Biochemical Printed Pharmcology . Vol.37. NO. 8, pp. 1425-1431. cmw2952/88 s3.00 + 0.00 IQ 1988. Pergamon Press plc 1988 in Great Britain. zyxwvutsrqp COMMENTARY PHYSIOLOGICALLY BASED MODELS AND STRATEGIC EXPERIMENTS IN HEPATIC PHARMACOLOGY LUDVIK BASS* and SUSANNEKEIDING Department of Mathematics, University of Queensland, Australia, and Medical Department A, Rigshospitalet, Copenhagen, Denmark The central problem of modelling hepatic pharmacokinetics quantitatively is to transpose the relevant test-tube kinetics (known or postulated) into the setting of hepatocellular anatomy and microcirculation and, hence, to derive relations between quantities observable on the intact organ. In such a transposition, the mathematical form of the concentration-dependence of local metabolic rates (such as the Michaelis-Menten form) is commonly preserved, but a change in the numerical values of the kinetic constants (such as V,,,, and K,) cannot be excluded a priori [l]. In general, a compromise between realism and parsimony is desirable: physiology must not be altogether lost in oversimplifications, but the interrelation between the model and data should not be trivialized by an excess of adjustable parameters [Z]. As models of steady-state processes are particularly parsimonious, we defer discussion of time-dependent experiments to the penultimate section. Substrates of liver enzymes are carried by blood through many (107-108) discontinuous capillaries (hepatic sinusoids) which are lined with hepatocytes containing the enzymes. Hepatic blood flow is manifolded through these capillaries from their common inlet and reunites in the hepatic vein. The activity of the enzymes depletes the substrates, so that concentration gradients develop along the flow, and these gradients in turn affect the global elimination rates of the substrates by the intact liver. Only at very high substrate concentrations is a flow-independent, biochemically determined elimination rate maintained by the saturation of the enzymes along all the capillaries. At the other extreme of low substrate concentrations and high enzyme densities, dach sinusoid acts as a perfect sink of the substrate, so that global elimination is controlled by blood flow independently of biochemical parameters. The first task of any physiologically based modelling is to quantify elimination occurring between these two extremes. A physiological approach: the undistributedperfmion model We consider the steady hepatic elimination of a blood-borne substrate, due to irreversible metabolic conversion by hepatocellular enzymes, excretion. Let the hepatic blood flow of rate F carry the substrate into the liver at the observed concentration ci, and out of it at the observed concentration c,. The steady rate of elimination, V, by the intact liver is then V= - Co). (1) For any saturation kinetics, increasing ci (and hence co) to sufficiently high values makes V tend to the maximum (saturated) value V,,,,, so that VmdF is the maximum attainable arterial-venous cancentration difference. Next, let Michaelis-Menten kinetics hold for the rate-determining step of the elimination process by each hepatocyte. Then, if all hepatocytes were presented with the same substrate concentration c, the hepatic elimination rate would be V,,,axc/(~ + K,,,) with the half-saturating concentration K,,,. In the intact organ, the substrate concentration presented to hepatocytes varies continuously from Ci at the inlet to c, at the outlet, the concentration profile being itself the result of the interplay of local elimination with blood flow. What is the counterpart, for the intact organ, of the Michaelis-Menten relation in the test tube? We consider a substrate rapidly equilibrated between blood and hepatocytes that are held in tied positions in the hepatic blood flow by the scaffolding of the hepatic parenchyma, so that they provide a spatially distributed sink of the substrate. We put the x-axis along the blood flow, with inlet at x = 0 and outlet at x = L, so that the steady concentration profile c(x) satisfies c(0) = Ci, c(L) = c,. The depletion of the substrate flux Fc by elimination in any interval n, x + a!x is expressed by applying equation 1 and Michaelis-Menten kinetics locally: Fdc = - [p(x)&]c/(c + K,). (2) Here p(x)& is the part of V,,,, in any interval n, x + dr, so that L I0 P(XW = vllxa,. (3) Separating equation 2, integrating from 0 to L and using equation 3 gives [3]: zyxwvutsrqponmlkjihgfedcbaZYXWV Ci or to biliary - K, * Correspondence: Prof. L. Bass, Department of Mathematics, University of Queensland, Brisbane, Q. 4067, Australia. F(Ci Co V $- lnCi=mar c, FK; This result may be interpreted as a generalization to saturation kinetics of the single-capillary model of first-order uptake in capillary physiology [4], initi- 1425 1426 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA L. BASS and S. KEIDING ated by Bohr [S] and exploited in hepatology by because V,,,JFK,,, 6 1, or because V approaches Brauer [6]. Let an organ comprise N capillaries V,,,, (saturation). If the series is broken off after the (hepatic sinusoids) acting in parallel, in the funcsecond (linear) term, and the resulting approxitional rather than the anatomical sense. If all have mation is combined with equation 1, one obtains V = V,, co/(c,, + Km): the elimination rate is the the same values of 4, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Km, maximum elimination same as if all hepatocytes were presented with the capacity V,,,,/N and flow rate F/N, then the mixed venous concentration c, is again given by equation 4 substrate concentration c,. In this approximation to because its right-hand side can be written as (VmaJ the perfusion model, one arrives at “venous equiN)/(F/N)K,,,, with N cancelling. The indifference of libration” [12,13] by the use of reasoning, without the resulting c,, to the manifolding of the blood flow postulating arbitrarily that the venous concentration c,, is actually uniform throughout the liver (with a through the parallel capillaries is characteristic of the concentration jump from ci to c, at the inlet). This undistributed (single capillary) model of elimination; consideration delimits quantitatively the circumit does not hold in the distributed model discussed stances under which the venous equilibration model below. We emphasize that the form of the function [12,13] predicts approximately the same phenomena p(x) does not affect equation 4: the enzyme distrias the undistributed perfusion model [3], and under bution can vary arbitrarily along the blood flow and even amongst sinusoids, so long as equation 3 is which it shares the rational basis of the perfusion model consistent with autoradiographic evidence satisfied. This important but often unrecognized [7,8] result increases the realism and robustness [III. When a series of pairs co, ci is measured in an of the undistributed perfusion model [3]: if spatial uniformity of enzyme distributions [p(x) = const = appropriately designed experiment, equations 1, 5 V,,,,/L] were a prerequisite, the model would be and 6 of the undistributed perfusion model permit the determination of the kinetic parameters V,,,,, refuted simply by noting the observed zonal distributions of liver function along the hepatic blood flow and K,,, of the intact organ using, for example, the Lineweaver-Burk plot of test-tube biochemistry for ]91. the variables l/V, l/t. This and other procedures If we define the logarithmic mean concentration zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED have been used, with satisfactory results, to quantify Ci - C, the kinetic parameters of a variety of enzyme-sube=c <e<c, lIl(Ci/C,J’ o ’ strate combinations in isolated perfused livers, intact animals and humans [3, 10, 14, 151. The values of and eliminate F from equations 1 and 4, we find the kinetic parameters so determined presuppose the [3, 101: validity of the underlying model equations. It is therefore desirable to test parameter-free predictions of the model equations by specially designed (6) ?fl experiments. This formulation of the model has the same mathA set of model- testing experiments ematical form as the test-tube MichaelieMenten We consider an isolated rat liver perfused in a rerelationship, but with concentration given by circulating system, eliminating steadily a substrate equation 5 in terms of the physiological observables Ci, c,. The elimination rate is the same as if all infused into the system at the steady rate 2 = V. The foregoing equations of the undistributed perfusion hepatocytes were presented with the same substrate concentration 2. While the Earth attracts bodies as model then hold with Z put in place of V. At some fixed I< V,,,,, let the flow rate F be changed from if its mass were concentrated at its centre, no one believes, contrary to tangible evidence, that it is one steady value to another. Then ci - c, = Z/F (from equation 1) must change accordingly, but e in fact so concentrated. Similarly, compartmental should not change (because equation 6 does not analysis should not be driven so far as to interpret equation 6 to mean that substrate actually has the involve fl, provided that V,,,, and K,,, remain uniform concentration e throughout the liver (with unchanged. That is: if concentration jumps at the inlet and outlet), contrary avIM to autoradiographic evidence of arterial-venous con-=(J and -0, (9) aF aFcentration gradients of a variety of substrates [ll]. Indeed, equation 2 shows that the concentration then zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH gradient dc/dx is the very reason for I? being the ae operative concentration in the context of Michaelis= 0, Menten uptake. aF* Combining equations 1 and 4, we find where the suffix Z indicates that the change in F is ci VnUX-v made at fixed I. The undistributed perfusion model -=e FK, (7) predicts that, in this experimental design, the change c0 in Ci and c,, with a change in F is such that I?, given for the inverse of the bioavailability co/cl. The series by equation 5, is flow-independent. No adjustable expansion parameters are involved in this prediction (a good model lives dangerously). V cizl+ mm The validity of equations 9 for the intact liver does FK,V+I[v~~V]Z+. . . (8) CO not follow from the biochemical meaning of V,, and K,,,, because at sufficiently low flow rates sinusoids converges rapidly if (If,,,,, - V)/FK,,, Q 1, either v=v,&-. wn (> Models and experiments in hepatic pharmacology 1427 not occur [16,17,21]. The high statistical significance collapse, depriving some hepatocytes of substrate, of these deviations (P < 0.001: [21,22]) leaves little oxygen and ATP [16]. Then zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA V,,,, as well as K,,, doubt that equation 11 and, hence, the undistributed may change with the flow rate. This is the familiar physiologic phenomenon of recruitment or de- perfusion model itself is at fault. What is the explanation of these deviations from recruitment of capillaries. To establish the validity of equations 9 within a range of flow rates high equation 11 at flow rates which are high enough to enough to avoid de-recruitment of sinusoids, it has preserve the numbers of perfused sinusoids? If D,, and f, respectively, are the contributions by a single been shown [17] that, under saturation with galactose, there is no statistically significant change in sinusoid to V,,,, and F of a whole liver comprising N galactokinase V,,,, and oxygen consumption of a set sinusoids in parallel, then the undistributed model asserts that for all sinusoids v,, = V- /N and f = of rat livers, until flow rates are reduced below F/N. Such regularity cannot be expected in real 0.9 ml.min-‘a (g liver)-’ on average. Furthermore, for elimination of alcohol by a set of rat livers, no capillary beds, as exemplified directly by quantitative studies of single capillaries [23]. We must expect statistically significant changes in V,,,,, and K,,, were statistical distributions of V,,,~, f and of zyxwvutsrqponmlkjihg v,$f about found when these parameters were determined (from pairs of infusion rates, and equation 6) at two their mean values over all sinusoids of a liver. Sinudifferent flow rates above 0.9 mlemin-‘.(g liver)-* soids with common ci and K,,, but different v,df will then have different output concentrations c,; see [18]. These results clear the way for interpreting unambiguously experimental tests of equation 10. equations 4 and 11 with-V,,,dF replaced by t.&/f. Then the mixed venous concentration E, will be the These were done on sets of rat livers eliminating galactose [19] and propranolol [20]. Changes in e flow-weighted mean of the c, values from individual sinusoids, different from the c, calculated from the with F [kept above 0.9 ml-min-‘.(g liver)-‘] were not statistically significant, while changes in ci and c, undistributed model of a liver with the same organ were highly significant. The latter change demonvalues V,, and F, which we denote by c,(V,,,d F). It is a general theorem [2] for steady saturable strated that, for interpreting these experiments, the elimination by capillary beds that expansion in equation 8 must be carried beyond the linear term: the “venous equilibration” approxiE, 2 G (V,,,/F) (12) mation is inadequate. The successful testing of equation 10 involves an holds for any dispersions of a,,, zyxwvutsrqponmlkjihgfed f and zyxwvutsrqponmlkjihgfed v-/f over interesting methodological trade-off. No numerical the set of parallel capillaries, with equality attained information about any kinetic parameters is obtained only at saturation: heterogeneity increases bioavailfrom the experiments; in exchange, confidence in ability. The distributed sinusoidal perfusion model the model is increased by its passing a particularly [24-261 retains the equations of the undistributed severe (parameter-free) experimental test. The nullmodel for the description of uptake by each sinusoid prediction made by equation 10, reminiscent of the but quantifies effects of distributions of sinusoidal methodology of physics, is particularly suited to statproperties upon the relations between the quantities istical evaluation (by slippage tests [18-201). ci, E,, F, and V observable on the organ. Envisage the set of sinusoids of a given liver classified into groups labelled 1, 2 . . . n . . . , each conSingle- pass experiments and the distributed perfusion sisting of sinusoids having the same v&f, a total model maximum elimination capacity V$,!jkand a total flow For single-pass (once-through) experiments with rate fl”) (with X V$,& = V,,, and X fl) = F). The isolated perfused rat livers, the input concentration groups may contain unequal numbers of sinusoids. Ciis fixed, and changes in perfusate flow change the A notional sub-organ made up of any one such group output concentration c, as well as the elimination of sinusoids would be properly described by the rate V. Experiments are often performed in the limit undistributed sinusoidal perfusion model. The coefof first-order kinetics (V Q V,,,,,). Then equation 7 ficient of variation, zyxwvutsrqponmlkjihgfedcbaZYXWVU E, of the distribution of v,df can be written as over all the notional sub-organs constituting the actual liver is given by [26] In(ci/cO) = 2, m 2 mwli?ax)* _ 1 (13) F(“)fF ’ predicting that the plot of ln(c,/cJ against l/Fshould ” be a straight line through the origin, with the slope E*vanishes only when all sub-organs have the same V,,,/K,,,. As sinusoids collapse with decreasing flow values of V&/F(“) ; it provides, in general, a measrates (increasing l/F), V,,,,, is reduced, the slope ure of organ heterogeneity in the context of elimV,,,,/K, falls, and ln(cJc,) falls further below the ination. Equation 13 shows that .s*is left unchanged straight line given by equation 11 at high flow rates. by any change in the organ flow rate F which is Brauer et al. [16] discussed such plots for the uptake associated with pro ortionate changes in suborgan of colloidal CrP04 by Kupffer cells of the rat liver flow rates fi) (fl) PF = const). and used the deviation of data from equation 11 to For a slightly heterogeneous liver (2 Q 1) elimquantify the fraction of sinusoids open at each flow inating substrate steadily by first-order kinetics in a rate. Similar concave plots have since been obtained single-pass system, equation 11 is now replaced by for other substrates, most recently for the elimination [251 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML of taurocholate by rat liver [21]. However, the concave deviation from equation 11 is commonly found ln(Ci/~o)=~-~&2[~]2, (14) from data at flow rates at which de-recruitment does m E* = 1428 L. BASSand S. KEIDING neglecting higher-order terms in a2 estimated by a this second assumption (a = 0 holding up to the remainder term [25]. For any given zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB V,,, F, ci and hepatocyte surface) is incredible, especially when E, the E0predicted by equation 14 is greater than the uptake is avid. If, nevertheless, it is adopted, the c, predicted by equation 11: that is a particular effect of protein-binding on steady elimination is realization of the general relation 12. As F is reduced simple. The left-hand side of equation 2 is (l/F increased), the second term in equation 14 unchanged, but on the right-hand side we must replace c by fcqc which is available for uptake: feqc/ produces the concave deviation from the linear relation between ln(ci/cO) and l/F predicted by (f&c + K,,,) = c/(c + K,,,/f, ). Consequently, throughout the foregoin mode ‘i equations, K,,, is to be equation 11. The distributed model thus accounts quantitatively [22,25] for the aforementioned obserreplaced with K,,,ff&. In particular, using equation 7 vations which are at variance with the undistributed so modified to express the extraction fraction E = model in the absence of de-recruitment, and in doing 1 - CJ Ci, w e obtain so it yields values of .sz characterizing the heteroV -V E = 1 - e-f, max geneity of the sinusoidal bed. While the values $ = (17) FK, 0.12 [22] and 0.14 [25] were readily detected in this Here K,,, is the Michaelis constant pertaining. to way, values of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Ed below 0.18 could not have been detected (if present) in experiments with rat livers unbound substrate: the enzyme is assumed not to interact with bound substrate in the hepatocyte, subjected to flow-changes in a re-circulating system while unbound substrate is assumed to be equi[19,27]. This interesting methodological difference extends the usefulness of the undistributed model in librated across the hepatocyte membrane. We will now focus on the difficulties of the traditional pharrecirculating systems such as the intact body. macokinetic approach by formulating two characteristic paradoxes. Paradoxical effects ofplasma albumins on elimination (A) The low- extraction paradox. At very low What hindrance to elimination arises from the extractions, arterial-venous concentration differbinding of a fraction of substrate to plasma proteins ences are small, and all models of organ elimination (especially albumins) in the perfusate? This question, fundamental for the delivery of some substrates and become the same. This is reflected in retaining only drugs to perfused organs, has another aspect. If the linear term in the expansion of the exponential in equation 17, with the result the answer were known quantitatively, experimental variations in the bound fraction could be used to test the validity of models of elimination by intact organs [28,29]. In an element of perfusate, the total concentration c of substrate consists of concentrations c, of unbound and cb of protein-bound substrate. For simplicity we consider total protein concentrations p such that the protein is so far from saturation by the substrate (p + cb) that p - cb can be replaced with p. Then the difference between local rates of formation and of decomposition of the protein-substrate complex is where we set V = I for steady uptake in a recirculating system. From equation 1 with V = I, we have the clearance I/ci = FE of the total substrate and the clearance Z/(fegci) = FE/feq of the unbound substrate (called “unbound clearance”). Note that our use of the concept of clearance is not confined to first-order kinetics. From equation 18 we see that the unbound clearance, (V,,,,, - Z)/K,,,, is independent of feq and hence of the protein concentration p (equation 16). However, the calculated unbound c, +cb =c (15) clearance of prazosin by perfused rat liver is o=k,,,pcu -koficbr where ken and kOfl are the relevant absolute rate increased by some 69% by adding to the perfusate 14 g/l albumin, and is reduced by 51% by adding constants, determining the equilibrium dissociation 0.85g/l a-l-acid glycoprotein [30]. The unbound constant Kd = k,H/k,. When the protein-substrate interaction is not per- clearance of a variety of substrates by hepatocyte turbed by cellular uptake, as in large vessels or in a suspensions or monolayers is increased substantially by adding albumin to the perfusate (see Ref. 31). As dialysis apparatus, we have (J= 0. Then equations 15 yield the equilibrium unbound fraction c,/c = fes the architecture of the liver is immaterial in all these situations, the low extraction paradox shows that this of substrate: difficulty of the traditional pharmacokinetic view lies feq = (1 + p/K&l 9 (16) at the level of single hepatocytes interacting with which is independent of substrate concentration, and perfusate. (B) The high- extraction paradox. In order to conwhich can be varied at will by varying p [28,29]. The traditional pharmacokinetic approach [12, sider high extraction fractions, we now turn to first28,291 makes two separate assumptions. First, it is order kinetics (V Q V,,,). Writing f, V,,,,/K,,, = (CL)i,, (“intrinsic clearance” of total substrate), assumed that only unbound substrate is involved in uptake leading to elimination (steady uptake of equation 17 becomes proteins is negligible in the present context). Second, E = 1 - e-(CLhntIF. (19) it is assumed that when the accessible surface of a hepatocyte is presented with the total concentration Accordingly, (CL),Jf& = Vmax/K,,, is the unbound c of substrate, then it is presented with the con- intrinsic clearance. For bromosulfophthalein (BSP) centration fesc of unbound substrate, with feq given in a perfusate containing the physiological conby equation 16. As only unbound substrate enters centration 5Og/l albumin, Baker and Bradley [32] the hepatocyte according to the first assumption, measured the very small unbound fraction, feq = 1429 Models and experiments in hepatic pharmacology 7.7 x 10d5, and asked, therefore, how observed extraction fractions of BSP of 0.91 in humans and 0.50 in dogs are possible. To devise the highest possible value of (CL)i,t, they assumed that the elimination rate was not limited by enzymatic elimination but by the diffusion of unbound BSP (with diffusion coefficient D,) across the combined thickness S of the perisinusoidal space and endothelial cells, into the accessible area A of hepatocytes (which sequestered all BSP on arrival): (CL)., = feq D,AIS* (20) With realistic values favourable to rapid uptake (in particular, 6 = 10e4 cm), equation 20 yielded at most E = 0.16. Hence the earliest suggestion [32] that dissociation of the BSP-albumin complex is somehow facilitated by the hepatocyte surface: that is, that the use of fcs in a perfusate compartment extending up to the hepatocyte surface is inappropriate in non-equilibrium conditions. Non-compartmental approach to the albumin paradoxes In avid uptake of unbound substrate, dissociation of substrate-protein complexes replenishes the depleted unbound substrate by mass-action. The rate of this replenishment varies continuously with the distance from the hepatocyte and cannot be calculated in terms of compartments. The key to understanding facilitation of the unbound clearance by albumin is in the identification of a new, albumindependent length [31] (denoted by l/J. in what follows) which characterises the thickness of the nonequilibrium layer (within which a# 0 in equation 15) adhering to the hepatocyte surface exposed to perfusate. That length is to be compared with the thickness 6 of the familiar unstirred layer adhering to the hepatocyte surface, as for example in [32], within which all transport is by diffusion only. If x is the perpendicular distance from the hepatocyte surface, then the steady transport equations in the unstirred layer (0 d x 6 S) are -D+= d2c -a,-Db-=o d2cb aY (21) where Db is the diffusion coefficient of the substratealbumin complex. The sink of c, and source of cb in equations 21 are agiven by equation 15. Multiplying the first of equations 21 by k,,,p/D,, the second by koff/Db and subtracting, we find d2a -g-A7=0, with A2 = &p/D, + koR/Db. (23) The solutions of equation 22 are exp(?Ar) but as u= 0 far from the hepatocyte, only the solution exp(-Ax) is of interest: l/n is the thickness of the non-equilibrium layer separating the hepatocyte from the equilibrated solution characterized by u = 0. The thickness l/A falls as p is increased (equation 23). If ci < l/L, stirring abolishes the nonequilibrium layer; but if 6 > l/n, it is diffusion across the non-equilibrium layer that limits avid uptake of the unbound substrate. Calculation shows that the intrinsic clearance of equation 20 must be replaced with [31] (Clint =fcq DuA A tanh(aa)* (24) For j16 4 1 (tanh(A6) = as), equation 20 is recovered, but for A6 > 2 we find tanh(l6) = 1 and equation 24 is reduced to (CL)i,, = IfesD,,A/(l/L): the fixed denominator 6 in equation 20 is replaced by the smaller l/J which is reduced asp increases (equation 23). The unbound intrinsic clearance (CL)Jfq is no longer albumin-independent but increases with p as A does. Using the best (though still provisional) values of BSP parameters appearing in equation 23 from published literature (see Ref. 31), one finds at p = SOgIl of albumin [32] the estimate l/2, = 0.38 x 10e4cm. Even if 6 were only 10-4cm, we haveld = 2.63, tanh()c@ = 0.99. The use in equation 19 of (CL)i,, from equation 24 (in place of (CL)i”, from equation 20) yields E = 0.39 (in place of 0.16). Considering the uncertainty of the parameter estimates, especially of k,,* for BSP [31], this closer approach to observed values of E gives an encouraging illustration of the role of the non-equilibrium layer. More detailed quantitative applications of this concept have been made [31] to observed enhancements by albumin of hepatic clearances of unbound oleate, palmitate and BSP. However, the reduction of the hepatic clearance of unbound prazosin in rat liver by cu-1-glycoprotein [30] cannot be explained by effects of the non-equilibrium layer at hepatocyte surfaces. It is to be noted that, whereas albumin enhances the unbound intrinsic clearance of albumin-bound substrates, it reduces their total intrinsic clearance (CL)i,l: asp is increased, the fall inf,, outweighs the rise in (CL)int/fes. Furthermore, the effects of the non-equilibrium layer are reduced, and the approximate validity of the traditional pharmacokinetic approach is restored, when the rate-determining step in elimination is the metabolism in the interior of the hepatocytes (rather than transport into hepatocytes). In the general case, both metabolism and transport influence the elimination rate. It is therefore inappropriate, in general, to test organ models of hepatic elimination by analysing observed effects of varying zyxwvutsrqpon f according to the traditional phannacokinetic &umptions [28,29,31]. Interpretations of time-dependent elimination Interpretations of elimination experiments with time-dependent concentrations require major extensions of the foregoing modelling. Consider an arterially injected bolus of substrate, fractionated by the vascular bed of the organ into elements which pass through the eliminating capillaries (sinusoids) and are re-united in the vein. There the mixed output concentration is seen with some time-dependence E,(t), which is the flow-weighted mean of the outputs c,(t) from individual vascular pathways before mixing. The observed form of E,(t), always more dispersed than the input Ci(t), is due partly to the variety of transit times of the bolus fractions through the set of parallel vascular pathways and partly to L. BASSand S. KEIDING 1430 elimination along some part of each vascular The forms of E(T) in equations 26 and 27 can pathway. describe various observed results of the multiple If a sinusoidal transit time during which elimindicator diffusion (single injection) method [4], but ination takes place (sometimes called contact time) existing experimental results cannot conversely is T, then zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA v,,Jf = a-r, where (Y= urnax/ is the determine the choice of tp(zjT), nor even give prefmaximum elimination rate per unit blood volume zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE ft erence to either the deterministic or probabilistic of the sinusoid perfused at the rate f of flow. If class of models. the density of relevant hepatocellular enzymes were Whereas interpretations of elimination transients uniform along each sinusoid, and if it were the same in real organs are burdened with the necessity to for each sinusoid, then @would be the same constant choose between hypothetical forms of 1+9(tfT),steady along all sinusoids for a given substrate. If that is elimination is unaffected by the extrasinusoidal tranassumed, then the distribution of v&f over the sit times T’ and hence by the form of q(dT). This is sinusoids is the same as the distribution of r. If the because equations 4 and 14 are independent of the distribution of r were known, the relation between form of the distribution p(x) of the eliminating the observed time-courses ci(f), c,,(t) could be interenzyme (equations 2 and 3): we can set p(x) = 0 preted quantitatively for each substrate in an organ along any non-eliminating part of any vascular pathrepresented by a set of sinusoids in parallel, each of way without effect on the relations between the which eliminates substrate according to the undissteady observable concentrations ci and c, or S,. We tributed perfusion model. note here another methodological trade-off: steadyUnfortunately, the distribution of r is not state experiments yield less information than tranobserved. The instruments sampling cl and E,, are sients (for example, on enzyme distributions along inevitably placed well upstream and downstream of the blood flow) but require fewer speculative the sinusoids, with arterioles, venules and the other assumptions. non-eliminating vascular domains included between We have assumed so far that transport of indicators them. When labelled inert indicators, suitably and substrates along the blood flow in capillaries and matched to the substrate [4], are used to determine sinusoids is predominantly convective [2-4,23, the distributions of transit times T between the 33,341: bolus fractions do not disperse appreciably instruments, each T includes substantial times T’ of while travelling through a capillary, so that the passage through regions in which substrate is not dispersion of E,(t) seen in the vein is due to distrieliminated. For any value of T, one must expect butions of transit times of the bolus fractions. This distributions of the times T’ and r(such that T’ + t = picture is supported directly by work with single T). Given an observed distribution of transit times capillaries [2,23]. In the intact liver it has been shown T, elimination depends therefore also on the con[35] that, after plausible corrections for volumes of ditional probability density &rjT), defined as fol- distribution, a variety of substances injected arterially in the same bolus have the same venous outflows lows [33]: given a transit time T between sampling E,,(t) as labelled red cells, despite having diffusion instruments, I@ is the probability that T includes a time between t and t + dt spent in eliminating coefficients differing by many orders of magnitude. For these reasons the modelling of substrate diffusion sinusoids. We note that, because of zones of metaalong blood flow in capillary beds [36] has not been bolic activity in the liver [9], the same vascular pathway may have different values of r for different influential: “the intravascular pathways traversed by tracer molecules vary enough to give practically all substrates. In the simplest case of a bolus input eliminated (without back-diffusion) by first-order the dispersion (of E,(t)) observed” [37]. zyxwvutsrqponmlkjihgfe Interest in diffusion-like dispersion of substrates kinetics, the extraction fraction E(T) at time T is along the flow through hepatic sinusoidal beds has given by [33] been revived recently by Roberts and Rowland [38]. T They envisage the molecular diffusion coefficient E(T) = 1 - 1 e-“q(tfT)dt. (25) [36,37] replaced by a much larger dispersion coef0 ficient representing effects of microscopic convective interconnections This cannot be evaluated without models of q(4 T), eddies related to intrahepatic between sinusoids. This dispersion coefficient is priwhich fall into two broad classes: deterministic and probabilistic. Deterministic models assume that to marily a property of the anatomy of the intact sinusoidal bed and of the dynamics of its perfusion and each value of T there corresponds some unique value depends only slightly on the molecular diffusion coefof t. An influential example [34] of a deterministic model is obtained by assuming that I$ = 0 unless t = ficients of indicators and substrates. In particular, a + bT, where the positive constants a, b depend on the dispersion coefficient must be expected to depend the state of vasodilation or vasoconstriction of the on the total hepatic blood flow in some essential but hitherto unspecified way [38]. Current attempts to vascular bed. In that case, equation 25 yields model this flow-dependence [22] are confined to E(T) = 1 - e-No+br). (26) limiting situations which do not cover the range of the strategic flow-change experiments discussed in The simplest example [33] of the class of probabilistic models is to assume that, at any T, all values of t the first part of this commentary. Interpretations of these experiments in terms of the revived dispersion between zero and Tare equiprobable. This quantimodel [38] must, therefore, await its completion. tative expression of our ignorance of the detailed situation yields, from equation 25, E(T) = 1 - (1 - PT)/cuT. (27) Conclusion Hepatic physiology and clinical pharmacokinetics 1431 Models and experiments in hepatic pharmacology have now both developed so far that the traditional gap between these disciplines should be bridged. This requires that pharmacokinetic models should be physiologically reasonable and, on the other hand, that physiologic models should be clinically useful. 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