Abstract
Despite its welldocumented importance, the mechanism for nitric oxide (NO) transport in vivo is still unclear. In particular, the effect of hemoglobinNO interaction and the range of NO action have not been characterized in the microcirculation, where blood flow is optimally regulated. Using a mathematical model and experimental data on NO production and degradation rates, we investigated factors that determine the effective diffusion distance of NO in the microcirculation. This distance is defined as the distance within which NO concentration is greater than the equilibrium dissociation constant (0.25 μM) of soluble guanylyl cyclase, the target enzyme for NO action. We found that the size of the vessel is an important factor in determining the effective diffusion distance of NO. In ∼30 to 100μmID microvessels the luminal NO concentrations and the abluminal effective diffusion distance are maximal. Furthermore, the model suggests that if the NOerythrocyte reaction rate is as fast as the rate reported for the in vitro NOhemoglobin reaction, the NO concentration in the vascular smooth muscle will be insufficient to stimulate smooth muscle guanylyl cyclase effectively. In addition, the existence of an erythrocytefree layer near the vascular wall is important in determining the effective NO diffusion distance. These results suggest that 1) the range of NO action may exhibit significant spatial heterogeneity in vivo, depending on the size of the vessel and the local chemistry of NO degradation, 2) the NO binding/reaction constant with hemoglobin in the red blood cell may be much smaller than that with free hemoglobin, and3) the microcirculation is the optimal site for NO to exert its regulatory function. Because NO exhibits vasodilatory function and antiatherogenic activity, the high NO concentration and its long effective range in the microcirculation may serve as intrinsic factors to prevent the development of systemic hypertension and atherosclerotic pathology in microvessels.
 mathematical model
 reaction kinetics
 hemoglobin
 endothelium
 mass transport
endothelial release of nitric oxide (NO) has been documented to play an important role in the regulation of vascular tone and permeability (30), platelet adhesion and aggregation (36), smooth muscle proliferation (14), and endothelial cellleukocyte interactions (18). NO in the microcirculation is of particular interest, since the majority of vascular resistance is found in <150μmdiameter microvessels. The transport of NO from the producing cell to the target cell is not well understood, since NO, as a free radical, can be degraded in a variety of reactions. It has been proposed that NO is transported from endothelial cells through diffusion and that cell membranes are readily permeable to NO (28, 32). Under this hypothesis, NO diffuses from the endothelium into the surrounding smooth muscle or into the vascular lumen. The NO that diffuses into the vascular smooth muscle cells targets the enzyme guanylyl cyclase (32) to exert its vasoregulatory function. NO may also react with superoxide anion (31), bind with hemecontaining proteins (5, 16), interact with enzymes containing ironsulfur centers (30), or be degraded in several other reactions (1). Most importantly, NO reacts with deoxy and oxyhemoglobin at a very high rate to form nitrosylhemoglobin [HbFe(II)NO] and methemoglobin [HbFe(III)] and nitrate, respectively. The bimolecular rate constants for these reactions are on the order of 25–50 μM^{−1}s^{−1}(5, 9), which gives a halflife of 1–3 μs for NO under physiological conditions (hemoglobin concentration is ∼2.3 mM in blood). If these reactions cause the concentration of NO to fall significantly below that needed to activate guanylyl cyclase, which has an equilibrium dissociation constant (k _{dis}) equal to 0.25 μM (41), then the biological function of NO will be diminished. Therefore, the effective diffusion distance of NO, which is defined as the distance within which NO concentration is greater than thek _{dis} of guanylyl cyclase, determines the functional range of NO action.
Several lines of in vivo and in vitro evidence suggest that hemoglobin is an effective NO scavenger that depletes NO. For example, injection of hemoglobin solution into experimental animals results in hypertension (15), most likely due to the oxidative reaction of NO with oxyhemoglobin in arterioles and surrounding tissue. Furthermore, 6 μM free hemoglobin can abolish NOmediated vasodilation in vitro (7). These results suggest that the effect of hemoglobin on NO diffusion distance is significant. However, exactly how far NO diffuses away from the blood vessel and how hemoglobin affects diffusion distance remain unclear.
This NO reactiondiffusion problem was studied by Lancaster (2123) using a modeling approach. He simplified the system by superimposing point sources of NO production and consumption. With his mathematical calculation, he concluded that NO could diffuse a relatively long distance from the source. This work (21), along with work reported by others (24, 43), represents the first generation of modeling effort on NO diffusion and reaction. However, three aspects need to be revised to examine the NO diffusion distance in the microcirculation. 1) The geometric factor was not considered. The source of NO production is, in fact, a surface source (proportional to vessel diameter), whereas NO consumption by hemoglobin in blood is a volumetric sink (proportional to the square of vessel diameter). Therefore, the interaction between NO source and sink cannot be evaluated using the Lancaster model.2) Superimposing point sources is a valid approximation of a surface source only if production and reaction follow the zeroth or firstorder kinetics with respect to NO. This approximation fails when the kinetic rate laws are nonlinear.3) The parameters used previously (2123) need to be reexamined for their effects on predicted NO concentration.
In this study we developed a mathematical formulation that is suitable for blood vessels and used mainly experimentally derived parameters to determine factors that affect the NO concentration and the effective diffusion distance, especially in the microcirculation. It should be emphasized that our goal is not to predict the actual concentration of NO in tissue. Rather, we are investigating the consequences of the diffusionreaction hypothesis. Discrepancy between the model output and physiological data suggests possible points of deficiency of this proposed mechanism.
METHOD
Model assumptions.
To model the NO concentration in blood vessels and parenchymal tissue, we divided this system into three compartments: the lumen, the endothelium, and the abluminal region (Fig.1). The system was modeled using cylindrical coordinates. Because endothelial constitutive NO synthase is partially membrane bound, we considered NO to be produced from the luminal and abluminal sides of the endothelial cell membrane. These two sides of the endothelial cell were modeled as two singular surface sources, displaced by the average thickness of an endothelial cell. Furthermore, to make the system tractable, we made five assumptions.1) The axial NO gradient along the vessel is small compared with the length of the region emitting NO, so NO transport by convection can be neglected.2) We considered only the steadystate case in this study, although our model simulation also considers the time factor (see ).3) The rate of NO production from the endothelial cells does not vary with the vessel size.4) Blood was treated as a continuum. In some cases, the particulate nature of blood was taken into account by recognizing the existence of a thin, erythrocytefree layer near the vessel wall. This arrangement was modeled by including an erythrocytefree region in the lumen. The thickness of this erythrocytefree zone depends on fluid mechanical considerations.5) NO diffuses freely across cell membranes, and the diffusion coefficients for NO in all regions were taken to be the same. Because NO is dilute, the diffusion coefficient is assumed independent of concentration.
Model equations.
With assumptions 1 and2, the system can be treated as a onedimensional problem, with NO concentration varying only in the radial direction (r). The balance between NO diffusion and reaction in all three compartments can be written for cylindrical coordinates as (see
for detailed derivation)
Boundary conditions.
In this system, NO is produced from two concentric surfaces separated by a distance of 2.5 μm, the approximate thickness of the endothelial cell (19). For a vessel of inner radiusR, the boundary conditions at the lumenendothelium interface (r =R) are given by (see
)
Model parameters.
The parameters in the model include the diffusion coefficient of NO (D), the NO production rate (q˙_{NO}), and the rate constant of NO degradation in each region. The diffusion coefficient of NO has been determined to be between 3,300 and 4,500 μm^{2} ⋅ s^{−1}. We used 3,300 μm^{2} ⋅ s^{−1}, which is consistent with our estimate from the data of Malinski et al. (28) and close to the diffusion coefficient of O_{2}, i.e., 1,300–2,000 μm^{2} ⋅ s^{−1}(12). According to assumption 5, the NO diffusion coefficients in all the regions are assumed to be the same.
The rate of NO production by the endothelium surfaces and the NO consumption rate constant in the endothelium and the abluminal region were estimated by fitting the data of Malinski et al. (28) to a similar model based on oblate spherical coordinates (42). Theq˙_{NO} was estimated to be 5.3 × 10^{−14}μmol ⋅ μm^{−2} ⋅ s^{−1}. The secondorder rate law for NO consumption provided the best fit to the data, but the difference between the first and secondorder rate law was insufficient to exclude the firstorder rate law. The reaction rate coefficient (k _{2,ab}) was estimated to be 0.05 μM^{−1} ⋅ s^{−1}if V _{NO} is assumed to be second order. These computations have been detailed elsewhere (42).
The rate constant for NO consumption by hemoglobin in the lumen (k _{1,lu}, according to Eq. 4 ) was initially taken to be 2.3 × 10^{5}s^{−1}, which was calculated from the in vitro rate constant between NO and free hemoglobin (6, 9), with the assumption that the hemoglobin concentration in blood is 2.3 mM (tetramer). Because of uncertainty, several values fork _{1,lu} were used for the computations that follow. These range from 2.3 × 10^{5} to 15 s^{−1}. The solution of the model equations is discussed in the .
RESULTS
Hemoglobin is an effective NO scavenger.
The above model was used to calculate the NO concentration in the vascular lumen and the abluminal region of an NOproducing vessel with a diameter of 100 μm. In this case, the secondorder reaction rate law was used to describe NO consumption in the endothelium and the abluminal region, and no erythrocytefree zone was considered. To examine the significance of NO reaction with hemoglobin, we used a wide range of rate constants,k _{1,lu}, in the calculation. Figure 2 shows the NO concentration in the vicinity of the vessel. When the reaction rate in the lumen is high (largek _{1,lu}), the blood acts as a sink for NO. For example, whenk _{1,lu} = 2.3 × 10^{5}s^{−1}, >99% of the NO produced flows into the lumen and <1% is available to diffuse into the vascular smooth muscle (results not shown). If this rate constant is estimated on the basis of the in vitro hemoglobinNO reaction,k _{1,lu} = 2.3 × 10^{5}s^{−1} (6, 9), the NO concentrations in all the regions are much less than thek _{dis} of soluble guanylyl cyclase. Under this condition, NO cannot effectively stimulate guanylyl cyclase.
Because of the uncertainty of the NOblood reaction rate, we considered other estimates ofk _{1,lu}. If the NOdeoxy erythrocyte reaction rate from Carlsen and Comroe (5) was used to estimatek _{1,lu}, 1,280 s^{−1} was obtained. However, this value still gives an NO concentration much lower than thek _{dis} of soluble guanylyl cyclase (Fig. 2). For NO concentration in the vascular smooth muscle to reachk _{dis},k _{1,lu} has to be <15 s^{−1}. Ifk _{1,lu} exceeds ∼15 s^{−1}, the NO concentration would not reach thek _{dis} of guanylyl cyclase, and thus the effective diffusion distance is zero. In these cases, the local NO concentration is not sufficient to activate guanylyl cyclase in the vascular smooth muscle cells. Therefore, these calculations provide boundaries for the effective NOerythrocyte reaction rate constant under the free diffusion hypothesis.
Erythrocytefree layer increases the mass transfer resistance.
In the above calculation, the blood is assumed to be a continuum. It can be argued, however, that if the particulate nature of the blood is taken into consideration, the erythrocytefree layer near the vascular wall may provide sufficient mass transfer resistance to reduce the NOscavenging effect of the blood. To analyze this situation, an erythrocytefree region adjacent to the luminal side of the endothelium is included in the model. In this region, NO is not consumed by hemoglobin in blood. Rather, it is likely to be consumed by O_{2} in plasma with a rate law second order in NO and first order in O_{2}. With a dissolved O_{2} concentration in plasma of ∼27 μM (35), the rate constant,k _{2,ef}, where the subscript indicates the erythrocytefree region, is estimated to be 0.002 μM^{−1} ⋅ s^{−1}. The thickness of the erythrocytefree zone (δ) depends on fluid mechanical considerations; we expected δ = 2.5 μm (38) for a 100μmdiameter vessel. The NO consumption rate constant (k _{1,lu}) in the erythrocyterich lumen is taken to be 1,280 s^{−1} as an illustration. Other parameters were the same as the above case. Figure3 shows that under these conditions the endothelial and abluminal NO concentration increased almost twofold for δ = 2.5 μm. However, the abluminal NO concentration still falls below the k _{dis} of guanylyl cyclase. Therefore, the existence of the erythrocytefree layer cannot prevent the NOscavenging effect if the NOhemoglobin reaction is already very fast. The impact of the erythrocytefree layer on the NO diffusion distance is diminished whenk _{1,lu} decreased (results not shown).
NO effective diffusion distance depends on the vessel diameter.
Because the vascular properties such as the sensitivities to adenosine, shear stress, and transluminal pressure exhibit significant variation with vessel diameter (20), we investigated effective diffusion distance for NO in microvessels of various sizes. Although the NO production rate per endothelial cell (or per unit surface area of the endothelium) is assumed to be constant regardless of vessel size, larger vessels have a lower surfacetovolume ratio, which may affect the NO diffusion distance. Moreover, smaller vessels are known to have a lower hematocrit (7), which reduces the NO consumption rate in the lumen. To investigate the effect of these factors, we considered three cases. The first simulates the condition where physiological solution is perfused. In this case, the NOhemoglobin reaction is absent in the lumen, and NO degradation in this region is assumed to be second order, withk _{2,lu} = 0.002 μM^{−1} ⋅ s^{−1}. Other parameters are as indicated in the legend of Figure4. The effective diffusion distance was calculated as a function of vessel diameter. As shown in Fig.4 A, the effective diffusion distance increases as diameter increases. This result is due to the geometry of the vessel: as vessel size increases, the total endothelial production of NO increases, which drives the NO diffusion further.
When blood is perfused in the vessels, the results are different. In the second case, we considered the situation where blood with constant hematocrit is perfused through vessels of different sizes. No erythrocytefree layer was considered here for simplicity. The luminal NO consumption rate constant,k _{1,lu}, was taken to be 15 s^{−1}, since higher values give zero effective diffusion distance. Interestingly, the effective diffusion distance exhibits a maximum as vessel diameter increases (Fig. 4 A). The optimal vessel size for NO diffusion is 30–100 μm in diameter. This phenomenon is attributed to the combined effect of NO production from the endothelium and NO scavenging by the blood. The total NO production increases linearly with the vessel diameter, whereas the luminal blood volume increases with the square of the vessel diameter.
In vivo, because of the Fähraeus effect, the hematocrit decreases as vessel diameter decreases (7). In the third case, we took this phenomenon into account using literature data for the correlation between hematocrit and vessel diameter (7). The change in hematocrit is reflected ink _{1,lu}, which is proportional to the total hemoglobin concentration in the vascular lumen. Figure 4 A shows that the peak of the effective NO diffusion distance is even more pronounced here than for the second case. The location of the peak in the bloodperfused cases suggests that the most effective region for NO to exert its function is in ∼20μmdiameter arterioles. The effective diffusion distance is sharply decreased in <20μmdiameter vessels. For small and large arteries (>200 μm), the effective NO diffusion distance may be less than the thickness of the smooth muscle layer. These results indicate that the arterioles, 20–100 μm in diameter, may be a primary action site for NO to exert its biological function in terms of regulating downstream pressure and perfusion.
The mean luminal NO concentrations for the above three cases as a function of vessel diameter are shown in Fig.4 B. The diameter dependence of the mean luminal NO concentration closely resembles that of the effective diffusion distance, although the effect of varying the hematocrit is much less pronounced. For larger vessels, the mean NO concentration does not depend strongly on vessel size. These results indicate that NO concentration and effective diffusion distance may exhibit spatial heterogeneity in a vascular network.
DISCUSSION
In the blood vessels the luminal release of NO from the endothelium has been shown to prevent platelet aggregation and to inhibit the adhesion of platelets, neutrophils, and lymphocytes to the endothelial surface (18, 36). The abluminal presence of NO inhibits smooth muscle contraction, proliferation, and migration (14, 30). Although the physiological and pathophysiological significance of NO has been well documented, the problem of NO transport has been unresolved.
In addition to the free diffusion hypothesis (28, 32) described above, a nitrosyl thiolmediated transport hypothesis (17) has been proposed. However, none of these hypotheses accounted for all the details of the NO transport in vivo. In particular, the role of hemoglobin as an NO scavenger and the effect of vessel diameter deserve further attention. We have explored the free diffusion hypothesis by synthesizing the diffusion and reaction processes of NO into a mathematical model with experimentally derived parameters.
NOhemoglobin reaction rate.
Because of the uncertainty of the NOhemoglobin reaction rate in vivo, we examined a wide range of rate constants for this reaction (Fig. 2). The highest k _{1,lu}(2.3 × 10^{5}s^{−1}) corresponds to the in vitro NOhemoglobin reaction rate constant (6, 9) combined with the average physiological hemoglobin concentration in the whole blood (2.3 mM tetramer). The in vitro NOhemoglobin reaction is first order in hemoglobin and NO, with a secondorder rate constant of 25 μM^{−1} ⋅ s^{−1}(5, 6). This rate constant, although evaluated for free deoxyhemoglobin, is similar to that for free oxyhemoglobin (9). Figure2 shows that, with thisk _{1,lu}, the NO concentration in the abluminal region is significantly below thek _{dis} of guanylyl cyclase.
For hemoglobin in deoxy erythrocytes, the NOerythrocyte reaction rate constant, measured to be 0.14 μM^{−1} ⋅ s^{−1}[the mean of the data of Carlsen and Comroe (5)], and an average hemoglobin concentration (2.3 mM) result ink _{1,lu} = 1,280 s^{−1}. Figure 2 shows that this rate constant still leads to an NO concentration much lower than that necessary for activating guanylyl cyclase.
Another estimate fork _{1,lu} may be obtained from the minimum free hemoglobin concentration that causes significantly increased vascular resistance, since the value ofk _{1,lu} for blood must be less than this value. For perfused rabbit hearts, Pohl and Lamontagne (34) reported this level to be 6 μM free hemoglobin. Thek _{1,lu}corresponding to this hemoglobin concentration is 150 s^{−1} (half time = 0.005 s). As expected, this value gives an NO concentration lower than thek _{dis} of guanylyl cyclase (Fig. 2).
Finally, we determined thek _{1,lu} that gives a mean free NO concentration in the blood corresponding to that measured experimentally: 0.0034 μM (40). If the blood was taken from a 4mmdiameter vessel and if the effect of the erythrocytefree layer is ignored, this corresponds tok _{1,lu} = 15 s^{−1} (halftime = 0.05 s) according to our model. Now, this value gives an abluminal NO concentration that is sufficient to activate guanylyl cyclase (Fig. 2). Although the estimation of this value is crude, it provides the order of magnitude of the reaction rate constant that gives an NO concentration exceeding thek _{dis} of guanylyl cyclase.
One possible explanation for the low NO concentration predicted from the in vitro NOfree hemoglobin reaction rate constant is that the erythrocytefree layer near the blood vessel may provide a mass transfer barrier to slow the NOerythrocyte reaction. For a flowing suspension, such as blood, the timeaveraged concentration of cells varies with position. Because of hydrodynamic interactions, cells migrate toward the center of the vessel, leaving a thin layer of fluid near the vessel wall in which there are few cells (38). We approximate this distribution by a uniform concentration in the interior of the vessel and a 2.5μmthick erythrocytefree layer adjacent to the endothelium. This layer introduces mass transfer resistance, which diminishes the effective reaction rate in the lumen of the vessel. In Fig. 3 we see that the erythrocytefree layer does raise the NO concentration appreciably. However, it cannot account for the difference between the NO concentration measured in vivo and the NO concentration predicted from the in vitro NOhemoglobin reaction rate. This and the previous results (Fig. 2) suggest that the NOerythrocyte reaction in vivo is much slower than the NOhemoglobin reaction in vitro or that the NO is transported by other mechanisms not accounted for in the model.
Free hemoglobin has been proposed to be used as a blood substitute. However, because of the NOhemoglobin interaction, the administration of free hemoglobin causes vasoconstriction and possibly hypertension (44). Sitedirected mutagenesis has been used to create mutant hemoglobins that have reduced the NOhemoglobin interaction by 37fold (9). Our model suggests that reduction of the NOhemoglobin interaction may be achieved by enclosing hemoglobin in erythrocytes. Thus free hemoglobininduced hypertension may be mitigated by packaging hemoglobin for blood substitutes in a celllike manner.
Vessel diameter dependency of the effective NO diffusion distance.
Among the findings, the effect of vessel size on the luminal NO concentration and the effective NO diffusion distance in the abluminal region are the most unexpected (Fig. 4). The model suggests that in the domain of the microcirculation an optimal range of vessel diameters exists, within which the effective NO diffusion distance is maximized. This conclusion is predicated on the free diffusionreaction mechanism of NO. On the basis of this mechanism, we examine the possible implications of this conclusion and discuss possible experimental supports.
The wall thickness of 10 to 100μmID microvessels is ∼5–8 μm (Ref. 2; L. Kuo, unpublished observations in isolated and pressurized coronary arterioles), which is less than the effective diffusion distance. However, for >400μmdiameter vessels, the effective NO diffusion distance may not cover the whole thickness of the vascular smooth muscle layer (Fig.4
A), and thus the vasodilation property of NO may be compromised. Indeed, >200μmdiameter vessels show little NOmediated vasodilation in response to increased shear stress (20). The existence of an optimal vessel diameter for NO diffusion is a result of the competition between increased NO production surface and increased NOscavenging blood volume in the lumen as the diameter increases. The effective diffusion distance can be enhanced by the reduction of hematocrit in microvessels in vivo. It is worth noting that luminal and perivascular
Our model also suggests that the microvessels are most sensitive to shear stress, provided that a given shear stress induces equal NO production per endothelial cell in vessels of different sizes. This is seen from the effective diffusion distance in Fig.4 A, which was calculated with a constant NO production rate. If the NO production rate decreases, e.g., by reducing shear stress, the curves in Fig.4 A will move down almost proportionally. For example, if the NO production is stimulated to 50% of the value used in Fig. 4 A, the effective NO diffusion distance (the solid line in Fig.4 A) can cover only the smooth muscle layers of ∼30 to 100μmdiameter vessels. The effective NO diffusion distance in vessels outside this diameter range will not cover the whole smooth muscle layer, and thus these vessels will not be sufficiently dilated by NO. Consequently, on the basis of the model calculation, 30 to 100μmID vessels will have a lower threshold for shearinduced dilation than smaller or larger vessels. This prediction is qualitatively supported by experimental results (20) which show that 60 to 100μmID vessels, in comparison to their downstream and upstream vessels, exhibit the lowest threshold for shearinduced dilation.
The diameter dependency of NO diffusion suggests that the microcirculation is optimally situated for NO to exert its regulatory function. Microvessels with different sizes have been shown to exhibit different regulatory properties (20), which may reflect the longitudinal integration of regulatory mechanism (25). Small arterioles (40–90 μm) and venules that are arranged in parallel may communicate and interact (11, 37), contributing to the integrated control of the microcirculation. It has been shown that NOmediated, shearinduced dilation in intermediate and large arterioles is particularly important for flow regulation (25). It appears that the circulation system takes full advantage of the size dependency of NO diffusion distance and builds the control mechanism accordingly.
Because our model predicts that 30 to 100μmdiameter microvessels exhibit the optimal NO efficiency, it is expected that inhibition of NO synthesis would produce a significant constriction of these vessels. Indeed, this view is supported by an in vivo study of skeletal muscle microcirculation showing that inhibition of NO synthesis predominantly increases vascular resistance in >25μmdiameter microvessels (10). Moreover, because the antiatherogenic property of NO has been demonstrated (27, 39), vessel size dependence of NO concentration may be one of the factors responsible for the difference in the vascular pathology between large vessels and microvessels during the development of atherosclerosis. For example, microvessels do not develop atherosclerotic lesions (29), possibly because of the higher NO concentration in these vessels.
Modeling studies.
Although previous workers have made important contributions to the analysis of NO diffusion (2124, 28, 43), this problem deserves further investigation. The approach outlined here considers the geometry of NO production and consumption regions and provides a more general analysis of NO reaction and diffusion. Although we considered only the steadystate behavior of the system, the numerical simulation did take into account the transient behavior, as discussed in the . More regions can be added to the model, if desired. For example, myoglobincontaining cells in muscle are expected to consume NO at a much higher rate than others. The model can be extended to account for this effect by considering a myoglobincontaining region. Our analysis here highlights the importance of analyzing diffusion and reaction quantitatively, which echoes the theme emphasized by many previous workers (3, 4, 13, 23, 26,33).
Acknowledgments
This work was sponsored by a Whitaker Foundation Biomedical Engineering Research Grant and National Science Foundation Grant BES9511737.
Footnotes

Address for reprint requests: J. C. Liao, Dept. of Chemical Engineering, University of California, Los Angeles, CA 900951592.
 Copyright © 1998 the American Physiological Society
Appendix
Governing Equations
General form.
Here the general form of the governing equations is stated. The concentration of a diffusing, reacting substance, such as NO, is described by the species mass balance (4a). For NO, this balance can be written as
Cylindrical coordinates.
For a trace component diffusing into a onedimensional cylindrical flow with no axial concentration gradients, ∇c_{NO} ⋅ v= 0, since onlyv_{z}
and dc_{NO}/drare nonzero. For firstorder NO consumption under steadystate conditions, ∂c_{NO}/∂t= 0 and V
_{NO} =k
_{1,NO}c_{NO}. With these simplifications the dimensionless form ofEq. EA1
becomes
For an nthorder reaction,V
_{NO} =k
_{1,NO}
Boundary conditions.
The NO distribution in each of the three regions, the lumen, the endothelium, and the abluminal region, is governed byEq. EA1 with the regions linked by conditions at their boundaries. Six boundary conditions and three initial conditions are needed to solve the complete timevarying problem. Here we are interested in the steadystate case,Eq. EA2 , which requires only the six boundary conditions and no initial conditions.
Production and mass transfer of a substance from the singular surfaces bounding the endothelium are described by the surface mass balance (sometimes called the jump mass balance) (4a). Because the NO consumption in the luminal region is different from that in the abluminal region (and possibly, the endothelial cell itself), the NO gradient and the flux of NO in each of the regions will also differ. The jump mass balance related the fluxes to the production rate. For NO produced at the interface between regions 1 and 2, this balance can be written as
The product c_{NO}
v
_{NO}≡
N
(mass transported per unit time per unit area) is denoted the mass flux vector. For a trace quantity like NO, it can be expressed in the form of Fick’s law
At the endotheliumlumen interface of a cylindrical vessel,Eq. A8 reduces to
At the endotheliumabluminal region interface, Eq. A8 reduces to
At these interfaces, continuity of NO concentrations supplies another two boundary conditions
When the NO production from the endothelium is uniform around the vessel, as we have stated in assumption 1, the concentration in the vessel is symmetrical about the axis
Solution
Analytic solution.
For firstorder NO decomposition, the reactiondiffusion system,Eq. EA2
with Eqs.EA12
and
EA13
and Eqs. EA14a
and
EA15b
, has an analytic solution. This solution is based on the general solution of Eq.EA2
, which is applied to each of the luminal, endothelial, and abluminal regions. For the luminal region
The complete form of the analytic solution is not presented here because of its length, but it can be obtained from the authors on request. This analytic solution was used to verify the numerical solution, which was used for most of the computations.
Numerical solution.
The diffusion equation (Eq. EA2
) can be solved by several different numerical techniques. For the work here, we solved the nonsteadystate form of Eq.EA2
To solve Eq. EA18 , the partial differential equation was transformed to a system of ordinary differential equations in time by discretizing the spatial derivative (e.g., Eq. EA3 ). We used secondorder centered differencing, because its implementation is straightforward. The derivatives in the flux boundary conditions at the cell surfaces were discretized using forward or backward secondorder accurate differencing. Because the concentration in the endothelium and close vicinity was of primary interest, variable grid spacing was used. This allowed the points near the endothelial surfaces, where the concentration gradients are steep, to be more closely spaced.
As is common for systems derived from parabolic partial differential equations, the system of ordinary differential equations was stiff. Our solution used the routines “odeint,” “stiff,” and “bstif” (3a); the routines “stiff” and “bstif” were modified to take advantage of the tridiagonal structure of the Jacobian matrix. We typically used a grid with 100–200 points. A finer grid does not materially change the result. With an error criterion for “stiff” of 3 × 10^{−4} and a grid of 100 points, the maximum difference between the computed solution and the analytic solution was <5%. Although the boundary condition (Eq. EA13 ) applies to an infinite domain, we solved them over a finite grid by applyingEq. EA13 to the outermost points, typically 2,000–4,000 μm from the vessel axis. Several maximum distances were used to ensure that the result did not vary appreciably with the distance where the boundary condition was applied.
Sensitivity to parameters.
In results we show the effect of luminal reaction rate and vessel diameter. Here, the sensitivity of the NO concentration to additional parameters is discussed. This effect is quantified by the sensitivity coefficientS defined by
NO concentration was most sensitive toq˙_{NO} for which
We have assumed that the NO production rate does not vary with time and that the NO production was equally distributed between the abluminal and luminal surface of the endothelium;f = 0.5, wheref is the fraction of total NO production from the abluminal surface of the endothelium
We also assumed that the luminal and abluminal diffusion coefficients were the same. The effect of the luminal diffusion coefficient can be characterized by
The sensitivity of the endothelial NO concentration to the abluminal diffusion coefficient (D
_{ab})