Paper submission : Model order reduction and sensitivity analysis for complex heat transfer simulations inside the human eyeball

[thomas-saigre]

Model order reduction and sensitivity analysis for complex heat transfer simulations inside the human eyeball

Overview

Heat transfer in the human eyeball, a complex organ, is significantly influenced by various pathophysiological and external parameters. Particularly, heat transfer critically affects fluid behavior within the eye and ocular drug delivery processes. Overcoming the challenges of experimental analysis, this study introduces a comprehensive three-dimensional mathematical and computational model to simulate the heat transfer in a realistic geometry.
This work includes an extensive sensitivity analysis to address uncertainties and delineate the impact of different variables on heat distribution in ocular tissues.
To manage the model’s complexity, we employed a very fast model reduction technique with certified sharp error bounds, ensuring computational efficiency without compromising accuracy. Our results demonstrate remarkable consistency with experimental observations and align closely with existing numerical findings in the literature.
Our findings underscore the significant role of blood flow and environmental conditions, particularly in the eye’s internal tissues.

Clinically, this model offers a promising tool for examining the temperature-related effects of various therapeutic interventions on the eye. Such insights are invaluable for optimizing treatment strategies in ophthalmology.

Three-dimensional parameter-dependant model

We model the heat transfer in the human eyeball. Due to the complexity of the organ, various regions with different physiological properties are involved.

Vertical cut of the 3D geometrical model of the human eyeball.

To model the thermal exchanges with the surrounding body, or the ambient air (convective exchange, radiative heat transfer, tear evaporation rate), boundary conditions are set.

In the model, many parameters are involved. We want to study the impact of these parameters on the resulting distribution of temperature.

Solutions for three values of parameters: $\bar{\mu}$ composed of nominal values for a human eyeball, $\mu_\min$ and $\mu_\max$ as extremal values.

Computational framework and certified reduced basis

This model can be computed numerically using the library Feel++, but a simulation for given values of parameters is long and costly (62 s for a numerical problem of size 1 580 932).
This high-fidelity model is also validated against experimental data and findings previously reported in the literature.
As the sensitivity analysis requires running this model for a large set of parameters, we develop a reduced model thanks to the reduced basis method (RBM).

This method allows to building of a surrogate model to simulate the distribution of the temperature over the eyeball, by means of an efficient and stable procedure: we replicate the solution while significantly reducing the time of execution (20,000 times faster than the finite element simulation).
The RBM ensures the quality of the reduced solution, providing a posteriori error-bound $\Delta_N$.

Error on RBM for various reduced basis sizes, represented with the error bound $\Delta_N$, for a sample of 100 parameters

Sensitivity analysis

Thanks to the RBM, we can compute the solution for a very large number of parameters, and perform an Uncertainty Quantification, such as compute the Sobol indices.
Sobol indices are indicators of the effect of a parameter on the output of interest.

Sensitivity analysis results: Sobol indices of the parameters on the temperature at the front of the eye $T_O$, and at the back of it $T_G$.

We notice a significant impact of external factors (ambient air temperature $T_\text{amb}$, tear evaporation $E$), and subject-specific data (such as the blood temperature $T_\text{bl}$) on the corneal temperature; while at the back of the eye, only the blood temperature $T_\text{bl}$ is influential.

Next steps

• Model: Couple thermal effect with aqueous humor dynamics in the anterior chamber,
• Non-intrusive reduced basis method with zoom in a zone of interest,
• Application: robust framework to simulate drug delivery in the eye.

References

Check out the publication on the HAL portal

Acknowledgements

This work was funded by the French Ministry of Higher Education, Research and Innovation.