Hybrid Models for Simulating Blood Flow in Microvascular Networks

Abstract

In this paper, we are concerned with the simulation of blood flow in microvascular networks and the surrounding tissue. To reduce the computational complexity of this issue, the network structures are modeled by a one-dimensional graph, whose location in space is determined by the centerlines of the three-dimensional vessels. The surrounding tissue is considered as a homogeneous porous medium. Darcy's equation is used to simulate flow in the extra-vascular space, where the mass exchange with the blood vessels is accounted for by means of line source terms. However, this model reduction approach still causes high computational costs, in particular, when larger parts of an organ have to be simulated. This observation motivates the consideration of a further model reduction step. Thereby, we homogenize the fine scale structures of the microvascular networks resulting in a new hybrid approach modeling the fine scale structures as a heterogeneous porous medium and the flow in the larger vessels by one-dimensional flow equations. Both modeling approaches are compared with respect to mass fluxes and averaged pressures. The simulations have been performed on a microvascular network that has been extracted from the cortex of a rat brain.

A multiscale subvoxel perfusion model to estimate diffusive capillary wall conductivity in multiple sclerosis lesions from perfusion MRI data

“…These models are formulated on the macroscale using averaging techniques. The relation of fluid‐mechanical models on the mesoscale (as considered in this work) and models formulated on the macroscale is yet to be better understood and is addressed in some recent studies . However, values for the diffusive wall permeabilities have been estimated from direct measurements with single or multiple capillaries from different tissues .…”

“…However, the computational cost can most likely be improved by applying model reduction techniques and machine learning algorithms. Likewise, homogenization techniques can be used for model reduction . However, such techniques are difficult to apply because of the hierarchical structure of the microcirculation.…”

“…The relation of fluid-mechanical models on the mesoscale (as considered in this work) and models formulated on the macroscale is yet to be better understood and is addressed in some recent studies. [74][75][76][77] However, values for the diffusive wall permeabilities have been estimated from direct measurements with single or multiple capillaries from different tissues. [78][79][80][81] The values and methods are F I G U R E 8 Histograms of model parameter distributions after learning from magnetic resonance (MR) voxel data from a lesion (see Figure 1, sample L).…”

“…Likewise, homogenization techniques can be used for model reduction. 74,75 However, such techniques are difficult to apply because of the hierarchical structure of the microcirculation. For all approaches, the presented model can be used as theoretical basis and as validation tool.…”

A multi-scale sub-voxel perfusion model to estimate diffusive capillary wall conductivity in multiple sclerosis lesions from perfusion MRI data

“…However, the 628 computational cost can most likely be improved by applying model reduction techniques 629 and machine learning algorithms. Likewise, homogenization techniques can be used for 630 model reduction[69,70]. However, such techniques are difficult to apply, due to the 631 hierarchical structure of the micro-circulation.…”