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1Department of Biomedical Engineering, University of California, Irvine; 2Department of Environmental Health, Cedars-Sinai Medical Center, Los Angeles; and 3Department of Radiological Sciences, University of California, Irvine, California
Submitted 21 July 2004 ; accepted in final form 17 February 2005
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ABSTRACT |
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2.2, which deviates from Murray’s prediction of 3.0. Furthermore, we found the total arterial equivalent resistance to be 0.93, 0.77, and 1.28 mmHg·ml–1·s–1·g–1 for the right coronary artery, left anterior descending coronary artery, and left circumflex artery, respectively. The significance of the present study is that it yields a predictive model that incorporates some of the factors controlling coronary blood flow. The model of normal hearts will serve as a physiological reference state. Pathological states can then be studied in relation to changes in model parameters that alter coronary perfusion.
vascular reconstruction; coronary morphometry; flow simulation; flow resistance; transit time
Over a decade ago, a program was initiated to provide the necessary details of the coronary vascular anatomy (vascular geometry and branching pattern) to enable anatomically based modeling of coronary circulation. In this approach, the vascular system comprised of millions of distensible vessel branches, strategically distributed and mostly embedded within the myocardium, must be modeled in as much detail as possible rather than "lumped." Although we are still several years away from accomplishing this goal, some important strides have been made. As a first step, Kassab and colleagues (13, 15–18) reconstructed the entire vascular anatomy of the porcine heart in the framework of a mathematical model of a tree structure, yielding data on the diameter, lengths, numbers, and connectivity of coronary arteries, capillaries, and veins for every order or generation number. Almost simultaneously, VanBavel and Spaan (32) provided morphometric data on the coronary arterial tree. The anatomic programs produced detailed anatomic data on the vascular geometry and branching pattern in a statistical framework. The sheer number of vessels involved and the hemodynamic computations require automation of reconstruction and analysis. In response, we recently developed (23) a computer program to automate the reconstruction of the full coronary arterial tree based on our previous morphometric measurements.
The objective of the present study was to use the reconstruction platform to carry out an analysis of blood flow in the entire coronary arterial tree under steady-flow conditions. This is only the first step in a program aimed toward understanding the spatial and temporal distribution of blood flow throughout the cardiac cycle. The three-dimensional branching pattern embedded in a realistic dynamic model of the beating heart that integrates the blood vessel elasticity and tissue mechanics can be built in the future, adding to the degree of sophistication and realism of the present model. The model’s limitations and physiological implications are discussed here.
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METHODS |
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A flow simulation was carried out in each of the coronary arterial trees recently reconstructed by Mittal et al. (23) based on measured morphometric data of Kassab et al. (16). A two-step approach was used in the reconstruction of the entire coronary arterial tree down to the capillary level (
8 µm in diameter). Briefly, portions of the arterial tree [right coronary artery (RCA), left anterior descending coronary artery (LAD), or left circumflex artery (LCx)] missing from the cast data were computationally reconstructed from anatomic data. Missing components of the tree, from broken vessel segments down to vessels of diameter of 40 µm, were reconstructed from the intact cast data. Portions of the tree made up of vessels with diameter of <40 µm were reconstructed based on histological data. Reconstructed networks were terminated at segments of diameter 8 µm. Any terminal vessels in the cast data with diameter >8 µm was treated as a broken vessel, and the reconstruction algorithm was applied to generate a subtree that branched down to the terminal arterioles of diameter 8 µm.
Flow Simulation
After the branching pattern and vascular geometry of the full arterial network were generated, a network analysis was performed similar to that of Kassab et al. (19). If we assume that the flow through a blood vessel is laminar, steady, and free from end effects, then the volumetric flow Qij in a vessel between any two nodes, represented by i and j, is given in terms of the pressure differential 
Pij and vessel conductance Gij by
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Pij = Pi – Pj, Gij = D
/µijLij, and Dij, Lij, and µij are the diameter, length, and viscosity, respectively, between nodes i and j. Data on the variation of viscosity with vessel diameter and hematocrit given by Pries et al. (26) was used in our model. They proposed a modified viscosity relationship based on a compilation of literature data on relative blood viscosity in tube flow in vitro and in vivo experimental measurements, which reflects the Fahraeus-Lindqvist effect as given by
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There are two or more vessels that emanate from the jth node anywhere in the tree, with the number of vessels converging at the jth node being mj. By conservation of mass we must have
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Variable Boundary Conditions
The previous results were implemented for an inlet pressure of 100 mmHg and outlet pressure (at the outlet of the first capillary segment) of 26 mmHg. To examine the variation of mean transit time with flow, we varied the inlet boundary condition to 30, 60, 100, 120, 140, 160, and 180 mmHg. We also examined the effect of changing the outlet boundary conditions. We maintained the inlet pressure at 100 mmHg and varied the outlet capillary pressure as a Gaussian distribution with a mean of 26 mmHg and a SD of 0, 2, 4, 6, or 8 mmHg. We observed flow reversal at the terminal capillary vessels for SD > 0. The frequency of flow reversal was quantified as the fraction of vessels with negative flow divided by the total number of vessels in the respective circuit.
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RESULTS |
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Figure 1 shows the relationship between normalized flow through a vessel segment and normalized segment diameter for the LAD arterial tree, excluding the capillaries. This is an isodensity plot showing five layers of frequency. As expected, the majority of vessels are the smaller-diameter arterioles. The diameter and flow are normalized with respect to the inlet, most proximal segment. The relationship obeys a power law relation as suggested by Murray’s law. However, the value of exponent is not 3, as predicted by Murray’s law. As determined by least-squares fits of the data, the exponent has values of 2.2 (R2 = 0.993), 2.1 (R2 = 0.995), and 2.1 (R2 = 0.994) for RCA, LAD, and LCx, respectively.
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PGart model where P, Q, and Gart are the pressure difference along the entire coronary arterial tree, inlet flow, and total arterial conductance, respectively. A linear least-squares fit of the data revealed Gart values of 0.0072 (R2 = 1.000), 0.0086 (R2 = 1.000), and 0.0052 (R2 = 0.999) ml·s–1·mmHg–1 for the RCA, LAD, and LCx arterial trees, respectively.
The mean ± SD arterial transit times were found to be 2.3 ± 0.87 (RCA), 1.5 ± 0.56 (LAD), and 1.9 ± 0.67 (LCx) s at inlet pressure of 100 mmHg; the respective maximum transit times were 8.8, 7.8, and 4.5 s. The transit times were calculated along all possible pathways in the arterial tree by adding the transit times through each individual segment. There were a total of 858,353, 936,014, and 572,632 pathways (equal to the number of first segment of capillaries) for the RCA, LAD, and LCx arterial trees, respectively. The probability density function for the transit times of the LAD arterial tree is shown in Fig. 5. The vertical dashed line represents the mean value. The decay of transit time frequency from the mean, h(t), was fitted by the form h(t) = 
t–
, where t represents the transit time. The empirical constants and were determined by using a nonlinear least-squares fit. We found the values of the exponent to be 3.4 (R2 = 0.937), 3.2 (R2 = 0.888), and 3.2 (R2 = 0.782) for the RCA, LAD, and LCx arterial trees, respectively. The least-squares fit curve for the LAD arterial tree is shown in Fig. 5.
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) was simulated by varying the inlet pressure at 30, 60, 100, 120, 140, 160, and 180 mmHg. The results can be summarized as hyperbolic relations between and inlet flow rate (Qin) as = Vt/Qin, where Vt represents the total arterial volume. The computed value of Vt equals to 1.3, 1.0, and 0.61 ml for the RCA, LAD, and LCx arterial trees, respectively.
Finally, we considered the effect of varying the SD of outlet boundary conditions at the first capillary segment. Figure 6A shows the probability density function for the flow at the capillary outlet for an SD of 6 mmHg. The relation between the SD of outlet Gaussian pressure distribution (with a mean of 26 mmHg) and the relative dispersion (or CV) of capillary outlet flow is summarized in Fig. 6B. It is clear that the CV of blood flow is smallest when the pressure boundary condition has no variability and increases with an increase in heterogeneity of capillary pressure. The change in CV is relatively modest over a significant change in SD of pressure. It was observed that the variability in capillary pressure can give rise to local flow reversal within the capillary bifurcation (Fig. 6A). We quantified this reversal as a percentage of capillary vessels with negative flow normalized with respect to the total number of capillary vessels. We found that the greatest frequency of negative capillary flow occurs for the case of SD = 10 mmHg (10%).
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DISCUSSION |
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Murray’s law. Nearly 80 years ago, Murray deduced a cubed relationship between flow rate and diameter for a vessel segment based on the minimization of power cost function (24). Numerous papers have been published in the past 80 years on Murray’s law and on the validation of the exponent. The studies show support, but with significant scatter (27). The results for the coronary arterial tree show exponents of 2.1–2.2 for all three arterial trees (Fig. 1). The deviation of the exponent from 3 was previously explained based on a modification of the Murray formulation that viewed the coronary arterial tree as an integrated system rather than individual vessel segments (34). It was found that the exponent is not a universal constant as required by Murray’s law but depends on a scaling parameter that is dictated by the equivalent resistance of the vascular system of interest. The exponent was determined on the basis of flow analysis of anatomic circuits (34) and in vivo experiments based on angiography (33). In the former, two anatomic models were considered: 1) a truncated model representing the complete anatomic data for vessels proximal to 0.5 mm in diameter and 2) a full symmetric model of the coronary arterial tree based on mean morphometric data. It was found that the exponent had values in the ranges of 2.2–2.5 and 2.1–2.2 for the truncated and symmetric models, respectively, for the coronary arterial trees. In the in vivo studies based on angiography, the exponent was found to have a value of 2.7 for the LAD tree proximal to 0.5 mm in diameter. The present study is the first to determine the exponent of the flow-diameter relation based on the entire asymmetric model of the coronary arterial tree.
Distribution of flow. It can be noted that the blood flow through the trunk and primary branches (Fig. 2B) shows either an abrupt or a gradual drop along the path to the capillary blood vessels. The shorter paths (from trunk to capillary vessels) with fewer branches show an abrupt drop, whereas the longer paths with more branches show a more gradual drop of blood flow. These flow profiles are novel and have not been described previously because of a lack of a detailed anatomic model. The results suggest that a tracer will experience very different flow depending on the path. The flow at the capillary segment of the various branches is connected by a dotted line in Fig. 2B. The flow dispersion into the capillary bed is obvious. The pressure and flow curves for the trunk and various primary branches reduce to a set of characteristic curves when we combine Fig. 2, B and C, into Fig. 3.
Longitudinal pressure distribution. When the pressure values at the outlet sections of various segments were considered (Fig. 4), the profile showed a flat region followed by a large drop in pressure for vessels <100 µm in diameter. This is in agreement with experimental epicardial and subendocardial pressure measurements (5, 11, 31). Furthermore, a very steep drop in pressures was observed at the smallest arteriolar diameters. We have verified that the steep decline is not due to the fixed capillary pressure (Fig. 4A); i.e., the same steep decline is observed when the capillary pressure is varied according to a Gaussian distribution with various SDs as shown in Fig. 4, B and C. Instead, the steep drop in pressure is due to the large asymmetry in subtrees. If we consider the flow through a vessel segment, the bifurcation will supply two subtrees. If the subtrees are very asymmetric, i.e., different segment diameters, different total number of vessels in each subtree, and hence very different equivalent resistance, it is expected that the flow and pressure distribution will be quite different. Indeed, we would expect that the subtree with a smaller total number of vessels will have a very abrupt pressure drop compared with a more gradual pressure drop for a subtree with many more vessels. Although the "wall" appearance is quite pronounced in Fig. 4, the total number of vessels that give rise to this appearance is only 5% of the total number of vessels. Interestingly, previous flow simulation models have not reported the steep drop (2, 32). We believe that the degree of anatomic detail in the present model was not present in the previous studies.
Arterial pressure-flow relation. It is well known that the majority of flow resistance resides in the arterial tree, particularly in small arterioles (9). Hence, the arterial tree constitutes the majority of coronary circulation resistance. The computed linear pressure difference-flow relationship, whose slope is the flow conductance or inverse of flow resistance, yielded values of total equivalent resistance of 139, 116, and 192 mmHg·ml–1·s–1 for the RCA, LAD, and LCx arterial trees, respectively. If we normalize these values by the total weight of the heart (150 g), we obtain 0.93 (RCA), 0.77 (LAD), and 1.28 (LCx) mmHg·ml–1·s–1·g–1. The linearity arises from the rigid vessel assumption and linear rheology (no shear rate-dependent viscosity, no flow-dependent distribution of red blood cells and plasma to flow pathways, etc.). It is well known, however, that the coronary vessels are distensible and blood rheology is nonlinear, which gives rise to the nonlinear pressure-flow relation (7). It turns out that the nonlinearity is second order, as can be predicted primarily from the distensibility of the coronary blood vessels as shown by Kassab (12).
Transit times.
The relation between and inlet flow rate obeys the classic Stewart-Hamilton relationship, which states that the of a fluid through a confined compartment is equal to the total volume of the compartment divided by the flow rate into the compartment (35). The theoretical basis for this relation was provided by Meier and Zierler (22). For the coronary arterial trees, each transit time-flow relation was constructed from seven different inlet pressures. Each inlet pressure yielded a different inlet flow rate depending on the equivalent resistance of the respective coronary arterial tree. The Stewart-Hamilton relationship suggests that the total volume of the RCA, LAD, and LCx are 1.3, 1.0, and 0.61 ml, respectively. These values are in good agreement with previous cast measurements of arterial volumes (16). These results provide some validity for our calculations of in the coronary arterial tree.
The probability density function of transit times shown in Fig. 5 is equivalent to the normalized outflow concentration-time curve under certain conditions. Bassingthwaighte and Beard (1) showed the downslope of the outflow curves of tracer-labeled water from the rabbit myocardium to be power law functions of the form t–, with equal to 3. The same authors later showed that the t–3 form is a general property of a heterogeneous vascular network (3). More recently, Beard and Bassingthwaighte (4) modeled the left coronary arterial tree based on the data of Kassab et al. (16), with the capillary and venous systems lumped, to show that the tails of washout of intravascular tracer have the form t–3.1. This is comparable to the present finding of t–3.2 for the LAD arterial tree. Hence, the arterial tree seems to be the major determinant of the washout characteristic.
The decreased only slightly, in a nearly linear fashion, when the inlet pressure was increased from 120 to 180 mmHg. However, it increased rapidly when the inlet pressure was decreased from 60 to 30 mmHg. The for the entire coronary arterial tree was 1–2 s at physiological pressure (100 mmHg) under steady-flow conditions in rigid vessels. These values of the arterial tree are approximately one-half those reported for the entire coronary circulation (4). It is apparent that, at a given flow rate, the for the three vessels is RCA > LAD > LCx. This relates in part to the path length, which is largest for the RCA, but also to the velocity distribution.
Comparison with Other Models
Numerous attempts have been made to simulate blood flow in various organs based on detailed anatomic data (13). To our knowledge, our analysis is the most comprehensive in the sense that it relies on the most extensive set of anatomic data to date (13). VanBavel and Spaan (32) presented a flow simulation based on a partial porcine coronary arterial tree model in the diameter range of 10–500 µm. More recently, the same group developed a mathematical model of coronary arterial blood flow based on VanBavel and Spaan’s data. The model considered the myogenic response with an idealized branching structure (6).
Kassab et al. (19) presented a model of the entire coronary arterial tree based on statistical data on diameters, lengths, and connectivity of vessels. The previous model, however, considered some of the parallel vessels as equivalent elements and hence reduced the number of vessels significantly to decrease the computational cost. The vascular network was reduced to a total of 100,000 vessels, and the nodal pressures were solved for such a matrix. In the present model, no such assumptions were made and the nodal pressures were determined for every arterial segment, which equaled 858,353, 936,014, and 572,632 segments for the RCA, LAD, and LCx arterial trees, respectively.
The models described above lack a three-dimensional branching structure. Bassingthwaighte et al. (2) developed an avoidance algorithm to distribute the morphometric measurements of Kassab et al. (15–18) in a cylindrical model of the heart. The coronary arterial system in the passive cylindrical model of the heart gave a remarkably complete description of the distribution of the flows in the heart. Smith et al. (30), also using Kassab’s morphometric data, developed a similar algorithm to map the distribution of vascular elements in the myocardium as a nonlinear optimization of individual branch angles based on the minimum shear stress hypothesis. With this algorithm, a finite-element model of the largest six generations of arterial coronary tree has been generated. The vascular network was geometrically embedded in the finite-element ventricular model of Nielsen et al. (25). In a more recent study, the same group carried out a pulsatile analysis of blood flow in the partial tree (29).
Model Limitations
Although our analysis is based on detailed measured morphometric data of coronary blood vessels, there are still a number of assumptions made. For example, the vessels are assumed to be rigid, which is untrue in reality. This issue becomes particularly important when considering the flow in a contracting myocardium because the elasticity of vessels gives rise to the vessel-muscle interaction. Furthermore, all vessels in the present model are assumed to be in a vasodilated state where coronary flow reserve is substantially reduced. It is important to eventually include vasoregulatory mechanisms to investigate many clinically relevant phenomena that relate to coronary flow reserve (8).
In the future, the present arterial circuit will be extended to the entire coronary vasculature; i.e., we will connect the capillaries to the entire venous system. Hence, we will replace the ad hoc pressure boundary condition on the capillary vessel with a measured boundary condition at the outlet of the venous system. Because the capillary boundary conditions are ad hoc, we examined the effect of variation of capillary pressure. Figure 6B shows a modest effect of variation of capillary outlet pressure on the CV of capillary flow. We previously showed (20) that the cross-connections may serve to homogenize the pressure and flow distribution in the capillary bed.
One effect of increase in capillary pressure dispersion is the creation of local flow reversal as seen in Fig. 6A. The flow reversal does not extend beyond the segment of a capillary vessel. This was due to the fact that as the outlet pressure distribution became broad, it was possible that the pressure gradient was reversed locally at the capillary segment. It can be seen that the frequency of such an occurrence is quite small, which is ensured by the homogenization function of the capillary network (20).
Significance of Model
The present study yields a predictive model that incorporates some of the detailed anatomic features that influence coronary blood flow. The accuracy of the model will be determined through experimental validation. The model of normal hearts will serve as a physiological reference state. Pathological states can then be studied in relation to changes in model parameters that alter coronary perfusion. The proposed study makes use of physical principles, with the help of anatomy, to explain and predict the physiology of the coronary arterial circulation in quantitative terms. The present model will serve as a foundation for future more sophisticated models that incorporate additional realism and that will serve to quantitatively test various hypotheses in the coronary circulation.
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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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. Section 1734 solely to indicate this fact.
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REFERENCES |
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