Download Leveraging Prior Knowledge in Guiding Drug Development and

Introduction to Population
Analysis
Joga Gobburu
Pharmacometrics
Office of Clinical Pharmacology
Food and Drug Administration
September 30, 2008
Introduction to Population Analysis
[email protected]
1
Pharmacometrics Training
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
2
Agenda
• Introduction to population PK-PD
– Application of population PK-PD in drug development and
regulatory decision making
– Pharmacometrics @ FDA
• Introduction to population modeling
– Linear and nonlinear regression
– Introduction to mixed effects modeling
• Mixed effects modeling applied to population PK
–
–
–
–
–
September 30, 2008
Different methods of analysis
Bayesian theory
Maximum likelihood
Sources of variability
Variance (Error) models
Introduction to Population Analysis
Joga Gobburu
3
Agenda
• Introduction to population PK-PD
– Application of population PK-PD in drug development and
regulatory decision making
– Pharmacometrics @ FDA
• Introduction to population modeling
– Linear and nonlinear regression
– Introduction to mixed effects modeling
• Mixed effects modeling applied to population PK
–
–
–
–
–
September 30, 2008
Different methods of analysis
Bayesian theory
Maximum likelihood
Sources of variability
Variance (Error) models
Introduction to Population Analysis
Joga Gobburu
4
Definition: Modeling
Mathematical (conceptual) modeling is
describing a physical phenomenon
by logical principles characterized
with quantitative relationships, e.g.,
formulas, whose parameters may
be measured (or experimentally
determined)
http://www.hawcc.hawaii.edu/math/Courses/Math100/Chapter0/Glossary/Glossary.htm
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Uses of Models
1.
2.
3.
4.
5.
6.
Conceptualize the system
Codify current facts
Test competing hypotheses
Identify controlling factors
Estimate inaccessible system variables
Predict system response under new
conditions
Yates FE (1975) On the mathematical modeling of biological systems: a
qualified “pro”, in Physiological Adaptation to the Environment (Vernberg FJ
ed), Intext Educational Publishers, New York.
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Model and its parts
DV  f ( IDV , P)
• Parametric or Mechanistic
model parameters reflect
biological processes
• Non-parametric or empiric
model parameters do NOT
reflect biological processes
•
•
Deterministic models do not
account for variability
Stochastic models account for
variability
 WTi 
Vi  V pop  

 70kg 
DV?
September 30, 2008
Introduction to Population Analysis
DV=Dependent variable
IDV=Independent variables
P=Parameters

C p ,i
DV
CLi
tij
Vi
Dosei

e
Vi
IDV
P
Whether a quantity is DV, IDV
or P depends on the context
Joga Gobburu
7
Model and its parts
Dosei
Cpij 
e
Vi
^

CLi
tij
Vi
CLi  CLpop  CL ,i
 WTi 
Vi  V pop  
 V ,i

 70kg 
^
Cpij  Cpij   ij
September 30, 2008
Introduction to Population Analysis
Structural Model
Structural Model (covariate)
Stochastic Model (BSV)
Stochastic Model
(Residual Var)
Joga Gobburu
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Variability versus Uncertainty
Confidence Interval
Confidence Interval
(Lower CI, Mean, Upper CI)
(Lower CI, Variance, Upper CI)
Point
estimate
Point
estimate
Confidence interval is a measure of the uncertainty
on the point estimate. We obtain point estimates of
both population means and variances.
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Mixed-effects concept
Between Subject Variability
Residual Variability
1
-
+
Cp
0
(Individual-Pop Mean CL,V)
-
0.75
ith patient
0.5
0.25
Pop Avg
ij
0
+
Pred-Obs Conc
i (CL,i & V,i)
0
0
5
10
15
Time
Between-occasion variability = zero
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Mixed-effects concept
Dosei
Cpij 
e
Vi
^

CLi
tij
Vi
CLi  CLpop  CL ,i
 WTi 
Vi  V pop  
 V ,i

 70kg 
Fixed effects
Fixed
effects
Random effects
Random
effects
^
Cpij  Cpij   ij
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Types of data
• Continuous
– A variable can take any value (physically possible).
– E.g.: concentrations, time, dose, glucose levels
• Discrete
– A variable can take one of many pre-specified values
– Binary, ordinal
• Binary – Yes or No type response (e.g.: death, pain/no pain)
• Ordinal – Graded response (e.g.: mild/severe pain,
minor/major bleeding)
– Frequency – how often does the event occur?
• E.g.: seizures, vomiting
– Time to event – when does the event occur?
• E.g.: time to death, time to MI
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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PKPD Data
• Experimental
– Rich data are collected under controlled
conditions, usually small
– Best data for building structural models
– Example: Dose-proportionality
• Observational
– Sparse data are collected under ‘real’ life
conditions, usually large
– Best data for building statistical models
– Example: Pivotal or registration trials
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Linear versus Nonlinear models
• Whether a model is linear or nonlinear will need to be
determined relative to the parameters NOT the variables. For
example:
– Which of the two is linear?
DV
• DV = a·IDV
• DV = a·IDV + b·IDV2
• Linear models
– Partial derivative of DV w.r.t parameters is
independent of parameters
– Estimate parameters using linear regression
• Nonlinear models
IDV
DV
– Partial derivative of DV w.r.t parameters is
NOT independent of parameters
– Estimate parameters using non-linear regression
IDV
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Estimation via optimization
• Linear regression: Goal is to find a line that
goes as close to the observations as
possible.
• Comment on the goodness-of-fit of red, blue
and black lines shown on the right.
• Linear models can be analytically solved for
intercept and slope estimates.
DV
IDV
^
Sum of Squared Re siduals   (Y ij  Yij ) 2
Ideal value of the SSR is zero
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Estimation via optimization
• Nonlinear models do not have analytical solutions, so we need to
solve them numerically.
Obj Fn
CL
Maximum
Likelihood
Estimate
0
^
Sum of Squared Re siduals   (Y ij  Yij ) 2
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Maximum Likelihood Estimation
•
Non-linear mixed effects model
Yi =f( ,i ,X i )+ i
•
i
N (0,  )
i
N (0,  )
2
2
Likelihood for individual i
(Yi  f( ,i ,Xi )) 2
Li ( ; Xi )  
exp(
)
2
2
2 2
i 2
1

exp( 2 )di
2
2
2
1
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Technical goals of Population analyses
• Estimate population mean and variance
– Population mean CL, V
– Between subject variability of CL, V
– Residual variability of concentrations
• Explain between subject variability using patient covariates such
as body size, age, organ function
• Estimate individual CL and V to impute concentrations to
perform PKPD analyses
– Sometimes PK and PD measurements are not performed at the
same time
– PD change could be delayed from PK
– Modeling PD using differential equations mandates a functional
form (model) for PK
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Population mean versus Typical value
• Population mean is the naïve overall mean of a parameter
– For example, the population mean CL is 10 L/h.
CLi  CLpop  CL ,i
• When there are influential covariates that explain meaningful
variability in PK parameters, then Typical value is the mean of a
group of similar subjects.
– For example, the typical value of CL for a 70 kg subject is 10 L/h.
Similarly, for a 35 kg it is 5 L/h.

 WTi 
CLi  CLpop  
 CL ,i

 70kg 
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Methods of Population analyses
• Naïve averaged
• Naïve pooled
• Two-Stage
• One-Stage
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Naïve Averaged
Cp
Time
Cp
• Average concentration at each time point is
calculated using all subjects’ observed
concentrations.
• Average calculation does not take into the
number of observations at each time point
are equal or not; also subjects’
characteristics (heavy/light) are not
considered – hence called ‘naïve’.
• Average time course of concentrations is
then modelled to obtain naïve average PK
parameters.
Time
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Naïve Pooled
Cp
Time
• Individual observations from each subject
are ‘pooled’ to obtain average PK
parameters.
• Estimation does not take into the number
of observations at each time point are
equal or not; also subjects’ characteristics
(heavy/light) are not considered – hence
called ‘naïve’.
Cp
Time
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Two-Stage
Cp
Time
• Individual observations from each subject
are modelled separately to obtain average
PK parameters for each subject.
• Uncertainty in individual parameter
estimates is ignored.
• Each subject’s covariates and PK
parameters are correlated to explain BSV.
• Population mean (or typical value) and
variance are calculated.
Cp
Cp
Time
September 30, 2008
Introduction to Population Analysis
Time
Joga Gobburu
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Two-Stage
Uncertainty in individual parameter
estimates is ignored.
Cp
Cp
Time
Time
Which subject’s PK parameters are estimated with more certainty Red or Blue? Say, CL = 10±5 L/h and 10±1 L/h. When calculating the mean
only the point estimate is considered, the two-stage analysis does not account
for the different uncertainty
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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One-Stage
Cp
Time
• Data from all subjects are simultaneously
modeled. Population mean and variance
are estimated simultaneously, including
covariate modeling.
• Individual subject’s PK parameters are
calculated subsequent to ‘one-stage’
estimation. There is no model
‘optimization’ in this step – hence called
‘post-hoc’ step.
Cp
Cp
Time
September 30, 2008
Introduction to Population Analysis
Time
Joga Gobburu
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Methods of Population analyses
Feature
Naïve
Averaged
Naïve Pooled
Two-Stage
One-stage
Uncertainty at
each obs level
(also missing
obs)
Ignores; So,
mean will be
close to
extreme obs
Ignores; So,
mean will be
close to
extreme obs
Accounts; Will
not be
influenced by
extreme obs
Accounts; Will
not be
influenced by
extreme obs.
Uncertainty at
each subject
level
Ignores
Ignores
Ignores;
Subjects with
more or few
obs are
weighed equal.
Accounts;
Subjects with
more are
weighted more
Covariate
exploration
Not easy; can
average
subjects by
groups
Not easy; can
force model
with covariates
Possible
Possible
Complexity
Low
Low
Low
High; Needs
training
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Bayes Theorem
Future = Past ·Present
Posterior = Prior · Likelihood
P( )  P(Y |  )
P( | Y ) 
p(Y )
P( ) Probability
 Model Parameter
y
Data
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Bayes Theorem – Uninformative Prior
-
+ -
0
Prior
+
0
-
Current
0
+
Posterior
P( | y ) ~ P( )  P( y |  )
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Bayes Theorem – Informative Prior
-
+ -
0
Prior
+
0
-
Current
0
+
Posterior
P( | y ) ~ P( )  P( y |  )
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Bayes Theorem – One-Stage analysis
• ML estimation (such as that in NONMEM) uses an empirical
approach in obtaining the individual PK estimates. It uses the
maximum likelihood estimates (population parameters: mean
and variance) as PRIOR and the individual observations as
LIKELIHOOD (CURRENT) to calculate POSTERIOR. For this
reason, these individual estimates are called – ‘post hoc’,
‘empiric bayesian’ estimates.
• According to pure Bayesian estimation, POSTERIOR is a
distribution. ML only estimates the MODE (central tendency) of
that POSTERIOR distribution. Newer versions of NONMEM are
able to estimate the POSTERIOR distribution (never used it
myself). WinBUGS is a full fledged bayesian estimation program.
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Bayes Theorem – One-Stage analysis
Posterior = Prior · Likelihood
Individual
‘post-hoc’
Parameters
Population
Parameters
Individual
Observations
Indv estimates
close to indv
Pop Estimates
Rich obs/subject
Indv estimates
close to pop
Pop Estimates
Few obs/subject
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Sources of ‘random’ variability
• Between subject variability (BSV)
– Signifies deviance among subjects
– For example, CL varies between two ‘clones’
• Between occasion variability (BOV)
– Signifies deviance between occasions within a subject
– For example, CL varies between day 1 and 14 for subject#1
• Residual (or within subject) variability (WSV)
– Signifies deviance between predicted and observed in each subject.
This is at the observation level. Usually not assumed to be different
at the subject level also.
– For example, predicted Cp at time=0 is 10 ug/L, obs Cp=12 ug/L.
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Sources of ‘random’ variability
• All variability is typically assumed to be centered at zero. This is
so because if the deviation from mean is truly random, then
when the experiment is performed enough number of times,
observations will be some times above mean, sometimes below
mean with equal probability.
• Random variability is also ‘modeled’. Variability models also
need to be carefully considered. Differences between individual
and mean are generally described using normal or lognormal
distribution models.
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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BSV, BOV Variability models
Residuals are normally
distributed with a mean of zero
CLi  CLpop  CL ,i
CL
0
Residuals are log-normally distributed
with a mean of one
CL,i
CL
0
ln(CL)
CLi  CLpop  e
1
Normal GFR = 120 mL/min
Is GFR=60 mL/min possible?
Is GFR=240 mL/min possible?
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Measured
Residual variability models
-Spread of ‘measured’ values is constant across
true value range
Measured
True
-Spread of ‘measured values is higher at higher
true values
True
What would be the SD at each
true value for both scenarios?
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Residual variability models
SD
-Variability (SD) is same at low and high true values
-Called “additive” model
^
Cpij  Cpij   ij
True
-Variability (SD) increases with true values
-Called “proportional” or “constant CV” model
SD
^
CV
Cpij  Cpij  e
True
^
True
September 30, 2008
Introduction to Population Analysis
 ij
^
Cpij  Cpij  Cpij   ij
Joga Gobburu
36
Residual variability models
SD
-Variability (SD) is constant at low true values, but
increases with true values at higher values
-Called “combined additive-prop” model
^
True
^
Cpij  Cpij  Cpij   ij
^
prop
Cpij  Cpij   ij
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
prop
  ij
add
  ij
add
37
Agenda
• Introduction to population PK-PD
– Application of population PK-PD in drug development and
regulatory decision making
– Pharmacometrics @ FDA
• Introduction to population modeling
– Linear and nonlinear regression
– Introduction to mixed effects modeling
• Mixed effects modeling applied to population PK
–
–
–
–
–
September 30, 2008
Different methods of analysis
Bayesian theory
Maximum likelihood
Sources of variability
Variance (Error) models
Introduction to Population Analysis
Joga Gobburu
38
Pharmacometrics
Pharmacometrics is the science that deals with
quantifying pharmacology and disease to influence drug
development and regulatory decisions
• Includes
– Population PK
– Exposure-Response (or PKPD) for
effectiveness, safety
– Clinical trial simulations
– Disease-drug-trial modeling
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
39
Regulatory Initiatives Dictating Pharmacometrics
• Guidances for Industry
– Population PK
– Exposure-Response
– Dose-Response
– Evidence for Effectiveness
– Pediatrics Clinical Pharmacology
– EOP2A Meetings (draft)
• Critical Path Initiative
• OCP Strategic Plan
• Internal CDER Deliverables
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Pharmacometrics Scope
Tasks
Decisions Influenced
• NDA Reviews
• Protocols
– Dose-Finding trials
– Registration trials
• QT Reviews
•
Central QT team
• EOP2A Meetings
• Disease Models
– Knowledge Management
• Evidence of
Effectiveness
• Labeling
• Quantify benefit/risk
– Dose optimization
– Dose adjustments
• Trial design
1. Bhattaram et al. AAPS Journal. 2005
2. Bhattaram et al. CPT. Feb 2007
3. Garnett et al. JCP. Jan 2008
September 30, 2008
4. Wang et al. JCP. 2008 (in press)
Introduction to Population Analysis
Joga Gobburu
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FDA Pharmacometrics
Demand Increasing, Focus Expanding
Demand
Resources
5
0
QT
250
10
Reviews
FTEs
15
1995 2000 2005 2006 2008
200
150
100
50
0
1995
2000
2005
2006
2007
Focus
Efforts
100%
75%
50%
25%
0%
September 30, 2008
Policy
Design
Approval
Labeling
1995 2000 2005 2007
Introduction to Population Analysis
Joga Gobburu
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Gobburu, Sekar, Int.J.Clin.Pharm., 2002
Integration of Knowledge
Effectiveness
Safety
Dose
Ranging
Studies
Model
DDI, Age
Gender, Disease
Smoking, Food
Bridging
Studies
Effectiveness
15
10
5
Probability of Pain Relief
0.8
0.7
0
42.0 46.3 50.6 54.9 59.2 63.5 67.8 72.1 76.4 80.7 85.0
0.6
0.5
8
0.4
0.3
0
1
10
100
Safety
Dose, mg
6
4
2
0
40.0 43.1 46.2 49.3 52.4 55.5 58.6 61.7 64.8 67.9 71.0
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Argatroban
• Synthetic Direct Thrombin Inhibitor
• Approved in Adults
– prophylaxis or treatment of thrombosis in
patients with heparin-induced
thrombocytopenia (HIT)
– Anticoagulant in PCI patiets with HIT or at
risk for HIT
• Dosing
– Initial dose in HIT: 2 mcg/kg/min
– Titrated to 1.5 – 3 times baseline aPTT (aPTT
not to exceed 100s) at steady-state (1 – 3 hrs)
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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PKPD in Adults
•
•
•
•
Mainly distributed in ECF
Predominantly hepatically (CYP3A4/5) metabolized
Elimination half-life is 39 – 15 min
Direct relationship between argatroban plasma
concentration and anticoagulant effects.
• Steady-state reached in 1-3 hrs
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
45
Pediatric PKPD Data
Age group
Total
Birth – 6 months
8 years -16 years
7
4
5
Total
16
6 months – 8 years
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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PKPD Data
• 15 of the 16 patients received 6-10 doses of argatroban over 14
days.
• Serial concentration and aPTT measurements were available in
each patient. In total, about 166 concentration and 329 aPTT
measurements were available over a concentration range of 100
to 10,000 ng/mL.
• Argatroban plasma concentration and aPTT data from 5 healthy
adult studies (N=52) were used for model development.
• Infusion doses range from 1µg/kg/min – 40µg/kg/min
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Body weight reduces the between-patient
variability from 70% to 41%
Clearance, L/hr
20
16
12
8
4
0
0
25
50
75
100
Body Weight, kg
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Patients with elevated bilirubin exhibit 75% lower CL
than normals
Variability reduces further to 30% upon adjusting for hepatic status, after body
weight
CL, L/hr/kg
Patients with
Patients with
normal bilirubin elevated bilirubin
(N=11)
(N=4)
0.17
0.04
Elevated bilirubin was manifested by cardiac complications
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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Effect on aPTT is concentration dependent
Concentration-aPTT relationship is similar between adults (healthy) and
pediatrics (patients)
200
Pediatric Patients - Old Data
aPTT, seconds
Healthy Adults
Mean
150
100
50
0
0.1
1
10
100
1000
10000
Argatroban Concentration, ug/L
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
50
Simulations to explore optimal dosing regimen
Models
PKPD
Demographics
Baseline aPTT
Starting Dose
Simulations
Generate conc. &
aPTT data in
10000 peds at
each dose
Dosing
0.25-10 ug/kg/min
in increments of
0.25 ug/kg/min
Analysis
Count % patients:
< Target
Achieving Target
Exceeding Target
Titration Scheme
Simulations
Patients < Target at
each dose are given
the next higher dose
Target: 1.5-3 times baseline aPTT and < 100 seconds.
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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0.75 µg/kg/min in pediatrics is a reasonable starting
dose
20
0.75 µg/kg/min - 1.15% exceeds
target and 58.86% reach target
80
15
60
10
40
5
20
0
0
0
1
2
Dose, ug/kg/min
3
% Exceeding target aPTT
% Reaching target aPTT
100
Target: 1.5-3
times baseline
aPTT and < 100
seconds.
In adults, the approved starting dose is 2 µg/kg/min and the max dose is 10 µg/kg/min. This
starting dose results in 1.92% exceeding & 66.9% reaching target aPTT.
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
52
0.25 µg/kg/min is a reasonable incremental dose
No additional advantage beyond 3 ug/kg/min
% Below target aPTT
100
First Dose
Next Dose
80
20 of the 39/100 non-responders
at 0.75 ug/kg/min respond when
titrated to 1.0 ug/kg/min.
60
40
20
0
0
1
2
3
Dose, ug/kg/min
September 30, 2008
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Joga Gobburu
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Summary
 0.75 ug/kg/min is a reasonable starting dose
in pediatrics
 0.25 ug/kg/min is a reasonable incremental
dose
What other approaches can you
think of for optimizing dosing?
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
54
Knowledge, Skills Requirement
• What knowledge and skills do you need to perform
the previous analysis?
Knowledge
-Clinical Pharmacology
-Population Analysis (PKPD, Stats)
Skills
-Data formatting
-Modeling software usage
-Graphics
-Communication
September 30, 2008
Introduction to Population Analysis
Joga Gobburu
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REST AREA
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