**Purpose**:
This study explored the relationship among microvascular parameters as delineated by optical coherence tomography angiography (OCTA) and retinal perfusion. Here, we introduce a versatile framework to examine the interplay between the retinal vascular structure and function by generating virtual vasculatures from central retinal vessels to macular capillaries. Also, we have developed a hemodynamics model that evaluates the associations between vascular morphology and retinal perfusion.

**Methods**:
The generation of the vasculature is based on the distribution of four clinical parameters pertaining to the dimension and blood pressure of the central retinal vessels, constructive constrained optimization, and Voronoi diagrams. Arterial and venous trees are generated in the temporal retina and connected through three layers of capillaries at different depths in the macula. The correlations between total retinal blood flow and macular flow fraction and vascular morphology are derived as Spearman rank coefficients, and uncertainty from input parameters is quantified.

**Results**:
A virtual cohort of 200 healthy vasculatures was generated. Means and standard deviations for retinal blood flow and macular flow fraction were 20.80 ± 7.86 µL*/*min and 15.04% ± 5.42%, respectively. Retinal blood flow was correlated with vessel area density, vessel diameter index, fractal dimension, and vessel caliber index. The macular flow fraction was not correlated with any morphological metrics.

**Conclusions**:
The proposed framework is able to reproduce vascular networks in the macula that are morphologically and functionally similar to real vasculature. The framework provides quantitative insights into how macular perfusion can be affected by changes in vascular morphology delineated on OCTA.

^{1}As a result, the retina is sensitive to small changes that may lead to loss of visual functions. In silico modeling has the potential to offer insight into the complex interactions between the retinal environment and the underlying causes of retinal diseases. In particular, virtual populations and in silico clinical trials are a promising way to enhance basic research and clinical trials.

^{1}Optical coherence tomography angiography (OCTA) is a non-invasive imaging modality that offers three-dimensional, high-resolution angiograms of the macula, which is the central-most area of the retina. Several microvascular metrics, such as vessel density and fractal dimension, have been suggested to quantify the quality of the microvasculature on OCTAs.

^{2}Using those metrics, several microvascular changes have been linked not only with aging and diseased retinae

^{3}

^{–}

^{7}but also with cerebrovascular changes and cardiovascular diseases.

^{8}

^{–}

^{10}For the brain and heart vasculatures, virtual populations have been developed, but similar work is yet to be done for the retinal vasculature or for the linked cerebral–retinal vasculature.

^{1}

^{11}Although angiogenesis and vascular endothelial growth factor (VEGF) are essential to the development of the vascular and in maintaining physiological conditions, the upregulation of VEGF is involved with the development of pathological neovasculature.

^{12}

^{,}

^{13}Hypoxia is a known upregulating factor of VEGF, and targeting ischemia-induced angiogenic pathways might improve treatment outcomes.

^{14}

^{1}

^{,}

^{15}

^{–}

^{24}Compartmental models

^{15}

^{,}

^{18}

^{,}

^{21}and/or symmetrical branching networks

^{18}

^{,}

^{22}

^{,}

^{23}are used in many of these models. These approaches are favored for their simplicity and adaptability to systems with limited information; however, they fail to reproduce the complexity and heterogeneity of the retinal vasculature.

^{24}In contrast, models based on vascular networks reconstructed from imaging data

^{16}

^{,}

^{20}are more faithful to the morphology of the retina, but the reconstruction of the network is arduous; therefore, only a limited number of eyes can be modeled. Space-filling algorithms offer a way to circumvent these problems by generating heterogeneous networks with characteristics similar to those of real vasculature.

^{17}

^{,}

^{25}

^{,}

^{26}For example, Causin et al.

^{17}used diffusion-limited aggregation because it creates structures with a fractal dimension similar to that of retinal vasculature. The class of space-filling algorithms referred to as constrained constructive optimization (CCO) algorithms is another approach that includes rules and constraints meant to reproduce the angiogenesis process.

^{25}

^{,}

^{26}It has been applied to the retinal vasculature

^{27}

^{,}

^{28}but only to create synthetic data for deep-learning applications.

^{29}

^{,}

^{30}including diabetic retinopathy

^{31}

^{,}

^{32}and age-related macular degeneration.

^{6}

^{,}

^{33}Changes in these metrics may indicate impairment to retinal or macular blood flow, which could contribute to development of the disease. However, quantifying these impairments in a sufficiently large population is challenging with conventional experimental techniques.

^{36}of the major temporal arcades was developed using a fundus photographs dataset. The remaining superficial temporal vasculature was partially generated with a CCO algorithm.

^{26}

*z*. In contrast, the macrovasculature is only generated in the SVP. This is because the ICP and DCP are composed of capillaries in the perifovea and merge with the SVP outside the macula.

^{37}

^{34}The major temporal arcade vessels correspond to the four vessels (two veins, two arteries) that branch directly from the CRA and CRV and extend toward the superotemporal and inferotemporal quadrants of the retina, as shown in Figure 1A. Vessel centerline segmentations, from their branching at the level of the optic disc to the boundary of the fundus photograph (Fig. 1A, orange area), were extracted from eight color fundus photographs of eyes without retinopathy and centered at the fovea. Segmentation was performed using the Freehand Line tool in ImageJ 1.48 (National Institutes of Health, Bethesda, MD, USA).

^{38}The pixel coordinates for each curve were extracted and translated so that the fovea was at the origin. This ensures that the model learns the distance between the optic disc and the fovea and between the arcades and the fovea, which may have importance in disease.

^{39}For images of right eyes, curves are reflected across the

*y*-axis so that all shapes correspond to left eyes (i.e., the optic disc will be on the left-hand side of the image). A simple principal component analysis–based statistical shape model learns the shape of all four temporal arcade vessels based on the location of inflexion points along the vessels (see Supplementary Material S1). The generated shapes are converted to length units using the rule of thumb for fundus photographs: 10° ≈ 5 mm, where the angle describes the field of view of the fundus camera. The generated vessels are linked to the CRV or CRA accordingly and assume initially the same radius as the central retinal vessels. The radius of the CRA is given in Table 1, and the radius of the CRV is larger by a factor of 1.1.1.

^{40}The radii of vessels other than the CRA and CRV are updated during the generative process, as described below.

^{26}With the CCO, trees are grown to minimize the total volume of the tree while, for each addition of vessel segments, keeping a constant pressure drop from inlet to outlet and satisfying several geometrical constraints. The hemodynamic model used by the CCO algorithm is similar to Equations 5 to 7 but with a different viscosity model.

^{26}Viscosity is computed with a fixed-point scheme with starting point η. The geometrical constraints affect the radius of branches,

^{41}the symmetry of branches, and the branching angle and aspect ratio of new segments.

^{26}Specifically, when

*r*,

_{p}*r*

_{1}, and

*r*

_{2}represent the radius of the parent vessel and the two daughter branches, respectively, and θ is the angle between the two branches, the following constraints apply:

^{2}with

*N*terminal vessels, the location for a new terminal vessel is rejected if it is less than

*l*is reduced by a factor

_{min}*l*.

_{fr}^{42}For a candidate terminal vessel located at

*x*, the CCO algorithm finds the most suitable vessel within a radius \({f_n} \times {( {\mathop \smallint \nolimits_{\rm{\Omega }} dA} )^{1/2}}\) of

_{new}*x*that satisfies the above geometrical constraints.

_{new}*r*(i.e., the entire computational domain), in an annulus with radii

_{retina}*r*and

_{parafovea}*r*, and finally in a disk of radius

_{perifovea}*r*(Fig. 1).

_{parafovea}*x*= (

*r*cosθ,

*r*sinθ), where θ is the opening angle and follows a uniform distribution over the interval [0, 2π] and

*r*is the distance to the center of the macula, as shown in Figure 1A, and follows a log-normal distribution (see Table 1). Arteriovenous networks of the superficial vascular plexus are generated in three steps: (1) the CCO algorithm is applied to create a backbone of larger arterioles and venules from the arterial and venous arcades. The arterial and venous backbones are grown separately in the first step. For each tree, the CCO algorithm requires volumetric blood flow at the root (CRA or CRV) and a pressure drop across the vasculature. Blood flow in the CRA is computed from its radius and blood flow velocity. From conservation of mass, blood flow is the same in the CRV. Ocular perfusion pressure (

*OPP*) refers to the pressure drop between the CRA and CRV, namely:

*p*–

_{pre-capillary}*p*for the venous tree and

_{CRV}*p*–

_{CRA}*p*for the arterial trees. The value of

_{pre-capillary}*p*is taken from pressure in pre- and postcapillary vessels in the theoretical model by Takahashi et al.

_{pre-capillary}^{22}

^{35}that are incompatible with the logic of the CCO algorithm. Therefore, we adopted the method proposed by Linninger et al.

^{25}to generate capillary beds connecting the arterial and venous trees. In short, a disk of the size of the macula (see Table 1) is meshed with a Delaunay triangulation generated from

*N*randomly sampled points within the disk. The centroids of the triangles are used to generate a Voronoi diagram. In brief, a Voronoi diagram partitions the plane into polygonal regions centered around input points. The edges of the polygons form the capillary bed. In the SVP, capillaries coexist with arterioles and venules but should not intersect them; therefore, capillaries intersecting with arterioles or venules are removed from the capillary bed. This also creates a capillary-free region that is found surrounding arterioles in the SVP.

_{seeds}^{35}

^{18}

^{,}

^{35}

^{,}

^{45}Because the ICP and DCP are modeled in the macula only, interplexi connections are based on the findings by An et al.

^{35}in the parafovea. Specifically, arterioles and venules within the macular area of the SVP bifurcate to the ICP, and those branches immediately bifurcate to the DCP. This corresponds to the most prevalent patterns in the histology study.

^{35}From the SVP, 30% of the arterioles and venules were selected for bifurcation to the ICP, and the bifurcation points were added in the middle of the selected vessels.

*r*) unless they are connected to an arteriole or venule, in which case their initial radius is twice that of other capillaries. Diameter transitions at bifurcations are smoothed using the method proposed by Linninger et al.

_{capillary}^{25}In short, the diameter of a segment becomes the average of the diameters of itself and of the parent and daughter branches.

*p*, vascular resistance

*R*, and volumetric blood flow

*Q*are related by

^{46}in the microcirculation. Non-Newtonian effects are accounted for by a diameter- and hematocrit-dependent, empirical, effective viscosity law

^{47}:

*D*is the vessel diameter in microns and

*H*is the discharge hematocrit:

_{D}^{47}

*R*. All terminal veins drain the vascular compartment through artificial vessels with the same resistance. For baseline simulations,

*R*= 1 × 10

^{6}mmHg s

*/*mL was selected such that the hemodynamic parameters’ distribution across the vasculature is similar to experimental data.

^{19}

^{,}

^{48}

- • Four of the indices proposed by Chu et al.
^{2}: vessel area density (VAD), vessel skeleton density (VSD), vessel diameter index (VDI), and vessel complexity index (VCI), computed according to Equation 10 - • Fractal dimension (FD), computed with a box-counting method
^{4} - • Intervessel distance (IVD), computed with Euclidean distance transform.
^{52}

*N*and

_{terms}*N*(see Table 1). Both IVD and FD require skeletonized, pixelized images of the vasculature to be computed. To create those, generated macular vessels are mapped to a white canvas, then saved as binary images.

_{seeds}*X*as follows:

^{2}

^{35}:

- • If the vessel has one branch of order
*i*and all other branches are of order less than*i*, then the order of the vessel is*i* - • If the vessel has two or more branches of order
*i*and*i*is the largest order among the branches, then the order of the vessel is*i*+ 1.

*retinal blood flow*, defined as the volumetric flow rate of blood entering the retina, and the

*macular flow fraction*, defined as the percentage of retinal blood flow entering the macula. Spearman correlation coefficients were calculated for both hemodynamics variables and against each morphological metric. We derived 95% confidence intervals (CIs) using bootstrapping (

*N*= 1000).

^{26}for CCO algorithm–specific parameters. The values for these parameters either are unknown (e.g.,

*N*) or are subject to uncertainty in their measurement (e.g., δ, γ). We performed a variance-based sensitivity analysis to decompose the variance in the output of the model (

_{terms}*Var*[

*Y*]). Sobol indices summarize the importance of sets of inputs

*X*with indices between 0 and 1.

_{i}^{53}In this work, we report first (

*S*) and total (

_{i}*S*) order indices, which are often enough to understand parameter importance.

_{T}_{i}^{53}In short,

*S*quantifies the contribution of

_{i}*X*alone, and the

_{i}*S*quantify its total contribution—namely, its first-order contribution plus all the higher order contributions.

_{T}_{i}^{53}Further details can be found in the Supplementary Material.

^{54}

^{,}

^{55}For the hemodynamics model, uncertainty stems from two parameters:

*OPP*(Equation 3) and

*R*. To quantify the uncertainty brought on by the correlation coefficients, the same experiment was reproduced for 45 different scenarios where

- • The resistance parameter
*R*was set to 5 × 10^{5}mmHg s*/*mL, 1 × 10^{6}mmHg s*/*mL, and 5 × 10^{6}mmHg s*/*mL. - •
*OPP*was set to 80%, 100%, and 120% of its baseline value - • The fraction α was varied between 0.2 and 0.6 by increments of 0.1.

*OPP*. The mean ± SD for

*OPP*was 45.2 ± 4.2 mmHg (range, 32.83–56.0). The parameter

*v*was not used in the hemodynamics simulations.

_{CRA}^{35}Mean diameters were smaller in the model but lay within the reported ranges for each order. The ratio of average arteriole diameter to average venule diameter increased from 0.92 ± 0.09 in order 5 vessels to 0.95 ± 0.05 in order 1 vessels. For all orders combined, the ratio was 0.937 ± 0.031, which is consistent with experimental measurements of 0.9 ± 0.1.

^{40}

^{19}

^{,}

^{48}Blood velocity in the CRA was also lower in the model compared to the average of 6.3 cm

*/*s reported by experimental work.

^{49}Figure 5A compares blood velocity along the vasculature with experimental studies.

^{19}

^{,}

^{48}Additionally, these studies reported volumetric blood flow rates against diameter. These distributions are compared with those of the model in Figure 5B. In the venous circulation, model predictions of flow and velocity were consistent with experimental data. On the arterial side, both velocity and flow were visually lower compared to the same studies.

*stands out as the most influential overall, explaining around 50% of the variance in FD and most of the variance in VSD (*

_{min}*S*=

_{i}*S*≈ 1) by itself. Other parameters of interest include

_{Ti}*N*for the SVP,

_{seeds}*N*or the second stage of the CCO algorithm, and, to a lesser extent,

_{terms}*l*. These results indicate that those four parameters are enough to produce a virtual population with interpopulation variability, at least in the SVP.

_{fr}*OPP*had almost no effects on macular flow fraction (Pearson's

*r*

^{2}< 10

^{–3}for all values of

*R*) but was linearly correlated with total retinal blood flow (

*r*

^{2}> 0.89 for all values of

*R*). The coefficients are given for those nine scenarios in Supplementary Table S2. Parameter α was not correlated with either of the hemodynamic variables (

*r*

^{2}< 10

^{–2}for all variations of

*OPP*and

*R*).

^{2}

^{,}

^{4}

^{,}

^{52}

^{,}

^{56}

^{,}

^{58}In the ICP and DCP, VAD was very close to values reported in a histology study,

^{59}but IVD was larger in both plexuses compared to OCTA data.

^{52}This was more marked in the ICP compared to the DCP. The morphology of the microcirculation delineated on OCTA is sensitive to several factors, including scan postprocessing

^{56}

^{,}

^{57}and segmentation of the different plexuses, which makes direct comparison complicated.

*N*, although reasonable bounds need to be defined.

_{seeds}^{35}However, capillaries were smaller in our model compared to the data. In the study by An et al.,

^{35}capillaries were any vessel with diameter smaller than 8 µm. In our model, capillaries were assigned a diameter of 5 µm or 10 µm if they were directly connected to an arteriole or venule. This strategy may be too simplistic to represent the spread of diameters in the vascular bed. Others have suggested updating the diameter of vessels based on blood pressure from a first hemodynamics simulation.

^{25}More in-depth analysis of the capillary beds is necessary in order to develop an appropriate strategy. In the meantime, sensitivity analysis and uncertainty quantification can help improve the reliability of the model.

^{19}

^{,}

^{48}Similarly, blood flow and velocity in the CRA were both lower in the model compared to experimental data.

^{19}

^{,}

^{48}

^{,}

^{49}

^{,}

^{63}As seen in Equations 5 and 6, blood flow and velocity are respectively proportional to the fourth and second power of vessel radius. Therefore, an increase in radius by a factor of \(\sqrt 2 \) ≈ 1.4 for velocity and \(\sqrt[4]{2}\) ≈ 1.18 for flow would be sufficient to double the predictions of the model. It is unclear whether vessel diameter measurement in experimental studies

^{19}

^{,}

^{48}has included the vessel wall in the measurement. We assumed that the diameters were those of vessel lumen, which could lead to an overestimate of lumen radii between 20% and 35% for larger temporal arteries.

^{64}

^{–}

^{66}In experimental studies, the same relation between flow and radius is assumed, and blood flow is estimated from velocity

*v*and diameter

*D*measurements as

*Q*=

*v*π

*D*

^{2}/4. Therefore, even a small measurement error in vessel diameter combined with error in measurement in velocity still results in large deviations from the true blood flow. Both measurements are challenging and prone to errors.

^{67}To test this hypothesis, we reduced by 20% the lumen diameter of arteries larger than 100 µm in diameter and ran the hemodynamics simulations for the entire population. The velocity and flow distributions for this experiment are provided in Supplementary Figure S1 and show improved agreement with the experimental data. In addition, all parameters in our virtual populations were sampled from independent normal distributions, which is likely an incorrect assumption as, for example, vessel diameter is likely to be correlated with arterial pressure and

*IOP*.

^{68}As discussed by Doblhoff-Dier et al.,

^{19}studies have reported average total retinal blood flow ranging from 30 to 80 µL/min.

^{19}

^{,}

^{48}

^{,}

^{63}Despite the uncertainty in measurements, most studies seem to agree on values in healthy eyes of around 30 to 40 µL/min.

^{19}

^{,}

^{48}Despite the difference in hemodynamics in the CRA, Figure 5 shows that the discrepancy with experimental data is reduced as the vessels branch out. Similar to the study by Doblhoff-Dier et al.,

^{19}blood velocity seems to scale linearly with diameter for larger vessels, but this trend is lost in smaller vessels. The overall lower blood velocity can be attributed, as discussed above, to the discrepancy in velocity in the CRA.

^{42}Assuming that regions of higher cell density require similarly higher blood flow, it can be estimated that the macula requires 15% to 30% of the total retinal blood flow, although these are rough estimates.

*R*is similar to gradually closing the connections to/from the vascular compartment; therefore, flow is shunted toward the macula, and macular flow fraction increases. Decreasing

*R*has an opposite effect. However, the macular flow fraction will eventually reach a plateau when the CRA reaches its maximum capacity in terms of blood flow: Regardless of the resistance of paths outside the macula, blood will flow through the macula, and total retinal blood flow is bounded by physical constraints. Indeed, in our model, for a given

*OPP*, flow in the CRA is theoretically bounded by the radius and length of the CRA according to Equation 5. The same effect explains the non-symmetrical changes in total retinal blood flow as

*R*is decreased (see Supplementary Fig. S2).

*OPP*,

*R*, and α does not have any effect on the Spearman correlation coefficients.

^{53}In this case, developing a surrogate model (e.g., polynomial chaos expansion) might be required.

^{53}Nonetheless, the first-order and total-order indices suggest that the parameter space can be reduced to as little as four parameters, depending on which morphological metric is deemed more important.

*appeared to be the most influential parameter. In particular, it was the sole parameter influencing VSD. This result supports the hypothesis by Xao et al.*

_{min}^{30}that branch geometry is correlated with the vessel perimeter index, which would be twice VSD when computed on artificial vessels. The associations between vascular structure and hemodynamics should be investigated further, perhaps with spatial metrics that can be compared with OCTA measurements.

^{30}

*MAP*,

*IOP*,

*r*, and

_{CRA}*v*) on the generated vasculature remain to be quantified. The number of hyperparameters is large, and global sensitivity analysis has shown that their effects on vascular metrics are non-local. Therefore, directly adapting the method to generate different virtual populations may prove challenging. Indeed, as stated by Allen et al.,

_{CRA}^{69}efficient generation of virtual populations requires knowledge of plausible ranges for the model parameters and optimizing over the set of model parameters. Reducing the number of parameters to the most influential ones appears necessary, and the sensitivity analysis presented in this study is a first step toward this goal.

*r*,

_{CRA}*MAP*,

*IOP*, and

*v*. These are likely to be strongly related, and ignoring these associations may create discrepancies in the output of our model when compared to experimental data. However, these joint distributions are not readily available, to the best of our knowledge, but might be inferred from different studies in the future.

_{CRA}^{46}

^{,}

^{47}Additionally, we assumed that there were no lateral connections between the ICP and DCP and the circulation outside the macula. This is similar to assuming that both plexi are connected in series with the SVP, which is now known to be only partially correct.

^{35}Finally, the parameter

*R*introduced in this model remains unknown, and its value was based on simple computation of the estimated macular blood flow. Also, uncertainty quantification showed that this parameter had a strong effect on the fraction of flow going to the macula. However, it had limited effect on the total retinal blood flow (at most 30% of variation compared to baseline for

*R*varying over an order of magnitude). In future work, its influence on other hemodynamics measurements should be thoroughly evaluated.

*, which, according to the sensitivity analysis results presented earlier, can affect microvascular morphology. Additional mechanisms that may have importance in disease, such as autoregulation*

_{min}^{21}

^{,}

^{43}or plasma skimming,

^{70}can easily be added to the current model. For example, autoregulation may be negatively affected by diabetes,

^{71}while simultaneously asymmetric branching causes heterogeneous distribution of oxygen due to plasma skimming. This way, our model can be used to understand the relationship between microvasculature and pathological angiogenesis, a symptom of several blinding diseases,

^{11}and provides a framework to build upon to achieve patient-specific treatment simulations.

**R.J. Hernandez**, None;

**S. Madhusudhan**, None;

**Y. Zheng**, None;

**W.K. El-Bouri**, None

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