Supplement 3 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 Rostami-Hodjegan1,3, 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. System-related model parameters
1.1. Volumes of model compartments
1.2. Blood flows of model compartments
1.3. Small intestinal lumen radius
1.4. Gastric emptying and small intestinal transit rates
2. Stochastic population variability in system-related parameters
3. Figures
4. Tables
5. References
1. System-related model parameters
1.1. Volumes of model compartments
The volumes of the several model compartments were not considered constant across the population but were dependent on the total body weight of each individual using the relationship in Eq.S3.1,
wherefi is the fractional volume of tissue iwith respect to total body weight (WTE), (assuming a density of 1 kg/L for the studied tissues). The fractional volumes for each model compartment are reported in Supplementary Table S3.1. With this approach the population variability in weight propagates to the volumes of each model compartment. Note that the sum of the fractional volumes (Σfi) add up to one in order that the whole body volume equals the total body weight of each individual. The volumes used for the stomach content and small intestinal lumen are also reported in Supplementary Table S3.1 and they correspond to a fasted state.
1.2. Blood flows of model compartments
The cardiac output (CO) of each individual was allometrically related to total body weight (WTE)[1] using the relationship in Eq.S3.2,
where CO is expressed in mL/min and WTE in kg. The blood flows of the model compartments were not considered constant across the population but were dependent on cardiac output of each individual using the relationship in Eq.S3.3,
whereqi is the fractional blood flow of organ i with respect to the cardiac output. With this approach the population variability in weight propagates to blood flows of each model compartment. The fractional blood flows for each model compartment are presented in Supplementary Table S3.1. Note that the sum of the fractional blood flows (Σqi) add up to one in order that the sum of blood flows present in the model equals the cardiac output of each individual.
1.3. Small intestinal lumen radius
Body surface area (BSA) of each individual was calculated with the Du Bois & Du Bois formula (Eq.S3.4),
where WTE and HTE are weight and height expressed in kg and cm respectively. Subsequently, the diameter and length of the duodenum, jejunum and ileum segments of the gastro-intestinal tract was related with the BSA of each individual with Eq.S3.5-3.10 (Simcyp v13),
whereD and L refer to the diameter and length respectively of the of the duodenum (du), jejunum (je) and ileum (il) segments and they are expressed in metres. The radius of the small intestinal lumen (Rsil) used for each individual was then calculated with Eq.S3.11 as the mean of the radius of the different segments weighted by their respective lengths,
With this approach variability in BSA propagates to the radius of the small intestinal lumen and subsequently to the absorption process of SV.
1.4. Gastric emptying and small intestinal transit rates
Gastric emptying and small intestinal transit were assumed to be first-order processes. The gastric emptying rate constant (kge) and the small intestinal transit rate constant (ksit) were calculated with Eqs.S3.12-S3.13 as the reciprocal of the gastric (GRT) and small intestinal (SIRT) residence times respectively.
2. Stochastic population variability in system-related parameters
Variability in system parameters can be incorporatedby linking them with observed variables such as total body weight or body surface area through the use of fractional multipliers or scaling coefficients. As reported above, such an approach was followed here to introduce population variability in the volumes and blood flows of model compartments (sections 1.1 and 1.2 respectively) and in the radius of the small intestinal lumen (section 1.3). An advantage of this approach is that it can capture the correlation between system parameters through their dependency on a well-defined physiological covariate, such as body weight. However, a clear distinction should be made here. Such an approach does not capture the stochastic-random variability but only the fraction of variability that can be explained through the dependency of system parameters on known covariates (e.g. weight). Therefore, in the current work the aim was to consider also stochastic-random variability for system-related parameters with the inclusion of inter-individual variability random effect terms.
Gastric residence time (GRT) and small intestinal residence time (SIRT) are parameters with well documented population variability [2, 3]. Therefore this variability was introduced here in order to assign different gastric emptying and small intestinal transit rate constants in each individual. Both GRT and SIRT were assumed to follow a generalisation of the logit-normal distribution (see NONMEM code in Supplement 9). The reason of applying this kind of distribution instead of the commonly assumed log-normal is that the former is much more flexible and has a bounded support between a lower and an upper limit. Therefore estimation or sampling of non-physiological parameter values is avoided. Further technical details regarding the implementation of this distribution are outside the focus of this manuscript, but are extensively described inTsamandouras et al (in preparation). Taking advantage of the available prior information[2-6]we assumed population distributionsfor GRT and SIRT, the probability density functions of which are illustrated in Supplementary Figure S3.1. In these assumed distributions GRT had an expected value of 0.25 h, coefficient of variation (CV) of 37% and lower and upper bounds of 0.05 h and 1 h respectively. Similarly, SIRT had an expected value of 214.1 mins, CV of 42% and lower and upper bounds of 29 mins and 999 mins respectively.
The inter-individual variability (IIV) random effect terms related to GRT and SIRT were not fixed during estimation. However, they were strongly supported by the assumed prior distributions reported above and the confidence in these priors was considered so high that practically these random effects behaved as almost fixed. More specifically, given the prior knowledge, the inter-individual variability variance terms related to GRT and SIRT are 0.60652 and 0.6172 (for derivation, see Tsamandouras et al (in preparation)) and the level of confidence related to them is assumed so high as they are known with a standard error of 0.001. Therefore, the hyperparameters of the inverse-Wishart (IW) distributed prior related to GRT are 0.60652 and 270620 (mode and degrees of freedom respectively). Similarly, the hyperparameters of the IW distributed prior related to SIRT are 0.6172 and 289850 (mode and degrees of freedom respectively). In general, a high value of degrees of freedom assigned to the IW prior, corresponds to a high level of confidence to the related prior. In the current work the exact values of degrees of freedom assigned to the IW prior given a specific level of confidence (uncertainty) on the IIV term, were calculated with the analytical expressions described by Dokoumetzidis and Aarons [7]. Given the model structure and the fact that only plasma SV/SVA concentrations are available, it is very unlikely that IIV terms in GRT and SIRT can be robustly and uniquely estimated from the data disentangled from IIV in other model parameters involved during dissolution / absorption. It was due to this reason that it was decided to assign a very high level of confidence (SE of 0.001) on prior knowledge of GRT and SIRT population variability. However, we avoided strictly fixing these IIV terms in order to illustrate the framework that prior knowledge can be used to support the estimation of variability terms in complex PBPK system parameters.
Incorporation of random-stochastic unexplained variability in the volumes and blood flows of model compartments is a very challenging task as their joint distributions are subject of physiological constraints. Specifically the sum of the compartmental blood flows in an individual should always equal to the cardiac output and the sum of the compartment volumes should equal to the total body volume. In order to overcome this difficulty, we proposed that a multivariate logistic normal distribution can be assigned to the fractional multipliers fi (Eq.S3.1) and qi (Eq.S3.3) in the population level, taking advantage of the convenient properties of this particular distribution. Further technical details regarding this distribution and its implementation in a hierarchical PBPK modelling framework are outside the focus of this manuscript. However, they are extensively provided in Tsamandouras et al(in preparation). Model development and parameter estimation was therefore performed by both including or omitting stochastic-random population variability in compartmental blood flows and volumes (a fixed 5% CV was assumed for the population variability of the fractional multipliers fi and qi). For both approaches the model parameter estimates were equivalent (the ratio between estimates when omitting compared to including this population variability was in the range of 0.984 – 1.030 for all fixed and 0.993 – 1.042 for all random effect parameters) and the model diagnostics were similar. These results reflected only a minor (non-important) inflation of the random effect IIV estimates related to drug-related parameters when this additional level of variability in blood flows and volumes is omitted. Therefore, for the sake of simplicity and computational efficiency the final model and results presented throughout this work refer to the scenario when random population variability in compartmental blood flows and volumes is omitted and hence variability in these parameters derive solely through their dependency on total body weight (Eqs.S3.1-S3.3). The detailed results of the scenario where random population variability in compartmental blood flows and volumes is taken into account can be provided upon request.
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3. Figures
Figure S3.1. Probability density functions (pdf) of the assumed population distributions for gastric (GRT) and small intestinal (SIRT) residence times (left and right respectively)
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4. Tables
Table S3.1: System-related model parametersParameter / Value / Reference
fsiw / 0.0091 / [8]
flt / 0.0257 / [8]
flv / 0.00296 / [9](a)
fbl / 0.0743 / [10]
fm / 0.4 / [8]
fspl / 0.0114 / [8] (b)
frob / 0.4765 / Calculated(c)
Rsil (cm) / 1.61 / Calculated(d)
kge (h-1) / 4 / [4, 5](e)
ksit (h-1) / 0.28 / [3](e)
qsiw / 0.105 / [11]
qha / 0.065 / [11]
qm / 0.145 / [11]
qspl / 0.095 / [11](f)
qrob / 0.590 / Calculated(g)
Vstom (L) / 0.05 / [4, 5]
Vsil (L) / 0.608 / [4]
fi represents the fractional volume of tissue-compartment (i)with respect to total body weight; qi represents the fractional blood flow of tissue-compartment (i) with respect to the cardiac output; subscripts bl, lt, lv, m, siw, ha, spl and rob are referringto systemic blood, liver tissue, liver vascular, muscle, small intestinal wall, hepatic artery, “rest of splachnic” and“rest of body” respectively. For example the muscle tissue in the model represents 40% of the total body weight and it is receiving 14.5% of the cardiac output of an individual. Vstom and Vsil refer to the volume of stomach content and small intestinal lumen respectively;Rsilrepresents the radius of small intestinal lumen;kge and ksit represent the gastric emptying and small intestinal transit rate constants respectively.
(a) The fractional volume of the liver vascular compartment has been calculated as 11.5% of the liver tissue fractional volume.
(b) The fractional volume of the “rest of splachnic” compartment(see Supplement 1) has been calculated as the sum of the fractional volumes of stomach, large intestine, pancreas and spleen which are 0.0021, 0.0053, 0.0014 and 0.0026 respectively.
(c) The fractional volume of the “rest of body” compartment has been calculated as 1 minus the sum of other fractional volumes in the model.
(d) The reported radius of small intestinal lumen has been calculated with Eqs.S3.4-S3.11 using the average weight and height of the individuals in the analysed datasets. Note that this is only a reference-typical value as this parameter is not fixed across the population.
(e) Note that the gastric emptying and the small intestinal transit rate constants reported here are only reference-typical values as these parameters are not fixed across the population.
(f) The fractional blood flow of the “rest of splachnic” compartment(see Supplement 1) has been calculated as the sum of the fractional blood flows of stomach, large intestine, pancreas and spleen which are 0.01, 0.045, 0.01 and 0.03 respectively.
(g) The fractional blood flow of the “rest of body” compartment has been calculated as 1 minus the sum of other fractional blood flows in the model.
5. References
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