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1 Department of Chemical Engineering, Nitric oxide (NO) produced
by the vascular endothelium is an important biologic messenger that
regulates vessel tone and permeability and inhibits platelet adhesion
and aggregation. NO exerts its control of vessel tone by interacting
with guanylyl cyclase in the vascular smooth muscle to initiate a
series of reactions that lead to vessel dilation. Previous efforts to
investigate this interaction by mathematical modeling of NO diffusion
and reaction have been hampered by the lack of information on the
production and degradation rate of NO. We use a mathematical model and
previously published experimental data to estimate the rate of NO
production, 6.8 × 10
mass transfer; kinetics; parameter estimation; diffusion
NITRIC OXIDE (NO) plays a diverse role as a biologic
messenger, regulator, and cytotoxic agent. In the circulatory system, NO produced by the endothelium is a major factor in preventing platelet
aggregation and in regulating microvascular tone and permeability (18).
This control is exercised through its interaction with soluble guanylyl
cyclase in the vascular tissue. However, the mechanism by which NO, a
reactive free radical, is transferred from the endothelium to the lumen
and vascular smooth muscle is still under debate.
Mathematical modeling can be a useful tool for investigating the
simultaneous diffusion and reaction of NO. However, these models must
be based on known mechanisms and employ physiologically relevant rate
constants. These rate constants are not well characterized, nor is it
clear how the NO production and decomposition rates affect the
diffusion distance of NO in the blood, the endothelium, and the
surrounding smooth muscle.
In vivo, NO diffuses from the endothelium into the luminal and
abluminal regions. The NO that diffuses into the abluminal region
travels into the vascular smooth muscle and binds with guanylyl
cyclase. Because of its reactivity, NO may be consumed by numerous
reactions before it reaches its target. In particular, NO may react
with O2 or free radicals (19), bind with heme-containing proteins (20), or interact with iron-sulfur centers (18), as well as
participate in several other reactions that result in its degradation
(1). Because of the complexity of the kinetics, the interaction of NO
in the smooth muscle is not well characterized and is usually ignored
or treated as a first-order reaction with a half-life
(t1/2). The value of t1/2 is
usually estimated from the decomposition of NO in aqueous solution,
where the reaction of NO with O2 is second order in NO (7,
17, 27), or from measurements made from perfused tissue (9). However,
t1/2 alone provides little insight when diffusion
and reaction occur simultaneously.
Several previous models of NO reaction and diffusion depended on the in
situ NO concentration measurements obtained by Malinski et al. (16),
who used an electrochemical technique to measure the NO concentration
at the endothelium and in the vascular smooth muscle of a rabbit aorta.
Although the primary goal of their work was to demonstrate the utility
of their porphyrinic microsensor (15), their data have had significant
impact on NO diffusion modeling efforts.
The first mathematical model of NO diffusion and reaction in a biologic
system was that of Lancaster (11-13). He investigated whether the
NO concentration in a cell was primarily influenced by the cell or the
surrounding cells and examined the influences of NO sinks. He modeled
the endothelium as a line of NO point sources and sinks and used an NO
production rate based on the maximum NO concentration from the
measurements of Malinski et al. (16). Lancaster's initial work was
closely followed by that of Wood and Garthwaite (28), who modeled
neuronal NO production and diffusion in the brain. They modeled a
neuron as a point source of NO and assumed that NO degraded in
the brain tissue as a first-order reaction with
t1/2 of 0.5-5 s. The data of
Malinski et al. were used as a guide in approximating the NO production
rate. These researchers used experimental data to justify their choices for the NO production rate, but the accuracy of the estimates and the
effects of other parameters were not examined. Furthermore, the finite
size of the NO-emitting region has not been taken into account.
The NO concentration in tissue depends on the diffusion coefficient and
the degradation (reaction) rate as well as the production rate. Our
purpose here is to provide a general procedure for estimating these
three parameters, to determine the error in these estimates, and to
investigate the relationships between the parameters. To accomplish
this, we set out a general mathematical model for NO production from
the endothelium and consumption in the surrounding tissue. This model
is then applied to the experiment of Malinski et al. (16). The
parameters and their variability are determined by applying nonlinear
methods to fit the model to the data.
System description.
We modeled NO production, diffusion, and reaction as a three-section
system composed of the luminal region, the endothelium, and the
abluminal region. NO is produced from the surfaces of the endothelium
and diffuses into each of these regions. The spatial variation of NO
concentration will be different in each section, reflecting the
diffusion coefficient and the reaction rate expression of that section.
ABSTRACT
Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References
14
µmol · µm2 · s1;
the NO diffusion coefficient, 3,300 µm2
s1; and the NO consumption rate coefficient in the
vascular smooth muscle, 0.01 s1 (1st-order rate
expression) or 0.05 µM1 · s1 (2nd-order rate
expression). The modeling approach is discussed in detail. It provides
a general framework for modeling the NO produced from the endothelium
and for estimating relevant physical parameters.
INTRODUCTION
Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References
ESTIMATING PARAMETERS FROM EXPERIMENTAL DATA
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Fig. 1.
NO-producing singular surfaces that represent endothelium. A:
microscopic planar source represented by a thin ellipsoid about
endothelium. Dotted lines, cross section of ellipsoid that approximates
luminal and abluminal surfaces. Interior of ellipsoid represents
endothelial cells. NO is produced at these surfaces and diffuses into
and away from enclosed endothelial cells. In limit of zero thickness,
ellipsoid collapses to a disk. B: close-up of an endothelial
cell showing flux of NO when there is a concentration gradient across
cell. Flux of NO from surface ( 
) is shown for a case
in which most of NO production flows into lumen. Subscripts lu, ab, and
ec are lumen, abluminal region, and endothelium, respectively.
NO). The NO diffuses into the vessel wall
as well as into the luminal region. In the physiological salt solution
in the lumen, NO reacts with O2 at a known rate. The
reaction rate in the vessel wall is unknown.
We approximated this microscopic source by a thin, flat
"pancake"-shaped ellipsoid with foci at ±a. This
geometry can be described using a system of oblate spheroidal
coordinates, as depicted in Fig. 2. The
ellipsoidal source is suited for modeling microscale NO distribution,
because it can represent a single cell or a finite group of endothelial
cells. In Fig. 2 the thickness of the ellipsoid (2
) is exaggerated
for illustration, typically a/ 
100 for the size of the
ellipsoidal source estimated below. As approaches 0, the ellipsoid
appears increasingly disklike, with a approaching the diameter
of the disk.
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Assumptions. The following assumptions were made to simplify the analysis.
1) NO production occurs on the cytosolic membrane of endothelial cells. One of the primary enzymes that produces NO, constitutive NO synthase (cNOS), is membrane associated (2, 10). This suggests that NO production by cNOS can be approximated by a surface reaction. This production rate fixes the flux of NO at the endothelial cell surface and the corresponding boundary conditions. Thus the endothelium is treated as two producing surfaces: one corresponding to the luminal surface and one corresponding to the abluminal surface. 2) The reaction of NO depends only on the NO concentration. The NO concentration in a stimulated endothelial cell in vitro is on the order of 1 µM (16). The concentration of O2 in tissue in vivo is ~27 µM (22), and the concentration in air-saturated buffer is about eight times higher. Because these concentrations are much larger than that of NO, we assume that the NO reaction rate depends only on the NO concentration (cNO). In this study, two reaction rate expressions are used: NO degradation is proportional to cNO (1st order) or to c2NO (2nd order). When referring to NO, we use the terms reaction and degradation interchangeably. 3) NO diffuses as if in a binary solution. We approximate the diffusion of NO by a binary system in which NO is a trace quantity. The diffusion coefficient (D) of NO in buffer at 37°C has been estimated as 3,300 (16) and 4,500 µm2/s (5). We expect the effective diffusion coefficient in the vascular smooth muscle to be less. The diffusion coefficient of NO in lipid membrane was ~10-40% of the value in buffer (5). Because NO is dilute, the effective diffusion coefficient is assumed to be independent of concentration. This assumption also means that we neglect the different properties of lipids and aqueous solution, so the NO concentration will be continuous throughout all regions. In general, only the chemical potential of NO is continuous. Because NO is dilute and uncharged, NO would be expected to behave ideally in solution, so partial pressure, rather than concentration, is continuous. However, there are few data about NO solution properties, so as a first approximation we will assume NO concentration to be continuous throughout.Governing equations and boundary conditions. Here the general form of the equations and boundary conditions are stated. They are valid for any coordinate system. In Oblate spheroidal coordinates, these equations are written for the coordinate system that describes the microscopic NO-producing source. Although not discussed in this article, the equations of this section could be used to determine the NO concentration in other systems of physiological interest.
The concentration of a diffusing, reacting substance, such as NO, is described by the species mass balance (26). For NO, this balance can be written as
|
(1) |
is the vector gradient operator,
2 is the Laplacian operator, cNO is NO
concentration, and 
NO 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). For a diffusing trace component in a
quiescent system or tissue, cNO · v = 0.
The NO distribution in each of the three regions, the lumen,
endothelium, and abluminal region, is governed by Eq. 1, with the regions linked through their boundaries. Six boundary conditions and three initial conditions are needed. The initial conditions are
|
(2) |
|
(3) |
)
to the production rate (see APPENDIX A). These conditions
can be written, at the luminal surface, as
|
|
(4) |
|
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|
(5) |
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lu is the flux of NO from
the producing surface into the lumen, ab is the
flux of NO from the producing surface into the abluminal region,
ec is the flux of NO from the producing surface
into the endothelial section, nlu is the unit
normal vector pointing into the lumen, nab is the
unit normal vector pointing into the abluminal region, and
NO(t) = NO,lu(t) + NO,ab(t) is the total NO
production rate per unit area.
At the NO-producing surface, the sum of fluxes from the singular
surface is equal to the surface production rate (see APPENDIX A for more detail). When the NO consumption rate on one side of
the endothelial cell (e.g., if the lumen contained a hemoglobin solution) is much greater than that on the other (abluminal) side, the
NO concentration will be higher on the abluminal side, causing a
concentration gradient directed toward the lumen. In this case, ec = 0 at the luminal surface and
lu is composed of
NO,lu(t) and a portion of
NO,ab(t). This is
depicted in Fig. 1B.
Oblate spheroidal coordinates.
As discussed above, we believe that the size of the source may be
important and should be determined in the analysis. For this analysis
the size is taken into account by approximating a group of endothelial
cells as a thin ellipsoid (Fig. 1A), which is modeled in
oblate spheroidal coordinates. This description would also be valid for
one cell. In this system the ellipsoidal NO-producing surface is
represented by a coordinate surface. The oblate spheroidal coordinate
system is shown in Fig. 2. The coordinates have the following ranges:
< , 0 
/2, and 0 
< 2 (out of
the plane of the page). The expressions for the gradient and the
Laplacian in this coordinate system can be determined from Happel and
Brenner (8) or can be found explicitly in Moon and Spencer (21). For
axisymmetric NO production, the general NO mass balance (Eq. 1)
in this coordinate system is given by
|
|
|
(6) |
sinh , a cos sinh ) are the
Cartesian coordinates of a point in the - plane. NO is produced
by the surface of the thin ellipsoidal region.
The boundary conditions, Eqs. 4 and 5, for the luminal
surface = 0 = sinh1 /a
are
|
|
(7) |
= 0
|
|
(8) |
-component of the NO flux at the surface is, at
= ±0 (the surface of the endothelium)
|
(9) |
and . As a partial differential equation in time and
two space dimensions, the solution requires significant computational
effort. We did solve this problem (the 2-dimensional problem), but for most of the following discussion, we use a simplified approximation of
the system. To estimate the parameters of the system, repeated solution
of the reaction-diffusion equation is required. The effort required to
repeatedly solve the two-dimensional problem, Eqs. 6-9, is
not justified by the limited data available.
For parameter estimation and to understand the character of the
solution, we want to simplify the problem in a way that allows a
one-dimensional solution. There are two limiting cases of oblate spheroidal coordinates: the sphere and the disk. In the limiting case
of a 
0 or a finite and sinh , the
spheroidal coordinate surfaces become spherical. In this case, the
concentration depends only on r = a sinh , so the
concentration dependence on may be neglected. In the other limiting
case, as sinh 0 the surface collapses to a disk. Near the
edges of the disk, the concentration depends strongly on . The
concentration data were obtained at points that were on the axis of the
NO-producing region, so, depending on the size of the region
(a), these measurements may be less influenced by the edges.
This suggests that we investigate the effect of neglecting the
concentration dependence on . Because this dependence enters through
the flux term Eq. 9, let us neglect and replace the
boundary condition Eq. 9 with the approximation
|
(10) |
0 1 for small 0. This boundary
condition is equivalent to requiring that the concentration not vary
with position over the surface of the emitting region, although it
still will vary with time. Again by drawing on the analogy of the disk
source, the error introduced by this estimate can be examined. For a
disk source in an infinite homogeneous medium, the steady-state
constant flux concentration profile can be obtained from Carslaw and
Jaeger (3). For this case, the ratio of NO concentration at the
centerline to NO concentration at the edge is 2/. Thus we can expect
this approximation to underestimate the true production rate, since the
average concentration for the constant flux case is less than the
centerline value. For larger values of a the approximation will
improve, since the effect of the edges will decrease. For small values
of a the approximation will be less reliable.
Because we have neglected the dependence of the concentration at
the endothelial surface, let us also simplify the NO mass balance by
neglecting the concentration dependence in the direction everywhere
|
(11) |
) is taken as 2.5 µm. The
remaining boundary conditions are given by Eq. 3
|
(12) |
NO(t) = NO, where the NO production per unit area is a constant. In general, the production rate would be expected to vary
with time, and for the data of Malinski et al. (16), it does appear to
decrease somewhat after ~30 s. For this analysis, we consider it
constant for the first 30 s and omit the subsequent data. Malinski et
al. show NO concentration over a much longer period, and it appears
that the simplest case, constant production, is a reasonable first
approximation.
NO reaction rates. The reaction in the smooth muscle is represented as a composite rate summing the contributions from numerous NO decomposition mechanisms. The simplest description for this composite reaction in the luminal or abluminal region is given by the empirical rate expression
|
(13) |
NO is the rate of NO
decomposition, n is the reaction order (for this study, we use
1 or 2), x refers to the luminal (lu) or abluminal (ab) region,
and k is the rate coefficient. The reaction rate within the
endothelium is taken to be the same as that in the abluminal region. In
what follows, we will often refer to the "first-order rate
expression" when the NO decomposition is described by Eq. 13
with n = 1 or the "second-order rate expression" when
n = 2.
NO diffusing into the lumen reacts with O2 in aqueous
solution. This reaction has been shown to have third-order kinetics (27)
|
(14) |
NO is the rate of
decomposition of NO by reaction with the surrounding media and
k3,lu 9 × 106
M2 · s1 (27). Similar values
of the rate constant have been given by Mayer et al. (17) and Ford et
al. (7). In analyzing experimental data, we will consider the bath to
be air saturated. Because the concentration of O2 is much
greater than that of NO, it will remain approximately constant. This
suggests the approximation
|
(15) |
1 · s1 (assumption
2). This is the rate of NO decomposition that is used in
the luminal region.
Numerical technique. The one-dimensional diffusion problem, Eqs. 7, 8, and 10-12, was solved numerically. We transformed each partial differential equation into a system of ordinary differential equations in time by discretizing the spatial derivative (6). A similar approach has been used to model the NO concentration profile in solution resulting from NO production by a monolayer of cultured cells (14).
We used second-order centered differencing, because it is straightforward to implement. Other discretization methods, such as orthogonal collocation, may be more efficient for computations. 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 stifbs (23); the routines stiff and stifbs 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. Because numerical derivatives are computed for the parameter estimation routines and the sensitivity analysis, a high-accuracy solution is required. The error criterion for stifbs was 3 × 106. Although the boundary condition, Eq. 12, applies to an infinite domain, we solved the differenced
equations over a finite grid by applying the boundary condition,
Eq. 12, to the outermost points. Several maximum distances were
used to ensure that the result did not vary appreciably.
The two-dimensional problem was solved similarly, but a uniformly
spaced grid of 51 × 10 in. (, ) was used. Because the Jacobian was no longer tridiagonal, time updates were obtained by
solving the resulting system using the conjugant gradient method linbcg (23).
Parameter estimation.
Our objective was to determine the values for the parameter set
p = (a, NO,
kn,lu, Dlu,
Dab), in the one-dimensional problem, Eqs. 7,
8, and 10-12. We did this by minimizing the
2 merit function.
|
(16) |
i.
When numerical parameter estimation routines are used, it is usually
desirable to scale the parameters so that they have a value on the
order of 1.0. Here the scaling was done by making the parameter set
nondimensional by using the following characteristic values: length
(l0) of 100 µm [distance from the endothelium
to the sensor in the smooth muscle in the experiment of Malinski et al.
(16)], concentration (c0) of 1 µM [order of NO
concentration measured by Malinski et al. (16)], and diffusion
coefficient (D0) of 1,500 µm2/s
(typical NO diffusion coefficient in lipid, assumption
3)
|
|
(17) |
3
amol · µm3 · µM1
is a unit conversion constant.
The 2 merit function, Eq. 16, was minimized
using the Levenberg-Marquardt method. This method is implemented in
marqcoef (23). All points were weighted equally. The experimental error
is unknown, so we arbitrarily selected a standard deviation of 5% of
the maximum reading; the estimated parameter values depended only
weakly on this value. The parameter error estimates were made using the formal covariance matrix of the fit, which is computed by the routine
marqcoef. The derivatives of the merit function with respect to the
parameters were computed numerically. Numerous simulations were
performed using a range of initial parameter values.
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RESULTS |
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Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References
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Parameter estimation.
The parameters were estimated for two different kinetic rate
expressions for the NO reaction in the vascular smooth muscle, first
and second order, and for different parameters held constant. In
cases 1 and 2, Dlu was fixed at
4,500 µm2/s, and the parameter set (a,
NO, kn,ab,
Dab) was fit for first- and second-order reaction
(Table 1). In cases 3 and 4, Dlu was added to the parameter set. The value for
Dlu was estimated to account for any enhancements
to luminal mass transfer that might have occurred during the
experimental measurements. Examples of fluid motion that increase NO
loss to the lumen include such effects as thermal gradients (which may
occur in experimental work at elevated temperature) and small
disturbances from the injection of the bradykinin bolus. If such
effects are present, then the value estimated for
Dlu should be considered an "effective" luminal diffusion coefficient. In case 5, we examined the
parameters that would be required to model the measured concentrations
if there were no NO degradation.
|
2 function, and many
initial values in parameter space were tried. The differences of the
fit between the cases in Table 1 were statistically insignificant.
The computed NO concentration profiles from the parameters in Table 1
are shown in Fig. 3, along with the
experimental data of Malinski et al. (16). The concentration profiles
in Fig. 3 were determined for cases 1 and 2. The
concentration curves computed for other cases in Table 1 were
essentially superimposed over cases 1 and 2 (data not
shown). The parameters were estimated using the one-dimensional model,
but the production rate used in the two-dimensional model, 6.8 × 1014
µmol · µm2 · s1,
was determined by scaling the one-dimensional production rate by 1.25, the difference between the one- and two-dimensional models, as shown in
APPENDIX B. It is important to know the effect of the
parameters on the model prediction, so the sensitivity of the
parameters on the one-dimensional model are also computed. Because the
approximation used to obtain the one-dimensional model primarily
affects the estimate of production rate, the sensitivity estimates for
the one-dimensional model should be valid approximations of the
two-dimensional model as well.
|
Confidence regions. When several parameters are estimated, there are usually interactions. For example, increasing the NO production rate, decreasing the NO reaction rate, or increasing the luminal diffusion coefficient increases the NO concentration in the smooth muscle. These interactions may be quantified by computing the formal covariance matrix of the fitted parameters (Table 2). The information from the covariance matrix can be displayed graphically as joint confidence regions (23). The confidence regions of pairs of parameters for case 1 of Table 1 are shown in Fig. 4. These regions depict the interdependence between pairs of estimated parameters at the 90% confidence level, with the other parameters held constant at their computed value listed in Table 1. The curves in Fig. 4 are normalized by the fitted value from Table 1.
|
|
NO). Its confidence regions (Fig. 4,
A-C) are narrow, and much of the regions are centered on
the normalized optimal value, 1. The "tilted" orientation of the
confidence regions of NO with a and
Dab (Fig. 4, A and C) suggests
that these pairs of parameters are correlated. However, the regions are
fairly compact. There is substantial uncertainty in the value of
k1,ab, as evidenced by its extended confidence
regions (Fig. 4, B, D, and E). The value of
k1,ab is correlated only slightly with a,
and it is nearly independent of Dab.
The confidence regions for case 2, the second-order rate
expression, are similar (Fig. 4, dashed curves). Figure 4, A
and F, is more elongated, reflecting larger uncertainty in
the value for a. Similarly, there is much uncertainty in
k2,ab. The major difference between the cases is
that the value of k2,ab depends quite strongly on
a, whereas in Fig. 4, B, D, and E,
k1,ab is essentially independent.
Parameter sensitivity. A measure of the sensitivity of the NO concentration to the parameters can be determined by computing the relative change in NO concentration caused by a small change in a parameter. This measure is called the sensitivity coefficient and is defined by
|
|
(18) |
NO,
k2,ab, Dlu,
Dab) and subscripts refer to the point at which the concentration is measured: the endothelium (ec) or the vascular smooth muscle (ab). These correspond to the two locations measured in
the experiment of Malinski et al. (16).
Over the time course of the experiment, the sensitivity coefficients
for the NO production rate are among the largest of all the sensitivity
coefficients (Figs. 5 and 6). They vary little with time or reaction
order. This high sensitivity explains why the NO production rate can be
estimated accurately. The reaction rate coefficients have the least
influence on NO concentration for the first-order rate expression. This
result explains the large variability in the estimates of
k1,ab and k2,ab.
The sensitivity to the diffusion coefficients is more complex.
Regardless of the reaction rate expression, the concentration in the
smooth muscle is very sensitive to Dab at early
time. This suggests that an NO probe with a fast response and
measurements of early time data would be useful in determining the
value for Dab. At the endothelial cell,
SD(ab) and SD(lu) are of
similar magnitude for both reaction rate expressions.
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DISCUSSION |
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Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References
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The constant NO production approximation provides a reasonable fit to the data of Malinski et al. (16), as shown in Fig. 3. Although the NO production rate probably varies with time because of the delay in cNOS activation and negative-feedback inhibition from NO (24), NO production rate has the least variability, and the value is consistent over the cases examined in Table 1. Because of the physical difficulties in stimulating a microscopic region of cells, there are several other potential sources of error. It is unlikely that the NO-emitting region is actually circular or that the production is precisely uniform over the region. Any of these factors could account for the difference between the experiment and the computed concentration. Despite these factors, we find that the models presented here approximate the magnitude and trend of the data reasonably well. The NO concentration predicted by the two-dimensional model tracks the data trend more closely than the one-dimensional approximation, as expected if the constant flux approximation is valid.
Estimation of NO production rate.
The NO production rate is required for any modeling study of NO
concentration in a biologic system. For our computations we assumed
this rate to be constant; however, as more is learned of endothelial NO
synthase and its regulation, it may become possible to predict the NO
production as a function of time,
NO(t). The theoretical
NO(t) could be tested
by modeling the production of NO in a manner analogous to that done
here.
NO will always depend on
the value selected for a and Dab. Despite this, NO varies over a fairly narrow range.
To obtain the actual production rate, 6.8 × 1014
µmol · µm2 · s1,
we use the difference between the two- and one-dimensional models (see
Fig. 8) to correct the estimate of Table 1.
For the first-order rate expression,
SNO is relatively constant, as seen in
Fig. 5, A and C, so the
cNO,ec and cNO,ab are directly proportional to
NO production rate over the course of the experiment. This might be
anticipated, since the boundary conditions and the mass balance
equations are linear in NO production rate. Therefore, a small change
in the NO production rate would proportionally change the NO
concentration at each point. The second-order rate expression is
nonlinear in NO concentration, and this is reflected by the decreased
dependence of the smooth muscle NO concentration on NO production rate,
as indicated in Fig. 6, A and
C.
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Estimation of NO degradation rate constant.
There was significant uncertainty in the values of the rate
coefficients that characterize the NO consumption. We were unable to
differentiate between the first- and second-order rate expressions from
these data. If the reaction is second order, then it may be inferred
that the interaction of NO with materials in tissue is more complicated
than simple binding, an interaction that would be characterized by
first-order degradation. One possible second-order reaction has been
suggested by Lancaster (11). The NO reaction in the tissue is not
primarily with O2, because the estimated value for
k2,ab is more than an order of magnitude larger
than it would be if the decomposition of NO were governed by Eq. 15 (k2,ab 2.0 × 103
µM1 · s1).
1. For a second-order reaction,
t1/2 is determined by t1/2 = 1/(c0k2), where c0 is the
initial concentration. By use of the estimated value for the
second-order rate coefficient (Table 1, cases 2 and 4),
k2,ab = 0.05 µM1 · s1, and by setting
cNO = 1 µM, t1/2 20 s. There is
additional NO loss through diffusion that is not included in
t1/2, so t1/2 should be used
with caution.
The possibility of complicated interactions between NO and tissue has
implications in modeling the reaction and diffusion of NO. If the
reaction of NO with the vascular smooth muscle is first order, then
Eq. 1 is linear, and the concentration in the tissue from
several sources can be summed (11, 28). This will not be the case if
the rate equation is nonlinear in NO concentration, such as the
second-order kinetics discussed here or Michaelis-Menten kinetics. That
the contribution from different sources cannot be directly added is a
complication resulting from the mathematics of the nonlinear
differential equations that represent the diffusion process, rather
than the physics of the diffusion process itself.
Estimation of the size of the NO-producing area. In the experiment of Malinski et al. (16), the NO concentration was measured at the endothelium and in the vascular smooth muscle 100 µm away. Because even a few endothelial cells are of this order of magnitude, a realistic model must take into account the size of the NO-producing region. Furthermore, even if the NO-producing region could be approximated by a point source or infinite plane, the validity of these approximations cannot be determined a priori, so an estimate of size is still necessary. The finite size is taken into account by solving the diffusion equation, Eq. 1, in oblate spheroidal coordinates. The values for the characteristic size a for this system are given in Table 1. They are the same order of magnitude as the distance. For this reason, a point-source approximation of NO production would not accurately describe the concentration profile. Assuming that NO is produced from an infinite region introduces additional error, as can be seen in the limiting behavior discussed in APPENDIX B.
The values for a were fairly consistent between the cases presented in Table 1. For the first-order rate expression, a was weakly correlated with Dab and k1,ab, as shown in Fig. 4D and Table 2. For the second-order rate expression, there was more uncertainty in a and the value was more strongly correlated with Dab and k2,ab, as shown in Fig. 4D and Table 2. A further effect of reaction order is that the first-order rate expression is more sensitive to a than the second-order rate expression, as can be seen in Figs. 5, A and C, and 6, A and C. The sensitivity of the concentration to a for the models is shown quantitatively by the slope of the curves in Fig. 8. For a > 150 µm the two-dimensional model is somewhat more sensitive to a than the one-dimensional model, as indicated by the slightly steeper slope, although the difference diminishes as a increases. Because of the similar character of these solutions, we expect that the preceding comments about the effect of a are also valid for the two-dimensional model. The computed size of the source is much larger than a single cell but is about what might be expected from the experiment. Malinski et al. (16) injected a 10 nM bolus of bradykinin. At 0.1 M, the bolus would have a volume of 0.1 µl, so a spherical bolus would have a radius of 280 µm. Even a smaller bolus would be dispersed by diffusion and convection from injection. Because there are only two sensors and they are on the axis of the ellipsoid, we have no direct information about the size of the region; it must be implied from the model. If off-axis data were available (i.e., there were more sensors), the one-dimensional model may be found to be too simple, in which case the two-dimensional model would have to be used.Estimation of diffusion coefficient.
Previous analyses (11, 14, 28) used the diffusion coefficient
determined by Malinski et al. (16) for tissue and aqueous systems. In
our analysis we estimated Dab while holding
Dlu fixed (Table 1, cases 1 and 2)
or including it with the variables estimated (Table 1, cases 3 and 4). When Dlu was estimated, we found it to be unexpectedly high. There are two reasons why
Dlu may be unrealistically high. First,
experimental difficulties may result from minute convection currents
caused by maintaining temperature control or the action of the
microinjector. This point underscores the importance of mass transfer
to the lumen. Second, it may be an artifact of the parameter estimation
algorithm. Because the lumen can act as a mass sink when
Dlu is not fixed, it can offset the effect of the
other parameters. Thus the parameters may be selected from a much wider
range by adjusting the mass transfer to the lumen accordingly. This is
why cases 3 and 4 have more variability in the
parameters. In either case, the effect of Dlu in
reducing the 2 was not very large, so cases 1 and 2 are considered to be the most reliable.
Comparison with previous mathematical models. This analysis differs from past analysis by accounting for the size of the NO-producing region, by considering NO production to be by surface reaction, by using the time course of the concentration data, and by allowing different properties in the tissue and luminal regions.
Lancaster (11) and Wood and Garthwaite (28) used the data of Malinski et al. (16) to estimate NO production rate. Wood and Garthwaite (28) assumed an NO point source surrounded by a 0.5-µm-radius sphere. They estimated the strength of the point source, 2.1 × 1017
mol/s, which gave an NO concentration of 1 µM at the surface of the
sphere. In terms of total NO production, our estimates (Table 1,
cases 1 and 2) are 1.3 × 1014 mol/s.
The difference is primarily the result of different assumptions of the
source: point or finite.
The phenomenological production rate constant given by Lancaster (11),
10.3 µM/s, is difficult to compare directly with the production rate
obtained here. However, the equation for NO diffusion from a
one-dimensional point source given by Lancaster (11) is identical to
that for NO diffusion from an infinite plane. Therefore, in matching
this equation to the steady-state data (see Fig. 2 in Ref. 11),
Lancaster effectively chose an NO production rate on the basis of NO
production from an infinite planar source. We have used an infinite
planar source to verify our computations and found that the production
rate was similar to that predicted by the oblate spheroidal geometry,
although the steady state was attained much more quickly. The
similarity is a consequence of the NO-producing region being larger
than the distance between NO sensors.
Laurent et al. (14) assumed NO production rates of 1 × 1010 to 1.6 × 108 M/s. If it is
assumed that the average thickness of the endothelium is 2.5 µm, this
would be equivalent to 2.5 × 1019 to 4 × 1017
µmol · µm2 · s1.
This production rate is three to five orders of magnitude less than
that computed here. Because Laurent et al. (14) were interested in the
NO production from activated inducible NO synthase, rather than
bradykinin-stimulated cNOS, using a much lower production rate is
reasonable.
Previous researchers considering the NO consumption rate in the tissue
have assumed it to be first order, with a t1/2 ranging from 0.5 to 5 s (k1,ab = 1-0.1
s1), which is much faster than the first-order result
from Table 1. This difference is probably due to two factors. First,
estimates of t1/2 in experimental systems usually
do not distinguish between reaction and diffusion; therefore, the rapid
diffusion of NO decreases the effective t1/2. In
Table 1, the two effects are separated. Second, the reaction rate may
vary with tissue location. The cardiac tissue for which Kelm and
Schrader (9) estimated t1/2 of 0.1 s is rich in
myoglobin, which reacts very rapidly with NO.
Wood and Garthwaite (28) observed that the reaction rate of NO is of
relatively little importance near the source and in short time scales.
This is consistent with Figs. 5, A and C, and 6,
A and C, which show that the sensitivity of NO
concentration to the rate coefficient is higher at later times.
However, NO is produced by the endothelium over a fairly long time
scale, so the reaction rate may be expected to be important at later times and far from the endothelium.
As pointed out by Lancaster (13), the concentration profile of a
three-dimensional point source falls off much too rapidly to simulate
the data of Malinski et al. (16). However, for an array of point
sources, the overlapping production broadens the concentration profile
(11, 13, 28). Similarly, the ellipsoidal macroscopic source presented
here follows the time course of the data at the endothelium and in the
smooth muscle. The similarity between multiple point sources and the
continuum case is because a continuous macroscopic source can be
considered a distribution of point sources, when the reaction kinetics
are linear. In fact, the equation for a disk-shaped point source
(APPENDIX B) is derived by integrating the point source
solution over the surface of the disk, effectively combining an
infinity of point sources.
Implications for further modeling and experimentation. Although not their original purpose, Malinski et al. (16) provided information that has proven quite useful in parameter estimation. As seen in Fig. 3, there is little difference in the fit or the concentration profiles between the first- and second-order reactions. Furthermore, there is considerable uncertainty in the values of the parameters, particularly in the reaction rate coefficient kn,ab. A knowledge of the tissue reaction mechanism is important for understanding NO interactions in tissue and for determining the range of NO influence. Consequently, it would be useful to have a technique that clearly shows a difference in mechanisms.
An experiment similar to that of Malinski et al. (16) could be modified to distinguish between reaction mechanisms. An additional experiment in which the rate of NO diffusion into the luminal region has been decreased, possibly by decreasing the diffusion rate by adding protein to the buffer, is required. The effect of reduced diffusion can be seen in Fig. 7. When the parameters are estimated from one experiment, Dlu = 4,500 µm2/s (Table 1, cases 1 and 2), the profiles for first- and second-order reactions are indistinguishable. By use of the same parameters, except with Dlu decreased to 1,000 µm2/s, the NO concentration profiles for the first- and second-order cases can be distinguished. Thus experiments using two different diffusion coefficients can be used to distinguish between reaction expressions. In either case, it is important to prevent convective luminal mass transfer. The presence of even a slight amount of convection can change the interpretation of experimental data significantly. As seen in Table 1, case 5, enhanced mass transfer to the lumen can make pure diffusion indistinguishable from diffusion-reaction.
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APPENDIX A |
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Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References
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Boundary conditions at the endothelial surface. 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) (26). The NO consumption in the luminal region is different from that in the abluminal region (and possibly the endothelial cell itself ). Therefore, the NO gradient and the flux of NO in each of these regions will also differ. The jump mass balance relates the fluxes to the production rate. For NO, this balance can be written
![[c<SUB>NO</SUB>(<B>v</B><SUB>NO</SUB> − <B>u</B>)⋅<B>n</B>] = <A><AC>q</AC><AC>˙</AC></A><SUB>NO</SUB>](/content/vol274/issue6/fulltext/H2163/img031.gif)
|
(19) |
A2 · n1, where
n1 is the normal vector pointing into phase
1. The NO concentration is denoted by cNO,
vNO is the molar average velocity of NO at the
interface, u is the velocity of the interface, and
NO is the rate of NO production by a unit of
surface area
(moles · time1 · area1).
This equation relates the velocity of NO at the interface to the
velocity of the interface and the production rate of NO. For the
parameter estimation problem in oblate spheroidal coordinate systems,
the system is quiescent and u = 0 at the endothelium.
An example of nonzero interface velocity u occurs during
changes in tone of cylindrical blood microvessels as part of the
integrated control of the microcirculation. As the vessel dilates and
contracts, the endothelium-lumen interface moves; therefore, u 
0.
The mass flux vector 
cNO
vNO (mass transported per unit time per unit area)
will, typically, vary with time. For a trace quantity like NO, it can
be expressed in the form of Fick's law
|
(20) |
|
(21) |
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APPENDIX B |
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Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References
|
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Comparison of oblate spheroidal models and other limiting cases. It would be helpful to have an analytic solution of the diffusion equation in oblate spheroidal coordinates to assist in verifying the numerical computations. Because predicting the concentration as a function of time is central to our parameter estimation, steady-state approximations, which are much simpler, are less interesting. However, even for the simplest case, i.e., no reaction, the solution is quite complicated, so we look for limiting cases to verify the computations. If the size of the source is neglected, there are two alternative treatments: 1) the extent of the source is very large, as an infinite plane, or 2) it is very small, as a point source. These treatments have the advantage that the NO diffusion is one dimensional for constant surface flux, perpendicular to the plane or in the radial direction, as from a point source.
Because oblate spheroidal coordinates are used to approximate a disk-shaped NO source, we can use this geometry to test the validity of our concentration profile. This is illustrated in Fig. 8, where we plot the diffusion-only (NO = 0) concentration at the
producing surface and 100 µm from the source for a range of source
sizes a by using the homogeneous disk source solution Eq. 24 and the one- and two-dimensional numerical models in oblate spheroidal coordinates. For each source, a production rate of 5.4 × 1014
µmol · µm2 · s1
is used, and the concentration was computed at 12 s after production began, about when experimental data (16) show the concentration at the
source reached a maximum. We see that the concentration profile for the
two-dimensional model for a thin ellipse is very near the theoretical
disk value. As expected, the simplified one-dimensional model predicts
a higher concentration for smaller values of a. For the range
of interest, a > 150 µm, the error between the simplified one-dimensional model and the two-dimensional model is a maximum of
25%. Because the difference in predicted concentration is most likely
due to the underestimate of production rate (see discussion in
Oblate spheroidal coordinates), this deviation is used to
adjust the production rate estimated by the one-dimensional model. The deviation could also be viewed as an underestimate in the estimated size a of the region, as can be seen by shifting the
one-dimensional model curves to the right. This interdependency between
a and NO can be expected from
covariance analysis (Fig. 4A).
|
NO = 0) computations, the adjustment
should be valid when NO 0. This is
because the NO concentration is determined primarily by diffusion. To
see this, note that the dimensionless form of the diffusion equation
for the abluminal region with first-order NO degradation is
|
(22) |
2*, c*NO = cNO/c0, t* = tDlu/a2, and k* = a2k1,ab/Dlu
are the dimensionless Laplacian, concentration, time, and rate
coefficient (Damköhler number), respectively. For our parameters
(Table 1), k* 0.1. The NO concentration
c*NO can be expanded in a perturbation series
c*NO = c*NO,0 + k* c
NO,1 + k*2c2NO,2 + higher-order
terms, where c*NO,0 is the solution to the
diffusion-only problem. Substituting this expansion into Eq. 22
and collecting terms containing like powers of k*, it can be
seen that the concentration at a point depends, to first order in
k*, on the concentration computed for the diffusion-only
problem (26). The second-order reaction problem can be treated
similarly. For the one- and two-dimensional problems the concentration
depends primarily on the diffusion-only solution, as long as k*
is small, so it is reasonable to use the diffusion-only production
rates to adjust NO.
The analytic solution for a disk-shaped source on a plane can be
obtained in the following manner. An ellipsoid with foci at
±a approaches a disk in the limit of vanishing thickness. The differential mass balance for a disk of radius a centered at
the origin of x-y plane is
|
(23) |
NO a2
units of mass per time can be obtained by integrating the instantaneous disk source solution from t' = 0 to t' = t.
[This is analogous to the treatment of a continuous point source
(3).] The resulting concentration profile for a disk with a constant
flux NO is
|
|
(24) |
( · ) is the generalized incomplete
gamma function. By comparing the computed concentration with the
experimental data, Eq. 24 can be used to estimate the
production rate in a diffusion-only system and serves as a limiting
case for the numerical model. It may also be useful in estimating the
NO concentration near a microscopic source at intermediate distance,
where neither the point source solution nor the plane source solution
is valid.
Equation 24 is valid for production from an infinitely thin
disk when there is no reaction. In the experimental system, NO would be
lost to reaction. Additionally, there may be a higher diffusive flux
into the lumen where D is higher. For a given D, Eq. 24 can be used to obtain a lower-limit estimate of
a and NO. For this estimate, we do a
simplified fit of the data of Malinski et al. (16) by using only
the maximum concentration points at the endothelium (1.3 µM, 12 s)
and in the smooth muscle 100 µm away (0.84 µM, 20.3 s). The results
are as follows: a = 270 µm and NO = 4.8 × 1014
µmol · µm2 · s1.
Had a large area of the endothelium been stimulated, the infinite plane
source solution (3) could have been used to estimate NO. With production estimated for the disk
source of a = 270 µm, the concentration at 20.3 s and 100 µm from an infinite planar source is 63% higher than the disk source
estimate. For a 
100 µm, the solutions rapidly converge,
and for a = 800 µm, they differ by only 2%.
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ACKNOWLEDGEMENTS |
|---|
This work was supported by a Whitaker Foundation Biomedical Engineering Grant.
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FOOTNOTES |
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Present address of M. W. Vaughn: Dept. of Chemical Engineering, University of California, Los Angeles, CA 90095.
Address for reprint requests: J. C. Liao, Dept. of Chemical Engineering, University of California, 405 Hilgard Ave., Los Angeles, CA 90095-1592.
Received 19 May 1997; accepted in final form 3 February 1998.
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REFERENCES |
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Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References
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