Journal of Pharmacokinetics and Biopharmaceutics, Vol. 20, No. 2, 1992
Application of the Axial Dispersion Model of Hepatic Drug Elimination to the Kinetics of Diazepam in the Isolated Perfused Rat Liver Juan M. Diaz-Garcia, ~'2 Allan M. Evans, t'3 and Malcolm Rowland t'4 Received August 16, 1991--Final February 5, 1992 The application of the axial dispersion model to diazepam hepatic elimination was evaluated using data obtained for several conditions using the single-pass isolated pesfused rat liver preparation. The influence of alterations in the fi'action unbound in perfusate (fu) and pelfusate 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 - I. In the absence of protein (fu = 1), diazepam availability was 0.011 4- 0.005 (~ 4- SD). As fu decreased, availability progressively increased and at a lISA concentration of 2% (g/ lO0 ml), when fu was 0.023, diazepam availability was 0.851 9 0.011. Application of the axial dispersion model to the relationship between fu and F provided estimates for the dispersion number (DN) of 0.3374-0.197, and intrinsic clearance (CL#,,) of 1324-34 ml min -1. The availability of diazepam during persuasion with protein-free media was also studied at three different flow rates (15, 22.5, and 30 ml min-I). Diazepam availability always progressively increased as perfusate flow increased, with the axial dispersion model yielding estimates for DN of 0.393• and CLi,,, of 144+38 ml min -t. 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~,,~estimates of 0.251 4- 0.093 and 135 4- 59 ml rain-, respectively. Hence, all methods provided similar estimates for DN and CL,,,. Furthermore, the magnitude Of DN is similar to that determined for nonetiminated substances such as eJTthrocytes, albumin, sucrose, and water. These findings suggest that the dispersion of diazepam in the pe~fused rat liver is determined primarily by the architecture of the hepatic microvasculature.
KEY WORDS: diazepam; hepatic elimination; physiologic models; dispersion model; isolated perfused rat liver; protein binding. This work was supported by the Commission of the European Communities and the Medical Research Council. One of us (A.M.E.) was partially supported by a Merck, Sharp & Dohme Fellowship. We are grateful to Roche (Switzerland) for the supply of diazepam and 2-[14C]diazepam, and Kabi AB (Sweden) for the supply of human serum albumin. ~Department of Pharmacy, University of Manchester, MI3 9PL, United Kingdom. 2present address: Facultad de Farmacia, Universidad de Navarra, 31008 Pamplona, Spain. 3present address: School of Pharmacy, University of South Australia, Adelaide 5000, South Australia. 4To whom correspondence should be addressed. 171 0090-466x/92/04000171506.50/09 1992PlenumPublishingCorporation
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INTRODUCTION The fraction of drug escaping extraction during a single passage through the liver (availability) depends on the fraction of drug unbound in blood, organ perfusion, hepatocellular enzyme activity, and, in some cases, the permeability of the hepatocyte membrane to the drug. Various models have been used to describe the influence of these physiological determinants on hepatic drug clearance and availability (1). The fundamental difference between these models lies in the assumption applied to the extent of axial spreading of substrate within the hepatic microvasculatory system. At the two extremes are the "venous equilibrium model" (well-stirred model) and the "undistributed parallel-tube model" (1,2). Because neither of these simplistic models is entirely compatible with the growing body of experimental data on the hepatic handling of drugs and with the known morphological features of the liver (3), models that apply a more realistic view of hepatic microvasculatory processes, such as the "distributed parallel-tube models" (4-6) and the "axial dispersion model" (7) may be more appropriate. These more complex models attempt to account for the known heterogeneity of the microvasculature, and in some cases the enzyme activity, of the liver. In the distributed parallel-tube models, the degree of heterogeneity is dictated by the variance of the statistical distribution chosen to represent the particular property. In the axial dispersion model, organ blood flow heterogeneity is reflected by the magnitude of the dispersion number, DN. The undistributed parallel-tube model and the venous equilibrium model are extreme forms of the axial dispersion model in which DN approaches 0 or infinity, respectively (3). The dispersion model has also been modified to incorporate transverse heterogeneity of enzyme activity (8). The magnitude of dispersion can be estimated either by reference to the concentration-time profile of a substance in the effluent from the liver after bolus input into the hepatic portal vein (impulse-response experiments) or by examining the influence of controlled changes in physiological determinants such as the blood (perfusate) flow rate to the liver or drug binding within blood, on drug availability (3,9). Analysis of the impulse-response data for a number of noneliminated reference markers in the isolated perfused rat liver have provided estimates for DN in the range of 0.2-0.5 (10,11 ). However, estimates of DN provided from considerations of drug availability at steady state appear to vary more widely, with values up to infinity reported for the endogenous compound taurocholic acid (12,13). The influence of altered protein binding on the availability of diazepam at steady state has been shown previously to conform to the predictions of the dispersion model with a DN of 0.29 (9). This present study extends these
Kinetics of Diazepam Elimination in Rat Liver
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previous findings by evaluating the consistency of the dispersion model in describing different data sets with diazepam. Namely, impulse-response data and steady-state availability data obtained under conditions of either altered protein binding, perfusate flow, or both, using the single-pass perfused in situ rat liver preparation. In addition, for the data obtained under steady state conditions the dispersion model is evaluated against the parallel-tube and well-stirred models of hepatic elimination.
THEORY Under linear conditions, the outflow profile of a substance injected into the hepatic portal vein is given by
c(t) =Q x(O * w(t)N. * w(t).
(1)
where C(t) is the concentration of material in the hepatic outflow at time t, D is the amount of drug administered, Q is the hepatic blood (perfusate) flow rate, x(t) is the function that describes the form of drug input (bolus, constant infusion, etc.), w(t)NH and w(t)H are the unit impulse-responses for the nonhepatic and hepatic regions of the experimental system, respectively, and 9 denotes the convolution integral. By expressing output as a fraction of the dose appearing per unit time (frequency output), y(t), and on taking Laplace transforms, Eq. (1) becomes
y(s) = x(s)" W(S)NH" W(S)H
(2)
When the substance is injected as a bolus, the Laplace transform of the input function, x(s), is equal to unity and Eq. (2) can be simplified to y(s) = w(s)~. . w(s).
(3)
where W(S)NHand W(S)Hare the transfer functions for nonhepatic and hepatic regions of the experimental system, respectively. In the present study, these transfer functions were defined using the axial dispersion model. This model, recently introduced into the field of pharmacokinetics (7), is widely used to represent nonideal flow behavior in chemical reactors (14). According to the theory of the axial dispersion model, when a substance is injected into the liver and distributes at a finite rate between the vascular and cellular spaces (two-compartment dispersion model), the concentration of the substance in blood (perfusate), CB, as a function of time (t) and distance along the liver length (z), assuming elimination from the cellular
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space, is given by (10,15)
OCB-D 02C~-v OCB-k12" Ca+k2) " C~ t3t Oz2 8z
(4)
with the corresponding mass-balance equation for the cellular compartment being ~C~_
Ot
VBk,2CB-k21" Cc-k23" Cc Vc
(5)
where D is the axial dispersion coefficient, which characterises the degree of axial dispersion of the injected substance in blood, v is the linear velocity of blood, k12 and kz! are the first-order rate constants for the transt~r of solute into and out of the cellular space, respectively, k23 is the first-order rate constant for the irreversible removal of solute from the cellular compartment, VB and Vc are the volumes of the vascular and cellular spaces, respectively, and Cc is the concentration of drug in the cell. Equations (4) and (5) comprise a set of second-order partial differential equations. Assuming mixed boundary conditions (see Discussion), the transfer function for the liver can be determined (10,15), and Eq. (3) becomes
2DN
(6)
where s is the Laplace operator, and DN=D/vL, where L is the liver length. When the radial transfer of a substance between the vascular and cellular spaces is instantaneous (one-compartment dispersion model), the corresponding solution is (3,11,15)
eXp['I'+4~ 2DN
(7)
Kinetics of Diazepam Elimination in Rat Liver
175
where VH is the volume of distribution of the drug in the liver. The term RN (the efficiency number) is given by RN = GLint" fu"
Q
p
(8)
where intrinsic clearance, CL~,t, is defined as the proportionality constant between the rate of elimination and the unbound substrate concentration within the cell, and p is given by P P + GLint
p=- -
(9)
where P is the permeability of the hepatocyte membrane to the drug. Thus, p represents the ratio of the concentration of unbound drug in the cell to that in blood. The transfer function for the nonhepatic regions in the experimental system can be defined using the one-compartment form of the dispersion model for a noneliminated solute (RN = 0) (1 1)
W(S)NH=exp I_I --( I + 4DN,NH " MRTNH " s)'/21 (10)
2DN,Nrt
where DN,NH and MRTNH are the dispersion number and the mean residence time of the substance in the nonhepatic regions of the experimental system. These parameters can be estimated by applying Eq. (10) to the ouflow profile determined by injecting the substance as a bolus into a system in which the hepatic inflow and outflow cannulas are connected. The fraction of a dose escaping extraction during a single pass through the liver (availability, F) is given by (16) F = limf(s) = lira [W(S)NH " w(s)n ] = lira [w(s)H ] s~0
s---~0
a'--*0
(1 1)
Therefore, for the one-compartment dispersion model F = e x p I 1 - ( 1 +4" DN" RN) 1/2] 2~
(12)
and for the two-compartment dispersion model i.
4VB" DN"
F=exp
k12" k23 (13)
2DN
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Diaz-Garcla, Evans, and Rowland
The first-order rate constants can also be defined as k,2 = P" f u / V ,
(14)
k2, = P "fuc/Vc
(15)
k23 GLint"fuc/Vc
(16)
-~
where fu~ is the fraction of drug in the cell which is unbound. Substituting these equalities into Eq. (13), the latter is transformed into Eq. (12). Thus, the steady state solutions for the one- and two-compartment forms of the dispersion model are identical (10,15). According to the venous equilibrium model, availability is given by (9) F=--
1
1 +RN
(17)
and according to the undistributed parallel-tube model, F is given by (9) F = exp (-RN)
(18)
METHODS Materials
Unlabeled diazepam and 2-[J4C]-diazepam (purity 99% TLC, 194 p C i / mg) were obtained from Roche (Basel, Switzerland) and human serum albumin (HSA) was obtained from Kabi AB (Sweden). HPLC solvents were purchased from British Drug Houses. All other reagents were of analytical grade. Protein Binding Determination
The binding of diazepam within perfusates containing varying concentrations of HSA was determined by equilibrium dialysis (Dianorm | Switzerland) with Spectra/Por-2 membranes (Spectrum Medical Industries Inc., CA). Briefly, 1 ml perfusate containing 1 mg L -1 of diazepam and 0.005 pCi of 14C-diazepam were dialyzed against 1 ml of protein-free perfusate at 37~ for 4 hr. 14C-diazepam was determined in a 0.2 ml aliquot of each compartment by radiochemical analysis (LKB Rackbeta liquid scintillation counter, Finland). The unbound fraction of diazepam was taken to be the ratio of the concentration of radiolabeled diazepam in the protein-free buffer to that in the perfusate compartment at the end of dialysis. Preliminary experiments showed that equilibrium was reached within 3 hr, that volume shifts were negligible, and that diazepam did not bind to the experimental system. To assess whether the binding of diazepam was influenced by passage through the liver, the fraction unbound of diazepam was also determined in a set of
Kinetics of Diazepam Elimination in Rat Liver
177
perfusate samples collected from the venous effluent, including protein-free samples.
Diazepam HPLC Assay Diazepam was assayed by HPLC using a modification of the method of Raisys et al. (17). All glassware was silanized. Briefly, 50 pl of internal standard solution (nitrazepam 3.75 pg ml-l; Sigma, USA) and 200 pl of acetonitrile (to precipitate HSA) were added to 200 pl of sample. After vortex mixing (I rain) and centrifugation (3400 rpm, 20 rain) 20 pl of the supernatant was injected into the HPLC system. Calibration curves were constructed over the range of 100 to 1500 ng ml -t and the intraday coefficient of variation (n=5) ranged between 7.42% (100ngm1-1) and 3.54% (1500 ng ml-l). Hepatic outflow samples obtained during perfusion with protein-free perfusate contained concentrations of diazepam that could not be measured accurately using the method described above. Therefore it was necessary to perform a simple extraction procedure prior to HPLC analysis of such samples. Briefly, 50 pl of internal standard solution (nitrazepam, 3.75 pg ml -I) and 8 ml of hexane:ethylacetate (8:2, v: v) were added to 4 ml of sample. After vortex mixing (1 rain) and centrifugation (3400 rpm, 20 rain) the upper organic layer was transferred to a clean test tube and evaporated to dryness (N2, 35~ Extraction efficiency for diazepam and nitrazepam was greater than 97%. The residue was reconstituted in 75 pl methanol and 20 pl was injected into the HPLC system. Calibration curves were constructed over the range of 5 to 150 ng ml -j and the intraday coefficient of variation (n= 5) ranged between 8.23% (5 ng ml -~) and 5.22% (150 ng ml-I). The HPLC system consisted of a Kontron analytic LC 410 Pump (Zurich, Switzerland) which delivered mobile phase (acetonitrile:water with 1% triethylamine adjusted to pH 3 with 85% orthophosphoric acid; 50:50, v:v) at a flow rate of 1.5 ml rain -~, to a C~8 chromatographic column (Spherisorb, S10, ODS1). Samples were injected using a 7010 Rheodyne manual injector (Cotati, CA) and column effluent was monitored using a Waters Lambdamax model 481 absorbance detector (Milford, USA) set to 254 nm. The retention times of diazepam and nitrazepam were 8.5 and 4.5 min, respectively. Known metabolites of diazepam (oxazepam, temazepam, and desmethyldiazepam: Sigma, USA) eluted at 5.2, 5.7, and 6.5rain, respectively.
Hepatic Perfusion The single-pass isolated perfused in situ rat liver preparation using male Sprague-Dawley rats (200-400 g), was essentially that described previously
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(11,18). The perfusate was freshly prepared and filtered (0.2 #m) Krebs bicarbonate buffer containing 3 g L -I of 10 mg L -j sodium taurocholate (Sigma), equilibrated to pH 7.4 with humidified O2/CO2(95/5) and maintained at 37~ Perfusate was delivered to the liver via a cannula inserted into the hepatic portal vein and liver outflow was collected via a cannula inserted into the vena cava through the right atrium. The tubing between the injector and cannula entering the portal vein was 3 cm; the cannula exiting from the superior vena cava was 6 cm. In each case, the liver was stabilized for 15 to 20 min using drug and protein-free perfusate. At the end of the experiment, the liver was removed and weighed. Diazepam did not bind to the perfusion apparatus. Bile flow was 0.6-1.5/~1 min -1 per gram of liver with a modest fall of 18% over the 2 hr of the experiment.
Viability of the Liver Preparation and Linearity of Diazepam Elimination Preliminary experiments (n = 4) were conducted to assess the viability of the perfused liver preparation. In each preparation, the perfusate was alternated between 0 and 1% (g/100 ml) HSA for 120 min after the stabilization period, while the total concentration of diazepam was maintained at 1 mg L -I. Hepatic outflow samples were obtained every 5 min and analyzed for diazepam. For these and subsequent experiments, the condition of the liver during perfusion was also assessed by monitoring the bile output and the recovery of the inflowing perfusate and by visual examination of the liver. Linearity with respect to diazepam elimination was tested with HSAfree perfusate containing three different diazepam concentrations (0.5, 1, and 5 mg L --t) perfused in random order for 20 min each.
Experimental Design
Impulse-Response Experiments In these experiments (n = 4), 50/11 of a solution containing 0.125pCi of t4C-diazepam in 1% HSA was rapidly injected into the hepatic portal vein during a constant perfusion with unlabeled diazepam (1 mg L -J) in 1% HSA. The total effluent from the liver was automatically collected at 2-see intervals using a motor-driven carousel with 57 sampling holes, and thereafter (into silanized test tubes) at increasing time intervals for up to 4 min. Perfusate samples were also collected before and after the bolus injection in order to determine the availability at steady state of unlabeled diazepam. The concentration of ~4C-diazepam in effluent samples was determined by radiochemical analysis and that of unlabeled diazepam by HPLC.
Kinetics of Diazepam Elimination in Rat Liver
179
Influence of Protein Binding on Availability of Diazepam at Steady State These experiments (n=4) were conducted to assess the influence of HSA perfusate concentration on the availability of diazepam at steady state. Preliminary experiments indicated that up to 10 min was required to achieve steady state with respect to diazepam availability (e.g., see Fig. 3). Therefore, availability was determined from samples collected between 10 and 20 rain (at 3- to 4-min intervals). In each liver preparation, four different perfusates containing 1 mg L -~ of diazepam were used; one was protein-free and the other three contained 0.1, 0.5~ I, or 2% HSA. After a period of stabilization with drug-free and HSA-free perfusate, diazepam in the absence of protein was perfused for 20 min. Subsequently, the liver was perfused with two different HSA concentrations for 20 min each. To assess for changes in the viability of the liver preparation with time, the initial protein-free perfusate was then used again for 20 min before the third protein-containing test perfusate. At the end of the experiment (120 min after cannulation of the hepatic portal vein and 100 min after commencing perfusion with diazepam) the liver was perfused with protein-free solution for a third time. A preparation was acceptable if the availability of diazepam during the second and third periods of perfusion with protein-free perfusate did not differ by more than 10% from that during the first run. The four preparations used met these conditions~ The scheme of a typical experiment is presented in Fig. 3.
Influence of Perfusate Flow on Availability of Diazepam at Steady State In another four perfused liver preparations, the influence of flow on diazepam availability was examined at a diazepam concentration of 1 mg L -1 using 3 different flow rates (15, 22.5, and 30 ml rain -I) at 0 and 1% HSA (6 different conditions for 15 min each). After the stabilization period, the liver was initially perfused with protein-free perfusate for 15 min at a particular flow rate and, to assess viability of the liver preparation, the experiment was always terminated under the same conditions. An example of a typical experiment procedure, in which the control flow rate was 15 ml min -~, is shown in Fig. 5. Hepatic outflow samples were collected into silanized test tubes every 1-2 min after 10 rain of perfusion in each condition.
Data Analysis Impulse-Response Experiments After bolus input into the hepatic potal vein, the concentration of 14Cdiazepam in the hepatic outflow at time t, C(t), was expressed as a fraction
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Diaz-Garcia, Evans, and Rowland
of the dose appearing per second, y(t), using the following expression
y(t) -
C(t) Q dose
(19)
where Q is the perfusate flow rate. Availability was taken to be the area under the frequency outflow versus midpoint time profile, calculated using the trapezoidal rule with extrapolation to infinity. In all cases, the extrapolated area accounted for less than 2% of the total area. Equations (6) and (7) were fitted to the experimental frequency output versus time profiles using a numerical inversion program (MULTI-FILT, version 2.0) (15,19), with a weighting scheme of 1/y(t)ob~e,,ed. However, it was first necessary to determine the transfer function for the nonhepatic regions of the experimental apparatus, W(S)N~, by applying Eq. (10) to the frequency output profile of ~4C-diazepam in the absence of a liver. This analysis provided reproducible estimates of MRTNn (5.3 sec) and DN,NH (0.041). These values were substituted into Eq. (10) which was then used to define W(S)NHin Eqs. (6) and (7). VB was assumed to represent the hepatic distribution volume for albumin, which is confined to the blood space and the space of Disse. This volume represents about 15% of wet liver weight (20-23).
Steady-State Experiments The availability of diazepam in the steady-state experiments was estimated as the ratio of the concentrations of diazepam in the perfusate at the exit (Cou,) from the liver to that in the inflowing perfusate (Ci.). The dispersion model [Eq. (12)], venous equilibrium model [Eq. (17)], and undistributed parallel-tube model [Eq. (18)] were fitted to the F vs. fu and F vs. Q data by nonlinear least squares regression using 1/Fpredictodas the weighting factor (Siphar, 3.3, SIMED). In all cases, it was assumed that P>> CL~nt, such that p ~ 1. The quality of the fit was assessed by the minimum sum of squared residuals, plot of the residuals, and the log likelihood estimates.
Statistical Analysis All data are presented as mean :t: standard deviation. The likelihood ratio test (24), a--0.05, using computer-generated log likelihood estimates was used to assess whether the dispersion model provided a significantly improved description of the steady-state data than the venous equilibrium model and the undistributed parallel-tube model. Analysis of variance (a = 0.05) was used to test for differences in ON and GLint values estimated by
181
Kinetics of Diazepam Elimination in Rat Liver
applying the dispersion model to impulse-response data, availability versus
fu data, and availability versus Q data, and to test for changes in DN and fLint with increasing flow rate.
RESULTS
Viability of the Liver Preparation and Linearity Figure 1 shows a representative plot of availability versus time when diazepam was perfused at a constant input rate (1 mg L-~; 15 ml rain -I) while alternating between protein-free perfusate and perfusate containing 1% HSA. In all cases (n = 4), the availability of diazepam did not change by more than 10% throughout the experiment for each condition. Experiment conducted in two perfused livers indicated that the availability of diazepam did not change with drug concentration up to 5 mg L -1. In one experiment, F was 0.0089 at 0.5 mg L -1, 0.0085 at 1 mg L -l, and 0.0090 at 5 mg L -~, and in the other experiment F was 0.0162, 0.0141, and 0.0160 at 0.5, 1, and 5 mg L -~, respectively.
HSA 0%
HSA 1%
HSA 0%
HSA 1%
HSA 0%
0.8 0
9
o
9
9
9
9
9
9
0.6
>., 0.4 ..Q t~
.m
t~
0.2
0
~ 0
20
40
60
80
100
120
Time (rain) Fig. 1. Availability of diazepam versus time in a representative liver while alternating between protein-free perfusate and perfitsate containing 1% HSA with constant rate drug infusion (1 mg L 1; 15 ml min ~). The design of the experment is also shown.
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Diaz-Garcla, Evans, and Rowland
Protein Binding Determination The fraction of diazepam unbound in fresh perfusate at different HSA concentrations is shown in Table I. At each HSA concentration, the binding of diazepam in hepatic outflow samples was almost identical to that determined in fresh perfusate suggesting that binding was not influenced by passage through the liver. The unbound fraction of diazepam in protein-free perfusate collected from the effluent of different liver preparations (n = 5) was 1.000 4- 0.031, indicating that the drug did not bind to material escaping from the liver into the perfusate during the experiment. Impulse--Response Experiments Figure 2 shows a representative frequency outflow versus time profile for diazepam together with the profiles predicted from the one- and twocompartment dispersion models [Eqs. (7) and (6), respectively]. In all cases, the outflow profile for diazepam appeared as a sharp peak which eluted over the first 25-see interval, followed by a slowly eluting tail. The availability of ~4C-diazepam, estimated from the area under the frequency outflow versus midpoint time profile, was 0.67 =t=0.04. The availability of diazepam, estimated as the ratio of Cout/Cin for unlabeled material using samples collected immediately before and after the bolus injection, was 0.68 4-0.03. The identical availability estimates for diazepam determined using nonspecific (radiochemical) and specific (HPLC)) methods suggest that the metabolites of 14Cdiazepam did not contribute to the total radioactivity in hepatic outflow. The one-compartment dispersion model could not adequately describe the outflow profile of ~4C-diazepam (see, e.g., Fig. 2). In contrast, the outflow profile was well described by the two-compartment dispersion model, providing estimates for DN, kl2, k21, and k23 (Table II) with a high degree of precision (coefficient of variation t'or the parameter estimate 0.98). In contrast, when fu=0.052 (HSA concentration of 1%) the extraction ratio of diazepam is about 0.35, and under such conditions clearance should be dependent mainly on fLint and fu and relatively independent on flow rate. Therefore, any change in clearance with flow rate will reflect a change in fLint. However, in all experiments performed under low extraction conditions, we found that the change in CL (estimated as Q. E) with increasing flow and marginal (Table IV) and consistent with a flow-independent intrinsic clearance. DN and GLint were also estimated in the experiments in which flow rate was varied, by applying the dispersion model to the availabilities of diazepam at the same flow rate with two different HSA concentrations (0 and 1%). The derived parameters were not influenced by variations in flow rate (a = 0.05). These findings suggest that diazepam intrinsic clearance was independent of perfusate flow rate. Hence, in all three approaches (impulse-response experiments and availability vs. fu or Q experiments) the dispersion model could adequately
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191
describe the kinetics of diazepam. It should be noted that, in this study, we used the solution of the dispersion model obtained by assuming mixed boundary conditions. Although Eq. (12) cannot be reduced to the expressions defining the relationship between F andfu or Q for the venous equilibrium extreme as DN approaches oo [Eq. (17)], we decided to use the mixed boundary solution because of its relative mathematical simplicity, particlarly when applied to transient data. Although the closed boundary conditions may be more appropriate mathematically, the choice of boundary conditions is relevant only at high dispersion numbers. In the range of DN values determined for diazepam, the predictions of availability for the mixed and closed boundary conditions are almost identical either at steady state or after a bolus dose (10). In fact, the parameters derived by appling the axial dispersion model to the steady-state data for all three boundary conditions (open, closed, and mixed) considered by Roberts and Rowland (3) were almost identical, as were the log likelihood estimates. In this study, three different approaches were used to evaluate the dispersion model and to estimate the dispersion number and intrinsic clearance for an eliminated substance. The magnitude of DN (overall range of 0.2-0.5) is similar to that reported for substances that are restricted to the vascular and extracellular spaces (10,20) such as erythrocytes and albumin, providing support for the idea that the dispersion number is a characteristic of the liver and relatively independent of the substance, and suggesting that the axial spreading of diazepam in the perfused rat liver is determined primarily by the heterogeneity of the flow in the liver. However, it should be noted that a much higher DN may be needed to account for data on the hepatic elimination of taurocholate (12,13)., lidocaine, meperidine (29), and diclofenac (Hussein et al., submitted for publication) under steady state conditions. Data for these substrates appear to conform closely to the predictions of the venous equilibrium model, possibly as a consequence of processes such as axial diffusion within hepatic tissue (30), protein-facilitated transfer of substrate across an unstirred fluid layer adjacent to the hepatocyte surface (31), or transverse enzyme heterogeneity (10). The fact that DN for diazepam is almost identical to that found for erythrocytes and albumin suggests that none of these factors is important in diazepam hepatic elimination.
REFERENCES 1. G. R. Wilkinson. Clearance concepts in pharmacology, Pharm. Rev. 39:1-47 (1987). 2. K. S. Pang and M. Rowland. Hepatic clearance of drugs. I. Theoretical considerations of a "well stirred" model and a "parallel-tube" model. Influence of hepatic blood flow, plasma and blood cell binding and the hepatocellular enzymatic activity on hepatic drug clearance. 3". Pharmacokin. Biopharm. 5:625-653 (1977).
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3. M. S. Roberts and M. Rowland. A dispersion model of hepatic elimination. 1. Formulation of the model and bolus considerations. J. Pharmacokin. Biopharm. 14:227-260 (1986). 4. L. Bass, P. Robinson, and A. J. Bracken. Hepatic elimination of flowing substances: The distributed model. J. Theoret. Biol. 72:161--184 (1978). 5. E. L. Forker and B. Luxon. Hepatic transport kinetics and plasma disappearance curves: Distributed modelling versus conventional approach. Am. J. Physiol. 235:E648-E660, (1978). 6. Y. Sawada, Y. Sugiyama, Y. Miyamoto, T. Iga, and M. Hanano. Hepatic drug clearance model: Comparison among the distributed, parallel-tube and well-stirred models. Chem. Pharm. Bull. 33:319-326 (1985). 7. M. S. Roberts and M. Rowland. Hepatic elimination. Dispersion model. J. Pharm. Sci. 74:585-587 (1985). 8. L. Bass, M. S. Roberts, and P. J. Robinson. On the relation between extended forms of the sinusoidal perfusion and the convection-dispersion models of hepatic elimination. J. Theoret. BioL 126:457-482 (1987). 9. M. S. Roberts and M. Rowland. A dispersion model of hepatic elimination. 2. Steadystate considerations. Influence of hepatic blood flow, binding within blood and hepatocellular enzyme activity. J. Pharmacokin. Biopharm. 14:261-288, (1986). 10. M. S. Roberts, J. D. Donaldson, and M. Rowland. Models of hepatic elimination: Comparison of stochastic models to describe residence time distribution and to predict the influence of drug administration, enzyme heterogeneity and systemic recycling on hepatic elimination. J. Pharmacokin. Biopharm. 16:41-83 (1988). 11. A.M. Evans, Z. Hussein, and M. Rowland. A two-compartment dispersion model describes the hepatic outflow profile of diclofenae in the presence of its binding protein. J. Pharm. Pharmacol. 43:709-714, ( 1991). 12. R. H. Smallwood, D. J. Morgan, G. W. Mihaly, D. B. Jones, and R. A. Smallwood. Effect of plasma protein binding on elimination of taurocholate by isolated perfused rat liver: Comparison of venous equilibrium, undistributed and distributed sinusoidal, and dispersion models. J. Pharmacokin. Biopharm. 16:377-396 (1988). 13. M. S. Ching, D. J. Morgan, and R. A Smallwood. Models of hepatic elimination: Implications from studies of the simultaneous elimination of taurocholate and diazepam by isolated rat liver under varying conditions of binding. J. Pharmacol. Exp. Ther. 250:1048-1054 (1989). 14. O. Levenspiel. Chemical Reaction Engineering, Wiley, New York, 1972, pp. 253-315. 15. Y. Yano, K. Yamaoka, Y. Aoyama, and H. Tanaka. Two-compartment dispersion model for analysis of organ perfusion system of drugs by fast inverse Laplace transform (FILT). J. Pharmacokin. Biopharm. 17:179-202 (1989). 16. K. Yamaoka, T. Nakagawa, and T. Uno. Statistical moments in pharmacokinetics. J. Pharmacokin. Biopharm. 6:547-558 (1978). 17. V. A. Raisys, P. N. Friel, P. R. Graaff, K. E. Opheim, and A. J. Wilensky. High performance liquid chromatography and gas liquid chromatographic determination of diazepam and nitrazepam in plasma. J. Chromatog. 183:441-448 (1980). 18. K. S. Pang and M. Rowland. Hepatic clearance of drugs. II. Experimental evidence for acceptance of the "well stirred" model over the "parallel-tube" model using lidocaine in the perfused rat liver in situ preparation. J. Pharmacokin. Biopharm. 5:655-680 (1977). 19. Y. Yano, K. Yamaoka, and H. Tanaka. A non-linear least squares program, MULTI(FILT), based on fast inverse Laplace transforms for microcomputers. Chem. Pharm. Bull. 37:1535-1538 (1989). 20. M. S. Roberts, S. Fraser, A. Wagner, and L. McLeod. Residence time distribution of solutes in the perfused rat liver using a dispersion model of hepatic elimination. 1. Effect of changes in perfusate flow and albumin concentration on sucrose and taurocholate. J. Pharmacokin. Biopharm. 18:209-234 (1990). 21. S. C. Tsao, Y. Sugiyama, Y. Sawada, S. Nagase, T. Iga, and M. Hanano. Effect of albumin on hepatic uptake of warfarin in normal and analbunemic mutant rats: Analysis by multiple indicator dilution method. J. Pharmacokin. Biopharm. 14:51-64 (1986).
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22. S. C. Tsao, Y. Sugiyama, Y. Sawada, T. Iga, and M. Hanano. Kinetic analysis of albumin-mediated uptake of warfarin by perfused rat liver. J. Pharrnacokin. Biopharm. 16:165-181 (1988). 23. C. A. Goresky. A linear method for determining liver sinusoidal and extravascular volumes. Am. J. Physiol. 204:626-640 (1963). 24. A. Buse. The likelihood ratio, Wald, and Lagrange multiplier tests: An expository note. Am. Statist. 36:153-157 (1982). 25. Y. Yano, K. Yamaoka, T. Minamide, T. Nakagawa, and H. Tanaka. Evaluation of protein binding effect on local disposition of oxacillin in rat liver by a two-compartment dispersion model. J. Pharm. Pharmacol. 42:632-636 (1990). 26. M. Rowland, D. Leitch, G. Fleming, and B. Smith. Protein binding and hepatic clearance: Discrimination between models of hepatic clearance with diazepam, a drug of high intrinsic clearance, in the isolated perfused rat liver preparation. J. Pharmacokin. Biopharm. 12:129-147 (1984). 27. J. F. Cumming and G. J. Mannering. Effect of phenobarbital administration on the oxygen requirement for hexobarbital metabolism in the isolated perfused rat liver preparation and the intact rat. Biochem. PharmacoL 19:973-978 (1970). 28. K. S. Pang, W-F. Lee, W. F. Cherry, V. Yuen, J. Accaputo, S. Fayz, A. J. Schwab, and C. A. Goresky. Effects of perfusate flow rate on measured blood volume, Disse space, intracellular water space, and drug extraction in the perfused rat liver preparation: Characterization by the multiple indicator dilution technique. J. Pharmacokin. Biopharm. 16:595-632 (1988). 29. A. B. Ahmad, P. N. Bennett, and M. Rowland. Models of hepatic drug clearance: discrimination between the "well-stirred" and "parallel-tube" models. J. Pharm. Pharmacol. 35:219-224 (1983). 30. L. P. Rivory, M. S. Roberts, and S. M. Pond. Axial tissue diffusion can account for the disparity between current models of hepatic elimination for lipophilic drugs. J. Pharmacokin. Biopharm. 20:19-62 (1992). 31. L. Bass and S. M. Pond. The puzzle of rates of cellular uptake of protein-bound ligands. In A. Pecile and A. Rescigno (eds.), Pharmacokinetics: Mathematical and Statistical Approaches to Metabolism and Distribution of Chemicals and Drugs, Plenum Press, London, 1988, pp. 245-269.
Application of the Axial Dispersion Model of Hepatic Drug Elimination to the Kinetics of Diazepam in the Isolated Perfused Rat Liver Juan M. Diaz-Garcia, ~'2 Allan M. Evans, t'3 and Malcolm Rowland t'4 Received August 16, 1991--Final February 5, 1992 The application of the axial dispersion model to diazepam hepatic elimination was evaluated using data obtained for several conditions using the single-pass isolated pesfused rat liver preparation. The influence of alterations in the fi'action unbound in perfusate (fu) and pelfusate 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 - I. In the absence of protein (fu = 1), diazepam availability was 0.011 4- 0.005 (~ 4- SD). As fu decreased, availability progressively increased and at a lISA concentration of 2% (g/ lO0 ml), when fu was 0.023, diazepam availability was 0.851 9 0.011. Application of the axial dispersion model to the relationship between fu and F provided estimates for the dispersion number (DN) of 0.3374-0.197, and intrinsic clearance (CL#,,) of 1324-34 ml min -1. The availability of diazepam during persuasion with protein-free media was also studied at three different flow rates (15, 22.5, and 30 ml min-I). Diazepam availability always progressively increased as perfusate flow increased, with the axial dispersion model yielding estimates for DN of 0.393• and CLi,,, of 144+38 ml min -t. 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~,,~estimates of 0.251 4- 0.093 and 135 4- 59 ml rain-, respectively. Hence, all methods provided similar estimates for DN and CL,,,. Furthermore, the magnitude Of DN is similar to that determined for nonetiminated substances such as eJTthrocytes, albumin, sucrose, and water. These findings suggest that the dispersion of diazepam in the pe~fused rat liver is determined primarily by the architecture of the hepatic microvasculature.
KEY WORDS: diazepam; hepatic elimination; physiologic models; dispersion model; isolated perfused rat liver; protein binding. This work was supported by the Commission of the European Communities and the Medical Research Council. One of us (A.M.E.) was partially supported by a Merck, Sharp & Dohme Fellowship. We are grateful to Roche (Switzerland) for the supply of diazepam and 2-[14C]diazepam, and Kabi AB (Sweden) for the supply of human serum albumin. ~Department of Pharmacy, University of Manchester, MI3 9PL, United Kingdom. 2present address: Facultad de Farmacia, Universidad de Navarra, 31008 Pamplona, Spain. 3present address: School of Pharmacy, University of South Australia, Adelaide 5000, South Australia. 4To whom correspondence should be addressed. 171 0090-466x/92/04000171506.50/09 1992PlenumPublishingCorporation
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INTRODUCTION The fraction of drug escaping extraction during a single passage through the liver (availability) depends on the fraction of drug unbound in blood, organ perfusion, hepatocellular enzyme activity, and, in some cases, the permeability of the hepatocyte membrane to the drug. Various models have been used to describe the influence of these physiological determinants on hepatic drug clearance and availability (1). The fundamental difference between these models lies in the assumption applied to the extent of axial spreading of substrate within the hepatic microvasculatory system. At the two extremes are the "venous equilibrium model" (well-stirred model) and the "undistributed parallel-tube model" (1,2). Because neither of these simplistic models is entirely compatible with the growing body of experimental data on the hepatic handling of drugs and with the known morphological features of the liver (3), models that apply a more realistic view of hepatic microvasculatory processes, such as the "distributed parallel-tube models" (4-6) and the "axial dispersion model" (7) may be more appropriate. These more complex models attempt to account for the known heterogeneity of the microvasculature, and in some cases the enzyme activity, of the liver. In the distributed parallel-tube models, the degree of heterogeneity is dictated by the variance of the statistical distribution chosen to represent the particular property. In the axial dispersion model, organ blood flow heterogeneity is reflected by the magnitude of the dispersion number, DN. The undistributed parallel-tube model and the venous equilibrium model are extreme forms of the axial dispersion model in which DN approaches 0 or infinity, respectively (3). The dispersion model has also been modified to incorporate transverse heterogeneity of enzyme activity (8). The magnitude of dispersion can be estimated either by reference to the concentration-time profile of a substance in the effluent from the liver after bolus input into the hepatic portal vein (impulse-response experiments) or by examining the influence of controlled changes in physiological determinants such as the blood (perfusate) flow rate to the liver or drug binding within blood, on drug availability (3,9). Analysis of the impulse-response data for a number of noneliminated reference markers in the isolated perfused rat liver have provided estimates for DN in the range of 0.2-0.5 (10,11 ). However, estimates of DN provided from considerations of drug availability at steady state appear to vary more widely, with values up to infinity reported for the endogenous compound taurocholic acid (12,13). The influence of altered protein binding on the availability of diazepam at steady state has been shown previously to conform to the predictions of the dispersion model with a DN of 0.29 (9). This present study extends these
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previous findings by evaluating the consistency of the dispersion model in describing different data sets with diazepam. Namely, impulse-response data and steady-state availability data obtained under conditions of either altered protein binding, perfusate flow, or both, using the single-pass perfused in situ rat liver preparation. In addition, for the data obtained under steady state conditions the dispersion model is evaluated against the parallel-tube and well-stirred models of hepatic elimination.
THEORY Under linear conditions, the outflow profile of a substance injected into the hepatic portal vein is given by
c(t) =Q x(O * w(t)N. * w(t).
(1)
where C(t) is the concentration of material in the hepatic outflow at time t, D is the amount of drug administered, Q is the hepatic blood (perfusate) flow rate, x(t) is the function that describes the form of drug input (bolus, constant infusion, etc.), w(t)NH and w(t)H are the unit impulse-responses for the nonhepatic and hepatic regions of the experimental system, respectively, and 9 denotes the convolution integral. By expressing output as a fraction of the dose appearing per unit time (frequency output), y(t), and on taking Laplace transforms, Eq. (1) becomes
y(s) = x(s)" W(S)NH" W(S)H
(2)
When the substance is injected as a bolus, the Laplace transform of the input function, x(s), is equal to unity and Eq. (2) can be simplified to y(s) = w(s)~. . w(s).
(3)
where W(S)NHand W(S)Hare the transfer functions for nonhepatic and hepatic regions of the experimental system, respectively. In the present study, these transfer functions were defined using the axial dispersion model. This model, recently introduced into the field of pharmacokinetics (7), is widely used to represent nonideal flow behavior in chemical reactors (14). According to the theory of the axial dispersion model, when a substance is injected into the liver and distributes at a finite rate between the vascular and cellular spaces (two-compartment dispersion model), the concentration of the substance in blood (perfusate), CB, as a function of time (t) and distance along the liver length (z), assuming elimination from the cellular
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space, is given by (10,15)
OCB-D 02C~-v OCB-k12" Ca+k2) " C~ t3t Oz2 8z
(4)
with the corresponding mass-balance equation for the cellular compartment being ~C~_
Ot
VBk,2CB-k21" Cc-k23" Cc Vc
(5)
where D is the axial dispersion coefficient, which characterises the degree of axial dispersion of the injected substance in blood, v is the linear velocity of blood, k12 and kz! are the first-order rate constants for the transt~r of solute into and out of the cellular space, respectively, k23 is the first-order rate constant for the irreversible removal of solute from the cellular compartment, VB and Vc are the volumes of the vascular and cellular spaces, respectively, and Cc is the concentration of drug in the cell. Equations (4) and (5) comprise a set of second-order partial differential equations. Assuming mixed boundary conditions (see Discussion), the transfer function for the liver can be determined (10,15), and Eq. (3) becomes
2DN
(6)
where s is the Laplace operator, and DN=D/vL, where L is the liver length. When the radial transfer of a substance between the vascular and cellular spaces is instantaneous (one-compartment dispersion model), the corresponding solution is (3,11,15)
eXp['I'+4~ 2DN
(7)
Kinetics of Diazepam Elimination in Rat Liver
175
where VH is the volume of distribution of the drug in the liver. The term RN (the efficiency number) is given by RN = GLint" fu"
Q
p
(8)
where intrinsic clearance, CL~,t, is defined as the proportionality constant between the rate of elimination and the unbound substrate concentration within the cell, and p is given by P P + GLint
p=- -
(9)
where P is the permeability of the hepatocyte membrane to the drug. Thus, p represents the ratio of the concentration of unbound drug in the cell to that in blood. The transfer function for the nonhepatic regions in the experimental system can be defined using the one-compartment form of the dispersion model for a noneliminated solute (RN = 0) (1 1)
W(S)NH=exp I_I --( I + 4DN,NH " MRTNH " s)'/21 (10)
2DN,Nrt
where DN,NH and MRTNH are the dispersion number and the mean residence time of the substance in the nonhepatic regions of the experimental system. These parameters can be estimated by applying Eq. (10) to the ouflow profile determined by injecting the substance as a bolus into a system in which the hepatic inflow and outflow cannulas are connected. The fraction of a dose escaping extraction during a single pass through the liver (availability, F) is given by (16) F = limf(s) = lira [W(S)NH " w(s)n ] = lira [w(s)H ] s~0
s---~0
a'--*0
(1 1)
Therefore, for the one-compartment dispersion model F = e x p I 1 - ( 1 +4" DN" RN) 1/2] 2~
(12)
and for the two-compartment dispersion model i.
4VB" DN"
F=exp
k12" k23 (13)
2DN
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Diaz-Garcla, Evans, and Rowland
The first-order rate constants can also be defined as k,2 = P" f u / V ,
(14)
k2, = P "fuc/Vc
(15)
k23 GLint"fuc/Vc
(16)
-~
where fu~ is the fraction of drug in the cell which is unbound. Substituting these equalities into Eq. (13), the latter is transformed into Eq. (12). Thus, the steady state solutions for the one- and two-compartment forms of the dispersion model are identical (10,15). According to the venous equilibrium model, availability is given by (9) F=--
1
1 +RN
(17)
and according to the undistributed parallel-tube model, F is given by (9) F = exp (-RN)
(18)
METHODS Materials
Unlabeled diazepam and 2-[J4C]-diazepam (purity 99% TLC, 194 p C i / mg) were obtained from Roche (Basel, Switzerland) and human serum albumin (HSA) was obtained from Kabi AB (Sweden). HPLC solvents were purchased from British Drug Houses. All other reagents were of analytical grade. Protein Binding Determination
The binding of diazepam within perfusates containing varying concentrations of HSA was determined by equilibrium dialysis (Dianorm | Switzerland) with Spectra/Por-2 membranes (Spectrum Medical Industries Inc., CA). Briefly, 1 ml perfusate containing 1 mg L -1 of diazepam and 0.005 pCi of 14C-diazepam were dialyzed against 1 ml of protein-free perfusate at 37~ for 4 hr. 14C-diazepam was determined in a 0.2 ml aliquot of each compartment by radiochemical analysis (LKB Rackbeta liquid scintillation counter, Finland). The unbound fraction of diazepam was taken to be the ratio of the concentration of radiolabeled diazepam in the protein-free buffer to that in the perfusate compartment at the end of dialysis. Preliminary experiments showed that equilibrium was reached within 3 hr, that volume shifts were negligible, and that diazepam did not bind to the experimental system. To assess whether the binding of diazepam was influenced by passage through the liver, the fraction unbound of diazepam was also determined in a set of
Kinetics of Diazepam Elimination in Rat Liver
177
perfusate samples collected from the venous effluent, including protein-free samples.
Diazepam HPLC Assay Diazepam was assayed by HPLC using a modification of the method of Raisys et al. (17). All glassware was silanized. Briefly, 50 pl of internal standard solution (nitrazepam 3.75 pg ml-l; Sigma, USA) and 200 pl of acetonitrile (to precipitate HSA) were added to 200 pl of sample. After vortex mixing (I rain) and centrifugation (3400 rpm, 20 rain) 20 pl of the supernatant was injected into the HPLC system. Calibration curves were constructed over the range of 100 to 1500 ng ml -t and the intraday coefficient of variation (n=5) ranged between 7.42% (100ngm1-1) and 3.54% (1500 ng ml-l). Hepatic outflow samples obtained during perfusion with protein-free perfusate contained concentrations of diazepam that could not be measured accurately using the method described above. Therefore it was necessary to perform a simple extraction procedure prior to HPLC analysis of such samples. Briefly, 50 pl of internal standard solution (nitrazepam, 3.75 pg ml -I) and 8 ml of hexane:ethylacetate (8:2, v: v) were added to 4 ml of sample. After vortex mixing (1 rain) and centrifugation (3400 rpm, 20 rain) the upper organic layer was transferred to a clean test tube and evaporated to dryness (N2, 35~ Extraction efficiency for diazepam and nitrazepam was greater than 97%. The residue was reconstituted in 75 pl methanol and 20 pl was injected into the HPLC system. Calibration curves were constructed over the range of 5 to 150 ng ml -j and the intraday coefficient of variation (n= 5) ranged between 8.23% (5 ng ml -~) and 5.22% (150 ng ml-I). The HPLC system consisted of a Kontron analytic LC 410 Pump (Zurich, Switzerland) which delivered mobile phase (acetonitrile:water with 1% triethylamine adjusted to pH 3 with 85% orthophosphoric acid; 50:50, v:v) at a flow rate of 1.5 ml rain -~, to a C~8 chromatographic column (Spherisorb, S10, ODS1). Samples were injected using a 7010 Rheodyne manual injector (Cotati, CA) and column effluent was monitored using a Waters Lambdamax model 481 absorbance detector (Milford, USA) set to 254 nm. The retention times of diazepam and nitrazepam were 8.5 and 4.5 min, respectively. Known metabolites of diazepam (oxazepam, temazepam, and desmethyldiazepam: Sigma, USA) eluted at 5.2, 5.7, and 6.5rain, respectively.
Hepatic Perfusion The single-pass isolated perfused in situ rat liver preparation using male Sprague-Dawley rats (200-400 g), was essentially that described previously
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(11,18). The perfusate was freshly prepared and filtered (0.2 #m) Krebs bicarbonate buffer containing 3 g L -I of 10 mg L -j sodium taurocholate (Sigma), equilibrated to pH 7.4 with humidified O2/CO2(95/5) and maintained at 37~ Perfusate was delivered to the liver via a cannula inserted into the hepatic portal vein and liver outflow was collected via a cannula inserted into the vena cava through the right atrium. The tubing between the injector and cannula entering the portal vein was 3 cm; the cannula exiting from the superior vena cava was 6 cm. In each case, the liver was stabilized for 15 to 20 min using drug and protein-free perfusate. At the end of the experiment, the liver was removed and weighed. Diazepam did not bind to the perfusion apparatus. Bile flow was 0.6-1.5/~1 min -1 per gram of liver with a modest fall of 18% over the 2 hr of the experiment.
Viability of the Liver Preparation and Linearity of Diazepam Elimination Preliminary experiments (n = 4) were conducted to assess the viability of the perfused liver preparation. In each preparation, the perfusate was alternated between 0 and 1% (g/100 ml) HSA for 120 min after the stabilization period, while the total concentration of diazepam was maintained at 1 mg L -I. Hepatic outflow samples were obtained every 5 min and analyzed for diazepam. For these and subsequent experiments, the condition of the liver during perfusion was also assessed by monitoring the bile output and the recovery of the inflowing perfusate and by visual examination of the liver. Linearity with respect to diazepam elimination was tested with HSAfree perfusate containing three different diazepam concentrations (0.5, 1, and 5 mg L --t) perfused in random order for 20 min each.
Experimental Design
Impulse-Response Experiments In these experiments (n = 4), 50/11 of a solution containing 0.125pCi of t4C-diazepam in 1% HSA was rapidly injected into the hepatic portal vein during a constant perfusion with unlabeled diazepam (1 mg L -J) in 1% HSA. The total effluent from the liver was automatically collected at 2-see intervals using a motor-driven carousel with 57 sampling holes, and thereafter (into silanized test tubes) at increasing time intervals for up to 4 min. Perfusate samples were also collected before and after the bolus injection in order to determine the availability at steady state of unlabeled diazepam. The concentration of ~4C-diazepam in effluent samples was determined by radiochemical analysis and that of unlabeled diazepam by HPLC.
Kinetics of Diazepam Elimination in Rat Liver
179
Influence of Protein Binding on Availability of Diazepam at Steady State These experiments (n=4) were conducted to assess the influence of HSA perfusate concentration on the availability of diazepam at steady state. Preliminary experiments indicated that up to 10 min was required to achieve steady state with respect to diazepam availability (e.g., see Fig. 3). Therefore, availability was determined from samples collected between 10 and 20 rain (at 3- to 4-min intervals). In each liver preparation, four different perfusates containing 1 mg L -~ of diazepam were used; one was protein-free and the other three contained 0.1, 0.5~ I, or 2% HSA. After a period of stabilization with drug-free and HSA-free perfusate, diazepam in the absence of protein was perfused for 20 min. Subsequently, the liver was perfused with two different HSA concentrations for 20 min each. To assess for changes in the viability of the liver preparation with time, the initial protein-free perfusate was then used again for 20 min before the third protein-containing test perfusate. At the end of the experiment (120 min after cannulation of the hepatic portal vein and 100 min after commencing perfusion with diazepam) the liver was perfused with protein-free solution for a third time. A preparation was acceptable if the availability of diazepam during the second and third periods of perfusion with protein-free perfusate did not differ by more than 10% from that during the first run. The four preparations used met these conditions~ The scheme of a typical experiment is presented in Fig. 3.
Influence of Perfusate Flow on Availability of Diazepam at Steady State In another four perfused liver preparations, the influence of flow on diazepam availability was examined at a diazepam concentration of 1 mg L -1 using 3 different flow rates (15, 22.5, and 30 ml rain -I) at 0 and 1% HSA (6 different conditions for 15 min each). After the stabilization period, the liver was initially perfused with protein-free perfusate for 15 min at a particular flow rate and, to assess viability of the liver preparation, the experiment was always terminated under the same conditions. An example of a typical experiment procedure, in which the control flow rate was 15 ml min -~, is shown in Fig. 5. Hepatic outflow samples were collected into silanized test tubes every 1-2 min after 10 rain of perfusion in each condition.
Data Analysis Impulse-Response Experiments After bolus input into the hepatic potal vein, the concentration of 14Cdiazepam in the hepatic outflow at time t, C(t), was expressed as a fraction
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Diaz-Garcia, Evans, and Rowland
of the dose appearing per second, y(t), using the following expression
y(t) -
C(t) Q dose
(19)
where Q is the perfusate flow rate. Availability was taken to be the area under the frequency outflow versus midpoint time profile, calculated using the trapezoidal rule with extrapolation to infinity. In all cases, the extrapolated area accounted for less than 2% of the total area. Equations (6) and (7) were fitted to the experimental frequency output versus time profiles using a numerical inversion program (MULTI-FILT, version 2.0) (15,19), with a weighting scheme of 1/y(t)ob~e,,ed. However, it was first necessary to determine the transfer function for the nonhepatic regions of the experimental apparatus, W(S)N~, by applying Eq. (10) to the frequency output profile of ~4C-diazepam in the absence of a liver. This analysis provided reproducible estimates of MRTNn (5.3 sec) and DN,NH (0.041). These values were substituted into Eq. (10) which was then used to define W(S)NHin Eqs. (6) and (7). VB was assumed to represent the hepatic distribution volume for albumin, which is confined to the blood space and the space of Disse. This volume represents about 15% of wet liver weight (20-23).
Steady-State Experiments The availability of diazepam in the steady-state experiments was estimated as the ratio of the concentrations of diazepam in the perfusate at the exit (Cou,) from the liver to that in the inflowing perfusate (Ci.). The dispersion model [Eq. (12)], venous equilibrium model [Eq. (17)], and undistributed parallel-tube model [Eq. (18)] were fitted to the F vs. fu and F vs. Q data by nonlinear least squares regression using 1/Fpredictodas the weighting factor (Siphar, 3.3, SIMED). In all cases, it was assumed that P>> CL~nt, such that p ~ 1. The quality of the fit was assessed by the minimum sum of squared residuals, plot of the residuals, and the log likelihood estimates.
Statistical Analysis All data are presented as mean :t: standard deviation. The likelihood ratio test (24), a--0.05, using computer-generated log likelihood estimates was used to assess whether the dispersion model provided a significantly improved description of the steady-state data than the venous equilibrium model and the undistributed parallel-tube model. Analysis of variance (a = 0.05) was used to test for differences in ON and GLint values estimated by
181
Kinetics of Diazepam Elimination in Rat Liver
applying the dispersion model to impulse-response data, availability versus
fu data, and availability versus Q data, and to test for changes in DN and fLint with increasing flow rate.
RESULTS
Viability of the Liver Preparation and Linearity Figure 1 shows a representative plot of availability versus time when diazepam was perfused at a constant input rate (1 mg L-~; 15 ml rain -I) while alternating between protein-free perfusate and perfusate containing 1% HSA. In all cases (n = 4), the availability of diazepam did not change by more than 10% throughout the experiment for each condition. Experiment conducted in two perfused livers indicated that the availability of diazepam did not change with drug concentration up to 5 mg L -1. In one experiment, F was 0.0089 at 0.5 mg L -1, 0.0085 at 1 mg L -l, and 0.0090 at 5 mg L -~, and in the other experiment F was 0.0162, 0.0141, and 0.0160 at 0.5, 1, and 5 mg L -~, respectively.
HSA 0%
HSA 1%
HSA 0%
HSA 1%
HSA 0%
0.8 0
9
o
9
9
9
9
9
9
0.6
>., 0.4 ..Q t~
.m
t~
0.2
0
~ 0
20
40
60
80
100
120
Time (rain) Fig. 1. Availability of diazepam versus time in a representative liver while alternating between protein-free perfusate and perfitsate containing 1% HSA with constant rate drug infusion (1 mg L 1; 15 ml min ~). The design of the experment is also shown.
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Protein Binding Determination The fraction of diazepam unbound in fresh perfusate at different HSA concentrations is shown in Table I. At each HSA concentration, the binding of diazepam in hepatic outflow samples was almost identical to that determined in fresh perfusate suggesting that binding was not influenced by passage through the liver. The unbound fraction of diazepam in protein-free perfusate collected from the effluent of different liver preparations (n = 5) was 1.000 4- 0.031, indicating that the drug did not bind to material escaping from the liver into the perfusate during the experiment. Impulse--Response Experiments Figure 2 shows a representative frequency outflow versus time profile for diazepam together with the profiles predicted from the one- and twocompartment dispersion models [Eqs. (7) and (6), respectively]. In all cases, the outflow profile for diazepam appeared as a sharp peak which eluted over the first 25-see interval, followed by a slowly eluting tail. The availability of ~4C-diazepam, estimated from the area under the frequency outflow versus midpoint time profile, was 0.67 =t=0.04. The availability of diazepam, estimated as the ratio of Cout/Cin for unlabeled material using samples collected immediately before and after the bolus injection, was 0.68 4-0.03. The identical availability estimates for diazepam determined using nonspecific (radiochemical) and specific (HPLC)) methods suggest that the metabolites of 14Cdiazepam did not contribute to the total radioactivity in hepatic outflow. The one-compartment dispersion model could not adequately describe the outflow profile of ~4C-diazepam (see, e.g., Fig. 2). In contrast, the outflow profile was well described by the two-compartment dispersion model, providing estimates for DN, kl2, k21, and k23 (Table II) with a high degree of precision (coefficient of variation t'or the parameter estimate 0.98). In contrast, when fu=0.052 (HSA concentration of 1%) the extraction ratio of diazepam is about 0.35, and under such conditions clearance should be dependent mainly on fLint and fu and relatively independent on flow rate. Therefore, any change in clearance with flow rate will reflect a change in fLint. However, in all experiments performed under low extraction conditions, we found that the change in CL (estimated as Q. E) with increasing flow and marginal (Table IV) and consistent with a flow-independent intrinsic clearance. DN and GLint were also estimated in the experiments in which flow rate was varied, by applying the dispersion model to the availabilities of diazepam at the same flow rate with two different HSA concentrations (0 and 1%). The derived parameters were not influenced by variations in flow rate (a = 0.05). These findings suggest that diazepam intrinsic clearance was independent of perfusate flow rate. Hence, in all three approaches (impulse-response experiments and availability vs. fu or Q experiments) the dispersion model could adequately
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describe the kinetics of diazepam. It should be noted that, in this study, we used the solution of the dispersion model obtained by assuming mixed boundary conditions. Although Eq. (12) cannot be reduced to the expressions defining the relationship between F andfu or Q for the venous equilibrium extreme as DN approaches oo [Eq. (17)], we decided to use the mixed boundary solution because of its relative mathematical simplicity, particlarly when applied to transient data. Although the closed boundary conditions may be more appropriate mathematically, the choice of boundary conditions is relevant only at high dispersion numbers. In the range of DN values determined for diazepam, the predictions of availability for the mixed and closed boundary conditions are almost identical either at steady state or after a bolus dose (10). In fact, the parameters derived by appling the axial dispersion model to the steady-state data for all three boundary conditions (open, closed, and mixed) considered by Roberts and Rowland (3) were almost identical, as were the log likelihood estimates. In this study, three different approaches were used to evaluate the dispersion model and to estimate the dispersion number and intrinsic clearance for an eliminated substance. The magnitude of DN (overall range of 0.2-0.5) is similar to that reported for substances that are restricted to the vascular and extracellular spaces (10,20) such as erythrocytes and albumin, providing support for the idea that the dispersion number is a characteristic of the liver and relatively independent of the substance, and suggesting that the axial spreading of diazepam in the perfused rat liver is determined primarily by the heterogeneity of the flow in the liver. However, it should be noted that a much higher DN may be needed to account for data on the hepatic elimination of taurocholate (12,13)., lidocaine, meperidine (29), and diclofenac (Hussein et al., submitted for publication) under steady state conditions. Data for these substrates appear to conform closely to the predictions of the venous equilibrium model, possibly as a consequence of processes such as axial diffusion within hepatic tissue (30), protein-facilitated transfer of substrate across an unstirred fluid layer adjacent to the hepatocyte surface (31), or transverse enzyme heterogeneity (10). The fact that DN for diazepam is almost identical to that found for erythrocytes and albumin suggests that none of these factors is important in diazepam hepatic elimination.
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