**Purpose**:
To study the relationship between the circumferential extent of angle closure and elevation in intraocular pressure (IOP) using a novel mechanistic model of aqueous humor (AH) flow.

**Methods**:
AH flow through conventional and unconventional outflow pathways was modeled using the unified Stokes and Darcy equations, which were solved using the finite element method. The severity and circumferential extent of angle closure were modeled by lowering the permeability of the outflow pathways. The IOP predicted by the model was compared with biometric and IOP data from the Chinese American Eye Study, wherein the circumferential extent of angle closure was determined using anterior segment OCT measurements of angle opening distance.

**Results**:
The mechanistic model predicted an initial linear rise in IOP with increasing extent of angle closure which became nonlinear when the extent of closure exceeded around one-half of the circumference. The nonlinear rise in IOP was associated with a nonlinear increase in AH outflow velocity in the open regions of the angle. These predictions were consistent with the nonlinear relationship between angle closure and IOP observed in the clinical data.

**Conclusions**:
IOP increases rapidly when the circumferential extent of angle closure exceeds 180°. Residual AH outflow may explain why not all angle closure eyes develop elevated IOP when angle closure is extensive.

**Translational Relevance**:
This study provides insight into the extent of angle closure that is clinically relevant and confers increased risk of elevated IOP. The proposed model can be utilized to study other mechanisms of impaired aqueous outflow.

^{1}Elevated intraocular pressure (IOP) is an important risk factor for glaucomatous optic neuropathy and contributes to irreversible damage of retinal ganglion cell axons.

^{2}Primary glaucoma is divided into two broad categories: primary open-angle glaucoma (POAG) and primary angle-closure glaucoma (PACG). PACG is characterized by narrowing and closure of the anterior chamber angle, the primary site of aqueous outflow from the eye. PACG carries a threefold increased risk for severe bilateral visual impairment compared to POAG and is associated with severe ocular morbidity and high rates of unilateral blindness on diagnosis.

^{3}

^{–}

^{6}

^{7}AH then flows between the lens and the iris, around the pupillary margin, and into the anterior chamber (AC). From the AC, AH exits the eye through either the trabecular meshwork (TM; conventional outflow) or through the CB face (unconventional outflow) (Fig. 1). The unconventional outflow pathway comprises approximately 50% of outflow in young healthy individuals and declines with age at a rate of approximately 3.5% per decade,

^{8}possibly due to a build-up of extracellular material, an increase in the number and thickness of collagen fibers, and a thickening of the elastic fibers, which results in reduced space between muscle bundles.

^{9}Angle closure, characterized by apposition between the peripheral iris and TM, leads to mechanical obstruction of both outflow pathways. This contributes to increased outflow resistance and increase in IOP, as AH production is mostly independent of IOP.

^{10}

^{–}

^{12}and extensive work has been done in this area, there is relatively sparse information about the relation between the extent of angle closure and elevated IOP in humans.

^{13}and the model was built using Gmsh 4.6.0. The entire model geometry is shown in Figure 2. Following Dautriche et al.,

^{14}the TM was modeled as a uniform body with a rectangular cross-section of 275 µm (height in the axial direction,

*z*) by 100 µm (length in the radial direction,

*r*) and with uniformly distributed permeability. Therefore, the height of the TM was set equal to the height of the Schlemm's canal (SC), or 275 µm. The SC was also modeled as a uniform body with a rectangular cross-section of 275 µm by 30 µm, based on the work of Gong et al.

^{16}Seven evenly distributed collector channels (CCs) with a diameter of 50 µm were placed along the outer wall of the SC.

^{16}It was verified that the distribution and location of the CCs did not affect outflow or IOP (see Appendix A). The unconventional outflow path consisted primarily of the CB face, which allowed the AH to exit the eye through the AC angle. In the model, the CB face was modeled adjacent to the TM in the AC angle with uniformly distributed permeability. The outflow pathway to the vitreous body,

^{17}which can be characterized as a porous domain,

^{18}was not accounted for in the model, as that flow occurs prior to the AH inflow in our model. The finite element mesh contained 336,516 tetrahedral finite elements. Figure 2 shows an overview of the mesh.

^{19}Additionally, the convective and diffusive transport of energy due to the temperature difference between the cornea and internal surfaces was modeled by solving a convection–diffusion equation for the temperature. The model was developed in the finite-element solver FEniCS 2019.1.0.

^{20}The model and the solver type chosen for this study allowed for an accurate representation of the AH outflow system and the ability to model angle closure.

*is the fluid velocity vector field,*

**u***p*is the scalar pressure field, and

*is the body-force term, defined as*

**f***is the gravitational acceleration vector, ρ*

**g**_{0}is the reference density, γ is the volume expansion coefficient,

*T*is the temperature, and

*T*is the reference temperature. It was verified that for this study the Re is sufficiently small (Re ≈ 0.15) and therefore the Stokes approximation is appropriate.

_{ref}*K*is the permeability of the medium. The Darcy model was modified to allow for decreasing permeability with increasing IOP:

^{19}In the TM, β is selected to simulate the mild decrease in permeability associated with increasing IOP (a drop of 19% as IOP increases from 15 to 30 mmHg). In the CB, β is selected to be sufficiently large so that \({\boldsymbol u} \approx - \frac{\alpha }{\beta }\nabla p/p = - \frac{\alpha }{\beta }\nabla \ln (p)\), which is much weaker than the linear dependence of velocity on pressure gradient in other regions. This makes the unconventional outflow in our model weakly dependent on IOP, a phenomenon that is consistent with experimental studies.

^{21}

^{,}

^{22}

*is also assumed to be zero), and in the Stokes flow regions the parameter*

**g***K*is set to zero. Thus, this single model captures both the Stokes and Darcy flow models.

**u**^{(1)}=

**u**^{(2)}. This is because we use continuous function spaces for the velocity field. Further, the Euler–Lagrange equations associated with the weak form of the Stokes–Brinkman equations yield:

^{(}

^{s}^{)}

*is the symmetric gradient of the velocity field, and*

**u**

**n**^{(1)}is the normal vector pointed away from the Stokes region. Equation 9 indicates that the pressure is continuous across the interface, and Equation 10 reveals that at shear stress it is zero.

*C*is the heat capacity, and

_{p}*k*is the thermal conductivity.

^{23}in enucleated human eyes (see Appendix B). The derived TM permeability parameter, α

*, was 4.5 × 10*

_{TM}^{−13}m

^{2}, and the permeability parameter of the CB face was of the same order of magnitude. These values are in accordance with the findings of other numerical studies.

^{12}

*x*= 0 mm and

*y*= 0 mm planes, a symmetric boundary condition was applied, wherein the normal velocity and temperature gradient were set to zero.

^{24}and even eyes with extensive gonioscopic iris–meshwork apposition sometimes have normal IOP levels,

^{25}indicating the presence of residual AH flow. Thus, by reducing permeability we allow for residual AH flow, which can be regulated with the magnitude of the outflow facility reduction factor.

^{26}Data from 2395 individual eyes from 2395 participants were used. Each CHES participant received a complete eye examination, including IOP checked using the Goldman Applanation Tonometer (Haag Streit USA, Mason, OH) and anterior segment OCT (AS-OCT) imaging using the CASIA SS-1000 (Tomey Corporation, Nagoya, Japan). Anterior segment parameters were measured in four AS-OCT images per eye.

^{27}Table 2 describes the presence of peripheral anterior synechiae (PAS), primary angle closure suspect (PACS), and PACG in the eyes in the clinical data. Angle width in the form of angle opening distance at 500 µm from the scleral spur (AOD500) was measured in eight sectors evenly spaced 45° apart. The lateral resolution of the CASIA SS-1000 is about 30 µm; therefore, a threshold of 0.04 mm (slightly more than 1 pixel) was used to detect angle closure, as manual detection of iridotrabecular contact (ITC) was not performed. A secondary sensitivity analysis was performed with thresholds of 0.02 and 0.06 mm. Additional information on the clinical assessment and AS-OCT image analysis in CHES is provided in Xu et al.

^{28}

^{29}which is captured by our model. Figure 5a shows that the temperature distribution in the AC uniformly varies from 35°C at the outer boundary surface (cornea) to 37°C at the inner surfaces, and Figure 5b shows the flow velocity (both direction and magnitude) at a cross-section of the AC with an inflow rate of 0.75 µL/min (3 µL/min overall over all four quadrants). We note that the section of the flow shown in Figure 5b is representative of the radially symmetric flow in an eye with no angle closure. The buoyancy-driven vortex shown in Figure 5b is comprised of a flow moving downward at the center of the AC, outward near the iris, and inward near the cornea. We note that the buoyancy-driven vortex has also been observed in other computational studies in counterclockwise

^{30}

^{–}

^{32}and clockwise directions.

^{12}

^{,}

^{33}

^{,}

^{34}It is likely that these two counter-rotating states are both stable solutions of the problem and that either may be attained by perturbations in the boundary and initial conditions or the mesh used to solve the problem.

^{−3}m/s) due to the narrow section of these conduits (see Fig. 5b). Other than at the CCs, the AH moves fast within the buoyancy-driven vortex in the AC (4 × 10

^{−5}m/s), and it is significantly slower at the TM (2.5 × 10

^{−6}m/s) and CB (5 × 10

^{−7}m/s). The flow velocities predicted by our model are in the same range as those of other studies.

^{12}

^{,}

^{34}

^{11}

^{,}

^{12}including a model for outflow through the unconventional pathway and a decrease in TM permeability with increasing IOP. These features allow us to accurately model changes in IOP related to different physiological conditions. Specifically, we have demonstrated that our model can simulate characteristics of AH flow and IOP related to angle closure. These findings on angle closure are consistent with clinical data suggesting that IOP rise occurs when one-half of the angle is obstructed and may provide an explanation as to why only some patients with angle closure develop elevated IOP and PACG.

*v*=

*q*/

*A*, where

*v*is the flow velocity,

*q*is the flow rate, and

*A*is the “open” outflow pathway area, which represents the “open” circumferential extent of the outflow pathways. In our model, the flow rate is fixed, and an increase in the extent of angle closure leads to a linear decrease in the “open” area. However, because this term appears in the denominator of the expression for the velocity in this region, it leads to a nonlinear increase in the velocity. Finally, through Darcy's equation this leads to a nonlinear increase in the IOP, as well, which is what is observed in Figure 6. In our model, TM permeability decreases with increases in IOP, and this dependence could accentuate the increase in the IOP. However, this effect plays a relatively small role in our model compared to angle closure, as permeability is only reduced by 19% in the range of pressures considered in this study. In Dvoriashyna et al.,

^{35}the effect of pupillary block (i.e., the closure of the gap between the lens and the iris) on the IOP was studied using a semi-analytical model. Remarkably, the authors observed the same nonlinear increase in IOP with pupillary block as we observed for anterior chamber angle closure, which is governed by a similar mechanistic model.

*single*, simulated eye. In contrast, in the clinical data, we considered this relationship across many different eyes, and then examined how some conditional statistic of the IOP (e.g., mean) is related to the extent of angle closure. The mechanistic model reveals a nonlinear increase in IOP with the extent of closure (

*e*) for all values of baseline permeability (

*K*

_{0}) and levels of severity of closure (

*S*). This implies that, if in the clinical study subjects were drawn from a distribution of the form, then

^{36}Knowledge about this relationship is sparse but crucial, as it could help clinicians determine when treatments that alleviate angle closure, including laser peripheral iridotomy and lens extraction, should be performed to prevent elevated IOP.

^{37}

^{,}

^{38}However, precisely measuring the anterior–posterior and circumferential extent of angle closure remains challenging even with modern angle assessment methods. Gonioscopic grades are weakly correlated with IOP and AS-OCT measurements of angle width in angle closure eyes, which makes them poorly suited for measuring the extent of angle closure.

^{39}

^{,}

^{40}Although AS-OCT can provide quantitative data on angle width, manual image analysis is time consuming and expertise dependent. Therefore, additional methods that support automated analysis of AS-OCT images are needed before structure–IOP relationships can be used to guide personalized treatment of angle closure eyes.

^{41}

^{,}

^{42}Second, not all angle closure eyes develop elevated IOP even when angle closure is extensive. Complex factors, such as residual permeability of the iris and ciliary body to AH, may explain why the median IOP remains lower than 21 mmHg in CHES eyes with complete angle closure (Fig. 7).

^{43}

^{,}

^{44}This finding serves as a reminder that the IOP cut-off of 21 mmHg to detect primary angle closure is somewhat arbitrary and that relative IOP elevations from angle closure confers higher glaucoma risk even when absolute IOP is under 21 mmHg.

^{45}

^{11}

^{,}

^{12}

^{,}

^{32}it is an approximation, as it circumvents the need to explicitly model complex mechanisms related to the outflow resistance such as the collapse of the SC. Also, the geometries of the TM and CB are simplified, and the permeability is considered uniform in the direction of the outflow (radial [

*r*] and axial [

*z*] directions). Hence, to study in detail the effects that the TM and CB have in the outflow facility, a more complex model of these tissues would be needed. Furthermore, all tissues in this model were treated as rigid. An improvement to this would be to treat them as elastic and replace the current model for the tissue with its poroelastic counterpart.

^{46}Additionally, there are several improvements that can be made to make the model more realistic. These include (1) allowing for some permeability through the iris so that we have some residual flow even when the angle is fully closed, (2) using patient-specific models for the anatomy of the eye, (3) modeling outflow through the vitreous, and (4) accounting for the periodic pressure oscillations induced by the ocular pulse.

**J. Murgoitio-Esandi**, None;

**B.Y. Xu**, None;

**B.J. Song**, None;

**Q. Zhou**, None;

**A.A. Oberai**, None

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^{47}

^{,}

^{48}which is believed to be caused by the morphology of the SC and CCs. It is known that CCs are randomly distributed around the circumference of the SC with higher concentrations in the inferior-nasal and temporal quadrants

^{49}and that segmental flow occurs toward quadrants with higher CC concentrations.

^{47}Hence, if angle closure occurs at sectors with higher outflow rates, there could be an additional increase of IOP. However, this behavior is not replicated by our model, as it considers a circumferentially continuous Schlemm's canal, which imposes near-zero flow resistance. For this reason, an uneven distribution of CCs does not affect our results. To show this, we have conducted additional simulations that verify that the distribution and location of the CCs does not affect outflow or IOP. Figure A.1a shows a geometry with evenly distributed CCs, and Figure A.1b shows the geometry with unevenly distributed CCs. In addition, Figure A.1c shows the increase in IOP for these two cases with increasing angle closure. This demonstrates that, if the SC is considered to be continuous, the distribution of CCs does not affect the IOP. Martínez Sánchez et al.

^{50}arrived at the same conclusion using a similar model. Hence, for simplicity, in this study we considered an even distribution of CCs.

^{22}which were reported for enucleated human eyes. This study was chosen because it is one of the seminal studies demonstrating the increase in outflow resistance with elevated IOP. Figure B.1 shows both the experimental results by Brubaker

^{22}and the results obtained using our mechanistic model. Figure B.1a is a plot of the outflow rate as a function of IOP, and Figure B.1b is a plot of the outflow facility as a function of IOP. In both cases, we observed that the model accurately represents the trend of the experimental data; therefore, the mechanistic model can represent physiologically accurate outflow facility. The permeability parameters used in the simulations shown in Figure B.1 are listed in Table B.1.