Bayesian Population
Pharmacokinetic/Pharmacodynamic
Modeling
Steven Kathman
GlaxoSmithKline
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?
R
k12
A2 = C2V2
A1 = C1V1
k21
k10
dA1/dt = R + k21A2 – k12A1 – k10A1
dA2/dt = k12A1 – k21A2
dose
 1t
 2 t
C (t ) 
( Ae  (1  A)e )
V1
1  1 / 2[k10  k12  k 21  {( k10  k12  k 21 ) 2  4k10k 21}1/ 2 ]
2  k10  k12  k21  1
1  k 21
A
1  2
CL = k10V1
Q = k12V1 = k21V2
Infusion
1T
2T

k0 1  e
1 e
1t
2t 
C (t )  
Ae 
(1  A)e 
V1   1
 2

k0 = zero order infusion rate
T=t during infusion, constant time infusion was stopped after infusion.
PK Model
Concij ~ N (ij , ij )
ij  C (t , i  [ln( CLi ), ln( Qi ), ln( V1i ), ln( V2i )])
i ~ MVN (,  )
1  1
2  2  6 ( BSAi  1.95)
3  3  4 ( BSAi  1.95)
 4  5
PK Model
µ~Vague MVN prior
 ~ Wish ( R,4)
R chosen based on CV=30%
If that was painful…
In mathematics you don't understand things. You
just get used to them.
Johann von Neumann (1903 - 1957)
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)
Actual Concentrations
1500
1000
500
0
50
300
550
800
1050
1300
1550
Predicted Concentrations from 2-comp model
1800
R
k12
A2=C2V2
k13
A1=C1V1
k31
k21
k10
dA1/dt = R + k21A2 + k31A3 – k12 A1 – k13A1 – k10 A1
dA2/dt = k12A1 – k21A2
dA3/dt = k13A1 – k31A3
A3=C3V3
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)
Model of Myelosuppression
Prol
ktr
Transit 1
ktr
Transit 2
ktr
kprol = ktr
EDrug = βConc
Transit 3
ktr
Circ
kcirc = ktr
 Circ 0 
Feedback  

 Circ 

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)
Vague prior.
Fairly informative
priors (Literature).
Actual ANC vs Model Fit (Posterior Mean)
Observed ANC
15
10
5
0
0.5
3.0
5.5
8.0
10.5
13.0
15.5
ANC predicted from Model (Posterior Mean)
18.0
Subject 14
6
ANC
4
2
0
0
100
200
300
Time
400
500
Subject 16
ANC
6
4
2
0
0
100
200
300
Time
400
500
Subject 18
8
ANC
6
4
2
0
0
100
200
300
Time
400
500
Subject 24
5
4
ANC
3
2
1
0
0
100
200
300
Time
400
500
600
Subject 118
5
4
ANC
3
2
1
0
0
100
200
300
Time
400
500
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.
ANC for Weekly Schedule - 7mg/m 2
median
25th and 75th percentile
6
ANC
4
2
0
0
100
200
300
400
Time
500
600
700
800
ANC (10 9/L)
11
7
3
-1
0
5
10
15
Time (Days)
20
25
30
35
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)
Subject 24
100
80
60
40
20
0
10
20
weeks
30
40
Subject 174
100
80
60
40
20
0
10
20
weeks
30
40
Subject 421
90
85
80
75
70
65
0
10
20
30
weeks
40
50
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)