# Mathematical-Statistical Modeling to Inform the Design of HIV Treatment Strategies and Clinical Trials

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1 Mathematical-Statistical Modeling to Inform the Design of HIV Treatment Strategies and Clinical Trials Eric S. Rosenberg Massachusetts General Hospital and Harvard Medical School Marie Davidian North Carolina State University Joint work with H. Thomas Banks, North Carolina State University, and our colleagues and postdocs Modeling for Design of HIV Treatment and Trials

2 Overview Multidisciplinary collaboration supported by a grant from NIAID Main players: Imunologist/infectious disease clinician (Eric, MGH), statistician (Marie, NCSU), applied mathematician/control theorist (H.T. Banks,NCSU) Big picture: Use mathematical-statistical modeling of disease progression and simulation to design HIV treatment strategies and clinical trials to study them Develop better strategies for use of existing antiretroviral therapies (ART) to manage HIV infection over time Design and carry out a clinical trial in subjects with acute HIV infection assisted by modeling and simulation Collect extensive data to inform refined modeling = sophisticated strategies and trials more Modeling for Design of HIV Treatment and Trials

3 Overview 1. Eric will give background on HIV, acute HIV infection, HIV therapy, and structured treatment interruption (STI) 2. Marie will follow with descriptions of our mathematical-statistical modeling and simulation framework, our approach to design of treatment strategies and clinical trials to study them, and the clinical trial we will carry out Modeling for Design of HIV Treatment and Trials

4 Mathematical-Statistical Modeling Marie will talk about: Recap: HIV therapy and STI HIV dynamic models and control Mathematical-statistical framework Design of a clinical trial in acute infection Next steps - design of STI strategies Closing remarks Modeling for Design of HIV Treatment and Trials 1

5 Recap: HIV therapy and STI Main points: Continuous ART is impossible for most patients This has inspired interest in the use of structured treatment interruption (STI) Non-adaptive (non-dynamic) strategies planned in advance, e.g., cycles of 8-weeks-on/8-weeks-off Adaptive (dynamic) strategies decisions to interrupt and re-initiate based on rules taking patient information as input, e.g., stop or start based on CD4+ T cell count or viral load Modeling for Design of HIV Treatment and Trials 2

6 Recap: HIV therapy and STI STI studies so far: Mixed results CPCRA Strategies for Management of Antiretroviral Therapy (SMART) trial (El-Sadr, Neaton, et al., 2006) in chronically- infected subjects Compared continuous ART to an adaptive STI strategy ( drug conservation ) on-off ART dictated by CD4+ T cell count Stopped early ( 5500 pts), drug conservation = 2x risk of primary endpoint (AIDS or death) Modeling for Design of HIV Treatment and Trials 3

7 Recap: HIV therapy and STI Drug conservation strategy in the SMART study: CD4 >250 OFF OFF CD4 <=250 ON CD4 > 350 OFF CD4 <=350 ON Modeling for Design of HIV Treatment and Trials 4

8 Recap: HIV therapy and STI Our premise: Strategies so far may have been unfortunately chosen Based on educated guesses expert opinion, pieced-together clinical evidence E.g., CD4+ thresholds in SMART chosen after much debate among experts......and decision rules did not include viral load (or other info) = it is premature to dismiss STI and adaptive treatment strategies for managing HIV infection A formal, evidence-based approach combining biological knowledge, data in a principled way is needed to design and evaluate strategies Modeling for Design of HIV Treatment and Trials 5

9 HIV dynamic models and control HIV dynamic models: Represent mathematically known and hypothesized mechanisms involved in the virus-immune system interaction taking place within a single subject Series of compartments characterizing different populations of virus and constituents of the immune system Interactions among compartments described by a system of (deterministic) nonlinear ordinary differential equations The solution to the system of ODEs yields a mechanistic model characterizing the joint behavior of the compartments over time (the dynamics ) = Viral load, CD4+ T cell count, etc, at any time Modeling for Design of HIV Treatment and Trials 6

10 HIV dynamic models and control Possible model for within-subject dynamics: Modeling for Design of HIV Treatment and Trials 7

11 HIV dynamic models and control Model for within-subject dynamics: s = 7 states T 1 = λ 1 d 1 T 1 {1 ǫ 1 u(t)}k 1 V I T 1 T 2 = λ 2 d 2 T 2 {1 fǫ 1 u(t)}k 2 V I T 2 T 1 = {1 ǫ 1 u(t)}k 1 V I T 1 δt1 m 2 ET1 T 2 = {1 fǫ 1 u(t)}k 2 V I T 2 δt2 m 2 ET2 V I = {1 ǫ 2 u(t)}10 3 N T δ(t1 + T2 ) cv I {1 ǫ 1 u(t)}ρ k 1 T 1 V I {1 fǫ 1 u(t)}ρ k 2 T 2 V I V NI = ǫ 2 u(t)10 3 N T δ(t1 + T2 ) cv NI Ė = λ E + b E(T1 + T2 ) (T1 + T 2 ) + K E d E(T1 + T2 ) b (T1 + T 2 ) + K E δ E E d θ = (λ 1, d 1, ǫ 1, k 1,...) T plus initial conditions Observable: CD4 count = T 1 + T 1, viral load = V I + V NI u(t) = ART input at t (0 u(t) 1, 0 = off, 1 = on) Modeling for Design of HIV Treatment and Trials 8

12 HIV dynamic models and control In general: HIV dynamic model with s states ẋ(t, θ) = g{t, x(t, θ), θ}, solution x(t, θ) (s 1) Embodies hypothesized mechanisms through model parameters θ θ includes cell and virus production, death, clearance rates; treatment efficacy parameters; etc θ dictates pattern of progression over time (deterministic) under any treatment pattern u(t) Control theory: Mathematical theory and techniques for modifying (controlling) the behavior of such systems Goal Optimize some objective function, e.g., drive viral load set point below a threshold while keeping cost of therapy low I.e., determine u(t) to achieve this objective Modeling for Design of HIV Treatment and Trials 9

13 HIV dynamic models and control Our ultimate goal: Use HIV dynamic models and control along with simulation to design treatment strategies u(t) for acute HIV infection and to design clinical trials to study them Find strategies that do well for individuals and for the population Need evidence supporting HIV dynamic model = data (e.g., measured CD4, VL, other stuff over time on lots of subjects) Intra-subject variation due to assay error, realization error; left-censoring of viral loads due to assay lower limits of quantification Substantial inter-subject variation heterogeneity in mechanisms θ across the subject population For both fitting to data and simulation: Must embed the (deterministic) mathematical model in a statistical framework that characterizes faithfully inter- and intra-subject variation Modeling for Design of HIV Treatment and Trials 10

14 HIV dynamic models and control Data: Eric has been collecting intensive longitudinal viral loads, CD4 counts on a cohort of 150 acutely-infected subjects for > 7 years CD4 + T cells / ul Patient #14 data fit w/half fit w/all virus copies/ml time (days) Modeling for Design of HIV Treatment and Trials 11

15 Mathematical-statistical framework Mathematical model: ẋ(t, θ) = g{t, x(t, θ), θ}, solution x(t, θ) (s 1) Observations not available on all s states x = Ox for observation operator O E.g., CD4 and VL only Statistical framework: Embed x in a hierarchical statistical model For each subject i in the population, conceive of a bivariate (CD4, VL) subject-specific stochastic process under u(t) Y i {t, u(t)} = [Y CD4 i {t, u(t)}, Yi V L {t, u(t)} ] T Depends on treatment strategy u(t) up through time t Modeling for Design of HIV Treatment and Trials 12

16 Mathematical-statistical framework Intra-subject model: Decompose into Y i {t, u(t)} = x{t, u(t), θ i } + e i {t, u(t)} e i {t, u(t)} is the deviation process realizations, assay errors e i {t, u(t)} average out to zero over all possible realizations, assay errors (conditional on θ i and the strategy imposed) Interpret x{t, u(t), θ i } as average trajectories over all possible realizations we could see on subject i under strategy u(t) Also, assumptions on (conditional) correlation (across t and among elements of e i {t, u(t)}), variances, probability distribution Inter-subject model: θ i is an inherent characteristic of subject i Probability distribution p(θ i ; θ, D), e.g., θ i N(θ, D) Could also be conditional on subject characteristics Modeling for Design of HIV Treatment and Trials 13

17 Mathematical-statistical framework Result: Full description of the hypothesized data-generation process in continuous time For individual subjects (randomly-chosen from the population) And thus for samples of such subjects drawn at random from the population For large enough sample = effectively, knowledge of the entire population Basis for simulation of virtual subjects Needed: Full characterization based on data Modeling for Design of HIV Treatment and Trials 14

18 Mathematical-statistical framework Data: For subject i, i = 1,...,N, observed at n i times t i1,...,t ini U i (t) = actual ART pattern over entire observation period (known) Y ij = (Yij CD4, Yij V L ) T at time t ij = Y i = (Y i1,...,y ini ) T Conceive Y ij = Y i {t ij, U i (t ij )} (similarly for e i ) Eric s data N 150, n i A i = possible subject characteristics Nonlinear mixed effects model (bivariate response): Fit to data Y ij = x{t ij, U i (t ij ), θ i } + e ij, j = 1,...,n i θ i p(θ i ; θ, D), i = 1,...,N Modeling for Design of HIV Treatment and Trials 15

19 Mathematical-statistical framework Challenges: Left-censoring of VL by lower assay limit dim(θ) > 25 and not all identifiable from CD4, VL only Components of x only calculable numerically by forward solution of ODEs Two-stage approach: For each i, estimate θ i via EM algorithm to handle censoring incorporating regularization to address identifiability and dimensionality Use resulting θ i as data to obtain θ, D using moment methods Modeling for Design of HIV Treatment and Trials 16

20 Mathematical-statistical framework Predictive capability: CD4 + T cells / ul Patient #14 data fit w/half fit w/all virus copies/ml time (days) Modeling for Design of HIV Treatment and Trials 17

21 Mathematical-statistical framework Simulation: Generate N sim virtual subjects by generating θ i, i = 1,...,N sim, from p(θ i ; θ, D) Generate inherent trajectories x{t, u(t), θi (continuous time) } under a given u(t) Add within-subject deviations according to intra-subject model to obtain virtual data Suitable p(θ i ; θ, D) determined by comparing virtual profile distributions (VL, CD4) to those from Multicenter AIDS Cohort Study (MACS, u(t) 0) and Eric s data (various u(t)) Mixture of normals Modeling for Design of HIV Treatment and Trials 18

22 Design of a clinical trial Armed with this framework: Use to design treatment strategies and clinical trials Our first step: Proof of principle can we use this capability to assist in addressing a question involving non-adaptive treatment strategies? Unresolved Whether or not individuals acutely-infected with HIV should be treated with ART More precisely Is it better to give ART for some period following acute infection ( train the immune system, self-vaccinate ) or is it better to give no treatment at all until later (delay drug resistance, etc) Primary endpoint VL set point at 12 months Modeling for Design of HIV Treatment and Trials 19

23 Design of a clinical trial Which strategies to study? u(t) 0 vs. strategies of the form u(t) = 1, 0 t τ = 0, t > τ for termination times τ = 3, 4,..., 12 months Approach: Evaluate effects of candidate strategies on the (virtual) population by simulation Insight into which strategies to study based on their anticipated effects on the entire population Modeling for Design of HIV Treatment and Trials 20

24 Design of a clinical trial Strategy u(t) with τ = 6: 100 virtual inherent viral load trajectories with ART terminated at 6 months, i.e., u(t) = 1, 0 t 6, u(t) = 0, t > virus copies/ml time (months) Modeling for Design of HIV Treatment and Trials 21

25 Design of a clinical trial Different termination times τ: Means of 15,000 virtual CD4 and viral load data profiles with u(t) = 1, 0 t τ, u(t) = 0, t > τ, τ =0,3,4,...,12 months Median CD Mean log10 VL months months Modeling for Design of HIV Treatment and Trials 22

26 Design of a clinical trial Summary: Based on this (simple) HIV dynamic model, no differences expected Simple model does not represent adequately the immune response Simulations with a refined model showed larger subpopulations with lowered VL set point for larger τ......but are less reliable (very little data on immune response) Result: Study ART under more than one termination time τ = 3 ( short-term ) and τ = 8 months ( long-term ) Modeling for Design of HIV Treatment and Trials 23

27 Design of a clinical trial Trial schema: 1/2 pts randomized to ART, 1/2 pts to no ART ACUTE HIV 1 INFECTION ARV THERAPY SHORT TERM TREATMENT THEN INTERRUPTION LONG TERM TREATMENT THEN INTERRUPTION OUTCOME NO THERAPY Modeling for Design of HIV Treatment and Trials 24

28 Design of a clinical trial Plan: 3 year accrual period (36 patients), 1 year follow-up Standard design considerations for primary VL comparison at 12 months Intensive visit schedule collect CD4, VL, CTLs, viral fitness, etc Data collection more frequent when dynamics are anticipated to be changing (e.g., in the weeks after ART termination) Modeling for Design of HIV Treatment and Trials 25

29 Design of STI strategies Next step: Armed with more informative data (e.g., measurements reflecting aspects of immune response) Develop and validate more realistic HIV dynamic models......and refine the entire mathematical-statistical framework...and use to develop and evaluate ( virtually ) potential adaptive treatment strategies Receding horizon control methods And design the next trial to study the most promising strategies... Modeling for Design of HIV Treatment and Trials 26

30 Design of STI strategies Clinical Data Input data: Clinical, immunological and virological data Individual Model Clinical Trial Design Clinical Practice Population Model Model Simulation & Robust Control Design Modeling for Design of HIV Treatment and Trials 27

31 Closing remarks Modeling and simulation have a significant role to play in design of HIV treatment strategies In principle could link dynamic models with models for PK, etc We envision cycles of smaller learning trials that provide richer information needed to develop more refined adaptive strategies that will then be evaluated in confirmatory trials We ll see how this turns out! Slides at: davidian Modeling for Design of HIV Treatment and Trials 28

32 References Adams, B.M., Banks, H.T., Davidian, M., and Rosenberg, E.S. (2007) Model fitting and prediction with HIV treatment interruption data. Bulletin of Mathematical Biology 69, Rosenberg, E.S., Davidian, M., and Banks, H.T. (2007) Using mathematical modeling and control to develop structured treatment interruption strategies for HIV infection. Drug and Alcohol Dependence special supplement issue on Customizing Treatment to the Patient: Adaptive Treatment Strategies 88S, S41-S51. Modeling for Design of HIV Treatment and Trials 29

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