Shear stress distribution in arterial tree models, generated by constrained constructive optimization

A constrained constructive optimization model of branching arteriolar networks in rat skeletal muscle

Blood flow regulation within the microvasculature reflects a complex interaction of regulatory mechanisms and varies spatially and temporally according to conditions such as metabolism, growth, injury, and disease. Understanding the role of microvascular flow distributions across conditions is of interest to investigators spanning multiple disciplines; however, data collection within networks can be labor-intensive and challenging due to limited resolution. To overcome these experimental challenges, computational network models that can accurately simulate vascular behavior are highly beneficial. Constrained constructive optimization (CCO) is a commonly used algorithm for vascular simulation, particularly well known for its adaptability toward vascular modeling across tissues. The present work demonstrates an implementation of CCO aimed to simulate a branching arteriolar microvasculature in healthy skeletal muscle, validated against literature including comprehensive rat gluteus maximus vasculature datasets, and reviews a list of user-specified adjustable model parameters to understand how their variability affects the simulated networks. Network geometric properties, including mean element diameters, lengths, and numbers of bifurcations per order, Horton's law ratios, and fractal dimension, demonstrate good validation once model parameters are adjusted to experimental data. This model successfully demonstrates hemodynamic properties such as Murray's law and the network Fahraeus effect. Application of centrifugal and Strahler ordering schemes results in divergent descriptions of identical simulated networks. This work introduces a novel CCO-based model focused on generating branching skeletal muscle microvascular arteriolar networks based on adjustable model parameters, thus making it a valuable tool for investigations into skeletal muscle microvascular structure and tissue perfusion.NEW & NOTEWORTHY The present work introduces a CCO-based algorithm for generating branching arteriolar networks, with adjustable model parameters to enable modeling in varying skeletal muscle tissues. The geometric and hemodynamic parameters of the generated networks have been comprehensively validated using experimental data collected previously in-house and from literature. This is one of few validated CCO-based models to specialize in skeletal muscle microvasculature and acts as a beneficial tool for investigating the microvasculature for hypothesis testing and validation.

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.

Mathematical synthesis of the cortical circulation for the whole mouse brain-part I. theory and image integration

Microcirculation plays a significant role in cerebral metabolism and blood flow control, yet explaining and predicting functional mechanisms remains elusive because it is difficult to make physiologically accurate mathematical models of the vascular network. As a precursor to the human brain, this paper presents a computational framework for synthesizing anatomically accurate network models for the cortical blood supply in mouse. It addresses two critical deficiencies in cerebrovascular modeling. At the microscopic length scale of individual capillaries, we present a novel synthesis method for building anatomically consistent capillary networks with loops and anastomoses (=microcirculatory closure). This overcomes shortcomings in existing algorithms which are unable to create closed circulatory networks. A second critical innovation allows the incorporation of detailed anatomical features from image data into vascular growth. Specifically, computed tomography and two photon laser scanning microscopy data are input into the novel synthesis algorithm to build the cortical circulation for the entire mouse brain in silico. Computer predictions of blood flow and oxygen exchange executed on synthetic large-scale network models are expected to elucidate poorly understood functional mechanisms of the cerebral circulation.

Towards organ printing: engineering an intra-organ branched vascular tree

IMPORTANCE OF THE FIELD

Effective vascularization of thick three-dimensional engineered tissue constructs is a problem in tissue engineering. As in native organs, a tissue-engineered intra-organ vascular tree must be comprised of a network of hierarchically branched vascular segments. Despite this requirement, current tissue-engineering efforts are still focused predominantly on engineering either large-diameter macrovessels or microvascular networks.

AREAS COVERED IN THIS REVIEW

We present the emerging concept of organ printing or robotic additive biofabrication of an intra-organ branched vascular tree, based on the ability of vascular tissue spheroids to undergo self-assembly.

WHAT THE READER WILL GAIN

The feasibility and challenges of this robotic biofabrication approach to intra-organ vascularization for tissue engineering based on organ-printing technology using self-assembling vascular tissue spheroids including clinically relevantly vascular cell sources are analyzed.

TAKE HOME MESSAGE

It is not possible to engineer 3D thick tissue or organ constructs without effective vascularization. An effective intra-organ vascular system cannot be built by the simple connection of large-diameter vessels and microvessels. Successful engineering of functional human organs suitable for surgical implantation will require concomitant engineering of a 'built in' intra-organ branched vascular system. Organ printing enables biofabrication of human organ constructs with a 'built in' intra-organ branched vascular tree.

Error estimation of geometrical data obtained by histomorphometry of oblique vessel sections: a computer model study

The errors of radius and wall thickness of a single vessel due to oblique sectioning in histomorphometry are expressed as a function of the circular shape factor (CSF) of the section's lumen, assuming cylindrical geometry and the absence of tissue deformation. Using computer model trees generated by constrained constructive optimization, mean errors are estimated for an ensemble of vessel segments. A geometrical exclusion criterion for segments cut too obliquely is defined on the basis of a CSF-cutoff value. It is shown that CSF-values ranging from 0.95 to 0.9 are reasonable choices for a cutoff and lead to mean errors of the same order of magnitude (9.6% [9.3%] to 15.4% [14.8%] for the radius [wall thickness]) as errors due to histological tissue processing.

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.

Heterogeneous perfusion is a consequence of uniform shear stress in optimized arterial tree models

Using optimized computer models of arterial trees we demonstrate that flow heterogeneity is a necessary consequence of a uniform shear stress distribution. Model trees are generated and optimized under different modes of boundary conditions. In one mode flow is delivered to the tissue as homogeneously as possible. Although this primary goal can be achieved, resulting shear stresses between blood and the vessel walls show very large spread. In a second mode, models are optimized under the condition of uniform shear stress in all segments which in turn renders flow distribution heterogeneous. Both homogeneous perfusion and uniform shear stress are desirable goals in real arterial trees but each of these goals can only be approached at the expense of the other. While the present paper refers only to optimized models, we assume that this dual relation between the heterogeneities in flow and shear stress may represent a more general principle of vascular systems.

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.

Structural and biophysical simulation of angiogenesis and vascular remodeling

The purpose of this report is to introduce a new computer model for the simulation of microvascular growth and remodeling into arteries and veins that imitates angiogenesis and blood flow in real vascular plexuses. A C++ computer program was developed based on geometric and biophysical initial and boundary conditions. Geometry was defined on a two-dimensional isometric grid by using defined sources and drains and elementary bifurcations that were able to proliferate or to regress under the influence of random and deterministic processes. Biophysics was defined by pressure, flow, and velocity distributions in the network by using the nodal-admittance-matrix-method, and accounting for hemodynamic peculiarities like Fahraeus-Lindqvist effect and exchange with extravascular tissue. The proposed model is the first to simulate interdigitation between the terminal branches of arterial and venous trees. This was achieved by inclusion of vessel regression and anastomosis in the capillary plexus and by remodeling in dependence from hemodynamics. The choice of regulatory properties influences the resulting vascular patterns. The model predicts interdigitating arteriovenous patterning if shear stress-dependent but not pressure-dependent remodeling was applied. By approximating the variability of natural vascular patterns, we hope to better understand homogeneity of transport, spatial distribution of hemodynamic properties and biomass allocation to the vascular wall or blood during development, or during evolution of circulatory systems.

Branching exponent heterogeneity and wall shear stress distribution in vascular trees

A bifurcating arterial system with Poiseuille flow can function at minimum cost and with uniform wall shear stress if the branching exponent (z) = 3 [where z is defined by (D(1))(z) = (D(2))(z) + (D(3))(z); D(1) is the parent vessel diameter and D(2) and D(3) are the two daughter vessel diameters at a bifurcation]. Because wall shear stress is a physiologically transducible force, shear stress-dependent control over vessel diameter would appear to provide a means for preserving this optimal structure through maintenance of uniform shear stress. A mean z of 3 has been considered confirmation of such a control mechanism. The objective of the present study was to evaluate the consequences of a heterogeneous distribution of z values about the mean with regard to this uniform shear stress hypothesis. Simulations were carried out on model structures otherwise conforming to the criteria consistent with uniform shear stress when z = 3 but with varying distributions of z. The result was that when there was significant heterogeneity in z approaching that found in a real arterial tree, the coefficient of variation in shear stress was comparable to the coefficient of variation in z and nearly independent of the mean value of z. A systematic increase in mean shear stress with decreasing vessel diameter was one component of the variation in shear stress even when the mean z = 3. The conclusion is that the influence of shear stress in determining vessel diameters is not, per se, manifested in a mean value of z. In a vascular tree having a heterogeneous distribution in z values, a particular mean value of z (e.g., z = 3) apparently has little bearing on the uniform shear stress hypothesis.

Form and function of fetal and neonatal pulmonary arterial bifurcations

Bifurcation is a basic form of vascular connection. It is composed of a parent vessel of diameter d(0), and two daughter vessels, d(1) and d(2), where d(0) > d(1) >/= d(2). Optimal values for the bifurcation area ratio, beta = (d(1)(2) + d(2)(2))/d(0)(2), and the junction exponent, x, in d(0)(x) = d(1)(x) + d(2)(x), are postulated to be universal in nature. However, we have hypothesized that the perinatal pulmonary arterial circulation is an exception. Arterial diameters were measured in pulmonary vascular casts of a fetal lamb (140 days gestation/145 days term) and a neonatal lamb (1 day old). The values for beta and x were evaluated in 10,970 fetal and 846 neonatal bifurcations sampled from the proximal and intermediate arterial regions. Mean values and confidence intervals (CI) for the fetus were beta = 0.890 (0.886-0.895 CI) and x = 1.75 (1.74-1.76 CI); and for the newborn were beta = 0.913 (0.90-0.93 CI) and x = 1. 79 (1.75-1.82 CI). These values are significantly different from Murray's law (beta > 1, x = 3) or the West-Brown-Enquist law (beta = 1, x = 2). Therefore, perinatal pulmonary bifurcation design appears to be distinctive and exceptional. The decreasing cross-sectional area with branching leads to the hemodynamic consequence of shear stress amplification. This structural organization may be important for facilitating vascular development at low flow rates; however, it may be the origin of unstable reactivity if elevated blood flow and pressure occurs.