## Visual Overview

## Abstract

Among the basic hepatic clearance models, the dispersion model (DM) is the most physiologically sound compared with the well-stirred model and the parallel tube model. However, its application in physiologically-based pharmacokinetic (PBPK) modeling has been limited due to computational complexities. The series compartment models (SCM) of hepatic elimination that treats the liver as a cascade of well-stirred compartments connected by hepatic blood flow exhibits some mathematical similarities to the DM but is easier to operate. This work assesses the quantitative correlation between the SCM and DM and demonstrates the operation of the SCM in PBPK with the published single-dose blood and liver concentration-time data of six flow-limited compounds. The predicted liver concentrations and the estimated intrinsic clearance (*CL _{int}*) and PBPK-operative tissue-to-plasma partition coefficient (

*K*) values were shown to depend on the number of liver sub-compartments (

_{p}*n*) and hepatic enzyme zonation in the SCM. The

*CL*and

_{int}*K*decreased with increasing

_{p}*n*, with more remarkable differences for drugs with higher hepatic extraction ratios. Given the same total

*CL*, the SCM yields a higher

_{int}*K*when the liver perivenous region exhibits a lower

_{p}*CL*as compared with a high

_{int}*CL*at this region. Overall, the SCM nicely approximates the DM in characterizing hepatic elimination and offers an alternative flexible approach as well as providing some insights regarding sequential drug concentrations in the liver.

_{int}**SIGNIFICANCE STATEMENT** The SCM nicely approximates the DM when applied in PBPK for characterizing hepatic elimination. The number of liver sub-compartments and hepatic enzyme zonation are influencing factors for the SCM resulting in model-dependent predictions of total/internal liver concentrations and estimates of *CL _{int}* and the PBPK-operative

*K*. Such model-dependency may have an impact when the SCM is used for in vitro-to-in vivo extrapolation (IVIVE) and may also be relevant for PK/PD/toxicological effects when it is the driving force for such responses.

_{p}## Introduction

The pharmacokinetic (PK) field has sought to elucidate the properties and mechanisms of drug distribution and elimination in the liver as it serves as the major clearance organ. Various liver-derived in vitro metabolic systems (microsomes, hepatocytes, homogenates, slices) have been used for prediction of in vivo hepatic clearance using the measured intrinsic clearance (*CL _{int}*) (viz. in vitro

*-*to-in vivo extrapolation, IVIVE). On the other hand, physiologically-based PK (PBPK) modeling allows for characterizing dynamic changes in hepatic concentrations in pre-clinical species and scaling-up to humans (Lee et al., 2020) where the

*CL*and tissue-to-plasma partition coefficient (

_{int}*K*) are needed. Both IVIVE and PBPK modeling require a structural liver model (Rane et al., 1977; Houston, 1994; Hallifax et al., 2010; Miller et al., 2019).

_{p}The primary hepatic models are the well-stirred (WSM), parallel tube (PTM), and dispersion (DM) models (Rowland et al., 1973; Pang and Rowland, 1977; Roberts and Rowland, 1986a; Roberts and Rowland, 1986b; Pang et al., 2019; Jusko and Li, 2021; Li and Jusko, 2022). Their assumed internal flow and mixing patterns differ, with the degree of longitudinal or axial dispersion of a solute passing through the liver characterized by the dispersion number (*D _{N}*). The WSM has infinite mixing of blood (

*D*= ∞) and uniform intrahepatic and outflow blood concentrations while the PTM exhibits no mixing (

_{N}*D*= 0) and a mono-exponential concentration decline. The DM features intermediate mixing or dispersion (0 <

_{N}*D*< ∞) rendering a continuous concentration decline lying between those of the WSM and PTM. The average intrahepatic blood concentration ranking is WSM<DM<PTM, with the corresponding

_{N}*CL*being opposite to achieve the same hepatic extraction ratio (

_{int}*ER*).

The utilization of these models in PBPK has been limited to the least physiologic WSM, largely due to its computational simplicity. The use of the more physiologically sound DM in PBPK is complicated as second-order partial differential equations are required (Oliver, 1995; Oliver et al., 2001). The DM is based on the residence time distribution of solutes reflecting the degree of dispersion (Levenspiel, 1999).

Another model to approximate complex flow and reaction systems is the “tanks-in-series” or “series compartment” model (SCM) where the liver is viewed as a cascade of identical well-stirred compartments connected by hepatic blood flow (Fig. 1). The SCM is mathematically similar to the gamma distribution function describing the residence time distribution of a tracer exiting an organ (Buffham and Gibilaro, 1968; Davenport, 1983) and therefore equally well represents indicator-dilution curves (Goresky et al., 1973) as does the DM (Roberts and Rowland, 1986a; Gray and Tam, 1987). The SCM functioning is intermediate between the WSM and PTM and mimics the DM but is mathematically simpler. The degree of mixing in the SCM is determined by the number of liver sub-compartments (*n*); the SCM predicts the same hepatic availability (*F*) as the WSM when *n* = 1 and gives approximately equal *F* values to the PTM when *n* > 30 (Gray and Tam, 1987).

A zonal-compartment model, a modified version of the SCM, considers enzyme zonal heterogeneity, where the liver is divided into three zones with different metabolic activities, i.e., periportal (PP), midzonal, and perivenous (PV) regions (Tirona and Pang, 1996; Abu-Zahra and Pang, 2000). The SCM was also extended to characterize the disposition of pravastatin where the liver was divided into five sequential units of extracellular and subcellular compartments to include transporters (Watanabe et al., 2009). The selection of five liver sub-compartments was primarily based on how it predicts the closest *F* value of pravastatin as given by the DM. This complex SCM was applied to other substrates of hepatic transporters (Jones et al., 2012; Li et al., 2014; Li et al., 2016; Morse et al., 2017). Recently, the SCM with varying *n* was used to assess the correlation between in vitro and in vivo unbound liver-to-plasma partition coefficients; however, their analysis was limited to algebraic equations derived under steady-state (SS) conditions (Li et al., 2019).

Although the SCM was claimed to closely approximate the DM, their quantitative correlations have not been explored. It remains unclear how the choice of *n* and hepatic enzyme zonation affect the estimation of *CL _{int}* and

*K*as well as model predictions of internal liver concentration-time profiles. We herein consider these aspects by incorporating the basic hepatic SCM (Gray and Tam, 1987) into PBPK with characterization of published single-dose blood/plasma and liver concentration-time data from rats for six flow-limited compounds primarily cleared by the liver. This report will also serve as a tutorial on operating the SCM in PBPK modeling.

_{p}## Methods

The SCM of hepatic elimination is displayed in Fig. 1, where the liver is divided into *n* well-stirred sub-compartments connected by hepatic blood flow. Each of the sub-compartments was assumed to equally share the same total tissue volume (*V _{h}*), availability (

*F*),

*ER*, and

*CL*. Thus, for the

_{int}*i*th (

*i*=1, 2, 3…

*n*) compartment: where

*C*is the total liver concentration, and

_{h}*f*is the unbound fraction in blood.

_{ub}The outflow blood concentration from the compartment *i* (*C _{outi}*,

*i*=1, 2, 3…

*n*) is the input function for the subsequent compartment

*i*+1 and is assumed to be in equilibrium with the tissue concentration throughout the

*i*th compartment (

*C*=1, 2, 3…

_{hi}, i*n*) with each of the liver sub-compartments sharing the same liver-to-plasma concentration ratio

*K*: where

_{p}*R*is the blood-to-plasma ratio.

_{b}Therefore:

The concentration changes in the 1^{st} and *i*th well-stirred liver compartments are described by:
and
where *C _{in}* is the input blood concentration into the 1

^{st}liver compartment.

The total liver concentration is assumed to be the average of *C _{hi}* (

*i*=1, 2, 3…

*n*) in all compartments:

#### Model Simulations of *ER versus CL*_{int}: Comparing the SCM with the Basic Hepatic Clearance Models

_{int}

The mathematical expression of *ER* for the SCM with *n* liver sub-compartments was derived from eqs. 2–5 as follows:

The relationships between *ER* and *CL _{int}* have been reported for the basic hepatic clearance models (Roberts and Rowland, 1986a; Roberts and Rowland, 1986b). For the DM based on the closed boundary conditions (0 < Z < 1, where Z is defined as the distance along the length of the liver):
where

*a*= (1 + 4

*D*)

_{N}R_{N}^{1/2}and

*R*, the efficiency number that measures the removal rate of substances by liver cells, is given by:

_{N}The mathematical expression of *ER* for the WSM can be derived from that of the SCM (eq. 11) by setting *n* = 1, and from that of the DM (eq. 12) with *D _{N} =* ∞:

Similarly, the mathematical expression of *ER* for the PTM is equivalent to that of the SCM (eq. 11) with *n* = ∞ and the DM (eq. 12) with *D _{N} =* 0:

To assess the quantitative similarities between the SCM and the basic hepatic clearance models, the relationships between *ER* and *CL _{int}* were simulated according to eqs. 11, 12, 14, and 15 with varying values of

*n*for the SCM and

*D*for the DM. For the DM, the commonly reported range of

_{N}*D*(0.1–0.6) (Diaz-Garcia et al., 1992; Chou et al., 1993; Evans et al., 1993; Oliver et al., 2001) was applied, and

_{N}*f*was assumed to be 1 in all simulations for simplicity.

_{ub}#### Assessing Liver *CL*_{int} and *K*_{p} by the SCM of Hepatic Elimination in PBPK

_{int}

_{p}

The measured blood or plasma and liver concentration-time data in rats after single intravenous (IV) bolus doses were obtained from the literature for six compounds, cyclosporine A (CyA) (Kawai et al., 1998), ethoxybenzamide (EB) (Lin et al., 1978), fingolimod (FTY720) (Meno-Tetang et al., 2006), diazepam (DZP) (Igari et al., 1983), verapamil (VEM) (Yamano et al., 2000), and diltiazem (DLZ) (Yamano et al., 2000). Concentration versus time data were digitized from the published graphs using GetData Graph Digitizer version 2.26 (http://getdata-graph-digitizer.com/). These model compounds were selected based on the following conditions:

Liver is the major eliminating organ

Extrahepatic clearances are known or assumed to be negligible

Tissue-to-plasma concentration ratios are linear

*ER*ranges from low to highTime courses of blood/plasma and liver concentrations are resolvable

Distribution into the liver and access to the hepatic enzymes are flow-limited (high permeability) with minor or negligible transporter involvement

As was similarly done previously (Ebling et al., 1994; Foster, 1998; Gueorguieva et al., 2004; Cheung et al., 2018; Li and Jusko, 2022), a piecewise open-loop approach was applied. Briefly, the blood concentration (*C _{b}*)-time profile was fitted first and then used as the forcing input function (i.e., replacing the

*C*term in eq. 8 by the fitted

_{in}*C*) to model the concentration-time data of the liver as a single organ. This method was advantageous for our purposes as we did not have to deal with involvement of organs/tissues other than the liver and it was also reported to generate comparable hepatic

_{b}*K*estimates as obtained by fitting all tissues simultaneously using a full PBPK model (Gueorguieva et al., 2004).

_{p}The exponential equations used to describe the *C _{b}*-time data are:
where

*A*,

*B*, and

*C*are the intercepts,

*α*,

*β*, and

*γ*are the slopes,

*C*is the initial blood concentration at time 0, and

_{b0}*V*is the average value of the reported blood volumes in the source literature (78 ml/kg).

_{b}Subsequently, the total blood clearance (*CL _{b}*) can be obtained from:

By assuming liver is the only clearance organ (*CL _{b}* = hepatic clearance),

*ER*is given by: where

*Q*= 60.82 ml/min/kg and

_{h}*V*= 36.6 ml/kg are published values for rats (Brown et al., 1997).

_{h}For the SCM, *f _{ub}CL_{inti}* can be calculated from

*ER*by rearranging eq. 11:

Due to the limitation of the open-loop approach that *CL _{int}* and

*K*are highly correlated yielding extremely large CV% values when estimated simultaneously, the

_{p}*CL*of the SCM containing different

_{int}*n*were first obtained using eqs. 17–19 with the

*CL*

_{b}and

*ER*estimated from fitting the blood PK data and then fixed in the subsequent fitting of liver concentration-time data to estimate the PBPK-operative

*K*according eqs. 7–10, with

_{p}*C*in eq. 8 being replaced by the pre-fitted

_{in}*C*in eq. 16. The outflow blood concentration (

_{b}*C*,

_{outi}*i*=n) from the last liver segment was calculated from the corresponding liver concentration (

*C*,

_{hi}*i*=n) according to eq. 7 and the concentration-time profile in each of the liver sub-compartments was also simulated based on eqs. 8 and 9.

#### Assessing Effects of Hepatic Enzyme Zonation on *K*_{p} Estimation and Prediction of Liver Concentrations by the SCM

_{p}

The hepatic zonal heterogeneity of key metabolizing enzymes has been reviewed and its potential effects on hepatic metabolism of xenobiotics have been assessed by various in vitro and in situ liver-derived systems (Jungermann and Katz, 1989; Jungermann, 1995; Tirona and Pang, 1996; Oinonen and Lindros, 1998; Abu-Zahra and Pang, 2000; Li et al., 2019; Tomlinson et al., 2019). To assess how such hepatic zonation affects the estimation of liver *K _{p}* and the prediction of liver concentrations by PBPK models, uneven zonal-enzyme distribution (i.e., different

*ER*and

_{i}*CL*) was assumed for each of the sub-compartments in the SCM with

_{inti}*n*= 5. The following scenarios were tested: (1) lower

*ER*/

*CL*at the PP region by assuming

_{int}*F*=

_{i+1}*F*

_{i}^{2}(

*i*=1, 2…4), and (2) lower

*ER*/

*CL*at the PV region by assuming

_{int}*F*=

_{i+1}*F*

_{i}^{1/2}(

*i*= 1, 2…4). The results were compared with those previously obtained by assuming equal

*ER*/

*CL*for all liver segments.

_{int}For testing Scenario 1, the availability of the 1^{st} and subsequent liver compartments are:

For testing Scenario 2, the availability of the 1^{st} and subsequent liver compartments are:

With the assumed *F _{i}* values, the corresponding

*CL*can be obtained by eqs. 3 and 4 for the subsequent model fittings of liver data and the estimation of

_{inti}*K*using eqs. 7–10. Provided the relationship of

_{p}*F*=

_{(i+1)}*F*(

_{i}^{a}*i*=1, …

*n*-1), larger values of

*a*produce steeper gradients of the metabolic clearance within the liver. By assigning a wide range of

*a*values from 1–1000, the impact of steepness in the internal metabolic clearance gradients on the estimation of total

*CL*and

_{int}*K*by the 2- and 5-compartment zonal SCM was further assessed under testing Scenario 1.

_{p}#### Model Fitting

The model fittings of blood and liver concentration-time data were performed by nonlinear regression using the maximum likelihood algorithm in ADAPT 5 (Biomedical Simulations Resources, Los Angeles, CA) (D'Argenio et al., 2009). The variance model was:
where V* _{i}* is the variance of the

*i*th data point, Y

*is the*

_{i}*i*th model-predicted concentration, σ

*and σ*

_{inter}*are the variance model parameters. Model selection was based on the goodness-of-fit criteria, which included the Akaike Information Criterion (AIC), visual inspection of the fitted profiles, and CV% of the parameter estimates.*

_{slope}The maximum predicted liver concentration (*C _{max}*), time to reach

*C*(

_{max}*T*), and area under the curve (

_{max}*AUC*) of the predicted liver concentration-time data were obtained by non-compartmental analysis performed using Phoenix WinNonlin version 6.4 (Certara USA, Inc., Princeton, NJ).

The ADAPT code for the SCM model for one compound (DLZ) is provided in the Supplemental Methods.

## Results

#### Simulations of *ER versus CL*_{int} as a Function of “*n*” and “*D*_{N}”: Comparing the SCM with Basic Hepatic Clearance Models

_{int}

_{N}

The simulated *ER versus CL _{int}* profiles using the SCM with varying values of

*n*and the three basic hepatic clearance models are displayed in Fig. 2. For the DM, the commonly reported

*D*range of 0.1–0.6 was used for the model simulations with only the results obtained from the boundary values (i.e., 0.1 and 0.6) presented as any

_{N}*D*value in between will yield intermediate profiles.

_{N}As expected, the *ER* increases with increasing *CL _{int}* for all models, with the SCM and DM exhibiting intermediate profiles that lie in between those of the PTM and WSM. With the same

*CL*,

_{int}*ER*values given by the SCM increase with increasing

*n*while the DM yields lower

*ER*as

*D*increases. To achieve the same

_{N}*ER*, the rank order of

*CL*is WSM>DM/SCM>PTM. The

_{int}*ER versus CL*profile of the SCM is identical to that of the WSM when

_{int}*n*= 1 and becomes approximately equal to that of the PTM when

*n*> 30, consistent with the previous report (Gray and Tam, 1987).

As can be seen from Fig. 2, the simulated *ER* profiles using the SCM containing two and five liver sub-compartments closely match those of the DM with *D _{N}* equal to 0.6 and 0.1. Therefore, the SCM containing one (viz. the basic WSM), two, and five liver sub-compartments were further used and compared in the subsequent fitting of liver data and estimation of

*CL*and

_{int}*K*.

_{p}#### Assessing Liver *CL*_{int} and *K*_{p} in PBPK Models by the SCM: Exploring the Impact of *n*

_{int}

_{p}

With the open-loop approach, the *C _{b}*-time data were first characterized by eq. 16 with model fittings displayed in Fig. 3. The estimates of intercepts and slopes are listed in Supplemental Table 1. Secondary parameters are shown in Table 1, where

*CL*and

_{b}*ER*were estimated by eqs. 17 and 18, and

*f*of the SCM with different values of

_{ub}CL_{int}*n*were obtained using eq. 19 with the

*ER*estimated from fitting the blood PK data. The

*C*-time profiles of all model compounds are well described by the exponential equations and parameter estimates exhibit low CV% values. In line with the model simulations in Fig. 2, the estimated

_{b}*f*decreases with increasing

_{ub}CL_{int}*n*given the same

*ER*especially for drugs with higher

*ER*values (Table 1).

Subsequently, with the estimated *f _{ub}CL_{int}* and the fitted

*C*-time profile as the forcing input function into the 1

_{b}^{st}liver sub-compartment (replacing

*C*in eq. 8) of the SCM, liver

_{in}*K*were obtained by fitting the measured

_{p}*C*-time data using the SCM with

_{h}*n*equal to 1, 2, and 5 (eqs. 7–10). The model fittings are presented in Fig. 3, and the estimated

*K*as well as the AIC values for each of the models are listed in Table 1. Overall, the fitted

_{p}*C*-time profiles by the SCM of different

_{h}*n*exhibit slight differences at early time points while later concentrations were fitted equally well as the basic WSM (i.e.,

*n*=1) (Fig. 3). As shown in Table 1, the selection of

*n*shows little influence on the estimation of

*K*for drugs with low

_{p}*ER*(e.g., EB and CyA), however, the model-dependencies in

*K*become more remarkable as

_{p}*ER*increases. For the same compound, the SCM with larger

*n*yields smaller

*K*values. Such model dependency is in accordance with that of the

_{p}*CL*(Fig. 2 and Table 1). The similar AIC values suggest that model performance of the SCM in describing the measured liver data are not significantly affected by the selection of

_{int}*n*. In line with the theoretical correlations (Fig. 2), the parameter estimates obtained by the 2-segment SCM are comparable to those yielded by the DM with

*D*= 0.6 from our previous analysis (Li and Jusko, 2022), viz., the

_{N}*CL*and

_{int}*K*values for DLZ are 211.9 ml/min/kg and 14.11 for the SCM, and 213 ml/min/kg and 13.96 for the DM.

_{p}The non-compartmental analysis was performed to obtain the maximum concentration (*C _{max}*), time to reach

*C*(

_{max}*T*), and area under the curve from time 0 to infinity (

_{max}*AUC*) of the fitted liver concentration-time data, and the relative changes in these parameters are provided to examine the differences in model predictions in relation to

_{inf}*n*(Table 2). As indicated by Fig. 3 and Table 2, the predicted liver

*C*of all compounds increases with increasing

_{max}*n*, while the

*T*are delayed for some of the compounds (e.g., FTY720, VEM, and DLZ); the

_{max}*AUC*values are mostly comparable except for that of DZP which slightly increased with increasing

_{inf}*n*.

To explore the characteristics of intrahepatic drug disposition with the SCM, the concentration (*C _{hi}*)-time profiles in each of the liver sub-compartments were simulated and displayed in Fig. 4. The SCM predicts an intrahepatic concentration gradient from the first to the last liver sub-compartment, which is quantitatively consistent with the theoretical expectations (Kashiwagi et al., 1981), and the magnitude of such concentration gradients was shown to be dependent on both the values of

*n*and

*ER*(Fig. 4). For both the 2- and 5-compartment SCM, the

*C*-time curves shift to the right (delayed

_{hi}*T*) with decreasing

_{max}*C*values as

_{max}*i*increases, exhibiting the typical profiles similar to those of transit compartment models (Sun and Jusko, 1998).

To assess whether the time delays in the intrahepatic concentrations are dependent on the selection of *n*, the concentration of DLZ in the venous blood leaving the last liver sub-compartment (*C _{outn}*,

*n*= 1, 2, and 5) was calculated from the corresponding liver concentration (

*C*,

_{hn}*n*= 1, 2, and 5) and the estimated

*K*(Table 1) using eq. 7. As shown in Fig. 5, the

_{p}*T*of the

_{max}*C*-time profile increases with increasing values of

_{outn}*n*, which is 0.97 minutes (

*n*=1), 2.42 minutes (

*n*=2), and 4.23 minutes (

*n*=5), suggesting similar features of the SCM in characterizing time delays as transit compartment models. Such delays in

*T*with increasing

_{max}*n*were also observed for all the other tested compounds (Supplemental Figure 1). Since

*C*from the last liver sub-compartment is one of the input functions into the venous blood pool, such delays associated with the selection of

_{out}*n*may be more relevant if the SCM is used in a full PBPK model even though the model fittings of total liver concentrations were similar (Fig. 3 and Table 2).

#### Effects of Metabolic Zonation on Characterizing Liver Concentration-Time Data and Estimating *CL*_{int} and *K*_{p} by the SCM in PBPK

_{int}

_{p}

The effects of hepatic enzyme zonation were assessed using the SCM containing five sub-compartments with the data for DLZ. To keep the *ER* of DLZ (Table 1) unchanged for different models, it is mathematically convenient to assign the assumed patterns of hepatic enzyme zonation based on the *F* value in each of the sub-compartments rather than the value of *CL _{int}* as was done previously (Li et al., 2019). Provided the relationship of

*F*=

_{i+1}*F*(

_{i}^{a}*i*= 1, …

*n*-1), the exponential of

*F*,

_{i}*a*was set to be 2 meaning increased metabolic clearance from the PP to PV region (Scenario 1) and 0.5 reflecting decreased metabolic clearance along the sinusoidal flow path (Scenario 2). This yielded an approximately 100-fold difference between the lowest and highest intrinsic clearances. In all testing scenarios, the

*f*value was first obtained using the total

_{ub}CL_{inti}*ER*based on eqs. 3 and 4 and 20–23. The calculated

*f*(Table 3) and the pre-estimated blood PK parameters of DLZ (Supplemental Table 1) were fixed in the subsequent model fitting of liver data according to eqs. 7–10. The model fittings utilizing two different testing scenarios were compared with those with an even distribution of enzymatic activity throughout the liver (Fig. 6). The estimated

_{ub}CL_{inti}*K*along with the

_{p}*T*,

_{max}*C*, and

_{max}*AUC*values of the fitted liver profiles are listed in Table 3.

_{inf}The estimated total *CL _{int}* are identical under both scenarios; however, the model fittings and the resulting

*K*are different because of the differing patterns of hepatic enzyme zonation. The predicted liver concentrations mainly differ at early time points and are almost identical after 20 minutes post dosing (Fig. 6). The SCM with lower

_{p}*ER/CLint*at the PP region (Scenario 1) predicted higher

*T*,

_{max}*C*, and

_{max}*AUC*, but a lower

_{inf}*K*value as compared with those predicted by the SCM with lower

_{p}*ER/CLint*at the PV region (Scenario 2). The model predictions and resulting parameters of the SCM with evenly distributed

*ER/CLint*lie in between those two testing scenarios (Fig. 6 and Table 3). To achieve the same

*ER*, the zonal SCM requires a higher total

*CLint*(183.6 ml/min/kg) than the SCM with an even distribution of enzymatic activity throughout all the liver segments (151.2 ml/min/kg) (Table 3). Nevertheless, the model performance is comparable regardless of the hepatic heterogeneity of metabolic enzymes as suggested by the AIC values shown in Table 3.

The effects of enzyme zonation on the intrahepatic concentration gradients as a function of time and tissue space were assessed using the 5-compartment SCM and the data for DLZ. The *C _{hi}*-time profiles of DLZ predicted by the zonal SCM are presented in Fig. 7, A–C. For all models, the

*C*of the subsequent compartment is always less than that of the previous one with the concentration-time curves shifting to the right as the drug moves from the PP region to the PV region and being metabolized. As compared with the case of evenly distributed metabolic clearance, the

_{max}*C*of each liver sub-compartment decreases slower with less delay in

_{max}*T*when metabolism primarily occurs at the PV region (Scenario 1), while a faster drop in

_{max}*C*and a more remarkable delay in

_{max}*T*were observed when metabolic clearance mainly locates at the PP region (Scenario 2).

_{max}The predicted tissue (*C _{h}*) and venous blood (

*C*) concentration gradients of DLZ from the inlet to the outlet of the liver at one time point after pseudo equilibrium (e.g., 180 minutes) are displayed in Fig. 7, D–F, where

_{out}*C*was calculated from

_{out}*C*and the estimated

_{h}*K*values (Table 3) using eq. 7 assuming

_{p}*R*= 1. As can be seen, different hepatic enzyme zonation results in differed intra-organ concentration gradients. Fig. 7D shows a slow initial decrease in both

_{b}*C*and

_{h}*C*at the PP region where the metabolic clearance was low so that the drug can accumulate before being exposed to the higher enzymatic activity at the PV region where a faster drop in both

_{out}*C*and

_{h}*C*was observed. When the metabolic clearance is evenly distributed, the drug is equally extracted (i.e., equal

_{out}*ER*,

*CL*, and

_{int}*F*) by all the liver sub-compartments so that

*C*and

_{h}*C*exhibit a constant rate of decline from the PP to the PV region (Fig. 7E). In contrast, there is an immediate initial drop in both

_{out}*C*and

_{h}*C*when the PP region has higher intrinsic clearance and the decreases in both concentrations become less remarkable as the drug moves toward the PV region where the intrinsic clearance is low (Fig. 7F).

_{out}As shown by Fig. 7, D–F, the zonal distribution of hepatic enzymes also has an impact on the tissue space-averaged blood concentration (red dashed lines), which is the highest (37.6 ng/ml) when metabolism was mainly at the PV region followed by that with an even distribution (22.7 ng/ml), and the lowest (13.2 ng/ml) when metabolism primarily occurs at the PP region.

The *C _{out}*-time profiles leaving the last liver segment predicted by the zonal SCM with

*n*=2 and 5 as a function of hepatic enzyme zonation are displayed in Fig. 8. Regardless of the value of

*n*, the SCM exhibiting lower metabolic clearance at the PV region (Scenario 2) yields a lower (2.3-fold decrease in

*C*) and later peak (3.1-fold increase in

_{max}*T*) of

_{max}*C*-time profile than that obtained under Scenario 1 (i.e., lower metabolic clearance at the PP region), with the SCM of even enzyme distribution showing the intermediate profile. Consistent with those shown in Fig. 5, a more significant delay in

_{outn}*T*is observed when there are more liver segments incorporated in the SCM given the same enzyme zonation. A 10-fold shallower gradient of metabolic clearances (i.e., when

_{max}*a*= 1.6 or 1/1.6) has been tested, but a twofold difference was still observed in both

*C*and

_{max}*T*of the

_{max}*C*-time profiles (Supplemental Table 2 and Supplemental Figure 2). For drugs with a narrow therapeutic window or requiring a delayed or immediate onset of effects, a 2- to 3-fold change in

_{out5}*C*or

_{max}*T*may result in significant impacts on predicting drug efficacy and/or safety.

_{max}For the zonal SCM, the difference between hepatic availabilities in two adjacent liver segments (*F _{i}* and

*F*,

_{(i+1)}*F*=

_{(i+1)}*F*) (

_{i}^{a}*i*=1, …

*n*-1) resulting from differed hepatic enzyme zonation also plays a role in the estimation of

*CL*and

_{int}*K*in PBPK. The changes in

_{p}*CL*and

_{int}*K*estimates with changing values of

_{p}*a*, the exponential of

*F*(

_{i}*i*=1, …

*n*-1), were examined and presented in Fig. 9. All model fittings were performed assuming lower metabolic clearances at the PP region with the

*ER*of DLZ remaining unchanged. As indicated in Fig. 9A, the estimated total

*CL*of the zonal SCM lies in between those of the two extreme cases (viz. WSM and PTM) regardless of the changes in the number of liver sub-compartments (

_{int}*n*) and the exponential of

*F*(

_{i}*a*), consistent with the previous demonstrations (Fig. 2). The total

*CL*of the zonal SCM decreases with increasing

_{int}*n*but increases and approaches the WSM-estimated intrinsic clearance (

*CL*) as the difference between the metabolic clearance in two adjacent liver segments becomes larger. The estimated

_{int,WSM}*K*show the same dependency on

_{p}*n*as

*CL*, i.e., the SCM with a larger

_{int}*n*produces a smaller

*K*given the same zonation of hepatic enzymes. However, in contrast to the relationship between

_{p}*CL*and

_{int}*a*, the estimated

*K*is negatively correlated with

_{p}*a*and becomes less than that obtained by the PTM (

*K*) as

_{p, PTM}*a*increases (Fig. 9B).

## Discussion

In PBPK modeling, the true *CLint* and *Kp* are an ‘unknowable’ mystery in the ‘black box’ of the liver and the differential equations that describe total liver concentrations require an assumed model with these parameters that are model-dependent (Li and Jusko, 2022). The SCM offers an alternative to the WSM, PTM, and DM with close resemblance to the latter and offering many flexibilities to better approximate the known or expected functioning and concentrations in the liver.

The SCM was initially used to describe the dilution curves of non-eliminating tracers from organs where the flow through a blood vessel was modeled as a series of identical well-stirred chambers with the same volumes (Davenport, 1983). Our studies confirm the SCM as an alternative to the complex DM as they share some mathematical similarities (Goresky et al., 1973; Roberts and Rowland, 1986a; Gray and Tam, 1987) with fewer computation complexities and offering advantages in describing a variety of phenomena, including PBPK profiles. It was not clear from prior work how closely the SCM and DM are quantitatively related. The major determinants differentiating the SCM and DM from the WSM and PTM, the two limiting cases of the basic hepatic clearance models, are the number of liver sub-compartments (*n*) and the *D _{N}*. In this work, the values of

*n*and

*D*were found to play a dominant role in comparing the SCM and DM. The changes in

_{N}*ER versus CL*profiles of the SCM with increasing

_{int}*n*exhibit the same trend as those of the DM with decreasing

*D*; specifically, the SCM with

_{N}*n*= 2 and 5 yield almost identical

*ER*as those predicted by the DM with

*D*= 0.6 and 0.1, the boundary values of the

_{N}*D*range (Obach et al., 1997; Oliver et al., 2001; Pang et al., 2019) given the same

_{N}*CL*(Fig. 2).

_{int}The SCM has had limited use as a model of hepatic elimination in PBPK. So far, such applications have mainly focused on the complex transporter SCM with *n* = 5 (Watanabe et al., 2009; Jones et al., 2012; Morse et al., 2017), and it was unclear how changing *n* had an impact on the liver concentration-time data. In the 5-segment transporter SCM, at least four clearance terms (i.e., active uptake and biliary efflux, passive diffusion, and metabolic clearance) are involved, causing parameter identifiability issues (biliary and metabolic clearances cannot be uniquely identified through model-fitting). Many assumptions are required and substrate- and transporter-dependent empirical scaling factors are needed, especially for IVIVE. To assess the impact of *n* on the model performance of SCM in PBPK without either these or other complexities, the in vivo time course data of six flow-limited substances were examined. Interestingly, the model-dependencies in the estimated *CL _{int}* and PBPK-operative

*K*as a function of

_{p}*n*were shown to be the same, i.e., SCM (

*n*=1) (WSM)>SCM (

*n*=2)>SCM (

*n*=5)>PTM, indicating that the assumed average unbound tissue/blood concentrations within the liver for the SCM are lower when

*n*is smaller given the same

*ER*. These findings are similar to our recent analysis where the rank order of

*CL*and

_{int}*K*was WSM>DM (

_{p}*D*=0.6)>PTM (Li and Jusko, 2022). In model fittings, the choice of

_{N}*n*mainly affects the prediction of early total liver concentrations as reflected by the higher

*C*(all compounds) and delayed

_{max}*T*(FTY720, VEM, and DLZ) with larger values of

_{max}*n*although minor differences in

*AUC*were observed. Such differences in model fittings suggest that the optimal

_{inf}*n*value of the SCM may not be readily identifiable by merely fitting in vivo time course data if early observations are missing, especially when lacking intravenous data.

Hepatic zonal heterogeneity of key metabolizing enzymes and its potential effects on hepatic metabolism of xenobiotics are known (Jungermann, 1986; Gebhardt, 1992; Tirona and Pang, 1996; Oinonen and Lindros, 1998; Cunningham and Porat-Shliom, 2021). For instance, the expression of major drug-metabolizing cytochrome P450 enzymes is higher in the downstream PV region of the liver than in the PP region, with a 30- to 60-fold difference reported for CYP2E1 (Buhler et al., 1992; Tachikawa et al., 2018). Among the phase II reactions, glucuronidation occurs preferentially in the PV cells, and sulphation is greater in the PP cells (Jungermann and Katz, 1989; Jungermann, 1995). Traditional hepatic clearance models assume uniform distribution of metabolic enzymes in the liver, which does not represent the true physiology. The SCM allows for hepatic enzyme zonation by assigning different *CL _{int}* values to each liver segment. For example, a three-zone SCM (i.e., PP-, mid-, and PV-zone) with differing metabolic clearances (i.e., PV/PP esterase activity = 6.6) obtained by examining the metabolic activities toward enalapril in S9 fractions of enriched rat PP and PV hepatocytes was applied to characterize the uneven ester hydrolysis of enalapril in rat liver based on in vitro and perfusion data (Abu-Zahra and Pang, 2000). The effect of zonal differences in metabolism on the translation of unbound liver-to-plasma partition coefficients from in vitro to in vivo was explored theoretically using a 5-segment model in which a 256-fold difference in metabolic clearances between the PP and PV regions was assigned (Li et al., 2019). However, none of these assessments were based on assessing in vivo PBPK-type time course data. By assigning liver segment-specific

*CL*values, we show that hepatic enzyme zonation produces more remarkable differences in the predicted total liver concentrations (Fig. 6) as compared with the effects of

_{int}*n*(Fig. 3). It is worth noting that even with the same total

*CL*, the model predictions and estimated

_{int}*K*can differ depending on how the metabolic clearances are distributed as do the internal concentration gradients as a function of time and tissue space (Fig. 7). The SCM with a lower

_{p}*CL*at the PP region predicts the highest tissue space-averaged blood concentration (red dashed lines in Fig. 7B); therefore, the tissue-to-plasma ratio is the smallest at the pseudo-equilibrium state given the comparable tissue space-averaged liver concentrations (black dashed lines in Fig. 7B) compared with the other models, consistent with the rank order of the estimated

_{int}*K*based on the full curves (Table 3). Additionally, the estimation of total

_{p}*CL*and

_{int}*K*by the zonal SCM was also affected by the steepness of the metabolic clearance gradients within the liver as reflected by the value of

_{p}*a*(

*F*=

_{(i+1)}*F*,

_{i}^{a}*i*=1, …

*n*-1) (Fig. 9). Similarly, hepatic transporters also exhibit zonal heterogeneity (Tachikawa et al., 2018), which may be included in the SCM if relevant information is available.

The hepatic WSM is often operated with PBPK modeling for IVIVE and to assess and predict drug-drug interactions. Such modeling with use of in silico tissue *K _{p}* values, when first optimized based on adjustments to capture known human plasma concentration versus time profiles (“top-down” or “middle-out”), is about 80% successful in predicting drug-drug interaction AUC values within 1.25-fold. An IVIVE approach (“bottom-up”) is used infrequently (Wagner et al., 2015) and requires a large scaling factor when employed de novo for predicting hepatic clearance based on in vitro

*CL*values (Tess et al., 2022). The SCM offers the possibility of application of a more flexible modeling approach for handling known complexities of the liver, such as transporters and zonal differences in metabolism. The application of SCM in PBPK may also have important pharmacological/toxicological implications. For example, oral dosing with rapid absorption may exacerbate the differences in early model predictions as the initial concentration in the 1

_{int}^{st}liver sub-compartment (

*Dose*/

*V*

_{h1}) will be higher compared with that of an intravenous dose that undergoes tissue distribution. This is relevant when a drug shows hepatic toxicity. In addition to zonal expression, the regulation of hepatic enzymes also exhibits zonal dependency and was shown to be partially responsible for the zonal pattern of certain liver injuries (Wojcik et al., 1988; Buhler et al., 1992; Lindros, 1997; Oinonen and Lindros, 1998; McEnerney et al., 2017). For instance, the regiospecific expression and induction of CYP2E1 play a major role in explaining the centrilobular damage caused by a high dose of acetaminophen (Anundi et al., 1993). This is worth consideration when assessing drug-drug interactions, especially if the liver is the target of efficacy or toxicity. When the systemic PK/exposure of the victim drug is not significantly affected by a perpetrator drug, it is possible that the concentrations of the victim drug at a certain zonal region of the liver might change significantly due to such position-dependent regulation and thereby result in suboptimal efficacy or potential local cell injury. Also, the predicted intrahepatic concentration gradient for the SCM differs appreciably with the value of

*n*, which may have an impact when assessing PK/pharmacodynamic (PD) relationships if the assumed hepatic concentrations relate to the pharmacological effects and/or toxicity. All these aspects may be better assessed with the SCM as it allows incorporation of zonal expression and regulation of metabolic enzymes/transporters and the prediction of concentrations in each of the liver sub-compartments. The intra-organ concentration gradients are also expected to exist in other eliminating tissues, indicating a potential wider application of the SCM in addition to describing hepatic disposition. The SCM shares similar construction as transit compartment models in which a series of compartments are connected by first-order processes in a catenary manner and described with serial differential equations (Sun and Jusko, 1998). Such models have been extensively used in accounting for various short to long time delays in PK and PK/PD when information is lacking about the intermediary steps causing delays. Such applications include delayed drug absorption and biliary excretion (Bischoff et al., 1971; Savic et al., 2007; Kagan et al., 2010; Lachi-Silva et al., 2015), drug movement along the spinal cord (Heetla et al., 2016), delayed pharmacological responses due to signal transduction processes (Byun et al., 2022), and for disease progression modeling (Earp et al., 2008). There is considerable flexibility and variety in the number of transit steps and the time constants. This is partly similar for hepatic SCM models, although the physiologic volume/flow rates control the time constants (eq. 9). The intra-organ concentrations decrease down the compartments in a stepwise fashion (Fig. 4). As a result, an SCM with more liver segments produces greater delays in the time profiles of outflow blood concentrations from the last liver segment (

*C*) (Fig. 5). On top of this, a higher

_{outn}*CL*at the PP region predicts even further delayed

_{int}*C*-time profiles, especially with larger values of

_{outn}*n*(Fig. 8). As

*C*enters the venous blood pool, such differences may have an impact on the kinetics in other tissues in full PBPK analyses. Furthermore, if the SCM is paired with the wrong

_{outn}*K*for simulation purposes, it could appreciably distort capturing the liver concentrations as shown in Supplemental Figure 3. One of the limitations of this study is that the gradient of intrinsic clearances existing between the PP and PV regions was arbitrarily assigned by setting

_{p}*a*value to 2 and 0.5 (i.e., about 100-fold difference in the gradient of metabolic activity). This may not be true for the compounds assessed herein; however, it should suffice for examining the trends of changes in model fittings and parameter estimates (e.g.,

*CL*and

_{int}*K*) resulting from the metabolic zonation as compared with even enzyme distribution that has been commonly assumed and applied.

_{p}In this work, the quantitative equivalency between the SCM and DM was demonstrated through theoretical simulations and fittings of plasma/liver data by PBPK modeling, and factors that need to be considered when applying the SCM were proposed. As was shown, five sub-compartments seem to be sufficient for the SCM to mimic the DM and will have to be accepted as reasonable. Without extremely frequent early liver data, the best *n*-SCM cannot be ascertained. Despite certain limitations, zone-specific intrinsic clearances may be obtained using isolated hepatocytes from various regions of the liver (Abu-Zahra and Pang, 2000) as a starting point for constructing more physiologically-based zonal SCM for IVIVE. With advances in technology, it may be possible to collect samples from different zonal regions of the liver to validate the predicted concentrations in liver sub-compartments by the SCM and further aid model selection and refinements. In conclusion, the model-dependencies in the predicted total/intrahepatic concentrations and the estimates of *CL _{int}* and PBPK-operative

*K*as a function of

_{p}*n*and hepatic enzyme zonation are most relevant when: 1) the SCM is used for IVIVE in that a model-dependent

*ER*will be expected from the same in vitro

*CL*, 2) different SCM are applied with

_{int}*K*from the same source (such as

_{p}*in silico*predictions) in PBPK that will likely yield different liver predictions, 3) assessing PK/PD/toxicity relationships if the assumed hepatic concentrations are the driving forces. In addition to those factors assessed herein, other complexities, such as transporter heterogeneity and nonlinear

*CL*or

_{int}*K*, may be further added to the SCM given suitable experimental information.

_{p}** Note Added in Proof:** A change was made to Equation 19 published in the Fast Forward version published March 7, 2023. Equation 19 has now been corrected.

## Authorship Contributions

*Participated in research design:* Li, Jusko.

*Performed data analysis:* Li.

*Wrote or contributed to the writing of the manuscript:* Li, Jusko.

## Footnotes

- Received October 27, 2022.
- Accepted January 24, 2023.
This work was supported by National Institutes of Health National Institute of General Medical Sciences [Grant R35-GM131800].

No author has an actual or perceived conflict of interest with the contents of this article.

↵This article has supplemental material available at dmd.aspetjournals.org.

## Abbreviations

- AIC
- Akaike Information Criterion
*AUC*- area under the curve
*CL*_{b}- total blood clearance
*CL*_{int}- hepatic intrinsic clearance
*C*_{max}- maximum concentration
- CyA
- cyclosporine A
- DLZ
- diltiazem
*D*_{N}- dispersion number
- DM
- dispersion model
- DZP
- diazepam
- EB
- ethoxybenzamide
- ER
- extraction ratio
*F*- hepatic availability
- FTY720
- fingolimod
*f*_{ub}- unbound fraction in blood
- IVIVE
- in vitro-to-in vivo extrapolation
*K*_{p}- tissue-to-plasma partition coefficient
*n*- number of liver sub-compartments
- PBPK
- physiologically-based pharmacokinetic
- PD
- pharmacodynamic
- PK
- pharmacokinetic
- PP
- periportal
- PTM
- parallel tube model
- PV
- perivenous
*R*_{b}- blood-to-plasma ratio
- SCM
- series compartment model
- SS
- steady state
*T*_{max}- time to reach
*C*_{max} - VEM
- verapamil
- WSM
- well-stirred model

- Copyright © 2023 by The American Society for Pharmacology and Experimental Therapeutics