Material publication: Mathematical modeling and simulation of coupled aqueous humor flow and temperature distribution in a realistic 3D human eye geometry

[thomas-saigre]

Overview

We present a comprehensive computational model that simulates the coupled dynamics of aqueous humor (AH) flow and heat transfer in the human eye. To manage the model's complexity, we employ advanced meshing techniques and leverage high-performance computing resources to efficiently solve the discrete problem. The model accurately captures the dynamics of AH within the anterior and posterior chambers, incorporating convective effects driven by temperature variations. The resulting fluid velocity, pressure, and temperature distributions align well with existing numerical findings in the literature. Additionally, we analyze the impact of postural changes and wall shear stress behavior, offering new insights into the mechanical forces exerted on ocular tissues. This work provides a detailed three-dimensional simulation that enhances our understanding of ocular physiology and may contribute to advancements in clinical research and treatment optimization in ophthalmology.

Preprint:

Geometry:

Meshes:

Model description

Our study develops a three-dimensional mathematical and computational model to simulate heat transfer across the entire human eyeball, coupled with the dynamic flow of AH in both the anterior (AC) and posterior (PC) chambers (see Fig. 1(b)).

Fig. 1: Structure of the human eye (left panel) and a detailed view of the anterior and posterior chambers (right panel).

The geometry is generated using a Computer-Aided Design (CAD) model based on a realistic human eye. The full workflow, from CAD generation to meshing, is documented and available as open-access data on Zenodo.

Following established literature, we model AH as an incompressible Newtonian fluid, assuming negligible density variations except in the buoyancy term, where we apply the Boussinesq approximation to account for temperature-induced buoyancy effects. The AH flow is governed by the incompressible Navier-Stokes equations coupled with heat transfer:

$$ \begin{alignat}{2} \rho (\vec{u}\cdot\nabla) \vec{u} - \nabla \cdot \underline{\underline{\sigma}} &= -\rho\beta(T-T_\text{ref})\vec{g} &\quad & \text{in $\Omega_\text{AH}$},\\ \nabla\cdot \vec{u} &= 0 & & \text{in $\Omega_\text{AH}$},\\ \rho C_p \vec{u}\cdot\nabla T - k\nabla^2 T &= 0 & & \text{in $\Omega$}. \end{alignat} $$

The model incorporates postural orientation by including the gravity vector $\vec{g}$ in the Boussinesq term.

Geometric discretization

The geometry in Fig. 1(a) is discretized with particular attention to the AC and PC domains, where the coupled fluid-thermal problem is solved.

All meshes supporting this study (simulations, convergence studies, etc.) are openly available on Zenodo.

Computational framework

We employ the finite element method to solve the coupled equations, utilizing the heatfluid toolbox from the Feel++ library. A significant effort is dedicated to preconditioning the system, specifically using a field-split strategy to enhance computational efficiency.

Numerical results

Fig. 2 illustrates the computed temperature distribution within the eyeball in a standing position, shown on a vertical cross-sectional plane.

Fig. 2: Computed temperature distribution within the eyeball in a standing position, with mesh discretization.

Fig. 3 presents simulation results for different postural orientations, depicting flow patterns and pressure distributions. In the standing position (Fig. 3(a)), gravity strongly influences the flow, leading to higher velocities and a pronounced downward movement of AH. This behavior contributes to characteristic patterns such as Krukenberg’s spindle and recirculation zones within the AC, aligning with prior observations in the literature.

Fig. 3: Simulation results for different postural orientations. Streamlines are colored by pressure, and arrows represent fluid velocity magnitude.

We also analyze the effects of position and ambient temperature on Wall Shear Stress (WSS), a key parameter for clinical applications, personalized medicine, and ocular device design. Fig. 4 presents these results, demonstrating that WSS is highly sensitive to postural orientation and ambient temperature.

Fig. 4: Average WSS magnitude as a function of ambient temperature for different postural orientations, measured on the internal corneal surface.

These findings provide valuable insights for clinicians and may help refine ocular treatments, such as endothelial cell injections to restore corneal function.

References

The preprint of this publication is available on HAL and arXiv.

• Chabannes, V., Prud'homme, C., Saigre, T., Lorenzo, S., Szopos, M., & Trophime, C. (2024). A 3D geometrical model and meshing procedures for the human eyeball (v1.0.0-preview.2). Zenodo.
• Saigre, T., Prud'homme, C., Szopos, M., & Chabannes, V. (2024). Mesh and configuration files to perform coupled heat+fluid simulations on a realistic human eyeball geometry with Feel++ [Data set]. Zenodo.
• Prud'homme, C., Chabannes, V., Saigre, T., Trophime, C., Berti, L., Samaké, A., Van Landeghem, C., Szopos, M., Giraldi, L., Bertoluzza, S., & Maday, Y. (2024). feelpp/feelpp: Feel++ Release V111 preview.10 (v0.111.0-preview.10). Zenodo.

Acknowledgments

We acknowledge the support of the Cemosis platform at the University of Strasbourg and the French Ministry of Higher Education, Research, and Innovation. This work was also funded by the France 2030 NumPEx Exa-MA (ANR-22-EXNU-0002) project, managed by the French National Research Agency (ANR).