Mathematical modeling of local perfusion in large distensible microvascular networks

Abstract

Microvessels -blood vessels with diameter less than 200 µm-form large, intricate networks organized into arterioles, capillaries and venules. In these networks, the distribution of flow and pressure drop is a highly interlaced function of single vessel resistances and mutual vessel interactions. Since, it is often impossible to quantify all these aspects when collecting experimental measures, in this paper we propose a mathematical and computational model to study the behavior of microcirculatory networks subjected to different conditions. The network geometry, which can be derived from digitized images of experimental measures or constructed in silico on a computer by mathematical laws, is simplified for computational purposes into a graph of connected straight cylinders, each one representing a vessel. The blood flow and pressure drop across the single vessel, further split into smaller elements, are related through a generalized Ohm's law featuring a conductivity parameter, function of the vessel cross section area and geometry, which undergo deformations under pressure loads. The membrane theory is used for the description of the deformation of vessel lumina, tailored to the structure of thick-walled arterioles and thin-walled venules. In addition, since venules can possibly experience negative values of transmural pressure (difference between luminal and interstitial pressure), a buckling model is also included to represent vessel collapse. The complete model including arterioles, capillaries and venules represents a nonlinear coupled system of PDEs, which is approached numerically by finite element discretization and linearization techniques. As an example of application, we use the model to simulate flow in the microcirculation of the human eye retina, a terminal system with a single inlet and outlet. After a phase of validation against experimental measurements of the correctness of the blood flow and pressure fields in the network, we compute the network response to different interstitial pressure values. Such a study is carried out both for global and localized variations of the interstitial pressure. In both cases, significant redistributions of the blood flow in the network arise, highlighting the importance of considering the single vessel behavior along with its position and connectivity in the network.

Reduced‐order modeling of hemodynamics across macroscopic through mesoscopic circulation scales

“…Only arteries, arterioles and capillaries are modeled, avoiding the need for complex venular tree modeling. [ 43 ] Despite spatial dimension simplifications, hemodynamics simulation across several scales is a very demanding computational task. The new implementation makes feasible the dynamical simulation of large compliant network with (10 4 ) vessels, without resorting to domain decomposition, thanks to the rapid convergence properties exhibited by the spectral elements method, with reasonable dimensionality: spatial number of degrees of freedom: 2 × × × ⌈ 3 2 ⌉ ∼ (10 5 ).…”

“…This simple thin membrane law has been successfully exploited in many applications, but is not well adapted for arteriolar wall bearing a thick muscularis layer designed to sustain high transmural pressures. In the following, we adopt the structural models described in [ 43 ] in order to address the specifics of the wall thickness-to-radius ratios. Each vessel segment is modeled as an elastic but incompressible cylindrical shell with its own rigidity and only moderate deformations are considered.…”

“…In practice, given a reference vessel diameter, is piecewise linearly interpolated from literature data. [ 43 ] It was checked that other types of interpolant (nonlinear polynomials, spline, shape-preserving piecewise cubic) did not significantly change the value of the Young modulus within the considered range (i.e.…”

“…where the parameter may encompass a vector of parameters describing the mechanical properties of the membrane, related to the distensibility of the vessel: = 2∕ √ . Causin et al [ 43 ] have identified two cases:…”

In silico model of the early effects of radiation therapy on the microcirculation and the surrounding tissues

“…We analyze here the vessels with a mechanical point of view, following the approach adopted in Causin et al [20]. The wall of a vessel consists in a single layer of endothelial cells, a basement membrane and a layer made of fibrils, like collagen.…”