Transcript Bayesian Concept Introduction

Pharmacometrics Introduction
Yaning Wang, Ph.D.
Team Leader, Pharmacometrics
Office of Clinical Pharmacology
Center for Drug Evaluation and Research
Food and Drug Administration
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Disclaimer: My remarks today do not necessarily reflect the official views of FDA
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Outline
• Issues and opportunities in drug development
• Model-based drug development
• Bayesian statistics and its relationship with
NONMEM
• NONMEM estimation methods
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Issues in Drug Development
• Low efficiency
–
–
–
–
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NME IND = NDA <20% of time
Reported >50% failure rate in Phase 3 (Carl Peck, CDDS)
Decreased NME NDAs despite increased INDs
Cost per NME approved estimated at $1.7B
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Low Success Rate Across Different Diseases
Kola I, Landis J.Can the pharmaceutical industry reduce attrition rates? Nat.Rev.Drug.Disc. Aug 2004.
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High Failure Rate Even in Late Development
Kola I, Landis J.Can the pharmaceutical industry reduce attrition rates? Nat.Rev.Drug.Disc. Aug 2004.
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10-Year Trends of Major Submissions to FDA
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Investment Escalation
Source: Windhover’s in Vivo: The Business & Medicine Report, Bain drug economics model, 2003
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Various Initiatives
• National Institutes of Health (NIH)
– Roadmap initiative
• National Cancer Institute (NCI)
– Specialized Programs of Research Excellence (SPOREs)
• European Organization for the Treatment of Cancer
(EORTC)
– make translational research a part of all cancer clinical trials
• National Translational Cancer Research Network
– facilitate and enhance translational research in the United
Kingdom
• FDA
– Critical Path Initiative
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2004 FDA Critical Path Initiative
Application of New Scientific
Knowledge to Drug Development：
•
•
•
Application of quantitative disease
models to drug development
Pharmacogenomics in drug
development
New imaging technologies may
contribute biomarkers in drug
development
http://www.fda.gov/oc/initiatives/criticalpath/
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2006 FDA Critical Path Update
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Model-Based Drug Development
“The concept of model-based drug development, in
which pharmaco-statistical models of drug efficacy and
safety are developed from preclinical and available
clinical data, offers an important approach to improving
drug development knowledge management and
development decision-making”
Adapted from Lewis B. Sheiner, “Learning vs Confirming in Clinical Drug
Development”, Clin. Pharmacol. Ther., 1997, 61:275-291.
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Terminology
• Model-based drug development
–
–
–
–
–
Pharmacokinetics/Pharmacodynamics (PK/PD)
Modeling and simulation
Exposure-response
Quantitative clinical pharmacology
Quantitative disease and drug models
• Pharmacometrics-Pharmacometricians
• Areas involved: clinical pharmacology, statistics,
pathophysiology, biology, bioengineering
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DRUG MODEL
PLACEBO/DISEASE MODEL
Surrogate
Morbidity#2
Mortality
Surrogate
0 2 4 6 8
6
30
Other races Male
0 10
140
200
TIME
80 120 160
Other races Female
Body Weight
TIME
% Compliance
0
20 40 60
0
80
Black Female
Surrogate
High dose
Low dose
PATIENT/CLINICAL
TRIAL MODEL
80 120 160
% Drop-out
2
4
150
0 50
400
200
0
100
200
Caucasian Female
80 120
180
Black Male
Disease
Drug
Trial
Models
Exposure
Surrogate
Exposure
TIME
100
200
Caucasian Male
Toxicity
Relative
Risk
Morbidity#1
QD
TID
Patient Population
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Applications in FDA Review
• Exposure-response analysis of efficacy and safety data in NDA
review for the choice of dosing regimen (s)
• Dose adjustment in special populations (hepatic, renal, gender, age
and drug interactions) based on intersubject variability and
risk assessment
• Routine use of population PK and PD data analysis to understand
variability and to provide evidence for label claims
• Issuing guidance to assist the industry in using these tools
• Case studies
– Leveraging Prior Quantitative Knowledge to Guide Drug Development Decisions
and Regulatory Science Recommendations: Impact of FDA Pharmacometrics
During 2004-2006, Journal of Clinical Pharmacology (2008) 48: 146-157
– Impact of Pharmacometric Reviews on New Drug Approval and Labeling Decisions
- A Survey of 31 New Drug Applications Submitted Between 2005-2006, Clinical
Pharmacology and Therapeutics (2007) 81: 213-221
– Impact of Pharmacometrics on Drug Approval and Labeling Decisions - A Survey
of 42 New Drug Applications, The AAPS Journal, 7(3): E503-E512, 2005
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Tools for Modeling and Simulation
•
•
•
•
•
•
•
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NONMEM (UCSF, Globomax)
SAS (SAS Institute Inc)
Splus (Insightful Corporation) or R (Free)
WinBUGS (MRC Biostatistics, Free)
ADAPT II (USC, Free)
WinNonLin/WinNonMix (Pharsight)
Trial Simulator (Pharsight)
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Basic Statistics
1.0
1.0
0.8
0.8
0.6
0.6
• Random Variable (Y)
0.4
0.4
– Discrete (gender, pain score)
– Continuous (body weight,
clearance)
0.2
0.2
0.0
0.0
0
1
0
1
0.020
rdn
0.02
0.015
• Distribution (histogram)
0.015
0.010
– Binomial
– Normal
– Lognormal
0.005
0.010
0.005
0.0
0.0
• Probability Function (P(Y=y))
20
40
40
60
60
80
80 100 120 140
100
120
140
160
rdn
0.4
0.5
0.6
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
– Probability Mass Function (PMF)
– Probability Density Function (PDF)
1
1
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2
3
3
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4
rdn
5
5
6
6
7
7
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Normal Distribution

1
P(Y  y) 
e
2 
Y ~ N ( , )
2 2
0.04
2
( y  )2
0.03
0.04
0.02
P(y)
dens[order(rdn)]
0.03
0.02
0.01
Y~N(100, 102)
0.0
0.0
0.01
80
80
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100
100
sort(rdn)
Y
120
120
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Joint, Marginal and Conditional Probabilities
Survival
(Y2=0)
Not survival
(Y2=1)
P(Y1)
Early stage
(Y1=0)
Late stage
(Y1=1)
0.72
0.02
0.74
0.18
0.08
0.26
0.90
0.10
1.00
P(Y2)
Joint probability: P(Y1=0,Y2=0)=0.72
Marginal probability: P(Y1=0)= 0.90
Conditional probability: P(Y2=0|Y1=0)= 0.72/0.90=0.80
P(Y1  0)  P(Y1  0,Y2  0)  P(Y1  0,Y2  1)
Conditiona l 
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P (Y2  0 | Y1  0) 
P (Y1  0, Y2  0)
P (Y1  0)
Joint
Marginal
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Marginal and Conditional Probabilities
Survival
(Y2=0)
Not survival
(Y2=1)
P(Y1)
Early stage
(Y1=0)
Late stage
(Y1=1)
0.72
0.02
0.74
0.18
0.08
0.26
0.90
0.10
1.00
P(Y2)
Conditional probability:
P(Y2  0 | Y1  0)  0.72 / 0.9  0.8, P(Y2  0 | Y1  1)  0.02 / 0.1  0.2
Marginal probability:
P(Y2  0)  P(Y2  0,Y1  0)  P(Y2  0,Y1  1)
 P(Y2  0 | Y1  0)  P(Y1  0)  P(Y2  0 | Y1  1)  P(Y1  1)
 0.8  0.9  0.2  0.1  0.74
For continuous variables: P (Y2 )   P (Y1 , Y2 )dY1   P (Y2 | Y1 )  P (Y1 )dY1
Marginal probability: weighted average of conditional probability
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Bayes’ Theorem
P (Y1 , Y2 )
P (Y2 | Y1 ) 
P (Y1 )
P (Y1 , Y2 )
P (Y1 | Y2 ) 
P (Y2 )
P(Y1 ,Y2 )  P(Y2 | Y1 )  P(Y1 )  P(Y1 | Y2 )  P(Y2 )
P (Y1 | Y2 )  P (Y2 )
P (Y2 | Y1 ) 
P (Y1 )
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Parameter
• Unknown but fixed (Frequentist)
Y ~ N ( ,  )
2
• Unknown and random (Bayesian)
Y ~ N ( ,  )
2
 ~ N (  , )
2
(Prior distribution of )
Hyperparameters
• Unknown hyperparameters
– Hyperprior (Full Bayesian)
– Estimation (Empirical Bayesian)
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Prior, Likelihood and Posterior
P (Y , )  P (Y |  )  P ( )  P ( | Y )  P (Y )
Likelihood
Posterior
Distribution
Marginal
Distribution
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P (Y |  )  P ( )
P ( | Y ) 
P (Y )
Prior
Distribution
P(Y )   P(Y |  )  P( )d
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Prior, Likelihood and Posterior
 ~ N (  , 2 )
Y |  ~ N ( ,  2 )
P (Y |  )  P ( )
P ( | Y ) 
P (Y )
1
1
2
2
1

/
n

 |Y ~ N(
y
,
)
1
1
1
1
1
1
 2
 2
 2
2
2
2
 /n 
 /n 
 /n 
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
Posterior
Likelihood
0.2
pdens
0.3
0.4
Shrinkage
0.1
Prior
n
0.0
2 
-10
0
ˆ | Y
Y
pdens
0.1
0.2
0.3
0.4
0.3
0.2
0.0
0.0
0.1
pdens
20
0.4

10
xp
-10
-10
0
10
0
10
20
20
xp
xp
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A Simple Example
• Unknown parameter (): long-term systolic blood
pressure (SBP) of one particular 60-year-old female
• 4 measurements with a mean Y  130 and a standard
deviation =5
• Survey of the same population (60-year-old female):
mean SBP =120 and standard deviation =10
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Estimation of 
• Frequentist
– Point estimate: ˆ  Y  130
– Interval estimate (95%CI)
Y  1.96

n
 130  1.96 * 2.5
• Bayesian
– Posterior distribution P(|Y)
– Posterior mean ˆ | Y  129.4
– 95% credible interval 129.4  1.96 * 2.4
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0.15
Prior, Likelihood and Posterior
Individual parameter
(posterior)
Population Distribution
(prior)
0.0
0.05
pdens
0.10
Individual data
(likelihood)
80
100
120
140
160
Long-term systolic blood pressure
(SBP) of 60-year-old woman
xp
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Population PK Variability
Sheiner, 1992
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NONMEM and Bayesian
Dose
Yij 
 exp( ki  t ij )   ij
Vi
 ki 
2
2
p( yij |  i ,  ) ~ N ( f (t ij , i ),  )  i   
  k2i
p( i |  ,  ) ~ MVN p (  ,  )   
 k V
 ii
ki  TVk  exp( ki ) Vi  TVV  exp(Vi )
2


 ki
p( i |  ) ~ MVN p (0,  )


  
p( i |  ) ~ MVLN p (  ,  )
 ki Vi
 Vi 
 k 
 kiVi 
 



V2i 
 V 
 
2
ki Vi
Vi

 log(TVk ) 
   


log(TVV ) 


i: Posthoc estimate (Empirical Bayesian Estimate)
because , , 2 are all estimated based on MLE
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A Simple Example
Dose ( k i )t ij
yij 
e
  ij  f (k ,i , t ij )   ij
V
 i ~ N (0,  2 )
 ij ~ N (0,  2 )
yij: the jth observation for the ith individual
Assume: only k is the unknown parameter to be estimated
Goal: search for the estimate, k̂ , that can minimize the following
objective function
OBJ ( k )  2 log Li ( k )
i
where Li(k) is the marginal likelihood of k for ith individual
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What Is Marginal Likelihood ?
nj

 ( yij  f ( k , ,t ij ))2
nj
 1  
2
Li ( k )   
 e
2  
 
P(Yi|k)
P(Yi|k,i)
j 1
2
 2

1
e
2 
P(i)
d 

 h ( , k )d
i

P(Yi,i|k)
4
This is just the area under the curve (AUC) of h(,k) versus  at a fixed k
2
0
1
h(ETA)
h ()
3
k=1
-2
-1
0
1
2
ETA

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• Each fixed k leads to different AUC
• Assuming only 1 subject, k̂ is
associated with maximum AUC. k̂ is
called the maximum likelihood
estimator (MLE) of k
• Even though AUC is a function of k, it
cannot be expressed as a closed-form
equation of k (difficulty of nonlinear
mixed effect modeling)
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k̂
Grid Search to Find
-0.5
0.0
0.5

1
2
h(ETA)
3
0
1
0
0
-1.0
k=1.1
3
k=0.7
2
h(ETA)
2
1
()
hh(ETA)
3
k=0.3
4
4
Imagine we can try every possible k and evaluate AUC with numerical
integration (similar to trapezoidal rule with very dense data)
4
•
1.0
-1.0
-0.5
0.0
0.5

ETA
-1.0
1.0
-0.5
0.0

0.5
1.0
ETA
0.8
1.0
ETA
(k)
AUC
AUC(ke)
yall
0.4
y
0.6
1.5
1.0
0.5
0.2
k=0.3
k=0.7
k=1.1
0
2
4
6
8
10
time (hr)
Time
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0.0
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0.0
0.3
0.5
0.8
1.0
1.3
1.5
1.8
2.0
k
ke
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Estimation Methods
• NONMEM
– Laplacian
– First order conditional estimate (FOCE)
– First order (FO)
• SAS
– Adaptive Gaussian Quadrature
– Importance sampling
– FO
• Splus
– Lindstrom and Bates Algorithm
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What is LAPLACIAN doing?
• Approximate h() with another function LAP() so that AUC
is a close-form function of k

 h ( , k )d   LAP ( , k )d  AUC
i
i

4
4
4
d 2 log hi ( )
g i " (ˆ ) 
d 2
 ˆ
k=1.1
0
0
1
1
2
h(ETA)
h(ETA)
3
3
k=0.7
3
2
1
0
or h ()
LAP ()
h(ETA)
k=0.3
-0.5
0.0

0.5
1.0
-1.0
-0.5
0.0

0.5
1.0
-1.0
-0.5
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0.0

0.5
1.0
ETA
ETA
ETA
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(k )

2
AUCi _ LAP ( k ) 
 hi (ˆ )
 gi " (ˆ )
-1.0
i _ LAP
2
Li ( k ) 

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What is FOCE doing?
• After Laplacian approximation, the second derivative, g i " (ˆ ) ,
is further approximated by H gi ' (ˆ ), a function of the first
derivative
d log hi ( )
gi ' (ˆ ) 
Li ( k ) 

 h ( , k )d   LAP ( , k )d  AUC
i
i


i _ LAP
4
4
4
k=0.7
k=1.1
h(ETA)
1
2
0
0
0
1
1
2
h(ETA)
3
3
3
k=0.3
-1.0
-0.5
0.0

0.5
1.0
-1.0
-0.5
0.0

0.5
1.0
-1.0
-0.5
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0.0

0.5
1.0
ETA
ETA
ETA
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( k )  AUCi _ FOCE ( k )
2
 hi (ˆ )
H gi ' (ˆ )
AUCi _ FOCE ( k ) 
() or h ()
FOCEh(ETA)
 ˆ
2

d
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What is FO doing?
• Approximate h() with another function FO() and also
approximate the second derivative, gi " (0) , with a function of
the first derivative H g ' (0)


i

 h ( , k )d   FO ( , k )d  AUC
i
i


-1.0
-0.5
0.0
0.5
1.0
h(ETA)
3
k=1.1
1
-1.0
-0.5
ETA
0.0
0.5
1.0
-1.0
ETA

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 gi '( 0 ) 2
2 H gi '( 0 )
0
1
2
h(ETA)
3
k=0.7
0
h(ETA)
2
1
0
FO () or h ()
3
k=0.3
(k )
4
4
4
AUCi _ FO ( k ) 
2
 hi (0)  e
H gi ' (0)
i _ FO
2
Li ( k ) 

0.0
0.5
1.0
ETA

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-0.5

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Which one is the best?
aucgq
True
auclap
LAP
aucfoce
FOCE
aucfo
FO
Marginal Likelihood
1.7
1.2
kˆFO  0.83
k̂True  kˆ LAP  0.74
kˆFOCE  0.72
0.7
LAP is the best even though occasionally FOCE, even FO, is
closer to the true marginal likelihood
0.2
0.2
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0.4
0.6
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0.8
k
1.0
1.2
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1.4
37
Proportional Residual Error Model
Dose ( k i )t ij
yij 
e
(1   ij )  f (k ,i , t ij )(1   ij )
V
 i ~ N (0,  2 )
 ij ~ N (0,  2 )
yij: the jth observation for the ith individual
Assume: only k is the unknown parameter to be estimated
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8
8
FOCE
FO
8
LAP
8
FOCEI**
8
LAPI*
0.5
1.0
0.5
1.0
6
2
0
8
1.0
6
4
h(ETA)
2
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
0.5
1.0
ETA
6
ETA
h(ETA)
6
2
h(ETA)
0.0

0.5
0
-0.5
2
-0.5
0.0
ETA
8
-1.0
0
-1.0
-0.5
4
1.0
4
h(ETA)
6
0.5
6
h(ETA)
0.0

ETA
4
2
6
h(ETA)
0.0
ETA
2
-0.5
-1.0
1.0
2
-0.5
0
-1.0
0.5
0
-1.0
6
1.0
0.0
ETA
0
1.0
-0.5
8
0.5
2
0.5
-1.0
1.0
2
0.0
ETA
0
0.0

ETA
0.5
0
-0.5
4
h(ETA)
6
4
2
-0.5
0.0
ETA
8
-1.0
k=1.1
0
-1.0
-0.5
6
h(ETA)
6
2
0
1.0
8
0.5
0
-1.0
8
1.0
4
0.5
4
0.0
ETA
4
h(ETA)
6
4
2
0.0
ETA
h(ETA)
6
2
0
-0.5
k=0.7
0
-0.5
8
-1.0
4
h(ETA)
4
-1.0
1.0
8
0.5
4
0.0
ETA
8
-0.5
8
-1.0
4
0
0
2
2
4
h(ETA)
6
6
k=0.3
-1.0
ETA
-0.5
0.0

0.5
ETA
1.0
-1.0
-0.5
0.0

ETA
*: LAPI, Laplacian with interaction is available in NONMEM VI
**: FOCEI, first-order conditional estimate with interaction
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Which one is the best?
2.2
aucgq
True
auclapi
LAPI
aucfocei
FOCEI
auclap
LAP
FOCE
aucfoce
FO
aucfo
Marginal Likelihood
1.8
1.4
ˆ  0.83
k
ˆ
FO
k̂True  k LAPI  0.69
kˆFOCEI  0.7 k̂ LAP  kˆFOCE  0.72
LAPI is the best and LAP is worse than FOCEI for
proportional error model
1.0
0.6
0.2
0.4
0.6
0.8
ke
11/13/2008
NONMEM Estimation Methods
k
1.0
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1.2
40
A Two-Dimensional Example
Dose ( k 1 i )t ij
yij 
e
  ij  f ( k ,V ,1i ,2 i , t ij )   ij
V  2i
2


0
i ~ N (0,  )
1

2
2
 ij ~ N (0,  )
 0 2 
yij: the jth observation for the ith individual
Assume: k and V are the unknown parameters to be estimated
Goal: search for the estimates, k̂ and V̂ , that can minimize the
following objective function
OBJ ( k ,V )  2 log Li ( k ,V )
i
where Li(k, V) is the marginal likelihood of k and V for ith individual
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What Is Marginal Likelihood ?
nj

nj
 1 
Li ( k )   
 e
2





 ( yij  f ( k ,V , ,t ij ))2

j 1
2
1
e
2 1

12
12
1
2  2

e
 22
 22
d1d 2 

 h ( ,
i
1
2
, k ,V )d1d 2

This is just the volume under the surface (VUS) of h(1, 2, , k, V) versus 1 and 2 at a fixed (k,V)
h (1, 2)
k=1 and V=1
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NONMEM Estimation Methods
• Each fixed pair (k,V) leads to different
VUS
• If k̂ and V̂ are associated with
maximum VUS. They are the
maximum likelihood estimators
(MLEs) of k and V
• Even though VUS is a function of k
and V, it cannot be expressed as a
closed-form equation of k and V
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LAPLACIAN, FOCE, FO Volumes vs True Volume
FOCE
FO
h (1, 2)
h (1, 2)
LAPLACIAN
2
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NONMEM Estimation Methods
2
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2
43
Summary for Estimation Methods
• The approximation of true marginal likelihood is analogous to
approximating an irregular shape with a symmetric shape whose
AUC or VUS can be easily calculated
• The difference among LAPLACIAN, FOCE and FO is mainly the
thinness and height of the symmetric shapes
• In general, the shape generated by LAPLACIAN method is the
closest to the true shape, but the shape from FOCE is almost
equally close
• For models with - interaction, taking the interaction into
account seems more important than avoiding first-order
approximation (FOCEI vs LAP)
Reference: Derivation of Various NONMEM Estimation Methods, Yaning Wang,
Journal of Pharmacokinetics and Pharmacodynamics, 34: 575-93, 2007
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Full Bayesian for PopPK
Dose
Yij 
 exp( ki  t ij )   ij
Vi
p( yij |  i ,  ) ~ N ( f (t ij , i ),  )
 ki 
 i   
 Vi 
  k2i
p( i |  ,  ) ~ MVN p (  ,  )   
 k V
 ii
k V
V2
2
p( 2 ) ~ IG (a, b)
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p(  ) ~ MVN p ( , H )
NONMEM Estimation Methods
k 


i i 
   

V 

i 
p( ) ~ IW ( R,  )
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Acyclic Structure

H

R
MVN
a
W

b
G

 1
MVN

Dose
Model
 2
Time
2
N
Data
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