A novel multiscale modelling for the hemodynamics in retinal microcirculation with an analytic solution for the capillary-tissue coupled system

TL;DR

This workdevelops a multiscale retinal hemodynamics framework that couples 1D arterial/venous trees with a capillary-tissue Darcy system. A key novelty is the analytic capillary-tissue solution obtained by decoupling pressures into a mean and an exchange component, enabling fast computation and clear interpretation of capillary-tissue exchange via a dynamic coupling condition. The model is validated against experimental data and used to explore how capillary permeability, drainage rate, and pulsatile inlet conditions shape retinal blood flow and pressures, revealing a saturating response to capillary permeability and regime-dependent sensitivity to fluid exchange. These results offer a tractable tool for understanding retinal pathophysiology and for patient-specific exploration, while highlighting avenues for extending the geometry, permeabilities, and osmotic effects in future work.

Abstract

Mathematical modelling of the microcirculatory hemodynamics in the retina is an essential tool for understanding various diseases of the retina, yet remains challenging due to the multiscale nature of the retinal vasculature and its coupling to surrounding tissue. To address this, we develop a multiscale model that couples retinal vasculature across scales with interstitial tissue. Our model combines the one-dimensional (1D) model for arteries and veins with the coupled Darcy equations for capillaries and tissue. The model uses an analytic solution for capillary-tissue coupled system that provides a simple interpretation of the results along with much faster computation. The analytic solution implies a dynamic coupling condition that links the capillary bed with upstream arterial and downstream venous flows. The model is mathematically robust, demonstrated through analysis of the solution's truncation error and convergence. Its predictive accuracy is verified against experimental data and other models, making it useful in interpreting experimental results. Finally, the role of various parameters in controlling retinal hemodynamics is explored.

Figures (19)

• Figure 1: Spatial discretization of vessel.
• Figure 2: Synthetic and real retinal vasculature. (a) The arterial (red) and venous (blue) trees in retinal tissue (black circle) generated using an L-system. (b) Fundus image from DRIVE dataset staal_ridge-based_2004. (c) The manually segmented retinal vasculature for (b). Arteries and veins are displayed in red and blue, respectively. The green denotes pixels classified as both artery and vein and the white denotes the uncertain segment.
• Figure 3: The dependence of relative ratio $\mathcal{E}_{m,m+1}$ on the order $m$ for $r_n=0.25R_t$. Solid lines denote the values of relative ratio while dashed lines are the upper bound. Different colours correspond to various values of radial coordinate. (a) Value of drainage rate is of $\alpha=2\times 10^{-9}~{\rm{cm}\cdot \rm{s}/\rm{g}}$. (b) Value of drainage rate is of $\alpha=2\times 10^{-8}~{\rm{cm}\cdot \rm{s}/\rm{g}}$. (c) Value of drainage rate is of $\alpha=2\times 10^{-7}~\rm{cm}\cdot \rm{s}/\rm{g}$. Other parameter values are $k_{cap}=2\times 10^{-9}~{\rm{cm}^2}$ and $k_t=2\times 10^{-12}~{\rm{cm}^2}$.
• Figure 4: The dependence of relative ratio $\mathcal{E}_{m,m+1}$ on the order $m$ for $r_n=0.25R_t$. Solid lines denote the values of relative ratio while dashed lines are the upper bound. Different colours correspond to various values of radial coordinate. The drainage rate is $\alpha=2\times 10^{-6}~{\rm{cm}\cdot \rm{s}/\rm{g}}$.
• Figure 5: Two different bifurcation boundaries.