Table 1.
Model parameters.
Fig 1.
(a) A cobweb vessel structure mimicking the main branches of retinal vessel networks. There are six inlets and six outlets for blood flow near the center of the network. The inlets and outlets are in a spaced arrangement. The arrows show the direction of blood flow. (b) A refined vessel network with 4306 blood vessels. The network structure is adapted from a real retinal vessel network measured in experiment by stretching and symmetrical extensions. There are four flow inlets and flow outlets near the center. (c)A single vessel embedded in a tissue cube. (d) A refined 3D vessel network with 7815 blood vessels. The intermediate and deep capillary plexi layers in retina are reconstructed by projecting the above 2D network onto two spherical shells. The diameter of the two spheres for the deep capillary plexi layer, the intermediate layer, and the choroid layer is 2 mm, 2.1 mm, and 2.2 mm, respectively. The widths of the lines in each figure show the radii of blood vessels. The unit for the axes is micrometer for (c), and millimeter for others.
Fig 2.
Oxygen partial pressure and its numerical error.
(a) Numerical solution of the oxygen partial pressure obtained with a 1024 × 1024 square mesh. (b) Numerical error of the oxygen partial pressure calculated by the difference between the solution obtained with a 1024 × 1024 mesh and a 2048 × 2048 mesh. The unit for oxygen partial pressure is mmHg.
Fig 3.
Blood oxygen partial pressure and oxygen fluxes from the vessels.
(a) Blood oxygen partial pressure Pb on the vessels. (b) Oxygen fluxes q from the vessels to tissue. (c-d) Pb and q on four marked vessel segments in (a). The x-axis denotes the distance from the inlet for each vessel (i.e., arc-length coordinate). The unit of Pb is mmHg. The unit of q is 10−9cm3O2/cm/s.
Fig 4.
Oxygen partial pressure obtained with the refined vessel network.
The partial pressure for normal tissue consumption and reduced tissue consumption are shown in (a) and (c), respectively. The tissue inside the white circles shown in (a) and (c) are used to statistically evaluate the area percentages of tissue with particular partial pressure of oxygen. The statistical results are shown in Figure (b) and (c), respectively. The unit of the x− and y− axes in (a) and (c) is mm. The unit of oxygen partial pressure is mmHg.
Fig 5.
The oxygen partial pressure at different layers of the retina.
The oxygen profiles on the layers corresponding to 0, 25, and 50 percent of the retina depth are shown in the Figure (a), (b), and (c), respectively. The unit of the axes is mm and the unit of oxygen partial pressure is mmHg.
Fig 6.
Comparison between retina oxygen partial pressure profiles simulated by our model for the mouse retina and those experimentally measured in mice by [47], in macaques by [41], in rats by [48] and in a human retina model [31].
DCP: deep capillary plexi, ICP: intermediate capillary plexi.
Fig 7.
Model validation and convergence analysis.
(a) The profiles of oxygen partial pressure on a line perpendicular to the vessel, where x = 0 represents cube center. The oxygen partial pressure inside the vessel (−10, 10) are set to be equal to the blood oxygen partial pressure Pb. The line “Secomb” is obtained from the method developed in Ref. [26] with a mesh size of 10μm. All simulations are performed with tissue consumption M0 = 2.0 × 10−3cm3O2/cm3/s and total blood inflow Q0 = 0.05nl/s. (b) The blood oxygen partial pressure Pb along the blood vessel.
Fig 8.
Numerical convergence analysis.
(a)Mesh refinement test. The stars show the numerical error. The fitted lines show a convergence order of 1.73 for the 2D case and 1.71 for the 3D case with respect to mesh size. The relative error is computed by the normalized L2-norm of PO. (b) The decay of the relative difference between two iteration steps. The relative difference is computed by the normalized L2-norm of q. (c) Iteration numbers for different mesh size.
Fig 9.
Numerical efficiency analysis.
(a) and (b) Average time cost for each iteration step in the 2-D and 3-D cases, respectively. The black solid lines, the blue dotted lines, and the red dashed lines show the average time cost for solving the PDE, the ODEs, and the post-processing in each Newton iteration step, respectively. For the 2-D case, the circled and squared lines show the time cost for the refined vessel network and the simple cobweb network, respectively; whereas for the 3-D case, the circled and squared lines show the time cost for the refined retina network and the single vessel system, respectively. (c) The time ratio between the OP-time and PDE-time for the 2-D and 3-D cases. (d) The normalized time ratio for different systems.
Fig 10.
(a) Evaluation of oxygen flux in 2D. The red solid lines represent the vessel wall. The red dashed line shows the center line of the blood vessel, the crosses on which show the discrete nodes. The blue (green) solid circle shows x+ (x−), whereas the blue (green) solid squares show the nearest mesh points used to fit the linear function and evaluate the flux. (b) Evaluation of oxygen flux in 3D. The red dashed line shows the center line of the blood vessel. The black circle shows the cross-section of blood vessel, whereas the blue (green) solid squares show the nearest mesh points used to fit the solution and evaluate the flux.
Fig 11.
The relative error δPk on a local mesh.
Fig 12.
Detailed oxygen field error analysis for the single vessel model.
(a) The oxygen profile on the plane of the vessel under the fine grid (h = 5μm). The numerical error for the h = 10, h = 20 and h = 40 are shown in (b-d), respectively, while the result of 5-μm mesh were used as a standard. The unit of the axes is μm and the unit of oxygen partial pressure is mmHg.
Table 2.