Vol. 273, Issue 6, H2832-H2842, December 1997
Longitudinal position matrix of the pig coronary vasculature
and its hemodynamic implications
Ghassan S.
Kassab,
Edith
Pallencaoe,
Amy
Schatz, and
Yuan-Cheng B.
Fung
Department of Bioengineering, University of California, San
Diego, La Jolla, California 92093-0412
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ABSTRACT |
Hemodynamic analysis of coronary blood flow
must be based on a statistically valid geometric model of the coronary
vasculature. We have previously developed a diameter-defined Strahler
model for the arterial and venous trees and a network model for the capillaries. A full set of data describing the geometric properties of
the porcine coronary vasculature was given. The order number, diameter,
length, connectivity matrix
[m,n] (CM),
and parallel-series features were measured for all orders of vessels of
the right coronary artery (RCA), left anterior descending artery (LAD), left circumflex artery (LCX), and coronary venous system. The purpose
of the present study is to present another feature of the branching
pattern of the coronary vasculature: the longitudinal position matrix
[m,n] (LPM), whose
component in row m and column n is the fractional longitudinal
position of the branch point on vessels of order
n at which vessels of order
m branch off
(m 
n). The LPM of the pig RCA, LAD and
LCX arterial trees, as well as the coronary sinusal and thebesian
venous trees, are presented. The hemodynamic implications of the LPM
are illustrated by comparing two kinds of circuits: one, the CM + LPM
model, simulates the mean data on the morphology (diameters, lengths,
and numbers), CM, and LPM of vessels, whereas the other, the CM model,
simulates the mean data on the morphology and CM without considering
the LPM. We found that the LPM affects the hemodynamics of coronary blood flow especially with regard to the nonuniformity or dispersion of
flow distribution.
heart; connectivity matrix; blood flow; flow dispersion; pressure
distribution
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INTRODUCTION |
THE LARGEST CORONARY ARTERIES and veins are small in
number, and their geometric characteristics can be recognized
individually. Flow in these large vessels can be analyzed in
conventional ways, e.g., by methods of computational fluid mechanics.
Small coronary arteries and veins are very large in number and are
organized topologically like trees except at the epicardial surface,
where arcades are found connecting the sinusal veins, and at the
endocardial surface, where arcades are found connecting thebesian
veins. Analysis of blood flow in the smaller treelike blood vessels is
best done on mathematical models of the vasculature substantiated by
statistical data. In the last three decades, the most popular
mathematical models have been the bifurcation model of Weibel (29) and
the geographical "rivulets-rivers" model of Strahler (27). The
latter has been used in the analysis of botanical trees, pulmonary
airways, and neural networks (for reviews see Refs. 12 and 30). Fenton and Zweifach (8) used Strahler's model to study the human bulbar conjunctiva and rabbit omentum, whereas Yen et al. (31, 32) used it to
describe the entire pulmonary arterial and venous trees in the cat.
Strahler's system has also been used to study the microcirculation of
the skeletal muscle (3, 5-7, 21), rat mesenteric microvessels
(23), rat pial arterial system (13), human retinal microvessels (24,
25), and pig coronary arteries (28).
Recently, three innovations in the mathematical modeling of trees have
been introduced (15, 20): 1) a
modified Strahler system that employs a rule for assigning the order
numbers of the vessels on the basis of diameter ranges,
2) a measurement of the fraction of
vessel segments connected in series in terms of a segments-to-elements
ratio (S/E), defined as the ratio of the total number of vessel
segments to the total number of vessel elements, and
3) a connectivity matrix
[m,n] (CM), whose
component in the mth row and
nth column is the ratio of the total
number of elements of order m that
spring directly from parent elements of order
n divided by the total number of
elements of order n, to describe the
asymmetric branching. It should be noted that the second innovation was
simultaneously proposed by Van Bavel and Spaan (28).
With these innovations, sets of complete morphometric data have been
obtained for the pig coronary arteries in normal and right ventricular
hypertrophy (18, 20), the rat pulmonary arteries (14), the pig coronary
veins (19), and the dog pulmonary veins (10). The purpose of the
present study is to introduce a fourth innovation: the longitudinal
position matrix [m,n]
(LPM), whose component in row m and
column n is the fractional
longitudinal position along the length of parent elements of order
n of the branch point at which
elements of order m spring off. The
LPMs of the pig right coronary artery (RCA), left anterior descending artery (LAD), and left circumflex artery (LCX) arterial trees, as well
as that of the sinusal and thebesian venous trees, are presented.
Recently, we have used the morphometric data on the branching pattern
and vascular geometry of the coronary arterial trees to construct a
tree circuit for hemodynamic analysis of coronary arterial blood flow
in the pig (16). We used the mathematical model to analyze the
distribution of pressure, flow, and volume. To date, the only other
mathematical model of coronary vasculature based on measured
morphometric data has been that of Van Bavel and Spaan (28). The
quantitative basis for their stochastic tree model was provided by
measuring the relation between the diameter and length of vessel
segments, the relation between diameters of parent and daughter
segments, and the relation between the area expansion ratio and
symmetry of vascular nodes and the diameter of the mother segment. On
the experimental side, however, there is a tremendous amount of data on
the pressure-flow distributions in the coronary circulation (see Refs.
11 and 26 for reviews).
Our model in Ref. 16 did not take into account the longitudinal
positions of elements of order n 
1, n 2, n 3, ... that arise directly
from elements of order n. A realistic
model of a vascular circuit set up for numerical analysis must have a
topology that is in agreement with the real measured anatomic connectivity and longitudinal position matrices. Hence, in the following, the hemodynamic implications of the LPM are illustrated by
comparing the blood flow in a fifth-order arteriole, whose mean
diameter is ~70 µm, via two kinds of circuits. One, called the CM
model, simulates the mean morphometric data (on diameters, lengths, and
numbers) and the CM of the arteriole. The other, called the CM + LPM
model, simulates the mean data on the morphology (on diameters,
lengths, and numbers), CM, and LPM of the arteriole.
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METHODS |
The methods of animal and specimen preparation, histological and cast
measurements, and morphometric analysis of the coronary arteries and
veins have been described in detail in Refs. 19 and 20. Briefly, a
KCl-arrested, adenosine-dilated pig heart was perfused with freshly
catalyzed silicone elastomer through its major coronary arteries (RCA,
LAD, and LCX). The arterial perfusion pressure was maintained at 80 mmHg until the elastomer hardened. The heart, after being refrigerated
in saline for several days to increase the strength of the polymer, was
corroded with a 30% KOH solution for several days. The RCA, LAD, and
LCX and the sinusal and thebesian veins were dissected and viewed with a stereodissection microscope and displayed on a video monitor through
a television camera as described in Ref. 20. The coronary arteries and
veins were reconstructed completely, with the lumen diameter and length
of each vessel segment measured. The blood vessels were classified into
sequential sets of successive order numbers according to the
diameter-defined Strahler system with the capillary blood vessels
defined as order 0, the smallest arteries as orders 1, 2, ..., (Refs.
17 and 20), and the smallest veins as orders 1, 2, ..., (Ref. 19). The data on the diameters and lengths of vessels of orders
1-4 were obtained from histological sections as previously
described (19, 20). The data on diameters of vessels of orders 3 and 4 were used to assign order numbers to the segments of the cast whose
diameters fell within the diameter ranges of orders 3 and 4. Diameters
of vessels of orders 4, 5, and higher were then obtained according to
the diameter-defined Strahler system (20). By this method, we have
obtained a complete set of morphometric data of the RCA, LAD, and LCX
arterial trees (20) and the coronary sinusal and thebesian venous trees
(19). The relationships between diameters, lengths, number of elements, CMs, and order numbers for the coronary arteries and veins are presented in Refs. 19 and 20, respectively.
From the same specimens that yielded the histological and cast data
named above, we determined the LPM as follows. In Fig. 1, a vessel element of order
n and its branches are shown. A local dimensionless coordinate x is
introduced along the vessel n, with x = 0 at the inlet and
x = 1 at the outlet. At a point
i on the vessel
n, a blood vessel element whose
diameter qualifies it to be of the order number
m branches out. The coordinate
x of the point
i is equal to
xmn, as
indicated in Fig. 1, and is called the fractional longitudinal position
of a vessel element of order m that
springs from a vessel of order n.
Similarly, the vessel of order m has a
branch that is of the order number k.
The branching point j has a
dimensionless coordinate
xkm that is the
fractional longitudinal position of a vessel of order
k springing from the vessel of order
m. The matrix of the numbers
xmn is the LPM.
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Fig. 1.
Schematic diagram of a vessel element of order
n and its branches. Direction of blood
flow is from inlet at left to outlet
on right. A local dimensionless
curvilinear coordinate x is shown
along the vessel elements with x = 0 at inlet and x = 1 at outlet. See
METHODS for details. fij, Rate of
volume flow from i to j.
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For each element of order n, we
introduced a nondimensional local longitudinal coordinate
x, with
x = 0 at the inlet of the vessel and
x = 1 at the exit end. The coordinates
of the branching points on the vessels of order
n giving rise to branches are
determined. The statistical data of these coordinates can then be
assembled into a column number n. A
listing of the results for all columns and rows yields the
LPM[m,n], whose component
in row m and column n is the fractional longitudinal
position of the branching point of the elements of order
m on the elements of order
n.
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RESULTS |
Tables
1-3 show
the LPM for the RCA, LAD, and LCX arterial trees, respectively. Tables
4 and 5 show
the LPM for the sinusal and thebesian venous trees, respectively. The
data shown are averaged over many elements and represent means ± SE. The number of measurements are also shown in Tables 1-5 for
the RCA, LAD, LCX, sinusal veins, and thebesian veins, respectively.
The probability distribution functions for several orders of the RCA
are shown in Fig. 2. It can be seen that
the skewness and the kurtosis of the distributions are a function of
the branching order. The probability distributions for the respective
orders of veins are very similar. The applications of the data are
shown below and discussed later in the
DISCUSSION.
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Fig. 2.
Probability frequency functions (histograms) of longitudinal position
matrix (LPM) components LPM(5,6)
(A), LPM(5,7)
(B), and LPM(5,8) (C) of
the right coronary artery.
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Applications to hemodynamics.
In the following, let us consider the blood flow in a left ventricular
(LV) fifth-order arteriole via two kinds of circuits: the CM model,
which satisfies the mean data on the morphology (diameters, lengths,
and numbers) and connectivity of the arteriole (20), and the CM + LPM
model, which satisfies the mean data on the morphology, connectivity,
and longitudinal position of the arteriole. Figure
3A shows a
schematic of the CM model that is consistent with its mean CM given in
Ref. 20. Figure 3B shows a schematic
of the CM + LPM model that is consistent with both the mean CM and the
LPM. Each of the branches arising from the trunk of the fifth-order
arteriole gives rise to further branches, and so on, down to the
arterial capillaries (order 0). Each element shown in Fig. 3,
A and
B, may represent several elements in
parallel, as determined by the total number of elements in each order.
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Fig. 3.
A: schematic diagram of a 5th-order
arteriole demonstrating all possible pathways to the capillaries as
prescribed by connectivity matrix (CM) model.
B: schematic diagram of a 5th-order
arteriole demonstrating all possible pathways to capillaries as
prescribed by the CM + LPM model.
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Parallel elements, by definition, have the same conductances and
boundary conditions. The total number of parallel elements of a given
order is distributed among the various pathways. To compute the mean
number of parallel elements through the various pathways, let us use
the symbol
Nijk...rst to
denote the number of parallel elements in pathway
ijk...rst. Let
C(t,s)
be a component of the
CM[m,n] given in Ref. 20,
and
NT(t)
be the mean total number of parallel elements of order
t. The
NT(t)
arising from a single order 5 LV arteriole are 2.82, 6.96, 22.0, 57.7, and 199 for orders 4, 3, 2, 1, and 0, respectively (20). Hence, referring to Fig. 3, A and
B, and using the
CM[m,n] of the LAD arterial tree of Ref. 20, we find
and
and
Analogously, vessels of orders 1 and 0 have 7 and 13 pathways,
respectively. In
general
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(1)
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Next, we shall formulate the hemodynamics of the blood flow in the
vessel of order m between nodes
i and
j. Assume that the vessel is so small
that both the Reynolds number and the Womersley number are much smaller
than 1 (with order number m
sufficiently small, e.g., m 5) so
that the flow is quasi-steady and the entrance and exit effect is
negligible compared with the resistance in the whole vessel; then
Poiseuille's formula applies. Let the pressure at node
j be denoted by
Pj. A vessel element joining nodes i and
j is denoted by vessel
ij, whose diameter is
Dij and length
is Lij. If there
were N vessel elements connecting
nodes i and
j, then the number of these elements
would be denoted by Nij. The rate of
volume flow in the vessel connecting the two nodes
i and
j is then given in terms of the
pressure differential (Pi Pj), vascular geometry
(Dij and
Lij), and
blood viscosity (µij), by
(see Ref.
9)
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(2)
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The
total flow between these two nodes,
Fij, is the sum of the flow in
the N parallel elements
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(3)
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Equations
2 and 3 can be
combined to yield
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(4)
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where
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(5)
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(6)
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The
length Lij in
Eq. 6 represents the total length of
an element joining nodes i and
j for the CM model of Fig.
3A. However, for the CM + LPM model of
Fig. 3B, the length
Lij in
Eq. 6 represents a fraction of the
total element length as dictated by the LPM. Figure 3,
A and
B, shows that there may be
mj vessels
converging at the jth node.
By the law of conservation of mass we must have
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(7)
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where
the volumetric flow into a node is considered positive and that out of
a node is negative for any branch. From Eqs. 4 and 7 we obtain a
set of linear algebraic equations for pressure for all the
nodes in the network, M in number,
namely
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(8)
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The
set of equations represented by the summation in Eq. 8 is reduced to a set of simultaneous
linear algebraic equations with the nodal pressures as unknowns. In
matrix form, the set of M equations
represented by Eq. 8
is
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(9)
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where
Geq is the
M × M matrix of equivalent conductances
(M is 15 for the CM model and 31 for
the CM + LPM model), P is a 1 × M column vector of the unknown nodal
pressures, and
G'eqP'
is the column vector of the boundary pressures times the conductances
of their attached vessels. The pressure at the inlet of the fifth-order
arteriole is taken as 46 mmHg as extrapolated from the epicardial
pressure measurements of Chilian et al. (4) in the cat, whereas the
pressure at the first bifurcation of the capillary network is set at 26 mmHg. The conductances can be computed from the mean diameter, length, number, and connectivity data of Ref. 20 with the assumption that the
coefficient of viscosity is 4 cP (0.004 Pa · s) in a vessel of order 5 and decreases linearly with order number to 1.5 cP in
the capillaries. We solved Eq. 9 by
numerical algorithms group subroutine F04AEF-NAG (Fortran library
routine).
To make this development more accessible to the reader, we now specify
the units of the hemodynamic variables used in the analysis. The
diameters and lengths are read from the input files in units of
centimeters, whereas the viscosity is read in units of poise, which
yields conductances in units of milliliters per second per Pascal.
However, because our pressure boundary conditions are in units of
millimeters Hg, we must multiply the conductances by a conversion
factor of 133.3 Pa/mmHg to yield pressures in units of millimeters Hg
and flows in units of milliliters per second.
The solution to Eq. 9 for the two
model circuits was obtained in the form of a column vector of nodal
pressures throughout the two models shown in Fig. 3. To study the
effect of variations in the LPM on the hemodynamics, we examined 100 runs of the model of Fig. 3B,
corresponding to different values of the LPM. Because we have the raw
data on the LPM, we used the data directly as input to the computation
to avoid an intermediate step of constructing a random number table
satisfying the means ± SE of Table 2 for an LV arteriole. Figures
4-6
show the mean values of longitudinal pressure distribution, the
pressure drop per vessel element, and the coronary blood flow per
vessel element in the CM and CM + LPM models, respectively.
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Fig. 4.
Relationship between mean blood pressure at outlet of a blood vessel
element and order number of blood vessel for arteriolar branches of CM
model and CM + LPM model of a left ventricular (LV) 5th-order
arteriole. Data for CM + LPM model are averaged (±SD) over 100 runs
of model, corresponding to different cases of LPM data.
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Fig. 5.
Relationship between mean pressure drop per vessel element and vessel
order number for arteriolar branches of CM model and CM + LPM
model of LV 5th-order arteriole. Data for CM + LPM model are averaged
(±SD) over 100 runs of the model, corresponding to different cases
of LPM data.
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Fig. 6.
Relationship between mean blood flow per vessel element at inlet of
element and order number for arterial branches of CM model and CM + LPM
model of an LV 5th-order arteriole. Data for the CM + LPM
model are averaged (±SD) over 100 runs of the model, corresponding
to different cases of LPM data.
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On recalling that the validity of Poiseuille's formula depends on the
smallness of the Reynolds and Womersley numbers, we computed the
Reynolds number UD/
, where
U is the mean velocity of flow,
D is the blood vessel lumen diameter,
and is the kinematic viscosity of blood, and the Womersley number
(D/2)(
/)1/2,
where is the radian frequency of pulsatile flow (taken to be 110 cycles/min for the pig). Figure 7 shows the
relation between the Reynolds and Womersley numbers and the order
number of coronary blood vessels, respectively. They are less than one
for all five orders of vessels. To verify that the entrance effect is
negligible at such Reynolds numbers, we consider the inlet length,
which is defined as a distance through which the velocity has
redistributed itself approximately into a parabolic profile for an
entry flow into a circular cylindrical tube with a uniform axial
velocity at the entry section. It has been shown that when the Reynolds number tends toward zero, the inlet length tends toward a constant, 0.65D (22). Hence, the inlet length is
found to be negligible for the models shown in Fig. 3 because the
various lengths are much greater than their corresponding diameters.
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Fig. 7.
Relationship between Reynolds and Womersley numbers and order number of
vessel elements for arterial branches of CM model and CM + LPM model of
LV 5th-order arteriole.
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The computed heterogeneity of blood flow can be expressed in terms of
its relative dispersion (RD = SD/mean) or coefficient of variance.
Figure 8 shows the relation of the RDs of
flow and the order number for the two models, respectively.
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Fig. 8.
Relationship between %flow dispersion [(SD/mean) × 100] at inlet of element and order number of vessel elements for
CM model and CM + LPM model of LV 5th-order arteriole. Data for CM + LPM model are averaged (±SD) over 100 runs of model, corresponding
to different cases of LPM data.
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DISCUSSION |
We have introduced an additional innovation in morphometry: the LPM,
which characterizes the longitudinal position of vessels of order
m along the length of vessels of order
n from which they arise. To explain
the significance of the LPM, we recall that the blood flow
fij in the vessel of order
m between nodes i and
j is given by Eq. 2. The length
Lij in
Eq. 2 depends on the element length of
vessels of orders m and
n and the longitudinal fractions
xmn and
xkm. Hence,
fij can be computed only if the
vessel's origin in order n and exit
into a vessel of order k are known.
Thus the connections between vessels of orders
m, n,
and k are important from the point of
view of blood vessel circuit modeling. The connectivity and
longitudinal position matrices offer such connection data.
To demonstrate the hemodynamic implications of the LPM, an analysis is
done for two specific circuits shown in Fig. 3: the CM model, which
satisfies the measured mean CM and mean diameter and length data, and
the CM + LPM model, which satisfies the additional data of the mean
LPM. The CM + LPM circuit is a bifurcating tree model and hence
approximately satisfies the statistics of the S/E reported in Kassab et
al. (20). The CM + LPM model shown in Fig.
3B assumes that S/E is 2.0, 3.0, 2.0, 2.0, and 2.0 for orders 5, 4, 3, 2 and 1, respectively. The
experimental measurements of Kassab et al. (20) have shown that the
mean S/E is 2.0, 2.3, 2.0, 1.8, and 2.3 for the respective orders.
The present simulation of the CM + LPM model of a fifth-order
arteriolar tree is based on 100 realizations of the LPM. A large number
of realizations was needed to resolve the variations in the network
itself. It has been found that the results of the mean hemodynamic data
averaged over the 100 runs were similar to the results of a single
simulation based on the mean morphometric data. It was further found
that the variations within a single run were larger than the variations
over the 100 runs.
The mean values of the blood pressures at the outlets of elements, the
pressure drop per element, and the flow at the inlets of elements in
the two models of the LV fifth-order arteriole are shown in Figs.
4-6, respectively. It is interesting that the CM + LPM model,
which incorporates the LPM, yields mean longitudinal pressures and
pressure drop per element values that are similar to those of the CM
model. Furthermore, because only the pressures at the inlet and outlets
are specified for the two circuits, the total flow of the CM and CM + LPM model circuits may be unequal. In fact, it is determined that the
CM + LPM model tree carries 15% more flow for the same pressure drop
than the CM model tree. Hence, increasing the number of nodes alone
without changing the diameters, the total lengths, the number of
vessels, or the CM seems to lower the tree resistance to flow. It also
increases the dispersion of flow, as shown in Fig. 8. In fact, the
dispersion of flow at order 1 vessels in the CM + LPM model is 39%
greater than that in order 1 vessels of the CM model. The dispersion of pressure drops per element, however, is similar for the two models, as
shown in Fig. 9. It should be noted that
the dispersions of flow and pressure drop at orders 4 and 5 are zero
because only a single arteriole is considered. In a whole arterial
tree, there are many fourth- and fifth-order arterioles and hence the
dispersions will be nonzero at those orders. Hence, the usefulness of
the present simulation on the dispersions is only for the comparison of
the two models.
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Fig. 9.
Relationship between %pressure drop [(SD/mean) × 100] per element and order number of vessel elements for CM model
and CM + LPM model of an LV 5th-order arteriole. Data for the
CM + LPM model are averaged (±SD) over 100 runs of model,
corresponding to different cases of LPM data.
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The major differences between the two models can be seen
in Tables 6 and
7, which show the mean nodal pressure
and flow along the elements of the CM and CM + LPM models,
respectively. The difference is that in the CM + LPM model, there are
more than two nodes along a given element, unlike the CM model, which
has exactly two nodes, at the inlet and outlet of an element. Hence the
pressure and flow may have greater variation along the elements of the
CM + LPM model.
In the above hemodynamic calculations, several simplifications warrant
further discussion. First, it should be emphasized that the vascular
geometry of the tree models corresponds to that of a fully vasodilated
coronary vasculature. Although it is well known that the patterns of
myocardial blood flow are different in the resting and maximally
vasodilated states (1), the description of coronary blood flow in the
maximally vasodilated state is the first essential step. The vasoactive
components of the arteriolar smooth muscles can be added later when the
experimental data become more abundant. Second, the present analysis
includes the assumption that a number of elements are grouped in
parallel between connected pairs of nodes to allow for the simple
definition of the equivalent conductance
Geq.
This assumption simplifies the computations considerably by reducing
the number of nodes significantly (the number of nodes is reduced by
approximately a factor of 3). Finally, the above analysis covers a
small range of orders and must be extended to the full range of orders
(from order 11 down to the capillaries). However, the rationale for
choosing this particular range of orders was prompted by the
observation that most of the pressure drop occurs across these
smaller orders (orders 1-4) (4), and hence, refinement of pressure
distribution along these smaller orders is most important. However, to
obtain a more realistic picture of flow dispersions, the whole tree
must be considered in the analysis.
Chilian et al. (4) measured the microvascular pressures in different
sizes of coronary microvessels during control conditions and
dipyridamole infusion in the cat. The size of the arteriolar and
venular vessels measured in that study was in the range of ~100-400 µm in diameter. The pressure data were curve-fitted
over the full range of arterioles to venules, extrapolating for the missing data on the 100-µm-diameter vessels. Because our simulated arteriolar tree has a diameter of ~70 µm, we cannot directly
compare our predicted pressure with their experimental data. However, our predicted longitudinal pressures (Fig. 4) are within their curve-fitted portion of the control and dipyridamole-dilated hearts.
As mentioned previously, Van Bavel and Spaan (28) reconstructed a
number of computer models of the coronary arterial trees stochastically, segment for segment, for vessels <500 µm in
diameter. Their reconstructed trees, which resemble our
Fig. 3B, were used to calculate the
longitudinal pressure in each segment. Our longitudinal pressure
distribution is within the scatter of the their computed data.
Furthermore, they found a very heterogeneous flow in their simulated
networks, which they described by a fractal dimension similar to that
obtained in our previous analysis (16) and in agreement with the
experimental measurements of Bassingthwaighte et al. (2).
The LPM complements our previous morphometric data and CM and adds a
greater degree of sophistication to the proposed mathematical tree
models for hemodynamic analysis. However, despite the recent abundance
of morphometric data of the coronary vasculature and hence the
sophistication of the resulting coronary vascular models, a rigorous
hemodynamic analysis of spatial perfusion of the myocardium is still
unattainable. The reason for the inaccessibility of such a model is the
lack of quantitative data on the spatial relation between the
myocardium and the vessel elements, i.e., the three-dimensional branching angles of the coronary vasculature. Previously, there has
been some angle measurements in the plane of the coronary bifurcations
for various caliber vessels of rat and human hearts (33, 34). These
data are two-dimensional, however, and cannot be used to reconstruct
the three-dimensional branching pattern of the coronary vessels for a
spatial analysis of coronary blood flow. Hence, as with the LPM, such
data on the three-dimensional branching angles would further increase
the sophistication of the mathematical models for a realistic analysis
of coronary blood flow.
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ACKNOWLEDGEMENTS |
We thank Kha N. Le and James M. Wu for excellent technical
expertise and effort in data analysis and computations.
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FOOTNOTES |
This research is supported by National Heart, Lung, and Blood Institute
Training Grants HL-07089 and HL-43026 and National Science Foundation
Grant BCS-89-17576.
Address for reprint requests: G. S. Kassab, Dept. of Bioengineering,
Univ. of California, San Diego, 9500 Gilman Dr., La Jolla, CA
92093-0412.
Received 15 April 1996; accepted in final form 29 July 1997.
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REFERENCES |
1.
Austin, R. E.,
G. S. Aldea,
D. L. Coggins,
A. E. Flynn,
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Abstract
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