AJP - Heart Calcium Transients and Cell-Sarcomere
HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS
 QUICK SEARCH:   [advanced]


     


Am J Physiol Heart Circ Physiol 274: H2163-H2176, 1998;
0363-6135/98 $5.00
This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Right arrow Citation Map
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Similar articles in PubMed
Right arrow Alert me to new issues of the journal
Right arrow Download to citation manager
Citing Articles
Right arrow Citing Articles via HighWire
Right arrow Citing Articles via Google Scholar
Google Scholar
Right arrow Articles by Vaughn, M. W.
Right arrow Articles by Liao, J. C.
Right arrow Search for Related Content
PubMed
Right arrow PubMed Citation
Right arrow Articles by Vaughn, M. W.
Right arrow Articles by Liao, J. C.
Vol. 274, Issue 6, H2163-H2176, June 1998

MODELING IN PHYSIOLOGY
Estimation of nitric oxide production and reaction
rates in tissue by use of a mathematical model

Mark W. Vaughn1, Lih Kuo2, and James C. Liao3

1 Department of Chemical Engineering, Texas A&M University, 2 Department of Medical Physiology, Texas A&M University Health Science Center, College Station, Texas 77843; and 3 Department of Chemical Engineering, University of California, Los Angeles, California 90095

    ABSTRACT
Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References

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 × 10-14 µmol · µm-2 · s-1; the NO diffusion coefficient, 3,300 µm2 s-1; and the NO consumption rate coefficient in the vascular smooth muscle, 0.01 s-1 (1st-order rate expression) or 0.05 µM-1 · s-1 (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.

mass transfer; kinetics; parameter estimation; diffusion

    INTRODUCTION
Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References

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.

    ESTIMATING PARAMETERS FROM EXPERIMENTAL 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.

To obtain realistic model predictions, the size of the NO-producing region, the NO production rate, the rate constants for the NO reaction, and the diffusion coefficient are needed for each region. For the buffer in the lumen, the NO reaction rate constant and the diffusion coefficient are known. In the tissue we obtain the parameters by analyzing the experimental data of Malinski et al. (16), who stimulated a microscopic portion of rabbit aortic endothelium by injecting a bolus of bradykinin at the surface of the endothelium and measured the NO concentration electrochemically. Our model of the NO production from this experiment is depicted in Fig. 1A. In the experiment of Malinski et al., one sensor was placed adjacent to an endothelial cell and another in a vascular smooth muscle cell 100 µm away. The luminal side of the endothelium is bathed in a physiological salt solution. A microscopic portion of the endothelium was then stimulated by using a microinjector to place a bolus of bradykinin in proximity to the endothelial sensor.


View larger version (24K):
[in this window]
[in a new window]
 
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 (&cjs1715;) 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.

The size of the microscopic region producing NO, characterized by a, is unknown and must be estimated. A single endothelial cell has a diameter of 10-20 µm, so the distance between the sensors was only 5-10 times the size of the source. This implies that the size of the producing region may be an important factor in determining the NO concentration at the sensor in the smooth muscle. In fact, we anticipate the producing region to be much larger than a single cell, because the size of the bolus of bradykinin and the effect of bradykinin diffusion are unknown, even though Malinski et al. (16) stated that a single cell was stimulated.

NO is produced from the endothelium at some rate per unit area (&qdot;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 (2epsilon ) is exaggerated for illustration, typically a/epsilon  approx  100 for the size of the ellipsoidal source estimated below. As epsilon  approaches 0, the ellipsoid appears increasingly disklike, with a approaching the diameter of the disk.


View larger version (35K):
[in this window]
[in a new window]
 
Fig. 2.   Oblate spheroidal coordinate system representing NO production from a microscopic portion of endothelium. Coordinate surface, xi  = constant, is a flattened ellipsoid. Movement along surface of ellipsoid from minor (vertical) axis to major (horizontal) axis is characterized by eta ; minor axis corresponds to eta  = 0, and major axis corresponds to eta  = pi /2. This example shows an ellipsoid with an exaggerated thickness compared with an endothelial cell: maximum thickness is 5.8 µm, and maximum radius is 20 µm, which corresponds to size of NO-producing region a = 19.7 µm. NO is produced at surface of ellipsoid and diffuses in xi -direction. Half-thickness of ellipsoid is epsilon , and distance between foci is 2a.

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
 <FR><NU>∂c<SUB>NO</SUB></NU><DE>∂<IT>t</IT></DE></FR> = <IT>D</IT>∇<SUP>2</SUP>c<SUB>NO</SUB> − ∇c<SUB>NO</SUB>⋅<B>v</B> − <A><AC>V</AC><AC>˙</AC></A><SUB>NO</SUB> (1)
where nabla  is the vector gradient operator, nabla 2 is the Laplacian operator, cNO is NO concentration, 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). For a diffusing trace component in a quiescent system or tissue, nabla 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
c<SUB>NO</SUB> = 0, at <IT>t</IT> = 0, for each of the 3 regions (2)

Two boundary conditions can be fixed far from the producing surface where the NO concentration changes slowly; therefore
∇c<SUB>NO</SUB> = 0, for all <IT>t</IT>, far from the endothelium (3)
Four boundary conditions can be determined at the NO-producing surfaces bounding the endothelium. Two are supplied by the continuity of NO concentration. The remaining two are obtained by relating the NO flux from the endothelial surface (&cjs1715;) to the production rate (see APPENDIX A). These conditions can be written, at the luminal surface, as
c<SUB>NO</SUB>‖<SUB>lu</SUB> = c<SUB>NO</SUB>‖<SUB>ec</SUB>
 <A><AC>q</AC><AC>˙</AC></A><SUB>NO,lu</SUB>(<IT>t</IT>) = &cjs1715;<SUB>lu</SUB>⋅<B>n</B><SUB>lu</SUB> − &cjs1715;<SUB>ec</SUB>⋅<B>n</B><SUB>lu</SUB> (4)
= −<IT>D</IT>∇c<SUB>NO</SUB>‖<SUB>lu</SUB>⋅<B>n</B><SUB>lu</SUB> + <IT>D</IT>∇c<SUB>NO</SUB>‖<SUB>ec</SUB>⋅<B>n</B><SUB>lu</SUB>
These equations are valid at all times. Similarly, at the abluminal surface
c<SUB>NO</SUB>‖<SUB>ec</SUB> = c<SUB>NO</SUB>‖<SUB>ab</SUB>
 <A><AC>q</AC><AC>˙</AC></A><SUB>NO,ab</SUB>(<IT>t</IT>) = &cjs1715;<SUB>ec</SUB>⋅<B>n</B><SUB>ab</SUB> − &cjs1715;<SUB>ab</SUB>⋅<B>n</B><SUB>ab</SUB> (5)
= <IT>D</IT>∇c<SUB>NO</SUB>‖<SUB>ec</SUB>⋅<B>n</B><SUB>ab</SUB> − <IT>D</IT>∇c<SUB>NO</SUB>‖<SUB>ab</SUB>⋅<B>n</B><SUB>ab</SUB>
where &cjs1715;lu is the flux of NO from the producing surface into the lumen, &cjs1715;ab is the flux of NO from the producing surface into the abluminal region, &cjs1715;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 &qdot;NO(t) = &qdot;NO,lu(t) &qdot;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, &cjs1715;ec = 0 at the luminal surface and &cjs1715;lu is composed of &qdot;NO,lu(t) and a portion of &qdot;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: -infinity  <=  xi  < infinity , 0 <=  eta  <=  pi /2, and 0 <=  phi < 2pi (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
 <FR><NU>∂c<SUB>NO</SUB></NU><DE>∂<IT>t</IT></DE></FR> = <FR><NU><IT>D</IT></NU><DE><IT>a</IT><SUP>2</SUP>(cosh<SUP>2</SUP> &xgr; − sin<SUP>2</SUP> &eegr;)</DE></FR>
· <FENCE>tanh &xgr; <FR><NU>∂c<SUB>NO</SUB></NU><DE>∂&xgr;</DE></FR> + <FR><NU>∂<SUP>2</SUP>c<SUB>NO</SUB></NU><DE>∂&xgr;<SUP>2</SUP></DE></FR> + cot &eegr; <FR><NU>∂c<SUB>NO</SUB></NU><DE>∂&eegr;</DE></FR> + <FR><NU>∂<SUP>2</SUP>c<SUB>NO</SUB></NU><DE>∂&eegr;<SUP>2</SUP></DE></FR></FENCE>
 − <A><AC>V</AC><AC>˙</AC></A><SUB>NO</SUB> (6)
where a is the distance from the origin to the foci and (a sin eta  sinh xi , a cos eta  sinh xi ) are the Cartesian coordinates of a point in the xi -eta plane. NO is produced by the surface of the thin ellipsoidal region.

The boundary conditions, Eqs. 4 and 5, for the luminal surface xi  = xi 0 = sinh-1 epsilon /a are
c<SUB>NO</SUB>‖<SUB>lu</SUB> = c<SUB>NO</SUB>‖<SUB>ec</SUB>
<A><AC>q</AC><AC>˙</AC></A><SUB>NO,lu</SUB> = −&cjs1715;<SUB>&xgr;</SUB>‖<SUB>lu</SUB> + &cjs1715;<SUB>&xgr;</SUB>‖<SUB>ec</SUB> (7)
and, for the abluminal surface, xi  = -xi 0
c<SUB>NO</SUB>‖<SUB>ec</SUB> = c<SUB>NO</SUB>‖<SUB>ab</SUB>
<A><AC>q</AC><AC>˙</AC></A><SUB>NO,ab</SUB> = &cjs1715;<SUB>&xgr;</SUB>‖<SUB>ab</SUB> − &cjs1715;<SUB>&xgr;</SUB>‖<SUB>ec</SUB> (8)
The xi -component of the NO flux at the surface is, at xi  = ±xi 0 (the surface of the endothelium)
&cjs1715;<SUB>&xgr;</SUB> = − <FR><NU><IT>D</IT></NU><DE><IT>a</IT><RAD><RCD>cosh<SUP>2</SUP> &xgr;<SUB>0</SUB> − sin<SUP>2</SUP> &eegr;</RCD></RAD></DE></FR> <FR><NU>∂c<SUB>NO</SUB></NU><DE>∂&xgr;</DE></FR> (9)

Equations 6-9 can be solved for concentration as a function of xi  and eta . 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 right-arrow 0 or a finite and sinh xi right-arrow infinity , the spheroidal coordinate surfaces become spherical. In this case, the concentration depends only on r = a sinh xi , so the concentration dependence on eta  may be neglected. In the other limiting case, as sinh xi right-arrow 0 the surface collapses to a disk. Near the edges of the disk, the concentration depends strongly on eta . 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 eta . Because this dependence enters through the flux term Eq. 9, let us neglect eta  and replace the boundary condition Eq. 9 with the approximation
&cjs1715;<SUB>&xgr;</SUB> = − <FR><NU><IT>D</IT></NU><DE><IT>a</IT></DE></FR> <FR><NU>∂c<SUB>NO</SUB></NU><DE>∂&xgr;</DE></FR> (10)
Here we have recognized that cosh2 xi 0 approx  1 for small xi 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/pi . 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 eta  dependence of the concentration at the endothelial surface, let us also simplify the NO mass balance by neglecting the concentration dependence in the eta  direction everywhere
 <FR><NU>∂c<SUB>NO</SUB></NU><DE>∂<IT>t</IT></DE></FR> = <FR><NU><IT>D</IT></NU><DE><IT>a</IT><SUP>2</SUP> cosh<SUP>2</SUP> &xgr;</DE></FR> <FENCE>tanh &xgr; <FR><NU>∂c<SUB>NO</SUB></NU><DE>∂&xgr;</DE></FR> + <FR><NU>∂<SUP>2</SUP>c<SUB>NO</SUB></NU><DE>∂&xgr;<SUP>2</SUP></DE></FR></FENCE> − <A><AC>V</AC><AC>˙</AC></A><SUB>NO</SUB> (11)
To verify this approximation (the 1-dimensional problem), the concentration profile along the z-axis predicted by this equation can be compared with the two-dimensional problem and with the two-dimensional analytic solution for a disk (see APPENDIX B). For a wide range of a, it provides a reasonable approximation of the analytic result. We therefore use Eq. 11 in estimating the size of the NO-producing region, the production rate, and the NO decomposition rate. If off-axis concentrations were needed, it would be especially important to test this approximation using the full two-dimensional model.

The maximum thickness of the ellipse (which corresponds to the approximate endothelial cell thickness, 2epsilon ) is taken as 2.5 µm. The remaining boundary conditions are given by Eq. 3
as &xgr; → ∞  <FR><NU>dc<SUB>NO</SUB></NU><DE>d&xgr;</DE></FR> = 0 (12)

For the problem solved here, the production term is described as &qdot;NO(t) = &qdot;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
 <A><AC>V</AC><AC>˙</AC></A><SUB>NO</SUB> = <IT>k</IT><SUB><IT>n</IT>,<IT>x</IT></SUB>c<SUP><IT>n</IT></SUP><SUB>NO</SUB> (13)
where VNO 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)
 <A><AC>V</AC><AC>˙</AC></A><SUB>NO</SUB> = <IT>k</IT><SUB>3,lu</SUB>c<SUP>2</SUP><SUB>NO</SUB>c<SUB>O<SUB>2</SUB></SUB> (14)
where VNO is the rate of decomposition of NO by reaction with the surrounding media and k3,lu approx  9 × 106 M-2 · s-1 (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
 <A><AC>V</AC><AC>˙</AC></A><SUB>NO</SUB> ≈ <IT>k</IT><SUB>2,lu</SUB>c<SUP>2</SUP><SUB>NO</SUB> (15)
where k2,lu = 0.002 µM-1 · s-1 (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 × 10-6. 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. (xi , eta ) 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, &qdot;NO, kn,lu, Dlu, Dab), in the one-dimensional problem, Eqs. 7, 8, and 10-12. We did this by minimizing the chi 2 merit function.
&khgr;<SUP>2</SUP> = <LIM><OP>∑</OP><LL><IT>i</IT>=1</LL><UL><IT>j</IT></UL></LIM> <FENCE><FR><NU>c<SUB><IT>i</IT></SUB> − c(<IT>t</IT><SUB><IT>i</IT></SUB>; <IT>p</IT><SUB>1</SUB>,…, <IT>p<SUB>m</SUB></IT>)</NU><DE>&sfgr;<SUB><IT>i</IT></SUB></DE></FR></FENCE><SUP>2</SUP> (16)
where c(ti; p1, ... , pm) is the NO concentration obtained by solving Eqs. 7, 8, and 10-12, with the m-component parameter set p1, ... , pm, and (ti, ci) is the ith data point. These data were obtained by picking 32 points from Malinski et al. (16) and scaling the endothelial cell data to a maximum NO concentration of 1.3 µM and the smooth muscle cell data to a maximum of 0.85 µM. The standard deviation of each experimental point (ti, ci) is sigma 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)
<IT>a</IT>* = <FR><NU><IT>a</IT></NU><DE><IT>l</IT><SUB>0</SUB></DE></FR>  <A><AC>q</AC><AC>˙</AC></A>*<SUB>NO</SUB> = <FR><NU><A><AC>q</AC><AC>˙</AC></A><SUB>NO</SUB><IT>l</IT><SUB>0</SUB></NU><DE>c<SUB>0</SUB><IT>f</IT><SUB>c</SUB></DE></FR>  <IT>k</IT>*<SUB>1,lu</SUB> = <FR><NU><IT>k</IT><SUB>1,lu</SUB><IT>l</IT><SUP>2</SUP><SUB>0</SUB></NU><DE><IT>D</IT><SUB>0</SUB></DE></FR>
<IT>k</IT>*<SUB>2,lu</SUB> = <FR><NU><IT>k</IT><SUB>1,lu</SUB><IT>l</IT><SUP>2</SUP><SUB>0</SUB>c<SUB>0</SUB></NU><DE><IT>D</IT><SUB>0</SUB></DE></FR>  <IT>D</IT>*<SUB>lu</SUB> = <FR><NU><IT>D</IT><SUB>lu</SUB></NU><DE><IT>D</IT><SUB>0</SUB></DE></FR>  <IT>D</IT>*<SUB>ab</SUB> = <FR><NU><IT>D</IT><SUB>ab</SUB></NU><DE><IT>D</IT><SUB>0</SUB></DE></FR> (17)
where fc = 10-3 amol · µm-3 · µM-1 is a unit conversion constant.

The chi 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.

    RESULTS
Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References

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, &qdot;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.

                              
View this table:
[in this window]
[in a new window]
 
Table 1.   Parameters that minimize chi 2 of the model and the data of Malinski et al. (16)

There were many local minima in the chi 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 × 10-14 µmol · µm-2 · s-1, 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.


View larger version (25K):
[in this window]
[in a new window]
 
Fig. 3.   NO concentration computed using oblate spheroidal coordinates and parameters from Table 1, cases 1 and 2. For 1-dimensional model (solid line), 1st- and 2nd-order reactions are nearly indistinguishable. For 2-dimensional model, 1st- (dotted lines) and 2nd-order reactions (dashed lines), production rate was 6.8 amol · µm-2 · s-1 and other parameters are as in Table 1, cases 1 and 2. Data are from Malinski et al. (16).

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.

                              
View this table:
[in this window]
[in a new window]
 
Table 2.   Formal covariance matrix of the fit when D*lu is fixed


View larger version (29K):
[in this window]
[in a new window]
 
Fig. 4.   Confidence ellipses from linearized covariance analysis for computed parameters a, rate of NO production (&qdot;NO), rate coefficient for abluminal region (k1,ab), and diffusion coefficient for abluminal region (Dab). Ellipses show regions from which each pair of parameters may be selected within 90% confidence region. Ellipses are computed from covariance matrix (Table 2, cases 1 and 2): solid line. 1st-order reaction; dashed line, 2nd-order reaction.

The value with the greatest certainty is the NO production (&qdot;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 &qdot;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
<IT>S</IT><SUB><IT>i</IT>,ec</SUB> = <FR><NU><IT>p<SUB>i</SUB></IT></NU><DE>c<SUB>NO,ec</SUB></DE></FR> <FR><NU>∂c<SUB>NO,ec</SUB></NU><DE>∂<IT>p<SUB>i</SUB></IT></DE></FR>
<IT>S</IT><SUB><IT>i</IT>,ab</SUB> = <FR><NU><IT>p<SUB>i</SUB></IT></NU><DE>c<SUB>NO,ab</SUB></DE></FR> <FR><NU>∂c<SUB>NO,ab</SUB></NU><DE>∂<IT>p<SUB>i</SUB></IT></DE></FR> (18)
where pi is the ith component of the parameter set (a, &qdot;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.

    DISCUSSION
Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References

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, &qdot;NO(t). The theoretical &qdot;NO(t) could be tested by modeling the production of NO in a manner analogous to that done here.

The estimated NO production rate was somewhat correlated with a and Dab (Fig. 4, A and C and Table 2), so these parameters cannot be estimated independently from this experiment. That is, &qdot;NO will always depend on the value selected for a and Dab. Despite this, &qdot;NO varies over a fairly narrow range.

To obtain the actual production rate, 6.8 × 10-14 µmol · µm-2 · s-1, 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, S&qdot;NO 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. 


View larger version (30K):
[in this window]
[in a new window]
 
Fig. 5.   Sensitivity coefficients for 1st-order reaction, case 1 of Table 1, in vascular smooth muscle (A and B) and endothelium (C and D). cNO, NO concentration; pi, component of parameter set.


View larger version (30K):
[in this window]
[in a new window]
 
Fig. 6.   Sensitivity coefficients for parameters for 2nd-order reaction, case 2 of Table 1, in vascular smooth muscle (A and B) and endothelium (C and D).

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 approx  2.0 × 10-3 µM-1 · s-1).

The reaction rate coefficient influences the concentration most at longer time scales, as shown in Figs. 5, A and C, and 6, A and C. Therefore, the steady-state NO concentration will be useful for estimating the reaction rate coefficient. Because the NO production began to fall at 20-30 s, only the data obtained before 30 s were used in the analysis. Had long-time steady-state data been available, the estimates for the rate coefficients would probably be more reliable. Because the reaction rate depends only on local concentration, we expect the sensitivity of the one- and two-dimensional models to be similar, if the production rates are such that the concentration profiles are similar.

It is common to characterize the reaction of NO by t1/2. For the first-order reaction (Table 1, cases 1 and 3), t1/2 = 30-115 s-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 µM-1 · s-1, and by setting cNO = 1 µM, t1/2 approx  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 chi 2 was not very large, so cases 1 and 2 are considered to be the most reliable.

NO diffuses through the vascular smooth muscle with surprising speed. Even the lowest estimate, Dab = 2,000 µm2/s (Table 1, case 3), is relatively high compared with the diffusion rate in lipid (5). The value for the highest estimate, Dab = 3,400 µm2/s, is approximately that measured in an aqueous salt solution by Malinski et al. (16).

The maximum NO concentration as measured by Malinski et al. (16) (Fig. 3) shows a decrease from 1.3 to 0.85 µM over 100 µm. This rapid drop in NO concentration with distance has been taken by some researchers (25) to imply that NO must be reacting with components of the cellular membrane. Although we agree that reactions do occur, this reasoning is misleading. Depending on the NO production rate and the mass transfer to the lumen, the concentration gradient may be just the result of mass transfer from a finite NO source, as seen in Table 1, case 5.

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 × 10-17 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 × 10-14 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 × 10-10 to 1.6 × 10-8 M/s. If it is assumed that the average thickness of the endothelium is 2.5 µm, this would be equivalent to 2.5 × 10-19 to 4 × 10-17 µmol · µm-2 · s-1. 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 s-1), 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.


View larger version (21K):
[in this window]
[in a new window]
 
Fig. 7.   Effect of modifying Dlu on concentration of different reaction mechanisms. NO concentration is computed using oblate spheroidal coordinates and parameters from Table