A Fast-Fourier Preconditioned Schur Complement Method for the Simulation of Cerebrocortical Oxygen Supply

The transfer of oxygen from blood to tissue involves mixed modes of transport between topologically distinct domains. To enable the high-resolution simulation of cortical oxygen supplied by an anatomically realistic microvascular network, we separate computational domains into a 1D graph representing the microvasculature and a rectangular array of 3D tissue voxels. Blood flow induces an orientation for which the vascular network is a directed acyclic graph, and this property enables efficient computation of RBC convection. Similarly, the 3D Cartesian array allows for extremely fast computation of oxygen diffusion with a fast-Fourier based Poisson equation solver. We find that the Schur complement method of domain decomposition decouples the tissue and vascular simulations without compromising the advantage of their respective fast solution methods, even with nonlinearity introduced by Michaelis-Menten reaction kinetics simulating metabolism in the tissue, and the Hill equation modeling dissociation of oxygen from RBCs into blood plasma before crossing the blood brain barrier. In particular, the discrete trigonometric transforms behind fast Poisson solvers efficiently precondition the Schur complement system for the tissue oxygen. With this method we are able to simulate oxygen supply both dynamically and at steady state for voxel resolutions approaching one micron side length without supercomputers.