pharmacokinetic pharmacodynamic modeling & simulation.pdf

8/10/2019 Pharmacokinetic Pharmacodynamic Modeling & Simulation.pdf

1/70

Introduction toPharmacokinetic/

Pharmacodynamic Modeling:Concepts and Methods

Alan Hartford

Agensys, Inc.An Affiliate of Astellas Pharma Inc


2/70

2

Outline Introduction to Pharmacokinetics

Compartmental Modeling

Maximum Likelihood Methodology

Pharmacodynamic Models Relevance of NONMEM

(A few examples fitting nonlinear mixedmodels with R included through-out astime allows)


3/70

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


4/70

4

Pharmacokinetics Endpoints

AUC, Cmax, Tmax, half-life (terminal),C_trough, Clearance, Volume

The effect of the drug is assumed to berelated to some measure of exposure.

(AUC, Cmax, C_trough)


5/70

5

PK/PD Modeling Procedure:

Estimate exposure and examine correlation betweenexposure and PD or other endpoints (including AErates)

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)


6/70

6

Cmax

Tmax

AUC

Figure 2

Time

Concentration

Concentration of Drug as a Function of Time

Model for Extra-vascular Absorption


7/70

7

Observed or Predicted PK?

Are you able to measure PK?

Concentration in blood is a biomarker forconcentration at site of action

PK parameters are not directly measured While you can measure C_trough in blood directly,

you cant measure Clearance and Volume


8/70

8

The Nonlinear Mixed Effects Model

Pharmacokineticists use the term population

model when the model involves random effects.


9/70

9

Compartmental Modeling A persons body is modeled with a system of differential

equations, one for each compartment

If each equation represents a specific organ or set oforgans with similar perfusion rates, then called

Physiologically Based PK (PBPK) modeling.

The mean function fis a solution of this system ofdifferential equations.

Each equation in the system describes the flow of druginto and out of a specific compartment.


10/70

10

Input

Elimination

Central

Vc

k10

First-Order 1-CompartmentModel (Intravenous injection)

Solution:


11/70

11

Choice of Parameterization

For making distribution assumptions for

parameters, it is more physiologicallyrelevant to assume that systemicclearance a random effect instead ofelimination rate.

Because clearance and volume are

assumed to be independent, this reducesthe number of parameters in thecovariance matrix.


12/70

12

Input

Elimination

Central

Vc

k10

First-Order 1-Compartment

Model (Intravenous injection)Parameterized with Clearance

Solution:

Another parameterization for the solution

uses Clearance = Cl = k10 Vc

Clearance = Volume of drug eliminatedper unit time


13/70

13

Input

Elimination

Central

Vc

k10

First-Order 1-CompartmentModel (Extravascular Administration)

ka

Solution:F = Bioavailability

(i.e., amount absorbed)

Absorption depot:

Central compartment:


14/70

14


Model (Extravascular Administration)Parameterized with Clearance

Input

Elimination

Central

Vc

k10

ka

Solution:

F = Bioavailability

(i.e., amount absorbed)


15/70

15

Parameterization ka, k10, V

Micro constant ka, Cl, V

Macro constant

Note that usually F, V, and Cl are not estimable(unless you perform studies with both IV andextravascular administration)

Instead, apparent V (V/F) and apparent Cl (Cl/F)are estimated when only extravascular data areavailable


16/70

16

Technical ConsiderationsOutline

General form of NLME

Parameterization

Error Models Model fitting

(Approximate) Maximum Likelihood

Fitting Algorithms


17/70

17

The Nonlinear Mixed Effects Model

Pharmacokineticists use the term populationmodel when the model involves random effects.


18/70

18

For simplification at this stage, assume

and


19/70

19

Error Models Error models used for PK modeling:

Additive error

Proportional error

Additive and Proportional error

Exponential error


20/70

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 beconstant (2)

Heteroscedastic variance is modeled, by

pharmacokineticists, using the proportional errorterm

Statisticians, in general, use the approach with

additive error model assuming a variancefunction R() where is an m x 1 vector whichcan incorporate , D and other parameters, e.g.,R()=2[f()]2, =[, ]


21/70

21

For the 1-compartment modelparameterized with Cl, V, ka

And cov(logCli, logVi) is assumed to be 0 bydefinition of the pharmacokinetic parameters.

Input

Elimination

Central

Vc

k10

ka


22/70

22

We obtain the maximum likelihood estimate by

maximizing

Where p(yi) is the probability distribution function(pdf) of y where now we use the notation of yias a vector of all responses for the ith subject

The problem is that we dont have thisprobability density function for y directly.

Use Maximum Likelihood


23/70

23

We use the following:

Where pand are normal probability density functions.

Maximization is in =[

, vech(D), vech(R)]T

.

Notation: the vech function of a matrix is equal to a vector of the

unique elements of the matrix.


24/70

24

Under Normal Assumptions


25/70

25

Approach: Approximate ML Use numerical approaches to

approximate the integral and thenmaximize the approximation

Some ways to do this are:

1. Approximate the integrand to somethingintegrable

2. Approximate the whole integral3. Gibbs sampler


26/70

26

Maximum LikelihoodGiven data yij, we use maximum likelihood to

obtain parameters estimates for , D, and2.

Because the mean function, f, is assumed tobe nonlinear in i in pharmacokinetics,

least squares does not result in equivalentparameter estimates.


27/70

27

Approximate Methods Options:

Approximate the integrand by something wecan integrate

First Order method (Taylor series)

Approximate the whole integral Laplaces approximation (second order

approximation)

Gaussian Quadrature

Use Bayesian methodology


28/70

28

Algorithms UsedApproximate integrand

Or approximate whole integral

First Order

First Order Conditional Estimation

Laplaces Approximation

Importance Sampling

Gaussian Quadrature

Spherical-Radial

Gibbs Sampler

Monolix Not covered in this presentation

Available in NONMEM


29/70

29

First Order Method Approximate with a first order Taylor series

expansion

If the model assumes

And Ri = 2I, then this is pretty straight-forward.

You use a Taylor series expansion about bi.


30/70

30

Taylor Series ExpansionWith a first order Taylor series approximation

expanded about , the mean of the i

Let this approximation be

You use this approximation in the integrand.


31/70

31

Substituting back in and simplifying

And now the exponent term is linear in bi and we canintegrate directly. Now we can maximize the likelihood.

See slide 23.


32/70

32

A second order approximation can be constructed

by using Laplaces approximation

Using Laplaces Approximation

In this manner, the whole integral is approximatedso no integration is needed.


33/70

33

Numerical Considerations for

Laplaces Approximation To guard against numerical overflow errors,

Laplaces approximation is programmed intosoftware in a way that is not intuitive.


34/70

34

Numerical Integration:

Importance SamplingConsider a function g(b).

To get a numerical solution to the integral simplyuse a random number generator to sample

many b and change the expectation to a samplemean.


35/70

35

Where h is the index for the sampling from

(bi).

and


36/70

36

Problem!If each evaluation of the likelihood surface requires

a resampling, then you introduce a randomnessto your likelihood surface.

The likelihood surface would have smallperturbations which would affect yourdetermination of a maximum.

Solution: sample once and re-use this sample foreach evaluation of the likelihood.


37/70

37

It turns out that importance sampling is notvery efficient. To improve on this method,another method takes advantage of the

normal assumption of distribution of bi. This method is called GaussianQuadrature. Instead of a random sample,

specific abscissas have been determinedto best evaluate the integral.

In particular, adaptive Gaussian

Quadrature is a preferred method (notcovered here).


38/70

38

Review of Approx Methods First order: biased, only useful for getting

starting values for better methods; convergesoften even if model is horrible. DONT RELY ONTHIS METHOD!

Laplacian: numerically cheap, reasonably

good fit Importance sampling: Need lots of abscissas, sonot useful

Gaussian Quadrature: GOLD STANDARD! But

when data set large, method is slow and difficultto get convergence.


39/70


40/70

40

Software NONMEM (industry standard, 1979,

FORTRAN) SAS

R and S-Plus Monolix

WinBugs (PKBugs)

Phoenix (new 2008)


41/70

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 PKFITpackage

http://www.pharmastatsci.com/pharmacokinetics.htm (provided by In-Sun Nam)


42/70

42

Objective Function and Gradient

Vector

The maximum likelihood solution is the

vector of parameters that minimize thenegative 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


43/70

43

Hessian Matrix The Hessian Matrix is the symmetric matrix of second

partial derivatives of the objective function

The 2nd derivative test can be use to confirmminimization 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 2nd deriv. test inconclusive

Also note, the variance matrix of the parameter

estimates is the inverse Hessian

Obj i F i f M d l


44/70

44

Objective Function for Model

Selection For nested models, the difference in the

objective function has a chi-squaredistribution with df=difference in thenumber parameters


45/70

45

Input

Elimination

CentralPeripheral

Vc (Vp)

k10

k12

k21


Model (Intravenous Dose)

Parameterized in terms ofMicro constants

Note that including Vp over-parameterizes the model since

Ac = Amount of drug in central compartment

Ap = Amount of drug in peripheral compartment


46/70

46

Web Demonstration http://vam.anest.ufl.edu/simulations/secon

dorderstochasticsim2.html#sim

(Requires installation of AdobeShockwave player.)

Fi O d 2 C


47/70

47

Input

Elimination

CentralPeripheral

Vc (Vp)

k10

k12

k21


Model (Intravenous Dose)

General form ofsolution:

Another, preferred


48/70

48

Input

Elimination

CentralPeripheral

Vc Vp

Cl

Q

Another, preferred

parameterization (macro constants)Q is the inter-compartmentaldistribution parameter

It is the amount of drugtransferred back to Vc per unittime.


49/70


50/70

50

Modeling CovariatesAssumed: PK parameters vary with respect to apatients weight or age.

Covariates can be added to the model in a secondarystructure (hierarchical model).


51/70

51

Nonlinear Mixed Effects ModelWith secondary structure for covariates:

xi is a vector of covariates which, for simplificationhere, is assumed to be constant over time j.Often, is a vector of log Cl, log V, and log ka


52/70

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 modelswritten in terms of ODEs instead of closed formsolution.

Multiple Dose Model


53/70

53

Multiple Dose Model

Daily Dose with Fast Elimination

Multiple Dose Model


54/70

54

Multiple Dose Model

Daily Dose with Slower Elimination

Super-positionprinciple


55/70

55

Super-position Principle Assume dosing every 24 hours Assume concentration for single dose is

Then concentration, C(t) is

Multiple Dose Model


56/70

56

Multiple Dose Model

Missed Third Dose

Dose Delayed by 3 Hours Every


57/70

57

Dose Delayed by 3 Hours Every

Other Day


58/70

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)


59/70

59

Emax Model

E=Emax * Conc

EC50+Conc

Mechanistic Models


60/70

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

Mechanistic Model Example


61/70

61


Multiple Binding Site Model

Effect = ________________kD + Conc + K2*Conc

2

RT * Conc

RT = total receptor contentkD = k-1/ k1K2 = k2/ k-2



62/70

62


Multiple Binding Site Model

K2=0

K2=0.001

K2=0.01

K2=0.05

K2=0.5


63/70

63

Which PD model? If mechanism is known, then choice of

model is more clear.

If mechanism not known, then tryingdifferent models leads to suggestionsabout mechanism.

Competitive Inhibition in a Tissue


64/70

64

p

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


65/70

65

Kinetics DiagramDose

PlasmaCompartment

V1

Excretion andMetabolism

EffectCompartment

V2

Elimination fromV2

k10

k12

k20


66/70

66

Kinetics EquationsParam Description

C1

Concentration in plasma cmpt. (amount/vol)

C2 Concentration in effect cmpt (amount/vol)

k10 Elimination rate (1/time)

k12 Rate of transfer to effect cmpt (1/time)

k20 Rate of elimination from effect cmpt (1/time)


67/70

67

Kinetics Equations (cont.)Param DescriptionE Measured effect

E0 Baseline effect

Emax Maximum possible effect of infinite protein

EC50,prot Concentration of half-maximal effect for protein

(amount/vol)

Cprotein

Concentration of the protein (amount/vol)

EC50,drug Concentration of half-maximal inhibition of theprotein by the drug at a particular protein

concentration. (amount/vol)


68/70

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 todiscriminate doses


69/70

69

AcknowledgementsThanks 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 BillJusko, SUNY Buffalo.


70/70

70

References Davidian, M. and D. Giltinan, Nonlinear Models for

Repeated Measurement Data, Chapman and Hall, New

York, 1995. Gabrielsson, J. and D. Weiner, Pharmacokinetic andPharmacodynamic Data Analysis: Concepts andApplications, Swedish Pharmaceutic, 2007.

Pinheiro, J.C. and D.M. Bates, Approximations to thelog-likelihood function in the nonlinear effects model, J.Comput. Graph. Statist., 4 (1995) 12-35.

Pinheiro, J.C. and D.M. Bates, Mixed-Effects Models inS and S-Plus, Springer, New York, 2004.

The Comprehensive R Network, http://cran.r-project.org/ Pharma Stat Sci, http://www.pharmastatsci.com/

pharmacokinetic pharmacodynamic modeling & simulation.pdf

Documents