Pharmacokinetic Pharmacodynamic Modeling &

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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