Chemicals

Williams Medium E (WME) without phenol-red and without L-glutamine and stabilized L-glutamine were obtained from Pan-Biotech-GmbH (Aidenbach, Germany). Penicillin/Streptomycin and ITS-X were obtained from Invitrogen (Karlsruhe, Germany). Bovine serum albumin (BSA) and dimethyl sulfoxide (DMSO) were purchased from Sigma-Aldrich Chemicals (Taufkirchen, Germany). Dexa Inject was obtained from Jenapharm (Jena, Germany). AS, its metabolites and deuterated standards were purchased from Toronto Research Chemicals Inc. (North York, Canada). PBS was obtained from Invitrogen (Karlsruhe, Germany) and Complete Mini, EDTA-free from Roche Diagnostics (Mannheim, Germany). NaPP, sucrose and LiChrosolv were purchased from Merck (Darmstadt, Germany); acetonitrile from Roth (Karlsruhe, Germany), formic acid from Fluka, (Germany), UGT1A3 monoclonal mouse antibody from Abcam (Cambridge, England). Trypsin was purchased from Promega (Mannheim, Germany). Synthetic peptides were purchased from Sigma-Genosys (Haverhill, UK).

Isolation and cultivation of primary human hepatocytes

Tissue samples from human liver resections were obtained from patients undergoing partial hepatectomy. Experimental procedures were performed according to the guidelines of the charitable state-controlled foundation HTCR (Human Tissue and Cell Research) Regensburg, Germany, and the institutional guidelines for liver resections of tumor patients with primary or secondary liver tumors, Technical University Munich, MRI, Munich, Germany. The use of human hepatocytes for research purposes was approved by the local ethics committees of the Ludwig-Maximilians-University of Munich [36] and the Charité, Humboldt University Berlin [37], Germany, and written informed consent was obtained from all patients. Hepatocytes were cultured on collagen gel precoated 6-well plates at a density of 1.5·10⁶ cells/well. Cells were allowed to attach to the collagen layer. After transport, culture media was disposed and attached cells were cultured 24 h at 37°C in a humidified chamber with 95%/5% air/CO2 in serum-free medium WME, supplemented with albumin (0.1% (v/v)), penicillin/streptomycin (100 U/ml), stabilized L-glutamine (2 mM), dexamethasone dihydrogenphosphate (0.025% (v/v)) and ITS-X (5 mg insulin, 3.35 μg natrium-selenit, 2.75 mg transferrin and 1 mg ethanolamine), further named SFM.

Time-series experiments

Incubation with AS was started by disposing the culture media and cultivation of the attached cells at 37°C in a humidified chamber with 95%/5% air/CO₂ in 2 ml SFM, supplemented with 10 μM AS, 0.1% (v/v) BSA and 0.1% DMSO. At specified time-points, three wells were further treated for the preparation of samples for the measurement of extracellular and intracellular metabolites, respectively. SFM media was collected and 50 μL formic acid and deuterated internal standard was added for the further measurement of extracellular metabolites. Cells were harvested in pre-cooled albumin-free SFM, disrupted by freeze/thaw and ultra-sonification and centrifuged. The supernatant was used for the determination of intracellular metabolites.

For the preparation of samples for the protein measurements, culture medium was disposed and cells were harvested in pre-cooled PBS, supplemented with Complete Mini EDTA-free (1 Tablet/10 ml Buffer). Cell suspensions were centrifuged 5 min (500 g) at 4°C and cell pellets were resuspended in 150 μL NaPP-buffer (0.1 M, pH 7.4), containing 250 mM sucrose and Complete Mini EDTA-free (1 Tablet/10 ml Buffer). Cells were disrupted by ultra-sonification and lyophilized for the analysis of total protein concentration, CYP3A4 and UGT1A3 content.

For cell number determination, cells from two wells were fixed with methanol-acetic acid fixative solution (10 min at 37°C and 4°C) and afterwards nuclei were stained for 15 min with Meyers Hämalaun (Sigma-Aldrich Chemie GmbH, Germany), rinsed with water and air-dried. Stained nuclei were counted in digital images (10 per well) at 40-fold Magnification (ImageJ Image Processing and Analysis Program).

Quantification of atorvastatin and its metabolites

AS and ASL, and their para- (ASpOH, ASLpOH) and ortho-hydroxy-metabolites (ASoOH, ASLoOH), were determined by LC-MS-MS analysis using the respective deuterium labeled analogues as internal standards, essentially as described [38]. HPLC separation was performed at 30°C on a XBridge Shield RP18 column (2.1 × 50 mm, 3.5 μm, Waters) using (A) 1 mM formic acid and (B) acetonitrile as mobile phases at a flow rate of 0.4 ml/min. Gradients were programmed as follows: 63% A for 4 min; linear decrease of A to 60% within 9 min; linear decrease of A to 55% within 2.5 min; 55% A for 1 min; increase of A to 63% in 0.2 min. Equilibration time of the column was 20 min. MS-MS analysis was performed on an Esquire HCT ultra ion trap mass spectrometer (Bruker Daltonics, Bremen, Germany) coupled to an HPLC 1100-System (Agilent, Waldbronn, Germany) consisting of binary pump G1312A, degasser G1379A, well-plate sampler G1367A and column thermostat G1330B. The ionization mode was electrospray (ESI), polarity positive, mass range mode ultrascan, and nitrogen was used as a drying and nebulizer gas. The following parameters were applied: nebulizer 45 psi, dry gas 10 l/min, dry temperature 300°C, capillary 4100 V, scan range 200 - 600 m/z.

Precursor and product ions (m/z) of analytes and internal standards, respectively, were ATV (559 and 440.2; 466.2), [²H₅]ATV (564 and 445.2; 471.2), ATV-L (541.2 and 448.2), [²H₅]ATV-L (546.2 and 453.2), p-OH-ATV (575 and 440.2; 466.2), [²H₅]p-OH-ATV (580 and 445.2; 471.2), p-OH-ATV-L (557 and 448.4), [²H₅]p-OH-ATV-L (562 and 448.4), o-OH-ATV (575 and 466.4), [²H₅]o-OH-ATV (580 and 471.2), o-OH-ATV-L (557 and 448.4), [²H₅]o-OH-ATV-L (562 and 448.4). Sample quantification was possible in a range from 0.5 to 500 pmol.

CYP3A4 and UGT1A3 protein quantification

Protein quantification of CYP3A4 and UGT1A3 in human liver microsomes and relative protein quantification of UGT1A3 in lyophilized samples of primary hepatocytes was performed by immunoblotting as described previously [38, 39].

Cell lysates for absolute quantification analysis of CYP3A4 were prepared from lyophilized human primary hepatocytes by sonification in the presence of glass beads in buffer containing Complete Mini-Protease Inhibitor Cocktail, and following homogenization. In the CYP3A4 quantification assay, three synthetic isotopically labeled peptides (13C/15N amino acid) were used as internal standard for calibration. These isotope labeled standard peptides represented sequence analogues to proteotypic peptides of CYP3A4, which arise from tryptic digestion. After acetone precipitation and resolving of the proteins in 8 M Urea, a definite amount of internal standard peptides was added. The sample mixture was reduced with 5 mM DTT and alkylated with 15 mM iodacetamide in 50 mM ammonium bicarbonate. Subsequently samples were digested with trypsin at 42°C for 4 h (enzyme/substrate ratio of 1:10). Efficiency of tryptic digestion was checked by SDS-PAGE followed by silver staining. The resulting peptides were purified using C18 OMIX^® Tips (Varian, Darmstadt, Germany) according to manufacturer's suggested protocol and separated on a nanoliter-flow Ultimate HPLC system (Dionex, Idstein, Germany). After injection (15 μl), peptides were trapped and desalted on a precolumn (0.3 mm I.D. × 5 mm PepMapTM, Dionex) at a flow rate of 30 μl/min in 0.1% TFA for 6 min. Peptides were transferred to the separation column (75 μm I.D. × 250 mm PepMapTM column, Dionex) and separated in a linear gradient of mobile phase (A: 0.1% formic acid, B: 84% acetonitrile/0.1% formic acid) from 5% B to 35% B over a period of 35 min with a flow of 290 nl/min. The column effluent was continuously directed into the NanoSpray II source of a 4000QTrap mass spectrometer (Applied Biosystems, Foster City, CA, USA). The MS was set up to run a multiple reaction monitoring experiment essentially, as described previously [40], including two to three parent-to-product ion transitions for each internal standard peptide as well as the corresponding transitions of native peptide of CYP3A4. The instrument settings were as follows: ion spray voltage, 3-4 kV; interface heater, 150°C; declustering potential, 50 V; collision energy, peptide specific; entrance potential, 10 V; collision cell exit potential, 10 V. MS data were processed by integrating the appropriate peak areas from extracted ion chromatograms by MultiQuantTM Software (Applied Biosystems). The absolute amount of CYP3A4 protein was calculated from the peak area ratio "internal standard peptide/native peptide". Total protein content of the samples were determined by amino acid analysis (AAA) on a Waters 2695 HPLC system using the AccQ•Tag derivatization method (Waters, Eschborn, Germany), according to manufacturer's instructions.

Identification of metabolic network structure

Numerous metabolic and physicochemical aspects about AS had to be considered in the initial model building. AS exists in two forms, a very lipophilic lactone (ASL) and a comparably hydrophilic hydroxyl-acid (AS). AS is converted enzymatically via an instable intermediate product into ASL, mediated by different UGT isoenzymes [41, 42]. Recent investigations by ourselves and others have shown that the most important contributor to UGT-driven lactonization is UGT1A3, whereas UGT1A1 plays an insignificant role [38, 42]. AS acids and lactones are inter-converted chemically into each other [43]. However, studies have indicated, that the chemical lactonization of AS to ASL can be neglected at physiological pH of 7.4 [43]. Recent studies highlight, that different PON enzymes might also be possible contributors to the lactone hydrolysis and that PON1 is present in liver [44–48].

Both AS and ASL are hydroxylated in human hepatocytes leading to para- and ortho-hydroxy-metabolites, ASpOH, ASoOH, ASLpOH and ASLoOH [49, 50], mainly catalyzed by CYP3A4 [51]. Recent studies have reported, that CYP2C8 and CYP3A5 also hydroxylate AS to a minor extent [50, 52].

Furthermore, AS is transported into the cell via organic anion transport polypeptides (OATP), and recent studies on recombinant systems showed, that OATP1B1 and OATP2B1 contribute to the AS import [53, 54], which are both expressed in the human liver [55, 56]

OATP1B3 is supposed to be also a main contributor to drug transport [54], since it shows high gene expression levels in liver [57], but its importance for AS has not been investigated kinetically so far.

OATP transporters have been reported to be bidirectional facilitated diffusion transporters, independent from ATP and Na+, K+ and H+, but with a possible involvement of reduced glutathione [58, 59]. However, previous investigations on transport mechanisms on rat hepatocytes, showed, that the intracellular concentrations of pitavastatin and other compounds are much higher than outside the cell [33, 34, 60]. Facilitated or passive diffusion do not allow a greater intracellular than extracellular concentration of the parent drug of interest, when the source is the initial extracellular concentration, because it is a concentration-gradient dependent mechanism. Therefore, the import mechanism should be rather considered as active mediated transport, which is not concentration-gradient dependent, rather than as the proposed facilitated diffusion process.

As shown with the recombinantly expressed transporters mentioned above [54], transport of the acidic metabolites, ASpOH and ASoOH, by OATP1B1 was similar to that of AS, and transport of the corresponding lactones was also mediated by this transporter, although at somewhat lower rates. We therefore assumed that the same OATP transporters are responsible for the import of AS and its hydroxylated and lactone metabolites into hepatocytes.

AS and its hydroxylated metabolites, ASpOH and ASoOH, are actively exported out of the hepatocytes into the bile by the ATP-dependent MDR1 transporter [61, 62]. In addition, acidic and lactone form of AS showed inhibitory effects in transport studies of substrates of MDR1 and MRP2 [63–65], pointing to the competitive transport mechanism of these substrates at this proteins. Further, the transporters MRP1, MRP3 and MRP6 are also reported to be responsible for the transport of organic compounds including AS from inside the hepatocytes into the plasma [56, 66].

Passive diffusion might play also an important role in the transfer of AS, ASL and the corresponding metabolites, ASpOH and ASoOH, ASLpOH and ASLoOH, respectively. Since the acidic forms of AS are rather hydrophilic and the lactone forms of AS are rather lipophilic, it can be assumed that passive diffusion plays a more important role for the lactone forms, and the transporter mediated active transport plays a more important role for the acidic forms, as reported earlier for statins [67].

Finally, lipophilic drugs have a high affinity to bind non-specifically to proteins, and previous studies concentrated on the modeling of drug binding in the intracellular and the extracellular space as well as on the surface of the cells [60, 68, 69].

Mathematical modeling

The mathematical model can be described by the system of non-linear ordinary differential equations

(1)

where the change in extracellular, intracellular or unspecific bound metabolite concentration c _j in the extra - or intracellular compartment with V _comp is effected by the conversion or production of contributing chemical or enzymatic reactions or by transport steps r _ij , respectively. The extracellular volume, , equals to the volume of the media used. The intracellular volume, , equals to the total volume of all cells used, and is determined by multiplying the cell number by the volume of a single hepatocyte, estimated to be 14.1 pL by the approximation of a spherical shape with a diameter of 30 μm [70].

Appropriate reaction kinetics r _ij are modeled for the CYP3A4 hydroxylation, the UGT1A3 lactonization, the chemical and enzymatic lactone hydrolysis and intracellular unspecific binding to macromolecules.

Previous studies determined substrate inhibition kinetics of the CYP3A4 mediated hydroxylation of AS on human microsomes [52]. However, inhibition effects contribute severely only at a concentration higher than 100 μM. Furthermore, our model approach considers the competitive nature of alternative substrates, by integrating the CYP3A4 hydroxylation of AS and ASL as reaction kinetics describing the competitive conversion of alternative substrates to alternative products, illustrated for the hydroxylation of AS to ASpOH

(2)

(see Additional file 1 - Derivation of Atorvastatin kinetics at CYP3A4).

The lactonization of AS to ASL is mediated by UGT1A3 enzymes and the reaction is formulated as substrate inhibition kinetics [41].

(3)

The lactone metabolites are either hydrolyzed chemically to the respective acid metabolites [43] inside (c) or outside (m) the cell, or enzymatically by the contribution of PON enzymes inside the cell. Both reactions are described as first order kinetics

(4)

Unspecific binding of intracellular metabolites to macromolecules is formulated as

(5)

with the dissociation coefficient k _dis and the intracellular fraction unbound [71]

(6)

which describes the ratio between the intracellular free concentration to the sum of intracellular free and bound concentration, and , in equilibrium (index eq) (see Additional file 2 - Derivation of kinetics of unspecific protein binding). However, the intracellular free and bound concentrations in equilibrium are not measurable; therefore, the fraction unbound fu _j is set as a parameter to be estimated in the parameter optimization procedure.

Transport steps include active import and export of the metabolites as well as passive diffusion steps. Both active import by OATP1B1 or OATP2B1 and export of AS are described as Michaelis-Menten-kinetics [53, 54]

(7)

whereas the active transport kinetics of the other metabolites are assumed to be of first-order [72].

(8)

Besides the active transport, metabolites undergo passive diffusion through the double-layer lipid-membrane. Passive diffusion, described as

(9)

is driven by the concentration difference between outside and inside the cell, , over the lipid-membrane with thickness d, through all cells with the total surface area A _cells , and controlled by the diffusion coefficient D _j and is comprised as the permeability coefficient P _j .

Optimization procedure for estimation of model parameters

The optimization procedure is based on evolutionary strategies which are implemented with JavaEva (WSI Computer Science Department, Center for Bioinformatics, University of Tübingen, Germany) and a MVA (main vector adaptation) mutation operator [73]. The optimization procedure estimates the parameters in equations (2) to (9) based on the optimization criterion

(10)

where the deviation of calculated and measured concentrations divided by the measurement standard deviation s _j , squared and summed over all metabolites J and all time points N, has to reach a minimum. Additional optimization constraints are fu _AS > fu _ASL , P _ASL > P _AS , P _AS > P _ASOH and P _ASL > P _ASLOH , because the lactones have a higher lipophilicity than the acids and the metabolites are supposed to be more hydrophilic than the corresponding parent lactone or acid drug.

The integration of the differential equations (1) using the reaction kinetics in equations (2) to (9) was performed by the differential algebraic equation solver LIMEX (Konrad-Zuse-Zentrum für Informationstechnik, Berlin) [74].

Relative abundance approach for prediction of r _max -parameters

For pharmacokinetic predictions taking inter-individual variability of CYP3A4 and UGT1A3 expression levels into account, maximal rate parameters are predicted via a relative abundance approach, which is based on the assumption that the maximal rate of the reaction is proportional to the enzyme concentration:

(11)

The maximal velocities of the respective enzyme e, here CYP3A4 or UGT1A3, in the conversion to the product P _j in the liver of interest li is estimated from the respective maximal rate and the enzyme concentration in the reference liver and the enzyme concentration in the liver of interest li.

Computational approach

The mathematical model of AS metabolism was coded in FORTRAN language and linked to the numerical integrator LIMEX, also written in FORTRAN. After compilation to the executable program, optimization was started by the call of JavaEva. The mathematical model is supplemented as SBML-file for review purpose.

Model-setup

The model structure of AS biotransformation in primary human hepatocytes is schematized in Figure 1 and comprises the description of the extracellular, intracellular and unspecifically bound metabolites, as well as corresponding reaction and transport steps and unspecific protein binding.

Since the calibration and quality samples were prepared in the same media as the experimental metabolite samples and because it can be assumed that the intracellular protein concentration is much higher compared to the outside of the cells, the unspecific protein binding was only considered for the intracellular space.Regarding the unspecific protein binding, it can be assumed that the intracellular protein concentration is much higher compared to the outside of the cells. This assumption is based on the estimation of the ratio of intracellular protein to extracellular media protein concentration. From the total cell protein measurement and the determination of total cell volume a total intracellular protein concentration of about 30 g/l can be estimated. The media includes as sole protein component 0.1% (v/v) albumin, corresponding 1 g/l. Therefore, this assumption is justified.

Because our hepatocyte culture model does not allow to distinguish experimentally import and export, single irreversible, actively mediated transport steps for both directions, as well as passive diffusion steps were considered for each compound, respectively, similar to mechanistic transport models of other studies [33, 34]. Only in the case of AS different transport steps for OATP1B1 and OATP2B1 were implemented in the model, because specific KMs had been determined in previous studies [53, 54].

Model verification

Model verification was performed by quantitative measurements of AS and all metabolites in the extra- and intracellular space during time-series experiments, performed first on a single batch of primary hepatocytes from a single individual 1. Special emphasis was given to the issue of parameter sensitivity and correlation between parameters to improve the quality of the model.

Analysis of the predicted concentration-time-profiles

The model predicted concentration-time-profiles are illustrated together with the measured concentrations (see Additional file 3 - Atorvastatin metabolite concentrations from the time-series experiment on primary human hepatocytes of individual 1) in Figure 2. Both intracellular AS and ASL are converted to the corresponding para- and ortho-hydroxy metabolites, ASpOH and ASoOH, and ASLpOH and ASLoOH, respectively. However, the acidic metabolites, ASpOH and ASoOH, show higher intracellular concentrations than the lactone metabolites, ASLpOH and ASLoOH. The ratio between intracellular AUC _{0-600 min} of the acidic form to that of the lactone form equals 20.1 in case of the para-hydroxy-metabolites and 23.6 in the case of the ortho-hydroxy-metabolites.

Accordingly, the extracellular concentrations of the acidic metabolites are higher than that of the lactone metabolites. The ratio between extracellular AUC _{0-600 min} of the acidic form to that of the lactone form equals 54.3 in case of the para-hydroxy-metabolites and 70.8 in the case of the ortho-hydroxy-metabolites.

Further, the results show that the components have higher concentrations in the intracellular space than outside the cell. The ratio of intracellular AUC _{0-600 min} to extracellular AUC _{0-600 min} equals minimally 4.5 in the case of ASoOH and maximally 38.2 in the case of ASLpOH. Similarly, the ratio of intracellular c _max to extracellular c _max equals minimally 2.5 in the case of ASoOH and maximally 34.5 in the case of ASLpOH.

Simultaneous model verification on different individual hepatocyte donors

The process of model verification so far has been applied on a single experiment on primary human hepatocytes of individual 1. In the next step it has to be proven, that the model is also capable to describe different individual metabolic profiles, especially being further able to reflect the inter-individual parameter variability, not only in the parameters r _max of the phase I and II reactions, but also in the transporters. Therefore, the model verification is performed based on AS biotransformation data on primary hepatocytes from three different individuals simultaneously.

However, the analysis of parameter sensitivity and identifiability (see Additional file 4 - Analysis of parameter sensitivity and following model reduction procedure) showed that the first order kinetics of OATP1B1 mediated import is favored over the zero-order kinetics of OATP2B1 import of AS, and that the first order kinetics are hard to distinguish from first order passive diffusion mechanisms in this evaluation system. Thus, in the following, mechanisms of active transport and passive diffusion are lumped, resulting in the apparent transport rate for import,

(12)

and in the apparent transport rate for export

(13)

described by the product of apparent rate constant κ _im/ex of import or export and extra- or intracellular concentration , respectively (Figure 3). Consequently, the rate constants κ of import and export are set individually, and were to be estimated in the optimization procedure.

The simultaneous model verification was performed based on the stimulus response data obtained from primary human hepatocytes of individual 1 (see Additional file 3 - Atorvastatin metabolite concentrations from the time-series experiment on primary human hepatocytes of individual 1), individual 2 (see Additional file 5 - Atorvastatin metabolite concentrations from the time-series experiment on primary human hepatocytes of individual 2) and individual 3 (see Additional file 6 - Atorvastatin metabolite concentrations from the time-series experiment on primary human hepatocytes of individual 3), respectively.

The model prediction is in satisfying agreement with experimental data of individual 2 (Figure 4). However, in case of individual 3, the model prediction is comparably poor, because there are relatively high deviations, especially with the intracellular metabolites ASL and ASoOH and extracellular ASpOH (Figure 5, solid line). One reason could be the extended contribution of beta-oxidation in AS metabolism. Beta oxidation at the heptanoic acid side chain is a typical transformation pathway for all statins, but is reported to play only a minor role in humans [75]. However, high activity of beta-oxidation of fatty acids, which is responsible for the supply of ATP for gluconeogenesis in type 2 diabetes mellitus [76, 77], which individual 3 was diagnosed with, may severely influence the AS metabolism. To test this hypothesis, respective reactions with the acid metabolites as substrates were considered in the model verification of individual 3, but the results showed no significant improvement (data not shown). The second reason could be, that CYP3A4 and UGT1A3 protein concentrations of individual 3 used in the estimation of corresponding r _max value via relative abundance approach (equation (11)) differ from the measured mean value (Table 4). Due to the fact that the r _max parameters of CYP3A4 and UGT1A3 show high sensitivities in the parameter sensitivity analysis (see Additional file 4 - Analysis of parameter sensitivity and following model reduction procedure), a variation in the values would have a high impact on the metabolic profiles. Therefore, in a further optimization step, the CYP3A4 protein concentrations are allowed to vary in the interval of mean ± standard deviation (Table 4), where the standard deviation of UGT1A3 protein concentration is assumed to be 30% of the mean value. Notably, an improvement in the model prediction on individual 3 could be achieved in the case of the intracellular metabolites AS and ASL and extracellular AS (Figure 5, dashed lines). But there are still major deviations in case of intracellular ASL and ASoOH and extracellular ASpOH, which could not be explained any further, but shows, that there must be some other reactions or effects in the system, which have not been discovered yet.

The mathematical model of AS metabolism in human hepatocytes of individual 1 is supplemented as SBML-file (see Additional file 7 - Model of atorvastatin metabolism in primary human hepatocytes of individual 1, and BioModels database).

Dynamic analysis of inter-individual CYP3A4 and UGT1A3 expression level variability

Based on the model version of optimized parameters, gained in the simultaneous model fit, the effect of inter-individual variability of CYP3A4 and UGT1A3 protein expression levels was investigated by linking the protein expression data of 150 liver samples (Figure 6) via the described relative abundance approach in equation (11), using individual 1 as reference. UGT1A3 and CYP3A4 protein concentrations of individual 1 were converted from based on total protein amount to based on microsomal protein amount by multiplying with the factor 0.22, determined in human liver homogenates and corresponding microsome fractions via Bradford test [78].

Simulations were performed with an initial extracellular AS concentration of 50 (pmol ml^-1), which is in the range of physiological plasma concentrations [54, 79], over a time period of 1200 min.

The most important question that arises from this dynamic analysis was, how the metabolic profiles of the intracellular metabolites AS, ASpOH and ASoOH are influenced by this variability, since they are considered to be the active drugs, which inhibit HMGCoA-reductase [80]. Therefore, AUC, c _max and t(c _max ) of the concentration-time-profiles of either AS alone or the sum of concentration of AS, ASpOH and ASoOH were calculated for each liver sample over a time period of 1200 min and the distributions over all liver samples were evaluated, respectively. Finally, appropriate probability density functions are fitted to the distributions (Figure 7). The probability density function characteristics are summarized in Table 6. Obviously, there are differences between the examination of only AS or the sum of concentrations of AS, ASpOH and ASoOH. AUC, c _max and t(c _max ) have lower values in case of AS alone compared to the sum of all acidic metabolites. The population mean of AUC is 75051 (pmol ml^-1 min) in the case of AS and 203617 (pmol ml^-1 min) in the case of the sum of AS, ASpOH and ASoOH. Also, c_max is lower in the case of AS alone, 201 (pmol ml^-1), compared to the sum of the acidic metabolites, 366 (pmol ml^-1). Further, the maximal concentration appears at a shorter time point, 48 min, in the case of AS alone, compared to the time point, 100 min, of the sum of AS, ASpOH and ASoOH. The results are quite explainable, since ASpOH and ASoOH are the hydroxylated products of AS and therefore their maximal concentrations event at a delayed time-point compared to AS.

Further, except for the AUC of AS, the probability density functions show a very narrow shape, regarding the relative standard deviation, when comparing to the sample standard deviations of the underlying distribution of CYP3A4 and UGT1A3, respectively. The probability density functions fitted to CYP3A4 and UGT1A3 distributions have a relative standard deviation of 259% and 137%, respectively, whereas the relative standard deviations of the probability density functions of AUC, c_max and t(c_max) are lower than 50%, except for the probability density function of AUC of intracellular AS, which equals 123%.