (Table 5.1).
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All = 6580 (156958 person-‐visit) Patients included after April 1996 Visits after April 2005 Visits before April 1996
All = 2100 (47444 person-‐visit)
Visits with ARV (no HAART) prescibed All = 1970 (36431 person-‐visit) Visits with missing values for CD4 Visits with missing values for Viral load All = 1900 (29601 person-‐visit) Patients with treatment before April 1996 All = 685 (8689 person-‐visit)
Patients without half-‐year visits +/-‐ 2.5 months Patients with less than 1 year follow-‐up (2 visits) All = 212 (1510 person-‐visit)
Figure 5.2 : Diagramme de s´election des patients GECSA pour la comparaison des mod`eles descriptifs et explicatifs. Caract´ eristiques ˆ Age moyen (ann´ees) Femmes(%) SIDA (%) cART (%) Nombre de CD4 (%) <200 200-350 >350 Nombre moyen de CD4 Niveaux de charge virale(%) <500 501-10000 >10000 Niveaux moyens en log10 de charge virale
Temps de base suivi n=212 personnes n= 1298 personnes-visites 36.7 40.6 30.2 28.5 6.1 12.3 0.0 61.2 15.1 11.3 73.1 490 [69 ;1047]
7.6 18.7 73.5 508 [168 ;1092]
15.1 33.5 51.4 4.0 [2.7 ;5.4]
55.1 26.0 18.9 2.7 [1.7 ;4.8]
Table 5.1 : Caract´eristiques des 212 patients infect´es par le VIH s´electionn´es dans la cohorte GECSA Anrs C03.
5.1.3
Aspects techniques et limites
Par la suite, nous avons impl´ement´e chacun des mod`eles descriptifs (avec SAS proc´edure GEE [Karim and Zeger, 1988]), explicatifs en temps discret (avec SAS proc´edure
5.1 : Comparaison des mod`eles descriptifs et dynamiques
97
NLMIXED [Wolfinger, 1999]) et dynamique (avec NIMROD [Prague et al., 2013a]). Nous avons ´etudi´e les diff´erences d’hypoth`eses sur lesquelles reposent ces mod`eles. Concernant les mod`eles descriptifs, nous avons mis en ´evidence une instabilit´e li´ee au choix de la matrice de variance-covariance de travail (Table 5.2). Ce d´efaut avait d´ej`a ´et´e not´e par [Sullivan Pepe and Anderson, 1994]. En effet, pour obtenir des estimateurs non biais´es lors d’une analyse GEE, il faut utiliser une matrice de corr´elation de travail diagonale si certaines des covariables sont d´ependantes du temps comme c’est le cas ici. Matrice de travail Effet traitement ≤ 1 an Mod`ele 4 moy. e.t. Z stat. Diagonale 63.4 37.2 1.71 ´ Echangeable 61.0 15.4 3.96 m-d´ependante 87.1 81.6 1.06 Non structur´ee Convergence
Effet traitement > 1 an moy. e.t. Z stat. 44.0 30.4 1.44 50.3 26.8 1.88 32.2 125.5 0.25 non atteinte
Table 5.2 : Effet du traitement sur le nombre de CD4 dans le mod`ele d´efini par [Cole et al., 2003], mod`ele 4 voir article Section 5.1.4. Impact du choix de la matrice de travail dans la proc´edure GEE pour les mod`eles MSM. L’analyse explicative a aussi montr´e des limites lorsque le mod`ele ODE est trop ´eloign´e de la r´ealit´e. Par exemple lorsque les composantes peuvent prendre des valeurs aberrantes par rapport aux probl´ematiques biologiques trait´ees, tel qu’un nombre de cellules n´egatif ou infini. Dans ces cas, la log-vraisemblance ne peut pas ˆetre calcul´ee par manque de pr´ecision. Si cela arrive pour un grand nombre de combinaisons de param`etres, cela empˆeche l’optimisation. L’aspect m´ecaniste des mod`eles explicatifs s’av`ere donc ˆetre un handicap lorsque les connaissances biologiques d’une dynamique sont mal cern´ees.
5.1.4
Un continuum entre mod` eles descriptifs et explicatifs
Nous montrons qu’il existe un continuum entre les mod`eles descriptifs et les mod`eles explicatifs bien que les capacit´es pr´edictives des mod`eles explicatifs semblent meilleures a` la fois a` court et long terme apr`es comparaison avec des donn´ees observ´ees et les connaissances sur l’´evolution des biomarqueurs apr`es mise sous cART. Ce travail fait l’objet d’un article soumis.
From descriptive to mechanistic dynamical models: effect of HAART on CD4 counts Daniel Commenges1,2 , M´elanie Prague1,2 , Anne G´egout-Petit3,4 and Rodolphe Thi´ebaut1,2 1 Univ. Bordeaux, ISPED, F-33000 Bordeaux, France 2 Centre INSERM U897-´ epid´ emiologie-Biostatistique, F-33000 Bordeaux, France 3 CNRS UMR5251, Univ. Bordeaux I/II, F-33000 Bordeaux, France 4 INRIA CQFD, Univ. Bordeaux I, F-33000 Bordeaux, France June 2013 Abstract The problem of assessing the effect of a treatment on a marker in observational studies raises the difficulty that attribution of the treatment may depend on the observed marker values. This problem has been treated using marginal structural models relying on the counterfactual/potential response formalism. Another approach to causality is based on dynamical models, and causal influence has been formalized in the framework of the Doob-Meyer decomposition of stochastic processes. Causal inference however needs assumptions that we detail in this paper and we call this approach to causality the ”stochastic system” approach. This approach allows incorporating biological knowledge naturally, so that a continuum can be established between descriptive and mechanistic modeling. The mechanistic models involve distinguishing the model for the system and the model for the observations. Indeed, biological systems live in continuous time, and mechanisms can be expressed in the form of a system of differential equations. Inference in mechanistic models is challenging, particularly from a numerical point of view, but these models can yield much richer and reliable results. Because of the difficulty of inference in models based on stochastic differential equations, most models that have been developed are based on ordinary differential equations. The different approaches are illustrated by estimating the effect of highly active antiretroviral treatment (HAART) on CD4+ lymphocytes counts in an observational study of HIV infected subjects.
1
Introduction
sions lead to biased estimates of the treatment effect, typically underestimating it and possibly indicating a negative effect. Robins et al. (2000) developed the marginal structural model (MSM) approach for treating this problem. Essentially the approach is based on choosing a causal model in terms of counterfactual responses to the treatment, the parameters of which are estimated through a weighted generalized estimating equation (GEE). The weights are determined in order to eliminate the bias due to the dependance of treatment attribution on covariates linked to the outcome and are obtained through a model of treatment attribution. The approach has been applied by Hern´ an et al. (2002) and by Cole et al. (2005) for estimating the effect on CD4 counts of zidovudine and of highly active antiretrovial therapy (HAART) respectively. Cole et al. (2007) and Cole et al. (2010) used it for estimating the effect of HAART on viral load
One problem that has appeared as particularly challenging in clinical research is that of assessing the effect of a dynamic treatment regime, that is, a regime where covariates history of a subject up to time t may influence treatment given after t and may also influence the process of interest. A motivating application has been to assess the effect of antiretroviral therapy in HIV infected subjects. As CD4+ lymphocytes are the main target cells of the virus, it is possible to assess this effect on the blood concentration of these cells: CD4 counts are measurements of this concentration, generally given in cells by mL. In observational studies however, the decision to start an antiretroviral therapy may depend on CD4 counts as well as on other covariates. In this setting it has been demonstrated that conventional regres-
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and on AIDS or death, respectively. Robins (1994) also introduced the structural nested models which can be applied to this problem. The dynamic approach to causality which does not use the counterfactual representation can also be applied to this problem. This approach has been pioneered by Granger (1969). Schweder (1970) and then Aalen (1987) generalized this approach and formalized the influence between stochastic processes using the Doob-Meyer decomposition. This was further developed by Didelez (2008); Commenges and G´egoutPetit (2009); G´egout-Petit and Commenges (2010); Eichler and Didelez (2010). Recently, Aalen et al. (2012) have suggested that a discrete-time dynamic model can be considered for studying HAART effect on CD4 counts and viral load. The aim of this paper is to give the assumptions that are needed in the dynamic approach for establishing the causal effect of a treatment in an observational study and to give examples of implementation in both discrete and continuous time. The concept of ”system” is fundamental to ground a causal meaning of influence between stochastic processes. We call the dynamic approach aimed at causal inference the ”stochastic system” approach. We examine models of different degrees of complexity, going from very simple descriptive models to a rather complex mechanistic model, and we compare this approach with the MSM approach. These models are applied for estimating the effect of HAART on CD4 counts. In Section 2, dynamic discrete-time models with different degrees of complexity are studied within the framework of the stochastic system approach. The central role of the Doob decomposition in causal interpretation is highlighted. The concept of ”perfect system” is defined. We begin with a simple perfect system involving only the marker and the treatment. It is shown that in this case, the dynamic approach can estimate the same parameters as the MSM. The assumptions needed for making causal inference in the stochastic system approach are precisely given. Then we explore systems where the marker of interest is influenced by another process, which may act as a confounder. In Section 3 it is argued that we should model separately the biological system which lives in continuous time and the observations which are taken at discrete times. Mechanistic models based on systems of stochastic differential equations (SDE) or ordinary differential equations (ODE) can be developed; in practice ODE-based models are more manageable. Application of these different models is presented in Section 4 for estimating the effect of HAART on CD4 counts, based on the data of a cohort of HIV infected patients (the Aquitaine cohort). Section 5 concludes.
2
2.1 2.1.1
The stochastic system approach to the dynamic treatment problem Introduction to the Doob decomposition Markers as stochastic processes
As in Hern´an et al. (2002), we denote the value of a physiological marker at time t by Y (t) and the value of a treatment given at time t by A(t). The problem is to assess the causal effect of the treatment on the marker in a situation where treatment attribution may be influenced by previous values of the marker. Here we immediately encounter a difficulty which comes from the discrete-time modeling of a phenomenon which actually evolves in continuous time: treatment attribution, if it can depend on Y (t), must be decided after t, at t + δ where δ is a small positive delay. To keep notation simple we will still call A(t) the treatment attributed just after t. However this raises a problem for the Doob decomposition: in a rigorous description of the problem time t and t + δ must be distinguished (see Appendix 6.1). The family of variables (Y (0), Y (1), . . .) and (A(0), A(1), . . .) form stochastic processes in discrete time. Thus it is natural to use the formalism and the results of the stochastic process theory. We will denote the processes themselves by Y and A and, in this context it is common to write Yt for the value of the process Y at the value t of the discrete ”time-parameter”. 2.1.2
Martingales and previsible processes
The Doob decomposition (Doob-Meyer in the case of continuous time parameter) is central to the theory of stochastic processes and can play a central role for formalizing causality. For understanding it, one must first understand the concept of filtration. An intuitive description is possible; for more rigorous definitions see Williams (1991). A filtration {Ft : t ≥ 0} is an increasing family of sigma-fields. In our example, the sigma-field F0 will be generated by A0 , Y0 , that is, the set of all events that we can form with these two variables. Similarly Ft will be generated by the variables (Y¯t , A¯t ), where Y¯t = (Y0 , . . . , Yt ) and A¯t = (A0 , . . . , At ). We note: Ft = σ(A¯t , Y¯t ). Observing Ft can be interpreted as getting the information about Y and A up to t. There is more information available at t than at t−1, so Ft−1 ⊂ Ft . Our filtration is a family of such sigma-fields: {Ft : t ≥ 0}. Next we have to define a previsible process in a given filtration. A process C is previsible if, for all t, the variable Ct is ”mea-
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surable” for Ft−1 . For instance if At = 1{Y (t−1)<300} the events {At = 1} and {At = 0} belong to Ft−1 . The process A defined by At = 1{Y (t−1)<300} for all t, is previsible. On the other hand, if attribution of treatment is randomized such that A(t) = 1 with probability 1/2, then A is not previsible. Finally, a martingale is a process M = (Mt : t ≥ 0) such that, the conditional expectation of Mt given the past is such that: E(Mt |Ft−1 ) = Mt−1 , for all t. 2.1.3
The Doob decomposition
The Doob decomposition of a process is an additive decomposition into a previsible process and a martingale; see again Williams (1991). Applied to Y this gives: Y = Y0 + ΛY + MY , (2.1) where ΛY is a previsible process null at 0 that we call the compensator, and MY is a martingale null at 0. This decomposition exists and is unique (modulo undistinguishability). In continuous time, the same kind of decomposition is called the Doob-Meyer decomposition. We also have the Doob decomposition for A: A = A0 + ΛA + MA , with ΛA a previsible process and MA a martingale. Equation (2.1) means that for all t we have Yt = Y0 + ΛY,t + MY,t . It is illuminating to look at the difference equation which gives the evolution of Y between t − 1 and t: Zt = Yt −Yt−1 = ΛY,t −ΛY,t−1 +MY,t −MY,t−1 . (2.2)
marker Y and a treatment A, we can consider the filtration {F} generated by (Y, A), as described in 2.1.2. {F } Denote by ΛY the compensator of Y in this filtration. In a richer filtration {G} generated by (Y, A, V ) where V is another process, the compensator will be {G} ΛY . However we will restrict to what we call ”physical filtrations”: {F} is a physical filtration if Ft contains only events that may have occurred before or at t (here, t is not just a mathematical parameter but represents physical time). A physical process is a process which generates a physical filtration. Definition 2.1 (Perfect system). A system specified by the filtration {F} is perfect for Y if under the true probability P∗ , the Doob decomposition of Yt is the same in {F} and in any physical filtration {G} which is finer than {F}, that is {F} ⊂ {G}.
2.2 2.2.1
= E(ΛY,t − ΛY,t−1 Ft−1 )
=
ΛY,t − ΛY,t−1 .
The physical law assumption
It is necessary to make explicit the fact that the processes at hand represent real events, occurring to subjects that we label i, i = 1, . . . , n. The events of interest for subject i are denoted by the process X i = (Y i , Ai ), which generates the filtration {F}i , and Y i has a Doob decomposition in this filtration under probability P∗i (the true probability law for subject i): Yti
We have, because of the martingale property and the previsibility of ΛY : E(Zt |Ft−1 )
The simple perfect system
= Y0i + ΛY i ,t + MY i ,t = Y i + φi (Y¯ i , A¯i ) + MY i ,t . 0
t
t−1
t−1
We make the following assumptions. Perfect system assumption: System represented by the filtration {F}i is perfect for Y i , for all i.
That is, if we know ΛY we can compute the expectation of the evolution of Y between t − 1 and t; or Independent innovation assumption: Under P∗ , in other terms, the evolution is equal to something for all i, j and t: ∆MY i ,t+1 ⊥ A¯jt and, for all ¯ Y j ,t . which can be computed from information at t − 1 j 6= i and all t: ∆MY i ,t+1 ⊥ M plus a random term of null expectation. Stability assumption: Under P∗ , φit (., .) = φt (., .) for all t and i; the martingales MY i , for all i, 2.1.4 Perfect system have the same law denoted P∗MY ; Y0i have the We must remember that the Doob decomposition of a same law for all i. process depends on both the probability law and the Note : We cannot say of course that ΛY i ,t and filtration. We can (and we shall) play with different 0 probability laws and filtrations, not completely freely, ΛY i0 ,t for i 6= i are the same process, not even that but for formalizing causality. First, t will mean the they have the same law: what is invariant is the funci physical time. The probability law which is interest- tion φt (., .). Also the Y do not have the same law i ∗ ing in this context is the ”true probability law” P , because the A may have different laws for different which gives rise to the realized events (which may be i’s. The conjunction of the perfect system, the indeobserved). We can consider more or less rich filtrapendent innovation and the stability assumptions are tions. For instance, for the problem of a physiological Page 3 of 14
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called the physical law assumption for Y . To com- difference equation: pletely specify the physical law for Y on horizon tf ∆φt (Y¯t−1 , A¯t−1 ) = β1∗ At−1 + β2∗ . (2.5) one must specify the law of Y0i , φt (., .) for t ≤ tf , ∗ i and the law of the martingale PMY . The laws of X , Summing equations (2.5) for t = 1 to tf we obtain P∗i , may be different because the treatment regimes φtf (Y¯tf −1 , A¯tf −1 ) = β1∗ cum(A¯tf −1 ) + β2∗ tf and thus may be different. The treatment process can also be described by a Doob decomposition Ytf = Y0 + β1∗ cum(A¯tf −1 ) + β2∗ tf + MY,tf , Ptf −1 where cum(A¯tf −1 ) = t=0 At . We can assess the = (2.3) causal effect at horizon t by: CE(a, a0 ) = Ea (Ytf ) − f 0 t t t−1 ∗ Ea0 (Ytf ) = β1 [cum(¯ atf −1 ) − cum(¯ a0tf −1 )]. i ¯ with again the slight difficulty that Yt (rather than Another possibility would be to use a ratio rather i Y¯t−1 ) appears in the compensator because we assume than a difference of expectations. that treatment attribution is decided just after observing Y . The martingales MAi and MY j are in- 2.2.3 Inference dependent for all i, j. Thus, the treatment regime ¯ ti = (Y¯ti , A¯it ), i = 1, . . . , n. up to tf is specified by Ai0 , ψti (.) and the law of Suppose that we observe X f f f ∗ the martingale PM i for t ≤ tf . Finally, the law One important feature of this approach is that we can A of X i = (Y i , Ai ), P∗i , is completely specified by the infer causal effects without knowing the law of Ai . Inwas a very large number of observations law of Y0i , the physical law for Y and the treatment deed, if there ¯i , Y¯ i took a finite number of values we and both A t−1 t−1 regime. i could estimate φ (A¯it−1 , Y¯t−1 ) by a simple mean of the t The treatment regimes may differ, in particular observations. An interesting factorisation of the likebetween subjects who are part of an observational lihood also shows that the distribution of Ai is not study and subjects enrolled in a randomized study. If one knows the physical law for Y , one knows the needed. We use light notations writing the likelihood causal effect of A on Y . For instance if by an in- for variable X f (X) rather than the more rigorous tervention on subject i we give the treatment regime fX (X), where fX (.) is the probability density functhe Ai = a (that is, Ait = at , t = 0, . . . , tf where the at are tion of X. Under the physical law assumption of ¯i fixed values), the law of X i is completely specified if perfect system, the likelihood of observation Xtf , or i i we know the specific physical law for Y ; we denote it equivalently (Z¯tf , A¯tf ) (see equation (2.2)), factors in: ∗a i by P . Note that if A = a, the Doob decomposition tf tf Y Y 2.3 still hold but is degenerate with a zero martingale. i ¯i i ¯ti , A¯it ) = ¯ f ( Z f (Z | Y , A ) f (Ait |Y¯ti , A¯it−1 ). t t−1 t−1 f f More generally, if we know the physical law, we know t=1 t=1 the probability law of Y for any treatment regime. (2.6) Ait
=
Ai0 + ΛAi ,t + MAi ,t Ai + ψ i (Y¯ i , A¯i ) + MAi ,t ,
This factorisation can be easily checked. For inference, provided the same parameters are not involved i ¯i i ¯ i ¯i ¯i It has been conventional to define a particular sum- in both f (Zt |Yt−1 , At−1 ) and f (At |Yt , At−1 ), we can mary of the treatment effect. There are many ways of use the partial likelihood (in the sense of Andersen defining such a summary: for contrasting the effects et al. (1993)): of values a and a0 given to A, any contrast between tf 0 Y i P∗a and P∗a would do. For simplicity and to match f (Zti |Y¯t−1 , A¯it−1 ). (2.7) with the summary used by Cole et al. (2005) we define t=1 the causal effect on horizon tf as It is clear that we do not need to model treatment ati , A¯it−1 ) gives the ”physCE(a, a0 ) = Ea (Ytf ) − Ea0 (Ytf ), (2.4) tribution. The term f (Zti |Y¯t−1 ical law” of the dynamics of Y i and can be expressed ∗a where Ea means expectation under probability P . in terms of φt (., .) and the law of the martingale MY i . More generally, denoting by P∗1 and P∗2 probabil- Specifically, the contribution to the likelihood of subQf i ity laws specified by two different treatment regimes, ject i is tt=1 , A¯it−1 )]. f∆MY,t [Zti − φt (Y¯t−1 one could contrast the causal effects of the two treatIn practice we must make assumptions to obtain ment regimes by CE(P∗1 , P∗2 ) = EP∗1 (Ytf )−EP∗2 (Ytf ). good estimates with a moderate number of observaAs an example, suppose that the physical law is tions. First a Markov assumption leads to : such that the compensator of Y is specified by the i i φt (A¯it−1 , Y¯t−1 ) = φt (Ait−1 , Yt−1 ). 2.2.2
Definition of causal effects
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A very simple parametric form, with (β1 , β2 ) ∈ <2 , is The physical law assumption here is the conjunc: tion of the following assumptions: i ∆φt (Ait−1 , Yt−1 ) = β1 Ait−1 + β2 , Perfect system assumption: System represented by This gives : the filtration {G}i is perfect for Y i , for all i. i φt (Ait−1 , Yt−1 ) = β1 cum(A¯it−1 ) + β2 t.
In terms of Yti this model implies that:
Independent innovation assumption: Under P∗ , for all j and t: ∆MY i ,t+1 ⊥ A¯jt , V¯tj and, for all ¯ j. j 6= i and all t: ∆MY i ,t+1 ⊥ M Y t
Ea (Yti ) = Ea (Y0i ) + β1 cum(¯ at−1 ) + β2 t,
Stability assumption: Under P∗ , φit (., ., .) = φt (., ., .), for all t and i; the MY i have the same law for which is the same model as in Hern´ an et al. (2002) all i. who call β1 the causal parameter. If one makes the assumption that all the martingale increments are inde- We also make the assumption that Y i −→ / XCi V i . The pendent (both in t and i), all the Zti are independent; i process V also admit a Doob decomposition: Vti = we may in addtion make a distribution assumption i V0i + χit (V¯t−1 ) + MV i ,t . For simplicity we make an for the martingale increments, for instance assume additional stability assumption on V . they are distributed as N (0, σ 2 ): this completes the model, up to the law of Ai . The maximum likeli- Stability assumption for the confounder: Under hood estimator βˆ1 of β1∗ can be computed by stanP∗ , χit (.) = χt (.) for all i and t and the martindard regression of Zti on Ait−1 ; it is consistent and gales MV i have the same law for all i. has asymptotic efficiency if the model is well specified. However it is important to note that we use With these assumptions, the Doob decompositions in fact the partial likelihood (2.7) which does not in- of the three components of X i are: C volve the distribution of Ai . That is, the mechanism i i i i of treatment attribution is ignorable (reminiscent of ) + MY i ,t , Yt = Y0 + φt (Y¯t−1 , A¯it−1 , V¯t−1 i i i i ignorable mechanisms of incomplete observations and Vt = V0 + χt (V¯t−1 , A¯t−1 ) + MV i ,t , i i ”reduced model” in Commenges et al. (2007)). ) + MAi ,t , At = Ai0 + ψti (Y¯ti , A¯it−1 , V¯t−1 If one does not want to make such assumptions on (2.8) the law of the martingale, β1∗ can be estimated using with again the slight difficulty that Y¯ti (rather than i an unweighted GEE for estimating the parameters Y¯t−1 )appears in ψti . We assume that all the martini of the model for the expectation of Zt conditional on gales MY i , MAj , MV k for any i, j, k are independent Ait−1 : E(Zti |Ait−1 ) = β1 Ait−1 +β2 . Even if Zti1 and Zti2 and the increments of these martingales are indepenare not strictly independent given A (for instance if dent of the past. Note that the φt and χt do not the martingales do not have independent increments), depend on i but the ψti do because different subjects the independence working assumption is likely to be may have different treatment regimes. better for Z i than for Y i . V can be a confounder or a mediator, of both. For studying the effect of Ai in this system we make that V i is WCLI of Ai , in 2.3 Perfect system with a confounder the further assumption i other words A does not influence V i ; this is noted: 2.3.1 The physical law assumption Ai −→ / XCi V i ; we do not investigate further the probWe examine the case where we can find a physical lem of mediation in the theoretical part of this paper. process V i which modifies the dynamics of Y i . We Thus we will write the second equation of the system 2.8: i i denote by ΛF Y the compensator of Y in the filtration i Vti = V0i + χt (V¯t−1 ) + MV i ,t . Gi i {F}i generated by X and by ΛY the compensator of Y i in {G}i where {G}i is the filtration generated by In the Doob decomposition 2.8 we have in general i i i i XCi = (Y i , Ai , V i ). If ΛGYi = ΛF i A and V −→XCi Y i , so that V i is Y we would say that that V −→XC weak conditional local independence (WCLI) holds a confounder. The corresponding graph of influences for Y i relative to V i (Commenges and G´egout-Petit, (assuming V i is WCLI of Ai ) is depicted in Figure 2009). Saying that X i = (Y i , Ai ) is not a perfect 1, where an arrow from one process to another repi system means that ΛGYi 6= ΛF Y (that is, they are not resents an influence and conversely, lack of an arrow undistinguishable). This can be noted : V i −→XCi represents a WCLI relation. As in the simple perfect system, if one knows the Y i , which reads: V i influences Y i in the system XCi . i i physical law for Y , one knows the causal effect of A ΛGYi can be written as φit (Y¯t−1 , A¯it−1 , V¯t−1 ). Page 5 of 14
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expectation of conditional causal effects: ACE(a, a0 ) = E{CE(a, a0 ; V )}. Note that ACE(a, a0 ) depends on the law of V. Example 1. Suppose that the physical law is such that ∆φt (Y¯t−1 , A¯t−1 , V¯t−1 ) = β1∗ At−1 + β2∗ + β3∗ Vt−1 . Under P∗a and conditional on V = v we obtain: i φt (Y¯t−1 , a ¯t−1 , v¯t−1 ) = β1∗ cum(¯ at−1 )+β2∗ t+β3∗ cum(¯ vt−1 ).
We find :
Figure 1: A system with confounder: Y is the process of interest, A is the treatment process, V is a process which influences both A and Y . Note the arrow from Y to A meaning that process Y influences the treatment process A. Thick arrows pointing toward Y pertain to the physical law for Y , while thin arrows pointing toward A pertain to treatment regime; in some treatment regimes the latter can be broken.
CE(a, a0 ; v) = β1∗ {cum(¯ atf −1 ) − cum(¯ a0tf −1 )}. Here, due to the linearity of the contrast and of the compensator in V , the causal effect does not depend on the value v of V . Thus ACE(a, a0 ) = CE(a, a0 ; v) for any v. Example 2. Suppose that the physical law is such that ∆φt (Y¯t−1 , A¯t−1 , V¯t−1 ) = β1∗ At−1 Vt−1 + β2∗ . Under P∗a and conditional on V = v we obtain: φt (Y¯t−1 , a ¯t−1 , v¯t−1 ) = β1∗
t−1 X
as vs + β2∗ t. on Y . For instance if by an intervention on subject i s=0 we give the treatment regime Ai = a, the law of XCi is completely specified if we know the specific physical We find : law for Y and the law of V i . This is true for any t−1 X treatment regime. CE(a, a0 ; v) = β1∗ vs (as − a0s ). s=0
2.3.2
Conditional, average, and marginal causal effects
Here, the causal effect depends on v. The average causal effect is
As in the simple system of section 2.2 we can sumt X marize the causal effect by well chosen contrasts. In ACE(a, a0 ) = E{CE(a, a0 ; V )} = β1∗ E(Vs )(as −a0s ). this section we omit the index i. In this more coms=0 plex situation however, we can define three types of There is another way to define an effect which gets effects: conditional, average, and marginal. rid of the value of V , and that we call the marginal Conditional causal effect: A conditional causal ef- causal effect. We can define the marginal casual effect as: fect of value of the trajectory A = a versus A = a0 on horizon tf conditional on V = v is: CE(a, a0 ; v) = Ea (Ytf |V = v) − Ea0 (Ytf |V = v).
Marginal causal effect: A marginal causal effect of value A = a versus A = a0 on horizon tf is the expectation of conditional causal effects:
Note that the expectation here does not depend on the treatment regime. The causal effects are defined for each value v of V , and when V is random we have a distribution of causal effects. It may be interesting to define the average causal effect.
MCE(a, a0 ) = Ea (Ytf ) − Ea0 (Ytf ).
Formally this is the same definition as that of the causal effect in the case of a simple perfect system. However here it is marginal and we need know the law of XC to compute it. Because of the linearity of the contrast, the average causal effect is equal to the marginal causal effect : MCE(a, a0 ) = ACE(a, a0 ). Average causal effect: An average causal effect of This would not hold if the contrast was not linear (like value A = a versus A = a0 on horizon tf is the a ratio). Page 6 of 14
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2.3.3
Inference
For inference, we still can use a partial likelihood which does not involve the distribution of A nor V : tf Y
t=1
f (Zt |Y¯t−1 , A¯t−1 , V¯t−1 ).
(2.9)
Like in the simple perfect system, inference can be made by maximum likelihood or GEE. Once we have estimated fZt |Y¯t−1 ,A¯t−1 ,V¯t−1 for all t, we can estimate conditional causal effects. For estimating average or marginal causal effects we must also estimate the probability law of V , which is easy if V is observed.
2.4
The unperfect system case
Assume that the perfect system is XC but we use X, and we observe subjects in an observational study where A is random; we denote by P∗O the probability law of XC . In the filtration {F} the comG pensator of Y under P∗O is φF t = EP∗O (φt |Ft−1 ) (the ”innovation theorem”). This compensator can be ”observed” and hence estimated. The question is whether knowledge of this compensator gives us the marginal causal effect or not. There are two cases and the answer to the question is in agreement with the intuition: in the case where V does not influence A (V −→ / XC A), the observed compensator gives the marginal causal effect; when V −→XC A, that is V is a confounder, the observed compensator does not give the marginal causal effect. In order to prove the result when V −→ / XC A we need to show that and under any P∗ which respect the physical laws assumption for Y and V the {F} compensators of Y are the same. For this we need the following lemma.
where the expectation is taken either with P∗a or P∗ . The computation of this expectation involves integrating using the kernel fV¯t−1 |A¯t−1 ,Y¯t−1 . φGt is the same function whatever the probability which respects the physical law; if the integrating kernel is the same the compensators of Y will be the same, which proves the Theorem. In particular, they are the same under P∗O and P∗a . Thus, estimating the compensator in an observational study allows to know the dynamic of Y on a subject for which A is fixed to the value a. Thus if we ignore processes that influence Y , we can still estimate the marginal causal effect, although not the average causal effect if it is different. For estimating the marginal causal effect there must be ”no unmeasured confounder”, a well known condition, although we speak in terms of processes rather than random variables. Consider the Example 1 of section 2.3. The {G}compensator is : φGt (Y¯t−1 , A¯t−1 , V¯t−1 ) = β1∗ cum(A¯t−1 )+β2∗ t+β3∗ cum(V¯t−1 ). The {F}-compensator is : ¯ ¯ ¯ φF t (Yt−1 , At−1 , Vt−1 )
=
β1∗ cum(A¯t−1 ) + β2∗ t +β ∗ E{cum(V¯t−1 )|Y¯t−1 , A¯t−1 }. 3
If V −→ / XC A, then in virtue of Lemma 2.1, the result of E{cum(V¯t−1 )|Y¯t−1 , A¯t−1 } is the same for all probability laws respecting the physical law, in particular it is the same under P∗O and P∗a . In that case the marginal causal effect is MCE(a, a0 ; v) = β1∗ {cum(¯ atf −1 ) − cum(¯ a0tf −1 )}. β1∗ can be estimated by maximum likelihood as described in Section 2.2.3 from observation of Y and A in an observational study. On the other hand, if V −→XC A, then Lemma 2.1. Let P1 and P2 two probability laws such i E{cum(V¯t−1 )|Y¯t−1 , A¯t−1 } depends on A¯t−1 in a way that: (i) V0 independent of (Y0 , A0 ); (ii) the associwhich depends on the probability law. Thus the obated pdf satisfy the physical law assumption: served {F}-compensator does not give us the marginal fY1t |A¯t−1 ,V¯t−1 ,Y¯t−1 = fY2t |A¯t−1 ,V¯t−1 ,Y¯t−1 , and fV1¯t = fV2¯t causal effect. for all 0 < t ≤ T ; (iii) V −→ / XC A. Then fV1t−1 |A¯t−1 ,Y¯t−1 = fV2t−1 |A¯t−1 ,Y¯t−1 . 3 Generalization to continuous The proof is given in Appendix 6.2. The main result now follows.
time and mechanistic models
3.1
Stochastic differential equations
Theorem 2.1. Under any P which respects the physmodels ical laws assumption for Y and V , the {F} compenA more realistic modeling of the dynamics of CD4 sators of Y are the same. and viral load acknowledges that the biological proBy the innovation theorem, the {F}-compensator cesses live in continuous time. However, since obseris given by : vations are in discrete time we must distinguish between the biological processes and their observations. G ¯ ¯ ¯ ¯ ¯ ¯ ¯ φF t (Yt−1 , At−1 ) = E[φt (Yt−1 , At−1 , Vt−1 )|Yt−1 , At−1 ], Thus we have the Doob-Meyer decomposition for the Page 7 of 14
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physiological process T , representing the concentration of CD4+ T-lymphocytes and another process V which may represent the concentration of viruses. This model must be completed by an equation for the observations. The differential form of the DoobMeyer decomposition is: dTt = dφT,t {T¯t− , V¯t− , A¯t− } + dMT,t , dVt = dφV,t {T¯t− , V¯t− , A¯t− } + dMV,t , ∆At = ∆φA,t {Tt− , Vt− , At− } + ∆MA,t . (3.1) V could play the role of a confounder if it influences both T and A; if V is influenced by A, part of the effect of A on T is mediated by V . The equation for A is given by the operator ∆ because the trajectory of A is not continuous and only makes jumps at visit times. Note that in the inference based on the partial likelihood, as in the discrete-time case, we do not need in fact the model for A, which is supposed to be completely observed. CD4 counts and viral load are observed at times tj , j = 1, . . . , m. For making a model with gaussian errors acceptable, transformations can be used. Noting Yˇj and Vˇj transforms by g1 and g2 of the CD4 counts and viral loads at tj respectively, the observational model could be: Yˇj = g1 (Ttj ) + ε1j , (3.2) Vˇj = g2 (Vtj ) + ε2j ,
and viruses die at a certain rate and so on. These considerations lead to rather complex non-linear differential equations. This kind of models have been initiated by Ho et al. (1995) and developed in further works (Wu, 2005). One advantage of these non-linear models is that they generally admit an equilibrium state, which is indeed observed. The models are made more flexible by introduction of random effects and this was developed by Huang et al. (2006) and Guedj et al. (2007). The ”activated cells model” used by Prague et al. (2012) proved good predictive ability. For describing this model, we distinguish the mathematical structure, the model for the interindividual variability and the model for the observations. The mathematical structure distinguishes quiescent (Qt ), activated (Tt ) and infected (Tt∗ ) CD4+ Tcells: dQt = (λ + ρTt − αQt − µQ Qt )dt, dTt = (αQt − γTt Vt − ρTt − µT Tt )dt, dTt∗ = (γTt Vt − µT ∗ Tt∗ )dt, dVt = (πTt∗ − µV Vt )dt. (3.3) Here, the parameters have biological meanings: λ is the production rate of new CD4+ T-cells, the µ’s are mortality rates, α and ρ are transition rates between quiescent and activated cells, π is the rate of production of virions by infected cells, and γ is the infectivity parameter. The model assumes that the rate of infection of activated T cells is γVt . The model for inter-individual variability of the parameters includes possibly time-dependent explanatory variables (such as the treatment) and random effects. Score tests or Akaike criterion are guides to choose how to model the effect of explanatory variables and where to put the random effects. The treatment effect can be modeled as an effect on the infectivity parameter γ that consequently depends on t through At :
where g1 (.) and g2 (.) are known functions and ε1j and ε2j are Gaussian errors terms. The effect of the treatment can still be assessed by a contrast applied to compensators for treatments a or a0 . More realistic models can be constructed on the basis of knowledge of biological mechanisms. For instance we know that only infected cells can produce viruses, so we should distinguish between uninfected and infected cells. In practice however, stochastic differential equations are difficult to treat statistically log γ(t) = γˇ0 + βAt , (3.4) as well as numerically, and methods are restricted for the moment to rather simple systems: see for instance Picchini et al. (2010), Delattre et al. (2012). So we where we expect β < 0. Random effects can be put on several parameters; restrict in the remaining of the paper to models based based on previous experience with this model we proon systems of ordinary differential equations (ODE). pose to put random effects on λ and µT ∗ as:
3.2
ˇ 0 + uλ ; log µT ∗ = µ Practical mechanistic models based log λ = λ ˇT ∗ 0 + uµT ∗ , (3.5) on systems of ordinary differential where uλ and uµT ∗ are independent normal variables. equations
We can use general principles and the specificity of the problem to develop a mechanistic model. We can take into account that viruses can infect T-cells, that infected T cells can produce viruses, that both T-cells
CD4 counts and viral loads are observations of total CD4+ T-cells and virus concentrations respectively made at times tj , j = 1, . . . , m. A model for the observations of the type of 3.2 can be used with and g1 (x) = (x)0.25 and g2 (x) = log10 (x). This allows
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making a normal measurement error distribution acceptable:
4.2
The models
We used several discrete-time models, close to those used by Hern´an et al. (2002) and Cole et al. (2005) tot 0.25 ˇ for analysing CD4 counts. Our first series of models Yj = (Ttj ) + 2j j = 1, ..., n, (3.6) involves only an equation for the evolution of CD4 ˇ Vj = log10 (Vtj ) + 1j j = 1, ..., n, counts (Y): where Tttot = Qt + Tt + Tt∗ is the total number of CD4 • Model 1: E(Yt ) = γ0 + γ1 cum(A¯t−1 ) + γ2 t; uncells at time t, and the kj are independent Gausweighted GEE sian with zero mean for all k and j. The observation mechanism is generally complicated by a detection • Model 2: E(Yt ) = β0 + β1 cum(A¯t−1 ) limit for the viral load, producing left-censoring for +β3 cum(A¯t−3 ) + β2 t; MSM, weighted GEE deVˇj Jacqmin-Gadda et al. (2000) pending only on CD4 counts In this mechanistic model, it is natural to consider that β in (3.4) is a causal parameter. A more com• Model 3: E(Yt − Yt−1 |Ft−1 ) = β2 + β1 At−1 + parable measure of effect would be the extension of a β3 At−3 ; unweighted GEE tot 0 (T contrast of the type Ea (Tttot ) − E ): here, the a tf f • Model 3’: E(Yt − Yt−1 |Ft−1 ) = β2 + β1 At−1 + expectation is degenerate since the equations are deβ3 At−3 ; maximum likelihood assuming normal terministic: it simply means the value of the solution 0 distribution. for the particular value of the treatment, a or a . We can also add random effects in this models, in which Thus, Model 1 is the naive one-slope regression case we can compute an effect conditional on the ranmodel treated by unweighted GEE; Model 2 has two dom effects and also the average of these conditional different slopes for cumulative treatment effect up to effects. one year and larger than one year as in Cole et al. (2005). We used a constant trend (β2 ) as in Hern´ an et al. (2002). In this MSM, we performed weighted 4 Application GEE with only baseline and time-varying CD4 count to compute stabilized probability-of-treatment weights; 4.1 The sample Model 3 is a dynamic model which is the same as We compared the estimation of the effect of HAART Model 2 for the expectation and which uses unweighted on CD4 counts obtained by the naive, the MSM and GEE, with the advantage that the working indepenthe dynamic approaches. We used real data from dence matrix should be more acceptable; Model 3’ the Aquitaine cohort (Thi´ebaut et al., 2000). Sim- further assumes conditional independence and a norilarly as in Cole et al. (2005) we took the sample of mal distribution which allows computation of the likepatients who were alive, HIV positive, yet untreated lihood. and under follow-up in April 1996 when HAART beCole et al. (2005) used a more complete model uscame available. All patients taking ARV in mono- ing for the weights both probability-of-treatment and or bi-therapy instead of HAART were excluded. For probability-of-censoring. These weights were comapplying the discrete-time models we need to have puted with sex, CD4 count in categories (< 200, 200− equally spaced observations without missing data. In 350 and > 350), viral load in categories (< 401, 401 − the Aquitaine cohort the patients were not regularly 10000 and > 10000) as baseline covariates. The subseen every six months, thus, we selected only patients set of time-varying covariates consisted of indicator for whom a visit with both CD4 count and viral load variable for detectable viral load, CD4 count and dewere available at least two months and a half around tectable log10 viral load. Our Model 4 was identical the theoretical half-year visit date. We selected 212 to the model of Cole et al. (2005) except that these patients and for each of them the follow-up began authors used splines for the effect of viral load in the with the first visit after April 1996 and ended with logistic model for the weights. 1) the last visit at which he or she was seen alive, • Model 4: E(Yt ) = β0 + β1 cum(A¯t−1 ) 2) the last visit before patient discontinued the twice +β3 cum(A¯t−3 ) + β2 t; MSM, weighted GEE for yearly check-up, or 3) April 2005, whichever came treatment and censoring, depending on several first. A total of 1510 person-visits were gathered. We baseline and time-varying covariates (including assumed that HAART was not interrupted until the viral load). end of the follow-up as defined above.
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Treatment Standard Test In the dynamic approach we have to jointly model Model Effect error Z-stat the dynamics of viral load together with that of CD4 counts. This is feasible and leads to a richer descripModel 1 -25.6 9.7 -2.64 tion of the physiological processes. Our second series of models are systems of two or more difference or Model 2 ≤ 1 year -52.1 19.6 -2.66 differential equations. Model 5 is a discrete-time sys> 1 year -10.7 8.4 -1.27 tem of difference equations. It is not necessary here to explicitly put two slopes since the solution of these Model 3 ≤ 1 year 40.5 9.5 4.27 linear difference equations is non-linear in t. The co> 1 year 27.1 6.8 3.95 efficients of the difference equations can be estimated by two separate regressions that can be done by eiModel 4 ≤ 1 year 63.4 37.2 1.71 ther unweighted GEE or, if we assume conditional > 1 year 44.0 30.4 1.44 independence and normal distribution, by maximum likelihood. In that case we can also use Akaike criteTable 1: Effect of treatment on CD4 count: Model 1: rion for model selection. naive regression; Model 2: MSM with simple weights; ˇ Model 3: simple dynamical model, unweighted GEE; • Model 5: Y : CD4 counts; V : log viral load. Model 3’: simple dynamical model, maximum liklei E(Y − Y |F ) = β + β A hood; Model 4: MSM with well-adjusted weights. t t−1 t−1 0 1 t−1 +β2 Yt−1 + β3 Vˇt−1 , E(Vˇt − Vˇt−1 |Ft−1 ) = α0 + α1 At−1 3 is close to the estimate found by Cole et al. (2005) +α3 Yt−1 + α4 Vˇt−1 . (2×40.5 = 81 cells per year) but larger for the second (4.1) slope. Model 4 finds even larger effects but with large Finally, we considered a mechanistic continuous- standard errors. Next, we turn to the multivariate dynamical modtime model. els. Table 2 shows the estimates for the effect of treat• Model 6: the mechanistic model (3.3), com- ment in Models 5 and 6. In model 5 there is a papleted by treatment effect (3.4), random effects rameter for the effect of the treatment on CD4 and on viral load; in Model 6, the effect of the treatment (3.5) and the observation model (3.2). concentrates on the infectivity parameter γ. The absolute values of these parameters are not comparable: 4.3 Results we look only at the direction of the effect and the Table 1 displays the results we obtained for the treat- degree of significance. ment effect on CD4 counts in the first five models. Treatment Standard Test For all these models but Model 3’, GEE was used. Model Effect error Z-stat Models 2 and 4 used weighted GEE with stabilized Model 5 CD4 4.96 9.70 0.51 weights. For these GEE, independence working corlog VL -0.12 0.05 -2.36 relation matrices were used since Pepe and Anderson (1994) showed that GEE using non-diagonal working Model 6 effect γ -2.18 0.29 -7.41 matrices could be biased in case of time-varying covariates, which is the case here. As expected, the naive Model 1 (not correcting Table 2: Effect of treatment on CD4, viral for treatment attribution) indicates a deleterious efload and infectivity: Model 5: bivariate dynamifect of the treatment. Moreover, the MSM Model 2 cal discrete-time model; Model 7: bivariate dynamiwhich computes weights based on only CD4 counts cal continuous-time model; Model 6: activated cells does not correct anything. The results for the dymodel namical Model 3 are consistent with a favourable effect of treatment with two slopes, and are close to In Model 5, we see that treatment has no signifithe results of the dynamical Model 3’ (which makes cant effect on CD4 count but has a significant effect on the conditional independence and normality assump- viral load, by decreasing it. Another interesting pations); only the estimates of the standard errors are rameter is the effect of viral load on CD4. We found different. Model 4 is also consistent with a favourable βˆ = −11.60 with standard error 4.29. This is consis2 two-slopes effect of the treatment, but fails to reach tent with the treatment decreasing the viral load and significance. The estimates of the first slope by Model having a positive effect on CD4 count entirely mediPage 10 of 14
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ated by the decrease of viral load; this fits biological knowledge. Biological knowledge is incorporated in Model 6 in which the treatment may change the infectivity parameter. Here, the effect on CD4 concentration (here we distinguish between CD4 concentration and CD4 count which is a measurement of it) is mediated by viral concentration and by the interaction between virus and infected CD4 cells. Model 6 was fitted by penalized likelihood thanks to the NIMROD program (Prague et al., 2013). The incorporation of biological knowledge in the mechanistic model leads to a large increase of power to detect the effect of treatment (Z = −7.41). Figure 4.3 shows the mean evolution of CD4 predicted by Models 1, 4, 5 and 6 for treated and untreated patients, starting at the mean value of treated patients at the day they start treatment, that is 400 CD4. The naive Model 1 yields a slight increase for treated and a much larger increase for untreated patients. The three other models yield a strong increase for treated patients, but with different shapes. The dynamical models tend to level of. The MSM Model 4 and the mechanistic Model 6 are in very good agreement for the first year, but while the mechanistic model has a set point, Model 4 gives a constant increase every year which is not very realistic. As for the untreated, Model 6 yields an increase which is not credible; Model 4 yields a slight decrease, while Model 6 imposes a constant value of the CD4 counts. Table 3 gives figures of the year-by-year changes for all the models. Note that in models 5 and 6 the changes decrease which is consistent with the CD4 counts reaching a set point.
5
Conclusion
We have stated the assumptions which are needed for grounding causal interpretation in the stochastic system approach. We focussed on the problem of estimating the effect of HAART on CD4 in an observational study. This approach can be extended to processes in continuous time and we argue that distinguishing the system which lives in continuous time and the observations, most often done at discrete times, allows moving towards more realistic representations, and hence more reliable causal interpretations. Describing the stochastic system approach in continuous time was out of the scope of this paper, but we did try such models on the real data of the Aquitaine cohort study and compared them to simpler discrete-time models and MSM. Also, we did not have space to develop the issue of mediation in the theoretical part of the paper, although there are mediations in the activated cell model, and the dynamic
approach seems to be better adapted to cope with this issue than the counterfactual approach (Aalen et al., 2012). As expected, the naive regression model indicated a deleterious effect of the treatment. The MSM model with simple weights did not correct this misleading result. The MSM with more elaborate weights (Model 4) did, but failed to reach significance. A problem with GEE was highlighted by Pan and Connett (2002). In case of time-varying covariates, which is the case here, Pepe and Anderson (1994) noted that GEE results could be biased with non-diagonal working correlation matrix; Fitzmaurice (1995) highlighted that GEE estimates could be very inefficient when using an independent working correlation matrix. See also Emond et al. (1997),Vansteelandt (2007). In contrast, the very simple dynamic model (Model 3) yielded consistent results. Model 5 which jointly models CD4 and viral load also gave consistent results (except for the non-credible increase of CD4 in untreated), with a richer interpretation since it indicates that the effect of HAART on CD4 is mediated by viral load. All these discrete-time models are easy to run. The mechanistic model directly incorporates biological knowledge; for instance we know that the effect of HAART on CD4 is mediated by virus concentration. This leads to a much more powerful test for the parameter of interest. Also the mechanistic model has an equilibrium point, which is indeed observed. In the mechanistic approach, we distinguish the system living in continuous time and observations taken at discrete times. One of the advantages of this distinction is that we would be able to treat a much larger data set: with discrete-time models we must have approximately equally-spaced observations, which rarely occurs in real observational studies. The mechanistic model which does not have this restriction has a much greater flexibility for treating different type of data. Finally the mechanistic model, thanks to random effects, allows representing the inter-subject variability. The drawback of this approach is that it is numerically challenging and requires special software. Another limitation is that we do not have always sufficient biological knowledge for constructing a mechanistic model. In that case we may still construct dynamical models with less (or no) mechanistic assumptions.
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800
Model 1 Model 4 Model 5 Model 6
● ●
●
600
● ●
● ●
● ●
●
●
200
400
Mean CD4 without HAART
600 400 200
Mean CD4 count on HAART
800
Model 1 Model 4 Model 5 Model 6
0
10
20
30
40
50
60
0
Time in months
10
20
30
40
50
60
Time in months
Figure 2: Mean evolution of CD4 predicted by Models 1, 4, 5 and 6 for treated and untreated patients. Predicted trends for each model are presented with different style lines. In the no-treatment group for the mechanistic model, the value is the without treatment equilibrium of the system.
Model Model Model Model Model Model
1 2 3 4 5 6
0 - 1 yr. 7.7 -53.5 49.9 119.2 44.8 111.2
HAART 1 - 2 yr. 2 - 3 yr. 7.7 7.7 29.3 29.3 23.1 23.1 80.2 80.2 42.2 33.0 37.6 5.3
3 - 4 yr. 7.7 29.3 23.1 80.2 23.8 0.3
0 - 1 yr. 58.9 50.7 -31.1 -7.7 34.3 0.0
NO HAART 1 - 2 yr. 2 - 3 yr. 58.9 58.9 50.7 50.7 -31.1 -31.1 -7.7 -7.7 31.7 24.6 0.0 0.0
3 - 4 yr. 58.9 50.7 -31.1 -7.7 17.8 0.0
Table 3: Mean CD4 count change in cells/mL across time in different modeling. Model 1: naive model; Model 2: simple MSM; Model 3: simple dynamical model; Model 4: MSM with well adjusted weights; Model 5: bivariate dynamical discrete-time model; Model 6: activated cells model
6
Appendix section
in this context it is common to write Yn for the value of the process Y at the discrete ”time-parameter” 6.1 Distinguishing measurement time n. In this approach we have to distinguish the disand time to treatment attribution crete time-parameter and the physical time, and we have the correspondence: t = [n/2] for n even and Treatment attribution, if it can depend on Y (t), must t = [n − 1)/2] + δ for n odd, n ≥ 1, so that Y = Y (t) 2t be decided after t, at t + δ where δ is a small positive and Y 2t−1 = Y (t − 1 + δ). The Doob decomposition delay. The family of variables (Y (0), Y (1), . . .) and (Doob-Meyer in the case of continuous time param(A(0), A(δ), A(1 + δ), . . .) form stochastic processes in eter) is central to the theory of stochastic processes discrete time. Thus it is natural to use the formalism and can play a central role for formalizing causality. and the results of the stochastic process theory. We For understanding it one must first understand the will denote the processes themselves by Y and A and, Page 12 of 14
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concept of filtration. An intuitive description is possiStatistical Society: Series A (Statistics in Socible; for more rigorous definitions see Williams (1990). ety) 175 (4), 831–861. A filtration {Fn : n ≥ 0} is an increasing family of sigma-fields. We will be interested in the sigma- Andersen, P., O. Borgan, R. Gill, and N. Keiding (1993). Statistical models based on counting profields generated by the variables Yn and An . Thus the cesses. Springer Verlag. sigma-field F0 will be generated generated by A0 , Y0 , that is, the set of all events that we can form with Cole, S., M. Hern´an, K. Anastos, B. Jamieson, and these two variables. Similarly Fn will be generated J. Robins (2007). Determining the effect of highly by the variables by (Y¯n , A¯n ), where Y¯n = (Y0 , . . . , Yn ) active antiretroviral therapy on changes in human and A¯n = (A0 , . . . , An ). Observing Fn can be interimmunodeficiency virus type 1 rna viral load uspreted as getting the information about Y and A up ing a marginal structural left-censored mean model. to n. Our filtration, which distinguishes information American journal of epidemiology 166 (2), 219–227. at t and information at t + δ, is: {Fn : n ≥ 0}, with Fn = σ(A¯n , Y¯n ) the sigma-field generated by (A¯n , Y¯n ) Cole, S. R., M. a. Hern´an, J. B. Margolick, M. H. Cohen, and J. M. Robins (2005, September). Marginal and A¯n = (A0 , . . . , An ); Y¯n = (Y0 , . . . , Yn ). structural models for estimating the effect of highly active antiretroviral therapy initiation on CD4 cell 6.2 Proof of Lemma 2.1 count. American journal of epidemiology 162 (5), The proof can be made by induction. The result holds 471–8. for t = 1 since fV1¯0 |A¯0 ,Y¯0 = fV2¯0 |A¯0 ,Y¯0 = fV0 . AssumCole, S. R., L. P. Jacobson, P. C. Tien, L. Kingsing fV1¯t−1 |A¯t−1 ,Y¯t−1 = fV2¯t−1 |A¯t−1 ,Y¯t−1 , we prove that we ley, J. S. Chmiel, and K. Anastos (2010, January). have fV1¯t |A¯t ,Y¯t = fV2t |A¯t ,Y¯t . Using marginal structural measurement-error modWe have first that fV1¯t |A¯t ,Y¯t = fV1t |V¯t−1 fV1¯t−1 |A¯t ,Y¯t els to estimate the long-term effect of antiretroviral therapy on incident AIDS or death. American jour(because A −→ / XC V and Y −→ / XC V ). Using the nal of epidemiology 171 (1), 113–22. decomposition (A¯t , Y¯t ) = (Yt , At , A¯t−1 , Y¯t−1 and letting all densities conditioned on (A¯t−1 , Y¯t−1 ), we have Commenges, D. and A. G´egout-Petit (2009, June). A from Bayes theorem: general dynamical statistical model with causal in1 terpretation. Journal of the Royal Statistical SocifV¯t−1 |A¯t ,Y¯t = ety: Series B (Statistical Methodology) 71 (3), 719– 1 1 736. fYt ,At |A¯t−1 ,V¯t−1 ,Y¯t−1 fV¯t−1 |A¯t−1 ,Y¯t−1 R 1 . fYt ,At |A¯t−1 ,V¯t−1 =¯vt−1 ,Y¯t−1 fV1¯t−1 |A¯t−1 ,Y¯t−1 d¯ vt−1 Commenges, D., P. Joly, A. G´egout-Petit, and B. Liquet (2007). Choice between semi-parametric esNext, we have: timators of markov and non-markov multi-state 1 1 1 fYt ,At |A¯t−1 ,V¯t−1 ,Y¯t−1 = fAt |A¯t−1 ,V¯t−1 ,Y¯t fYt |A¯t−1 ,V¯t−1 ,Y¯t−1 . models from coarsened observations. Scandinavian Journal of Statistics 34 (1), 33–52. Because V −→ / XC A we have that At ⊥ A¯t−1 ,Y¯t V¯t−1 . Delattre, M., V. Genon-Catalot, and A. SamThus fA1 t |A¯t−1 ,V¯t−1 ,Y¯t = fA1 t |A¯t−1 ,Y¯t−1 . Since this term son (2012). Maximum likelihood estimation for does not depend on v¯t−1 it can be taken out of the instochastic differential equations with random eftegral in the denominator and deleted from numerator fects. Scandinavian Journal of Statistics. and denominator. It is easy to see that all the other terms are equal under P1 and P2 using the assump- Didelez, V. (2008, January). Graphical models for marked point processes based on local indepentions of the Lemma and our inductive assumption. dence. Journal of the Royal Statistical Society: SeHence the Lemma. ries B (Statistical Methodology) 70 (1), 245–264.
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Fitzmaurice, G. M. (1995). A caveat concerning independence estimating equations with multivariate binary data. Biometrics, 309–317. G´egout-Petit, A. and D. Commenges (2010, January). A general definition of influence between stochastic processes. Lifetime data analysis 16 (1), 33–44. Granger, C. W. J. (1969, August). Investigating Causal Relations by Econometric Models and Cross-spectral Methods. Econometrica 37 (3), 424.
Prague, M., D. Commenges, J. Guedj, J. Drylewicz, and R. Thi´ebaut (2013). NIMROD: A Program for Inference via Normal Approximation of the Posterior in Models with Random effects based on Ordinary Differential Equations. Computer Methods and Programs in Biomedicine in press. Robins, J. (1994). Correcting for non-compliance in randomized trials using structural nested mean models. Communications in Statistics-Theory and methods 23 (8), 2379–2412.
Guedj, J., R. Thi´ebaut, and D. Commenges (2007). Robins, J. M., M. a. Hern´an, and B. Brumback (2000, Maximum likelihood estimation in dynamical modSeptember). Marginal structural models and causal els of HIV. Biometrics 63 (4), 1198–1206. inference in epidemiology. Epidemiology (Cambridge, Mass.) 11 (5), 550–60. Hern´ an, M. a., B. a. Brumback, and J. M. Robins (2002, June). Estimating the causal effect of zi- Schweder, T. (1970). Composable markov processes. dovudine on CD4 count with a marginal strucJournal of applied probability 7 (2), 400–410. tural model for repeated measures. Statistics in medicine 21 (12), 1689–709. Thi´ebaut, R., P. Morlat, H. Jacqmin-Gadda, D. Neau, P. Merci´e, F. Dabis, G. Chˆene, et al. (2000). ClinHo, D., A. Neumann, A. Perelson, W. Chen, ical progression of hiv-1 infection according to the J. Leonard, M. Markowitz, et al. (1995). Rapid viral response during the first year of antiretroviral turnover of plasma virions and cd4 lymphocytes in treatment. Aids 14 (8), 971–978. hiv-1 infection. Nature 373 (6510), 123–126. Vansteelandt, S. (2007). On confounding, prediction Huang, Y., D. Liu, and H. Wu (2006). Hierarchiand efficiency in the analysis of longitudinal and cal Bayesian methods for estimation of parameters cross-sectional clustered data. Scandinavian Jourin a longitudinal HIV dynamic system. Biometnal of Statistics 34 (3), 478–498. rics 62 (2), 413–423. Williams, D. (1991). Probability with martingales. Jacqmin-Gadda, H., R. Thi´ebaut, G. Chene, and Cambridge university press. D. Commenges (2000). Analysis of left-censored longitudinal data with application to viral load in Wu, H. (2005). Statistical methods for HIV dynamic hiv infection. Biostatistics 1 (4), 355–368. studies in AIDS clinical trials. Statistical Methods in Medical Research 14 (2), 171–192. Pan, W. and J. E. Connett (2002). Selecting the working correlation structure in generalized estimating equations with application to the lung health study. Statistica Sinica 12 (2), 475–490. Pepe, M. S. and G. L. Anderson (1994). A cautionary note on inference for marginal regression models with longitudinal data and general correlated response data. Communications in StatisticsSimulation and Computation 23 (4), 939–951. Picchini, U., A. Gaetano, and S. Ditlevsen (2010). Stochastic differential mixed-effects models. Scandinavian Journal of Statistics 37 (1), 67–90. Prague, M., D. Commenges, J. Drylewicz, and R. Thi´ebaut (2012). Treatment monitoring of hivinfected patients based on mechanistic models. Biometrics 68, 902–911.
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Perspectives
Dans cette comparaison des mod`eles descriptifs aux mod`eles m´ecanistes, nous avons montr´e que, pour utiliser des mod`eles MSM, il est n´ecessaire de faire une s´election des patients tr`es importante ; elle r´eduit l’´echantillon de 6580 a` 212 patients. S´electionner les patients en fonction de leurs fr´equences de visites et l’absence de donn´ees manquantes conduit probablement a` exclure les patients pour lesquels le cART est le plus actif et qui ne ` l’inverse, les patients avec des visites tr`es r´eguli`eres font pas de visite tous les six mois. A pourraient ˆetre des patients connaissant de nombreux effets secondaires importants ou n´ecessitant des visites plus r´eguli`eres, car ils sont en ´echec de traitement. Ainsi, cette s´election obligatoire pour pouvoir adopter une approche causale par MSM introduit un biais de s´election si le design de la cohorte n’est pas pr´ealablement ´etabli pour effectuer ce type d’analyse (comme dans la MACS o` u les visites sont ´equir´eparties tous les 6 mois). L’approche par mod`eles dynamiques permet de prendre en compte beaucoup plus de donn´ees et donc d’avoir une estimation plus puissante des effets traitement. Cependant, comme nous avons pu le voir dans la Section 4, les temps de calcul sont importants dans les NLME-ODE et rendent un ´echantillon de taille 6580 difficile `a prendre en compte. De plus, la grande h´et´erog´en´eit´e des individus dans un tel ´echantillon peut entraˆıner un besoin de plus d’effets al´eatoires ; or, pour des raisons de pr´ecision num´erique et de temps de calcul, nous sommes limit´es a` environ 5 effets al´eatoires. En effet, rajouter un effet al´eatoire ´equivaut `a faire une int´egration num´erique suppl´ementaire (voir ´equation 3.7 Section 3.2.2.3). Pour ne pas avoir `a rajouter d’effet al´eatoire suppl´ementaire, mais appliquer une variabilit´e individuelle `a tous les param`etres, nous pourrions prendre en compte une matrice d’effets al´eatoires partag´es par plusieurs param`etres. L’effet al´eatoire sur un param`etre serait une combinaison lin´eaire des ´el´ements de cette matrice d’effets al´eatoires. En reprenant le mod`ele statistique ´equation 3.4 Section 3.2.1.2, nous red´efinissons les effets al´eatoires par :
ui = bv i ,
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o` u ui , le vecteur d’effets al´eatoires sur tous les param`etres est de dimension np ; v i , le vecteur d’effets al´eatoires a` estimer, est de dimension nq ; et, pour des raisons d’identifiabilit´e, b est une matrice triangulaire inf´erieure de np × nq coefficients a` estimer. Cette m´ethode demande l’estimation d’un plus grand nombre de param`etres : (np + 1) × nq au lieu de nq . Cependant, elle a l’avantage que tous les param`etres ont un effet al´eatoire sans pour autant augmenter la multiplicit´e de l’int´egrale dans le calcul de la quadrature adaptative gaussienne. Son impl´ementation permettrait plus de souplesse sans pour autant rendre les calculs trop coˆ uteux en temps et en pr´ecision. Pour prendre en compte un ´echantillon de tr`es grande taille, il est aussi possible de faire une synth`ese bay´esienne de l’information [Ades and Sutton, 2006, Presanis et al., 2011]. En effet, en coupant en sous-groupes l’´echantillon global il est possible de faire plusieurs analyses en prenant comme nouveau a priori, l’a posteriori de l’´etude pr´ec´edente. Cette approche ne pose pas de probl`emes dans un cadre bay´esien classique, soit un ´echantillon compos´e de s sous ´echantillons y¯S = {y 1 , y 2 , . . . y S } : p(θ|y 1 , . . . , y s ) ∝ p(θ|y s )p(θ|y 1 , . . . , y s−1 ), s Y ∝ p(θ|y j )p(θ|¯ y j−1 ) , j=1
avec la convention que p(θ|y0 ) = π(θ) l’a priori sur θ. En revanche, lorsque nous faisons une approximation normale de l’a posteriori, l’hypoth`ese de normalit´e faite successivement sur les lois a posteriori de θ conditionnellement aux observations peut perturber les propri´et´es de l’analyse s´equentielle. Nous avons cherch´e `a savoir dans quelle mesure cela pouvait impacter les r´esultats pour le mod`ele `a cellules cibles. Pour cela nous avons utilis´e les donn´ees de l’essai clinique ALBI [Molina et al., 1999] compos´e de 148 patients que nous avons coup´e en trois sous-´etude de tailles diff´erentes (nA = 50, nB = 25 et nC = 73). Nous avons effectu´e des analyses s´equentielles pour toutes les s´equences des trois sous-´etudes et v´erifi´e la coh´erence des r´esultats obtenus (voir Table 5.3 pour l’analyse s´equentielle A− > B− > C). Nous avons compar´e les courbes des biomarqueurs moyens entre les valeurs `a convergence de l’analyse globale ({A, B, C}) et celles a` convergence de la synth`ese
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d’information bay´esienne (A− > B− > C). Les courbes sont quasi confondues, ainsi nous concluons que les diff´erences de valeurs des param`etres a` convergence s’explique en grande part par une non-identifiabilit´e r´esiduelle (Figure 5.3). L’approximation normale de l’a posteriori ne semble pas trop perturber les propri´et´es bay´esiennes d’une analyse s´equentielle ; cependant une validation th´eorique permettrait de fournir un cadre plus formel `a cette approche.
Figure 5.3 : Courbes moyennes des biomarqueurs : pointill´e analyse globale ({A, B, C}), trait plein synth`ese d’information bay´esienne (A− > B− > C). Pour finir, nous avons analys´e les donn´ees de l’essai clinique PUZZLE [Raguin et al., 2004] avec pour a priori 1) ceux de la litt´erature 2) l’a posteriori de l’analyse d’ALBI avec des a priori de la litt´erature. La valeur de la vraisemblance a` convergence montre la sup´eriorit´e de l’approche par synth`ese d’information bay´esienne. La Figure 5.4 illustre ce gain dans la qualit´e d’ajustement.
Figure 5.4 : Exemple d’ajustement des donn´ees : pointill´e analyse PUZZLE avec les a priori de la litt´erature, trait plein synth`ese d’information bay´esienne avec ALBI.
A posteriori {A, B, C} A B C ( L θ, y) -1066.5 -418.0 -230.1 -475.0 P -1070.8 -414.5 -225.7 -475.2 L (θ, y) It´erations 403 57 415 238 Param` etres biologiques : Param`etre moy (et.) moy (et.) moy (et.) moy (et.) moy (et.) ˜ 2.55 (1.90) 1.70 (0.10) 2.02 (0.27) 2.13 (0.12) 1.96 (0.17) λ µ ˜T ∗ -0.05 (0.68) -1.61 (0.05) -1.04 (0.16) -0.78 (0.07) -1.30 (0.11) µ ˜Q -9.00 (1.00) -9.34 (0.99) -8.98 (0.99) -9.00 (0.99) -9.31 (0.99) α ˜ -4.00 (2.00) -4.07 (0.13) -3.80 (0.36) -3.34 (0.24) -3.70 (0.22) ρ˜ -4.34 (1.38) -7.88 (1.34) -5.74 (1.08) -4.63 (1.15) -5.96 (1.08) µ ˜T -2.59 (0.34) -3.53 (0.03) -3.07 (0.17) -3.07 (0.05) -3.43 (0.13) γ˜0 -5.76 (4.02) -5.14 (0.12) -6.05 (0.19) -6.27 (0.04) -5.76 (0.09) π ˜ 4.04 (2.66) 2.75 (0.67) 3.94 (0.72) 3.96 (0.59) 3.41 (0.68) µ ˜V 2.90 (0.68) 3.16 (0.66) 3.0 (0.66) 2.91 (0.43) 3.40 (0.66) Effets traitement : Param`etre moy (et.) moy (et.) moy (et.) moy (et.) moy (et.) βAZT +3T C -1.10 (0.37) -1.13 (0.10) -0.96 (0.15) -0.73 (0.03) -0.78 (0.06) βd4T +ddI -1.10 (0.37) -1.45 (0.13) -1.02 (0.16) -0.80 (0.03) -1.05 (0.07) ´ Ecarts-types des effets al´ eatoires : Param`etre m´ediane moy (et.) moy (et.) moy (et.) moy (et.) ωλ 0.10 0.26 (0.02) 0.25 (0.06) 0.22 (0.01) 0.24 (0.04) ωµT ∗ 0.37 0.25 (0.02) 0.24 (0.06) 0.21 (0.01) 0.25 (0.03) ´ Ecarts-types des erreurs de mesures : Param`etre moy (et.) moy (et.) moy (et.) moy (et.) σCV 0.58 (0.01) 0.61 (0.02) 0.46 (0.004) 0.51 (0.02) σCD4 0.20 (0.005) 0.20 (0.008) 0.21 (0.003) 0.20 (0.01)
A priori A->B->C -477.6 -480.1 45 moy (et.) 1.94 (0.11) -1.23 (0.07) -9.21 (0.99) -3.74 (0.13) -5.22 (0.56) -3.33 (0.08) -5.73 (0.07) 3.46 (0.37) 3.29 (0.37) moy (et.) -0.78 (0.05) -1.04 (0.06) moy (et.) 0.24 (0.04) 0.23 (0.03) moy (et.) 0.50 (0.02) 0.20 (0.007)
B− > A -171.7 -166.3 27 moy (et.) 2.01 (0.17) -0.95 (0.12) -8.98 (0.99) -3.54 (0.24) -4.99 (0.88) -3.11 (0.12) -6.06 (0.13) 3.84 (0.51) 3.05 (0.50) moy (et.) -0.81 (0.10) -0.91 (0.10) moy (et.) 0.23 (0.09) 0.21 (0.06) moy (et.) 0.47 (0.06) 0.21 (0.01)
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Table 5.3 : Analyse s´equentielle de l’information pour trois sous-groupes de l’essai ALBI.
PARTIE III : Vers une m´ edecine personnalis´ ee dans le domaine du VIH
Chapitre 6 Optimisation de la dose de traitement R´ esum´ e : Dans ce chapitre, nous rappelons quelles sont les propri´et´es d’ajustement aux donn´ees et de pr´ediction que doit poss´eder un mod`ele afin d’ˆetre valide comme outil pour la m´edecine personnalis´ee. Nous proposons la construction d’intervalles d’encadrement des pr´edictions pour prendre en compte l’incertitude et les erreurs de mesure sur les observations faites ou a` venir. Puis, apr`es avoir introduit diff´erentes approches statistiques disponibles pour optimiser les traitements, nous montrons que les propri´et´es des syst`emes dynamiques sont un avantage pour s’affranchir de fonctions de coˆ ut habituellement utilis´ees. Pour finir, nous proposons une strat´egie adaptative de l’individualisation de la dose de traitement bas´ee sur une approche bay´esienne. Mots-cl´ es : Bernstein von-Mises ; cART ; choix de mod`eles ; dose cible ; erreur de mesure ; fonction de coˆ ut ; incertitude ; individualisation ; intervalle de cr´edibilit´e ; m´edecine personnalis´ee ; MCMC ; M´etropolis-Hastings ; NMLE-ODE ; nombre de reproduction de base (R0 ) ; optimisation des traitements.
Abstract : In this chapter, we recall the properties of quality of fit and prediction that a model needs to possess in order to be a valid tool for personalized medicine. We build confidence and credible intervals for predictions to account for uncertainty and measurment error on past and future observed data. Then, we introduce different statistical approaches available to optimize treatments and we show that dynamical systems properties are an advantage to be free of a cost function usually used in those problematics. Finally, we propose an adaptive treatment strategy for drug dose individualization based on bayesian approach. Key words : Bernstein von-Mises ; cART ; cost function ; credibility interval ; individualization ; MCMC ; measurement error ; M´etropolis-Hastings ; model choice ; NLME-ODE ; personalized medicine ; targeted dose ; treatment optimization ; reproductive number (R0 ) ; uncertainty.
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Les probl´ ematiques li´ ees ` a l’optimisation des traitements
Dans les parties pr´ec´edentes, nous avons vu qu’il ´etait possible de manipuler les mod`eles dynamiques et d’estimer, grˆace `a NIMROD, l’ensemble des param`etres sous r´eserve d’identifiabilit´e du syst`eme. Nous avons aussi vu que les mod`eles descriptifs sont moins adapt´es a` l’analyse des biomarqueurs du VIH que les mod`eles explicatifs dynamiques. Cependant, pour pouvoir envisager de faire de l’optimisation de traitement il faut s’assurer des bonnes propri´et´es pr´edictives des mod`eles.
6.1.1
´ Evaluation des qualit´ es des mod` eles
6.1.1.1
Qualit´ e d’ajustement
La qualit´e d’ajustement peut ˆetre ´evalu´ee a` l’aide de plusieurs indicateurs et permet de choisir le mod`ele le plus ad´equat aux donn´ees en prenant en compte des arguments de d´eviance et de complexit´e. Des crit`eres, par exemple le coefficient de d´etermination adapt´e aux cas non lin´eaires [Dougherty et al., 2000], l’erreur quadratique moyenne [Wang and Bovik, 2009] ou encore l’observation des crit`eres d’Akaike sur la vraisemblance (AIC et BIC [Akaike, 1998]), permettent d’avoir une id´ee de l’erreur r´esiduelle entre la courbe d’ajustement et les donn´ees. Cependant, ils ne prennent pas en compte la dimension bay´esienne de notre approche. Un crit`ere bay´esien classique pour ´evaluer la qualit´e d’ajustement et de pr´ediction du mod`ele est le facteur de Bayes [Kass and Raftery, 1995]. Cet indicateur est critiqu´e, car il est tr`es sensible aux choix des a priori et n’est pas correctement d´efini pour des a priori non informatifs. De plus, il est souvent difficile ou impossible a` calculer. Ainsi, le crit`ere d’information de d´eviance (DIC) qui en est une approximation a ´et´e propos´e par [Spiegelhalter et al., 2002]. En notant D(θ) la d´eviance d´efinie comme deux fois l’inverse de la log vraisemblance (−2L(θ, y)), le DIC s’´ecrit : DIC(θ) = E [D(θ)|y] + {E [D(θ)|y] − D (E [θ|y])} .
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L’interpr´etation du DIC, sous cette formulation, est critiqu´ee dans des mod`eles non lin´eaires et dans les mod`eles avec des donn´ees manquantes, car il est trop variable d’un mod`ele `a un autre [Celeux et al., 2006]. En alternative, [Liquet et al., 2007] propose d’utiliser le LCVa (approximate Likelihood cross validation criterion en anglais) qui se base sur la vraisemblance elle-mˆeme et sur une approximation de la distance de Kullback-Leibler. De plus, il pr´esente l’avantage d’ˆetre rapide `a calculer. Pour deux mod`eles dont les vraisemblances sont proches, le plus parcimonieux sera choisi ; le LCVa doit ˆetre aussi petit que possible. Nous notons H L (resp. H LP ) la d´eriv´ee seconde de la vraisemblance non p´enalis´ee (resp. p´enalis´ee), en reprenant les notations de la section 3.2.2.3, le LCVa s’´ecrit :
LCVa (θ) = −
1 L(y, θ) − T r(H −1 (θ)H (θ)) . P L L n
Ce crit`ere sera utilis´e pour s´electionner le mod`ele qui a un meilleur ajustement aux donn´ees. Cela peut permettre de faire un choix entre diff´erents mod`eles m´ecanistes dynamiques en accord avec les donn´ees et revient a` tester la v´eracit´e d’une hypoth`ese biologique en confrontant un mod`ele test aux mod`eles pr´ec´edents. Par exemple, cela peut permettre de pr´eciser la voie m´etabolique d’action d’un traitement comme nous avons pu le faire dans le cadre de travaux annexes a` cette th`ese o` u nous avons ´etudi´e l’effet de l’Interleukine-7 (IL7) sur la restauration du syst`eme immunitaire 1 .
6.1.1.2
Param` etres individuels pour la pr´ ediction
Il est important de bien distinguer la qualit´e d’ajustement et la capacit´e pr´edictive d’un mod`ele. En effet, un bon ajustement n’implique pas toujours de bonnes qualit´es 1. Nous avons compar´e diff´erents mod`eles d’effet d’injections d’IL7 (sur la cr´eation, la prolif´eration et la survie des CD4 na¨ıfs et m´emoire). Par le LCVa , nous avons mis en valeur un effet significatif de l’IL7 sur l’augmentation de la survie des CD4 et l’augmentation de leur prolif´eration apr`es chaque cure d’IL7. Dans la suite de ce travail, nous avons simul´e des sc´enarios de variation des effets de l’IL7 pour des traitements r´ep´et´es lorsque le patient passe en dessous de 500 cellules CD4 par mm3 afin d’´evaluer la possibilit´e de maintenance du syst`eme immunitaire. Les r´esultats sont encourageants. En effet, mˆeme dans les sc´enarios les plus pessimistes, il serait possible de maintenir les patients 50% du temps au-dessus de 500 CD4 pendant deux ans avec seulement une injection tous les 135 jours soit environ 4 injections en 24 mois. Ces chiffres sont ` a mettre en perspective avec 12% du temps pass´e en dessus de 500 CD4 en deux ans pour des patients sans cure successive d’IL7. Ce travail fait l’objet d’un article soumis
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de pr´ediction : pour nous repr´esenter ce ph´enom`ene, nous pouvons citer le cas des mod`eles sur param´etr´es [Reichert and Omlin, 1997]. Comme nous l’avons vu au Chapitre 3, nous nous pla¸cons dans le cadre de mod`eles NLME-ODE, nous nous int´eressons aux pr´edictions individuelles. Pour chaque patient, en utilisant un sous-ensemble de ses donn´ees, nous calculons ses param`etres individuels a` l’aide d’estimateurs param´etriques empiriques de Bayes [Morris, 1983, Kass and Steffey, 1989] (PEB). En reprenant les notations de la Section 3.2.1.2, l’estimateur PEB s’´ecrit : ˆ + βˆT z(t)i + uˆi |F i , ξˆi |Fji = φ j
(6.1)
ˆ et β ˆ sont les valeurs MAP (voir Section 3.2.2.3) des param`etres fixes estim´es sur un o` uφ ´ ´echantillon de population excluant les valeurs que l’on cherche `a pr´edire. Eventuellement toutes les donn´ees du patient i peuvent ˆetre exclues pour ne pas avoir a` faire de r´eactualisation en fonction de l’horizon de la pr´ediction. Les effets al´eatoires individuels uˆi |Fji sont obtenus en maximisant la vraisemblance p´enalis´ee individuelle du patient i en utilisant j observations de ses biomarqueurs (Yi1 , . . . , Yij ) : uˆi |Fji = argmaxu∈Rnq ln p(yi1 , . . . , yij |θ M AP , zi , u) − J(θ M AP , u) . Ces valeurs peuvent ˆetre obtenues par un algorithme d’optimisation de type Marquardt∗
Levenberg. Nous notons ξ i les vraies valeurs des param`etres du patient i. La preuve th´eorique de la convergence de la loi a posteriori de l’estimateur PEB, not´ee p(ξ i |Fji ), vers les vraies valeurs des param`etres du patient fait intervenir une double asymptotique o` u le nombre de patients et le nombre de mesures du patient pour lequel nous faisons les pr´edictions tendent vers l’infini. Nous ne d´emontrons pas ici th´eoriquement la convergence, cependant nous l’´etayons dans la partie perspective de cette Section. Nous avons v´erifi´e empiriquement cette propri´et´e et nous admettons : p(ξ i |Fji ) −→ δξi ∗ . j→+∞
(6.2)
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Qualit´ e de pr´ ediction
Dans l’ensemble de cette Section, nous utilisons comme exemple fil conducteur la pr´ediction individuelle des trajectoires de 10 patients s´electionn´es de mani`eres al´eatoires parmi les patients de l’essai clinique ALBI [Molina et al., 1999], exclus de l’analyse principale. La mod´elisation utilis´ee pour l’ajustement aux donn´ees et les pr´edictions est celle pr´esent´ee dans l’article section 6.2.3 [Prague et al., 2012a]. Pour les pr´edictions, une, deux ou les cinq premi`eres observations de chaque biomarqueur ont ´et´e utilis´ees.
Figure 6.1 : Graphe des r´esidus pour l’´evaluation de la qualit´e des pr´edictions : (A,B) couples valeurs observ´ees / valeurs pr´edites pour la charge virale d´etectable et le nombre de CD4 transform´es (C,D) Tendance temporelle de qualit´e de pr´ediction pour la charge virale et le nombre de CD4 transform´es. Les graphes sont `a mettre en relation avec les erreurs de mesures observ´ees sur chacun des biomarqueurs repr´esent´ees en pointill´es.
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Les param`etres ξˆi |Fji sont utilis´es dans un solveur d’´equations diff´erentielles afin de
pr´edire la trajectoire de chaque composante de chaque patient not´ee X(t, ξˆi |Fji ). Notons h i ˆi i ) = g X(t, ξˆi i ) . La validit´e i la pr´ediction de la ki`eme composante observ´ee Yˆk|F (t, ξ i k |Fj |Fj j
des pr´edictions peut ˆetre ´evalu´ee visuellement par un graphe des r´esidus montrant les couples valeurs observ´ees / valeurs pr´edites (voir Figure 6.1(A) et Figure 6.1(B)) :
ˆi i ); Yi (t ) i Yˆk|F (t , ξ i ij |Fj k ij j
i=1,...,n,j=1,...,ni ,k=1,...,K 0
.
De mˆeme, pour v´erifier qu’il n’y a pas de tendance temporelle dans la qualit´e des pr´edictions, un second graphe des r´esidus peut ˆetre trac´e et compar´e `a une ligne horizontale passant par l’origine (Figure 6.1(C) et Figure 6.1(D)) :
h i ˆi i i tij , Yˆk|F i (tij , ξ |F i ) − Yk (tij ) j j
i=1,...,n,j=1,...,ni ,k=1,...,K 0
.
Une alternative consiste a` tracer des intervalles pour ´evaluer la fiabilit´e des pr´edictions. La Figure 6.2 pr´esente une illustration. Il s’agit de prendre en compte l’incertitude statistique et d’erreur de mesure. – Intervalle de pr´ ediction de l’erreur de mesure : Nous faisons l’hypoth`ese que la variabilit´e entre les observations et les pr´edictions n’est issue que des erreurs de mesure dans le mod`ele suppos´ees gaussiennes. Cet intervalle de pr´ediction ne prend ` 95%, pour k = 1, . . . K 0 , pas en compte l’incertitude qui existe sur l’estimation. A cet intervalle s’´ecrit : h i ˆi i ) ± 1.96σˆ . i Yki (t) ∈ Yˆk|F (t , ξ i ij k |Fj j
– Intervalle de cr´ edibilit´ e : Aucune solution analytique n’existe de mani`ere g´en´erale dans les probl`emes NLME-ODE, nous passons donc par une phase de r´e´echantillonage. Avec l’approximation normale des a posteriori, il est possible de tirer au hasard L r´ealisations des effets fixes et des effets al´eatoires sur leurs lois normales
6.1 : Les probl´ematiques li´ees `a l’optimisation des traitements
125
approximantes. En effet, utiliser des m´ethodes MCMC classiques serait beaucoup plus long pour un gain de pr´ecision faible. Ainsi, par sommation il est possible d’obo n l , de la loi a posteriori p(ξ i |Fji ) des param`etres tenir un L-´echantillon, ξ i |Fji l=1,...,L
individuels du patient i sachant l’information Fji : l
l
l
ξ i |Fji = φl + β T z(t)i + ui |Fji , avec, l
(φl , β T ) ∼ N (θ M AP , H −1 (θ M AP )), I −1 uˆi |F i l j ui |Fji ∼ N uˆi |Fji , . j Il est donc possible d’´echantillonner les L trajectoires a posteriori associ´ees, cet o n il i (t , ξ ) et constitue des r´ealisations de la loi ´echantillon est not´e Yˆk|F i i ij |Fj j
l=1,...,L
a posteriori des trajectoires du patient i sachant l’information Fji . Les quantiles `a 5% et 95% de cette s´erie de donn´ees sont les intervalles `a 95% de cr´edibilit´e, pour k = 1, . . . K 0 , cet intervalle s’´ecrit : Yki (t)
∈ q5%
n
il i Yˆk|F i (tij , ξ |F i ) j j
o
l=1,...,L
n o il i ; q95% Yˆk|F i (tij , ξ |F i ) j j
l=1,...,L
.
Cependant, faire de l’extrapolation avec un mod`ele trop simple et mal sp´ecifi´e peut amener a` des r´esultats trop optimistes. En effet, les intervalles a` 95% de pr´ediction n’englobera que les points pour lesquels il n’y a pas ou peu d’erreur de mesure. – Intervalle pr´ edictif : Afin de prendre en compte un intervalle de pr´edictibilit´e global, nous prenons en compte l’erreur commise sur la pr´ediction (`a l’aide de l’intervalle de cr´edibilit´e) et l’erreur de mesure possible (`a l’aide l’intervalle de pr´ediction de l’erreur de mesure). En supposant l’ind´ependance entre ces deux sources d’erreur, n o il ˆ i l’´ecart-type de Yk|F i (tij , ξ |Fji ) not´e κˆik|F i est calcul´e empiriquement sur cet j
l=1,...,L
j
Chapitre 6 : Optimisation de la dose de traitement
126
´echantillon. En faisant une approximation gaussienne des trajectoires a posteriori, a` 95%, pour k = 1, . . . K 0 , cet intervalle s’´ecrit : "
r
2
#
ˆi i Yki (t) ∈ Yˆk|F σˆk 2 + κˆik |F i . i (tij , ξ |F i ) ± 1.96 j j
j
Figure 6.2 : Intervalles pour ´evaluer la fiabilit´e des pr´edictions `a 95% pour les patients 316 et 15 de l’essai clinique ALBI en fonction du nombre d’observations utilis´ees pour les pr´edictions. Les intervalles a` 95% d’erreurs de mesure sont mat´erialis´es par la zone gris´ee. Les intervalles a` 95% de cr´edibilit´e sont en tirets. Les intervalles a` 95% de pr´edictibilit´e sont en pointill´es. Les triangles repr´esentent les observations de la charge virale (`a gauche) et du nombre de CD4 (`a droite). De mani`ere g´en´erale l’intervalle pr´edictif, sommant l’incertitude de mesure et d’estimation, sera plus grand que les intervalles de cr´edibilit´e et d’erreur de mesure. Par contre, la relation entre l’intervalle de cr´edibilit´e et celui d’erreur de mesure d´epend du niveau d’information recueillie pour le patient. Plus elle est grande plus l’intervalle de cr´edibilit´e
6.2 : Adaptation de la dose
127
aura tendance a` devenir plus petit que l’intervalle d’erreur de mesure, comme le montre le patient 15 de la Figure 6.2. Cette propri´et´e montre bien le b´en´efice que peut apporter une mod´elisation par rapport a` l’observation seule des biomarqueurs.
6.1.2
Optimisation et th´ eorie de la commande
Sous r´eserve d’une consistance et d’une robustesse suffisante, c.-`a-d. que le mod`ele pr´edictif soit suffisamment r´ealiste pour imiter les donn´ees observ´ees, nous voulons construire une strat´egie adaptative de traitement. Selon la d´efinition de [Lavori and Dawson, 2008], c’est une r`egle pour adapter le traitement d’un patient en prenant en compte a` la fois son historique de traitement et son historique de r´eponses aux traitements. Nous sommes dans le cadre d’un probl`eme de d´ecision : il est n´ecessaire de d´efinir le type d’actions pouvant jouer sur le syst`eme et une fonction objectif pour ´evaluer l’impact de chaque action. La fonction objectif peut ˆetre d´efinie de mani`ere explicite par exemple un seuil `a atteindre sur les biomarqueurs. Dans ce cas, les m´ethodes de simulations par ´echantillonnage sont particuli`erement indiqu´ees, car elles permettent de simuler les cons´equences attendues d’une action [Dorazio and Johnson, 2003]. Une alternative consiste `a d´efinir une fonction objectif aussi appel´ee fonction de coˆ ut (resp. gain) et de chercher a` la minimiser (resp. maximiser) [Joshi, 2002, Kirschner et al., 1997]. Cependant, la d´efinition de cette fonction d´epend de la question pos´ee : pond´eration entre toxicit´e et efficacit´e [Agur et al., 1988], mise en avantage d’interruptions de cART [Bajaria et al., 2004, Adams et al., 2004], meilleur temps d’initiation de la cART [Kirschner and Webb, 1996].
6.2 6.2.1
Adaptation de la dose D´ efinition de la fonction objectif
La fonction objectif d´epend de l’enjeu principal de la strat´egie : pour la r´eduction de dose, il s’agit de chercher l’am´elioration de la qualit´e de vie du patient en r´eduisant les effets secondaires de la cART sans diminuer son efficacit´e. Or, d´efinir une fonction de coˆ uts
Chapitre 6 : Optimisation de la dose de traitement
128
en effets secondaires est difficile, car cela demande de les quantifier. Cette quantification d´epend de l’opinion de chaque patient ou de chaque clinicien ce qui limite les possibilit´es de g´en´eralisation de la m´ethode. Pour s’affranchir de la d´efinition d’une fonction de coˆ ut analytique, il est possible de la d´efinir de mani`ere intuitive : – Plus la dose administr´ee est importante plus le coˆ ut li´e aux effets secondaires est important, voir Figure 6.3.C1. – Plus la charge virale est grande plus le coˆ ut de l’infection est important, voir Figure 6.3.C2 ; or, pass´ee une certaine dose la charge virale devrait d´efinir nulle. – Il est plus coˆ uteux pour le patient d’avoir une infection non contrˆol´ee que des effets secondaires li´es `a la dose de traitement administr´ee. La meilleure dose (not´ee dcrit ) est donc la plus faible dose pour laquelle l’infection est contrˆol´ee, voir Figure 6.3C.
Figure 6.3 : Caract´erisation graphique de la meilleure dose de traitement (dcrit ) par une fonction de coˆ ut d´efinie de mani`ere implicite.
6.2.2
Exploitation des propri´ et´ es dynamiques
L’´etude des trajectoires dans de pr´ec´edentes mod´elisations a montr´e qu’un ´equilibre de la charge virale et du nombre de CD4 ´etait pratiquement atteint dans un d´elai de 15 `a 21 jours apr`es la mise sous traitement. De plus, il y compl`ete stabilisation apr`es 24 semaines a` l’´etat d’´equilibre [Haase, 1999, Pakker et al., 1998], ordre de grandeur retrouv´e `a l’aide des mod`eles dynamiques par [Guedj et al., 2007a]. Ainsi, il nous est permis d’´eviter le calcul complexe de la commande optimale en temps continu comme cela peut ˆetre fait en automatique et a ´et´e adapt´e, dans le domaine du VIH, a` l’aide de fonctions de r´etrocontrˆole par [Brandt and Chen, 2001]. Les changements de dose seront donc faits en temps discrets en travaillant sur les ´etats d’´equilibre. Notre but est donc d’adapter la dose du traitement
6.2 : Adaptation de la dose
129
administr´ee en la mettant a` jour visite apr`es visite. Ces visites doivent ˆetre suffisamment espac´ees pour que l’´etat d’´equilibre soit atteint entre l’une et l’autre. ¯ T¯, T¯∗ , V¯ ), peut se calculer de ma¯ = (Q, L’´equilibre d’un syst`eme dynamique, not´e X ni`ere analytique en r´esolvant dX(t)/dt = 0. Lorsqu’aucune solution analytique n’existe, il peut aussi ˆetre calcul´e num´eriquement par un solveur d’´equations diff´erentielles. Le nombre de reproduction de base R0 caract´erise l’existence et la stabilit´e des ´equilibres du syst`eme. Dans le cas du mod`ele a` cellules cibles d´efini par l’´equation 3.9 Section 3.2.3.3, R0 poss`ede une expression analytique. D’un point de vue biologique, chaque cellule infect´ee produit des virus au taux π. Pendant cette production de virus, les cellules infect´ees vivent en moyenne un temps 1/µT ∗ . Ainsi, chaque cellule infect´ee produit un total de π/µT ∗ virions. Comme la clearance des virus est faite au taux µV par virion, chaque virion survit un temps moyen de 1/µV . Pendant cette p´eriode, chaque virion infecte en moyenne (γ T¯)/µV cellules saines, o` u T¯ est le nombre de cellules cibles `a l’´equilibre. Ainsi, R0 =
παλγ µV µT ∗ (αµT +ρµQ +µT µQ )
et est le nombre de nouvelles cellules infect´ees caus´ees
en moyenne par une cellule infect´ee `a l’´etat d’´equilibre [Gran et al., 2008]. Si R0 > 1, cela signifie qu’une cellule infect´ee T ∗ va produire plus d’une cellule infect´ee et V¯ > 0. En revanche, si R0 ≤ 1, l’infection sera par opposition sous contrˆole (V¯ = 0). C’est donc ce crit`ere qui caract´erise la fonction objectif C2 (Figure 6.3). Par ailleurs, R0 est une fonction strictement d´ecroissante de la dose, en effet, plus la dose de traitement est forte plus R0 est faible et plus l’infection est contrˆol´ee. Ainsi, trouver la meilleure dose pour le patient revient a` trouver la plus petite dose pour laquelle R0 ≤ 1, c’est a` dire la dose pour laquelle R0 = 1.
6.2.3
Strat´ egie adaptative d’optimisation de la dose de traitement
Lorsque les param`etres de l’individu (ξ i∗ ) sont suppos´es connus, une r´esolution analytique de R0 = 1 pour trouver dcrit peut ˆetre faite. Dans la pratique, les valeurs des param`etres sp´ecifiques `a un patient sont inconnues. Nous pouvons cependant consid´erer que
Chapitre 6 : Optimisation de la dose de traitement
130
nous avons une certaine connaissance a priori issue des a posteriori des effets fixes pour des ´etudes pr´ec´edentes. Par ailleurs, les observations concernant le patient avant et apr`es le d´ebut de l’initiation de la cART permettent une accumulation d’informations concernant les valeurs des effets al´eatoires pour le patient. Il serait possible d’utiliser un estimateur PEB (´equation 6.1). Cependant nous nous int´eressons aux queues de distribution o` u l’approximation normale de l’a posteriori est moins vraisemblable. Nous mettons donc en oeuvre un algorithme bay´esien de M´etropolis-Hastings [Metropolis et al., 1953, Hastings, 1970] afin d’´echantillonner le R0 sp´ecifique au patient et ainsi d´efinir conditionnellement `a l’information disponible la meilleure dose pour le patient. Une preuve math´ematique et des simulations permettent de v´erifier les bonnes propri´et´es de cette strat´egie adaptative d’optimisation de la dose de traitement. Ce travail fait l’objet d’un article publi´e dans Biometrics , Volume 68, Issue 3, septembre 2012, Pages 902-911 [Prague et al., 2012a]. Le mat´eriel suppl´ementaire ´electronique de cet article se trouve en annexe B.
Biometrics 68, 902–911 September 2012
DOI: 10.1111/j.1541-0420.2012.01749.x
Treatment Monitoring of HIV-Infected Patients based on Mechanistic Models M´ elanie Prague,1,2,∗ Daniel Commenges,1,2 Julia Drylewicz,3 and Rodolphe Thi´ ebaut1,2 1 ´ emiologie-Biostatistique, F-33000 Bordeaux, France University of Bordeaux, ISPED, Centre INSERM U897-Epid´ 2 ´ INSERM, ISPED, Centre INSERM U897-Epid´emiologie-Biostatistique, F-33000 Bordeaux, France 3 Department of Immunology, University Medical Center Utrecht, The Netherlands ∗ email:
[email protected]
Summary. For most patients, the HIV viral load can be made undetectable by highly active antiretroviral treatments highly active antiretroviral therapy: the virus, however, cannot be eradicated. Thus, the major problem is to try to reduce the side effects of the treatment that patients have to take during their life time. We tackle the problem of monitoring the treatment dose, with the aim of giving the minimum dose that yields an undetectable viral load. The approach is based on mechanistic models of the interaction between virus and the immune system. It is shown that the “activated cells model,” allows making good predictions of the effect of dose changes and, thus, could be a good basis for treatment monitoring. Then, we use the fact that in dynamical models, there is a nontrivial equilibrium point, that is with a virus load larger than zero, only if the reproductive number R0 is larger than one. For reducing side effects, we may give a dose just above the critical dose corresponding to R0 equal to 1. A prior distribution of the parameters of the model can be taken as the posterior arising from the analysis of previous clinical trials. Then the observations for a given patient can be used to dynamically tune the dose so that there is a high probability that the reproductive number is below one. The advantage of the approach is that it does not depend on a cost function, weighing side effects and efficiency of the drug. It is shown that it is possible to approach the critical dose if the model is correct. A sensitivity analysis assesses the robustness of the approach. Key words: control.
Bayes; Differential equations; Epidemiology; HIV; Metropolis–Hastings algorithm; Monitoring; Optimal
1. Introduction In developed countries, most HIV-infected patients are treated with highly active antiretroviral therapies highly active antiretroviral therapy: which include a combination of several antiretroviral drugs. For many patients, such treatments succeed to control viral replication (i.e., the viral load becomes undetectable) and to restore the immune system sufficiently to dramatically decrease the risk of opportunistic diseases and AIDS (Egger et al., 2002). The virus is, however, still present in sanctuaries like resting memory cells (Chomont et al., 2009). Several side effects have been described in patients having long-term therapy (Carr and Cooper, 2000). To reduce these side effects, treatment interruptions have been attempted. However, these trials have been stopped at interim analysis showing an increased risk of opportunistic diseases (Ananworanich et al., 2006; Danel et al., 2006; Lundgren et al., 2008). Attempts have been made to decrease the number of drugs but did not succeed either (Girard et al., 2009). Reducing the doses of treatment may be a flexible way to alleviate side effects while maintaining efficiency (S´anchez-Conde et al., 2005; Milinkovic et al., 2007; Meynard et al., 2010). These studies, however, proposed an arm-specific reduction (same reduction for every patient from the same arm). A subject-specific dose adaptation (dose individualization) will probably lead to more significant results.
902
Recently, adaptive treatment strategies have been proposed in the statistical literature (Murphy, 2003; Murphy and McKay, 2004; Moodie, Richardson, and Stephens, 2007; Lavori and Dawson, 2008; Henderson, Ansell, and Alshibani, 2010). These approaches adapt some concepts of control theory, in particular dynamic programming, developed in engineering (Sage and White, 1977). In engineering, dynamical models on which optimal control is based are more complex than those generally used for adaptive treatment strategies and they are often expressed through a system of differential equations. Kirschner, Lenhart, and Serbin (1997) proposed to use control theory based on mechanistic models of the interaction of HIV and the immune system for adapting the dose of antiretroviral treatment. Indeed, since the pioneering work of Ho et al. (1995) and Wei et al. (1995), several mechanistic models have been developed. However, treatment monitoring raises different issues than control problems in engineering and the estimation of parameters in HIV dynamics models is challenging. Recent statistical approaches use random effect models with the aim to estimate the parameters of complex models using rather large samples from clinical trials: see Wu (2005) for a review and Putter et al. (2002); Huang, Liu, and Wu (2006); Guedj, Thi´ebaut, and Commenges (2007); Huang, Wu, and Acosta (2010) among others for original developments. C
2012, The International Biometric Society
Treatment Monitoring and Mechanistic Models The aim of this article is to examine the possibility of using such sophisticated mechanistic models, with parameters that can be estimated on previous clinical trial data, for monitoring the treatment dose of an individual patient. A Bayesian approach seems here natural (Berger, 1985). Previous data give a prior for the parameters of the model. Observations for a new patient give additional information, especially on parameters varying between patients, and permit to update the priors. Then the dose can be adapted so as to minimize a risk function. One key idea in this article is that, for reasonable risk functions, the target dose is the critical dose which makes the reproductive number R0 equal to (or just below) one. After presenting the method, we shall examine whether it can work in real life. This is a “proof of concept” article: real patients have not yet taken doses recommended by our method but we give arguments to convince that mechanistic models, and this particular method, can be used for treatment monitoring. We first examine, using real data, whether our model can detect an influence of the treatment dose; then we examine the predictive ability of the model, still using real data. Then, we apply the control method on simulated data. The article is organized as follows. Section 2 recalls the mathematical and statistical models developed by Guedj et al. (2007) and Drylewicz et al. (2010). Section 3 tackles the control problem first treating the case with known parameters, then the case with unknown parameters. The optimal dose is defined as the dose which controls the probability that R0 < 1 and this can be computed by use of a MCMC algorithm. In Section 4, a detailed analysis of the ALBI clinical trial is presented. In this trial, patients received different treatments or doses (patients in the third arm shifted treatment and some changed their dose) allowing us to study the predictive ability of the model. In Section 5, we present a simulation study of the proposed strategy. The posterior distribution of the parameters is used as a prior for a new patient whose data are generated from the model. We examine the sensitivity of the method to misspecification of the prior. A conclusion and perspectives are given in Section 6. 2. Models of Interaction Between HIV and Immune System 2.1 Mathematical Models Perelson et al. (1996) proposed a three-dimensional Ordinary Differential Equation (ODE) system featuring uninfected CD4+ T cells, infected CD4+ T cells (T ∗ ) and viruses (V ) concentrations. We shall work with an extension of this model called the “activated cells model,” which distinguishes among uninfected cells those which are quiescent (Q) and those which are activated (T ). This model has been shown to fit the ALBI trial data much better (Commenges et al., 2008). The definition of the parameters can be found in Table 1. The model can be written as:
⎧ dQ ⎪ ⎪ = λ + ρT − αQ − μQ Q, ⎪ ⎪ dt ⎪ ⎪ ⎪ dT ⎪ ⎨ = αQ − γT V − ρT − μT T , dt
dT ∗ ⎪ ⎪ = γT V − μT ∗ T ∗ , ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dV = πT ∗ − μV V. dt
(1)
903
Table 1 Biological parameters for the “activated cells model” Parameter
Meaning Activation rate of Q cells (day−1 ). Death rate of T ∗ cells (day−1 ). −1 Rate of Q cells production (μ−1 L day ). −1 Death rate of T cells (day ). Rate of virions per T ∗ cell (day−1 ). Rate of reversion to the Q state (day−1 ). Infectivity: Infection rate of T cells per virion (day−1 μL ). Death rate of Q cells (day−1 ). Death rate of free virions (day−1 ).
α μT ∗ λ μT π ρ γ μQ μV
The basic reproductive number R0 (Gran et al., 2008) is given by: R0 =
γπαλ . μT ∗ μV (ρμQ + αμT + μQ μT )
(2)
If R0 is lower than one, the only equilibrium point is the trivial one with V = 0, otherwise the trivial equilibrium is unstable and after introduction of the virus, the system stabilizes to a nontrivial equilibrium with V > 0 (see the formula in Web Appendix A ). 2.2 Statistical Models For the interindividual variability of the parameters, we use a statistical model with the same structure as proposed by Guedj et al. (2007). For i = 1 . . . N , we denote by ξ i the vector of the nine individual biological parameters and by ξ˜i the vector of the log-transformed parameters:
˜i , μ ˜i , μ ˜ iT ∗ , λ ˜ iT , π ˜ i , ρ˜i , μ ˜ iQ , μ ˜ iV , γ˜ i ξ˜i = ξ˜li , l = 1 . . . 9 = α
T
.
The possible between-subjects variability and time variability of the parameters is modeled as: i ξ˜l (t) = φl + zli (t)βl + ωli ui ,
zli
l = 1 . . . 9,
(3)
ωli
are the vectors of (possibly where φl is the intercept, and time-dependent) explanatory variables associated to the fixed and random effects respectively of the lth biological parameter. The βl are vectors of regression coefficients associated to the fixed effects; ui is the individual vector of random effects. We assume ui ∼ N (0, Σ) with Σ a diagonal matrix. Specifically, in the following, we shall use as explanatory variable only the treatments doses dij (t), where j is the treatment index. In the application, there are two treatments (j = 1, 2); for sake of simplicity, we present the theory for only one treatment. We assume that the treatment acts by diminishing the infectivity (this is indeed the case for the reverse transcriptase inhibitors used in the application). The model for the infectivity parameter is assumed to be: γ˜ i = γ˜0 + βψ{di (t)},
(4)
with β < 0 and where ψ(.) is a known increasing positive “pharmaco-dynamic” function. We assume in addition ψ(0) = 0 and ψ(.) unbounded.
Biometrics, September 2012
904
Figure 1. Flow chart for the control strategy procedure: individualized dose monitoring. Let X = (Q, T , T ∗ , V ) in model (1). We used g1 (X) = log10 (V ) and g2 (X) = (Q + T + T ∗ )0. 25 as transformations to achieve normality and homoscedasticity of measurement error distributions (Thi´ebaut et al., 2003) of the observed two compartments: the viral load (number of virions per μL) and the total CD4 count (number of cells per μL). Let Yjim denote the jth measurements of the mth observable component for subject i at time tij m ; we assume that:
+ ij 1 Yji1 = g1 X tij 1 , ξ˜i Y i = g2 X ti , ξ˜i + i j2
j2
j2
j = 1, . . . , ni1 j = 1, . . . , ni2 ,
(5)
where ij 1 and ij 2 are independent Gaussian with zero mean 2 2 and σCD4 , respectively. The observation and variances σVL mechanism is generally complicated by a detection limit for the viral load, producing left-censoring for Yji1 (JacqminGadda et al., 2000). 3. Control Strategy 3.1 Targeting R0 , Known Parameters Our aim is to adapt the treatment dose at visits spaced by several weeks. Study of mathematical models and observed trajectories shows that the equilibrium state is essentially reached in a very short period of time. So, we can avoid the complexity of optimal control theory where the control is in continuous time and the cost function is an integral over time of a function of state and control. Instead, we define a cost ¯ function which depends only on the equilibrium state X(d; ξi ) ¯ that is reached for a dose d: C{X(d; ξ i ), d}, and more specifically C{V¯ (d; ξ i ), d}. The best dose for the patient minimizes C{V¯ (d; ξ i ), d} on [0, dmax ]. The function C(., .) is strictly increasing in both arguments. Then, we can use the fact that in our HIV dynamic model if R0 (d, ξ i ) < 1 there is only one equilibrium state which is the trivial one with V¯ = 0. The critical dose dcrit (ξ i ) is defined as the solution of R0 (d, ξ i ) = 1. Because ψ(.) is an unbounded and increasing function in the dose, R0 is a one-to-one function strictly decreasing with the dose. Thus, for any given ξ i , a unique critical dose exists. We obtain the critical dose, dcrit (ξ i ) from (2) and (4): i
dcrit (ξ ) = ψ
−1
1 ln β
μiT ∗ μiV
ρi μiQ + αi μiT + μiQ μiT π i αi λi γ0i
(6) .
Because C(., .) is increasing in its second argument, the optimal dose cannot be higher than dcrit (ξ i ). If C(., .) rises
very sharply in its first argument, the optimal dose will be close to dcrit (ξ i ). If C(x, y) presents a discontinuity in x = 0, the optimal dose is dcrit (ξ i ). This is a reasonable requirement from the medical point of view: dcrit (ξ i ) is the minimum dose which achieves virus extinction (in practice, a very low viral load). This relieves us from the burden of specifying a cost function which would weigh the toxicity and the efficiency of the drug in an arbitrary way. 3.2 Unknown Parameters: Controlling the Probability that R0 < 1 In practice, the values of the parameters of a particular subject are unknown. We have, however, a certain amount of knowledge about ξ i . We make observations of both viral load and CD4 count at times before and after the initiation of the control strategy: t0 , t−1 , t−2 . . ., and t1 , t2 , . . . , respectively; this gives essentially information about the values of the random effects ui for this subject. Given Ftik , the information at time tk , R0 (d, ξ i ) has a posterior distribution. At time tk , we k , the readjusted optimum dose, as the propose to choose dtopt minimum dose which gives a high posterior probability (ω) that R0 is below 1: P
Fti
k
t
k R0 dopt (ω), ξ i
< 1 = ω.
(7) t
k We may take ω = 90%; in the following, we write dopt for the optimal dose at time tk . R0 is a decreasing one-to-one tk is unique. We denote by ξ i ∗ the true pafunction, thus, dopt rameters value for patient i and dcrit (ξ i ∗ ) his critical dose. If we assume that the information increases in time so that Doob’s consistency theorem can be applied (Van der Vaart, 2000), tk will tend to the critical dose (see Web Appendix B for dopt the proof):
t
k dopt
Fi
P
−→
t k →∞
dcrit (ξ i ∗ ). Fi
We have that P t k {R0 (d, ξ i ) < 1} = P t k {dcrit (ξ i ) < d} (see tk is the ω-quantile of the posterior Web Appendix B). Thus dopt distribution of dcrit (ξ i ) which can be computed by MCMC using formula (6). Figure 1 displays a flow chart for the control strategy. Time after time, observations of CD4 counts and viral loads are collected and an optimal dose given the reaction to the previous doses is provided. Optimal dose greater than dmax make the patient leave the control strategy because of treatment failure.
Treatment Monitoring and Mechanistic Models 4. Dose Effect and Predictive Ability in the ALBI Trial 4.1 The ALBI Trial We use the data of the ALBI ANRS 070 trial (Molina et al., 1999). In this three arms controlled trial, 151 antiretroviralnaive patients (with viral loads between 10,000 and 100,000 copies/ml and CD4 counts greater than 200 cells/mm3 ) received 24 weeks of treatment. The 51 patients from arm 1, received the d4T + ddI treatment, that is stavudine (d4T)/didanosine (ddI) whose dosage depended on their weight; d4T 250 mg plus ddI 60 mg for patients less than 60 kg, otherwise d4T 400 mg plus ddI 80 mg. For arm 2, 51 patients received the AZT + 3TC treatment, that is 500 mg of zidovudine (AZT) and 300 mg of lamivudine (3TC). The 49 patients from arm 3, the switch arm, received the d4T + ddI treatment for 12 weeks followed by AZT + 3TC, with the same doses as in arm 1 and 2. Blood samples were collected every 4 weeks until 24 weeks. CD4 counts and viral loads (with a lower quantification limit of 50 copies/ml) were measured. Two patients dropped out before the first blood sample collection time, so 149 patients were available for the analysis. For each patient, changes in dose were either self reported or declared by the clinician. Twenty-seven patients (18%) changed their doses during the trial; some of them even interrupted the treatment during a period of time. Molina et al. (1999) found that the d4T + ddI treatment was significantly better than the AZT + 3TC treatment in an intent-to-treat (ITT) analysis. 4.2 Model For the ALBI Trial We performed a thorough analysis of the ALBI trial using the “activated cells model,” taking into account all the complexities of the design and the doses actually taken by the patients. Doses were included in the analysis, by considering treatment as a time-varying covariate. We took into account the doses of the two ALBI treatments, denoted {di11 (t), di12 (t)} and {di21 (t), di22 (t)}. To overcome unities problem, we took as reference dose the median dose for each drug, denoted (d∗11 , d∗12 ) and (d∗21 , d∗22 ). We constructed an indicator of the doses for treatment j = 1 (d4T + ddI) or 2 (AZT + 3TC) as: dij (t) =
1 2
dij 2 (t) dij 1 (t) + ∗ d∗j 1 dj 2
.
Moreover, we smoothed the dose indicators by taking the mean of the three doses taken in previous days, as this provided a better fit. Therefore, the dose indicator for treatment j of subject i at time t can be written:
d¯ij (t) = dij (t − 2) + dij (t − 1) + dij (t) /3. As for the choice of ψ in (4), we first tried a linear function, but this achieved a poor fit of the data. We obtained a much better fit with power functions, with a power around 0.1. However, such a simple power function does not have an acceptable shape in the sense that the effect is still high for very low doses (for which we had no data). We tried sigmoid functions as in Shen et al. (2008) but this yielded shapes very similar to the power functions. To overcome this problem, we constructed a function which was a power function in the range where information was available [0.6; 1], connected to
905
zero by a linear function. Finally, the “pharmaco-dynamical” function for patient i, treatment j at time t was:
ψ dij (t) =
dij (t)κ d (0.6)κ d −1 dij (t)
if dij (t) > 0.6, otherwise.
(8)
We also tried to introduce the weights of the subjects as explanatory variable but did not find any effect of it; so these results are not shown. 4.3 Elicitation of a Prior, Algorithm, and Posterior We denote by θ = (φ, β, σα , σλ , σμ ∗T , σVL , σCD4 ), the vector of the parameters. Here, φ = (φl , l = 1 . . . 9), appearing in (3), is the vector of the nine baseline biological parameters and β the two-dimensional vector of regressors for treatment doses. There are also three standard deviations of random effects (on α, λ and μT ∗ as suggested in Guedj et al. (2007)) and the two standard deviations of errors measurements. Normal independent priors were assumed for the components of φ with expectation and variance elicited in accordance with the literature. Essentially, the rule was to define the lowest and the largest values proposed in the literand φupper and take the mean and standard deviature φlower l l + φlower )/2 and (φupper − φlower )/4, respectively. ation as (φupper l l l l The literature was not precise enough to elicit these bounds for the infectivity parameter γ˜0 ; we took a very wide range: = log(10−6 ) = −13.8 and φupper = log(10) = 2.3. For the φlower 9 9 effects of the treatment doses, we also assumed normal priors with expectation corresponding to dividing the infectivity by 3 for the standard dose, and standard deviation equals to about the third of the absolute value of this expectation: this expresses the fact that it is unlikely that treatments increase the infectivity while yielding rather weakly informative priors. The priors did not value one treatment over the other. Half-Cauchy priors were taken for the variance of the random effects, as recommended by Gelman (2006). For the variances of the measurement errors, conventional Jeffreys’s improper noninformative priors were taken. Table 2 summarizes the priors for the different parameters. In view of the complexity of the problem, MCMC methods would be too time consuming and the INLA method (Approximate Bayesian Inference for Latent Gaussian Models; Rue, Martino, and Chopin, 2009) can not be applied to this problem. We turned to a normal approximation of the posterior which is justified by the Bernstein–Von Mises Theorem (Van der Vaart, 2000). Numerically, this amounts to compute the maximum a posteriori estimator (MAP); this is identical to penalized likelihood maximization. Specifically, the function that must be maximized is pl = L − J (θ), where the penalty term J (θ) is: J (θ) =
9 [φj − E0 (φj )]2 j =1
2var0 (φ
j)
+
2 [βj − E0 (βj )]2 j =1
2var0 (βj )
+ log(σα 2 + sα 2 ) + log(σμ T ∗ 2 + sμ T ∗ 2 ) + log(σVL ) + log(σCD4 ), 0
where E and var0 stand for the means and variances of the priors. Here, L is the log likelihood which is described in Guedj et al. (2007). The maximization (or rather the
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Table 2 Priors and posteriors for the “activated cells model” parameters estimated from ALBI trial data Priors
Posteriors
Biological parameter (Normal priors): Parameter Mean (SD) α ˜ μ ˜T ∗
−4.00 (2.00) −0.05 (0.68)
˜ λ μ ˜T π ˜ ρ˜ γ˜0 μ ˜Q μ ˜V
2.55 (1.90) −2.59 (0.34) 4.04 (2.66) −4.34 (1.38) −5.76 (4.02) −9.00 (1.00) 2.90 (0.68)
References
Mean
SD
Ribeiro et al. (2002) Althaus, De Vos, and De Boer (2009) Brandin et al. (2006) Mohri et al. (1998) Ribeiro et al. (2002) Chen et al. (2007) Ribeiro et al. (2002) See Section 4.3 Vrisekoop et al. (2008) Ramratnam et al. (1999)
−3.19 −0.52
0.14 0.12
2.52 −2.57 2.49 −5.13 −5.38 −11.2 1.70
0.10 0.10 0.54 0.54 0.03 0.99 0.59
Regressors for treatments doses (Normal priors): Parameter Mean (SD)
References
Mean
SD
See Section 4.3 See Section 4.3
−0.97 −1.03
0.09 0.09
References
Mean
SD
Guedj et al. (2007) Guedj et al. (2007) Guedj et al. (2007)
0.38 0.03 0.03
0.03 0.01 0.01
−1.10 (0.37) −1.10 (0.37)
β1 β2
Standard deviation for random effects (Half-Cauchy priors): Parameter Median σα σμ T ∗ σλ
0.53 0.37 0.10
Standard deviation for error measurment (Jeffrey’s priors): Parameter σC V σC D 4
– –
– –
minimization of −pl) can be done using the algorithm described in Guedj et al. (2007), modified to take into account the penalty brought by the prior. This algorithm uses the so-called RVS algorithm which approximates the Hessian using first derivatives. For penalized likelihood, the way the Hessian is approximated must be modified as described in section 4.5 of Commenges et al. (2006). There is, however, a sign error in the original version. The correct formula for the Hessian of −pl in θk (at iteration k) is: G(θk ) =
N i =1
Ui (θk )UiT (θk ) − n−1 U (θk )U T (θk ) +
∂ 2 J (θ) , (9) ∂θ 2
where U (θk ) = − ∂∂pθl |θ k and the Ui (θk )’s are the individual
N
∂ 2 J (θ ) ∂θ2
gradients (U (θk ) = i =1 Ui (θk )). The addition of (here a diagonal matrix with positive diagonal) generally improves the condition number of the approximate Hessian. G(θk ) is 2 close to − ∂∂ θp2l (θk ) near the maximum. It is generally easy to 2
compute ∂ ∂Jθ(θ2 ) . We evaluated κd in (8) by profile likelihood, leading to κd = 0.13. For other parameters, the posterior was approximated by a normal distribution with expectation given by the MAP and variance given by the inverse of the approximation of the Hessian of −pl (9). We present their means and standard deviations of posteriors in Table 2. Finally, we checked reproducibility by starting from 10 different initial values drawn at random at one standard
Mean
SD
0.45 0.20
0.01 0.01
deviation from the mean of the prior. The algorithm converged toward the same region obtaining the same two significant digits for the standard deviations of measurement errors and main random effect, one significant digit for most other parameters, except π ˜ , μ˜Q , and μ˜P , for which we only had the order of magnitude. This is probably due to lack of practical identifiably: actually, predictions are not impacted by this relative lack of precision. In Web Appendix C, we show that the fits for three stopping points with rather different values of π ˜, μ˜Q , and μ˜P are nearly indistinguishable. Because we use the model essentially for prediction, the problem is not too severe. 4.4 Results: Predictive Ability for Treatment Change The ALBI design offers a very good opportunity to assess the predictive ability of our model, especially thanks to the switch arm. We can use the first period to assess the random effects for these patients and then predict how they react to the next treatment. We selected patients from the switch arm in an objective manner: we took them at the quartile values of the distribution of the viral load at the end of the study. We computed the MAP estimates removing these patients. Then, we estimated the parameters with random effects thanks to a parametric empirical Bayes (PEB) procedure (Kass and Steffey, 1989) based on the first 12 weeks; viral loads and CD4 counts were computed for the value of the parameters using MAP estimates for fixed effects and PEB for parameters with random effects, taking into account the
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Figure 2. Viral load (log10 copies/ml) et CD4 count (cells/L) predictions for patient 642 from the switch arm who took full dose of d4T + ddI then switched after 85 days to full dose AZT + 3TC. are observations. Dashed lines represent 95% “measurement error predictive interval.” Vertical line materializes when the treatment switch occurred. Left side: fit; right side: predictions. Treatment adherence is presented on the lower part of the graph. adherence for the remaining 12 weeks. Using notations in (5), we also computed a 95% “measurement error predictive interval” as [Yˆji1 ± 1.96σVL ] and [Yˆji2 ± 1.96σCD4 ]. For space concern, we only present the median patient, but predictions for Q1 and Q3 patients at quartiles are available in Web Appendix D, others on request. Figure 2 presents the viral load and the total CD4 count with a fit in the first 12 weeks and predictions for the last 12 weeks after the treatment switch. One can see that we are able to predict quite well the viral load rebound after the treatment change (because the AZT + 3TC treatment is less efficient). 4.5 Results: Predictive Ability for Dose Change We selected analyzed patients in an objective manner among the 27 patients with dose changes: we took those at each quartile values in function of the distribution of the viral load at the end of the study. We estimated again all the parameters excluding this new subset of patients and did step-by-step prediction. Each time we had an additional observation, we updated the knowledge about random effect and computed the predicted value taking into account the reported adherence. For sake of illustration, we present a patient who particularly well filled his adherence information and who had a large range of behaviors (drug dose reduction, stop, switch . . . ). Figure 3 presents predicted of viral loads and total CD4 counts for each time taking into account the increase of information in time. The 95% “measurement error predictive interval” is represented. Predictions taking into account drugs doses are much better than the ITT fits. This can be quantified by computing the mean square error (MSE) over all the patients with dose changes. The MSE for the viral load is 0.23 for the prediction versus 0.61 for the ITT fit. For the CD4 counts, the MSE are 7768 versus 7962. The predictions together with the MSE for selected patients are available in Web Appendix E.
5. Simulation Study For Dose Optimization 5.1 Principle of the Data Simulation In this section, we aim at analyzing the efficiency of our drug dose reduction algorithm in term of precision and safety (recommended doses should be higher than critical doses). Data were simulated using the MAP estimates from the analysis of the ALBI trial (Table 2). We simulated samples of 100 patients (n). Parameters with random effects, α, μT ∗ and λ were drawn from the multinormal Gaussian posterior for every patient. Fixed parameters were drawn with the conˆ − θ) ˆ = 1 and were the same for all the ˆ T G(θ)(θ straint (θ − θ) patients of a sample. We only selected patients who had a baseline R0 (that is without treatment) higher than 1 and a critical dose lower than the standard dose; thus, we excluded long-term nonprogressors and only kept patients needing a dose reduction. We assumed a plausible observation schedule: observation times were at 0, 7, and 14 days whereas dose readjustments were performed every 15 days. The initial dose was taken as the reference, d0 = 1. We simulated the viral loads and CD4 counts with measurement errors variances equal to (0.45)2 and (0.22 ; Table 2). Finally, as ALBI treatments are under-efficient compared to those given in 2011, we took a treatment effect equal to twice the MAP estimate of treatment effect for d4T + ddI (β2 ), that is β = −2.06. 5.2 Illustration of the Monitoring Method We ran the control strategy algorithm described in Section 3.2 with a burn-in phase consisting in a 100,000 updates, a sampling phase of 50,000 updates, dmax = 2 and ω = 90%. All the nine biological parameters were updated even if significant updates were only noticeable for parameters with random effects. It never recommended a dose greater than dmax . We first present the results for a particular patient drawn at random as visual illustration (10 updates were performed). His critical dose was 0.306. We can see on Figure 4 (left) that at the first readjustment time we could advise a reduction by
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Figure 3. Viral load (log10 copies/ml) et CD4 count (cells/L) predictions for patient 316 who took 0.7% of d4T + ddI during 4 days, then decreased by about two during 8 days and then stopped until day 59. He then took again his treatment until day 65 but with reference posology (probably because he gained weight during the first treatment part and then passed the threshold of 60 kg). He stopped again the treatment and then from day 91 to the end of the study, he switched to AZT + 3TC treatment. are observations. Each color represents the predictions knowing the information up to the previous observation time. Dashed lines represent 95% “measurement error predictive interval.” Treatments are presented on the lower part of the graph. The gray plain line presents the fit without taking into account adherence.
together with the percentage of doses lower than the critical dose for the patient and the percentage of doses greater than dmax . In a well-specified model, the MSE decreases while information about the patient increases and we effectively control the probability to give a dose below the critical dose.
from the same population as in ALBI trial. The value of the safety margin was determined such that our key proposition Fi k (ω), ξ i } < 1] < 10% hold in the worst ∀k ∈ 0..5, P t k [R0 {dtopt simulated case for k = 2. We ran again the control procedure, applying the safety margin, over a new validation sample of 1000 patients with misspecified priors. Table 3 shows that MSE are increased but are still reasonable while, as expected, tk < dcrit is smaller than 10%. the percentage of dopt To finish, we tried a misspecified model. Data were generated from a “productive cell model” (Drylewicz et al., 2010) which has five compartments instead of four. To do so, we took a “productive cell model” with the same equilibrium state as in the “activated cell model.” Productive cell production rate was fixed at −0.12, the estimates found by Drylewicz et al. (2010). For all the readjustment times, the percentage of optimal doses lower than the critical dose was always lower than 10%. MSE ranged from 1.21 at time t1 to 0.12 at time t5 .
5.3 Sensitivity Analysis and A Safer Monitoring Method We performed a sensitivity analysis to evaluate the robustness of our control strategy in misspecified models. First, we investigated the effect of priors misspecification. In the same fashion as in part 5.1, we simulated samples of ˆ − θ) ˆ = ˆ T G(θ)(θ 100 patients with parameters such that (θ − θ) k, k = 1.5, 2, 2.5, 3, and 5. For each k, 10 different samples were taken to explore different profiles of prior misspecification. This extensive simulation study was made possible by using parallel computing. Results are presented in Table 3. MSE were good up to k = 2.5 but the percentage tk < dcrit reached 31% in average at the fifth readjustof dopt ment. To fix this problem, we propose to add a safety margin to the readjusted dose. We focused on k = 2 because priors should not be too badly specified if patients come
6. Discussion In this article, we have proposed a method of dose individualization of HAART for HIV infected patients. We have attempted to demonstrate its feasibility. First, a pharmacodynamic model was developed for the ALBI trial and we showed that it could be applied to real data. The model fitted the viral loads and CD4 counts data quite well, providing an in vivo estimation of the treatment efficacy. More importantly, the model had good prediction abilities. Then, the simulation study showed that when information is rich enough, the critical dose can be found in a rather small number of readjustments, while staying most of the time above the critical dose.
more than 55% of the dose given to the patient. It is noticeable that the final dose was close to the critical dose. Moreover, this result was obtained while remaining above the critical dose during the tuning period. Furthermore, we can check on Figure 4 (right) that neither the viral load nor the CD4 count are impacted by drug reduction. Table 3 presents the MSEs between the optimal dose and the critical dose for each readjustment time computed as: MSE(tk ) =
n 2 1 tk dopt (i ) − dcrit (ξ i ∗ ) , n i =1
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Figure 4. Simulation of 10 doses readjustments spaced by 15 days after three observation times: (up) Viral load (+) and k become closer by above to dcrit = 0.306 CD4 count (∗) simulated observations (down) Dose readjustment simulation, dtopt (horizontal line). An issue is the possible lack of information due to leftcensoring when the treatment is very effective. Then, Doob’s consistency theorem invoked in Section 3.2, would not apply. In that case, more intensive schedules and/or the measurements of more compartments would be needed. Other concerns could arise from model misspecification. A misspecification could lead to biased results and wrong recommendations. We have made a sensitivity analysis to test the robustness of the procedure to misspecification of the sampling distribution and to misspecification of the prior (which happens if the new patients are different from those used for defining the prior). In both cases, the convergence toward the critical dose is slower but the procedure, which can be improved by using a safety margin, generally works. A major issue is development of drug resistance. If this happens during the monitoring, then viral load will raise leading to higher recommended dose until we reach the maximum admissible dose. This is a failure of the treatment and another treatment has to be prescribed. The search of the optimal
dose is relevant in a time period where no major resistance mutation develops. We conclude that the use of such an adaptive scheme is to be further tested on more informative data (especially with well-documented adherence); however, this work shows that dose individualization is possible. The further step would be to validate the procedure in a clinical trial before using it in clinical practice. Such a clinical trial would typically have two arms, one in which the proposed procedure of dose monitoring would be applied, the other in which standard clinical practice would be applied; the endpoints would be the doses given at the end of the trial, measures of adverse effects and the proportion of virological failures.
7. Supplementary Materials Web Appendices referenced in Sections 2.1, 3.2, 4.3, 4.4, and 4.5 are available with this article at the Biometrics website on Wiley Online Library.
Biometrics, September 2012 0.04 (0.2%–0%) 0.05 (1%–0%) 0.07 (2%–0%) 0.09 (2%–0.9%)
0.002 (25%–0%) 0.01 (31%–0%) 0.02 (40%–0%) 0.08 (47%–0%) 0.55 (37%–10%) 0.01 (20%–0%) 0.02 (27%–0%) 0.02 (37%–0%) 0.08 (45%–0%) 0.56 (39%–11%) 0.01 (14%–0%) 0.02 (19%–0%) 0.03 (28%–0%) 0.11 (42%–0%) 0.61 (40%–12%) 0.03 (9%–0%) 0.04 (14%–0%) 0.06 (20%–0.1%) 0.18 (37%–1.3%) 0.74 (40%–15%)
0.001 (9%–0%) 0.005 (4%–0%) 0.01 (1%–0%)
Well specification 0.62 (0%–0%) 0.04 (0%–0%) Prior Misspecification k = 1.5, Act. T cell 0.59 (0%–0%) 0.04 (9%–0%) k = 2, Act. T cell 0.66 (0%–0%) 0.09 (15%–0.5%) k = 2.5, Act. T cell 0.69 (0%–0%) 0.13 (22%–1.5%) k = 3, Act. T cell 0.67 (0%–0%) 0.24 (39%–4%) k = 5, Act. T cell 0.74 (0%–0%) 0.71 (41%–16%) A safer monitoring method : “safety margin” = 0.07 k = 2, Act. T cell 0.71 (0%–0%) 0.18 (3%– 0%)
0.03 (0%–0%)
t5 t4 t3 t2 t1 t0
k k k MSE(dtopt ; % dtopt < dcrit – % dtopt > dmax )
Table 3 tk )k =0. . . 5 toward dcrit Analysis of sensitivity to prior misspecification: evaluation of the convergence of (dopt
910
Acknowledgements The authors would like to thank the investigators of the ALBI ANRS 070 trial, particularly J. M. Molina (principal investigator), G. Chˆene (methodologist) and V. Journot for data management help. Parallel computing was used thanks to the computing facilities MCIA (M´esocentre de Calcul Intensif Aquitain) of the Universit´e de Bordeaux and of the Universit´e de Pau et des Pays de l’Adour.
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Received September 2011. Revised December 2011. Accepted January 2012.
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Perspectives
Nous avons montr´e que cette strat´egie d’adaptation de la dose poss`ede de bonnes propri´et´es pour r´eduire la dose de traitement des patients infect´es par le VIH. Nous avons fait de nombreuses ´etudes de robustesse et de sensibilit´e, comme par exemple mesurer l’impact d’un plus grand nombre d’observations du patient, de la censure a` gauche ou de la mauvaise sp´ecification du mod`ele. Cela nous a permis de mettre en lumi`ere les limites de l’algorithme de M´etropolis-Hastings `a marche al´eatoire simple que nous utilisons. En effet, pour un mod`ele bien sp´ecifi´e, le taux d’acceptation commence `a environ 60% puis diminue de presque 15% a` chaque nouvelle observation utilis´ee pour la mise `a jour de la ` partir de 15 observations utilis´ees, le taux de rejet est sup´erieur `a 99% rendant dose. A les calculs tr`es longs. Dans certains cas o` u le mod`ele est mal sp´ecifi´e, comme lorsque les donn´ees sont simul´ees avec un mod`ele a` cellules productives [Drylewicz et al., 2010] et l’algorithme de mise a` jour de la dose se base sur un mod`ele a` cellules cibles (´equation 3.9), M´etropolis-Hastings ne parvient pas `a proposer de nouvelles valeurs dans un temps raisonnable. Ainsi, un axe d’am´elioration serait d’impl´ementer un algorithme bay´esien plus conservateur, par exemple des filtres particulaires [Del Moral et al., 2006]. Par ailleurs, il serait int´eressant de mettre au point des indicateurs pouvant r´ev´eler une mauvaise sp´ecification du mod`ele, qu’elle soit intrins`eque (mauvais nombre de composantes) ou dynamique (terme manquant dans les ODE) des mod`eles. Bien que dans nos mod`eles, la magnitude des variances des effets al´eatoires peut ˆetre vue comme un indice de mauvaise sp´ecification, il serait int´eressant de le formaliser. En effet, nous avons observ´e que lorsque des effets al´eatoires deviennent tr`es grands le mod`ele n’est souvent plus ad´equat. Il serait aussi int´eressant d’introduire un param`etre d’adh´esion du mod`ele au syst`eme ODE comme propos´e par [Jaeger and Lambert, 2012] dans une approche non param´etrique bas´ee sur celle de [Wang et al., 2012]. Ainsi, nous pourrions travailler sur un terme de p´enalit´e qui, lorsque l’´ecart devient trop grand entre donn´ees observ´ees et trajectoires pr´edites, avertit le mod´elisateur. Par ailleurs, la mauvaise sp´ecification peut aussi ˆetre ´evalu´ee en introduisant de l’al´ea dans les ODE via des termes stochastiques appliqu´es
Chapitre 6 : Optimisation de la dose de traitement
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directement sur les composantes ou par des drifts sur les param`etres. Il s’agit ensuite d’´evaluer l’impact sur la qualit´e d’ajustement avec pour outil les crit`eres de choix de mod`ele. En particulier, il serait int´eressant de reprendre les travaux de [Commenges et al., 2012] afin d’´etendre le crit`ere LCVa afin de pourvoir comparer des mod`eles ne prenant pas en compte la mˆeme quantit´e d’information.
Un aspect plus th´eorique a` d´evelopper consiste en la d´emonstration de la convergence de la loi a posteriori des param`etres vers la vraie valeur des param`etres individuels ´enonc´ee ´equation 6.2. En reprenant les notations de la Section 6.1.1.2 et en supposant les observations faites du patient ind´ependantes les unes des autres, nous pouvons appliquer le th´eor`eme de Bernstein von-Mises aux effets al´eatoires individuels ce qui nous donne la convergence en probabilit´e : I −1 ˆ i u |F i i i ˆi j p(u |Fj ) − N u |Fji , −→ 0 j→+∞ j
ˆ et β) ˆ ´etant des constantes, nous pouvons ´ecrire : Ainsi, pour n fix´e, (φ I −1 ˆ i u |F i i i ˆi j p(ξ |Fj ) − N ξ |Fji , −→ 0 j→+∞ j
Nous nous pla¸cons ensuite dans l’asymptotique o` u n est grand et o` u nous pouvons consiˆ = φ∗ et β ˆ = β ∗ . Alors, il est possible d’´ecrire : d´erer que φ n→+∞
p(ξ i |Fji ) −→ δξi ∗ j→+∞
Cependant, cette d´emonstration repose sur deux hypoth`eses fortes, probablement non v´erifi´ees sur le type de donn´ees dont nous disposons : les observations (Yi1 , . . . , Yij ) ne sont pas i.i.d. et la validit´e de la double asymptotique n’est pas assur´ee, car pour n fini ˆ β) ˆ sont des estimateurs. (φ,
6.3 : Perspectives
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Pour finir, l’analyse de patients r´eels issus de l’essai ALBI donne une id´ee de ce que la strat´egie adaptative de traitement pourrait permettre en pratique. Ainsi, les patients pour lesquels l’infection ´etait contrˆol´ee avec la dose standard de l’´etude auraient vu leur dose r´eduite ; d’autres patients auraient vu leurs doses augment´ees. Par exemple, le patient 538 aurait dˆ u prendre 53% de sa dose alors que la dose du patient 33 aurait dˆ u ˆetre augment´ee de 2.5%. Cette bidirectionnalit´e conf`ere `a cet algorithme la capacit´e de mettre en ´evidence des ´echecs th´erapeutiques. En effet, si la dose de traitement propos´ee est sup´erieure `a une dose maximale admissible de traitement (dmax ), le patient pourra ˆetre d´eclar´e en ´echec th´erapeutique, car aucune nouvelle dose ne peut lui ˆetre propos´ee pour le traitement en cours. Ainsi, un d´eveloppement compl´ementaire possible serait le d´eveloppement d’une strat´egie adaptative de changement de traitement comme pr´esent´e Figure 6.4. Cependant, ceci demande de prendre en compte les aspects de pharmacologie et de r´esistances au traitement d’o` u les travaux suivants.
Figure 6.4 : Protocole d’optimisation du traitement pour le choix d’une nouvelle cART apr`es un ´echec th´erapeutique mis en valeur par une strat´egie adaptative de r´eajustement de la dose de traitement.
Chapitre 7 Mod´ elisation des effets des cART, vers une optimisation R´ esum´ e : Certains mod`eles dynamiques pour le VIH, en particulier le mod`ele a` cellules cibles, poss`edent comme nous l’avons vu au chapitre pr´ec´edent de bonnes propri´et´es pr´edictives, le rendant utile pour construire des strat´egies de surveillance des traitements. Dans ce chapitre, nous passons en revue l’ensemble des ph´enom`enes biologiques qui doivent ˆetre consid´er´es dans le contexte des cART : les m´ecanismes d’action des mol´ecules, leurs pharmacocin´etiques, leurs pharmacodynamiques ainsi que les caract´eristiques g´en´etiques du virus et de l’hˆote. Nous pr´esentons une mod´elisation permettant la quantification in vivo d’effets d’antir´etroviraux et de cART bas´e sur des donn´ees in vitro. Pour finir, nous introduisons les probl´ematiques d’int´erˆet dans les travaux futurs comme la compr´ehension des m´ecanismes d’adh´erence ou de r´eservoirs latents. Mots-cl´ es : activit´e antivirale ; adh´erence ; antir´etroviraux ; cART ; in vitro ; in vivo ; m´edecine personnalis´ee ; NLME-ODE ; pharmacocin´etique ; pharmacodynamique ; pharmacog´en´etique ; potentiel inhibiteur instantan´e (IIP) ; r´eservoir latent ; surveillance des traitements ; VIH. Abstract : As shown in the previous chapter, some dynamical models for HIV, the target cells model in particular, may have good predictive abilities, and are useful in the design of treatment monitoring strategies. In this chapter, we review the biological aspects that have to be considered to understand the effect of cART : drug types of action, pharmacokinetics, pharmacodynamics and genetic caracteristics of patients and HIV. We present a modeling that allows quantifying in vivo the effect of antiretroviral drugs and cART based on in vitro knowledge. Finally, we introduce future concerns and prospects such as the understanding of adherence or latent reservoirs mechanisms. Key words : adherence ; antiretroviral drugs ; antiviral activity ; cART ; instantaneous inhibitory potential (IIP) ; in vitro ; in vivo ; HIV ; latent reservoir ; NLME-ODE ; personalized medicine ; pharmacokinetics ; pharmacodynamics ; pharmacogenomics ; treatment monitoring.
Chapitre 7 : Mod´elisation des effets des cART, vers une optimisation
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Les facteurs d’int´ erˆ et dans les mod` eles dynamiques appliqu´ es au VIH
Dans l’id´ee de la m´edecine personnalis´ee, comme d´evelopp´ee Section 2.2 et Figure 7.1, chaque patient est trait´e par une strat´egie th´erapeutique pr´ecis´ement adapt´ee `a ses besoins. La r´ealisation de cette id´ee repose in´evitablement sur l’utilisation r´eussie d’une approche syst´emique par laquelle les biomarqueurs et les param`etres clinico-pathologiques sont int´egr´es aux profils g´en´etiques et mol´eculaires [Mias and Snyder, 2013]. Ce travail passe par la r´ealisation d’une revue de la litt´erature des diff´erents facteurs pouvant avoir un impact sur la r´eponse au traitement d’un patient. Cette ´etape est essentielle, car si certaines sources de forte variabilit´e ne sont pas identifi´ees, les performances d’optimisation des traitements seront largement d´egrad´ees, comme cela a ´et´e montr´e en ´ecologie [Williams, 1996, Ludwig, 1996].
Figure 7.1 : Abstract graphique : Sch´ematisation du protocole de m´edecine personnalis´ee dans le domaine du VIH bas´ee sur des mod`eles dynamiques Ce travail fait l’objet d’un article publi´e dans Advanced Drug Delivery Reviews , Volume 65, Juin 2013, Pages 954-965 [Prague et al., 2013b].
Advanced Drug Delivery Reviews 65 (2013) 954–965
Contents lists available at SciVerse ScienceDirect
Advanced Drug Delivery Reviews journal homepage: www.elsevier.com/locate/addr
Dynamical models of biomarkers and clinical progression for personalized medicine: The HIV context☆ M. Prague, D. Commenges, R. Thiébaut ⁎ University of Bordeaux, ISPED-Epidémiologie-Biostatistique, F-33000 Bordeaux, France INSERM, ISPED, Centre INSERM U897, F-33000 Bordeaux, France
a r t i c l e
i n f o
Article history: Accepted 10 April 2013 Available online 18 April 2013 Keywords: Mechanistic models Dynamic models Personalized medicine HIV infection Biomarkers
a b s t r a c t Mechanistic models, based on ordinary differential equation systems, can exhibit very good predictive abilities that will be useful to build treatment monitoring strategies. In this review, we present the potential and the limitations of such models for guiding treatment (monitoring and optimizing) in HIV-infected patients. In the context of antiretroviral therapy, several biological processes should be considered in addition to the interaction between viruses and the host immune system: the mechanisms of action of the drugs, their pharmacokinetics and pharmacodynamics, as well as the viral and host characteristics. Another important aspect to take into account is clinical progression, although its implementation in such modelling approaches is not easy. Finally, the control theory and the use of intrinsic properties of mechanistic models make them very relevant for dynamic treatment adaptation. Their implementation would nevertheless require their evaluation through clinical trials. © 2013 Elsevier B.V. All rights reserved.
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Models for HIV biomarker dynamics and their predictive ability . . . . . 2.1. A tool: the dynamical/mechanistic models . . . . . . . . . . . . 2.2. An example of field of application: HIV infection . . . . . . . . . 2.3. Statistical challenges in parameters estimation in dynamical models 2.4. Predictive ability of the “activated cell” model in HIV infection . . . 3. Including in vivo/in vitro drug efficacy and compliance . . . . . . . . . 3.1. The type of drug mechanism . . . . . . . . . . . . . . . . . . 3.2. Pharmacokinetics/dynamics . . . . . . . . . . . . . . . . . . . 3.3. Compliance . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Viral mutations . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Host genetics . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Including clinical progression . . . . . . . . . . . . . . . . . . 4. Toward the use of mechanistic models for personalized medicine . . . . . 4.1. Treatment monitoring using mechanistic models . . . . . . . . . 4.2. Implementing the TDM in clinical settings . . . . . . . . . . . . 5. The challenge of the approach . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Abbreviations: HIV, Human Immunodeficiency Virus; HAART, Highly Active antiretroviral Therapy; TDM, Therapeutic Drug monitoring; STI, Structured Treatment Interruptions; ODE, Ordinary Differential Equation; PK, Pharmacokinetics; PD, Pharmacodynamics; PG, Pharmacogenomics. ☆ This review is part of the Advanced Drug Delivery Reviews theme issue on “Mathematical modeling of systems pharmacogenomics towards personalized drug delivery”. ⁎ Corresponding author. Tel.: +33 557574521. E-mail address:
[email protected] (R. Thiébaut). 0169-409X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.addr.2013.04.004
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1. Introduction Guidelines for medical treatments are usually based on empirical results of clinical trials. For instance, for defining the treatment of an HIV-infected subject, these recommendations pertain to four questions [1]: when to start antiretroviral therapy? What to start? When to change? What to change?
M. Prague et al. / Advanced Drug Delivery Reviews 65 (2013) 954–965
The guidelines are defined at a population level and are to be applied to all patients in a standardized way. Personalized medicine aims at defining more specific criteria for each patient. A usual approach is to define subgroups that would more likely benefit from starting or changing a treatment than others because of some individual characteristics based on clinical, biological or genetic markers. Once the treatment is started, another possibility is to observe the initial response of a given patient in order to adapt his/her treatment dynamically: this is often referred to as Treatment Drug Monitoring (TDM) or adaptive treatment strategy [2–4]. Although mathematical models could be useful for this purpose, their use is actually limited by the complexity of their implementation. A well-known example is the regulation of insulin injection based on glycaemia measurements [5]. In a model-based approach, the idea is to learn from the data collected from an individual to forecast the best choices likely to produce the best clinical outcomes. This relies on considering the patient as a “system” that is changing over time [2,6,7]. It then becomes possible to apply models to regulate this system and thereby achieve more desirable outcomes based on control of the input and feedback [8]. The main objective of this review is to show how useful mechanistic models can be for personalized medicine, with a special emphasis on HIV infection. This is introduced in Section 2, where we also present a model of HIV dynamics which has already proven its predictive ability. In Section 3, we then review the potential of using additional information relating to the individual's clinical data related to the HIV infection such as patient compliance, pharmacokinetics (PK)/dynamics data, viral and host genetics. In Section 4, we discuss attempts that have been made for treatment monitoring and Section 5 raises challenges that remain to be addressed. 2. Models for HIV biomarker dynamics and their predictive ability 2.1. A tool: the dynamical/mechanistic models In medicine and especially in clinical research, mathematical models can be useful for dealing with the growing amount of information. More and more factors are recorded for each patient, and more and more biomarkers are measured repeatedly over time. The aim of modelling is usually to summarize multi-dimensional datasets, as well as trying to predict outcomes in future patients. A summary may be a set of independent factors associated to the occurrence of a disease, such as in conventional epidemiological studies. Regarding models, two types can be distinguished, i.e. “descriptive” and “mechanistic” models. Descriptive models are designed using a standard structure that will best fit all available data, with a view to capture association between factors and the evolution of the disease. For instance, if a continuous marker is repeatedly measured over time and has a non-linear evolution, a model based on non-linear mixed effect could then be used. Once the estimates have been obtained, the interpretation of the results might then lead to biological hypotheses explaining the link between factors and the change in the marker. In contrast, the “mechanistic” approach starts from biological knowledge. Biological knowledge is translated into a set of mathematical equations, generally a system of differential equations. Here, we will restrict our discussion to the case of ordinary differential equations (ODE), although models have also been based on the more complex partial differential equations or stochastic differential equations. The ODE system defines a dynamical system for which the behaviour can be studied. There generally are, however, several unknown parameters involved in such a system of differential equations, and its behaviour will of course depend on the value of all parameters, including the unknown ones. Mechanistic models based on differential equations are commonly used in physics. Using this type of models in biology is challenging because the biological mechanisms are different from physical laws and the biological systems are generally more complex than physical
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systems. In other words, models are always a simplification of reality, but an acceptable simplification is often more difficult to achieve in biology than in physics. The gap between model and reality may be reduced by increasing the model complexity, or by introducing stochasticity [9]. Moreover, in physics many parameters are assumed to be known while in biological models, nearly all the parameters are either estimated, or unknown. Thus, the issue of parameter estimation, which is a statistical issue, is a major challenge in biological models. As a consequence, in order to be able to estimate the parameters, a compromise has to be found to design a reasonably simple model which correctly fits the observed data, an issue coined in statistics as the parsimony principle. In mechanistic models we have the additional requirement that the structure of the model must represent important characteristics of the underlying biological mechanisms. Using a mechanistic model presents the advantage of bringing external information to the analysis, i.e. information coming from the knowledge of the biological mechanisms at work, which may lead to a comparison of models reflecting different biological mechanisms [10]. Interaction between various biomarkers may be better captured by the mechanistic modelling than by a complex descriptive model. For instance, several descriptive models were suggested for jointly modelling two markers in HIV infection such as viral load and lymphocytes T CD4 + count [11–14]. However, none were fully satisfactory because none gave both a good fit of the data and an easy biological interpretation of the parameters. The mechanistic nature of the model should ensure two fundamental properties: (i) the parameters should be universal (at least approximately) in the sense that their value should be consistent across any datasets (e.g. cell survival rate); (ii) the predictive ability should be better than that of a descriptive model, that is in situations with data different from those used for learning the model. Once a model has been obtained and shown to have a good predictive ability, it becomes possible to envisage using it for controlling the response, for example for adapting and monitoring drug treatments. 2.2. An example of field of application: HIV infection The HIV infection provides a good example of the use of mathematical mechanistic modelling. The basic biological knowledge is as follows. The HIV virus uses the CD4 receptor to enter into a cell that it will infect: thus the target cells are those presenting a large number of CD4 receptors, essentially the CD4 + T lymphocytes. It is the use of mechanistic models that helped to reveal that a high turn-over of virus and cells is a fundamental phenomenon in the HIV disease [15]. When infected subjects are given a potent antiretroviral drug, the viral load decreases sharply. Assuming there is no more viral replication, the half-life of the virus can be estimated by fitting a simple exponential model (which is the solution of the differential equation dV ¼ −μ V V, where V is the viral load and μV the death rate of the dt virus). Further refinements provided updated estimations of the infected cells half-life [16]. This revealed that the half-life of the virus was short [17]. Hence, the equilibrium state observed during the asymptomatic phase of the HIV disease was actually associated to a constant renewing of both the virus and target cells with a very high turnover rate. One implication of these results was the very high probability of mutations of the virus in just one day. This understanding constituted one of the bases for the rationale of combined antiretroviral therapies that is now the standard of care (SOC). These models, as any model, made several strong assumptions, including a perfect efficacy of the antiretroviral treatment (see [18,19] for a review). This assumption was acceptable as a first approximation because the antiretroviral treatments were highly potent on a short term. However, researchers have also used these models to estimate the antiviral potency of the antiretroviral regimen [20]. Although this objective is clinically highly relevant, several methodological issues arise when embarking in such studies. This is presented
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in part in the Xiao's paper of the present issue (see [21] in this journal) and summarized in the following paragraphs. A wide variety of models exists. As mentioned above, in the pioneering work of Wei et al. [17], only the viral decay was modelled. A model that also incorporated viral production with separate analysis of the viral load and CD4 count was then developed [15]. Later, many proposals were made by various authors, leading to models including more compartments. First, distinguishing between uninfected and infected CD4 + T lymphocytes (in short “CD4”) led to detect that there are heterogeneous half-lives for infected cells and that more complex non-linear models are needed to accurately describe long-term viral decay [16,22]. Several other models were then proposed, assuming exponential life-span of infected CD4 [23], different maturation stages of CD4, CD8 modelling [24] and mutations, most of the time modelled by a finite number of mutants in a discrete form [25–27]. Most of these models were associated to new breakthroughs in the understanding of the infection, and were designed to include the action of antiretroviral drugs. The effects of different drug classes, however, have typically been aggregated in these modelling approaches, with some notable exceptions [20,28–30]. Nevertheless, most of these works are burdened by problems of identifiability: see [31,32] for a description of this issue in the HIV setting. One of the main problems is that, while it is useful to distinguish several types of cells, leading to several compartments, only very few compartments can be observed clinically (typically two in the HIV setting: CD4 count and viral load). Because of the difficulty of estimating parameters, most authors used mathematical modelling as a tool for studying possible dynamics of the system when all the parameters are fixed in advance. With carefully chosen values of parameters, this type of approach can indeed be valuable, for example to examine hypotheses on different biological mechanisms [33], to perform simulations of mutations occurrence [34] or to compare designs of therapies [35]. In a particularly interesting instance, Smith and Wahl [30] proposed an impulsive differential equations model accounting for kinetics of drug actions to evaluate the effect of the duration of the interval between administrations. It was found that a high frequency and a short interval dosing interval is compulsory to maintain a “normal” T cell count and a threshold defined per individual can result in optimized dosing intervals. However the behaviour of dynamical systems strongly depends on the values of the parameters. This type of approach is thus limited to an exploratory phase. We believe that the strength of the mechanistic models approach for medicine individualization can be exploited only if the parameters have been estimated from data. Estimation is a statistical problem, and this represents a particularly challenging one for mechanistic models. This partly arises from the fact that we want to fit the model not on a single patient but using a sample of patients in order to have as much information as possible. However, the assumption that the parameters are the same for all patients does not hold. Treating some parameters as random allows taking into account inter-patient variability. Statistical approaches using random effect models have been developed and applied to data from large clinical trials samples. One should note, however, that fitting ODE-based models with random effects is numerically challenging (see [36] for a review). 2.3. Statistical challenges in parameters estimation in dynamical models In this section, we introduce various methods to estimate parameters in ODE-based models. Classical non-linear mixed effects models [37] based on the exact analytic formula of the likelihood cannot be applied since most of the time the ODEs used to model the dynamics between HIV and the immune system do not have analytic solutions. To get rid of this problem, one may adopt functional data analysis where data are fitted by splines curve smoothing while a penalty term maintains the fidelity to ODE [38]. Although, ODEs do not have
to be solved in this approach, computations are heavy whereas numerical solving of ODEs is computationally possible. Thus, the classical method is to use an ODE solver for computing the likelihood and adopt classical maximization techniques. But, individual hierarchical level in the modelling leads to multiple integration for exact computation of the likelihood and most of the time no close-form exists for the integral. Then, estimation with an approximation of the likelihood has been proposed and implemented in the NONMEM software [39] but this may lead to biased estimations [40]. An alternative is the Stochastic Approximation Expectation Maximization (SAEM) algorithm implemented in the MONOLIX software [41–43] which is a spinoff of the EM algorithm [44] extended to non-linear models. Alternatively, numerical computation of integrals is possible and highly effective in particular by Adaptive Gaussian Quadrature methods [45], thus the exact numerical likelihood computation can be computed instead of making approximations. Complementary to the frequentist approaches presented above, Bayesian techniques have made great improvements in parameters estimation because information from noisy data is often sparse due to practical rather than theoretical lack of identifiability [31]. The most used techniques of estimation are via stochastic simulation known as Markov chain Monte Carlo (MCMC) [46–48]. These approaches can be implemented in the freeware Winbugs [49]. This is particularly appreciated since the direct numerical integrations for the computation of the likelihood are avoided. However, these algorithms are highly time-consuming and convergence is often difficult to assess. A valuable alternative based on penalized maximum likelihood can be found in [50]. This method has been implemented in the NIMROD program (Normal approximation Inference in Models with Random effects based on ODE) which is available for download [51]. In short, the penalized likelihood is computed as the sum of the exact numerical log-likelihood, computed thanks to Adaptive Gaussian Quadrature and the ODEs numerical solver, plus a penalization term coming from the external information that we have on the parameters of the model. Then, a Newton–Raphson like algorithm (the Robust Variance Scoring algorithm [52]) maximizes the penalized likelihood. A significant gain of time is made possible by the approximation of gradients and Hessian by numerical computation of the derivatives of the penalized log-likelihood. The penalized likelihood approach can be thought of as an approximate Bayesian approach: the approximation consists in approximating the posterior distribution of the parameters by a normal distribution (which is asymptotically justified by the Bernstein–Von Mises theorem). This method takes advantage of newly widespread high performance computation clusters thanks to parallel computing of the likelihood for each patient. Moreover, specific attention has been paid to the definition of good convergence criteria. Satisfactory bias and root mean squared errors are reached compared to MCMC technique, see [51] for quantitative details. 2.4. Predictive ability of the “activated cell” model in HIV infection In this section, we illustrate the good predictive ability that can be achieved with a particular mathematical model applied to a HIV clinical trial. We used the data of the controlled randomized open-label ALBI ANRS 070 trial [53] in which three antiretroviral strategies based on two nucleosidic reverse transcriptase inhibitors (NRTI) were compared in 151 previously untreated patients over 24 weeks. The 51 patients from arm 1 received the d4T (stavudine) + ddI (didanosine) treatment, with drug doses adjusted to their weight: d4T 250 mg plus ddI 60 mg for patients under 60 kg, and d4T 400 mg plus ddI 80 mg for those above 60 kg. For arm 2, 51 patients received 500 mg of zidovudine (AZT) and 300 mg of lamivudine (3TC). In the third arm (49 patients), patients were given the same treatment as in arm 1 (D4T + ddI) for 12 weeks and then switched to the treatment given to arm 2 patients (AZT + 3TC) for the remaining 12 weeks. CD4 counts and viral loads
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were measured every 4 weeks until 24 weeks. Two patients dropped out before the time of the first blood sample collection, so 149 patients were available for the Intent-To-Treat analysis. For each patient, changes in dose were either self-reported or declared by the clinician. The details of the design and the collecting methods have been described in details elsewhere [53]. The “activated cell” model is represented by a four-dimensional ODE system, featuring quiescent (Q) and activated (T) uninfected CD4+ T cells, infected CD4 + T cells (T*) and viruses (V) concentrations (see Fig. 1 and Table 1 for parameters description): 8 dQ > > ¼ λ þ ρT−αQ−μ Q Q ; > > > dt > > > dT > < ¼ αQ −γVT−ρT−μ T T; dt dT > > > ¼ γVT−μ T T ; > > dt > > dV > > : ¼ πT −μ V V: dt Based on this mathematical structure, a statistical model was developed for the logarithm of all biological parameters. This transformation was chosen to ensure positivity of all these rates. Let us denote θ˜ ¼ logðθÞ. To take inter-patient variability into account, parameters can be modelled as the sum of a population (fixed) parameter θ˜ 0 and a random effect bi having independent standard normal distributions: θ˜ i ¼ θ˜ 0 þ bi leading to a mixed effects model. After a for˜ and μ˜ T were found to ward selection strategy [54], parameters α˜ ; λ vary substantially among patients and were selected for having random effects, while the other parameters were assumed constant in the population. Because the antiretroviral drugs considered were reverse transcriptase inhibitors (RTI), the treatment was assumed to act on the infectivity parameter γ [55]. For a specific treatment, the standardized dose d was introduced as a covariate with a power funcκ tion: γ˜ i ¼ γ˜ 0 þ βdi , where β is the treatment effects. This can be easily extended to treatment regimens that may change over time by considering as many treatment effects and power functions as the number of drugs in combination. As an illustration, if two treatments are 1 2 considered then the infectivity will be γ˜ i ¼ γ˜ 0 þ β1 di κ 1 þ β2 di κ 2 with two distinct treatment effects estimated. Although the model distinguishes between quiescent, non-infected and infected CD4, only the total number of CD4 is observed. We built an observational model for the log10 of the viral load and the fourth root of total CD4 count (written Y = [log10(V);(Q + T + T*) 0.25]) plus a Gaussian measurement error. From the ALBI data, we constructed a training dataset for estimating the model parameters and a validation dataset on which we performed predictions. To avoid over-fitting, we excluded the patients for whom we made predictions from the learning set. A description of the analysis can be found in [56]. The individual predicted trajectories of viral load and CD4 count were computed by plugging individual estimates of the parameters into the ODE model and by computing the solutions numerically. We assessed
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Table 1 Biological parameters for the “Activated cells model”. Parameter
Meaning
α μT* λ μT π ρ γ
Activation rate of Q cells (day−1) Death rate of T⁎ cells (day−1) Rate of Q cells production (μl−1 day−1) Death rate of T cells (day−1) Rate of virions per T⁎ cell (day−1) Rate of reversion to the Q state (day−1) Infectivity: Infection rate of T cells per virion (μl day−1) Death rate of Q cells (day−1) Death rate of free virions (day−1)
μQ μV
two endpoints: prediction of treatment changes and dose changes. In the third arm of ALBI, patients switched treatment after 12 weeks. To examine whether it was possible to forecast their response to the new treatment given during the remaining 12 weeks, we selected patients at each decile of the distribution of HIV viral load at the end of the study and excluded them from the learning dataset. For each patient belonging to these different deciles, we used the first 12 weeks associated with the first 4 measurements to compute individual parameters and predict the patient responses to the treatment modification. Fig. 2 presents the individual prediction of the markers for a patient (the median patient for viral load) who switched from d4t + ddI to AZT + 3TC. The predictions thus obtained were quite good: only 4.9% of the observations were out of the 95% measurement error predictive intervals. This high quality of prediction suggests that this arm could actually have been avoided since we were able to predict the marker dynamics reliably for each individual. The same type of analysis was performed to assess the predictive ability of our model to dose changes. Predictions of the response to treatment changes were also found to be quite good (Figs. 3 and 4a). Taken together, Fig. 4a and b illustrate that, as expected, these predictions are better if many observations from the patients are used to build the predictions. In other words, short-term predictions are better than long-term predictions. In this particular example, CD4 counts can be correctly predicted using three observations of biomarkers from the same patient (both viral load and CD4 count), whereas one is not sufficient. The AZT + 3TC regimen appeared to have a lower potency in this trial. Indeed, the prescribed AZT dose of 300 mg/d was probably too low if we consider that the current dose prescribed nowadays is 600 mg/d. Additional virus genotype data did not allow to conclude for a higher incidence of resistance mutations in AZT + 3TC group [57]. In conclusion, the individual predictions of the mechanistic model in this trial were found to be very good, thanks to the relevance of the model used in this application and a full estimation of all the parameters jointly. If the mechanistic model does not include a biological mechanism playing an important role in a given circumstance, however, the predictions will probably not be as good as those we observed in the ALBI trial. Further work is currently ongoing to look at the robustness of such models with more up-to-date antiretroviral therapies. Moreover, since it is generally accepted that this kind of model is by definition misspecified, other components will have to be considered to go further. 3. Including in vivo/in vitro drug efficacy and compliance 3.1. The type of drug mechanism
Fig. 1. Graphical representation of the “Activated cells model”.
A potential power of mechanistic models is to distinguish between the mechanisms of action of different antiretroviral agents. Nowadays, five classes of antiretroviral agents exist which do not act on the same pathways (see [58] for a review). Entry inhibitors, interfere with the binding, fusion and entry of the HIV virus. RTIs impair the synthesis of the HIV DNA genome from the RNA strands. In mechanistic models,
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Fig. 2. Viral load (log10 copies/ml) and CD4 count (cells/l) predictions for patients at median of the final HIV viral load distribution for patients in ALBI switch arm (#642). Triangles are observations, red diamonds are observations outside the CI; black lines are fits, blue line on the left of the horizontal line is prediction. Shaded zone represents the 95% measurement error predictive interval. Horizontal dashed line represent left censoring threshold for viral load. Bottom part of graphs gives treatment posology.
we may consider that drugs belonging to these two classes decrease the rate of infection of the CD4. Consequently, the infectivity parameter (γ) should be reduced. Integrase inhibitors block the action of the integrase, a viral enzyme that inserts the viral genome into the cell nucleus. Different stages of infection of the CD4 cells can be modelled, the last stage being virus production. Thus, integrase inhibitors diminish the rate of conversion from early stage infection to late-stage infection. It has been argued that the steep viral load decrease observed with integrase inhibitors is due to the action of the drug at a rather late stage of the viral life cycle [29]. Finally, protease inhibitors (PIs) prevent the viral replication by inhibiting the activity of proteases enzymes used by the virus to cleave nascent proteins for the final assembly of new virions. This can be represented in the model by assuming that part of the produced viruses are defective and non-infectious [59]. Altogether, antiretroviral drugs do not act at the same stage of the viral life cycle; this has a significant impact on the dynamics of the system and must be adequately represented in the models developed. Of note, the ability to distinguish the biological mechanism of any antiretroviral drug will depend on the precision of the measurement (e.g. infected cell, integrated DNA…). In this regard, the models used in most previously published
studies have assumed known and/or constant effects of the drugs [59–61].
3.2. Pharmacokinetics/dynamics PK of antiretroviral drugs are also important factors, but they are difficult to use directly because of the high variability between subjects in the drugs absorption, distribution, metabolism and excretion [62]. Moreover, bioavailability and protein binding can complicate the in vitro drugs quantification and cause confusions [63]. Although the antiretroviral activity should be linked with plasma concentration, this is only correlated with the drugs intracellular concentrations, which are the truly significant variable [64]; this is mainly because antiretroviral agents act mostly at intracellular sites. Quantification techniques thus need to improve before PK values can be used to evaluate treatment efficacy and clinical endpoints. Thus, the PK data may be insufficient to guide the adjustment of administered doses in patients appropriately and must be combined with considerations about pharmacodynamics (PD).
Fig. 3. Viral load (log10 copies/ml) and CD4 count (cells/l) predictions for patients at quartiles (Q1 and median) of the final HIV viral load distribution for patients in ALBI experiencing dose changes (#660 and #562). Triangles are observations, red diamonds are observations outside the CI; black lines are fits, blue line on the left of the horizontal line is prediction. Shaded zone represents the 95% measurement error predictive interval. Horizontal dashed line represent left censoring threshold for viral load. Bottom part of graphs gives treatment posology.
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Fig. 4. Viral load (log10 copies/ml) and CD4 count (cells/l) predictions for patient at third quartile (Q3) of the final HIV viral load distribution for patients in ALBI experiencing dose changes (#664). Triangles are observations, red diamonds are observations outside the CI; black lines are fits, blue line on the left of the horizontal line is prediction. Shaded zone represents the 95% measurement error predictive interval. Horizontal dashed line represent left censoring threshold for viral load. Bottom part of graphs gives treatment posology. Two graphs at the lower part of the figure represent predictions for the same patient (Q3) demanding on the number of observations used to construct predictions. Left is on a long-term (1 observation used to predict a 6 points trajectory) and on a short-term (3 observations used to predict a 3 points trajectory). (a) Only the baseline observation is used to build long-term predictions. (b) First three observations are used to build shorter-term predictions.
Several factors are important to determine PK–PD relations, ranging from inter-patients differences to heterogeneity of viral responses to drugs, as well as interactions between the two. Table 2 presents a summary of some of the most used indicators. Most of the time, PD is evaluated in vitro since it is easier to measure viral replication in the presence of various concentrations of the drug. The most commonly used indicator is the IC50, defined as the concentration of drug required to achieve 50% inhibition for a specific strain of virus. This is a static indicator that does not indicate what the impact of lower or higher drug concentrations would be. Moreover, it is a poor reflection of antiviral activity since it depends on the intrinsic properties of the drug, the kinetics of the stage of the viral life cycle and the kinetics of the infected cells [18]. Inhibitory quotients (IQs) are associated with the virological response since they are defined as the ratio between drug concentration (Cthrough, plasma concentration before the next dose) and IC50 [65]. IQs are, however, constant factors and do not describe dynamic variations according to the administrated doses. Hence, using dose–response curves from infectivity assays has been suggested for quantifying the antiretroviral
potency [66]. The instantaneous inhibitory potential (IIP) is a function of the dose (D). It is dynamic thanks to the Hill coefficient (m) which tunes the slope of the dose–response curve. This is particularly interesting because it allows to use the number of infected cells as a direct readout for viral replication. In practical terms, IIP is the logarithm of the number of single-round infection events reduced by a drug [67]: D m IIPðDÞ ¼ log10 1 þ : IC50 Thus, the effectiveness of an antiretroviral in vivo is determined by m and IC50. It then becomes possible to introduce time variations by using the PK for a particular dosing schedule leading to D(t) (where t is the time). This time-varying ratio between the dose and IC50 in IIP has an interpretation comparable to IQs [67]. Moreover, apart from the dynamical advantages, IIPs are similar to IQs for predicting HIV responses to antiretroviral agents [68]. Furthermore, IIPs are sensitive to PK-boosting effects and interactions between drugs [69]. Empirically observed interactions of antiretroviral drugs in clinical trials
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Table 2 PK–PD indicators definition. PK–PD markers
Meaning
C
Plasma Drug concentration observed after a drug administration and just prior to the administration of a subsequent dose. Drug concentration needed to obtain 50% inhibition of the HIV replication in culture cells.
through
IC50
IQ
vP
vG vIQ
gIQ
D
IIP
C through/IC50 The Inhibitory Quotient measures how much the drug exceeds the IC50. The Virtual Phenotype is the change-fold in IC50 for a mutant variant compared to a WT. The Virtual Genotype is number of expressed listed mutations linked to the considered drug. IQ/vP The Phenotypic Inhibitory Quotient is the IQ for a specific virus strain compared to WT. C through/vG The Genotypic Inhibitory Quotient evaluates the effective concentration of the drug. Drug concentration achieved with a standard dosing. log10(1 + (D/IC50)m) Instantaneous Inhibitory potential measures the antiviral activity accounting for the slope of the inhibition curve (m, the Hill coefficient) depending on the dose.
Advantages
Drawbacks
⁎ Easy to sample, ⁎ Recognized as a standard measure. ⁎ Intuitive interpretation considered as the minimum required level, ⁎ Can quantify phenotypic resistances. ⁎ Include virus susceptibility and individual drug absorption information. ⁎ Evaluate individual virus susceptibility, ⁎ Account for a mixture of HIV variants. ⁎ Evaluate individual virus genotype, ⁎ Fast to run. ⁎ Easy to interpret for a clinician, ⁎ No understanding of mutations effects. ⁎ Avoid use of IC50, ⁎ Sensitive to detect new resistances. ⁎ Dynamic modeling of drug intake.
⁎ Interpretation regarding the replication, ⁎ Possible to account for cooperation between drugs, ⁎ Differentiate antiretroviral agents classes with m.
are reviewed in [70]. There is no metric to describe the combined effect of multiple drugs, but, four categories of interactions can be distinguished: synergism when one drug enhances the effect of the other: Bliss independence when each drug inhibits a portion of the targets; Loewe additivity that assumes that two inhibitors act on a target through a similar mechanism; antagonism when action of a drug inhibits the action of the other [71]. Combined IIP can be evaluated in vitro and reflects these interactions. An exhaustive listing for more than 166 pair-wise and 1892 three-ways combinations over 20 treatments is available in [72]. This provides a large dataset of in vitro potency assessment and a quantitative basis to compare and analyse treatment effects. However, extrapolating from in vitro data to in vivo situations is complex. First, in vitro susceptibility assays do not follow the viral dynamics during days or weeks but at a specific time point in specific conditions [73]. Second, variations of several fold of these PK–PD indicators can occur during particular clinical situations such as acute infections, advanced HIV disease or pregnancy [74]. This can, however, be partially corrected by introducing explanatory variables, or by introducing a conversion parameter that has to be estimated from the data [75]. This is analogous to introducing a scaling factor that is reflective of the uncertainty between the drug measurements of drug potency in vitro and the specific model-based drug potency in vivo [76]. 3.3. Compliance Compliance to treatment is essential for achieving efficient concentration at the target tissues. Lack of compliance can be represented by dose-delay or dose-omission. Although many methods exist, compliance remains very difficult to assess [77]. Poor compliance is a key factor for explaining viral load variations and is strongly associated with treatment failure [78,79] as it favours viral mutations and therefore viral resistance to treatment (see next section). Nevertheless, the impact of compliance varies according to the type of antiretroviral regimen [80]. For estimation concerns, poor or unmeasured
⁎ ⁎ ⁎ ⁎ ⁎
Spatiotemporal, Few data, Inter individual variations. In vivo determination is difficult, Depend on the virus strains.
⁎ ⁎ ⁎ ⁎ ⁎
Based on in vitro results, Not optimal in case of mutations, Assume linear impact of the dose. Based on in vitro results, High costs and long to run.
⁎ Need a full understanding and knowledge of virus position of mutations. ⁎ Related with the drugs cut-off, ⁎ Low sensitivity to emerging strains. ⁎ Values difficult to interpret, ⁎ Need to understand gene pathways due to mutations. ⁎ Spatiotemporal variations, ⁎ Difficult to establish which dose is of interest (oral, plasma, intracellular…). ⁎ It is unclear whether it is better than IQ in vivo to correlate with virologic outcomes
compliance can result in an underestimation of drug effectiveness leading to overestimation of drug dosage [81]. A classical and convenient way to take compliance into account in models is to include a compliance rate A(t) and to replace the dose D(t) by the product A(t)D(t) in the PD model [48,82–86].
3.4. Viral mutations Genotypic and phenotypic characteristics of the virus can also greatly influence the response to antiretroviral treatment. Genetic barrier for resistance refers to the number of nucleotide changes a virus needs to accumulate to become resistant to a given antiretroviral drug. Information on patterns of resistances and cross-resistances is important for making decisions on how to change and combine drugs to achieve an optimum antiviral effect. Hence, genotype information is increasingly used for deciding which antiretroviral drugs should be included in a given regimen. Genotypic and phenotypic assays based either on the detection of specific point mutations or on complete DNA sequencing are increasingly available [87]. In vitro assessment of HIV fitness has been extensively studied by different techniques; see [88] for a complete review of this topic. In short, resistance is typically measured by the virtual phenotype which is the fold change in IC50 relative to the wild-type (WT) virus [89,90]. A derivation of this is the virtual IQ (vIQ), which is the IQ divided by the virtual phenotype and has been shown to have a significant link with the viral response to treatment [91]. An improvement is to introduce vIQs in IIPs (described in Section 2.1 multiplying the IC50 by the virtual phenotype leading to virtual IIPs) because this allows accounting for the steepness of the dose–response curves (represented by Hill coefficient m). This is particularly important for PIs whose IC50 is usually the same in mutant variants and WT but for which a higher dose is needed to achieve viral extinction because the Hill coefficient m is smaller [92]. In vivo, the viral response to a new antiretroviral regimen is studied according to the genotype measured at the time of treatment change. Various approaches have been proposed
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to predict the response to antiretroviral regimen according to the baseline genotype [93]. In mathematical modelling, mutations are introduced by adding virus compartments to account for different strains of virus (for an example, see the pioneering work of Nowak et al. [94]). One approach consists in considering that several genotypes of the virus, V = (V1…VK) exist in each patient. They are called quasi-species. The different genotypes may have different fitness, which is efficiency of replication. In real life, the number of genotypes can be large, but numerical complexity limits the number of possible genotypes that can be represented in a model [95]. Another approach which is easier to build focuses on vIIPs instead of IIPs, and modulates the treatment effect depending on the mutant variant k. ODE-based models with immune cells and multiple strains of viruses have allowed the comparison of scenarios; as an example of results, it has been calculated that the probability that resistant mutants pre-exist to treatment is higher than the probability that resistant mutants are generated during therapy [96]. In similar models, Rong et al. introduced the effects of compliance to short-term [86] and long-term [97] treatment, and showed that many variants of the virus live as quasi-species in the host. Mutant variants that are selected have a lower sensitivity to the antiretroviral drugs and are associated with a rebound of the viral load [98]. The resistant viruses are capable of replicating better than the sensitive viruses and will therefore be positively selected [99]. The viral fitness of a mutant is, however, generally lower than that of a wild type; consequently, replication, virulence, and transmission of the selected mutant viruses may be reduced compared with those of the WT [100]. Specific models have been suggested for estimating these differences in viral fitness [101,102]. An integrated model has been recently proposed by Rosenbloom et al. [85]. The authors suggested that this approach might be more relevant than the use of a genotype analysis for evaluating the antiviral response. Indeed, genotype information is based on meta-analysis of data obtained from various regimens in patients with various characteristics, and this is not reliable because the effect of mutations is probably highly dependent on the context. In practice the detection of resistant mutants is commonly used to choose or modify the drug regimen given to a patient; however, this kind of knowledge has not yet been incorporated in a mechanistic model for personalized medicine. 3.5. Host genetics It is not common to use pharmacogenomic (PG), which is the patient genetic characteristics, for determining the choice of antiretroviral drug but this may be relevant in some very specific indications. For instance, the HLA-B*5701 allele has been shown to lead to a risk of about 50% of hypersensitivity with abacavir, making the screening of this haplotype systematic before prescribing this drug [103]. Drug–drug interactions and genetic polymorphism in drug-metabolizing enzymes and drug transporters (MDR1, CYP3A5, CYP2C19…) contribute to wide variability in drug PK, response to therapy and toxicity [104–107]. For instance, using a population PK model including genetic characteristics, several CYP2B6 polymorphisms have been identified as being associated with steady-state nevirapine clearance among HIV-infected patients [108]. A dynamic approach is valuable to model gene interaction in a quantitative manner [109]. A modelling strategy with linear ODEs has been proposed by Lu et al. [110] to capture gene regulatory network. The perspective of introducing biomarkers and antiretroviral drugs as new compartments in these systems is appealing and can bridge the gap to combine PK–PD–PG to explain HIV dynamics under treatment. To date, however, the information on individual susceptibility to antiretroviral drugs remains sparse [105]. The inclusion of any PG data in mechanistic models, provided that the information is available, can be expressed by modelling some parameters as a function of individual values. The remaining inter-individual variability is then taken into account through random effects on parameters, especially those representing treatment effects. Including PG information would decrease
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the unexplained variability captured by the variance of the random effect. 3.6. Including clinical progression The first outcome of interest when modelling the response to antiretroviral treatment is the viral rebound [85] followed by the CD4 dynamics since both these markers are very good prognostic markers [111,112]. These markers, however, are still imperfect surrogate markers of clinical outcome, that is, it is not possible to predict the exact clinical effect of an antiretroviral treatment by just observing an increase in CD4 or a decrease of HIV viral load [113,114]. This is even more the case nowadays because AIDS-classifying diseases are less common, whereas the proportion of deaths due to cardiovascular diseases and cancer is growing [115]. Also, depending on the type of therapeutic approach used, the ability of these markers to reflect the clinical benefit is not guaranteed. For instance, the administration of the IL-2 cytokine results in a substantial increase of CD4 but without any significant clinical benefit [116]. Therefore, it may be important to model the dynamics of biomarkers and clinical progression jointly. Many joint models have been proposed in the literature, including applications in the HIV-AIDS field [13,14,117–120]. Wu et al. [121] proposed a joint model including a non-linear (bi-exponential) mixed effect model for the viral dynamics with a proportional hazards model for the time-todisease progression, and a complete mechanistic model based on ODE including a time-to-event process has subsequently been proposed by Guedj et al. [122]. Starting from a previous ODE-based model [50], the time-to-disease progression has been linked to the marker predictions at any time by a proportional hazard model. Such joint models of longitudinal markers and time-to-event processes take into account the informative missing data process due to the censoring of observations after the occurrence of disease progression. Furthermore, an easier two steps approach, where the longitudinal model is fitted first and predictions are then used in a survival model, ignores the variability of the parameters of the longitudinal model estimated in the first step [118]. Finally, in the context of adaptive treatment, one needs dynamical models from which predictions are available at any time, thus requiring joint models. What is interesting with such models is that one can obtain predictions of changes in markers (CD4 count, viral load) but also the probability of disease progression (according to the change in markers). A treatment monitoring based on this type of models sounds very useful. However, the wide spectrum of clinical diseases observed today in HIV-infected patients is due to various processes, from antiretroviral side effects [123] to the consequences of long-term HIV infection (cell activation, inflammation) [124]. Therefore, including all types of clinical events may result in a highly complex model. On the other hand, this type of model could be devoted to specific questions. For instance, with the objective of finding an optimal dose of antiretroviral, a model that includes the effect on the viral dynamics as well as the probability of side effects could serve at finding a good compromise between viral efficacy and side effects. 4. Toward the use of mechanistic models for personalized medicine 4.1. Treatment monitoring using mechanistic models HAART succeeds in reducing viral load in most HIV infected patients but does not eradicate the virus. Thus the treatment has to be taken life-long. To reduce the burden of side effects it has been proposed to interrupt the treatment under certain conditions or to find an optimal dose. Structured treatment interruptions (STI), where antiretroviral treatments are stopped either during a given period or according to CD4 counts has been a first attempt to optimize drug administration [28,125,126]. These periods of treatment interruption
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were supposed to boost the immune system, to offer pill burden vacancy and decrease side effects. Some studies have also evaluated strategies where the immune system was boosted with vaccine and/or interleukin 2 before interruption [127,128]. Unfortunately, the evaluation of the effect of STI on clinical outcomes through randomized clinical trials has concluded that STI were harmful [129]. Control theory (taken from engineering literature) has been suggested to find the optimal interventions that reduce a given cost function [130]. Quadratic cost functions that weighted system response (viral load or CD4 count) and side-effects of the drug have been proposed [9]. Although conceptually interesting, this approach is not realistic because (i) the model is not known, (ii) the choice of the cost function is debatable, (iii) the treatment cannot be continuously adapted. More recent approaches tend to get free from cost functions [131] and do not aim at adapting the dose continuously [132]. Prague et al. [56] proposed to give the lowest dose for which there is infection control. This is based on ODE stability criterion, the reproductive number R0 also used in SIR models: if R0 b 1 there is extinction of the infection. Thus the idea is to give the lowest possible dose that ensures R0 b 1. Such an approach was applied to the ALBI clinical trial. The patients in the most potent group (d4T + ddI) were selected to evaluate the best dose change required. A specific patient with undetectable viral load (below 50 copies/ml) during all the study after treatment initiation was proposed a dose reduction up to 53% of her/his initial dose whereas another patient with default in infection control during antiretroviral therapy was assigned a moderate dose increase. Then, observing how the patient reacts to the new dose it should be possible to update it in an iterative manner. Such recommendations should be tested in order to be validated and in order to observe how patients react. 4.2. Implementing the TDM in clinical settings Before the efficiency of TDM on HIV disease control can be tested, a first step should be to assess the capacity of TDM to influence the concentrations of drugs or biomarkers values into a predefined range so as to validate or invalidate the model used, and the TDM approach. This could initially be applied to all antiviral drugs, with a view to focus subsequently on more “targetable” drugs. This was already done by Duval et al. [133], who showed that, among PIs, dose TDM is more likely to be useful for indinavir and lopinavir than for nelfinavir. Variability should also exist between treatment regimens: on this subject, Back et al. [134] argue in favour of TDM for PIs instead of NRTIs. It is still difficult to build an acceptable design to test TDM. To date, TDM trials are difficult to interpret and statistically underpowered [135]. The method of choice could be a randomized clinical trial comparing at least two strategies: SOC and TDM. One of the simplest treatment adaptations in HIV infection is based on the use of viral resistance information. Several randomized clinical trials have been performed to evaluate a strategy based on resistance information compared to SOC [136–138]. Specific designs could, however, be considered for the evaluation of more complex approaches for dynamic treatment. Murphy [139] advocated for the use of adaptive design, where the term “adaptive” refers to the experimental design that could be flexible: changing strategy allocation according to the responses in previous patients. An example of this is the sequential multiple assignment randomized (SMAR) trials where each individual may be randomized multiple times and multiple randomizations occur sequentially through time. Finally, an easily acceptable way of testing, which can be adapted retrospectively by adjusting on confounders, should be to analyse a previously designed clinical trial in two groups: after the forecast of the best treatment according to mechanistic models, comparing results between patients who had treatments below TDM targets and
others. This kind of comparison was partly used by Best et al. [140] to assess antiviral drug dose TDM and this study showed that patients below TDM forecast tended toward a greater viral load. 5. The challenge of the approach In this paper, we have shown that, in HIV patients, mechanistic models could be very useful to perform individual prediction of the response to an intervention based on antiretroviral drugs and could therefore be a useful tool for treatment monitoring. This could also be the case for other interventions such as immune based therapy [128,141]. However, the mechanistic model needs to (i) include relevant biological processes involved in the response (ii) to be fed with enough data (that is with relevant markers measured repeatedly). In some circumstances such as in the ALBI trial this could be achieved, but more developments are probably needed to cope with the numerous types of drugs available today. The implementation of processes such as the occurrence of genotypic mutation or the effect of known host genotype on the treatment efficacy could be achieved without too much complexity. In such cases, however, the restriction might come from the absence of measured information. Also, in line with the parsimony principle that maximizes the external validity of the predictions, simple models should be prioritized. Then, if a good model proves relevant for current treatments, the next step would be to evaluate the potential of such approaches for treatment monitoring by well-designed clinical trials. Acknowledgments We thank the ANRS/Vaccine Research Institute for grant support. Parallel computing used in illustrations was used thanks to the computing facilities MCIA (Mésocentre de Calcul Intensif Aquitain) of the Université de Bordeaux and of the Université de Pau et des Pays de l'Adour. References [1] S.M. Hammer, M.S. Saag, M. Schechter, J.S.G. Montaner, R.T. Schooley, D.M. Jacobsen, M.A. Thompson, C.C.J. Carpenter, M.A. Fischl, B.G. Gazzard, et al., Treatment for adult HIV infection, JAMA 296 (7) (2006) 827–843. [2] P.W. Lavori, R. Dawson, Adaptive treatment strategies in chronic disease, Annu. Rev. Med. 59 (2008) 443–453. [3] D. Back, G. Gatti, C. Fletcher, R. Garaffo, R. Haubrich, R. Hoetelmans, M. Kurowski, A. Luber, C. Merry, C.F. Perno, Therapeutic drug monitoring in HIV infection: current status and future directions, AIDS 16 (2002) S5–S37. [4] J.G. Gerber, E.P. Acosta, Therapeutic drug monitoring in the treatment of HIVinfection, J. Clin. Virol. 27 (2) (2003) 117–128. [5] D.J. Lunn, C. Wei, R. Hovorka, Fitting dynamic models with forcing functions: application to continuous glucose monitoring in insulin therapy, Stat. Med. 30 (18) (2011) 2234–2250. [6] S. Murphy, Optimal dynamic treatment regimes, J. R. Stat. Soc. Ser. B (Stat. Methodol.) 65 (2) (2003) 331–355. [7] S.A. Murphy, J.R. McKay, Adaptive treatment strategies: an emerging approach for improving treatment effectiveness, Clin. Sci. 12 (3) (2004) 7–13. [8] D.E. Rivera, M.D. Pew, L.M. Collins, Using engineering control principles to inform the design of adaptive interventions: a conceptual introduction, Drug Alcohol Depend. 88 (2007) S31–S40. [9] R.F. Stengel, R. Ghigliazza, Stochastic optimal therapy for enhanced immune response, Math. Biosci. 191 (2) (2004) 123–142. [10] A. Yates, J. Stark, N. Klein, R. Antia, R. Callard, Understanding the slow depletion of memory CD4+ T cells in HIV infection, PLoS Med. 4 (5) (2007) e177. [11] W.J. Boscardin, J.M.G. Taylor, N. Law, Longitudinal models for AIDS marker data, Stat. Methods Med. Res. 7 (1) (1998) 13–27. [12] R. Thiébaut, H. Jacqmin-Gadda, C. Leport, C. Katlama, D. Costagliola, V.L. Moing, P. Morlat, G. Chêne, A.S. Group, Bivariate longitudinal model for the analysis of the evolution of HIV RNA and CD4 cell count in HIV infection taking into account left censoring of HIV RNA measures, J. Biopharm. Stat. 13 (2) (2003) 271–282. [13] R. Thiébaut, H. Jacqmin-Gadda, A. Babiker, D. Commenges, Joint modelling of bivariate longitudinal data with informative dropout and left-censoring, with application to the evolution of CD4+ cell count and HIV RNA viral load in response to treatment of HIV infection, Stat. Med. 24 (1) (2005) 65–82. [14] E.R. Brown, J.G. Ibrahim, V. DeGruttola, A flexible B-spline model for multiple longitudinal biomarkers and survival, Biometrics 61 (1) (2005) 64–73.
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7.2 : De l’in vitro a` l’in vivo
7.2
159
De l’in vitro ` a l’in vivo
Dans les mod`eles prenant en compte la pharmacologie et la g´enomique, il y a un grand ` cause de probl`emes d’identifiabilit´e, tous les param`etres ne nombre de param`etres. A peuvent pas ˆetre estim´es directement. C’est pourquoi, dans ce chapitre, nous d´eveloppons une approche pour inclure des donn´ees issues d’essais in vitro et des donn´ees issues de combinaisons de plusieurs essais in vivo.
7.2.1
L’information in vitro
Dans le domaine du VIH, pour mesurer les capacit´es antir´etrovirales d’un traitement, les exp´erimentations in vitro sont r´ealis´ees en mesurant le nombre de nouvelles infections en pr´esence du traitement a` diff´erents niveaux de concentration. Les technologies d’essais in vitro ont ´evolu´e permettant des quantifications de plus en plus pr´ecises. Initialement, il existait des kits d’essai multi ´etapes (multiround assays en anglais) o` u des cellules cibles humaines ´etaient mise en pr´esence de virus et incub´ees pendant plusieurs semaines [Japour et al., 1993]. Cependant, les essais in vitro multi-´etapes sont difficilement reproductibles [Kellam and Larder, 1994] et sous-estiment potentiellement la capacit´e antivirale [Ferguson et al., 2001]. Depuis le d´ebut des ann´ees 2000 sont apparues des alternatives beaucoup plus rapides et permettant d’avoir un indicateur de la capacit´e antivirale instantan´ee, ce sont les kits d’essai une-´etape (single round infectivity assays an anglais) [Zhang et al., 2004], voir Figure 7.2 pour une description. Des cellules cibles g´en´etiquement modifi´ees par la prot´eine verte fluorescente GFP qui ne fluorescent que si la cellule devient productrice de VIH, sont infect´ees par spinoculation (inoculation par centrifugeuse). Les cellules pour lesquelles le traitement est actif n’expriment pas la GFP et ne sont donc pas d´etect´ees par les techniques de quantification par cytom´etrie de flux. Pour diff´erents niveaux et types de traitements (`a l’image de ce qui est fait dans les tests ph´enotypiques voir Section 2.2.1), il est alors possible de quantifier la fraction non affect´ee par le traitement (fu ) et d’obtenir des courbes instantan´ees d’activit´e antivirale. Une m´ethode usuelle pour ensuite analyser
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160
ces r´esultats est d’utiliser une sigmo¨ıde ce qui am`ene la capacit´e antivirale a` d´ependre de trois facteurs [Chou, 1976] : la dose (D), un coefficient d’inhibition a` 50% (IC50 ) et la pente de la courbe dose-r´eponse (m), aussi appel´ee coefficient de Hill [Hill et al., 1910].
Figure 7.2 : Essai in vitro une ´etape (single round infectivity assays) pour la quantification instantan´ee de capacit´e antivirale des traitements VIH Des courbes doses-r´eponses ont ´et´e ´evalu´ees in vitro pour diff´erents traitements antir´etroviraux [Shen et al., 2008] puis ´etendues pour des cART en prenant en compte des interactions [Jilek et al., 2012]. Pour finir, la question de l’impact de mutations sp´ecifiques du VIH sur ces courbes a ´et´e quantifi´e par [Sampah et al., 2011, Rosenbloom et al., 2012]. Ainsi, il existe une base de donn´ees in vitro pour la majorit´e des cART donn´es en routine chez les patients infect´es par le VIH que nous pouvons exploiter pour augmenter l’information contenue dans des jeux de donn´ees cliniques.
7.2.2
De l’in vitro ` a l’estimation d’effets traitement in vivo
La diversit´e du VIH et des cellules cibles ne peut pas ˆetre reproduite dans toute sa complexit´e dans des conditions d’exp´erience in vitro. De plus, [Orenstein, 2007] a montr´e qu’il existe de nombreuses disparit´es entre ce qui peut ˆetre observ´e in vitro et in vivo lors de la r´eplication du VIH. Par exemple, alors que les macrophages sont facilement infect´es in vitro, ceci est moins commun in vivo. Quand l’int´erˆet se porte sur l’´evaluation des
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161
effets traitement, la dynamique est encore mal comprise, en particulier, il n’existe pas de relation claire entre activit´e antivirale et efficacit´e chez le patient [McMahon et al., 2009]. L’information in vitro, n’est donc pas facile `a prendre en compte de mani`ere directe mˆeme lorsqu’elle est disponible c’est pourquoi il est n´ecessaire d’introduire des facteurs de conversion in vitro-in vivo [Huang et al., 2003]. Nous proposons une nouvelle mod´elisation dynamique afin d’ˆetre capables de pr´edire la r´eaction d’un patient a` un nouveau traitement. Ce travail fait l’objet d’un article en cours d’´ecriture et d’une collaboration avec l’´equipe de Robert Siliciano (Department of Medicine, Johns Hopkins University School of Medicine, Howard Hughes Medical Institute, Baltimore, Maryland, USA.), Daniel Rosenbloom (Program for Evolutionary Dynamics, Department of Mathematics, Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts, USA.) et Martin Nowak (Program for Evolutionary Dynamics, Department of Mathematics, Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts, USA.). Nous introduisons ici une note d’´etape r´esumant la mod´elisation, les principaux r´esultats et les perspectives de travaux futurs. Ce document ne constitue pas un travail soumis mais une pr´esentation prospective des voies de recherche possibles.
Work in progress (not submitted)
From in vitro to in vivo quantification of antiretroviral drugs effects based on dynamical models of HIV Prague M´elanie1,2,3 , Commenges Daniel1,2,3 and Rodolphe Thi´ebaut1,2,3,∗ 1Univ. Bordeaux, ISPED, F-33000 Bordeaux, France 2 Centre INSERM U897-´epid´emiologie-Biostatistique, F-33000 Bordeaux, France 3 INRIA, SISTM, F-33000 Bordeaux, France ∗ E-mail:
[email protected] Abstract. Population dynamics of HIV and CD4+ T cells can be modeled with Ordinary Differential Equations (ODE). We aim at quantifying the in vivo effect of combinations of antiretroviral drugs treatments (cARTs) by a function of the effects of the antiretroviral drugs in the combination. To estimate the antiretroviral drugs effects we must have a large dataset and it is desirable to add external information to ensure identifiability. An adequate modeling of in vitro assays yields such information. Recent single-round infectivity assays allowed quantifying the dose-response curves in vitro: the instantaneous inhibitory potential (IIP) has been established as a measure of antiretroviral drugs activity. The IIPs of cARTs can be viewed as a function of antiretroviral drug’s IIPs based on known interactions. Bliss independence is a convenient assumption. Random effects account for inter-individual variability of IIPs that may result from host and virus genetics. Finally, more flexibility is provided by estimating an in vitro to in vivo conversion factor. We used a Bayesian approach for estimating the antiretroviral drugs in vivo effects: cARTs in vivo effects follow by computation. We demonstrate that this model has good fit abilities and that our approach opens the perspective to get a deeper understanding of treatment action regarding adherence and latent reservoirs and to consider treatment choice optimization. This analysis is applied to a dataset of 350 patients taking 7 different antiretroviral drugs (AZT, D4T, ddI, 3TC, LPV, APV, DRV) from two stand-alone clinical trial: ALBI, PUZZLE, and two clinical trials nested in the ANRS CO3 Aquitaine Cohort: ZEPHIR and PREDIZISTA.
T *
T
T
T*
20
EFV TDF FTC
RAL ABC 3TC
cART
c
In vitro to In vivo
Drug-specific in vitro to in vivo estimated conversion factor Time-dependant Adherence % [average]
Q
10
IIP [average]
20 3TC ABC AZT d4t ddI FTC TDF EFV NVP DLV ATV APV DRV IDV LPV NFV SQV TPV T20 RAL
Q
APV 3TC AZT
Drugs
In vivo dynamic ”target cells model”
d
In vitro IIP (cART)
10
IIP [average]
In vitro IIP (drugs)
0
b
Drugs
NRTI: AZT (600mg) , 3TC (300mg), d4T (60mg), ddI (400mg) PI: APV(1400mg), DRV(800mg), SQV(1000mg)
0
a
150% 100%
V
V
50% 0% 0
1
2
Years
3
Figure 1: Graphical abstract. (a) the 7 drugs studied in the application (b) have intrinsic antiviral activities called IIPs evaluated in vitro. Assuming Bliss independence IIPs of cART can be computed from drugs IIPs by summation. (c) Estimated conversion factor and patients’ adherence information allow us to use in vitro IIPs in (d) the in vivo “target cells model” of the dynamics between HIV and the immune system. Page 1 of 10
Work in progress (not submitted)
Biological Model
in vivo IIP of a cART composed of ncART antiretroviral drugs is :
Mathematical model.
IIPcART (dX1 (t),
We recall the “target cells model” (see Figure 1.d). This is exactly similar to the “activated cells model” used in this PhD thesis but was renamed to avoid over interpretation regarding the term “activation”. dQ = λ + ρT − αQ − µQ Q, dt dT = αQ − γT V − ρT − µT T, dt dT ∗ = γT V − µT ∗ T ∗ , dt dV = πT ∗ − µV V. dt
In vitro pharmacodynamics.
Shen et al. (2008) designed the instantaneous inhibitory potential (IIP) as a measure of antiviral activity. It is the number of logs single-round infection events that are prevented by the treatment. Thus, it is related to the fraction of cells unaffected by an antiretroviral drug X, fuX , also called fraction of infectible cells. Then, IIP depends on the drug intake dX (t), the concentration producing 50% inhibition of viral replication (IC50X ) and the Hill coefficient (mX ), see Chou (1976) for definitions. IIPX (dX (t)) = −log(fuX (t)), mX dX (t) = log 1 + . IC50X
...
, dXncART (t))
=
−log(fucART (t)),
=
=
XncART
(t)),
nX cART
βXi IIPXi (dXi (t)).
Dynamical modeling of in vivo effects of cART. NRTI, NNRTI, FI and II directly act on the fraction of infectible T cells: under such antiretroviral drug X, only fuX T cells can be infected. PI acts on the number of T ∗ cells producing viruses, thus only fuX V viruses can infect T cells. In both cases the number of infected cells T ∗ created in a time dt is γ0 fuX V T , where γ0 is the infectivity without treatment. Thus it is equivalent to consider that fu modify the infectivity parameter in a multiplicative way. Figure 2 shows the impact of fucART on the law mass action of production of T ∗ cells depending on V and T. WITHOUT TREATMENT T
0TV
T*
WITH TREATMENT T
0 f ucART TV
Unaffected by the cART fraction of cells.
T*
V
Figure 2: Effect of cART on the infectivity parameter. Altogether, we denote γ˜ the log-infectivity under treatment and γ˜0 the base log-infectivity without treatment : γ˜ = γ˜0 −
−log(fuX1 (t) × · · · × fu nX cART IIPXi (dXi (t)).
, dXncART (t))
i=1
V
Jilek et al. (2012) showed that drug-drug interaction in combination of antiretroviral therapies (cART) obeys most of the time Bliss independence (Bliss, 1939). In other words, drugs act through competitive phenomenon on the same receptors, thus, fucART is the product of the fraction of infectible cells for each antiretroviral in the cART. The in vitro IIP of a cART composed of ncART antiretroviral drugs is : IIPcART (dX1 (t),
...
nX cART
βXi IIPXi (dXi (t))
i=1
Evaluation of drug intakes.
We define the adherence as a time-dependent drug-specific covariate AX (t), which is the percentage i=1 of the drug taken by the patient compared to what was prescribed by the clinician. Thus, drug intake dX (t) at time t can be defined as dX (t) = AX (t)DX , In vivo pharmacodynamics. where DX is the standard daily dose given in stanWe introduce a drug-specific conversion factors dard of care routine for a drug X whatever the daily (β’s) between in vitro and in vivo IIPs. This gives the number of intake (see Figure 1 a for details on these model more flexibility compared to conversion factors values). As instance, if an antiretroviral drug X must common to all drugs (Wu et al., 2005; Huang et al., be taken twice daily and is forgotten every morning 2006) or class-specific (Fang and Jadhav, 2012). If in AX (t) = 50%. vitro and in vivo effects are equivalent β = 1.0. The =
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Statistical Model Inter-individual variability. We use a mixed effect model, with the structure proposed by Guedj et al. (2007), for the logtransformed ODE parameters. Normally distributed random effects are put on immune system cells in˜ and death rate of infected cells (˜ put (λ) µT ∗ ), which is consistent with biological mechanisms knowledge (Vieillard et al., 2010; Mogensen et al., 2010). An additional random effect is put on γ˜ to model variability in subjects responses to treatment potentially due to pharmacogenomics (Michaud et al., 2012) or virus mutations (Gardner et al., 2009; Harrigan et al., 2005). This modeling is consistent with in vitro knowledge. Actually, Sampah et al. (2011) showed that for the same concentration of cART, IIPs may change depending on virus mutations; generally wildtype viruses lead to greater IIPs (Rosenbloom et al., 2012). Observational Model. In HIV studies, longitudinal data collected in routine are the total CD4 count in cells/µL and viral load in copies RNA/µL. To model measurement errors, we use transformations to achieve normality and homoscedasticity of noises. We assume that combinations of compartments are observed, to say (Q + T + T ∗ )0.25 and log10 (V ). The standard deviation of measurement errors are respectively denoted by σCD4 and σV L . Statistical methodology.
Briefly, we used the pooled data of these 4 clinical trials with an overall study population of 350 patients. Studies sizes, number of available data, lengths of follow-up, antiretroviral drugs repartition and patients characteristics are available in Table 1. We considered eight main antiretroviral drugs given in cART, other peripheral drugs were omitted (less than 5% of the patients). We studied zidovudine (AZT), stavudine (D4T), didonasine (ddI) and lamivudine (3TC) as Reverse Transcriptase Inhibitors (RTIs), and lopinavir (LPV), amprenavir (APV), darunavir (DRV) as Protease Inhibitors (PIs). Ritonavir (RTV) was given as a pharmacokinetic boost but always in the same cART. Thus, to avoid identifiability problems we did not estimate it individual effect. As a consequence, we consider the estimation of boosted DRV/r, APV/r and SQV/r effects. Daily adherence documented by self-reported questionnaires was available for 42.5% of the population (patients from ALBI Study); basically AX (t) can take every value (even more than 100%). For patients in other studies we had information about treatment doses, regimen switches and interruptions reported during visits to clinician; basically AX (t) = 0% or AX (t) = 100%. Statistical analysis. For the Bayesian analysis in NIMROD, we put informative priors on biological parameters, see Prague et al. (2012) for a description of the choice of these priors. We put weakly informative priors for the conversion factor, to say wide normal laws with mean 1. We took non-informative uniforme priors for variance parameters. Priors and posteriors are presented Table 2. Biological parameters estimates are consistent with the literature. Measurement errors have the expected magnitude, however it remains high for σCD4 . Altogether, data adjustment quality is acceptable, see fits examples (Figure 3, Figure 4 and Figure 5) and visual checks (Figure 6).
A total of 22 parameters have to be estimated on clinical data : biological parameters, in vitro to in vivo conversion factors, standards deviation of random effects and measurement errors. We used the NIMROD program for estimation (Normal approximation Inference in Models with Random effects based on Ordinary Differential equations). This is a bayesian approach based on maximization of a penal- Conversion factors analysis. ized likelihood, see (Prague et al., 2013) for details. A Wald significance test found that, as expected, conversion factors were positive (to say, the treatment decrease the infectivity γ). This is significant except Descriptive analysis of the for 3TC (p-value=40.8%). We have a high variability dataset : Albi, Puzzle, Zephir of β’s values between drugs and drug classes (NRTI and PI). Moreover, the conversion factors β’s are sigand Predizista nificantly different from one. Thus conversion between doses used in vitro and in vivo is not straightforward. For instance, the in vivo effect of ddI is Study Population. significantly higher than it in vitro effect. This may The ALBI, PUZZLE, ZEPHIR and PREDIZISTA suggest that the dose used in vitro were too low comstudies have been described elsewhere (Molina et al., pared to standard in vivo dose. The reverse reasoning 1999; Raguin et al., 2004; Pellegrin et al., 2007, 2008). applies for DRV/r. Page 3 of 10
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Table 1: Studies and patients characteristics. 350 patients from ALBI, PUZZLE, ZEPHIR and PREDIZISTA studies. Studies characteristics Number of patients Average Duration (weeks) Drugs characteristics Use of AZT (%) Use of 3TC (%) Use of d4T (%) Use of ddI (%) Use of LPV (%) Use of APV (%) Use of DRV (%) Patients characteristics Age (year) Median follow-up (days) log10 HIV-RNA baseline (copies/mL) log10 HIV-RNA 4 months (copies/mL) CD4 count baseline (cells/mL) CD4 count 4 months (cells/mL)
ALBI
PUZZLE
ZEPHIR
PREDIZISTA
ALL
148 24
22 26
116 12
64 12
350 19
66 66 68 68 0 0 0
23 86 46 68 86 86 0
22 56 3 11 10 100 100
3 17 3 20 2 0 0
37 55 33 41 9 39 18
36 173 4.5 2.5 406 521
42 200 4.8 2.5 198 341
44 168 4.3 2.9 294 369
45 165 4.6 2.7 183 198
42 171 4.5 2.6 317 459
Table 2: Priors and posteriors for the “target cells model” parameters. Estimation from ALBI, PUZZLE, ZEPHIR and PREDIZISTA datasets Priors Posteriors log-parameters Mean s.d. Mean s.d. Biological Parameters — λ 2.55 1.90 4.15 0.572 µT ∗ -0.05 0.68 0.49 0.124 γ -5.76 4.02 -5.95 0.128 α -4.00 2.00 -3.76 0.170 ρ -4.34 1.38 -6.22 1.188 µT -2.59 0.34 -3.40 0.071 µQ -9.00 1.00 -8.99 0.999 π 4.04 2.66 0.85 0.120 µV 2.90 0.68 -1.27 0.039 Standard deviations of random effects σλ 0.76 0.027 σµT ∗ 1.18 0.053 σγ 1.19 0.05
Priors log-parameters Mean s.d. Conversion factors βAZT 1.00 2.00 β3T C 1.00 2.00 βd4T 1.00 2.00 βddI 1.00 2.00 βLP V 1.00 2.00 βAP V 1.00 2.00 βDRV 1.00 2.00
Posteriors Mean s.d. 2.71 0.01 4.81 0.33 0.23 0.20 0.16
0.114 0.031 0.225 0.166 0.012 0.018 0.014
Standard deviations of measurement error σCV 0.57 0.011 σCD4 0.41 0.001
Figure 3: Biomarkers adjustment according to adherence for patient 39 in ALBI. Observations (triangle), fits (plain), 95% predictivity intervals (dashed)
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Figure 4: Biomarkers adjustment according to adherence for patient 230 in PUZZLE. Observations (triangle), fits (plain), 95% predictivity intervals (dashed)
Figure 5: Biomarkers adjustment according to adherence for patient 348 in PREDIZISTA. Observations (uncensored triangle, censored round), fits (plain), 95% predictivity intervals (dashed)
a
b
4 3 2 1 0 -1
1
2
3
4
5
6
-2
-1
0
1
2
d
-3
c
3
Observed values of Viral load Difference between Predicted and observed values of CD4 count
Predicted values of Viral load
5
6
Figure 6: Visual diagnostic plot for data fits of viral load (a,c) and CD4 count (b,d). In scatter plot (a,b), model trajectories values are plotted against observed values. Lines represent the first bisector and acceptable error regarding estimated measurement error for each biomarker. For viral load, shaded square highlight points that are not misclassified since true viral load is unknown because of detection limit complication in data. In (c,d) Differences between model trajectories and observation are plotted against time together with acceptable deviance threshold of 95% confidence interval regarding measurement error.
0
50
100
150 Time (days)
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200
250
300
Work in progress (not submitted)
In vivo quantification and forecast of treatment effects
of antiretroviral drugs. Finally, if forecasts are possible we could consider the possibility of optimizing the treatment change as suggested by Figure 8. Based on individual observations of the patient, we could forecast the responses Antiretroviral drugs in vivo effect. It is interesting to look at the in vivo effect of to various cARTs and take the best cART according each antiretroviral (βX IIPX ), presented in Figure 7a. to criteria such as to have the highest probability of Mean value and 95% credible intervals can be di- infection control. We currently build collaborations to gather more rectly found by computation from estimates in Tadata with a wider variety of antiretroviral drugs and ble 2. DRV/r is the most effective (both in vitro cARTs. and in vivo). It effect in mono-therapy is comparable with the in vivo effects of some weak cART (let say AZT+3TC Figure 7b), which is in accordance with Figure 7: In vivo estimated treatment effects previous studies (Katlama et al., 2010; Phung and for antiretroviral drugs and selected cART standard daily doses. (a) βXi IIPXi (b) Yeni, 2011). Moreover, we observe that drugs may be for PncART βXi IIPXi . ranked. Whereas the in vitro ranking is DRV > LPV i=1 > APV > 3TC > ddI > AZT > d4T, the in vitro ranking is DRV > AZT > d4T > LPV > APV > ddI > 3TC. RTIs could be missclassified due to phosphorylation that makes dose-response curves difficult to evaluate. The cART in vivo effect. As an illustration, we selected four commonly used cART (AZT+3TC, d4T+ddI, DRV+ddI and AZT+3TC+APV) and computed their effects in vivo PncART ( i=1 βXi IIPXi ) which are displayed on Figure 7b. 95% credible intervals are not significantly different. However, AZT+3TC+APV and DRV+ddI seem to have a greater effect than old fashion NRTI bitherapies. This guess is consistent with knowledge on the benefit to add a PI in cART (Deeks et al., 1997). Moreover, this ranking in these cARTs is consistent with the average observed viral load and CD4 count for patients. Regimens AZT+3TC+APV and DRV+ddI lead to a greater decrease of the viral load (about 2 log10 in 6 months) and increase of CD4 count (about 100 cells in 6 months) compared to AZT+3TC and d4T+ddI. Limitation and perspectives. These preliminary results are obtained from samples with moderate size regarding the quantification of treatment effects: most of caRTs are given in less than 20 patients, some are only given in less than 5 patients. Moreover, results are mainly driven by ALBI sub study (60% of our dataset). Forecasts of various cARTs reThen, concerning the validation of our model, it Figure 8: sponses. Patient 50 from ALBI Study. would be interesting to estimate in vivo effects of drugs on a learning dataset and test it on a validation dataset. Although cross-validation remains a solution to assess the forecast quality, we expect to do the same analysis on a larger dataset. Moreover, we aim at estimating the in vivo effect of a larger number Page 6 of 10
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Figure 9: Probability of virologic (a) and immunologic (b) failure as a function of adherence levels. Observed proportions of failures in the studied population are indicated for comparison purpose.
Dynamical models are valuable to investigate the matter of adherence
detectability (resp. 500 copies) is reached 35% (resp. 60%) of the time. Thus, our model correctly mimics observed data where only 27% (resp. 61%) of the population reached 50 copies/mL (resp. 500 copies/mL) undetectability after 100 days. Figure 9b shows that an adherence greater than 80% should be sufficient Simulation of adherence in dynamical models. to ensure less than 25% of immunologic failure which Adherence is very difficult to assess accurately but is pessimistic compared to observed data where only is a main issue in HIV (Paterson et al., 2002). In our 15% of the patients have less than 250 CD4 cells/mL model we can account for adherence through AX (t). after 100 days. Thus, at each time t, we may simulate a stochastic term AX (t) which equal 100% with the probability p Limitation and perspectives. and 0% with the probability (1 − p). This is the adIn this investigation, we tried to see if we were able herence pattern of a patient who is randomly adherence 100p% of the time. Then, parametric empirical to assess the impact of adherence patterns. This could bayes predictions (Kass and Steffey, 1989) are used to be extended to the analysis of all adherence patterns forecast how patients would have responded to cART including intermittent treatments (x days on, (7 − x) days off). However, before answering precisely quesaccording to their simulated adherence pattern. tions on adherence, estimation must be done on a Random adherence and treatment failures dataset where adherence is correctly reported. ActuWe tried to investigate the effect of 100p% random ally, the major weakness in this analysis is that adheradherence on virologic and immunologic failure after ence information is sparse and the number of 100% 100 days. Actually this length of study should let the adherent patients is probably overestimated. Durtime to the mechanistic model to reach equilibrium ing estimation, we think that this poor measurement state. resulted in an underestimation of drug effectiveness Impact of adherence seems to be more percepti- (Ickovics and Meisler, 1997). This may explain why ble on viral load than on CD4 count (Figure 9a). We we showed that an adherence of more than 80% is found that there is a treshold at 80% for adherence requested which is slightly higher than what is realisbelow which there is a high probability of treatment tic, to say around 60-70% (Arnsten et al., 2001), but failure whatever the real adherence level. Figure 9a not impossible (Paterson et al., 2000). We currently shows that the probability of experiencing virologic build collaborations to gather studies with good refailure is greater than 90% for all adherence level ported adherence data and possibly MEMS (Medicalower than 70%. For a full adherence 50 copies un- tion Events Monitoring System) information. Page 7 of 10
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Additional discussions and concerns The issue of mutations. We made the choice to account for mutations by including random effects, especially on γ˜ , so that drugs can have an effect on some patients but be less effective for others. This is justified because drug resistance mutations reduce the virus fitness relatively to wild-type virus. There is no trend in time on biomarkers predictions (Figure 6), thus we can guess that our model remains valid on our data. Probably, there is no major resistance occurrence. However, genotypic susceptibility scores (GSS) were available for all the patients except those from ALBI. We showed that high GSS was correlated with high values of random effects on γ˜ . Thus, we conclude that the use of resistance testing data (genotypic and phenotypic test) could help in accounting for preexisting mutations and probably significantly improve the results. Methodological concerns to build a likelihood and optimization procedure that account for the fact that information is not the same in all dataset is a work in progress.
now on, two research leads are considered : - to enhance the identifiability by fixing some parameters (for example µQ = µQ∗ ), - to find data where viral DNA is measured to have a proxy to measure the number of latent cells (T ∗ + Q∗ ). Other pharmacological models. The model we used in this preliminary note sheet is highly parameterized, particularly because of the conversion factors. We currently compare it with a model with a power function to assess and validate the contribution of using in vitro data. Moreover, if data are not rich enough and we plan to estimate the in vivo effects of a greater number of antiretroviral drugs, we may surely encounter identifiability problems. Instead of using IIPs, we will propose to compare our approach with the use of inhibitory factors (IF) that may be computed and consist in time-averaged IIPs over all antiretroviral drugs in the cART. In other words, the pharmacodynamical function will not be drug-specific anymore but cARTspecific (Figure 11). Thus, the number of conversion factor to be estimated is reduced.
The issue of latent reservoirs.
Figure 11: Inhibitory Factors may be computed according to cART administration and adherence inFigure 10: Target cells with latent reservoirs formation (source : D. Rosenbloom 8th International model : we extended the “target cells model” by Conference on HIV treatment and prevention adheradding a latent infected cells (Q∗ ) compartment. ence).
One of the actual major concern in HIV treatment and cART dynamics is to assess their impact on latent reservoirs. For this purpose, we begin to use the “Target cells with latent reservoirs model” (Figure 10). First investigations on ALBI data showed that this model improves the log-likelihood (-950 vs -1293) as well as model choice criterion LCVa (6.3 vs 7.9). Thus, it provides a better adjustment to data. Regarding parameters estimates, µQ ' µQ∗ , α α∗ and ρ ρ∗ . This is what was expected and order of magnitudes are similar to those found by Ribeiro et al. (2002). Nevertheless, convergence criteria are difficult to reach because identifiability is weak. From Page 8 of 10
Work in progress (not submitted)
Acknowledgments
Brumme, Z. L., Mo, T., Alexander, C. S., et al. (2005). Predictors of HIV drug-resistance mutations in a large antiretroviral-naive cohort initiating triple antiretroviral therapy. Journal of Infectious Diseases, 191(3):339–347.
The authors would like to thank the investigators of the CO3 ANRS Aquitaine Cohort and ´ emiologie Clinique du Sida en the Groupe d’Epid´ Aquitaine (GECSA), particularly M. Bruyand, G. Chˆene and F. Dabis. We also thank the investigators Huang, Y., Liu, D., and Wu, H. (2006). Hierarchical Bayesian methods for estimation of parameters in of PUZZLE ANRS 104 trial particularly G. Raguin a longitudinal HIV dynamic system. Biometrics, and the investigators of the ALBI ANRS 070 trial par62(2):413–423. ticularly J.M. Molina. Parallel computing was used thanks to the computing facilities MCIA (M´esocentre Ickovics, J. R. and Meisler, A. W. (1997). Adherde Calcul Intensif Aquitain) of the Universit´e de Borence in AIDS clinical trials: a framework for clindeaux and of the Universit´e de Pau et des Pays de ical research and clinical care. Journal of clinical l’Adour. epidemiology, 50(4):385–391.
Jilek, B., Zarr, M., Sampah, M., Rabi, S., Bullen, C., Lai, J., Shen, L., and Siliciano, R. (2012). A quantitative basis for antiretroviral therapy for HIV-1 Arnsten, J. H., Demas, P. A., Farzadegan, H., Grant, infection. Nature Medicine. R. W., Gourevitch, M. N., Chang, C.-J., Buono, D., Eckholdt, H., Howard, A. A., and Schoenbaum, Kass, R. and Steffey, D. (1989). Approximate E. E. (2001). Antiretroviral therapy adherence and bayesian inference in conditionally independent hiviral suppression in HIV-infected drug users: comerarchical models (parametric empirical bayes modparison of self-report and electronic monitoring. els). Journal of the American Statistical AssociaClinical Infectious Diseases, 33(8):1417–1423. tion, 87(407):717–726.
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Conclusion et perspectives g´ en´ erales Nous avons d´evelopp´e dans cette th`ese une m´ethodologie g´en´erale pour la mise au point de strat´egies adaptatives de traitement dans le domaine du VIH. Notre objet d’int´erˆet a ´et´e les mod`eles dynamiques qui sont bas´es sur des NLME-ODE. L’une des r´ealisations de ce travail de th`ese a ´et´e l’´elaboration et le perfectionnement d’un algorithme et d’un programme d’estimation des param`etres dans les mod`eles dynamiques. Les m´ethodes bay´esiennes d´evelopp´ees, bas´ees sur une maximisation de la distribution a posteriori des param`etres, ont ´et´e valid´ees et compar´ees avec des algorithmes d´ej`a propos´es dans la litt´erature : elles poss`edent des propri´et´es comparables en terme de biais et de pr´ecision des estimateurs mais permettent des calculs plus rapides que des m´ethodes bas´ees sur des simulations bay´esiennes. Par ailleurs, nous avons effectu´e des travaux sur la comparaison des mod`eles pour la prise en compte des effets traitement. Nous avons montr´e la sup´eriorit´e des mod`eles dynamiques par rapport aux mod`eles descriptifs de type structuraux marginaux pour d´ecrire la trajectoire de biomarqueurs lorsque la mise sous traitement d´epend de la quantit´e d’int´erˆet. D’un point de vue appliqu´e, nous nous sommes int´eress´es a` l’interaction entre le VIH et le syst`eme immunitaire. Pour cela, nous avons analys´e les trajectoires des biomarqueurs de routine, la charge virale et le nombre de CD4, des patients infect´es par le VIH avant et apr`es mise sous traitement. Nous avons consid´er´e et compar´e, sur des jeux de donn´ees o` u les patients et les traitements avaient des caract´eristiques vari´ees, les qualit´es d’ajustement et de pr´edictions de nombreux mod`eles dynamiques : nous avons s´electionn´e le mod`ele `a cellules cibles qui prend en compte l’activation et l’infection des CD4 par le VIH. Nous avons estim´e les param`etres sur diff´erentes donn´ees d’essais cliniques et de cohortes (ALBI, PREDIZISTA,
Conclusion et perspectives g´en´erales
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PUZZLE, ZEPHIR, donn´ees GECSA . . .). Cela nous a permis de valider les connaissances qualitatives sur le m´ecanisme de l’interaction entre le VIH et le syst`eme immunitaire et d’obtenir une quantification des diff´erents taux de cr´eation, taux de d´ec`es et effets traitement pour les cART. Ainsi, nous avons pu d´evelopper des strat´egies d’optimisation du traitement, par exemple une individualisation de la dose de traitement en fonction des r´eactions pass´ees du patient aux doses pr´ec´edentes. Nous avons montr´e que, sous la condition forte d’accroissement de l’information sur le patient a` chaque visite (malgr´e notamment la censure `a gauche de certains biomarqueurs), il ´etait possible de mettre `a jour de mani`ere it´erative la dose de traitement jusqu’`a donner au patient une dose optimale. Cependant, des ´etudes de sensibilit´e ont montr´e que la mauvaise sp´ecification des mod`eles et leur extrˆeme simplicit´e rendent difficile une mise en pratique. Nous avons donc propos´e une mod´elisation plus complexe faisant intervenir des approches de synth`ese bay´esienne pour combiner diff´erentes donn´ees in vivo avec des donn´ees in vitro. Cela permet d’avoir un mod`ele plus proche de la r´ealit´e m´ecaniste de l’infection tout en assurant l’identifiabilit´e pour envisager la quantification de l’effet de chaque antir´etroviral.
Il existe de nombreuses perspectives de d´eveloppement dans les mod`eles NLME-ODE. D’un point de vu statistique, les m´ethodes pr´esent´ees dans cette th`ese ont ´et´e valid´ees empiriquement mais certains travaux th´eoriques pourraient ˆetre continu´es. Nous avons ´etudi´e en simulation par MCMC le comportement des distributions a posteriori : pour un nombre de patients n interm´ediaire, l’hypoth`ese de normalit´e semble v´erifi´ee. De plus, pour n tr`es grand, nous avons montr´e que les propri´et´es asymptotiques rendent l’estimation valide [van der Vaart, 2000]. Cependant, un approfondissement th´eorique de l’approximation normale de la distribution a posteriori permettrait une validation de la m´ethode. En particulier, il serait int´eressant d’´etudier les vitesses de convergence pour savoir `a partir de quelle taille d’´echantillon (n) et quel nombre de param`etres (np + ne + nq + K 0 ) estim´es simultan´ement ces approximations restent vraisemblables (voir Section 6.3). Par ailleurs, la complexification des mod`eles, qui est n´ecessaire `a la meilleure description m´ecaniste des ph´enom`enes, entraˆıne des probl`emes d’identifiabilit´e. La non-identifiabilit´e pratique
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pourrait ˆetre r´egl´ee par l’utilisation des techniques biologiques plus puissantes (kit de charge virale avec sensibilit´e a` 1 copie en routine, meilleur d´enombrement des cellules immunitaire, r´eduction des erreurs de mesure. . .) et par l’acquisition de donn´ees exhaustives (en terme de nombre de mesures et de nombre de biomarqueurs observ´es). Si la non-identifiabilit´e est th´eorique, le mod`ele devra ˆetre reformul´e ou certains param`etres devront ˆetre fix´es ou avoir une distribution a priori tr`es informative. Dans les deux cas, limiter autant que possible cette probl´ematique passe par l’utilisation de toute l’information et de toutes les donn´ees disponibles dans plusieurs jeux de donn´ees ; or, cela alourdit la mod´elisation et les calculs. Les sources de variabilit´e seront tr`es importantes et il faudra inclure de nombreux effets al´eatoires ce qui pose de nombreux probl`emes num´eriques (voir Section 4.2 et Section 5.2) ; mˆeme s’il faut toutefois souligner que cette difficult´e num´erique est partiellement corrig´ee par la disponibilit´e de surpercalculateurs. Par ailleurs, si les covariables disponibles pour chaque jeu de donn´ees ne sont pas les mˆemes, il faudra approfondir la possibilit´e de faire une synth`ese bay´esienne de l’information pour des mod`eles avec des donn´ees manquantes (voir Section 5.2).
D’un point de vue biologique, il est int´eressant de regarder la coh´erence des estimateurs avec les connaissances immunologiques et virologiques de l’infection. Cette comparaison est a` mettre en perspective avec une analyse de sensibilit´e et de robustesse des estimations (voir en Annexes B). Nous mettons en valeur que plus le mod`ele est complexe, moins la pr´ecision est grande sur la valeur des estimateurs. Bien que ces valeurs seront d’autant plus pr´ecises que le mod`ele repr´esente bien la r´ealit´e, nous pensons que l’apport des mod`eles dynamiques repose en particulier sur leurs capacit´es de pr´ediction (voir Section 6.1.1.3). L’approche par estimateur empirique de Bayes est celle que nous avons favoris´ee, d’autres m´ethodes de pr´ediction bay´esienne, comme celles bas´ees sur l’´echantillonnage de la distribution a posteriori, pourraient ˆetre compar´ees. Par ailleurs, les combinaisons antir´etrovirales sont de plus en plus efficaces, cependant il serait important de dissocier l’effet de chaque mol´ecule antir´etrovirale afin de pouvoir pr´edire l’effet de nouvelles combinaisons (voir Section 7.2.2). Ceci permettrait d’´eviter de monter des essais cliniques comparant
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de nouvelles strat´egies dont nous pouvons pr´edire avec ces mod´elisations qu’elles seront inf´erieures aux strat´egies standards. Par exemple, dans l’essai clinique ALBI, s’il avait exist´e des ´etudes sur les effets traitements des antir´etroviraux AZT, 3TC, d4T et ddI (comme nous l’avons fait Section 6.2.3), il aurait pu ˆetre pr´edit que le bras “switch” ´etait sous-optimal. Par ailleurs, si nos travaux permettent de mettre `a jour une relation claire entre les effets in vitro et in vivo des antir´etroviraux, une ´etude par mod`eles dynamiques pourrait ˆetre envisag´ee pour pr´ealablement confirmer la sup´eriorit´e d’une nouvelle mol´ecule. Certaines question m´edicales actuelles pourraient aussi ˆetre abord´ees. Par exemple, les nouveaux traitements permettraient une cure fonctionnelle o` u seules certaines cellules infect´ees restent cach´ees dans les r´eservoirs latents : mod´eliser ce ph´enom`ene reste une piste `a exploiter (voir Section 7.2.2). Cependant, le mod´elisateur devra garder a` l’esprit que la mod´elisation de nouveaux traitements et de nouveaux m´ecanismes peut amener a` la n´ecessit´e d’un r´evision globale de la mod´elisation, voire sortir du contexte NLMEODE de cette th`ese. Par exemple, cela a ´et´e le cas pour l’h´epatite virale (HCV) o` u la consid´eration de de nouvelles th´erapies (comme le daclatasvir) contraint a` utiliser des mod`eles dynamiques multi-´echelles NLME-PDE. Finalement, un ph´enom`ene peu mod´elis´e dans ces travaux, mais d’int´erˆet majeur dans le domaine du VIH est l’apparition de mutations de r´esistance. Ce dernier aspect est la piste privil´egi´ee a` exploiter pour enrichir les mod`eles m´ecanistes. En conclusion, ces travaux peuvent ˆetre poursuivis dans le but de trouver un mod`ele dynamique o` u tous les param`etres peuvent ˆetre estim´es individuellement afin de guider une strat´egie th´erapeutique globale prenant en compte la diversit´e des patients, des virus, des traitements et de leurs interactions. Par ailleurs, ce travail a aussi ´et´e l’opportunit´e d’am´eliorer la technique d’estimations et de pr´edictions dans les mod`eles NLME-ODE qui pourrait maintenant ˆetre utilis´ee dans un tout autre contexte que l’optimisation des traitements dans l’infection a` VIH.
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Annexes
Annexe A Mat´ eriel suppl´ ementaire Comp. Meth. and Prog. in Biomedecine
Annexe B Mat´ eriel suppl´ ementaire Biometrics
1
2
3
1
*email:
[email protected]
Department of Immunology, University Medical Center Utrecht, The Netherlands
´ emiologie-Biostatistique, F-33000 Bordeaux, France INSERM, ISPED, Centre INSERM U897-Epid´
´ emiologie-Biostatistique, F-33000 Bordeaux, France Univ. Bordeaux, ISPED, Centre INSERM U897-Epid´
M´elanie Prague1,2,∗ , Daniel Commenges1,2 , Julia Drylewicz 3 and Rodolphe Thi´ebaut1,2
based on mechanistic models.
Web-based Supplementary Materials : Treatment monitoring of HIV infected patients
, .
λγπα−ρµV µT ∗ µQ −αµV µT ∗ µT −µV µT ∗ µQ µT γπµT ∗ (α+µQ ) λγπα−ρµV µT ∗ µQ −αµV µT ∗ µT −µV µT ∗ µQ µT γµT ∗ µV (α+µQ )
parameters; here we omit the superscript i. We denote Pξ
Ft k
the posterior law of ξ knowing the
Let ξ be the random vector of biological parameters for a patient and ξ ∗ the true value of these
theorem applies
k toward dcrit when Doob’s consistency 2. Web Appendix B: Proof of the convergence of dtopt
Otherwise the trivial equilibrium (1) is asymptotically reached.
T¯∗ = V¯ =
the system stabilizes to a new non-trivial equilibrium: ¯ = λγπ+ρµT ∗ µv , Q γπ(α+µQ ) T¯ = µT ∗ µV , γπ
1
(1)
If the basic reproductive number R0 is higher than or equal to one, after the introduction of virions,
T0∗− = 0, V = 0. 0−
trivial equilibrium without infection is: (ρ+µT )λ , Q0− = αµT +ρµ Q +µQ µT αλ T 0− = , αµT +ρµQ +µQ µT
Figure 1. Graphical representation of the “activated cells model”.
in the main article is available in Figure 1. We distinguish two different equilibrium points. The
A graphical representation of the “activated cell model” presented as a differential equation system
1. Web Appendix A: “Activated cell model”
Web-based Supplementary Materials : Treatment monitoring and mechanistic models
tk →∞
(2)
R0 {ξ, dcrit (ξ)} = 1.
(3)
Ft
k k |Ftk = P R0 (ξ, dtopt ) < 1 |Ftk = ω, dcrit (ξ) < dtopt
(4)
Viral load and total CD4 count fits for the patient with median viral load at the end of
We selected patients in the switch arm at each quartile values of the distribution of the viral load at the final visit (week 24) in the ALBI trial. The median patient (Q2 ) is presented in the article. The patients for the other quartiles (Q1 and Q3 ) are presented Figures 3 and 4. The organization of the figures is the same as in the article. We present the viral load and the total CD4 count: this is a fit for the first treatment phase (d4t+ddI) and a prediction after the treatment switch to AZT+3TC.
We ran our optimization algorithm from 10 different starting points drawn at random at one
standard deviation of the mean of the prior. The average coefficient of variation of the values
at convergence is 0.3. In particular for π ˜ , µ˜V and µ˜Q we had only the order of magnitude. Thus
the accuracy on the values of the parameters is not high. However we checked that the different
4. Web Appendix D: Predictive ability for treatment change (quartile patients)
line), P1 (red point-dashed line) and P2 (blue dashed line).
study for 3 different stopping points: the one presented in Table 2 in the main article (black plain
Figure 2.
same trajectories for both viral load and CD4 count.
P2 : (4.97, 4.18, −10.72)). It is the case that these three sets of parameter values yield nearly the
3
3. Web Appendix C: Assessment of convergence reproducibility
tk →∞
k dtopt −→ dcrit (ξ ∗ ).
P
dcrit (ξ ∗ ) any quantile of the law tends to dcrit (ξ ∗ ):
k k This entails that dtopt is the ω-quantile of Pdcrit (ξ) . Since the law of dcrit (ξ) tends to a Dirac in
P
{R0 (ξ, d) < 1}. Thus we have:
The properties of R0 and (3) lead to a perfect match between the events {dcrit (ξ) < d} and
tk →∞
to convergence in probability, so we have : dcrit (ξ) −→ dcrit (ξ ∗ ).
P
dcrit (ξ ∗ ) holds by composition. Since dcrit (ξ ∗ ) is a constant, convergence in law is here equivalent
Since the application dcrit (ξ) : Rp 7→ R is continuous, convergence in law of dcrit (ξ) toward
defined as:
the critical dose is unique. Therefore, we can write the critical dose as a function of ξ implicitly
R0 is a one-to-one function in the dose, thus, for a specific value of the biological parameters ξ,
Pξ
L
extreme values for the combination of parameters π ˜ , µ˜V and µ˜Q (P1 : (1.48, 0.65, −9.35) and
the posterior distribution tends to a Dirac in the true value:
−→ δξ∗ .
different stopping points: the one presented in Table 2 in the main article and stopping points with
these assumptions, Doob’s consistency theorem (see Van der (2000) p. 149 for details) ensures that
Ft k
median viral load at the end of the study (labeled 663) and compared the fits (Figure 2) for three
∗
tion increases infinitely with time and the version of the posterior for ξ is essentially unique). With
Web-based Supplementary Materials : Treatment monitoring and mechanistic models
values obtained at convergence yielded essentially the same fit. We selected the patient with
Biometrics 2012
available information Ftk at time tk . We also assume that we have identifiability (i.e., the informa-
2
Biometrics 2012
Prediction of patient 881 (Q1 ) viral load and CD4 count after treatment change. We
0.15 0.32
316 (D7 ) 1.25 664 (Q3 )
the figures is the same as in the article. We present the viral load and the total CD4 count: the fit
load than on CD4 count.
error predictive interval for both viral load and CD4 count. We expect a better prediction on viral
is labeled by a color change). Most of the time, all the observations are in the 95% measurement
as it is shown Table 1.
The patients for the quartiles (Q1 ,Q2 and Q3 ) are presented Figures 5 to 7. The organization of
of the patient with random effects are updated each time we have an additional observation; this
5479
6062
476
2163
Dose model
one with the most reduced MSE compared to the ITT fit (without taking into account adherence)
5354
4982
2364
2389
ITT
for the ITT analysis and the step by step predictions taking into account dose changes (parameters
0.39
0.10
0.17
660 (Q2 )
0.15
0.18
562 (Q1 )
Dose model
ITT
Patient
MSE for Viral Load MSE for CD4 Count
Comparison of mean MSE between ITT model and our model accounting for dose changes.
Table 1
Figure 4. Prediction of patient 748 (Q3 ) viral load and CD4 count after treatment change.
Web-based Supplementary Materials : Treatment monitoring and mechanistic models
main article as an illustration because of a large range of behaviors. Nevertheless, this is not the
at the final visit (week 24) in the ALBI trial. Patient 316 (at the 6th deciles) is presented in the
We selected patients with dose changes at each quartile values of the distribution of the viral load
5. Web Appendix E: Predictive ability for dose change (quartile patients)
a reporting artefact.
ignored a dose change from day 97 to day 99 because it only lasted 2 days and was considered as
Figure 3.
95% measurement error predictive interval for both viral load and CD4 count.
viral loads and CD4 counts after treatment switch. Most of the time, all the observations are in the
We only used the first 12 weeks observations to update the parameters of the patients and predicted
4
5
Biometrics 2012
changes.
Figure 6. Prediction of patient 660 (Q2 median) viral load and CD4 count accounting for dose
Figure 5. Prediction of patient 562 (Q1 ) viral load and CD4 count accounting for dose changes.
6
Prediction of patient 664 (Q3 ) viral load and CD4 count accounting for dose changes.
7
Accepted January 2012.
Received September 2011. Revised December 2011.
Mathematics, Cambridge University Press, UK.
Van der, A. (2000). Asymptotic statistics. Cambridge Series in Statistical and Probabilistic
References
Figure 7.
Web-based Supplementary Materials : Treatment monitoring and mechanistic models
[c]
´ Individualisation de la dose chez les patients infectes par le VIH : un algorihme de ´ Robbins-Monro-Metropolis-Hastings ´ Melanie Prague 1,2, Daniel Commenges1,2
1
´ ´ INSERM, U897 Centre de Recherche en Epid emiologie et Biostatistiques, Bordeaux, France 2 Universite´ de Bordeaux 2, Bordeaux, France
Contexte
´ ` predictif ´ Resultats : Capacite´ du modele
´ par le HIV, la charge virale peut etre ˆ ´ Pour la plupart des patients infectes rendue indetectable par des ´ ´ traitements antiretroviraux hautement actifs. L’enjeux principal est alors la qualite´ de vie : il faut reduire les effets secondaires des traitements. ´ des modeles ` ´ ´ ˆ optimal afin de Dans la lignee mechanistiques, nous appliquons la theorie de controle proposer au patient la dose efficace de traitement la plus faible possible. Objectifs I
I
´ e´ realis ´ ´ sur un jeux de donnees ´ issu d’un essai clinique ALBI ANRS 070 : Les estimations ont et ees compose´ de 3 bras ; AZT+3TC sur 24 semaines, d4T+ddI sur 24 semaines, AZT+3TC sur 12 semaines suivi de d4T+ddI sur 12 semaines (Molina et al., J. of Inf. Diseases, 1999) ´ ´ ´ Predictions aux changements de molecules de traitement : Nous sommes capable de predire le ` le changement de traitement duˆ au fait qu’AZT+3TC est inferieur ´ rebond de la charge virale apres a` d4T+ddI.
´ ´ predictives ´ ´ Evaluer les capacites de notre modelisation pour les changements de traitements et les changements de doses. ´ Proposer une strategie adaptative de traitement permettant l’individualisation de la dose de traitement, diminuer la dose de traitement en controlant le risque de donner une dose de traitement trop faible au patient. ´ ´ ` Modelisation de l’interaction VIH - systeme immunitaire
´ ´ ´ ees ´ : les CD4 quiesModelisation mechanistique a` 4 compartiments. 4 types de cellules sont modelis ´ (T ), les CD4 actives ´ infectes ´ (T ∗) et les virus (V ). cents (Q), les CD4 actives
` Parametre λ µQ α ρ µT γ µT ∗ π µV
dQ dt dT dt dT ∗ dt dV dt
= λ − µQ Q − αQ + ρT
= αQ − ρT − µT T − γVT
= γVT − µT ∗ T ∗
´ ´ Predictions aux changements de dose de traitement : Les prediction sont plus proches des observations (triangles) que les fits ITT (gris).
= πT ∗ − µV V
Signification ´ Taux de creation de cellules Q ´ es ` des cellules Q Taux de dec Taux d’activation des cellules Q ´ Taux de desactivation des cellules T ´ es ` des cellules T Taux de dec Infectivite´ : Taux d’infection des cellules T par les virus ´ es ` des cellules T ∗ Taux de dec ´ Taux de creation des virus ´ es ` des virus Taux de dec
´ Modelisation pharmaco-dynamique : prise en compte du traitement et de sa dose. Le traitement agit ´ L’absence de donnees ´ en diminuant l’infectivite. pour des doses ≤ 0.6 nous a contraint a` faire une ´ approximation lineaire pour les doses faibles. γ˜0 + β(ψ(d(t)))κd si d(t) ≥ 0.6 γ˜ = log(γ) = γ˜0 + β(0.6)κd −1d(t) sinon
´ Resultats : Individualisation de la dose de traitement ´ ´ de 15 jours. (haut) Charge virale et Illustration : Simulation de 11 reajustements de doses espaces tk ´ (bas) dopt CD4 observes devient de plus en plus proche de dcrit .
´ ` mixte. Posons Modelisation statistique : prise en compte de la variabilite´ inter-individu avec un modele ` ´ ´ sur les parametres ` ξ˜i le vecteur des parametres a` estimer. Les effets aleatoires sont places α, λ et µT ∗ i i ξ˜li = φ zli (t)β}l + ω l = 1 ... 9 | l + {z | l (t)u {z } Effets fixes
` Effets aleatoires
` d’observation : prise en compte d’erreurs de mesures et transformation pour assurer la norModele ´ ´ malite´ et l’homoscedasticit e. Charge virale : Yij1 = log10(V (tij , ξ˜i )) + ij1 Nombre total de CD4 : Yij2 = (Q(tij , ξ˜i ) + T (tij , ξ˜i ) + T ∗(tij , ξ˜i ))0.25 + ij2 ` ´ Estimation MAP : les parametres biologiques (9), les effets aleatoires (3), les effets traitements et ´ par un algorithme Newton-like par maximisation a Posteriori les erreurs de mesures (2) sont estimes (Guedj et al., Biometrics, 2007). ´ ` ˆ ´ Prediction : Calcul des parametres individuels grace aux estimateurs parametriques empiriques de Bayes (Kass et Steffey, JASA, 1989). ´ Caracterisation de la dose critique de traitement
Trouver la dose critique qui minimiser une fonction de cout ˆ raisonnable pour le patient : Trouver la dose ´ de traitement minimale (dcrit ) pour laquelle le virus est indetectable
tk Simulation pour 100 patients : MSE (distance entre dopt et dcrit ) et pourcentage de d’administration de dose trop faibles (toujours ≤ ω = 90%)
Temps t0 = 0 t1 = 29 t2 = 44 t3 = 59 t4 = 74
´ ˆ (R0) : R0 est le nombre de cellules moyen infectees ´ par une cellule infectee. ´ Caracterisation du controle ´ ´ Ainsi, si R0 ≤ 1 alors V est indetectable. R0 est strictement decroissant de la dose. παλγ R0 = µV µT ∗ (αµT + ρµQ + µT µQ ) ˆ ´ directement si les parametres ` La dose critique est telle que R0(ξ, dcrit ) = 1. Elle peut etre calculee ξi du patient sont connus.
I
´ Algorithme Robbins-Monro-Metropolis-Hastings (RMMH) : (Robbins et Monro, Ann. Stat. Math., 1951 - Hastings, Biometrika, 1970) ´ ´ 1 Tirer avec un algorithme de Metropolis-Hastings (MH) une realisation des ξ i sachant Ftik ´ ´ ´ 2 En deduire une realisation a posteriori de R0(ξ i , dntk ) sachant Ftik tk 1 3 Injecter dans l’algorithme de Robbins-Monro (RM) : dn+1 = dntk + n0.65 (ω − IR (ξ i ,d tk )<1) 0 n 4 Tant que non convergence retour en 1
tk % dopt < dcrit 0% 0% 1% 3% 2%
Conclusion I
´ Strategie adaptative de recherche de dose optimale pour le patient ˆ la probabilite´ de donner une dose de traitement trop faible : Les parametres ` Controler ξ i sont inconnus. Nous disposons seulement d’observations de biomarqueurs au temps tk (Ftik ). Par exemple, ω = 90%. tk P R0(ξ i , dopt ) < 1 Ftik = ω
MSE (ecart-type) 0.631 (0.113) 0.050 (0.095) 0.027 (0.017) 0.011 (0.010) 0.006 (0.007)
I
I
I I
´ ´ ´ ` Notre modelisation est acceptable d’un point de vue predictif. Introduire la dose ameliore le modele. ´ La strategie adaptative de traitement RMMH permet de diminuer la dose de traitement sans impact significatif sur la charge virale et le nombre de CD4. tk ´ dopt est proche de dcrit en un petit nombre d’iterations. Le risque d’administrer une dose trop faible ˆ e. ´ est control
´ Les limitations evidentes sont le manque d’information sur le patient (censure a` gauche de la charge ´ ` ´ ` et un mauvais choix de la virale, derive des parametres . . . ), la mauvaise specification du modele fonction pharmaco-dynamique. ´ reelles. ´ La preuve de concept est faite : Extension possible a` des donnees ´ ´ ´ Une perspective interessante serait de tester la procedure RMMH sur des donnees d’essai avec ´ ´ administration de dose faibles et une adhe´ erence rigoureusement renseignee.
References tk ´ es ´ asymptotiques : Nous prouvons mathematiquement ´ Propriet que dopt est unique pour tout temps tk et converge vers dcrit lorsque l’information sur le patient devient infinie. P
tk dntk −→ dopt n→∞ P
tk dopt −→ dcrit (ξ i ) tk →∞
´ ISPED equipe Biostatistique, Bordeaux-France
´ [1] Guedj, J., Thiebaut,, R., and Commenges, D. (2007) Biometrics 63(4), 1198–1206. [2] Kass, R.E. and Steffey, D. (1989) Journal of American Statistical Association 717–726. [3] Hastings, W.K. (1970) Biometrika 57(1), 97–109. [4] Robbins, H. and Monro, S. (1951) The annals of Mathematical Statistics 400–407. ˆ [5] Molina, J.M., Chene, G., Ferchal, F., Journot, V., et al. (1999) J. of Infectious Diseases 180(2), 351.
http://www.isped.u-bordeaux2.fr/
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R´ esum´ e Utilisation de mod` eles dynamiques pour l’optimisation des traitements des patients infect´ es par le VIH De nos jours, la plupart des patients infect´es par le VIH ont une charge virale qui peut ˆetre rendue ind´etectable par des combinaisons antir´etrovirales hautement actives (cART) ; cependant, il existe des effets secondaires de ces traitements qui doivent ˆetre pris `a vie. L’utilisation des mod`eles m´ecanistes dynamiques bas´es sur des ´equations diff´erentielles ordinaires (ODE) a consid´erablement am´elior´e les connaissances de la dynamique HIV-syst`eme immunitaire et permet d´esormais d’envisager une personnalisation du traitement. L’objectif de ces travaux de th`ese est d’am´eliorer les techniques statistiques d’estimation de param`etres dans les mod`eles m´ecanistes dynamiques afin de proposer des strat´egies de surveillance et d’optimisation des traitements chez les patients infect´es par le VIH. Dans une premi`ere partie, nous pr´esentons les probl´ematiques d’inf´erence dans les mod`eles ODE avec effets mixtes sur les param`etres. Nous introduisons un algorithme d’estimation bay´esienne bas´e sur une maximisation de la vraisemblance p´enalis´ee puis un programme associ´e NIMROD. Nous montrons la puissance des approches m´ecanistes dynamiques concernant l’´evaluation des effets traitements par rapport aux m´ethodes descriptives d’analyse des trajectoires des biomarqueurs. Dans une deuxi`eme partie, nous d´efinissons le mod`ele `a cellules cibles : un syst`eme ODE `a 4 compartiments d´ecrivant la dynamique du VIH, des CD4 quiescents, activ´es et activ´es-infect´es. Nous l’ajustons sur des donn´ees cliniques et nous montrons qu’il poss`ede de bonnes capacit´es pr´edictives. Nous proposons une preuve de concept de la possibilit´e de contrˆoler individuellement la dose de traitement. Cette strat´egie adaptative tire parti des crit`eres de stabilit´e des mod`eles ODE et r´eajuste la dose du patient en fonction de sa r´eaction ` a la dose pr´ec´edente par une proc´edure bay´esienne. Pour finir, nous introduisons les diff´erents facteurs, en particulier g´en´etiques et pharmacologiques, `a prendre en compte pour envisager l’individualisation des changements de cART. Ce travail passe par la quantification in vivo d’effets de cART en utilisant des indicateurs d’activit´e antivirale ´etablis in vitro. Nous discutons la validit´e des r´esultats et les ´etapes m´ethodologiques encore n´ecessaires pour l’int´egration de ces m´ethodes dans les pratiques cliniques. ´ : antir´etroviraux ; approximation normale de l’a posteriori ; approche bay´esienne ; choix de MOTS-CLES mod`ele ; ´equation diff´erentielles ordinaire (ODE) ; in vitro ; in vivo ; m´edecine personnalis´ee ; mutations ; mod`eles causaux ; mod`eles dynamiques ; mod`eles `a effets mixtes ; optimisation num´erique ; pharmacologie ; pr´edictions ; SIDA ; VIH.
Abstract Use of dynamical models for treatment optimization in HIV infected patients Today, most HIV-infected patients viral loads can be made undetectable by highly active combination of antiretroviral therapy (cART), but there are side effects of these treatments taken for life. The use of dynamic mechanistic models based on ordinary differential equations (ODE) has greatly improved the knowledge of the dynamics of HIV and of the immune system and can now be considered for personalization of treatment. The aim of these PhD works is to improve the statistical techniques for estimating parameters in dynamic mechanistic models so as to elaborate strategies for monitoring and optimizing treatments in HIV-infected patients. In the first part, we present the problem of inference in ODE models with mixed effects on parameters. We introduce a Bayesian estimation procedure based on the maximization of the penalized likelihood. We present the related program called NIMROD. Then, we show the power of dynamic mechanistic approaches for the evaluation of treatment effects compared to methods based on the descriptive analysis of the biomarkers trajectories. In a second part, we build the « target cells model » which is a four compartments ODE system with HIV, quiescent, activated and activated-infected CD4. We adjust it to clinical data and we demonstrate it has good predictive capabilities. We build a proof of concept for drug dose individualization. This is an adaptive strategy based on the equilibrium criteria of ODE models. It consists in tuning the dose of the patient based on his reaction to the previous doses using a Bayesian update procedure. Finally, we introduce the various factors, including genetic and pharmacological considerations, that are important to design an individualized change of cART. This work involves the quantification of in vivo effects of cART using antiviral activity indicators established in vitro. We discuss the validity of the results and the further steps needed for the integration of these methods in clinical practice. KEY WORDS : AIDS ; antiretroviral drugs ; Bayesian approach ; causal models ; dynamical models ; HIV ; in vitro ; in vivo ; mixed effects models ; model choice ; normal approximation of the a posteriori ; mutations ; numerical optimization ; ordinary differential equation (ODE) ; personalized medicine ; pharmacology ; predictions.
DISCIPLINE : Sant´e publique option Biostatistiques LABORATOIRE : Unit´e INSERM U897 - Universit´e Victor S´egalen Bordeaux 2 - ISPED 146, rue L´eo Saignat 33076 Bordeaux, FRANCE
