Structural quantification and bifurcation symmetry in arterial tree models generated by constrained constructive optimization

Computational models for generating microvascular structures: Investigations beyond medical imaging resolution

Angiogenesis, arteriogenesis, and pruning are revascularization processes essential to our natural vascular development and adaptation, as well as central players in the onset and development of pathologies such as tumoral growth and stroke recovery. Computational modeling allows for repeatable experimentation and exploration of these complex biological processes. In this review, we provide an introduction to the biological understanding of the vascular adaptation processes of sprouting angiogenesis, intussusceptive angiogenesis, anastomosis, pruning, and arteriogenesis, discussing some of the more significant contributions made to the computational modeling of these processes. Each computational model represents a theoretical framework for how biology functions, and with rises in computing power and study of the problem these frameworks become more accurate and complete. We highlight physiological, pathological, and technological applications that can be benefit from the advances performed by these models, and we also identify which elements of the biology are underexplored in the current state-of-the-art computational models. This article is categorized under: Cancer > Computational Models Cardiovascular Diseases > Computational Models.

Parallel generation of extensive vascular networks with application to an archetypal human kidney model

Given the relevance of the inextricable coupling between microcirculation and physiology, and the relation to organ function and disease progression, the construction of synthetic vascular networks for mathematical modelling and computer simulation is becoming an increasingly broad field of research. Building vascular networks that mimic in vivo morphometry is feasible through algorithms such as constrained constructive optimization (CCO) and variations. Nevertheless, these methods are limited by the maximum number of vessels to be generated due to the whole network update required at each vessel addition. In this work, we propose a CCO-based approach endowed with a domain decomposition strategy to concurrently create vascular networks. The performance of this approach is evaluated by analysing the agreement with the sequentially generated networks and studying the scalability when building vascular networks up to 200 000 vascular segments. Finally, we apply our method to vascularize a highly complex geometry corresponding to the cortex of a prototypical human kidney. The technique presented in this work enables the automatic generation of extensive vascular networks, removing the limitation from previous works. Thus, we can extend vascular networks (e.g. obtained from medical images) to pre-arteriolar level, yielding patient-specific whole-organ vascular models with an unprecedented level of detail.

Adaptive constrained constructive optimisation for complex vascularisation processes

Mimicking angiogenetic processes in vascular territories acquires importance in the analysis of the multi-scale circulatory cascade and the coupling between blood flow and cell function. The present work extends, in several aspects, the Constrained Constructive Optimisation (CCO) algorithm to tackle complex automatic vascularisation tasks. The main extensions are based on the integration of adaptive optimisation criteria and multi-staged space-filling strategies which enhance the modelling capabilities of CCO for specific vascular architectures. Moreover, this vascular outgrowth can be performed either from scratch or from an existing network of vessels. Hence, the vascular territory is defined as a partition of vascular, avascular and carriage domains (the last one contains vessels but not terminals) allowing one to model complex vascular domains. In turn, the multi-staged space-filling approach allows one to delineate a sequence of biologically-inspired stages during the vascularisation process by exploiting different constraints, optimisation strategies and domain partitions stage by stage, improving the consistency with the architectural hierarchy observed in anatomical structures. With these features, the aDaptive CCO (DCCO) algorithm proposed here aims at improving the modelled network anatomy. The capabilities of the DCCO algorithm are assessed with a number of anatomically realistic scenarios.

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.

Modeling of Kidney Hemodynamics: Probability-Based Topology of an Arterial Network

Through regulation of the extracellular fluid volume, the kidneys provide important long-term regulation of blood pressure. At the level of the individual functional unit (the nephron), pressure and flow control involves two different mechanisms that both produce oscillations. The nephrons are arranged in a complex branching structure that delivers blood to each nephron and, at the same time, provides a basis for an interaction between adjacent nephrons. The functional consequences of this interaction are not understood, and at present it is not possible to address this question experimentally. We provide experimental data and a new modeling approach to clarify this problem. To resolve details of microvascular structure, we collected 3D data from more than 150 afferent arterioles in an optically cleared rat kidney. Using these results together with published micro-computed tomography (μCT) data we develop an algorithm for generating the renal arterial network. We then introduce a mathematical model describing blood flow dynamics and nephron to nephron interaction in the network. The model includes an implementation of electrical signal propagation along a vascular wall. Simulation results show that the renal arterial architecture plays an important role in maintaining adequate pressure levels and the self-sustained dynamics of nephrons.

By maintaining the volume and composition of the body fluids within narrow ranges, and by producing a set of hormones that affect the blood vessels, the kidneys provide important long-term regulation of blood pressure. Disturbances of kidney function can cause hypertension, a prevalent disease in modern societies. The kidneys protect their own function against short-term variations in blood pressure at the level of the individual unit (the nephron). In recent years, it has become clear that there is an interaction between nephrons, and that this interaction is mediated through the arterial network of the kidney. The renal vacular network has a complex topology, and at present there are no computational models of this topology, precluding a computational assessment of the consequences of nephron-nephron interactions for renal blood flow control. In this work we focus on understanding how kidney specific vascular structure affects blood flow patterns and nephron-to-nephron interaction in kidney. The paper presents an approach to constructing realistic models of the renal vascular architecture. We developed a computational approach to reproduce the architecture and to examine its consequences for the operating regime of the nephrons.

Toward an optimal design principle in symmetric and asymmetric tree flow networks

Fluid flow in tree-shaped networks plays an important role in both natural and engineered systems. This paper focuses on laminar flows of Newtonian and non-Newtonian power law fluids in symmetric and asymmetric bifurcating trees. Based on the constructal law, we predict the tree-shaped architecture that provides greater access to the flow subjected to the total network volume constraint. The relationships between the sizes of parent and daughter tubes are presented both for symmetric and asymmetric branching tubes. We also approach the wall-shear stresses and the flow resistance in terms of first tube size, degree of asymmetry between daughter branches, and rheological behavior of the fluid. The influence of tubes obstructing the fluid flow is also accounted for. The predictions obtained by our theory-driven approach find clear support in the findings of previous experimental studies.

Assessing tumor cytoarchitecture using multiecho DSC-MRI derived measures of the transverse relaxivity at tracer equilibrium (TRATE)

PURPOSE

In brain tumor dynamic susceptibility contrast (DSC)-MRI studies, multiecho acquisition methods are used to quantify the dynamic changes in T1 and T2 * that occur when contrast agent (CA) extravasates. Such methods also enable the estimation of the effective tissue CA transverse relaxivity. The goal of this study was to evaluate the sensitivity of the transverse relaxivity at tracer equilibrium (TRATE) to tumor cytoarchitecture.

METHODS

Computational and in vitro studies were used to evaluate the biophysical basis of TRATE. In 9L, C6, and human brain tumors, TRATE, the apparent diffusion coefficient (ADC), the CA transfer constant (K(trans) ), the extravascular extracellular volume fraction (ve ), and histological data were compared.

RESULTS

Simulations and in vitro results indicate that TRATE is highly sensitive to variations in cellular properties such as cell size and density. The histologic cell density and TRATE values were significantly higher in 9L tumors as compared to C6 tumors. In animal and human tumors, a voxel-wise comparison of TRATE with ADC, ve , and K(trans) maps showed low spatial correlation.

CONCLUSION

The assessment of TRATE is clinically feasible and its sensitivity to tissue cytoarchitectural features not present in other imaging methods indicate that it could potentially serve as a unique structural signature or "trait" of cancer.

An efficient computational approach to characterize DSC-MRI signals arising from three-dimensional heterogeneous tissue structures

The systematic investigation of susceptibility-induced contrast in MRI is important to better interpret the influence of microvascular and microcellular morphology on DSC-MRI derived perfusion data. Recently, a novel computational approach called the Finite Perturber Method (FPM), which enables the study of susceptibility-induced contrast in MRI arising from arbitrary microvascular morphologies in 3D has been developed. However, the FPM has lower efficiency in simulating water diffusion especially for complex tissues. In this work, an improved computational approach that combines the FPM with a matrix-based finite difference method (FDM), which we call the Finite Perturber the Finite Difference Method (FPFDM), has been developed in order to efficiently investigate the influence of vascular and extravascular morphological features on susceptibility-induced transverse relaxation. The current work provides a framework for better interpreting how DSC-MRI data depend on various phenomena, including contrast agent leakage in cancerous tissues and water diffusion rates. In addition, we illustrate using simulated and micro-CT extracted tissue structures the improved FPFDM along with its potential applications and limitations.

Dynamics of nephron-vascular network

The paper presents a modeling study of the spatial dynamics of a nephro-vascular network consisting of individual nephrons connected via a tree-like vascular branching structure. We focus on the effects of nonlinear mechanisms that are responsible for the formation of synchronous patterns in order to learn about processes not directly amenable to experimentation. We demonstrate that: (i) the nearest nephrons are synchronized in-phase due to a vascular propagated electrical coupling, (ii) the next few branching levels display a formation of phase-shifted patterns due to hemodynamic coupling and mode elimination, and (iii) distantly located areas show asynchronous behavior or, if all nephrons and branches are perfectly identical, an infinitely long transient behavior. These results contribute to the understanding of mechanisms responsible for the highly dynamic and limited synchronization observed among groups of nephrons despite of the fairly strong interaction between the individual units.

Analysis and algorithmic generation of hepatic vascular systems

A proper geometric model of the vascular systems in the liver is crucial for modeling blood flow, the connection between the organ and the rest of the organism. In vivo imaging does not provide sufficient details, so an algorithmic concept for extending measured vascular tree data is needed such that geometrically realistic structures can be generated. We develop a quantification of similarity in terms of different geometric features. This involves topological Strahler ordering of the vascular trees, statistical testing, and averaging. Invariant features are identified in human clinical in vivo CT scans. Results of the existing "Constrained Constructive Optimization" algorithm are compared to real vascular tree data. To improve bifurcation angles in the algorithmic results, we implement a postprocessing step calibrated to the measured features. This framework is finally applied to generate realistic additional details in a patient-specific hepatic vascular tree data set.

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.

Coronary structure and perfusion in health and disease

Blood flow is distributed through the heart muscle via a system of vessels forming the coronary circulation. The perfusion of the myocardium can be hampered by atherosclerosis creating localized obstructions in the epicardial vessels or by microvascular disease. In early stages of the disease, these impediments to blood flow are offset by dilation of the resistance vessels, which normally compensates for a decrease in perfusion pressure or increased metabolism. However, this dilatory reserve can become exhausted, which in general occurs first at the deeper layers of the heart wall where intramural vessels are subjected to compressive forces related to heart contraction. In the catheterization laboratory, guide wires of 0.33 mm diameter are available that are equipped with a pressure and flow velocity sensor at the tip, which can be positioned distal to the stenosis. These signals provide information about the impediment of the stenosis on coronary flow and allow for the evaluation of the status of the microcirculation. However, the interpretation of these signals is strongly model-dependent and therefore it is of paramount importance to develop realistic models reflecting the anatomy and unique physiology of the coronary circulation.

Retinal image analysis: concepts, applications and potential

As digital imaging and computing power increasingly develop, so too does the potential to use these technologies in ophthalmology. Image processing, analysis and computer vision techniques are increasing in prominence in all fields of medical science, and are especially pertinent to modern ophthalmology, as it is heavily dependent on visually oriented signs. The retinal microvasculature is unique in that it is the only part of the human circulation that can be directly visualised non-invasively in vivo, readily photographed and subject to digital image analysis. Exciting developments in image processing relevant to ophthalmology over the past 15 years includes the progress being made towards developing automated diagnostic systems for conditions, such as diabetic retinopathy, age-related macular degeneration and retinopathy of prematurity. These diagnostic systems offer the potential to be used in large-scale screening programs, with the potential for significant resource savings, as well as being free from observer bias and fatigue. In addition, quantitative measurements of retinal vascular topography using digital image analysis from retinal photography have been used as research tools to better understand the relationship between the retinal microvasculature and cardiovascular disease. Furthermore, advances in electronic media transmission increase the relevance of using image processing in 'teleophthalmology' as an aid in clinical decision-making, with particular relevance to large rural-based communities. In this review, we outline the principles upon which retinal digital image analysis is based. We discuss current techniques used to automatically detect landmark features of the fundus, such as the optic disc, fovea and blood vessels. We review the use of image analysis in the automated diagnosis of pathology (with particular reference to diabetic retinopathy). We also review its role in defining and performing quantitative measurements of vascular topography, how these entities are based on 'optimisation' principles and how they have helped to describe the relationship between systemic cardiovascular disease and retinal vascular changes. We also review the potential future use of fundal image analysis in telemedicine.

Optimized arterial trees supplying hollow organs

Computer models of arterial trees can be generated from optimization principles using the algorithm of constrained constructive optimization (CCO). Up to now this algorithm could handle only tissue areas of convex shape, without concavities. CCO is now generalized to cope also with non-convex organ shapes, possibly featuring external as well as internal concavities. This allows the modeling of a much larger class of interesting real arterial systems. The concept of a generalized domain-potential was developed to represent arbitrary non-convex shapes mathematically and incorporate them as boundary conditions to optimization. Domain-potentials may be derived from analytical representations as well as from finite element triangulations obtained from organ images. To demonstrate the feasibility of the concept, the optimized growth of an arterial tree model is confined to some part of an elliptical shell, representing the free wall of the left ventricle of the heart.

Physiologically based modeling of 3-D vascular networks and CT scan angiography

In this paper, a model-based approach to medical image analysis is presented. It is aimed at understanding the influence of the physiological (related to tissue) and physical (related to image modality) processes underlying the image content. This methodology is exemplified by modeling first, the liver and its vascular network, and second, the standard computed tomography (CT) scan acquisition. After a brief survey on vascular modeling literature, a new method, aimed at the generation of growing three-dimensional vascular structures perfusing the tissue, is described. A solution is proposed in order to avoid intersections among vessels belonging to arterial and/or venous trees, which are physiologically connected. Then it is shown how the propagation of contrast material leads to simulate time-dependent sequences of enhanced liver CT slices.

On the principles of the vascular network branching

We propose an explanation of Murray's law without applying the minimality principles. The model deals with a "delivering" artery system of an organ that is characterized, first, by the space-filling embedding into the organ tissue and, second, by the uniform distribution of the blood pressure drop over it. The latter assumption is justified using the available physiological data and the idea about conditions needed for perfect self-regulation. Based on the two statements we get Murray's law, and so, demonstrate that it can be also regarded as a direct consequence of the organism's capacity for controlling finely the blood flow redistribution over peripheral vascular networks.

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.Key words: arteries, optimization, volume, power, geometry.

A three-dimensional model for arterial tree representation, generated by constrained constructive optimization

The computational method of constrained constructive optimization (CCO) has been generalized in two important respects: (1) arterial model trees are now grown within a convex, three-dimensional piece of tissue and (2) terminal flow variability has been incorporated into the model to account for the heterogeneity of blood flow observed in real vascular beds. Although no direct information from topographic anatomy enters the model, computer-generated CCO trees closely resemble corrosion casts of real arterial trees, both on a visual basis and with regard to morphometric parameters. Terminal flow variability was found to induce transitions in the connective structure early in the trees' development. The present generalization of CCO offers--for the first time--the possibility to generate optimized arterial model trees in three dimensions, representing a realistic geometrical substrate for hemodynamic simulation studies. With the implementation of terminal flow variability the model is ready to simulate processes such as the adaptation of arterial diameters to changes in blood flow rate or the formation of different patterns of angiogenesis induced by changing needs of blood supply.