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

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Introduction toPharmacokinetic/

Pharmacodynamic Modeling:Concepts and Methods

Alan Hartford

Agensys, Inc.An Affiliate of Astellas Pharma Inc

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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)

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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

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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)

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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)

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Cmax

Tmax

AUC

Figure 2

Time

Concentration

Concentration of Drug as a Function of Time

Model for Extra-vascular Absorption

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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

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The Nonlinear Mixed Effects Model

Pharmacokineticists use the term population

model when the model involves random effects.

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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.

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Input

Elimination

Central

Vc

k10

First-Order 1-CompartmentModel (Intravenous injection)

Solution:

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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.

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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

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Input

Elimination

Central

Vc

k10

First-Order 1-CompartmentModel (Extravascular Administration)

ka

Solution:F = Bioavailability

(i.e., amount absorbed)

Absorption depot:

Central compartment:

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First-Order 1-Compartment

Model (Extravascular Administration)Parameterized with Clearance

Input

Elimination

Central

Vc

k10

ka

Solution:

F = Bioavailability

(i.e., amount absorbed)

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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

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Technical ConsiderationsOutline

General form of NLME

Parameterization

Error Models Model fitting

(Approximate) Maximum Likelihood

Fitting Algorithms

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The Nonlinear Mixed Effects Model

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

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For simplification at this stage, assume

and

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Error Models Error models used for PK modeling:

Additive error

Proportional error

Additive and Proportional error

Exponential error

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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, =[, ]

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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

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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

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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.

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Under Normal Assumptions

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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

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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.

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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

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Algorithms UsedApproximate integrand

Or approximate whole integral

First Order

First Order Conditional Estimation

Laplaces Approximation

Importance Sampling

Gaussian Quadrature

Spherical-Radial

Gibbs Sampler