Uptake of lactate by the liver: effect of red blood cell
carriage
Carl A.
Goresky
,1,
Glen G.
Bach2,
André
Simard1,
Andreas J.
Schwab2, and
Adelar
Bracht3
1 McGill University Medical Clinic, Montreal
General Hospital, Montreal H3G 1A4 and
2 Department of Mechanical Engineering, McGill
University, Montreal, Quebec, Canada H3A 2T5; and
3 Department of Biochemistry, University of
Maringá, Maringá 87020-900, Brazil
 |
ABSTRACT |
Multiple-indicator dilution experiments with labeled
lactate were performed in the livers of anesthetized dogs. A mixture of
51Cr-labeled erythrocytes,
[3H]sucrose, and
L-[1-14C]lactate or a mixture of
51Cr-labeled erythrocytes,
[14C]sucrose, and
L-[2-3H]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 of
L-[1-14C]lactate, metabolism to
H[14C]O
3
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 (PinS, in
ml · s
1 · g
1)
varied with total flow rate (F, in ml
s
1 · g
1)
according to PinS = (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
 |
INTRODUCTION |
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:
Na251Cr2O7 solution (6 Ci/mmol; Merck Frosst, Montreal, QC),
[fructose-3H(N)]sucrose (20 Ci/mmol;
NEN, Boston, MA), [14C(U)]sucrose (10 mCi/mmol;
ICN, Costa Mesa, CA),
L-[1-14C]lactate (63 mCi/mmol;
ICN), L-[2- 3H(N)]lactate
prepared as outlined below, and
Na2H[14C]O3 (1 mCi/mmol; NEN).
For synthesis of L-[2-3H]lactate,
[1,1-3H]ethanol was first prepared from sodium
boro[3H]hydride and acetaldehyde as follows.
Fourteen micromoles of boro[3H]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
[1,1-3H]ethanol was stored frozen. The
L-[2-3H]lactate was prepared
enzymatically from this solution and pyruvate, as described by Vind and
Grunnet (51). One-half milliliter of the frozen solution of
[1,1-3H]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: 51Cr-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, [3H]sucrose and
L-[1-14C]lactate or
[14C]sucrose and
L-[2-3H]lactate. The approximate
amounts of activities injected were 51Cr-labeled
erythrocytes, 20 µCi; 14C-labeled activity, 15 µCi; and
3H-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-[1-14C]lactate,
the anaerobic fraction collector was cooled with ice to prevent loss of
L-[2-3H]lactate and formation of
[3H]OH in the samples between collection and analysis.
Previous experiments have shown that the products, labeled bicarbonate
and CO2, ordinarily distribute rapidly into liver cells and
that their space of distribution is smaller than that available to
labeled water (43). In the experiments with
L-[1-14C]lactate, a second run was therefore
carried out using an injection mixture containing
NaH[14C]O3,
51Cr-labeled erythrocytes, and
[3H]HO, to determine the space of distribution
available to the labeled bicarbonate and CO2.
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[14C]O
3,
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[14C]O
3
to [14C]O2, 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
51Cr-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. In
L-[2-3H]lactate
experiments, the possible metabolic products containing 3H
label are [3H]HO and
[3H]glucose. Because these products were not
analyzed separately, it was not possible to include them in the
modeling of lactate metabolism. In
L-[1-14C]lactate experiments,
metabolism is expected to yield
[14C]O2 through oxidation of
pyruvate by pyruvate dehydrogenase and in the citric acid cycle (41).
At physiological pH, [14C]O2 is
converted to
H[14C]O
3
by carbonate dehydratase (carbonic anhydrase) within erythrocytes and
hepatocyte mitochondria and at the surface of endothelial cells to
attain
[14C]O2/H[14C]O
3
equilibrium (43, 50).
To determine combined 14C activity of bicarbonate and
CO2 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 14C- and 3H-labeled lactate
experiments. For the 14C experiments, this allowed the
assessment of nonionic labeled products such as glucose. For the
3H 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
[3H]HO from other tritiated metabolites such as
carbohydrates. For the 14C experiments, the contents of the
opened flasks used for the determination of 14C activity in
bicarbonate and CO2 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
3H 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,
PO2, and
PCO2 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 combination
L-Lactic Acid, Boehringer Mannheim)
In vitro kinetics in erythrocytes.
Lactate dehydrogenase present in erythrocytes catalyzes rapid
interconversion between lactate and pyruvate. The 14C label
undergoes exchange between lactate and pyruvate, whereas the
3H label appears as 3HOH. To ascertain the rate
of the exchange reaction and the composition of the equilibration
mixture, L-[14C]lactate or
[14C]pyruvate was mixed with dog blood at
37°C. Alternatively,
L-[2-3H]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 [1-14C]lactate
experiments, the
H[14C]O
3
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 CO2 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 (
) and of the common nonsinusoidal transit time,
t0 (APPENDIX A). Similarly,
simultaneous analysis of outflow profiles obtained for second runs was
used for estimating t0 and the ratios of
extravascular (combined interstitial and parenchymal) to vascular
(combined plasma and erythrocyte) space ratio for
bicarbonate/CO2 (
m) and water
(
w).
The description of lactate tracer disposition in mathematical form is
developed in APPENDIX B 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 51Cr-labeled
erythrocytes, whereas lactate in plasma travels with the lower velocity
of [14C]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),
, becomes 0.65 × 107.21
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).

View larger version (32K):
[in this window]
[in a new window]
|
Fig. 1.
Schematic illustration of rationale underlying mathematical analysis of
outflow dilution curves. cp,
ch, cr, Blood plasma,
hepatocyte, and erythrocyte lactate concentration, respectively;
kph, khp,
krp, kpr, rate constants for
hepatocyte influx and efflux and erythrocyte efflux and influx,
respectively; kseq, rate constant for metabolism;
m, concentration of
bicarbonate/CO2.
|
|
The plasma lactate exchanges with that in the hepatocytes via
carrier-mediated transport (12, 14, 25, 35). Metabolic conversion to
CO2 occurs within the mitochondrial matrix of hepatocytes (50). Product [14C]O2 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 [14C]O2
and
H[14C]O
3
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 B for erythrocyte
efflux, hepatocyte influx and efflux, and metabolism,
krp , kph,
khp, and kseq, respectively,
were varied until an optimal simultaneous fit for lactate and
bicarbonate/CO2 was found. Erythrocyte influx (rate
constant kpr) was not fitted because it is directly related to krp (Eq. B3) 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/CO2 profile is available for
analysis, values for
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/CO2, linear superposition according to the
delayed-wave model with catheter correction (16, 56) was carried out as outlined in APPENDIX A. For the lactate and
bicarbonate/CO2 curves, a similar procedure was followed,
as explained in APPENDIX B.
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-[14C]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 14C label was introduced as
[14C]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 of
L-[2-3H]lactate, a net loss of
3H label occurred with time, the label appearing as
[3H]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, PCO2,
PO2, hemoglobin concentration in
blood, and O2 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).

View larger version (29K):
[in this window]
[in a new window]
|
Fig. 2.
Representative sets of outflow dilution curves, from experiments with
low and high hepatic blood flow rate. Dashed lines represent
extrapolation of erythrocyte and sucrose profiles according to terminal
slope.
|
|
Parameters derived from primary analysis of the data are shown in Table
2. Because of technical reasons, second-run
data were not available for 3 of 16 [1-14C]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 Table
3. Also shown are the parameters from the
simultaneous fit of the transformed erythrocyte profiles to the water
and bicarbonate/CO2 profiles from second runs (only for
experiments with [1-14C]lactate injection in
the first run). Values for t0 have been corrected
for catheter transit time.
Parameters from the fit of superimposed erythrocytes and sucrose
profiles to lactate and bicarbonate/CO2 profiles are shown in Table 4. For the
[1-14C]lactate experiments that had no
second-run data, the value of
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/CO2 data were not used. However, the resulting
standard deviations of kseq (estimated using the
matrix of sensitivities) were too large to allow for a reliable
kseq estimate. Therefore, the outflow profiles for lactate and its metabolite, bicarbonate/CO2, were fitted
simultaneously to lactate and bicarbonate/CO2 data. This
provided for adequate reliability of the kseq
estimates. Because [2-3H]lactate experiments
had no product bicarbonate or CO2 identified, the
kseq value used was the average value obtained from
the analysis of [1-14C]lactate experiments.
Fits to first-run labeled lactate and bicarbonate/CO2 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.

View larger version (24K):
[in this window]
[in a new window]
|
Fig. 3.
Top: fits to labeled lactate outflow curves (solid line),
illustrating erythrocyte throughput, total (erythrocytes + plasma)
throughput, and returning components. Bottom: fits to labeled
bicarbonate/CO2 outflow curves.
|
|
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, kseq 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 kseq.

View larger version (22K):
[in this window]
[in a new window]
|
Fig. 4.
Variation of calculated outflow curves with varying parameter values.
For each pair of panels, one parameter was doubled or halved, and the
others were subsequently optimized by refitting the outflow curves to
the data.
|
|
The change in the vascular volume with flow rate is illustrated in Fig.
5A. The best fit from linear
regression is Vvasc = (6.9 ± 0.9)F + (0.05 ± 0.02), where Vvasc 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).

View larger version (15K):
[in this window]
[in a new window]
|
Fig. 5.
A: variation in vascular volume with flow rate. B:
variation of influx permeability-surface area product
(PinS) with flow rate. Linear fits to data
are also shown.
|
|
The variation in the lactate influx permeability-surface area product
with flow rate is shown in Fig. 5B. The equation for the fitted
line is PinS = (3.1 ± 0.5)F + (0.021 ± 0.014), where PinS 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 PinS 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/CO2 (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/CO2, water, or glucose
(APPENDIX C).
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,
krp, 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
-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 of L-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 Km
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 PinS 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
khp to kseq; 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
[3H]2O for
[2-3H]lactate and mainly
[14C]bicarbonate and
[14C]O2 for
[1-14C]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
[1-14C]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). [14C]pyruvate
contained in the cytosol of hepatocytes can thus be combined with
cytosolic [14C]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 14C label in
bicarbonate/CO2 is most probably due to the action of
pyruvate dehydrogenase. It has been shown in the rat liver that labeled
bicarbonate/CO2 is formed from
[1-14C]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/CO2 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[14C]O
3
reflects the activity of pyruvate dehydrogenase. However, no attempt
was made to quantify the rate of the pyruvate dehydrogenase reaction.
When [2-3H]lactate is used as a tracer,
sequestration and formation of labeled products occur in a different
manner. Lactate dehydrogenase will mediate the transfer of
3H to the
-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 14C 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 3HOH
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 3H 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.
 |
APPENDIX A |
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, CRBC(t), is the convolution of the
organ erythrocyte transport function (catheter-corrected outflow
profile or impulse response), hRBC(t), with
the outflow profile obtained from the apparatus in the absence of a
liver, Ccath(t)
|
(A1)
|
where
* is the convolution operator. Similarly, for sucrose
|
(A2)
|
The 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,
cath, combined with a simple delay,
cath.
The outflow profile of the collection catheter alone was represented by
the function
|
(A3)
|
The
impulse response of the liver was then obtained by deconvolution using
the following relation
|
(A4)
|
The
erythrocyte curve, CRBC, was approximated by a
piecewise third-order polynomial (8), which is easily differentiated.
The organ sucrose transport function,
hsuc(t), was calculated from the organ red
blood cell transport function, hRBC(t),
using linear superposition according to the flow-limited model of
Goresky (16) with uniform large-vessel transit time
|
(A5)
|
where
is the interstitial to vascular distribution spaces and
t0 is the common large-vessel transit time. The
sucrose outflow profile, Csuc(t), was then
calculated by convolution according to Eq. A1, 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
and t0.
 |
APPENDIX B |
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 cp, lactate in erythrocytes, with concentration
cr, and a combined bicarbonate/CO2 pool
representing bicarbonate and CO2 contained in blood plasma,
erythrocytes, and hepatocytes, with a plasma concentration m.
The combined pool was formulated based on the observation that
bicarbonate and CO2 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 ch.
The behavior of tracers is described by the following system of partial
differential equations
|
(B1a)
|
|
(B1b)
|
|
(B1c)
|
|
(B1d)
|
where
t is time; x is the quotient of the distance along the
length and the velocity of flow;
is the ratio of the Disse space to
that in sinusoidal plasma;
= hematocrit/(1-hematocrit) is the
erythrocytes-to-sinusoidal plasma volume ratio;
is the hepatocyte-to-sinusoidal plasma volume ratio; and
m is
the ratio of the stationary to the mobile part of the product. The
latter is given by
|
(B2)
|
where
m and
mh are partition coefficients for
bicarbonate/CO2 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
krp 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,
PrpSr/VRBC. Because
the distribution space ratio of lactate in the Disse space to that in
sinusoidal plasma,
, is the same as that for the reference
indicator, sucrose, the transfer coefficient kpr is
determined by the relation
|
(B3)
|
where
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
|
(B4a)
|
|
(B4b)
|
where
PphS and PhpS
are the permeability-surface area products for the exchange of lactate
across the hepatocyte cell membranes in the inward and outward
direction, respectively; Vp is the sinusoidal plasma
volume; and Vcell is the hepatocellular volume. The
transfer coefficient kseq 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)
|
(B5a)
|
|
(B5b)
|
where
q0 is the amount of tracer initially applied to the
entrance of the sinusoid, Fs is sinusoidal blood
flow rate, and
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 C for precursor concentrations
cr and cp. However, no solution
was found for the product plasma concentration m. Moreover, the
numerical evaluation of the solution detailed in APPENDIX C
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
|
(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 +
), and that of
total bicarbonate/CO2 is 1/(1 +
m). The
relative velocities are collected in the diagonal matrix W
|
(B7)
|
Concentration terms are collected in a vector, u, whose
elements are amounts per unit sinusoidal space, normalized to the
injected dose
|
(B8)
|
With these definitions, the system of partial differential equations
can then be written concisely as
|
(B9)
|
The
initial conditions become
|
(B10)
|
where
|
(B11)
|
The 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)dx is
the proportion of the flow with transit times between x and
x + dx], 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
|
(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.
|
(B13)
|
where
S is the Laplace variable. Equation B9 then becomes
|
(B14)
|
which
has the solution
|
(B15)
|
For
the definition of an exponential with a matrix-valued exponent, see
APPENDIX D. The transit time distribution was approximated
by a sum of n exponential terms, as follows
|
(B16)
|
where
the parameters
i and
i
are arbitrary and have no physical meaning. Thus the parameters
i and
i may be determined
from the reference curve.
Substitution of Eq. B16 into Eq. B12 yields
|
(B17)
|
and
substitution of Eq. B13 yields
|
(B18)
|
Further
substitution of Eq. B15 yields
|
(B19)
|
When the elements of the matrices A and W and the
values for
i and
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
|
(B20)
|
We
use the following multiexponential approximation for the erythrocyte
outflow profile
|
(B21)
|
where
'i are empirical coefficients. Because the
upslope is smooth, the following restriction is introduced
|
(B22)
|
To
account for the influence of the injection and collection devices,
Eq. B21 is substituted into Eq. A4 from APPENDIX
A. The result of this substitution is of the form of Eq. B20 with the following coefficients
|
(B23)
|
The
observable outflow profile of lactate CLac(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), hp(t) and
hr(t)
|
(B24)
|
According
to Eq. B19, h1(t) and
h2(t) are sums of exponential functions,
and according to Eq. A3, Ccath(t)
is a single exponential function. CLac(t)
can then be evaluated analytically using the convolution formula for
exponential functions
|
(B25)
|
Similarly,
for the product, bicarbonate/CO2
|
(B26)
|
where
hm(t) is the third element of
h(t).
The hepatocyte influx permeability-surface area product is obtained by
multiplying the value obtained for kph with the
space of distribution of lactate in the plasma
|
(B27)
|
where
F is blood flow rate, Hct is hematocrit, and
suc is sucrose mean
transit time.
 |
APPENDIX C |
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
|
(C1)
|
b) in the time interval x
t
(1 +
)x
|
(C2a)
|
|
(C2b)
|
where I0 and
I1 are modified Bessel functions of the first kind
and order 0 and 1, respectively.
c) when t > (1 +
)x
|
(C3a)
|
|
(C3b)
|
where
|
(C4a)
|
|
(C4b)
|
|
(C5)
|
|
(C6)
|
|
(C7)
|
|
(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
B, with reasonable accuracy.
Special cases.
If erythrocytes are considered impermeable, kpr = 0 and hence
= 0 and
r0, therefore all
terms in Eqs. C1a-C2b vanish except the first term of Eq.
C1a and the first term of Eq. C2a. This
yields
|
(C9)
|
where Fp = F/(1 +
) is
plasma flow rate. This equation corresponds to the one describing
hepatic galactose uptake (17) and, with
= 0 and
kseq = 0, to the original equation by Sangren and
Sheppard (39).
If hepatocytes are considered impermeable, kph = 0 and hence
= 0, therefore Eqs. C3a and C3b 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
is interpreted as the ratio of the total extravascular
distribution space to the vascular space (18).
 |
APPENDIX D |
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
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.
 |
ACKNOWLEDGEMENTS |
The authors thank Eva Ibrahim, Bruce Ritchie, and Kay Lumsden for
superb technical assistance.
 |
FOOTNOTES |
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.
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).
Received 3 April 1999; accepted in final form 1 January 2000.
 |
REFERENCES |
1.
Allard, MF,
Kamimura CT,
English DR,
Henning SL,
and
Wiggs BR.
Regional myocardial capillary erythrocyte transit time in the normal resting heart.
Circ Res
72:
187-193,
1993[Abstract].
2.
Bracht, A,
Schwab AJ,
and
Scholz R.
Untersuchung von Flußgeschwindigkeiten in der isolierten perfundierten Rattenleber durch Pulsmarkierung mit radioaktiven Substraten und mathematischer Analyse der Auswashkinetiken.
Hoppe-Seylers Z Physiol Chem
361:
357-377,
1980[ISI][Medline].
3.
Braun, W.
Bestimmung der Geschwindigkeit der Puruvatdehydrogenase-Reaktion in der perfundierten Rattenleber (Dissertation). Munich: Ludwig-Maximilians-Universität, 1976.
4.
Brooks, GA.
Lactate production under fully aerobic conditions: the lactate shuttle during rest and exercise.
Fed Proc
45:
2924-2929,
1986[ISI][Medline].
5.
Burczynski, FJ,
Luxon BA,
and
Weisiger RA.
Intrahepatic blood flow distribution in the perfused rat liver: effect of hepatic artery perfusion.
Am J Physiol Gastrointest Liver Physiol
271:
G561-G567,
1996[Abstract/Free Full Text].
6.
Chinard, RP,
Goresky CA,
Enns T,
Nolan MF,
and
House RW.
Trapping of urea by red cells in the kidney.
Am J Physiol
209:
253-263,
1965[ISI].
7.
Chow, FS,
Piekoszewski W,
and
Jusko WJ.
Effect of hematocrit and albumin concentration on hepatic clearance of tacrolimus (FK506) during rabbit liver perfusion.
Drug Metab Dispos
25:
610-616,
1997[Abstract/Free Full Text].
8.
De Boor, C.
A Practical Guide to Splines. New York: Springer, 1978.
9.
De Bruijne, AW,
Vreeburg H,
and
Van Steveninck J.
Kinetic analysis of L-lactate transport in human erythrocytes via the monocarboxylate-specific carrier system.
Biochim Biophys Acta
732:
562-568,
1983[ISI][Medline].
10.
Deuticke, B.
Monocarboxylate transport in erythrocytes.
J Membr Biol
70:
89-103,
1982[ISI][Medline].
11.
Deuticke, B,
Rickert I,
and
Beyer E.
Stereoselective, SH-dependent transfer of lactate in mammalian erythrocytes.
Biochim Biophys Acta
507:
137-155,
1978[ISI][Medline].
12.
Edlund, GL,
and
Halestrap AP.
The kinetics of transport of lactate and pyruvate into rat hepatocytes. Evidence for the presence of a specific carrier similar to that in erythrocytes.
Biochem J
249:
117-126,
1988[ISI][Medline].
13.
Enns, T,
Chinard FP,
and
Schepard RH.
A simple device for rapid serial collection of anaerobic blood samples.
J Appl Physiol
13:
513-514,
1958[ISI].
14.
Fafournoux, P,
Demigne C,
and
Remesy C.
Carrier-mediated uptake of lactate in rat hepatocytes. Effects of pH and possible mechanisms for L-lactate transport.
J Biol Chem
260:
292-299,
1985[Abstract/Free Full Text].
15.
Garcia, CK,
Brown MS,
Pathak RK,
and
Goldstein JL.
cDNA cloning of MCT2, a second monocarboxylate transporter expressed in different cells than MCT1.
J Biol Chem
270:
1843-1849,
1995[Abstract/Free Full Text].
16.
Goresky, CA.
A linear method for determining liver sinusoidal and extravascular volumes.
Am J Physiol
204:
626-640,
1963[ISI].
17.
Goresky, CA,
Bach GG,
and
Nadeau BE.
On the uptake of materials by the intact liver. The transport and net removal of galactose.
J Clin Invest
52:
991-1009,
1973[ISI][Medline].
18.
Goresky, CA,
Bach GG,
and
Nadeau BE.
Red cell carriage of label: its limiting effect on the exchange of materials in the liver.
Circ Res
36:
328-351,
1975[Abstract].
19.
Goresky, CA,
and
Groom AC.
Microcirculatory events in the liver and the spleen.
In: Handbook of Physiology, Section 2: The Cardiovascular System, edited by Renkin E. M.,
and Michel C. C.. Bethesda: American Physiological Society, 1984, p. 689-780.
20.
Goresky, CA,
Simard A,
and
Schwab AJ.
Increased hepatocyte permeability surface area product for 86Rb with increase in blood flow.
Circ Res
80:
645-654,
1997[Abstract/Free Full Text].
21.
Gutmann, I,
and
Wahlefeld AW.
L-(+)-Lactate, determination with lactate dehydrogenase and NAD.
In: Method of Enzymatic Analysis (2nd ed.), edited by Bergmayer H. U.. Weinheim, Germany: Verlag Chemie, 1974, p. 1464.
22.
Honig, CR,
Feldstein ML,
and
Frierson JL.
Capillary lengths, anastomoses, and estimated capillary transit times in skeletal muscle.
Am J Physiol Heart Circ Physiol
233:
H122-H129,
1977[ISI][Medline].
23.
Hughes, D,
Rozanski ER,
Shofer FS,
Laster LL,
and
Drobatz KJ.
Effect of sampling site, repeated sampling, pH, and PCO2 on plasma lactate concentration in healthy dogs.
Am J Vet Res
60:
521-524,
1999[ISI][Medline].
24.
Hung, HC,
and
Hoberman HD.
Influence of steric specificity on the rates of hydrogen exchange between substrates of NAD-coupled dehydrogenases.
Biochem Biophys Res Commun
46:
399-405,
1972[ISI][Medline].
25.
Jackson, VN,
and
Halestrap AP.
The kinetics, substrate, and inhibitor specificity of the monocarboxylate (lactate) transporter of rat liver cells determined using the fluorescent intracellular pH indicator, 2',7'-bis(carboxyethyl)-5(6)-carboxyfluorescein.
J Biol Chem
271:
861-868,
1996[Abstract/Free Full Text].
26.
Landaw, EM,
and
DiStefano JJ, III.
Multiexponential, multicompartmental, and noncompartmental modeling. II. Data analysis and statistical considerations.
Am J Physiol Regulatory Integrative Comp Physiol
246:
R665-R677,
1984[ISI][Medline].
27.
Large, V,
Soloviev M,
Brunengraber H,
and
Beylot M.
Lactate and pyruvate isotopic enrichments in plasma and tissues of postabsorptive and starved rats.
Am J Physiol Endocrinol Metab
268:
E880-E888,
1995[Abstract/Free Full Text].
28.
Lin, RY,
Vera JC,
Chaganti RS,
and
Golde DW.
Human monocarboxylate transporter 2 (MCT2) is a high affinity pyruvate transporter.
J Biol Chem
273:
28959-28965,
1998[Abstract/Free Full Text].
29.
Masters, C,
and
Holmes R.
Peroxisomes: new aspects of cell physiology and biochemistry.
Physiol Rev
57:
816-882,
1977[Free Full Text].
30.
Müllhofer, G,
Kuntzen O,
Hesse S,
and
Bücher T.
Isotope equilibration measurements in perfused rat liver synthesizing glucose form L-lactate-2-T-2-C14.
FEBS Lett
4:
47-49,
1969[ISI][Medline].
31.
Palm, D.
Kinetische Untersuchungen zum Mechanismus der Lactatdehydrogenase mit Tritium-Isotopeneffekten.
Eur J Biochem
5:
270-275,
1968[ISI][Medline].
32.
Pang, KS,
Barker F,
Simard A,
Schwab AJ,
and
Goresky CA.
Sulfation of acetaminophen by the perfused rat liver: the effect of red blood cell carriage.
Hepatology
22:
267-282,
1995[ISI][Medline].
33.
Pang, KS,
Sherman IA,
Schwab AJ,
Geng W,
Barker F, 3rd,
Dlugosz JA,
Cuerrier G,
and
Goresky CA.
Role of the hepatic artery in the metabolism of phenacetin and acetaminophen: intravital microscopic and multiple-indicator dilution study in perfused rat liver.
Hepatology
20:
672-683,
1994[ISI][Medline].
34.
Piekoszewski, W,
Chow FS,
and
Jusko WJ.
Disposition of tacrolimus (FK 506) in rabbits. Role of red blood cell binding in hepatic clearance.
Drug Metab Dispos
21:
690-698,
1993[Abstract].
35.
Poole, RC,
and
Halestrap AP.
Transport of lactate and other monocarboxylates across mammalian plasma membranes.
Am J Physiol Cell Physiol
264:
C761-C782,
1993[Abstract/Free Full Text].
36.
Quintana, I,
Felipe A,
Remesar X,
and
Pastor-Anglada M.
Carrier-mediated uptake of L-(+)-lactate in plasma membrane vesicles from rat liver.
FEBS Lett
235:
224-228,
1988[ISI][Medline].
37.
Romijn, JA,
Chinkes DL,
Schwarz JM,
and
Wolfe RR.
Lactate-pyruvate interconversion in blood: implications for in vivo tracer studies.
Am J Physiol Endocrinol Metab
266:
E334-E340,
1994[Abstract/Free Full Text].
38.
Rose, CP,
and
Goresky CA.
Vasomotor control of capillary transit time heterogeneity in the canine coronary circulation.
Circ Res
39:
541-554,
1976[Abstract].
39.
Sangren, WC,
and
Sheppard CW.
Mathematical derivation of the exchange of a labeled substance between a liquid flowing in a vessel and an external compartment.
Bull Math Biol
15:
387-394,
1953.
40.
Schober, KE.
Investigation into intraerythrocytic and extraerythrocytic acid-base and electrolyte changes after long-term ammonium chloride administration in dogs.
Am J Vet Res
57:
743-749,
1996[ISI][Medline].
41.
Scholz, R,
Olson MS,
Schwab AJ,
Schwabe U,
Noell C,
and
Braun W.
The effect of fatty acids on the regulation of pyruvate dehydrogenase in perfused rat liver.
Eur J Biochem
86:
519-530,
1978[Abstract].
42.
Schwab, AJ.
Extension of the theory of the multiple-indicator dilution technique to metabolic systems with an arbitrary number of rate constants.
Math Biosci
71:
57-79,
1984[ISI].
43.
Schwab, AJ,
Goresky CA,
and
Rose CP.
Handling of tracer bicarbonate by the liver. The relative impermeability of hepatocyte cell membranes to the ionic species.
Circ Res
65:
1646-1656,
1989[Abstract].
44.
Schwab, AJ,
Zwiebel FM,
Bracht A,
and
Scholz R.
Transport and metabolism of L-lactate in perfused rat liver studied by multiple pulse labelling.
In: Carrier Mediated Transport of Solutes from Blood to Tissue, edited by Yudilevich DL,
and Mann GE.. London: Longman, 1985, p. 339-344.
45.
Sies, H,
Ta ST,
Brauser B,
and
Bücher T.
Direct measurement of the state of the lactate dehydrogenase system in hemoglobin-free perfused rat liver.
Adv Enzyme Regul
10:
309-322,
1972[Medline].
46.
Silverstein, E,
and
Boyer PD.
Equilibrium reaction rates and the mechanisms of bovine heart and rabbit muscle lactate dehydrogenases.
J Biol Chem
239:
3901-3907,
1964[Free Full Text].
47.
Skelton, MS,
Kremer DE,
Smith EW,
and
Gladden LB.
Lactate influx into red blood cells of athletic and nonathletic species.
Am J Physiol Regulatory Integrative Comp Physiol
268:
R1121-R1128,
1995[Abstract/Free Full Text].
48.
Smith, EW,
Skelton MS,
Kremer DE,
Pascoe DD,
and
Gladden LB.
Lactate distribution in the blood during steady-state exercise.
Med Sci Sports Exerc
30:
1424-1429,
1998[ISI][Medline].
49.
Stein, WD.
Channels, Carriers, and Pumps. San Diego: Academic, 1990.
50.
Taguchi, Y,
Ono Y,
Lin L,
Storey BT,
Dodgson SJ,
and
Forster RE.
Mechanism of the acceleration of CO2 production from pyruvate in liver mitochondria by HCO3.
Am J Physiol Cell Physiol
273:
C92-C100,
1997[Abstract/Free Full Text].
51.
Vind, C,
and
Grunnet N.
The reversibility of cytosolic dehydrogenase reactions in hepatocytes from starved and fed rats. Effect of fructose.
Biochem J
222:
437-446,
1984[ISI][Medline].
52.
Vind, C,
Hunding A,
and
Grunnet N.
Pathways of reducing equivalents in hepatocytes from rats. Estimation of cytosolic fluxes by means of 3H-labelled substrates for either A- or B-specific dehydrogenases.
Biochem J
243:
625-630,
1987[ISI][Medline].
53.
Wasserman, DH,
Connolly CC,
and
Pagliassotti MJ.
Regulation of hepatic lactate balance during exercise.
Med Sci Sports Exerc
23:
912-919,
1991[ISI][Medline].
54.
Williamson, DH,
Lund P,
and
Krebs HA.
The redox state of free nicotinamide-adenine dinucleotide in the cytoplasm and mitochondria of rat liver.
Biochem J
103:
514-527,
1967[ISI][Medline].
55.
Wilson, O,
and
Dixon E.
Erythrocyte cation content and sodium transport in Siberian huskies.
Am J Vet Res
52:
1427-32,
1991[ISI][Medline].
56.
Yoshimura, T,
Schwab AJ,
Tao L,
Barker F,
and
Pang KS.
Hepatic uptake of hippurate: a multiple-indicator dilution, perfused rat liver study.
Am J Physiol Gastrointest Liver Physiol
274:
G10-G20,
1998[Abstract/Free Full Text].
Am J Physiol Gastrointest Liver Physiol 278(5):G775-G788
0193-1857/00 $5.00
Copyright © 2000 the American Physiological Society