WHAT’S IN BETWEEN DOSE Outline AND RESPONSE?davidian/pkpd.pdf · 2006-05-31 · † Characterizing dose-response relationship in the population is not informative enough † One

WHAT’S IN BETWEEN DOSEAND RESPONSE?

Pharmacokinetics, Pharmacodynamics, and

Statistics

Marie Davidian

Department of Statistics

North Carolina State University

http://www.stat.ncsu.edu/∼davidian

Greenberg Lecture I: PK, PD, and Statistics 1

Outline

1. Introduction

2. What is pharmacokinetics?

3. What is pharmacodynamics ?

4. Population PK/PD and statistics – the history

5. An example

6. PK/PD today

7. Concluding remarks

Warning: There are very few equations in this talk!


1. Introduction

What do we want in a drug?

• Safety

• Efficacy

Can people take it, and does it work?

The usual paradigm: Look at “what goes in ” and “what comes out,”

often by asking

• If we were to administer this drug at some dose to a population of

interest, what would the mean response be?

• . . . And how does it compare to that for other drugs or other doses

of this drug?


1. Introduction

Key message, part I: Understanding what goes on between dose

(administration) and response can yield insight on

• How best to choose doses at which to evaluate a drug

• How best to use a drug in a population

• How best to use a drug to treat individual patients or

subpopulations of patients

• . . . And a lot more

Key concepts:

• Pharmacokinetics (PK) – “what the body does to the drug”

• Pharmacodynamics (PD) – “what the drug does to the body”


1. Introduction

PK -

concentration

PD- -dose response

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1. Introduction

Key message, part II: Understanding what goes on between dose

(administration) and response for both individuals and the population

• Relies critically on combining physiological (mathematical) modeling

with STATISTICAL MODELING

• “Population PK/PD ”

• Statistical modeling is a integral part of the science

Key message, part III: Combining mathematical and statistical

modeling is becoming more generally recognized as a critical tool in the

study of treatment of disease


2. What is Pharmacokinetics?

“What the body does to the drug”

Goals of drug therapy: From a pharmacologist’s point of view, for an

individual patient or type of patient

• Achieve therapeutic objective (cure disease, mitigate symptoms,etc.)

• Minimize toxicity

• Minimize difficulty of administration

• “Optimize ” dose regimen to address these issues



Implementation of drug therapy: To achieve this, must determine

• How much ? How often ?

• To whom ? Different for different patients ? ages? genders?

• Under what conditions (or not)?

Information on this: Pharmacokinetics

• Study of how the drug moves through the body and the processes

that govern this movement



What goes on inside: ADME

Routes of drug administration: Intravenously, Intramuscularly,

Subcutaneously, Orally, . . .



Basic assumptions and principles:

• There is a “site of action ” where drug will have its effect

• Magnitudes of response, toxicity are functions of drug concentration

at the site of action

• Drug cannot be placed directly at site of action, must move there

• Concentrations at site of action are determined by how drug is

absorbed, distributed to tissues/organs, metabolized, excreted

(eliminated) (how it moves over time)

• Concentrations must be kept high enough to produce response, low

enough to avoid toxicity =⇒ Therapeutic window

• Cannot measure concentration at site of action directly, but can

measure in blood/plasma/serum; reflect those at site



Result:

• ADME dictates concentration at site of action, but can not be

observed directly

• Plasma concentrations have information about ADME =⇒ monitor

concentration over time

• Understanding ADME allows manipulation of concentrations



Data for 4 subjects given same oral dose of anti-asthmatic

theophylline:

The

ophy

lline

con

c. (

mg/

L)

0 5 10 15 20 25

02

46

810

12

Subject 1

0 5 10 15 20 25

02

46

810

12

Subject 6

Time (hr)

The

ophy

lline

con

c. (

mg/

L)

0 5 10 15 20 25

02

46

810

12

Subject 10

Time (hr)

0 5 10 15 20 25

02

46

810

12

Subject 12



Time (hr)

Co

nce

ntr

atio

n (

mg

/L)

Absorption Elimination



Time (hr)

Co

nce

ntr

atio

n (

mg

/L)

Absorption Elimination

Therapeutic Window

Duration of Effect



Multiple dosing: Ordinarily, sustaining doses are given to replace drug

eliminated, maintain concentrations in therapeutic window over time

• Steady state : amount lost = amount gained

Frequency, amount for multiple-dose regimen governed by:

• ADME

• Width of therapeutic window



Principle of superposition:



Effect of different frequency: Same dose and ADME characteristics



Effect of different elimination characteristics: Same dose and

frequency



Need a way to deduce ADME from plasma concentrations. . .

Compartmental modeling: Represent the body by a system of

compartments depending on ADME processes

• Can be grossly simplistic, but often sufficient approximation

• Compartments may or may not have physical meaning



One-compartment model with first-order absorption, elimination:

oral dose D X(t) --

keka

dX(t)

dt= kaXa(t) − keX(t), X(0) = 0

dXa(t)

dt= −kaXa(t), Xa(0) = FD

F = bioavailability, Xa(t) = amount at absorption site

C(t) =X(t)

V=

kaDF

V (ka − ke){exp(−ket) − exp(−kat)}, ke = Cl/V

V = “volume ” of compartment, Cl = clearance



Two-compartment model, IV bolus injection: Dose D

(instantaneous)

X(t)

-k12

¾k21

Xtis(t)D :

?ke

dX(t)

dt= k21Xtis(t) − k12X(t) − keX(t), X(0) = D

dXtis(t)

dt= k12X(t) − k21Xtis(t), Xtis(0) = 0

C(t) = A1 exp(−λ1t) + A2 exp(−λ2t)



Extensions:

• More compartments (e.g. peripheral tissues), nonlinear kinetics

(saturation at high concentrations)

• Physiologically-Based Pharmacokinetic (PBPK) models

Result: Deterministic model for time-concentration within a subject

• Based on (albeit simplified) “physiological ” considerations

• Depends on PK parameters characterizing ADME processes

for that subject

• Multiple doses : Apply superposition principle, e.g.

C(t) =∑

d:td<t

Dd

Vexp

{Cl

V(t − td)

}



Only half the battle!

• What is a “good ” concentration?

• What is the “therapeutic window ?” Is it the same for everyone ?

Further complicating matters: Recall theophylline

• Identical dose =⇒ substantial variation in drug concentrations

among people. . .

• . . . due to substantial variation in ADME among people =⇒ each

subject may have same model but with different PK parameters


3. What is Pharmacodynamics?

“What the drug does to the body”

Idea:

• Characterizing dose-response relationship in the population is

not informative enough

• One reason: inter-subject variation in PK

• I.e., Inter-subject variation in concentrations for same dose

=⇒ inter-subject variation in response for same dose

• Understanding concentration-response for individuals provides more

precise information for deciding how to dose

Pharmacodynamics: Relationship of response (drug effect) to drug

concentration



Furthermore: Inter-subject variation in concentration due to different

PK is only part of the reason subjects vary in their responses

• Response varies across subjects who achieve the same concentrations

=⇒ Study concentration-response within subjects and how it varies

across subjects

• Understanding inter-subject variation in concentrations and

responses gives insight on width and placement of therapeutic

window and how it varies across subjects



PD models: Model concentration-response within a given subject

• Empirical rather than physiological in basis

• E.g. the so-called “Emax model ” for continuous response

R = E0 +Emax − E0

1 + EC50/C, C = concentration

• Each subject has his/her own PD parameters

• Ideally : Concentration at site of action

• Realistically : Concentration in plasma

Complications:

• Choice of R, measurement error in C

• Time lag : difference between concentration in blood and at site



Pharmacokinetics: Learn about PK parameters in a suitable

compartment model

• For individual subjects and how they vary in the population

• . . . In order to understand how to dose individual subjects and

develop guidelines for dosing certain types of subjects (e.g. elderly)

• . . . To achieve desired concentrations

Pharmacodynamics: Completing the story

• Learn about concentrations eliciting desired responses and

inter-subject variation in how this happens. . .

• . . . In order to gain understanding of the width and variation (among

subjects) of the therapeutic window

• . . . And use this knowledge to refine dosing strategies



Ultimate objectives:

• Improve drug development process

• Inform better drug use in routine clinical care

PK variation -concentration

PD variation- -dose response

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AAK


4. Population PK/PD and Statistics

The story begins: In the early 1970s with. . .



Lewis B. Sheiner, M.D.



Traditional PK studies: Often in Phase I, II

• Get basic information, e.g., average concentrations achieved, insight

into toxicities

• Healthy volunteers, different from patient population, homogeneous

• Small number of subjects

• Lots of blood samples from each following single, multiple doses

• Might randomize according to a single factor, e.g. fed vs. fasting

state, evaluate effect on PK parameters

Lewis: These can provide

• Good info on appropriate compartment model. . .

• Some info on PK parameters and how they vary, but not much



Lewis: Can learn a lot more – “Population ” studies

• Study PK in target population of heterogeneous patients undergoing

chronic dosing as part of routine clinical care

• Large number of subjects

• Sparse, haphazard sampling of each subject

• Lots of demographic, physiological, behavioral characteristics

recorded for each subject, e.g. weight, age, renal function, race,

ethnicity, disease status, smoking, . . .

Population PK: Learn about variation of PK parameters in population

• Associated with subject characteristics (and their interactions)

• Unexplained by these (“inherent variation ?” unmeasured factors ?)



How to do this? Statistical modeling !

Data: Repeated concentration measurements on each of m subjects

from the population of interest

Yij plasma concentration at time tij , j = 1, . . . , ni

Y i (Yi1, . . . , Yini)T

ui dosing history for subject (conditions of measurement)

ai subject characteristics (covariates)

(Y i, ui, ai) independent across i = 1, . . . , m

Perspective:

• Focus not on population mean concentration, but on population of

individual PK parameters in the PK mathematical model

• Need to embed the PK mathematical model in a statistical model. . .



Statistical model: Sheiner, Rosenberg, and Melmon (1972); Sheiner,

Rosenberg, and Marathe (1977)

• What is now known as a nonlinear mixed effects (hierarchical ) model

Stage 1: Intra-subject model

• Assumption : Observed concentrations equal deterministic

mathematical PK model plus deviation due to assay error,

“realization variance ” (and model misspecification)

• The PK model and superposition principle give an expression for

concentration at time t under dosing history u

f(t, u, β), β = PK parameters (p × 1)

• E.g., β = (ka, Cl, V )T in the one compartment model



Subject i: Subject-specific PK parameters , e.g., βi = (kai, Cli, Vi)T

0 5 10 15 20

02

46

810

12

time

conc

entr

atio

n

Yi(t) = f(t, ui, βi) + ei(t), Yij = Yi(tij)



Stage 1: Intra-subject model

• Result:Yij = f(tij , ui, βi) + eij

• Possible intra-subject correlation (usually assumed negligible )

• Intra-subject variance about f often small (CV ≈ 10-30%), not

constant , dominated by assay error

• Standard : Yij |ui, βi ∼ normal or lognormal with

E(Yij |ui, βi) = f(tij , ui, βi), var(Yij |ui, βi) = σ2f2(tij , ui, βi)

• Compactly : Y i = f i(ui, βi) + ei with

E(Y i|ui, βi) = f i(ui, βi), var(Y i|ui, βi) = Ri(ui, βi, ξ)

Y i = f i(ui, βi) + R1/2

i (ui, βi, ξ)εi︸︷︷︸

ei



Stage 2: Inter-subject population model

βi = d(ai, θ, bi) (p × 1)

• bi (k × 1) random effects ∼ H, mean 0

• Standard assumption: H is k-variate Nk(0, D)

• E.g. Different parameterizations

Cli = θ1 + θT2ai + bCl,i, Vi = θ3 + θT

4ai + bV,i

log Cli = θ1 + θT2ai + bCl,i, log Vi = θ3 + θT

4ai + bV,i

• Often βi = Aiθ + Bibi

• Moderate inter-subject variation in PK parameters (CV ≈ 30–70%)



Together: Two-stage hierarchy

• Intra-subject model (Stage 1 ): Substitute for βi

E(Y i|ui, ai, bi) = f i{ui, d(ai, θ, bi)}, var(Y i|ui, ai, bi) = Ri{ui, d(ai, θ, bi), ξ}

Y i = f i{ui, d(ai, θ, bi)} + R1/2

i {d(ai, θ, bi), ξ}εi

=⇒ (Y i|ui, ai, bi) has density py|b(yi|ui, ai, bi; θ, ξ)

• Inter-subject population model (Stage 2 ):

βi = d(ai, θ, bi), bi ∼ H, E(bi) = 0

Subject-matter and statistical principles combined in one

framework:

• Stage 1 : physiological + empirical statistical modeling

• Stage 2 : empirical statistical modeling



Objectives of analysis:

Determine d Relationship between PK, covariates

Estimate θ Relationship between PK, covariates

Estimate H “Unexplained” variation in population

“Estimate” βi Characterize individuals =⇒ individualized dosing

Likelihood (conditional on covariates ui, ai): Maximize

m∏

i=1

py(yi|ui, ai) =

m∏

i=1

ì(θ, ξ, H; yi) =

m∏

i=1

∫

py|b(yi|ui, ai, bi; θ, ξ) dH(bi)

• Complex dosing histories =⇒ complex PK model f

• Intractable integral in general (nonlinear in bi)



Lewis & Co.: “First-Order method ” (Beal and Sheiner, 1982)

• Assume py|b is normal, H is Nk(0, D), approximate about bi = 0:

Y i = f i{ui, d(ai, θ, bi)} + R1/2

i {ui, d(ai, θ, bi), ξ}εi

≈ f i{ui, d(ai, θ,0)} + Zi(ui, θ,0)bi + R1/2

i {ui, d(ai, θ,0)}εi

• Approximate ì by ni-variate normal with

E(Y i|ui, ai) ≈ f i{ui, d(ai, θ,0)},

var(Y i|ui, ai) ≈ Ri{ui, d(ai, θ,0)} + Zi(ui, θ,0)DZTi (ui, θ,0)

• Implemented in the FORTRAN program NONMEM (FO method)

(University of California, San Francisco)

• Obvious bias (but worked pretty well in simulations)

• Generated huge excitement in PK community



Meanwhile: Statisticians were just beginning to pay attention. . .



Stumpy Giltinan (R.I.P.) Ed Vonesh, Ph.D.

Mary Lindstrom, Ph.D. Doug Bates, Ph.D.



Main catalyst for statistical research in nonlinear mixed models. . .

Better approximations to the integral:

• Assume H is Nk(0, D), py|b normal with Ri(ui, βi, ξ) = Ri(ui, ξ)

• Use Laplace’s approximation or a Taylor series

• =⇒ ì ≈ ni-variate normal with

E(Y i|ui, ai) ≈ f i{ui, d(ai, θ, bi)} − Zi{ui, θ, bi}bi

var(Y i|ui, ai) ≈ Zi(ui, θ, bi)DZTi (ui, θ, bi) + Ri(ui, ξ)

• bi = “empirical Bayes estimate ” of bi maximizing pb|y(bi|ui, ai, yi)



Remarks:

• Lindstrom and Bates (1990), Wolfinger (1993), Vonesh (1996), . . .

• Implementation : Iterate between updating bi and fitting

approximate model

• Approximation works remarkably well for sparse (small ni)

population PK data as long as intra-subject variation is “small ”

• Variations : R/Splus nlme(), SAS %nlinmix,

• NONMEM (FOCE method), big advantage – PK models built-in !



This motivated lots more. . .

Computational work: Why not just “do ” the integral? Deterministic

and stochastic numerical integration, e.g.

• Variants of quadrature

• Importance sampling

• Monte Carlo EM

• Pinheiro and Bates (1995), Walker (1996)

• Implementation : SAS proc nlmixed



Model refinements: Assumption on H – why should bi be normal ?

• Rather than assume a parametric form, estimate the distribution of

βi directly nonparametrically (Mallet, 1986, and others)

• USC*PACK-NPEM (University of Southern California)

• Or assume H has a “nice” density and estimate it (Davidian and

Gallant, 1993)

• FORTRAN nlmix (user-unfriendly)

• Inspect estimates to identify subpopulations , omitted covariates



This was also a natural area for Bayesians. . .



Jon Wakefield, Ph.D. Gary Rosner, Sc.D.

Peter Muller, Ph.D. Joe Ibrahim, Ph.D.



Bayesian view: Add

• Stage 3 : Hyperprior (β, ξ, D) ∼ pβ,ξ,D(β, ξ, D)

Implementation: To do the intractable integration us MCMC

techniques

• An early showcase for these methods

• Wakefield (1996), Muller and Rosner (1997)

• Gelman, Bois, Jiang (1996), Mezzetti, Ibrahim, et al. (2003)

• Parametric (normality) or more flexible models for bi, βi

• PKBugs, a WinBUGS interface with built-in PK models

• MCSim, for systems of differential equations (PBPK models)



What about pharmacodynamics? PK is only part of the full story

• Population PK/PD study : Collect PK/PD data on same subjects

• PD responses Rij at times t∗ij (categorical, continuous, “surrogate”)

• Intra-subject PD model : True plasma concentration cij at t∗ij

Rij = g(cij , αi) + e∗ij e.g. g(c, αi) = E0i +Emaxi − E0i

1 + EC50/c

Joint PK/PD model: Describe cij by PK model

• Intra-subject joint PK/PD model

Yij = f(tij , ui, βi) + eij , Rij = g{f(t∗ij , ui, βi), αi} + e∗ij

• βi = d(ai, θ, bi), αi = d∗(ai, γ, b∗i ), (bT

i , b∗Ti ) ∼ H

• Often : Incorporate a lag between PK and PD in the joint model



Extensions and by-products:

• Individual estimation : Use posterior modes to “estimate”

βi (and αi)

• “Bayesian dosage adjustment :” Use these for current or future

subjects given a few observations =⇒ Individual dosing regimen

• Inter-occasion variation – parameters may vary within the same

individual, implications for dosing

• Non/semiparametric population models

• Censored concentrations/response

• Missing/mismeasured covariates

• Etc.



Summary: By the late-1990s

• Lots of statistical research

• Nonlinear mixed effects models became standard tools

• Exploited, enhanced by PK/PD community =⇒ specialized software

implementing one or more of these methods (and built-in catalogs of

PK/PD models)

• NONMEM (UCSF, now GloboMax)

• ADAPT II (University of Southern California)

• WinNonMix (Pharsight Corporation)

• Etc.


5. Example

World-famous example: Population PK of phenobarbital

• m = 59 pre-term infants treated for seizures

• ni = 1 to 6 concentration measurements per subject, total of 155

measurements

• Birth weight wi and 5-minute Apgar score δi = I[Apgar < 5]

• IV multiple doses; one-compartment model

f(tij , ui, βi) =∑

d:tid<t

Did

Viexp

{

−CliVi

(t − tid)

}

Objectives: Characterize PK and its variation (typical Cli, Vi? do

covariates matter? extent of biological variation?)


5. Example

Dosing history and concentrations for one subject:

Time (hours)

Phe

noba

rbita

l con

c. (

mcg

/ml)

0 50 100 150 200 250 300

020

4060


5. Example

Inter-subject models: βi = (Cli, Vi)T

• Without covariates

log Cli = θ1 + b1i, log Vi = θ2 + b2i

• Final model with covariates

log Cli = θ1 + θ3wi + b1i, log Vi = θ2 + θ4wi + θ5δi + b2i


5. Example

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5. Example

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5. Example

Density estimates:

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6h

(a)

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(d)


6. PK/PD Today

Population PK/PD analysis:

• Is an important component of the drug development process

• Recognized benefit: Identifying differences in drug safety and

efficacy among population subgroups that can be addressed by dose

modification. . .

• . . . particularly when intended population is quite heterogeneous and

typical therapeutic window is narrow

• Is an important component of the regulatory process. . .


6. PK/PD Today

Guidance for IndustryPopulation Pharmacokinetics

U.S. Department of Health and Human ServicesFood and Drug Administration

Center for Drug Evaluation and Research (CDER)Center for Biologics Evaluation and Research (CBER)

February 1999CP 1


6. PK/PD Today

Current interest:

• Clinical Pharmacology Subcommittee of the FDA Advisory

Committee for Pharmaceutical Science

• A population PK/PD guidance

• Incorporation of genetic information

• Special design/model considerations for pediatric populations


6. PK/PD Today

Clinical trial simulation: Use PK, PD info to target, design trials

• Pharsight Corporation (and others)

Ingredients: Based on prior PK/PD investigation

• Covariate distribution model : A model for the target population

• PK model : Hierarchical model incorporating covariates impacting

PK =⇒ concentrations

• PD model : Hierarchical model incorporating concentrations,

covariates impacting PD =⇒ responses (also placebo model)

• Hazard model : Relating responses to a clinical endpoint

Simulation: Generate samples of patients under different designs

(numbers, inclusion criteria, dose regimens, etc)

• “End-of-Phase-2a Meetings ”


7. Concluding Remarks

Population PK/PD analysis:

• A statistical success story

• Statistical modeling is central to the subject-matter science

A model for other biomedical research. . .



Currently: Great interest in combining mathematical and statistical

modeling to address other questions, e.g., treatment of HIV infection

• Potent antiretroviral drugs cannot be taken continually

• What is the best strategy for treatment ?

• A promising tool: Within-subject HIV dynamical systems models

• Describe the interplay between virus and immune system over time,

incorporates effects of treatment

• Can these models be used to develop dynamic treatment regimes for

HIV infection?

• Tomorrow afternoon !





0 200 400 600 800 1000 1200 1400 1600

100

101

102

103

104

105

time (days)

viru

s co

pies

/ml

Model Fits to the Clinical Data

data

model fit

censored data



Where to get a copy of these slides:

http://www.stat.ncsu.edu/∼davidian

Where to find a great intro course on PK on the web:

http://www.boomer.org/c/p1/

Thanks to David Bourne at University of Oklahoma for some of the

pictures in this talk!

Some books about PK/PD:

Rowland, M. and Tozer, T.N., Clinical Pharmaockinetics: Concepts

and Applications (nth edition)

Gibaldi, M. and Perrier, D., Pharmacokinetics (2nd edition)

Journal with lots of statistical content: Journal of

Pharmacokinetics and Pharmacodynamics (formerly Journal of

Pharmacokinetics and Biopharmaceutics)


Dedication

This talk is dedicated to the memory of

Lewis B. Sheiner, M.D.

1940–2004


Documents

WHAT’S IN BETWEEN DOSE Outline AND RESPONSE?davidian/pkpd.pdf · 2006-05-31 · † Characterizing dose-response relationship in the population is not informative enough † One