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
the high solubility of O_{2} in perfluorocarbon (PFC) compared to blood plasma has led to the development of fluorocarbonbased additives designed to enhance the O_{2}carrying capacity and delivery characteristics of blood. These fluorocarbon emulsions are designed to enhance O_{2}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 O_{2} 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 O_{2} delivery in spite of decreased O_{2} content (19). If PFC is found to be a useful intravascular additive, transient infusion of fluorocarbon emulsions during hemodilution may further supplement O_{2} transport (16). Other applications of PFC include cardioplegia, O_{2} delivery distal to the balloon in coronary angioplasty (17), and liquid ventilation (10), to name a few.
The amount of O_{2} carried by the PFC is linearly related to the local Po _{2}, unlike the sigmoidal O_{2} dissociation curve of hemoglobin. When PFC was added to the blood, an increase in tissue Po _{2} was observed (4). It has been shown theoretically that low O_{2}solubility in plasma is a major determinant of the intracapillary transport resistance to oxygen (8, 9). The increase in tissue Po _{2} 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 O_{2} content in the plasma region is responsible for enhancing O_{2} transport, they performed experiments on electrically stimulated isolated dog gastrocnemius muscle preparations under control conditions [with plasma solubility (α_{p}) = 3 × 10^{−5} ml O_{2} ⋅ ml^{−1} ⋅ torr^{−1}] and with 6 g/70 ml blood perfluorooctylbromide (α_{p} = 5 × 10^{−5} ml O_{2} ⋅ ml^{−1} ⋅ torr^{−1}); the hemoglobin concentration was reduced to 8.7 g/dl [corresponding to systemic hematocrit (H_{sys}) = 0.26] in these experiments to increase the effect of this PFC. Increasing α_{p} from 0.003 to 0.005 ml O_{2} ⋅ ml^{−1} ⋅ torr^{−1} did not affect whole muscle diffusivity (Do
_{2}). Do
_{2} is defined by analogy to Fick’s law of diffusion
Keipert et al. (16) measured O_{2} delivery in dogs ventilated with air and 100% oxygen. About 8–10% of total O_{2}content was dissolved in the PFC emulsion, but 25–30% ofV˙o _{2} was delivered by the PFC. Correspondingly, hemoglobinbound O_{2} accounted for 46 and 15% of V˙o _{2} for control and PFC cases, respectively. Keipert et al. (16) suggested that by serving as a first dispenser of O_{2}, PFC leaves more O_{2} bound to hemoglobin to act as an O_{2} reserve. Vaslef and Goldstick (26) used a capillary tube oxygenator in a steadystate closed loop to study the effect of PFC addition to bovine blood on O_{2}uptake. It was found that for a 2.1% volume of PFC the outlet Po _{2} increased by 10–20%. A mathematical model of O_{2} transport in tubes with PFC additives was developed by Shah and Mehra (22). The uptake of O_{2} by a pure PFC emulsion and by blood is calculated as the respective fluid flows through a tube with a constant Po _{2} along the tube wall to compare the O_{2} transport properties of each fluid. Emulsions with varying PFC concentrations and blood of different hematocrits were considered, and O_{2} 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 O_{2}carrying capacity of blood when the wall Po _{2} corresponds to values for ambient air; however, the O_{2} content of a PFC emulsion can match that of blood at normal conditions when the wall Po _{2} corresponds to a 100% O_{2}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 (57, 18, 23,28). The mathematical model developed here includes plasma O_{2} 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 α_{p} into the model. The model can be used to predict the enhancement of O_{2} transport (characterized by a reduction in critical endcapillary Po _{2} 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 O_{2} transport conductance (mass transfer coefficient). To test our hypothesis, the O_{2} content of the erythrocyte is held constant and the mass transfer coefficient is calculated as a function of plasma O_{2} solubility.
MATHEMATICAL MODEL
This section describes a model of O_{2} transport that includes a radial plasma solubility distribution to assess the effect of intravascular fluorocarbon on Po _{2}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 Po _{2} 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 .
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 Po _{2} differences between adjacent erythrocytes are not considered. Plasma convection is neglected because its effect on O_{2} 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 O_{2} 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 3mm diameter has been observed (15). In a smaller capillary tube of 200μm 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 nearwall 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. O_{2}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 O_{2} 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 (V_{rbc}) = 69.3 × 10^{−12}cm^{3} (21) remains constant in all our simulations. Microscopic observations of single capillaries provided average erythrocyte length (L _{rbc}) = 8.16 μm (5). Because this value was found to depend on hamster age, we used an interpolated value for 34dayold hamsters, the average age of hamsters considered by Ellsworth et al. (7).
Erythrocyte velocity (v _{rbc}) 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 usedv _{rbc} = 4.67 × 10^{−2} 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 (r _{p}) = 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 Po _{2} corresponding to 50% hemoglobin saturation (P_{50}) = 29.3 torr (corrected for pH and Pco _{2}), 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 Po _{2} (P_{c}) = 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 P_{c}; 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/mm^{2}. 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 O_{2} ⋅ 100 g^{−1} ⋅ min^{−1} 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 (V˙o _{2 max}) would be greater than the resting consumption by a factor of 21 (6).
Facilitation of O_{2} diffusion is modeled using a myoglobin concentration (N _{Mb}) of 0.4 mM in hamster retractor measured by Meng et al. (18) and a myoglobin diffusion coefficient (D _{Mb}) of 1.73 × 10^{−7} cm^{2}/s reported by Jürgens et al. (14).
RESULTS
Po_{2} 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 α_{0} = 2.82 × 10^{−3} ml O_{2} ⋅ ml^{−1} ⋅ 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, α_{p} would retain its normal value between the erythrocytes, but the fluorocarbon concentration near the endothelium would be increased by a factor ξ
The results demonstrate an increase in Po _{2} in the entire plasma domain with increases in plasma solubility. Figure2 shows the radial Po _{2} 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 α = α(r) is also shown in Fig. 2; the overall Po _{2} is higher compared with the case in which the solubility enhancement is equally distributed. The same trend is evident in the Po _{2} profiles through the center of the plasma gap (Fig. 3). This is further illustrated in Fig. 4, which shows the Po _{2} 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.
Mass transfer coefficient.
With the Po
_{2} distribution in the entire domain, we can calculate a number of derived quantities in addition to determining the radial and axial Po
_{2}distributions through various sections of the domain. The flux of oxygen leaving the erythrocyte (J
_{rbc}; mol/s) is calculated from
The intracapillary mass transfer coefficientk
_{cap} is defined in terms of P_{c} and the Po
_{2} and flux at the capillary wall [P_{p}(z) andJ
_{p}(z)]
For the reference case,
The absolute values of
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
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 V˙o
_{2 max}. Only the parameters that affect the intracapillary mass transfer coefficient are discussed here. It was found that if the oxyhemoglobin dissociation rate constant (κ_{d}) was altered from the standard κ_{d} = 44 s^{−1} to κ_{d} = 22 s^{−1} and κ_{d} = 88 s^{−1},
The sensitivity of the present model to variations in velocity and capillary radius was tested for an increase in O_{2}solubility in plasma of α = 1.7α_{0} at H = 0.43. The velocity was minimized (stationary case) and increased to twice and five times the reference velocity. The changes in
DISCUSSION
The solubility of O_{2} in plasma was increased ∼70% at H_{sys} = 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 O_{2} conductance of 9% at H = 0.25, assuming no heterogeneity in the capillaries. The increase in the whole muscle O_{2} 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 O_{2}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 O_{2} 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 P_{c}, and therefore the O_{2}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 O_{2} in the cell (P_{c}). 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 O_{2} transfer at zero hematocrit as a function of PFC concentration were presented previously (22).
The model of O_{2} transport developed has been used to study the effects of plasma O_{2} solubility on the transport resistance of O_{2} from the cell to the capillary wall. This is only one aspect of the transport of O_{2} from the blood to the tissue. Increasing plasma O_{2} solubility might have an effect on other aspects of O_{2} delivery, such as O_{2} uptake and change in blood viscosity, all of which act together to determine the amount of O_{2} delivered to the tissue.
An idealized model of a capillary vessel has been used to simulate the transport of O_{2}. The results are limited by the simplifying assumptions but serve as a tool for predicting the dominant physical mechanisms responsible for enhancing O_{2} transport to the tissue. These simulations predict that Po _{2} in the plasma and tissue surrounding the capillary increases with increasing fluorocarbon concentration. The simulations show that at fixed erythrocyte saturation, higher levels of Po _{2} in the tissue are related to higher Po _{2} 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 O_{2}supplied to the tissue the drop in Po _{2} 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, leadingorder effects of perfluorocarbon additives can be understood by considering a constant increase in plasma solubility.
Acknowledgments
This study was supported by National Heart, Lung, and Blood Institute Grant HL18292 and postdoctoral Training Grant HL52864.
Footnotes

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.

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.
 Copyright © 1998 the American Physiological Society
Appendix
Model Equations
The describes a model of O_{2} transport that includes a radial plasma solubility distribution to assess the effect of intravascular fluorocarbon on Po _{2}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 Po _{2}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 (L
_{rbc}) are measured experimentally; these values yield the length of the plasma gap (L
_{p}) from
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)
Po
_{2} 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
In the plasma
Extracapillary transport.
O_{2} diffusion inside the capillary wall was modeled by
In the interstitial fluid layer
In the tissue region
The boundary condition at the edge of the tissue cylinder is
Numerical method.
The above equations were solved in dimensionless form using finitedifference approximations in a finitevolume formulation. Timedependent terms were added, and time marching was used to find the steadystate solution with the CrankNicolson method. The resulting set of linear equations was solved iteratively at each time step using GaussSeidel line relaxation. The grid size was 315 × 96 for the normal hematocrit case (H = 0.43) and 315 × 166 for the lowhematocrit 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. EA5
forq(P_{c}, P) was used for values of P close to P_{c} because the finitedifference equations contain an expression with the term dq/dP, which is singular at P = P_{c}. Initial values for the Po
_{2}profile in the domain were generated by using the onedimensional equations for radial diffusion in each region. The equations were solved with specified Po
_{2} 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 <10^{−4}. 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. EA4. The final maximum relative difference between time steps for all runs was <5 × 10^{−5}. The calculated flux of O_{2} (moles/time) out of each erythrocyte and out of each layer (inner and outer capillary wall, interstitial fluid) agreed with the total consumption