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)