Vascular metabolic dissipation in Murray's law

A multiscale framework for defining homeostasis in distal vascular trees: applications to the pulmonary circulation

Pulmonary arteries constitute a low-pressure network of vessels, often characterized as a bifurcating tree with heterogeneous vessel mechanics. Understanding the vascular complexity and establishing homeostasis is important to study diseases such as pulmonary arterial hypertension (PAH). The onset and early progression of PAH can be traced to changes in the morphometry and structure of the distal vasculature. Coupling hemodynamics with vessel wall growth and remodeling (G&R) is crucial for understanding pathology at distal vasculature. Accordingly, the goal of this study is to provide a multiscale modeling framework that embeds the essential features of arterial wall constituents coupled with the hemodynamics within an arterial network characterized by an extension of Murray's law. This framework will be used to establish the homeostatic baseline characteristics of a pulmonary arterial tree, including important parameters such as vessel radius, wall thickness and shear stress. To define the vascular homeostasis and hemodynamics in the tree, we consider two timescales: a cardiac cycle and a longer period of vascular adaptations. An iterative homeostatic optimization, which integrates a metabolic cost function minimization, the stress equilibrium, and hemodynamics, is performed at the slow timescale. In the fast timescale, the pulsatile blood flow dynamics is described by a Womersley's deformable wall analytical solution. Illustrative examples for symmetric and asymmetric trees are presented that provide baseline characteristics for the normal pulmonary arterial vasculature. The results are compared with diverse literature data on morphometry, structure, and mechanics of pulmonary arteries. The developed framework demonstrates a potential for advanced parametric studies and future G&R and hemodynamics modeling of PAH.

Tracking the Adaptation and Compensation Processes of Patients' Brain Arterial Network to an Evolving Glioblastoma

The brain's vascular network dynamically affects its development and core functions. It rapidly responds to abnormal conditions by adjusting properties of the network, aiding stabilization and regulation of brain activities. Tracking prominent arterial changes has clear clinical and surgical advantages. However, the arterial network functions as a system; thus, local changes may imply global compensatory effects that could impact the dynamic progression of a disease. We developed automated personalized system-level analysis methods of the compensatory arterial changes and mean blood flow behavior from a patient's clinical images. By applying our approach to data from a patient with aggressive brain cancer compared with healthy individuals, we found unique spatiotemporal patterns of the arterial network that could assist in predicting the evolution of glioblastoma over time. Our personalized approach provides a valuable analysis tool that could augment current clinical assessments of the progression of glioblastoma and other neurological disorders affecting the brain.

The role of vascular complexity on optimal junction exponents

We examine the role of complexity on arterial tree structures, determining globally optimal vessel arrangements using the Simulated AnneaLing Vascular Optimization algorithm, a computational method which we have previously used to reproduce features of cardiac and cerebral vasculatures. In order to progress computational methods for growing arterial networks, deeper understanding of the stability of computational arterial growth algorithms to complexity, variations in physiological parameters (such as metabolic costs for maintaining and pumping blood), and underlying assumptions regarding the value of junction exponents is needed. We determine the globally optimal structure of two-dimensional arterial trees; analysing how physiological parameters affect tree morphology and optimal bifurcation exponent. We find that considering the full complexity of arterial trees is essential for determining the fundamental properties of vasculatures. We conclude that optimisation-based arterial growth algorithms are stable against uncertainties in physiological parameters, while optimal bifurcation exponents (a key parameter for many arterial growth algorithms) are affected by the complexity of vascular networks and the boundary conditions dictated by organs.

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.

Development of a globally optimised model of the cerebral arteries

The cerebral arteries are difficult to reproduce from first principles, featuring interwoven territories, and intricate layers of grey and white matter with differing metabolic demand. The aim of this study was to identify the ideal configuration of arteries required to sustain an entire brain hemisphere based on minimisation of the energy required to supply the tissue. The 3D distribution of grey and white matter within a healthy human brain was first segmented from magnetic resonance images. A novel simulated annealing algorithm was then applied to determine the optimal configuration of arteries required to supply brain tissue. The model was validated through comparison of this ideal, entirely optimised, brain vasculature with the structure and properties of real arteries. This analysis established that the human cerebral vasculature is highly optimised; closely resembling the most energy efficient arrangement of vessels. In addition to local adherence to fluid dynamical optimisation principles, the optimised vasculature reproduced expected brain perfusion territories, featuring well-defined boundaries between anterior, middle and posterior regions. This validated brain vascular model and algorithm can be used for patient-specific modelling of stroke and cerebral haemodynamics, identification of sub-optimal conditions associated with vascular disease, and optimising vascular structures for tissue engineering applications and artificial organ design.

Fundamental principles of vascular network topology

The vascular system is arguably the most important biological system in many organisms. Although the general principles of its architecture are simple, the growth of blood vessels occurs under extreme physical conditions. Optimization is an important aspect of the development of computational models of the vascular branching structures. This review surveys the approaches used to optimize the topology and estimate different geometrical parameters of the vascular system. The review is focused on optimizations using complex cost functions based on the minimum total energy principle and the relationship between the laws of growth and precise vascular network topology. Experimental studies of vascular networks in different species are also discussed.

Simulated annealing approach to vascular structure with application to the coronary arteries

Do the complex processes of angiogenesis during organism development ultimately lead to a near optimal coronary vasculature in the organs of adult mammals? We examine this hypothesis using a powerful and universal method, built on physical and physiological principles, for the determination of globally energetically optimal arterial trees. The method is based on simulated annealing, and can be used to examine arteries in hollow organs with arbitrary tissue geometries. We demonstrate that the approach can generate in silico vasculatures which closely match porcine anatomical data for the coronary arteries on all length scales, and that the optimized arterial trees improve systematically as computational time increases. The method presented here is general, and could in principle be used to examine the arteries of other organs. Potential applications include improvement of medical imaging analysis and the design of vascular trees for artificial organs.

A Modification of Murray's Law for Shear-Thinning Rheology

This study reformulates Murray's well-known principle of minimum work as applied to the cardiovascular system to include the effects of the shear-thinning rheology of blood. The viscous behavior is described using the extended modified power law (EMPL), which is a time-independent, but shear-thinning rheological constitutive equation. The resulting minimization problem is solved numerically for typical parameter ranges. The non-Newtonian analysis still predicts the classical cubic diameter dependence of the volume flow rate and the cubic branching law. The current analysis also predicts a constant wall shear stress throughout the vascular tree, albeit with a numerical value about 15–25% higher than the Newtonian analysis. Thus, experimentally observed deviations from the cubic branching law or the predicted constant wall shear stress in the vasculature cannot likely be attributed to blood's shear-thinning behavior. Further differences between the predictions of the non-Newtonian and the Newtonian analyses are highlighted, and the limitations of the Newtonian analysis are discussed. Finally, the range and limits of applicability of the current results as applied to the human arterial tree are also discussed.

Extension of Murray's law including nonlinear mechanics of a composite artery wall

A goal function approach is used to derive an extension of Murray’s law that includes effects of nonlinear mechanics of the artery wall. The artery is modeled as a thin-walled tube composed of different species of nonlinear elastic materials that deform together. These materials grow and remodel in a process that is governed by a target state defined by a homeostatic radius and a homeostatic material composition. Following Murray’s original idea, this target state is defined by a principle of minimum work. We take this work to include that of pumping and maintaining blood, as well as maintaining the materials of the artery wall. The minimization is performed under a constraint imposed by mechanical equilibrium. We derive a condition for the existence of a cost-optimal homeostatic state. We also conduct parametric studies using this novel theoretical frame to investigate how the cost-optimal radius and composition of the artery wall depend on flow rate, blood pressure, and elastin content

Digital reconstruction and morphometric analysis of human brain arterial vasculature from magnetic resonance angiography

Characterization of the complex branching architecture of cerebral arteries across a representative sample of the human population is important for diagnosing, analyzing, and predicting pathological states. Brain arterial vasculature can be visualized by magnetic resonance angiography (MRA). However, most MRA studies are limited to qualitative assessments, partial morphometric analyses, individual (or small numbers of) subjects, proprietary datasets, or combinations of the above limitations. Neuroinformatics tools, developed for neuronal arbor analysis, were used to quantify vascular morphology from 3T time-of-flight MRA high-resolution (620 μm isotropic) images collected in 61 healthy volunteers (36/25 F/M, average age=31.2 ± 10.7, range=19-64 years). We present in-depth morphometric analyses of the global and local anatomical features of these arbors. The overall structure and size of the vasculature did not significantly differ across genders, ages, or hemispheres. The total length of the three major arterial trees stemming from the circle of Willis (from smallest to largest: the posterior, anterior, and middle cerebral arteries; or PCAs, ACAs, and MCAs, respectively) followed an approximate 1:2:4 proportion. Arterial size co-varied across individuals: subjects with one artery longer than average tended to have all other arteries also longer than average. There was no net right-left difference across the population in any of the individual arteries, but ACAs were more lateralized than MCAs. MCAs, ACAs, and PCAs had similar branch-level properties such as bifurcation angles. Throughout the arterial vasculature, there were considerable differences between branch types: bifurcating branches were significantly shorter and straighter than terminating branches. Furthermore, the length and meandering of bifurcating branches increased with age and with path distance from the circle of Willis. All reconstructions are freely distributed through a public database to enable additional analyses and modeling (cng.gmu.edu/brava).

Design of vascular networks: a mathematical model approach

In this paper, methods for the modeling of a realistic vascular tree in 3D space and for hemodynamic calculating throughout the simulated tree structure have been developed. Vascular trees are generated based on the power law relationship. Variations in branching asymmetry and segment length of the vasculature are precisely controlled by the designed gaussian distributions. The resolution limit of current imaging techniques for vessel detectability is simulated by designed pruning technique. On the basis of the generated diameters and lengths, the space locations of the vessel segments are calculated by optimizing the out-of-plane angles of two daughter branches. The generated vascular tree not only follows the power law relationship, but also maximizes the filling volume of the tree structure in 3D space. From the hemodynamic calculation in the simulated vasculature, the processes for which structural changes affect hemodynamic distributions are studied in detail. And also, the fractal nature and resistance of the vascular trees are quantified and compared. The developed method provides some insight into the design of the vascular trees in biology and may be used as a reference for the study of vascular diseases.

3D Imaging of vascular networks for biophysical modeling of perfusion distribution within the heart

One of the main determinants of perfusion distribution within an organ is the structure of its vascular network. Past studies were based on angiography or corrosion casting and lacked quantitative three dimensional, 3D, representation. Based on branching rules and other properties derived from such imaging, 3D vascular tree models were generated which were rather useful for generating and testing hypotheses on perfusion distribution in organs. Progress in advanced computational models for prediction of perfusion distribution has raised the need for more realistic representations of vascular trees with higher resolution. This paper presents an overview of the different methods developed over time for imaging and modeling the structure of vascular networks and perfusion distribution, with a focus on the heart. The strengths and limitations of these different techniques are discussed. Episcopic fluorescent imaging using a cryomicrotome is presently being developed in different laboratories. This technique is discussed in more detail, since it provides high-resolution 3D structural information that is important for the development and validation of biophysical models but also for studying the adaptations of vascular networks to diseases. An added advantage of this method being is the ability to measure local tissue perfusion. Clinically, indices for patient-specific coronary stenosis evaluation derived from vascular networks have been proposed and high-resolution noninvasive methods for perfusion distribution are in development. All these techniques depend on a proper representation of the relevant vascular network structures.

Dynamic adaption of vascular morphology

The structure of vascular networks adapts continuously to meet changes in demand of the surrounding tissue. Most of the known vascular adaptation mechanisms are based on local reactions to local stimuli such as pressure and flow, which in turn reflects influence from the surrounding tissue. Here we present a simple two-dimensional model in which, as an alternative approach, the tissue is modeled as a porous medium with intervening sharply defined flow channels. Based on simple, physiologically realistic assumptions, flow-channel structure adapts so as to reach a configuration in which all parts of the tissue are supplied. A set of model parameters uniquely determine the model dynamics, and we have identified the region of the best-performing model parameters (a global optimum). This region is surrounded in parameter space by less optimal model parameter values, and this separation is characterized by steep gradients in the related fitness landscape. Hence it appears that the optimal set of parameters tends to localize close to critical transition zones. Consequently, while the optimal solution is stable for modest parameter perturbations, larger perturbations may cause a profound and permanent shift in systems characteristics. We suggest that the system is driven toward a critical state as a consequence of the ongoing parameter optimization, mimicking an evolutionary pressure on the system.

Slackness between vessel and myocardium is necessary for coronary flow reserve

Tone regulation in coronary microvessels has largely been studied in isolated vessels in the absence of myocardial tethering. Here, the potential effect of radial tethering and interstitial space connective tissue (ISCT) between coronary microvessels and the surrounding myocardium was studied. We hypothesized that rigid tethering between microvessels and the myocardium would constrain the active contraction of arterioles and is not compatible with the observed tone regulation. The ISCT between coronary microvessels and myocardium in five swine was found to increase exponentially from 0.22 ± 0.02 μm in capillaries (modified Strahler order 0) of the endocardium to 34.9 ± 7.1 μm in epicardial vessels (order 10). Microvessels with both soft tethering and ISCT gap were capable of significant changes in vessel resistance (up to an ∼1,600% increase), consistent with experimental measurements of high coronary flow reserve. Additionally, the mechanical energy required for myogenic contraction was estimated. The results indicate that rigid tethering requires up to four times more mechanical energy than soft tethering in the absence of a gap. Hence, the experimental measurements and model predictions suggest that effectiveness and efficiency in tone regulation can be achieved only if the vessel is both softly tethered to and separated from the myocardium in accordance with the experimental findings of ISCT gap. These results have fundamental implications on future simulations of coronary circulation.

Theoretical models for coronary vascular biomechanics: progress & challenges

A key aim of the cardiac Physiome Project is to develop theoretical models to simulate the functional behaviour of the heart under physiological and pathophysiological conditions. Heart function is critically dependent on the delivery of an adequate blood supply to the myocardium via the coronary vasculature. Key to this critical function of the coronary vasculature is system dynamics that emerge via the interactions of the numerous constituent components at a range of spatial and temporal scales. Here, we focus on several components for which theoretical approaches can be applied, including vascular structure and mechanics, blood flow and mass transport, flow regulation, angiogenesis and vascular remodelling, and vascular cellular mechanics. For each component, we summarise the current state of the art in model development, and discuss areas requiring further research. We highlight the major challenges associated with integrating the component models to develop a computational tool that can ultimately be used to simulate the responses of the coronary vascular system to changing demands and to diseases and therapies.

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.

Shear stress regulation of nitric oxide production in uterine and placental artery endothelial cells: experimental studies and hemodynamic models of shear stresses on endothelial cells

Hemodynamic shear stress is the most powerful physiological regulator of endothelial Nitric Oxide Synthase (eNOS), leading to rapid rises in nitric oxide (NO). The substantial increases in uterine and placental blood flows throughout gestation rely heavily on the action of NO. We and others have investigated endothelial function in response to shear stress with cell culture models of shear stress. In order to apply the results of these studies more effectively, we need a more complete understanding of the origin and coupling of the hemodynamic forces and vascular tissue behavior. For example, equations commonly used to calculate in vivo shear stress incorporate assumptions of steady (non-pulsatile) blood flow and constant viscosity of blood (Newtonian fluid). Using computational models, we can estimate a waveform of shear stress over a cardiac cycle and the change in blood viscosity with shear rate and hematocrit levels, two variables that often change with size of vessel and location within a vascular tree. This review discusses hemodynamics as they apply to blood flow in vessels, in the hope that an integration of these fields can lead to improved in vitro shear stress experiments and understanding of NO production in uterine and placental vascular physiology during gestation.

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.

A scaling law of vascular volume

Vascular volume is of fundamental significance to the function of the cardiovascular system. An accurate prediction of blood volume in patients is physiologically and clinically significant. This study proposes what we believe is a novel volume scaling relation of the form: V(c)=K(v)D(s)(2/3)L(c), where V(c) and L(c) are cumulative vessel volume and length, respectively, in the tree, and D(s) is the diameter of the vessel segment. The scaling relation is validated in vascular trees of various organs including the heart, lung, mesentery, muscle, and eye of different species. Based on the minimum energy hypothesis and volume scaling relation, four structure-function scaling relations are predicted, including the diameter-length, volume-length, flow-diameter, and volume-diameter relations, with exponent values of 3/7, 1(2/7), 2(1/3), and 3, respectively. These four relations are validated in the various vascular trees, which further confirm the volume scaling relation. This scaling relation may serve as a control reference to estimate the blood volume in various organs and species. The deviation from the scaling relation may indicate hypovolemia or hypervolemia and aid diagnosis.

Surrounding tissues affect the passive mechanics of the vessel wall: theory and experiment

The stress and strain in the vessel wall are important determinants of vascular physiology and pathophysiology. Vessels are constrained radially by the surrounding tissue. The hypothesis in this work is that the surrounding tissue takes up a considerable portion of the intravascular pressure and significantly reduces the wall strain and stress. Ten swine of either sex were used to test this hypothesis. An impedance catheter was inserted into the carotid or femoral artery, and after mechanical preconditioning pressure-cross-sectional area relations were obtained with the surrounding tissue intact and dissected away (untethered), respectively. The radial constraint of the surrounding tissue was quantified as an effective perivascular pressure on the outer surface of the vessel, which was estimated as 50% or more of the intravascular pressure. For carotid arteries at pressure of 100 mmHg, the circumferential wall stretch ratio in the intact state was approximately 20% lower than in the untethered state and the average circumferential stress was reduced by approximately 70%. For femoral arteries, the reductions were approximately 15% and 70%, respectively. These experimental data support the proposed hypothesis and suggest that in vitro and in vivo measurements of the mechanical properties of vessels must be interpreted with consideration of the constraint of the surrounding tissue.