Supplement 4 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.

Contents1. Additional notes on setting the prior distributions on drug-related parameters22. Drug-related model parameters42.1. Dissolution-related parameters42.2. Absorption rate constant of SV from the intestinal lumen into the epithelium42.3. SV and SVA blood to plasma ratios72.4. SV and SVA unbound fractions in different model compartments82.5. Tissue to blood partition coefficients92.6. Parameters related to hepatic permeability112.7. Parameters related to CYP3A metabolism of SV122.8. Parameters related to CYP3A metabolism of SVA152.9. SV to SVA hydrolysis related parameters.162.9.1. SV to SVA hydrolysis in systemic blood.172.9.2. SV to SVA hydrolysis in the liver vascular compartment.172.9.3. SV to SVA hydrolysis in the liver tissue.172.9.4. SV to SVA hydrolysis in the muscle tissue.182.9.5. SV to SVA hydrolysis in the small intestinal wall and rest of body compartment.182.10. SVA to SV lactonisation related parameters.203. Notes on parameter identifiability234. Random population variability in drug-related model parameters245. Figures276. References31

1. Additional notes on setting the prior distributions on drug-related parameters

As the available prior knowledge is in the domain of the original untransformed parameter (assumed log-normal) the hyperparameters assigned to the respective normal prior (referring to the natural logarithm of the parameter) were calculated by using normal/log-normal reverse algebra [1], which is described below:

Assuming that a variable follows log-normal distribution with mean (M) and variance (V), then the mean () and standard deviation () of the respective normally distributed variable can be calculated with Eqs.S4.1 and S4.2 respectively.

Similarly, assuming that a variable follows normal distribution with mean () and standard deviation (), then the mean () and variance (V) of the respective log-normally distributed variable can be calculated with Eqs.S4.3 and S4.4 respectively.

Finally, it should be noted that as the available prior information derives from in vitro and in silico experiments, sometimes certain transformations need to be applied either in terms of unit conversion or in terms of scaling in order to obtain the desirable parameterisation. However, usually these are linear transformations (multiplication with a real positive constant a) of the variable for which prior information is available and therefore this information can be easily translated in the domain of the transformed/corrected variable. Two basic distribution properties that allow such a translation and were applied in this work are provided below:

, where a is a real number and , refer to the mean and standard deviation of the normally distributed

, where a is a real number and , refer to the mean and standard deviation of the normally distributed natural logarithm of

However, although transformations of the prior knowledge are possible in order to re-parameterise the model, such a process should be performed with care due to the following reasons. Firstly, unlike maximum likelihood estimation, MAP estimates are not invariant under re-parameterisation [2, 3]. The second and most important reason for PBPK models informed from IVIVE is that the translation of the prior knowledge to the desirable parameterisation incorporates an additional level of uncertainty. This argument is illustrated with the following example. Assume that the available prior information related to SV hepatic clearance comes from an in vitro experiment performed in human liver microsomes and therefore is in the domain of intrinsic clearance (CLint) expressed per mg of microsomal protein. It is tempting to transform this prior knowledge (CLint estimate and the related uncertainty) in the domain of the total hepatic clearance and perform parameter estimation with the latter parameterisation that is more easily interpretable. However, such a transformation involves scaling factors such as the amount of microsomal protein in the total liver which is not known with absolute certainty. Therefore, if the in vitro -derived prior is translated to the domain of total hepatic clearance (e.g. with the relationships S4.5 or S4.6) by treating these scaling factors as real constants, then this prior uncertainty that will be reported related to total hepatic clearance will be misleading. For this reason, in the current work we tried as much as possible to avoid transformations of the prior distributions and ideally parameterise the model in the domain where the prior information is.

2. Drug-related model parameters2.1. Dissolution-related parameters

SV solubility in stomach content (Solstom) was considered to be 14.5 g/ml, based on literature reports of simvastatin solubility in pH 1.2 buffer [4]. SV solubility in small intestinal lumen contents (Solsil) was considered to be 16.4 g/ml, based on literature reports of simvastatin solubility in Fasted State Simulated Intestinal Fluid (FaSSIF), pH=5 [4]. The diffusion coefficient (D) of SV was predicted from the molecular weight (418.6) within the Simcyp v13 [5] prediction toolbox to be 4.04 10-4 cm2/min. SV particles were considered to have a particle radius (r) of 9.18 calculated as the mean value reported in three different publications that used scanning electron microscopy to analyse particle size [6-8]. SV was assumed to have a particle density () of 1.2 g/ml, which is the default value in Simcyp v13. Finally the diffusion layer thickness (h) was set to 9.18 based on the assumption that for particles with radius less than 30, the diffusion layer thickness equals the radius [9]. All the above parameter values were utilized in the models differential equations (see Supplement 1) that describe the dissolution process of simvastatin.

2.2. Absorption rate constant of SV from the intestinal lumen into the epithelium

The absorption rate constant from the intestinal lumen into the small intestinal epithelium (ka) was determined with Eq.S4.7 [10].

,where Peff is the effective permeability of the compound, Rsil is the radius of the small intestinal lumen (see Supplement 3) and Lsil is the length of the small intestinal lumen. Eq.S4.7 assumes that the small intestinal lumen is a perfect cylinder in order to calculate the intestinal surface area and volume. Although it has to be acknowledged that this assumption may introduce some bias, Eq.S4.7 is generally considered up to now the gold standard approach to determine ka using in vitro-in vivo extrapolation (IVIVE).

The effective permeability (Peff) of SV has been reported to be 4.3 m/s [11], as calculated from apparent permeability (Papp) data of SV (215 nm/s) determined in MDCK-MDR1 cells in vitro [12]. The experimentally determined Papp is converted to Peff using a regression equation (Eq.S4.8) developed from a set of 20 different passively permeable compounds for which both in vivo Peff and in vitro Papp were available [12].

Usually, the effective permeability (Peff) calculated from Papp data is considered as a fixed parameter in PBPK modelling without accounting for any uncertainty associated with such a prediction.

The publication where the regression equation for Peff was developed [12] reports only relative standard errors (RSE) associated with the parameter estimates of slope (15%) and intercept (17%). However, of more interest is the standard error associated with a future individual prediction of Peff given an experimentally determined Papp value. In order to determine this error the data used to develop Eq.S4.8 (in vitro Papp values in MDCK-MDR1 cells [12] and in vivo Peff values in the upper jejunum [13] for a set of 20 passively permeable compounds) were extracted from the literature and the linear regression model was re-fitted (Figure S4.1). The determined regression model was exactly the same with the regression equation that has been already reported (Eq.S4.8). However, re-fitting of the data made feasible the calculation of the standard error for a future individual prediction, SE(Ypred), (Eq.S4.9).

, where s is the square root of the mean-squared error, n is the number of observations, x represents the sample of the independent variable, the mean of x and the value of the independent variable that we want to predict for. Therefore, in simvastatin example for the in vitro determined simvastatin Papp of 215 nm/s, the predicted log10(Peff) is 0.6336 and the standard error associated with such a prediction is 0.4715 (based on Eqs.S4.8 and S4.9) . Using this calculated standard error it can be shown that simvastatin Peff is predicted to be 4.3 m/s but with wide 95% prediction intervals of (0.43, 41.56), (Figure S4.1). This uncertainty propagates to the calculation (Eq.S4.7) of SV ka, which subsequently is predicted (using average value of the small intestinal lumen radius) to be 1.92 h-1, but with 95% prediction intervals of (0.19, 18.59). Sensitivity analysis performed indicated that the impact of this uncertainty on the plasma concentration profiles of SV and SVA is more than substantial (Figure S4.2).

In this work, the uncertainty related to the prediction of Peff is summarised in terms of a prior distribution. Assuming that the log10(Peff) parameter follows a normal distribution N (, 2), with and derived from Eqs.S4.8 and S4.9 respectively, it follows from the relationship in Eq.S4.5 that

Finally as up to now Peff is expressed in m/s, and in the model is applied in cm/h, it follows from the relationship in Eq.S4.5 that after applying the unit conversion factor of 0.36 and calculating the mean and variance of the linearly transformed variable, that the normal prior assigned to the ln(Peff) parameter (in cm/h) has a mean of 0.4372 and a variance of 1.1787. Note that the normal prior is supplied in the ln(Peff) parameter domain, in order that the actual Peff will follow a log-normal distribution and sampling of negative values is avoided.

However, the described prior distribution is relatively diffuse and as shown above the Peff values that can be sampled from such a prior have a very wide range leading to very different plasma concentration profiles (Figure S4.2). Therefore, although this relatively diffuse prior (prior A) will be primarily used to assist parameter estimation (as it represents the true uncertainty related to the performed in vitro-in vivo extrapolation), an alternative more informative prior distribution (prior B) will be used as a secondary option in the case where the parameter estimation process using the original relatively diffuse prior is unstable. This alternative prior distribution was constructed by treating the slope and intercept of the developed regression equation (Eq.S4.8) as independent parameters and assuming that the uncertainty in the prediction of Peff arises solely as a consequence of the standard errors associated with the slope (RSE = 15%) and intercept (RSE = 17%) parameter estimates. Although such a prior derives from oversimplification and does not correctly reflect the true uncertainty related to the performed IVIVE, it allows the provision of a more informative prior to aid parameter estimation and avoids the fixation of the Peff parameter value in the model. Using derivations similar to those reported above for the original primary prior (prior A), it follows that the alternative normal prior (prior B) assigned to the ln(Peff) parameter (in cm/h) has a mean of 0.4372 and a variance of 0.703.

2.3. SV and SVA blood to plasma ratios

The blood to plasma ratio (BP) of SV was considered to be 0.57 based on literature reports [11, 12]. The blood to plasma ratio of SVA (BP) is not available in the literature and therefore it was predicted to be 0.56 with the following procedure: The erythrocyte to plasma unbound SVA partition coefficient (KPuT:P,rbc) was predicted with in silico mechanistic equations [14], assuming that erythrocytes have no extracellular space and albumin or lipoprotein content [15]. Then blood to plasma ratio was calculated with Eq.S4.10, assuming that the hematocrit fraction in the blood (H) is 0.45 and the fraction unbound of SVA in plasma (fup) is 0.0548 [16].

The predicted value of 0.56 is in line with the common assumption that the blood to plasma ratio of acidic compounds is around 0.55 [17].

2.4. SV and SVA unbound fractions in different model compartments

The unbound fractions in plasma of SV (fup) and SVA (fup) were considered to be 0.0134 and 0.0548 respectively, as reported in the literature [16]. SV and SVA fractions unbound in blood (fubl) were calculated with Eq.S4.11.

The fractions unbound in the liver vascular compartment were assumed to be the same as in blood. The unbound fractions of both SV and SVA in the small intestinal wall compartment (fusiw) were assumed to be 1 (see Supplement 1). The unbound fractions of SV in muscle and liver tissues were calculated with Eq.S4.12 which assumes equilibrium conditions and well stirred distribution in these tissues (no active uptake or efflux),

, where k refers to muscle or liver tissue and KPT:B, k is the k tissue to blood partition coefficient. The fraction unbound of SV in the rest of body compartment was predicted with Eq.S4.13 [18],

, where R refers to the average value of the tissue interstitial fluid to plasma ratio of albumin and lipoproteins (considered to be 0.5 [18], as if the rest of body compartment was a lean tissue). The unbound fraction of SVA in the liver tissue cannot be determined with Eq.S4.12, as this hydrophilic compound is subject of active uptake into the hepatocytes mediated mainly by OATP1B1 [19]. Therefore the unbound fraction of SVA in the liver tissue was determined with the regression equation (Eq.S4.14), developed by Menochet et al using experimental data for 7 OATP1B1 substrates in rat hepatocytes [20].

, where the logD7.4 of SVA was set to 1.45 (SciFinder Database, American Chemical society).

2.5. Tissue to blood partition coefficients

Tissue to plasma unbound partition coefficients (KPuT:P) were predicted with in silico mechanistic equations, developed by Rodgers and Rowland [14]. These equations need input parameters that refer to human tissue composition which were extracted from [18]. In addition input parameters referring to the physicochemical properties of the compound are also needed. For the neutral SV, a logP of 4.68 (Drugbank) was used. For the acidic SVA, a logP of 4.54 (SciFinder Database, American Chemical society) and pKa of 4.31 (SciFinder Database, American Chemical society) were used. With this approach tissue to plasma unbound partition coefficients (KPuT:P) were predicted for the muscle and liver tissues for SV and only for the muscle tissue for SVA. These predicted values can be converted with Eq.S4.15 to the tissue to blood partition coefficients (KPT:B) that are used in the model differential equations.

The in silico predictions were used to construct informative priors for the tissue to plasma unbound partition coefficients (KPuT:P), which will aid their estimation from the clinical data. However, summarising these in silico predictions in terms of appropriate statistical prior distributions is difficult, as they produce only point estimates which are not accompanied with any measure of uncertainty. Previously, in a Bayesian PBPK model of diazepam [21] this was addressed by assuming an arbitrary 20% CV on the Poulin and Theil [22] in silico partition coefficient predictions. In the current work the prior distributions for the partition coefficients were constructed as explained below. Assume that the true tissue to plasma unbound partition coefficient (KPuT:P) equals to the in silico predicted tissue to plasma unbound partition coefficient (), multiplied by a random error factor variable (RF), Eq.S4.16.

Subsequently a log-normal distribution can be assumed for RF that will have the following characteristics: 1) The expected value of RF should equal to 1 (Eq.S4.17). This condition derives from the fact that the mechanistic equations developed by Rodgers and Rowland were not reported to produce systematically biased either upwards or downwards predictions when compared to experimentally determined partition coefficients for a wide set of compounds [14]. 2) The area under the probability density function of RF, f(RF), between 1/3 and 3 should equal to 0.84 (Eq.S4.18). This condition derives from the fact that in the work mentioned above [14], 84% of the in silico predicted values were found to agree with the experimentally determined values within a factor of 3. Therefore from Eqs.S4.16 and S4.18, it follows that the partition coefficient in silico predictions will be within a 3-fold error of the true value with 84% probability.

It can be easily shown using normal-log normal reverse algebra (section 1) and numerical integration, that a log-normal distribution for RF (RF ~ logN(, 2)) that satisfies both the above conditions (Eqs.S4.17, S4.18) has and parameters ( and are referring accordingly to the mean and standard deviation of the normally distributed natural logarithm of RF) of -0.266 and 0.7343 accordingly. The probability density function (pdf) of this log-normal distribution that has been assumed for RF is presented in Figure S4.3. Taking advantage of the relationships in Eqs.S4.6 and S4.16, we can easily derive the prior distribution referring to actual partition coefficient KPuT:P:

The specific priors assigned to each partition coefficient of the model are reported in Table 2 of the manuscript. With the described approach the partition coefficients used in the model will not be fixed to the respective in silico predictions, but instead the latter were used to construct priors which aid the parameter estimation process. Finally, as noted in Supplement 1, these mechanistic equations were not used to provide a prior relating to the partition coefficient for the empirical rest of body compartments of SV and SVA, as tissues are not properly lumped [23]. Therefore, these parameters (SV and SVA rest of body tissue to blood partition coefficients, KPT:B,rob and KPT:B,rob respectively) were solely estimated from the data (completely uninformative priors were assigned). Hence, these estimated rest of body partition coefficients are not of physiological interpretation per se. However, the product of the estimated partition coefficient and the known physiological volume of the rest of body compartment will be indicative of the extent of the compounds distribution in these empirical peripheral spaces.

2.6. Parameters related to hepatic permeability

As stated in Supplement 1, SV is assumed to have perfusion limited distribution into the liver. Therefore, the permeability surface products for unbound SV influx (PSuinf) and efflux (PSueff) across the basolateral membrane were considered to be equal and 10,000 times greater than the hepatic blood flow in order to satisfy these perfusion limited assumptions [11]. On the contrary SVA is suggested to enter the hepatocytes through a combination of active uptake and passive diffusion processes. The passive diffusion clearance across the basolateral membrane for unbound SVA (PSudif) is not reported in the literature. Therefore this parameter was predicted from the lipophilicity of the compound (logD7.4=1.45), using a regression equation (Eq.S4.19) which was developed from in vitro experimental data of 7 OATP1B1 substrates in human hepatocytes [24].

Note that PSudif in Eq.S4.19 is expressed in L/min/106 cells and therefore it was scaled up assuming the standard hepatocellularity of 120x106 cells/g of liver [25] and average liver weight of 1718 g (average liver volume of 1.591 L, corrected for liver density of 1.08 kg/L [26, 27]). Using Eq.S4.19 the passive diffusion clearance of unbound SVA was predicted to be 4 L/min/106 cells. This predicted value was comparable to the passive diffusion clearance of unbound atorvastatin acid (similar logP and pKa with SVA [28]) from in house experimental data in human hepatocytes (5.91 L/min/106 cells, unpublished data). The SVA passive diffusion clearance was fixed to the predicted value in the model and was not used to construct an informative prior. This was performed as previous work has shown that simultaneous accurate estimation of both passive diffusion and active uptake clearance (see below) is difficult using solely plasma concentration data [29].

To our knowledge the active uptake clearance of unbound SVA (CLuact) has not been reported in the literature from human hepatocyte experiments. Therefore, in the present study information for this parameter derived solely from the clinical data (a completely uninformative prior has been assigned). However, even in the cases where such in vitro data were available, their direct use in PBPK modelling (as fixed parameters or informative priors) may be questionable, as a systematic under-prediction of hepatic clearance has been reported [30]. This trend of under-prediction is consistent regardless of the cellular in vitro system used [24]. The under-prediction has been speculated to be attributed to several reasons (e.g. discrepancy in the expression of the transporters in the in vitro systems compared to in vivo). However, until these reasons are firmly determined and PBPK models are adapted accordingly, a middle-out approach as the one applied here (and in [29]) will be restricted to rely solely on clinical data for the estimation of this parameter, as a biased prior will affect model fit. Nevertheless, this implies that at least good informative priors are available for related model parameters such as passive diffusion and hepatic metabolic clearance to support identifiability of active uptake clearance given solely plasma data.

2.7. Parameters related to CYP3A metabolism of SV

Several reports can be found in the literature with regard to the intrinsic clearance of SV based on hepatic microsomal experiments [12, 31, 32], but the results are variable. An additional limitation is that the acid form (SVA) is commonly not monitored simultaneously with the lactone. Therefore, clear distinction between CYP3A-mediated metabolism and esterase metabolism of SV in microsomes is difficult. In the current work, the results of the 3 in vitro studies available in the literature [12, 31, 32], were combined in order to provide a composite informative prior that will facilitate the estimation of this parameter. These 3 in vitro studies report their results in different units (per mg of microsomal protein or per pmol of enzyme) while also some of them have applied a correction for non-specific binding in the microsomes and others not. Therefore in order to combine the information of these studies in a single prior, all the results were expressed in L/min/pmol of CYP3A and were corrected for microsomal binding. An enzyme abundance of 155 pmols of CYP3A per mg of hepatic microsomal protein was used [33]. The SV fraction unbound in microsomes (fumic) was predicted to be 0.122 and 0.36 for the in vitro studies [31] and [32] respectively, determined by Eq.S4.20,

, where Ka is the microsomal binding constant for SV (17.9 from [12]) and C is the microsomal protein concentration used in the in vitro study (0.4 mg/ml in [31] and 0.1 mg/ml in [32]). In addition two of the three in vitro studies repeated the experiment in more than one liver microsomal preparation so they report not only the mean but also the standard deviation () of the intrinsic clearance estimate. Therefore when the unit and binding corrections described above were performed on the intrinsic clearance estimates, the effect of the correction was also accounted for on the standard deviation of the estimates, using the relationship in Eq.S4.5. Finally the combined mean estimate (c) and combined variance (Vc) of SV in vitro intrinsic clearance using the transformed means (CLinti) and standard deviations (SDi) of the 3 in vitro studies were calculated as follows:

(q.S4.22)

, where ni is the number of repeats of each experiment using different microsomal preparations. With this approach it was calculated that the in vitro determined SV intrinsic clearance has a combined mean of 27.04 L/min/pmol of CYP3A and a combined standard deviation of 17.377. As this parameter is assumed to follow a log-normal distribution (to avoid negative sampled values), using reverse normal log-normal algebra (section 1) it follows that the respective normal prior should have mean of 3.1245 and variance of 0.3457 (Manuscript Table 2).

The intrinsic clearance for the whole hepatic CYP3A metabolism of SV (CLintCYP3A,lt) was calculated by scaling up the in vitro intrinsic clearance described above with Eq.S4.23.

, where CLintCYP3A, vitro is the in vitro unbound intrinsic clearance of SV in human liver microsomes (expressed in L/min/pmol of CYP3A), ACYP3A/MMP is the amount of CYP3A (155 pmols) [33] per mg of hepatic microsomal protein, MPPGL refers to mg microsomal protein/ g of liver for which the standard microsomal recovery of 40 mg/g [11] was used and LW refers to the average liver weight for which the value of 1718 g was used (average liver volume of 1.591 L, corrected for liver density of 1.08 kg/L [26, 27]).

The intrinsic clearance for the whole small intestinal wall CYP3A metabolism of SV (CLintCYP3A,siw) was calculated by scaling up the in vitro intrinsic clearance described above with Eq.S4.24.

, where CLintCYP3A, vitro is again the in vitro unbound intrinsic clearance of SV in human liver microsomes (expressed in L/min/pmol of CYP3A) and ACYP3A/siw is the total amount of CYP3A in the small intestinal wall (70,500 pmols) [34]. It should be noted that Eq.S4.24 assumes that the intrinsic clearance per pmol of CYP3A is the same in both small intestinal wall and liver. This is a common assumption as the intrinsic activities of gut and liver CYP3A (once normalised for the respective enzyme abundance) have been reported to be similar [35, 36]. In addition this assumption is of particular practical importance in terms of parameter estimation, as it abolishes the need to estimate two separate parameters regarding to the small intestinal wall and liver SV clearance [37].

2.8. Parameters related to CYP3A metabolism of SVA

The intrinsic clearance of SVA has been reported in the literature based on experiments in human liver microsomes and the measurement of formation of its 3 major oxidative metabolites [38]. In the current work, the results of this in vitro study were used in order to provide an informative prior that will facilitate the estimation of this parameter. The means and standard deviations of the intrinsic clearance to each of the CYP3A oxidative metabolites (M1: 0.02 0.01, M2: 0.02 0.01, M3: 0.015 0.01, all expressed in ml/min/mg of microsomal protein [38]) were combined in order to determine a prior distribution for the in vitro intrinsic CYP3A clearance of SVA that has a mean of 0.055 ml/min/mg of microsomal protein and a standard deviation of 0.0173. Note that the intrinsic clearances for the formation of each metabolite were treated as independent variables (no covariance), so that the variance of their sum equals the sum of their variances. As this parameter is assumed to follow a log-normal distribution (to avoid negative sampled values), using reverse normal log-normal algebra (section 1) it follows that the respective normal prior should have mean of -2.9476 and variance of 0.0944 (Manuscript Table 2).

The intrinsic clearance for the whole hepatic CYP3A metabolism of SVA (CLintCYP3A,lt) was calculated by scaling up the in vitro intrinsic clearance described above with Eq.S4.25.

, where CLintCYP3A, vitro is the in vitro intrinsic clearance of SVA in human liver microsomes (expressed in mL/min/mg of microsomal protein); MPPGL refers to mg of microsomal protein per g of liver for which the standard microsomal recovery of 40 mg/g [11] was used; LW refers to the average liver weight for which the value of 1718 g was used and fumic is the SVA fraction unbound in microsomes as the in vitro results [38] had not been corrected for non-specific binding. The fraction unbound in microsomes was predicted to be 0.9496 using the Hallifax and Houston equation [39], (Eq.S4.26).

, where C is the microsomal protein concentration used in the in vitro study (0.4 mg/ml) [38] and the logD7.4 of SVA is 1.45 (SciFinder Database, American Chemical society).

The intrinsic clearance for the whole small intestinal wall CYP3A metabolism of SVA (CLintCYP3A,siw) was calculated by scaling up the in vitro intrinsic clearance described above with Eq.S4.27.

, where CLintCYP3A, vitro is again the in vitro intrinsic clearance of SVA in human liver microsomes (expressed in mL/min/mg of microsomal protein); fumic is the SVA fraction unbound in microsomes (0.9496, see above); ACYP3A/siw is the total amount of CYP3A in the small intestinal wall (70,500 pmols) [34] and ACYP3A/MMP is the amount of CYP3A (155 pmols) [33] per mg of hepatic microsomal protein. As explained in the previous section it has been assumed also here that the intrinsic clearance per pmol of CYP3A is the same in both small intestinal wall and liver.

2.9. SV to SVA hydrolysis related parameters.

The rate of SV to SVA hydrolysis formation has been in vitro determined previously in buffer (pH=7.4), human plasma (pH=7.4) and human liver S9 (pH=7.4) [40]. These in vitro results were used in order to construct informative priors that will facilitate the parameter estimation process. The SV to SVA hydrolysis rate constants have been determined to be approximately 0.113 h-1 ( 0.03) in human plasma, 0.026 h-1 ( 0.001) in buffer (pH=7.4) and 0.064 h-1 ( 0.005) in human liver S9 (values are mean SD of triplicate determinations) [40]. As these parameters are assumed to follow a log-normal distribution (to avoid negative sampled values), using reverse normal log-normal algebra (section 1) it follows that the respective normal priors should have mean of -2.2144, -3.6504, -2.7519 and variance of 0.0681, 0.0015, 0.0061 for plasma, buffer and liver S9 respectively (Manuscript Table 2).

2.9.1. SV to SVA hydrolysis in systemic blood.

The SV to SVA hydrolysis intrinsic clearance in the systemic blood compartment (CLinthydr,bl) was calculated by scaling up the in vitro determined hydrolysis rate constant in plasma (khydr,pl) described above (section 2.9), using equation Eq.S4.28,

, where H is the hematocrit fraction in the blood (0.45), Vbl is the volume of the systemic blood compartment, fup is the SV fraction unbound in plasma that needs to be incorporated as the in vitro results [40] have not been corrected for plasma binding and BP is the SV blood to plasma ratio. It should be noted that any potential SV to SVA hydrolysis inside red blood cells has been neglected due to limited information.

2.9.2. SV to SVA hydrolysis in the liver vascular compartment.

The SV to SVA hydrolysis intrinsic clearance in the liver vascular compartment (CLinthydr,lv) was calculated by scaling up the in vitro determined hydrolysis rate constant in plasma (khydr,pl) described above (section 2.9), using equation Eq.S4.29,

, where everything is defined as in Eq.S4.28 and Vlv is the volume of the blood in the liver vascular compartment.

2.9.3. SV to SVA hydrolysis in the liver tissue.

The SV to SVA hydrolysis intrinsic clearance in the liver tissue compartment (CLinthydr,lt) was calculated by scaling up the in vitro determined hydrolysis rate constant in liver S9 (khydr,S9) described above (section 2.9), using equation Eq.S4.30,

, where Vinc and AS9 is the volume and the amount of S9 fraction respectively in the incubation (2 mg/mL) [40], S9PGL is the amount of S9 protein per g of liver (96.1 mg) [41, 42] , LW is the average liver weight for which the value of 1718 g was used and fuS9 is the SV fraction unbound in the S9 fraction that needs to be incorporated as the in vitro results [40] have not been corrected for protein binding. This unbound fraction was calculated with Eq.S4.20 to be 0.027, using the S9 protein concentration from the in vitro study (2mg/mL) [40] and assuming that the binding in microsomal and S9 protein is similar at the same protein concentrations [42].

2.9.4. SV to SVA hydrolysis in the muscle tissue.

The SV to SVA hydrolysis intrinsic clearance in the muscle compartment (CLinthydr,m) was calculated by scaling up the in vitro determined hydrolysis rate constant in buffer, pH=7.4, (khydr,buff) described before (section 2.9), using equation Eq.S4.31,

, where Vm is the volume of the muscle compartment. As discussed in Supplement 1, it is assumed here that the SV to SVA hydrolysis in the muscle is mediated only chemically, without the involvement of esterases, based on the fact that human carboxylesterases 1 (hCE1) and 2 (hCE2) are not known to be significantly expressed in the muscle tissue [43].

2.9.5. SV to SVA hydrolysis in the small intestinal wall and rest of body compartment.

SV to SVA hydrolysis has been assumed to take place both in the small intestinal wall and the rest of body compartment (see Supplement 1). The expression of carboxylesterases in the small intestine is substantial (especially hCE2) [43-45] and the formation of SVA from SV hydrolysis in the small intestinal wall is acknowledged in the literature [46, 47]. However, the rate of this conversion has never been reported to be determined in a tissue specific in vitro system (e.g. intestinal S9 fraction), partly because the preparation of such a system usually involves serine protease inhibitors that might interfere with esterase assays. As discussed in Supplement 1, the hydrolysis of SV to SVA in the rest of body compartment has also been allowed. This conversion was assumed to be mediated both chemically and enzymatically, as in several of the tissues (e.g. heart, kidney) informally lumped in this empirical peripheral eliminating compartment, carboxylesterases are also expressed [43]. Similarly to the conversion rate in the small intestinal wall, no prior information is available for this parameter apart from the rational realisation that this rate should be higher than the conversion rate observed in the buffer pH=7.4 (where hydrolysis is only chemical) and smaller than the hydrolysis in the liver tissue (the tissue considered to have the most abundant expression of carboxylesterases). Estimation of two separate conversion rates for the small intestinal wall and the rest of body compartment solely from the available plasma data and without any prior information was not feasible. Therefore, a single hydrolysis rate constant (khydr,hybrid) was estimated which was scaled up to SV to SVA hydrolysis intrinsic clearance in the small intestinal wall compartment (CLinthydr,siw) and to SV to SVA hydrolysis intrinsic clearance in the rest of body compartment (CLinthydr,rob) with Eqs.S4.32 and S4.33 respectively.

, where Vsiw and Vrob represent the volumes of the small intestinal wall and rest of body compartment respectively. A completely uninformative prior was assigned for this hybrid hydrolysis rate constant (khydr,hybrid), which was therefore solely estimated from the information in the clinical data. Finally, it should be noted that it is a model assumption that the unknown hydrolysis rate constant in the small intestinal wall and the empirical rest of body compartment are similar. Therefore any attempts for physiological interpretation of this hybrid constant estimate or extrapolation outside the studied population should be performed with care because of this assumption. The most rigid way to validate such an estimate is: firstly, to assess if it is physiological plausible compared to other hydrolysis rate constants in the model (which are more robustly known from experimental work) and secondly, to assess the ability of the model to predict the effect of DDIs that take place in the small intestinal wall, not only on SV but also on the hydrolysed metabolite (SVA) [37].

2.10. SVA to SV lactonisation related parameters.

Although the formation of SVA from SV is the favoured predominant process [40], SVA can convert back to the lactone form (SV) either chemically (minor) or enzymatically (major) via an acyl glucuronide intermediate and its spontaneous cyclisation [48] or via a CoASH-dependent pathway [49]. Whereas, the CoASH-dependent pathway is important for the lactonisation of SVA in rodents [49, 50], it seems that the acyl-glucuronidation is the enzymatic process that plays the most significant role for the lactonisation of SVA in dogs and humans in vivo [50].

Information regarding the lactonisation of SVA to SV was extracted from two different in vitro studies. In the first in vitro study [40] , the rate of the lactonisation (SV formation) was determined in human liver S9 (pH=7.4) incubated with SVA. However, in this in vitro work UDPGA had not been added as a co-factor in the SVA incubation. Therefore, any SV measured in the experiment is formed either solely chemically or in conjunction with any enzymatic process other than glucuronidation. In order to also account for the well-described mechanism of SV lactonisation through an acyl-glucuronide intermediate, the results of a second in vitro study were considered where the glucuronidation of SVA is quantified in human liver microsomes in the presence of UDPGA [48]. It has been assumed in the current model that any SVA glucuronidation not accounted within lactonisation is negligible (Supplement 1); in other words that any formed SVA acyl-glucuronide will undergo cyclisation to SV. This assumption is supported by experimental evidence that shows that the glucuronide conjugate of SVA readily undergoes spontaneous cyclisation to SV at physiological pH [48] and therefore was barely detectable in human hepatocytes incubated with SVA (SV formation was used as indicator of SVA glucuronidation) [51]. The total lactonisation intrinsic clearance in the liver tissue compartment (CLintlact,lt) was therefore calculated as the sum (Eq.S4.34) of the lactonisation clearance mediated through other than acyl-glucuronidation processes (CLint1lact,lt) and the lactonisation clearance mediated through acyl-glucuronidation (CLint2lact,lt).

The lactonisation clearance mediated through other than acyl-glucuronidation processes (CLint1lact,lt) was calculated by scaling up the lactonisation rate constant (klact,S9) determined in the first in vitro experiment described above [40], using Eq.S4.35. This lactonisation rate constant was determined to be approximately 0.0024 0.0001 h-1 (mean SD of triplicate determinations) [40]. This parameter was fixed to the reported above mean value and was not a subject of parameter estimation assisted by prior information. This assumption was considered valid as the uncertainty on the experimental result was minor (CV 4%); in addition, it was not possible to estimate components of both CLint1lact,lt and CLint2lact,lt simultaneously, as it is only their sum that affects the output of the model.

, where Vinc and AS9 is the volume and the amount of S9 fraction respectively in the incubation (2 mg/mL) [40], S9PGL is the amount of S9 protein per g of liver (96.1 mg) [41, 42] , LW is the average liver weight for which the value of 1718 g was used and fuS9 is the SVA fraction unbound in the S9 fraction that needs to be incorporated as the in vitro results [40] have not been corrected for protein binding. This unbound fraction was calculated with the Hallifax and Houston equation [39] (Eq.S4.26) to be 0.79, using the S9 protein concentration from the in vitro study (2mg/mL) [40], a SVA logD7.4 of 1.45 (SciFinder Database, American Chemical society) and assuming that the binding in microsomal and S9 protein is similar at the same protein concentrations [42].

The lactonisation clearance mediated through the acyl-glucuronidation processes (CLint2lact,lt) was calculated by scaling up the in vitro intrinsic UDPGA-dependent glucuronidation clearance of SVA in human liver microsomes (CLintgluc,vitro) determined in the second in vitro study described above [48]. The results of this in vitro study were used in order to construct an informative prior that will facilitate the parameter estimation process. Mean estimates and standard deviations can be extracted from the in vitro report [48] for both Vmax and Km. However, standard deviations on the Vmax/Km ratio (CLintgluc,vitro) were not reported, therefore the mean and standard deviation of CLintgluc,vitro were calculated as following:

Suppose Vmax and Km are random variables with non-zero means Vmax and Km respectively and variance-covariance matrix . The assumption that Vmax and Km have zero covariance terms has been made due to limited information. Let g be a function of Vmax and Km, as defined in Eq.S4.36.

It can be then derived using the multivariate generalisation of the delta method that:

, where D is the matrix of partial derivatives of the function g, illustrated in Eq.S4.39

After algebraic manipulations it follows from Eqs.S4.38, S4.39 that:

With the approach above it was calculated that the in vitro intrinsic glucuronidation clearance of SVA in human liver microsomes (CLintgluc,vitro) has a mean of 0.389 L/min/mg of microsomal protein and standard deviation of 0.1316. As this parameter is assumed to follow a log-normal distribution (to avoid negative sampled values), using reverse normal log-normal algebra (section 1) it follows that the respective normal prior should have mean of -0.9972 and variance of 0.1081 (Manuscript Table 2).

The lactonisation clearance mediated through the acyl-glucuronidation processes (CLint2lact,lt) was calculated by scaling up CLintgluc,vitro with Eq.S4.41.

, where CLintgluc,vitro is the in vitro intrinsic UDPGA-dependent glucuronidation clearance of SVA in human liver microsomes described above (expressed in L/min/mg of microsomal protein); MPPGL refers to mg of microsomal protein per g of liver for which the standard microsomal recovery of 40 mg/g [11] was used; LW refers to the average liver weight for which the value of 1718 g was used and fumic is the SVA fraction unbound in microsomes, as the in vitro results [48] had not been corrected for non-specific binding. The fraction unbound in microsomes was predicted to be 0.8341 with the Hallifax and Houston equation [39] (Eq.S4.26), using the microsomal protein concentration from the in vitro study (1.5 mg/mL) [48] and a SVA logD7.4 of 1.45 (SciFinder Database, American Chemical society).

3. Notes on parameter identifiability

A structural identifiability analysis of the developed physiological model was attempted using DAISY (Differential Algebra for Identifiability of SYstems) [52]. However, the program was not able to complete the computations related to this analysis (even after several days) probably due to the high complexity of the evaluated system. Utilisation of prior knowledge with regard to model parameters [53] can help to break down any possible identifiability issues [37]. Therefore, informative priors were provided for the majority of the structural model parameters to aid their estimation. With this approach even in the case where there is no information in the fitted data to inform the estimation of a model parameter, its estimate is expected to shrink towards the prior, supporting the identifiability of the model. Only 4 structural model parameters were estimated solely from the data without the provision of any prior. It was considered that these parameters can be at least numerically/deterministically identifiable as the observed output (intensively sampled SV/SVA plasma concentrations) is very sensitive to numerical perturbations of their values (Figure S4.4). However, this sensitivity does not affirm that unique estimates can be identified for these parameters (structurally globally identifiable). Therefore, identifiability was also at least indirectly assessed by carefully examining for the incidence of any of the practical indications of identifiability problems: failure of the optimisation procedure to converge; sensitivity of results to the initial estimates used for optimisation; highly correlated parameters; high standard errors of the parameter estimates as assessed both from NONMEM covariance step and a bootstrapping procedure.

4. Random population variability in drug-related model parameters

In the current work population variability in structural model parameters was modelled with the incorporation of random effect terms. The consideration of population variability in the system-related parameters of the model was discussed in Supplement 3. Population variability was also applied in the drug-related model parameters. Note that the variability in the latter also derives from system differences between individuals. However, as the model (although mechanistic) represents an oversimplification of the actual system, inter-individual differences related to the system (e.g., variability in enzyme abundance) that are not accounted for in the model is manifested as variability in drug related parameters (SV clearance).

As the developed complex model includes numerous drug-related parameters the development of the statistical-stochastic level of the model is challenging. The incorporation of random effect (variability) terms in all the drug-related parameters of the model may lead to over-parameterisation of the statistical model which is manifested by convergence problems and sensitivity to the initial estimates of the random effect terms. In addition, it seems irrational to expect that the estimation of all these variability terms in the several complex model parameters can be informed from the observed data which are limited to plasma SV/SVA concentrations. In addition, there is no available prior information with regard to the magnitude of these variability terms, to support estimation. Therefore, in the current work the stochastic-statistical model with regard to drug-related parameters was decided to be kept as simple as possible in order to avoid over-parameterisation. Indeed practical experience during model development indicated that the higher the complexity of the stochastic model (in terms of number of random effects and covariance structures), the higher the likelihood to get unstable models with convergence problems (at least with FOCE-I) and failure of the covariance step.

In the final model that is presented here population variability terms were incorporated in only 7 drug-related model parameters summarised in the Manuscript Table 2. This parameterisation of the stochastic model was selected by taking also into consideration which model parameters were naturally associated with a substantial population variability that can also propagate to the observed output (SV/SVA plasma concentrations). The 7 drug-related model parameters incorporating variability terms were: the SV/SVA tissue to blood partition coefficients for the rest of body compartments (surrogate for the variability in the overall SV/SVA volume of distribution); the permeability of SV from the lumen to small intestinal wall (surrogate for the variability in SV absorption); the hybrid estimated hydrolysis rate constant (section 2.9.5) for the hydrolysis of SV to SVA (surrogate for the variability in SV-SVA interconversion); the SV/SVA intrinsic clearances for CYP3A metabolism; and finally the active uptake clearance of SVA from the liver vascular to the liver tissue compartment. However, the identified population variability terms related to these parameters should be treated with caution. As population variability in the other drug-related parameters has been omitted, the identified variability terms may represent an admixture formed by the variability of several model parameters. This can be illustrated by taking the SV absorption part of the model (Manuscript Figure 1) as an example. This part is described by 4 processes, 2 system-related (gastric emptying and small intestinal transit) and 2 drug-related (dissolution and permeability from the lumen to small intestinal wall). Population variability informed by prior knowledge was incorporated for the system-related processes (Supplement 3) and in this way we avoid the propagation of this system variability to the estimated variability term in the permeability parameter. In contrast, the population variability in the dissolution process has been neglected and if it exists, it will propagate to the estimate of variability related to permeability. Therefore, this estimate should be treated with caution considering this assumption of the stochastic model. Although this assumption can be avoided with the incorporation of an additional random effect related to dissolution the ability to uniquely and precisely estimate variability terms for both the dissolution and permeability (in-series) processes solely from plasma data is questionable.

In the developed stochastic model inter-individual variability on all drug-related model parameters was modelled exponentially and no covariance was assumed between the inter-individual variability random effect terms (see NONMEM code in Supplement 9). Uninformative priors were designated for the random effects related to the drug-related parameters by setting the degrees of freedom of the inverse-Wishart distributed prior equal to the dimension of each omega block [54] (see NONMEM code in Supplement 9).

4

5. Figures

Figure S4.1. Relationship between in vitro Papp obtained in MDCK-MDR1 cells and in vivo Peff for 20 passively permeable drugs [12, 13]. The black solid line represents the line of best of fit (Eq.S4.8), the light black dotted lines represent the 95% confidence intervals for the mean Peff, while the dashed black lines (wider) represent the 95% prediction intervals for a future individual Peff prediction. Papp and Peff refer to apparent and effective permeability respectively.

Figure S4.2. Sensitivity analysis with respect the propagation of Peff (effective permeability) uncertainty on the plasma concentration profiles of SV and SVA. The model predictions using the lower and upper bounds of the 95% Peff prediction intervals (0.43 m/s, 41.56 m/s) are plotted with black solid lines. The area between these two concentration-time profiles is highlighted with grey.

Figure S4.3. The probability density function (pdf) of the log-normal distribution assigned to the error factor variable RF. The area under the pdf of RF between 1/3 and 3 equals to 0.84 (dark grey area). RF has an expected value of 1 and a variance of 0.72.

Figure S4.4. Sensitivity of SV (left) and SVA (right) plasma concentration profiles to numerical perturbations of the 4 model parameters for which prior information was not provided. (a) KPT:B,rob: tissue to blood partition coefficient for SV rest of the body compartment (see section 2.5); (b) KPT:B,rob: tissue to blood partition coefficient for SVA rest of the body compartment (see section 2.5); (c) CLuact: active uptake clearance for unbound SVA across the hepatic basolateral membrane (see section 2.6); (d) khydr,hybrid: SV to SVA hybrid hydrolysis rate constant in the small intestinal wall and rest of body compartment (see section 2.9.5). In (b) and (c) only SVA plasma concentrations are affected as these parameters are related to the disposition of the metabolite.

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