A Model of the Mechanisms Underpinning Unconventional Aqueous Humor Outflow

Abstract

Purpose: To develop a mathematical model of the unconventional outflow pathway.

Methods: The unconventional pathway is modeled as having two key components: the uveo-vortex and the trans-scleral pathways. The uveo-vortex pathway is modeled using Starling's law and the trans-scleral flow using predominately hydrostatic forces. We include transcytosis from the choriocapillaris (CC) and collapsibility of the suprachoroidal space (SCS) as particular features. There is considerable uncertainty in a number of model parameter values, and we identify the most significant ones using sensitivity analysis.

Results: The model successfully generates a fluid flow from anterior to posterior in the choroidal tissue and the SCS, which also demonstrates many of the known physiological features, including the insensitivity of the unconventional flow to fluctuations in the IOP, albumin removal by the trans-scleral flow, and the CC as a net absorber of fluid from, and supplier of albumin to, the choroidal tissue. The model supports the two previously proposed mechanisms of the action of prostaglandin F2α analogues.

Conclusions: We have developed a theoretical model of the unconventional aqueous outflow pathway that successfully captures its physiological features and elucidates the actions of prostaglandin F2α analogues and other drugs.

Figure 1.

(Left) Schematic of the anterior segment of the eye. Aqueous humor flow in the posterior and anterior chambers is shown with cyan arrows. The conventional outflow is indicated by the green arrow, and the unconventional pathway is marked by black arrows. (Right) Zoom of the choroid (based on Fig. 4 by Wajer et al.⁸⁸). (Bottom) Cross-section, showing the layers of different types of vessels; top: view of apical surface of the CC.

Figure 2.

Schematic diagram of the eye highlighting the setup and parameters used in the mathematical model. The anterior of the eye is at the top of the diagram and the model has rotational symmetry about the vertical axis θ = 0. The inset zooms into the region we focus on in the model, which is the outer part of the posterior eye.

Figure 3.

Model results in the reference physiological case, showing the IF flow and albumin concentration plotted against distance from the inlet. (A) Pressure in the choroid–SCS, p, CC, $p_{\mathcal{CC}}$, orbit, $p_{\mathcal{O}}$, and anterior chamber, IOP. (B) Albumin concentration in the choroid–SCS, c, CC, $c_{\mathcal{CC}}$, and orbit, $c_{\mathcal{O}}$. (C) IF flow rate posteriorly in the choroid–SCS, showing also the components of this flow in the choroidal tissue and SCS. (D) As (C), but for the albumin flow rate. (E) IF flow rate per unit area of surface into (positive) and out of (negative) the choroid–SCS. (F) As (E), but for albumin (with the same colors). The flow of albumin out of the CC is the sum of the contributions due to transcytosis (out of the CC, positive) and to advection by the crystalloid fluid (into the CC, negative).

Figure 4.

Results of sensitivity analysis showing the first-order (blue) and total-order (red) sensitivities for (A) the unconventional flow rate, and (B) the average albumin concentration across the choroid–SCS. The directions of influence for the most sensitive parameters are indicated by the black arrows above the bars, that is, an upward (downward) arrow indicates that the flow rate/albumin concentration increases (decreases) as the given parameter is increased. The label $\mathsf{d}$ is the dummy variable, which is a parameter that does not appear in the model and hence represents a ‘negative control’ for the sensitivity analysis. In the simulations, $k_{\mathcal{S}}$ is varied independently of $k_{\mathcal{C}}$.

Figure 5.

Dependence of the unconventional flow rate on the IOP. (A) Comparison of conventional and unconventional flow rates. Solid dots: Measurements of the conventional (green) and unconventional (black) flow rates by Bill on living monkeys with best fit dashed lines added. Blue curve: Predictions of the model. (B) Detailed graph of predictions of the model accounting for various mechanisms. In the table in the figure inset, the column Albumin indicates the presence or absence of albumin exchange with the vessels: + indicates that there is exchange; − that there is not (we set ${\sigma }_{\mathcal{CC}}={\sigma }_{\mathcal{S}}=0$ in the model). The column SCS refers to the second mechanism: The symbols + and − indicate the presence or absence of a collapsible SCS, respectively. The black dashed line and blue solid curve are the same as those appearing in (A). In both figures, the black vertical line corresponds with the episcleral venous pressure of 8.4 mm Hg and the black horizontal line marks zero flux. Note that, as the IOP is varied, we also vary the capillary pressure $p_{\mathcal{CC}}=\mathrm{IOP}+5$ mm Hg and apical RPE pressure $p_{\mathcal{R}}=\mathrm{IOP}$.

Figure 6.

Parts (A) and (B) investigate four possible mechanisms of unconventional flow increase with the use of PGF_2α analogues; (C) shows albumin concentration in one of the cases (increasing $L_{p,\mathcal{CC}}$); (D) shows a summary of the four mechanisms. In all figures, we consider a departure from the reference physiological case (which is shown on the left sides of A–C), that is, the SCS height is allowed to deform. (A) Top axis (red dashed): Increase in unconventional flow rate with decreasing iris root resistance, R_icm. The x-axis shows the factor by which R_icm is reduced, with 1 being the reference physiological case and 0.5 corresponding with the reduction of R_icm by a factor of 2. Bottom axis (blue solid): Unconventional flow rate increase with increasing scleral hydraulic conductivity, $k_{\mathcal{S}}$. The x-axis is the factor by which $k_{\mathcal{S}}$ is increased. (B) Top axis (red dashed): Unconventional flow rate increase as pressure in the capillaries, $p_{\mathcal{CC}}$, is reduced (extreme left is the reference physiological case, $p_{\mathcal{CC}}=\mathrm{IOP}+5$ mm Hg; extreme right is $p_{\mathcal{CC}}=\mathrm{IOP}+2.5$ mm Hg). Bottom axis (blue solid): Unconventional flow rate increase with increasing hydraulic conductance of the vessel walls, $L_{p,\mathcal{CC}}$. The x-axis is the factor by which we increase $L_{p,\mathcal{CC}}$ from the reference physiological case. (C) Albumin concentration with increasing hydraulic conductance (x-axis is the same as the bottom one in B). The solid line is the albumin concentration at the inlet and the punctured line is the spatially averaged albumin concentration in the choroid–SCS. (D) Summary of all four mechanisms of relative increase of the flow rate: decreasing R_icm by a factor of 2, doubling $k_{\mathcal{S}}$, reducing $p_{\mathcal{CC}}$ by 2.5 mm Hg and doubling $L_{p,\mathcal{CC}}$. The colors of the bars correspond with the colors of the lines in A and B.

Figure A1.

(A) Flow rate through the inlet as the proportion of flow through the choroidal tissue is varied. Instead of requiring that the flow rates through the choroidal tissue and SCS are equal at the inlet, we demand that the flow rate through the choroidal tissue is a fixed percentage of the total unconventional flow rate. For each value of this percentage, we calculate values of R_icm, k_C and $h_{\mathcal{P}0}$ that also fix the correct pressures at the inlet and posterior pole (more details in text). We show the flow rates through the choroidal tissue and SCS as well as the total unconventional flow rate. (B) Dependence of the unconventional flow rate on the IOP for three different values of ${\lambda }_{\mathcal{P}}$ (legend, with ${\lambda }_{\mathcal{P}}=100$ Pa (blue curve) being the baseline physiological value). The pressures $p_{\mathcal{CC}}=\mathrm{IOP}+5$ mm Hg and $p_{\mathcal{R}}=\mathrm{IOP}$ vary with the IOP. The blue dashed line shows the case $p_{\mathcal{R}}=\mathrm{IOP}-0.5$ mm Hg (with $p_{\mathcal{CC}}=\mathrm{IOP}+5$ mm Hg, ${\lambda }_{\mathcal{P}}=100$ Pa).