Bayesian Population Pharmacokinetic/Pharmacodynamic Modeling

1. Bayesian Population Pharmacokinetic/Pharmacodynamic Modeling Steven Kathman GlaxoSmithKline

2. Half of the modern drugs could well be thrown out of the window, except that the birds might eat them. Dr. Martin Henry Fischer

3. Outline • Introduction • Population PK modeling • Population PK/PD modeling • Modeling the time course of ANC • Other examples • Conclusions

4. Introduction • KSP inhibitor (Ispinesib) being developed for the treatment of cancer. • Blocks assembly of a functional mitotic spindle and leads to G2/M arrest. • Causes cell cycle arrest in mitosis and subsequent cell death. • Leads to a transient reduction in absolute neutrophil counts (ANC).

5. Introduction • KSP10001 was the FTIH study. • Ispinesib dosed once every three weeks. • PK data collected after first dose. • ANC assessed on Days 1 (pre-dose), 8, 15, and 22 (C2D1 pre-dose). More frequent assessments done if ANC < 0.75 (109/L). • Prolonged Grade 4 neutropenia (> 5 days) most common DLT.

6. Objectives • Determine a suitable PK model. - Examine 2 vs 3 compartment models. • Determine a suitable model for PD endpoint (i.e., time course of absolute neutrophil counts). - Using Nonlinear mixed models. - Using Bayesian methods.

7. Pharmacokinetics The action of drugs in the body over a period of time, including the processes of absorption, distribution, localisation in tissues, biotransformation and excretion. Simple terms – what happens to the drug after it enters the body. What is the body doing to the drug over time?

8. R A2 = C2V2 A1 = C1V1 k12 k21 k10 dA1/dt = R + k21A2 – k12A1 – k10A1 dA2/dt = k12A1 – k21A2

9. CL = k10V1 Q = k12V1 = k21V2

10. Infusion k0 = zero order infusion rate T=t during infusion, constant time infusion was stopped after infusion.

11. PK Model

12. PK Model µ~Vague MVN prior R chosen based on CV=30%

13. If that was painful… In mathematics you don't understand things. You just get used to them. Johann von Neumann (1903 - 1957)

14. Bayesian Results • Typical Bayesian analysis (via MCMC) involves estimation of the joint posterior distribution of all unobserved stochastic quantities conditional on observed data. • Generating random samples from the joint posterior distribution of the parameters. • Marginal distribution of each parameter is completely characterized (numerical integration). P(individual specific PK parameters, population PK parameters | PK data)

16. R k12 k13 A2=C2V2 A1=C1V1 A3=C3V3 k31 k21 k10 dA1/dt = R + k21A2 + k31A3 – k12 A1 – k13A1 – k10 A1 dA2/dt = k12A1 – k21A2 dA3/dt = k13A1 – k31A3

17. Pharmacodynamics The study of the biochemical and physiological effects of drugs and the mechanisms of their actions, including the correlation of actions and effects of drugs with their chemical structure, also, such effects on the actions of a particular drug or drugs. What is the drug doing to the body?

18. Modeling the Time Course: Absolute Neutrophil Counts When you are curious, you find lots of interesting things to do. The way to get started is to quit talking and begin doing. – Walt Disney (1901-1966)

19. Model of Myelosuppression Prol Circ Transit 1 Transit 2 Transit 3 ktr ktr ktr ktr kprol = ktr kcirc = ktr EDrug = βConc

20. Features of Model • Proliferating compartment – sensitive to drug. • Three transit compartments – represent maturation. • Compartment of circulating blood cells. • System parameters: MTT, baseline, and feedback. • Drug specific parameter: Slope.

21. Feedback • Account for rebound phase (overshoot). • Negative feedback from circulating cells to proliferative cells. • G-CSF levels increase when circulating neutrophil counts are low. • G-CSF stimulates proliferation in bone marrow.

22. Model of Myelosuppression • dProl/dt = kprol*Prol*(1-EDrug)*(Circ0/Circ)-ktr*Prol • dTransit1/dt = ktr*Prol-ktr*Transit1 • dTransit2/dt = ktr*Transit1-ktr*Transit2 • dTransit3/dt = ktr*Transit2-ktr*Transit3 • dCirc/dt = ktr*Transit3-kcirc*Circ

23. ANCij~t(Meanij(MTTi, Circ0(i),,βi; Concij), ij, 4) Mean = Solution of the differential equation (Circ) MTTi = 4/(ktr(i)) = Mean transit time. Fairly informative priors (Literature). ln(MTTi)~N(MTT, MTT) ln(Circ0(i))~N(circ, circ) ln(βi)~N(β, β) Vague prior.

30. Simulate New Schedule • Using mechanistic/semi-physiological models allows for simulation of new schedules. • Simulate dosing on days 1, 8, and 15 repeated every 28 days. • PK/PD model accurately predicted the observed severity and duration of neutropenia.

33. Why Bayesian? • Incorporate prior information (MTT and baseline). • Better integration algorithm (Monte Carlo vs Taylor Series or Quadrature). • Posterior distribution vs MLE: More informative, avoids potentially problematic maximization algorithms. • Better individual estimates: Bayesian vs Empirical Bayesian (which usually fail to account for estimated population parameters?).

34. Tumor Growth Models • dC/dt = KL*C(t) – KD*C(t)*D(t)*exp(-t) where KL = Tumor growth rate KD = Drug constant kill rate D(t) = Dose or PK measure  = rate constant for resistance • dC/dt = exp(1t) *C(t) – KD*C(t)*D(t)*exp(-2t)

38. Preclinical PK • Concentrations in plasma. • Concentrations in a tumor. • Relate the two: • Plasma: two-compartment model. • Tumor: dCT(t)/dt = (KP/VT)AP(t)-KTCT(t)

39. More PK • Compound given through iv infusion. • Should be 1-hr infusion. • Reason to believe that the infusion time is less for some subjects. • Making the infusion times a parameter to be estimated, with informative priors.

40. Software • WinBugs (Pharmaco and WBDiff) - Pharmaco: Built in PK functions. - WBDiff: Differential Equation Solver • NONMEM • SAS macro • R: nlmeODE library and function

41. Conclusions • PK/PD modeling often involves interesting and complicated models. • Models can serve many useful functions in drug development. • Bayesian methods help with: • Better algorithms • More flexibility • Incorporating outside information

42. General Remarks • PK/PD modeling involves different skills coming together (medical, pharmacokinetics, pharmacology, statistics, etc.). • As a statistician, helps to develop knowledge in areas outside of statistics.

43. References Knowledge is of two kinds. We know a subject ourselves, or we know where we can find information on it. Samuel Johnson (1709 - 1784), quoted in Boswell's Life of Johnson

44. References • Gibaldi, M. and Perrier, D. (1982) Pharmacokinetics. • Friberg, L. et. al. (2002). Model of Chemotherapy-Induced Myelosuppression with Parameter Consistency Across Drugs. JCO 20:4713-4721. • Friberg, L. et. al. (2003). Mechanistic Models for Myelosuppression. Investigational New Drugs 21:183-194. • Lunn, D. et. al. (2002). Bayesian Analysis of Population PK/PD Models: General Concepts and Software. Journal of PK and PD 29:271-307. • PK Bugs User Guide. • Christian, R. and Casella, G. (2005) Monte Carlo Statistical Methods. • Gelman, A. et. al. (2003) Bayesian Data Analysis. • Gabrielson, J. and Weiner, D. (2006) Pharmacokinetic and Pharmcodynamic Data Analysis: Concepts and Applications

45. Questions The outcome of any serious research can only be to make two questions grow where only one grew before. Thorstein Veblen (1857 - 1929)