Vol. 277, Issue 6, H2158-H2166, December 1999
A hemodynamic analysis of coronary capillary blood flow based
on anatomic and distensibility data
Ghassan S.
Kassab,
Kha N.
Le, and
Yuan-Cheng B.
Fung
Department of Bioengineering, University of California, San
Diego, La Jolla, California 92093-0412
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ABSTRACT |
An understanding
of cardiac health and disease requires knowledge of the various factors
that control coronary capillary blood flow. An analysis of coronary
capillary blood flow based on a complete set of actual data on the
capillary anatomy and elasticity does not exist. Previously, a complete
set of data on the branching pattern and the vascular geometry of the
pig coronary capillary network were obtained in our laboratory. In the
present study, we obtained distensibility data on the coronary
capillary blood vessels on the epicardial surface in the form of a
pressure-diameter relationship using intravital microscopy. A
mathematical model of the coronary capillary blood flow was then
constructed on the basis of measured anatomic and elasticity data of
the coronary capillary network, rheology of blood, physical laws
governing blood flow, and appropriate boundary conditions. The
constructed model was used to examine the heterogeneity of the spatial
distribution of coronary blood flow, which is an important issue in
coronary physiology. One interesting result of the model is that the
dispersions of pressure and flow are significantly reduced in the
presence of capillary cross-connections, and the resistance to flow is reduced as well. Finally, we found that the compliance of the epicardial surface capillary vessels is so small that its effect on the
blood pressure drop is negligible in the diastolic state. However, the
compliance of the intramyocardial capillaries remains unknown, and the
interaction of the muscle contraction and blood vessel elasticity in
systole remains to be studied.
capillary cross-connections; pressure distribution; flow
distributions; heterogeneity; compliance
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INTRODUCTION |
THE SPATIAL DISTRIBUTION of blood flow into the
coronary capillaries has obvious physiological significance because the
nutrition of the heart muscle depends on the blood flow in the
capillaries. Our hypothesis is that the branching pattern and vascular
geometry of the coronary capillary network are important determinants
of coronary capillary blood flow that, in turn, influence the transport of oxygen and nutrients to the myocardium. We have previously stressed
that the topological structure of the coronary arteries and
intramyocardial veins are treelike but that the coronary capillary blood vessels have a non-treelike topology (9-11). The capillaries not only branch but also cross-connect along their lengths
(9). The presence of cross-connections in the myocardial
capillaries may make the pressure and flow distributions in the
capillary bed more uniform.
A network simulation of the coronary capillary blood flow has been
previously done by Wieringa et al. (25) in a model of hexagonally
stacked parallel capillaries with randomly distributed interconnections
based on the experimental data of Bassingthwaighte et al. (2). Wieringa
et al. (25) assumed that all capillaries have uniform diameter and obey
a linear pressure-flow relationship that did not take into account the
capillary distensibility and the non-Newtonian blood rheology.
We wanted to improve the existing analysis of the coronary capillary
blood flow in four ways: 1) use the topology and vascular dimensions of the coronary capillary network as determined by morphometry; 2) measure the
compliance or distensibility of the coronary capillaries and use the
results in the network analysis; 3) incorporate the
non-Newtonian blood rheology in the analysis; and 4) embed
the capillary networks between coronary arterial and venous trees
realistically according to morphometric data. With regard to
1 and
4 above, we refer to our previously
measured morphometric data (9). With regard to
3 above, we refer to the work of
Lingard (13), Lipowsky et al. (14), and Pries et al. (15). For
2 above, a new morphometric study on
the distensibility of epicardial capillaries of the pig left ventricle
at the diastolic state is described here. With these data we can use
the laws of physics (conservation of mass and momentum) and the
appropriate boundary conditions to formulate well-posed boundary value
problems for the hemodynamics of the coronary capillary network.
Extension to systolic state and the beating heart awaits future work.
In the context of the present analysis, our main goals are to determine the effect of capillary cross-connections and the elasticity of vessels
on the pressure and flow distributions in the coronary capillary network.
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METHODS |
Isolated heart preparation.
The studies were performed on healthy farm pigs weighing 28-32 kg.
Surgical anesthesia was induced with ketamine (33 mg/kg im) and
atropine (0.05 mg/kg im) and maintained with pentobarbital sodium (30 mg/kg iv, in an ear vein). A midline sternotomy was performed,
ventilation with room air was provided with a respiratory pump, and
anticoagulation was induced with heparin (100 U/kg). An incision was
made in the pericardium, and the heart was supported in a pericardial
cradle. The heart was arrested with a saturated KCl solution given
through a jugular vein. The heart was then excised, with the ascending
aorta clamped to keep air bubbles out of the coronary vessels, and
placed in a cold (0°C) saline bath. The right coronary artery, left
anterior descending coronary artery (LAD), left circumflex artery
(LCX), and coronary sinus artery (CS) were cannulated under saline to
avoid air bubbles. These coronary arteries were then immediately
perfused with an isosmotic, cardioplegic rinsing solution as described
in Kassab et al. (11) to maintain the myocardium relaxed and the
vasculature vasodilated.
Pressure-diameter relationship.
To examine the distensibility of the epicardial surface capillaries,
the surface of the isolated heart preparation was transilluminated and
viewed with an intravital microscope. The coronary arteries were
perfused with a colored microfil (inert, fluid silicone) to visualize
the epicardial surface microvessels. The silicone microfil is a
water-immiscible, long-chain polymeric material that does not
extravasate the vessel. Also, no catalyst was added to the microfil
polymer, so it remained a viscous fluid throughout the experiment. Once
the microfil was observed in the CS and in the heart chambers,
fast-hardening, catalyzed microfil (curing time 3-5 min) was
poured into the heart chambers. The cast chambers seal off any arterial
luminal or Thebesian drainage. Hence, the pressure was regulated in the
entire vasculature by clamping off the CS and establishing a static
pressure throughout the vasculature. Epicardial surface capillaries
were identified and their elastic pressure-diameter relationship
determined. Initially, the vessels were preconditioned with loading and
unloading pressure ramps over several cycles. Subsequently, images were
recorded in the pressure range from 0 to 60 mmHg in increments of 10 mmHg for the capillaries. Typical images of the coronary epicardial
surface vessels perfused with microfil are shown in Kassab et al. (10). Diameter measurements were made using a ×40 objective (NA = 0.77) with a resolution of 0.5 µm. An image grabber was used to grab the
frames of a given capillary vessel at different pressures. The images
were stored on a video floppy disk and analyzed at a later time. The
grabbed frames were input into our image processing system, where the
diameters of the vessels were digitized. The mean diameter, for each
pressure, was computed from several diameter measurements made over a
20- to 30-µm length of the capillary vessel.
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RESULTS |
Compliance of capillary vessels.
Figure
1A shows
the pressure-diameter relationship of the 12 capillary vessels measured
at the epicardial surface from 3 pig hearts (4 vessels from each
heart). The measurements were made at the base of the heart near the
LAD-LCX artery bifurcation. Figure 1B
shows that, in the pressure range (10-50 mmHg), the elastic
deformation can be described by the equation
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(1)
|
where
D is the diameter at a given
intravascular pressure P, D* is the
diameter corresponding to the physiological pressure P* (30 mmHg), and
is the compliance constant of the vessel. The mean ± SD of over the 12 measurements was found to be 1.7 ± 0.91 × 10
6 cm/mmHg by a linear
least-squares fit of the data with the intercept set at zero, as shown
in Fig. 1B, in the 10-50 mmHg
pressure range with a mean correlation coefficient of 0.92.
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Fig. 1.
A: pressure-diameter relationship for
12 capillary vessels measured at epicardial surface.
B: relationship between pressure (P)
minus physiological pressure (P*, 30 mmHg) and diameter
(D) minus diameter at physiological
pressure (D*) for the 12 capillary
vessels in A.
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Simulation analysis of coronary capillary blood flow based on
morphometric, rheological, and compliance data.
Kassab and Fung (9) designated all capillaries as blood vessels of
order number zero; we further designated the capillaries as those fed
directly by arterioles (C0a),
those drained directly into venules
(C0v), and those connected to
C0a and
C0v
(C00). The capillaries branch in
patterns identified as Y, T, H, or HP (hairpin) on the basis of their
geometric shape and anastomose through capillary cross-connections
(Ccc) (2, 9). The
Ccc vessels may connect adjacent
capillaries or capillaries originating from different arterioles. A
branching pattern of the coronary capillary bed was constructed on the
basis of these patterns (the frequency of Y, T, H, and HP patterns
simulated the measurements in Ref. 9), whereas the vascular dimensions
(diameters and lengths) were prescribed by the morphometric data of
Kassab and Fung (9). An idealized case is shown in Fig.
2.
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Fig. 2.
A schematic of an idealized capillary network. Asterisk indicates
capillaries that connect to capillary vessels above and below the
capillary plane.
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The vascular geometry and the flow condition justify the assumptions
that the Reynolds and Womersley numbers of the flow are very small
(<<1) and that the length-to-diameter ratio of each capillary
vessel is large. Under these assumptions, the classic Poiseuille's law
can be used to describe the local pressure-flow relationship in a
circular cylindrical capillary tube as
|
(2)
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where
Q is the volume-flow rate,
x is the axial coordinate measured
from the entrance section of each capillary, and
D and µ are the local diameter and
coefficient of viscosity, respectively (7).
At the capillary dimension, the particulate nature of the blood cells
becomes important and the blood properties become non-Newtonian. The
viscosity in Poiseuille's law (Eq. 2) is no longer constant and should be considered as
an apparent viscosity (µapp).
In general, the apparent viscosity is a function of vessel diameter,
hematocrit, and shear strain rate. The dependence of apparent viscosity
on vessel diameter and shear rate was given by Pries et al. (15). They
proposed a modified viscosity law based on a compilation of literature
data on relative blood viscosity in in vitro and in vivo tube flow
experimental measurements. In our range of capillary diameters
(2.5-9.0 µm), the apparent viscosity computed from Pries et
al.'s equation (Eq. 9 of Ref. 15,
with 
= 0.5 and W = 0.1) can be fitted by a cubic relation
(R2 = 1.00) as
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(3a)
|
where
a0,
b1,
c2, and
d3 are constants
with values equal to 9.24, 
1.03, 0.11, and 0.0045,
respectively. We initially computed the apparent viscosity on the basis
of the diameter of the capillary vessels in accordance with
Eq. 3a. We then solved the network
equations to obtain the shear strain rate in the various capillary
vessels. The influence of shear rate on viscosity appears to be small
in the shear range >50 s1
(14), whereas significant effects of shear rate on viscosity are
expected at substantially lower shear rates. Hence, for those vessels
with a shear rate <50 s1,
we considered a Casson-like empirical formula for the apparent viscosity that accounts for the shear rate dependence as
![&mgr;<SUB>app</SUB> = [<IT>k</IT><SUB>1</SUB> + <IT>k</IT><SUB>2</SUB>(<IT>U</IT>/<IT>D</IT>)<SUP>−1/2</SUP>]<SUP>2</SUP>](/content/vol277/issue6/fulltext/H2158/img004.gif)
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(3b)
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where
U is the mean velocity of blood and
D is the diameter of the vessel (16).
k1 and
k2 are constants
whose values depend on the vessel diameter, hematocrit, and the shear
rate. We chose k1 = 1.81 cP1/2 and
k2 = 1.04 (cP/s)1/2, corresponding to a tube
diameter of 5.34 µm and a tube hematocrit of 39% (13). Hence, we
computed the apparent viscosity using Eq. 3b to obtain an updated apparent viscosity. This
process was iterated until the viscosity converged for those vessels
with a shear rate <50 s1.
The hydrodynamic law (Eq. 2) can be
combined with the elasticity (Eq. 1)
and rheology of blood (Eq. 3)
relationships to
yield
![+ <IT>D</IT>*]<SUP>3</SUP>}[&agr;(P − P*) <IT>D</IT>*]<SUP>−4</SUP> <IT>Q</IT>](/content/vol277/issue6/fulltext/H2158/img007.gif)
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(4a)
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for vessels with a shear rate >50
s1
and
![+ <IT>D</IT>*]<SUP>3</SUP>/4<IT>Q</IT>}<SUP>1/2</SUP>)<SUP>2</SUP>][&agr;(P − P*) + <IT>D</IT>*]<SUP>−4</SUP><IT>Q</IT>](/content/vol277/issue6/fulltext/H2158/img009.gif)
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(4b)
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for vessels with a shear rate <50
s1. These are the governing
equations for non-Newtonian viscous blood flow in an elastic vessel at
steady-state conditions that can be integrated for specific boundary conditions.
Effect of Ccc on the hemodynamics of the
capillary network.
To examine the effect of Ccc on
the pressure and flow distributions in the capillary bed, we assumed
that the compliance was zero ( = 0) and considered a special case of
Eq. 4. The problem was formulated as
follows. In a network of capillaries (see example in Fig. 2), the nodes
were numbered as 1, 2, ..., M. In a
vessel connecting two nodes represented as
i and
j, the flow from node i to node
j was denoted as
Qij, whereas the
differential of pressures at nodes i
and j was denoted as
Pij. The pressure-flow relationship was then written as
|
(5)
|
where
and
is
a quantity called the vascular conductance of the vessel
ij, which is a function of
Dij,
Lij, and
Uij, the
diameter, length, and velocity, respectively, between nodes
i and
j. The expression for
µapp ij is given by
Eq. 3.
Figure 2 shows that there are three vessels emanating from the
jth node anywhere in the network. By
denoting these three vessels as ij
with i = 1, 2, and 3, and by
conservation of mass, we obtained
|
(6)
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in
which the volumetric flow into a node is considered positive and that
out of a node is negative. From Eqs. 5 and 6 we obtained a set of nonlinear
algebraic equations in pressure for M
nodes in the network (M = 136 nodes
for the simulated network in Fig. 2), namely
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(7)
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The
set of equations represented by Eq. 7
can be reduced to a set of simultaneous algebraic equations that are
solved iteratively for the nodal pressures once the conductances are
evaluated from the geometry and suitable boundary conditions are
specified. The boundary conditions are simulated by a random uniform
distribution with the
range
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(8a)
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These boundary conditions represent a mean pressure drop of 10 mmHg,
which is in agreement with the micropressure measurements of Klassen et
al. (12) on the epicardial surface of the dog left ventricle. The
arterioles and venules were chosen randomly, always maintaining a
ratio of four arterioles to seven venules, consistent with our previous
morphometric measurements (10, 11). In matrix form, this set of
equations is
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(8b)
|
where
G is the M × M matrix of conductances, P
is a 1 × M column
vector of the unknown nodal pressures, and G'P' is the column vector of the
boundary pressures times the conductances of their attached vessels.
Equations 8a and 8b are solved by the Gaussian
elimination method. The capillaries indicated by asterisks in Fig. 2
connect to capillary vessels above and below the capillary plane. We
initially assigned these vessels an arbitrary pressure value and then
updated their values according to the mean values of pressures in
nearby capillaries. We found that these pressures converge within
several runs of the model.
Because we have the raw data for the diameters and lengths of the four
orders of capillaries (C0a,
C00,
C0v, and
Ccc), we input them directly to
the computation to avoid an intermediate step of constructing a random
number table (satisfying the means and standard deviations of Table 2 in Ref. 9). We numbered the entries in our data file, for each order,
in positions 1, 2, 3, ..., N, where
N is the total number of measurements
for that order. If the total number of elements in the computational model exceeded N, then the program
used the data sets repeatedly in sequence until the required number of
elements was obtained. This algorithm is simple and as accurate as that
using the raw data. To account for the randomness by which the measured
data are arranged in sequence, and to use the whole distributions, we
ran the program repeatedly and varied the starting point in sequence
for each order.
The solution satisfying Eq. 8 is
obtained in the form of a column vector of the nodal pressures
throughout the arterial network. The pressure drops as well as the
corresponding flows were computed. Figure
3,
A-D, shows the log-transformed,
median-normalized flow distribution in capillaries of orders
C0a,
C00,
C0v, and
Ccc, respectively. These
distributions correspond to data obtained over 100 runs of the model.
We examined the effect of varying the number of runs of the
model and found that the median flows obtained from 100 runs were
within 1.5% of the median flows obtained from 1,000 runs. Hence,
we opted to run the numerical program 100 times to obtain all the
results shown here. Figures
4A and 5A show
the effect of Ccc on the
median flows and pressure drops per capillary segment,
respectively, for all orders of capillaries. Figures
4B and
5B show the effect of
Ccc on the coefficient of variance
(CV; SD/mean, %) of the flows and pressure drops, respectively. The
effect of Ccc on the total
flow into the capillary network (QTin) is
shown in Fig. 6.
QTin is
normalized with respect to the total flow into the network in the
presence of all Ccc.
Finally, the effect of cross-connections on the median velocity of
various capillary orders is shown in Fig. 7.
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Fig. 3.
Log-transformed, median-normalized flow distributions in capillaries
fed directly by arterioles (C0a;
A), capillaries connected to
C0a and
C0v
(C00;
B), capillaries drained directly
into venules (C0v;
C) and capillary cross-connections
(Ccc;
D).
Q0a,
Q00,
Q0v, and
Qcc, flow in
C0a,
C00,
C0v, and
Ccc, respectively;
Qmedian, median
flow.
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Fig. 4.
A: relationship between median flow
per capillary segment and capillary order number, with and without
Ccc.
B: relationship between coefficient of
variance (CV; SD/mean, %) of blood flow and capillary order number,
with and without Ccc.
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Fig. 5.
A: relationship between median
pressure drop (P) per capillary segment and capillary order number,
with and without Ccc.
B: relationship between CV of pressure
drop and capillary order number, with and without
Ccc.
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Fig. 6.
Relationship between median total flow into capillary network
(QTin) and
number of Ccc. Flow is normalized
with respect to
QTin with
presence of all Ccc.
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Fig. 7.
Relationship between median flow velocity per capillary segment and
capillary order number, with and without
Ccc.
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Effect of vessel compliance on the hemodynamics of the capillary
network.
With the distensibility of the blood vessels known, the mechanics of
the blood vessel were coupled to the mechanics of blood flow to yield a
pressure-flow relationship for each vessel segment. This can be
demonstrated as follows. In a stationary, nonpermeable tube,
Q is a constant throughout the length
of the tube, whereas the tube diameter and the apparent coefficient of
viscosity are functions of x because
of the elastic deformation. The elastic deformation of the coronary
capillaries can be described by Eq. 1,
which can be differentiated to yield
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(9)
|
By
substituting Eq. 9 into
Eq. 2 and assuming
µapp to be a constant for each
individual order (5.7 cP for the order
C0a, 5.0 cP for
C00, 5.5 cP for
C0v, and 5.1 cP for
Ccc as the computed medians from
our model), we obtained
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(10)
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Because
the right-hand term is a constant independent of
x, we obtained the integrated result
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(11)
|
The
integration constant is determined by the boundary condition stating
that when x = 0, D(x) = D(0). Solving Eq. 2 for P(x) with
D = D(x)
given by Eq. 11 yielded a nonlinear
pressure-flow relationship for each capillary segment that takes the
form (see APPENDIX)
![&Dgr;<IT>P</IT><SUB>n</SUB> = [−<IT>D<SUB>n</SUB> + </IT>(<IT>D</IT><SUP>2</SUP><SUB><IT>n</IT></SUB> + 8&agr;<SUB><IT>n</IT></SUB>&Dgr;P<SUP>p</SUP><SUB><IT>n</IT></SUB>/<IT>D</IT><SUB><IT>n</IT></SUB>)<SUP>1/2</SUP>]/4&agr;<SUB><IT>n</IT></SUB>](/content/vol277/issue6/fulltext/H2158/img021.gif)
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(12)
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where
Pp is Poiseuille's pressure
drop as given by Eq. 2 and
n is the order number of the
capillary. Equation 12 can be used instead of Poiseuille's law (Eq. 2)
in the various segments to synthesize the pressure and flow
distributions in the capillary network.
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DISCUSSION |
Effect of Ccc on the hemodynamics of the
capillary network.
One of the main goals of the present study was to examine the effect of
Ccc on the pressure and flow
distributions in the capillary network. We investigated this
aspect by constructing a typical capillary network based on measured
branching pattern and vascular geometry (diameters and lengths) and
applied the laws of physics and the appropriate boundary conditions to
analyze the pressure and flow distributions with and without
Ccc.
Ccc were removed from the network
by setting their conductances equal to zero. Our results show that the
median flow and pressure drop increased in the presence of
Ccc, as shown in Figs.
4A and
5A. The total flow into the capillary
network also increased in the presence of
Ccc, as shown in Fig. 6. The
increase in total flow occurred for the same pressure drop across the
capillary network. Hence, the resistance to flow was decreased by the
presence of Ccc. Furthermore,
Ccc also homogenized the flow and
pressure distributions. Figures 4B and
5B show that the relative dispersion
(or CV) of flow and pressure drop was reduced in the presence of
Ccc. Although these dispersions
were reduced in the presence of
Ccc, the dispersions at the venous
capillaries were still greater than those at the arterial capillaries,
that is, the capillary outlet flow and pressure are more heterogeneous
than the inlet flow and pressure with or without
Ccc. This result may stem from the
fact that the number of venous capillaries is greater than the number
of arterial capillaries (10). Hence, an increase in the number of
possible pathways may lead to an increase in the variability of the
hemodynamic parameters.
The median blood velocity shown in Fig. 7 was computed from the flows
and cross-sectional areas of the capillary vessels. The median
velocities range from ~300 µm/s in capillaries of order C00 to 1,400 µm/s in capillaries
of order C0a. Tillmanns et al. (24) found average diastolic capillary red cell velocities of 909 and
1,428 µm/s on the epicardial surface of turtle and dog left
ventricles, respectively. Direct comparison cannot be made, however,
because the order of capillary vessels is unspecified in the
measurements of Tillmanns et al. Wieringa et al. (25) found a mean
value of 1,428 µm/s in their capillary network simulation.
Once the hemodynamics of the network have been determined,
Poiseuille's hypothesis can be reexamined. Poiseuille's law applies only when the flow has low Reynolds and Womersley numbers. The former
is defined by the formula UD/
,
where U is the mean velocity of flow,
D is the blood vessel lumen diameter,
and is the kinematic viscosity of blood. The latter is defined as
(D/2)(
/)1/2,
where (in radians/s) is the circular frequency of pulsatile flow
and is computed for a heart rate of 100 cycles/min. We have found that
the median Reynolds and Womersley numbers are <0.002 and <0.006,
respectively, for all orders of capillary vessels and hence justify the
steady-state assumption.
Effect of vessel compliance on the hemodynamics of the capillary
network.
The pressure drop in Eq. 12 can be
plotted as a function of the compliance for the various orders of
capillary vessels as shown in Fig. 8. When
the compliance is zero (rigid vessel), the pressure drop corresponds to
that given by Poiseuille's equation. However, when the compliance is
nonzero, the pressure drop is smaller than that given by Poiseuille's
equation and varies for the different orders of capillary vessels.
For our measured values of compliance (7.2 × 107-3.5 × 106 cm/mmHg), the
pressure drop predicted by Eq. 12 is
nearly equal to Poiseuille's pressure drop as shown in Fig. 8. Hence,
we concluded that the effect of epicardial surface capillary compliance
on the hemodynamics of the capillary network is negligible in the physiological range of pressures. However, these are only the epicardial surface vessels, and it is unknown whether the
intramyocardial vessels are equally stiff.
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Fig. 8.
Relationship between pressure drop per vessel segment (P) and
logarithm of compliance constant for various orders of capillary
vessels. Pressure drop is normalized with respect to Poiseuille's
pressure drop (Pp).
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Comparison with other capillary distensibility data.
The distensibility of capillary blood vessels was previously determined
using several methods: airtight pressure chamber for the bat wing (4),
microannulation and injection of oil drops (23), microocclusion within
the limits of a pulse pressure range (18), and elastomer perfusion
under known hydrostatic pressures (20). Table
1 summarizes the data for the
distensibility of the capillary vessels in various organs and species.
It can be seen that the epicardial coronary capillaries are among the
least distensible vessels in various organs. Unlike coronary and
systemic capillaries, pulmonary capillaries are very distensible
because they receive little support from the surrounding tissues (6).
Model limitations.
Although our analysis is based on measured morphometric and elasticity
data for coronary capillary blood vessels, there were still a number of
assumptions made. For example, the distensibility data were obtained
from the epicardial surface only because intramural vessels could not
be readily visualized with the present technique. Furthermore, the
topology of the capillary branching at the epicardial surface is
different from that of intramural layers, where the morphometric data
were measured. Hence, we combined morphometric data from intramural
capillaries with distensibility data from epicardial surface
capillaries. Furthermore, we did not measure the distensibility of
various orders of capillaries. The findings in the bat wing suggest the
existence of a longitudinal gradient of distensibility in the capillary
compartment (3). We also restricted our hemodynamic analysis to the
diastolic state of the myocardium. Hence, we did not consider the
muscle-vessel interaction that is very important in systole. Muscle
tension and contraction may have a large effect on capillary diameters
and may change the capillary transmural pressure so that the
pressure-diameter relationship shown in Fig. 1 is no longer applicable.
Furthermore, we did not consider the time-varying vasoactive components
of the arterioles that would make the boundary conditions time
dependent. These important effects of muscle-vessel interactions must
be investigated in the future. The present approach is only a logical first step.
Fibich et al. (5) formulated a continuum mechanics model of the
coronary capillary vessel throughout the cardiac cycle. They modeled
the capillary network as a single long tube with Pa and
Pv as the inlet arteriole and
outlet venous pressures, respectively. Their model considers the effect
of permeability, distensibility, and tissue stress on the basis of a
number of hypotheses. Their results show that under physiological
conditions ultrafiltration is of minor importance, and its effect was
neglected in the present study. Their analysis also predicts regional
differences in capillary flow. It would be interesting to combine their
time- and space-dependent flow equation with our network model using measured capillary diameters, lengths, and compliance. This would yield
a system of nonlinear partial differential equations for pressure whose
solution would vary transmurally and throughout the cardiac cycle.
In conclusion, our network analysis has shown that the capillary
cross-connections tend to homogenize the pressure and flow distributions and reduce the flow resistance of the capillary network.
Hence, the analysis clarifies an important hemodynamic function of the
capillary cross-connections, showing that the cross-connections play an
important role in the structure-function relationship. Furthermore, our
measured compliance of the epicardial surface coronary capillary
vessels is relatively small in the physiological pressure range. Our
analysis has shown that the effect of the measured epicardial surface
vessel compliance on the hemodynamics of the coronary capillary is
negligible in the diastolic state of the heart. The compliance of
intramyocardial blood vessels remains unknown, however, and the
interaction of the muscle contraction and blood vessel elasticity in
systole remains to be studied.
| |
APPENDIX |
The mathematical steps between Eqs. 11 and 12 are given below. We let
x = 0 be the entry section and
x = L
be the exit section of a capillary. Letting
x = L
in Eq. 11 yielded
|
(13)
|
We
next sought an approximate expression of Eq. 13 for which
D(L) D(0) is small. By letting
D(L) = D(0) + 
, expanding the left-hand
side of Eq. 13 in a power series of
, and retaining only terms up to
2, we obtained the
approximation
![= (−128&mgr;<SUB>app</SUB>&agr;<IT>LQ</IT>)<IT>/</IT>[<IT>&pgr;D</IT><SUP>4</SUP>(0)]](/content/vol277/issue6/fulltext/H2158/img024.gif)
|
(14)
|
Using Eq. 1 first at
x = L
and then at x = 0 and subtracting, we
obtained
![<IT>D</IT>(<IT>L</IT>) − <IT>D</IT>(0) = &agr;[P(<IT>L</IT>) − P(0)]](/content/vol277/issue6/fulltext/H2158/img025.gif)
|
(15)
|
By
combining Eqs. 14 and 15 and writing
D0 for
D(0), we obtained
|
(16)
|
where
P = P(L) P(0). The
solution to Eq. 16 is
Eq. 12.
| |
ACKNOWLEDGEMENTS |
We thank Chris Feezor and Edith Pallencaoe for excellent technical assistance.
| |
FOOTNOTES |
This research was supported by National Heart, Lung, and Blood
Institute Grants HL-43026 and 5-R29-HL-55554. G. S. Kassab is a
recipient of the National Institutes of Health First Independent Research Support and Transition Award, and K. N. Le is a recipient of
the McNair scholarship.
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: G. S. Kassab,
Dept. of Bioengineering, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA 92093-0412 (E-mail:
kassab{at}bioeng.ucsd.edu).
Received 17 February 1999; accepted in final form 30 June 1999.
| |
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Am J Physiol Heart Circ Physiol 277(6):H2158-H2166
0002-9513/99 $5.00
Copyright © 1999 the American Physiological Society
TOP
ABSTRACT
INTRODUCTION
![dP/d<IT>x</IT> = 128/&pgr;{<IT>a</IT><SUB>0</SUB> + <IT>b</IT><SUB>1</SUB>[&agr;(P − P*) + <IT>D</IT>*]](/content/vol277/issue6/fulltext/H2158/img005.gif)
![+ <IT>c</IT><SUB>2</SUB>[&agr;(P − P*) + <IT>D</IT>*]<SUP>2</SUP> + <IT>d</IT><SUB>3</SUB>[&agr;(P − P*)](/content/vol277/issue6/fulltext/H2158/img006.gif)
![[<IT>D</IT>(<IT>L</IT>) − <IT>D</IT>(0)]{1 + 2[<IT>D</IT>(<IT>L</IT>) − <IT>D</IT>(0)]/<IT>D</IT>(0)}](/content/vol277/issue6/fulltext/H2158/img023.gif)
