A Mathematical Model of Aqueous Humor Production and Composition

Abstract

Purpose: We develop a mathematical model that predicts aqueous humor (AH) production rate by the ciliary processes and aqueous composition in the posterior chamber (PC), with the aim of estimating how the aqueous production rate depends on the controlling parameters and how it can be manipulated.

Methods: We propose a compartmental mathematical model that considers the stromal region, ciliary epithelium, and PC. All domains contain an aqueous solution with different chemical species. We impose the concentration of all species on the stromal side and exploit the various ion channels present on the cell membrane to compute the water flux produced by osmosis, the solute concentrations in the AH and the transepithelial potential difference.

Results: With a feasible set of parameters, the model predictions of water flux from the stroma to the PC and of the solute concentrations in the AH are in good agreement with measurements. Key parameters which impact the aqueous production rate are identified. A relevant role is predicted to be played by cell membrane permeability to \(\text{K}^+\) and \(\text{Cl}^-\), by the level of transport due to the Na+-H+ exchanger and to the co-transporter of Na+/K+/2Cl-; and by carbonic anhydrase.

Conclusions: The mathematical model predicts the formation and composition of AH, based on the structure of the ciliary epithelium. The model provides insight into the physical processes underlying the functioning of drugs that are adopted to regulate the aqueous production. It also suggests ion channels and cell membrane properties that may be targeted to manipulate the aqueous production rate.

Figure 1.

Sketches of a cross section of the human eye (A), of the ciliary processes (B) and of the CE (C).

Figure 2.

(A) Ion channels in the CE. PE basolateral membrane: nkcc: co-transporter of ${\text{Na}}^{+}$/${\text{K}}^{+}$/2${\text{Cl}}^{-}$; ${ae}_s$: ${\text{Cl}}^{-}$-${\text{HCO}}_3^{-}$ exchanger; nhe: ${\text{Na}}^{+}$-${\text{H}}^{+}$ exchanger; $nb{c}_s$: ${\text{Na}}^{+}$- ${\text{HCO}}_3^{-}$ cotransporter, presumed to be of stoichiometry 1:2; ${\text{K}}^{+}$- channels. NPE basolateral membrane: $pump$: ${\text{Na}}^{+}$-${\text{K}}^{+}$ ATPase; $a{e}_p$: ${\text{HCO}}_3^{-}$-${\text{Cl}}^{-}$ exchanger; $nb{c}_p$: ${\text{Na}}^{+}$- ${\text{HCO}}_3^{-}$ cotransporter; ${\text{K}}^{+}$ and ${\text{Cl}}^{-}$ channels. (B) A sketch of the compartmental model with water and ion fluxes (not to scale). The symbols $s$, $c$ and $p$ denote stroma, intracellular space and PC respectively. The symbols $\tilde{p},\tilde{s}$ and $tj$ denote the PC and stromal cell membranes and the tight junction. In the direction from region $m$ to region $k$, $J_i^{mk}$ denotes flux of solute $i$ and $Q^{mk}$ the water flux (see main text).

Figure 3.

Total sensitivity index for (A) production rate $Q$ and (B) concentrations in the PC. The arrows indicate how the given parameter influences the model output and are based on the total sensitivity index, which is an output of the eFAST method. The vertical dashed line separates the intensities of ion channels, $P_k$ in Equation (2), from the permeabilities $P$ in Equation (3).

Figure 4.

Change of (A) production rate $Q$ and (B) concentrations in the PC with CA inhibition.

Figure 5.

Change of ion fluxes through each individual channel with CA inhibition (see Appendix for detailed expression of ion channel fluxes). The value is positive if the flux is directed outside the cell. For anion exchangers, the value is positive if ${\text{HCO}}_3^{-}$ goes out of the cell and flux through nhe is positive if ${\text{H}}^{+}$ is transported out of the cell.