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Abstract

Background and aims: This work aims at constructing a computational model that can faithfully predict cerebral blood flow (CBF) in realistic vascular networks. We propose a vascular graph (VG) model, based on simple fluid dynamic principles, that comprises an upscaling algorithm, which both reduces the computational cost and allows CBF simulations even when the discrete topology of the capillary bed is unknown.
Methods: The cerebral vasculature can be represented by a graph of resistive elements. Each vessel bifurcation or end-point is a vertex, the blood vessels themselves designate the graph's edges that connect pairs of vertices. By requiring mass conservation at the vertices and providing appropriate boundary conditions, the resulting linear system of equations can be solved to yield pressure and flow values for the entire graph. This procedure has been thoroughly described for the analogous problem in electrical circuits (Weinberg, 1962), was adapted to vascular networks by Lipowsky and Zweifach (Lipowsky et al, 1974), and was recently extended by Boas and coworkers (Boas et al, 2008). We have further enhanced the model with focus on three-dimensionality and applicability to complex realistic vascular networks. By embedding the vasculature in a computational grid representative of brain tissue, the interaction between the two compartments can be captured in a truly three-dimensional fashion. The proposed upscaling algorithm replaces the discrete topology of the capillary bed by a coarser-scale network with similar fluid dynamical properties. This is achieved by dividing the computational domain into sub-volumes for each of which effective conductance values of the capillary bed are computed.
Results: The VG model was applied to high-resolution angiography data of the rat cortex obtained with synchrotron radiation based x-ray microscopy (Weber et al, 2006). Our simulations reproduce the well-established finding that the dilation of an arteriole leads to an increase in CBF in the feeding vessel, as well as the capillary bed it irrigates and the corresponding draining veins. We also find that flow decreases in all other arterioles that supply that same region of the capillary bed. This decrease, however, is not as pronounced as the flow increase in the dilated vessel. Our simulations further suggest that the spatial specificity of a local dilation depends both on the cortical depth of the dilation site as well as the type of vessel (Duvernoy et al, 1981) whose diameter is modified. Our simulations of occluded penetrating arterioles demonstrate that CBF recovers downstream of the occlusion with increasing number of bifurcations. Approximately 350 micron away from the occlusion site, the effect of the constriction is negligible. This distance is consistent with the spatial distribution of the penetrating arterioles and compares well with the recent findings of Nishimura and coworkers (Nishimura et al, 2007).
Conclusions: We have presented a modeling framework that can compute blood pressure and flow, as well as scalar transport and exchange between vasculature and tissue. The introduction of an upscaling algorithm extends the model's applicability to very large networks and eliminates the need for detailed knowledge of the capillary bed topology.