Fig 1.
Matrix structure of diffusion problem with different meshing approaches in finite volume discretization.
(A) The tetrahedral mesh matrix has a large bandwidth with no apparent block structure. In FEV discretization of unstructured domains, unknown nodal values typically occur in dozens of balance equations which augments matrix bandwidth thus substantially hampering the solvability of the underlying transport problem. (B) The structured Cartesian (cuboid) mesh covering the same domain with similar characteristic edge length has a much narrower bandwidth and has block structure which is ideally suited for block diagonalization preconditioning. Tetrahedral meshes also require almost ten times (in our example 9.2 times) more grid cells (nz) than the analogue cuboid mesh.
Fig 2.
Illustration of the masking algorithm for labeling at different resolutions in the cortical microdomain.
A) The cortical network (E1.1) is labeled by pressure visualizing arteries (red), veins (blue) and connecting capillaries. B-C) This cortical network has been labeled a mesh visualized at many resolutions. Coloration of mesh is tri-color; red = blood, grey = endothelial, and blue = extravascular cells. Note, most of the cells belong to the extravascular space (blue). The blood lumen is marked by intravascular elements (red). Mass transfer occurs at the interface elements (grey). D-F) Visualization of mesh labeling at three resolutions reflects the capability of the labeling to resolve the connectivity between vasculature and tissue at all length scales. F) The distribution of endothelial layer, intravascular, and extravascular elements can be seen at the finest resolution.
Table 1.
Statistical properties of numerical performance of blood flow and oxygen simulations.
The small loss terms demonstrate divergence free simulations.
Fig 3.
Tracing of oxygen tension across 25 rays penetrating the tissue domain (raytracing analysis) for an empirical network (specimen E4.1).
A) 3D Cartesian tissue mesh and location of 25 rays to examine the variability of tissue tension as a function of grid resolution. B) Statistics of deviations of the computed oxygen profiles from the reference solution (= the finest grid with a resolution of 305x305x305 elements per volume corresponding to ~29M cells). C) Simulated oxygen profiles as a function of 15 grid resolutions along 25 rays through the domain. The oxygen simulations stabilize well below the highest resolution. Specifically, results become mesh independent at 205x205x205 domain (resolution = 205) which corresponds to ~9.115M unknowns.
Fig 4.
Oxygen distribution throughout a microvascular network specimen (E4.1) from the vibrissa primary sensory cortex in mouse.
A) Visualization of a portion of the extravascular space with a block cutout defined by 5 observation planes at different cortical depths (120, 280, 480, 680, and 880μm below the cortical surface). Color coding corresponds to oxygen tension in blood vessels. B-F) The visualization of oxygen distribution at the individual layers. C) Steep oxygen gradients form around penetrating arterioles–marked with red arrows. E) Oxygen tension is also elevated around veins–marked with blue arrows. In the deeper regions E-F) a hypoxic valley begins to form seen as deep blue basins.
Fig 5.
Validation of oxygen perfusion simulations against three sets of experimental data.
The experimental results were obtained with 2-photon phosphorescence lifetime oxygen imaging. The simulations were performed in planes with two different resolutions; a coarse 20x20 and a dense 200x200. There is reasonable agreement between experimental and simulated oxygen fields (2.9±1.4% error). Manual matching between the concentration profiles was used to determine tissue diffusivity, reaction rate, and mass transfer coefficient. The 3D perspective illustrates the steep concentration gradients along two penetrating arterioles.
Table 2.
Parameters used in the prediction of oxygen tension throughout the murine cortex.
Fig 6.
Simulations of age-related changes in oxygen tension at different cortical depths: 120, 280, 480, 680, and 880μm.
(Top Row) young normal brain, (Second Row) aged brain with capillary micro-occlusions. (Third Row) aged brain with reduced systemic hematocrit, and (Bottom Row) aged brain with the combined aging effect of reduced hematocrit and micro-occlusions. The region of tissue containing low oxygen content greatly increase in the aged models. Hypoxic pockets form in the lower cortical region when the effect of hematocrit and micro-occlusions combine (aged Hct+Occl at 880μm). The white dotted regions delineate the formation of hypoxic regions.
Fig 7.
Comparison between oxygen tension in young mouse and aged mouse suffering from reduced hematocrit in a large size synthetic vascular network spanning 3x3x1.2mm3 of the mouse cortex.
A) The vascular structure colored by diameter (red = thick, blue = thin) shows a dense set of tortuous vessels connected through a complete capillary bed. B) The tabulated occurrence rate of low-oxygen mesh elements in a portion (= a 1x1x0.8mm3 tissue subset from the interior of the domain) of the predicted oxygen field shows the same trends as in the experimental data. C) Predicted pressure distribution throughout the network matches the topology of arteries, veins, and capillary closure. D) A cut through the dense structure reveals regions of hypoxia and pockets of hypoxia occur more frequently in the lower regions of the cortex. D) A zoomed view of oxygen tension in the interior of the vascular network. There are some segments with low oxygen content (but above the hypoxic threshold) especially in deeper regions). E) The oxygen levels in the tissue reveal the lower regions are more susceptible to lower oxygen tension then regions near the cortical surface. A detailed view compares oxygen tension in young to aged specimen. In the young specimen, no hypoxic areas form. In the simulated age mouse, hypoxic regions and their spatial extent were identified; hypoxic pockets are indicated in red and are larger and more prevalent in the aged brain.
Table 3.
Comparison of simulation methods for blood and oxygen microperfusion.
Fig 8.
Vascular masking of the voxelized Cartesian mesh and vessel partitioning.
A) A penetrating arterial tree with the overlay of a Cartesian mesh reflects a network whose straight segments span many cuboid cell volumes. B) The vasculature is re-segmented (partitioned) until all segments span no more than two adjacent volumes. C) A labeled vessel inside a Cartesian mesh shows the extravascular elements in blue, intravascular segments in red, and endothelial segments in grey. D) The original centerline before and after partitioning for a penetrating arteriole sub-tree. In this illustrative case, the algorithm requires 8 steps to fully partition the tree.