Nonlinear mixed effects models
in pharmacokinetic modeling
Lecture notes
Pyry Välitalo 1.10.2009
Pharmacokinetics: Traditional,
standard two-stage* approach
• Recruit subjects from a homogenous
(healthy?) population. Collect lots of blood
samples from each patient.
• Estimate pharmacokinetic parameters for
each patient separately.
• Calculate means and variances for
pharmacokinetic parameters. Use regression
to investigate the effects of covariates. **
*Not to be confused with the term ”two-stage estimation procedures” in statistics
** Riviere, J. Comparative Pharmacokinetics: Principles, Techniques and Applications, page 260. Iowa State Press, 1999.
Pyry Välitalo 1.10.2009
Problems associated with the
traditional approach:
• Parameter variabilities are inflated.*
• Since recruitment is usually done from a
homogenous (healthy?) population, it is
harder to extrapolate into target population.
• Difficult to study special populations who
would not handle the blood loss well
(neonates, AIDS/cancer patients, critical care
patients, etc).
*Sheiner LB, Beal SL. J Pharmacokinet Biopharm. 1980 Dec;8(6):553-71.
Pyry Välitalo 1.10.2009
A solution: population
pharmacokinetics
( = Pharmacokinetics using nonlinear mixed effects
modeling)
• Build a pharmacokinetic model with fixed effects,
between-subject variabilities, and residual
variabilities.
• Differences to traditional pharmacokinetic modeling:
– One model explains all data
– Between-subject variability is included in the model as a
new kind of parameter: A random effect that varies
between patients but stays constant within the patient.
Pyry Välitalo 1.10.2009
Advantages of a population
approach to pharmacokinetics 1/2
• Less blood samples per patient are needed
– Special patient groups can be studied (children,
cancer patients, etc)
– Samples taken during routine treatment can be
used in studies.
• Cost-effectiveness increases
• The results naturally reflect the patient group that is
usually receiving the drug.
Pyry Välitalo 1.10.2009
Advantages of a population
approach to pharmacokinetics 2/2
• Because data are combined into a single
model, more detailed models can be used.
• E.g. nonlinearity can be detected better than
with standard two-stage approach*
• Easier to design future clinical trials with one
single model.
*EN Jonsson, JR Wade, MO Karlsson. AAPS PharmSci. 2000;2(3):E32
Pyry Välitalo 1.10.2009
Advantages of a population approach to
pharmacokinetics: an example
• Docetaxel, a chemotherapeutic agent
• Main problem: Patients with poor liver function
– FDA: do not dose due to unpredictable PK!
• Aim: Build a clinically relevant model to predict docetaxel clearance in
patients with poor liver function. Evaluate CYP3A activity spesifically as a
predictor of docetaxel clearance.
• A population PK model was built with following covariates:
– Liver functioning classification based on a few markers
– Plasma protein binding
– CYP3A liver enzyme activity
• After these covariates had been accounted for, the unexplained variability
in clearance actually became lower for patients with poor liver function
than with normal-liver-function patients.
Hooker et al. Clin Pharmacol Ther. 2008 Jul;84(1):111-8.
Pyry Välitalo 1.10.2009
Using the parameters of
NLME models in pop-PK
• Fixed-effects parameters (Thetas, θ)
– Can be for example a typical value for volume of distribution, typical
value for clearance or the effect of a covariate (e.g. sex) on a
parameter.
– A capital theta (Θ) denotes a vector of all fixed-effects parameters in
model.
– A lowercase theta (θ) denotes an element of Θ, one specific
parameter.
Pyry Välitalo 1.10.2009
Using the parameters of
NLME models in pop-PK
• Between-subject variability (Etas, η)
(Empirical Bayes Estimates)
– These describe unexplained differences in parameter values between
individuals.
– E.g. the individual value for clearance could be described as θCL*eη(CL).
– Can also be used to describe inter-occasion variability, between-study
variability.
– It is expected that etas are distributed N(0, ω2)
Pyry Välitalo 1.10.2009
Using the parameters of
NLME models in pop-PK
• Residual random effects (Epsilons, ε)
– The unexplained residual error (different for each observation).
– Sources: Misrecording the time of sampling, mistreatment of samples,
error induced by analytical methods, model misspesification, etc.
– It is expected that epsilons are distributed N(0, σ2)
– E.g. observation=prediction+ε
Pyry Välitalo 1.10.2009
In summary: The population
model
yij=f(Θi)+εij
Where
• yij is the jth observation of ith individual
• f is a model that describes all observations
• Θi is a vector of individual i’s parameter values
• εij is residual error of individual i’s jth observation
The elements of Θi are usually θi= θ*eη, where
• θ is the typical value for a parameter
• ω2 is the variance of η values
Pyry Välitalo 1.10.2009
Components of a population
pharmacokinetic model
Fixed
effects
Mostly random
effects
Pyry Välitalo 1.10.2009
A hypothetical example: Building
a 2-compartment model
• Let’s say we have pharmacokinetic data from 40
individuals, of which 20 received oral dosing and 20
received intravenous injection.
• A total of 200 plasma concentrations, 2-8 per
individual
• Known covariates: Weight, sex (SEX=0 to indicate
male, SEX=1 to indicate female)
• This example takes many shortcuts and should not
be viewed as a reference of how to build a pop-PK
model.
Pyry Välitalo 1.10.2009
Let’s start by building a single-compartment
IV model (and ignore the oral treatment
group for now)
We use three parameters:
•Volume of distribution for central compartment (V)= θV
•Clearance (CL) = θCL
•Residual error (with standard deviation σ)
The starting amount (A) in central compartment is Dose. After
dosing, amount (A) of drug in central compartment starts to get
eliminated.
At0=Dose
dA/dt=-A*CL/V
IPRED=A/V
The prediction (PRED) is compared to observation (Y)
Y=IPRED+ε
;σ is standard deviation
;of all epsilons
Pyry Välitalo 1.10.2009
The model failed to converge. Let’s add
between-subject variabilities to CL and V.
We estimate five parameters:
V= θV*eη(V)
;Volume of distribution and BSV for it
CL= θCL*eη(CL)
;Clearance and BSV for it
Residual error (σ)
At0=Dose
dA/dt=-A*CL/V
IPRED=A/V
Y=IPRED+ε
; σ is standard deviation
;of all epsilons
…Success! The model now converges.
Pyry Välitalo 1.10.2009
The ”weighted residuals vs time” graph hints that
a 2-compartment model might perform better.
Weighted residuals vs. Time
On the left: WRES vs Time. Notice the red
line (locally weighted scatterplot smoothing)
starting above zero, falling below and rising
back above zero.
6
Weighted residuals
4
Below: An illustration of what it usually
means if WRES vs time has a shape like
that on the left. The black line represents
predictions and red dotted line represents
observations.
2
0
-2
-4
-6
2
4
6
Time
8
Pyry Välitalo 1.10.2009
Let us try the twocompartment model.
Vcentral= θV*eη(V)
Vperipheral= θV2
Q= θQ
CL= θCL*eη(CL)
;Peripheral compartment volume
;Intercompartmental clearance
At0,central=Dose
dAcentral/dt=-Acentral*CL/Vcentral – Acentral*Q/Vcentral + Aperiph*Q/Vperiph
dAperiph= Acentral*Q/Vcentral – Aperiph*Q/Vperiph
IPRED=Acentral/Vcentral
Y=IPRED+ε
Pyry Välitalo 1.10.2009
Let us try adding weight as a
covariate into the model now.
Vcentral= θV*(WT/70)*eη(V)
Vperipheral= θV2*(WT/70)
Q= θQ
CL= θCL*(WT/70) θscale *eη(CL)
; Linear scaling
; Allometric scaling
At0,central=Dose
dAcentral/dt=-Acentral*CL/Vcentral – Acentral*Q/Vcentral + Aperiph*Q/Vperiph
dAperiph= Acentral*Q/Vcentral – Aperiph*Q/Vperiph
IPRED=Acentral/Vcentral
Y=IPRED+ε
When adding covariates, we should also check if all the BSV’s are still necessary. It might
be that covariates can explain most of the between-subject variability.
Pyry Välitalo 1.10.2009
Is sex a covariate? Let’s find out (remember,
SEX=0 for male, SEX=1 for female).
Vcentral= θV*(WT/70)* (1+SEX*θsex1)*eη(V) ;affects only
;females
Vperipheral= θV2*(WT/70)* (1+SEX* θsex2)
Q= θQ
CL= θCL*(WT/70) θscale * (1+SEX* θsex3)*eη(CL)
At0,central=Dose
dAcentral/dt=-Acentral*CL/Vcentral – Acentral*Q/Vcentral + Aperiph*Q/Vperiph
dAperiph= Acentral*Q/Vcentral – Aperiph*Q/Vperiph
IPRED=Acentral/Vcentral
Y=IPRED+ε
The testing of covariate relationships should be done one at a time.
In this example we didn’t find any significant improvement when adding sex as a covariate in
any of the parameters.
Pyry Välitalo 1.10.2009
Once satisfied with the IV model,
we add the oral treatment group
Ka= θKa
;Rate of absorption
F= θF
;Oral bioavailability
Vcentral= θV*(WT/70)*eη(V)
Vperipheral= θV2*(WT/70)
Q= θQ
CL= θCL*(WT/70) θscale *eη(CL)
At0,depot=F*Doseoral
At0,central=DoseIV
dAdepot/dt= - Adepot*Ka
dAcentral/dt= -Acentral*CL/Vcentral – Acentral*Q/Vcentral + Aperiph*Q/Vperiph
+Adepot*Ka
dAperiph= Acentral*Q/Vcentral – Aperiph*Q/Vperiph
IPRED=Acentral/Vcentral
Y=IPRED1+ε
Pyry Välitalo 1.10.2009
Another example: Modeling flurbiprofen
pharmacokinetics in children (real case)
• Data from 64 patients, 1-7 samples per patient
• Oral dose given to 37, intravenous dose to 27
patients.
Pyry Välitalo 1.10.2009
Flurbiprofen pharmacokinetics:
At the beginning…
Observations of flurbiprofen CSF
concentrations (60), and observations of
both total (304) and unbound (62)
flurbiprofen concentrations in plasma
Prior knowledge: The doses given for each
patient and the volume of CSF
compartment.
Pyry Välitalo 1.10.2009
Flurbiprofen pharmacokinetics:
What we ended up with
All the parameters were estimated to best
describe concentrations in central compartment
and CSF. The number of parameters in the final
model was:
•13 fixed-effect parameters
•4 between-subject variability parameters
•2 residual error variability parameters
Pyry Välitalo 1.10.2009
Flurbiprofen pharmacokinetics:
Most significant findings
•Bioavailability of oral flurbiprofen syrup for
children was estimated.
•The model includes children from 3
months to 13 years (previous study:
children aged 6-12 years). There was no
impairment of clearance seen in infants.
•Flurbiprofen distributes into cerebrospinal
fluid very effectively.
Pyry Välitalo 1.10.2009
Conclusion
• With flurbiprofen data, population pharmacokinetic approach
yielded several benefits:
– Estimating bioavailability of oral flurbiprofen syrup was
made possible.
– More credibility when describing CSF kinetics with model
parameters than with a summary of raw data (e.g. mean
ratio of unbound flurbiprofen in plasma versus flurbiprofen
in CSF).
– More thoroughly investigated covariate model.
– Only one model: Possible to use in simulations in future.
Pyry Välitalo 1.10.2009
Resources
•
•
•
•
•
•
NONMEM. Currently the ”golden standard” in population pharmacokinetic modeling. Requires license.
– http://www.icondevsolutions.com/nonmem.htm
Xpose: An R package that helps in deciphering the output of NONMEM. Free.
– http://xpose.sourceforge.net/
R: A program needed by Xpose to operate. Free.
– http://www.r-project.org/
Census. A helpful program for keeping record of NONMEM runs. Free.
– http://census.sourceforge.net/
PsN (Perl-speaks-Nonmem). A collection of helpful Perl scripts for NONMEM that make life easier in a lot of ways. Free.
MONOLIX. Another population pharmacokinetic program. Free.
– Advantages: Shorter runtimes than in NONMEM, provides also graphical output by itself
– Disadvantages: Currently not as flexible as NONMEM.
– http://software.monolix.org/sdoms/software/
Pyry Välitalo 1.10.2009
EXTRA: An example of simulatory model
diagnostics: Visual Predictive Check
Visual Predictive Check results
Total flurbiprofen, IV group
Total flurbiprofen, oral group
Observations
Observations
15
10
5
0
10
5
0
0
5
10
0
5
10
Time
20
Time
Unbound flurbiprofen
Flurbiprofen in CSF
Observations
0.004
Observations
15
0.003
0.002
0.03
0.02
0.01
0.001
0.00
0
5
10
Time
15
Using the model parameters (including
random effects), simulate a number of
observations, e.g. 200 simulated
observations for every true
observation.
Calculate the prediction intervals for
these simulated observations
-> see if they agree with real
observations.
Blue dots: Real observations
Black lines: 95th percentile prediction
intervals
0
5
10
Time
15
In this text, prediction interval means
an interval, inside which a certain
percentile of simulated observations
fall.
Pyry Välitalo 1.10.2009
EXTRA: An example of simulatory model
diagnostics: Visual Predictive Check
Usually a better alternative is to plot
confidence intervals for prediction
intervals and see if the intervals for real
observations fall inside the confidence
intervals for prediction intervals.
Red lines: Intervals of real observations
Blue area: Confidence intervals for
prediction intervals
Pyry Välitalo 1.10.2009
EXTRA: Features of the flurbiprofen model:
Intravenous infusion
The intravenous dosage had to have an
absorption rate constant. The reason for this is
that the intravenously given drug is a prodrug
and takes some time to hydrolyze into active
flurbiprofen (see figure below).
An example: Conc vs Time in five IV-dosed patients
Observations / Predictions
DV
IPRE
PRED
0 2 4 6 8
ID:18
ID:19
0 2 4 6 8
ID:20
ID:26
ID:27
10
5
0
0 2 4 6 8
0 2 4 6 8
Time
0 2 4 6 8
Pyry Välitalo 1.10.2009
EXTRA: Features of the flurbiprofen model:
Implementing unbound observations
Central compartment included two kinds of
observations:
•Total flurbiprofen concentrations
•Unbound flurbiprofen concentrations
If the observation was marked as unbound
observation, the prediction was multiplied by
fraction unbound (FU) before it was compared to
the observation.
FU= θFU*eη(FU)
…
IPRED=A(2)/V2
;TOTAL
IF (FLAG.EQ.3) IPRED=A(2)/V2*FU ;UNBOUND
Pyry Välitalo 1.10.2009
EXTRA: Features of the flurbiprofen model:
CSF kinetics
Modeling the distribution of flurbiprofen into CSF
was challenging.
Only unbound flurbiprofen can enter CSF.
However, in CSF the concentrations of
flurbiprofen were circa sevenfold compared to
unbound flurbiprofen in plasma.
This happened probably because of protein
binding in CSF (the CSF observations reflect the
total amount of flurbiprofen in CSF).
Pyry Välitalo 1.10.2009
EXTRA: Features of the flurbiprofen model:
CSF kinetics
An intercompartmental clearance QCSF was
estimated to describe the movement between
central compartment and cerebrospinal fluid
(CSF). The rate from central to CSF was
adjusted by fraction unbound and an uptake
factor (UPTK).
QCSF= θQCSF
UPTK= θUPTK
K25=QCSF*FU/V2*UPTK
K52=QCSF/V5
;K25 and K52 represent rate constants from
;central compartment to CSF and from CSF to
;central compartment
Pyry Välitalo 1.10.2009