Introduction to Pharmacokinetic/ Pharmacodynamic Modeling: Concepts and Methods. Alan Hartford Agensys, Inc. An Affiliate of Astellas Pharma Inc

Transcription

1 Introduction to Pharmacokinetic/ Pharmacodynamic Modeling: Concepts and Methods Alan Hartford Agensys, Inc. An Affiliate of Astellas Pharma Inc

2 Outline Introduction to Pharmacokinetics Compartmental Modeling Maximum Likelihood Methodology Pharmacodynamic Models Relevance of NONMEM (A few examples fitting nonlinear mixed models with R included through-out as time allows) 2

3 Introduction Pharmacokinetics is the study of what a body does with a dose of a drug kinetics = motion Absorbs, Distributes, Metabolizes, Excretes Pharmacodynamics is the study of what the drug does to the body dynamics = change 3

4 Pharmacokinetics Endpoints AUC, Cmax, Tmax, half-life (terminal), C_trough, Clearance, Volume The effect of the drug is assumed to be related to some measure of exposure. (AUC, Cmax, C_trough) 4

5 PK/PD Modeling Procedure: Estimate exposure and examine correlation between exposure and PD or other endpoints (including AE rates) Use mechanistic models Purpose: Estimate therapeutic window Dose selection Aids in identifying mechanism of action Model probability of AE as function of exposure (and covariates) 5

6 Concentration of Drug as a Function of Time Model for Extra-vascular Absorption C max Concentration AUC T max Time 6 Figure 2

7 Observed or Predicted PK? Are you able to measure PK? Concentration in blood is a biomarker for concentration at site of action PK parameters are not directly measured While you can measure C_trough in blood directly, you can t measure Clearance and Volume 7

8 The Nonlinear Mixed Effects Model Pharmacokineticists use the term population model when the model involves random effects. 8

9 Compartmental Modeling A person s body is modeled with a system of differential equations, one for each compartment If each equation represents a specific organ or set of organs with similar perfusion rates, then called Physiologically Based PK (PBPK) modeling. The mean function f is a solution of this system of differential equations. Each equation in the system describes the flow of drug into and out of a specific compartment. 9

10 First-Order 1-Compartment Model (Intravenous injection) Input Central V c Elimination k 10 Solution: 10

11 Choice of Parameterization For making distribution assumptions for parameters, it is more physiologically relevant to assume that systemic clearance a random effect instead of elimination rate. Because clearance and volume are assumed to be independent, this reduces the number of parameters in the covariance matrix. 11

12 First-Order 1-Compartment Model (Intravenous injection) Parameterized with Clearance Input Central V c Another parameterization for the solution uses Clearance = Cl = k 10 V c Elimination Clearance = Volume of drug eliminated per unit time k 10 Solution: 12

13 First-Order 1-Compartment Model (Extravascular Administration) Input k a Absorption depot: Central Central compartment: V c Elimination k 10 Solution: F = Bioavailability 13 (i.e., amount absorbed)

14 First-Order 1-Compartment Model (Extravascular Administration) Parameterized with Clearance Input k a Central V c Solution: Elimination k 10 F = Bioavailability 14 (i.e., amount absorbed)

15 Parameterization k a, k 10, V Micro constant k a, Cl, V Macro constant Note that usually F, V, and Cl are not estimable (unless you perform studies with both IV and extravascular administration) Instead, apparent V (V/F) and apparent Cl (Cl/F) are estimated when only extravascular data are available 15

16 Technical Considerations Outline General form of NLME Parameterization Error Models Model fitting (Approximate) Maximum Likelihood Fitting Algorithms 16

17 The Nonlinear Mixed Effects Model Pharmacokineticists use the term population model when the model involves random effects. 17

18 For simplification at this stage, assume and 18

19 Error Models Error models used for PK modeling: Additive error Proportional error Additive and Proportional error Exponential error 19

20 Distribution of Error In each case, the errors are assumed to be normally distributed with mean 0 In PK literature, the variance is assumed to be constant (σ 2 ) Heteroscedastic variance is modeled, by pharmacokineticists, using the proportional error term Statisticians, in general, use the approach with additive error model assuming a variance function R(θ) where θ is an m x 1 vector which can incorporate β, D and other parameters, e.g., R(θ)=σ 2 [f(β)] 2, θ=[σ, β Τ ] Τ 20

21 For the 1-compartment model parameterized with Cl, V, ka Input k a Central V c Elimination k 10 And cov(logcl i, logv i ) is assumed to be 0 by definition of the pharmacokinetic parameters. 21

22 Use Maximum Likelihood We obtain the maximum likelihood estimate by maximizing Where p(y i ) is the probability distribution function (pdf) of y where now we use the notation of y i as a vector of all responses for the i th subject The problem is that we don t have this probability density function for y directly. 22

23 We use the following: Where p and π are normal probability density functions. Maximization is in φ=[β Τ, vech(d), vech(r)] T. Notation: the vech function of a matrix is equal to a vector of the unique elements of the matrix. 23

24 Under Normal Assumptions 24

25 Approach: Approximate ML Use numerical approaches to approximate the integral and then maximize the approximation Some ways to do this are: 1. Approximate the integrand to something integrable 2. Approximate the whole integral 3. Gibb s sampler 25

26 Maximum Likelihood Given data y ij, we use maximum likelihood to obtain parameters estimates for β, D, and σ 2. Because the mean function, f, is assumed to be nonlinear in β i in pharmacokinetics, least squares does not result in equivalent parameter estimates. 26

27 Approximate Methods Options: Approximate the integrand by something we can integrate First Order method (Taylor series) Approximate the whole integral Laplace s approximation (second order approximation) Gaussian Quadrature Use Bayesian methodology 27

28 Algorithms Used First Order First Order Conditional Estimation Laplace s Approximation Importance Sampling Gaussian Quadrature Available in NONMEM Approximate integrand Spherical-Radial Gibb s Sampler Or approximate whole integral Monolix Not covered in this presentation 28

29 First Order Method Approximate with a first order Taylor series expansion If the model assumes And R i = σ 2 I, then this is pretty straight-forward. You use a Taylor series expansion about b i. 29

30 Taylor Series Expansion With a first order Taylor series approximation expanded about β, the mean of the β i Let this approximation be You use this approximation in the integrand. 30

31 Substituting back in and simplifying See slide 23. And now the exponent term is linear in b i and we can integrate directly. Now we can maximize the likelihood. 31

32 Using Laplace s Approximation A second order approximation can be constructed by using Laplace s approximation In this manner, the whole integral is approximated so no integration is needed. 32

33 Numerical Considerations for Laplace s Approximation To guard against numerical overflow errors, Laplace s approximation is programmed into software in a way that is not intuitive. 33

34 Numerical Integration: Importance Sampling Consider a function g(b). To get a numerical solution to the integral simply use a random number generator to sample many b and change the expectation to a sample mean. 34

35 Where h is the index for the sampling from π(b i ). and 35

36 Problem! If each evaluation of the likelihood surface requires a resampling, then you introduce a randomness to your likelihood surface. The likelihood surface would have small perturbations which would affect your determination of a maximum. Solution: sample once and re-use this sample for each evaluation of the likelihood. 36

37 It turns out that importance sampling is not very efficient. To improve on this method, another method takes advantage of the normal assumption of distribution of b i. This method is called Gaussian Quadrature. Instead of a random sample, specific abscissas have been determined to best evaluate the integral. In particular, adaptive Gaussian Quadrature is a preferred method (not covered here). 37

38 Review of Approx Methods First order: biased, only useful for getting starting values for better methods; converges often even if model is horrible. DON T RELY ON THIS METHOD! Laplacian: numerically cheap, reasonably good fit Importance sampling: Need lots of abscissas, so not useful Gaussian Quadrature: GOLD STANDARD! But when data set large, method is slow and difficult to get convergence. 38

39 Additional Note When your model does not converge, often it s because you have the wrong model. Don t switch algorithms just because of nonconvergence. First plot data and scrutinize choice of model. 39

40 Software NONMEM (industry standard, 1979, FORTRAN) SAS R and S-Plus Monolix WinBugs (PKBugs) Phoenix (new 2008) 40

41 Using R Nonlinear mixed effects fitting function: nlme (provided by Pinheiro and Bates) (You also need the lattice package.) Pre-written PK models available in PKFIT package cs.htm (provided by In-Sun Nam) 41

42 Objective Function and Gradient Vector The maximum likelihood solution is the vector of parameters that minimize the negative of the log likelihood function (a.k.a. the objective function). The gradient of the objective function (vector of partial derivatives of the objective function w.r.t. the parameters) should be a vector of zeros 42

43 Hessian Matrix The Hessian Matrix is the symmetric matrix of second partial derivatives of the objective function The 2 nd derivative test can be use to confirm minimization If the Hessian is positive definite (equivalently, have all positive eigenvalues) then the objective function has been minimized at the solution However, not a necessary condition. If any of the eigenvalues are zero then 2 nd deriv. test inconclusive Also note, the variance matrix of the parameter estimates is the inverse Hessian 43

44 Objective Function for Model Selection For nested models, the difference in the objective function has a chi-square distribution with df=difference in the number parameters 44

45 First-Order 2-Compartment Model (Intravenous Dose) Input Central k 12 Peripheral V c (V p ) Parameterized in terms of Micro constants Note that including V p overparameterizes the model since k 21 Elimination k 10 A c = Amount of drug in central compartment A p = Amount of drug in peripheral compartment 45

46 Web Demonstration dorderstochasticsim2.html#sim (Requires installation of Adobe Shockwave player.) 46

47 First-Order 2-Compartment Model (Intravenous Dose) Input Central k 12 Peripheral V c (V p ) k 21 Elimination k 10 General form of solution: 47

48 Another, preferred parameterization (macro constants) Input Q is the inter-compartmental distribution parameter Central V c Cl Elimination Q Peripheral V p It is the amount of drug transferred back to V c per unit time. 48

49 Do we have time now for an example using R? 49

50 Modeling Covariates Assumed: PK parameters vary with respect to a patient s weight or age. Covariates can be added to the model in a secondary structure (hierarchical model). 50

51 Nonlinear Mixed Effects Model With secondary structure for covariates: x i is a vector of covariates which, for simplification here, is assumed to be constant over time j. Often, φ is a vector of log Cl, log V, and log k a 51

52 Why is NONMEM gold standard? Software needs easy input of PK models. More challenging for multiple dose settings. Functional form dependent on data. Not many software packages allow for models written in terms of ODEs instead of closed form solution. 52

53 Multiple Dose Model Daily Dose with Fast Elimination 53

54 Multiple Dose Model Daily Dose with Slower Elimination Superposition principle 54

55 Super-position Principle Assume dosing every 24 hours Assume concentration for single dose is Then concentration, C(t) is 55

56 Multiple Dose Model Missed Third Dose 56

57 Dose Delayed by 3 Hours Every Other Day 57

58 Pharmacodynamic Model PK: nonlinear mixed effect model PD: now assume predicted PK parameters are true less PD data per subject (or more, e.g. EKG data) nonlinear fixed effect model (mechanistic) 58

59 Emax Model E= Emax * Conc EC50+Conc 59

60 Mechanistic Models (from Bill Jusko course 2007) Reversible Direct (example: Emax model) Rapid (CNS, CV) Slow (Ab, Ca-Ch-BI) Indirect Synthesis, secretion Cell trafficking Enzyme induction Irreversible Chemotherapy Enzyme Inactivation William Jusko, Pharmaceutical Sciences, SUNY Distinguished Professor 60

61 Mechanistic Model Example Multiple Binding Site Model Effect = R T * Conc k D + Conc + K 2 *Conc 2 R T = total receptor content k D = k -1 / k 1 K 2 = k 2 / k -2 61

62 Mechanistic Model Example Multiple Binding Site Model K2=0 K2=0.001 K2=0.01 K2=0.05 K2=0.5 62

63 Which PD model? If mechanism is known, then choice of model is more clear. If mechanism not known, then trying different models leads to suggestions about mechanism. 63

64 Competitive Inhibition in a Tissue Compartment Example with the following properties: One compartment IV observed kinetics Competitive inhibition (the binding of an endogenous molecule or protein is competing for the same site on the molecule as the drug) The competitive inhibition occurs in a compartment that does not affect the PK, but does affect the PD readout 64

65 Kinetics Diagram Dose Plasma Compartment V1 k 12 Effect Compartment V2 k 10 Excretion and Metabolism k 20 Elimination from V2 65

66 Kinetics Equations Param C 1 C 2 k 10 k 12 k 20 Description Concentration in plasma cmpt. (amount/vol) Concentration in effect cmpt (amount/vol) Elimination rate (1/time) Rate of transfer to effect cmpt (1/time) Rate of elimination from effect cmpt (1/time) 66

67 Kinetics Equations (cont.) Param E E 0 E max EC 50,prot C protein EC 50,drug Description Measured effect Baseline effect Maximum possible effect of infinite protein Concentration of half-maximal effect for protein (amount/vol) Concentration of the protein (amount/vol) Concentration of half-maximal inhibition of the protein by the drug at a particular protein concentration. (amount/vol) 67

68 Next Step: Simulations Using the PK/PD model, clinical trial simulations can be performed to: Inform adaptive design Determine good dose or dosing regimen for future trial Satisfy regulatory agencies in place of additional trials????? (Controversial topic.) Surrogate for trials for testing biomarkers to discriminate doses 68

69 Acknowledgements Thanks to Huafeng Zhou, Bill Denney and Banmeet Anand for help with concepts and examples! Thanks also to Yao Huang for reviewing slides. Huafeng Zhou, Gilead, Biostatistician Bill Denney, Pfizer, Pharmacokineticist Banmeet Anand, Agensys, Pharmacokineticist Yao Huang, Agensys, Biostatistician Also referenced was a PD Modeling short course by Bill Jusko, SUNY Buffalo. 69

70 References Davidian, M. and D. Giltinan, Nonlinear Models for Repeated Measurement Data, Chapman and Hall, New York, Gabrielsson, J. and D. Weiner, Pharmacokinetic and Pharmacodynamic Data Analysis: Concepts and Applications, Swedish Pharmaceutic, Pinheiro, J.C. and D.M. Bates, Approximations to the log-likelihood function in the nonlinear effects model, J. Comput. Graph. Statist., 4 (1995) Pinheiro, J.C. and D.M. Bates, Mixed-Effects Models in S and S-Plus, Springer, New York, The Comprehensive R Network, Pharma Stat Sci, 70