Red Cell Carriage of Label
ITS LIMITING EFFECT ON THE EXCHANGE OF MATERIALS IN THE LIVER
By Carl A. Goresky, Glen G. Bach, and Brita E. Nadeau
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ABSTRACT
The red cell membrane is a permeability barrier that limits the equilibration of a
variety of solutes between red cell and plasma water. We utilized the multiple indicator
dilution technique to investigate the effect of this barrier on the exchange in the liver of
a group of tracer substances that are not removed in net fashion from the hepatic
circulation: thiourea, urea, and chloride. We demonstrated that, after preequilibration
of the label with red cells, a red cell carriage effect appeared (the trapping and
translocation of label in the red cells), that this effect was most marked when the
permeability of the red cell was relatively low for the substance under consideration
(thiourea), and that the effect became small when the permeability of the red cells was
large for the exchanging substance (urea and chloride). We developed a theoretical
description of the retarding effect of the red cell permeability barrier on the extravascular exchange of label and were able to use this description to obtain estimates of the red
cell permeability from the in vivo dilution curves. We examined the effect of plasma
injection, of changing the input in such a fashion that the label was not preequilibrated
with red cells, and found, both experimentally and theoretically, that for substances of
low permeability the transit time from these experiments, if multiplied by the total
water flow or solute flux, gave an overestimate of both the apparent total volume of
distribution and the mass of traced material in the system. This last effect is of great
importance for the practical design of many biological experiments. Reliable volume
and mass estimates can be made only when the labeled material has been preequilibrated with red cells.
KEY WORDS
red cell permeability
thiourea
urea and chloride
multiple indicator dilution studies
hepatic microcirculation
theoretical estimates of volume and mass
dog red blood cells
permeability barrier of the red cell membrane
• The membrane of the red blood cell constitutes
a permeability barrier that is significant for a
variety of low molecular weight substances. When
tracer is added to blood, this barrier limits the rate
of equilibration of label between the red cells and
the plasma. Through this effect, the barrier
changes the manner in which the label is distributed into tissue during the blood's passage through
the microcirculatory bed of an organ. In similar
fashion, if there is net removal of material by the
organ, the presence of the red cell permeability
barrier limits the accessibility of the material
within the red cells to the removal mechanism.
The phenomena which result from limited permeaFrom the McGill University Medical Clinic in the Montreal General Hospital and the Departments of Medicine and
Mechanical Engineering, McGill University, Montreal, Canada.
This work was supported by the Medical Research Council
of Canada and by the Quebec Heart Foundation.
Dr. Goresky is a Medical Research Associate of the Medical
Research Council of Canada.
Received September 14, 1973. Accepted for publication
November 4, 1974.
328
bility across the red cell membrane have not been
extensively explored in a quantitative fashion in
mammalian systems. Our purpose is to provide
such an exploration.
We selected the liver as the organ most suitable
for this exploration. The hepatic parenchymal cells
are rather more permeable to low molecular
weight substances than are the red cells, and this
permeability difference provides a unique opportunity for exploring the effects of limited permeability of the red cells on the distribution of labeled
materials. We selected, for this initial exploration,
a group of substances that are not removed in net
fashion by the liver thiourea, urea, and chloride
ions.
We carried out both an experimental and a
theoretical study. We found that the latter provided not only insight into data from the specific
experiments but also a general set of arguments
that are of unusual interest because of their importance for experimental design. We demonstrated
that, in the presence of a limited permeability at
the red cell membrane, the manner in which
Circulation Research, Vol. 36, February 1975
RED CELL CARRIAGE OF LABEL
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labeled material traverses the liver depends on the
conditions under which the label is introduced.
When the label is preequilibrated in the red cells,
the outflow pattiern includes a moiety which does
not escape the red cells during passage through the
liver. In contrast, when the label is introduced in
the plasma phase only, when none of it has previously been allowed to enter the red cells, the
outflow pattern is usually quite different. The
component trapped in the red cells is missing, and
the mean transit time is larger, although the space
of distribution available in the hepatic parenchyma is unchanged. For substances entering red
cells, this change in mean transit time with a
change in the method of introduction of tracer has
not been widely recognized previously. It affects
computation of spaces of distribution and masses of
traced material within tissue in a major way. We
have therefore not only described the general effects of the limitation in red cell permeability, but
we have also emphasized the manner in which the
effects must be considered in designing experiments.
Methods
General Experimental Procedure.—In this study we
used the rapid single injection, multiple indicator dilution technique to carry out an examination of the
behavior of labeled substances in the microcirculatory
bed of the liver. We carried out these examinations in
the dog, since blood samples of significant size can be
obtained in this species without disturbing the overall
hemodynamic stability of the animal. In each experiment we simultaneously injected three substances: (1)
51
Cr-labeled red cells, a vascular indicator which marks
out the transit pattern of red cells, (2) the substance
under study, one which enters the extravascular space
of the liver freely but is limited in its passage across
the red cell membrane, and (3) a second reference
substance, one which freely enters both the red cells
and the same extravascular space as does the study
substance or one which enters the same extravascular
space as the study substance but does not enter the red
cells at all.
In most experiments a single injection mixture was
constituted with a hematocrit matching that of peripheral blood. The 5lCr-labeled red cells were added to
plasma to which the other reference substance and the
study substance had been added. The mixture was
allowed to incubate for more than 2 minutes at 37°C
and was then placed in a single syringe for injection.
We had previously ascertained, through preliminary
experiments in which each of the study substances was
added to blood and the red cells were rapidly separated,
that this period of time was adequate for equilibration
between red cells and plasma. The group of experiments in which the injection mixture was prepared in
this fashion will be termed preincubated or preequilibrated. In a few instances, a second variety of experiment was carried out. Labeled red cells and plasma
Circulation Research, Vol. 36, February 1975
329
containing the study and second reference substances
were separated until the instant of injection. The two
mixtures were placed in separate syringes, and an
injection block was used which provided for continuous
equal mixing of the materials from the two syringes at
the point of injection. Unavoidably, a common injection
nipple was utilized and a small proportion of the
material was preincubated in this nipple. The hematocrit values were matched in such a fashion that the
hematocrit of the material injected was identical to that
of the dog. The hematocrit in the syringe containing the
red cells was thus twice that of the dog. It would have
been impossible to inject this mixture rapidly without
producing hemolysis of the red cells if the hematocrit of
the dogs had been allowed to remain at normal values.
The hematocrit in these dogs was therefore reduced by
bleeding coupled with simultaneous dextran and saline
volume replacement. This variety of experiment will be
termed non-preincubated or non-preequilibrated.
In both types of experiment the materials were
injected into the portal vein as rapidly as possible, and
serial effluent hepatic venous blood samples were collected over the next 30-60 seconds.
Special Materials.—The following labeled materials
were used: Na25ICr04 4 c/mmole (Charles E. Frosst,
Montreal); uC-thiourea 23 mc/mmole, 14C-sucrose 4.7
mc/mmole, and rPHO 2 c/mmole (New England Nuclear Corp., Boston); 14C-urea 5 mc/mmole (Merck,
Sharp, and Dohme, Montreal); and Na36Cl 0.14 me/
mmole (Radiochemical Center, Amersham).
Operative Procedure and Collection of Samples.—The
experiments were carried out on mongrel dogs anesthetized with sodium pentobarbital (25 mg/kg, iv). An
injection catheter was placed in the common portal
vein, and a collection catheter was placed in the left
hepatic venous reservoir, as described previously (1).
The dogs were given 4 mg/kg of heparin at the time of
catheter insertion. The hepatic venous samples were
pumped from the outflow catheter with a Sigmamotor
finger pump into sample tubes containing dry heparin.
The collecting system had a volume of 2.89 ml, and the
sampling rate was set at about 70 ml/min.
Analysis of Samples.—Standards were prepared from
each injection solution by addition, in serial dilution, of
blood obtained from the hepatic venous catheter prior
to collection of the samples. An 0.5-ml aliquot of sample
or diluted standard was pipetted into saline and centrifuged. The sample pellet was assayed in a Nuclear
Chicago well-type scintillation crystal gamma-ray spectrometer for the gamma rays originating from the 51Crlabeled red cells. The supernatant fluid was then prepared for liquid scintillation counting and subsequently
assayed for appropriate beta activity in a three-channel
Nuclear Chicago liquid scintillation counter. Appropriate standards and a set of simultaneous equations
were used to determine the activity due to each species.
Results
THIOUREA
In these experiments labeled red cells, water,
and thiourea were the substances injected. Dilution patterns corresponding to each substance were
detected in the hepatic venous blood. To normalize
330
GORESKY, BACH, NADEAU
TIME(XK)
FIGURE 1
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Sets of typical dilution curves derived from red blood cell
(RBC) preincubated thiourea experiments. Outflow fraction
per milliliter is plotted on the ordinate; a logarithmic scale
was used on the left and a linear scale on the right. Time
(seconds) is plotted on the abscissa; the time delay of the input
and collection systems was 2.5 seconds. Experimental values
are plotted at midintervals. The hematocrit values are given in
each case.
these patterns (and hence to provide a basis for
comparison within each experiment), the outflowing activity was expressed as a fraction of the
injected amount per milliliter of blood (i.e., a
reciprocal volume).
Red Cell Preincubated Experiments.—Figure 1
illustrates a set of experiments in which the labeled red cells were preincubated in the injection
mixture and a single syringe was used for the
injection. Results typical of experiments in dogs
with a high and a low hematocrit are illustrated.
The outflow fraction per milliliter for labeled red
cells rose to the highest and earliest peak and
decayed rapidly until recirculation occurred. For
the labeled water the initial outflow delay was
larger. It extended measurably beyond the time of
outflow appearance of the labeled red cells. The
values on the upslope then climbed slowly to a
much lower and later peak; from this peak, the
values of the downslope decayed very slowly. The
labeled thiourea curves were bimodal. A first
higher peak occurred in relation to the labeled red
cell curve, and a second later and lower component
reached its peak at a time substantially later than
did labeled water. The proportion of the thiourea
label in the first peak was much higher at the
higher hematocrit, and the first peak greatly resembled the red cell curve in the semilogarithmic
plot (Fig. 1). In fact, it was almost a scaled reproduction of the labeled red cell curve. Thus, there
was, even on inspection, substantial evidence that
a proportion of the labeled thiourea was carried
through the liver in the labeled red cells. This
impression was strengthened even further by inspection of the cumulative outflow profiles (Fig. 2).
The curves in Figure 2 show, in even more graphic
form, the influence of hematocrit on the form of the
outflow curve for thiourea. In this series the lower
hematocrit values were obtained by bleeding coupled with dextran and saline volume replacement.
The downslopes of the labeled red cell and water
curves were extrapolated in linear fashion on semilogarithmic plots to correct for recirculation, as
suggested by Hamilton et al. (2). The extrapolated
curves were taken to be the primary first passage
dilution curves, and from them estimates of blood
flow and mean transit times for the labeled red
cells and the labeled water were obtained. These
values are assembled in Table 1. In general, we
considered it important to obtain a fairly precise
definition of the early part of the bimodal thiourea
curves, and thus in some of the experiments the
labeled water curve was not followed far enough to
observe reliable indications of recirculation or to
provide an accurate value for the area under the
primary dilution curve (the experiments displayed
in Figure 1 are in this category). However, we
have previously shown, in similar dilution studies
(1), that the area under the labeled water curve is
equivalent to that under the labeled red cell curve,
and that, by assuming that labeled water undergoes delayed wave flow-limited distribution, we
can quite accurately predict the shape of the labeled water curve from that of the labeled red cell
curve. For the few prematurely terminated experiments we used these properties to extrapolate the
0
5
10
15 20 25 30 35 40 45 50
5
10 15 20 25 30 35 40 45 SO
TIMEUc)
FIGURE 2
Cumulative outflows of each indicator versus time. The broken
parts of the curves for labeled red blood cells (RBC) and
labeled water correspond to the extrapolated values. The
cumulative thiourea curves have also been arbitrarily extrapolated for the purposes of the illustration to indicate their
expected pattern of increase. All values are plotted at end
intervals.
Circulation Research, Vol. 36, February 1975
331
RED CELL CARRIAGE OF LABEL
TABLE 1
Red Cell Preincubated Thiourea Experiments: Transit Time and Flow Data and Parameters Derived from Their Analysis
Corrected curves
Original curves
Hepatic
Body Liver perfusion
weight weight (ml/sec HematoExpt. (kg)
crit
(g)
g"1)
1
2
3
4
5t
6
7t
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8
9
10
11
12
14
18
14
15
17
15
17
16
16
18
15
16
278
619
432
560
588
390
495
305
617
418
338
321
0.026
0.025
0.020
0.049
0.029
0.031
0.031
0.021
0.043
0.046
0.022
0.024
0.54
0.52
0.48
0.47
0.42
0.31
0.25
0.25
0.23
0.19
0.17
0.13
l
RBC
7
l
THO
*
(sec)
(sec)
9.08
7.87
15.27
5.75
10.16
5.85
6.18
8.62
5.37
8.68
6.60
8.55
33.91
36.53
49.33
23.34
31.46
31.26
32.50
45.69
19.57
26.40
32.93
32.13
y
(sec)
0.87
0.81
0.69
0.66
0.54
0.33
0.25
0.25
0.22
0.17
0.15
0.11
3.01
2.52
4.65
1.67
3.33
1.75
2.14
1.93
0.62
2.43
1.20
3.28
8.72
13.57
5.65
7.96
5.83
11.47
9.18
8.74
4.76
3.94
7.40
6.72
*•
to
(sec"1)
(sec)
0.077
0.057
0.087
0.048
0.080
0.070
0.054
0.044
0.069
0.057
0.073
0.063
2.52
2.18
4.11
1.76
2.94
1.70
1.65
1.48
0.43
1.92
0.96
2.89
7
9.09
14.40
5.79
10.82
6.21
14.79
9.86
9.26
5.40
4.11
8.32
7.24
*.
(sec-1)
0.081
0.061
0.88
0.066
0.084
0.090
0.056
0.046
0.078
0.059
0.081
0.067
i/tBc = red cell mean transit time, tTH0 = labeled water mean transit time, /3 = ratio of red cell water space to plasma water
space, t0 = large vessel transit time, y = ratio of extravascular water space to sinusoidal water space, and £, = red cell influx
coefficient.
*Mean transit time was corrected for catheter transit time.
tThese experiments are illustrated in Figures 1-4.
curves and estimate the mean transit time of the
labeled water.
The hematocrit and the water contents of the
two phases, red cells and plasma, were then used
to estimate the total water flow (for this estimation
we used the following values: a specific gravity of
1.094 for dog red cells [3] and a fractional water
content for these cells of 0.644 g/g [4], the product
of which gives a red cell water content, fr, of 0.70
ml/ml and a plasma water content, fp, of 0.94 ml/
ml [4]). From this flow and the transit time for
labeled water, the total water content of each liver
was then estimated. The values included the water
contained in the blood in the large draining and
feeding vessels. For comparison the water content
of the postmortem liver was obtained from its
weight and the fractional water content (average
0.75 ml/g) of a tissue sample. The dilution estimates averaged 1.09 times the value obtained from
the specimen (the large vessels of the postmortem
liver were, however, empty of blood). The agreement was quite satisfactory and indicated that the
dilution estimates were of the right order of magnitude.
There is no analogous simple way to complete
the characterization of the primary thiourea dilution curves. However, since thiourea is expected to
distribute in an equilibrious fashion into the water
space of liver tissue, it appears appropriate to
expect its mean transit time to equal that of
Circulation Research, Vol. 36, February 1975
labeled water. This hypothesis will be examined in
detail later in this paper.
Use of Modeling to Analyze the Dilution Experiments.—It is impossible to gain further useful
insight into the factors shaping the thiourea outflow dilution curves without utilizing modeling.
We therefore have developed, in the Appendix, a
kinetic model to describe both the simultaneous
red cell carriage of label and the flow-limited
distribution of thiourea into the liver tissue. The
modeling is distributed in space, and the outflow
curves are viewed as the result of events occurring
upstream from the point of observation.
The characteristics of the modeling are best
understood by reference to Figure 10 in the Appendix. For a single sinusoid-tissue unit, the outflow
profile for a substance undergoing red cell carriage
and delayed wave flow-limited distribution into
the tissue consists of four parts: two moieties
representing material originally present in the red
cells and the plasma, respectively, which emerge
without ever having left, these phases, and two
exchanging moieties, which emerge either in the
red cell or the plasma phase. The first component
to emerge consists of material that was originally
present in the red cells and did not leave them
during passage through the system. In form, it is a
damped impulse function. The last component to
emerge consists of that material in the plasma
phase which never entered the red cells and propa-
332
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gated along as a delayed wave in the plasma phase
of the sinusoids and in the surrounding hepatic
parenchyma. In form, it is a delayed damped
impulse function. The velocity of this component in
relation to that of the red cells is described by the
ratio of the water content of the sinusoidal plasma
to the sum of the water contents of the extravascular tissue and the sinusoidal plasma. Between
these two extremes, the other two components
emerge as spread out functions.. The exchanging
component emerging in the red cells consists of
material that entered the red cells from the plasma
phase during transit along the sinusoid; this component originated both from material originally in
the plasma phase at the origin and from material
that left the red cells only later to reenter them.
Conversely, the exchanging component emerging
in the intermediate plasma component consists of
both material that originated in the red cell phase
at the origin and material that entered the red
cells and later reentered the plasma phase during
passage along the sinusoid. As the permeability of
the red cells increases, the proportion in the two
impulse functions (the red cell and delayed plasma
waves) decreases until finally, in the limit, when
the red cell membrane is completely permeable to
the material, the label propagates as an intermediate delayed wave. In this case, which corresponds
to that for labeled water, the velocity is related to
that of the labeled red cells by the ratio of the total
water content of the sinusoid (red cells and plasma)
to the sum of the water contents of the extravascular tissue and the sinusoid. Thus, from the modeling, we expect the labeled water curve to be
located in time at a position intermediate between
the two components of the thiourea curve.
After the blood has left the sinusoids, a process
of postsinusoidal equilibration occurs. The processes leading to a progressive separation between
the labeled thiourea in red cells and plasma are no
longer operative. The label emerging in the red
cells undergoes equilibration with the surrounding
plasma, and the label in the plasma undergoes
equilibration with the red cells. This process takes
place in the large vessels, the collecting system,
and the collected samples themselves. The continuation of the process during recirculation means
that much of the label emerging early in red cells
is re-presented for recirculation in the plasma
phase. Despite this phenomenon, some of the thiourea activity is re-presented in the red cells and
reemerges at the outflow before the labeled water
has begun to recirculate. This component is not
easily distinguishable. The continuation of the
GORESKY, BACH, NADEAU
equilibration process in the collected samples results in equal concentrations of label in red cell
and plasma water. In these experiments we sampled the plasma phase of the samples long after
equilibration was complete, and the thiourea activity curve constructed from these measurements
therefore represents a summation of the label
outflow in both phases.
Utilizing the modeling developed in the Appendix, we analyzed the outflow activity-time curves.
Five natural parameters arise in the course of the
modeling: /3 (the ratio of the red cell water space to
the plasma water space in the blood), t0 (the large
vessel transit time), y (the ratio of the extravascular water space to the sinusoidal plasma water
space), kt (the red cell influx coefficient with the
dimensions sec"1), and kjk^ (the ratio of the influx
coefficient to the efflux coefficient). The factors /3
and kjk2 can immediately be evaluated. The value
for /3 can be calculated directly from the hematocrit and the water contents of the two phases of
blood (in this series of experiments it varied from
0.87 at a hematocrit of 0.54 to 0.11 at a hematocrit
of 0.13). The use of a value for /3 calculated in this
way implies that, in utilizing the modeling, we
have assumed the value for the sinusoidal hematocrit to be the same as that for the peripheral blood.
Now consider the second parameter, the ratio k1/k2.
The process of entry of thiourea into red cells
results in equal concentrations in red cell and
plasma water (5); thus, for this substance, the ratio
kjk2 = 1. For labeled water, the ratio is also unity.
The processes of evaluating the three remaining
parameters are somewhat more complex. Prior to
describing these procedures, it must be emphasized
that there are two other implicit assumptions
underlying this analysis: (1) thiourea and the other
solutes being evaluated do not damage the erythrocytes or change their permeability characteristics
and (2) trace amounts of solutes are being used so
that there is no appreciable net water movement
or change in the volume of the erythrocytes during
their passage along the sinusoids.
We first used the labeled red cell and labeled
water curves to derive estimates of (1 + /3 + y)/(l
+ /3) and ^ in the same manner described previously (1, 6). We expected from the Appendix, that
the ratio of the normalized labeled red cell peak
height to the normalized labeled water peak height
would provide an approximate value for the ratio
(1 + /3 + y)/(l + /3); if this ratio and t0 are optimal,
then, when each of the values of the labeled water
curve is multiplied by this ratio and the time of
each of these values beyond t0, t - to, is divided by
Circulation Research, Vol. 36, February 1975
333
RED CELL CARRIAGE OF LABEL
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this ratio, the adjusted labeled water curve will
superimpose on the labeled red cell curve. Utilizing
a digital computer, we developed continuous thirdorder spline fits to the cumulative data to take into
account the mechanical integration of each sample
over its collection period, and with these fits we
developed an iterative procedure to optimize the
superimposition of the labeled water curve on the
labeled red cell curve. This procedure provided
best-fit estimates of the ratio (1 +/3 +-y)/(l +0)
and of t0 and hence of y. The average relative
coefficient of variation of the fit was 0.093.
A second iterative procedure was then developed
to provide an optimal estimate of the red cell influx
coefficient A,. The relations developed in the Appendix (Eqs. 12-14) were utilized to derive a computed thiourea curve. First a hypothetical curve
with a volume of distribution corresponding to the
plasma component of the thiourea curve, one with
an extravascular space of distribution y times the
plasma water space, was generated from the labeled water curve. From this curve, the labeled red
cell curve, the values already obtained for /3, t0, y,
and kjk^, and a selected value for A,, a computed
thiourea curve was generated. In each case the
coefficient kx was then varied until the best possible fit between the computed curve and the experimental data was attained. This procedure resulted
in a computed curve, the second component of
which systematically deviated from the experimental data in a fashion which indicated that the
value selected for y was too small. This lack of
agreement was expected, since the distortion in the
curves due to the large vessels and the collecting
system should reduce the magnitude of the most
rapidly changing curve, that for red cells, most and
the magnitude of the least rapidly changing curve,
that for the second or delayed component of the
thiourea curve, least (7). The iterative procedure
for optimizing the value for &, was therefore
changed to one which simultaneously optimized
values for both kt and y. The fit obtained was
better and no longer displayed the systematic
deviation observed earlier. The values obtained are
presented in Table 1. The average relative coefficient of variation of the fit between the computed
and experimental curves was 0.069, and the new
value for y was, on the average, 1.13 times that
which would have been predicted from the relation
between the labeled water and labeled red cell
curves.
This first set of parameters was generated from
the raw experimental data. These data have, however, been altered by the delay in and the distorCirculation Research, Vol. 36, February 1975
Hema»ocrit..41
5
10 15 20 25 30 35 40 45 50 55 6 0 65 70
Hematocrit..25
O
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70
TIME (sec)
FIGURE 3
Components of the fitted curves. The third and fourth components (the intermediate profiles emerging in red cells and
plasma, respectively) are no longer distinguishable one from
the other after sample equilibration.
tion of the collecting system (7). We therefore
removed these factors from the data by a process of
deconvolution and then utilized the procedure just
outlined to obtain numerical estimates of the fitting parameters from the corrected curves (Table
1). The new values obtained for y and kt were
somewhat larger. The average coefficient of variation of the iterative fits to the corrected thiourea
curves was 0.072, and the value computed for y
was now, on the average, 1.09 times the value
which would have been predicted from the corrected labeled water and red cell curves.
Figure 3 demonstrates the results of the analysis
of the corrected curves arising from the experiments illustrated in Figure 1. The two major
components making up the curves are: (1) the first
shaded peak, that part which has come through
the sinusoids without leaving the red cells, and (2)
the second shaded peak, that part which has come
through the sinusoids in the delayed wave without
entering the red cells. The latter has propagated
along both in the water of the plasma phase of the
334
GORESKY, BACH, NADEAU
sinusoids and in the water of the extravascular
tissue. The reduction in magnitude of the delayed
component, its damping, by loss of label to the
intermediate components, is much more marked in
the high-hematocrit experiment. The third and
fourth components (those emerging in the red cells
and the plasma, respectively) fill in the valley
between the initial and final peaks. These intermediate components, which are not easily separable
one from the other, form a larger part of the total
response when the hematocrit is higher. The kind
of fit obtained between the computed and corrected
curves for our two example experiments is illustrated in Figure 4.
In the Appendix (Eqs. 9 and 10) we have shown
that the first component of the thiourea curve, the
label carried in the red cells, is reduced by the
term exp( - k^xiW), where x/W is a transit time,
the ratio of the distance traveled to the velocity of
blood flow, and that the last component of the
curve, the final delayed concentration wave, is
reduced by the term exp(- kfalW). We have plotted the computed values of the damping constants,
k2 andfej/3,both for the raw experimental and the
corrected data as a function of the hematocrit in
Figure 5. The values obtained for k2 scatter but
show no consistent change with hematocrit; indeed, ki appears to be independent of hematocrit.
The average of the values obtained from the raw
experimental data was 0.068 ± 0.031 (SD) sec"1,
and that obtained from the data corrected for the
distortion imposed by the collecting system was
0.072 ± 0.029 sec"1. In view of the lack of change
of k2 with hematocrit, and because we know that kt
= k2 in this particular instance, Jfe,/3 must vary
Hematocrit=.41
9
u
°0
5
10 15 20
3<T~35 40 45 50 55 60
11.
I
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° 3
Hematocrit..25
0
5
10
15 20 25 30 35 40 45 50 55
60
TIME (sec)
FIGURE 4
Best-fit calculated thiourea curves and the experimental data.
The solid circles represent the experimental data, and the open
circles represent the computed values. The values predicted
from the computer program for times beyond that of the last
sample are also included. The coefficient of variation for the
fit in the high-hematocrit experiments (top) was 0.036 and
that in the low-hematocrit experiments (bottom) was 0.034.
09
ce.
&.
Ob.
.05
.03.
.02
J01
A
0
HEMATOCRIT
FIGURE 5
Relation between the damping constants, kj and k,/3, and the hematocrit. The solid circles
represent values arising from the raw experimental curves, and the open circles represent those
from the corrected curves. The solid line in the right section corresponds to the average of the
values for k, derived from the raw experimental curves, and the broken line corresponds to the
average derived from the corrected curves.
Circulation Research, Vol. 36, February 1975
335
RED CELL CARRIAGE OF LABEL
,5>ORBC
5
3
"a
35
0
TIME (ax)
FIGURE 6
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Dilution curves from a thiourea experiment in which the red blood cells (RBC) were not
preincubated prior to injection. The mean transit times, corrected for that of the collecting system,
were 3.79 seconds for red blood cells and 20.69 seconds for thiourea; that for the extrapolated
thiourea curve was 23.76 seconds. The hematocrit was 021, and the corresponding value forfi was
020; the fitted values for y and t,, were 8.03 and 0.12 seconds. Perl has suggested in an
accompanying article (9) that the value for the red cell transfer coefficient could be more simply
calculated by use of the mean transit times in the non-preequilibrated case. The value obtained by
use of his expression is of the same order of magnitude as that derived by the more complex fitting
procedure.
with hematocrit in the same way that /3 varies
with hematocrit. This parameter has the value
zero when the hematocrit is zero, and it increases
asymptotically without limit as the blood approaches the state in which it is composed entirely
of red cells. Thus, the red cell wave loses its
contained label at a rate which is independent of
hematocrit, whereas the delayed plasma wave
loses its contained label to the more rapidly advancing red cells in a way which varies with
hematocrit in the same way that /3 varies with
hematocrit. The latter effect becomes increasingly
larger at high hematocrit and underlies the large
reduction of the second component which is evident
in our example experiment at the higher hematocrit.
The corrected data still contain a residual distortion imposed by the large vessels, and hence our
final estimates of/3 and k, may still be slightly too
low. However, since the change in the estimates of
k, and &2 with removal of catheter distortion was
relatively small in these experiments, we expect
that the deviation between our estimates of A, and
k2 and the true values is also quite small. We have
therefore treated the values obtained from the
corrected curves as true values.
One minor set of observations should also be
recorded. In three experiments Evans blue dye, T1824, was included in the injection mixture. The
curves arising from these experiments had the
same general shape as those just presented, but an
adequate curve fit could not be obtained. We found
on inspection of red cells suspended in correspondCirculation Research, Vol. 36, February 1975
ing concentrations of T-1824 that the red cells had
become both cup-shaped and spherocytic. These
chemically misshapen erythrocytes (8) may have
had exchange properties different from those of
normal cells. These experiments were therefore not
included in the data presented in this paper.
Red Cell Non-Preincubated Experiments.—We
will now examine what occurs when the red cells
are not preincubated, i.e., when the bulk of the
thiourea label is separated from the red cells until
the instant of injection. In Figure 6, the set of
outflow dilution curves arising from such an experiment is portrayed. The hematocrit of this dog was
0.21. The experiment suitable for comparison is
that illustrated in the bottom sections of Figure 1,
where the hematocrit of the dog was 0.25. The
labeled red cell and labeled water curves in the
two illustrations are similar. In contrast, the labeled thiourea curve in Figure 6 differs quantitatively from that in Figure 1. There is a very much
reduced initial component in relation to the labeled
red cell curve. This component, that due to carriage of label through the liver via red cells, has
become exceedingly small. The curve then grades
over into a later component, substantially higher
in magnitude, which peaks somewhat later in time
and only slightly lower in magnitude than does the
labeled water curve. The corresponding cumulative
outflow curves are illustrated in Figure 7. The
curve for labeled thiourea has the form expected,
in comparison with the preloaded experiment with
a similar hematocrit (Fig. 2, right). The initial rise
in the curve is exceedingly small. The bulk of the
336
GORESKY, BACH, NADEAU
1.0
The cumulative outflow curves can also be compared with those for the single sinusoid illustrated
in Figure 11 in the Appendix. In the present case,
the expected differences are seen. The first component of the thiourea curve is almost absent, and
the bulk of the label emerges delayed with respect
to the labeled water curve.
In the Appendix we examined the expected effect
of the red cell non-preincubated kind of input on
the mean transit time for a substance like thiourea
which undergoes red cell carriage. We demonstrated that, because of the separation of the label
from the cellular phase of the blood at the input,
the mean transit time for thiourea is larger than
the value which, when multiplied by the total
water flow, provides an accurate estimate of the
total space of distribution accessible to this substance (see Fig. 13). This phenomenon is of importance when qualitative (and quantitative) judgments are made from experiments of this kind.
The modeling developed in the Appendix was
also utilized to gain estimates of the red cell
transfer coefficient. The case which we examined
was one in which a non-preequilibrated bolus was
introduced directly into the sinusoid. In the experimental situation there are two additional features.
The red cells in the non-preequilibrated injection
bolus begin to take up the thiourea label immediately in the input catheter and the portal vein,
and a small proportion of the injection bolus is
preequilibrated in the injection nipple prior to
input. This portion was carefully measured. To
deal with the non-preequilibrated portion, we assumed that the rate constant for thiourea exchange across the red cell membrane was the same
during bulk flow in the presinusoidal portal vessels
as it was during the process of bolus flow in the
sinusoid.
Analysis of the data from the experiment illustrated in Figure 6, after correction for catheter
distortion, yielded a kt estimate of 0.087 sec"1; for a
second experiment, the value was 0.073 sec "'.
These values correspond closely to those arising
from the preequilibrated experimental data.
Again, in these two experiments, the values computed for y were larger than (1.06 and 1.02 times)
the values which would have been predicted from
the corrected labeled water and red cell curves.
.9
B
fc .6
° .5
LLJ
Hct-210
I
»5lCr
_ _•RBC
o3HHO
14
9 C Thiourea,
non-pre-equilibrated
I '3
I
^ .2
.1
K)
20
30
40
TIME (sec)
50
60
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FIGURE 7
Cumulative outflow curves for the experiment illustrated in
Figure 6. In this format it is obvious that the mean transit
time for the labeled thiourea is larger than that for the labeled
water. RBC = red blood cells.
cumulative outflow curve occurs as a later component that is delayed in relation to the labeled
water curve.
We again utilized the modeling developed in the
Appendix to gain insight into the factors shaping
the labeled thiourea curve in this new case and, in
particular, to gain insight into the specific differences imposed by the different manner of input of
the labeled thiourea. The characteristic features of
the new response are again best understood by
reference to Figure 10 in the Appendix. When no
labeled material is allowed to enter the red cells
prior to input into the sinusoids, the part of the
outflow that would have represented material originally present in the red cells, which came through
the sinusoids without leaving these cells, is missing. The component in the plasma phase, which
sweeps along to emerge at the outflow in delayed
fashion without entering the red cells is, of course,
much larger, and this component of the dilution
curves is consequently greatly emphasized. The
two remaining exchanging components, which
emerge either in the red cells or the plasma, are
modified in the manner expected: that arising from
entry of label into red cells from the plasma phase
is larger and that arising in the plasma from, this
time, only reentry into the plasma phase from the
red cells is smaller. The major grossly recognizable
changes were expected to be the absence of the
early red cell component and the emphasis of the
later wave, which is delayed with respect to labeled water. These are the changes which we
observed.
UREA
Urea is a much smaller molecule than thiourea,
and it permeates the red cell much more quickly
(5). If our general kinetic hypothesis is correct, the
urea curve will approach that for labeled water,
and the two components most evident in the thioCirculation Research, Vol. 36, February 1975
337
RED CELL CARRIAGE OF LABEL
FIGURE 8
Dilution curves from an experiment in which labeled urea was incorporated in the injection
mixture. The red blood cells (RBC) were preincubated prior to injection.
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urea curves, that traveling in the red cells and that
delayed with respect to the labeled water, will
disappear.
A typical set of dilution curves from a urea
experiment is illustrated in Figure 8. Prior to
injection the labeled urea was allowed to equilibrate in the red cells. As expected, the labeled urea
curve approached the labeled water curve in form,
and, as would have been predicted from Figure 10
in the Appendix, the labeled urea curve rose
slightly more quickly on the upslope.
The general procedure outlined in the preceding
section for the analysis of the thiourea experiments
was followed for the urea experiments, and an
estimate of the red cell influx coefficient, kx, was
obtained. We again assumed that the ratio kjk^ =
1.0. Two experiments were analyzed. The values
obtained for kt from the raw experimental curves
were 13.00 and 13.95 sec"1, and those obtained
from the data corrected for catheter distortion were
13.87 and 17.15 sec"1. The average of the two
values obtained from the corrected curves was
15.51 sec"1. This value is 215 times as large as the
average value for thiourea, and the permeability of
the red cell membrane would be expected to be
that much larger. The coefficient of variation of the
fit to the urea curve was, in each case, 0.09.
Analysis of these experiments, in which kt was
very high, required a great deal of computing time,
even though we utilized those asymptotic forms of
the imaginary Bessel functions which are designed
for numerical evaluation when the values of the
argument are large. It should be noted that in this
range the change which occurs in the form of the
computed curves with fairly large changes in the
value of &, is small (the curve is approaching the
form characteristic of an infinite kt value).
The shape of the urea curves and their susceptibility to quantitative recapitulation indicate that
the general outline of our hypothesis is adequate.
Circulation Research, Vol. 36, February 1975
It should however be noted that in this analysis we
have neglected the miniscule net production of
urea by the liver during a single passage.
CHLORIDE
The chloride anion penetrates the red cells
quickly (10). Because the red cell is filled with
hemoglobin, the equilibrium concentration of chloride in the red cells is lower than that in the
plasma: expressed in terms of red cell and plasma
water, we found it to be 0.70 times the plasma
concentration when the red cells were fully oxygenated. The relative exclusion of the chloride
anions from the red cells appears to arise as part of
two phenomena: (1) an ordinary Donnan equilibrium, which occurs because the hemoglobin is
above its isoelectric point, at ordinary pH values,
and (2) a cooperative interaction between the hemoglobin molecules at high concentration, which
produces an exclusion of anions even larger than
otherwise expected (11). During passage of blood
through the liver, the hemoglobin loses its oxygen
and becomes less ionized, and the pH of the injectate decreases somewhat; there is a minor but
detectable net increase in the chloride and bicarbonate contents of the red cells (12). In our present
analysis we neglected the last phenomenon and set
the ratio kt/k2 = 0.70.
In these experiments labeled red cells, labeled
sucrose, and labeled chloride were introduced together after preequilibration. A typical set of outflow dilution curves is illustrated in Figure 9. The
36
C1- curve is very different from that for labeled
sucrose. The upslope values begin earlier and rise
more rapidly, the peak is higher and earlier, and
the downslope decreases earlier and slightly more
rapidly. The chloride curve is, like the urea curve,
unimodal, and it lies between the labeled red cell
and labeled sucrose curves in time and magnitude.
Prior to analysis of the data in the fashion outlined
338
GORESKY, BACH, NADEAU
• 51 ORBC
o ^ C S«ro>e
e 3 * Chloride
50
20
30
TIME (sec)
FIGURE 9
Typical set of dilution curves from a labeled chloride experiment. RBC = red blood cell.
Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017
previously, we carried out a preliminary evaluation of whether the labeled sucrose curve was an
appropriate extracellular or second reference.
From the Appendix we expect that, in the preequilibrated case, the mean transit time for labeled
chloride (^-,) will be independent of the value of
the rate constant for exchange across the red cell
membrane and that
1
[i + (
•RBC
[1 +
the two values indicates that no important amount
of the chloride anion enters the liver cells during
the time of a single passage.
The values for A, computed from the dilution
curves corrected for catheter distortion were 3.36,
5.56, 4.66, and 3.67 sec"1; the average of these
values was 4.31 sec"1. On a comparative basis, this
value is 60 times as large as the average for
thiourea and approximately 0.25 times that for
urea. The average coefficient of variation of the fit
to the chloride curves was 0.09. For this substance
the curve once again approached the intermediate
form characteristic of a substance with an infinite
k{ value. In this regime, where kl is large, only a
small change occurs in the form of the computed
curve with a relatively large change in the value of
•t EV>
where tRBC is the mean transit time for labeled red
cells and tEV is the appropriate extravascular reference for the chloride curve. We determined the
transit times for labeled red cells, chloride, and
sucrose from the extrapolated curves, estimated the
transit time for the appropriate extracellular reference, and compared this value to that for labeled
sucrose. The data are displayed in Table 2. For the
four chloride experiments the average ratio between the transit time computed for the appropriate
extravascular reference and that for labeled sucrose
was 1.01. The value is so little different from unity
that we utilized the sucrose curve as the appropriate index for the initial extravascular space of
distribution of labeled chloride. The coincidence of
Discussion
Form of the Curves.—The outflow curves for
thiourea, after preequilibration, exemplify the pattern typical of a substance undergoing substantial
red cell carriage through the liver because of the
limiting effect of the exchange at the red cell
membrane. Two distinct major nonexchanging
components are clearly evident: that carried in red
TABLE 2
Data from Preequilibrated Labeled Chloride Experiments
Liver
weight
Expt.
Body
weight
(kg)
1
2
3
4
13
16
18
19
tRBC
trlCT
ta
(g)
Hepatic
perfusion
(ml/sec g~')
Hematocrit
(sec)
(sec)
(sec)
406
399
457
378
0.030
0.038
0.040
0.037
0.43
0.42
0.41
0.42
13.89
7.78
11.22
14.68
24.47
15.56
18.77
23.30
21.96
13.91
16.62
21.38
tRBc ~ r e d cell mean transit time, <„,.,. = sucrose mean transit time, la = chloride mean transit time, and tEV
time for the appropriate extravascular reference for the chloride curve.
1.01
1.04
0.98
1.02
=
mean transit
Circulation Research, Vol. 36, February 1975
RED CELL CARRIAGE OF LABEL
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cells and that delayed with respect to labeled
water, a later component which is undergoing
delayed wave flow-limited distribution into the
tissue from the plasma space. In contrast, the two
minor exchanging components (those leaving the
sinusoid in red cells and plasma, respectively) that
fill the valley between the two major peaks are not
so easily discernible. The variation in the proportion emerging in each of the major components
with a change in hematocrit clearly illustrates the
volume component of the red cell carriage phenomenon; the increase in the rate of loss of material
from the delayed wave to the more rapidly advancing red cells with an increase in /3 illustrates the
surface component of the phenomenon. These
changes are the ones expected on the basis of our
analysis of the phenomenon.
When the red cells have not been preincubated
in the injection mixture prior to injection and when
exchange at the red cell membrane is limiting, the
major part of the outflow curve is the delayed
component arising from flow-limited distribution of
the plasma label into the extravascular space. The
initial component resulting from carriage through
the organ by the red cells is lacking. The two
exchanging components arising from the passage
of materials across the red cell membranes during
their passage through the organ continue to contribute to the outflow in a minor fashion in the
interval between the vascular reference and the
delayed component.
When the rate constant for exchange becomes
very large, the intermediate components increase
in magnitude and finally dominate the outflow
patterns for both cases, preincubated and nonpreincubated. In the extreme in which the red cell
permeability is unlimited, the outflow responses
become identical. The total response then consists
of a unimodal peak, intermediate between the red
cell reference curve and the delayed wave plasma
reference curve.
Effect of the Carrying Capacity of the Red Cell
Volume Fraction on the Red Cell Carriage Phenomenon.—The effect of the carrying capacity of the
red cell volume fraction, the hematocrit, on the red
cell carriage phenomenon has previously been encountered and described. Chinard et al. (13) examined renal venous outflow patterns after the intraarterial injection, into the plasma phase, of labeled
urea and labeled creatinine. In this organ the
outflow patterns for a large group of substances
converge on the outflow pattern for creatinine
when this substance has not been preequilibrated
with red cells, and the transit pattern for the
Circulation Research, Vol. 36, February 1975
339
creatinine appears to mark out the flow-limited
pattern of distribution for plasma low molecular
weight substances into the extracellular space of
the renal cortex (14). The outflow pattern for
creatinine is delayed with respect to the labeled
red cells in the manner that would be expected if
the flow limitation is of the delayed wave variety
(1). At exceedingly low hematocrit values the
transit pattern of labeled urea approximates that
of creatinine, but as the hematocrit is increased an
increasing proportion of the labeled urea is carried
along ahead of the creatinine and appears at the
outflow between labeled red cells and the creatinine, even though the labeled urea has not been
preequilibrated with the red cells. The urea precession with respect to the creatinine is quantitatively
related to the hematocrit. The observed labeled
urea outflow curves are unimodal and are of the
form expected if the rate constant for exchange of
urea is high in relation to the time of transit.
Under these circumstances significant labeled urea
entry into red cells should be expected in the
preglomerular arterial system. That this entry
does occur is demonstrated by a progressive reduction in the glomerular filtration of the labeled urea
in relation to labeled creatinine with an increase
in hematocrit.
Non-preincubated thiourea exhibits a similar
precession over creatinine in the renal circulation
but to a lesser degree. In this original study these
findings raised the suspicion of red cell carriage;
the presence of the phenomenon was confirmed by
carrying out experiments in which the red cells
were incorporated into the injection mixture and
preincubated prior to input. As expected, an early
shoulder emerged on the thiourea curve, concomitant with the vascular label and prior to the
outflow emergence of labeled urea. Changes in the
downslope of the curve were not expected and so
the surface effect associated with an increase in
hematocrit remained unappreciated.
Effect on Red Cell Carriage of the Increase in
Surface Associated with an Increase in Hematocrit.—The proportional increase in the entry of
label into red cells from the plasma phase of blood,
which we have observed with an increase in hematocrit, is logical and expected. The effect is emphasized in that part of the outflow response which
consists chiefly of material originally introduced in
the plasma phase, the trailing part of the outflow.
In the thiourea experiments, it is illustrated by the
increased "clear space" at long time in the highhematocrit section of Figure 3. This hematocritsurface effect is less obvious but also present in the
GORESKY, BACH, NADEAU
340
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modeling which Paganelli and Solomon (15) developed to analyze their flow tube study of the exchange of tritiated water across the membrane of
the human red cell. In that analysis they developed a parameter k/vq, which is equivalent to our
kx. This parameter was then used in a fashion
which automatically accommodated for the hematocrit effect.
The average value which they found for kt for
tritiated water was 119 sec"1. The permeability of
human red cells to labeled water thus appears to
be at least eight times as large as the permeability
of dog red cells to urea.
Conversion of Our Rate Constants for Red Cell
Exchange to Permeability Coefficients.—The physical parameters describing dog blood (5.95 x 106
erythrocytes/mm3 at a hematocrit of 0.441, a mean
corpuscular volume of 74 x 10~12 cm3, a surface
area of 126 x 10"8 cm2, and a water content of 0.70
ml/ml) can be used to convert our measurements
to permeability coefficients (16). We find, from the
measurements outlined for dog red cells, that the
ratio of the surface area to the water content (Sr/
V ra ) is 24,324, where the dimensions of the measurement are cm"1. Now the parameter which we
have determined from our experimental data, kt, is
equal to k^iSrlVn,), where kx' is the membrane
permeability coefficient, or, in some cases, transport coefficient in cm/sec. This value, in turn, can
be converted to the thermodynamic form of the
permeability coefficient a> by use of the expression
at = ki 'IRT, where R is the universal gas constant
and T is the absolute temperature (17). The values
resulting from our experiments are displayed in
Table 3.
The value that we obtained for urea can be
compared with the value obtained for human blood
by Savitz and Solomon (17) by use of the isotope
exchange flow tube method. They reported an
average value for the thermodynamic form of the
permeability coefficient, w, of 13 x 10~15 moles/
dyne cm"1, which corresponds to a &,' value of 320
x 10~6 cm/sec. This value is of the same order of
TABLE 3
Red Cell Permeability Coefficients
Substance
Thiourea
Urea
Chloride
103 x k,'
(cm/sec)
1015 x o>
(moles/dyne sec"1)
2.9 ±0.6
640+90
180 + 40
0.11 ± 0.02
24.70 ±3.70
6.70 ±0.06
All values are means ± SD.
magnitude as that which we found for dog red
cells, as expected from the similar hemolysis times
in these species (5). The ratio of the permeability of
the dog red cells to urea and thiourea is also of the
order expected on the basis of their differing hemolysis times (5). The permeability coefficient that
we obtained for chloride can be compared with that
which arises from the data obtained by Tosteson
(10). From Figure 4 in reference 10, where the
half-time of exchange and the hematocrit are
given, we find, from human blood, by use of Eqs.
l l a and l i b and the physical parameters of
human red cells (16), that fc,' is 70 x 10"6 cm/sec
and that w, for influx, is 3 x 10~15 moles/dyne
sec"1. The chloride permeability values obtained
for dog red cells from the in vivo dilution curves
therefore appear reasonable in terms of their
magnitude.
Effect on Computed Volumes of Non-Preincubation of the Red Cells in the Injected Materials.—
When the red cells have not been preincubated in
the injection mixture and the exchange at the red
cell membrane is limiting, the labeled substance is
relatively excluded from the red cells and its
transit is unduly delayed by its proportionately
larger distribution into the tissue. The volume of
distribution, computed as the product of both red
cell and plasma flows and the outflow transit time,
is too large. This fact has important connotations
for experimental design. Whenever one of the aims
of an experiment is to estimate the space of distribution of a label in tissue from its mean transit
time, a preincubated injection mixture should be
used. In this case, when there is slow equilibration
between red cells and sinusoidal plasma, the rapid
early exit of tracer contained in the red cells is
compensated for by delayed return of the tracer in
the plasma space so that the mean transit time
obtained by whole blood sampling is always correct. If the preincubated injection mixture is not
used, the space of distribution derived from the
transit time will be accurate only if the rate of
exchange of materials across the red cell membrane is infinitely rapid (so that the injection bolus
becomes preincubated in the presinusoidal or precapillary vessels). This will be the case, for instance, in the design of experiments of the sort
carried out by Effros and Chinard (3), in their
estimation of the in vivo pH of the extravascular
space of the lung. Here the procedure of using an
injection of test substances initially confined to the
plasma phase entails choosing a set of substances
which either penetrate the red cells exceedingly
rapidly or not at all during the duration of the
experiment.
Circulation Research, Vol. 36, February 1975
341
RED CELL CARRIAGE OF LABEL
Implications of These Studies.—These studies
provide the necessary background for the examination of substances which undergo both red cell
carriage and parenchymal cellular uptake and
metabolism. Now the behavior of substances of this
group which are important metabolic substrates
for the liver and the behavior of labeled oxygen
can be investigated.
Appendix
RED CELL CARRIAGE IN A SINGLE SINUSOID
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Consider a sinusoid of length L in which blood flows
with a velocity W, which is enfolded by an extracellular
space of Disse, with which free communication occurs,
and beyond this by a sheet of cells. The dimensions of
the space of Disse are such that soluble substances in
the sinusoidal plasma will undergo flow-limited distribution into the space (1), and the depth of the cellular
space is such that if substances penetrate the cell
membrane freely lateral diffusion equilibration will
occur as well (1). Now consider a substance that penetrates freely into an extravascular space (either extracellar or extracellular plus cellular) but that penetrates
the red cell membrane with much less facility. Let
u(x,t) = concentration in the sinusoidal plasma water at
some point along the length x at time t, v(x,i) =
corresponding concentration in the adjacent extracellular space, y{x,t) = concentration in red cell water, AJi,C
= volume per unit length for the sinusoidal plasma
space, the extracellular space, and the red cell space,
respectively (they are regarded as constant along the
length), BIA = y, and CIA = /3.
If we assume that the only mechanism for transport
along the length is vascular flow (that lengthwise
diffusion can be neglected within the time periods being
considered) and that red cell and plasma velocities are
equal, then it follows from mass continuity that the
change in quantity of material in the system (including
red cell, sinusoidal plasma, and extracellular space)
between any x and x + Ax during any fixed interval of
time At is exactly the change in the sinusoid alone
between the time t and t + At over the volume element
between x and x + Ax. The rate of change of material in
the sinusoid per unit length between x and x + Ax (the
flow of each element of the blood through the sinusoid
multiplied by its gradient in concentration across the
interval) is
...,d"
_ dy ,,
. e ,
.
-AW
CW—; the rate of change of
dx
dx
material in the whole volume element per unit length
is
du
dv
dy
_
1- B
r C—.
Equating these, we find
A
dt
dt
dt
(dy
dy\
du
du
dv
+ /3 — + W-) = 0.
— + W— +
\dt
dxl
dt
dx
du
/I3y
dy\
n
— +— +
ft
+ —• I = 0 .
(lb)
W dt dx P\Wdt
dx
We must now formulate an equation describing the
exchange of material between the red cells and the
plasma. The equilibrium concentration of material in
red cell water may equal that in plasma water or, if the
partition coefficient \ for the substance in red cell
water is less than one or if there is a Donnan effect at
the membrane excluding it, it may be less than one. We
will therefore develop an expression in which the rate
constants across the red cell membrane are potentially
unequal. This expression can be transformed to the
equilibrious case simply by equating these rate constants. Let ki be the membrane permeability coefficient or transport coefficient for red cell uptake per unit
area (with dimensions cm/sec) and kt = &,'GSr/V'rm),
where Sr/V^ is the ratio of the red cell surface area to
its contained water content, and let k^ 7\ be the permeability-type rate constant for red cell efflux divided by
the partition coefficient A and k^ = (&j 7\) (SrlV^. Note
that both kt and k2 should be independent of the
hematocrit of the blood, since the ratio Sr/VTO for red
cells does not vary with hematocrit. Utilizing these expressions we find that the average rate of gain of material by red cells over the volume element between x and
x + Ax over the time between t and t + At is equal to the
sum of the net delivery of red cells containing label and
the difference between the fluxes across the red cell
membrane, i.e.,
dy
dy
— + W— +
dt
dx
(2)
= 0.
SOLUTION OF THE EQUATIONS FOR A SINGLE SINUSOID
Now introduce at the origin (x = 0) of the initially
empty sinusoid, with flow Fs, the quantity of material
q0 at the time t = 0. Eqs. lb and 2 must then be solved
according to the initial conditions
u(0,t) =cl8(x),
y(O,t) = CjSOc),
u(x,0) = y(x,0) = 0,
where c, and c2 are constants with the dimensions
amount/cm2, which will later be used to define the
manner in which the label is initially distributed between red cells and plasma.
Now if we substitute Eq. 2 into Eq. la and apply to
these equations the Laplace operator with respect to
time
L\fbc,t)]= I
f{x,t)e'"dt=F(x,s),
we find
dTj(x,s) . _
dx
(la)
Now consider that the material under consideration is
undergoing flow-limited distribution from the sinusoidal to the extracellular space. Then, v(x,t) = u{x,t)
and dvldt = duldt, and so
Circulation Research, Vol. 36, February 1975
l+ydu
W
W
and
w\
•c,8(x)
w
Y(x,s)
(3)
342
GORESKY, BACH, NADEAU
x
From these equations, if we transform x' = —,
&U
dU
-— + [s(2 + y) + k2 + /a*,]—ax
&e
—
y) + s(l + y)]t/
s)]
and
dY
ax
—
y) + s(l + y)]Y
J—-
~— + [s(2
ox
where S' (*') is the unit doublet, the derivative of the
Dirac delta function 6Cx'). Hence,
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+ y)
V(x,s) =
r t )(l + y)
y)]
y)
y)]
where r, has the positive sign and r2 has the negative
sign in the expression
s(2 + y) - *, - /Sfc, ± V[ys - (k2
4/Sfc,*,
Then find, if the inverse transform is defined as
i r
after successively substituting
_
x,i)=-—
e"F(x,s)As
2m Jc_lm
+ T,
S =
dr = dp/y,
that
u(x,t) = E
-
P>
i
2m
\2c2
\Vp2
+ 4/3
* ^ " p) \
/p" + 10k,k, + p) [2c
Circulation Research, Vol. 36, February 1975
343
RED CELL CARRIAGE OF LABEL
and
i
y(x,t) = E • —
2m Jc.
_
-P>
p 2W
2c,
+ 4/afe.A,
w h e r e E = ( l / 2 y ) e x p i - (k, - Bk,)t - (k, - Bk, +
1-yL
- p) J
dp,
yk,)—
W
Now when
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F(p) =
we find (18) the inverse transform
fit) = Jo(a
= 0
- b2) when t > b,
when t < b,
where Jo is the zero-order Bessel function of argument
2. Now if we define S(t - a) as a unit step function at
time a, we find from the inverse transform and the
preceding equations that the outflow plasma label concentration is
u(x,t) =
S ( <
8[t -
- ^ -
- (1 + y)—] > - 1 '
c, ( 1 + y)
W
y
Vd+ y)(x/W)-t
8[t -
i
+ 1 S(t
{
Bk,ki [t
Circulation Research, Vol. 36, February 1975
i> - t] I •
(9)
344
GORESKY, BACH, NADEAU
And the outflow red cell label concentration is
X
.
y < x
'
t )
=
^
W'
"
- ] ( * , - pk,)t
- (k, - I
A^l.
%
W
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. - / » . +7*. £ l
}
\ — - k l + Yy) + —• —'
W y
[Wy
W
(n + 1)
where I0(z) and 7,(2) are the modified zero-order and
first-order Bessel functions of argument z.
c2 =
INPUT CONDITIONS AND IMPULSE RESPONSE
The problem which now arises is that of defining c,
and c2. There are two input conditions: (1) red cells are
preequilibrated in the suspending phase so that the
labeled material which enters the red cells with an
impediment at the surface comes to its equilibrium
value prior to injection and (2) the label is added to the
plasma phase which is separated from the red cells
until the moment of injection. The two cases will be
considered in sequence.
The Preequilibrated Case.—Suppose that the amount
q0 is added to the injection. At equilibrium the concentration of added material in the red cell water will be
kjk2 times that in plasma water. The volume of distribution in each of the phases will be defined by the
hematocrit and the water content of each of the phases
(19). For each milliliter of the injection mixture the
total water content will be
Hct fr+(I
- Hct)fp = fb,
where Hct is the hematocrit, fr is the water content of
the red cells (0.70 ml/ml), fp is the water content of the
plasma (0.94 ml/ml), and fb is the water content of the
blood. Note that /3 = Hct frl(l - Hct) fp. Then we find
q0W(l + 0)
F,fb(l
-1/
«2
where F, is the flow of blood in the sinusoid.
With these input conditions, the outflow profile from
the single sinusoid now becomes defined. To provide
insight into the manner in which the rate of emergence
of material changes as the red cell permeability
changes, we have computed a set of numerical illustrations (Fig. 10, left) for the case in which Hct = 0.40, y =
5.0,x = L (the length of the sinusoid), and L/W = r (the
sinusoidal transit time for red cells). When the rate
constant is zero, the material in the red cells emerges
with these cells, and the material in the plasma
emerges as a delayed wave at th = 6.0. As the red cell
permeability is increased further, the material introduced with the red cells emerges in two parts: (1) a
proportion which emerges as an impulse function of
reduced area with the red cells, and (2) a remainder
which appears as spread out material intermediate
between the red cells and the delayed plasma wave.
The material introduced in the plasma emerges in two
parts in similar fashion: (1) intermediate material
spread out in time, appearing between the red cells and
the delayed plasma wave, and (2) material which has
not left the delayed plasma wave and which therefore
emerges as a delayed impulse function of reduced area.
When the red cell permeability becomes very large,
Circulation Research, Vol. 36, February 1975
345
RED CELL CARRIAGE OF LABEL
i 50
8-5.0
.i -
4—J—t
a
i i
i
.9
AIM
93
k.O
— i - - 4 - -J-
'—.
4
1
i—
A I M -S76
to
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TIME UNITS ( l A )
function of unit area emerges at an intermediate time.
When A, = k2 (the case illustrated in the figure), this
delayed impulse function, intermediate in time, corresponds to the outflow pattern expected for labeled
water.
More insight into the characteristics of the system
can be gained by examining the cumulative outflow
profiles. For this particular case, these profiles are
displayed in Figure 11 (left). These curves, as are those
in Figure 10, are computed by summing the red cell
and plasma outputs according to the proportion flowing
in each phase, i.e.,
u(L,t),
1+/8
where F,fb is the flow of blood water and yiLjt) and
u(L,t) are expressed as amounts per milliliter of water.
The manner in which the profile changes with an
increase in permeability is clearly displayed. There is a
steady progression from a bimodal to a unimodal response as the red cell permeability increases. The
response is, in every case, centered around the outflow
response to the non-red cell-limited extreme, the delayed wave expected for a substance distributed into
the red cell water and tissue water spaces infinitely
rapidly.
The Non-Preequilibrated Case.—Once again suppose
that the amount q0 of tracer is added to the injection
mixture but that it is confined to the plasma phase,
that it is added right at the origin of the sinusoid, and
that it has not previously exchanged with the red cells.
Then, in this instance, we find
FIGURE 10
Outflow profiles from a sinusoid through which blood with a
hematocrit of 0.40 is flowing at a rate F,. Left: Preequilibrated case. Right Non-preequilibrated case. For each of these
cases the input is an impulse function and the total introduced
is the amount q>. In the former case, the input label has been
partitioned between red cells and plasma according to their
water contents; in the latter, all of the label has been confined
to the plasma at the instant of input. It is difficult to display,
as part of the outflow profile, an impulse function, which
theoretically has an infinitely large magnitude and an infinitesimally short duration. We have used vertical lines at the
appropriate times for impulse functions both in red cells and
in plasma and have indicated numerically the area associated
with each. A solid line has been used for the total profile, a
broken line for the part of the intermediate profile emerging in
the red cells, and a dotted line for the part of the intermediate
profile emerging in plasma phase. In this illustration, T is the
mean transit time through the sinusoid for red cells, and y is
the ratio of the extrauascular plasma space to the sinusoidal
plasma space of distribution for label. In this case, where the
equilibrium concentration of label in red cell and plasma
water is taken to be the same, the entrance and exit red cell
transfer coefficients are equal (k, = kj = k). It should be noted
that the ordinate scale in this illustration is logarithmic.
none of the area is associated with either impulse
function. All of the material emerges as a spread out
function that peaks in concentration at an intermediate
time. Finally, in the extreme (A,, k? —> ^), an impulse
Circulation Research, Vol. 36, February 1975
C
'
F.fba+y)'
c2 = 0.
With these input conditions, the outflow profile from
the single sinusoid will come to consist of three components: (1) that material which has entered the red cells
from the plasma and which emerges in the red cells, (2)
that material which has entered the red cells from the
delayed plasma wave and has been "pulled" through
the sinusoid more rapidly but which emerges in the
plasma phase, and (3) that material which has not left
the plasma wave and which therefore emerges as a
delayed impulse function of reduced area. Figure 10
(right) illustrates the progressive change in the shape
of the outflow profile for this case. All of the material is
initially confined to the delayed plasma wave. As the
red cell permeability is increased, more of this material
is peeled off and emerges in time between the red cell
and the delayed plasma waves. Again, as the permeability increases further, all of the material comes to
reside in the intermediate functions which peak in time
between the red cell and the delayed plasma waves.
Finally, in the extreme (k,, k2 -> »), an impulse
function of unit area emerges at an intermediate time
(in the illustration, A, = h^, and this function corresponds in time to the delayed wave expected for labeled
water).
The cumulative outflow is illustrated in Figure 11
(right). At low levels of permeability the profiles are
dramatically different from those of the preequilibrated
cases. Initially all of the material resides in the delayed
346
GORESKY, BACH, NADEAU
TIME UNTTS ( l / j )
FIGURE 11
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Cumulative outflow profiles, representing the manner in which accumulated outflowing material
emerges. The curves are the time integrals of the profiles displayed in Figure 10. The ordinate is
expressed as a fraction of the amount introduced, q,. Left Preequilibrated case. Right Nonpreequilibrated case. These curves also represent the response of the system to a step input, a steady
infusion.
plasma wave. As the permeability is increased a larger
proportion is pulled forward via the intermediate profiles of Figure 10. Finally, at high levels of red cell
permeability, the outflow profiles for the two kinds of
input come to resemble one another, and in the non-red
cell-limited extreme (A,, k^ —* *0 they become identical.
It is evident, from inspection of the change in form of
the cumulative outflow profiles, that, in the red cell
non-preequilibrated case, the mean transit time will
vary with the red cell permeability.
Now the input description used in this case will
clearly be unreal from the physiological point of view.
When injected materials are introduced only in the
plasma at the level of the portal vein, they travel along
through its ramifications before reaching the sinusoids.
A partial prehepatic red cell equilibration will occur in
the large vessels. If we assume that there is no important separation between the labeled red cells and
plasma in the large vessels (20), then we may use the
transforms (Eqs. 7 and 8) to describe this phenomenon,
setting 7 = 0 . We find, where the label was initially
introduced in the plasma phase, that the input to each
sinusoid is
- kv),
LV
(lla)
and
qa
"'
+
F,fb k2 + pk
(lib)
where utv sxidyLV are the concentrations in plasma and
red cell phases, respectively, and tLV is the time spent in
the large vessels. Since the system is linear, we can
regard the input to the sinusoid as if it were composed
of two fractions: (1) one part in which red cell equilibration has already taken place, and (2) another part in
which there has as yet been no entry of label into the
red cells. These expressions can then be used to describe the output from the sinusoid under consideration.
RECOVERIES AND TRANSIT TIMES
We attempted to directly evaluate the integrals
needed to derive expressions for the outflow recoveries
and mean transit times. For the latter determinations
the integrals proved intractable. We therefore utilized
instead a property of the Laplace transform for their
evaluation. We will illustrate its use for both purposes.
If
U(J,,s)=
u(L,t)e-"dt
-st+
s2*2
— - .. .)dt,
then, if we expand U(Z,,s) in the form
IJd.s) = A - Bs + Cs2 - . ..,
we find
f
u(L,t)dt = A a n d I u(L,t)tdt
Jo
The Red Cell Preequilibrated
=—
u(L,t)stdt
ds Jo
= B.
Case.—We find, after
suitable expansion, that
U(L,s) =
go
Circulation Research, Vol. 36, February 1975
347
RED CELL CARRIAGE OF LABEL
where CKs2) are terms of the order of s. From this
equation,
Jo
„1
q
f (L
°
F.fb (1 +
Jo
w
(i +
and
J u(L,t)tdt
T
- (A. * B * . >
_
_ g
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Also,
From this equation,
go (1
y(
0
''
•
F sf b ( 1
•
"
>
!
.
and
Jo ^ ' ' ^ ~F~Jb{\
+ (hjkjp)
1 ( 1 + (fc./^W
W ~ (fej + /3fe,
_
Hence,
y) L
From the foregoing, the total expression for conservation at the outflow,
'f» -r—o f
u(L t)dt + F
-
i. + p Ja
'f» T^—O \
1 + p •>(,
is complete, as expected.
From the transit times, when these equations are
combined according to the proportion of the potential
mass being traced in each phase (21), we find
y)L
(1 + (*,/*2)j3)
W
The weighted transit time of the whole is independent
of the permeability of the red cells. This finding conforms to the expectations arising from Figure 10. When
A, = Aj, the corresponding volume of distribution, F,fb,
times the weighted transit time is the total water
space. When kt * !%, the volume of distribution is an
apparent volume expressed in terms of an equivalent
plasma water space. The use of the expression M = 11
evades this semantic difficulty by relating the transit
times to the mass of the material in the system (21).
Here M is the total mass of material in the system, / is
the rate of input of the material being traced (the
product of the total water flow and the mean concentraCirculation Research, Vol. 36, February 1975
tion in blood water), and t is the weighted mean transit
time for the whole. The computed mass of material in
the system is independent of the red cell permeability.
The Red Cell Nonequilibrated Case.—In this case we
find
U(JL,S) =
-
S
+
(1+ P)(hilfh) -<*, + /»,4
w
e
'£; "1/1
(kjkjp
_ UkJ^pXl + p)~l fy(l - (*,/fc,)jB)1
~ I 1 + (A,/^)/3 JL (1 + (kjkjp)!
(i + (kjkjprOh + /»,)
W
348
GORESKY, BACH, NADEAU
From this equation,
Jo
F ^ L l + Ofe,/*,)^
l + ( MW/8
and
if
1 + p Jo
l + (Jtjkjp
+
1+
The outflow recovery in the plasma phase at the end of
the sinusoid will vary from the complete amount injected q0 when kt —* 0 but ^Ik^ remains constant to the
equilibrium proportion q o [l/(l + (fc]/^)/?)] in the
plasma water phase when kx —» » and kjki remains
constant. Now
I +
P
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(1 + (jfe,/*,)^)
w
w
w
[
n,
and
_ _ / ! + (k1!k1)p+ y\L
C
"~\
1+ (fcj/^J/S
V
y
2(k,/k2)p[l-e-ik^Bk'^]
f
( 1 + (A,/*2)j3)M*!!+ j8*,)[l+ (A,/A2)/3e-(*' + 8 * ^ :
- ( * . -f s * . >
[1+ (fe.
When £, -» 0, <„ ^ (1 + y)-±-, and when k^ «> and A,/
remains constant
(1 + (^,/^,)/3 + y) L
(l+ (hjkjp)
W
Also,
(i
1+ {kjkt)p+
y
(1 + (ki/k2)p+
— s
y
L (i + {kjkjpnh
+ /at,)
_
Circulation Research, Vol. 36, February 1975
349
RED CELL CARRIAGE OF LABEL
From the foregoing the total expression for conservation at the outflow,
1
F.fbF,fb
y(L,t)At = q0,
From this equation,
T
and
The outflow recovery in tiio red cell phase at the end of
the sinusoid will vary from zero when A, —» 0 but Ai/A2
remains constant to the equilibrium proportion qo[(kj
k^fiKl + (Aj/A2)/3)] in the red cell water phase when
A, —» oo and k,/k2 remains constant. From this transform,
f
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(A,/A,)j8y) -<*,
1 + (kjkjfi
4
1+
/3
yd
'f
y(L,t)dt or (F,fblq0)
•Jn
j8*,) JL
and
=
/ ! + (A1
\ 1+ (
d +
- (A./AJ/B)
(1 + (A./A^/SXAj. + jSfe,)'
From this equation when A, —» 0 and A,//^ remains
constant
.
(1+ (fe1/fe2)/3+ y) L_ _ y d - (fe./fe^ffl £
'" ~* (1 + flfe,/*,)/© W d + (A,/*,)^) 2! W'
and when A, —» oo
.
(1+ (
t
W"
f
i JLcomplete, as expected. However, the mean
is once again
transit times cannot be combined in this instance in
such a fashion as to yield a constant volume. Even the
combination of the transit times according to the potential mass being traced in each phase of the blood will
fail to produce a constant value, because the input has
not been totally labeled (21, 22). The tracer has been
separated at the input from a proportion of the potential vascular material being traced, and a whole new
set of kinetic phenomena has emerged.
The recoveries and mean transit times for the moieties emerging in the two phases, red cells and plasma,
behave in a very different manner in the preequilibrated and the non-preequilibrated cases. We have
portrayed the difference in Figure 12. We have, for each
moiety, plotted an index of the recovery in that phase,
u(LMt,
against the
normalized transit time. We selected this recovery
index for the illustration, because it gives a locus
which is continuous over the range of the two moieties.
In the preequilibrated case the recovery index is 1.0,
and it does not change for either moiety as the red cell
permeability changes. As the red cell permeability
increases, the transit times of the two moieties approach one another. In the non-preequilibrated case all
of the label remains in the plasma phase when A, = 0,
and the transit time of the outflowing moiety is
(1 + y)LIW. None of the label emerges in the red cell
phase, and the limiting transit time for the red cell
moiety as &, -> 0 is not L/W, because none of the
material has been present in the red cells at the time
of entry into the sinusoids. When the red cell permeability increases, less of the material emerges in the
plasma phase, more emerges in the red cell phase, and
finally in the limit as kt -» oo the behavior corresponds
exactly to that seen in the preequilibrated case.
In the description of the transit times, we have
treated the label emerging in the red cells and plasma
.
Non-pre-equilibrated
Hd=Q40
)f-5.0
ok'O
• k-Q3
-8
»k.ao
•k=30.Q
ik.1000
ak=oo
krk2=k
2_
3
LA
FIGURE 12
Relation between the recovery and the mean transit time for the label emerging in the red cell and plasma phases. Left:
Preequilibrated case. Right: Non-preequilibrated case. The locus for the red cell moiety is solid, and that for the plasma moiety is
dotted. A broken line is used to indicate the common value for Me k -> » case.
Circulation Research, Vol. 36, February 1975
350
GORESKY, BACH, NADEAU
equations will not however take into account the difference in the phenomena occurring in the presinusoidal
or precapillary vessels; therefore, we have not pursued
this point.
OUTFLOW RESPONSE FROM THE WHOLE LIVER
I5
We will continue to use the same simplifying assumptions as we have in the past (6), i.e., that the large
vessel transit times are uniform and that the distribution of outflow arrival times of the vascular reference
substance occurs chiefly as the result of the distribution
of sinusoidal transit times. If we define Q(t) = quantity
of injected material arriving at the outflow per unit
time, F = total flow through the system, C(t) = Q(t)/F,
concentration of material, L = common sinusoidal
length (a consequence of the regular structure of the
liver), n(JJWs)d0jW,) = proportion of sinusoids with
transit times from (L/Ws) to (L/W,) + &QJW,), arising
from the distribution of values for the sinusoidal velocity. W,, qFJF = proportion of q, the total amount of
material injected, which enters a sinusoid with flow Fs,
and t0 = common large vessel transit time, then for the
vascular reference substance
ok=0
• k=0.3
AkO.O
-; 4
ok=3.0
Ak=10.0
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M E A N TRANSIT TIME
FIGURE 13
Relation between the mean transit time for the total outflowing material and the rate constant for red cell exchange for the
non-preequilibrated case analyzed in Figure 10. The broken
line corresponds to the k —* <*> value. The two dotted lines
correspond
to the values L/W and (1 +y) L/W.
In the preequilibrated case, in contrast, the transit time is
And if we set V = t —
described by the broken line for all values of the rate constant
for red cell exchange.
as material emerging as two separate entities. However, the phenomenon of postcapillary equilibration
blurs this distinction completely, and, in fact, the experimental information obtained consists of a weighted
sample of the total outflowing material. The accessible
experimental transit time is that of the total, t,ot, and is
represented as follows:
[1/(1
u(L,t)tdt
[1/(1+ 0)]J u(L,*)d< +
]j y(L,t)tdt
+ 0)] I y(L,t)dt
(12)
Now consider the case in which the red cells have
been preequilibrated in the plasma and in which label
undergoes both red cell carriage and flow-limited distribution into an extravascular space in the liver. The
observed outflow concentration of label will be described by the convolution of the single sinusoidal
response for label in each of the two phases, red cells
and plasma (Eqs. 9 and 10), with the distribution of
sinusoidal transit times. Each of these experiments
must, in turn, be weighted by the proportion of the flow
in each phase because of the later equilibration of label
between the red cells and the plasma in the large
vessels and the blood samples. Hence,
(1 + (ft,/ft2)/3+ y) L
(1 + (ft./ft^)
W
•Kfe,/fe2)j8
(1 + (ft,/ft,)/3)(ft, + /3ft
The changes in this expression with the rate constant k
(i.e., when ft, = k?) are recorded graphically in Figure
13 for the non-preequilibrated case analyzed in Figure
10 (right). When ft -» 0, the transit time is (1 + y)L/W;
when ft -» oo, it approaches [(1 + /3 + y)/(l + 0)] LJW.
Only in the limit when ft —» °= does the product of the
water flow and the transit time yield a value which is
identical to the accessible geometrical volume. Otherwise, the value is larger.
If the non-preequilibrated label were introduced with
materials for which ft —» 0 and ft —* », an estimate of ft
could theoretically be determined from the interrelations of the transit times for the three substances. Such
(*,/*, )0
iqF,
F~F
iqF,
/3
ft,
F F 1 + (kJkJPty
er
V-^,"f''
Jr.
i
y
n\ n\
(n + 1)
yk,<
ft,
F F 1 + (ftt/ftJ/S y e
f
ft' -ft,ft'
n\
n!
T)1
+ 1)
1)J
y(nn +
lqFs
1
i**'*7^'f t' \
F ~F 1 + {kjkjp 1 + yn\l + y /'
Circulation Research, Vol. 36, February 1975
RED CELL CARRIAGE OF LABEL
where ta = (LIW2\,in - <o or t'Kl + -y), whichever is
smaller.
In the limit, when &,-»<» and kjf^ remains constant,
this equation becomes
= C[(l + ( k ^ & t ' K l + folk,)? + y)]RBC
(14)
y))
[(l
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When k, = *j and if the space of distribution encompasses the total cell water space, the expression describes
the outflow appearance of labeled water. The major
assumptions underlying • the development of this last
equation are that, in each vascular lobule, the velocities
of flow in adjacent sinusoids are identical, so that the
diffusional interactions between adjacent sinusoids cancel and that, therefore, the heterogeneity of perfusion
exists chiefly at a lobular level.
We have developed an expression similar to Eq. 13
for the red cell non-preequilibrated case. It is not
outlined in this paper since it is, in principle, analogous
to that presented for the preequilibrated case.
351
materials by the intact liver Concentrative transport
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Am J Physiol 207:883-892, 1964
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agents and changes in ionic environment. Biochim
Biophys Acta 163:494-500, 1968
9. PERL W: Red cell permeability effect on the mean transit
time of an indicator transported through an organ by
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Acknowledgment
We wish to especially express our appreciation to Dr. W.
Perl for his critical and stimulating correspondence and to
Dr. J. B. Bassingthwaighte for his incisive comments. We
wish to thank Miss Heather MacDonald and Mrs. Denise
Joubert for their technical assistance and to acknowledge the
patience and care with which Miss Margaret Mulherin typed
this demanding manuscript.
transit times and distribution volumes of T-1824, creatinine and water. Am J Physiol 209:243-252, 1965
15. PAGANELLJ CV, SOLOMON AK: Rate of exchange of triti-
ated water across the human red cell membrane. J
Gen Physiol 41:259-277, 1957
16. CHIEN S, USAMI S, DELLENBACK RJ, BRYANT CA: Com-
parative hemorheology: Hematological implications of
species differences in blood viscosity. Biorheology
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man red cell membrane permeability to small electrolytes. J Gen Physiol 58:259-266, 1971
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C A Goresky, G G Bach and B E Nadeau
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Circ Res. 1975;36:328-351
doi: 10.1161/01.RES.36.2.328
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