Vol. 275, Issue 6, H2250-H2257, December 1998
Predictions of capillary oxygen transport in the presence of
fluorocarbon additives
Charles D.
Eggleton,
Tuhin K.
Roy, and
Aleksander S.
Popel
Department of Biomedical Engineering and Center for Computational
Medicine and Biology, Johns Hopkins University School of Medicine,
Baltimore, Maryland 21205
 |
ABSTRACT |
A mathematical model
of capillary oxygen transport was formulated to determine the effect of
increasing plasma solubility, e.g., by the addition of an intravascular
fluorocarbon emulsion. The effect of increased plasma solubility is
studied for two distributions of fluorocarbon, when the fluorocarbon
droplets are uniformly distributed throughout the plasma and when the
fluorocarbon droplets are concentrated in a layer adjacent to the
endothelium. The model was applied to working hamster retractor muscle
at normal and lowered hematocrit. The intracapillary mass transfer
coefficient was found to increase by 18% as the solubility was
increased by a factor of 1.7 at a hematocrit of 43%. An additional
increase of 6% was predicted when the solubility increase was
concentrated in the layer adjacent to the endothelium. At a hematocrit
of 25%, the intracapillary mass transfer coefficient increased 14%
when the solubility was increased by a factor of 1.7.
mass transfer coefficient; mathematical model; hamster retractor
muscle; microcirculation; blood substitute
| |
INTRODUCTION |
THE HIGH SOLUBILITY of O2 in
perfluorocarbon (PFC) compared to blood plasma has led to the
development of fluorocarbon-based additives designed to enhance the
O2-carrying capacity and delivery characteristics of blood.
These fluorocarbon emulsions are designed to enhance O2
transport in critically ill patients and patients who have been
hemodiluted after acute blood loss or during an operative procedure
(29). One technique for improving O2 transport in patients
undergoing surgery is intraoperative hemodilution, in which the
patient's hematocrit is intentionally reduced by removing blood that
will be transfused to the patient postoperatively. The lost volume is
replenished with a crystalloid solution. Hemodilution also occurs as a
result of volume repletion after acute blood loss. In either case, this
has the effect of reducing total peripheral (vascular) resistance
because the blood viscosity is decreased, which may improve
O2 delivery in spite of decreased O2 content (19). If PFC is found to be a useful intravascular additive, transient
infusion of fluorocarbon emulsions during hemodilution may further
supplement O2 transport (16). Other applications of PFC
include cardioplegia, O2 delivery distal to the balloon in
coronary angioplasty (17), and liquid ventilation (10), to name a few.
The amount of O2 carried by the PFC is linearly related to
the local PO2, unlike the sigmoidal
O2 dissociation curve of hemoglobin. When PFC was added to
the blood, an increase in tissue PO2 was observed (4). It has been shown theoretically that low O2
solubility in plasma is a major determinant of the intracapillary
transport resistance to oxygen (8, 9). The increase in tissue
PO2 with addition of PFC to the plasma may be
the result of decreasing intracapillary resistance.
Hogan et al. (12) specifically investigated the use of PFC emulsions to
decrease intracapillary transport resistance. To test the hypothesis
that an increased O2 content in the plasma region is
responsible for enhancing O2 transport, they performed experiments on electrically stimulated isolated dog gastrocnemius muscle preparations under control conditions [with plasma solubility (
p) = 3 × 10
5 ml
O2 · ml1 · torr
1] and with 6 g/70 ml blood perfluorooctylbromide
(p = 5 × 105 ml
O2 · ml1 · torr
1); the hemoglobin concentration was reduced to 8.7 g/dl [corresponding to systemic hematocrit
(Hsys) = 0.26] in these experiments to increase the effect of this PFC. Increasing p from 0.003 to 0.005 ml
O2 · ml1 · torr
1 did not affect whole muscle diffusivity
(DO2). DO2 is defined by analogy to Fick's law of diffusion
|
(1)
|
where 
O2 is
O2 uptake, Pcap is mean capillary
PO2, and Pmt is mean tissue
PO2. Under maximal O2 uptake
conditions (O2 max), tissue
PO2 is low and can be considered negligible
compared to Pcap (12). DO2
represents a whole organ mass transfer coefficient. Hogan et al. (12)
concluded that elevated plasma O2 solubility increases O2 max in proportion to
the increase in convective O2 delivery and suggested that
increasing the diffusion coefficient for O2 in plasma does
not increase the whole muscle diffusivity.
Keipert et al. (16) measured O2 delivery in dogs ventilated
with air and 100% oxygen. About 8-10% of total O2
content was dissolved in the PFC emulsion, but 25-30% of
O2 was delivered by the PFC.
Correspondingly, hemoglobin-bound O2 accounted for 46 and
15% of O2 for control and PFC
cases, respectively. Keipert et al. (16) suggested that by serving as a
first dispenser of O2, PFC leaves more O2 bound
to hemoglobin to act as an O2 reserve. Vaslef and Goldstick
(26) used a capillary tube oxygenator in a steady-state closed loop to
study the effect of PFC addition to bovine blood on O2
uptake. It was found that for a 2.1% volume of PFC the outlet
PO2 increased by 10-20%. A mathematical
model of O2 transport in tubes with PFC additives was
developed by Shah and Mehra (22). The uptake of O2 by a
pure PFC emulsion and by blood is calculated as the respective fluid
flows through a tube with a constant PO2 along
the tube wall to compare the O2 transport properties of
each fluid. Emulsions with varying PFC concentrations and blood of
different hematocrits were considered, and O2 flux density
(moles/area/time) and content as a function of the distance from the
entrance were determined. A shorter entrance length was required for
blood saturation, in comparison to the PFC emulsions. Unreasonably high
concentrations of PFC are required to match the O2-carrying
capacity of blood when the wall PO2 corresponds to values for ambient air; however, the O2 content of a PFC
emulsion can match that of blood at normal conditions when the wall
PO2 corresponds to a 100% O2 environment.
In this paper, we use characteristics of working hamster retractor
muscle to assess the potential reduction in intracapillary resistance
by the addition of PFC in anticipation of the development of an
experimental model. Characteristics of this muscle have been studied
extensively in parallel with mathematical modeling (5-7, 18, 23,
28). The mathematical model developed here includes plasma
O2 solubility as a parameter so that the effect of PFC can
be simulated. There is evidence from in vitro studies that the PFC may
accumulate near the capillary wall (15); thus we incorporated a radial
variation of p into the model. The model can be used to
predict the enhancement of O2 transport (characterized by a
reduction in critical end-capillary PO2 or an
increased mass transfer coefficient) by the addition of PFC.
Our hypothesis is that the presence of PFC in the plasma will increase
the intracapillary O2 transport conductance (mass transfer coefficient). To test our hypothesis, the O2 content of the
erythrocyte is held constant and the mass transfer coefficient is
calculated as a function of plasma O2 solubility.
| |
MATHEMATICAL MODEL |
This section describes a model of O2 transport that
includes a radial plasma solubility distribution to assess the effect of intravascular fluorocarbon on PO2
distributions and the intracapillary mass transfer coefficient. The
model uses morphologically observed parameters and assumes that no
heterogeneity among blood capillaries is present.
The model considers a single capillary surrounded by a cylindrical
volume of tissue and includes both intra- and extracapillary regions. A
schematic for this model is shown in Fig.
1. The equations are written in the frame
of reference of a single erythrocyte. Thus the capillary wall,
interstitial fluid, and tissue regions move relative to the erythrocyte
and its surrounding plasma. Periodic boundary conditions are imposed at
the axial ends of the domain, and the PO2 in
the core of the erythrocyte is held constant. This allows us to
estimate the capillary mass transfer coefficient. The features of the
model are described here, and the equations and their descriptions are
given in the APPENDIX.
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Fig. 1.
Combined intra- and extracapillary model geometry. Arrow indicates the
movement of tissue, interstitial fluid (ISF), and capillary wall
regions relative to the erythrocyte. Model domain consists of a
capillary section from center of 1 plasma gap to the next with
erythrocyte at center and periodic boundary conditions at ends.
PO2 values (Pp, Pw,
Pi, and Pt) vary with axial position
z. Mb, myoglobin; Hb, hemoglobin. Pc, core
PO2.
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|
Intracapillary transport.
The intracapillary geometry is identical to the model considered in a
previous paper (20), in which the capillary lumen is assumed to contain
plasma and equally spaced erythrocytes modeled as cylinders containing hemoglobin.
The axisymmetric equations are solved in a domain containing a single
erythrocyte with periodic boundary conditions, meaning that
PO2 differences between adjacent erythrocytes
are not considered. Plasma convection is neglected because its effect
on O2 transport has been shown to be small (1). Thus the
erythrocyte and its surrounding plasma are stationary (no convection)
relative to the moving capillary wall, interstitial fluid, and tissue
regions. A capillary mass transfer coefficient is calculated to account for O2 diffusion within the erythrocyte and plasma and the
nonequilibrium concentration boundary layer in the erythrocyte
resulting from oxyhemoglobin dissociation kinetics. The
intraerythrocyte transport resistance is calculated using the results
of the kinetic boundary layer analysis of Clark et al. (3).
Fluorocarbon accumulation near the wall of a capillary tube with a 3-mm
diameter has been observed (15). In a smaller capillary tube of
200-µm diameter, platelets and microspheres of similar dimensions
were seen to accumulate near the wall (25). Braun et al. (2) proposed that because the fluorocarbon droplets have a median diameter of 0.25 µm, a near-wall excess may occur in capillary vessels. Although it
has not been observed in the capillary vessels within the tissue, a
radial plasma solubility distribution is considered here to simulate
fluorocarbon accumulation near the wall to investigate its possible effects.
Extracapillary transport.
The capillary wall and interstitium are modeled as annular regions of
finite thickness with appropriate transport properties. O2
consumption in these regions is not considered, because they occupy
only a small volume compared to the tissue region. The muscle fibers
are assumed to contain myoglobin and to consume O2 at a
constant rate corresponding to working hamster retractor muscle.
| |
PARAMETERS |
The parameters used in this study were chosen to represent working
hamster retractor muscle. Most of the parameters specific to this
muscle were taken from Ellsworth et al. (7) except where noted. Values
for most other parameters are those used by Roy and Popel (20).
Intracapillary parameters.
Erythrocyte volume (Vrbc) = 69.3 × 1012
cm3 (21) remains constant in all our simulations.
Microscopic observations of single capillaries provided average
erythrocyte length (Lrbc) = 8.16 µm (5).
Because this value was found to depend on hamster age, we used an
interpolated value for 34-day-old hamsters, the average age of hamsters
considered by Ellsworth et al. (7).
Erythrocyte velocity (vrbc) was obtained by
averaging the mean velocities observed in arteriolar and venular
capillaries at rest (7) and using a factor of 5 to estimate erythrocyte
velocity in working hamster retractor. This factor for increase is
based on velocity measurements in rat skeletal muscle at rest and
during contractions (13). For working muscle, we used
vrbc = 4.67 × 102 cm/s.
The average of the mean linear densities observed in arteriolar and
venular capillaries (632 cells/cm; Ref. 7) is used to obtain the
reference capillary hematocrit (H = 0.43). The mean radius observed
for arteriolar and venular capillaries
(rp) = 1.8 µm (7) is used in our
model. These values are consistent with subsequent measurements in this
muscle (23).
For consistency, we used values of the Hill coefficient
(n) = 2.2 and PO2 corresponding to
50% hemoglobin saturation (P50) = 29.3 torr
(corrected for pH and PCO2), for the Hill
equation cited by Ellsworth et al. (7). With the use of these
parameters in the Hill equation and the average of the observed
saturation values in arteriolar and venular capillaries
(S = 0.5035; Ref. 7), we obtained an erythrocyte core
PO2 (Pc) = 29.5 torr. The effect
of erythrocyte saturation has been studied (8, 27), and it was found
that the mass transfer coefficient was only weakly dependent on
Pc; thus it is not varied in this study.
Extracapillary parameters.
On the basis of in vivo microscopic intercapillary distances (7),
capillary density was set to 1,435 capillaries/mm2. This
value for resting muscle is used in our simulation of working muscle on
the basis of our assumption that capillary recruitment is small in
skeletal muscles of animals of this size (13).
The working muscle consumption assumed by Ellsworth et al. (7) as 10 times the resting muscle consumption of 0.89 ml
O2 · 100 g1 · min1 measured by
Sullivan and Pittman (24) is uniformly distributed throughout the
muscle tissue. Estimates for this muscle based on mitochondrial volume
density predict that the maximum consumption (O2 max) would be greater than the
resting consumption by a factor of 21 (6).
Facilitation of O2 diffusion is modeled using a myoglobin
concentration (NMb) of 0.4 mM in hamster
retractor measured by Meng et al. (18) and a myoglobin diffusion
coefficient (DMb) of 1.73 × 107 cm2/s reported by Jürgens et al.
(14).
| |
RESULTS |
PO2 distribution.
To simulate the effect of fluorocarbon, we used the model described
here to assess the impact of an increase in plasma solubility. The
reference value 0 = 2.82 × 103 ml
O2 · ml1 · torr
1 was increased by a factor of 1.7 as in the experiments
of Hogan et al. (12).
If, however, the amount of fluorocarbon required to produce such an
increase in solubility were concentrated in the plasma adjacent to the
endothelium, p would retain its normal value between the
erythrocytes, but the fluorocarbon concentration near the endothelium
would be increased by a factor 
|
(2)
|
where Ltot is total length of the tissue
cylinder. The value of for the intracapillary dimensions specified
in PARAMETERS was 3.4, resulting in a value of
p = 0 for r < rrbc and p = 3.40 for rrbc 
r < rp. An additional simulation is
performed with p = 3.4 0 in the entire domain.
The results demonstrate an increase in PO2 in
the entire plasma domain with increases in plasma solubility. Figure
2 shows the radial
PO2 profiles in the plasma through the
erythrocyte center as the plasma solubility is increased to 1.7 and 3.4 times its normal value. The effect of concentrating the solubility
increase in the plasma near the endothelium = (r) is
also shown in Fig. 2; the overall PO2 is higher
compared with the case in which the solubility enhancement is equally
distributed. The same trend is evident in the
PO2 profiles through the center of the plasma gap (Fig. 3). This is further illustrated
in Fig. 4, which shows the
PO2 profiles in the axial direction at the
inner capillary wall. Note the "zone of influence" in the regions
of the domain close to the erythrocyte.
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Fig. 2.
Midcell radial PO2 profile as a function of
plasma solubility () at hematocrit (H) = 43%.
PO2 values at z = 0 are shown as a
function of normalized radial position
r/rp, where r ranges from
0 to rt. Vertical dotted lines from
left to right indicate positions of
rrbc, rp,
rw, ri, and
rt, respectively.
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Fig. 3.
Midgap radial PO2 profile as a function of at H = 43%. PO2 values at z =  Ltot/2, where Ltot is
total tissue cylinder length, are shown as a function of normalized
radial position r/rp, where r ranges
from 0 to rt. Vertical dotted lines from
left to right indicate positions of
rp, rw,
ri, and rt,
respectively.
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Fig. 4.
Axial PO2 profiles as a function of at
capillary wall at H = 43%. PO2 values are
shown as a function of normalized axial position
z/rp, where z ranges from
Ltot/2 to + Ltot/2.
Vertical dotted lines indicate boundaries of erythrocyte
(Lrbc/2 and Lrbc/2,
where Lrbc is erythrocyte length).
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Mass transfer coefficient.
With the PO2 distribution in the entire domain,
we can calculate a number of derived quantities in addition to
determining the radial and axial PO2
distributions through various sections of the domain. The flux of
oxygen leaving the erythrocyte (Jrbc; mol/s) is
calculated from
|
(3)
|
where JLb is the
total flux leaving the left erythrocyte basal surface,
JRb is the total flux
leaving the right basal surface, and J
is the total flux leaving the lateral surface. Values of
JLb and
JRb are calculated from
|
(4)
|
where P is PO2 at the erythrocyte
membrane, z is the axial position, and
J is given by
|
(5)
|
The intracapillary mass transfer coefficient
kcap is defined in terms of Pc and
the PO2 and flux at the capillary wall
[Pp(z) and
Jp(z)]
|
(6)
|
where the overbar indicates the mean value,
with
|
(7)
|
|
(8)
|
Here we have defined the average mass transfer coefficient
per cell (
cap)
based on the average flux and PO2 along the
capillary wall. The mass transfer coefficient per unit length of the
capillary wall (
cap) is given by
|
(9)
|
For the reference case,
JLb/Jrbc = 0.08 (leading edge),
JRb/Jrbc = 0.08 (trailing edge), and
J/Jrbc = 0.84, i.e., 84% of the O2 leaves the erythrocyte through its
lateral surface. The reference value of Jrbc was
the same in all cases because tissue O2 consumption was the
same, but the value of J was found to
increase when fluorocarbon was introduced in the plasma sleeve region.
The reference value of the intracapillary mass transfer coefficient
cap was calculated to
be 6.3 × 1012 ml
O2 · s1 · torr
1.
The absolute values of
cap and
cap are shown in Fig. 5, A
and B; at the reference
hematocrit (H = 0.43), increasing plasma solubility by a factor of
1.7 increased cap by a factor of 1.18, and increasing p by 3.4 increased
cap by a factor of 1.40. The effect of concentrating the plasma solubility enhancement in the
layer near the endothelium was to increase cap by 6% over the
evenly distributed case.
Also shown in Fig. 5 is the effect of increased plasma solubility at a
lower hematocrit (H = 0.25). Although the trends are similar, the
magnitude of the increases in the mass transfer coefficient are
smaller. Increasing plasma solubility by a factor of 1.7 increased cap by a factor of
1.14, and increasing p by 3.4 increased cap by a factor of
1.31. The effect of concentrating the plasma solubility enhancement in
the layer near the endothelium was to increase
cap by 4.5%. The
corresponding increases are the same for
cap.
It is also instructive to examine the intracapillary transport
resistance as a fraction of the total resistance along the pathway from
the erythrocyte to the mitochondria, as defined previously (20). The
model predicts a decrease in intracapillary resistance fraction with
increasing solubility. The intracapillary resistance as a fraction of
the total resistance between the erythrocyte and the mitochondria at
the reference hematocrit is 0.43 for normal plasma, 0.40 for a relative
solubility increase of 1.7, 0.36 for a relative solubility increase of
3.4, and 0.39 for the case where the PFC solubility increase is
concentrated in the sleeve between the erythrocyte and the endothelium.
The associated changes in the total transport conductance from the
erythrocyte to the mitochondria, relative to the total conductance with
normal plasma are 1.08 for a relative solubility increase of 1.7, 1.17 for a relative solubility increase of 3.4, and 1.11 for the case where
the PFC solubility increase is concentrated in the sleeve between the erythrocyte and the endothelium. At H = 0.25 the intracapillary resistance as a fraction of the total resistance between the
erythrocyte and the mitochondria is 0.64 for normal plasma, 0.61 for a
relative solubility increase of 1.7, 0.58 for a relative solubility
increase of 3.4, and 0.60 for the case where the PFC solubility
increase is concentrated in the sleeve between the erythrocyte and the endothelium. The associated changes in the total transport conductance relative to the total conductance with normal plasma are 1.09 for a
relative solubility increase of 1.7, 1.19 for a relative solubility
increase of 3.4, and 1.13 for the case where the PFC solubility
increase is concentrated in the sleeve between the erythrocyte and the endothelium.
Sensitivity analysis.
The predictions of the model depend on the physiological mechanisms
represented and the physiological properties that determine their
magnitude. Although well characterized, some physiological properties
of the hamster retractor muscle have yet to be measured. The
sensitivity of the mathematical model to several input parameters was
tested previously (20), using a different solution method, for the dog
vastus medialus muscle at O2 max.
Only the parameters that affect the intracapillary mass transfer
coefficient are discussed here. It was found that if the oxyhemoglobin
dissociation rate constant (
d) was altered from the
standard d = 44 s1 to d = 22 s1 and d = 88 s1,
cap decreased 17%
and increased 22%, respectively. When the value of P50 was
lowered from 37.2 to 30.8 torr,
cap increased by
10%.
The sensitivity of the present model to variations in velocity and
capillary radius was tested for an increase in O2
solubility in plasma of = 1.70 at H = 0.43. The
velocity was minimized (stationary case) and increased to twice and
five times the reference velocity. The changes in
cap are 4.7, +2.2,
and 6.1%, respectively. Recall that the reference velocity is 5 times
the resting velocity so that the highest velocity considered is 25 times the resting velocity. The capillary radius used was the average
of the mean values measured at the arteriolar and venular ends, 1.8 µm. The average of the standard deviations in the radial measurements
at both ends of the capillary was 0.175 µm. The capillary radius was
varied by one and two standard deviations, and
cap was calculated
for the new geometry. In each the case the geometry was altered so that
the hematocrit remained fixed at H = 0.43 and the plasma sleeve, the
gap between the lateral wall of the erythrocyte and the capillary wall,
remained at 0.18 µm. The mass transfer coefficient changed by 2 and
0%, respectively when the radius was decreased and increased by one
standard deviation and changed by 8 and 1% when the radius was
decreased and increased by two standard deviations.
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DISCUSSION |
The solubility of O2 in plasma was increased ~70% at
Hsys = 0.25 in the measurements made on the dog
gastrocnemius muscle (12). No appreciable difference in the whole
muscle diffusivity was found. The same increase in solubility in the
current model of the hamster retractor muscle predicts an increase in
the intracapillary mass transfer coefficient of 14% and the whole
muscle O2 conductance of 9% at H = 0.25, assuming no
heterogeneity in the capillaries. The increase in the whole muscle
O2 conductance at the higher reference hematocrit, H = 0.43, was roughly the same. Therefore, the predicted increase is fairly
small. It should be kept in mind that these calculations pertain to the
hamster retractor muscle, whereas the results of Hogan et al.
(12) are for dog gastrocnemius muscle. We are not able to
repeat our calculations for the gastrocnemius muscle because most
morphological and biophysical parameters are not available.
It was also shown through measurements in the dog gastrocnemius muscle
by Hogan et al. (11) that whole diffusivity depends on hematocrit. The
relationship between hematocrit and increased plasma O2
solubility can be examined from the results of this study. It was
expected that the mass transfer coefficient would show a larger
proportional increase with increased plasma O2 solubility at lower hematocrit. This was not the case, although the mass transfer
coefficient did indeed increase in every case. Federspiel and Popel (8)
showed that increased erythrocyte spacing decreases the mass transfer
coefficient. By fixing Pc, and therefore the O2
content of the core of the erythrocyte, we have isolated the diffusional characteristics of increased plasma solubility. The mass
transfer coefficient (Eq. 6) is defined in terms of the
partial pressure of O2 in the cell (Pc). The
presence of a cell in the capillary is implied so that, with this
definition and the geometry of our model, hematocrit is taken to zero
by extending the axial length of the domain toward infinity. Estimates
of the O2 transfer at zero hematocrit as a function of PFC
concentration were presented previously (22).
The model of O2 transport developed has been used to study
the effects of plasma O2 solubility on the transport
resistance of O2 from the cell to the capillary wall. This
is only one aspect of the transport of O2 from the blood to
the tissue. Increasing plasma O2 solubility might have an
effect on other aspects of O2 delivery, such as
O2 uptake and change in blood viscosity, all of which act
together to determine the amount of O2 delivered to the tissue.
An idealized model of a capillary vessel has been used to simulate the
transport of O2. The results are limited by the simplifying assumptions but serve as a tool for predicting the dominant physical mechanisms responsible for enhancing O2 transport to the
tissue. These simulations predict that PO2 in
the plasma and tissue surrounding the capillary increases with
increasing fluorocarbon concentration. The simulations show that at
fixed erythrocyte saturation, higher levels of
PO2 in the tissue are related to higher
PO2 levels in the plasma caused by increased
solubility. Increasing the plasma solubility increases the
intracapillary mass transfer coefficient (decreases transport
resistance). The result is that for the same flux of O2
supplied to the tissue the drop in PO2 from the
erythrocyte to the capillary wall is smaller. Concentrating the
increase in plasma solubility in the plasma sleeve between the
erythrocyte and the endothelium results in only a small increase
compared with the case in which the solubility enhancement is evenly
distributed. Therefore, for the muscle and physiological conditions
considered in this work, leading-order effects of perfluorocarbon
additives can be understood by considering a constant increase in
plasma solubility.
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APPENDIX |
Model Equations
The APPENDIX describes a model of O2 transport that
includes a radial plasma solubility distribution to assess the effect
of intravascular fluorocarbon on PO2
distributions and the intracapillary mass transfer coefficient. The
model uses morphologically observed parameters and assumes that no
heterogeneity among blood capillaries is present. The axisymmetric
equations are solved in a domain containing a single erythrocyte with
periodic boundary conditions, meaning that PO2
differences between adjacent erythrocytes are not considered.
Intracapillary transport.
Values of the erythrocyte linear density (LD), defined as the number of
cells per unit length along the capillary, and erythrocyte length
(Lrbc) are measured experimentally; these
values yield the length of the plasma gap (Lp)
from
|
(A1)
|
and define the total length of the tissue cylinder,
Ltot = Lrbc + Lp. The specified erythrocyte volume
Vrbc is used to calculate erythrocyte radius
rrbc from
|
(A2)
|
Given a specified value of the inner capillary radius
rp, the capillary hematocrit
(Hcap) can then be calculated from
|
(A3)
|
The local flux density out of the erythrocyte ( j) is
calculated using the results of the kinetic boundary layer analysis of
Clark et al. (3)
|
(A4)
|
where Pc is the PO2 in
the core of the erythrocyte, P is the PO2 at
the erythrocyte membrane, Drbc is the diffusion
coefficient of free O2 inside the erythrocyte,
rbc is the intraerythrocyte O2 solubility
coefficient, d is the oxyhemoglobin dissociation rate
constant, P50 is the PO2
corresponding to 50% hemoglobin (Hb) saturation,
NHb is the heme concentration inside the
erythrocyte, and q is a dimensionless flux density given
by
|
(A5)
|
The saturation Sc = S(Pc) is determined from Hill's equation for
the equilibrium oxyhemoglobin dissociation curve
|
(A6)
|
where n is the Hill coefficient.
PO2 varies continuously over the erythrocyte
surface; j at each point on the erythrocyte is determined by
continuity with the plasma flux density at the erythrocyte surface,
calculated from
|
(A7)
|
where N is the surface normal.
In the plasma
|
(A8)
|
Extracapillary transport.
O2 diffusion inside the capillary wall was modeled by
|
(A9)
|
In the interstitial fluid layer
|
(A10)
|
where vrbc is the erythrocyte
velocity; the convective term appears in this equation because it is
written in the erythrocyte frame of reference.
In the tissue region
|
(A11)
|
where M represents a constant consumption rate. As in the
previous model (20), free and myoglobin bound O2 are
assumed to be in equilibrium, SMb = SMb(P).
The boundary condition at the edge of the tissue cylinder is
|
(A12)
|
At the erythrocyte-plasma interface,
PO2 is determined by Eq. A7.
Periodic boundary conditions for PO2 were used
for all regions in the axial direction. The core erythrocyte
PO2 (Pc) was assumed to remain constant.
Numerical method.
The above equations were solved in dimensionless form using
finite-difference approximations in a finite-volume formulation. Time-dependent terms were added, and time marching was used to find the
steady-state solution with the Crank-Nicolson method. The resulting set
of linear equations was solved iteratively at each time step using
Gauss-Seidel line relaxation. The grid size was 315 × 96 for the
normal hematocrit case (H = 0.43) and 315 × 166 for the
low-hematocrit case (H = 0.25). Hematocrit was decreased by keeping
the erythrocyte dimensions fixed and increasing the axial dimensions of
the entire domain. A time step corresponding to the
characteristic diffusion time in the plasma was used for all runs.
A linearized version of Eq. A5 for
q(Pc, P) was used for values of P close to
Pc because the finite-difference equations contain an
expression with the term dq/dP, which is singular at P = Pc. Initial values for the PO2
profile in the domain were generated by using the one-dimensional
equations for radial diffusion in each region. The equations were
solved with specified PO2 boundary conditions
for the plasma adjacent to the erythrocyte until the maximum relative
difference in the profile from one time step to the next was
<104. This profile was used as the initial
condition to solve the full set of equations given above using the
boundary condition for the flux density at the erythrocyte surface
expressed by Eq. A4. The final maximum relative difference
between time steps for all runs was <5 × 105. The
calculated flux of O2 (moles/time) out of each erythrocyte and out of each layer (inner and outer capillary wall, interstitial fluid) agreed with the total consumption
|
(A13)
|
to within 4% in each case.
| |
ACKNOWLEDGEMENTS |
This study was supported by National Heart, Lung, and Blood
Institute Grant HL-18292 and postdoctoral Training Grant HL-52864.
| |
FOOTNOTES |
Present address of C. D. Eggleton: Dept. of Mechanical Engineering,
UMBC, Baltimore, MD 21250.
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.
Address for reprint requests: A. S. Popel, Dept. of Biomedical
Engineering and Ctr. for Computational Medicine and Biology, Johns
Hopkins Univ. School of Med., Baltimore, MD 21205.
Received 26 Jan 1998; accepted in final form 1 September 1998.
| |
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Am J Physiol Heart Circ Physiol 275(6):H2250-H2257
0002-9513/98 $5.00
Copyright © 1998 the American Physiological Society
Top
Abstract
Introduction
![−<IT>v</IT><SUB>rbc</SUB> <FR><NU>∂</NU><DE>∂<IT>z</IT></DE></FR> [&agr;<SUB>t</SUB>P + <IT>N</IT><SUB>Mb</SUB><IT>S</IT><SUB>Mb</SUB>(P)] = <IT>D</IT><SUB>t</SUB>&agr;<SUB>t</SUB><FENCE><FR><NU>1</NU><DE><IT>r</IT></DE></FR> <FR><NU>∂</NU><DE>∂<IT>r</IT></DE></FR> <FENCE><IT>r</IT> <FR><NU>∂P</NU><DE>∂<IT>r</IT></DE></FR></FENCE> + <FR><NU>∂<SUP>2</SUP>P</NU><DE>∂<IT>z</IT><SUP>2</SUP></DE></FR></FENCE>](/content/vol275/issue6/fulltext/H2250/img023.gif)
