Investigation of the structure of human coronary vasculature

Some results of a morphometric study of the parameters of coronary arteries are presented. The parameters that characterize the structure of the arterial vasculature as an optimal branching system have been calculated. Statistically reliable correlations between the diameter of the bigger of two daughter vessels in a bifurcation with the diameter of the parent vessel as well as between the diameter of the smaller daughter vessel and the asymmetry coefficient have been obtained. Differences in the structural parameters of the two types of coronary arteries that provide blood delivery and distribution have been revealed. The relationships between the lengths and diameters of the arteries of different subsystems have been obtained. It is shown that asymmetrical branching is characteristic of the coronary vasculature, and self-similar asymmetric tree-like systems may be used for its modeling.     

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.     

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.     

Human coronary arteries. II. Branching anatomical pattern and arterial wall microarchitecture

A study on 22 pairs, including subjects of similar age, sex and cause of death but with different branching anatomical patterns, shows, that the amount and structural organization of elastic and muscular tissue of the thickened intima are strongly influenced by (a) the size, length, mode of ramification of the parent vessels of the coronary tree; (b) the number of branch orifices in a given arterial segment, and (c) the collateral or terminal character of certain daughter branches.     

Simulation of 3D Solid Tumour Angiogenesis Including Arteriole,Capillary and Venule

In this paper, a 3D mathematical model of tumour angiogenesis is developed, to generate a functional tumour vasculature for blood microcirculation. The model follows that of Anderson and Chaplain (1998) ^[1] with three exceptions: (a) extending the model from 2D to 3D, one arteriole and one venule is induced as two parent vessels to form an intact circulation network for blood flow; (b) generating networks able to penetrate into the tumour interior rather than the exterior only; (c) considering branching generations with different diameters, based on which three groups of vessels, such as arterioles, venules and capillaries are classified. The present study contains four steps: 1. Generation of 3D angiogenic vasculature induced from one arteriole and one venule, with branching generations considered. 2. Examination of vessel connectivity among each other to construct a functional network for blood circulation, investigation of sensitivity of network architectures to changes in some model parameters. 3. Simulation of blood flow in the developed vasculatures. 4. Comparisons of blood flow calculated on the networks induced from an arteriole-venule system and from a single parent vessel.
The networks from simulations could present basic geometric and morphological features of tumour vasculatures. The sensitivity analysis indicates the controllability of the created networks, which could construct architectures of some specific geometric features to suit different types of tumours. The comparisons of blood flow mentioned above demonstrate the validity of the present vasculature, which could be served as a more realistic network structure for research of microcirculation, drug delivery in solid tumors.     

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.     

In Utero Intra-cardiac Tomato-lectin Injections on Mouse Embryos to Gauge Renal Blood Flow

The formation and perfusion of developing renal blood vessels (apart from glomeruli) are greatly understudied. As vasculature develops via angiogenesis (which is the branching off of major vessels) and vasculogenesis (de novo vessel formation), perfusion mapping techniques such as resin casts, in vivo ultrasound imaging, and micro-dissection have been limited in demonstrating the intimate relationships between these two processes and developing renal structures within the embryo. Here, we describe the procedure of in utero intra-cardiac ultrasound-guided FITC-labeled tomato lectin microinjections on mouse embryos to gauge the ontogeny of renal perfusion. Tomato lectin (TL) was perfused throughout the embryo and kidneys harvested. Tissues were co-stained for various kidney structures including: nephron progenitors, nephron structures, ureteric epithelium, and vasculature. Starting at E13.5 large caliber vessels were perfused, however peripheral vessels remained unperfused. By E15.5 and E17.5, small peripheral vessels as well as glomeruli started to become perfused. This experimental technique is critical for studying the role of vasculature and blood flow during embryonic development.     

VEGFR1 (Flt1) Regulates Rab4 Recycling to Control Fibronectin Polymerization and Endothelial Vessel Branching

The cell's main receptor for VEGF, VEGFR2 (Kdr) is one of the most important positive regulators of new blood vessel growth and its downstream signalling is well characterized. By contrast, VEGFR1 (Flt1) and the mechanisms by which this VEGF receptor promotes branching morphogenesis in angiogenesis remain relatively unclear. Here we report that engagement of VEGFR1 activates a Rab4A-dependent pathway that transports αvβ3 integrin from early endosomes to the plasma membrane, and that this is required for VEGF-driven fibronectin polymerization in endothelial cells. Furthermore, VEGFR1 acts to promote endothelial tubule branching in an organotypic model of angiogenesis via a mechanism that requires Rab4A and αvβ3 integrin. We conclude that a recycling pathway regulated by Rab4A is a critical effector of VEGFR1 during branching morphogenesis of the vasculature.     

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.     

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.     

Pulsatile blood flow in the entire coronary arterial tree: theory and experiment

The pulsatility of coronary circulation can be accurately simulated on the basis of the measured branching pattern, vascular geometry, and material properties of the coronary vasculature. A Womersley-type mathematical model is developed to analyze pulsatile blood flow in diastole in the absence of vessel tone in the entire coronary arterial tree on the basis of previously measured morphometric data. The model incorporates a constitutive equation of pressure and cross-section area relation based on our previous experimental data. The formulation enables the prediction of the impedance, the pressure distribution, and the pulsatile flow distribution throughout the entire coronary arterial tree. The model is validated by experimental measurements in six diastolic arrested, vasodilated porcine hearts. The agreement between theory and experiment is excellent. Furthermore, the present pulse wave results at low frequency agree very well with previously published steady-state model. Finally, the phase angle of flow is seen to decrease along the trunk of the major coronary artery and primary branches toward the capillary vessels. This study represents the first, most extensive validated analysis of Womersley-type pulse wave transmission in the entire coronary arterial tree down to the first segment of capillaries. The present model will serve to quantitatively test various hypotheses in the coronary circulation under pulsatile flow conditions.     

Tree Branching: Leonardo da Vinci's Rule versus Biomechanical Models

This study examined Leonardo da Vinci''s rule (i.e., the sum of the cross-sectional area of all tree branches above a branching point at any height is equal to the cross-sectional area of the trunk or the branch immediately below the branching point) using simulations based on two biomechanical models: the uniform stress and elastic similarity models. Model calculations of the daughter/mother ratio (i.e., the ratio of the total cross-sectional area of the daughter branches to the cross-sectional area of the mother branch at the branching point) showed that both biomechanical models agreed with da Vinci''s rule when the branching angles of daughter branches and the weights of lateral daughter branches were small; however, the models deviated from da Vinci''s rule as the weights and/or the branching angles of lateral daughter branches increased. The calculated values of the two models were largely similar but differed in some ways. Field measurements of Fagus crenata and Abies homolepis also fit this trend, wherein models deviated from da Vinci''s rule with increasing relative weights of lateral daughter branches. However, this deviation was small for a branching pattern in nature, where empirical measurements were taken under realistic measurement conditions; thus, da Vinci''s rule did not critically contradict the biomechanical models in the case of real branching patterns, though the model calculations described the contradiction between da Vinci''s rule and the biomechanical models. The field data for Fagus crenata fit the uniform stress model best, indicating that stress uniformity is the key constraint of branch morphology in Fagus crenata rather than elastic similarity or da Vinci''s rule. On the other hand, mechanical constraints are not necessarily significant in the morphology of Abies homolepis branches, depending on the number of daughter branches. Rather, these branches were often in agreement with da Vinci''s rule.     

FOG-2, a cofactor for GATA transcription factors, is essential for heart morphogenesis and development of coronary vessels from epicardium

We disrupted the FOG-2 gene in mice to define its requirement in vivo. FOG-2(-/-) embryos die at midgestation with a cardiac defect characterized by a thin ventricular myocardium, common atrioventricular canal, and the tetralogy of Fallot malformation. Remarkably, coronary vasculature is absent in FOG-2(-/-) hearts. Despite formation of an intact epicardial layer and expression of epicardium-specific genes, markers of cardiac vessel development (ICAM-2 and FLK-1) are not detected, indicative of failure to activate their expression and/or to initiate the epithelial to mesenchymal transformation of epicardial cells. Transgenic reexpression of FOG-2 in cardiomyocytes rescues the FOG-2(-/-) vascular phenotype, demonstrating that FOG-2 function in myocardium is required and sufficient for coronary vessel development. Our findings provide the molecular inroad into the induction of coronary vasculature by myocardium in the developing heart.     

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.     

Cerebral Neovascularization and Remodeling Patterns in Two Different Models of Type 2 Diabetes

We previously reported intense pial cerebral collateralization and arteriogenesis in a mild and lean model of type 2 diabetes (T2D), Goto-Kakizaki (GK) rats. Increased cerebral neovascularization differed regionally and was associated with poor vessel wall maturity. Building upon these findings, the goals of this study were to determine whether a) glycemic control prevents this erratic cerebral neovascularization in the GK model, and b) this pathological neovascularization pattern occurs in Lepr^db/db model, which is the most commonly used model of T2D for studies involving cerebral complications of diabetes. Vascular volume, surface area and structural parameters including microvessel/macrovessel ratio, non-FITC (fluorescein) perfusing vessel abundance, vessel tortuosity, and branch density were measured by 3D reconstruction of FITC stained vasculature in GK rats or Lepr^db/db mice. GK rats exhibited an increase in all of these parameters, which were prevented by glycemic control with metformin. In Lepr^db/db mice, microvascular density was increased but there was no change in nonFITC-perfusing vessels. Increased PA branch density was associated with reduced branch diameter. These results suggest that T2D leads to cerebral neovascularization and remodeling but some structural characteristics of newly formed vessels differ between these models of T2D. The prevention of dysfunctional cerebral neovascularization by early glucose control suggests that hyperglycemia is a mediator of this response.     

On connecting large vessels to small. The meaning of Murray's law

A large part of the branching vasculature of the mammalian circulatory and respiratory systems obeys Murray's law, which states that the cube of the radius of a parent vessel equals the sum of the cubes of the radii of the daughters. Where this law is obeyed, a functional relationship exists between vessel radius and volumetric flow, average linear velocity of flow, velocity profile, vessel-wall shear stress, Reynolds number, and pressure gradient in individual vessels. In homogeneous, full-flow sets of vessels, a relation is also established between vessel radius and the conductance, resistance, and cross- sectional area of a full-flow set.     

Bim is responsible for the inherent sensitivity of the developing retinal vasculature to hyperoxia

Apoptosis plays an important role in development and remodeling of vasculature during organogenesis. Coordinated branching and remodeling of the retinal vascular tree is essential for normal retinal function. Bcl-2 family members, such as bim not only influence apoptosis, but also cell adhesive and migratory properties essential during vascular development. Here we examined the impact of bim deficiency on postnatal retinal vascularization, as well as retinal neovascularization during oxygen-induced ischemic retinopathy (OIR) and laser-induced choroidal neovascularization. Loss of bim expression was associated with increased retinal vascular density in mature animals. This was mainly attributed to increased numbers of pericytes and endothelial cells. However, the initial spread of the superficial layer of retinal vasculature and, the appearance and density of the tip cells were similar in bim+/+ and bim−/− mice. In addition, hyaloid vessel regression was attenuated in the absence of bim. Furthermore, in the absence of bim retinal vessel obliteration and neovascularization did not occur during OIR. Instead, normal inner retinal vascularization proceeded independent of changes in oxygen levels. In contrast, choroidal neovascularization occurred equally well in bim+/+ and bim−/− mice. Together our data suggest bim expression may be responsible for the inherent sensitivity of the developing retinal vasculature to changes in oxygen levels, and promotes vessel obliteration in response to hyperoxia.     

Computer-assisted measurement of vessel shape from 3T magnetic resonance angiography of mouse brain

Blood vessel morphology (vessel radius, branching pattern, and tortuosity) is altered by a multitude of diseases. Although murine models of human pathology are important to the investigation of many diseases, there are few publications that address quantitative measurements of murine vascular morphology. This report outlines methods of imaging mice in vivo using magnetic resonance angiograms obtained on a clinical 3T unit, of defining mouse vasculature from these images, and of quantifying measures of vessel shape. We provide examples of both healthy and diseased vasculature and illustrate how the approach can be used to assess pathology both visually and quantitatively. The method is amenable to the assessment of many diseases in both human beings and mice.     

Distribution of Cell Adhesion Molecules (CAM) along the Branching Neurite Surface: Model-Inherited Effects of the Branch Diameters and Mode of CAM Transport

In many cases, an increase in the surface density of cell adhesion molecules (CAM) in the distal parts of a growing neurite is favorable for the neurite elongation. This increase is attained by exocytotic insertion of CAM-containing vesicles into the growth cones with subsequent redistribution of CAM along the cell surface due to lateral diffusion and endocytosis. Using a mathematical model describing these processes, we quantitatively describe conditions providing two qualitatively different profiles in a branching neurite: (i) the CAM surface density increases along both daughter branches, which would be in favor of further outgrowth of both branches, i.e., successful branching, or (ii) the CAM surface density increases along one daughter branch and decreases along another branch, which could lead to the retraction of the latter. The geometric factors and mechanisms underlying the intracellular CAM transport to the daughter growth cones were proved to determine the profile of CAM surface density. A similarity in the diameters of daughter branches, their short lengths, a high value of the lateral transfer constant, and partitioning of CAM transport at the branching point proportionally to the surface areas of daughter branches are in favor of an increase in the CAM surface density along both daughter branches. Asymmetric branching can lead to a decrease in the CAM surface density along the thinner or thicker daughter branch, if CAM trafficking was equally partitioned or was proportional to the branch cross-sectional areas, respectively. The proposed model helps to understand possible relationships between the intracellular CAM trafficking, CAM surface distribution, and geometry of branching of the neurites.