## Abstract

Multiple-indicator dilution experiments with labeled lactate were performed in the livers of anesthetized dogs. A mixture of^{51}Cr-labeled erythrocytes, [^{3}H]sucrose, andl-[1-^{14}C]lactate or a mixture of^{51}Cr-labeled erythrocytes, [^{14}C]sucrose, andl-[2-^{3}H]lactate was injected into the portal vein, and samples were obtained from the hepatic vein. Data were evaluated using a model comprising flow along sinusoids, exchange of lactate between plasma and erythrocytes and between plasma and hepatocytes, and, in the case ofl-[1-^{14}C]lactate, metabolism to H[^{14}C]
within hepatocytes. The coefficient for lactate efflux from erythrocytes was 0.62 ± 0.24 s^{−1}, and those for influx into and efflux from hepatocytes were 0.44 ± 0.13 and 0.14 ± 0.07 s^{−1}, respectively. The influx permeability-surface area product of the hepatocyte membrane for lactate (*P*
_{in}
*S*, in ml ⋅ s^{−1} ⋅ g^{−1}) varied with total flow rate (*F*, in ml s^{−1} ⋅ g^{−1}) according to *P*
_{in}
*S* = (3.1 ± 0.5)*F*+ (0.021 ± 0.014). Lactate in plasma, erythrocytes, and hepatocytes was close to equilibrium, whereas lactate metabolism was rate limiting.

- multiple-indicator dilution
- membrane permeability
- biological transport
- erythrocytes
- mathematical models
- hepatic microcirculation

lactate plays a key role in intermediate metabolism and serves as a shuttle for oxidizable substrate between organs (4). In the liver, it is both a substrate for gluconeogenesis and a product of glycolysis. In many tissues, lactate is generated and metabolized simultaneously. The overall rate of lactate turnover in the whole body is, however, difficult to evaluate because formation and metabolism can occur simultaneously inside the same cell (27). Quantitative assessment of these processes and of their regulation thus depends on an appraisal of the involvement of membrane transport. It has been inferred that the [lactate]-to-[pyruvate] ratio in the medium and the cytosolic [NADH]-to-[NAD^{+}] ratio are generally near equilibrium (45, 51, 52, 54). This implies a high activity of lactate dehydrogenase as well as a high permeability of the sinusoidal membranes of hepatocytes for lactate and pyruvate.

Saturable stereospecific transport of lactate has been demonstrated in isolated hepatocytes (12, 14, 25). Erythrocytes compose a substantial proportion of blood, and lactate and pyruvate are also rapidly transported into these cells (10, 11, 35, 47). Cloning experiments using hamster and rat hepatocytes suggest that the mammalian liver contains a specific monocarboxylate transporter (MCT2), which is distinct from that present in erythrocytes (MCT1) (15, 25). In contrast, MCT2 expression in human liver seems to be of minor importance (28).

This multitude of tissue-specific monocarboxylate transporters with different properties suggests a decisive role of transport kinetics in lactate metabolism. Studies with single cells (11, 12, 14, 25, 35) or membrane vesicles (36) have been instrumental in unraveling transport kinetics. Because conditions in vivo may differ from those in vitro, in vivo studies are expected to be more useful to explore the relative importance of the various processes involved in lactate metabolism and to elucidate the role of carriage of lactate by erythrocytes.

For this goal, we designed a set of in vivo multiple-indicator dilution tracer experiments in the anesthetized dog structured to elucidate hepatic l-lactate transport and metabolism and erythrocyte effects. Transient tracer l-lactate studies were carried out in a situation in which endogenous steady-state lactate levels were not being changed. Our expectations were that, apart from transport into hepatocytes, what would be observed at the outflow would depend on the rate of exchange between erythrocytes and plasma.

## MATERIALS AND METHODS

#### Radioactive materials.

The following radioactive material was used: Na_{2}
^{51}Cr_{2}O_{7} solution (6 Ci/mmol; Merck Frosst, Montreal, QC), [*fructose*-^{3}H(N)]sucrose (20 Ci/mmol; NEN, Boston, MA), [^{14}C(U)]sucrose (10 mCi/mmol; ICN, Costa Mesa, CA),l-[1-^{14}C]lactate (63 mCi/mmol; ICN), l-[2- ^{3}H(N)]lactate prepared as outlined below, and Na_{2}H[^{14}C]O_{3} (1 mCi/mmol; NEN).

For synthesis of l-[2-^{3}H]lactate, [1,1-^{3}H]ethanol was first prepared from sodium boro[^{3}H]hydride and acetaldehyde as follows. Fourteen micromoles of boro[^{3}H]hydride (specific activity 7.2 Ci/mmol; Amersham International, Little Chalfont, UK) were dissolved in 0.5 ml of water placed on ice. Acetaldehyde (0.75 μl) dissolved in 0.5 ml of water was added in a dropwise manner. After 15 min, 50 μl of 1 M hydrochloric acid were added. The resultant solution of [1,1-^{3}H]ethanol was stored frozen. Thel-[2-^{3}H]lactate was prepared enzymatically from this solution and pyruvate, as described by Vind and Grunnet (51). One-half milliliter of the frozen solution of [1,1-^{3}H]ethanol was neutralized with 20 μl of 1 N sodium hydroxide and 100 μl of Tris, to obtain a pH of 7.5. Then 50 μl of 1 M pyruvate in water, lactate dehydrogenase (rabbit muscle, Boehringer Mannheim), alcohol dehydrogenase (yeast, Boehringer Mannheim), and 70 μl (50 μmol) of NAD (grade II, Boehringer Mannheim) were added. Lactate was separated from the resulting solution by anion-exchange chromatography (Dowex 2 × 5, chloride form) using a linear elution gradient of 0–0.1 M hydrochloric acid.

#### Multiple-indicator dilution experiments.

Mongrel dogs were anesthetized with pentobarbital (25 mg/kg) with supplementary doses as necessary, intubated, and allowed to breathe room air naturally with occasional assistance with a resuscitator bag (Ambu International, Glostrup, Denmark). The abdomen was opened, and after administration of 4 mg/kg heparin, a catheter was placed in the portal vein for injection of tracers and a sampling catheter in the left main hepatic venous reservoir in such a fashion that no outflow obstruction resulted (16). The abdomen was closed to allow its contents to return to a normal temperature. At the end of each experiment an overdose of pentobarbital followed by a saturated magnesium sulfate solution (∼1 ml/kg) was used for euthanasia.

The injection mixture contained three tracers: ^{51}Cr-labeled erythrocytes, a vascular indicator that does not leave the microcirculation and that was used as a reference; labeled sucrose, a second reference indicator that enters the interstitial space (the space of Disse) freely but does not enter erythrocytes or the liver cells; and labeled l-lactate, which enters both liver cells and erythrocytes. Either of two combinations of labeled sucrose and labeled lactate was used, [^{3}H]sucrose andl-[1-^{14}C]lactate or [^{14}C]sucrose andl-[2-^{3}H]lactate. The approximate amounts of activities injected were ^{51}Cr-labeled erythrocytes, 20 μCi; ^{14}C-labeled activity, 15 μCi; and^{3}H-labeled activity, 60 μCi. The injection mixture, constituted with a hematocrit matching that of the peripheral blood, was incubated for 2 min at 37°C. This time period proved adequate for equilibration of the l-lactate between erythrocytes and plasma, as judged from in vitro experiments (11, 27, 37, 47). The mixture was then placed in a syringe and introduced as rapidly as possible into the portal vein to produce cross-sectional mixing. Part of the hepatic venous outflow was pumped through the sampling catheter at a rate of ∼75 ml/min. Samples were collected anaerobically with a mercury-through-syringe-type anaerobic fraction collector (13). In the experiments with l-[1-^{14}C]lactate, the anaerobic fraction collector was cooled with ice to prevent loss ofl-[2-^{3}H]lactate and formation of [^{3}H]OH in the samples between collection and analysis.

Previous experiments have shown that the products, labeled bicarbonate and CO_{2}, ordinarily distribute rapidly into liver cells and that their space of distribution is smaller than that available to labeled water (43). In the experiments withl-[1-^{14}C]lactate, a second run was therefore carried out using an injection mixture containing NaH[^{14}C]O_{3},^{51}Cr-labeled erythrocytes, and [^{3}H]HO, to determine the space of distribution available to the labeled bicarbonate and CO_{2}.

Standards were prepared from each injection solution by the addition, in serial dilution, of blood obtained from the hepatic venous catheter before the collection of samples. Both the injection solution and the diluent blood were cooled to 0°C.

#### Analysis of samples.

Total radioactivity contained in the samples, excluding H[^{14}C]
, was assessed as follows. A 0.1-ml aliquot of each sample or diluted standard was pipetted into 1.5 ml of saline, and 0.2 ml of 25% trichloroacetic acid was added to precipitate proteins. This treatment also leads to conversion of H[^{14}C]
to [^{14}C]O_{2}, which is then lost to the atmosphere. The samples were assayed in a Cobra well-type scintillation crystal gamma ray spectrometer (Nuclear Chicago, Des Plaines, IL) or a three-channel Cobra gamma ray spectrometer (Packard Instrument, Meriden, CT) for gamma radiation originating from the^{51}Cr-labeled erythrocytes. Of the supernatant from these samples, 0.4 ml were transferred to scintillation vials containing 7 ml of Scinti Verse II (Fisher Scientific, Fairlawn, NJ) or Ready Safe (Beckman Instruments, Fullerton, CA) and assayed for beta radiation in a Beckman LS5801 multi-channel liquid scintillation spectrometer. Appropriate counting standards and a set of simultaneous linear equations were used to determine the activity due to each species.

Labeled metabolic products were different depending on which variety of labeled lactate was injected. Inl-[2-^{3}H]lactate experiments, the possible metabolic products containing ^{3}H label are [^{3}H]HO and [^{3}H]glucose. Because these products were not analyzed separately, it was not possible to include them in the modeling of lactate metabolism. Inl-[1-^{14}C]lactate experiments, metabolism is expected to yield [^{14}C]O_{2} through oxidation of pyruvate by pyruvate dehydrogenase and in the citric acid cycle (41). At physiological pH, [^{14}C]O_{2} is converted to H[^{14}C]
by carbonate dehydratase (carbonic anhydrase) within erythrocytes and hepatocyte mitochondria and at the surface of endothelial cells to attain [^{14}C]O_{2}/H[^{14}C]
equilibrium (43, 50).

To determine combined ^{14}C activity of bicarbonate and CO_{2} in the blood sample, 0.5-ml aliquots were withdrawn and immediately injected into closed 25-ml conical flasks containing 1 ml of 1.5 M perchloric acid, equipped with a center well (Kontes Glass, Vineland, NJ) containing 0.1 ml of phenylethylamine. The flasks were opened the next day, and the radioactivity contained in the center wells was assayed by liquid scintillation spectrometry.

Anion chromatography was used to separate nonionic and ionic species for both the ^{14}C- and ^{3}H-labeled lactate experiments. For the ^{14}C experiments, this allowed the assessment of nonionic labeled products such as glucose. For the^{3}H experiments, this provided for the separation of the activity in lactate from that in metabolic products. However, the chromatographic system used did not allow separation of [^{3}H]HO from other tritiated metabolites such as carbohydrates. For the ^{14}C experiments, the contents of the opened flasks used for the determination of ^{14}C activity in bicarbonate and CO_{2} were neutralized with 50 μl of 1 M Tris base and 40 μl of 5 M KOH, resulting in a pH of ∼5. For the^{3}H experiments, 0.5 ml of blood was precipitated with 1.0 ml of 1.5 M perchloric acid, and the sample was neutralized as above. In either instance, the solids were removed by centrifugation and chromatographic separations were carried out on the supernatant.

Dowex 2 × 5 (200 mesh; Sigma) was washed in 1 M formic acid, suspended in 1 M sodium formate (to obtain the formate form), stirred overnight, and then washed three times with distilled water. Columns were set up in glass pipettes with 5-mm inner diameter, which were plugged at the tip with glass wool and then filled with the formate form of the anion exchange resin to a height of 6 cm. Flow in the columns was maintained by gravity.

An aliquot of 0.5 ml from the neutralized sample was applied to the column. The column was then flushed with 2.5 ml of water, of which two aliquots of 1 ml each were collected into vials containing 6 ml of scintillation fluid. Water and the nonionic species (sucrose and glucose) were found in this solution. To elute lactate, each column was flushed with 3.5 ml of 1 M formic acid, and the last 2 ml were collected in equal aliquots into vials containing 6 ml of scintillation fluid (Ready Safe, Beckman Instruments) and analyzed for beta radiation. To elute pyruvate, the column was flushed with 3.5 ml of 1 M hydrochloric acid and the last 2 ml were collected and analyzed as above.

Hemoglobin content, oxygen saturation, , and were measured in samples of venous blood collected through the portal and hepatic vein catheters and samples of arterial blood collected from a cannula inserted into the femoral artery. Unlabeled lactate was measured in plasma obtained from these samples by centrifugation, using a standard spectrophotometric technique (Ref. 21; test combinationl-Lactic Acid, Boehringer Mannheim)

#### In vitro kinetics in erythrocytes.

Lactate dehydrogenase present in erythrocytes catalyzes rapid interconversion between lactate and pyruvate. The ^{14}C label undergoes exchange between lactate and pyruvate, whereas the^{3}H label appears as ^{3}HOH. To ascertain the rate of the exchange reaction and the composition of the equilibration mixture, l-[^{14}C]lactate or [^{14}C]pyruvate was mixed with dog blood at 37°C. Alternatively,l-[2-^{3}H]lactate was added to a suspension of dog erythrocytes (45%) in Krebs-Henseleit buffer solution (pH = 7.4) at 37°C. At varying intervals after mixing, the reaction was stopped with perchloric acid. The labeled lactate and pyruvate or lactate and water were then separated chromatographically, as outlined above.

#### Model analysis and parameter identification.

The data needed to carry out the model analysis are the outflow dilution curves for labeled erythrocytes, labeled sucrose, and labeled lactate. In the case of [1-^{14}C]lactate experiments, the H[^{14}C]
curve was also used to parameterize a full precursor-product model. The outflow dilution curves were normalized in relation to the amount of tracer injected. Equivalent areas under the curves therefore signify equivalent outflow recoveries. For a substance completely recovered at the outflow (labeled erythrocytes or labeled sucrose), hepatic blood flow rate is equal to the reciprocal of the area under the curve (16). The liver vascular space was calculated as the product of blood flow rate and the mean transit time (the time integral of the product of time and concentration divided by the area under the curve) for the labeled erythrocytes. This procedure does not take into account the unlabeled input through the hepatic artery. The error introduced by this simplification was considered minor because in experiments with perfused rat livers, injection of indicators into the hepatic artery and the portal vein yielded similar distribution volumes for vascular indicators (5, 33).

The model analysis was carried out in two steps. In the first step, the relation between the labeled erythrocyte and sucrose curves was analyzed. In the second step, the relation among the labeled erythrocyte, labeled sucrose, and labeled lactate curves was analyzed.

Sucrose is distributed into the interstitial space (the space of Disse) in a flow-limited fashion but does not enter the liver cells. It enters the interstitial space as rapidly as it is presented because, with the fenestrae perforating the sinusoidal lining cells, there is no resistance to exchange between the sinusoidal plasma and the very shallow interstitial space. As a consequence, the labeled sucrose impulse propagates along the sinusoid in both the sinusoidal plasma and interstitial spaces; it travels less rapidly than the labeled erythrocytes, which are carried along the sinusoids by flow, so that the labeled sucrose impulse emerges later (16, 19). Labeled sucrose thus marks out the interstitial space, which would be available to labeled lactate if it did not enter hepatocytes or erythrocytes. The distribution of transit times in the large vessels and sinusoids is such that virtually all of the heterogeneity occurs in the sinusoidal bed (this is the predominant part of the contained blood, in the liver), and the transit times of the nonsinusoidal (“large”) vessels were assumed to be uniform. In a similar way, labeled water, bicarbonate, and CO_{2} enter the interstitial space as well as the erythrocyte and hepatocyte space in a flow-limited fashion (16,43). Simultaneous analysis of the outflow profiles of labeled erythrocytes and sucrose according to the delayed-wave model (16) allowed estimation of the ratios of extravascular to vascular space of sucrose (γ) and of the common nonsinusoidal transit time,*t*
_{0} (appendix
). Similarly, simultaneous analysis of outflow profiles obtained for second runs was used for estimating *t*
_{0} and the ratios of extravascular (combined interstitial and parenchymal) to vascular (combined plasma and erythrocyte) space ratio for bicarbonate/CO_{2} (γ_{m}) and water (γ_{w}).

The description of lactate tracer disposition in mathematical form is developed in appendix and is presented in schematic form in Fig. 1. Because the data demonstrate a limiting erythrocyte exchange of label, plasma and erythrocytes are represented by two compartments interconnected by the exchange of tracer between plasma and erythrocytes. Lactate in erythrocytes travels along the sinusoid with the same velocity as ^{51}Cr-labeled erythrocytes, whereas lactate in plasma travels with the lower velocity of [^{14}C]sucrose. Unlabeled lactate is assumed to be in a Donnan equilibrium between the plasma and the erythrocyte space, as was found for human erythrocytes (9). With pH values of 7.21 (40) and 7.30 (see Table 1) for dog erythrocytes and dog plasma, respectively, and a ratio of erythrocyte to plasma water content of 0.65:1 (55), the equilibrium lactate concentration ratio (erythrocytes:plasma), λ, becomes 0.65 × 10^{7.21 − 7.3} = 0.53. This value is the same as that previously assumed for chloride (18) and is similar to the value of 0.58 found for human blood (48).

The plasma lactate exchanges with that in the hepatocytes via carrier-mediated transport (12, 14, 25, 35). Metabolic conversion to CO_{2} occurs within the mitochondrial matrix of hepatocytes (50). Product [^{14}C]O_{2} equilibrates rapidly across the mitochondrial and plasma membranes by simple diffusion and with labeled bicarbonate by the action of carbonate dehydratases present in hepatocyte mitochondria, erythrocyte matrix, and the surface of endothelial cells (43, 50). Therefore, the products are represented as combined [^{14}C]O_{2}and H[^{14}C]
radioactivity that is shared between liver cells, plasma, and erythrocytes (16, 43) and undergoes flow-limited distribution between these phases (the tracer equilibrates as rapidly as it is presented).

The rate constant for transfer across each membrane is defined as the respective permeability-surface area product divided by the volume of the compartment from which the flux originates. For the metabolic conversion, it is the conversion rate divided by the amount of unlabeled lactate in the compartment in which the conversion is occurring.

The rate constants defined in appendix
for erythrocyte efflux, hepatocyte influx and efflux, and metabolism,*k*
_{rp} , *k*
_{ph},*k*
_{hp}, and *k*
_{seq}, respectively, were varied until an optimal simultaneous fit for lactate and bicarbonate/CO_{2} was found. Erythrocyte influx (rate constant *k*
_{pr}) was not fitted because it is directly related to *k*
_{rp} (*Eq. EB3
*) according to the erythrocyte-plasma equilibrium lactate concentration ratio, which was estimated separately (see above). Because only part of the upslope of the product bicarbonate/CO_{2} profile is available for analysis, values for γ_{m} were taken from the analysis of second-run data.

Input and collection catheters impose delay and distortion on the outflow curves. It is therefore necessary to obtain parameter values from data derived by deconvolution of the catheter transfer function. For the flow-limited indicators, sucrose, water, and bicarbonate/CO_{2}, linear superposition according to the delayed-wave model with catheter correction (16, 56) was carried out as outlined in appendix
. For the lactate and bicarbonate/CO_{2} curves, a similar procedure was followed, as explained in appendix
.

In carrying out the fitting of the modeling to the experimental data, a modified Levenberg-Marquardt nonlinear least-squares routine (Visual Numerics, Houston, TX) was used. Parameter estimates were evaluated statistically (26). The Jacobian matrix (matrix of sensitivities) obtained from the fitting program was used to calculate the variances and covariances of the fitted parameters. The square roots of the variances, the standard deviations of the fitted parameter, were calculated for each experiment, representing the uncertainty in the determination of the parameter from the data of the experiment (as opposed to that for parameters from different experiments, representing interindividual variability).

## RESULTS

#### Effect of lactate dehydrogenase in erythrocytes on lactate label in injection mixture.

When l-[^{14}C]lactate was mixed with dog blood with a hematocrit of 0.40, the exchange reaction occurring on the enzyme resulted within 30 s in a mixture in which 98% of the label was in lactate and 2% in pyruvate; the relative composition did not change thereafter. The change was too small to permit assessment of the rate. When the ^{14}C label was introduced as [^{14}C]pyruvate, the same equilibrium mixture resulted: 98% lactate and 2% pyruvate. The half-time for the exchange was on the order of 5–10 s. The exchange was faster than that observed in human or rat blood (27, 37). In the case ofl-[2-^{3}H]lactate, a net loss of^{3}H label occurred with time, the label appearing as [^{3}H]HO. The rate was found to be ∼1% per min at 37°C and a hematocrit of 0.40.

#### Physiological parameters.

Table 1 summarizes physiological parameters measured in blood or plasma of the animals used in the experiments. Values for pH,
,
, hemoglobin concentration in blood, and O_{2} saturation of hemoglobin were in the physiological range. Hepatic blood flow rate was variable, with values between 1 and 3 ml ⋅ min^{−1} ⋅ g liver^{−1}. This spontaneous variation was used to assess the flow dependence of kinetic parameters. Plasma concentrations of unlabeled lactate were between 0.44 and 6 mM. There was no significant difference among arterial, venous, or portal concentrations. In 4 of 13 cases the venous lactate concentration exceeded the reference range of 0.3–2.5 mM reported for healthy mongrel dogs (23). Assuming that 25% of hepatic blood flow is arterial, net production of lactate was observed in 7 of 10 animals. In the other three animals, the calculated net hepatic extraction ratio of lactate was <0.2.

#### Multiple-indicator dilution experiments.

Representative sets of data from experiments with low and high hepatic blood flow rate are presented in Fig. 2. First-run data are represented in the upper panels, and second-run data are in the lower panels. First-run outflow fraction for labeled erythrocytes rises to an early and high peak. The peak of the sucrose profile is lower, delayed, and followed by a slow decrease compared with erythrocyte profile. Single-pass erythrocyte and sucrose dilution curves were obtained by truncating the late portion of outflow profiles where recirculation is apparent and applying exponential extrapolation (16). The areas under the labeled erythrocyte and sucrose reference curves are higher at the lower flow rate and smaller at the higher flow rate, as expected. The peak of the labeled lactate profile approximately coincides in time with the erythrocyte peak, but is low in magnitude, and is followed by a very slow decay. Second-run data curves are very similar to the data presented previously (43).

Parameters derived from primary analysis of the data are shown in Table2. Because of technical reasons, second-run data were not available for 3 of 16 [1-^{14}C]lactate experiments. Mean transit times have been corrected for catheter transit time.

With appropriate correction for catheter distortion, erythrocyte and sucrose data were superimposable according to the delayed-wave model (fits not shown). This confirms previous studies (16, 18, 20, 43) with comparable quality of the fit (median Pearson's *r* = 0.999). Parameters obtained from the fit of transformed first-run erythrocyte profile to sucrose profile are shown in Table3. Also shown are the parameters from the simultaneous fit of the transformed erythrocyte profiles to the water and bicarbonate/CO_{2} profiles from second runs (only for experiments with [1-^{14}C]lactate injection in the first run). Values for *t*
_{0} have been corrected for catheter transit time.

Parameters from the fit of superimposed erythrocytes and sucrose profiles to lactate and bicarbonate/CO_{2} profiles are shown in Table 4. For the [1-^{14}C]lactate experiments that had no second-run data, the value of γ_{m} used to fit first-run precursor and product profiles was the average value obtained by the fit of second-run profiles in the other experiments. In a first attempt, only the lactate profile was fitted to data, and the product bicarbonate/CO_{2} data were not used. However, the resulting standard deviations of *k*
_{seq} (estimated using the matrix of sensitivities) were too large to allow for a reliable*k*
_{seq} estimate. Therefore, the outflow profiles for lactate and its metabolite, bicarbonate/CO_{2}, were fitted simultaneously to lactate and bicarbonate/CO_{2} data. This provided for adequate reliability of the *k*
_{seq}estimates. Because [2-^{3}H]lactate experiments had no product bicarbonate or CO_{2} identified, the*k*
_{seq} value used was the average value obtained from the analysis of [1-^{14}C]lactate experiments.

Fits to first-run labeled lactate and bicarbonate/CO_{2} data of the representative experiments are shown in Fig.3. Also shown in the illustration are the following components of the total predicted outflow response: tracer present in erythrocytes at the time of injection and that passes through the microcirculation without being released in the plasma (erythrocyte throughput), tracer in erythrocytes and plasma that never enters the liver cells (total throughput component, which includes erythrocyte throughput), and tracer that returns from hepatocytes to the vascular space and then exits the liver (returning component). The components show that, although the magnitude of erythrocyte throughput is significant, the lactate early peak is due to plasma as well as erythrocyte throughput.

Reliability of the fitted parameters was further established by assessing the effect of changes in parameter values on calculated outflow profiles (Fig. 4). For one parameter at a time, its value was set to a fixed value equal to either one-half or double the optimal value. With highly correlated parameters, changing several parameters simultaneously can have a smaller effect on the calculated outflow profile than changing each parameter separately. Therefore, the remaining parameters were optimized by refitting the calculated outflow profiles. In all four cases examined, the parameter values had a definite effect on the shapes of the calculated outflow profiles, confirming their pertinence. It is noted that the permeability of red blood cell membranes influences only the peak of the lactate profile, whereas that of the hepatocyte membrane has an impact on the whole profile. These parameters have only a small effect on the product profile, especially considering that the scatter of product data is larger than that of precursor data. In contrast, *k*
_{seq} has no effect on the lactate profile and thus cannot be determined from lactate data alone (see above); it has, however, a strong effect on the product profile: the magnitude of the latter is roughly proportional to the value of *k*
_{seq}.

The change in the vascular volume with flow rate is illustrated in Fig.5
*A*. The best fit from linear regression is V_{vasc} = (6.9 ± 0.9)*F* + (0.05 ± 0.02), where V_{vasc} is the vascular volume in milliliters per gram of liver and *F* is flow rate in milliliters per second per gram of liver. The values for the slope and the ordinate intercept are approximately one standard deviation away from the values obtained in a previous study for normal hematocrit data (20).

The variation in the lactate influx permeability-surface area product with flow rate is shown in Fig. 5
*B*. The equation for the fitted line is *P*
_{in}
*S* = (3.1 ± 0.5)*F* + (0.021 ± 0.014), where *P*
_{in}
*S* is the influx permeability-surface area product in milliliters per second per gram of liver and *F* is flow rate in milliliters per second per gram of liver.

There was no significant correlation between plasma lactate concentration and *P*
_{in}
*S* for lactate or the sequestration constant.

## DISCUSSION

#### Lactate carriage by erythrocytes.

Modeling analysis of the results suggests that the sinusoidal membrane of hepatocytes and the membrane of erythrocytes constitute barriers for lactate transport. Lactate is thus partially trapped within erythrocytes during transit through the sinusoidal bed, and this part is not available for exchange with plasma and uptake by hepatocytes. This results in an early peak portion in the outflow profiles for lactate. In some experiments, the peak is diminished to a shoulder (curves not displayed).

Depending on the rate of tracer exchange between erythrocytes and plasma, three different cases generally must be considered. If the permeability of the erythrocyte is high such that the inverse of the efflux rate constant is very large compared with the sinusoidal transit time of the erythrocytes through the acinus, erythrocytes can be viewed as an extension of the plasma compartment, resulting in an erythrocyte capacity effect. This case has been previously investigated theoretically and experimentally for water, urea, chloride, and bicarbonate/CO_{2} (6, 16, 18, 43). If the efflux rate constant is very small compared with the inverse of the transit time of erythrocytes through the acinus, a substance trapped within erythrocytes will not be removed to a significant extent during one passage through the sinusoidal bed. In this case, the kinetics of erythrocyte exchange between plasma and erythrocytes cannot be studied using the multiple-indicator dilution method. For example, the efflux rate coefficient for the immunosuppressant tacrolimus from human erythrocytes was <1 min^{−1} and thus much lower than the inverse of the sinusoidal transit time. Hepatic uptake of this drug was investigated with recirculating perfusion of rabbit liver. These experiments were evaluated using compartmental analysis, with consideration of tracer exchange between plasma and erythrocytes (7,34). However, this evaluation does not contribute to elucidation of the mechanism of hepatic uptake, because most of the exchange takes place outside the liver within the reservoir of the perfusion apparatus.

In the present case, the efflux rate constant is of the same order of magnitude as the inverse of transit time of erythrocytes through the acinus, resulting in “red blood cell carriage,” i.e., partial trapping of the indicator within erythrocytes. This phenomenon has originally been observed with thiourea in the dog kidney (6) and analyzed quantitatively in the dog liver, where thiourea enters hepatocytes rapidly (18). A similar carriage phenomenon was described more recently with acetaminophen in rat liver perfused with human erythrocytes (32). In this study, lactate was found to undergo barrier-limited uptake into hepatocytes where it is metabolized, a situation similar to that previously described for acetaminophen (32). Analytical solutions have been found for the pertinent differential equations, and they have been used for modeling. A system of six differential equations was used to describe hepatic acetaminophen disposition. Numerical solutions were sought to calculate outflow profiles of the precursor, acetaminophen, and the metabolic product, acetaminophen sulfate. Investigation of the present case led to a system of four differential equations, which are very similar to the case described for acetaminophen. An analytical solution could be formulated for the outflow profile for the precursor, lactate, but not of the products, bicarbonate/CO_{2}, water, or glucose (appendix
).

Lactate has previously been found to be transported across canine erythrocyte membranes primarily via a monocarboxylate carrier, with minor contributions of capnophorin (band 3 protein) and nonionic diffusion of lactic acid (11, 47). Because unlabeled lactate is near Donnan equilibrium between plasma and erythrocyte matrix, the rate coefficients reported here represent equilibrium exchange. Direct measurements of equilibrium exchange rates at physiological temperature have not been reported, because this process seems to be very fast and difficult to measure. Deuticke et al. (11) measured equilibrium exchange rates at 10°C and found a rate constant for efflux of 0.004 s^{−1}. If this value is extrapolated to 37°C, using data for human erythrocytes and assuming that the ratio of the activities at the two temperatures are the same for the two species, an estimated rate constant of 0.04 s^{−1} at 37°C is obtained, which is one order of magnitude slower than the fitted efflux coefficients obtained from the present experiments. However, attempts to fit to the data a model with a fixed value of 0.04 s^{−1} for the coefficient of efflux from erythrocytes,*k*
_{rp}, did not result in acceptable fits. This discrepancy between the fitted and the in vitro equilibrium exchange values does not have an obvious explanation. Possibly, dog plasma contains unknown factors that enhance lactate transport into erythrocytes.

Skelton et al. (47) measured zero-trans influx into erythrocytes at 37°C and found, at the lowest lactate concentration of 1.6 mM, influx rates of 1.7 ± 0.14 μmol ⋅ min^{−1} ⋅ ml cellular volume^{−1}. Because at Donnan equilibrium, the intracellular concentration is about one-half the extracellular concentration, zero-trans influx can therefore be expressed as 1.7/0.8 = 2.13 μmol ⋅ min^{−1} ⋅ μmol intracellular lactate^{−1} or 0.035 μmol ⋅ s^{−1} ⋅ μmol intracellular lactate^{−1}. According to membrane carrier kinetics, equilibrium exchange is generally expected to be faster than zero-trans uptake (10, 12, 49). The rate constant for equilibrium exchange of 0.04 s^{−1} could thus be considered a minimal value.

Rates of lactate transport into erythrocytes have been shown to vary widely between species (11, 47). In “athletic” species (dogs and horses) with high maximal oxygen uptake, lactate transport is up to 160 times faster than in “nonathletic” species (cattle and goats). Presumably, rapid partition of lactate into erythrocytes increases the capacity of blood to remove excess lactate from the musculature during heavy exercise. At normal hematocrit (0.42 for dog) and Donnan equilibrium, 27% of blood lactate is present in erythrocytes, so the capacity would increase by ∼36%. The rates reported in the literature, however, are not sufficiently high to ensure that this mechanism is operative, because the characteristic times for filling the erythrocyte lactate pool exceed typical transit times through the capillary beds of exercising skeletal or cardiac muscles, which are on the order of a few seconds (1, 22, 38). The present analysis yielded much higher activities of lactate transport across the erythrocyte membranes, and it provides more direct evidence that, in the liver, lactate within the erythrocytes can partly exchange with lactate in plasma and in hepatocytes during the hepatic transit time of blood.

#### Lactate transport into hepatocytes.

Hepatic lactate transport has previously been studied only in the rat. Near-equilibrium exchange rates were measured in the isolated, perfused rat liver using the multiple-indicator dilution method in the absence of erythrocytes (2, 44). In agreement with our results in dogs, saturation was not observed with plasma concentrations up to 8 mM. A carrier-mediated transport mechanism was nevertheless postulated on the basis of inhibition by α-cyanocinnamate. At 8 mM extracellular lactate, the transport rate was 240 μmol ⋅ min^{−1} ⋅ ml cell^{−1}. With a cellular volume of 0.8 ml/g, this value yields a influx permeability-surface area product of 0.4 ml ⋅ s^{−1} ⋅ g^{−1}, which is about twice the maximal value obtained in the dog liver. In experiments with isolated hepatocytes, zero-trans influx ofl-lactate was found to be carrier mediated (12, 14, 25). The most reliable values are those by Jackson and Halestrap (25) using intracellular pH indicators. At 25°C, the *K*
_{m}value was 4.5 mM and the maximal influx rate was 20 μmol ⋅ min^{−1} ⋅ ml cell^{−1}. Extrapolation to 37°C yields a maximal influx rate of 46 μmol ⋅ min^{−1} ⋅ ml cell^{−1}. This is slower than equilibrium exchange rates, as also observed for monocarboxylate transport in erythrocytes (10, 12), and as expected for exchange transport systems (49). Lactate transport across the basolateral membranes of canine hepatocytes is fast (values of *P*
_{in}
*S* are, on average, 4 times those of hepatic blood flow rate) and carrier mediated. A large proportion of intracellular lactate returns into the plasma space, and a much smaller proportion is metabolized, as judged from comparison of*k*
_{hp} to *k*
_{seq}; thus the intracellular lactate is maintained near equilibrium with plasma lactate, and the available intracellular lactate does not limit overall lactate metabolism. Also, this means that rates of disappearance of labeled lactate from plasma provide acceptable estimates for cellular lactate metabolism (27).

The observed correlation of the influx permeability-surface area product for lactate with hepatic blood flow rate is analogous to that previously described for rubidium (20). This phenomenon is tentatively attributed to a portion of the sinusoids that are intermittently stagnant at low flows such that they are not accessible to the tracers; at higher flows, the proportion of sinusoids available for tracer exchange at any time would increase.

#### Hepatic lactate metabolism.

The initial step in hepatic lactate metabolism is oxidation to pyruvate by the action of lactate dehydrogenase. This reaction is reversible, and is probably near equilibrium under physiological conditions (45,51, 54). The final products are mainly [^{3}H]_{2}O for [2-^{3}H]lactate and mainly [^{14}C]bicarbonate and [^{14}C]O_{2} for [1-^{14}C]lactate. The only other known pathway of lactate metabolism is peroxysomal oxidation by glycolate oxidase, which is negligible (29).

The role of lactate dehydrogenase in sequestration of labeled lactate depends on the kind of labeling. In the case of [1-^{14}C]lactate, the label is retained in pyruvate formed by oxidation (46). This pathway is a dead end in erythrocytes, which lack mitochondria and therefore the ability to further oxidize pyruvate. Label contained in pyruvate within erythrocytes and plasma will not exceed a few percentage points, as shown by the reported in vitro experiments with dog blood. In the majority of the animals, net release of unlabeled lactate by the liver was observed despite sequestration of labeled lactate. Labeled lactate is thus at least partly replaced by unlabeled lactate produced from other intrahepatic metabolites. This can be attributed to production of lactate by glycolysis and its simultaneous consumption by oxidation and gluconeogenesis (53).

In hepatocytes, the proportion of tracer pyruvate is also expected to be insignificant because of equilibration with lactate with a similar lactate-to-pyruvate ratio (50:1). [^{14}C]pyruvate contained in the cytosol of hepatocytes can thus be combined with cytosolic [^{14}C]lactate to form a single pool. Tracer exchange between lactate and pyruvate is expected to be faster than maximal net rates of lactate production or consumption, because it can take place without dissociation of the enzyme-coenzyme complex (46). Appearance of the ^{14}C label in bicarbonate/CO_{2} is most probably due to the action of pyruvate dehydrogenase. It has been shown in the rat liver that labeled bicarbonate/CO_{2} is formed from [1-^{14}C]pyruvate in two different ways that can be distinguished kinetically: the part that is formed directly from pyruvate by the action of pyruvate dehydrogenase appears in the perfusate very quickly, whereas bicarbonate/CO_{2} formed indirectly (e.g., via oxaloacetate or tricarboxylic acid intermediates) appears after a lag time of 2 min (3, 41). Because in our experiments, the sampling time was shorter than that lag time, it may be presumed that formation of H[^{14}C]
reflects the activity of pyruvate dehydrogenase. However, no attempt was made to quantify the rate of the pyruvate dehydrogenase reaction.

When [2-^{3}H]lactate is used as a tracer, sequestration and formation of labeled products occur in a different manner. Lactate dehydrogenase will mediate the transfer of^{3}H to the α-position of NADH, from where it can be transferred to other metabolic intermediates by the action of other NADH-dependent dehydrogenases (24, 30). This transfer is expected to be slower than that of the ^{14}C label, because it requires the dissociation of the coenzyme from the enzyme. Moreover, the lactate dehydrogenase reaction shows a considerable kinetic isotope effect (31). Part of the label will eventually appear as ^{3}HOH because of oxidation reactions and hydrogen exchange with cellular water. Another part may be incorporated into metabolites released by the liver, such as into glucose formed by gluconeogenesis (30, 52). A detailed analysis of the fate of the ^{3}H label is beyond the scope of this work.

#### General conclusions.

Lactate transport across the sinusoidal membranes of hepatocytes and across the erythrocyte membrane is bidirectional under in vivo conditions, resulting in an approximate equilibrium between lactate in plasma and in the cytosol of hepatocytes. Lactate carried in erythrocytes can exchange with that in plasma and is available in part for hepatic metabolism.

## Acknowledgments

The authors thank Eva Ibrahim, Bruce Ritchie, and Kay Lumsden for superb technical assistance.

## Appendix

### Linear Superposition According To Delayed Wave Model With Catheter Correction

To evaluate the experimentally obtained outflow profiles, the dispersion of the injected bolus by the injection apparatus and the inflow and outflow catheters must be considered, as previously described in detail (12, 13). For example, the experimental erythrocyte curve, *C*
_{RBC}(*t*), is the convolution of the organ erythrocyte transport function (catheter-corrected outflow profile or impulse response), *h*
_{RBC}(*t*), with the outflow profile obtained from the apparatus in the absence of a liver, *C*
_{cath}(*t*)
Equation A1where * is the convolution operator. Similarly, for sucrose
Equation A2The experimental outflow profiles are distorted by the effect of the catheter and pump used for sample collection from the hepatic vein. The response of the latter has been shown to be adequately approximated by a monoexponential decay with delay constant, α_{cath}, combined with a simple delay, δ_{cath}. The outflow profile of the collection catheter alone was represented by the function
Equation A3The impulse response of the liver was then obtained by deconvolution using the following relation
Equation A4The erythrocyte curve, *C*
_{RBC}, was approximated by a piecewise third-order polynomial (8), which is easily differentiated.

The organ sucrose transport function,*h*
_{suc}(*t*), was calculated from the organ red blood cell transport function, *h*
_{RBC}(*t*), using linear superposition according to the flow-limited model of Goresky (16) with uniform large-vessel transit time
Equation A5where γ is the interstitial to vascular distribution spaces and*t*
_{0} is the common large-vessel transit time. The sucrose outflow profile, *C*
_{suc}(*t*), was then calculated by convolution according to *Eq. EA1
*, using a numerical integration algorithm (QDAGS from Visual Numerics, Houston, TX). The result was fitted to the experimental outflow profile for sucrose by a nonlinear least-squares procedure to find optimal values for γ and *t*
_{0}.

## Appendix

### Matrix Approach To Multiple-Indicator Dilution Equations

The evaluation of the multiple-indicator dilution data is similar to that previously used for experiments with acetaminophen (32). It is based on a published mathematical representation of transport and metabolism in the liver (42).

Labeled material is contained in various pools, as shown in Fig. 1. Mobile pools include lactate in blood plasma, with concentration*c*
_{p}, lactate in erythrocytes, with concentration*c*
_{r}, and a combined bicarbonate/CO_{2} pool representing bicarbonate and CO_{2} contained in blood plasma, erythrocytes, and hepatocytes, with a plasma concentration *m*. The combined pool was formulated based on the observation that bicarbonate and CO_{2} exchange very rapidly between these spaces (43). The only stationary pool is the intracellular lactate pool, which corresponds to the lactate content of hepatocytes, with concentration *c*
_{h}.

The behavior of tracers is described by the following system of partial differential equations
Equation B1a
Equation B1b
Equation B1c
Equation B1d where*t* is time; *x* is the quotient of the distance along the length and the velocity of flow; γ is the ratio of the Disse space to that in sinusoidal plasma; β = hematocrit/(1–hematocrit) is the erythrocytes-to-sinusoidal plasma volume ratio; θ is the hepatocyte-to-sinusoidal plasma volume ratio; and γ_{m} is the ratio of the stationary to the mobile part of the product. The latter is given by
Equation B2 where λ_{m} and λ_{mh} are partition coefficients for bicarbonate/CO_{2} between plasma and erythrocytes and between plasma and hepatocytes, respectively (43).

Exchange between pools is determined by transfer coefficients, with the dimension of reciprocal time, related to permeabilities and enzymatic activities in the following fashion. The transfer coefficient*k*
_{rp} is the turnover number for lactate within erythrocytes. The transfer coefficient is equal to the permeability of the erythrocyte membrane multiplied by the ratio of the surface area to the volume of the erythrocyte,*P*
_{rp}
*S*
_{r}/V_{RBC}. Because the distribution space ratio of lactate in the Disse space to that in sinusoidal plasma, γ, is the same as that for the reference indicator, sucrose, the transfer coefficient *k*
_{pr} is determined by the relation
Equation B3 where λ is the equilibrium partition coefficient between lactate concentration in erythrocytes and the unbound lactate concentration in plasma. The following equations relate the transfer coefficients for hepatocellular membrane passage of lactate to membrane permeabilities
Equation B4a
Equation B4b where*P*
_{ph}
*S* and *P*
_{hp}
*S*are the permeability-surface area products for the exchange of lactate across the hepatocyte cell membranes in the inward and outward direction, respectively; V_{p} is the sinusoidal plasma volume; and V_{cell} is the hepatocellular volume. The transfer coefficient *k*
_{seq} represents the irreversible hepatic biotransformation activity and is given by the ratio of the reaction rate to the amount of intracellular precursor.

The dose introduced at the origin (*x* = 0) of the initially tracer-free sinusoid had previously been equilibrated between plasma and erythrocytes. The system of differential equations, therefore, must be solved with the following initial conditions (18)
Equation B5a
Equation B5b where*q*
_{0} is the amount of tracer initially applied to the entrance of the sinusoid, *F*
_{s} is sinusoidal blood flow rate, and δ is the impulse function.

An analytical solution of the system of partial differential equations*B1a–B1d* with initial conditions *B5a–B5b* is presented in appendix
for precursor concentrations*c*
_{r} and *c*
_{p}. However, no solution was found for the product plasma concentration *m*. Moreover, the numerical evaluation of the solution detailed in appendix
is expensive in terms of computer time. We therefore use an eigenvalue method previously developed by Schwab (42) for our calculations. For this purpose, the transfer coefficients are collected into the compartmental matrix **A**
Equation B6 Each pool travels along the sinusoids with a relative velocity that is defined to be 1 for erythrocytes and 0 for hepatocytes. The average relative velocity of lactate in plasma contained in the combined sinusoidal and the interstitial spaces is 1/(1 + γ), and that of total bicarbonate/CO_{2} is 1/(1 + γ_{m}). The relative velocities are collected in the diagonal matrix **W**
Equation B7Concentration terms are collected in a vector, **u**, whose elements are amounts per unit sinusoidal space, normalized to the injected dose
Equation B8With these definitions, the system of partial differential equations can then be written concisely as
Equation B9 The initial conditions become
Equation B10 where
Equation B11The response of the whole organ will be obtained by integrating over all flow paths with different transit times. If *n(x*) is the distribution of flow paths [that is, *n(x*)d*x* is the proportion of the flow with transit times between *x* and*x* + d*x*], then the concentrations at the outflow of the whole liver will be the flow-weighted average of the responses of single sinusoidal paths (18), according to the integrals
Equation B12 where**h**(*t*) is a vector function whose elements are the total amounts per unit sinusoidal plasma space.

We now perform Laplace transformation with respect to *x*.
Equation B13where*S* is the Laplace variable. *Equation EB9
* then becomes
Equation B14 which has the solution
Equation B15For the definition of an exponential with a matrix-valued exponent, seeappendix
. The transit time distribution was approximated by a sum of *n* exponential terms, as follows
Equation B16 where the parameters α_{i} and ε_{i}are arbitrary and have no physical meaning. Thus the parameters α_{i} and ε_{i} may be determined from the reference curve.

Substitution of *Eq. EB16
* into *Eq. EB12
* yields
Equation B17 and substitution of *Eq. EB13 *yields
Equation B18 Further substitution of *Eq. EB15
* yields
Equation B19When the elements of the matrices **A** and **W** and the values for α_{i} and ε_{i} are known, the outflow profiles contained in **h** can be calculated.

For labeled erythrocytes as the vascular reference, we obtain the single-element matrices **A** = (0) and **W** = (1). The response of the whole liver is then simply
Equation B20 We use the following multiexponential approximation for the erythrocyte outflow profile
Equation B21 where α′_{i} are empirical coefficients. Because the upslope is smooth, the following restriction is introduced
Equation B22 To account for the influence of the injection and collection devices,*Eq. EB21
* is substituted into* Eq. EA4 *from appendix
. The result of this substitution is of the form of* Eq.EB20
* with the following coefficients
Equation B23 The observable outflow profile of lactate *C*
_{Lac}
*(t)*(outflow concentration normalized to dose) is the sum of the plasma and erythrocyte contents. When calculated per milliliter of blood, the latter becomes the sum of the first two elements of**h**(*t*), *h*
_{p}(*t*) and*h*
_{r}(*t*)
Equation B24 According to *Eq. EB19
*, *h*
_{1}(*t*) and*h*
_{2}(*t*) are sums of exponential functions, and according to *Eq. EA3
*, *C*
_{cath}(*t*) is a single exponential function. *C*
_{Lac}(*t*) can then be evaluated analytically using the convolution formula for exponential functions
Equation B25 Similarly, for the product, bicarbonate/CO_{2}
Equation B26 where*h*
_{m}(*t*) is the third element of**h**(*t*).

The hepatocyte influx permeability-surface area product is obtained by multiplying the value obtained for *k*
_{ph} with the space of distribution of lactate in the plasma
Equation B27 where*F* is blood flow rate, Hct is hematocrit, and
_{suc} is sucrose mean transit time.

## Appendix

### Analytical Solutions for a Single Sinusoid

An analytical solution of the system of partial differential equations*B1a–B1d* with initial conditions *B5a–B5b *was found by using Laplace transformations. The strategy was similar to that used in the case of no barrier between plasma and parenchymal cells (18). Details of the solution procedure have been deposited.^{1} The solution, representing the concentrations at time *t *at the outflow of a sinusoid of length*x*, is given below *a*) when *t* < *x*
Equation C1
*b*) in the time interval *x* ≤ *t* ≤ (1 + γ)*x*
Equation C2a
Equation C2b

where *I*
_{0} and*I*
_{1} are modified Bessel functions of the first kind and order 0 and 1, respectively.*c*) when *t*> (1 + γ)*x*
Equation C3a
Equation C3b

where Equation C4a Equation C4b Equation C5 Equation C6 Equation C7 Equation C8

Starting from this solution, the evaluation of the response of the whole liver is analogous to that previously reported for the case of no barrier between plasma and parenchymal cells (18). The numerical results were similar to those obtained as shown in appendix , with reasonable accuracy.

#### Special cases.

If erythrocytes are considered impermeable, *k*
_{pr} = 0 and hence ζ = 0 and *c̃*
_{r0}, therefore all terms in *Eqs. C1a–C2b* vanish except the first term of *Eq. C1a* and the first term of *Eq. EC2a
*. This yields
Equation C9where *F*
_{p} = *F*/(1 + β) is plasma flow rate. This equation corresponds to the one describing hepatic galactose uptake (17) and, with γ = 0 and*k*
_{seq} = 0, to the original equation by Sangren and Sheppard (39).

If hepatocytes are considered impermeable, *k*
_{ph} = 0 and hence φ = 0, therefore *Eqs. EC3a
* and *
EC3b *and the terms in *Eqs. C1a* and *C1b* containing integrals vanish. The resulting equations are equivalent to those formulated previously for the case of thiourea, an indicator that penetrates hepatocytes rapidly, if γ is interpreted as the ratio of the total extravascular distribution space to the vascular space (18).

## Appendix

### Exponentials With Matrix-Valued Exponents

The usual definition for an exponential with a matrix-valued exponent**X** is
with an identity matrix **I** that has the same dimension as **X**. Assuming all the eigenvalues of **X** are distinct, the following relations hold
and
where Λ is a diagonal matrix containing the eigenvalues of **X**, and**M** is a matrix whose columns are the eigenvectors of **X**. These expressions can be evaluated after finding the eigenvalues and the eigenvectors of the matrix **X**, using standardized numerical methods for finding the eigenvalues and eigenvectors.

## Footnotes

Address for reprint requests and other correspondence: A. J. Schwab, Rm. C10 157, Univ. Medical Clinic, Montreal General Hospital, 1650 Cedar Ave., Montreal, QC, Canada H3G 1A4 (E-mail:mchw{at}musica.mcgill.ca).

↵† Deceased 21 March 1996. The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

*advertisement*” in accordance with 18 U.S.C. §1734 solely to indicate this fact.This work was supported by grants from the Medical Research Council of Canada, the Heart and Stroke Foundation of Quebec, and the Fast Foundation. Adelar Bracht was a visiting researcher (

*Chercheur Invité)*at the Montreal General Hospital supported by the*Fonds de la recherche en santé du Québec*.↵1 Deposited with the National Auxiliary Publications Service (NAPS), c/o Micfrofiche Publications, PO Box 3513, Grand Central Station, New York, NY 10017.

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