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.     

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.     

The blood-brain barrier: an alternative hypothesis

Brain blood vessels, unlike most vessels elsewhere in the body, exhibit a blood-brain barrier (BBB) to certain substances, e.g. trypan blue. Under some circumstances this barrier is no longer effective and the permeability of the vessels increases. Although capillarization is much less in the brain than in many other organs, e.g. heart muscle, total cerebral blood flow per minute is enormous. Consequently, to accommodate a large blood volume with a limited capillary bed, the velocity of blood through brain vessels must be extremely fast. The hypothesis presented in this paper is that this rapid flow results in a low or negative pressure on the endothelium, and plasma and trypan blue are prevented from passing through the wall. The tight junctions of cerebral endothelial cells may be able to withstand only a limited amount of pressure on their luminal surface. If the velocity of blood in brain capillaries decreases, pressure on the endothelium should increase, and brain vessels, like blood vessels elsewhere in the body, become permeable to vital dyes. Other conditions also increase capillary permeability, e.g. acute arterial hypertension or venous congestion. Although brain vessels can adapt to a moderate, gradual change in systemic pressure, when a significant rise in cerebral arterial pressure is abrupt, the compensatory changes in the postcapillary venous bed may be inadequate and consequently intracapillary pressure and vascular permeability are increased. Venous congestion increases intracapillary pressure by restricting capillary outflow as well as by reducing velocity through capillary beds. Under such conditions increased capillary permeability may be indicated by cerebral edema, and even, on occasion, by petechial hemorrhages. In short, if the flow is fast and unimpeded the BBB will be effective; if the velocity decreases, or intracapillary pressure increases for whatever reason, the permeability of the brain endothelium will be abnormally increased.     

A hybrid one-dimensional/Womersley model of pulsatile blood flow in the entire coronary arterial tree

Using a frequency-domain Womersley-type model, we previously simulated pulsatile blood flow throughout the coronary arterial tree. Although this model represents a good approximation for the smaller vessels, it does not take into account the nonlinear convective energy losses in larger vessels. Here, using Womersley's theory, we present a hybrid model that considers the nonlinear effects for the larger epicardial arteries while simulating the distal vessels (down to the 1st capillary segments) with the use of Womersley's Theory. The main trunk and primary branches were discretized and modeled with one-dimensional Navier-Stokes equations, while the smaller-diameter vessels were treated as Womersley-type vessels. Energy losses associated with vessel bifurcations were incorporated in the present analysis. The formulation enables prediction of impedance and pressure and pulsatile flow distribution throughout the entire coronary arterial tree down to the first capillary segments in the arrested, vasodilated state. We found that the nonlinear convective term is negligible and the loss of energy at a bifurcation is small in the larger epicardial vessels of an arrested heart. Furthermore, we found that the flow waves along the trunk or at the primary branches tend to scale (normalized with respect to their mean values) to a single curve, except for a small phase angle difference. Finally, the model predictions for the inlet pressure and flow waves are in excellent agreement with previously published experimental results. This hybrid one-dimensional/Womersley model is an efficient approach that captures the essence of the hemodynamics of a complex large-scale vascular network. The present model has numerous applications to understanding the dynamics of coronary circulation.     

Direct observation of epicardial coronary capillary hemodynamics during reactive hyperemia and during adenosine administration by intravital video microscopy

Using high-resolution intravital charge-coupled device video microscopy, we visualized the epicardial capillary network of the beating canine heart in vivo to elucidate its functional role under control conditions, during reactive hyperemia (RH), and during intracoronary adenosine administration. The pencil-lens video-microscope probe was placed over capillaries fed by the left anterior descending artery in atrioventricular-blocked hearts of open-chest, anesthetized dogs paced at 60-90 beats/min (n = 17). In individual capillaries under control conditions, red blood cell flow was predominant during systole or diastole, indicating that the watershed between diastolic arterial and systolic venous flows is located within the capillaries. Capillary flow increased during RH and reached a peak flow velocity (2.1 +/- 0.6 mm/s), twice as high as control (1.2 +/- 0.5 mm/s), with enhancement of intercapillary cross-connection flow and enlargement of diameter (by 17%). With adenosine, capillary flow velocity significantly increased (1.8 +/- 0.7 mm/s). However, the increase in volumetric capillary flow with adenosine estimated from red blood cell velocity and diameter was less than the increase in arterial flow, whereas that during RH was nearly equivalent to the increase in arterial flow. There was a time lag of approximately 1.5 s for refilling of capillaries during RH, indicating their function as capacitance vessels. In conclusion, the coronary capillary network functions as 1) the major watershed between diastolic-dominant arterial and systolic-dominant venous flows, 2) a capacitor, and 3) a significant local flow amplifier and homogenizer of blood supply during RH, but with adenosine the increase in capillary flow velocity was less than the increase in arterial flow.     

Effect of synthetic physalaemin on splanchnic circulation in dogs

Physalaemin has been reported as one of the most potent vasodilator and hypotensive peptides (1-4). In spite of these studies, however, the effect of the peptide on splanchnic circulation is not known precisely. In the present study, the effect of synthetic physalaemin on superior mesenteric arterial blood flow, portal venous blood flow and pancreatic capillary blood flow was investigated in dogs. Dose dependent increases of superior mesenteric arterial blood flow and portal venous blood flow were induced in response to physalaemin (0.1-10.0 ng/kg). Superior mesenteric arterial blood flow and portal venous blood flow attained maximal increases of 77 +/- 8.9% and 70 +/- 8.6%, respectively, at a dose of 5 ng/kg. Physalaemin caused a dose-related decrease in systemic arterial blood pressure. Pancreatic capillary blood flow did not show significant change with the administration of physalaemin. These data suggest that physalaemin may play some physiological roles in the regulation of splanchnic 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.     

Effect of hemodilution on RBC velocity, supply rate, and hematocrit in the cerebral capillary network.

The effect of isovolemic hemodilution on the circulation of red blood cells (RBCs) in the cerebrocortical capillary network was studied by intravital videomicroscopy with use of a closed-cranial-window technique in the rat. Velocity and supply rate of RBCs were measured by tracking the movement and counting the number of fluorescently labeled cells. Arterial blood was withdrawn in increments of 2 ml and replaced by serum albumin. Arterial blood pressure was maintained constant with an infusion of methoxamine. Both velocity and supply rate of RBCs increased, by approximately equal amounts, as arterial hematocrit was reduced from 44 to 15%. The maximum increase in RBC velocity was 4.6 and in RBC supply rate was 5.2 times the baseline value. Calculated lineal density of RBC, an index of capillary hematocrit, did not change with hemodilution. The results suggest that RBC flow and oxygen supply in the cerebral capillary network are maintained during isovolemic hemodilution. The "optimal hematocrit" is as low as 15%.     

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.     

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.     

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.     

The coronary circulation in exercise training

Exercise training (EX) induces increases in coronary transport capacity through adaptations in the coronary microcirculation including increased arteriolar diameters and/or densities and changes in the vasomotor reactivity of coronary resistance arteries. In large animals, EX increases capillary exchange capacity through angiogenesis of new capillaries at a rate matched to EX-induced cardiac hypertrophy so that capillary density remains normal. However, after EX coronary capillary exchange area is greater (i.e., capillary permeability surface area product is greater) at any given blood flow because of altered coronary vascular resistance and matching of exchange surface area and blood flow distribution. The improved coronary capillary blood flow distribution appears to be the result of structural changes in the coronary tree and alterations in vasoreactivity of coronary resistance arteries. EX also alters vasomotor reactivity of conduit coronary arteries in that after EX, α-adrenergic receptor responsiveness is blunted. Of interest, α- and β-adrenergic tone appears to be maintained in the coronary microcirculation in the presence of lower circulating catecholamine levels because of increased receptor responsiveness to adrenergic stimulation. EX also alters other vasomotor control processes of coronary resistance vessels. For example, coronary arterioles exhibit increased myogenic tone after EX, likely because of a calcium-dependent PKC signaling-mediated alteration in voltage-gated calcium channel activity in response to stretch. Conversely, EX augments endothelium-dependent vasodilation throughout the coronary arteriolar network and in the conduit arteries in coronary artery disease (CAD). The enhanced endothelium-dependent dilation appears to result from increased nitric oxide bioavailability because of changes in nitric oxide synthase expression/activity and decreased oxidant stress. EX also decreases extravascular compressive forces in the myocardium at rest and at comparable levels of exercise, mainly because of decreases in heart rate and duration of systole. EX does not stimulate growth of coronary collateral vessels in the normal heart. However, if exercise produces ischemia, which would be absent or minimal under resting conditions, there is evidence that collateral growth can be enhanced. While there is evidence that EX can decrease the progression of atherosclerotic lesions or even induce the regression of atherosclerotic lesions in humans, the evidence of this is not strong due to the fact that most prospective trials conducted to date have included other lifestyle changes and treatment strategies by necessity. The literature from large animal models of CAD also presents a cloudy picture concerning whether EX can induce the regression of or slow the progression of atherosclerotic lesions. Thus, while evidence from research using humans with CAD and animal models of CAD indicates that EX increases endothelium-dependent dilation throughout the coronary vascular tree, evidence that EX reverses or slows the progression of lesion development in CAD is not conclusive at this time. This suggests that the beneficial effects of EX in CAD may not be the result of direct effects on the coronary artery wall. If this suggestion is true, it is important to determine the mechanisms involved in these beneficial effects.     

Targeted O2 delivery by low-P50 hemoglobin: a new basis for O2 therapeutics

To assess O2 delivery to tissue by a new surface-modified, polyethylene glycol-conjugated human hemoglobin [MP4; Po2 at 50% saturation of hemoglobin (P50); 5.4 mmHg], we studied microcirculatory hemodynamics and O2 release in golden Syrian hamsters hemodiluted with MP4 or polymerized bovine hemoglobin (PolyBvHb; P50 54.2 mmHg). Comparisons were made with the animals' hemodiluted blood with a non-O2 carrying plasma expander with similar solution properties (Dextran-70). Systemic hemodynamics (arterial blood pressure and heart rate) and acid-base parameters were not correlated with microhemodynamics (arteriolar and venular diameter, red blood cell velocity, and flow). Microscopic measurements of Po2 and the O2 equilibrium curves permitted analysis of O2 release in precapillary and capillary vessels by red blood cells and plasma hemoglobin separately. No significant differences between the groups of animals with respect to arteriolar diameter, flow, or flow velocity were observed, but the functional capillary density was significantly higher in the MP4-treated animals (67%) compared with PolyBvHb-treated animals (37%; P < 0.05) or dextran-treated animals (53%). In the PolyBvHb-treated animals, predominant O2 release (both red blood cells and plasma hemoglobin) occurred in precapillary vessels, whereas in MP4 animals most of the O2 was released from both red blood cells and plasma hemoglobin in capillaries. Base excess correlated directly with capillary O2 release but not systemic O2 content or total O2 release. Higher O2 extraction of both red blood cell and plasma hemoglobin in capillaries represents a new mechanism of action of cell-free hemoglobin. High O2 affinity appears to be an important property for cell-free hemoglobin solutions.     

Mechanics of blood supply to the heart: wave reflection effects in a right coronary artery.

Mechanics of blood flow in the coronary circulation have in the past been based largely on models in which the detailed architecture of the coronary network is not included because of lack of data: properties of individual vessels do not appear individually in the model but are represented collectively by the elements of a single electric circuit. Recent data from the human heart make it possible, for the first time, to examine the dynamics of flow in the coronary network based on detailed, measured vascular architecture. In particular, admittance values along the full course of the right coronary artery are computed based on actual lengths and diameters of the many thousands of branches which make up the distribution system of this vessel. The results indicate that effects of wave reflections on this flow are far more significant than those generally suspected to occur in coronary blood flow and that they are actually the reverse of the well known wave reflection effects in the aorta.     

Angiogenic network formation in the developing vertebrate trunk

We have used time-lapse multiphoton microscopy of living Tg(fli1:EGFP)y1 zebrafish embryos to examine how a patterned, functional network of angiogenic blood vessels is generated in the early vertebrate trunk. Angiogenic vascular sprouts emerge from the longitudinal trunk axial vessels (the dorsal aorta and posterior cardinal vein) in two spatially and temporally distinct steps. Dorsal aorta-derived sprouts form an initial primary network of vascular segments, followed by emergence of vein-derived secondary vascular sprouts that interact and interconnect dynamically with the primary network to initiate vascular flow. Using transgenic silent heart mutant embryos, we show that the gross anatomical patterning of this network of vessels does not require blood circulation. However, our results suggest that circulatory flow dynamics play an important role in helping to determine the pattern of interconnections between the primary network and secondary sprouts, and thus the final arterial or venous identity of the vessels in the functional network. We discuss a model to explain our results combining genetic programming of overall vascular architecture with hemodynamic determination of circulatory flow patterns.     

Microvascularization of the thyroid glands in larval and adult Xenopus laevis‐ histomorphology and scanning electron microscopy of vascular corrosion casts

This study demonstrates gross arterial supply, venous drainage and microvascular patterns of larval and adult thyroid glands in the African Clawed Frog, Xenopus laevis by scanning electron microscopy of vascular corrosion casts and light microscopy of stained serial tissue sections. Results confirm published findings gained by microscopical dissections with respect to gross arterial supply. However, in adult frogs one rather than two thyroid veins are found. This study reveals for the first time that bilaterally located thyroid glands in premetamorphosis have immature capillary networks, lack a clear hierarchy of blood vessels, and show many signs of sprouting angiogenesis. During metamorphic climax, blood vessels gain a clear hierarchy and capillaries form closed networks around thyroid follicles. From climax onwards, non‐sprouting angiogenesis (intussusceptive microvascular growth) becomes the prevailing mode of angiogenesis intensifying follicle capillarization. Due to narrow interfollicular spaces, thyroidal arterioles remain superficial while draining venules are located interfollicularly. In contrast with the mammalian thyroid gland where most thyroid follicles have their own capillary beds, most thyroid follicles in Xenopus share their capillary beds with neighbouring follicles. Consequently, the concept of individual morphological and functional angiofollicular units applicable to adult mammalian and human thyroid glands is not applicable to larval and adult amphibian thyroid glands.     

Coronary artery flow measurement using navigator echo gated phase contrast magnetic resonance velocity mapping at 3.0 T

A validation study and early results for non-invasive, in vivo measurement of coronary artery blood flow using phase contrast magnetic resonance imaging (PC-MRI) at 3.0T is presented. Accuracy of coronary artery blood flow measurements by phase contrast MRI is limited by heart and respiratory motion as well as the small size of the coronary arteries. In this study, a navigator echo gated, cine phase velocity mapping technique is described to obtain time-resolved velocity and flow waveforms of small diameter vessels at 3.0T. Phantom experiments using steady, laminar flow are presented to validate the technique and show flow rates measured by 3.0T phase contrast MRI to be accurate within 15% of true flow rates. Subsequently, in vivo scans on healthy volunteers yield velocity measurements for blood flow in the right, left anterior descending, and left circumflex arteries. Measurements of average, cross-sectional velocity were obtainable in 224/243 (92%) of the cardiac phases. Time-averaged, cross-sectional velocity of the blood flow was 6.8+/-4.3cm/s in the LAD, 8.0+/-3.8cm/s in the LCX, and 6.0+/-1.6cm/s in the RCA.     

Homogenization modeling for the mechanics of perfused myocardium

Altered coronary perfusion can change the apparent diastolic stiffness of ventricular myocardium--the ‘garden hose’ effect. Our recent findings showed that myocardial strains are reduced during ventricular filling, primarily along the directions transverse to the coronary microvessels. In this article, we review hypotheses and theoretical models regarding the role that regional wall stress plays in the mechanical interaction between myocardium and coronary circulation. Various mechanisms have been used to explain the effects of the tissue stress on coronary flow, as well as the effect of coronary dynamics on myocardial mechanics. Many models of coronary pressure-flow relations using lumped parameter circuit analogs. Poroelasticity and swelling theories have been used to model the mechanics of perfused muscle. Here, we describe a new mathematical model of the mechanics of perfused myocardium derived using homogenization theory. In this model, perfused myocardium is treated as a nonlinear anisotropic elastic solid embedded with cylindrical vessels of known distensibility. The solid compartment is incompressible but the vascular compartment may change volume according to a simple relation between vessel diameter and perfusion pressure. The work done by the perfusion pressure in changing vascular volume contributes to the macroscopic strain energy and hence affects the stress and stiffness of the composite. Conversely, the stress in the tissue affects microvessel diameter and volume, since tractions transverse to the vessel axis oppose the internal blood pressure. Finite element simulations of passive filling show good agreement of model with experimental results.     

Coronary three-vessel disease with occlusion of the right coronary artery: What are the most important factors that determine the right territory perfusion?

With progressive occlusion of a coronary main artery, some anastomotic vessels are recruited in order to supply blood to the ischemic region. This collateral circulation is an important factor in the preservation of the myocardium until reperfusion of the area at risk. An accurate estimation of collateral flow is crucial in surgical bypass planning as it alters the blood flow distribution in the coronary network and can influence the outcome of a given treatment for a given patient. The evaluation of collateral flow is frequently achieved using an index based on pressure measurements. It is named collateral flow index (CFI) and defined as: (P_w − P_v)/(P_ao − P_v), where P_w is the pressure distal to the thrombosis, P_ao the aortic pressure and P_v the central venous pressure. In the present work, we study patients with severe coronary disease (stenoses on the left branches and total occlusion of the right coronary artery). Using a mathematical model that describes the coronary hemodynamics in that situation, we demonstrate that the dependence of the collateral circulation to the pressure values is not as simple as it is commonly believed: using pressures alone as an index of collateral flow is likely to result in misinterpretation of the collateral flow contribution, because collateral flow depends on many other factors related to the status of the native stenosed arteries and to the microvascular resistances (capillary and collateral resistances, and the proportion between them).