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 diameterdefined 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 parallelseries 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 columnn is the fractional longitudinal position of the branch point on vessels of ordern at which vessels of orderm 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
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 “rivuletsrivers” 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, 57, 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 segmentstoelements ratio (S/E), defined as the ratio of the total number of vessel segments to the total number of vessel elements, and3) a connectivity matrix [m,n] (CM), whose component in the mth row andnth column is the ratio of the total number of elements of order m that spring directly from parent elements of ordern 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 ordern 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 pressureflow 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 fifthorder 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.
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 KClarrested, adenosinedilated 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 diameterdefined 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 diameterdefined 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 ordern and its branches are shown. A local dimensionless coordinate x is introduced along the vessel n, withx = 0 at the inlet andx = 1 at the outlet. At a pointi on the vesseln, a blood vessel element whose diameter qualifies it to be of the order numberm branches out. The coordinatex of the pointi is equal tox_{mn} , 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 coordinatex_{km} that is the fractional longitudinal position of a vessel of orderk springing from the vessel of orderm. The matrix of the numbersx_{mn} is the LPM.
For each element of order n, we introduced a nondimensional local longitudinal coordinatex, withx = 0 at the inlet of the vessel andx = 1 at the exit end. The coordinates of the branching points on the vessels of ordern 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 columnn is the fractional longitudinal position of the branching point of the elements of orderm on the elements of ordern.
RESULTS
Tables1–3 show the LPM for the RCA, LAD, and LCX arterial trees, respectively. Tables4 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 15 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 thediscussion.
Applications to hemodynamics.
In the following, let us consider the blood flow in a left ventricular (LV) fifthorder 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. Figure3 A shows a schematic of the CM model that is consistent with its mean CM given in Ref. 20. Figure 3 B 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 fifthorder arteriole gives rise to further branches, and so on, down to the arterial capillaries (order 0). Each element shown in Fig. 3,A andB, may represent several elements in parallel, as determined by the total number of elements in each order.
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 symbolN_{ijk...rst}
to denote the number of parallel elements in pathwayijk...rst. LetC(t,s) be a component of the CM[m,n] given in Ref. 20, andN_{T}
(t) be the mean total number of parallel elements of ordert. TheN_{T}
(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 andB, and using the CM[m,n] of the LAD arterial tree of Ref. 20, we find
Next, we shall formulate the hemodynamics of the blood flow in the vessel of order m between nodesi andj. Assume that the vessel is so small that both the Reynolds number and the Womersley number are much smaller than 1 (with order number msufficiently small, e.g., m ≤ 5) so that the flow is quasisteady 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 nodej be denoted by P_{j}. A vessel element joining nodes i andj is denoted by vesselij, whose diameter isD_{ij}
and length is L_{ij}
. If there were N vessel elements connecting nodes i andj, then the number of these elements would be denoted byN_{ij}
. The rate of volume flow in the vessel connecting the two nodesi andj is then given in terms of the pressure differential (P_{i} − P_{j}), vascular geometry (D_{ij}
andL_{ij}
), and blood viscosity (μ_{ij}), by (see Ref.9)
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. 3 B, 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. Figures4–6 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.
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/η, whereU is the mean velocity of flow,D is the blood vessel lumen diameter, and η is the kinematic viscosity of blood, and the Womersley number (D/2)(ω/η)^{½}, where ω 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.
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.
DISCUSSION
We have introduced an additional innovation in morphometry: the LPM, which characterizes the longitudinal position of vessels of orderm along the length of vessels of ordern from which they arise. To explain the significance of the LPM, we recall that the blood flow f_{ij} in the vessel of orderm between nodesi andj is given by Eq.2 . The lengthL_{ij} inEq. 2 depends on the element length of vessels of orders m andn and the longitudinal fractionsx_{mn} andx_{km} . Hence, f_{ij} 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 ordersm, 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.3 B 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 fifthorder 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 fifthorder arteriole are shown in Figs.46, 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 fifthorder 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.
The major differences between the two models can be seen in Tables 6 and7, 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 conductanceG _{eq}. 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 curvefitted over the full range of arterioles to venules, extrapolating for the missing data on the 100μmdiameter 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 curvefitted portion of the control and dipyridamoledilated 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. 3 B, 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 threedimensional 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 twodimensional, however, and cannot be used to reconstruct the threedimensional branching pattern of the coronary vessels for a spatial analysis of coronary blood flow. Hence, as with the LPM, such data on the threedimensional branching angles would further increase the sophistication of the mathematical models for a realistic analysis of coronary blood flow.
Acknowledgments
We thank Kha N. Le and James M. Wu for excellent technical expertise and effort in data analysis and computations.
Footnotes

Address for reprint requests: G. S. Kassab, Dept. of Bioengineering, Univ. of California, San Diego, 9500 Gilman Dr., La Jolla, CA 920930412.

This research is supported by National Heart, Lung, and Blood Institute Training Grants HL07089 and HL43026 and National Science Foundation Grant BCS8917576.
 Copyright © 1997 the American Physiological Society