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
Monolix Not covered in this presentation
Available in NONMEM
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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.
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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.
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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.
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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.
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Numerical Considerations for
Laplaces Approximation To guard against numerical overflow errors,
Laplaces approximation is programmed intosoftware in a way that is not intuitive.
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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.
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Where h is the index for the sampling from
(bi).
and
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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.
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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).
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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.
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Software NONMEM (industry standard, 1979,
FORTRAN) SAS
R and S-Plus Monolix
WinBugs (PKBugs)
Phoenix (new 2008)
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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)
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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
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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
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Objective Function for Model
Selection For nested models, the difference in the
objective function has a chi-squaredistribution with df=difference in thenumber parameters
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Input
Elimination
CentralPeripheral
Vc (Vp)
k10
k12
k21
First-Order 2-Compartment
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
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Web Demonstration http://vam.anest.ufl.edu/simulations/secon
dorderstochasticsim2.html#sim
(Requires installation of AdobeShockwave player.)
Fi O d 2 C
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Input
Elimination
CentralPeripheral
Vc (Vp)
k10
k12
k21
First-Order 2-Compartment
Model (Intravenous Dose)
General form ofsolution:
Another, preferred
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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.
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Modeling CovariatesAssumed: PK parameters vary with respect to apatients weight or age.
Covariates can be added to the model in a secondarystructure (hierarchical model).
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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
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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
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Multiple Dose Model
Daily Dose with Fast Elimination
Multiple Dose Model
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Multiple Dose Model
Daily Dose with Slower Elimination
Super-positionprinciple
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Super-position Principle Assume dosing every 24 hours Assume concentration for single dose is
Then concentration, C(t) is
Multiple Dose Model
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Multiple Dose Model
Missed Third Dose
Dose Delayed by 3 Hours Every
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Dose Delayed by 3 Hours Every
Other Day
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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)
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Emax Model
E=Emax * Conc
EC50+Conc
Mechanistic Models
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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
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Mechanistic Model Example
Multiple Binding Site Model
Effect = ________________kD + Conc + K2*Conc
2
RT * Conc
RT = total receptor contentkD = k-1/ k1K2 = k2/ k-2
Mechanistic Model Example
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Mechanistic Model Example
Multiple Binding Site Model
K2=0
K2=0.001
K2=0.01
K2=0.05
K2=0.5
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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
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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
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Kinetics DiagramDose
PlasmaCompartment
V1
Excretion andMetabolism
EffectCompartment
V2
Elimination fromV2
k10
k12
k20
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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)
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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)
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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
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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.
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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/