Studies on the structure of human coronary vasculature

Based on morphometric data, we calculate the structural parameters of the coronary vasculature as an optimal branching bed. We show (i) significant correlations between the diameters of the larger daughter and the parent vessel and between the diameter of the smaller daughter vessel and the asymmetry coefficient; (ii) differences in the structural parameters for two types of artery that deliver and distribute blood in the cardiac muscle; and (iii) the length-diameter relationships for different arteries. The coronary vasculature is characterized by asymmetrical branching and thus should be modeled with self-similar asymmetrical tree-like systems.     

Bifurcation asymmetry of the porcine coronary vasculature and its implications on coronary flow heterogeneity

The branching pattern of the coronary arteries and veins is asymmetric, i.e., many small vessels branch off of a large trunk such that the two daughter vessels at a bifurcation are of unequal diameters and lengths. One important implication of the geometric vascular asymmetry is the dispersion of blood flow at a bifurcation, which leads to large spatial heterogeneity of myocardial blood flow. To document the asymmetric branching pattern of the coronary vessels, we computed an asymmetry ratio for the diameters and lengths of all vessels, defined as the ratio of the daughter diameters and lengths, respectively. Previous data from silicone elastomer cast of the entire coronary vasculature including arteries, arterioles, venules, and veins were analyzed. Data on smaller vessels were obtained from histological specimens by optical sectioning, whereas data on larger vessels were obtained from vascular casts. Asymmetry ratios for vascular areas, volumes, resistances, and flows of the various daughter vessels were computed from the asymmetry ratios of diameters and lengths for every order of mother vessel. The results show that the largest orders of arterial and venous vessels are most asymmetric and the degree of asymmetry decreases toward the smaller vessels. Furthermore, the diameter asymmetry at a bifurcation is significantly larger for the coronary veins (1.7-6.8 for sinus veins) than the corresponding arteries (1.5-5.8 for left anterior descending coronary artery) for orders 2-10, respectively. The reported diameter asymmetry at a bifurcation leads to significant heterogeneity of blood flow at a bifurcation. Hence, the present data quantify the dispersion of blood flow at a bifurcation and are essential for understanding flow heterogeneity in the coronary circulation.     

Diameter asymmetry of porcine coronary arterial trees: structural and functional implications

The coronary vasculature is characterized by highly asymmetric diameters at bifurcations, which may be an important determinant of flow distribution. To facilitate accurate reconstruction of the coronary network for hemodynamic analysis, we introduce a statistical data set of the diameter asymmetry at bifurcations based on morphometric data of the porcine coronary arterial and venous trees. The bifurcation asymmetry data were represented by the diameter ratio of the daughters relative to mother vessel and by an area expansion ratio (AER) at each bifurcation. A novel asymmetry ratio matrix was introduced to describe the diameter asymmetry of daughters to mother vessels. The relations between AER and flow velocity, and asymmetry ratio matrix and flow distribution, were considered. The results indicate that the ratio of large daughter to mother vessel has a minimum value at order 5 (mean diameter of approximately 70 microm), whereas the ratio of small daughter to mother vessel decreases monotonically with increase in order number. The AER was found to be fairly uniform for larger vessels and to increase from order 5 toward the capillaries. At order 5, we observe a transition in asymmetric bifurcation pattern that may mark a hemodynamic transition from transmural to perfusion subnetworks. The functional implications of these structural transitions are considered.     

Readdressing the issue of thermally significant blood vessels using a countercurrent vessel network

A physiologically realistic arterio-venous countercurrent vessel network model consisting of ten branching vessel generations, where the diameter of each generation of vessels is smaller than the previous ones, has been created and used to determine the thermal significance of different vessel generations by investigating their ability to exchange thermal energy with the tissue. The temperature distribution in the 3D network (8178 vessels; diameters from 10 to 1000 microm) is obtained by solving the conduction equation in the tissue and the convective energy equation with a specified Nusselt number in the vessels. The sensitivity of the exchange of energy between the vessels and the tissue to changes in the network parameters is studied for two cases; a high temperature thermal therapy case when tissue is heated by a uniformly distributed source term and the network cools the tissue, and a hypothermia related case, when tissue is cooled from the surface and the blood heats the tissue. Results show that first, the relative roles of vessels of different diameters are strongly determined by the inlet temperatures to those vessels (e.g., as affected by changing mass flow rates), and the surrounding tissue temperature, but not by their diameter. Second, changes in the following do not significantly affect the heat transfer rates between tissue and vessels; (a) the ratio of arterial to venous vessel diameter, (b) the diameter reduction coefficient (the ratio of diameters of successive vessel generations), and (c) the Nusselt number. Third, both arteries and veins play significant roles in the exchange of energy between tissue and vessels, with arteries playing a more significant role. These results suggest that the determination of which diameter vessels are thermally important should be performed on a case-by-case, problem dependent basis. And, that in the development of site-specific vessel network models, reasonable predictions of the relative roles of different vessel diameters can be obtained by using any physiologically realistic values of Nusselt number and the diameter reduction coefficient.     

Scaling of myocardial mass to flow and morphometry of coronary arteries.

There is no doubt that scaling relations exist between myocardial mass and morphometry of coronary vasculature. The purpose of this study is to quantify several morphological (diameter, length, and volume) and functional (flow) parameters of the coronary arterial tree in relation to myocardial mass. Eight normal porcine hearts of 117-244 g (mean of 177.5 +/- 32.7) were used in this study. Various coronary subtrees of the left anterior descending, right coronary, and left circumflex arteries were perfused at pressure of 100 mmHg with different colors of a polymer (Microfil) to obtain rubber casts of arterial trees corresponding to different regions of myocardial mass. Volume, diameter, and cumulative length of coronary arteries were reconstructed from casts to analyze their relationship to the perfused myocardial mass. Volumetric flow was measured in relationship with perfused myocardial mass. Our results show that arterial volume is linearly related to regional myocardial mass, whereas the sum of coronary arterial branch lengths, vessel diameters, and volumetric flow show an approximately 3/4, 3/8, and 3/4 power-law relationship, respectively, in relation to myocardial mass. These scaling laws suggest fundamental design principles underlying the structure-function relationship of the coronary arterial tree that may facilitate diagnosis and management of diffuse coronary artery disease.     

On fractal properties of arterial trees

The question of fractal properties of arterial trees is considered in light of data from the extensive tree structure of the right coronary artery of a human heart. Because of the highly non-uniform structure of this tree, the study focuses on the purely geometrical rather than statistical aspects of fractal properties. The large number of arterial bifurcations comprising the tree were found to have a mixed degree of asymmetry at all levels of the tree, including the depth of the tree where it has been generally supposed that they would be symmetrical. Cross-sectional area ratios of daughter to parent vessels were also found to be highly mixed at all levels, having values both above and below 1.0, rather than consistently above as has been generally supposed in the past. Calculated values of the power law index which describes the theoretical relation between the diameters of the three vessel segments at an arterial bifurcation were found to range far beyond the two values associated with the cube and square laws, and not clearly favoring one or the other. On the whole the tree structure was found to have what we have termed "pseudo-fractal" properties, in the sense that vessels of different calibers displayed the same branching pattern but with a range of values of the branching parameters. The results suggest that a higher degree of fractal character, one in which the branching parameters are constant throughout the tree structure, is unlikely to be attained in non-uniform vascular structures.     

Epicardial coronary blood flow including the presence of stenoses and aorto-coronary bypasses--I: Model and numerical method

A computer model and numerical method for calculating left epicardial coronary blood flow has been developed. This model employs a finite-branching geometry of the coronary vasculature and the one-dimensional, unsteady equations for flow with friction. The epicardial coronary geometry includes the left main and its bifurcation, the left anterior descending and left circumflex coronary arteries, and a selected number of small branches. Each of the latter terminate in an impedance, whose resistive component is related to intramyocardial compression through a linear dependence on left ventricular pressure. The elastic properties of the epicardial arteries are taken to be non-linear and are prescribed by specifying the local small-disturbance wave speed. The model allows for the incorporation of multiple stenoses as well as aorto-coronary bypasses. Calculations using this model predict pressure and flow waveform development and allow for the systematic investigation of the dependence of coronary flow on various parameters, e.g., peripheral resistance, wall properties, and branching pattern, as well as the presence of stenoses and bypass grafts. Reasonable comparison between calculations and earlier experiments in horses has been obtained.     

Analysis of blood flow in the entire coronary arterial tree

A hemodynamic analysis of coronary blood flow must be based on the measured branching pattern and vascular geometry of the coronary vasculature. We recently developed a computer reconstruction of the entire coronary arterial tree of the porcine heart based on previously measured morphometric data. In the present study, we carried out an analysis of blood flow distribution through a network of millions of vessels that includes the entire coronary arterial tree down to the first capillary branch. The pressure and flow are computed throughout the coronary arterial tree based on conservation of mass and momentum and appropriate pressure boundary conditions. We found a power law relationship between the diameter and flow of each vessel branch. The exponent is approximately 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.     

Diameter-dependent axial prestretch of porcine coronary arteries and veins

The pressure-diameter relation (PDR) and the wall strain of coronary blood vessels have important implications for coronary blood flow and arthrosclerosis, respectively. Previous studies have shown that these mechanical quantities are significantly affected by the axial stretch of the vessels. The objective of this study was to measure the physiological axial stretch in the coronary vasculature; i.e., from left anterior descending (LAD) artery tree to coronary sinus vein and to determine its effect on the PDR and hence wall stiffness. Silicone elastomer was perfused through the LAD artery and coronary sinus trees to cast the vessels at the physiologic pressure. The results show that the physiological axial stretch exists for orders 4 to 11 (> 24 μm in diameter) arteries and orders -4 to -12 (>38 μm in diameter) veins but vanishes for the smaller vessels. Statistically, the axial stretch is higher for larger vessels and is higher for arteries than veins. The axial stretch λ(z) shows a linear variation with the order number (n) as: λ(z) = 0.062n + 0.75 (R(2) = 0.99) for artery and λ(z) = -0.029n + 0.89 (R(2) = 0.99) for vein. The mechanical analysis shows that the axial stretch significantly affects the PDR of the larger vessels. The circumferential stretch/strain was found to be significantly higher for the epicardial arteries (orders 9-11), which are free of myocardium constraint, than the intramyocardial arteries (orders 4-8). These findings have fundamental implications for coronary blood vessel mechanics.     

Predicting root biomass from branching patterns of Douglas-fir root systems

There are many examples of branching networks in nature, such as tree crowns, river systems, arteries and lungs. These networks have often been described as being self-similar, or following scale-invariant branching rules, and this property has been used to derive several scaling laws. In this paper we model root systems of Douglas-fir ( Pseudotsuga menziesii var. glauca (Beissn.) Franco) as branching networks following several simple branching rules. Our objective is to establish a relationship between trunk diameter and root biomass. We explore the effect of the self-similar branching assumption on this relationship. Using data collected from a mature stand in British Columbia, we find that branching asymmetry and the rate of root taper change with root size, thereby violating the assumption of self-similarity. However, the data are in general agreement with Leonardo da Vinci's area-preserving branching hypothesis. We use the field data to parameterize two models, one assuming self-similar branching and a second incorporating the measured size dependencies of branching parameters. The two models differ by only a small amount (≈8%) in their predictions. For both models, the predicted relationship between trunk diameter and root biomass is in good concordance with previously published empirical data. We conclude that the assumption of self-similar branching, although violated by the data, nevertheless provides a useful tool for predicting the allometric relationship between trunk diameter and root biomass. Finally, we use our models to show that the geometric properties of individual bifurcations fundamentally change the root biomass cost of different root topologies.     

Modeling of intraorgan arterial vasculatures. II. Propagation of pressure waves

The dependences of the wave conductivity of self-similar dichotomous branching models of intraorgan arterial vasculatures on the model parameters have been calculated. It was found that, with different sets of parameters, it is possible to model the sucking effect connected with negative wave reflection at the arterial branching as well as the resonant properties of arterial beds. It was shown that the selection of an adequate model for a given intraorgan vasculature should be based on the agreement between the biophysical characteristics of the model and the vasculature that reflect the propagation and reflection of pulse waves.     

Dynamics of human coronary arterial motion and its potential role in coronary atherogenesis

Mechanical forces have been widely recognized to play an important role in the pathogenesis of atherosclerosis. Since coronary arterial motion modulates both vessel wall mechanics and fluid dynamics, it is hypothesized that certain motion patterns might be atherogenic by generating adverse wall mechanical forces or fluid dynamic environments. To characterize the dynamics of coronary arterial motion and explore its implications in atherogenesis, a system was developed to track the motion of coronary arteries in vivo, and employed to quantify the dynamics of four right coronary arteries (RCA) and eight left anterior descending (LAD) coronary arteries. The analysis shows that: (a) The motion parameters vary among individuals, with coefficients of variation ranging from 0.25 to 0.59 for axially and temporally averaged values of the parameters; (b) the motion parameters of individual vessels vary widely along the vessel axis, with coefficients of variation as high as 2.28; (c) the LAD exhibits a greater axial variability in torsion, a measure of curve "helicity," than the RCA; (d) in comparison with the RCA, the LAD experiences less displacement (p = 0.009), but higher torsion (p = 0.03). These results suggest that: (i) the variability of certain motion parameters, particularly those that exhibit large axial variations, might be related to variations in susceptibility to atherosclerosis among different individuals and vascular regions; and (ii) differences in motion parameters between the RCA and LAD might relate to differences in their susceptibility to atherosclerosis.     

A model of growing vascular structures

Increasing attention is being paid to the configuration and development of vascular structures and their possible correlations with physiological events. The study of angiogenesis in normal and pathological states as well as in the embryo and adult has provided new insights into the mechanism of vessel growth and organization of the vasculature. Various mathematical branching models have been developed. These constructions are mainly geometrical and only involve a branching phenomenon. We propose the use of a deterministic non-linear model based on physiological laws and hydrodynamics. Growth, branching and anastomosis, the three actual main events occurring in vascular growth, are included in this model. Space growth, including cells and vessels, is defined by a decreasing transformation. Space density and the length of new sprouts are controlled by a set of parameters. The conditions on these parameters are well established, which allows the production of realistic patterns.     

Distributing and delivering vessels of the human heart

The branching characteristics of the right coronary artery, acute marginal, posterior descending, left anterior descending, circumflex, and obtuse marginal arteries are compared with those of diagonal branches, left and right ventricular branches, septal, and higher-order branches, to test a newly proposed functional classification of the coronary arteries in which the first group rank as distributing vessels and the second as delivering vessels. According to this classification, the function of the first type is merely to convey blood to the borders of myocardial zones, while the function of the second is to implement the actual delivery of blood into these zones. This functional difference is important in the hemodynamic analysis of coronary heart disease, as it provides an assessment of the role of a vessel within the coronary network and hence an assessment of the functional importance of that vessel in a particular heart. Measurements from casts of human coronary arteries are used to examine the relevant characteristics of these vessels and hence to test the basis of this classification.     

As human pial arteries (internal diameter 200-1000 microm) get smaller, their wall thickness and capacity to develop tension relative to their diameter increase.

Pial arteries play a key role in the regulation of human cerebral blood flow. However, many of the features and mechanisms that regulate the tone and diameters of these vessels cannot be studied in situ. One approach is to study in vitro segments of arteries obtained during neurosurgical procedures. The ratios of arterial media thickness to lumen diameter and of the capacity to develop wall force to lumen diameter have important functional consequences and are known to change in disease. Experiments were carried out on pial arteries from normotensive humans to determine the way in which these parameters vary with vessel size. Vessel dimensions--media thickness and lumen diameter were derived from fixed sections using quantitative morphometry. Wall force was measured using a resistance artery myograph. The ratio of media thickness to lumen diameter and of maximum tension developed to lumen diameter both increased as vessel diameter decreased. These ratios do not change over the age range of 15-75 years. These findings show that although in vivo intralumenal pressure falls as human pial arteries become smaller, their media thickness and capacity to develop tone increase.     

Estrogen replacement increases coronary artery distensibility in ovariectomized rats.

The effect of estrogen on the passive characteristics of arteries is not known. We hypothesized that estrogen would increase arterial distensibility as part of its protective effect on the vasculature. Female Sprague-Dawley rats were ovariectomized at 11 weeks of age. One group received a placebo (n = 6), while two other groups (n = 5 each) of rats received a 17beta-estradiol pellet (0.15 mg or 0.5 mg with 60-day release). After 4 weeks of estrogen replacement, coronary and mesenteric arteries (<200 microm diameter) were dissected and mounted on a dual-chamber arteriograph. Lumen diameter and wall thickness were measured in pressurized arteries. The relative changes in diameter (distensibility) as well as wall thickness per unit change in pressure were significantly increased (p < 0.05) in the coronary arteries of the 0.5 mg estradiol replaced rats compared with the ovariectomized control animals and the 0.15 mg estradiol replaced rats. Surprisingly, in the mesenteric arteries from the same animals, there was no difference in distensibility or pressure - wall thickness among the groups. This study provides experimental data of a novel hypothesis that estrogen may afford part of its protection through vascular remodeling of the coronary circulation.     

Anatomically based finite element models of the human pulmonary arterial and venous trees including supernumerary vessels.

Studies of the origin of pulmonary blood flow heterogeneity have highlighted the significant role of vessel branching structure on flow distribution. To enable more detailed investigation of structure-function relationships in the pulmonary circulation, an anatomically based finite element model of the arterial and venous networks has been developed to more accurately reflect the geometry found in vivo. Geometric models of the arterial and venous tree structures are created using a combination of multidetector row X-ray computed tomography imaging to define around 2,500 vessels from each tree, a volume-filling branching algorithm to generate the remaining accompanying conducting vessels, and an empirically based algorithm to generate the supernumerary vessel geometry. The explicit generation of supernumerary vessels is a unique feature of the computational model. Analysis of branching properties and geometric parameters demonstrates close correlation between the model geometry and anatomical measures of human pulmonary blood vessels. A total of 12 Strahler orders for the arterial system and 10 Strahler orders for the venous system are generated, down to the equivalent level of the terminal bronchioles in the bronchial tree. A simple Poiseuille flow solution, assuming rigid vessels, is obtained within the arterial geometry of the left lung, demonstrating a large amount of heterogeneity in the flow distribution, especially with inclusion of supernumerary vessels. This model has been constructed to accurately represent available morphometric data derived from the complex asymmetric branching structure of the human pulmonary vasculature in a form that will be suitable for application in functional simulations.     

Vessel network extraction and analysis of mouse pulmonary vasculature via X-ray micro-computed tomographic imaging

In this work, non-invasive high-spatial resolution three-dimensional (3D) X-ray micro-computed tomography (μCT) of healthy mouse lung vasculature is performed. Methodologies are presented for filtering, segmenting, and skeletonizing the collected 3D images. Novel methods for the removal of spurious branch artefacts from the skeletonized 3D image are introduced, and these novel methods involve a combination of distance transform gradients, diameter-length ratios, and the fast marching method (FMM). These new techniques of spurious branch removal result in the consistent removal of spurious branches without compromising the connectivity of the pulmonary circuit. Analysis of the filtered, skeletonized, and segmented 3D images is performed using a newly developed Vessel Network Extraction algorithm to fully characterize the morphology of the mouse pulmonary circuit. The removal of spurious branches from the skeletonized image results in an accurate representation of the pulmonary circuit with significantly less variability in vessel diameter and vessel length in each generation. The branching morphology of a full pulmonary circuit is characterized by the mean diameter per generation and number of vessels per generation. The methods presented in this paper lead to a significant improvement in the characterization of 3D vasculature imaging, allow for automatic separation of arteries and veins, and for the characterization of generations containing capillaries and intrapulmonary arteriovenous anastomoses (IPAVA).     

Impact of coronary vasa vasorum functional structure on coronary vessel wall perfusion distribution

Noncoronary vasa vasorum have been described as networks of microvessels in the wall of arteries and veins. However, we have shown, using microcomputerized tomography (micro-CT) imaging methods, that porcine coronary vasa vasorum have a tree-like branching structure similar to the vasculature in general. In this study, we elucidate functional aspects of coronary vasa vasorum perfusion territories. Three pig hearts were injected with radiopaque Microfil via the coronary sinus to fill the left anterior descending coronary arteries (LADs) retrogradely at atmospheric pressure. In three other hearts, LADs were injected antegradely at 100-mmHg pressure via the left main carotid artery. Additionally, six LADs were injected in vivo with a suspension of 100- or 300-microm-diameter microspheres before harvesting of the hearts and injection of the LADs with Microfil. All harvested LADs were scanned intact with micro-CT (20 microm cubic voxels). The spatial density of vasa vasorum (no. of vasa/mm2) was measured in 20-microm-thick cross sections (at 0.4-mm intervals). Retrogradely injected LADs showed high and uniformly distributed vasa vasorum densities in the adventitia (means +/- SE; 5.38 +/- 0.09 vs. 3.58 +/- 0.1 vasa/mm2 in antegradely prepared LADs; P < 0.001). Antegradely prepared LADs showed patchy distributed, low-vasa-vasorum-density territories especially on the myocardial side of the coronary artery wall (epicardial density: 4.29 +/- 0.13 vasa/mm2 vs. myocardial density: 2.80 +/- 0.1 vasa/mm2, P < 0.001). Microembolization reduced vasa vasorum densities significantly (100-mum-diameter microspheres: 3.26 +/- 0.07 vasa/mm2, P < 0.05; 300-microm-diameter microspheres: 2.66 +/- 0.07 vasa/mm2, P < 0.001 vs. antegrade controls) and increased the size of low-vasa-vasorum-density territories. We conclude that coronary vasa vasorum are functional endarteries not connected via a plexus. This characteristic may have a significant impact on the spatial distribution of perfusion and drainage of the coronary vessel wall.     

A new fundamental bioheat equation for muscle tissue--part II: Temperature of SAV vessels

In this study, a new theoretical framework was developed to investigate temperature variations along countercurrent SAV blood vessels from 300 to 1000 microm diameter in skeletal muscle. Vessels of this size lie outside the range of validity of the Weinbaum-Jiji bioheat equation and, heretofore, have been treated using discrete numerical methods. A new tissue cylinder surrounding these vessel pairs is defined based on vascular anatomy, Murray's law, and the assumption of uniform perfusion. The thermal interaction between the blood vessel pair and surrounding tissue is investigated for two vascular branching patterns, pure branching and pure perfusion. It is shown that temperature variations along these large vessel pairs strongly depend on the branching pattern and the local blood perfusion rate. The arterial supply temperature in different vessel generations was evaluated to estimate the arterial inlet temperature in the modified perfusion source term for the s vessels in Part I of this study. In addition, results from the current research enable one to explore the relative contribution of the SAV vessels and the s vessels to the overall thermal equilibration between blood and tissue.