Supplement 2 for the manuscript: “Development and application of a
mechanistic pharmacokinetic model for simvastatin and its active
metabolite simvastatin acid using an integrated population PBPK
approach”
Nikolaos Tsamandouras1, Gemma Dickinson2, Yingying Guo2, Stephen Hall2, Amin RostamiHodjegan1,3, Aleksandra Galetin1, Leon Aarons1
1. Centre for Applied Pharmacokinetic Research, Manchester Pharmacy School, University of
Manchester, Manchester, UK.
2. Eli Lilly and Company, Indianapolis, IN, USA
3. Simcyp Limited, Blades Enterprise Centre, Sheffield, UK.
Contents
1. Practical details regarding the application of the IMPMAP estimation method ................................. 2
2. Results regarding the parameter estimation with IMPMAP ............................................................... 3
3. Tables .................................................................................................................................................. 5
4. References ........................................................................................................................................... 6
1
1. Practical details regarding the application of the IMPMAP estimation method
The first order conditional estimation method with interaction (FOCE-I) was primarily used for the
current population analysis. Briefly, this method approximates the analytically intractable loglikelihood function (a measure of how likely is to observe the obtained data, given a specific set of
population model parameters) with a first order Taylor series linearization around the conditional
estimates of the individual random effects [1].
However, in this work it was aimed to also evaluate the performance of an alternative estimation
algorithm, the Monte-Carlo importance sampling method assisted by mode a posteriori estimation [2],
namely IMPMAP in NONMEM 7 (originally MC-PEM algorithm with pmethod=4 in S-ADAPT).
IMPMAP is a Monte-Carlo expectation-maximisation (EM) method and therefore has the advantage
of not using the linearized approximation to the analytically intractable log-likelihood. Instead the
latter is evaluated by Monte Carlo integration (sampling) during the expectation step of the algorithm
followed by a single iteration maximisation step that advances population parameters towards the
maximum likelihood estimates. The Monte Carlo samples during the expectation step are generated
from a multivariate normal proposal density, then weighted according to the posterior distribution
(importance part of the algorithm) and their conditional means and variances are evaluated. For the
first iteration conditional modes and conditional first-order variances are evaluated as in FOCE and
these are used as the parameters of the proposal density. For the subsequent iterations the standard
importance sampling algorithm (IMP in NONMEM 7) uses the Monte Carlo evaluated conditional
mean and variance of each subject from the previous EM iteration as parameters of the sampling
density. On the other hand, the IMPMAP method differs in the fact that conditional modes and
conditional first-order variances are evaluated (as in FOCE) and used as the parameters of the
sampling density not only on the first (as in IMP) ,but at each iteration of the algorithm [2]. Due to
this difference the IMPMAP method is more robust than the standard IMP for complex highly
dimensioned PK/PD problems as it assures that for each iteration the proposal density is placed at an
informative position [3].
2
The evaluation of this algorithm, as an alternative to FOCE-I was motivated in the current work from
the need for decreased computation times for convergence in such a complex mechanistic model.
There is some evidence that this Monte Carlo EM algorithm is robust for complex high-dimensional
PK/PD problems [3] and for such models can be faster than FOCE (due to the efficient maximisation
step) without compromising the robustness of parameter estimates [4, 5]. In addition in such complex
models the computational burden of the expectation step can be alleviated with parallel computing. In
the current work the reported results from FOCE-I are considered as the reference parameter
estimates. However, estimation was also performed with IMPMAP, parallelised in 7 cores of an Intel ®
Xeon® X5570 processor (2.93 GHz, 24 GB of RAM). The convergence tester properties for IMPMAP
were CTYPE=2, CALPHA=0.05, CITER=10, CINTERVAL=1. This configuration practically means
that at each iteration the program performs a linear regression on the objective function, thetas, sigmas
and diagonals of omegas obtained during the last 10 iterations to assess if they are stationary. If the
linear regression slope is not statistically different from 0 (at the 0.05 significance level) for all
parameters tested then convergence is considered achieved and the estimation is terminated [6].
During the expectation step of the algorithm, 3000 random samples per subject (ISAMPLE=3000)
were drawn in order to obtain IMPMAP results with a low level of Monte Carlo noise. However,
estimation was also performed with the default IMPMAP configuration of 300 random samples per
subject (ISAMPLE=300) in order to assess if this configuration achieves faster convergence without
affecting parameter estimates. Finally, in order to implement IMPMAP as efficiently as possible
according to the recommendations in [4, 6], all fixed effect parameters were mu-referenced [6] and
the fixed effects with no random effects in the FOCE-I run were assigned a random effect referring to
inter-individual variability variance fixed to 0.0001.
2. Results regarding the parameter estimation with IMPMAP
The parameter estimation process with the parallelised IMPMAP estimation method was also
completed successfully. A comparison between the parameter estimates between the FOCE-I and
IMPMAP methods is presented in Table S2.1. An IMPMAP optimisation with 3000 drawn random
3
samples per subject (ISAMPLE=3000) was initially performed in order to obtain IMPMAP results
with a low level of Monte Carlo noise. The estimation procedure with the IMPMAP method
converged after only 5.48 hours. It can be observed that both the methods converged to a fairly similar
objective function (Table S2.1) taking into account that the FOCE-I method provides an
approximation of the log-likelihood and that the model used with the IMPMAP method had for
technical reasons (see above) some additional inter-individual variability variance terms which were
fixed to be negligible. The parameter estimates between FOCE-I and this IMPMAP run were very
similar (Table S2.1). More specifically all the IMPMAP parameter estimates of the population model
including both fixed and random effects were within 2 SE of the FOCE-I estimates which were
treated as reference (Table S2.1) ,where SE is the standard error of estimate obtained from the FOCEI covariance step. Subsequently an IMPMAP optimisation was also performed with the default
IMPMAP configuration of 300 random samples per subject (ISAMPLE=300) in order to assess if the
faster convergence of this configuration did not affect parameter estimates. The latter configuration
achieved convergence in only 3.90 hours and this improvement in computation time did not
incorporate any additional bias on the estimates (Table S2.1). Specifically all parameter estimates
were almost identical with the previous IMPMAP run (ISAMPLE=3000) and still within 2 SE of the
FOCE-I estimates, with the single exception of the problematic inter-individual variability variance
associated with SVA CYP3A intrinsic clearance (CLintʹCYP3A,vitro, further discussed in Supplement 7,
section 1).
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3. Tables
Table S2.1: Comparison of parameter estimates between FOCE-I and IMPMAP methods
Model parameter
Structural model (d)
Peff (cm/h)
CLintCYP3A,vitro (μL/min/pmol CYP3A)
KPuT:P,m
KPT:B,rob
KPuT:P,lt
CLintʹCYP3A,vitro (mL/min/mg of MP)
KPuʹT:P,m
KPʹT:B,rob
CLuact (L/h)
khydr,pl (h-1)
khydr,buff (h-1)
khydr,S9 (h-1)
CLintgluc,vitro (μL/min/mg of MP)
khydr,hybrid (h-1)
Inter-individual variability (e)
GRT
SIRT
Peff
CLintCYP3A,vitro
KPT:B,rob
CLintʹCYP3A,vitro
KPʹT:B,rob
CLuact
khydr,hybrid
Residual variability (f)
epsSV
epsSVA
Objective function
Time to converge (h) (g)
FOCE-I (range) (a)
IMPMAP 3000 (b)
IMPMAP 300 (c)
0.24
2.63
8.37
2.95
7.97
-2.97
0.33
-0.05
9.02
-2.04
-3.65
-2.75
-0.99
1.02
(-0.15 , 0.64)
(2.47 , 2.79)
(7.90 , 8.84)
(2.80 , 3.10)
(7.60 , 8.34)
(-3.22 , -2.72)
(-0.04 , 0.70)
(-0.40 , 0.29)
(8.60 , 9.44)
(-2.22 , -1.86)
(-3.65 , -3.65)
(-2.76 , -2.74)
(-1.01 , -0.98)
(0.74 , 1.30)
-0.02
2.56
8.44
2.90
7.61
-2.94
0.31
0.03
8.69
-2.12
-3.65
-2.74
-1.00
0.97
-0.03
2.55
8.45
2.91
7.61
-2.94
0.31
0.07
8.67
-2.12
-3.64
-2.74
-1.00
0.98
0.37
0.38
0.88
0.14
0.13
0.14
0.40
0.41
0.34
(0.37 , 0.37)
(0.38 , 0.38)
(0.27 , 1.49)
(0.02 , 0.25)
(0.06 , 0.19)
(0.00 , 0.28)
(0.07 , 0.73)
(0.09 , 0.72)
(0.12 , 0.57)
0.37
0.38
0.83
0.17
0.16
0.23
0.39
0.42
0.39
0.37
0.38
0.83
0.17
0.16
0.38
0.34
0.44
0.38
0.27
0.12
9517.66
5.48
0.27
0.12
9518.74
3.90
0.27 (0.19 , 0.35)
0.12 (0.08 , 0.15)
9525.32
18.7
When any of the above parameters is relevant to both SV and SVA, the parameter abbreviation referring to SVA is followed
by a prime. Abbreviations are exactly as listed in Table 2 (Manuscript). Additional abbreviations used here are: MP:
microsomal protein; GRT and SIRT: gastric and small intestinal respectively residence time.
(a) Estimate from the FOCE-I method followed by a range which corresponds to (estimate - 2SE, estimate + 2SE), where SE
is the standard error of estimate obtained from the FOCE-I covariance step.
(b) Estimate from the IMPMAP method when 3000 (ISAMPLE=3000) random samples per subject was requested to be
drawn during the expectation step of the algorithm.
(c) Estimate from the IMPMAP method when 300 (ISAMPLE=300) random samples per subject was requested to be drawn
during the expectation step of the algorithm.
(d) Typical population parameter estimates for the structural model are reported in the domain of the estimated logtransformed parameter.
(e) Inter-individual variability is reported in the domain of the estimated variance term.
(f) Residual variability is reported in the domain of the estimated variance term (an additive error model was applied on
the log-transformed data). epsSV and epsSVA correspond to the residual error variance associated with SV and SVA
plasma concentrations respectively.
(g) Note that estimation with FOCE-I was not parallelised, while estimation with IMPMAP was parallelised in 7 cores.
5
4. References
1.
2.
3.
4.
5.
6.
Wang Y. Derivation of various NONMEM estimation methods. J Pharmacokinet
Pharmacodyn. 2007;34(5):575-593.
Bauer JR. Technical guide on the Expectation-Maximization population analysis methods in
the NONMEM 7 program. Ellicot City, Maryland: ICON Development Solutions; 2013.
Bulitta J, Landersdorfer C. Performance and robustness of the Monte Carlo importance
sampling algorithm using parallelized S-ADAPT for basic and complex mechanistic models.
AAPS J. 2011;13(2):212-226.
Gibiansky L, Gibiansky E, Bauer R. Comparison of Nonmem 7.2 estimation methods and
parallel processing efficiency on a target-mediated drug disposition model. J Pharmacokinet
Pharmacodyn. 2012;39(1):17-35.
Bauer R, Guzy S, Ng C. A survey of population analysis methods and software for complex
pharmacokinetic and pharmacodynamic models with examples. AAPS J. 2007;9(1):E60-E83.
Bauer JR. NONMEM users guide: Introduction to NONMEM 7.2.0. Ellicot City, Maryland:
ICON Development Solutions; 2011.
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