INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS OF ANALYSIS RESULTS DISCUSSION APPENDIX REFERENCES

ENDOTHELIUM-DERIVED relaxing factor (EDRF), a vasodilator released by arterial endothelial cells, has been identified as either free nitric oxide (NO) or a closely related compound (19). The role of NO as a ubiquitous intercellular messenger is well documented (1, 10, 52). NO is generated in vivo by the enzymatic conversion of L-arginine (L-Arg) to L-citrulline, which is catalyzed by nitric oxide synthase (NOS). The constitutive membrane-bound isoform, endothelial NOS (eNOS), produces NO in vascular endothelial cells, resulting in the dilation of blood vessels. NO activates the soluble isoform of the allosteric enzyme guanylate cyclase (sGC), which catalyzes the conversion of guanosine 5'-triphosphate to cGMP. The subsequent rise in cGMP concentration ultimately results in the dilation of smooth muscle. sGC activity is partially characterized in terms of its apparent Michaelis constant (k_m) value [the equilibrium NO concentration ([NO]) at which sGC is 50% activated]. Stone and Marletta (44) reported an upper limit of 250 nM for k_m. However, recent data suggest that k_m is most likely an order of magnitude lower (~23 nM; Refs. 9, 55). If k_m is on the order of 23 nM, previous estimates for the effective distance over which NO can influence the activation of sGC (49) need to be reevaluated.

Vaughn et al. (48, 49) demonstrated that the theoretical "effective diffusion distance" (defined as the distance away from its production source within which [NO] exceeds the k_m of sGC) is strongly dependent on the geometry of its source. Their results suggest that vascular endothelial cells cannot produce free NO at high enough levels to activate sGC in adjacent smooth muscle cells without the protection of an additional cofactor because of the consumption of NO by oxyhemoglobin (Hb) in blood. Their analysis considered a semi-infinite system at steady state, with k_m = 250 nM.

Expired human breath contains 4-100 ppb NO (14, 42, 43, 50). Exhaled NO has been proposed as a noninvasive biomarker for disease states characterized by inflammation, such as bronchial asthma and allergies (41-43). The potential of endothelium-derived NO to contribute significantly to the levels appearing in expired breath remains to be investigated.

EDRF is hypothesized to be either free NO or NO bound to a protective cofactor (13, 19). On the basis of in vitro data, the half-life of NO within red blood cells is ~1 µs (11, 17, 29). Thus the rapid reaction of NO with Hb present in erythrocytes suggests that other physical or chemical factors are required to activate sGC in smooth muscle cells. If free endothelium-derived NO is EDRF, how does it escape the abyss of Hb in blood vessels to perform its physiological function? One proposed hypothesis is that an erythrocyte-free zone (EFZ) is formed because of the tendency of erythrocytes to migrate away from the blood vessel wall under flow conditions. The EFZ provides a diffusion barrier between erythrocytes and the inner blood vessel wall (5, 29, 30, 48, 49). Recent studies suggest that erythrocytes regulate NO consumption via Hb by means of an intrinsic diffusion barrier at their cell membranes (18, 47). In either case, NO uptake by Hb is limited by diffusion resistance.

We present here a simplified analytical model for small NO-producing blood vessels within the human bronchial circulation, which are bounded by the airway lumen gas space. We consider both steady-state and transient behavior for a finite geometry, predict NO concentration profiles and diffusion rates at in vivo conditions, and hypothesize possible mechanisms for free NO-modulated smooth muscle dilation. Our goal is to identify potential pathways that may govern the fate and physiological activity of endothelium-derived NO in the human bronchial circulation.

Glossary

C	NO concentration (nM)
Ci	C(t, r = Ri) = NO concentration at surface; i = 1, 2, 3 (nM)
Ci,ss	Steady-state concentration at surface; i = 1, 2, 3 (nM)
C(x)	Initial, steady-state NO concentration distribution (nM)
C	Bulk fluid phase NO concentration (nM)
C1,ss	C(t, y = 1) = steady-state NO concentration at surface; i = 1 (nM)
C2,ss	C(t, y2) = steady-state NO concentration at surface; i = 2 (nM)
C'1,ss	C/y| at steady state = C1,ss/[y1 ln (y1/y0)]
C'2+,ss	C/y| at steady state = (f'12/f12)C2,ss  (g'12  g12f'12/f12)Q1,ss  Q2,ss
D	Diffusivity of NO within tissue (cm2/s or µm2/s)
F1(y)	Known function of y and input parameters; i = 1, 2
fij	fi(yj)
f'j(y)	dfj(y)/dy
f'ij	f'i(yj)
gi(y)	Known function of y and input parameters; i = 1, 2
gij	gi(yj)
g'j(y)	dgj(y)/dy
g'ij	g'i(yj)
hi	Mass transfer coefficient at boundary; i = 1, 2, 3 (µm/s)
h3	Mass transfer coefficient between adventitial boundary and external medium (µm/s)
H3	h3R1/D = dimensionless mass transfer coefficient
Ji	Molar diffusion flux at boundary i (µM · µm · s1)
k1	First-order rate constant for NO consumption in pulmonary tissue (s1)
km	Apparent Michaelis constant for sGC with NO as substrate (nM)
Km	Modified Bessel function of the second kind of order m
qi()	[Qi()  Q]/QMax = dimensionless endothelial NO production rate; i = 1, 2
q	(Q  Q)/QMax = maximum dimensionless NO production rate; i = 1, 2
Qi()	NO,i(t)R1/D = scaled endothelial NO production rate; i = 1, 2 (nM)
Q	R1/D = scaled endothelial NO production rate at t = 0; i = 1, 2 (nM)
Q	R1/D = maximum scaled endothelial NO production rate; i = 1, 2 (nM)
r	Radial space coordinate (µm)
reff	Effective diffusion radius (µm)
R0	Radius of hypothetical red blood cell core (µm)
R1	Blood vessel radius at inner membrane surface (µm)
R2	Blood vessel radius at outer membrane surface (µm)
R3	Radial distance from blood vessel center to outer adventitial boundary (µm)
RNO(C)	NO consumption rate in pulmonary tissue (nM/s) = k1C [RNO(C) = 0 in EFZ]
NO,i(t)	Endothelial NO production rate per unit surface; i = 1, 2 (µM · µm · s1)
	Basal endothelial NO production rate per unit surface; i = 1, 2 (µM · µm · s1)
	Maximum endothelial NO production rate per unit surface; i = 1, 2 (µM · µm · s1)
[species]	Concentration of species in tissue; species = NO, O2 (nM)
t	Time (s)
V	Equivalent sGC activity level (cGMP formation rate) at steady state (nM/s)
V/Vmax	Equivalent relative sGC activity level at steady state, V/VMax
VMax	Maximum possible cGMP formation rate (nM/s)
xj	Input parameter j for linear regression analysis
y	r/R1 = dimensionless radial space coordinate (relative to blood vessel radius)
yeff	reff/R1 = dimensionless effective diffusion radius
j (i)	Sensitivity coefficient (of i to input j) for linear regression analysis
	Equilibrium distribution coefficient between adventitial region and external medium
C	Driving force term (product of  and C) (nM)
Ci	Concentration driving force for mass transfer at boundary; i = 1, 2 (nM)
Q()	Q1() + Q2()  Q  Q
QMax	Q + Q  Q  Q
i	Fractional flux at surface; i = 1, 2, 3
2	Theile modulus = k1R/D
	Dt/R1 = dimensionless time
T	Time duration of simultaneous pulse changes in NO production (s)
T1	On-time for a continuous (square wave) pulse in NO production (s)
T2	Off-time for a continuous (square wave) pulse in NO production (s)

Subscripts and Superscripts

0	Initial or basal condition
i	Index corresponding to r = R1 (i = 1), r = R2 (i = 2), and r = R3 (i = 3) or general integer
i+	Evaluation at the outer surface of a boundary
i	Evaluation at the inner surface of a boundary
j	Index corresponding to general integer; j = 1, 2, 3
Max	Maximum
ss	Steady state

	METHODS OF ANALYSIS

TOP ABSTRACT INTRODUCTION METHODS OF ANALYSIS RESULTS DISCUSSION APPENDIX REFERENCES

Endothelial NO production near an arteriole. A typical geometry for a blood vessel of the bronchial circulation is depicted in Fig. 1, with the radial coordinate denoted by r. Erythrocyte(s) are assumed to be clustered at the center of the blood vessel (r < R₀). R₁ and R₀ denote the radii of the blood vessel and its hypothetical red blood cell (RBC) core, respectively, with the region R₀ < r < R₁ comprising an EFZ.

View larger version (25K): [in this window] [in a new window]   Fig. 1.   Geometry and boundary conditions for a typical bronchial blood vessel. A: adventitial region, endothelium, erythrocyte-free zone (EFZ), and red blood cell (RBC) core. B: detail of blood vessel showing endothelial membranes. See Glossary for definitions.

Governing equation and boundary conditions. We assume that, over limited regions of the blood vessel circumference, angular and axial diffusion rates are small compared with the radial diffusion rate. Hence, the one-dimensional diffusion equation in cylindrical coordinates applies to each region

(1)

where t is time, r is the radial coordinate, C is the concentration of NO, R_NO(C) is the NO consumption rate per unit volume, and D is the diffusivity of NO within tissue. D is estimated as the diffusivity of NO within water (i.e., D = 3.3 × 10⁵ cm²/s) (25, 26, 48).

(2)

(3)

3

Fractional mass transfer flux. We define the fractional flux, _i, as the mass flux across a boundary i relative to the total endothelial NO production rate. For a cylindrically shaped blood vessel, where _NO,i is uniform over each membrane surface (i = 1, 2)

(4)

where y_i = R_i/R₁ (i = 1, 2, 3) and Q_i = _NO,iR₁/D (i = 1, 2) denote the scaled production rates. Equation 4 assumes that J_i is directed normal to surface i and away from the endothelial cell. If the diffusive flux is directed into the endothelial cell, _i < 0. There is no accumulation of NO within the endothelial cell at steady state; therefore ₁ + ₂₊ = 0. At the outer boundary of the adventitial region (i = 3 and r = R), computation of ₃ from Eq. 4 is valid over a limited region of the blood vessel circumference, where the angular flux contribution is negligible. Henceforth, we denote ₁, ₂₊, and ₃ as ₁, ₂, and ₃ or _i (i = 1, 2, 3), respectively.

Steady-state analysis. When both the external conditions and the NO production rates remain unchanged for a long time, the time derivative in Eq. 1 vanishes and the steady-state solution can be derived by integration of the resulting ordinary differential equation (see APPENDIX). Although this yields a complex system of algebraic expressions, values of _i are readily computed from Eq. 4.

1

Transient analysis. The transient solution of Eq. 1 is obtained by the method of separation of variables, which leads to an infinite series of Bessel functions in the radial space coordinate (7, 15, 38). The solution is expressed as the sum of steady-state and transient parts (see APPENDIX).

1

1

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS OF ANALYSIS RESULTS DISCUSSION APPENDIX REFERENCES

Steady-state analysis. Table 2 summarizes the results of LHS linear regression analysis in terms of the P values and regression coefficients, which are indices of the significance and sensitivity, respectively, with respect to the corresponding input parameters. Both ₂ and ₃ are more sensitive to y₀, y₂, and y₃ than the other parameters, as demonstrated by their low P values and high regression coefficients (see Table 2); ₂ is weakly dependent on both R₁ and k₁. Unlike ₂, ₃ decreases with k₁ and is not significantly dependent on R₁.

View larger version (35K): [in this window] [in a new window]   Fig. 2.   Fractional flux 2 at the outer blood vessel surface with inner blood vessel radius (R1) = 10 µm, mass transfer coefficient (h3) = 1 × 106 µm/s, and driving force term (C) = 0.16 nM.

View larger version (35K): [in this window] [in a new window]   Fig. 3.   Fractional flux 3 at the airway lumen boundary with R1 = 10 µm, h3 = 1 × 106 µm/s, and C = 0.16 nM. y, Dimensionless radial space coordinate.

View larger version (8K): [in this window] [in a new window]   Fig. 4.   Effective diffusion distance, yeff, at steady state.

Transient analysis. The results presented in Figs. 5-7 were computed with all input parameters set at their mean values (see Table 1) and the airway lumen outer adventitial boundary condition, unless otherwise indicated. We express transient NO concentration profiles in terms of equivalent relative sGC activity level at steady state, V/V_Max. For step changes in the NO production at R₁ = 5 and 20 µm, the dependence of V/V_Max on R₁ is significant (compare Fig. 5, A and B). Figure 5 shows two V/V_Max peaks for both 0.01 and 0.1 ms (Fig. 5A; R₁ = 5 µm) and 0.1 and 1 ms (Fig. 5B; R₁ = 20 µm). These peaks correspond to the two NO sources at the inner and outer endothelial membranes and have a "Gaussian" appearance, because at these small times very little NO has reached the RBC core and outer adventitial boundary sinks. Within 50 ms, sGC activity levels in the vascular smooth muscle region reach 45-50% (r  6-7 µm) for R₁ = 5 µm (Fig. 5A) and 70-80% (r  24-28 µm) for R₁ = 20 µm (Fig. 5B). Because sGC activity level is dependent on , this analysis should be revisited when more information pertaining to in vivo NO production rates becomes available.

View larger version (28K): [in this window] [in a new window]   Fig. 5.   Radial profiles of soluble guanylate cyclase (sGC) activity, V/Vmax, with time, t, for a step increase in nitric oxide (NO) production rate. k1, y0 = R0/R1, y2 = R2/R1, y3 = R3/R1, h3 = 1 × 106 µm/s, and C are at the mean values listed in Table 1, and the maximum NO production rates are  =  = 26.5 µM · µm · s1. A: R1 = 5 µm. B: R1 = 20 µm.

View larger version (13K): [in this window] [in a new window]   Fig. 6.   Fractional fluxes at blood vessel and airway lumen surfaces. R1 = 20 µm; k1, y0 = R0/R1, y2 = R2/R1, y3 = R3/R1, h3 = 1 × 106 µm/s, and C are at the mean values listed in Table 1, and  =  = 26.5 µM · µm · s1.

View larger version (20K): [in this window] [in a new window]   Fig. 7.   Radial profiles of sGC activity, V/Vmax, with time, t, for both single pulse [pulse duration time (T) = 0.5 s] and square-wave pulse [pulse on-time (T1) = 0.5 s; pulse off-time (T2) = 5 s] increases in NO production rate. R1 = 20 µm; k1, y0 = R0/R1, y2 = R2/R1, y3 = R3/R1, h3 = 1 × 106 µm/s, and C are at the mean values listed in Table 1, and  =  = 26.5 µM · µm · s1.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS OF ANALYSIS RESULTS DISCUSSION APPENDIX REFERENCES

We have computed theoretical concentration profiles and diffusion rates around NO-producing blood vessels based on a mathematical model applicable to the human pulmonary system. Our results confirm that endothelium-derived free NO is capable of modulating vascular smooth muscle tone by activation of sGC. We hypothesize that intermittent generation of NO by eNOS may minimize losses to blood and surrounding tissue. In addition, we cannot rule out a potential contribution of eNOS to the levels of NO appearing in expired breath.

Several mathematical models have been used to study in vivo NO diffusion. The point source model (24-26, 53) assumes that the physical dimensions of each source (NO-producing cell) are small compared with the surrounding medium and sums up all contributions from multiple sources. For a single point source, which generates NO for 1-10 s, this model predicts that the "physiological sphere of influence" is ~200 µm away from its source, provided the half-life of NO within tissue is <5 s. These results led to the hypothesis that, with the exception of blood, chemical consumption of NO in most tissues is slow compared with its diffusion rates.

Vaughn et al. (48, 49) estimated effective diffusion distances for endothelial NO production in blood vessels. Their model accounted for the finite geometry of the production source. However, their analysis was limited to the steady-state behavior of a semi-infinite system, and they assumed k_m = 250 nM for sGC activation (44). Their results demonstrate the significant influence of blood vessel geometry and rapid binding of NO to Hb in blood. They suggested that free NO produced by vascular endothelial cell(s) cannot escape consumption by Hb and still activate sGC in vascular smooth muscle cells without the protection of an additional cofactor.

Under flow conditions, NO consumption by erythrocytes proceeds at a slower rate than that observed in the presence of free Hb (29, 47). Proposed explanations for this slower consumption rate include the diffusion resistance of the EFZ (devoid of erythrocytes) around the RBC core (30) and formation of relatively stagnant (unstirred) plasma layers around individual erythrocytes within the RBC core (32). Alternately, the possibility that a cytoskeletal network of proteins adjacent to the cell membranes of erythrocytes provides additional resistance to NO diffusion has also been proposed (18). Each of the above hypotheses suggests that an additional diffusion barrier limits consumption of NO by erythrocytes. Our model approximates this additional mass transfer resistance as an annular ring of erythrocyte-free plasma around a central RBC core.

Recent evidence suggests that k_m 23 nM, an order of magnitude lower than the upper limit of 250 nM reported by Stone and Marletta (Ref. 44; see Refs. 9, 55). On the basis of these data, we expressed NO concentrations in terms of sGC activity level, V/V_Max (Figs. 5 and 7), and reevaluated the effective diffusion distance at steady state (Fig. 4). The incorporation of a finite external boundary provides additional impetus for mass transport away from the blood vessel. Although our model assumes zero [NO] at the outer boundary of a central RBC core, we include additional diffusion resistance resulting from the EFZ thickness, which is consistent with current experimental data. Our results suggest that the EFZ substantially limits NO consumption by Hb and is potentially a major contributor to the effectiveness of free NO as a vasodilator.

Our results suggest that the above considerations are sufficient for free NO to perform its physiological role of vasodilation in smooth muscle. However, although 45-80% of full sGC activity is achieved in vascular smooth muscle at steady state, ~80% of the produced NO diffuses into the blood vessel (see Figs. 5 and 6). The time dependence of NO production provides a potential mechanism for enhanced utilization of NO for smooth muscle dilation (Figs. 6 and 7). At short times, these transient concentration profiles have Gaussian shapes as NO accumulates within the endothelium. Under these conditions, large concentration changes take place over a very thin region. Thus initial NO diffusion rates, both into and away from the blood vessel, are roughly equal and are virtually unaffected by external boundaries (Figs. 5 and 6). Transient concentration profiles strongly depend on the blood vessel radius, R₁ (Fig. 5). For a near-optimal blood vessel radius, R₁ = 20 µm and  =  = 26.5 µM · µm · s¹ (48), 60-80% of full sGC activation can be achieved in vascular smooth muscle within 50-100 ms (Fig. 5B).

With R₁ = 20, we compared V/V_Max for step, single-pulse, and continuous (square wave) pulse changes in NO production rate. Our results show that pulsatile generation of NO via eNOS results in enhanced utilization of NO for smooth muscle dilation. Square-wave and single-pulse changes in NO production achieve sGC activity levels of amplitude equal to that of the corresponding step change case (compare Figs. 5B and 7). However, the time-weighted average NO production rate for the square-wave case is only 10% of that for the corresponding step change case. The enhanced utilization efficiency results from reduced NO losses to Hb in the blood vessel.

Analysis of in vitro data demonstrates that activation of sGC by NO is rapid compared with its NO dissociation from sGC. Dissociation proceeds with a half-life of ~1-2 min (3, 9, 19, 22, 23, 36). However, sGC activation is at least an order of magnitude faster. Despite limited capacity, the initial binding rate of NO to sGC in smooth muscle is nearly as rapid as its binding rate with Hb in blood (9, 55). Thus NO binds to sGC, present in smooth muscle (at high [NO] with maximum NO production rate, ), much faster than it dissociates (at low [NO] with basal NO production, ). Therefore, if NO is generated in brief, intermittent bursts, diffusion rates away from the source are virtually identical in both directions and comparable amounts of NO bind to both sGC and Hb. When [NO] is high, sGC is rapidly converted into to the activated state. However, when NO production is "turned off", [NO] decreases, but dissociation is so slow that most of the sGC remains in the activated state and does not approach the basal state for 1-2 min. Our model results depict a scenario in which V/V_Max reaches ~80% within 500 ms and the next "pulse" of NO production reaches adjacent smooth muscle 5 s later, thereby maintaining sGC in the activated state. Thus T₁ = 500 ms and T₂ = 5 s represent a near-optimal NO utilization efficiency.

Temporal changes in shear stress and circumferential strain typically occur over time scales on the order of 1 s. These changes impose dynamic forces on vascular endothelial cells and impact NO production rates (35, 37). Therefore, cyclic changes in vascular stress and strain could trigger endothelial NO production according to the scenario described above. In addition, muscular contraction and relaxation have been shown to exhibit cyclic behavior with periods of <1 s. (21). Because NO remains bound to sGC for at least several seconds, a periodic NO signal would eventually force sGC into the activated state. Therefore, we hypothesize that pulsatile changes in blood flow actuate bursts of NO production over time scales on the order of seconds, which are capable of essentially full sGC activation.

At steady state, the fractional fluxes at the endothelial membranes are dependent on the endothelial and EFZ thicknesses (see Table 2 and Fig. 2). If the distance of the blood vessel from the outer adventitial boundary (characterized by y₃ and the blood vessel radius) is large, ₂ is also dependent on k₁ but essentially independent of y₃. Conversely, if the distance from the adventitial boundary is small, this dependence reverses (₂ is independent of k₁ but dependent on y₃). At fixed y₃, as R₁ increases the distance from the outer adventitial boundary, R₃ = y₃R₁, also increases. Hence, the further the blood vessel is from the outer adventitial boundary, the more time a diffusing molecule of NO has to react with scavengers in pulmonary tissue. As R₃ increases, increased NO consumption in the adventitial region results in a steeper concentration gradient and therefore a higher molar flux at the source (r = R₂), which leads to a higher value for ₂. Thus, for small values of R₁, ₂ is more sensitive to y₀, y₂, and y₃ than the rate constant, k₁. As R₁ increases the dependence of ₂ on k₁ becomes more significant, and ₂ becomes independent of y₃ as R₃ becomes large.

The molar flux estimated at the external boundary (characterized by ₃) is lower than the flux determined at the generating source (characterized by ₂). In contrast to ₂, ₃ is essentially independent of R₁ and exhibits greater dependence on k₁ than ₂. In addition, as k₁ increases, ₂ increases whereas ₃ decreases with increased adventitial NO consumption (compare P values and regression coefficients of 7.4 × 10⁶ and 0.016 for ₃ vs. 3.0 × 10³ and 0.009 for ₂, respectively, in Table 2). Thus NO consumption enhances mass transport at the production source (r = R₂) and attenuates mass transport at the outer adventitial boundary (r = R₃). Hence, as one moves further from the NO source, both the NO concentration and the molar flux become attenuated as a result of NO consumption by scavengers.

Table 2 also shows that both ₂ and ₃ are very sensitive to y₀, y₂, and y₃ (i.e., the ratios R₀/R₁, R₂/R₁, and R₃/R₁, respectively) but not appreciably sensitive to R₁. Hence, ₂ and ₃ are determined primarily by the relative differences 1  y₀ = (R₁  R₀)/R