A new approach to blood flow simulation in vascular networks

A proper analysis of blood flow is contingent upon accurate modelling of the branching pattern and vascular geometry of the network of interest. It is challenging to reconstruct the entire vascular network of any organ experimentally, in particular the pulmonary vasculature, because of its very high number of vessels, complexity of the branching pattern and poor accessibility in vivo. The objective of our research is to develop an innovative approach for the reconstruction of the full pulmonary vascular tree from available morphometric data. Our method consists of the use of morphometric data on those parts of the pulmonary vascular tree that are too small to reconstruct by medical imaging methods. This method is a three-step technique that reconstructs the entire pulmonary arterial tree down to the capillary bed. Vessels greater than 2 mm are reconstructed from direct volume and surface analysis using contrast-enhanced computed tomography. Vessels smaller than 2 mm are reconstructed from available morphometric and distensibility data and rearranged by applying Murray's laws. Implementation of morphometric data to reconstruct the branching pattern and applying Murray's laws to every vessel bifurcation simultaneously leads to an accurate vascular tree reconstruction. The reconstruction algorithm generates full arterial tree topography down to the ?rst capillary bifurcation. Geometry of each order of the vascular tree is generated separately to minimize the construction and simulation time. The node-to-node connectivity along with the diameter and length of every vessel segment is established and order numbers, according to the diameter-de?ned Strahler system, are assigned. In conclusion, the present model provides a morphological foundation for future analysis of blood flow in the pulmonary circulation     

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.     

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.     

Network analysis of an arterial tree

The arterial tree of a Sprague-Dawley rat was casted and carefully mapped with the aim of comparing its network characteristics with those suggested by the classical model of an arterial tree. It is shown that if the tree is to be measured accurately, the concept of 'whole vessels' on which the classical model is based must be abandoned since such vessels do not actually exist in the network, nor can they be accurately defined. The concept of 'vessel segments' is proposed instead and its use is demonstrated. A total of 1313 vessel segments in the arterial tree of the rat are mapped and divided into well defined 'levels'. The length and diameter of each segment are measured and the distribution and averages of these at different levels are presented as indicators of the branching characteristics of the tree.     

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.     

Heterogeneous mechanics of the mouse pulmonary arterial network

Individualized modeling and simulation of blood flow mechanics find applications in both animal research and patient care. Individual animal or patient models for blood vessel mechanics are based on combining measured vascular geometry with a fluid structure model coupling formulations describing dynamics of the fluid and mechanics of the wall. For example, one-dimensional fluid flow modeling requires a constitutive law relating vessel cross-sectional deformation to pressure in the lumen. To investigate means of identifying appropriate constitutive relationships, an automated segmentation algorithm was applied to micro-computerized tomography images from a mouse lung obtained at four different static pressures to identify the static pressure–radius relationship for four generations of vessels in the pulmonary arterial network. A shape-fitting function was parameterized for each vessel in the network to characterize the nonlinear and heterogeneous nature of vessel distensibility in the pulmonary arteries. These data on morphometric and mechanical properties were used to simulate pressure and flow velocity propagation in the network using one-dimensional representations of fluid and vessel wall mechanics. Moreover, wave intensity analysis was used to study effects of wall mechanics on generation and propagation of pressure wave reflections. Simulations were conducted to investigate the role of linear versus nonlinear formulations of wall elasticity and homogeneous versus heterogeneous treatments of vessel wall properties. Accounting for heterogeneity, by parameterizing the pressure/distention equation of state individually for each vessel segment, was found to have little effect on the predicted pressure profiles and wave propagation compared to a homogeneous parameterization based on average behavior. However, substantially different results were obtained using a linear elastic thin-shell model than were obtained using a nonlinear model that has a more physiologically realistic pressure versus radius relationship.     

Morphometry of the human pulmonary vasculature

Huang, W., R. T. Yen, M. McLaurine, and G. Bledsoe.Morphometry of the human pulmonary vasculature.J. Appl. Physiol. 81(5):2123-2133, 1996.The morphometric data on the branching patternand vascular geometry of the human pulmonary arterial and venous treesare presented. Arterial and venous casts were prepared by the siliconeelastomer casting method. Three recent innovations are used to describethe vascular geometry: the diameter-defined Strahler ordering model isused to assign branching orders, the connectivity matrix is used todescribe the connection of blood vessels from one order to another, anda distinction between vessel segments and vessel elements is used toexpress the series-parallel feature of the pulmonary vessels. A totalof 15 orders of arteries were found between the main pulmonary arteryand the capillaries in the left lung and a total of 15 orders of veinsbetween the capillaries and the left atrium in the right lung. Theelemental and segmental data are presented. The morphometric data arethen used to compute the total cross-sectional areas, blood volumes, and fractal dimensions in the pulmonary arterial and venous trees.

     

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.     

Functional hierarchy of coronary circulation: direct evidence of a structure-function relation

The heart muscle is nourished by a complex system of blood vessels that make up the coronary circulation. Here we show that the design of the coronary circulation has a functional hierarchy. A full anatomic model of the coronary arterial tree, containing millions of blood vessels down to the capillary vessels, was simulated based on previously measured porcine morphometric data. A network analysis of blood flow through every vessel segment was carried out based on the laws of fluid mechanics and appropriate boundary conditions. Our results show an abrupt change in cross-sectional area that demarcates the transition from epicardial (EPCA) to intramyocardial (IMCA) coronary arteries. Furthermore, a similar pattern of blood flow was observed with a corresponding transition from EPCA to IMCA. These results suggest functional differences between the two types of vessels. An additional abrupt change occurs in the IMCA in relation to flow velocity. The velocity is fairly uniform proximal to these vessels but drops significantly distal to those vessels toward the capillary branches. This finding suggests functional differences between large and small IMCA. Collectively, these observations suggest a novel functional hierarchy of the coronary vascular tree and provide direct evidence of a structure-function relation.     

Bronchial collateral vessel micropuncture pressure in postobstructive pulmonary vasculopathy.

Postobstructive pulmonary vasculopathy, produced by chronic ligation of one pulmonary artery, markedly increases bronchial blood flow. Previously, using arterial and venous occlusion, we determined that bronchial collaterals enter the pulmonary circuit at the distal end of the arterial segment. In this study, we tested the hypothesis that pressure in bronchial collaterals (Pbr) closely approximates that at the downstream end of the arterial segment (Pao). We pump perfused [111 +/- 10 (SE) ml/min] left lower lobes of seven open-chest live dogs 3-15 mo after ligation of the left main pulmonary artery. Bronchial blood flow was 122 +/- 16 ml/min. We measured pulmonary arterial and venous pressures and, by arterial and venous occlusion, respectively, Pao and the pressure at the upstream end of the venous segment (Pvo). Pbr was obtained by micropuncture of 34 pleural surface bronchial vessels 201 +/- 16 microns in diameter. We found that Pbr (14.4 +/- 1.0 mmHg) was similar to Pao (15.0 +/- 0.8 mmHg) but differed significantly (P < 0.01) from Pvo (11.3 +/- 0.5 mmHg). In addition, Pbr was independent of systemic arterial pressure and bronchial vessel diameter. Light and electron microscopy revealed that, in the lobes with the ligated pulmonary artery, the new bronchial collaterals entered the thickened pleura from the parenchyma via either bronchovascular bundles or interlobular septa and had sparsely muscularized walls. We conclude that, in postobstructive pulmonary vasculopathy, bronchial collateral pressure measured by micropuncture is very close to the pressure in precapillary pulmonary arteries and that most of the pressure drop in the bronchial collaterals occurs in vessels > 350 microns in diameter.     

Dominance of geometric overelastic factors in pulse transmission through arterial branching

The branching characteristic of the arterial system is such that blood pressure pulses propagate with minimum loss. This characteristic depends on the geometric and elastic properties of branching vessels. In the current investigation, mathematical relations of branching geometry and elastic properties are formulated and their relative contributions to pulse reflection at an arterial junction are analyzed. Results show that alteration of pulse transmission through the junction is more significantly affected by changes in branching vessel radii and wall thickness than by corresponding percentage changes in vessel wall elastic moduli.     

Morphometry of cat's pulmonary arterial tree

Morphometic data of the pulmonary artery in the cat's right lung are presented. Silicone elastomer casts of cat's right lung were made, and measured, counted and analyzed. The Strahler system is used to describe the branching pattern of the arterial vascular tree. These data are needed for any quantitative approach to the study of the pulmonary circulation. For all the pulmonary blood vessels of the cat lying between the main pulmonary artery and the capillary beds, there are a total of 10 orders of vessels in the right upper lobe, 9 orders of vessels in the right middle lobe and 11 orders of vessels in the right lower lobe. The ratio of the number of branches in successive orders of vessels or the branching ratio, is 3.58. The corresponding average diameter ratio is 1.72, whereas the average length ratio is 1.81.     

Wave Transmission through an Assembly of Randomly Branching Elastic Tubes

Calculations are presented of the transmission of oscillations through an assembly of randomly branching elastic tubes, as a model of not only the major arteries, but also a peripheral vascular bed. It appears that the viscosity of the arterial wall must be the major source of attenuation in the larger arteries, while the viscosity of the blood plays a significant role only in the smaller vessels. In all situations, variations of cross-sectional area have a considerable effect on wave transmission, causing a general decrease in amplitude and an accentuation of reflection from the terminations. The effects of variation in cross-sectional area are sufficiently great to indicate that they should be included in future models of the arterial system. Finally, it is argued that because of the presence of random branching and elastic nonuniformity, the determination of the reflection coefficient for a system such as the arterial tree may be quite misleading.     

Noninvasive assessment of pulmonary arterial hypertension by MR phase-mapping method.

We describe a magnetic resonance (MR) imaging method that emphasizes pressure wave velocity to noninvasively assess pulmonary arterial hypertension. Both the blood flow and the corresponding vessel cross-sectional area (CSA) were measured by MR phase mapping in the main pulmonary artery (MPA) in 15 patients. MPA pressures were also measured, in the same patients, by right-side heart catheterization. Two significant relationships were established: 1) between the pressure wave velocity in the MPA and the mean pressure in the MPA (Ppa) writing pressure wave velocity = 9.25 Ppa - 202.51 (r = 0.82) and 2) between the ratio of pressure wave velocity to the systolic blood velocity peak in the MPA (R) and the mean pressure in the MPA writing R = 0.68 Ppa - 4.33 (r = 0.89). Using these relationships, we estimated two pressure values to frame the actual Ppa value in each patient from the present series with a reasonable reliability percentage (87%).     

Shear stress distribution in arterial tree models, generated by constrained constructive optimization.

Models of arterial trees are generated by the algorithm of Constrained Constructive Optimization (CCO). Straight cylindrical, binary branching tubes are arranged in an optimized fashion so as to convey blood to the terminal sites of the tree, which are distributed over a predefined area, representing the tissue to be perfused. All terminal segments supply equal flows at a unique terminal pressure, and the radii of parent and daughter segments are related via a bifurcation law. The connective structure and geometry of the model are optimized according to a target function such as total intravascular volume. The shear rate between blood and the vessel walls is computed in each segment and a new method is presented for rescaling a given CCO tree to a desired value of shear rate in the root segment. The effect of viscosity varying with shear rate is evaluated and a new method is presented for rescaling a CCO-tree segment by segment to consistent values of radii and variable viscosity. Shear stress is evaluated for its deviation from being proportional to shear rate and then subjected to various types of analyses. Usually both, shear stress and its variability, are found to be larger in the smaller than in the larger segments of the CCO-model trees. However, it is shown how the shear-stress distribution can be reshuffled between small and large segments when rescaling a CCO tree to obey a different bifurcation law, while its whole geometry remains unchanged and all boundary conditions remain fulfilled. The selection of optimization target is found to drastically affect shear-stress variability within bifurcations, which reaches a distinct minimum if the model is optimized according to intravascular volume. Finally, a rank-analysis of shear stress within each bifurcation shows that only two out of six possible rank patterns actually occur: the parent segment always experiences medium shear stress while minimum shear stress resides mostly in the larger, less frequently in the smaller daughter.     

The subscapular arterial tree as a source of microvascular arterial grafts

The subscapular arterial tree may be used as a source of microvascular grafts to replace damaged or diseased portions of arteries, particularly in the hand and forearm. By studying cadaver dissections, it is possible to estimate the number of branches that may be found at different arterial segment lengths from the origin of the subscapular artery. Fifty-five preserved cadaver subscapular arterial trees were dissected, and the branching patterns were documented. Three major arterial branching patterns of the subscapular artery were observed with one, two, and three major branches to the serratus anterior in 60 percent, 29 percent, and 9 percent of the cases, respectively. The authors determined the number of 1-mm-diameter, 1-cm-long branches arising from each of six 3-cm regions of the arterial tree measured from the origin of the subscapular artery to the end of the longest terminal branch. The probability of finding at least one usable terminal branch that is at least 12.0 cm in length was found to be 98 percent. Typically, there are two to five useful branches at this distance. Such information may help surgeons fine tune their process of selecting an appropriate arterial donor site for a particular arterial defect and supports the use of the subscapular arterial tree as a donor site for microvascular arterial grafts.     

Arterial branching within the confines of fractal L-system formalism

Parametric Lindenmayer systems (L-systems) are formulated to generate branching tree structures that can incorporate the physiological laws of arterial branching. By construction, the generated trees are de facto fractal structures, and with appropriate choice of parameters, they can be made to exhibit some of the branching patterns of arterial trees, particularly those with a preponderant value of the asymmetry ratio. The question of whether arterial trees in general have these fractal characteristics is examined by comparison of pattern with vasculature from the cardiovascular system. The results suggest that parametric L-systems can be used to produce fractal tree structures but not with the variability in branching parameters observed in arterial trees. These parameters include the asymmetry ratio, the area ratio, branch diameters, and branching angles. The key issue is that the source of variability in these parameters is not known and, hence, it cannot be accurately reproduced in a model. L-systems with a random choice of parameters can be made to mimic some of the observed variability, but the legitimacy of that choice is not clear.     

Pulmonary acinus: geometry and morphometry of the peripheral airway system in rat and rabbit

The geometry and morphometry of intraacinar airways in rat and rabbit lungs were studied from silicone rubber casts. Acini, defined as the complex of alveolated airways distal to the "terminal" bronchiole, were trimmed off the bronchial tree. In both species, the acinar volume followed a log-normal distribution over a range in size of one order of magnitude. At an inflation level of 60% total lung capacity, their mean volume was 1.86 mm3 in the rat and 3.46 mm3 in the rabbit. On a representative sample of acini of different volumes, the branching pattern was characterized as irregular dichotomy, and the segment length and inner and outer diameters were measured. The average acinus had a mean of six generations in the rat and seven in the rabbit. Both showed a decrease in segment length and inner diameter with each generation. The mean longitudinal pathway length--that is, the distance from the initial acinar segment to the terminal sacs--was found to depend on the cube root of the acinar volume in both species. It was calculated at 1.46 and 1.95 mm for rat and rabbit, respectively.     

Relation between diameter and flow in major branches of the arch of the aorta.

In the analysis of arterial branching the classical "cube law' has provided a working model for the relation between the diameter of a blood vessel and the flow which the vessel carries on a long-term basis. The law has shown good agreement with biological data, but questions remain regarding its applicability to all levels of the arterial tree. The present study tests the hypothesis that the cube law may not be valid in the first few generations of the arterial tree, where vessel capacitance and gross anatomy may play important roles. Biological data have shown some support for this hypothesis in the past but the heterogeneity characteristic of past data has not allowed a conclusive test so far. We present new data which have been obtained from the same location on the arterial tree and in sufficient number to make this test possible for the first time. Also, while past tests have been based primarily on correlation of the measured data with an assumed power law, we show here that this can be misleading. The present data allow a simpler test which does not involve correlation and which leads to more direct conclusions. For the vessels surveyed, the results show unequivocally that the relation between diameter and flow is governed by a 'square law' rather than the classical cube law. Coupled with past findings this suggests that the square law may apply at the first few levels of the arterial tree, while the cube law continues from there to perhaps the precapillary levels.