Bayesian Population Pharmacokinetic/Pharmacodynamic

ModelingSteven KathmanGlaxoSmithKline

Half of the modern drugs could well be thrown out of the window, except that the birds might eat them. Dr. Martin Henry Fischer

Outline

• Introduction• Population PK modeling• Population PK/PD modeling

– Modeling the time course of ANC• Other examples• Conclusions

Introduction

• KSP inhibitor (Ispinesib) being developed for the treatment of cancer.

• Blocks assembly of a functional mitotic spindle and leads to G2/M arrest.

• Causes cell cycle arrest in mitosis and subsequent cell death.

• Leads to a transient reduction in absolute neutrophil counts (ANC).

Introduction

• KSP10001 was the FTIH study.• Ispinesib dosed once every three weeks.• PK data collected after first dose.• ANC assessed on Days 1 (pre-dose), 8,

15, and 22 (C2D1 pre-dose). More frequent assessments done if ANC < 0.75 (109/L).

• Prolonged Grade 4 neutropenia (> 5 days) most common DLT.

Objectives

• Determine a suitable PK model. - Examine 2 vs 3 compartment models.

• Determine a suitable model for PD endpoint (i.e., time course of absolute neutrophil counts).

- Using Nonlinear mixed models. - Using Bayesian methods.

Pharmacokinetics

The action of drugs in the body over a period of time, including the processes of absorption, distribution, localisation in tissues, biotransformation and excretion.

Simple terms – what happens to the drug after it enters the body.

What is the body doing to the drug over time?

A2 = C2V2 A1 = C1V1

R

k12

k21

k10

dA1/dt = R + k21A2 – k12A1 – k10A1

dA2/dt = k12A1 – k21A2

))1(()( 21

1

tt eAAeVdosetC

]}4){([2/1 2/12110

22112102112101 kkkkkkkk

12112102 kkk

21

211

kA

CL = k10V1

Q = k12V1 = k21V2

tT

tT

eAeAeeVktC 2

21

1

)1(11)(211

0

Infusion

k0 = zero order infusion rate

T=t during infusion, constant time infusion was stopped after infusion.

),(~ ijijij NConc

)])ln(),ln(),ln(),[ln(,( 21 iiiiiij VVQCLtC

),(~ MVNi

11

)95.1(622 iBSA

)95.1(433 iBSA

54

PK Model

µ~Vague MVN prior

)4,(~ RWish

R chosen based on CV=30%

PK Model

In mathematics you don't understand things. You just get used to them. Johann von Neumann (1903 - 1957)

If that was painful…

Bayesian Results• Typical Bayesian analysis (via MCMC) involves

estimation of the joint posterior distribution of all unobserved stochastic quantities conditional on observed data.

• Generating random samples from the joint posterior distribution of the parameters.

• Marginal distribution of each parameter is completely characterized (numerical integration).

P(individual specific PK parameters, population PK parameters | PK data)

50 300 550 800 1050 1300 1550 1800

0

500

1000

1500

Predicted Concentrations from 2-comp model

Act

ual C

once

ntra

tions

R

k10

k12

k21

k13

k31

A1=C1V1A2=C2V2 A3=C3V3

dA1/dt = R + k21A2 + k31A3 – k12 A1 – k13A1 – k10 A1

dA2/dt = k12A1 – k21A2

dA3/dt = k13A1 – k31A3

Pharmacodynamics

The study of the biochemical and physiological effects of drugs and the mechanisms of their actions, including the correlation of actions and effects of drugs with their chemical structure, also, such effects on the actions of a particular drug or drugs.

What is the drug doing to the body?

Modeling the Time Course: Absolute Neutrophil Counts

When you are curious, you find lots of interesting things to do.

The way to get started is to quit talking and begin doing.

– Walt Disney (1901-1966)

Prol CircTransit 1 Transit 2 Transit 3ktr ktr ktr ktr

kcirc = ktrkprol = ktr

EDrug = βConc

CircCircFeedback 0

Model of Myelosuppression

Features of Model

• Proliferating compartment – sensitive to drug.

• Three transit compartments – represent maturation.

• Compartment of circulating blood cells.• System parameters: MTT, baseline, and

feedback.• Drug specific parameter: Slope.

Feedback

• Account for rebound phase (overshoot).• Negative feedback from circulating cells to

proliferative cells.• G-CSF levels increase when circulating

neutrophil counts are low.• G-CSF stimulates proliferation in bone

marrow.

Model of Myelosuppression• dProl/dt = kprol*Prol*(1-EDrug)*(Circ0/Circ)-ktr*Prol

• dTransit1/dt = ktr*Prol-ktr*Transit1

• dTransit2/dt = ktr*Transit1-ktr*Transit2

• dTransit3/dt = ktr*Transit2-ktr*Transit3

• dCirc/dt = ktr*Transit3-kcirc*Circ

ANCij~t(Meanij(MTTi, Circ0(i),, βi; Concij), ij, 4)

Mean = Solution of the differential equation (Circ)

MTTi = 4/(ktr(i)) = Mean transit time.

ln(MTTi)~N(MTT, MTT)ln(Circ0(i))~N(circ, circ)ln(βi)~N(β, β)

Fairly informative priors (Literature).

Vague prior.

0.5 3.0 5.5 8.0 10.5 13.0 15.5 18.0

0

5

10

15

ANC predicted from Model (Posterior Mean)

Obs

erve

d A

NC

Actual ANC vs Model Fit (Posterior Mean)

0 100 200 300 400 500Time

0

2

4

6

AN

CSubject 14

0 100 200 300 400 500Time

0

2

4

6

AN

CSubject 16

0 100 200 300 400 500Time

0

2

4

6

8

AN

CSubject 18

0 100 200 300 400 500 600Time

0

1

2

3

4

5

AN

CSubject 24

0 100 200 300 400 500Time

0

1

2

3

4

5

AN

CSubject 118

Simulate New Schedule

• Using mechanistic/semi-physiological models allows for simulation of new schedules.

• Simulate dosing on days 1, 8, and 15 repeated every 28 days.

• PK/PD model accurately predicted the observed severity and duration of neutropenia.

0 100 200 300 400 500 600 700 800

0

2

4

6

Time

AN

CANC for Weekly Schedule - 7mg/m2

median25th and 75th percentile

0 5 10 15 20 25 30 35

-1

3

7

11

AN

C (1

09 /L)

Time (Days)

Why Bayesian?• Incorporate prior information (MTT and

baseline).• Better integration algorithm (Monte Carlo vs

Taylor Series or Quadrature).• Posterior distribution vs MLE: More informative,

avoids potentially problematic maximization algorithms.

• Better individual estimates: Bayesian vs Empirical Bayesian (which usually fail to account for estimated population parameters?).

Tumor Growth Models

• dC/dt = KL*C(t) – KD*C(t)*D(t)*exp(-t)

where KL = Tumor growth rate

KD = Drug constant kill rate

D(t) = Dose or PK measure = rate constant for resistance

• dC/dt = exp(1t) *C(t) – KD*C(t)*D(t)*exp(-2t)

0 10 20 30 40weeks

20

40

60

80

100

Subject 24

0 10 20 30 40weeks

20

40

60

80

100

Subject 174

0 10 20 30 40 50weeks

65

70

75

80

85

90

Subject 421

Preclinical PK

• Concentrations in plasma.• Concentrations in a tumor.• Relate the two:

– Plasma: two-compartment model.– Tumor: dCT(t)/dt = (KP/VT)AP(t)-KTCT(t)

More PK

• Compound given through iv infusion.• Should be 1-hr infusion.• Reason to believe that the infusion time is

less for some subjects. • Making the infusion times a parameter to

be estimated, with informative priors.

Software

• WinBugs (Pharmaco and WBDiff) - Pharmaco: Built in PK functions. - WBDiff: Differential Equation Solver• NONMEM• SAS macro• R: nlmeODE library and function

Conclusions

• PK/PD modeling often involves interesting and complicated models.

• Models can serve many useful functions in drug development.

• Bayesian methods help with:– Better algorithms– More flexibility– Incorporating outside information

General Remarks

• PK/PD modeling involves different skills coming together (medical, pharmacokinetics, pharmacology, statistics, etc.).

• As a statistician, helps to develop knowledge in areas outside of statistics.

References

Knowledge is of two kinds. We know a subject ourselves, or we know where we can find information on it. Samuel Johnson (1709 - 1784), quoted in Boswell's Life of Johnson

References• Gibaldi, M. and Perrier, D. (1982) Pharmacokinetics.• Friberg, L. et. al. (2002). Model of Chemotherapy-Induced

Myelosuppression with Parameter Consistency Across Drugs. JCO

20:4713-4721. • Friberg, L. et. al. (2003). Mechanistic Models for Myelosuppression.

Investigational New Drugs 21:183-194.• Lunn, D. et. al. (2002). Bayesian Analysis of Population PK/PD Models:

General Concepts and Software. Journal of PK and PD 29:271-307.• PK Bugs User Guide.• Christian, R. and Casella, G. (2005) Monte Carlo Statistical Methods.• Gelman, A. et. al. (2003) Bayesian Data Analysis.• Gabrielson, J. and Weiner, D. (2006) Pharmacokinetic and

Pharmcodynamic Data Analysis: Concepts and Applications

Questions

The outcome of any serious research can only be to make two questions grow where only one grew before. Thorstein Veblen (1857 - 1929)