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.     

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.     

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.     

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.     

Modeling the large-scale geometry of human coronary arteries

Two principles suffice to model the large-scale geometry of normal human coronary arterial networks. The first principle states that artery diameters are set to minimize the power required to distribute blood through the network. The second principle states that arterial tree geometries are set to globally minimize the lumen volume. Given only the coordinates of an arterial tree's source and "leaves", the model predicts the nature of the network connecting the source to the leaves. Measurements were made of the actual geometries of arterial trees from postmortem healthy human coronary arteriograms. The tree geometries predicted by the model look qualitatively similar to the actual tree geometries and have volumes that are within a few percent of those of the actual tree geometries. Human coronary arteries are therefore within a few percent of perfect global volume optimality. A possible mechanism for this near-perfect global volume optimality is suggested. Also, the model performs best under the assumption that the flow is not entirely steady and laminar.     

Mapping Tropical Forest Trees Using High‐Resolution Aerial Digital Photographs

The spatial arrangement of tree species is a key aspect of community ecology. Because tree species in tropical forests occur at low densities, it is logistically challenging to measure distributions across large areas. In this study, we evaluated the potential use of canopy tree crown maps, derived from high‐resolution aerial digital photographs, as a relatively simple method for measuring large‐scale tree distributions. At Barro Colorado Island, Panama, we used high‐resolution aerial digital photographs (~0.129 m/pixel) to identify tree species and map crown distributions of four target tree species. We determined crown mapping accuracy by comparing aerial and ground‐mapped distributions and tested whether the spatial characteristics of the crown maps reflect those of the ground‐mapped trees. Nearly a quarter (22%) of the common canopy species had sufficiently distinctive crowns to be good candidates for reliable mapping. The errors of commission (crowns misidentified as a target species) were relatively low, but the errors of omission (missed canopy trees of the target species) were high. Only 40 percent of canopy individuals were mapped on the air photographs. Despite failing to accurately predict exact abundances of canopy trees, crown distributions accurately reproduced the clumping patterns and spatial autocorrelation features of three of four tree species and predicted areas of high and low abundance. We discuss a range of ecological and forest management applications for which this method can be useful.     

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.     

Defining the landscape of adaptive genetic diversity

Whether they are used to describe fitness, genome architecture or the spatial distribution of environmental variables, the concept of a landscape has figured prominently in our collective reasoning. The tradition of landscapes in evolutionary biology is one of fitness mapped onto axes defined by phenotypes or molecular sequence states. The characteristics of these landscapes depend on natural selection, which is structured across both genomic and environmental landscapes, and thus, the bridge among differing uses of the landscape concept (i.e. metaphorically or literally) is that of an adaptive phenotype and its distribution across geographical landscapes in relation to selective pressures. One of the ultimate goals of evolutionary biology should thus be to construct fitness landscapes in geographical space. Natural plant populations are ideal systems with which to explore the feasibility of attaining this goal, because much is known about the quantitative genetic architecture of complex traits for many different plant species. What is less known are the molecular components of this architecture. In this issue of Molecular Ecology, Parchman et al. (2012) pioneer one of the first truly genome-wide association studies in a tree that moves us closer to this form of mechanistic understanding for an adaptive phenotype in natural populations of lodgepole pine (Pinus contorta Dougl. ex Loud.).     

Toward a noninvasive subject-specific estimation of abdominal aortic pressure

A method for estimation of central arterial pressure based on linear one-dimensional wave propagation theory is presented in this paper. The equations are applied to a distributed model of the arterial tree, truncated by three-element windkessels. To reflect individual differences in the properties of the arterial trees, we pose a minimization problem from which individual parameters are identified. The idea is to take a measured waveform in a peripheral artery and use it as input to the model. The model subsequently predicts the corresponding waveform in another peripheral artery in which a measurement has also been made, and the arterial tree model is then calibrated in such a way that the computed waveform matches its measured counterpart. For the purpose of validation, invasively recorded abdominal aortic, brachial, and femoral pressures in nine healthy subjects are used. The results show that the proposed method estimates the abdominal aortic pressure wave with good accuracy. The root mean square error (RMSE) of the estimated waveforms was 1.61 +/- 0.73 mmHg, whereas the errors in systolic and pulse pressure were 2.32 +/- 1.74 and 3.73 +/- 2.04 mmHg, respectively. These results are compared with another recently proposed method based on a signal processing technique, and it is shown that our method yields a significantly (P < 0.01) lower RMSE. With more extensive validation, the method may eventually be used in clinical practice to provide detailed, almost individual, specific information as a valuable basis for decision making.     

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.     

A Novel Optimal Load Distribution Algorithm for Divisible Loads

A new model for divisible load problem is introduced. Its characteristics are analyzed. Optimal load distribution algorithms on the new model are presented for the tree-network and linear network. Applications that fit our model are briefly described. We show that our model outperforms the existing model such as Cheng–Robertazzi model. We show that the linear model is equivalent to a single-level tree network if the intermediate processors do not follow the store-and-forward communication model, but they follow the store-and-bypass model. This paper introduces the concept of store-and-bypass for divisible load theory.     

Low Reynolds number turbulence modeling of blood flow in arterial stenoses.

Moderate and severe arterial stenoses can produce highly disturbed flow regions with transitional and or turbulent flow characteristics. Neither laminar flow modeling nor standard two-equation models such as the kappa-epsilon turbulence ones are suitable for this kind of blood flow. In order to analyze the transitional or turbulent flow distal to an arterial stenosis, authors of this study have used the Wilcox low-Re turbulence model. Flow simulations were carried out on stenoses with 50, 75 and 86% reductions in cross-sectional area over a range of physiologically relevant Reynolds numbers. The results obtained with this low-Re turbulence model were compared with experimental measurements and with the results obtained by the standard kappa-epsilon model in terms of velocity profile, vortex length, wall shear stress, wall static pressure, and turbulence intensity. The comparisons show that results predicted by the low-Re model are in good agreement with the experimental measurements. This model accurately predicts the critical Reynolds number at which blood flow becomes transitional or turbulent distal an arterial stenosis. Most interestingly, over the Re range of laminar flow, the vortex length calculated with the low-Re model also closely matches the vortex length predicted by laminar flow modeling. In conclusion, the study strongly suggests that the proposed model is suitable for blood flow studies in certain areas of the arterial tree where both laminar and transitional/turbulent flows coexist.     

Inlet boundary conditions for blood flow simulations in truncated arterial networks

In the context of patient-specific cardiovascular applications, hemodynamics models (going from 3D to 0D) are often limited to a part of the arterial tree. This restriction implies the set up of artificial interfaces with the remaining parts of the cardiovascular system. In particular, the inlet boundary condition is crucial: it supplies the impulsion to the system and receives the reflected backward waves created by the distal network. Some aspects of this boundary condition need to be properly defined such as the treatment of backward waves (reflected or absorbed) and the value of the imposed hemodynamic wave (total or forward component). Most authors prescribe as inlet boundary condition (BC) the total measured variable (pressure, velocity or flow rate) in a reflective way. We show that with this type of inlet boundary condition, the model does not produce physiological waveforms. We suggest instead to prescribe only the forward component of the prescribed variable in an absorbing way. In this way, the computed reflected waves superpose with the prescribed forward waves to produce the total wave at the inlet. In this work, different inlet boundary conditions are implemented and compared for a 1D blood flow model. We test our boundary conditions on a truncated arterial model presented in the literature as well as on a patient-specific lower-limb model of a femoral bypass. We show that with this new boundary condition, a much better fitting is observed on the shape and intensity of the simulated pressure and velocity waves.     

Phylogenetic Reconstruction Using an Unsupervised Growing Neural Network That Adopts the Topology of a Phylogenetic Tree

We propose a new type of unsupervised, growing, self-organizing neural network that expands itself by following the taxonomic relationships that exist among the sequences being classified. The binary tree topology of this neutral network, contrary to other more classical neural network topologies, permits an efficient classification of sequences. The growing nature of this procedure allows to stop it at the desired taxonomic level without the necessity of waiting until a complete phylogenetic tree is produced. This novel approach presents a number of other interesting properties, such as a time for convergence which is, approximately, a lineal function of the number of sequences. Computer simulation and a real example show that the algorithm accurately finds the phylogenetic tree that relates the data. All this makes the neural network presented here an excellent tool for phylogenetic analysis of a large number of sequences. Received: 14 May 1996 / Accepted: 6 August 1996     

Quantitative models of the rat pulmonary arterial tree morphometry applied to hypoxia-induced arterial remodeling.

Little is known about the constituent hemodynamic consequences of structural changes that occur in the pulmonary arteries during the onset and progression of pulmonary arterial remodeling. Many disease processes are known to be responsible for vascular remodeling that leads to pulmonary arterial hypertension, cor pulmonale, and death. Histology has been the primary tool for evaluating pulmonary remodeling, but it does not provide information on intact vascular structure or the vessel mechanical properties. This study is an extension of our previous work in which we developed an alternative imaging technique to evaluate pulmonary arterial structure. The lungs from Sprague-Dawley rats were removed, perfusion analysis was performed on the isolated lungs, and then an X-ray contrast agent was used to fill the arterial network for imaging. The lungs were scanned over a range of intravascular pressures by volumetric micro-computed tomography, and the arterial morphometry was mapped and measured in the reconstructed isotropic volumes. A quantitative assessment of hemodynamic, structural, and biomechanical differences between rats exposed for 21 days to hypoxia (10% O(2)) or normoxia (21.0% O(2)) was performed. One metric, the normalized distensibility of the arteries, is significantly (P < 0.001) larger [0.025 +/- 0.0011 (SE) mmHg(-1)] (n = 9) in normoxic rats compared with hypoxic [0.015 +/- 0.00077 (SE) mmHg(-1)] (n = 9). The results of the study show that these models can be applied to the Sprague-Dawley rat data and, specifically, can be used to differentiate between the hypoxic and the control groups.     

Hemodynamic Analysis in an Idealized Artery Tree: Differences in Wall Shear Stress between Newtonian and Non-Newtonian Blood Models

Development of many conditions and disorders, such as atherosclerosis and stroke, are dependent upon hemodynamic forces. To accurately predict and prevent these conditions and disorders hemodynamic forces must be properly mapped. Here we compare a shear-rate dependent fluid (SDF) constitutive model, based on the works by Yasuda et al in 1981, against a Newtonian model of blood. We verify our stabilized finite element numerical method with the benchmark lid-driven cavity flow problem. Numerical simulations show that the Newtonian model gives similar velocity profiles in the 2-dimensional cavity given different height and width dimensions, given the same Reynolds number. Conversely, the SDF model gave dissimilar velocity profiles, differing from the Newtonian velocity profiles by up to 25% in velocity magnitudes. This difference can affect estimation in platelet distribution within blood vessels or magnetic nanoparticle delivery. Wall shear stress (WSS) is an important quantity involved in vascular remodeling through integrin and adhesion molecule mechanotransduction. The SDF model gave a 7.3-fold greater WSS than the Newtonian model at the top of the 3-dimensional cavity. The SDF model gave a 37.7-fold greater WSS than the Newtonian model at artery walls located immediately after bifurcations in the idealized femoral artery tree. The pressure drop across arteries reveals arterial sections highly resistive to flow which correlates with stenosis formation. Numerical simulations give the pressure drop across the idealized femoral artery tree with the SDF model which is approximately 2.3-fold higher than with the Newtonian model. In atherosclerotic lesion models, the SDF model gives over 1 Pa higher WSS than the Newtonian model, a difference correlated with over twice as many adherent monocytes to endothelial cells from the Newtonian model compared to the SDF model.