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Departments of 1 Chemical Engineering and 2 Medical Physiology, Texas A & M University, College Station, Texas 77843; and 3 Department of Chemical Engineering, University of California, Los Angeles, California 90095-1592
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ABSTRACT |
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Top
Abstract
Introduction
Results
Discussion
Appendix
References
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Despite its well-documented importance, the mechanism for nitric oxide (NO) transport in vivo is still unclear. In particular, the effect of hemoglobin-NO 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 100-µm-ID microvessels the luminal NO concentrations and the abluminal effective diffusion distance are maximal. Furthermore, the model suggests that if the NO-erythrocyte reaction rate is as fast as the rate reported for the in vitro NO-hemoglobin 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 erythrocyte-free 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, and 3) 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
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INTRODUCTION |
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Top
Abstract
Introduction
Results
Discussion
Appendix
References
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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 cell-leukocyte interactions
(18). NO in the microcirculation is of particular interest, since the
majority of vascular resistance is found in <150-µm-diameter
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 heme-containing proteins (5, 16), interact with enzymes containing iron-sulfur 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
1s1
(5, 9), which gives a half-life 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
(kdis) 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 the
kdis 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 NO-mediated 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 reaction-diffusion problem was studied by Lancaster (21-23) 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 first-order kinetics with respect to NO. This approximation fails when the kinetic rate laws are nonlinear. 3) The parameters used previously (21-23) 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 diffusion-reaction hypothesis. Discrepancy between the model output and physiological data suggests possible points of deficiency of this proposed mechanism.
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METHOD |
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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 steady-state case in this study, although our model simulation also considers the time factor (see APPENDIX). 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, erythrocyte-free layer near the vessel wall. This arrangement was modeled by including an erythrocyte-free region in the lumen. The thickness of this erythrocyte-free 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.
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Model equations. With assumptions 1 and 2, the system can be treated as a one-dimensional 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 APPENDIX for detailed derivation)
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(1) |
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(2) |
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(3) |
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(4) |
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 radius R, the boundary conditions at the lumen-endothelium interface (r = R) are given by (see APPENDIX)
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(5) |
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(6) |
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(7) |
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(8) |
NO = NO,lu + NO,ab, where
NO,lu and
NO,ab are the NO
production rates from the luminal and abluminal sides of the endothelial membrane, respectively. From the data of Malinski et al.
(28), we assume that
NO,lu = NO,ab. Another
boundary condition is implied by the symmetry of the vessel
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(9) |
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(10) |
Model parameters.
The parameters in the model include the diffusion coefficient of NO
(D), the NO production rate
(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 µm2 · s1.
We used 3,300 µm2 · s1,
which is consistent with our estimate from the data of Malinski et al.
(28) and close to the diffusion coefficient of
O2, i.e., 1,300-2,000
µm2 · s1
(12). According to assumption 5, the
NO diffusion coefficients in all the regions are assumed to be the
same.
NO was estimated to
be 5.3 × 1014
µmol · µm2 · s1.
The second-order rate law for NO consumption provided the best fit to
the data, but the difference between the first- and second-order rate
law was insufficient to exclude the first-order rate law. The reaction
rate coefficient
(k2,ab) was
estimated to be 0.05 µM1 · s1
if VNO is assumed
to be second order. These computations have been detailed elsewhere
(42).
The rate constant for NO consumption by hemoglobin in the lumen
(k1,lu, according
to Eq. 4) was initially taken to be
2.3 × 105
s1, 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 for
k1,lu were used
for the computations that follow. These range from 2.3 × 105 to 15 s1. The solution of the
model equations is discussed in the
APPENDIX.
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RESULTS |
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Top
Abstract
Introduction
Results
Discussion
Appendix
References
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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 NO-producing vessel with
a diameter of 100 µm. In this case, the second-order reaction rate
law was used to describe NO consumption in the endothelium and the
abluminal region, and no erythrocyte-free zone was considered. To
examine the significance of NO reaction with hemoglobin, we used a wide
range of rate constants,
k1,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 (large
k1,lu), the blood acts as a sink for NO. For example, when
k1,lu = 2.3 × 105
s1, >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 hemoglobin-NO reaction,
k1,lu = 2.3 × 105
s1 (6, 9), the NO
concentrations in all the regions are much less than the
kdis of soluble
guanylyl cyclase. Under this condition, NO cannot effectively stimulate
guanylyl cyclase.
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1 was obtained. However,
this value still gives an NO concentration much lower than the
kdis of soluble
guanylyl cyclase (Fig. 2). For NO concentration in the vascular smooth
muscle to reach
kdis, k1,lu has to be
<15 s1. If
k1,lu exceeds
~15 s1, the NO
concentration would not reach the
kdis 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 NO-erythrocyte reaction rate constant under the free diffusion hypothesis.
Erythrocyte-free 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 erythrocyte-free layer near the vascular
wall may provide sufficient mass transfer resistance to reduce the
NO-scavenging effect of the blood. To analyze this situation, an
erythrocyte-free 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
O2 in plasma with a rate law
second order in NO and first order in
O2. With a dissolved
O2 concentration in plasma of
~27 µM (35), the rate constant,
k2,ef, where the subscript indicates the erythrocyte-free region, is estimated to be
0.002 µM1 · s1.
The thickness of the erythrocyte-free zone (
) depends on fluid mechanical considerations; we expected = 2.5 µm (38) for a 100-µm-diameter vessel. The NO consumption rate constant
(k1,lu) in the
erythrocyte-rich lumen is taken to be 1,280 s1 as an illustration.
Other parameters were the same as the above case. Figure
3 shows that under these conditions the
endothelial and abluminal NO concentration increased almost twofold for
= 2.5 µm. However, the abluminal NO concentration still falls below the kdis of
guanylyl cyclase. Therefore, the existence of the erythrocyte-free
layer cannot prevent the NO-scavenging effect if the NO-hemoglobin
reaction is already very fast. The impact of the erythrocyte-free layer
on the NO diffusion distance is diminished when
k1,lu decreased
(results not shown).
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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 surface-to-volume 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 NO-hemoglobin reaction is absent in the lumen, and NO degradation in this region is assumed to be second order, with k2,lu = 0.002 µM1 · s1.
Other parameters are as indicated in the legend of Figure
4. The effective diffusion distance was
calculated as a function of vessel diameter. As shown in Fig.
4A, 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.
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1, since higher
values give zero effective diffusion distance. Interestingly, the
effective diffusion distance exhibits a maximum as vessel diameter
increases (Fig. 4A). 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 in
k1,lu, which is
proportional to the total hemoglobin concentration in the vascular
lumen. Figure 4A 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
blood-perfused cases suggests that the most effective region for NO to
exert its function is in ~20-µm-diameter arterioles. The effective
diffusion distance is sharply decreased in <20-µm-diameter 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.
4B. 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.
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DISCUSSION |
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Top
Abstract
Introduction
Results
Discussion
Appendix
References
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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 thiol-mediated 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.
NO-hemoglobin reaction rate.
Because of the uncertainty of the NO-hemoglobin reaction rate in vivo,
we examined a wide range of rate constants for this reaction (Fig. 2).
The highest k1,lu
(2.3 × 105
s1) corresponds to the in
vitro NO-hemoglobin reaction rate constant (6, 9) combined with the
average physiological hemoglobin concentration in the whole blood (2.3 mM tetramer). The in vitro NO-hemoglobin reaction is first order in
hemoglobin and NO, with a second-order rate constant of 25 µM1 · s1
(5, 6). This rate constant, although evaluated for free deoxyhemoglobin, is similar to that for free oxyhemoglobin (9). Figure
2 shows that, with this
k1,lu, the NO
concentration in the abluminal region is significantly below the
kdis of guanylyl cyclase.
1 · s1
[the mean of the data of Carlsen and Comroe (5)], and an
average hemoglobin concentration (2.3 mM) result in
k1,lu = 1,280 s1. Figure 2 shows that
this rate constant still leads to an NO concentration much lower than
that necessary for activating guanylyl cyclase.
Another estimate for
k1,lu may be
obtained from the minimum free hemoglobin concentration that causes
significantly increased vascular resistance, since the value of
k1,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. The
k1,lu corresponding to this hemoglobin concentration is 150 s1 (half time = 0.005 s).
As expected, this value gives an NO concentration lower than the
kdis of guanylyl
cyclase (Fig. 2).
Finally, we determined the
k1,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 4-mm-diameter vessel and if the effect of the erythrocyte-free layer
is ignored, this corresponds to
k1,lu = 15 s1 (half-time = 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 the
kdis of guanylyl
cyclase.
One possible explanation for the low NO concentration predicted from
the in vitro NO-free hemoglobin reaction rate constant is that the
erythrocyte-free layer near the blood vessel may provide a mass
transfer barrier to slow the NO-erythrocyte reaction. For a flowing
suspension, such as blood, the time-averaged 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 2.5-µm-thick erythrocyte-free 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 erythrocyte-free 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 NO-hemoglobin reaction rate.
This and the previous results (Fig. 2) suggest that the NO-erythrocyte
reaction in vivo is much slower than the NO-hemoglobin 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 NO-hemoglobin interaction, the administration
of free hemoglobin causes vasoconstriction and possibly hypertension
(44). Site-directed mutagenesis has been used to create mutant
hemoglobins that have reduced the NO-hemoglobin interaction by 37-fold
(9). Our model suggests that reduction of the NO-hemoglobin interaction
may be achieved by enclosing hemoglobin in erythrocytes. Thus free
hemoglobin-induced hypertension may be mitigated by packaging
hemoglobin for blood substitutes in a cell-like 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 diffusion-reaction 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 100-µm-ID 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 >400-µm-diameter vessels, the effective NO diffusion distance may not cover the whole thickness of the vascular smooth muscle layer (Fig. 4A), and thus the vasodilation property of NO may be compromised. Indeed, >200-µm-diameter vessels show little NO-mediated 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 NO-scavenging 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 PO2 in microcirculation decreases as vessel size decreases (8, 33). The reduction in PO2 is likely to reduce the NO degradation rate in the blood and in the abluminal region and thus increases the effective diffusion distance of NO. This phenomenon may enhance the diameter dependency of the effective NO diffusion distance in the microcirculation. 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. 4A, 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. 4A will move down almost proportionally. For example, if the NO production is stimulated to 50% of the value used in Fig. 4A, the effective NO diffusion distance (the solid line in Fig. 4A) can cover only the smooth muscle layers of ~30- to 100-µm-diameter 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 100-µm-ID vessels will have a lower threshold for shear-induced dilation than smaller or larger vessels. This prediction is qualitatively supported by experimental results (20) which show that 60- to 100-µm-ID vessels, in comparison to their downstream and upstream vessels, exhibit the lowest threshold for shear-induced 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 NO-mediated, shear-induced 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 100-µm-diameter 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 >25-µm-diameter 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 (21-24, 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 steady-state behavior of the system, the numerical simulation did take into account the transient behavior, as discussed in the APPENDIX. More regions can be added to the model, if desired. For example, myoglobin-containing 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 myoglobin-containing 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).
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APPENDIX |
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Top
Abstract
Introduction
Results
Discussion
Appendix
References
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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
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(A1) |
is the vector gradient operator,
2 is the Laplacian operator,
cNO is the concentration of NO,
and VNO is the
rate at which NO is consumed by reaction. Two processes are involved in
the transfer of NO: the first term on the right-hand side represents the diffusion of NO; the second represents the transport of NO by a
molar averaged velocity v.
Cylindrical coordinates.
For a trace component diffusing into a one-dimensional cylindrical flow
with no axial concentration gradients,
cNO · v = 0, since only
vz and
dcNO/dr
are nonzero. For first-order NO consumption under steady-state
conditions,
cNO/t = 0 and VNO = k1,NOcNO.
With these simplifications the dimensionless form of
Eq. A1 becomes
|
(A2) |
),
and the dimensionless reaction rate coefficient (k*, the Damköhler number) are
defined by
|
(A3) |
|
(A4) |
|
(A5) |
Boundary conditions. The NO distribution in each of the three regions, the lumen, the endothelium, and the abluminal region, is governed by Eq. A1 with the regions linked by conditions at their boundaries. Six boundary conditions and three initial conditions are needed to solve the complete time-varying problem. Here we are interested in the steady-state case, Eq. A2, 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
|
(A6) |
NO is the rate
at which NO is produced by a unit of surface area
(mol · time1 · area1).
The notation |i means that
the concentration and velocities are evaluated in
phase i, as the interface is
approached. Equation A6 related the
velocity of NO at an interface to the velocity of the interface and the
production rate of NO. At steady state the vessel diameter no longer
changes with time, so u = 0.
The product
cNOvNO
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
|
(A7) |
|
(A8a) |
|
(A8b) |
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|
(A9) |
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|
(A10) |
NO(t) = NO,lu(t) + NO,ab(t)
is the total NO production rate per unit area of endothelium, which was
estimated from the data of Malinski et al.
(2A). Without evidence to the
contrary, we assume
NO,lu(t) = NO,ab(t).
At these interfaces, continuity of NO concentrations supplies another
two boundary conditions
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|
(A11a) |
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|
(A11b) |
|
(A12) |
|
(A13) |
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(A14a) |
|
(A14b) |
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(A15a) |
|
(A15b) |
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|
(A16) |
Solution
Analytic solution. For first-order NO decomposition, the reaction-diffusion system, Eq. A2 with Eqs. A12 and A13 and Eqs. A14a and A15b, has an analytic solution. This solution is based on the general solution of Eq. A2, which is applied to each of the luminal, endothelial, and abluminal regions. For the luminal region
|
(A17) |
. This fixes
C1 and
C6 at 0, since
K0(r)
as r 0 and
I0(r)
as r .
The remaining constants are obtained by substituting the three
equations for cNO into the four
boundary conditions (Eqs. A14 and A15). Solving the resulting system
yields the four unknown constants.
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. A2) can be solved by several different numerical techniques. For the work here, we solved the non-steady-state form of Eq. A2
|
(A18) |
/t = 0. This method has the advantage that transient behavior can be
examined. Alternately, Eq. A2 could be
discretized in r and solved by a
finite-difference technique, or a shooting method could be used.
To solve Eq. A18, the partial
differential equation was transformed to a system of ordinary
differential equations in time by discretizing the spatial derivative
(e.g., Eq. A3). We used second-order centered differencing, because its implementation is straightforward. The derivatives in the flux boundary conditions at the cell surfaces were discretized using forward or backward second-order 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 × 104 and a grid
of 100 points, the maximum difference between the computed solution and
the analytic solution was <5%. Although the boundary condition
(Eq. A13) applies to an infinite
domain, we solved them over a finite grid by applying
Eq. A13 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 coefficient S defined by
|
(A19) |
NO for
which 
=
1.0. This is expected, since the boundary condition is linear in
NO.
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, where
f is the fraction of total NO
production from the abluminal surface of the endothelium
|
(A20) |
|
(A21) |
|
|
(A22) |
|
|
(A23) |
|
| |
ACKNOWLEDGEMENTS |
|---|
This work was sponsored by a Whitaker Foundation Biomedical Engineering Research Grant and National Science Foundation Grant BES-9511737.
| |
FOOTNOTES |
|---|
Address for reprint requests: J. C. Liao, Dept. of Chemical Engineering, University of California, Los Angeles, CA 90095-1592.
Received 12 September 1997; accepted in final form 7 January 1998.
| |
REFERENCES |
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Top
Abstract
Introduction
Results
Discussion
Appendix
References
|
|---|
1.
Albina, J. E.,
and
R. B. Mateo.
Nitric oxide.
In: Amino Acid Metabolism and Therapy in Health and Nutritional Disease. Boca Raton, FL: CRC, 1995, p. 99-123.
2.
Baez, S.
Simultaneous measurements of radii and wall thickness of microvessels in the anesthetized rat.
Circ. Res.
25:
315-329,
1969[Abstract].
3.
Bassingthwaighte, J. B.,
C. Y. Wang,
and
I. S. Chan.
Blood-tissue exchange via transport and transformation by capillary endothelial cells.
Circ. Res.
65:
997-1020,
1989
4.
Baxter, L. T.,
and
R. K. Jain.
Transport of fluid and macromolecules in tumors. IV. A microscopic model of the perivascular distribution.
Microvasc. Res.
41:
252-272,
1991[Medline].
5.
Carlsen, E.,
and
J. H. Comroe, Jr.
The rate of uptake of carbon monoxide and of nitric oxide by normal human erythrocytes and experimentally produced spherocytes.
J. Gen. Physiol.
42:
83-107,
1958
6.
Cassoly, R.,
and
Q. Gibson.
Conformation, co-operativity and ligand binding in human hemoglobin.
J. Mol. Biol.
91:
301-313,
1975[Medline].
7.
Chien, S.,
S. Usami,
and
R. Skalak.
Blood flow in small tubes.
In: Handbook of Physiology. The Cardiovascular System. Microcirculation. Bethesda, MD: Am. Physiol. Soc., 1984, sect. 2, vol. IV, pt. 1, chapt. 6, p. 217-249.
8.
Duling, B. R.,
W. Kuschinsky,
and
M. Wahl.
Measurements of the perivascular PO2 in the vicinity of the pial vessels of the cat.
Pflügers Arch.
383:
29-34,
1979[Medline].
9.
Eich, R. E.,
T. Li,
D. D. Lemon,
D. H. Hoherty,
S. N. Curry,
A. J. Matthews,
K. A. Johnson,
R. D. Smith,
G. N. Phillips, Jr.,
and
J. S. Olson.
Mechanism of NO-induced oxidation of myoglobin and hemoglobin.
Biochemistry
35:
6976-6983,
1996[Medline].
10.
Ekelund, U.,
and
S. Mellander.
Role of endothelium-derived nitric oxide in the regulation of tonus in large-bore arterial resistance vessels, arterioles and veins in cat skeletal muscle.
Acta Physiol. Scand.
140:
301-309,
1990[Medline].
11.
Falcone, J. C.,
and
H. G. Bohlen.
EDRF from rat intestine and skeletal muscle venules causes dilation of arterioles.
Am. J. Physiol.
258 (Heart Circ. Physiol. 27):
H1515-H1523,
1990
12.
Fischkoff, S.,
and
J. M. Vanderkooi.
Oxygen diffusion in biological and artificial membranes determined by the fluorochrome pyrene.
J. Gen. Physiol.
65:
663-676,
1975
13.
Fung, Y. C.
Biomechanics: Motion, Flow, Stress, and Growth. New York: Springer-Verlag, 1990.
14.
Garg, U. C.,
and
A. Hassid.
Nitric oxide generating vasodilators and 8-bromo-cyclic guanosine monophosphate inhibit mitogenesis and proliferation of cultured rate vascular smooth muscle cells.
J. Clin. Invest.
88:
4651-4655,
1989.
15.
Hess, J. R.,
V. W. MacDonald,
C. S. Gomez,
and
V. Coppers.
Increased vascular resistance with hemoglobin-based oxygen carriers.
Artif. Cells Blood Substit. Immobil. Biotechnol.
22:
361-372,
1994[Medline].
16.
Ignarro, L. J.
Biosynthesis and metabolism of endothelium-derived nitric oxide.
Annu. Rev. Pharmacol. Toxicol.
30:
535-560,
1990[Medline].
17.
Jia, L.,
C. Bonaventura,
J. Bonaventura,
and
J. S. Stamler.
S-nitrosolhaemoglobin: a dynamic activity of blood involved in vascular control.
Nature
380:
221-226,
1996[Medline].
18.
Kubes, P.,
M. Suzuki,
and
D. N. Granger.
Nitric oxide: an endogenous modulator of leukocyte adhesion.
Proc. Natl. Acad. Sci. USA
88:
4651-4655,
1991
19.
Kuo, L.,
W. M. Chilian,
and
M. J. Davis.
Coronary arteriolar myogenic response is independent of endothelium.
Circ. Res.
66:
860-866,
1990
20.
Kuo, L.,
M. J. Davis,
and
W. M. Chilian.
Longitudinal gradients for endothelium-dependent and -independent vascular responses in the coronary microcirculation.
Circulation
92:
518-525,
1995
21.
Lancaster, J. R.
Simulation of the diffusion and reaction of endogenously produced nitric oxide.
Proc. Natl. Acad. Sci. USA
91:
8137-8141,
1994
22.
Lancaster, J. R.
Diffusion of free nitric oxide.
Methods Enzymol.
268:
31-50,
1996[Medline].
23.
Lancaster, J. R.
A tutorial on the diffusivity and reactivity of free nitric oxide.
Nitric Oxide
1:
18-30,
1997.[Medline]
24.
Laurent, M.,
M. Lepoivre,
and
J.-P. Tenu.
Kinetic modelling of the nitric oxide gradient generated in vitro by adherent cells expressing inducible nitric oxide synthase.
Biochem. J.
314:
109-113,
1996.
25.
Liao, J. C.,
and
L. Kuo.
Interaction between adenosine and flow-induced dilation in coronary microvascular network.
Am. J. Physiol.
272 (Heart Circ. Physiol. 41):
H1571-H1581,
1997
26.
Lightfoot, E. N.
Transport Phenomena and Living Systems. New York: Wiley, 1974.
27.
Lloyd-Jones, D. M.,
and
K. D. Bloch.
The vascular biology of nitric oxide and its role in atherogenesis.
Annu. Rev. Med.
47:
365-375,
1996[Medline].
28.
Malinski, T.,
Z. Taha,
S. Grunfeld,
S. Patton,
M. Kapturczak,
and
P. Tomboulian.
Diffusion of nitric oxide in the aorta wall monitored in situ by porphyrinic microsensors.
Biochem. Biophys. Res. Commun.
193:
1076-1082,
1993[Medline].
29.
McGill, H. C.
The Geographic Pathology of Atherosclerosis. Baltimore, MD: Williams & Wilkins, 1968, p. 1-193.
30.
Moncada, S.,
and
A. Higgs.
The L-arginine-nitric oxide pathway.
N. Engl. J. Med.
329:
2002-2012,
1993
31.
Moncada, S.,
and
A. Higgs.
Molecular mechanisms and therapeutic strategies related to nitric oxide.
FASEB J.
9:
1319-1330,
1995[Abstract].
32.
Moncada, S.,
M. J. Palmer,
and
E. A. Higgs.
Nitric oxide: physiology, pathophysiology, and pharmacology.
Pharmacol. Rev.
43:
109-142,
1991[Medline].
33.
Pitman, R. N.
Influence of microvascular architecture on oxygen exchange in skeletal muscle.
Microcirculation
2:
1-18,
1995[Medline].
34.
Pohl, U., and D. Lamontagne. Impaired tissue perfusion after
inhibition of endothelium-derived nitric oxide. Basic
Res. Cardiol. 86, Suppl. 2: 97-105, 1991.
35.
Popel, A. S.
Theory of oxygen transport to tissue.
Crit. Rev. Biomed. Eng.
17:
257-321,
1989[Medline].
36.
Radomski, M. W.,
R. M. J. Palmer,
and
S. Moncada.
An L-arginine/nitric oxide pathway present in human platelets regulates aggregation.
Proc. Natl. Acad. Sci. USA
87:
5193-5197,
1990
37.
Saito, Y.,
A. Eraslan,
V. Lockard,
and
R. L. Hester.
Role of venule endothelium in control of arteriolar diameter during functional hyperemia.
Am. J. Physiol.
267 (Heart Circ. Physiol. 36):
H1227-H1231,
1994
38.
Schmidt-Schönbein, H.,
T. Fisher,
G. Driessen,
and
H. Rieger.
Microcirculation. Baltimore, MD: University Park, 1979, p. 353-418.
39.
Shimokawa, H.,
and
A. Takeshita.
Endothelium-dependent regulation of the cardiovascular system.
Intern. Med.
34:
939-946,
1995[Medline].
40.
Stamler, J. S.,
O. Jaraki,
J. Osborne,
D. I. Simon,
J. Keaney,
J. Vita,
D. J. Singel,
C. R. Valeri,
and
J. Loscalzo.
Nitric oxide circulates in mammalian plasma primarily as an S-nitroso adduct of serum albumin.
Proc. Natl. Acad. Sci. USA
89:
7674-7677,
1992
41.
Stone, J. R.,
and
M. A. Marletta.
Spectral and kinetic studies on the activation of soluble guanylate cyclase by nitric oxide.
Biochemistry
35:
1093-1099,
1996[Medline].
42.
Vaughn, M. W., L. Kuo, and J. C. Liao.
Estimation of nitric oxide production and reaction rates in tissue
using a mathematical model. Am. J. Physiol. In press.
43.
Wood, J.,
and
J. Garthwaite.
Models of the diffusional spread of nitric oxide: implications for neural nitric oxide signalling and its pharmacological properties.
Neuropharmacology
33:
1235-1244,
1994[Medline].
44.
Zuck, T. F.,
and
J. G. Riess.
Current status of injectable oxygen carriers.
Crit. Rev. Clin. Lab. Sci.
31:
295-324,
1994[Medline].
1a.
Finlayson, B. A.
Nonlinear Analysis in Chemical Engineering. New York: McGraw-Hill, 1980, p. 214.
2a.
Malinski, T.,
Z. Taha,
S. Grunfeld,
S. Patton,
M. Kapturczak,
and
P. Tomboulian.
Diffusion of nitric oxide in the aorta wall monitored in situ by porphyrinic microsensors.
Biochem. Biophys. Res. Commun.
193:
1076-1082,
1993.
3a.
Press, W. H.,
B. P. Flannery,
S. A. Teukolsky,
and
W. T. Vetterling.
Numerical Recipes in C (2nd ed.). Cambridge, UK: Cambridge University Press, 1992.
4a.
Slattery, J. C.
Momentum, Energy, and Mass Transfer in Continua (2nd ed.). Malabar, FL: Kreiger, 1981, p. 482, 451.
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