Microcirculation (2003) 10, 479–495
© 2003 Nature Publishing Group 1073-9688/03 $25.00
www.nature.com/mn
A Model of Nitric Oxide Capillary Exchange
NIKOLAOS M. TSOUKIAS* AND ALEKSANDER S. POPEL*
*Department of Biomedical Engineering, School of Medicine, Johns Hopkins
University, Baltimore, MD, USA
ABSTRACT
Objective: Our aim was to develop a mathematical model that describes the
nitric oxide (NO) transport in and around capillaries. The model is used to make
quantitative predictions for (1) the contribution of capillary endothelium to the
nitric oxide flux into the parenchymal tissue cells; (2) the scavenging of arteriolar
endothelium-derived NO by capillaries in the surrounding tissue; and (3) the role
of myoglobin in tissue cells and plasma-based hemoglobin on NO diffusion in
and around capillaries.
Methods: We used a finite element model of a capillary and surrounding tissue
with discrete parachute-shape red blood cells (RBCs) moving inside the capillary
to obtain the NO concentration distribution. An intravascular mass transfer
coefficient is estimated as a function of RBC membrane permeability and capillary hematocrit. A continuum model of the capillary is also formulated, in
which blood is treated as a homogeneous fluid; it uses the mass transfer coefficient and provides a closed-form analytic solution for the average exchange rate
of NO in a capillary-perfused region.
Results: The NO concentration in the parenchymal cells depends on parameters
such as RBC membrane permeability and capillary hematocrit; the concentration
is predicted for a wide range of parameters. In the absence of myoglobin or
plasma-based hemoglobin, the average tissue concentration generally ranges
between 20 and 300 nM. In the presence of myoglobin or after transfusion of a
hemoglobin-based blood substitute, there is minimal NO penetration into the
tissue from the capillary endothelium.
Conclusions: The model suggests that NO originating from the capillary wall
can diffuse toward the parenchymal cells and potentially sustain physiologically
significant concentrations. The model provides estimates of NO exchange and
concentration level in capillary-perfused tissue, and it can be used in models of
NO transport around arterioles or other NO sources.
Microcirculation (2003) 10, 479–495. DOI:10.1038/sj.mn.7800210
KEY
tion
WORDS:
diffusion, transport, myoglobin, blood substitutes, vascular regula-
INTRODUCTION
Nitric Oxide (NO) is a signal transduction molecule
of extreme physiologic importance. NO is produced
endogenously in the body through the enzymatic
degradation of L-arginine by several isoforms of the
Supported by the National Institutes of Health Grant HL-18292
and The Eugene and Mary B. Meyer Center for Advanced Transfusion Practices and Blood Research.
For reprints of this article, contact Nikolaos Tsoukias, PhD, Department of Biomedical Engineering, Johns Hopkins University
School of Medicine, 613 Traylor Bldg., 720 Rutland Avenue,
Baltimore, MD 21205; e-mail: tsoukias@bme.jhu.edu.
Received 4 October 2002; accepted 21 February 2003
enzyme nitric oxide synthase (NOS). NO is a relatively reactive molecule with a short half-life in vivo.
It can be degraded by a number of reactions, but
under physiologic conditions, NO concentrations are
submicromolar, and it is the first-order reactions
with superoxide and heme-containing proteins such
as hemoglobin (Hb), myoglobin (Mb), guanylate cyclase (GC), and cytochrome c oxidase that should
dominate its chemistry in vivo (4,6,31).
NO is a highly diffusible molecule that enables it to
act both in an autocrine and in a paracrine fashion.
It is well established that NO released by the vascular endothelium diffuses to the nearby smooth
muscle to induce vasodilation. In addition, NO of
Capillary exchange of NO
NM Tsoukias and AS Popel
480
vascular origin can potentially diffuse to nearby parenchymal tissue cells and regulate various cellular
functions through the activation of cyclic GMPdependent and GMP-independent signal transduction pathways. In the heart, NO originating from the
coronary endothelium has been implicated in the
regulation of myocardial contractile function and in
myocardial energetics (33). It is also plausible that
NO of vascular origin plays a role in neurotransmission (11,34,35).
A number of experimental and theoretical studies
have been performed to investigate the diffusional
spread of NO away from its site of production in the
vascular endothelium (8,25,29,44,48). Theoretical
studies of NO transport have recently been reviewed
by Buerk (7). Prompted by the role of NO in the
regulation of blood flow, most of these studies focus
on the diffusional spread of NO around arterioles.
Theoretical simulations by Lancaster (25) showed
that physiologic amounts of Hb (2 mM) flowing in
the lumen of a 20-␮m arteriole scavenge significant
amounts of NO, leading to a dramatic reduction of
the NO concentration in the arteriolar smooth
muscle. The result questioned the previous notion
that free NO is the endothelium-derived relaxing
factor (EDRF) (18,23,30). Butler et al. (8) included
in their theoretical model a layer free of red blood
cells (RBCs) next to the endothelium. They demonstrated that despite the significant scavenging of NO
by Hb, a substantial amount of NO diffuses toward
the smooth muscle. By use of a detailed mathematical model, Vaughn et al. (44) pointed out the importance of the RBC-free layer and the size of the
vessel in the NO diffusion toward the smooth muscle.
They concluded that the uptake of NO by RBCs has
to be several orders of magnitude smaller than the
uptake by an equivalent amount of free Hb in solution for the concentration in the smooth muscle to
reach physiologically significant levels. Earlier work
had suggested a significant reduction in the rate of
NO uptake by RBCs compared with that by free Hb
(9). In subsequent studies, they estimated the rate of
uptake of NO by RBCs to be almost three orders of
magnitude slower than that of free Hb (42,43). They
attributed this reduction to RBC membrane and
cytoskeleton-associated NO-inert proteins that provide a barrier for NO diffusion (22). The exact contribution of the RBC membrane to the rate of NO
uptake by RBCs at this point remains controversial
(27,28,39).
Although these studies pointed out the importance of
intra-arteriole NO consumption in the diffusional
spread of NO toward the smooth muscle, little atten-
tion was paid to the effect of extra-arteriolar NO
consumption. We have recently presented theoretical
results accounting for significant scavenging of NO
by Mb in the tissue surrounding arterioles (24,38).
This scavenging can significantly affect the predicted NO concentration in the smooth muscle. This
result is consistent with experimental observations in
isolated perfused hearts showing MetHb formation
after infusion of NO or bradykinin-induced release
of NO (16). Mb-knockout mice also showed increased vasodilatory sensitivity in response to NO
release relative to wild-type mice (16,46). These experimental findings suggested the possibility for a
new role for Mb: in addition to facilitating oxygen
diffusion and serving as an oxygen reservoir, Mb
might serve as an NO scavenger in protecting heart
and skeletal muscle respiration (5,6). Somewhat
contradictory for the catabolic fate of NO were the
results of Pearce et al. (31): experimental data in
electrically stimulated cardiac myocytes suggest
minimal consumption of NO by Mb. The authors
proposed cytochrome c oxidase as the major route of
NO deactivation.
In addition to substrate such as Mb and cytochrome
c oxidase, Hb in capillaries surrounding an arteriole
might present a significant sink for arteriolarderived NO. The rate of NO uptake by the Hb in
capillaries should depend on parameters such as
capillary density (CD), capillary tube hematocrit
(Ht), and RBC membrane permeability (Pm). At this
point, there is no quantitative description for the rate
of uptake of NO by Hb flowing in a capillary.
There is also no quantitative information about the
diffusive transport of NO produced in capillary endothelium. Thus, it is unclear whether capillaries
represent a significant source of NO for the surrounding parenchymal cells. Although a significant
fraction of O2 consumed by the parenchymal cells is
exchanged through the capillaries, it is not clear
whether NO follows the same pattern. The close
proximity of capillaries to parenchymal cells makes
them a favorable site for NO release. However, the
RBCs might scavenge most of the NO produced because of their proximity to the capillary wall, thus
diminishing release to the tissue.
Administration of Hb-based oxygen carriers
(HBOCs) holds promise as an alternative to blood
transfusion. The hypertensive effects often seen after
administration, however, are considered a significant obstacle to the use of HBOCs (1,40,47). This
phenomenon has been attributed to the scavenging
of NO by the plasma-based Hb. NO consumption by
Capillary exchange of NO
NM Tsoukias and AS Popel
481
plasma-based Hb in the lumen of an arteriole should
be increased relative to the consumption by an
equivalent concentration of Hb “packed” in RBCs
(26,27,39,43). In addition, plasma-based Hb might
extravasate to the interstitial space, which would
further decrease the NO concentration in the arteriolar smooth muscle. We have previously presented
a theoretical model that examines the effect of
HBOC transfusion and extravasation on NO transport around arterioles (24). There is no information,
however, for the effect of HBOCs on NO transport
around capillaries.
The purpose of this article is first to seek a quantitative description for the rate of NO exchange in a
capillary-perfused tissue in the vicinity of an NO
source. Such a description can be used in modeling
NO transport around arterioles for a more accurate
prediction of smooth muscle NO concentration. Our
second purpose is to examine the diffusive transport
of NO released from the capillary wall and investigate the conditions under which capillaries represent
a significant source of NO for parenchymal tissue
cells. The effect of Mb and plasma-based Hb on NO
transport will also be examined.
METHODS
Previously, a simple expression was presented to describe the NO exchange rate in a perfused tissue region (24); a homogeneous NO production and consumption in the entire tissue region was assumed, as
if the RBCs and the capillary endothelium cells were
uniformly dispersed throughout the tissue (dispersed
cells model, presented in Appendix A). In this article,
we formulate a discrete cells finite element model to
provide a more accurate description of the problem’s
geometry and account for the extravascular and
intravascular diffusional resistance to NO transport.
The model is used to predict the NO concentration profile and the average exchange rate (i.e., net
production or consumption) of NO over the entire
capillary/tissue region. In addition, an effective
intravascular first-order reaction rate constant
will be estimated, and an empirical correlation for
its dependence on Ht and RBC membrane permeability will be obtained. Finally, a Krogh-type
approach will be used to provide a closed-form analytic solution for the average NO consumption/
production rate over the entire region. To accomplish that we will treat blood as a homogeneous
phase with an effective first-order reaction rate
constant as estimated from the finite element analysis.
Discrete Cells Model
A finite element axisymmetric discrete cells model of
gas transport in a capillary has been previously developed and used to estimate O2 transport from the
RBC to the surrounding tissue (13,41). The model is
extended here to describe NO transport. In the
analysis that follows, we consider NO transport in a
single cylindric capillary segment surrounded by a
tissue layer (Fig. 1). Equidistant parachute-shaped
RBCs are moving in a single file inside the capillary.
The shape of the RBC is reproduced from reference
32 for the corresponding capillary radius (Rc) and
RBC velocity (Urbc). The total RBC volume and surface area are in close agreement with the average
values for a human RBC under physiologic conditions (15,17).
The model includes regions for the RBC, plasma,
capillary wall, interstitial space, and surrounding
tissue. The effective radius of tissue (Rt) depends on
the tissue capillary density:
Rt = 共␲ CD兲−1 Ⲑ 2
(1)
NO can be consumed by a number of substrates in
the tissue layer, including superoxide, thiols, Mb, cytochrome c oxidase, and possibly by free Hb after
extravasation of transfused HBOC. An overall firstorder rate expression is used to model the NO consumption inside the tissue (kt). Negligible consump-
Figure 1. Discrete cells model. Equidistant parachuteshaped RBCs are moving in single file inside a capillary. A
tissue layer surrounds the capillary. The radius of the tissue is a function of capillary density. The model includes
regions for RBCs, plasma, capillary wall, interstitial space,
and parenchymal tissue.
Capillary exchange of NO
NM Tsoukias and AS Popel
482
tion of NO is assumed in the capillary wall. Inside
the RBC, NO consumption is dominated by the reaction with the Hb. In the plasma, small consumption of NO occurs through reactions with substrates
such as superoxide and thiols. Significant consumption of NO in the plasma might occur through the
reaction with free Hb after HBOC transfusion. Overall first-order rate expression is used to model the
NO consumption inside the RBC (krbc) and the
plasma (kpl). Second-order reactions of NO such as
the reaction with oxygen are considered negligible.
The computational domain with a representative finite element mesh is shown in Fig. 2A and includes
half of the axial cross-section (geometry is axisymmetric). We did not seek a solution inside the RBCs.
Instead, we specified the NO flux at the RBC surface
boundary as a function of the local NO concentration
(see Eq. 3). To eliminate possible edge effects caused
by boundary conditions imposed at the two ends of
the domain, a train of three RBCs was considered.
The spacing of the RBCs is set according to the local
Ht. In the computational domain, the RBCs are stationary. This is accomplished by setting a frame of
reference that is moving with the RBC. Hence, in this
reference frame, the tissue, vascular wall, and interstitial space are moving with a velocity equal and
opposite of the RBC velocity—Urbc.
The steady-state diffusion-convection equation with
first-order chemical reaction is formulated in each of
the four regions of the computational domain:
Djⵜ2C − v ⭈ ⵜC − kjC + Sj = 0
(2)
where j ⳱ pl, w, i, t for plasma, wall, interstitial
space, and tissue, respectively; Dj is the diffusivity of
NO in each region; v is the velocity vector; and Sj is
the endogenous production rate of NO per unit volume. Previous models have assumed NO production
on the luminal and abluminal surface of the endothelium and constant and equal production rates per
unit surface area (Qlu and Qab, respectively). For the
discrete cells model, we simplify the approach by
assuming uniform NO production during the volume
of the capillary wall (mostly endothelium). This assumption is justified, because the wall is thin compared with the capillary lumen and tissue regions.
Thus, Sj is equal to 2(RcQlu + RwQab)/(Rw2 − Rc2) for
the capillary wall region and zero everywhere else;
Rc and Rw are the inner and outer radii of the capillary wall.
Boundary conditions ensure continuity of flux and
partial pressure of NO across interfaces between the
different regions of the domain. A zero flux boundary condition was imposed on the axis of the capillary and at the outer boundary of the tissue when
simulating NO transport in capillaries far away from
arterioles. A constant concentration, C ⳱ Ct, boundary condition was imposed at the outer boundary of
the tissue when examining capillaries in the vicinity
of an arteriole or other NO sources. Periodic boundary conditions were imposed at the inlet and outlet
cross-sectional areas of the cylinder. At the RBC surface, the flux normal to the surface is described as a
linear function of the local NO concentration:
n ⭈ DplⵜC =
Figure 2. Finite element mesh (A) and plasma velocity
vector plot (B) from a representative FEM simulation using the discrete cells model. The shape of the RBC affects
the plasma velocity profile. However, simulations show
that convecting mixing has little contribution to NO transport.
1
C
1
1
+
Pm 公krbc Drbc
(3)
where n is the normal outer vector at a point on the
RBC surface, Pm is the RBC membrane permeability,
krbc and Drbc are the first-order reaction rate constant and the diffusivity of NO inside the RBC, respectively, and C is the local plasma NO concentration. For the derivation of Eq. 3, we assumed small
penetration of NO into the RBC relative to the curvature of the RBC (Appendix B). The velocity field in
Capillary exchange of NO
NM Tsoukias and AS Popel
483
the plasma in Eq. 2 was obtained by solving the
Navier-Stokes equations using the finite element
method as described in reference 41. Figure 2B depicts a velocity vector plot from a representative
simulation.
For any given set of parameter values, simulations
were performed for a range of values for Ct. At any
given value of Ct, the average exchange rate of NO
(Stis) (i.e., average production or consumption rate)
and the average NO concentration (CNO) were estimated. Stis was estimated from the outward flux
across the outer surface of the tissue region; at steady
state, the flux at the outer surface is equal to the net
NO exchange rate in the simulation domain (i.e.,
capillary and surrounding tissue). (Note: the periodic boundary condition at the ends of the domain
guarantees no net flux of NO across these two surfaces):
Stis =
CNO =
兰
L
0
⭸C
− 2␲Rt Dt
dz
⭸R R=Rt
␲Rt2L
兰兰
L
Rt
0
0
(4)
2␲rCdRdz
␲Rt2 L
(5)
The linearity of the system (equations and boundary
conditions) results in a linear relationship between
Stis and CNO for every value of Ct examined. An
apparent production rate and apparent reaction rate
constant (Sapp and kapp) were identified as the intercept and negative slope of this linear function
Stis = Sapp − kapp CNO
(6)
A capillary mass transfer coefficient was defined as:
kcap =
Jw
Cw
(7)
where kcap has units (length/time), Jw is the average
inward NO flux, and Cw is the surface average NO
concentration at the luminal side of the capillary
wall. The mass transfer coefficient kcap describes the
intravascular resistance to NO transport and depends on a number of model’s parameters, including
the hematocrit and the RBC membrane permeability. The finite element method (FEM) analysis
and numerical integration was performed using
FLEXPDE software (PdeSolutions, Antioch, CA).
Continuum Model
In the analysis that follows, we treat blood as a homogeneous phase. Thus, we assume an “effective”
first-order rate constant (kc) for the NO consumption inside the capillary. We assume constant surface
NO production rates at the luminal and abluminal
side of capillary endothelium (Qlu, Qab), a negligible
axial NO concentration gradient, and no convective
facilitation to NO transport and steady state.
Then differential steady-state mass balances in each
layer yield:
Dt
Dw
Dc
冉 冊
1 ⭸
⭸C
R
− ktC = 0
R ⭸R
⭸R
with Rw < R < Rt
冉 冊
1 ⭸
⭸C
R
=0
R ⭸R
⭸R
冉 冊
共8兲
with Rc < R < Rw 共9兲
1 ⭸
⭸C
R
− kcC = 0
R ⭸R
⭸R
with 0 < R < Rc 共10兲
where Dt, Dw, and Dc are the diffusivity coefficients
of NO in tissue, wall, and lumen of the capillary,
respectively. To simplify the solution, we do not consider the interstitial space explicitly. The calculations will show that this assumption does not significantly affect the results (see comparison of the discrete cells and continuum models in the “Results”
section). We approximate Dc as the diffusivity in the
plasma (Dpl); kt and kc are the first-order reaction
rate constants of NO consumption in the tissue and
lumen, respectively. Continuity of concentration
(i.e., we assume same solubility of NO in different
regions) and flux at the interfaces yield the following
boundary conditions:
C共R = Rt兲 = Ct
(11)
C共R = Rw+兲 = C共R = Rw−兲
(12)
C共R = Rc+兲 = C共R = Rc−兲
(13)
冋 册
冋 册
冋 册
−Dc
−Dt
⭸C
⭸R
⭸C
⭸R
⭸C
⭸R
−
R=Rc
R=Rw
+
=0
冋 册
冋 册
+ Qlu = −Dw
= −Dw
⭸C
⭸R
⭸C
⭸R
−
+
(14)
R=Rc
+ Qab
(15)
R=Rw
(16)
R=0
Ct is the concentration at the outer boundary of the
tissue layer and is assumed constant. The model’s
equations and boundary conditions are nondimen-
Capillary exchange of NO
NM Tsoukias and AS Popel
484
sionalized by introducing the following dimensionless variables:
r=
R
,
Rw
冑
kc
,
Dc
␳ = Rw
冑
kt
C共r兲
, ␰ = Rw
,
Ct
Dt
Rc
Rt
␧= , ␦=
Rw
Rw
⌽共r兲 =
⌽共r兲 =
ln共r兲
共⌽共␧+兲 − ⌽共1−兲兲
ln共␧兲
+ ⌽共1−兲,
⌽共r兲 = C2I0 共␰r兲 + C3K0 共␰r兲,
0ⱕrⱕ␧
(17a)
␧ⱕrⱕ1
(17b)
1ⱕrⱕ␦
(17c)
where I0(x) and K0(x) are the modified Bessel function of zero-order and C1, C2, C3 are integration
constants, which can be determined from Eqs.
12–16.
The average concentration in the entire volume CNO
will be:
兰 2␲r⌽共r兲dr
=A+BC
=C
兰 2␲rdr
␦
CNO
0
t
␦
t
(18)
0
and the total consumption/production of NO per
unit perfused tissue volume:
冋 册
冋 册
2Dt ⭸C
Rt ⭸R R=Rt
2DtCt ⭸⌽
=−
= E − F Ct
RtRw ⭸r r=␦
Stis = −
(19)
Algebraic expressions for the determination of constants A, B, E, and F are presented in Appendix C.
The linearity of the system gives rise to a linear dependence of Stis and CNO on Ct. As a result, the
dependence of Stis on CNO is also linear
冉
Stis = E +
冋 册
DcCt ⭸⌽
Rw ⭸r
r=␧
= 公kcDc
I1共␳␧兲
C共R = Rc兲
I0共␳␧兲
(21)
Because Cw and C(R ⳱ Rc) represent the same concentration at the inner surface of the capillary wall,
Eqs. 7 and 22 can be combined to give:
The solution of Eqs. 9–11 describes the radial concentration profile of NO in the capillary and surrounding tissue:
⌽共r兲 = C1I0共␳r兲,
Jw =
冊
AF
F
− CNO = Sapp − kapp CNO
B
B
(20)
Differentiation of Eq. 17a yields the inward flux of
NO at the luminal side of the capillary wall Jw:
kS = 公kcDc
I1 共␳␧兲
I0共␳␧兲
(22)
where kc is an effective intravascular reaction rate
constant that accounts for the extracellular diffusion
and RBC membrane resistance in addition to the
intracellular reaction. Eq. 22 can be solved for kc
iteratively, by use of kcap values from the discrete
cells model. In this way, we are able to incorporate
information on the intravascular NO transport resistance acquired from the discrete cells model to the
continuum model. The agreement of the results from
the continuum model (Eq. 20) with the discrete cells
model (Eq. 6) will be tested in the “Results” section.
Parameter Values
Values used in calculations are presented in Table 1.
The parachute-shaped RBCs used in the simulations
have a surface area of 93 ␮m2 and a volume of 135
␮m3. These values are in close agreement with the
measured values for the volume (90–98 ␮m3) and
the surface area (130–144 ␮m2) of human erythrocytes (2,17). The diffusivity of NO in the plasma,
wall, interstitial space, and tissue (Dpl, Dw, Di, Dt,)
was assumed the same and set to 3.3 × 10−5 cm2s−1
on the basis of the estimated value for NO diffusivity
in the aorta wall from Malinski et al. (29). The diffusivity of NO inside the RBC (Drbc) should be decreased compared with the plasma because of the
high concentration of Hb present. We set Drbc to half
the value of Dpl (i.e., 1.6 × 10−5 cm2s−1) on the basis
of the ratio of the extracellular and intracellular diffusivities for O2 from experimental measurements
(20,36) and assuming a similar dependence for NO.
A range of values has been previously reported for
the reaction rate constant of NO with oxyhemoglobin
(kHb) and oxymyoglobin (kMb). Cassoly and Gibson
(10) determined a reaction rate of NO with deoxyHb by stopped-flow spectroscopy of 25 ␮M−1s−1
(per heme) at 20 °C and pH 7.0. Eich et al. (14)
reported reaction rate constants of approximately 50
and 30 ␮M−1s−1 for oxy- or deoxy-Hb and oxy- or
deoxy-Mb, respectively, under the same conditions,
Capillary exchange of NO
NM Tsoukias and AS Popel
485
Table 1. Parameter values
Parameter
SRBC
VRBC
Urbc
Rc
Rw-Rc
Ri-Rw
CD
Drbc
Dj (j ⳱ pl,w,i,t)
Qlu, Qab
Sw
Pm
rbc
CHb
pl
CHb
CMb
kHb
kMb
kpl
krbc
kt
Value
Units
135
93
100
3.5
0.3
0.35
1500
1.6 × 10−5
3.3 × 10−5
2.65 × 10−3
176.7
0.1–40
20,300
0, 3000
0, 200
100
55
1 kHb Cpl
Hb
rbc
kHb CHb
1 kMb CMb
␮m2
␮m3
␮m s−1
␮m
␮m
␮m
mm−2
cm2/s
cm2/s
nmol⭈cm−2s−1
nmol⭈cm−3s−1
cm/s
␮M
␮M
␮M
␮M−1 s−1
␮M−1 s−1
s−1
s−1
s−1
Description
RBC surface area
RBC volume
RBC velocity
Capillary inner radius
Capillary wall thickness
Interstitial space thickness
Capillary density
NO diffusivity (RBC)
NO diffusivity
Surface NO production rate (lum., ablum.)
Volumetric NO production rate
Membrane permeability of RBC
RBC heme concentration
Plasma heme concentration
Tissue Mb concentration
NO-oxyHb rate const. @ 37 °C
NO-oxyMb rate const. @ 37 °C
Plasma first-order reaction constant
RBC first-order reaction constant
Tissue first-order reaction constant
and in a recent study, Herold et al. (21) suggested
reaction rate constants of 89 ␮M−1s−1 for kHb and 44
␮M−1s−1 for kMb. The temperature dependence of
the reaction is not known. Carlsen and Comroe (9),
and Cassoly and Gibson (10) suggested a temperature coefficient of 1.25 and 1.4, respectively, per 10 °C
for the reaction of CO with deoxy-Hb. If we assume
a temperature coefficient of 1.4 per 10 °C and extrapolate the values proposed by Herold et al. (21),
we obtain kHb and kMb at 37 °C as high as 160 and
78 ␮M−1s−1, respectively. For our simulations, we
assume a value for kHb of 100 ␮M−1s−1 and for kMb
of 55 ␮M−1s−1.
The overall reaction rate constants kpl and krbc can
be estimated from the product of kHb with the heme
pl
rbc
) and RBC (CHb
),
concentration in the plasma (CHb
respectively. kt is estimated from the product of kMb
with the Mb concentration in the tissue (CMb). In
addition, we add a small value (1 s−1) to kpl and kt
to account for the consumption of NO by other substrates present in the plasma or tissue. Such a value
is justified on the basis of the reaction rate of NO
with O2− (4300 ␮M−1s−1 [19]) and a concentration
of O2− in the plasma in the subnanomolar range.
This value also provides a half-life of NO in tissue on
the order of seconds, which is in agreement with the
observed rate of conversion of NO to NO2− in cultured cardiomyocytes (31). Thus, the consumption
of NO in the plasma or tissue is dominated by the
Reference
(2,17)
(2,17)
—
—
—
—
—
(20,29,36)
(29)
(29,44)
(29,37,39,42)
(2)
—
—
(10,14,21)
(10,14,21)
—
—
—
reaction with free-Hb or Mb, and in the absence of
plasma-based Hb or Mb, the model accounts for
consumption of NO in these regions through reaction
with substrates such as O2− and cytochrome c oxidase.
The exact value of Pm remains at this point controversial. On the basis of the solubility and diffusivity
of NO in lipid bilayers, Pm should be in the order of
40 cm s−1 (12,27,29,37). Recent studies suggested,
however, significant resistance to NO transport in
the membrane of erythrocytes and a Pm value 1000
times smaller (22,42,43). We have recently reviewed
these studies and examined the effect of Pm on the
predicted NO consumption rate by RBCs (39). We
found that values in the range 0.1 to 40 cm s−1 are
consistent with available experimental data if we incorporate the experimental observed range of kHb as
described previously.
The production rate of NO in the capillary wall is not
known. Here we use a value of 2.65 × 10 −3
nmol·cm−2s−1 for both the luminal and the abluminal side of the capillary wall on the basis of experimental data by Malinski in rabbit aorta (29,44,45);
in other words, we assume that NO production per
unit surface area by the capillary endothelium is the
same as for large-artery endothelium. The release
rate of NO in capillaries might be different because
of the different expression of the enzyme (eNOS) (3)
Capillary exchange of NO
NM Tsoukias and AS Popel
486
and different physiologic stimuli (shear stress, release of agonists).
The capillary tube hematocrit, Ht, should be reduced
compared with the discharge hematocrit, Hd, because of the Fahraeus effect and the effect of the
glycocalyx. For the simulations following, we assume
control parameter values of 33% for Ht and 1500
capillaries per mm2 for capillary density (CD). A
wide range of values for Ht will also be examined.
Simulations are performed in the presence and absence of 0.2 mM of Mb in the tissue and 3 mM of
free-Hb in the plasma.
RESULTS
Figure 3 depicts the NO concentration distribution
in and around a capillary from representative FEM
simulations (discrete cells model). Control parameter values are used, and simulations are performed
in the absence (Fig. 3A) and presence (Fig. 3B) of
0.2 mM of Mb in the tissue. For the simulations in
Fig. 3, a zero flux boundary condition is used at the
outer boundary of the tissue and a Pm value of 40 cm
s−1. Once the NO concentration distribution is
known, numeric integration can be used to estimate
average NO concentration or flux across the outer
surface of tissue.
NO Transport in Capillaries Far Away from
Arterioles or Other NO Sources
The zero flux boundary condition at the outer
boundary of the tissue is used in the FEM analysis
when examining NO transport in capillaries far away
from arterioles. In this case, there is no net flux of
NO into the simulation volume, and the NO supply
of the tissue region originates exclusively from the
capillary. Simulations were performed under different scenarios for parameter values for Ht and Pm and
in the presence and absence of Mb in the tissue or
plasma-based free Hb (Fig. 4). Figure 4A presents
results for the average NO concentration in the parenchymal tissue region (CNO,tis) for each of the examined scenarios. Control conditions include an Ht
of 33%, absence of Mb or free Hb, and a CD of 1500
capillaries per mm2. Simulations are performed for
Pm of 40 cm s−1 (solid bars) and 0.1 cm s−1 (open
bars). The simulation scenarios include an increase
in capillary density (twice the control value), a reduced Ht (10%), the presence of 0.2 mM of Mb in
the tissue, and the presence of 3 mM of free Hb in the
plasma in addition to a reduced Ht. In the absence of
Mb or plasma-based free Hb, the model predicts that
a significant amount of NO accumulates in the tis-
Figure 3. NO concentration distribution in and around a
capillary from representative FEM simulations using the
discrete cells model with zero flux boundary condition at
the outer surface of the tissue. Simulations were performed using control parameter values (A) and in the
presence of 0.2 mM of Mb in the tissue (B).
sue. The tissue NO concentration can be significantly
increased when the Ht or Pm is decreased, reaching
280 nM when Ht ⳱ 10% and Pm⳱ 0.1cm s−1. In the
presence of Mb in the tissue or free Hb in the plasma
(e.g., after HBOC transfusion), the model predicts
negligible tissue NO concentration regardless of the
value of Pm or Ht. The amount of NO that diffuses
abluminally from the endothelium to the parenchymal tissue is presented in Fig. 4B as a percent of total
NO produced. Same scenarios of parameter values
were examined. Under control parameter values,
most NO produced is consumed by the erythro-
Capillary exchange of NO
NM Tsoukias and AS Popel
487
Figure 4. Discrete cells model simulations for NO distribution around capillaries far away from arterioles. The
average NO concentration in the tissue CNO,tis (A) and the
percentage of NO produced by the endothelium that diffuses abluminally (B) are estimated for different scenarios
of parameter values. Control conditions include an Ht of
33%, a CD of 1500 capillaries per mm2, and absence of
Mb or plasma-based Hb. Simulations are performed for a
Pm of 40 cm s−1 (solid bars) and 0.1 cm s−−1 (open bars).
cytic Hb. Only a small percentage of NO produced in
the capillary wall diffuses abluminally. The amount
of abluminal NO flux is inversely related to CD, Ht,
and HBOC concentration. When 0.2 mM of Mb is
present in the tissue, the model predicts that most
NO (up to 85% depending on Pm) will diffuse abluminally.
NO Transport in Capillaries in the Vicinity of
Arterioles or Other NO Sources
For capillaries located close to arterioles, NO of arteriolar origin might contribute to the NO concentration in the tissue surrounding the capillary. In this
case, a nonzero flux of NO might enter the simulation volume. To simulate this condition, we replace
the zero flux condition at the outer boundary of the
tissue with a constant concentration (Ct) boundary
condition. We vary Ct during a wide range of values
(0–500 nM) in an effort to examine different
amounts of NO flux at the tissue boundary. For any
given value of Ct, the model yields the NO concentration distribution. Then Eqs. 4, 5, and 7 are used
to predict Stis, CNO, and kcap. Figure 5 depicts results
Figure 5. Representative results for control parameter
values from discrete cells model simulations of NO transport in capillaries with a constant concentration boundary
condition at the outer surface of the tissue (Ct), imitating
the effect of arterioles or other NO source. Different values
of Ct are used. For every value of Ct, FEM analysis yields
the NO concentration distribution. Numeric integration is
used to provide the average NO exchange rate (Stis), the
average NO concentration (CNO), and the intravascular
mass transfer coefficient (kcap) (Eqs. 4, 5, 7). Figure
presents Stis as a linear function of CNO. The slope and
intercept of this line are identified as an apparent production rate (Sapp) and an apparent reaction rate constant
(kapp) for NO exchange. Ct is presented in the secondary
x-axis.
from representative simulations for control parameter values. Stis and kcap are plotted as a function of
CNO; Ct is presented in the secondary x-axis. As expected from the linearity of the problem, there is a
linear dependence between Stis and CNO. The slope
and intercept of this line are identified as the apparent reaction rate constant (kapp) and the apparent
production rate (Sapp) for NO exchange during the
entire simulation volume (i.e., tissue and capillary).
The intravascular mass transfer coefficient kcap is
independent of Ct. Thus, the parameters Sapp, kapp,
kcap are identified for the system independently of
the value chosen for Ct.
In Fig. 6, kcap is plotted as a function of Ht. FEM
simulations using the discrete cells model were performed for Ht values between 5% and 40% and Pm
values between 0.1 and 40 cm s−1. The other parameters were held at the control values. kcap was
estimated as described previously (Eq. 7). Least
squares minimization was used to fit an empirical
correlation to the computed values (solid lines) and
provide a description for kcap as a function of Ht
and Pm:
Capillary exchange of NO
NM Tsoukias and AS Popel
488
Figure 6. Intravascular mass transfer coefficient (kcap) as
a function of Pm. Results are from FEM simulations (discrete cells model) using different values of Ht. All other
parameters are held at their control values. Least squares
fitting (lines) provided an empirical correlation for kcap as
a function of Pm and Ht (Eq. 23).
kcap 共cm s−1兲
共−5.70 × 10−6 Ht3 + 4.99 × 10−4 Ht2
− 9.26 × 10−4 Ht + 1.32 × 10−2兲Pm
=
Pm −2.64 × 10−6 Ht3 + 1.14 × 10−4 Ht2
+ 1.02 × 10−2 Ht + 9.71 × 10−2
(23)
where Pm is in cm s−1. An iterative algorithm and
Eq. 22 were used to predict the corresponding value
for kc for every kcap estimated in Fig. 6. Least
squares minimization yields the following empirical
correlation:
kc共s−1兲 =
共1.44 ⳯ 10−2 Ht3 + 5.65 Ht2
− 50.11 Ht + 2.30 ⳯ 102兲Pm
Pm + 6.69 ⳯ 10−6 Ht3 − 4.23 ⳯ 10−4
Ht2 + 3.64 ⳯ 10−2 Ht − 8.45 ⳯ 10−2
(24)
Comparison of Discrete Cells, Continuum, and
Dispersed Cells Models
In Fig. 7, Sapp and kapp from simulations using the
discrete cells model (symbols) are compared with the
predictions from the continuum model, Eq. 20
(lines), under different scenarios of parameter values. Control conditions (solid circles) include a Pm of
40 cm s−1, absence of Mb or free Hb, and a CD of
1500 capillaries per mm2. Simulations are also performed for Pm of 0.1 cm s−1 (open triangles), a capillary density twice the control value (open dia-
Figure 7. Comparison of the discrete cells and the continuum model. Prediction of Sapp (A) and kapp (B) from
the discrete cells model (symbols) and the continuum
model (lines) are presented as a function of capillary Ht.
Different scenarios of parameter values are used. Control
conditions (solid circles) include a Pm of 40 cm s−1, a CD
of 1500 capillaries per mm2, and absence of Mb or
plasma-based Hb. Simulations are also performed for a
Pm of 0.1 cm s−1 (open triangles), a CD of 3000 capillaries
per mm2 (open diamonds), in the presence of 0.2 mM of
Mb (solid squares), and in the presence of 3 mM of
plasma-based Hb (solid diamonds). CNO,eq, defined as the
ratio of Sapp and kapp, is presented in (C).
monds), in the presence of 3 mM of free Hb in the
plasma (solid diamonds) and in the presence of 0.2
mM of Mb (solid squares). All the predictions using
the continuum model use kc from Eq. 24 with the
exception of predictions in the presence of HBOC,
where kc is assumed equal to kpl. The results from
the continuum model are essentially equivalent to
Capillary exchange of NO
NM Tsoukias and AS Popel
489
results from the discrete cells model. A small difference between the two models occurs in the presence
of Mb and can be attributed to the absence of interstitial space in the continuum model. The ratio of
Sapp to kapp (Fig. 7C) represents the average concentration in the perfused tissue when production is
equal to consumption
CNO,eq = Sapp Ⲑ kapp
(25)
so that Eq. 6 becomes
Stis = −kapp共CNO − CNO,eq兲
(26)
CNO should approach this equilibrium concentration
as we move away from the arteriole. Interestingly,
kapp changes in a nonlinear fashion with CD; however, the ratio of Sapp to kapp remains the same independent of CD.
In Fig. 8, we compare the discrete cells model (Eq.
20) with the dispersed cells model in Appendix A.
The same scenarios of parameter values as before
were examined. The dispersed cells model overestimates both Sapp and kapp in every scenario examined. Significant difference (almost 400 times) occurs between the predictions of the two models for
kapp in the presence of plasma-based free Hb. The
discrete cells model predicts a CNO,eq between 20
and 95 nM in the absence of Mb and HBOC, depending on the value of Pm used. The corresponding
prediction of the dispersed cells model is 45 nM independently of the value for Pm. This is due to the
effective reaction rate of RBCs with NO, kRBC, used
in the model, which was set to a constant value independent of Pm (see Appendix A). In the presence
of 3 mM of HBOC, the discrete cells model yields a
CNO,eq ⳱ 4 nM versus the dispersed cells model prediction of 0.2 nM; thus, in this case, both models
predict low values of tissue NO concentration.
DISCUSSION
A discrete cells mathematical model was developed
to examine NO transport in and around capillaries.
The model was used to test to what extent and under
what conditions the capillary wall represents a significant source of NO for parenchymal tissue. The
model also provided a description for the NO intravascular mass transfer coefficient, kcap, as a function
of Ht and Pm. The closed-form analytical solution of
the continuum model uses the values of kcap obtained
in the discrete cells model to make predictions for
the rate of NO consumption in capillary-perfused
tissue. This closed-form solution can be used in
models of NO transport around arterioles to provide
a better description for the extra-arteriolar NO con-
Figure 8. Comparison of the discrete and the dispersed
cells model. Predictions of Sapp (A) and kapp (B) from the
discrete cells model (solid bars) and the dispersed cells
model (open bars) are presented for different scenarios of
parameter values. Control conditions include an Ht of
33%, a Pm of 40 cm s−1, a CD of 1500 capillaries per mm2,
and absence of Mb or plasma-based Hb. Simulations are
also performed for a Pm of 0.1 cm s−1, a CD of 3000
capillaries per mm2, and in the presence of 3 mM of
plasma-based Hb. CNO,eq is also presented (C).
sumption and should result in a better estimate for
the NO concentration in the arteriolar smooth
muscle.
The shape of the RBC affects the plasma velocity
profile in the capillary (Fig. 2B). However, our calculations show that the contribution of convecting
mixing to NO transport relative to diffusion is negligible (data not shown), and thus the convection
term in Eq. 2 can be omitted without compromising
Capillary exchange of NO
NM Tsoukias and AS Popel
490
the accuracy of the model. The high concentration of
Hb in the RBC allows the use of the periodic boundary condition at the entrance and exit of the capillary
segment; the Hb available to scavenge NO at the
entrance and exit of the capillary segment is essentially the same, and thus the NO concentration
should be the same. This is not valid in the case of
oxygen delivery by the RBC during its passage
through the capillary. The oxygen saturation of the
Hb at the outlet of the capillary would decrease compared with the one at the inlet, because a significant
amount of oxygen would be unloaded. This generates an axial gradient for oxygen along the capillary,
whereas the gradient is minimal for NO.
The high erythrocytic Hb concentration results in a
high consumption rate of NO, which generates a
steep NO concentration gradient inside the RBC. To
improve numeric accuracy and computational time,
we did not seek a solution inside the RBC. The
boundary condition (Eq. 3) gives a description of the
NO flux at the surface of the RBC, provided that the
NO penetration is small relative to the curvature of
the RBC (Appendix B). Numeric considerations also
suggested the use of volumetric NO production rate
rather than a surface source. However, the agreement between the discrete and the continuum model
(when we use surface NO source) suggests that this
does not introduce significant error.
NO Exchange in Capillaries Far Away from
Arterioles or Other NO Sources
Model simulations suggest that a significant amount
of NO originating from capillary wall can diffuse
toward the parenchymal cells, where it can sustain
physiologically relevant concentrations (Fig. 3A).
The predicted NO concentration and flux depend
significantly on a number of model parameters, for
some of which the values are not well known (e.g.,
Pm, Qlu, and Qab). For example, depending on the
value of Pm under control conditions, the NO concentration in the tissue predicted by the model can
vary within the range of 25 and 100 nM (Fig. 4A).
The tissue NO concentration changes proportionally
with the NO production rate (data not shown). This
is a result of the linearity of the system and suggests
that a 10-fold increase in Qlu and Qab would result in
a 10-fold increase in CNO,tis. Despite the significant
uncertainty regarding the absolute values predicted
by the model, relative comparisons suggest that the
NO concentration and flux toward the parenchymal
cells are significantly affected by tissue-dependent
parameters such as CD and CMb (Fig. 4). There is
also a significant increase in both the flux and the
average concentration in the tissue when the Ht is
decreased, because more NO can escape abluminally. Finally, after HBOC transfusion, most of the
capillary NO is scavenged by the plasma-based Hb.
As a result, only a small amount can diffuse abluminally, and the tissue NO concentration is significantly decreased relative to the levels predicted in
the absence of Mb. This suggests a potential NOrelated problem for HBOCs (in addition to causing
vasoconstriction); at least for tissues with small concentrations of Mb the regulatory role of endothelium-derived NO for parenchymal cells might be
compromised after transfusion.
The Catabolic Fate of NO in the Presence of Mb
The model suggests that when Mb is present, it competes with Hb for a significant amount of the total
NO produced. At a myoglobin concentration of 0.2
mM (a typical value for red muscle or myocardium),
the model predicts that more than half of the NO
produced diffuses abluminally, where it is taken up
by Mb, and depending on the value of Pm, the percentage can reach 85% (Fig. 4B). Despite the significant flux of NO abluminally, however, the penetration of NO into the Mb-containing tissue is minimal (Figs. 3B and 4A). The NO concentration
decreases rapidly as we move into the Mb-containing
tissue. In Fig. 3B at 2, 5, and 10 ␮m from the capillary wall, the NO concentration is 0.15, 5 × 10−4
and 8 × 10−8 nM, respectively. This suggests that to
sustain a physiologically important NO concentration (i.e., in the nM range) inside a cell at a location
10 ␮m away from the capillary wall, the NO production has to increase to supraphysiologic levels
(i.e., increase by a factor of 108). This result raises a
question about the bioavailability of NO in Mbcontaining tissues.
The role of Mb as a scavenger of NO is supported by
recent experimental studies in isolated hearts of Mb
knockout transgenic mice (16). Flogel et al. (16)
reported MetHb formation in wild-type mice and increased sensitivity to vasodilation for Mb knockout
mice relative to wild-type after infusion of exogenous
NO or bradykinin-induced release of NO. More recently, Wegener et al. (46) showed the effects on
contractility in myocardial strips of Mb knockout
but not wild-type mice after exogenously applied NO
at concentrations up to 10 ␮M. These experimental
results and our theoretical simulations question the
regulatory role of vascularly derived NO in Mbcontaining tissue. Co-localization and compartmentalization of the NO producing enzyme (eNOS) with
its target molecules in the parenchymal cells might
Capillary exchange of NO
NM Tsoukias and AS Popel
491
be necessary for NO to be able to able to play a
physiologically important role in Mb-containing tissue.
The simulations predict that when Mb is present, it
represents the major catabolic pathway for NO in the
parenchymal tissue. This is based on the assumption
that Mb and NO react in the presence of oxygen (i.e.,
oxyMb) through an irreversible fast first-order reaction, with a reaction constant on the order of 108
M−1s−1 as determined from in vitro measurements
(14,21). However, in a recent study, Pearce et al.
(31) propose cytochrome c oxidase and not oxyMb
as the major catabolic pathway for NO. Their experimental data on electrically stimulated cardiac
myocytes suggest no metmyoglobin (metMb) formation and conversion of NO to NO2− rather than
NO3−. This can be explained by reaction of NO with
cytochrome c rather than oxyMb. Because the oxyMb
concentration is much greater than cytochrome c
oxidase, they suggest that there must be an additional mechanism in vivo that either increases the
reaction of NO with cytochrome c oxidase or decreases the reaction with oxyMb. The second scenario is probably more likely based on a time constant on the order of seconds for the NO disappearance and NO2− formation in their experimental data.
The half-life of NO (t ⁄ ⳱ ln(2)/(kMb CMb)) in 0.2
mM of Mb based on the in vitro rate constant of 55
× 106M−1s−1 is 0.06 ms. For a half-life for NO consumption on the order of 0.5 seconds as it seems in
reference 31, the reaction rate of NO with oxyMb
should be 10,000 times slower than the in vitro one.
A mechanism for such a drastic reduction of the in
vivo reaction rate of NO with Mb has not been determined.
might underestimate the abluminal NO consumption. The model presented here provides a prediction
for the rate of NO consumption in a capillaryperfused tissue.
From Eq. 26 follows that at high NO concentrations
(i.e., CNO Ⰷ CNO,eq) the rate of NO consumption in
capillary-perfused tissue (Stis≈ kappCNO) is limited
by extravascular diffusion. This is supported by the
fact that increasing the Hb concentration in the capillary by either increasing the Ht (above 20%) or
adding plasma-based Hb has small effect on kapp
(Fig. 7B). On the contrary, kapp increases when CD
increases (i.e., extravascular diffusion distance decreases). The predicted rate of NO consumption is
generally between one and three orders of magnitude
higher than the previous estimates for the abluminal
NO consumption rate (8,29,45).
At low NO concentrations, the perfused tissue can
act either as a sink or as a source of NO if CNO is
greater or less than the equilibrium concentration
CNO,eq, respectively. Contrary to kapp, CNO,eq is independent of CD, suggesting that the extravascular
diffusion does not affect the equilibrium concentration far away from arterioles or other NO sources.
Model Comparison
12
NO Exchange in Capillaries in the Vicinity of
Arterioles or Other NO Sources
Some previous mathematical models for NO transport around arterioles have assumed an infinite
smooth muscle layer surrounding the arteriole
(8,44). In the smooth muscle, NO should be consumed predominately through the reaction with
soluble guanylate cyclase (sGC). The reaction is fast;
however, the relatively small concentration of sGC
yields a half-life for NO in the smooth muscle on the
order of seconds (first-order reaction rate constant
1.7 s−1 [8]). Analysis of experimental data for NO
decay in rabbit aortic rings yields a lower first-order
consumption rate constant (0.01 s−1) (29,45). The
small thickness of the arteriolar wall relative to the
aorta and the presence of perfused capillaries in the
vicinity of the wall suggest that these reaction rates
Predictions of the continuum and discrete cells models are in close agreement (Fig. 7). The discrete cells
model is necessary, because the continuum model
uses an effective intravascular reaction rate constant, kc, as predicted from the discrete cells analysis. The continuum model, however, provides a
closed-form analytic solution for the NO exchange
rate, which can be easily used.
The difference in kapp between the dispersed and the
discrete cells models is attributed to the extravascular diffusion limitation to NO transport that was neglected in the former analysis. kapp becomes independent of Hb concentration in the capillary at high
Ht, and thus plasma-based Hb should not increase it
significantly, provided that no extravasation occurs.
Despite the observed difference in kapp and thus in
the NO consumption rate at high NO concentrations,
CNO,eq seems to be less model dependent (Fig. 8C).
The prediction for CNO,eq on the basis of the dispersed cells model (45 nM) falls within the range of
predictions by the discrete cells model (25–90 nM;
Pm ⳱ 40-0.1 cm s−1). The dispersed cells model
provides an acceptable approximation of the NO exchange rate for NO concentrations close to CNO,eq.
Thus, this work is consistent with our earlier results
for NO transport around arterioles in the presence of
Capillary exchange of NO
NM Tsoukias and AS Popel
492
blood substitutes (24). The simplified dispersed cells
model, which was used in this earlier work, provides
an acceptable approximation of the NO exchange in
the perfused tissue region at least for the cases examined (M. Kavdia, personal communication).
CONCLUSIONS
Theoretical simulations suggest that under certain
conditions NO originating from the capillary wall
can diffuse toward the parenchymal cells and accumulate in physiologically significant concentrations
to regulate cellular functions. In the presence of Mb,
a significant part of the total NO produced should
diffuse abluminally and be taken up by Mb. Simulations predict, however, that the NO penetration is
minimal in Mb-containing tissue, suggesting perhaps
the need for co-localization of eNOS with its targets
for NO to play a physiologic role in the presence of
Mb. Extravascular diffusion limits the uptake of NO
in capillary-perfused tissue at high NO concentrations. The closed form solution for the rate of NO
consumption that is derived from the continuum
model can be easily used in models of NO transport
around arterioles to provide a better description of
extra-arteriolar NO consumption and smooth muscle
NO concentration, as well as around other NO
sources.
ACKNOWLEDGMENTS
Appendix B
In the analysis below, we assume that the penetration depth of NO into the RBC is small relative to the
radius of curvature of the cell. Then, a onedimensional analysis in cartesian coordinates can be
used to describe the flux of NO at the RBC surface
(Fig. 9).
Inside the RBC, the NO transport is described by the
following equation:
Drbc
⭸2Crbc
⭸x2
− krbcCrbc = 0
(B1)
with boundary conditions:
Crbc共x = 0兲 = Cin
(B2)
Crbc共x = ⬁兲 = 0
(B3)
Solution of Eq. B1 yields:
冑
−
Crbc 共x兲 = Cine
krbc
Drbc
x
(B4)
Differentiation of Eq. B4 yields the NO flux entering
the RBC:
冋 册
Flux = −Drbc
⭸Crbc
⭸x
x=0
= 公DrbckrbcCin
(B5)
We thank Drs. Roland Pittman and Mahendra
Kavdia for helpful discussions and review of the
manuscript.
APPENDIX
Appendix A: Dispersed Cells Model
For the derivation below, we assume that NO production and consumption is distributed homogeneously throughout the tissue; the NO sources and
sinks correspond to the number of capillary endothelial cells and RBCs per unit volume:
Stis = 2␲CD共RcQlu + RwQab兲
− 共␲Rc2CDHt kRBC兲 CNO
(A1)
The first term in Eq. A1 represents NO production
by the endothelial cells per unit tissue volume and
the second term NO consumption by the RBCs flowing in the capillaries per unit tissue volume. kRBC is
the reaction rate of NO with RBCs defined on a per
RBC volume and average NO concentration basis. A
detailed description for kRBC and its dependence on
Ht has been described elsewhere (39). Here we use a
value of 2300 s−1 for the simulations.
Figure 9. One-dimensional analysis of NO uptake by an
RBC. Cin and C, concentrations at the inner and outer
surface of the RBC membrane. Crbc(x), NO concentration
decay as a function of distance x from the membrane.
Capillary exchange of NO
NM Tsoukias and AS Popel
493
From the definition of membrane permeability we
have:
(B6)
Flux = Pm共C-Cin兲
Combining Eqs. 5 and 6
Flux =
1
1
1
+
Pm 公krbcDrbc
(B7)
C
The continuity of flux at the RBC surface yields:
n ⭈ DplⵜC =
1
1
1
+
Pm 公krbcDrbc
(B8)
C
Based on the values for krbc and Drbc from Table 1,
Eq. B4 suggests that the concentration of NO drops
by more than 97% for every 0.1 ␮m traveled into the
RBC. Thus, the assumption of minimal NO penetration depth relative to the RBC radius of curvature
and therefore Eq. B8 should introduce minimal error.
Appendix C: Constants in Eqs. 19 and 20:
A=
B=
2
共共a1m1 + d1兲共s2t1 + t2兲
␦ 共t1s1 − 1兲
+ 共b1n1 + c1o1 + e1兲共s1t2 + s2兲兲
2
2
␦2
冉
a1m1t3 + b1n1s1t3
+ c1o1s1t3 + d1t3 + e1s1t3
+ b1n2 + c1o1
共1 − s1t1兲
2Dt 共s1t2 + s2兲共␨1n1 + ␪1o1兲
E=
RtRw
共1 − s1t1兲
F=
冉
2Dt
共f1n1 + g1o1兲s1t3
f1n2 + g1o2 +
RtRw
共1 − s1t1兲
where:
m1 =
1
Io共␳␧兲
n1 =
Ko共␰␦兲
Io共␰兲Ko共␰␦兲 − Io 共␰␦兲Ko共␰兲
n2 = −
Ko共␰兲
Io共␰兲Ko共␰␦兲 − Io 共␰␦兲Ko共␰兲
o1 = −
Io共␰␦兲
Io共␰兲Ko共␰␦兲 − Io 共␰␦兲Ko共␰兲
o2 =
Io共␰兲
Io共␰兲Ko共␰␦兲 − Io 共␰␦兲Ko共␰兲
冊
冊
s1 = 1 −
s2 =
Dc
m ␳␧I 共␳␧兲
PmRw 1 1
␧Q1u
Pm
t1 = 1 −
Dt␰
共n I 共␰兲 − o1K1共␰兲兲
PmRw 1 1
t1 =
Qab
Pm
t3 =
Dt␰
共−n2I1共␰兲 + o2K1共␰兲兲
PmRW
a1 =
␧I1共␳␧兲
␳
b1 =
␦I1共␰␦兲 − I1共␰兲
␰
c1 =
K1共␰兲 − ␦K1共␰␦兲
␰
␭共␧2 − 2␧2 ln共␧兲 − 1
d1 =
4 ln共␧兲
␭共1 − ␧2 + 2 ln共␧兲兲
e1 =
4 ln共␧兲
f1 = ␰I1 共␰␦兲
g1 = −␰K1共␰␦兲
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