Axial tissue diffusion can account for the disparity between current models of hepatic elimination for lipophilic drugs

A Complete Extension of Classical Hepatic Clearance Models Using Fractional Distribution Parameter fd in Physiologically Based Pharmacokinetics

The classical organ clearance models have been proposed to relate the plasma clearance CLp to probable mechanism(s) of hepatic clearance. However, the classical models assume the intrinsic capability of drug elimination (CLu,int) that is physically segregated from the vascular blood but directly acts upon the unbound drug concentration in the blood (fubCavg), and do not handle the transit-time delay between the inlet/outlet concentrations in their closed-form clearance equations. Therefore, we propose unified model structures that can address the internal blood concentration patterns of clearance organs in a more mechanistic/physiological manner, based on the fractional distribution parameter fd operative in PBPK. The basic partial/ordinary differential equations for four classical models are revisited/modified to yield a more complete set of extended clearance models, i.e., the Rattle, Sieve, Tube, and Jar models, which are the counterparts of the dispersion, series-compartment, parallel-tube, and well-stirred models. We demonstrate the feasibility of applying the resulting extended models to isolated perfused rat liver data for 11 compounds and an example dataset for in vitro-in vivo extrapolation of the intrinsic to the systemic clearances. Based on their feasibilities to handle such real data, these models may serve as an improved basis for applying clearance models in the future.

Are There Any Experimental Perfusion Data that Preferentially Support the Dispersion and Parallel-Tube Models over the Well-Stirred Model of Organ Elimination?

In reviewing previously published isolated perfused rat liver studies, we find no experimental data for high-clearance metabolized drugs that reasonably or unambiguously support preference for the dispersion and parallel-tube models versus the well-stirred model of organ elimination when only entering and exiting drug concentrations are available. It is likely that the investigators cited here may have been influenced by: 1) the unphysiologic aspects of the well-stirred model, which may have led them to undervalue the studies that directly test the various hepatic disposition models for high-clearance drugs (for which model differences are the greatest); 2) experimental assumptions made in the last century, which are no longer valid today, related to the predictability of in vivo outcomes from in vitro measures of drug elimination and the influence of albumin in hepatic drug uptake; and 3) a lack of critical review of previously reported experimental studies, resulting in inappropriate interpretation of the available experimental data. The number of papers investigating the theoretical aspects of the dispersion, parallel-tube, and well-stirred models of hepatic elimination greatly outnumber the papers that actually examine the experimental evidence available to substantiate these models. When all experimental studies that measure organ elimination using entering and exiting drug concentrations at steady state are critically reviewed, the simple but unphysiologic well-stirred model is the only model that can describe all trustworthy published available data. SIGNIFICANCE STATEMENT: Although the dispersion model of hepatic elimination more adequately reflects physiologic reality, there are no convincing experimental data that unambiguously favor this model. The well-stirred model can describe all well-designed perfusion studies with high-clearance drugs and nondrug substrates, but the field has not recognized this because of hesitation to accept a nonphysiologic model and flawed attempts to utilize in vitro-in vivo extrapolation approaches.

Interconnected-tubes model of hepatic elimination: steady-state considerations

In the interconnected-tubes model of hepatic transport and elimination, intermixing between sinusoids was modelled by the continuous interchange of solutes between a set of parallel tubes. In the case of strongly interconnected tubes and for bolus input of solute, a zeroth-order approximation led to the governing equation of the dispersion model. The dispersion number was expressed for the first time in terms of its main physiological determinants: heterogeneity of flow and density of interconnections. The interconnected-tubes model is now applied to steady-state hepatic extraction. In the limit of strong interconnections, the expression for output concentrations is predicted to be similar in form to those predicted by the distributed model for a narrow distribution of elimination rates over sinusoids, and by the dispersion model in the limit of a small dispersion number D(N). More generally, the equations for the predicted output concentrations can be expressed in terms of a dimensionless 'heterogeneity number'H(N), which characterizes the combined effects of variations in enzyme distribution and flow rates between different sinusoids, together with the effects of interconnections between sinusoids. A comparative analysis of the equations for the dispersion and heterogeneity numbers shows that the value of H(N)can be less than, greater than or equal to the value of D(N)for a correlation between distributions of velocities and elimination rates over sinusoids, anticorrelation between them, and when all sinusoids have the same elimination rate, respectively. Simple model systems are used to illustrate the determinants of H(N)and D(N).

Modeling of hepatic elimination and organ distribution kinetics with the extended convection-dispersion model

The conventional convection-dispersion (also called axial dispersion) model is widely used to interrelate hepatic availability (F) and clearance (Cl) with the morphology and physiology of the liver and to predict effects such as changes in liver blood flow on F and Cl. An extended form of the convection-dispersion model has been developed to adequately describe the outflow concentration-time profiles for vascular markers at both short and long times after bolus injections into perfused livers. The model, based on flux concentration and a convolution of catheters and large vessels, assumes that solute elimination in hepatocytes follows either fast distribution into or radial diffusion in hepatocytes. The model includes a secondary vascular compartment, postulated to be interconnecting sinusoids. Analysis of the mean hepatic transit time (MTT) and normalized variance (CV2) of solutes with extraction showed that the discrepancy between the predictions of MTT and CV2 for the extended and unweighted conventional convection-dispersion models decreases as hepatic extraction increases. A correspondence of more than 95% in F and Cl exists for all solute extractions. In addition, the analysis showed that the outflow concentration-time profiles for both the extended and conventional models are essentially identical irrespective of the magnitude of rate constants representing permeability, volume, and clearance parameters, providing that there is significant hepatic extraction. In conclusion, the application of a newly developed extended convection-dispersion model has shown that the unweighted conventional convection-dispersion model can be used to describe the disposition of extracted solutes and, in particular, to estimate hepatic availability and clearance in both experimental and clinical situations.

Relative dispersions of intra-albumin transit times across rat and elasmobranch perfused livers, and implications for intra- and inter-species scaling of hepatic clearance using microsomal data

It is recognized that vascular dispersion in the liver is a determinant of high first-pass extraction of solutes by that organ. Such dispersion is also required for translation of in-vitro microsomal activity into in-vivo predictions of hepatic extraction for any solute. We therefore investigated the relative dispersion of albumin transit times (CV2) in the livers of adult and weanling rats and in elasmobranch livers. The mean and normalized variance of the hepatic transit time distribution of albumin was estimated using parametric non-linear regression (with a correction for catheter influence) after an impulse (bolus) input of labelled albumin into a single-pass liver perfusion. The mean+/-s.e. of CV2 for albumin determined in each of the liver groups were 0.85+/-0.20 (n = 12), 1.48+/-0.33 (n = 7) and 0.90+/-0.18 (n = 4) for the livers of adult and weanling rats and elasmobranch livers, respectively. These CV2 are comparable with that reported previously for the dog and suggest that the CV2 of the liver is of a similar order of magnitude irrespective of the age and morphological development of the species. It might, therefore, be justified, in the absence of other information, to predict the hepatic clearances and availabilities of highly extracted solutes by scaling within and between species livers using hepatic elimination models such as the dispersion model with a CV2 of approximately unity.

Application of the dispersion model for description of the outflow dilution profiles of noneliminated reference indicators in rat liver perfusion studies

The dispersion model (DM) is a stochastic model describing the distribution of blood-borne substances within organ vascular beds. It is based on assumptions of concurrent convective and random-walk (pseudodiffusive) movements in the direction of flow, and is characterized by the mean transit time (t) and the dispersion number (inverse Peclet number), DN. The model is used with either closed (reflective) boundary conditions at the inflow and the outflow point (Danckwerts conditions) or a closed condition at the inflow and an open (transparent) condition at the outflow (mixed conditions). The appropriateness of DM was assessed with outflow data from single-pass perfused rat liver multiple indicator dilution (MID) experiments, with varying lengths of the inflow and outflow catheters. The studies were performed by injection, of bolus doses of 51Crlabeled red blood cells (vascular indicator), 125I-labeled albumin and [14C] sucrose (interstitual indicators), and [3H]2O (whole tissue indicator) into the portal vein at a perfusion rate of 12 ml/ min. The outflow profiles based on the DM were convolved with the transport function of the catheters, then fitted to the data. A fairly good fit was obtained for most of the MID curve, with the exception of the late-in-time data (prolonged tail) beyond 3 x [symbol: see text]. The fitted DNS were found to differ among the indicators, and not with the length of the inflow and outflow catheters. But the differences disappeared when a delay parameter, t0 = 4.1 +/- 0.7 sec (x +/- SD), was included as an additional fitted parameter for all of the indicators except water. Using the short catheters, the average DN for the model with delay was 0.31 +/- 0.13 for closed and 0.22 +/- 0.07 for mixed boundary conditions, for all reference indicators. Mean transit times and the variances of the fitted distributions were always smaller than the experimental ones (on average, by 6.8 +/- 3.7% and 58 +/- 19%, respectively). In conclusion, the DM is a reasonable descriptor of dispersion for the early-in-time data and not the late-in-time data. The existence of a common DN for all noneliminated reference indicators suggests that intrahepatic dispersion depends only on the geometry of the vasculature rather than the diffusional processes. The role of the nonsinusoidal ("large") vessels can be partly represented by a simple delay.

Effect of altered tissue binding on the disposition of barbital in the isolated perfused rat liver: application of the axial dispersion model

To examine the dependence of hepatic dispersion on tissue binding, the distribution kinetics of barbital under varying conditions of barbiturate perfusate concentrations was studied in the isolated perfused rat liver preparation (n = 5). The in situ liver was perfused in a single-pass mode with protein-free Krebs bicarbonate medium (15 mL/min). During steady-state infusion with various barbiturate concentrations (barbital, 1 g/L; butethal, 0.1, 1 g/L), a bolus containing [3H]water (cellular space marker) and [14C]barbital was injected into the portal vein. The recoveries of [3H]water and [14C]barbital were complete. The mean transit time and hence the volume of distribution for barbital in the absence of bulk barbiturate concentration (56 s and 1.24 mL/g) were about 2-fold higher than those for water (29 s and 0.58 mL/g), and they decreased progressively as the perfusate barbiturate concentration increased, indicating a decrease in tissue binding. However, the relative dispersion values (CV2H) of water (0.60) and barbital (0.66) were about the same magnitude and independent of the bulk concentration of barbiturate. The one-compartment dispersion model adequately described the data of barbital with a constant DN (dispersion number) value of 0.35. The results indicate that varying the tissue binding of barbital does not change the magnitude of DN; as such it offers a new experimental approach to examine the hepatic dispersion of solutes with a large distribution volume.

Metabolite mean transit times in the liver as predicted by various models of hepatic elimination

Predicted area under curve (AUC), mean transit time (MTT) and normalized variance (CV2) data have been compared for parent compound and generated metabolite following an impulse input into the liver. Models studied were the well-stirred (tank) model, tube model, a distributed tube model, dispersion model (Danckwerts and mixed boundary conditions) and tanks-in-series model. It is well known that discrimination between models for a parent solute is greatest when the parent solute is highly extracted by the liver. With the metabolite, greatest model differences for MTT and CV2 occur when parent solute is poorly extracted. In all cases the predictions of the distributed tube, dispersion, and tanks-in-series models are between the predictions of the tank and tube models. The dispersion model with mixed boundary conditions yields identical predictions to those for the distributed tube model (assuming an inverse gaussian distribution of tube transit times). The dispersion model with Danckwerts boundary conditions and the tanks-in series models give similar predictions to the dispersion (mixed boundary conditions) and the distributed tube. The normalized variance for parent compound is dependent upon hepatocyte permeability only within a distinct range of permeability values. This range is similar for each model but the order of magnitude predicted for normalized variance is model dependent. Only for a one-compartment system is the MTT for generated metabolite equal to the sum of MTTs for the parent compound and preformed metabolite administered as parent.

Effect of erythrocyte binding on elimination of harmol by the isolated perfused rat liver

The effect on the hepatic elimination rate of drug bound to erythrocytes and to albumin was compared with harmol, a relatively hydrophilic drug of high hepatic intrinsic clearance, in the single-pass isolated perfused rat liver preparation (n = 12). The steady-state hepatic extraction ratio (E) of harmol (50 microM) was measured during three consecutive 35-min periods with three different perfusates: Krebs-Henseleit buffer, buffer containing bovine serum albumin (2%), and buffer containing washed human erythrocytes (10%) perfused at 5 mL/min/g liver in randomized order. The mean unbound fraction (fu) of harmol in the latter two perfusates was 0.55 +/- 0.07 and 0.62 +/- 0.08, respectively, and the mean E for the three perfusates were 0.85 +/- 0.06, 0.62 +/- 0.07, and 0.71 +/- 0.08, respectively. The sinusoidal model fitted the relationship between E and fu better than the venous equilibrium model. Four further experiments, with perfusates of buffer, buffer + 2% albumin, and buffer + 4% albumin, confirmed that harmol elimination conformed to the sinusoidal model. For each of the 12 experiments that used erythrocyte perfusate, E and fu data from each of the two non-erythrocyte perfusates were used to predict E for the erythrocyte perfusate at the observed fu of 0.62, with the sinusoidal model. There was no significant difference between the observed (0.71 +/- 0.08) and predicted (0.68 +/- 0.10) E values (p > 0.05). This result suggests that release of harmol from erythrocytes is not a rate-limiting factor in the hepatic elimination of harmol, and that plasma membrane permeability does not contribute readily to a red cell carriage effect, at least with moderately polar and small molecules.

Physiologic models of hepatic drug clearance: influence of altered protein binding on the elimination of diclofenac in the isolated perfused rat liver

The single-pass perfused rat liver preparation was used to assess the influence of binding to human serum albumin on the steady-state hepatic extraction of diclofenac (n = 8). In the absence of binding protein, the extraction ratio of diclofenac approached unity (range, 0.975-0.992), such that its clearance was perfusion-rate limited. As the binding of diclofenac to protein was increased by the addition of human serum albumin to the perfusion medium, its extraction ratio decreased dramatically, and clearance eventually became capacity limited. The relationship between diclofenac availability and fraction unbound was analyzed with various physiologic models of hepatic drug clearance. The dispersion model, which contains a parameter (the dispersion number) that quantifies the axial spreading of a substrate as it passes along the liver length, provided a significantly better description of the data (p < 0.05) than the undistributed parallel-tube model, which assumes that an eliminated substrate travels through the liver as an undispersed plug, and the well stirred (venous equilibrium) model, which assumes that substrate undergoes infinite mixing as soon as it enters the liver. The dispersion number estimated for diclofenac (mean, 3.03; range, 0.89-7.56) was significantly greater than that predicted from considerations of the transverse heterogeneity of blood flow within the hepatic sinusoidal bed, suggesting that additional factors influenced the relationship between availability and fraction unbound for this compound. Such factors may include transverse heterogeneity of the metabolizing enzyme system(s), axial flux of substrate created by diffusion within hepatic tissue, and protein-facilitated transfer of substrate across an unstirred fluid layer adjacent to the hepatocyte surface.

Application of the axial dispersion model of hepatic drug elimination to the kinetics of diazepam in the isolated perfused rat liver

The application of the axial dispersion model to diazepam hepatic elimination was evaluated using data obtained for several conditions using the single-pass isolated perfused rat liver preparation. The influence of alterations in the fraction unbound in perfusate (fu) and perfusate flow (Q) on the availability (F) of diazepam was studied under steady conditions (n = 4 in each case). Changes in fu were produced by altering the concentration of human serum albumin (HSA) in the perfusion medium while maintaining diazepam concentration at 1 mg L-1. In the absence of protein (fu = 1), diazepam availability was 0.011 +/- 0.005 (mean +/- SD). As fu decreased, availability progressively increased and at a HSA concentration of 2% (g/100 ml), when fu was 0.023, diazepam availability was 0.851 +/- 0.011. Application of the axial dispersion model to the relationship between fu and F provided estimates for the dispersion number (DN) of 0.337 +/- 0.197, and intrinsic clearance (CL(int)) of 132 +/- 34 ml min-1. The availability of diazepam during perfusion with protein-free media was also studied at three different flow rates (15, 22.5, and 30 ml min-1). Diazepam availability always progressively increased as perfusate flow increased, with the axial dispersion model yielding estimates for DN of 0.393 +/- 0.128 and CL(int) of 144 +/- 38 ml min-1. The transient form of the two-compartment dispersion model was also applied to the output concentration versus time profile of diazepam after bolus input of a radiolabeled tracer into the hepatic portal vein (n = 4), providing DN and CL(int) estimates of 0.251 +/- 0.093 and 135 +/- 59 ml min-1, respectively. Hence, all methods provided similar estimates for DN and CL(int). Furthermore, the magnitude of DN is similar to that determined for noneliminated substances such as erythrocytes, albumin, sucrose, and water. These findings suggest that the dispersion of diazepam in the perfused rat liver is determined primarily by the architecture of the hepatic microvasculature.