On the principles of the vascular network branching

A growth model for water distribution networks with loops

Water distribution networks (WDNs) expand their service areas over time. These growth dynamics are poorly understood. One facet of WDNs is that they have loops in general, and closing loops may be a functionally important process for enhancing their robustness and efficiency. We propose a growth model for WDNs that generates networks with loops and is applicable to networks with multiple water sources. We apply the proposed model to four empirical WDNs to show that it produces networks whose structure is similar to that of the empirical WDNs. The comparison between the empirical and modelled WDNs suggests that the empirical WDNs may realize a reasonable balance between cost, efficiency and robustness in terms of the network structure. We also study the design of pipe diameters based on a biological positive feedback mechanism. Specifically, we apply a model inspired by Physarum polycephalum to find moderate positive correlations between the empirical and modelled pipe diameters. The difference between the empirical and modelled pipe diameters suggests that we may be able to improve the performance of WDNs by following organizing principles of biological flow networks.

On the preservation of vessel bifurcations during flow-mediated angiogenic remodelling

During developmental angiogenesis, endothelial cells respond to shear stress by migrating and remodelling the initially hyperbranched plexus, removing certain vessels whilst maintaining others. In this study, we argue that the key regulator of vessel preservation is cell decision behaviour at bifurcations. At flow-convergent bifurcations where migration paths diverge, cells must finely tune migration along both possible paths if the bifurcation is to persist. Experiments have demonstrated that disrupting the cells’ ability to sense shear or the junction forces transmitted between cells impacts the preservation of bifurcations during the remodelling process. However, how these migratory cues integrate during cell decision making remains poorly understood. Therefore, we present the first agent-based model of endothelial cell flow-mediated migration suitable for interrogating the mechanisms behind bifurcation stability. The model simulates flow in a bifurcated vessel network composed of agents representing endothelial cells arranged into a lumen which migrate against flow. Upon approaching a bifurcation where more than one migration path exists, agents refer to a stochastic bifurcation rule which models the decision cells make as a combination of flow-based and collective-based migratory cues. With this rule, cells favour branches with relatively larger shear stress or cell number. We found that cells must integrate both cues nearly equally to maximise bifurcation stability. In simulations with stable bifurcations, we found competitive oscillations between flow and collective cues, and simulations that lost the bifurcation were unable to maintain these oscillations. The competition between these two cues is haemodynamic in origin, and demonstrates that a natural defence against bifurcation loss during remodelling exists: as vessel lumens narrow due to cell efflux, resistance to flow and shear stress increases, attracting new cells to enter and rescue the vessel from regression. Our work provides theoretical insight into the role of junction force transmission has in stabilising vasculature during remodelling and as an emergent mechanism to avoid functional shunting.

When new blood vessels are created, the endothelial cells that make up these vessels migrate and rearrange in response to blood flow to remodel and optimise the vessel network. An essential part of this process is maintaining the branched structure of the network; however, it is unclear what cues cells consider at regions where vessels branch (i.e., bifurcations). In this research, we present a computer model of cell migration to interrogate the process of preserving bifurcations during remodelling. In this model, cells at bifurcations are influenced by both flow and force transmitted from neighbouring cells. We found that both cues (flow-based and collective-based) must be considered equally in order to preserve branching in the vessel network. In simulations with stable bifurcations, we demonstrated that these cues oscillate: a strong signal in one was accompanied by a weak signal in the other. Furthermore, we found that these cues naturally compete with each other due to the coupling between blood flow and the size of the blood vessels, i.e. larger vessels with more cells produce less flow signals and vice versa. Our research provides insight into how forces transmitted between neighbouring cells stabilise and preserve branching during remodelling, as well as implicates the disruption of this force transmission as a potential mechanism when remodelling goes wrong as in the case of vascular malformation.

Fundamental constraints of vessels network architecture properties revealed by reconstruction of a rat brain vasculature

The studies of mammalian vasculature are an essential part of biomedical research, enabling the development of physiological understanding and forming the background of medical techniques and therapy. Despite the fact that the basic principles of vessel network description were established in the first quarter of the twentieth century, a digital model describing the vasculature in full accordance with experimental data has not yet been created. In the present study, we combine the determined structure design of basic arterial vessels with the stochastic creation of small vessel networks. By the example of rat brain arterial network model it was shown that the arterial blood volume and the magnitude of the blood flow impose a limitation on the network architecture. In particular, the bifurcation exponent (γ) should not be less than 2.7, and the optimal value of this parameter lies in the range of 2.9-3.0. Although the networks with a low γ appear as branched and complex, they do not fill out the phantom properly. Thus, the architecture of the vasculature is fundamentally determined by topological geometrical parameters.

Travel in city road networks follows similar transport trade-off principles to neural and plant arbors

Both engineered and biological transportation networks face trade-offs in their design. Network users desire to quickly get from one location in the network to another, whereas network planners need to minimize costs in building infrastructure. Here, we use the theory of Pareto optimality to study this design trade-off in the road networks of 101 cities, with wide-ranging population sizes, land areas and geographies. Using a simple one parameter trade-off function, we find that most cities lie near the Pareto front and are significantly closer to the front than expected by alternate design structures. To account for other optimization dimensions or constraints that may be important (e.g. traffic congestion, geography), we performed a higher-order Pareto optimality analysis and found that most cities analysed lie within a region of design space bounded by only four archetypal cities. The trade-offs studied here are also faced and well-optimized by two biological transport networks-neural arbors in the brain and branching architectures of plant shoots-suggesting similar design principles across some biological and engineered transport systems.

Interspecific scaling of blood flow rates and arterial sizes in mammals

This meta-study investigated the relationships between blood flow rate (Q̇; cm³ s^-1), wall shear stress (τ_w; dyn cm^-2) and lumen radius (r _i; cm) in 20 named systemic arteries of nine species of mammals, ranging in mass from 23 g mice to 652 kg cows, at rest. In the dataset, derived from 50 studies, lumen radius varied between 3.7 µm in a cremaster artery of a rat and 11.2 mm in the aorta of a human. The 92 logged data points of [Formula: see text] and r _i are described by a single second-order polynomial curve with the equation: [Formula: see text] The slope of the curve increased from approximately 2 in the largest arteries to approximately 3 in the smallest ones. Thus, da Vinci's rule ([Formula: see text]) applies to the main arteries and Murray's law ([Formula: see text]) applies to the microcirculation. A subset of the data, comprising only cephalic arteries in which [Formula: see text] is fairly constant, yielded the allometric power equation: [Formula: see text] These empirical equations allow calculation of resting perfusion rates from arterial lumen size alone, without reliance on theoretical models or assumptions on the scaling of wall shear stress in relation to body mass. As expected, [Formula: see text] of individual named arteries is strongly affected by body mass; however, [Formula: see text] of the common carotid artery from six species (mouse to horse) is also sensitive to differences in whole-body basal metabolic rate, independent of the effect of body mass.

Numerical study on the hemodynamics of patient-specific carotid bifurcation using a new mesh approach

The definition of a suitable mesh to simulate blood flow in the human carotid bifurcation has been investigated. In this research, a novel mesh generation method is proposed: hexahedral cells at the center of the vessel and a fine grid of tetrahedral cells near the artery wall, in order to correctly simulate the large blood velocity gradients associated with specific locations. The selected numerical examples to show the pertinence of the novel generation method are supported by carotid ultrasound image data of a patient-specific case. Doppler systolic blood velocities measured during ultrasound examination are compared with simulated velocities using 4 different combinations of hexahedral and tetrahedral meshes and different fluid dynamic simulators. The Lin's test was applied to show the concordance of the results. Wall shear stress-based descriptors and localized normalized helicity descriptor emphasize the performance of the new method. Another feature is the reduced computation time required by the developed methodology. With the accurate combined mesh, different flow rate partitions, between the internal carotid artery and external carotid artery, were studied. The overall effect of the partitions is mainly in the blood flow patterns and in the hot-spot modulation of atherosclerosis-susceptible regions, rather than in their distribution along the bifurcation.

Link between deviations from Murray's Law and occurrence of low wall shear stress regions in the left coronary artery

Murray developed two laws for the geometry of bifurcations in the circulatory system. Based on the principle of energy minimization, Murray found restrictions for the relation between the diameters and also between the angles of the branches. It is known that bifurcations are prone to the development of atherosclerosis, in regions associated to low wall shear stresses (WSS) and high oscillatory shear index (OSI). These indicators (size of low WSS regions, size of high OSI regions and size of high helicity regions) were evaluated in this work. All of them were normalized by the size of the outflow branches. The relation between Murray's laws and the size of low WSS regions was analysed in detail. It was found that the main factor leading to large regions of low WSS is the so called expansion ratio, a relation between the cross section areas of the outflow branches and the cross section area of the main branch. Large regions of low WSS appear for high expansion ratios. Furthermore, the size of low WSS regions is independent of the ratio between the diameters of the outflow branches. Since the expansion ratio in bifurcations following Murray's law is kept in a small range (1 and 1.25), all of them have regions of low WSS with similar size. However, the expansion ratio is not small enough to completely prevent regions with low WSS values and, therefore, Murray's law does not lead to atherosclerosis minimization. A study on the effect of the angulation of the bifurcation suggests that the Murray's law for the angles does not minimize the size of low WSS regions.

Phase transitions in Pareto optimal complex networks

The organization of interactions in complex systems can be described by networks connecting different units. These graphs are useful representations of the local and global complexity of the underlying systems. The origin of their topological structure can be diverse, resulting from different mechanisms including multiplicative processes and optimization. In spatial networks or in graphs where cost constraints are at work, as it occurs in a plethora of situations from power grids to the wiring of neurons in the brain, optimization plays an important part in shaping their organization. In this paper we study network designs resulting from a Pareto optimization process, where different simultaneous constraints are the targets of selection. We analyze three variations on a problem, finding phase transitions of different kinds. Distinct phases are associated with different arrangements of the connections, but the need of drastic topological changes does not determine the presence or the nature of the phase transitions encountered. Instead, the functions under optimization do play a determinant role. This reinforces the view that phase transitions do not arise from intrinsic properties of a system alone, but from the interplay of that system with its external constraints.

Radius exponent in elastic and rigid arterial models optimized by the least energy principle

It was analyzed in normal physiological arteries whether the least energy principle would suffice to account for the radius exponent x. The mammalian arterial system was modeled as two types, the elastic or the rigid, to which Bernoulli's and Hagen-Poiseuille's equations were applied, respectively. We minimized the total energy function E, which was defined as the sum of kinetic, pressure, metabolic and thermal energies, and loss of each per unit time in a single artery transporting viscous incompressible blood. Assuming a scaling exponent α between the vessel radius (r) and length (l) to be 1.0, x resulted in 2.33 in the elastic model. The rigid model provided a continuously changing x from 2.33 to 3.0, which corresponded to Uylings' and Murray's theories, respectively, through a function combining Reynolds number with a proportional coefficient of the l - r relationship. These results were expanded to an asymmetric arterial fractal tree with the blood flow preservation rule. While x in the optimal elastic model accounted for around 2.3 in proximal systemic (r >1 mm) and whole pulmonary arteries (r ≥0.004 mm), optimal x in the rigid model explained 2.7 in elastic-muscular (0.1 < r ≤1 mm) and 3.0 in peripheral resistive systemic arteries (0.004 ≤ r ≤0.1 mm), in agreement with data obtained from angiographic, cast-morphometric, and in vivo experimental studies in the literature. The least energy principle on the total energy basis provides an alternate concept of optimality relating to mammalian arterial fractal dimensions under α = 1.0.

Procedure for the development of multi-depth circular cross-sectional endothelialized microchannels-on-a-chip

Efforts have been focused on developing in vitro assays for the study of microvessels because in vivo animal studies are more time-consuming, expensive, and observation and quantification are very challenging. However, conventional in vitro microvessel assays have limitations when representing in vivo microvessels with respect to three-dimensional (3D) geometry and providing continuous fluid flow. Using a combination of photolithographic reflowable photoresist technique, soft lithography, and microfluidics, we have developed a multi-depth circular cross-sectional endothelialized microchannels-on-a-chip, which mimics the 3D geometry of in vivo microvessels and runs under controlled continuous perfusion flow. A positive reflowable photoresist was used to fabricate a master mold with a semicircular cross-sectional microchannel network. By the alignment and bonding of the two polydimethylsiloxane (PDMS) microchannels replicated from the master mold, a cylindrical microchannel network was created. The diameters of the microchannels can be well controlled. In addition, primary human umbilical vein endothelial cells (HUVECs) seeded inside the chip showed that the cells lined the inner surface of the microchannels under controlled perfusion lasting for a time period between 4 days to 2 weeks.

The influence of network structure on the transport of blood in the human cerebral microvasculature

In this article, we explore how the structural properties of miniature networks influence the transport of blood through the human cerebral microvasculature. We propose four methods for generating such networks, and investigate both how the resulting network properties match available experimental data from the human cortex and how these properties affect the flow of blood through the networks. As the nature of such microvascular flow patterns is inherently random, we run multiple simulations. We find that the modified spanning tree method produces artificial networks having characteristics closest to those of the microvasculature in human brain, and also allows for high network flow passage per unit material cost, being statistically significantly better than three other methods considered here. Such results are potentially extremely valuable in interpreting experimental data acquired from humans and in improving our understanding of cerebral blood flow at this very small length scale. This could have a significant impact on improving clinical outcomes for vascular brain diseases, particularly vascular dementia, where localized flow patterns are very important.

Model combining hydrodynamics and fractal theory for analysis of in vivo peripheral pulmonary and systemic resistance of shunt cardiac defects

The fractal state of the arterial vascular tree is considered to have a universal dimension related to the principle of minimum work rate, but can demonstrate the capacity to adapt to other dimensions in disease states such as congenital high-flow pulmonary hypertension (PH) by a process that is incompletely understood. To document and interpret fractal adaptation in patients with different degrees of PH, pulmonary and systemic vascular resistance was analyzed by a model that evaluated the fractal dimension, x, of the Poiseuille resistance contribution of the arterial vessel radius between 10 and 100μm, via the proportionality Q∝(R(peri)/BL)(-x/4), with Q, R(peri), and BL clinically observed variables representing total pulmonary or systemic blood flow, its peripheral arterial resistance, and body length, respectively. Identification of x in the pulmonary (P) and systemic (S) beds was evaluated from hemodynamic data of 213 patients, categorized into 7 groups by PH grade. In controls without PH, x(P)=2.2 while the dimension increased to 3.0, with the systemic dimension constant at x(S)=3.1. Our model predicts that severe grades of PH are associated with: a more elongated and hindered vessel in the periphery, and reductions in vessel numbers, as unit pulmonary resistive arterial trees (N(1)) and their component intra-acinar arteries (N(W)). These model network changes suggest a complex adaptive process of arterial network reorganization in the pulmonary circulation to minimize the work rate of high-flow congenital heart defects.

Development of optimized vascular fractal tree models using level set distance function

Using the concepts of fractal scaling and constrained constructive optimization (CCO), a branching tree model, which has physiologically meaningful geometric properties, can be constructed. A vascular branching tree model created in this way, although statistically correct in representing the vascular physiology, still does not possess a physiological correct arrangement of the major arteries. A distance-function based technique for "staged growth" of vascular models has been developed in this work to address this issue. Time-dependent constraints based on a signed-distance level set function have been added, so that the tree models will first be grown near the designated surface(s) and, then, gradually allowed to penetrate into the enclosed volume. The proposed technique has been applied to construct a model of the human cerebral vasculature, which is characterized by the above-mentioned distribution of the arteries.

Dynamics of pulsatile flow in fractal models of vascular branching networks

Efficient regulation of blood flow is critically important to the normal function of many organs, especially the brain. To investigate the circulation of blood in complex, multi-branching vascular networks, a computer model consisting of a virtual fractal model of the vasculature and a mathematical model describing the transport of blood has been developed. Although limited by some constraints, in particular, the use of simplistic, uniformly distributed model for cerebral vasculature and the omission of anastomosis, the proposed computer model was found to provide insights into blood circulation in the cerebral vascular branching network plus the physiological and pathological factors which may affect its functionality. The numerical study conducted on a model of the middle cerebral artery region signified the important effects of vessel compliance, blood viscosity variation as a function of the blood hematocrit, and flow velocity profile on the distributions of flow and pressure in the vascular network.

Extension of Murray's law using a non-Newtonian model of blood flow

Background

So far, none of the existing methods on Murray's law deal with the non-Newtonian behavior of blood flow although the non-Newtonian approach for blood flow modelling looks more accurate.

Modeling

In the present paper, Murray's law which is applicable to an arterial bifurcation, is generalized to a non-Newtonian blood flow model (power-law model). When the vessel size reaches the capillary limitation, blood can be modeled using a non-Newtonian constitutive equation. It is assumed two different constraints in addition to the pumping power: the volume constraint or the surface constraint (related to the internal surface of the vessel). For a seek of generality, the relationships are given for an arbitrary number of daughter vessels. It is shown that for a cost function including the volume constraint, classical Murray's law remains valid (i.e. ΣR^c = cste with c = 3 is verified and is independent of n, the dimensionless index in the viscosity equation; R being the radius of the vessel). On the contrary, for a cost function including the surface constraint, different values of c may be calculated depending on the value of n.

Results

We find that c varies for blood from 2.42 to 3 depending on the constraint and the fluid properties. For the Newtonian model, the surface constraint leads to c = 2.5. The cost function (based on the surface constraint) can be related to entropy generation, by dividing it by the temperature.

Conclusion

It is demonstrated that the entropy generated in all the daughter vessels is greater than the entropy generated in the parent vessel. Furthermore, it is shown that the difference of entropy generation between the parent and daughter vessels is smaller for a non-Newtonian fluid than for a Newtonian fluid.

Large variations in absolute wall shear stress levels within one species and between species

Wall shear stress (WSS), the frictional force between blood and endothelium, is an important determinant of vascular function. It is generally assumed that WSS remains constant at a reference value of 15 dyn/cm(2). In a study of small rodents, we realized that this assumption could not be valid. This review presents an overview of recent studies in large and small animals where shear stress was measured, derived from velocity measurements or otherwise, in large vessels. The data show that large variations exist within a single species (human: variation of 2-16 N/m(2)). Moreover, when we compared different species at the same location within the arterial tree, an inverse relationship between animal size and wall shear stress was noted. When we related WSS to diameter, a unique relationship was derived for all species studied. This relationship could not be described by the well-known r(3) law of Murray, but by the r(2) law introduced by Zamir et al. in 1972. In summary, by comparing data from the literature, we have shown that: (i) the assumption of a physiological WSS level of approximately 15 dyn/cm(2) for all straight vessels in the arterial tree is incorrect; (ii) WSS is not constant throughout the vascular tree; (iii) WSS varies between species; (iv) WSS is inversely related to the vessel diameter. These data support an "r(2) law" rather than Murray's r(3) law for the larger vessels in the arterial tree.

Optimal planar flow network designs for tissue engineered constructs with built-in vasculature

Convective delivery of nutrients is important to enhance mass transport within tissue engineered (TE) products. Depending on the target tissue, an ideal TE product will have an integrated microvasculature that will eliminate mass transport limitations that can occur during product growth in vitro and integration in vivo. A synthetic approach to develop microvasculature involves development of network designs with efficient mass transfer characteristics. In this paper, utilizing a planar bifurcating network as a basis, we develop an approach to design optimal flow networks that have maximum mass transport efficiency for a given pressure drop. We formulated the optimization problem for a TE skin product, incorporating two types of duct flow, rectangular and square, and solved using a generalized reduced gradient algorithm. Under the conditions of this study, we found that rectangular ducts have superior mass transport characteristics than square ducts. Microvascular area per volume values obtained in this work are significantly greater than those reported in the literature. We discuss the effect of network variables such as porosity and generations on the optimal designs. This research forms the engineering basis for the rational development of TE products with built-in microvasculature and will pave the way to design complex flow networks with optimal mass transfer characteristics.

Flows in large, self-similar tree networks and their control

We consider flow in large, regular networks configured in the form of a self-similar branching tree, in which each branch of the tree splits into two with a constant diameter and length ratio. Where appropriate, the limit of an infinite network is also considered. The laminar or turbulent nature of the flow at the inlet may be different from that at the outlet as a consequence of the change in Reynolds number from one generation to the next. A steady flow is shown to be globally stable for a friction law that is non-decreasing with respect to the flow rate. A difference between critical Reynolds numbers for transition and relaminarization may cause the flow in the entire network to oscillate. Control of branch flow rates and junction pressures is investigated using valves and pumps as the input variables. Fault detection due to blockage is discussed. The network hydrodynamics are only partially controllable if individual pumps are used to control the pressure at the junctions.

A design principle for vascular beds: the effects of complex blood rheology

We propose a design principle that extends Murray's original optimization principle for vascular architecture to account for complex blood rheology. Minimization of an energy dissipation function enables us to determine how rheology affects the morphology of simple branching networks. The behavior of various physical quantities associated with the networks, such as the wall shear stress and the flow velocity, is also determined. Our results are shown to be qualitatively and quantitatively compatible with independent experimental observations and simulations.

The multifractal structure of arterial trees

Fractal properties of arterial trees are analysed using the cascade model of turbulence theory. It is shown that the branching process leads to a non-uniform structure at the micro-level meaning that blood supply to the tissue varies in space. From the model it is concluded that, depending on the branching parameter, vessels of a specific size contribute dominantly to the blood supply of tissue. The corresponding tissue elements form a dense set in the tissue. Furthermore, if blood flow in vessels can get obstructed with some probability, the above set of tissue elements may not be dense anymore. Then there is the risk that, spread out over the tissue, nutrient and gas exchange fall short.