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Blood flow pulsatility is an important determinant of macro- and microvascular physiology. Pulsatility is damped largely in the microcirculation, but the characteristics of this damping and the factors that regulate it have not been fully elucidated yet. Applying computational approaches to real microvascular network geometry, we examined the pattern of pulsatility damping and the role of potential damping factors, including pulse frequency, vascular viscous resistance, vascular compliance, viscoelastic behavior of the vessel wall, and wave propagation and reflection. To this end, three full rat mesenteric vascular networks were reconstructed from intravital microscopic recordings, a one-dimensional (1D) model was used to reproduce pulsatile properties within the network, and potential damping factors were examined by sensitivity analysis. Results demonstrate that blood flow pulsatility is predominantly damped at the arteriolar side and remains at a low level at the venular side. Damping was sensitive to pulse frequency, vascular viscous resistance and vascular compliance, whereas viscoelasticity of the vessel wall or wave propagation and reflection contributed little to pulsatility damping. The present results contribute to our understanding of mechanical forces and their regulation in the microcirculation., (Copyright © 2021. Published by Elsevier Inc.)

In recent years, biological imaging techniques have advanced significantly and it is now possible to digitally reconstruct microvascular network structures in detail, identifying the smallest capillaries at sub-micron resolution and generating large 3D structural data sets of size >106 vessel segments. However, this relies on ex vivo imaging; corresponding in vivo measures of microvascular structure and flow are limited to larger branching vessels and are not achievable in three dimensions for the smallest vessels. This suggests the use of computational modelling to combine in vivo measures of branching vessel architecture and flows with ex vivo data on complete microvascular structures to predict effective flow and pressures distributions. In this paper, a hybrid discrete-continuum model to predict microcirculatory blood flow based on structural information is developed and compared with existing models for flow and pressure in individual vessels. A continuum-based Darcy model for transport in the capillary bed is coupled via point sources of flux to flows in individual arteriolar vessels, which are described explicitly using Poiseuille's law. The venular drainage is represented as a spatially uniform flow sink. The resulting discrete-continuum framework is parameterized using structural data from the capillary network and compared with a fully discrete flow and pressure solution in three networks derived from observations of the rat mesentery. The discrete-continuum approach is feasible and effective, providing a promising tool for extracting functional transport properties in situations where vascular branching structures are well defined., (© Crown copyright 2019.)

Hemodynamic pulsatility has been reported to regulate microcirculatory function. To quantitatively assess the impact of flow pulsatility on the microvasculature, a mathematical model was first developed to simulate the regulation of NO production by pulsatile flow in the microcirculation. Shear stress and pressure pulsatility were selected as regulators of endothelial NO production and NO-dependent vessel dilation as feedback to control microvascular hemodynamics. The model was then applied to a real microvascular network of the rat mesentery consisting of 546 microvessels. As compared to steady flow conditions, pulsatile flow increased the average NO concentration in arterioles from 256.8±93.1nM to 274.8±101.1nM (P<0.001), with a corresponding increase in vessel dilation by approximately 7% from 27.5±10.6% to 29.4±11.4% (P<0.001). In contrast, NO concentration and vessel size showed a far lesser increase (about 1.7%) in venules under pulsatile flow as compared to steady flow conditions. Network perfusion and flow heterogeneity were improved under pulsatile flow conditions, and vasodilation within the network was more sensitive to heart rate changes than pulse pressure amplitude. The proposed model simulates the role of flow pulsatility in the regulation of a complex microvascular network in terms of NO concentration and hemodynamics under varied physiological conditions., (Copyright © 2017 Elsevier Inc. All rights reserved.)

Objective: Recent developments in high-resolution imaging techniques have enabled digital reconstruction of three-dimensional sections of microvascular networks down to the capillary scale. To better interpret these large data sets, our goal is to distinguish branching trees of arterioles and venules from capillaries., Methods: Two novel algorithms are presented for classifying vessels in microvascular anatomical data sets without requiring flow information. The algorithms are compared with a classification based on observed flow directions (considered the gold standard), and with an existing resistance-based method that relies only on structural data., Results: The first algorithm, developed for networks with one arteriolar and one venular tree, performs well in identifying arterioles and venules and is robust to parameter changes, but incorrectly labels a significant number of capillaries as arterioles or venules. The second algorithm, developed for networks with multiple inlets and outlets, correctly identifies more arterioles and venules, but is more sensitive to parameter changes., Conclusions: The algorithms presented here can be used to classify microvessels in large microvascular data sets lacking flow information. This provides a basis for analyzing the distinct geometrical properties and modelling the functional behavior of arterioles, capillaries, and venules., (© 2014 The Authors. Microcirculation published by John Wiley & Sons Ltd.)

The adequate and efficient functioning of the microcirculation requires not only numerous vessels providing a large surface area for transport but also a structure that provides short diffusion distances from capillaries to tissue and efficient distribution of convective blood flow. Theoretical models show how a combination of angiogenesis, remodeling, and pruning in response to hemodynamic and metabolic stimuli, termed "angioadaptation," generates well organized, functional networks., (©2014 Int. Union Physiol. Sci./Am. Physiol. Soc.)

Techniques that model microvascular hemodynamics have been developed for decades. While the physiological significance of pressure pulsatility is acknowledged, most of the microcirculatory models use steady flow approaches. To theoretically study the extent and transmission of pulsatility in microcirculation, dynamic models need to be developed. In this paper, we present a one-dimensional model to describe the dynamic behavior of microvascular blood flow. The model is applied to a microvascular network from a rat mesentery. Intravital microscopy was used to record the morphology and flow velocities in individual vessel segments, and boundaries are defined according to the experimental data. The system of governing equations constituting the model is solved numerically using the discontinuous Galerkin method. An implicit integration scheme is adopted to increase computing efficiency. The model allows the simulation of the dynamic properties of blood flow in microcirculatory networks, including the pressure pulsatility (quantified by a pulsatility index) and pulse wave velocity (PWV). From the main input arteriole to the main output venule, the pulsatility index decreases by 66.7%. PWV obtained along arterioles declines with decreasing diameters, with mean values of 77.16, 25.31, and 8.30 cm/s for diameters of 26.84, 17.46, and 13.33 μm, respectively. These results suggest that the 1D model developed is able to simulate the characteristics of pressure pulsatility and wave propagation in complex microvascular networks.

Estimation of the boundary condition is a critical problem in simulating hemodynamics in microvascular networks. This paper proposed a boundary estimation strategy based on a particle swarm optimization (PSO) algorithm, which aims to minimize the number of vessels with inverted flow direction in comparison to the experimental observation. The algorithm took boundary values as the particle swarm and updated the position of the particles iteratively to approach the optimization target. The method was tested in a real rat mesenteric network. With random initial boundary values, the method achieved a minimized 9 segments with an inverted flow direction in the network with 546 vessels. Compared with reported literature, the current work has the advantage of a better fit with experimental observations and is more suitable for the boundary estimation problem in pulsatile hemodynamic models due to the experiment-based optimization target selection.

Formation of functionally adequate vascular networks by angiogenesis presents a problem in biological patterning. Generated without predetermined spatial patterns, networks must develop hierarchical tree-like structures for efficient convective transport over large distances, combined with dense space-filling meshes for short diffusion distances to every point in the tissue. Moreover, networks must be capable of restructuring in response to changing functional demands without interruption of blood flow. Here, theoretical simulations based on experimental data are used to demonstrate that this patterning problem can be solved through over-abundant stochastic generation of vessels in response to a growth factor generated in hypoxic tissue regions, in parallel with refinement by structural adaptation and pruning. Essential biological mechanisms for generation of adequate and efficient vascular patterns are identified and impairments in vascular properties resulting from defects in these mechanisms are predicted. The results provide a framework for understanding vascular network formation in normal or pathological conditions and for predicting effects of therapies targeting angiogenesis.

Relative to normal tissues, tumor microcirculation exhibits high structural and functional heterogeneity leading to hypoxic regions and impairing treatment efficacy. Here, computational simulations of blood vessel structural adaptation are used to explore the hypothesis that abnormal adaptive responses to local hemodynamic and metabolic stimuli contribute to aberrant morphological and hemodynamic characteristics of tumor microcirculation. Topology, vascular diameter, length, and red blood cell velocity of normal mesenteric and tumor vascular networks were recorded by intravital microscopy. Computational models were used to estimate hemodynamics and oxygen distribution and to simulate vascular diameter adaptation in response to hemodynamic, metabolic and conducted stimuli. The assumed sensitivity to hemodynamic and conducted signals, the vascular growth tendency, and the random variability of vascular responses were altered to simulate 'normal' and 'tumor' adaptation modes. The heterogeneous properties of vascular networks were characterized by diameter mismatch at vascular branch points (d(3) (var)) and deficit of oxygen delivery relative to demand (O(2def)). In the tumor, d(3) (var) and O(2def) were higher (0.404 and 0.182) than in normal networks (0.278 and 0.099). Simulated remodeling of the tumor network with 'normal' parameters gave low values (0.288 and 0.099). Conversely, normal networks attained tumor-like characteristics (0.41 and 0.179) upon adaptation with 'tumor' parameters, including low conducted sensitivity, increased growth tendency, and elevated random biological variability. It is concluded that the deviant properties of tumor microcirculation may result largely from defective structural adaptation, including strongly reduced responses to conducted stimuli.

Theoretical modeling approaches have made important contributions to research in the biological sciences, including the microcirculation, and their value is increasingly recognized. However, misconceptions about the nature and role of theoretical models persist, and such work is often presented in a way that does not maximize its value, especially for scientists who are not themselves using such models. In this review, a categorization of models as phenomenological, qualitative conceptual, quantitative conceptual, or predictive is proposed, and the characteristics of each type are discussed. Recommendations are made for the presentation of models and for the future development of modeling approaches. The concepts discussed are generally applicable to the theoretical modeling of biological systems and are illustrated by using examples of modeling in microcirculation research.

The functional properties of microcirculation crucially depend on its angioarchitecture, (i.e., vessel arrangement and morphology). The microcirculation is subject to continuous dynamic structural adaptation (i.e., remodeling) controlled by hemodynamic and metabolic stimuli. Due to the complexity of the interactions among stimuli, reactions, and functional properties, an adequate understanding of structural adaptation requires mathematical models in addition to experimental investigations. Mathematical models have been developed that allow the prediction of realistic vascular properties, based on generic patterns of vascular responses. These models can be used to investigate and predict distributions of vessel morphology consistent with certain putative adaptation principles of terminal vascular beds in response to local hemodynamic and metabolic conditions. They have suggested new hypotheses, including the importance of conducted responses in network adaptation, and can explain the mechanisms underlying observed structural and functional network properties. In the future, the value of such models can be enhanced by including the effects of longitudinal stretch and pulsatility, the relationship between acute tone and structural adaptation, and the description of molecular and cellular mechanisms underlying structural responses of microvessels.

A theoretical method is used to simulate the motion and deformation of mammalian red blood cells (RBCs) in microvessels, based on knowledge of the mechanical characteristics of RBCs. Each RBC is represented as a set of interconnected viscoelastic elements in two dimensions. The motion and deformation of the cell and the motion of the surrounding fluid are computed using a finite-element numerical method. Simulations of RBC motion in simple shear flow of a high-viscosity fluid show "tank-treading'' motion of the membrane around the cell perimeter, as observed experimentally. With appropriate choice of the parameters representing RBC mechanical properties, the tank-treading frequency and cell elongation agree closely with observations over a range of shear rates. In simulations of RBC motion in capillary-sized channels, initially circular cell shapes rapidly approach shapes typical of those seen experimentally in capillaries, convex in front and concave at the rear. An isolated RBC entering an 8-mum capillary close to the wall is predicted to migrate in the lateral direction as it traverses the capillary, achieving a position near the center-line after traveling a distance of about 60 mum. Cell trajectories agree closely with those observed in microvessels of the rat mesentery.

Vascular functions, including tissue perfusion and peripheral resistance, reflect continuous structural adaptation (remodeling) of blood vessels in response to several stimuli. Here, a theoretical model is presented that relates the structural and functional properties of microvascular networks to the adaptive responses of individual segments to hemodynamic and metabolic stimuli. All vessels are assumed to respond, according to a common set of adaptation rules, to changes in wall shear stress, circumferential wall stress, and tissue metabolic status (indicated by partial pressure of oxygen). An increase in vessel diameter with increasing wall shear stress and an increase in wall mass with increased circumferential stress are needed to ensure stable vascular adaptation. The model allows quantitative predictions of the effects of changes in systemic hemodynamic conditions or local adaptation characteristics on vessel structure and on peripheral resistance. Predicted effects of driving pressure on the ratio of wall thickness to vessel diameter are consistent with experimental observations. In addition, peripheral resistance increases by approximately 65% for an increase in driving pressure from 50 to 150 mm Hg. Peripheral resistance is predicted to be markedly increased in response to a decrease in vascular sensitivity to wall shear stress, and to be decreased in response to increased tissue metabolic demand. This theoretical approach provides a framework for integrating available information on structural remodeling in the vascular system and predicting responses to changing conditions or altered vascular reactivity, as may occur in hypertension.

The goals of this study were to determine the time course and spatial dependence of structural diameter changes in the mouse gracilis artery after a redistribution of blood flow and to compare the observations with predictions of computational models for structural adaptation. Diameters were measured 1, 2, 7, 14, 21, 28, and 56 days after resection of one of the two blood supplies to the artery. Overall average diameter, normalized with respect to diameters in untreated vessels, increased slightly during the first 7 days, then increased more rapidly, reaching a peak around day 21, and then decreased. This transient increase in diameter was spatially nonuniform, being largest toward the point of resection. A previously developed theoretical model, in which diameter varies in response to stimuli derived from local metabolic and hemodynamic conditions, was extended to include effects of time-delayed remodeling stimuli in regions of reduced perfusion. Predictions of this model were consistent with observed diameter changes, including the transient increase in diameters near the point of resection, when a remodeling stimulus with a time delay of approximately 7 days was included. The results suggest that delayed stimuli significantly influence the dynamic characteristics of vascular remodeling resulting from reduced blood supply.