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     

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.     

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.     

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.     

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.

     

Vascular metabolic dissipation in Murray's law

The metabolic dissipation in Murray's minimum energy hypothesis includes only the blood metabolism. The metabolic dissipation of the vascular tree, however, should also include the metabolism of passive and active components of the vessel wall. In this study, we extend the metabolic dissipation to include blood metabolism, as well as passive and active components of the vessel wall. The analysis is extended to the entire vascular arterial tree rather than a single vessel as in Murray's formulation. The calculations are based on experimentally measured morphological data of coronary artery network and the longitudinal distribution of blood pressure along the tree. Whereas the model includes multiple dissipation sources, the total metabolic consumption of a complex vascular tree is found to remain approximately proportional to the cumulative arterial volume of the unit. This implies that the previously described scaling relations for the various morphological features (volume, length, diameter, and flow) remain unchanged under the generalized condition of metabolic requirements of blood and blood vessel wall.     

Transit time dispersion in the pulmonary arterial tree

Knowledge of the contributions of arterialand venous transit time dispersion to the pulmonary vascular transittime distribution is important for understanding lung function and forinterpreting various kinds of data containing information aboutpulmonary function. Thus, to determine the dispersion of blood transittimes occurring within the pulmonary arterial and venous trees, imagesof a bolus of contrast medium passing through the vasculature ofpump-perfused dog lung lobes were acquired by using an X-ray microfocalangiography system. Time-absorbance curves from the lobar artery andvein and from selected locations within the intrapulmonary arterial tree were measured from the images. Overall dispersion within the lunglobe was determined from the difference in the first and second moments(mean transit time and variance, respectively) of the inlet arterialand outlet venous time-absorbance curves. Moments at selected locationswithin the arterial tree were also calculated and compared with thoseof the lobar artery curve. Transit times for the arterial pathwaysupstream from the smallest measured arteries (200-µm diameter) wereless than ~20% of the total lung lobe mean transit time. Transittime variance among these arterial pathways (interpathway dispersion)was less than ~5% of the total variance imparted on the bolus as itpassed through the lung lobe. On average, the dispersion that occurredalong a given pathway (intrapathway dispersion) was negligible. Similar results were obtained for the venous tree. Taken together, the resultssuggest that most of the variation in transit time in theintrapulmonary vasculature occurs within the pulmonary capillary bedrather than in conducting arteries or veins.

     

Pulmonary vascular remodelling in a high-altitude Aymara Indian

A histological study of the pulmonary vasculature in a young male high-altitude Aymara Indian revealed four aspects of interest. There was muscularization of the terminal portion of the pulmonary arterial tree to involve pulmonary arterioles as small as 15 m in diameter, thus forming a basis for the slightly increased pulmonary vascular resistance of native highlanders. Intimal longitudinal muscle was found in pulmonary arteries and arterioles and thought to be due to chronic alveolar hypoxia. Inner muscular tubes similar to those found in chronic obstructive lung disease were present. Pulmonary veins and venules also showed intimal muscularization suggesting that alveolar hypoxia affects vascular smooth muscle cells per se irrespective of their situation. The nature of the remodelling in a pulmonary blood vessel depends on a combination of hypoxia and haemodynamics.     

Vessel distensibility and flow distribution in vascular trees

 In a class of model vascular trees having distensible blood vessels, we prove that flow partitioning throughout the tree remains constant, independent of the nonzero driving flow (or nonzero inlet to terminal outlet pressure difference). Underlying assumptions are: (1) every vessel in the tree exhibits the same distensibility relationship given by $D/D_0 = f(P)$ where $D$ is the diameter which results from distending pressure $P$ and $D_0$ is the diameter of the individual vessel at zero pressure (each vessel may have its own individual $D_0$). The choice of $f(P)$ includes distensibilities often used in vessel biomechanics modeling, e.g., $f(P) = 1 + \alpha P$ or $f(P) = b + (1-b) \exp(-c P)$, as well as $f(P)$ which exhibit autoregulatory behavior. (2) Every terminal vessel in the tree is subjected to the same terminal outlet pressure. (3) Bernoulli effects are ignored. (4) Flow is nonpulsatile. (5) Blood viscosity within any individual vessel is constant. The results imply that for a vascular tree consistent with assumptions 2–5, the flow distribution calculations based on a rigid geometry, e.g., $D=D_0$, also gives the flow distribution when assuming the common distensibility relationships. Received: 30 October 2001 / Published online: 14 March 2002     

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.     

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.     

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.     

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.     

Caliber and media thickness of intrahepatic arteries in a normal human liver. A morphometric study

Morphometrical data on the intrahepatic arterial wall were studied over a wide range of vascular diameters (100-1708 microns) in one human liver. The liver tissue containing Araldite-filled blood vessels was embedded in the water-soluble glycolmethacrylate (GMA). The calculation of the correction factor for the morphometric data obtained from 3 microns sections is described. The influence of formaldehyde fixation on the volume of the liver proves to be negligible. During dehydration a linear shrinkage of about 3% occurs. After infiltration and embedding in GMA only 1% of this shrinkage remains. During the drying phase about 4.3% further linear stretching occurs. No significant "residual compression factor" was found. Araldite plays a significant role in the retention of the dimensions of the liver specimen during histologic processing, while the dimensional changes of the Araldite itself are negligible. A positive linear correlation was found between the media thickness and the radius of the vessel. The physiological consequences are discussed. It is concluded that in the morphometric analysis of the arterial wall it is essential to apply a standardized procedure in histologic processing and in the measuring of the inner vascular diameter. The advantages of our method, in which blood vessels are perfused with Araldite under physiological pressure, are discussed.     

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.     

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.     

Flow and pressure distributions in vascular networks consisting of distensible vessels

We examine the influence of vessel distensibility on the fraction of the total network flow passing through each vessel of a model vascular network. An exact computational methodology is developed yielding an analytic proof. For a class of structurally heterogeneous asymmetric vascular networks, if all the individual vessels share a common distensibility relation when the total network flow is changed, this methodology proves that each vessel will continue to receive the same fraction of the total network flow. This constant flow partitioning occurs despite a redistribution of pressures, which may result in a decrease in the diameter of one and an increase in the diameter of the other of two vessels having a common diameter at a common pressure. This theoretical observation, taken along with published experimental observations on pulmonary vessel distensibilities, suggests that vessel diameter-independent distensibility in the pulmonary vasculature may be an evolutionary adaptation for preserving the spatial distribution of pulmonary blood flow in the face of large variations in cardiac output.     

Fractal dimensions and multifractility in vascular branching

A definition for the fractal dimension of a vascular tree is proposed based on the hemodynamic function of the tree and in terms of two key branching parameters: the asymmetry ratio of arterial bifurcations and the power law exponent governing the relation between vessel diameter and flow. Data from the cardiovascular system, which generally exhibit considerable scatter in the values of these two parameters, are found to produce the same degree of scatter in the value of the fractal dimension. When this scatter is explored for a multifractal pattern, however, it is found that the required collapse onto a single curve is achieved in terms of the coarse H?lder exponent. Thus, the presence of multifractility is confirmed, and the legitimacy of the defined dimension is affirmed in the sense of the theoretical Hausdorff limit in as much as this limit can be reached with experimental data.     

An integrated geometric modelling framework for patient-specific computational haemodynamic study on wide-ranged vascular network

Patient-specific haemodynamic computations have been used as an effective tool in researches on cardiovascular disease associated with haemodynamics such as atherosclerosis and aneurysm. Recent development of computer resource has enabled 3D haemodynamic computations in wide-spread arterial network but there are still difficulties in modelling vascular geometry because of noise and limited resolution in medical images. In this paper, an integrated framework to model an arterial network tree for patient-specific computational haemodynamic study is developed. With this framework, 3D vascular geometry reconstruction of an arterial network and quantification of its geometric feature are aimed. The combination of 3D haemodynamic computation and vascular morphology quantification helps better understand the relationship between vascular morphology and haemodynamic force behind 'geometric risk factor' for cardiovascular diseases. The proposed method is applied to an intracranial arterial network to demonstrate its accuracy and effectiveness. The results are compared with the marching-cubes (MC) method. The comparison shows that the present modelling method can reconstruct a wide-ranged vascular network anatomically more accurate than the MC method, particularly in peripheral circulation where the image resolution is low in comparison to the vessel diameter, because of the recognition of an arterial network connectivity based on its centreline.     

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.