Structure-function relationships in the pulmonary arterial tree

Knowledge of the relationship between structure and function ofthe normal pulmonary arterial tree is necessary for understanding normal pulmonary hemodynamics and the functional consequences of thevascular remodeling that accompanies pulmonary vascular diseases. In aneffort to provide a means for relating the measurable vascular geometryand vessel mechanics data to the mean pressure-flow relationship andlongitudinal pressure profile, we present a mathematical model of thepulmonary arterial tree. The model is based on the observation that thenormal pulmonary arterial tree is a bifurcating tree in which theparent-to-daughter diameter ratios at a bifurcation and vesseldistensibility are independent of vessel diameter, and although theactual arterial tree is quite heterogeneous, the diameter of eachroute, through which the blood flows, tapers from the arterial inlet toessentially the same terminal arteriolar diameter. In the model theaverage route is represented as a tapered tube through which the bloodflow decreases with distance from the inlet because of the diversion offlow at the many bifurcations along the route. The taper and flowdiversion are expressed in terms of morphometric parameters obtainedusing various methods for summarizing morphometric data. To help putthe model parameter values in perspective, we applied one such methodto morphometric data obtained from perfused dog lungs. Modelsimulations demonstrate the sensitivity of model pressure-flowrelationships to variations in the morphometric parameters. Comparisonsof simulations with experimental data also raise questions as to the"hemodynamically" appropriate ways to summarize morphometric data.     

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.     

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.     

Mapping 3-D functional capillary geometry in rat skeletal muscle in vivo

We have developed a novel mapping software package to reconstruct microvascular networks in three dimensions (3-D) from in vivo video images for use in blood flow and O2 transport modeling. An intravital optical imaging system was used to collect video sequences of blood flow in microvessels at different depths in the tissue. Functional images of vessels were produced from the video sequences and were processed using automated edge tracking software to yield location and geometry data for construction of the 3-D network. The same video sequences were analyzed for hemodynamic and O2 saturation data from individual capillaries in the network. Simple user-driven commands allowed the connection of vessel segments at bifurcations, and semiautomated registration enabled the tracking of vessels across multiple focal planes and fields of view. The reconstructed networks can be rotated and manipulated in 3-D to verify vessel connections and continuity. Hemodynamic and O2 saturation measurements made in vivo can be indexed to corresponding vessels and visualized using colorized maps of the vascular geometry. Vessels in each reconstruction are saved as text-based files that can be easily imported into flow or O2 transport models with complete geometry, hemodynamic, and O2 transport conditions. The results of digital morphometric analysis of seven microvascular networks showed mean capillary diameters and overall capillary density consistent with previous findings using histology and corrosion cast techniques. The described mapping software is a valuable tool for the quantification of in vivo microvascular geometry, hemodynamics, and oxygenation, thus providing rich data sets for experiment-based computational models.     

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.     

Branching pattern of pulmonary arterial tree in anesthetized dogs

The geometry of the pulmonary arterial tree of six adult dogs was measured by a high-speed, volume-scanning, X-ray tomographic technique. After the dogs were anesthetized a catheter was advanced to the right ventricular outflow tract and 2 mL/kg Renovist contrast agent injected rapidly. During the subsequent pulmonary arterial phase of the angiogram the dogs were scanned. Three-dimensional geometry of the pulmonary arterial tree was measured in terms of vessel segment cross-sectional area, branching angles and interbranch segment lengths along axial pathways. The effect of lung inflation and phase of the cardiac cycle on geometry was shown to be most marked on vessel cross-sectional area. The geometric branching patterns in all dogs were similar. The observed, in-vivo branching pattern behaved somewhat like the branching pattern predicted from optimized models proposed by Murray, Zamir, and Uylings.     

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.     

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.     

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.     

A fractal continuum model of the pulmonary arterial tree.

The extant morphometric data from the intrapulmonary arteries of dog, human, and cat lungs produce graphs of the log of the vessel number, (N) or length (l) in each level vs. the log of the mean diameter (D) in each level that are sufficiently linear to suggest that a scale-independent self-similar or fractal structure may underlie the observed relationships. These data can be correlated by the following formulas: Nj = a1Dj-beta 1, and lj = a2Dj beta 2, where j denotes the level (order or generation) number measured from the largest vessel at the entrance to the arterial tree to the smallest vessel at the entrance to the capillary bed. With the hemodynamic resistance (R) represented by Rj = 128 microliterj/(Nj pi Dj4) and the vascular volume (Q) by Qj = Nj pi Dj2lj/4, the continuous cumulative distribution of vascular resistance (Rcum) vs. cumulative vascular volume (Qcum) (where Rcum and Qcum represent the total resistance or volume, respectively, upstream from the jth level) can be calculated from [formula: see text] where r = Dj/Dj+1 is a constant independent of j. Analogous equations are developed for the inertance and compliance distributions, providing simple formulas to represent the hemodynamic consequences of the pulmonary arterial tree structure.     

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.

     

Nanoparticle transport and delivery in a heterogeneous pulmonary vasculature

Quantitative understanding of nanoparticles delivery in a complex vascular networks is very challenging because it involves interplay of transport, hydrodynamic force, and multivalent interactions across different scales. Heterogeneous pulmonary network includes up to 16 generations of vessels in its arterial tree. Modeling the complete pulmonary vascular system in 3D is computationally unrealistic. To save computational cost, a model reconstructed from MRI scanned images is cut into an arbitrary pathway consisting of the upper 4-generations. The remaining generations are represented by an artificially rebuilt pathway. Physiological data such as branch information and connectivity matrix are used for geometry reconstruction. A lumped model is used to model the flow resistance of the branches that are cut off from the truncated pathway. Moreover, since the nanoparticle binding process is stochastic in nature, a binding probability function is used to simplify the carrier attachment and detachment processes. The stitched realistic and artificial geometries coupled with the lumped model at the unresolved outlets are used to resolve the flow field within the truncated arterial tree. Then, the biodistribution of 200 nm, 700 nm and 2 µm particles at different vessel generations is studied. At the end, 0.2–0.5% nanocarrier deposition is predicted during one time passage of drug carriers through pulmonary vascular tree. Our truncated approach enabled us to efficiently model hemodynamics and accordingly particle distribution in a complex 3D vasculature providing a simple, yet efficient predictive tool to study drug delivery at organ level.     

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.     

A model for renal arterial branching based on graph theory

The kidney is one of the most complicated organs in terms of structure and physiology, in part because it is highly vascularized. The renal vascular development occurs through two mechanisms that sometimes overlap: vasculogenesis and angiogenesis. Here, we consider angiogenesis to model the renal arterial tree with the two processes of vascular angiogenesis: sprouting and splitting. We recognize the vessels are not tubes with ends that get glued but physiological factors are relevant into the vascular development. Our contribution integrates the graph theory and physiological information to derive a quantitative model for the vascular tree in the sense that the vertices and edges represent, respectively, a branching point and a vessel. From such a premise, development of the arterial vascular tree of the kidney is mathematically expressed, including physiological processes as the effect of the vascular endothelial growth factor (VEGF) on the vessel length. A definition of the graph is used to visualize the topology of vascular tree in kidney providing physiological information into the edges. Thus, renal arterial branching is modeled as a graph where edges are labeled and oriented.     

Cat lung hemodynamics: comparison of experimental results and model predictions

Commonly, attempts have been made to learn about the structure and function of the pulmonary vascular bed from measurements of arterial and venous pressures and blood flow rate under steady-state conditions (e.g., from pressure vs. flow data) or dynamic conditions (e.g., from vascular occlusion data). Zhuang et al. (J. Appl. Physiol. 55: 1341-1348, 1983) have presented a detailed model of steady-state cat lung hemodynamics based on direct measurements of anatomical and elasticity data. This model provides an opportunity to better understand the information content of the hemodynamic data. Therefore, in the present study we carried out a series of steady-state and dynamic experiments on isolated cat lungs. We then compared the results with those predicted by the model. We found that the model provided a good fit to the steady-state data. However, to fit the dynamic data, some modifications were necessary to account for the viscous behavior of the vessel walls and to move the first moment of the distribution of vascular resistance toward the arterial end of the vascular bed relative to that of the distribution of vascular compliance. Due to the sensitivity of the vascular resistance to small changes in vessel diameters and branching ratio, the modifications in morphometry represent small changes in morphometric data and are probably within the range of uncertainty in such data. The modifications had little effect on the steady-state model simulations but substantially improved the dynamic model simulations, suggesting that the dynamic data are quite sensitive to small changes in the relative distributions of vessel diameters and elasticity.     

Matrix GLA protein, an inhibitory morphogen in pulmonary vascular development

Deficiency of matrix GLA protein (MGP), an inhibitor of bone morphogenetic protein (BMP)-2/4, is known to cause arterial calcification and peripheral pulmonary artery stenosis. Yet the vascular role of MGP remains poorly understood. To further investigate MGP, we created a new MGP transgenic mouse model with high expression of the transgene in the lungs. The excess MGP led to a disruption of the pulmonary pattern of BMP-4, and resulted in significant morphological defects in the pulmonary artery tree. Specifically, the vascular branching pattern lacked characteristic side branching, whereas control lungs had extensive side branching accounting for as much as 40% of the vascular endothelium. The vascular changes could be explained by a dramatic reduction of phosphorylated SMAD1/5/8 in the alveolar epithelium, and in epithelial expression of the activin-like kinase receptor 1 and vascular endothelial growth factor, both critical in vascular formation. Abnormalities were also found in the terminal airways and in lung cell differentiation; high levels of surfactant protein-B were distributed in an abnormal pattern suggesting lost coordination between vasculature and airways. Ex vivo, lung cells from MGP transgenic mice showed higher proliferation, in particular surfactant protein B-expressing cells, and conditioned medium from these cells poorly supported in vitro angiogenesis compared with normal lung cells. The vascular branching defect can be mechanistically explained by a computational model based on activator/inhibitor reaction-diffusion dynamics, where BMP-4 and MGP are considered as an activating and inhibitory morphogen, respectively, suggesting that morphogen interactions are important for vascular branching.     

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.     

Microfocal X-ray CT imaging and pulmonary arterial distensibility in excised rat lungs

The objective of this study was to develop an X-ray computed tomographic method for measuring pulmonary arterial dimensions and locations within the intact rat lung. Lungs were removed from rats and their pulmonary arterial trees were filled with perfluorooctyl bromide to enhance X-ray absorbance. The lungs were rotated within the cone of the X-ray beam projected from a microfocal X-ray source onto an image intensifier, and 360 images were obtained at 1 degrees increments. The three-dimensional image volumes were reconstructed with isotropic resolution using a cone beam reconstruction algorithm. The vessel diameters were obtained by fitting a functional form to the image of the vessel circular cross section. The functional form was chosen to take into account the point spread function of the image acquisition and reconstruction system. The diameter measurements obtained over a range of vascular pressures were used to characterize the distensibility of the rat pulmonary arteries. The distensibility coefficient alpha [defined by D(P) = D(0)(1 + alphaP), where D(P) is the diameter at intravascular pressure (P)] was approximately 2.8% mmHg and independent of vessel diameter in the diameter range (about 100 to 2,000 mm) studied.