WHAT S IN BETWEEN DOSE AND RESPONSE?

WHAT S IN BETWEEN DOSE AND RESPONSE? Pharmacokinetics, Pharmacodynamics, and Statistics Marie Davidian Department of Statistics North Carolina State University Outline 1 Introduction 2 What is pharmacokinetics? 3 What is pharmacodynamics? 4 Population PK/PD and statistics the history 5 An example 6 PK/PD today 7 Concluding remarks Warning: There are very few equations in this talk! http://wwwstatncsuedu/ davidian Greenberg Lecture I: PK, PD, and Statistics 1 Greenberg Lecture I: PK, PD, and Statistics 2 1 Introduction 1 Introduction What do we want in a drug? Safety Efficacy Can people take it, and does it work? The usual paradigm: Look at what goes in and what comes out, often by asking If we were to administer this drug at some dose to a population of interest, what would the mean response be? And how does it compare to that for other drugs or other doses of this drug? Key message, part I: Understanding what goes on between dose (administration) and response can yield insight on How best to choose doses at which to evaluate a drug How best to use a drug in a population How best to use a drug to treat individual patients or subpopulations of patients And a lot more Key concepts: Pharmacokinetics (PK) what the body does to the drug Pharmacodynamics (PD) what the drug does to the body Greenberg Lecture I: PK, PD, and Statistics 3 Greenberg Lecture I: PK, PD, and Statistics 4 1 Introduction 1 Introduction dose PK concentration PD response Key message, part II: Understanding what goes on between dose (administration) and response for both individuals and the population Relies critically on combining physiological (mathematical) modeling with STATISTICAL MODELING Population PK/PD Statistical modeling is a integral part of the science Key message, part III: Combining mathematical and statistical modeling is becoming more generally recognized as a critical tool in the study of treatment of disease Greenberg Lecture I: PK, PD, and Statistics 5 Greenberg Lecture I: PK, PD, and Statistics 6

2 What is Pharmacokinetics? 2 What is Pharmacokinetics? What the body does to the drug Goals of drug therapy: From a pharmacologist s point of view, for an individual patient or type of patient Achieve therapeutic objective (cure disease, mitigate symptoms,etc) Minimize toxicity Minimize difficulty of administration Optimize dose regimen to address these issues Implementation of drug therapy: To achieve this, must determine How much? How often? To whom? Different for different patients? ages? genders? Under what conditions (or not)? Information on this: Pharmacokinetics Study of how the drug moves through the body and the processes that govern this movement Greenberg Lecture I: PK, PD, and Statistics 7 Greenberg Lecture I: PK, PD, and Statistics 8 2 What is Pharmacokinetics? 2 What is Pharmacokinetics? What goes on inside: ADME Routes of drug administration: Intravenously, Intramuscularly, Subcutaneously, Orally, Basic assumptions and principles: There is a site of action where drug will have its effect Magnitudes of response, toxicity are functions of drug concentration at the site of action Drug cannot be placed directly at site of action, must move there Concentrations at site of action are determined by how drug is absorbed, distributed to tissues/organs, metabolized, excreted (eliminated) (how it moves over time) Concentrations must be kept high enough to produce response, low enough to avoid toxicity = Therapeutic window Cannot measure concentration at site of action directly, but can measure in blood/plasma/serum; reflect those at site Greenberg Lecture I: PK, PD, and Statistics 9 Greenberg Lecture I: PK, PD, and Statistics 10 2 What is Pharmacokinetics? 2 What is Pharmacokinetics? Result: ADME dictates concentration at site of action, but can not be observed directly Plasma concentrations have information about ADME = monitor concentration over time Understanding ADME allows manipulation of concentrations Data for 4 subjects given same oral dose of anti-asthmatic theophylline: Theophylline conc (mg/l) 0 2 4 6 8 10 12 Subject 1 0 2 4 6 8 10 12 Subject 6 0 5 10 15 20 25 0 5 10 15 20 25 Subject 10 Subject 12 Theophylline conc (mg/l) 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 5 10 15 20 25 Time (hr) 0 5 10 15 20 25 Time (hr) Greenberg Lecture I: PK, PD, and Statistics 11 Greenberg Lecture I: PK, PD, and Statistics 12

2 What is Pharmacokinetics? 2 What is Pharmacokinetics? Absorption Elimination Absorption Elimination Concentration (mg/l) Concentration (mg/l) Duration of Effect Therapeutic Window Time (hr) Time (hr) Greenberg Lecture I: PK, PD, and Statistics 13 Greenberg Lecture I: PK, PD, and Statistics 14 2 What is Pharmacokinetics? 2 What is Pharmacokinetics? Multiple dosing: Ordinarily, sustaining doses are given to replace drug eliminated, maintain concentrations in therapeutic window over time Steady state : amount lost = amount gained Principle of superposition: Frequency, amount for multiple-dose regimen governed by: ADME Width of therapeutic window Greenberg Lecture I: PK, PD, and Statistics 15 Greenberg Lecture I: PK, PD, and Statistics 16 2 What is Pharmacokinetics? 2 What is Pharmacokinetics? Effect of different frequency: Same dose and ADME characteristics Effect of different elimination characteristics: Same dose and frequency Greenberg Lecture I: PK, PD, and Statistics 17 Greenberg Lecture I: PK, PD, and Statistics 18

2 What is Pharmacokinetics? 2 What is Pharmacokinetics? Need a way to deduce ADME from plasma concentrations One-compartment model with first-order absorption, elimination: Compartmental modeling: Represent the body by a system of compartments depending on ADME processes Can be grossly simplistic, but often sufficient approximation Compartments may or may not have physical meaning oral dose D X(t) k a k e dx(t) dt dx a(t) dt = k ax a(t) k ex(t), X(0) = 0 = k ax a(t), X a(0) = F D F = bioavailability, X a(t) = amount at absorption site C(t) = X(t) = kadf {exp( ket) exp( kat)}, ke = Cl/V V V (k a k e) V = volume of compartment, Cl = clearance Greenberg Lecture I: PK, PD, and Statistics 19 Greenberg Lecture I: PK, PD, and Statistics 20 2 What is Pharmacokinetics? 2 What is Pharmacokinetics? Two-compartment model, IV bolus injection: Dose D (instantaneous) D : dx(t) dt dx tis(t) dt X(t) ke k 12 k21 Greenberg Lecture I: PK, PD, and Statistics 21 X tis(t) = k 21X tis(t) k 12X(t) k ex(t), X(0) = D = k 12X(t) k 21X tis(t), X tis(0) = 0 C(t) = A 1 exp( λ 1t) + A 2 exp( λ 2t) Extensions: More compartments (eg peripheral tissues), nonlinear kinetics (saturation at high concentrations) Physiologically-Based Pharmacokinetic (PBPK) models Result: Deterministic model for time-concentration within a subject Based on (albeit simplified) physiological considerations Depends on PK parameters characterizing ADME processes for that subject Multiple doses : Apply superposition principle, eg C(t) = { } D d Cl V exp (t td) V d:td<t Greenberg Lecture I: PK, PD, and Statistics 22 2 What is Pharmacokinetics? 3 What is Pharmacodynamics? Only half the battle! What is a good concentration? What is the therapeutic window? Is it the same for everyone? Further complicating matters: Recall theophylline Identical dose = substantial variation in drug concentrations among people due to substantial variation in ADME among people = each subject may have same model but with different PK parameters Idea: What the drug does to the body Characterizing dose-response relationship in the population is not informative enough One reason: inter-subject variation in PK Ie, Inter-subject variation in concentrations for same dose = inter-subject variation in response for same dose Understanding concentration-response for individuals provides more precise information for deciding how to dose Pharmacodynamics: Relationship of response (drug effect) to drug concentration Greenberg Lecture I: PK, PD, and Statistics 23 Greenberg Lecture I: PK, PD, and Statistics 24

3 What is Pharmacodynamics? 3 What is Pharmacodynamics? Furthermore: Inter-subject variation in concentration due to different PK is only part of the reason subjects vary in their responses Response varies across subjects who achieve the same concentrations = Study concentration-response within subjects and how it varies across subjects Understanding inter-subject variation in concentrations and responses gives insight on width and placement of therapeutic window and how it varies across subjects PD models: Model concentration-response within a given subject Empirical rather than physiological in basis Eg the so-called E max model for continuous response Emax E0 R = E 0 + 1 + EC, C = concentration 50/C Each subject has his/her own PD parameters Ideally : Concentration at site of action Realistically : Concentration in plasma Complications: Choice of R, measurement error in C Time lag : difference between concentration in blood and at site Greenberg Lecture I: PK, PD, and Statistics 25 Greenberg Lecture I: PK, PD, and Statistics 26 3 What is Pharmacodynamics? 3 What is Pharmacodynamics? Pharmacokinetics: Learn about PK parameters in a suitable compartment model For individual subjects and how they vary in the population In order to understand how to dose individual subjects and develop guidelines for dosing certain types of subjects (eg elderly) To achieve desired concentrations Pharmacodynamics: Completing the story Learn about concentrations eliciting desired responses and inter-subject variation in how this happens In order to gain understanding of the width and variation (among subjects) of the therapeutic window And use this knowledge to refine dosing strategies Ultimate objectives: dose Improve drug development process Inform better drug use in routine clinical care PK variation concentration PD variation response Greenberg Lecture I: PK, PD, and Statistics 27 Greenberg Lecture I: PK, PD, and Statistics 28 4 Population PK/PD and Statistics 4 Population PK/PD and Statistics The story begins: In the early 1970s with Lewis B Sheiner, MD Greenberg Lecture I: PK, PD, and Statistics 29 Greenberg Lecture I: PK, PD, and Statistics 30

4 Population PK/PD and Statistics 4 Population PK/PD and Statistics Traditional PK studies: Often in Phase I, II Get basic information, eg, average concentrations achieved, insight into toxicities Healthy volunteers, different from patient population, homogeneous Small number of subjects Lots of blood samples from each following single, multiple doses Might randomize according to a single factor, eg fed vs fasting state, evaluate effect on PK parameters Lewis: These can provide Good info on appropriate compartment model Some info on PK parameters and how they vary, but not much Lewis: Can learn a lot more Population studies Study PK in target population of heterogeneous patients undergoing chronic dosing as part of routine clinical care Large number of subjects Sparse, haphazard sampling of each subject Lots of demographic, physiological, behavioral characteristics recorded for each subject, eg weight, age, renal function, race, ethnicity, disease status, smoking, Population PK: Learn about variation of PK parameters in population Associated with subject characteristics (and their interactions) Unexplained by these ( inherent variation? unmeasured factors?) Greenberg Lecture I: PK, PD, and Statistics 31 Greenberg Lecture I: PK, PD, and Statistics 32 4 Population PK/PD and Statistics 4 Population PK/PD and Statistics How to do this? Statistical modeling! Data: Repeated concentration measurements on each of m subjects from the population of interest Y ij Y i u i a i plasma concentration at time t ij, j = 1,, n i (Y i1,, Y ini ) T dosing history for subject (conditions of measurement) subject characteristics (covariates) (Y i, u i, a i) independent across i = 1,, m Perspective: Focus not on population mean concentration, but on population of individual PK parameters in the PK mathematical model Need to embed the PK mathematical model in a statistical model Greenberg Lecture I: PK, PD, and Statistics 33 Statistical model: Sheiner, Rosenberg, and Melmon (1972); Sheiner, Rosenberg, and Marathe (1977) What is now known as a nonlinear mixed effects (hierarchical ) model Stage 1: Intra-subject model Assumption : Observed concentrations equal deterministic mathematical PK model plus deviation due to assay error, realization variance (and model misspecification) The PK model and superposition principle give an expression for concentration at time t under dosing history u f(t, u, β), β = PK parameters (p 1) Eg, β = (k a, Cl, V ) T in the one compartment model Greenberg Lecture I: PK, PD, and Statistics 34 4 Population PK/PD and Statistics 4 Population PK/PD and Statistics Subject i: Subject-specific PK parameters, eg, β i = (k ai, Cl i, V i) T concentration 0 2 4 6 8 10 12 0 5 10 15 20 time Y i(t) = f(t, u i, β i ) + e i(t), Y ij = Y i(t ij) Greenberg Lecture I: PK, PD, and Statistics 35 Stage 1: Intra-subject model Result: Y ij = f(t ij, u i, β i ) + e ij Possible intra-subject correlation (usually assumed negligible ) Intra-subject variance about f often small (CV 10-30%), not constant, dominated by assay error Standard : Y ij u i, β i normal or lognormal with E(Y ij u i, β i ) = f(t ij, u i, β i ), var(y ij u i, β i ) = σ 2 f 2 (t ij, u i, β i ) Compactly : Y i = f i (u i, β i ) + e i with E(Y i u i, β i ) = f i (u i, β i ), var(y i u i, β i ) = R i(u i, β i, ξ) Greenberg Lecture I: PK, PD, and Statistics 36 Y i = f i (u i, β i ) + R 1/2 i (u i, β i, ξ)ɛ i }{{} e i

4 Population PK/PD and Statistics 4 Population PK/PD and Statistics Stage 2: Inter-subject population model β i = d(a i, θ, b i) (p 1) b i (k 1) random effects H, mean 0 Standard assumption: H is k-variate N k(0, D) Eg Different parameterizations Cl i = θ 1 + θ T 2 a i + b Cl,i, V i = θ 3 + θ T 4 a i + b V,i log Cl i = θ 1 + θ T 2 a i + b Cl,i, log V i = θ 3 + θ T 4 a i + b V,i Often β i = A iθ + B ib i Moderate inter-subject variation in PK parameters (CV 30 70%) Greenberg Lecture I: PK, PD, and Statistics 37 Together: Two-stage hierarchy Intra-subject model (Stage 1 ): Substitute for β i E(Y i u i, a i, b i) = f i {u i, d(a i, θ, b i)}, var(y i u i, a i, b i) = R i{u i, d(a i, θ, b i), ξ} Y i = f i {u i, d(a i, θ, b i)} + R 1/2 i {d(a i, θ, b i), ξ}ɛ i = (Y i u i, a i, b i) has density p y b (y i u i, a i, b i; θ, ξ) Inter-subject population model (Stage 2 ): β i = d(a i, θ, b i), b i H, E(b i) = 0 Subject-matter and statistical principles combined in one framework: Stage 1 : physiological + empirical statistical modeling Stage 2 : empirical statistical modeling Greenberg Lecture I: PK, PD, and Statistics 38 4 Population PK/PD and Statistics 4 Population PK/PD and Statistics Objectives of analysis: Determine d Relationship between PK, covariates Estimate θ Relationship between PK, covariates Estimate H Unexplained variation in population Estimate β i Characterize individuals = individualized dosing Likelihood (conditional on covariates u i, a i): Maximize m m m p y(y i u i, a i) = l i(θ, ξ, H; y i ) = p y b (y i u i, a i, b i; θ, ξ) dh(b i) i=1 i=1 i=1 Complex dosing histories = complex PK model f Intractable integral in general (nonlinear in b i) Greenberg Lecture I: PK, PD, and Statistics 39 Lewis & Co: First-Order method (Beal and Sheiner, 1982) Assume p y b is normal, H is N k(0, D), approximate about b i = 0: Y i = f i {u i, d(a i, θ, b i)} + R 1/2 i {u i, d(a i, θ, b i), ξ}ɛ i f i {u i, d(a i, θ, 0)} + Z i(u i, θ, 0)b i + R 1/2 i {u i, d(a i, θ, 0)}ɛ i Approximate l i by n i-variate normal with E(Y i u i, a i) f i {u i, d(a i, θ, 0)}, var(y i u i, a i) R i{u i, d(a i, θ, 0)} + Z i(u i, θ, 0)DZ T i (u i, θ, 0) Implemented in the FORTRAN program NONMEM (FO method) (University of California, San Francisco) Obvious bias (but worked pretty well in simulations) Generated huge excitement in PK community Greenberg Lecture I: PK, PD, and Statistics 40 4 Population PK/PD and Statistics 4 Population PK/PD and Statistics Meanwhile: Statisticians were just beginning to pay attention Stumpy Giltinan (RIP) Ed Vonesh, PhD Mary Lindstrom, PhD Doug Bates, PhD Greenberg Lecture I: PK, PD, and Statistics 41 Greenberg Lecture I: PK, PD, and Statistics 42

4 Population PK/PD and Statistics 4 Population PK/PD and Statistics Main catalyst for statistical research in nonlinear mixed models Better approximations to the integral: Assume H is N k(0, D), p y b normal with R i(u i, β i, ξ) = R i(u i, ξ) Use Laplace s approximation or a Taylor series = l i n i-variate normal with E(Y i u i, a i) f i {u i, d(a i, θ, ˆb i)} Z i{u i, θ, ˆb i}ˆb i var(y i u i, a i) Z i(u i, θ, ˆb i)dz T i (u i, θ, ˆb i) + R i(u i, ξ) Remarks: Lindstrom and Bates (1990), Wolfinger (1993), Vonesh (1996), Implementation : Iterate between updating ˆb i and fitting approximate model Approximation works remarkably well for sparse (small n i) population PK data as long as intra-subject variation is small Variations : R/Splus nlme(), SAS %nlinmix, NONMEM (FOCE method), big advantage PK models built-in! ˆb i = empirical Bayes estimate of b i maximizing p b y (b i u i, a i, y i ) Greenberg Lecture I: PK, PD, and Statistics 43 Greenberg Lecture I: PK, PD, and Statistics 44 4 Population PK/PD and Statistics 4 Population PK/PD and Statistics This motivated lots more Computational work: Why not just do the integral? Deterministic and stochastic numerical integration, eg Variants of quadrature Importance sampling Monte Carlo EM Pinheiro and Bates (1995), Walker (1996) Implementation : SAS proc nlmixed Model refinements: Assumption on H why should b i be normal? Rather than assume a parametric form, estimate the distribution of β i directly nonparametrically (Mallet, 1986, and others) USC*PACK-NPEM (University of Southern California) Or assume H has a nice density and estimate it (Davidian and Gallant, 1993) FORTRAN nlmix (user-unfriendly) Inspect estimates to identify subpopulations, omitted covariates Greenberg Lecture I: PK, PD, and Statistics 45 Greenberg Lecture I: PK, PD, and Statistics 46 4 Population PK/PD and Statistics 4 Population PK/PD and Statistics This was also a natural area for Bayesians Jon Wakefield, PhD Gary Rosner, ScD Peter Müller, PhD Joe Ibrahim, PhD Greenberg Lecture I: PK, PD, and Statistics 47 Greenberg Lecture I: PK, PD, and Statistics 48

4 Population PK/PD and Statistics 4 Population PK/PD and Statistics Bayesian view: Add Stage 3 : Hyperprior (β, ξ, D) p β,ξ,d(β, ξ, D) Implementation: To do the intractable integration us MCMC techniques An early showcase for these methods Wakefield (1996), Müller and Rosner (1997) Gelman, Bois, Jiang (1996), Mezzetti, Ibrahim, et al (2003) Parametric (normality) or more flexible models for b i, β i PKBugs, a WinBUGS interface with built-in PK models MCSim, for systems of differential equations (PBPK models) Greenberg Lecture I: PK, PD, and Statistics 49 What about pharmacodynamics? PK is only part of the full story Population PK/PD study : Collect PK/PD data on same subjects PD responses R ij at times t ij (categorical, continuous, surrogate ) Intra-subject PD model : True plasma concentration c ij at t ij R ij = g(c ij, α i) + e ij Joint PK/PD model: Describe c ij by PK model Intra-subject joint PK/PD model Y ij = f(t ij, u i, β i ) + e ij, Emaxi E0i eg g(c, α i) = E 0i + 1 + EC 50/c R ij = g{f(t ij, u i, β i ), α i} + e ij β i = d(a i, θ, b i), α i = d (a i, γ, b i ), (b T i, b T i ) H Often : Incorporate a lag between PK and PD in the joint model Greenberg Lecture I: PK, PD, and Statistics 50 4 Population PK/PD and Statistics 4 Population PK/PD and Statistics Extensions and by-products: Individual estimation : Use posterior modes to estimate β i (and α i) Bayesian dosage adjustment : Use these for current or future subjects given a few observations = Individual dosing regimen Inter-occasion variation parameters may vary within the same individual, implications for dosing Non/semiparametric population models Censored concentrations/response Missing/mismeasured covariates Etc Summary: By the late-1990s Lots of statistical research Nonlinear mixed effects models became standard tools Exploited, enhanced by PK/PD community = specialized software implementing one or more of these methods (and built-in catalogs of PK/PD models) NONMEM (UCSF, now GloboMax) ADAPT II (University of Southern California) WinNonMix (Pharsight Corporation) Etc Greenberg Lecture I: PK, PD, and Statistics 51 Greenberg Lecture I: PK, PD, and Statistics 52 5 Example 5 Example World-famous example: Population PK of phenobarbital Dosing history and concentrations for one subject: m = 59 pre-term infants treated for seizures n i = 1 to 6 concentration measurements per subject, total of 155 measurements Birth weight w i and 5-minute Apgar score δ i = I[Apgar < 5] IV multiple doses; one-compartment model f(t ij, u i, β i ) = { } D id exp Cli (t t id) V i V i d:tid<t Objectives: Characterize PK and its variation (typical Cl i, V i? do covariates matter? extent of biological variation?) Phenobarbital conc (mcg/ml) 0 20 40 60 0 50 100 150 200 250 300 Time (hours) Greenberg Lecture I: PK, PD, and Statistics 53 Greenberg Lecture I: PK, PD, and Statistics 54

5 Example 5 Example Inter-subject models: β i = (Cl i, V i) T Without covariates log Cl i = θ 1 + b 1i, log V i = θ 2 + b 2i Final model with covariates log Cl i = θ 1 + θ 3w i + b 1i, log V i = θ 2 + θ 4w i + θ 5δ i + b 2i Clearance random effect -05 00 05 10 15 Clearance random effect -05 00 05 10 15 05 10 15 20 25 30 35 Birth weight Apgar<5 Apgar>=5 Apgar score Volume random effect -05 00 05 10 Volume random effect -05 00 05 10 05 10 15 20 25 30 35 Birth weight Apgar<5 Apgar>=5 Apgar score Greenberg Lecture I: PK, PD, and Statistics 55 Greenberg Lecture I: PK, PD, and Statistics 56 5 Example 5 Example Clearance random effect -05 00 05 05 10 15 20 25 30 35 Birth weight Volume random effect -02 00 01 02 03 05 10 15 20 25 30 35 Birth weight Density estimates: 0 1 2 3 4 5 6 h 04 02 0-02 Volume -04 (a) -05 0 05 Clearance h 0 2 4 6 8 10 06 04 02 0-02 -04 Volume -06 (b) -05 0 05 Clearance Clearance random effect -05 00 05 Apgar<5 Apgar score Apgar>=5 Volume random effect -02 00 01 02 03 Apgar<5 Apgar score Apgar>=5 Volume -06-04 -02 00 02 04 06 (c) -05 00 05 Clearance Volume -06-04 -02 00 02 04 06 (d) -05 00 05 Clearance Greenberg Lecture I: PK, PD, and Statistics 57 Greenberg Lecture I: PK, PD, and Statistics 58 6 PK/PD Today 6 PK/PD Today Population PK/PD analysis: Is an important component of the drug development process Recognized benefit: Identifying differences in drug safety and efficacy among population subgroups that can be addressed by dose modification particularly when intended population is quite heterogeneous and typical therapeutic window is narrow Is an important component of the regulatory process Guidance for Industry Population Pharmacokinetics US Department of Health and Human Services Food and Drug Administration Center for Drug Evaluation and Research (CDER) Center for Biologics Evaluation and Research (CBER) February 1999 CP 1 Greenberg Lecture I: PK, PD, and Statistics 59 Greenberg Lecture I: PK, PD, and Statistics 60

6 PK/PD Today 6 PK/PD Today Current interest: Clinical Pharmacology Subcommittee of the FDA Advisory Committee for Pharmaceutical Science A population PK/PD guidance Incorporation of genetic information Special design/model considerations for pediatric populations Greenberg Lecture I: PK, PD, and Statistics 61 Clinical trial simulation: Use PK, PD info to target, design trials Pharsight Corporation (and others) Ingredients: Based on prior PK/PD investigation Covariate distribution model : A model for the target population PK model : Hierarchical model incorporating covariates impacting PK = concentrations PD model : Hierarchical model incorporating concentrations, covariates impacting PD = responses (also placebo model) Hazard model : Relating responses to a clinical endpoint Simulation: Generate samples of patients under different designs (numbers, inclusion criteria, dose regimens, etc) End-of-Phase-2a Meetings Greenberg Lecture I: PK, PD, and Statistics 62 7 Concluding Remarks 7 Concluding Remarks Population PK/PD analysis: A statistical success story Statistical modeling is central to the subject-matter science A model for other biomedical research Currently: Great interest in combining mathematical and statistical modeling to address other questions, eg, treatment of HIV infection Potent antiretroviral drugs cannot be taken continually What is the best strategy for treatment? A promising tool: Within-subject HIV dynamical systems models Describe the interplay between virus and immune system over time, incorporates effects of treatment Can these models be used to develop dynamic treatment regimes for HIV infection? Tomorrow afternoon! Greenberg Lecture I: PK, PD, and Statistics 63 Greenberg Lecture I: PK, PD, and Statistics 64 7 Concluding Remarks 7 Concluding Remarks 10 5 Model Fits to the Clinical Data 10 4 data model fit censored data virus copies/ml 10 3 10 2 10 1 10 0 0 200 400 600 800 1000 1200 1400 1600 time (days) Greenberg Lecture I: PK, PD, and Statistics 65 Greenberg Lecture I: PK, PD, and Statistics 66

7 Concluding Remarks Dedication Where to get a copy of these slides: http://wwwstatncsuedu/ davidian This talk is dedicated to the memory of Where to find a great intro course on PK on the web: http://wwwboomerorg/c/p1/ Thanks to David Bourne at University of Oklahoma for some of the pictures in this talk! Some books about PK/PD: Rowland, M and Tozer, TN, Clinical Pharmaockinetics: Concepts and Applications (nth edition) Gibaldi, M and Perrier, D, Pharmacokinetics (2nd edition) Journal with lots of statistical content: Journal of Pharmacokinetics and Pharmacodynamics (formerly Journal of Pharmacokinetics and Biopharmaceutics) Greenberg Lecture I: PK, PD, and Statistics 67 Lewis B Sheiner, MD 1940 2004 Greenberg Lecture I: PK, PD, and Statistics 68