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Population Pharmacokinetics, Parameter Estimation and
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Disclosures, Affiliations, and
Acknowledgements
MODELING OF
POPULATION PK-PD DATA
Paolo Vicini, PhD, MBA
MedImmune, Cambridge, UK
University of Warwick School of Engineering
Population Pharmacokinetics, Parameter Estimation and Model
Validation Vacation School
21st – 25th September 2015
Derived from material previously presented at:
2010-11 SAMSI Program on Semiparametric Bayesian Inference: Applications in Pharmacokinetics
and Pharmacodynamics
Raleigh, NC, July 13, 2010
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Suggested additional reading
Additional reading has been hyperlinked in the slides
Accessing the hyperlink will take the reader to the NCBI/PubMed
link or to the original source
Contributors to the ideas presented today include:
•
All the authors of the work that is cited in the presentation
• Disclosures
•
PV is an employee of MedImmune, a wholly owned subsidiary of
AstraZeneca
•
PV is on the Editorial Advisory Board of the Journal of
Pharmacokinetics and Pharmacodynamics and is an Associate
Editor of Clinical Pharmacology & Therapeutics: Pharmacometrics
& Systems Pharmacology
•
PV past employment includes Pfizer and the University of
Washington
•
PV receives royalties from the University of Washington Center for
Commercialization
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Model Validation Vacation School – Paolo Vicini
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4
Pharmacokinetics (PK) and
Pharmacodynamics (PD)
Dose
Pharmacokinetics
Pharmacokinetics
• “What the body does to the
drug”
• Fairly established
• Key: concentration
Concentration
Drug Concentration
7
6
5
4
3
2
1
0
0
5
10
15
20
25
Time
Pharmacodynamics
Pharmacodynamics
• “What the drug does to the body”
• Less certain
• Key: effect
3.5
3
Drug Effect
2.5
Effect
2
1.5
1
0.5
0
Front. Pharmacol., 28 July 2014
http://dx.doi.org/10.3389/fphar.2014.00174
0
2
4
6
8
10
Drug Concentration
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Modeling and Simulation During Drug
Development
Population Pharmacokinetics, Parameter Estimation and
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What Is Pharmacometrics?
• “Pharmacometrics can be defined as that branch of
science concerned with mathematical models of biology,
pharmacology, disease, and physiology used to describe
and quantify interactions between xenobiotics and
patients, including beneficial effects and side effects
resultant from such interfaces.”
• Pharmacometrics is the science that quantifies drug,
disease and trial information to aid efficient drug
development and/or regulatory decisions.
• Drug models describe the relationship between exposure (or
pharmacokinetics), response (or pharmacodynamics) for both
desired and undesired effects, and individual patient
characteristics. Disease models describe the relationship between
biomarkers and clinical outcomes, time course of disease and
placebo effects.
• The trial models describe the inclusion/exclusion criteria, patient
dis-continuation and adherence.
Population Pharmacokinetics,
CPT: Pharmacometrics & Systems Pharmacology
Parameter Estimation and Model
Volume
1, Issue 9, pages 1-14, 26 SEP 2012 DOI: 10.1038/psp.2012.4
9/22/2015
Validation Vacation School – Paolo
http://onlinelibrary.wiley.com/doi/10.1038/psp.2012.4/full#psp420124-fig-0001
Vicini
5
Pharmacometrics: a multidisciplinary field to facilitate critical thinking in drug development and translational research settings. Barrett JS, Fossler MJ,
Cadieu KD, Gastonguay MR. J Clin Pharmacol. 2008 May;48(5):632-49. doi: 10.1177/0091270008315318.
Pharmacometrics at FDA: http://www.fda.gov/AboutFDA/CentersOffices/OfficeofMedicalProductsandTobacco/CDER/ucm167032.htm
1
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9/22/2015
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PK-PD Models
(Mechanistic or Descriptive)
Pharmacodynamics: Direct Effect
• There is a large variety of nonlinear models (usually
• Pharmacodynamic effect E is directly linked to the
based on ordinary differential equations) to describe PK
and PD data
• The unknown parameters of these models are thought to
reflect the biological processes at the basis of drug
disposition and efficacy
• Challenge: how to estimate these parameters from
measurements gathered in the practice of drug
development
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E(C) 
9/22/2015
A Graphical Representation of Direct
Effects
Emax  C 
E(C) 
plasma concentration C
• Model: Emax model or Hill function
Emax  C
EC50  C
E(C) 
Emax  C 

EC50  C 
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Pharmacodynamic Relationships

EC50  C 
Measured effect
100
80
60
40
20
0
0
20
40
60
80
100
Drug Plasma concentration
Gamma=0.5
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Gamma=1
Gamma=1.5
Gamma=2
Population Pharmacokinetics, Parameter Estimation and
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Direct vs. Indirect Pharmacodynamics
Dose dependent PK and PD observed in a rat
model of diabetes
CPT: Pharmacometrics & Systems Pharmacology
Volume 3, Issue 1, pages 1-16, 2 JAN 2014 DOI: 10.1038/psp.2013.71
http://onlinelibrary.wiley.com/doi/10.1038/psp.2013.71/full#psp4201371-fig-0003
Interpreting Pharmacodynamic Plots
Effect vs. plasma concentration of a drug
candidate following a single oral dose to tumor
bearing mice
Population Pharmacokinetics,
Front. Pharmacol., 28 July 2014
http://dx.doi.org/10.3389/fphar.2014.00174
CPT: Pharmacometrics & Systems Pharmacology
Parameter Estimation and Model
Volume
3, Issue 1, pages 1-16, 2 JAN 2014 DOI: 10.1038/psp.2013.71
9/22/2015
Validation Vacation School – Paolo
http://onlinelibrary.wiley.com/doi/10.1038/psp.2013.71/full#psp4201371-fig-0002
12
Vicini
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• When the effect is delayed with respect to observed
plasma concentration
CL
D Vt
e
V
Plasma
Concentration
dCe( t )
Effect Site
 k e0 Cp( t )  Ce( t ) Concentration
dt
E  Ce 
Pharmacodynamic
E(Ce)  max
Effect
EC50  Ce 
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Delay (Link) Model in Practice
Delay (Link) Models
Cp( t ) 
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Indirect Response (Turnover) Models
Plasma or Effect Site
Concentration
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3.5
3
2.5
2
1.5
1
0.5
0
0
2
4
6
8
10
Time
Cp(t)
Ce ke0=10
Ce ke0=1
Ce ke0=0.1
Ce ke0=0.01
Cp(t) = Plasma concentration; Ce(t) = Effect site concentration
Holford NH, Sheiner LB, “Kinetics of pharmacologic response”
Pharmacol Ther. 1982;16(2):143-66. Review. http://www.ncbi.nlm.nih.gov/pubmed/6752972
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Types of Indirect Response Models
(Intuition: Balance between Production and Loss)
• Based on the drug eliciting (changes in) the homeostasis
of endogenous substances (we will revisit this later)
Response
Production P(t)
Inhibition or
Stimulation
Response
Loss L(t)
Response
R(t)
dR( t )
 P( t )  L( t )
dt
+

Inhibition or
Stimulation
• Stimulation of production

I  Cp( t ) 
dR( t )
 k in 1  max
  k outR( t )
dt
 IC50  Cp( t ) 

S  Cp( t ) 
dR( t )
 k in 1  max
  k outR( t )
dt
 SC50  Cp( t ) 
• Inhibition of loss
• Stimulation of loss

I  Cp( t ) 
dR( t )
 k in  k out 1  max
R( t )
dt
 IC50  Cp( t ) 

S  Cp( t ) 
dR( t )
 k in  k out 1  max
R( t )
dt
 SC50  Cp( t ) 
+

Drug
Pharmacokinetics
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• Inhibition of production
Sharma A, Jusko WJ, “Characteristics of indirect pharmacodynamic models and applications to clinical drug responses”
Br J Clin Pharmacol. 1998 Mar;45(3):229-39. Review. http://www.ncbi.nlm.nih.gov/pubmed/9517366
Population Pharmacokinetics, Parameter Estimation and
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Drug Effects in Turnover Models
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The Role of Variability
• Biological variability, superimposed to typical values, is

I  Cp( t ) 
dR( t )
 k in 1  max
  k outR( t )
dt
 IC50  Cp( t ) 

S  Cp( t ) 
dR( t )
 k in 1  max
  k outR( t )
dt
 SC50  Cp( t ) 

I  Cp( t ) 
dR( t )
 k in  k out 1  max
R( t )
dt
 IC50  Cp( t ) 

S  Cp( t ) 
dR( t )
 k in  k out 1  max
R( t )
dt
 SC50  Cp( t ) 
always present when dealing with PK and PD
measurements
• It must be distinguished from uncertainty
• First and foremost is quantification of variability from
available data
• Once quantified, variability can then be used to simulate
(extrapolate or predict) unobserved experiments
Population Pharmacokinetics,
CPT: Pharmacometrics & Systems Pharmacology
Parameter Estimation and Model
Volume
3, Issue 1, pages 1-16, 2 JAN 2014 DOI: 10.1038/psp.2013.71
9/22/2015
Validation Vacation School – Paolo
http://onlinelibrary.wiley.com/doi/10.1038/psp.2013.71/full#psp4201371-fig-0006
17
Vicini
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19
Estimating Variability with Nonlinear
Models
“Typical Data”
Data (arbitrary units)
1.2
Data (arbitrary units)
exp(0.05t )
0.8
Fast
Intermediate
Slow
exp(0.5t )
0.6
0.4
exp( 0.05t )
1
exp(0.5t )
0.6
0.4
exp( 5.0t )
0.2
0
0
0
0
0.5
1
1.5
2
2.5
0.5
1
1.5
Time (arbitrary units)
Mean of the Data 
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Data (arbitrary units)
Population Pharmacokinetics, Parameter Estimation and
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exp( 0.05t )
22
A Population
An Individual
0.8
3
How To Estimate Parameters from Sparse Data?
Statistical Modeling Is Needed
“Typical
Subject”
1.2
Fast
Intermediate
Slow
Parameter Mean
exp( 0.5t )
0.6
2.5
exp(0.05t )  exp(0.5t )  exp(5.0t )
3
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Averaging Model Parameters
1
2
Time (arbitrary units)
3
What would be the “right typical value”?
0.4
exp( 5.0t )
0.2
Fast
Intermediate
Slow
Data Mean
0.8
exp(5.0t )
0.2
20
Averaging Measurements
1.2
1
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0
0
0.5
1
1.5
2
2.5
3
• Many (sparsely sampled) individuals
Time (arbitrary units)
population rather than individual response
  0.05  0.5  5.0 
Mean of the Parameters  exp
t
3


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• Statistical issues become paramount
• Goal: characterize (by modeling) variability sources
23
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Sources of Hierarchical Variability
Sources of Hierarchical Variability
No Variation
Between–Subject Variation
2.5
Concentration (mg/L)
Concentration (mg/L)
2.5
24
2
1.5
1
0.5
0
2
Physiological variability
between individual subjects
1.5
1
0.5
0
0
6
12
Time (hours)
18
24
0
6
12
18
24
Time (hours)
4
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Population Pharmacokinetics, Parameter Estimation and
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Sources of Hierarchical Variability
Sources of Hierarchical Variability
Residual Unknown Variation
Actual Measurements
2.5
Between-occasion,
measurement error and
other sources of biological
and experimental variation
2
1.5
1
0.5
Concentration (mg/L)
Concentration (mg/L)
2.5
How to quantify relevant
biological variation
sources?
1.5
1
0.5
0
0
6
12
18
0
24
6
12
18
24
Time (hours)
Time (hours)
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Individual vs. Population
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Parameter Estimation
• Population data are necessary to extract biological
• The idea is to “match” model prediction to
variability information
• Individual parameter estimation is the cornerstone of
population analysis
• It turns out that it is often preferable to extract
population parameters directly (from the ensemble of
population data) as opposed to deriving them from
individual estimates
• From a didactic standpoint, we will start with
individual analysis and now we will move on to
parametric population analysis
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Data: cadralazine pharmacokinetics, from Wakefield, Racine-Poon et al,
Applied Statistics 43,No 1, pp201-221,1994
2
0
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Population Pharmacokinetics, Parameter Estimation and
Model Validation Vacation School – Paolo Vicini
29
measurements in some unambiguous,
meaningful way and in doing so quantify
parameters that relate to biology and drug action
• Most often, parameters are estimated via some
kind of maximum likelihood approach; the
approach is usually termed “extended least
squares”
• Optimization (most often minimization) of an
objective function matching model to data is used
to choose the “best” parameters
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Individual Extended Least Squares
• Extended least squares (ELS) was introduced by Stuart
Population Pharmacokinetics, Parameter Estimation and
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A simple population pharmacokinetics
model
Beal (UCSF) in the context of development of the
software NONMEM
• It arises from Gaussian maximum likelihood and the
hypothesis of prediction-dependent measurement errors
ˆ,
ˆ  argmi n l ogL(, )

 ,
2
1  y  s ()  1
 logL(, )    j j   ln Vj (, )
2 j1  Vj (, )  2
n


CPT: Pharmacometrics & Systems Pharmacology
Volume 1, Issue 9, pages 1-14, 26 SEP 2012 DOI: 10.1038/psp.2012.4
http://onlinelibrary.wiley.com/doi/10.1038/psp.2012.4/full#psp420124-fig-0002
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From Individual to Population Analysis
2.5
How to quantify relevant
biological processes at the
root of observed
variability?
1.5
1
0.5
Concentration (mg/L)
2.5
2
32
Averaging or Naïve Pooling
This Is Where We Start From
Concentration (mg/L)
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Averaging of individual data points
Single exponential disappearance fit to data
Parameters: CL, V
2
s(t,) = D e-(Cl/V)t/V
1.5
1
0.5
0
0
0
6
12
18
24
0
6
12
Time (hours)
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18
24
Time (hours)
33
Averaging or Naïve Pooling
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Average of Individual Estimates
Subject
• It treats all the subjects as if they were a single subject,
either by
1
2
3
4
5
6
7
8
9
10
2.5
Concentration (mg/L)
• (biased for the reasons we discussed) Averaging the
measurements over time, or
• (slightly preferable) fitting the ensemble of all individual
measurements as if they came from a single subject
• Not recommended since it neglects and cannot quantify
between-subject variation
2
1.5
1
Clearance Volume
0.18
19.72
0.08
15.13
0.15
13.77
0.13
17.13
0.19
19.10
0.23
19.13
0.16
22.21
0.21
15.06
0.22
17.22
0.27
14.01
0.5
0
0
6
12
18
24
Time (hours)
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Population Pharmacokinetics, Parameter Estimation and
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“Standard Two–Stage”
Mixed Effects Models
• Population parameters are calculated by sample
• Two-Stage methods require the availability of individual
averaging:
Typical value (sample mean)
N
1
ˆ   ˆ i
N i1
Variability (sample variance)
ˆ

1 N ˆ ˆ ˆ ˆT
 (i  )(i  )
N i1
parameter estimates for every subject; at the same time,
they may postulate a statistical (additive) model for the
BSV (more later)
• Further analytical needs are:
• The individual estimates may not be available (for some or all
individuals)
• The statistical BSV model may need to be more flexible
• Potential issues:
• All individual estimates must be available
• They all have the same weight in mean and variance
• Uncertain estimates may inflate measured variability
6
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Mixed Effects Models
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The Population Approach
• Parameters in the model are represented as the
combination of other parameters that:
Clearance ~ Gaussian (0.182, 30%)
Volume ~ Gaussian (17.24, 16%)
2.5
Concentration (mg/L)
• do not vary in the population (fixed effects)
• vary in the population (random effects)
• An example of fixed effect is the mean clearance in the
population, or its variance
• An example of random effect is the clearance of a particular
individual, or rather the deviation of that clearance from the
population mean
2
1.5
1
0.5
0
0
6
12
18
24
Time (hours)
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Hierarchical Variability
40
Back To The Standard Two-Stage
Cl and V typical values: 
h ~ N(0, )
Typical value (sample mean)
Variability (sample variance)
1 N
ˆ   ˆ i
N i1
Between-individual (BSV)
Cl and V ~ N(, )
s(t,) = D e-(Cl/V)t/V
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ˆ

1 N ˆ ˆ ˆ ˆT
 (i  )(i  )
N i1
120
Frequency
100
y = s(t,,)
e ~ N(0, )
80
60
40
20
0
CL
Residual unknown (RUV)
,  and  are fixed effects
h and e are random effects
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Gaussian BSV distribution for the parameters:
e.g., for CL: CL= CL + h, h ~ N(0, CL)
then CL ~ N(CL, CL)
y = s(t,,) + e
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• STS results can be described as arising from a
41
BSV for Mixed Effects Models
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Population Distributions
• Between–subject variability in the model parameters
120
CL =  + h, h ~ N(0, )
then CL ~ N(, )
Frequency
100
• Normal Distribution:
80
60
40
20
0
CL
• Log–Normal Distribution:
CL =  exp(h), h ~ N(0, )
then CL ~ LN(, )
200
Frequency
can be explicitly modeled
• BSV random effects are assumed Normal, zero mean
and variance , thus if:
CL =  + h, h ~ N(0, )
then CL ~ N(, ) (Gaussian BSV)
• If on the other hand:
CL =  exp(h), h ~ N(0, )
then CL ~ LN(, ) (log-normal BSV)
42
150
100
50
0
CL
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A Primer on Statistical Distributions
Summary: Typical Temporal Profile
Clearance = 0.182
Volume = 17.24
CPT: Pharmacometrics & Systems Pharmacology
Volume 2, Issue 4, pages 1-14, 17 APR 2013 DOI: 10.1038/psp.2013.14
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BSV On Parameters
BSV At Work
Clearance ~ Gaussian (0.182, 30%)
Volume ~ Gaussian (17.24, 16%)
* 0.16)
= 17.24
= 0, sdsd
mean
dnorm(x,
* 0.16)
= 17.24
= 17.24,
mean
dnorm(x,
0.14
0.14
Concentration (mg/L)
2.5
0.20-0.1 0.25
0.30
0.0
0.35
0.10.40
0.06 0.08
0.08 0.10
0.10 0.12
0.12
0.020.020.040.040.06
22
4
4
66
0.182)
0.3* *0.182)
sd==0.3
dnorm(x,
0, sd
mean==0.3,
dnorm(x,mean
00
0.15
-0.2
12-6
0.450.2
14
-4
-216
0 18
2
20 4
22
6
hSubj#1 hSubj#2
hSubj#1 hSubj#2
CL=0.182
V=17.24
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Concentration (mg/L)
1
0.5
6
12
18
24
Time (hours)
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BSV and RUV Combined
Clearance ~ Gaussian (0.182, 30%)
Volume ~ Gaussian (17.24, 16%)
Measurement error ~ Gaussian (0, 0.01)
s(t,) = D e-(Cl/V)t/V
Cl and V ~ N(, )
e ~ N(0, )
1.5
1.5
0
Clearance ~ Gaussian (0.182, 30%)
Volume ~ Gaussian (17.24, 16%)
2
s(t,) = D e-(Cl/V)t/V
Cl and V ~ N(, )
0
RUV: Unexplained Variability
2.5
46
Clearance ~ Gaussian (0.182, 30%)
Volume ~ Gaussian (17.24, 16%)
2
xx
xx
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ePoint#1
ePoint#2
s(t,) = D e-(Cl/V)t/V
Cl and V ~ N(, )
e ~ N(0, )
ePoint#4
ePoint#3
0.5
ePoint#5
0
0
6
12
18
24
Time (hours)
8
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Residual Unknown Variation Structures
mean and variance , thus we can have e.g.:
• y(t, , , e) = D e-(Cl/V)t/V + e, h ~ N(0, ), e ~ N(0, )
(Additive RUV)
• y(t, , , e) = D e-(Cl/V)t/V (1 + e), h ~ N(0, ), e ~ N(0,
)
(Proportional RUV)
• y(t, , , e) = D e-(Cl/V)t/V exp(e), h ~ N(0, ), e ~ N(0,
)
(Log-normal or exponential)
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• A very general RUV model is:
y(t, , , e) = D e-(Cl/V)t/V (1 + e1) + e2
h ~ N(0, )
e1 ~ N(0, 1)
e2 ~ N(0, 2)
(Additive and Proportional RUV)
• Note that we are using s() for the model
of the data, and y for the data
themselves
52
How Is Parameter Estimation Done?
• In brief: maximum likelihood estimation based on the
• These are the terms we will consistently use:
• Typical Values ()
• Between-Subject Variation () (BSV)
• Residual Unknown Variation () (RUV)
marginal probability density of the population data
• Since the population likelihood cannot be explicitly
calculated for our models, which are nonlinear in the
parameters, various approximations are often
employed
• All the data are analyzed together, i.e. the estimate for
every individual builds on the population parameters
• Other commonly used terms include:
• Inter- and intra-subject variability
• Fixed and random effects
• Typical values and population means
• Inter-occasion or between-occasion variability
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54
A Major Obstacle
• We assume that the system response for the i-th
subject (N subjects total) is given by:
yi = si(bi) + ei,
i=1, …, N
where:
• y is the random vector of measurements
• s() is a known (nonlinear) model function
• bi is the vector of model parameters (e.g. CL and
V), which are functions of fixed () and random (hi)
effects
bi = bi(, hi)
• ei is the measurement error random vector
• The individual model prediction
y i  si (bi )  ei  si (, hi )  ei
implies the joint population likelihood:
N
L(, hi )   L i (, hi , ei )
i1
• However, L depends on the function s()!
• The integral needed to express the likelihood as a
function of fixed effects only cannot be analytically solved:
L(, , )  
Reference: M Davidian, DM Giltinan.
Nonlinear Models for Repeated Measurement Data.
Chapman & Hall/CRC (1995).
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Population Pharmacokinetics, Parameter Estimation and
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Summary: Model Assumptions
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Common Terms
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Residual Unknown Variation Structures
• RUV can also be flexibly modeled, just like BSV
• RUV random effects are also assumed Normal, zero
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 N

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L (, h , e )dh
i1
i
i
i
i
54
9
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First–Order Approximation
• Given the linearization:
yi = si(bi) + ei,
i=1, …, N
• Given a relationship between the random and the
fixed effects and the distribution of the random
effects: bi = bi(, hi), with hi~N(0, )
• The First-Order (FO) method is based on the
linearization of the model function around the
mean random effects of the population (i.e. 0)
 s [b (, h)] 
y i  si [bi (,0)]   i i
hi  ei
h

h 0 

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First–Order Method
• Given the model of the i–th individual:
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 s [b (, h)] 
y i  si [bi (,0)]   i i
hi  ei
h

h 0 

we can also write:
y i  si [bi (,0)]  Z i (,0)hi  ei
where Zi(,0) is a sensitivity matrix which depends
on the model parameters – the integration can now
be done. Expectation and covariance of Y are:
E[y i ]  si [bi (,0)]
Cov[y i ]  Z i (,0)  Z i (,0)T  Cov[ei ]  Vi (,0, , )
55
57
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Population Pharmacokinetics, Parameter Estimation and
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Maximum Likelihood (FO)
Other Approximations
• If we make the assumption that Y is Gaussian, from the
• Increase in accuracy especially for large BSV
expectation and covariance of Y we can write an
extended least squares estimator:
E[Yi ]  si [bi (,0)]
Cov[Yi ]  Z i (,0)  Z i (,0)T  Cov[ei ]  Vi (,0, , )
56
58
• Linearization around individual random effects
• Lindstrom and Bates (1988)
• Beal and Sheiner, First-Order Conditional Estimation (in NONMEM)
• Second-order approximations
• Laplacian
• Gaussian quadrature
which is (over all the subjects):
1

N  l og2 detVi (,0 , , )



 l ogL(, , )    2

1
T
T
1
i1 





Y

s
(

,
0
)
V
(

,
0
,

,

)
Y

s
(

,
0
)
i
i
i
i
i


2


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59
Change in Emphasis
from Individual Analysis
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Key Diagnostic Plots
• With population analysis, the focus shifts to the
population parameters, as opposed to the
individual
• The unknowns in population analysis are:
• Expected values (means, typical values) of the
parameters ()
• BSV (variances and covariances) of the parameters ()
• RUV (variances and covariances) of the model ()
• Challenge: find good starting values for , , 
• Note: Individual parameters can be calculated as
a byproduct, once population parameters are
known
CPT: Pharmacometrics & Systems Pharmacology
Volume 2, Issue 4, pages 1-14, 17 APR 2013 DOI: 10.1038/psp.2013.14
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Balance Between Information at the
Individual and Population Level
# of subjects
(N)
Few
Many
Reliable
individual and
population
information
Reliable individual
information only
Few
Reliable
population
information only
Unreliable
individual and
population
information
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Balance Between Information at the
Individual and Population Level
Many
# of data
per subject (n)
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Models for Covariate Functions
CPT: Pharmacometrics & Systems Pharmacology
Volume 1, Issue 9, pages 1-14, 26 SEP 2012 DOI: 10.1038/psp.2012.4
http://onlinelibrary.wiley.com/doi/10.1038/psp.2012.4/full#psp420124-fig-0003
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Effect of Covariates on Variability
• Linear function
• Centering (to a reference value)
• Centering (using allometry)
CPT: Pharmacometrics & Systems Pharmacology
Volume 1, Issue 9, pages 1-14, 26 SEP 2012 DOI: 10.1038/psp.2012.4
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CPT: Pharmacometrics & Systems Pharmacology
Volume 1, Issue 9, pages 1-14, 26 SEP 2012 DOI: 10.1038/psp.2012.4
http://onlinelibrary.wiley.com/doi/10.1038/psp.2012.4/full#psp420124-fig-0004
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Graphical Evaluation of Covariates
CPT: Pharmacometrics & Systems Pharmacology
Volume 2, Issue 4, pages 1-14, 17 APR 2013 DOI: 10.1038/psp.2013.14
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Simulated Drug Exposures for
Different Age Groups
Population Pharmacokinetics,
CPT: Pharmacometrics & Systems Pharmacology
Parameter Estimation and Model
Volume
1, Issue 9, pages 1-14, 26 SEP 2012 DOI: 10.1038/psp.2012.4
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Validation Vacation School – Paolo
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Vicini
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Additional Hierarchical Structure
List (Timeline) of Approaches
Hyperprior on  Hyperprior on  Hyperprior on 
(e.g. flat Gaussian)
(e.g. inverse Wishart)
(e.g. inverse Gamma)
Cl and V typical values: 
h ~ N(0, )
Between-individual (BSV)
Cl and V ~ N(, )
s(t,) = D e-(Cl/V)t/V
e ~ N(0, )
y = s(t,,)
Residual unknown (RUV)
,  and  are fixed effects
h and e are random effects
CPT: Pharmacometrics & Systems Pharmacology
Volume 1, Issue 9, pages 1-14, 26 SEP 2012 DOI: 10.1038/psp.2012.4
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y = s(t,,) + e
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Bayesian
Methods
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“Deterministic” vs. “Stochastic”
Simulations
Introduction to Simulation
• The purpose of estimating parameters describing typical
values and variability in the “subject” and “measurements”
dimensions is to use the model and parameters to
simulate unobserved conditions
• These may include: dosing schemes, sampling strategies, trials of
• One can envision two kinds of simulation:
• Deterministic (“engineering”)
• One realization of the simulation
• No variability included
• Stochastic (“statistical”)
• Multiple realizations of the simulation
different sizes, etc.
• Once parameters for the model are available, all the
elements that are needed to use the model to simulate
are in place
• Variability is used to drive the multiple realizations
• Variability can be either present or not
• Both kinds can be useful depending on the specifics of
the question to be addressed
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Example Simulation
(Posterior Predictive Check)
Population pharmacokinetic model of the pregabalin-sildenafil interaction in rats: application of simulation to preclinical PK-PD study design.
Bender G, Gosset J, Florian J, Tan K, Field M, Marshall S, DeJongh J, Bies R, Danhof M. Pharm Res. 2009 Oct;26(10):2259-69. doi:
10.1007/s11095-009-9942-y. Epub 2009 Aug 11.
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