Fig 1.
Clinical background of diabetic retinopathy.
(A) Fundus image of patient with a 10 year history of type 2 diabetes mellitus showing hard exudates in the macula indicating edema [15]. Dot/blot hemorrhages, and some laser burns are also present. Image is taken with a standard color fundus camera. Resolution is adequate to see arterioles and venules but not the capillaries connecting them. For perspective the disk diameter is approximately 1450 microns. (B) Fluorescein angiogram of the diabetic fundus. The image shows areas that are ‘dark’ and therefore lack patent capillaries. These occur in various areas of the image. Certain vessels appear to lack sharpness because they show some leakage of the fluorescein dye through their walls; walls that would not leak if the normal blood retinal barrier were intact. This image shows the phenomenon this paper is primarily addressing, the occurrence of contiguous areas of capillary loss both in the macula and in the periphery in diabetic retinopathy.
Fig 2.
Model abstraction and construction of human perifoveal capillary network from an Adaptive Optics Scanning Laser Ophthalmology (AOSLO) image.
On the right, AOSLO image shows juxtafoveal capillaries adjacent to the foveal avascular zone (FAZ). This is a normal capillary map in a patient without diabetes. On the left, a model schematic shows the reconstruction of the capillary network framed in the AOSLO image with cells filling in empty space between vessels uniformly. Capillary network has arteriole (A) and venule (V) termini marked. Boundary blood pressures are assigned for A and V termini. Boundary oxygen tensions are assigned for A terminus and for FAZ whereas venous oxygen tension is model dependent. Model objects in red, green, brown are capillary blocks (CAP), Mueller cells (MC) and other retinal cells (OT) respectively. Yellow pixels surrounding objects are object borders, which are muted in other figures of the manuscript. Note that this example shows a macular capillary network from a different subject than that shown in CASE 1. Diameter of the FAZ in this ASOLO image is approximately 500 microns. The scale bar is 100 microns.
Fig 3.
Schematic of oxygen and VEGF fluxes.
Colored blocks represent model objects: Capillary block (CAP in red), Fluid portion (FP in cyan), Mueller cell (MC in green) and Other retinal cells (OT in brown). Markers in the form arrows, triangles and squares are used to represent modeled fluxes. On the left are the oxygen fluxes including advection, diffusion and metabolism. Oxygen advection is modeled for object CB. Oxygen diffusion is modeled for object pairs among CAP, MC, OT and FP. Oxygen metabolism is modeled only for objects MC and OT. On the right are the VEGF fluxes including synthesis, diffusion and decay. VEGF synthesis is modeled for object MC again as the model’s only VEGF source. VEGF diffusion is modeled for object pairs among CAP, MC, OT and FP. VEGF decay is modeled for objects CAP, MC, OT and FP. The arrangement of model objects do not necessarily reflect detailed configurations constructed in the simulations.
Fig 4.
Schematic of conveyor-belt model of oxygen advection.
The schematic shows a small capillary network with five segments and two junctions. Along the blood flow direction, marked as arrow with empty triangular head, the first junction has merging blood flow and the second branching blood flow. Blocks in red are CAPs, building blocks, or structural elements (visually present in model configuration), of capillary segments. Blocks in yellow are CBs, functional elements for advection. Size of a CAP is fixed throughout a simulation, so each capillary segment has fixed number of CAPs. In contrast, the size of a CB is always proportional to flow velocity on the capillary segment, so flow velocity and accordingly the number of CBs on a patent capillary segment can vary following an occlusion elsewhere as flows in the network are adjusted to the changed network resistance structure. During each time step of advection, a CB passes its oxygen volume to the next CB. Importantly, the size of a CB is equal to flow velocity of host capillary segment multiplied by time step of advection, which means the “conveying” speed of oxygen from one CB to next is exactly equivalent to the flow velocity in that capillary segment. At a merging junction, the upstream capillary segments add the oxygen volumes in their last CBs and pass the total to the first CB of the down stream capillary segment. At a branching junction, the parent capillary segment distributes the oxygen volume in its last CB into the first CBs of the daughter segments according to conservation of blood flow volume. Mathematical descriptions were detailed as equations (9)-(11) in S1 Text. Each CB is associated with, or mapped to, a host CAP. The modules of advection (involving CBs) and diffusion (involving CAPs) are connected with processes that convert oxygen volumes between the host CAP and its associated CBs. (see Fig 5).
Fig 5.
Procedure of sequential simulation of advection and diffusion illustrated with an example.
During each time step, simulation of advection precedes simulation of other processes including diffusion. After advection (involving the CBs delivering the blood containing oxygen) and before diffusion (involving CAPs), a CAP always sums up oxygen volumes in all its associated CBs and update its pre-diffusion oxygen volume. Simulation of diffusion updates the CAP to have its post-diffusion oxygen volume. After diffusion and before advection at the next time step, the associated CBs must have the same relative change of oxygen volumes as the CAP gains or loses during diffusion, namely, percent change from pre-diffusion to post-diffusion values. This mathematically recognizes the conservation of oxygen. Three consecutive time steps are shown in the example. A naïve assumption here for this illustration is that the first CB always receives 1 unit oxygen volume from upstream (not drawn). Another simplification made for this illustration is that only diffusion between the CAP (shown in red) and nearby OT (shown in light brown) are considered. In the actual model OT to OT diffusion is also treated. Note that in a certain step, boldface numbers represent values being changed or updated. During the first time step from t0 to t0 + Δt, while the second CAP and its associated CBs still have a zero oxygen volume, the first CAP and associated CBs undergo (1) the process of advection that passes 1 unit oxygen volume to first CB, while CAPs are not involved; (2) an intermediate step that updates CAP’s pre-diffusion oxygen volume by adding 1 (its first associated CB) and 0 (its second associated CB); (3) the process of diffusion delivers 0.2 to OT in contact (amount assumed for convenience in this example, and again OT-OT diffusion is ignored in this example) and CAP’s post-diffusion oxygen volume becomes 0.8, while CBs are not involved; (4) a last step in the time period that updates CAP’s associated CBs’ oxygen volumes by subtracting 0.2/1 = 20% (diffused/pre-diffusion), the first CB thus having 0.8 oxygen volume now. During the second time step from t0 + Δt to t0 + 2Δt, similar verbal “simulation” goes. (1) process of advection goes as another 1 oxygen volume is passed to first CB and 0.8, previously held by the first CB, is passed to the second CB; (2) an intermediate step adds 1 and 0.8 to first CAP, still none added to the second CAP; (3) the process of diffusion updates first CAP’s oxygen volume to 1.44, with 0.36 diffused out; (4) last step updates oxygen volumes in both of the host CAP’s two CBs, again by subtracting diffused fraction 0.36/1.8 = 20%. During the third time step from t0 + 2Δt to t0 + 3Δt, advection now passes oxygen volume 0.64, previously held by the second CB, into the third CB, which is associated with the second CAP. An intermediate step updates both CAPs by summing up oxygen volumes in their associated CBs. Process of diffusion now changes the oxygen volumes of both CAPs, with the first and second diffusing out 0.36/1.8 = 20% and 0.1/0.64 = 15.625% respectively. The last step subtracts the oxygen volumes of their asscociated CBs’ with the percent change.
Fig 6.
Simulation is initialized with input of the original capillary network topology and structure as well as the component cells. Computation of flow velocities and simulation of oxygen and VEGF fluxes occurs next. Each model week the model addresses capillary leakiness and each model month capillary occlusion is evaluated. (1) If no new occlusion occurs, the model simply repeats checking for edema formation and capillary occlusion until occlusion occurs or until a pre-assigned simulated time (3 years at our temporal conversion rate) is arrived at. (2) Once a new occlusion occurs, the network topology is changed and its structure is adapted to eliminate flow in the irreversibly occluded capillary, followed by a new iteration of previous steps. The process stops if either no more effective flow paths exist, invalidating flow velocity calculations, or the pre-assigned simulated time is arrived at.
Fig 7.
(CASE1) system configurations and field pattern under normal condition.
(A) Shown in the 2D XY cross section involving capillary network, Mueller cells (green) and other retina cells (brown) are uniformly initialized between capillary segments composed of capillary blocks (red). The 3D image shows the simulated section during initialization (in the 3D completely initialized configuration, the capillary network is covered by MC and OT cells and visually inaccessible). The dimension of the simulated retinal section is 510μm × 600μm × 50μm. (B) Flow velocity map includes a primary arteriolar entrance (bottom right in the (B)), a main venular exit (top right), side traffic extending outside region of interest, and interconnected pathways. Capillaries closer to the FAZ (left in the figure) carry blood flow with relatively smaller velocity. The unit for velocity is μm/s. (C) Oxygen tension is highest near capillary segments and the FAZ. The more distant a cell is from irrigating capillaries, the lower is its oxygen level. The unit for oxygen tension is mmHg. (D) VEGF levels are initially assumed to be at a low but nonzero basal concentration across in the whole area. VEGF level has arbitrary unit. “FAZ refers to foveal avascular zone, “A” refers to arteriole, and “V” refers to venule.
Fig 8.
(CASE1) flow velocity pattern following initial capillary occlusion.
(A) The flow velocity map captures loss of a flow pathway due to capillary occlusion in week 0. (B) The second capillary spatially close to initial occlusion site became occluded in week 72. (C) Several capillaries near FAZ and venule became occluded in week 124. (D) More than a quarter of the capillary network was obstructed by week 152. Color and the direction of arrows reflect the magnitude and orientation of velocities respectively. The redder the color is the greater the flow velocity. The unit for velocity is μm/s. “FAZ” in the figure refers to foveal avascular zone, “A” in red refers to arteriole, and “V” refers to venule.
Fig 9.
(CASE1) oxygen tension pattern following initial capillary occlusion.
(A) Oxygen tension map shows a localized hypoxic region near the occluded capillary in week 0. (B) Hypoxic area of cells broadened to enclose a second occlusion site in week 72, but it’s still restricted and confined spatially to the arteriole-venule sector. (C) Local insult by hypoxia propagated to break the terminal capillary near the venule and extended to the neighboring AV sector in week 124. (D) Large area of hypoxia was observed in week 152. Color reflects magnitude of oxygen tension. The redder the color the higher the oxygen tension is. The unit for oxygen tension is mmHg. “FAZ” in the figure refers to foveal avascular zone, “A” in red refers to arteriole, and “V” refers to venule.
Fig 10.
(CASE1) VEGF level pattern following initial capillary occlusion.
(A) VEGF level map shows localized synthesis of VEGF by Mueller cells in response to hypoxia in week 0. (B)—(D) Increasing number of Mueller cells actively produced VEGF in weeks 72, 124, and 152, where the pattern of regions with high VEGF reproduced that of the area with low oxygen tension. Color reflects magnitude of VEGF level. The redder the color the higher the VEGF level is. VEGF level has arbitrary unit (A.U.). “FAZ” in the figure refers to foveal avascular zone, “A” in red refers to arteriole, and “V” refers to venule.
Fig 11.
(CASE 1) retinal thickness over time.
Thickness of the retinal layer is represented by a color map at the end of year 0.5 (A), year 1 (B), year 2 (C) and year 3 (D). Color represents the magnitude in Z axis. The bluer the color, the thicker a local retinal area is. Fluid appeared between year 1 and year 2, at the venous edge of the area of occlusion. The flow network is overlaid upon the color map to present patent flow paths at the time point of observation. Color bar only represents the thickness of tissue but not the flow velocities. “FAZ” in the figure refers to foveal avascular zone, “A” in red refers to arteriole, and “V” refers to venule.
Fig 12.
(CASE 1) quantitative development of model properties with time.
(A) Average oxygen tension decreased with increasing occlusion events despite occasional small increases, and had an overwhelming drop near the end of simulation. (B) The hypoxic fraction of Mueller cells was observed to grow rapidly after the capillary occlusions induced by locally elevated VEGF, and near the end of simulation about one third of the Mueller cells were ischemic. (C) Total volume inflow rate established a rising trend before a major decline in week 124, after that it remained at a lower level than the starting point and eventually descended to below 90% of initial rate. (D) Average minimal cell-to-vessel distance maintained an increasing trend to reach 3.5 aMC (Mueller cell diameter) units, which qualitatively reproduced the temporal pattern observed for hypoxic fraction of Mueller cells.
Fig 13.
(CASE1) cellular oxygen distribution over time.
Fraction of cells in each 4 mmHg oxygen bin is shown for the normal condition and 3 additional times. The distribution of oxygen tension within all cells exhibited an essentially unimodal shape under the normal condition (normal-red bars) where most cells had oxygen tensions of 10 to 25 mmHg, a small portion of cells located near vessels had higher levels ranging from 35 to 40 mmHg and no cells had an oxygen tension less than 4 mmHg O2. Capillary occlusions induced by VEGF gradually altered the distribution (week 0-blue bars, week 72-yellow bars, week 124-cyan bars). An increasing number of cells turned hypoxic. The broad peak of cells at moderate levels of oxygen decreased and broadened with more cells both at lower oxygen levels with each successive interval and more cells from about 25–30 mmHg in each successive interval. The cell oxygenation distribution gradually morphs from a unimodal distribution to a bimodal oxygenation distribution. Inset figure shows a comparison between normal condition and week 124 using a line-connecting-dot presentation, with finer oxygen tension spacing between two consecutive data points. The inset figure also strikingly shows the transition from unimodal to bimodal distribution as well as peak decreasing and broadening pattern at moderate levels.
Fig 14.
(CASE 1) Flow-Oxygen phase diagram of 362 replicate simulations.
Flow-Oxygen phase diagram describes how total volume inflow rate and oxygen tension evolve with time in all replicate simulations. Color represents simulation stages where bluish circles stand for early stages and reddish ones correspond to later stages. (A) Many simulations show clustering of flow-oxygen states through all monitored time points. Those clusters, quite close to the equilibrium state under normal condition, remained overlapped post the first occlusion. On the contrary, a few simulations displayed a scattered pattern of flow-oxygen states near the end of the simulation with either reduced flow or oxygen tension or both. (B) Temporal trajectories of the scattered points shown in A whose states had either less than 75% initial inflow rate or oxygen tension in year 3, were considered progressive. Simulations that lead to the most severe propagation of capillary occlusions tend to end with both lower than normal flow rates and lower oxygen tension.
Fig 15.
Patency map of the macular capillary network.
The patency map illustrates the probability of each capillary segment remaining open given a specific initial capillary closure. These data are 2 subsets of the 362 replicate simulations in which 2 specific capillaries were occluded. (A) Patency map following CASE1 initial occlusion site shows that a high probability of secondary occlusions is limited to adjacent capillaries, largely those closer to the FAZ (n = 45). (B) Patency map following CASE2 initial occlusion site shows greater probability of broader propagation of injury (n = 30). Color represents the frequency of a capillary segment being patent after 3 years of simulated time since the initial occlusion. Warmer color corresponds to less vulnerability to occlusion, or higher capillary patency.