Introduction to PK/PD Modeling for Statisticians

Part 1

Alan Hartford, AbbVie

ASA Biopharm FDA-Industry Statistics WorkshopSeptember 16, 2015

Objectives and Outline for Part 1

• Provide statisticians with the main concepts of PK/PD modeling (a.k.a., Pharmacometrics) (see outline below)

• Encourage statisticians to support pharmacometricians in their modeling efforts

• Provide an appreciation for Pharmacometrics• We won’t be reviewing step by step modeling methods using

AIC or p-values• Outline: PK Basic Concepts, PK Compartmental Models,

Technical Considerations, Software Considerations, Adding covariates to PK models

2

Introduction – PK and PD

Pharmacokinetics is the study of what an organism does with a dose of a drug

– kinetics = motion– Absorbs, Distributes, Metabolizes, Excretes

PK Endpoints– AUC, Cmax, Tmax, half-life (terminal), Cmin

The effect of the drug is assumed to be related to some measure of exposure. (AUC, Cmax, Cmin)Pharmacodynamics is the study of what the drug does to the body (dynamics = change)

3

PK/PD ModelingProcedure:

– Fit a model to exposure data and estimate exposure at the same time points where we have PD data.

– Examine correlation between estimated exposure and PD (or other endpoints, e.g., AE rates).

– Might need to fit a mechanistic model with exposure data as explanatory variable and PD as response.

Purpose: – Estimate therapeutic window– Dose selection– Identify mechanism of action– Model probability of AE as function of exposure (and covariates)

4

Cmax

Tmax

AUC

Time

Con

cent

ratio

n

Concentration of Drug as a Function of TimeModel for Extra-vascular Absorption

5

Observed or Predicted PK?

• Exposure endpoints are not measured – only modeled, i.e., estimated

• Concentration in blood or plasma is a biomarker for concentration at site of action

• PK parameters are not directly measured

6

Compartmental Modeling

• A person’s body is modeled with a system of differential equations, one equation 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.

7

Helpful Reminder …

8

Input

Elimination

Central

Vc

k10

First-Order 1-CompartmentModel (Intravenous injection)

Dose0

/

10

tA

VAC

Akdt

dA

c

ccc

cc

tk

cc e

V

DosetC 10)( Solution:

9

Input

Elimination

Central

Vc

k10

First-Order 1-CompartmentModel (Intravenous injection)

Parameterized with Clearance

VClt

cc e

V

DosetC /)( Solution:

Another parameterization for the solution uses Clearance = Cl = k10 Vc

Clearance = Volume of drug eliminated per unit time

10

Input

Elimination

Central

Vc

k10

First-Order 1-CompartmentModel (Extravascular Administration)

00

dose0

/

10

tA

tA

VAC

AkAkdt

dA

Akdt

dA

c

a

ccc

caac

aaa

ka

tktk

ac

ac

aeekkV

DoseFktC

10

10

)(Solution: F = Bioavailability (i.e., fraction absorbed)

Absorption depot:

Central compartment:

11

First-Order 1-CompartmentModel (Extravascular Administration)

Parameterized with ClearanceInput

Elimination

Central

Vc

k10

10kVCl c ka

tktVCl

ac

ac

ac eeClkV

DoseFktC

/)(Solution:

F = Bioavailability (i.e., amount absorbed)12

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 and extravascular administration)

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

13

First-Order 2-CompartmentModel (Intravenous Dose)

14

)exp()exp( tBtAtCc

00

Dose Bolus0

/

/

2112

101221

tA

tA

VAC

VAC

AkAkdt

dA

AkkAkdt

dA

p

c

ppp

ccc

pcp

cpc

Input

Elimination

Central Peripheral

Vc(Vp)

k10

k12

k21

General form of solution:

First-Order 2-CompartmentModel (Intravenous Dose)

15

cp Vk

kV

21

12

Input

Elimination

Central Peripheral

Vc(Vp)

k10

k12

k21

Parameterized in terms of “Micro constants”

Note that including Vp over-parameterizes the model since

Ac = Amount of drug in central compartment

Ap = Amount of drug in peripheral compartment

Web Demonstration

• http://vam.anest.ufl.edu/simulations/simulationportfolio.php

• http://vam.anest.ufl.edu/simulations/stochastictwocompartment.php

• (Requires installation of Adobe Shockwave player.)

16

Technical Considerations

Outline• General form of NLME

– Parameterization– Error Models

• Model fitting – (Approximate) Maximum Likelihood– Fitting Algorithms

17

The Nonlinear Mixed Effects Model

18

ii

i

ijiiijij

RN

DN

dtfy

,0~

,~

),,(

matrix covariance an is

matrix covariance a is D

error residual is

to1 from ranges

dose ssubject' i theis

subject i for the timej theis

vectorparameter 1 a is

in nonlinear function scalar a is

subject i for the response j theis

th

thth

thth

iii

ij

i

i

ij

ij

nnR

kk

nj

d

t

k

f

y

Pharmacokineticists use the term ”population” model when the model involves random effects.

19

• For simplification at this stage, assume

DNb

b

i

ii

,0~

ii IN 2,0~

and

Assay Variability

• Assays for measuring PK concentrations are validated for specific concentration ranges.

• If the concentration is higher than the upper limit of the validated assay, then the sample is diluted so the resulting diluted sample has PK concentration within the validated limits.

• If the concentration is lower than the lower limit of the validated assay, then the concentration is reported to be “below the limit of quantitation” (“BLOQ” or “<LLOQ”).

20

Assay Variability (cont.)

• The result of the assay specifications and the needed dilutions is that additional error is added into the measurement system.

• These errors can be accounted for in the statistical model.

21

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 (s2)

• Heteroscedastic variance is modeled using a proportional error term

• Another option is to use the additive error model assuming a variance function R(q) where q is an m x 1 vector which can incorporate b, D and other parameters, e.g. R(q)=s2[f(b)]2, =[ , q s bT]T

22

Error Models used for PK modeling

ijiijtf ,

ijiij etf 1,

ijijiij etf 1,

Additive error

Proportional error

Additive and Proportional error

ijetf iij, Exponential error

23

For the 1-compartment model parameterized with Cl, V, ka

24

logV

logCl

logk

,a

10 VkCl

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

Input

Elimination

Central

Vc

k10

ka

Maximum Likelihood Approach Is Used

25

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 yi as a vector of all responses for the ith subject

The problem is that we don’t have this probability density function for y directly.

N

iiyp

1

26

• We use the following:

ii

N

iii

N

ii bdbbypyp

11

|

Where p and p are normal probability density functions. Maximization is in =[bT, vech(D), vech(R)] T

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

Under Normal Assumptions

i

iTii

iiin

iii

iii

i

iiiT

iiii

bBB

bDbBb

dbtf

dbtf

dbtf

f

fyRfyBbyp

i

w.r.t. constantand

21

12

2

1

11

2

1exp

,,

,,

,,

2

1exp|

27

Maximum Likelihood

Given data yij, we use maximum likelihood to obtain parameters estimates for b, D, and s2.

Because the mean function, f, is assumed to be nonlinear in bi in pharmacokinetics, least squares does not result in equivalent parameter estimates.

28

Approximate Methods• Use numerical approaches to approximate the

integral and then maximize the approximation• Options:

– Approximate the integrand by something we can integrate

• First Order method (Taylor series)

– Approximate the whole integral• Laplace’s approximation (second order approximation)• Importance Sampling• Gaussian Quadrature

– Use Bayesian methodology29

Algorithms UsedFirst Order

First Order Conditional Estimation

Laplace’s Approximation

Importance Sampling

Gaussian Quadrature

Bayesian (Gibb’s Sampler; Not covered in this presentation)

30

ii

M

iii

bdbbyp 1

|

Approximate just the integrand

Or approximate whole integral

Available in NONMEM

(NONMEM is the gold standard software package for PKPD modeling.)

First Order Method

31

Approximate with a first order Taylor series expansion

If the model assumes

And Ri = s2I, then this is pretty straight-forward.

You use a Taylor series expansion about bi.

iiijiiij dbtfdtf ,,,,

Taylor Series Expansion

32

With a first order Taylor series approximation expanded about b, the mean of the bi

Let this approximation be

You use this approximation in the integrand.

ixiijiij

iiij

bdxtfxdtf

dbtf

|,, /,,

,,

iiijTay dbtf ,,,

Substituting back in and simplifying …

i

iTii

iiinTay

iiiTay

iiiTay

i

Tay

i

Tay

ii

T

i

Tay

iii

bBB

bDbBb

dbtf

dbtf

dbtf

f

fyRfyBbyp

i

.r.t.constant w and

2

1exp

,,,

,,,

,,,

2

1exp|

21

12

2

1

11

33

And now the exponent term is integrable and now we can maximize the likelihood.

See slide 26.

34

A second order approximation can be constructed by using Laplace’s approximation

iii

bnii

k

iiiii

bb

ebn

dbbn iii

ˆ

ˆ2exp

ˆ2/12/

maximizes

where

Using Laplace’s Approximation

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

iiii

ii bbypn

b |log1

Numerical Integration: Importance Sampling

35

Consider a function g(b), then

dbbbgbgE )()())((

To get a numerical solution to the integral simply use a random number generator to sample many b from the distribution (b) and change the expectation to a sample mean.

iiT

iiii

hihiNN

ii

iiii

bfyRbfybfy

bfyN

bypE

dbbbypyp

1

2/

where

~

2

1exp

2

11

|

|

Where h is the index for the sampling from p(bi).

iih bb ~

sampled the areand

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.

37

38

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 bi.

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).

Review of Approximate 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.

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.

40

Software

R – PKFIT packageNONMEM (industry standard, 1979, FORTRAN)Monolix (Bayesian)WinBugs (PKBugs)Phoenix (windows program incorporating methods from NONMEM)SAS and R can be used to fit very simple PK models but, in general, not very useful

41

Why is NONMEM the gold standard?

Software needs easy input of PK models.

Not many software packages allow for models written in terms of ODEs instead of closed form solution.

More challenging for multiple dose settings.

Functional form dependent on data.

42

Multiple Dose ModelDaily Dose with Fast Elimination

43

Multiple Dose ModelDaily Dose with Slower Elimination

44

Super-position principle

Super-position Principle

45

Assume dosing every 24 hoursAssume concentration for single dose isThen concentration, C(t) is

),,( iiij dtf

7248),,48(),,24(),,(

4824),,24(),,(

24),,(

),,(tdtfdtfdtf

tdtfdtf

tdtf

dtCiiiiii

iiii

ii

ii

Multiple Dose Model, Missed Third Dose

46

Dose Delayed by 3 Hours Every Other Day

47

Modeling Covariates

48

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).

“Population Pharmacokinetics” refers specifically to these mixed effects models with covariates included in the secondary, hierarchical structure

Nonlinear Mixed Effects Model

ii

i

iiiji

ijiiijij

RN

BNb

baxg

dtfy

,0~

,0~

),(

),,(

49

With secondary structure for covariates:

Part 1 Summary

• PK Basic Concepts (Cmax, Tmax, AUC …)• PK Compartmental Models (derived mean function

from differential equations)• Technical Considerations (approximate maximum

likelihood approach for fitting nonlinear mixed effects model)

• Software Considerations (many complications!)

50