A Mixed Approach for Modeling Blood Flow in Brain Microcirculation
Abstract:We have previously demonstrated  that the vascular system of the healthy human brain cortex is a superposition of two structural components, each corresponding to a different spatial scale. At small-scale, the vascular network has a capillary structure, which is homogeneous and space-filling over a cut-off length. At larger scale, veins and arteries conform to a quasi-fractal branched structure. This structural duality is consistent with the functional duality of the vasculature, i.e. distribution and exchange. From a modeling perspective, this can be viewed as the superposition of: (a) a continuum model describing slow transport in the small-scale capillary network, characterized by a representative elementary volume and effective properties; and (b) a discrete network approach  describing fast transport in the arterial and venous network, which cannot be homogenized because of its fractal nature.
This problematic is analogous to modeling problems encountered in geological media, e.g, in petroleum engineering, where fast conducting channels (wells or fractures) are embedded in a porous medium (reservoir rock). An efficient method to reduce the computational cost of fractures/continuum simulations is to use relatively large grid blocks for the continuum model. However, this also makes it difficult to accurately couple both structural components. In this work, we solve this issue by adapting the “well model” concept used in petroleum engineering  to brain specific 3-D situations. We obtain a unique linear system of equations describing the discrete network, the continuum and the well model coupling. Results are presented for realistic geometries and compared with a non-homogenized small-scale network model of an idealized periodic capillary network of known permeability.
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