## Abstract

Hepatic clearance of long-chain fatty acids is substantially faster in females than in males, a fact that may underlie known gender-related differences in lipoprotein metabolism and associated disease states. To further investigate the transport steps responsible for this difference, we used a novel method combining multiple-indicator dilution and steady-state measurements of palmitate extraction from albumin solutions. We found that cytoplasmic transport of palmitate is sufficiently slow (diffusion constants 9.0 and 5.9 × 10^{−9}cm^{2}/s for male and female liver, respectively) that the steady-state concentration of palmitate in the center of the cell should be ∼0.5 of that found in the cytoplasm just beneath the plasma membrane. Previous studies in cultured liver cells using nonphysiological fatty acids have shown more rapid cytoplasmic transport in females. This sex difference reflects higher concentrations of cytosolic fatty acid-binding protein, which acts as a carrier system to transport fatty acids across cell water layers. The current study confirmed slow cytoplasmic diffusion rates in intact perfused rat liver using a physiological fatty acid and found a similar female-to-male ratio. Female liver also had a greater influx rate constant and a larger vascular volume than male liver but had a similar rate of metabolism. Rapid cytoplasmic diffusion enhances movement of palmitate into deeper layers of the cell cytoplasm, thus reducing efflux. The larger sinusoidal volume in females not only permits more dissociation of palmitate from albumin within the sinusoids but also may generate a greater permeability-surface area product. These multiple sex-related differences combine to produce a nearly twofold greater steady-state uptake rate by female liver.

- cytoplasmic transport
- diffusion gradients
- fatty acid-binding proteins
- long-chain fatty acids
- sex factors
- carrier proteins
- membrane transport
- multiple-indicator dilution
- hepatic uptake
- rate constants
- mathematical models
- kinetics

hepatic uptake of long-chain fatty acids from plasma is nearly two times as fast for females as for males in both humans (11,24) and other mammalian species (12, 22, 25). Although this sex difference has been known for many years, its molecular basis is incompletely understood (2, 4, 9, 25, 29, 30, 32). Prior work has demonstrated gender-related differences in several factors related to fatty acid transport, both in isolated hepatocytes and whole livers. Sorrentino and co-workers (30) found a similar density of plasma membrane fatty acid carriers in males and females but found that the carriers from females had a higher affinity for oleate. Similarly, Noy and co-workers (20, 42) found greater binding of palmitate to plasma membranes isolated from female rat livers. These studies focused on the plasma membrane because it was assumed to be the principal barrier to the overall removal process. However, potentially rate-limiting steps in fatty acid removal from plasma also include dissociation from albumin in the extracellular space, diffusion across the extracellular unstirred plasma layer, diffusion through the cytoplasm, and metabolism (33, 36). Sex differences in the rates of these uptake steps could also contribute to the observed sex differences in fatty acid removal rates.

Ockner and co-workers (21) found that the cytosolic concentration of fatty acid-binding protein (FABP) is approximately two times larger in female than in male liver (21) and that the overall rate of hepatic fatty acid utilization correlated with the level of this binding protein (22). However, these investigators were unable to determine whether the higher concentration of FABP was the proximate cause or simply a result of the higher fatty acid flux in females.

We have shown that diffusion of a fluorescent fatty acid analog through liver cell cytoplasm is quite slow, such that the average time required for a fatty acid molecule to cross the liver cell is several minutes (19). Female hepatocytes had substantially greater intracellular transport rates than male cells (15, 19). Finally, the cytoplasmic transport of fatty acids has been shown to be carrier mediated; FABP and other cytosolic binding proteins catalyze transport through the cytoplasm with all the properties associated with a true carrier system (35). These results suggest that cytoplasmic transport could be an important factor determining the overall rate of hepatic utilization of fatty acids.

Extrapolation of these results to intact liver is limited by the unknown effect of the fluorescent side chain on disposition of the fatty acid probe and by the fact that single cells were studied in the absence of a steady uptake flux. Thus intracellular transport must be investigated using physiological fatty acids and intact livers. The current study was designed not only to fill that need but also to assess the relative importance of cytoplasmic transport in determining the observed steady uptake flux of fatty acids by the liver.

To achieve these ends, we used an extension of the traditional multiple-indicator dilution (MID) model to estimate the apparent influx, efflux, and cytoplasmic diffusion constants for palmitate in perfused livers from female and male rats (18). This “diffusion MID model” explicitly incorporates cytoplasmic diffusion and has been successfully used to study the intracellular transport of thyroid hormone (17).

Our results represent the first measurement of intracellular transport rates for native fatty acids in whole liver, a parameter that is not directly observable. The slow rate of cytoplasmic transport reported in this study suggests that cytoplasmic transport represents a substantial barrier in the uptake and metabolism of amphipathic molecules, a class that includes not only long-chain fatty acids but also bilirubin and hydrophobic bile acids.

## MATERIALS AND METHODS

*Sources of materials*. Bovine serum albumin (BSA, essentially fatty acid free, product no. A3782) was obtained from Sigma (St. Louis, MO). The fluorocarbon emulsion (Oxypherol FC-43) was obtained from Alpha Therapeutics (Los Angeles, CA). [^{3}H]palmitate, specific activity 57 mCi/mmol, and [^{14}C]sucrose were purchased from New England Nuclear (Boston, MA). All other reagents were of the highest grade commercially available.

*Preparation of tracer doses*. For each dose, tracer [^{14}C]sucrose (8 × 10^{6} counts/min) and 1 μCi [^{3}H]palmitate were added to 100 μl of perfusion media containing 2 g/dl (2%) BSA. The resulting mixture was equilibrated with 95% O_{2}-5% CO_{2}, drawn into a 1-ml tuberculin syringe, and maintained at 37°C until used.

*Liver perfusion*. Livers were isolated as previously described (17, 37). Briefly, livers from 55- to 65-day-old male (*n* = 5) or female (*n* = 7) Sprague-Dawley rats were removed and perfused via the portal vein with recirculating Oxypherol solution at a flow rate of 25–30 ml/min. After a 30-min equilibration period, the system was changed to single-pass perfusion with protein-free buffer for 5 min. The perfusion rate was then decreased to 20 ml/min, and a custom-made, high-speed fraction collector was positioned under the effluent tube.

Three studies were performed on each liver. Starting 60 s before collection of data, 2 g/dl BSA were included in the perfusate. The radioactive dose containing [^{14}C]sucrose and [^{3}H]palmitate in buffer otherwise identical to the perfusate was then injected into the portal vein catheter (defined as *time 0*). Effluent samples from the hepatic vein cannula were then collected automatically for 90 s at increasing intervals of 1–4 s. After a further 10 min of perfusion with protein-free perfusion buffer, the amount of radioactivity in the effluent was indistinguishable from background. A second indicator dilution curve was then obtained using an identical technique. Ten minutes after the final sample of the second indicator dilution curve, the inflow to the liver was changed to perfusate containing 2 g/dl BSA with tracer levels of [^{14}C]sucrose and [^{3}H]palmitate. Effluent samples (1 ml) were obtained at 30-s intervals from 0 to 240 s for determination of the steady-state extraction. Concentrations of [^{14}C]sucrose and [^{3}H]palmitate were expressed as a fraction of those present in the inflow perfusate.

*Sample processing*. Effluent samples from indicator dilution studies (0.3–1.3 ml each) and aliquots from the steady-state perfusions (0.1 ml each) were mixed with 10 ml scintillant (Redi-Solv; Beckman) and counted for^{3}H and^{14}C activity using a dual channel liquid scintillation counter. Channels were optimized for separation of^{14}C and^{3}H. Standards of^{14}C and^{3}H prepared using the same perfusion media were included to correct for counting efficiency and cross-channel spill.

*Estimation of metabolism rate*. The rate constants for metabolism of [^{3}H]palmitate by male and female liver cells were determined separately using suspended hepatocytes isolated by collagenase perfusion (19). Freshly isolated cells were suspended in Krebs buffer (1–3 × 10^{6} cells/ml, 10 ml, viability >90% by Trypan blue exclusion, 37°C, pH 7.4), and a trace amount of [^{3}H]palmitate (∼10^{6} disintegrations/min) was added at *time 0*. Preliminary studies indicated that uptake by the cells was virtually complete within the first 10 s. Samples (1 ml) of the suspension were then removed at intervals for 180 s. Cells were captured by filtration through a 1-μm Millipore filter and washed with ice-cold stop solution [100 mM NaCl, 100 mM sucrose, and 10 mM*N*-2-hydroxyethylpiperazine-*N*′-2-ethanesulfonic acid-KOH, pH 7.5]. The fraction of unmetabolized [^{3}H]palmitate remaining in the cells at each time point was determined by thin-layer chromatography of the total lipid extract as previously described (6). Less than 5% of the total radioactivity was found in the filtrate, which was not further analyzed. The metabolism rate constant was estimated from a plot of the cell-associated unmetabolized palmitate versus time.

*Data analysis*. The outflow concentrations of [^{14}C]sucrose and [^{3}H]palmitate were expressed as the fraction of the amount of tracer in the injected bolus appearing per second. The mean transit time for [^{14}C]sucrose was determined after extrapolating the terminal portion of its outflow curve to infinite time, assuming a monoexponential decay. In all cases, the area under the extrapolated portion of the reference curve contributed <4% of the total. Delay due to the catheter and nonexchanging vessels was estimated by extrapolating the ratio of the sucrose and palmitate curves to *time 0*as previously described (17, 37). The sinusoidal volume was calculated as the product of the flow rate and the mean transit time for the reference curve, after subtracting the delay volume of the catheters.

Outflow curves for each study were analyzed using an extension of the traditional MID model known as the diffusion MID model (17, 18). In addition to the usual rate constants for influx, efflux, and metabolism, this model (Fig. 1) includes the characteristic time required for palmitate to equilibrate across the cytoplasm, a measure of cytoplasmic transport. We have previously shown that this model can accurately recover the cytoplasmic diffusion constant from both simulated and experimental data (17, 18). Where possible, these rate constants are defined using permeability-surface area (*PS*) products*PS*
_{1} and*PS*
_{2}, which relate the unidirectional fluxes to their respective driving concentrations rather than masses. When normalized per cubic centimeter of liver volume, these *PS* products are clearances (units of cm^{3} ⋅ g^{−1} ⋅ s^{−1}), thus allowing direct comparison of the values for different steps.

Influx is related to the total palmitate concentration in the plasma immediately adjacent to each liver cell. Any effects due to the unstirred plasma layer or binding (including slow dissociation of palmitate from albumin) should be constant at constant albumin concentration (36) and are thus incorporated into the apparent influx constant. The efflux step is driven by the total cytoplasmic concentration just inside the plasma membrane. Movement of solute from this cytoplasm to deeper portions of the cytoplasm is assumed to occur solely by diffusion. The rapidity of the diffusional process is governed by a diffusional relaxation time,*t*
_{diff}, which combines the diffusion constant and the diffusional distance (17, 18). As with the traditional MID model, irreversible removal of solute (i.e., metabolism) is assumed to occur equally at all locations within the cell cytoplasm.

#### Fitting of [^{14}C]sucrose outflow curves.

[^{14}C]sucrose was used as a nontransported reference compound. Before the reference curve can be used, it must be converted to a continuous function. After the large vessel transit time (7, 17) was subtracted, the [^{14}C]sucrose output curve was therefore approximated as a sum of eight exponentials by curve fitting using the method of moments with exponential depression (17, 27, 28). The average coefficient of determination for this fit was >0.999. From this, we concluded that a sum of eight complex exponentials adequately described the reference outflow curves. Increasing the number of exponential terms to 10 or 12 did not further improve the fit, as the amplitudes of the additional terms were uniformly set to zero by the fitting algorithm.

*Data analysis*. To provide an additional constraint on the model, we also measured the steady-state extraction, determined as the average extraction fraction measured at 30-s intervals between 120 and 240 s. The specifics of the diffusion MID model and the relationship between the rate constants and the steady-state extraction fraction are contained in the
. The fitting algorithm then iteratively adjusted values of the unknown parameters to minimize the sum of squared residuals for both the transient outflow curve and steady-state data simultaneously. Studies in which the two different MID injections differed significantly were considered unstable preparations and were discarded before analysis.

Our fitting regime employs a derivative-free algorithm to minimize the sum of residuals using inverse variance weighting (26). Convergence to the best fit usually occurred in four to five iterations. The fitting algorithm returns estimates of the individual parameters and their 95% confidence intervals for each experiment. Goodness of fit is reported as the coefficient of determination. Differences in values between individual parameters were determined using the Student’s*t*-test, with*P* < 0.05 being judged as significant.

## RESULTS

*MID outflow curves*. Typical outflow curves for [^{14}C]sucrose and [^{3}H]palmitate are shown in Fig. 2. Delay due to the catheters and nonexchanging vessels averaged 1.1 s and was a small fraction of the mean sucrose transit time for the whole system (typically 7–8 s). Female livers had 37% larger volumes accessible to sucrose compared with males [0.26 ± 0.01 vs. 0.19 ± 0.01 (SE) ml/g, *n* = 5–7,*P* < 0.001].

*Steady-state extraction*. Extraction of [^{3}H]palmitate reached a steady value within 120 s after perfusion of a steady concentration of [^{3}H]palmitate and remained constant during the 120-s observation period (Fig.3). Consistent with previous observations, [^{14}C]sucrose was not measurably extracted (17). Female livers removed [^{3}H]palmitate from the 2 g/dl albumin solution more efficiently than male livers (extraction fraction 0.28 ± 0.03 vs. 0.17 ± 0.03,*P* < 0.05) despite comparable flow rates (both 0.17 ± 0.03 ml ⋅ g^{−1} ⋅ min^{−1}).

*Metabolism rate constants*. The amount of cytoplasmic palmitate remaining unmetabolized versus time is shown in Fig. 4. No significant differences were seen between male and female hepatocytes. The calculated rate constant for metabolic removal was the same for male and female livers (0.014 ± 0.001 vs. 0.013 ± 0.001 s^{−1},*P* > 0.5).

*Estimation of transport rate constants*. We analyzed each paired set of outflow curves and steady-state extractions two times. In the first analysis, we allowed the fitting algorithm to adjust four unknown parameters (the rate constants for influx, efflux, and metabolism plus the*t*
_{diff}). In the second case, we assigned an experimentally determined value to the metabolism rate constant and fitted the remaining three parameters to the data.

Each of the 24 fitting procedures converged within eight iterations. The average coefficient of determination for these fits as a whole was 0.992 (range 0.986–0.999). If a four-parameter fit was attempted (metabolism constant not fixed), the fitting algorithm converged; however, the estimate for the metabolism rate constant had a large coefficient of variation (typically >100%). Moreover, estimates for this rate constant (only) were dependent on the initial parameter values. This result is consistent with the relatively limited information about the removal process in the indicator dilution curve (16, 18, 28). These fitting difficulties led us to assign the value of the metabolism constant as a known parameter determined separately. The resulting three-parameter fit converged quickly to parameter values that were independent of the initial estimates and had small coefficients of variation (mean values for influx 15%; efflux 22%;*t*
_{diff} 55%;*n* = 24). The mean values ± SE of the influx and efflux rate constants (s^{−1}) and*t*
_{diff} (in seconds) for [^{3}H]palmitate are shown in Table 1. A typical fitted [^{3}H]palmitate curve is shown in Fig. 2.

*Apparent cytoplasmic diffusion coefficient*. The average*t*
_{diff} was 111 ± 38 and 170 ± 70 s in female and male livers, respectively (mean ± SE). As detailed in the
, this relaxation time is inversely related to the apparent diffusion constant, *D*
_{eff}. This relationship is governed by the equation*t*
_{diff} =*L*
^{2}/*D*
_{eff}, where *L* is the distance over which the diffusional relaxation occurs. If we assume that the hepatocyte plate within the liver acinus is ∼20 mm across and use one-half of this distance as the diffusional path length (because the plate is diffused on both sides), then the corresponding diffusion constant values for female and male cells are 9.0 and 5.9 × 10^{−9}cm^{2}/s, respectively. Although this difference was not statistically significant in the current study, the male-to-female ratio of 1.53 is not statistically different from the value of 1.65 reported earlier for cytoplasmic diffusion of a fluorescent fatty acid in male and female hepatocytes (19). In this earlier study, the sex difference was statistically significant at the*P* < 0.05 level (19).

## DISCUSSION

This study demonstrates that transport of palmitate (a typical amphipathic compound) within the cytoplasm of liver cells is not instantaneous as assumed by most current transport models. Instead, palmitate equilibrated across the liver cell with a characteristic relaxation time of 2.7–4.3 min. If we assume a mean equilibration distance of 10 mm (approximately one-half the thickness of the liver cell plate), then the calculated cytoplasmic diffusion constant is 9.0 and 5.9 × 10^{−9}cm^{2}/s for female and male liver, respectively. Although cytoplasmic diffusion of low molecular weight molecules is usually very rapid (13, 14), it is much slower for amphipathic and hydrophobic molecules due to extensive binding of these molecules to cytoplasmic membranes (15, 19, 35, 36). Cytoplasmic diffusion of palmitate is somewhat faster than values reported in our previous study (5.0 and 3.1 × 10^{−9}cm^{2}/s, respectively) in which we used fluorescence repolarization after photobleaching to measure the diffusion constant of*N*-methyl-7-nitrobenz-2-oxa-1,3-diazol-amino stearate (NBD stearate) in cultured rat liver cells (19). The slower diffusion rates for NBD stearate may reflect its greater binding to cell membranes compared with the less hydrophobic palmitate. The ratio of female to male cytoplasmic diffusion constants for the two experimental systems was not different.

The more rapid relaxation of cytoplasmic diffusion gradients in females could also be explained if female liver cells had a shorter diffusional path. However, we have previously failed to find any difference in the size of hepatocytes from male and female rats, suggesting that the diffusional path should be similar (25). This possibility also seems unlikely in view of our previous finding using direct methods that cytoplasmic transport of NBD stearate is more rapid in female rats (19).

Our data suggest that cytoplasmic diffusion mediated by cytosolic FABP could help to regulate the rate of fatty acid utilization by the liver. This hypothesis is supported by three arguments. First, the rate of cytoplasmic fatty acid transport has been shown to be regulated by the concentration of cytosolic FABP, which serves as a formal carrier system to transport poorly soluble fatty acids through cytosolic water (35). Second, the rate of fatty acid metabolism by intact livers correlates closely with the concentration of FABP in the cytosol (4). Finally, the rate of cytoplasmic transport appears to be slow enough to partly limit the overall rate of fatty acid utilization by the liver. Because cytoplasmic transport and metabolism occur simultaneously in cytoplasm, it is not possible to assess the degree to which cytoplasmic transport limits the uptake rate by simply comparing the rate constants of the two steps. New methods will be needed to properly address this question.

Table 2 lists experimental values for the rates of different steps in the hepatic uptake and metabolism of palmitate under physiological conditions. To permit direct comparison among the different steps, all rate constants are expressed as clearances (see Table 2 for details). Surprisingly, all rate constants turn out to be quite similar, clustering around 0.01 cm^{3} ⋅ g^{−1} ⋅ s^{−1}(range 0.005–0.036). Because no single step is markedly slower than all others, we conclude that no single uptake step determines the rate of fatty acid utilization. Instead, all steps contribute to determining this rate. We are aware of no other examples of such a “balanced” transport pathway in the biological literature. The presence of a balanced pathway suggests that nature has favored the most rapid possible clearance pathway for fatty acids and related amphipathic metabolites and toxins. If a high hepatic first-pass clearance of fatty acids and amphipathic toxins provides a survival advantage, then evolutionary pressures might be expected to increase the rate of the slowest step in the pathway until all have approximately equal rates.

The extracellular volume accessible to sucrose was 37% larger in female livers. The rate of dissociation of fatty acids from albumin within the liver is proportional to this volume. This difference maintains the balance of the pathway by providing a greater rate of dissociation to match the greater rates of the other transport steps in female liver. The larger sinusoids may also provide a greater plasma membrane surface area for uptake. If we assume that both female and male sinusoids have a similar geometry, the 37% larger volume corresponds to a 23% increase in the surface area available for influx. On the other hand, if male sinusoids are more flattened than female sinusoids, the larger sinusoidal volume in females could be associated with a lesser increase in surface area. Combined with the greater affinity of the membrane transporters for palmitate previously reported by Sorrentino and co-workers (30), the greater surface area could contribute to the 56% larger influx rate constant in female livers.

The sex difference in the influx rate constant reported in the current study (56%) is less than that found in previous studies of initial uptake of long-chain fatty acids by isolated hepatocytes (24, 30, 44). Thus, Sorrentino and co-workers (30) reported a 2.1-fold larger maximal velocity-to-Michaelis constant (*K*
_{m}) ratio for uptake of oleate by female rat hepatocytes, whereas Pond and co-workers (24, 25) reported twofold greater uptake of palmitate by female hepatocytes in both rats and humans. This suggests that factors other than plasma membrane permeability are contributing to the greater initial uptake reported in female hepatocyte suspensions (24, 25, 30,44). Differences in the geometry of the sinusoids cannot be responsible because the sinusoids are disrupted during liver suspension. As discussed in the next two paragraphs, we suggest that initial rate measurements may be affected to some degree by slow cytoplasmic transport.

This discrepancy might be resolved if initial rate measurements reflect not only membrane permeability but also cytoplasmic permeability. It is well known that diffusion barriers outside the plasma membrane can raise the apparent*K*
_{m} of a transport process well above its true value (1). In a like manner, diffusion barriers inside cells may also systematically affect measurements of membrane transport processes. We propose that initial rates of fatty acid uptake measured using suspended cells may not be true initial rates (defined as measurements obtained before efflux becomes important). Instead, they may reflect the presence of cytoplasmic concentration gradients that promote substantial efflux even at the earliest measured time points (35). The magnitude of such effects is unknown but is probably small for most transported substrates. The effect would be expected to be most pronounced for substrates with the slowest cytoplasmic diffusion constants. Further studies are needed to investigate this possibility.

The “linear” portion of the time-concentration curve for uptake of fatty acids by isolated hepatocytes typically begins within 5 s and lasts ∼30 s (31). The mean distance that a molecule can diffuse in a given time (*t*) may be calculated as
(see
). Assuming a cytoplasmic diffusion constant (*D*
_{eff}) of 6 × 10^{−9}cm^{2}/s (19), the average palmitate molecule will have penetrated only 1.6 μm into the liver cell after 5 s and <4 μm after 30 s. Because many more molecules will be near the inside surface of the plasma membrane, efflux could significantly reduce the uptake rate from the very beginning. The approximately twofold sex difference in initial uptake rates by isolated hepatocytes previously reported by others may thus reflect not only the 56% greater membrane permeability but also the 53% more rapid cytoplasmic transport of palmitate away from the plasma membrane. This explanation is consistent with the observations of Goresky and co-workers (7), who found increased efflux of palmitate after inhibiting intracellular binding sites using α-bromopalmitate.

*Efflux rate constants*. The sex ratio for the efflux rate constant was 1.64 (female/male), which is not statistically different from the ratio of the influx rate constants (1.56). This finding agrees with data of Sorrentino and co-workers (30), who found that oleate effluxed more rapidly from female hepatocytes, and suggests that the same transport mechanism is responsible for both influx and efflux.

*Formation of cytoplasmic concentration gradients*. When a diffusional process is rate limiting, diffusion gradients develop. Gradients predicted to exist at steady state within male and female liver cell cytoplasm are shown in Fig.5. These curves were generated by numerically simulating the uptake process for a single cell perfused on two opposing sides until the intracellular concentrations became constant, using methods previously described (35) and the model shown in Fig. 1. Details may be found in the legend of Fig. 1.

This analysis indicates that steady-state palmitate concentrations are approximately two times as large in the peripheral cytoplasm as in the center of the cell, suggesting that most metabolism of unesterified fatty acids occurs in the outer layers of liver cells.

Others have speculated that FABP might preferentially target fatty acids for either oxidation or esterification (9). If the enzymes for these metabolic pathways were to have different distributions within the liver cell, then cytoplasmic gradients would favor delivery of fatty acids to enzymes located in the peripheral cytoplasm. This effect would be more marked for male than for female cells because of the more pronounced gradients in male liver (Fig. 5). This mechanism could explain gender-related differences in fatty acid utilization by the liver and is further supported by the observation that higher levels of cytosolic FABP are associated with a shift in the utilization of fatty acids from oxidation to esterification (5, 21, 22, 41, 43).

*Limitations of the current approach*. The model used to analyze the data is based on a number of simplifying assumptions. We have assumed that diffusion is isotropic, i.e., diffusion occurs at the same rate in every direction and location within the cell cytoplasm. In our previous work using laser photobleaching to study the movement of NBD stearate in isolated hepatocytes, we could find no evidence to the contrary (19). We were also unable to detect alternate transport processes such as cytoplasmic streaming or convection and so have not included them in the current model. Hence, we feel that the assumption that transport occurs solely by isotropic diffusion is reasonable.

Our model also assumes that metabolism occurs equally at all locations within the cell. Enzymes that convert long-chain fatty acids to triglyceride esters or oxidative products are found throughout the cytoplasm (43). Although the precise distribution of these enzymes remains poorly characterized, their distribution turns out to have little effect on the diffusional equilibration time determined using our method. In preliminary studies, we analyzed the data using a previously published model in which metabolism was assumed to occur only at the center of the cell (17). Results were quite similar to those reported in Table 1, suggesting that the model is not highly sensitive to the distribution of metabolizing enzymes.

We have not modeled saturation of enzyme or FABP binding sites or nonequilibrium protein binding. By using only tracer amounts of palmitate, we have attempted to minimize these effects and assure first-order binding conditions. It is unclear from these studies or previous ones using simulated data how violation of these assumptions would effect the estimated*t*
_{diff} (18).

We had hoped to use the fitting algorithm to estimate the metabolism rate constant. However, the data set proved to contain insufficient information to identify all four unknown parameters, despite the fact that the model fit the data very well. Fortunately, independent measurement of the metabolism rate constant was possible by simply incubating suspended hepatocytes with labeled palmitate and assessing how rapidly the unmetabolized palmitate disappeared from the system. No albumin was present in the incubation mix, thus assuring rapid and irreversible uptake of the palmitate into the cells. Once inside the cell, metabolism began immediately and followed first-order kinetics (Fig. 4). Although fatty acids bound to plasma membrane may not be accessible for metabolism, plasma membranes make up <7% of the total membrane of the liver cell (3). In previous studies, we did not see enhanced fluorescence associated with the plasma membrane after loading cells with NBD stearate (19). Significant binding of fatty acids to the plasma membrane would lead to underestimation of the metabolism rate constant and, consequently, underestimation of the magnitude of cytoplasmic concentration gradients shown in Fig. 5.

Other models must certainly exist that could fit these data equally well. Nevertheless, our model appears appropriate to the questions asked. First, it is physiologically based, incorporating known structural and metabolic features of the liver. Second, it is based on a well-tested modification of the traditional MID method, which has been used successfully to study liver transport for more than 30 years. Third, the model fit the data very well, with only three unknown parameters. Fourth, the three unknown parameters appeared to have been adequately determined by the data. Finally, the estimated cytoplasmic diffusion constant and the male-to-female ratio of this constant for palmitate are similar to those obtained by more direct methods for NBD stearate, suggesting that both methods measure the same process.

*Implications for fatty acid transport*. Most prior studies of the uptake of amphipaths by liver cells have focused on transport across the plasma membrane. The emphasis on membrane transport has been justified by the traditional belief that this process is the crucial, rate-determining step in the overall uptake and utilization of amphipathic molecules and is further supported by a large body of data indicating that uptake in isolated cells is governed by saturable, presumably membrane-associated, transporters. We acknowledge and accept the importance of membrane transport in determining uptake rates and kinetics. Recently, however, it has become clear that a variety of other steps can contribute to limiting the fatty acid transport rate at steady state (34, 39). The current study adds cytoplasmic transport to the list of potentially rate-limiting steps. Full understanding of fatty acid transport requires understanding the roles of all rate-limiting steps in determining the overall transport rate.

## Acknowledgments

We acknowledge expert technical assistance by Wei-Lan Ma and Linda Kendrick.

## Appendix

This section outlines the origin of the model equations used. A more detailed presentation of the general approach used has been published (17, 18). Briefly, we use the technique of Laplace transformation to reduce the governing system of first-order linear partial differential equations to a single readily solvable ordinary differential equation for a single sinusoid. We then extend the single sinusoid solution to incorporate an arbitrary distribution of sinusoids, each identical except for differing sinusoidal transit times of the sucrose marker. Finally, we use the Laplace transform of the solution to derive expressions for the steady-state extraction fraction for the single and multiple sinusoid models. The following solution for the model used in these studies (Fig. 1) has not previously been published.

*Diffusion MID model equations*. As in the traditional MID model, we envision the liver as an array of identical vascular channels (sinusoids), each surrounded by a plate of liver cells (Fig. 1). Each sinusoid is identical except for the transit time, *T*. The distribution of vascular transit times is determined by a probability density function*Y*(*T*), which is defined experimentally by the reference outflow curve. Transfer of solute within a vascular channel is assumed to occur solely by convection at velocity *W*. We thus ignore diffusion in the axial direction in the vascular channel. This simplification is justified, since the rapid flow through the channel produces a convective flux that far exceeds the diffusional flux. The concentration of each injected solute within plasma flowing in the vascular channel,*u*(*x*,*t*), varies with axial position (*x*) but not in the direction perpendicular to the flow. Thus presentation of solute to the cell surface is determined solely by plasma flow. Here*t* denotes time since the bolus injection entered the sinusoid, and *x*runs from *x* = 0 at the inlet to*x* = R at the outlet. Transport of solute across the basolateral (sinusoidal) cell membrane separating plasma from cytoplasm is bidirectional. For influx, the unidirectional flux is given by*J*
_{1} =*P*
_{1}
*u*(*x*,*t*), where *P*
_{1} is the inward permeability of the membrane. In contrast to the plasma concentration, the concentration within the cytoplasm,*v*(*x*,*z*,*t*), depends both on the axial and radial positions. The radial location variable *z* runs from*z* = 0 at the canalicular membrane to*z* = *L*at the basolateral (sinusoidal) membrane. The unidirectional efflux per unit area from the cell to plasma,*J*
_{2}, is given by*J*
_{2} =*P*
_{2}
*v*(*x*,*z*=*L*,*t*) where*v*(*x*,*z*= L,*t*) is the concentration in the cytoplasm adjacent to the cell membrane. Transport of material within the cytoplasm is assumed to occur solely by diffusion with diffusion constant *D*. Metabolism is assumed to be irreversible with the flux given by*J*
_{3} =*P*
_{3}
*v*(*x*,*z*,*t*).

The system of partial differential equations governing the single sinusoid model is derived using conservation of mass. For the vascular channel we get
Equation 1where V_{u} and*A* are the volume and surface area of the vascular channel per unit length. The corresponding equation in the cell is given by the diffusion operator
Equation 2The form of ▿*v* depends on the geometry assumed. In the liver, the sheet of liver cells surrounding a sinusoid is approximately planar, and so we proceed here on that assumption. In this case, ▿*v* = (∂^{2}
*v*/∂*z*
^{2}).

*Equations 1
* and *
2
* are to be solved with the following initial and boundary conditions (determined experimentally). In the typical MID experiment, a bolus injection is given into the plasma inflow at *t* = 0, and both the plasma and cytoplasm are initially free of the solute of interest. Hence
Equation 3
Equation 4Two mixed-type boundary conditions describe the conservation of fluxes across the sinusoidal membrane (*z* =*L*) and canalicular membrane (*z* = 0). At*z* = *L*we have
Equation 5 and at *z* = 0 we have
Equation 6Following the steps outlined previously (17, 18), the Laplace transform of the solution to the diffusion model equation for a single sinusoid is
Equation 7
Equation 8 We define *k*
_{1} =*P*
_{1}
*A*/V_{u},*k*
_{2} =*P*
_{2}
*A*/V_{v},and*k*
_{3} =*P*
_{3}
*A*/V_{v}to be the rate constants for influx, efflux, and removal, respectively, and define a time constant for diffusional equilibration,*t*
_{diff} =*L*
^{2}/*D*. Because it is very difficult to measure the sinusoidal volume and area parameters, V_{u} and*A*, or the corresponding cell volume, V_{v}, we have formulated the solution (7) using rate constants *k*
_{1},*k*
_{2}, and*k*
_{3}. Here,*ū* is the Laplace transform of the concentration at the outflow of the sinusoid and *s* is the transform variable. In deriving *Eq. 7
*, we have assumed that the plasma concentration at the cell surface is equal to that at the center of the sinusoid. In formulating*k*
_{2} and*k*
_{3}, we have assumed a planar geometry so that V_{v} /*A* =*L*.

If the distribution of transit times of an extracellular marker (in our case [^{14}C]sucrose) is given by*Y*(*t*), then the Laplace transform of the whole organ solution,*c*(*t*), is given by
Equation 9 In*Eqs. 7
* and *
9
*, we have used
,
, and
to represent the Laplace transforms of *u*,*Y*, and*c*, respectively. In the case where*Y*(*t*) is a sum of *N* exponentials, we have
Equation 10
Equation 11 Combining*Eqs. 9
* and *
11
*yields
Equation 12 *Equation12
* was inverted numerically (23) to fit the palmitate outflow curves, having first set *N* = 8 and determined the*a _{i}
*and

*b*from the [

_{i}^{14}C]sucrose curve using the method of moments (27, 28).

Our choice of a sum of complex exponentials to describe the [^{14}C]sucrose curve is based on mathematical convenience, rather than on any physiological basis. As shown by *Eq. 11
*, this assumption yields a particularly simple expression for the Laplace transform of the palmitate outflow curve. We previously used the inverse normal distribution to characterize a theoretical [^{14}C]albumin curve (16). Although this function also is easily transformed, it does not present as much flexibility in fitting experimental outflow curves because it is characterized using only two parameters (a mean and a dispersion number). We therefore opted to use the sum of eight complex exponentials. Because each exponential has an amplitude and a characteristic time (exponent), a choice of 8 terms in the sum dictates finding 16 parameters. However, using the method of moments with exponential damping to determine these 16 values was computationally fast and efficient.

#### Determination of T_{0}.

Each of the above formulations requires estimating the delay due to the intrahepatic nonexchanging vessels and catheters (*T*
_{0}). As described previously (17), we noted that, for early time, the ratio of the sucrose to the palmitate outflow curve is a simple exponential function with respect to time and hence will yield a straight line if plotted using semilogarithmic axes. This approach avoids the need to include a third indicator as was initially described by Goresky et al. (7). Using an indicator that is confined to the vascular space, e.g.,^{51}Cr-labeled red blood cells, along with an extracellular marker such as [^{14}C]sucrose, allows one to estimate the nonexchanging delay and the extravascular, extracellular, and the vascular volume. Because we were not interested in these components of the extracellular space, we chose to eliminate the vascular indicator. This removes the difficulties associated with determining the radioactivities of three isotopes in small volume samples. It should be stressed, however, that our method of determining*T*
_{0} is based on an equally rigorous theoretical basis. It also utilizes multiple data points, rather than relying on the “first appearance” time as others have done (40).

*Calculation of steady-state extraction fraction (E).* For a single sinusoid with transit time*T*, the fraction of the solute recovered at the outflow is given by (18)
Equation 13
For multiple sinusoids
Equation 14
In deriving the last equality, we have assumed that
is sufficiently smooth (e.g., continuous) to transpose the limit and the function composition. Note that, as expected, if*t*
_{diff} = 0, then the single sinusoid expression yields
Equation 15 which is the Kety-Renkin-Crone model of a sinusoid (10). If*t*
_{diff} > 0, the extraction fraction falls monotonically with*t*
_{diff}, reaching a minimum value of zero as*t*
_{diff} becomes infinite. *Equation 8
* was used by the fitting procedure to predict the extraction fraction as a function of the adjustable parameters.

*Choice of numerical method*. Previous reports using the MID method have given explicit solutions to the model equation, both for the single and multiple sinusoid cases (28, 40). In the case of multiple sinusoids, the solution is given in terms of a convolution integral that utilizes an extracellular reference curve to define the probability density function of transit times. In using this expression to fit experimental outflow curves, the integral is evaluated using a numerical integration scheme. For the diffusion model, an analytical solution for even the single sinusoid case is not possible, as the solution exists in closed form only as its Laplace transform. For this reason, we performed the necessary convolution (integration) in transform space and then used numerical transform inversion to calculate values of the outflow curves at discrete time points. We used various numerical schemes to compute the predicted outflow curves (17). Although the fitted rate constants and*t*
_{diff} differed slightly (<5%) depending on which numerical scheme was used, no systematic trend in parameter values was noted. We have therefore chosen to use the fastest numerical scheme, in which a complex exponential sum is used to describe the [^{14}C]sucrose curve and the convolution is done in transform space.

## Footnotes

Address for reprint requests: R. A. Weisiger, Div. of Gastroenterology and the Liver Center, Univ. of California, San Francisco, San Francisco, CA 94110-0538.

This study was supported by National Institute of Diabetes and Digestive and Kidney Diseases Grants DK-32898 (to R. A. Weisiger), DK-46922 (to B. A. Luxon), and Liver Core Center Grant DK-26743. B. A. Luxon is a recipient of the Mary Richards Liver Scholar Award from the American Liver Foundation.

- Copyright © 1998 the American Physiological Society