Physiological modeling of inter-individual variability: Combining
PBPK modeling and Markov-Chain-Monte-Carlo approaches
Markus Krauß1,2,*, Michaela Meyer1, Lars Kuepfer1, Martin Hobe1, Thomas
Eissing1, Linus Goerlitz1
* [email protected]
1 Computational
2 Aachen
Systems Biology, Bayer Technology Services GmbH, Leverkusen, Germany
Institute for Advanced Study in Computational Engineering Sciences, RWTH Aachen, Aachen, Germany
Motivation
Results
The assessment of inter-individual variability is a key aspect in physiology-based pharmacokinetic
(PBPK) modeling. Physiological differences like age or blood protein content have to be considered
since these factors contribute to the pharmacokinetic (PK) variation. Only if the population wide
variation of parameters in reference populations is known in detail, reliable extrapolations to other
populations can be performed. Markov-Chain-Monte-Carlo (MCMC) approaches provide a state of
the art method to determine such variations for better predictions of individualized PK [1, 2].
Moreover, PBPK models simulate the behavior of exogenous and endogenous compounds under
consideration of a highly realistic parameterization. Thus, our combined PBPK-MCMC approach
supports the identification of relevant patient subgroups by statistical inference of physiological
differences such as enzyme activities or renal excretion. Such integrative approaches may therefore
have significant implications for the development of individualized therapeutic strategies with a
favorable risk-benefit profile in the future.
The combined PBPK-MCMC approach was performed to generate about 200,000 samples. The first
100,000 points were discarded because the chain had not converged, yet (Fig. 4). From the
remaining samples 200 parameter vectors are randomly chosen to obtain independent draws from
the Markov chain. The pravastatin PBPK model was parameterized with every parameter vector and
a PK simulation was performed. The results proof the reliability of the presented approach since the
experimental data from [11] can be correctly described by the simulated PK curves (Fig. 5 A), taking
the inter-individual variability between the 10 patients into account. By adding prior genotype
information of liver influx transporter OATP1B1 [11,12] onto the single patients, the PK curves can be
divided into three subgroups (Fig. 5 B).
-2.5
0
1
0
0
2
GET
200
1
5
0
0
1
2
400
200
0
0
1
2
Liver
0
0
1
Gall Bladder
Heart
1
2
5
x 10
2
5
x 10
1.5
1
0.5
0
1
x 10
100
50
0
0
0.9
0.8
0.7
2
5
km OAT3
Muscle
1
x 10
Venous Blood
Kidney
0
0
2
5
B
10
5
0
0
sd
1000
fu
Portal Vein
4
x 10
0.5
1
2
0
km OATP1B1
Intestine
5
x 10
2000
2
x 10
5
x 10
lip
Stomach
fac OATP1B1
5
1
2
Fat
220
200
180
0
1
5
2
5
x 10
x 10
Figure 4: Sampling traces. All in all,
209,000 samples were generated with
the combined PBPK-MCMC approach.
A) Traces of all samples are
monitored for all seven individual
parameters
and
measurement
uncertainty σ2 (sd) for one patient
exemplarily. B) Traces of all samples
are monitored for all five global
parameters.
Global
parameters
contain substance-specific parameters
and remain the same for all patients.
20
10
0
0
1
2
5
5
x 10
x 10
Skin
Bone
A
0.6
5 % - 95 % quantile
median
exp. data of all patients
Brain
Spleen
0.5
Lung
B
pravastatin [µM]
Gonads
Figure 1: PBPK model structure
implemented in PK-Sim®. The basic
model structure can be extended by
additional processes such as active
transporters
or enzyme-mediated
metabolization [3].
0.3
0.2
0.1
0
0
100
200
300
400
time [min]
500
600
700
pravastatin [µM]
Arterial Blood
pravastatin [µM]
0.4
pravastatin [µM]
PBPK models describe the mechanisms underlying the
absorption, distribution, metabolism and excretion (ADME) of a
substance within the body at a high level of detail. In contrast to
basic compartmental models, PBPK models intend to represent
all important physiological processes mechanistically. The
models are based on a large amount of prior physiological and
anatomical information. Beside these parameters taken from
huge data collections, parameters are calculated from drugdependent properties such as lipophilicity and protein binding.
The resulting models can then be used to quantitatively and
mechanistically simulate drug concentration profiles in various
organs and tissues (Fig. 1). Furthermore, the introduction of
additional mechanisms such as active transporters and
clearance processes is also possible [3-5].
A PBPK model of pravastatin was created with the software tools
PK-Sim® and MoBi®. The model represented hepatic drug uptake
by OATP1B1 and renal uptake by OAT3 as well as active efflux
by MRP2 and metabolization by sulfotransferases in either liver,
kidney and intestine (Fig. 2). Enzyme activity was quantified by
using expression data for the respective organs. To this manner,
tissue specific vmax was divided into a global vmax* which remains
constant in every organ and the organ specific relative
expression [6].
0.1
1
x 10
fac MRP2
CLlum
Physiology-based pharmacokinetic modelling
0
0
5
x 10
0.2
100
2
fac OAT3
-2
200
km MRP2
Pint
A
400
ITT
-1.5
0.6
5 % - 95 % quantile
median
exp. data of patients with TT genotype
0.4
0.2
0
0
100
200
300
400
500
600
700
0.2
5 % - 95 % quantile
median
exp. data of patients with TC genotype
0.1
0
0
100
200
300
400
500
600
700
0.6
5 % - 95 % quantile
median
exp. data of patients with CC genotype
0.4
0.2
0
0
100
200
300
400
time [min]
500
600
700
Figure 5: Pharmacokinetics of Pravastatin after performing the MCMC approach. 200 parameter vectors are
randomly chosen of the resulting posterior distribution samples to obtain independent draws. For every parameter vector
the PBPK model is parameterized and the respective PK is simulated. A) PK of all 10 patients is shown (5%- 95%)
together with the median and experimental data from [11]. B) PK is shown after dividing the patients corresponding to
the genotype of OATP1B1 (red – TT genotype, blue – TC genotype, green – CC genotype).
We next examined, whether the three genotypes can be identified by data-driven separation of the
vmax*,OATP1B1. Therefore, 5,000 parameter vector samples are chosen randomly from the posterior.
The vmax* of all three included transporters are depicted for all 10 patients (Fig. 6 A).
Figure 2: Integrated enzymes and transporters which are involved in the metabolization and distribution of
pravastatin. The transporters are integrated in different organs as illustrated [6].
A
Markov-Chain-Monte-Carlo approach
Well-known optimization-based approaches are usually point estimators which are used to identify
the optimal parameter vector representing the best fit to experimental data. But often only a region
can be identified which includes the true parameter vector θ. The Bayesian framework provides a
state of the art method which can handle such uncertainty about parameter vectors and which at the
same time can be used for the investigation of the variability and co-variability of parameters.
Furthermore, prior information of the parameters can be integrated. Bayes’ theorem
𝑝(𝐷|𝜃)∙𝑝(𝜃)
𝑝 𝜃𝐷 =
(1)
B
www.systems-biology.com
𝑝(𝐷)
combines the likelihood of the measurements p(D|θ) with prior information of the parameters p(θ) to
calculate the resulting posterior distribution p(θ|D) [7]. Since the direct calculation of the posterior
distribution is often impossible in high-dimensional problems we used the MCMC approach for
estimating the posterior [8,9,10]. MCMC generates random samples of the posterior by setting up a
Markov chain which has the posterior distribution as its long-run distribution [7].
We performed block-wise Metropolis-Hastings MCMC in combination with PBPK modeling to asses
the variability of seven individual and five global parameters together with unknown measurement
accuracy after pravastatin administration of 40 mg in a group of ten patients (Fig. 3). Experimental
PK data came from [11]. All in all, 85 unknown parameters have to be sampled from the posterior.
Prior information is subdivided into three parts: (1) informative (p(𝜃I)) and (2) non-informative (p(𝜃N))
distributions as well as (3) a prior distribution for measurement uncertainty (p(σ)):
𝑝 𝜃 = 𝑝(𝜃 𝐼 ) ∙ 𝑝(𝜃 𝑁 ) ∙ 𝑝(𝜎).
(2)
Figure
3:
MCMC
approach
in
combination with PBPK modeling. Prior
information can be integrated into the
approach. Resulting posterior distribution
takes the prior information, experimental
data and the simulation of the underlying
model structure into account. 85
parameters are varied. Five parameters
are global parameters (lipophilicity,
unbound fraction of the drug in plasma,
Km,OAT3, Km,OATP1B1, Km,MRP2), which have to
be the same for all patients. Eight
parameters are individual parameters
(intestinal permeability, intestinal transit
time, gastric emptying time, luminal
clearance rate, vmax*,OAT2, vmax*,OATP1B1,
vmax*,MRP2, σ2), which are varied in all ten
patients. We here use a block-wise
Metropolis-Hastings MCMC approach:
After sampling and accepting/rejecting the
global
parameters
which
include
compound specific parameters, the individual parameters are sampled and accepted/rejected. Acceptance criteria depend on a least-square error model which
assumes that errors are independent and lognormal distributed.
Figure 6: Dependencies of the
three included transporter
activities. A) The vmax* of the
three
included
transporters
MRP2, OATP1B1 and OAT3 of
all patients are monitored by
randomly
chosen
5000
parameter vectors from the
posterior samples. B) Genotype
information
of
the
liver
transporter OATP1B1 is mapped
onto the 5000 points also shown
in A. CC genotype (green points)
ranges
from
0
to
2400
µmol/L/min for OATP1B1 and
from 84 to 310 µmol/L/min for
MRP2. TC genotype (blue points)
and TT genotype (red points) are
distributed over the entire range.
Here, a clear separation is difficult to
identify for vmax*,OATP1B1. Additionally,
co-variation to the other transporter
activities is also hard to monitor.
Therefore, genotype information for
OATP1B1 was mapped onto the data
using the same color code as seen
before in Fig. 4 B (Fig. 6 B) [11,12].
Differentiation between the TC and TT
genotype is still difficult. However, the
parameter combinations corresponding
to the CC genotype can be slightly
separated from the TC and TT
genotype, since the CC genotype
appear only in combinations with
smaller vmax*,OATP1B1 and vmax*,MRP2.
Conclusions/Outlook
Overall the results constitute that the PBPK-MCMC approach offers a promising tool for the
prediction of inter-individual variability in groups of patients. Furthermore, the results suggest the
possibility of patient subgroup stratification only based on transporter activity by mapping genotype
information onto the received posterior samples of parameter vectors. For an even clearer datadriven separation of the genotypes, more prior information may be added to the approach which
reinforce the physiologically constraint of the parameter space. That implies additional PK data such
as urinary excretion data as well as the integration of patient meta data such as age, weight or sex.
Consequently, the combined PBPK-MCMC approach may lead to inferences about variation within
whole physiology, to determine the PK of real patients in the future.
References
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[2] Bois, F.Y. et al., 2010, Toxicology 278: 256-267
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[3] Eissing, T. et al., 2011, Front. Physio. 2
[4] Willmann, S. et al., 2003, Biosilico 1: 121-124
[5] PK-Sim® 4.2, 2010, www.systems-biology.com
[6] Meyer, M. et al., 2012, Drug Metab. Dispos. 40: 892-901
[7] Bolstad, W., 2010, John Wiley & Sons, Inc.
[8] Goerlitz, L. et al., 2011, Environ. Sci. Technol. 45: 4429–4437
[9] Hastings W. K., 1970, Biometrika 57: 97-109
[10] Geman S. and B. Geman, 1984, IEEE Trans. Pat. Mach. Intell. 6: 721-741
Acknowledgements
The authors acknowledge financial support by VirtualLiver (0315747) funded by the German Federal
Ministry of Education and Research..