Introduction

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There has been major progress made in the past three decades in the field of modeling of the liver for the processing of drugs and metabolites. All of the models feature the physiological determinants of clearance, i.e. organ blood flow, vascular and tissue binding, membrane permeability, and enzymatic activity (V_max) and affinity (K_M) of intracellular enzyme systems and/or excretory apparati for the removal of substrates. Several useful mathematical models that inter-relate the deterministic variables have been developed to describe temporal events and, in particular, steady-state events, wherein drug and metabolite binding to liver tissue is completed and does not contribute to drug loss. Accordingly, the liver has been viewed as a well-mixed compartment ("well-mixed" model) (Rowland et al., 1973; Pang and Rowland, 1977) or as an organ receiving a series of nonsegregated parallel flows surrounded by identical single sheets of hepatocytes of uniform enzymatic activity ("parallel tube" model) (Winkler et al., 1973). These models have been construed as being too extreme and idealized; because there is either infinite (well-stirred model) or no (parallel tube model) mixing, they are unable to describe the asymmetrical outflow profiles or RTDs¹ observed after bolus dosing. The inadequacy has been explained on the basis of heterogeneities in flow (Miller et al., 1979; Bronikowski et al., 1987) and a high degree of geometric branching within the liver, which provide an intermediate mixing or dispersion. This necessitates the reselection, refinement, or development of models that could describe the observations.

The barrier-limited, distributed, capillary transit time model of Goresky (Goresky, 1963; Goresky et al., 1973) and the adaptation (Roberts and Rowland, 1985, 1986) of the DM of Perl and Chinard (1968) with closed boundary conditions (Danckwerts, 1953) are such model developments. These describe curve models that resemble the outflow dilution profile observed after pulse injection, and they recognize that the shape and magnitude of the solute outflow concentration-time curve (the dilution profile) are complex functions of the interrelationships among blood and tissue binding, blood flow, vascular micromixing and flow heterogeneity, and cellular influx, efflux, and sequestration (metabolism or excretion). Many established principles have evolved through modeling of outflow data from experiments that encompass bolus injection of a dose containing a mixture of multiple noneliminated indicators and labeled tracer substrate; in these instances, differentiation among the injected species is enabled by the use of differential labeling and analyses (multichannel  and  spectrometry). The kinetics of the diffusible tracer reflect those under steady-state conditions for the bulk compound, as mandated in tracer methodology. This MID technique, which was first introduced to assess the behavior of noneliminated reference substances for the understanding of liver physiology (Goresky, 1963), has become a useful experimental tool to provide mechanistic insight into the processes of transport and irreversible loss of the tracer-labeled substrate, especially in single-pass, rat liver-perfusion experiments, wherein the recirculation of solutes, which presents an added complication in vivo, is avoided (Wolkoff et al., 1979; Schwab et al., 1990; Geng et al., 1995).

The GM and the mathematical formulations have a long history, first being used for descriptions of the handling of endogenous compounds by dog liver in vivo (Goresky et al., 1973) and then being extended to analyses of xenobiotics (Schwab et al., 1990; Geng et al., 1995) and metabolic processing (Pang et al., 1994) in perfused rat liver. The approach involves the construction of a vascular reference curve based on an initial consideration of the binding and distribution properties of the diffusible solute in the vasculature and then appraisal of the outflow curve for the labeled tracer with respect to this reference (Schwab et al., 1990). In the past decade, the DM has been reintroduced and modified to describe substrate processing within the liver (Yano et al., 1989, 1990, 1991; Evans et al., 1991, 1993; Díaz-García et al., 1992; Hussein et al., 1994; Chou et al., 1993, 1995; Yasui et al., 1995a; Ohata et al., 1996; Nishimura et al., 1996; Ueda et al., 1997). The model has been extended to the two-compartment DM, with recognition that the sinusoidal membrane of the liver is a transport barrier to solutes, producing compartmentalization of the vascular and cellular compartments.

Close examination of the GM and the DM reveals both similarities and differences; the models share common mathematical descriptions of cellular events such as transmembrane transfer, cellular drug metabolism, and biliary excretion (fig. 1), as first noted by Rowlett and Harris (1976). These models typically characterize cellular events by the use of influx (k₁₂), efflux (k₂₁), and sequestration (k_e) coefficients (see definitions in table 1). However, the GM and the DM differ in their descriptions of the shapes of the transit time distributions of their respective vascular reference curves.

View larger version (13K): [in this window] [in a new window]   Fig. 1.   Schematic representation of solute handling at the level of a single sinusoid. The solute enters the liver with flow Q and is processed in a distributed-in-space fashion. At each point, drug in plasma (concentration, CP) exchanges with that in tissue (Ct). Sinusoidal membrane transfer coefficients are denoted by k12 and k21, the influx and efflux coefficients, respectively, whereas the cellular sequestration coefficient is represented by ke (see table 1 for definitions).

Because of the pervasive use of both models in the interpretation of MID data to provide mechanistic insight into hepatic drug processing, there is the need to fully understand the consistency of both models with the data. The intent of this report was, therefore, to compare the ramifications of the GM and the DM to gain a better understanding of the two methods and their implications for hepatic drug processing. From a practical standpoint, it is apparent that the GM is experimentally and computationally more demanding than the DM because of the requirement for noneliminated reference indicators. We therefore became interested in assessing the similarities and differences in the analysis of dilution data by the DM compared with the GM because the solutions are more readily available, with inclusion of fewer or no reference indicators. Data that had been processed by the GM with deconvolution of the sham experiments (dilution experiments without liver) were reexamined with the DM, to allow for comparison of the two models. We compared the fitted values of the common parameters (volumes and transfer coefficients) used by the GM and the DM to describe the hepatic disposition of two compounds that are bound only to albumin and that exhibited saturable sinusoidal (carrier-mediated) transport. These compounds were BSPGSH (Geng et al., 1995) and HA, the glycine conjugate of benzoate (Yoshimura et al., 1998).

	Theoretical Considerations

The GM. For the GM, a set of noneliminated reference indicators ordinarily accompanies the tracer-labeled substrate for injection into the liver. The noneliminated reference indicators include ⁵¹Cr-labeled RBCs (the vascular reference, which is distributed only in the sinusoids), ¹²⁵I-labeled albumin and ¹⁴C- or ³H-labeled sucrose (which occupy the sinusoidal plasma and the interstitial or Disse space), and D₂O [a kinetic equivalent of water (Pang et al., 1991) that is distributed throughout the vascular, interstitial, and accessible cellular water spaces of the organ]. The outflow concentrations are normalized to the injected dose (fractional recovery per milliliter), with observations of the progressive dispersion of labeled RBCs, albumin, sucrose, and then water, based on their increasing spaces of distribution. Another well-known property of the model is the ability of the dilution curves for the diffusible noneliminated reference indicator to be superimposed on the labeled RBC curve, after appropriate correction of the time and the concentration for the difference in distribution volumes (Goresky, 1963). The procedure revealed the existence of a transit time delay (t₀), occurring in the nondispersing region or the large vessels, after correction for the transit times of the input and output catheters. The transit time profiles for albumin, sucrose, and water are superimposed on the RBC curve using the following relationship (Yoshimura et al., 1998),

(1)

where h(t) is the transfer function that describes the RTD of the reference indicator after deconvolution of the distortion caused by the inflow and outflow catheters and the injection device. The superposition procedure, when carried out by a fitting procedure, provides the optimized fitted values of t₀ and , a value denoting the space ratios [stationary to moving spaces, i.e. Disse space/sinusoidal plasma for albumin and sucrose and (cellular water + Disse space)/(sinusoidal plasma water + RBC water) for D₂O; see table 1 for definition].

(2)

(3)

(4)

(5)

1

1

1

The DM. The DM conceptualizes that the blood flowing through a labyrinth of ramified interconnecting sinusoids is nonideal and causes a dispersion of noneliminated and eliminated substances (Roberts and Rowland, 1985, 1986). There are two parameters that characterize the model, i.e. D_N, or the inverse of the Peclet number, which describes the degree of mixing or dispersion resulting from heterogeneous blood flow through the microvasculature, and the efficiency number, R_N, or the term for removal

(6)

where f_u is the unbound fraction, CL_int is the intrinsic clearance or the inherent ability for removal of the unbound substrate, and , as defined in earlier publications (Roberts and Rowland, 1985, 1986), is given by P/(P + CL_int), where P is the permeability. The use of  = 1 is justified when permeabilities far exceed the intrinsic clearance.

(7)

(8)

(9)

(10)

(11)

	Materials and Methods

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Data Set. Data from single-pass, rat liver-perfusion/MID experiments were used in the present analysis. The data had been previously analyzed using the GM. In the first set of studies (N = 12), a bolus injection of [³H]BSPGSH was introduced simultaneously with the reference indicators (⁵¹Cr-labeled RBCs, ¹²⁵I-albumin, [¹⁴C]sucrose, and D₂O) (Geng et al., 1995), under steady-state conditions, with various concentrations of BSPGSH (20-214 µM). Similarly, in another set of studies (N = 19), [³H]HA was injected, together with the set of noneliminated reference indicators, with a background concentration of HA (1-930 µM), under steady-state conditions, with or without the presence of lactate (20 mM) or benzoate (10-873 µM), compounds that were found to inhibit HA uptake (Yoshimura et al., 1998). The outflow of [³H]BSPGSH or [³H]HA was expressed as the fraction of the administered dose eluting per second [C(t)/dose] or as the frequency output [f(t) or C(t)Q/dose].

Data Analysis. The frequency output was analyzed according to the DM (Evans et al., 1991). Data from sham experiments, in which labeled RBCs were injected into the inflow and outflow catheters, were used to evaluate w(s)_NH (eq. 8). Preliminary fits indicated that eq. 8 was unable to describe the observed exponentially multiphasic decline in the sham experiments. Therefore, w(s)_NH was empirically fitted to the two-compartment DM for noneliminated references where there is no elimination

(12)

where D_N,NH is the nonhepatic D_N, V_NH is the volume of the nonhepatic experimental system, Q is the perfusate flow rate, and k₅₆ and k₆₅ are the parameters accounting for the biphasic decline. Least-squares fitting of the sham experiment data to eq. 12 yielded estimates of 0.0163, 1.04 ml, 0.105 sec¹, and 0.388 sec¹ for D_N,NH, V_NH, k₅₆, and k₆₅, respectively.

(13)

Fitting. The frequency outputs were fitted to eq. 7 or 13 with the use of a weighted least-squares minimization procedure, using a Levenberg-Marquardt algorithm, found in the program SCIENTIST (Micromath Scientific Software, Salt Lake City, UT). Numerical inversion of the Laplace transforms was accomplished using the Piessens-Huysmans algorithm used by the program. All parameters (D_N, V_B, the transfer coefficients, and, where appropriate, t₀) were estimated during the fitting procedure with the weighting chosen at 1/observed value. For each experiment, the frequency output data were modeled with the two-compartment DM without inclusion of t₀ (set A), with t₀ obtained from GM (set B), and with fitted t₀ (set C).

Statistical Analysis. All data are presented as mean ± SD. The paired t statistic was used for the comparison of paired data of sets A or C of the DM with those of the GM. Data of set A were compared with those of set C, because these are data fits of the DM with and without t₀, respectively. The MSC (SCIENTIST for Windows manual; Micromath Scientific Software), a modified Akaike information criterion, was used to compare the goodness of fit to the model; a larger value for MSC suggests a superior model.

	Results

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BSPGSH Data. Good correlation was obtained between the estimated apparent V_B from sets A, B, and C in the DM and the vascular volume for the reference in the GM (R > 0.90) (fig. 2 and table 2). However, these values (approximately 3-3.4 ml) were significantly greater than the volume for the vascular reference according to the GM (2.7-3 ml). Similarly, an excellent correlation (R > 0.91) was found between the influx coefficients determined by the DM (k₁₂) and those determined by the GM (f_uk₁') (fig. 3). Paired analysis of the k₁₂ values estimated in sets A, B, and C failed to show differences between the DM and the GM (p > 0.05) (table 2), although clearly the correlation improved with inclusion of t₀ (fig. 3). In contrast, values for k₂₁ and k_e obtained for the DM did not correspond well to the respective efflux and sequestration coefficients obtained from the GM (figs. 4 and 5 and table 2). The DM values for k₂₁ and k_e were higher than those from the GM (set C compared with set A, p < 0.01) (table 2). The addition of t₀ to the DM improved the correlation between the respective parameters k₂₁ and k_e (compare fig. 4A with fig. 4, B and C, and compare fig. 5A with fig. 5, B and C) and increased D_N significantly (almost 3-fold) (table 2).

View larger version (20K): [in this window] [in a new window]   Fig. 2.   Relationship between the vascular volume (VB) of the DM and the volume for the reference [VSuc(1 + rel)] of the GM for sets A, B, and C for BSPGSH (data from Geng et al., 1995). The dashed lines were established by linear regression (equations shown, with correlation coefficients, R), whereas the solid lines are the lines of identity.

View larger version (21K): [in this window] [in a new window]   Fig. 3.   Relationship between influx coefficients (k12) determined by the DM and the GM for BSPGSH (data from Geng et al., 1995). The dashed lines were established by linear regression (equations shown, with correlation coefficients, R), whereas the solid lines are the lines of identity.

View larger version (17K): [in this window] [in a new window]   Fig. 4.   Relationship between efflux coefficients (k21) determined by the DM and the GM for BSPGSH (data from Geng et al., 1995).

View larger version (20K): [in this window] [in a new window]   Fig. 5.   Relationship between sequestration coefficients (ke) determined by the DM and the GM for BSPGSH (data from Geng et al., 1995).

View larger version (27K): [in this window] [in a new window]   Fig. 6.   Model fit to a representative MID experiment conducted with [3H]BSPGSH (214 µM). A, reference curves for the DM (set C) and the GM; B, the fitted line and throughput and returning components for the DM (set C) and the GM.

HA Data. Fitting was successful for all experimental data in sets A and C. However, convergence was obtained for only 14 of the 19 experiments in set B, because of the unexpectedly large t₀ obtained with the GM. Hence, data from only the 14 experiments were reported among all data sets. The estimates of V_B for sets A, B, and C correlated well with the volume for the reference determined by the GM (3.0 ± 0.5 ml) (fig. 7 and table 2). The k₁₂ of the DM was moderately correlated with the influx coefficient of the GM in sets A and C (fig. 8). Inclusion of t₀ in the DM did not improve the correspondence for k₁₂ between the DM and GM estimates (compare fig. 8A with fig. 8, B and C). However, improved correspondence in the efflux parameter (k₂₁) was observed for sets B and C with GM values when t₀ was included in the DM (compare fig. 9A with fig. 9, B and C). Again, a significant increase of the D_N value was observed with the inclusion of t₀ (table 2).

View larger version (19K): [in this window] [in a new window]   Fig. 7.   Relationship between VB of the DM and VSuc(1 + rel) of the GM for sets A, B, and C for HA (data from Yoshimura et al., 1998). The dashed lines were established by linear regression (equations shown, with correlation coefficients, R), whereas the solid lines are the lines of identity.

View larger version (19K): [in this window] [in a new window]   Fig. 8.   Relationship between influx coefficients (k12) determined by the DM and the GM for HA (data from Yoshimura et al., 1998). The dashed lines were established by linear regression (equations shown, with correlation coefficients, R), whereas the solid lines are the lines of identity.

View larger version (20K): [in this window] [in a new window]   Fig. 9.   Relationship between efflux coefficients (k21) determined by the DM and the GM for HA (data from Yoshimura et al., 1998). The dashed lines were established by linear regression (equations shown, with correlation coefficients, R), whereas the solid lines are the lines of identity.

View larger version (27K): [in this window] [in a new window]   Fig. 10.   Model fit to a representative MID experiment conducted with [3H]HA (930 µM). A, reference curves for the DM (set C) and the GM; B, the fitted lines and throughput and returning components for the DM (set C) and the GM.

	Discussion

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In this report, we compared the fitted values for the common parameters that characterize the hepatic disposition of two solutes with saturable uptake (BSPGSH and HA) in the GM and the DM. From the various parameter estimates of the two models, several conclusions can be drawn. The first regards the similarities between the volume for the appropriate reference in the GM and the apparent volume of distribution in the DM and among the influx constants; the second, the greater than predicted dispersion of the vascular reference curves; and the third, the influence of the large-vessel transit time delay on D_N and the rate coefficients.

Results on the fitted volumes of the vasculature (V_B) for the basic DM (set A, without t₀) were similar to those for the reference from the GM [V_Suc(1 + _rel)]. The closeness of the estimates suggests that the fitted value derived from the DM provides a sound estimate of the apparent volume of distribution for the solute in the vasculature (defined as "the reference" in the GM). This volume is not the combined volumes of the sinusoidal blood and the Disse space, as originally conceived. However, the difference is expected to be small, because the apparent volume of distribution for the solute approximates the total albumin space or the total sucrose space for drugs that are totally bound or unbound to albumin, respectively. Moreover, the a priori assignment of V_B as 15% of the wet liver mass, as adopted in many reports (Evans et al., 1991, 1993), underestimates the volume for the reference in the DM and leads to poor fits (data not shown).

The influx coefficient (k₁₂) of the DM was found to be highly correlated with that (f_uk₁') of the GM (figs. 3 and 8). The observation was the result of the similarities of the respective reference curves at the early times (upswing portions of figs. 6 and 10), such that the predictions of the two models coalesced, as noted by Rowlett and Harris (1976). However, the respective reference curves eventually deviated, mostly for the later-in-time segments or the "tails" of the dilution profiles, such that the efflux and sequestration coefficients that characterize the returning component for the labeled tracer were also dissimilar. For this reason, the fit to the GM is superior (compare CD values in table 2) to that to the DM, although the fit to the DM is generally quite good, especially for the early data.

To investigate whether the large-vessel transit time could account for these differences, we included the lag time parameter t₀ in the DM, either as an assigned parameter (value of t₀ from the GM, set B) or as a fitted parameter (set C). Inclusion of t₀ in the DM (sets B and C) slightly improved the correspondence of influx, efflux, and sequestration coefficients between the DM and the GM. The most significant change occurred with D_N, which increased approximately 2-3-fold as a direct result of the incorporation of t₀ (table 2). This was also found in an analysis of the DM with closed- or mixed-boundary conditions, with or without t₀ as a fitted parameter, in rat liver-perfusion/MID studies conducted with catheters of varying lengths, in which different D_N estimates were observed for the noneliminated reference indicators in the absence of t₀ (Schwab et al., 1998). The inclusion of t₀ in the DM provided a common D_N (~0.22) for the noneliminated indicators (labeled RBCs, albumin, sucrose, and water), as expected with the dispersive capability of the liver. The universality of this value provides full justification for including the time delay. The finding suggests that, after the initial delay, the same extent of dispersion exists for the noneliminated references, with a common D_N (varying from 0.16 to 0.31 among preparations), regardless of the distribution volumes or catheter sizes (Schwab et al., 1998). Interestingly, the goodness of fit was also improved when t₀ was considered in the DM in the present analysis (higher values for the CD and MSC) (table 2). The hypothesis that the large nonexchanging vessels would not add to the dispersion of solutes in the liver thus appears valid. Further support for the use of t₀ is provided by the excellent superposition of the curves for the diffusible noneliminated indicators onto the labeled RBC curve (scaling the time and concentration terms by the ratio of the distribution volumes) when the elapsed time is corrected for t₀ (Goresky, 1963). Audi et al. (1994, 1995) have demonstrated that, in the lung, virtually all of the dispersion occurs in the capillary beds and only an undetectable amount of dispersion exists in the arterial and venous trees. Taking these findings together, we conclude that the addition of t₀ for fitting is an appropriate refinement of the DM for hepatic modeling. However, there appears to be no consistent relationship for the t₀ values between the DM and the GM (fig. 11); the t₀ values for BSPGSH were quite similar for the DM and the GM, but the fitted t₀ values for the DM were consistently lower than those obtained by the superposition procedure of Goresky.

View larger version (18K): [in this window] [in a new window]   Fig. 11.   Plots of the fitted t0 for the DM vs. the t0 for the GM, obtained with the superposition procedure, for BSPGSH (A) and HA (B).

Because the large-vessel transit time could not explain the major differences in the estimates for the transfer coefficients obtained with the GM and the DM, differences in the vascular reference curves for BSPGSH or HA might be the reason for the disparities. For the DM, the reference curve is an inverse Gaussian distribution of transit times and is described by a rapid upslope followed by a monoexponential decay; for the GM, the reference curve is molded by the shapes of curves for the appropriate reference indicators, i.e. albumin and sucrose and their dispersions within the liver, because of the partial binding of the solute to albumin (figs. 6 and 10). It was readily apparent that the reference curves for the DM and the GM were similar at early time points but diverged after 2 or 3 MTTs. The small difference in the early reference curves reflected the similarities in estimates for k₁₂ determined by the two models. Whereas the reference curve for the DM declined monoexponentially, the corresponding profile for the GM decayed in a fashion that closely resembled that of sucrose, with a slightly protracted tail. This difference in the downslopes of the reference curves brought about uncertainties in the throughput and returning components. There was lack of a definitive pattern observed for BSPGSH and HA with regard to the throughput and returning components (figs. 6 and 10). The throughput component is highly correlated with the magnitude of k₁₂ and the shape and downslope of the reference curve. When the estimated k₁₂ for the DM is greater than that for GM, the throughput component for the DM is greater than that for the GM and, conversely, the returning component is lower. A greater disparity exists for the returning component, which is obtained as a difference curve, especially when it is small. This is manifested as highly variable estimates for k₂₁ and k_e with the DM and the GM.

It must be reemphasized that, for the GM, the reference curve is constructed from the full complement of noneliminated reference indicators, and its basis should be firmly established. Comparison of the outflow profile for the diffusible labeled tracer with that for the reference would, in essence, correct for the heterogeneities in capillary transit time and account for micromixing or geometric tortuosity, as claimed by some investigators of liver physiology (King et al., 1996; Weiss, 1997). The suitability of the model is apparent when the late-in-time data are well fitted and displayed in semilogarithmic plots (high CD values) (table 2). This was seldom performed with the DM for noneliminated reference indicators and the eliminated tracer solute. The late-in-time observations that highlight the dispersion within the system showed a systematic deviation from the fitted curve for the DM. For this reason, the MTT and CV² for the DM are underestimates (table 3).

However, in some instances, the tailing profile for tracers such as enalaprilat (Schwab et al., 1990) and BSPGSH (in Eisai hyperbilirubinemic mutant rats) (Geng et al., 1998) could not be explained by the present DM and GM but could be modeled by the presence of a deep intracellular pool (equivalent to three-compartment DM fitting; fits not shown). In other instances, the influence of tight intracellular binding or slow diffusion (Luxon and Weisiger, 1993; Yasui et al., 1995b) could explain the unexpected RTDs of these solutes. For the DM, it should also be recognized that parameter identification was poor for efflux and removal, especially when the presence of a deep intracellular pool was evoked (fits not shown). Moreover, one must recognize the strong interrelationships between V_B and k₁₂, in addition to V_B and D_N, with the fitting procedure.

The comparison, however, reveals the utility of the DM. To reiterate a few points, DM is simpler with respect to computation and experimental strategy than the GM. The volume for the reference and the influx coefficient (and therefore uptake clearance) are well estimated, and this should provide insight into the uptake mechanism for solutes. Also, the exact amount of the dose can be verified independently using the area of the reference curve, thus circumventing experimental errors in its volumetric assessment. For optimization of the DM, however, model fitting with t₀ is necessary to provide improved estimates of V_B and k₁₂. Furthermore, proper application of tracer methodology necessitates the use of tracers in the injection dose under steady-state conditions for the bulk unlabeled compound. The appropriateness of the curve function for the sham experiment (without liver) must be demonstrated, because it is of paramount importance in the deconvolution procedure for the tracer outflow profiles. If the simple transfer function with monoexponential decay (eq. 8) were used instead of the biphasic decay curve model (eq. 12) in the analysis of the sham experiment data for the DM in the present study, D_N, V_B, and t₀ would be overestimated, whereas k₁₂, k₂₁, and k_e would be underestimated in the labeled tracer data (comparison not shown). Experimentally, it is necessary that data collection be extended beyond 3 MTTs, and it is imperative that the radiolabels quantified are identical to the quantities of the named drug or its metabolites. These recommendations ensure the proper use of indicator dilution profiles in providing mechanistic insight into drug uptake processes.