Models and Algorithms for the Refinement of Therapeutic Approaches for Retinal Diseases

Diagnostics · 3 authors, 2 centres

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TL;DR

This paper presents a 3D mathematical model and finite element simulations of drug distribution in the vitreous body to optimize anti-VEGF injection therapy for age-related macular degeneration. The main findings from in silico experiments suggest that injection angle and position significantly affect drug delivery to the macula, with optimal angles around 25-50 degrees and central vitreous injection being beneficial for longer-acting drugs. The study is limited by its computational nature, lacking direct clinical validation and relying on idealized parameters.

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BACKGROUND

Age-related macular degeneration (AMD) is a leading cause of vision loss, with anti-VEGF injections as standard treatment. The study aims to refine therapy using computational models to simulate drug distribution in the vitreous body. Methods: The authors developed a 3D mathematical model coupling a time-dependent convection-diffusion equation for drug transport with a steady-state Darcy equation for aqueous humor flow. The model includes anisotropic diffusion from collagen fibers and gravity effects. Simulations were performed using finite element methods over a 30-day period, with functionals to quantify drug concentration at the macula. Key Results: Simulations showed that gravity has a negligible effect on drug distribution (maximum difference <1% in drug concentration at the macula). Optimal injection angles were found to be between 25 and 50 degrees, with larger angles (e.g., 75 degrees) resulting in up to 38% less drug reaching the macula. Injection into the center of the vitreous was optimal for longer-acting drugs, while injection closer to the macula provided higher initial concentrations. A smaller diffusion coefficient (e.g., 2×10⁻¹¹ m²/s) resulted in more drug reaching the macula over 30 days compared to larger coefficients (e.g., 8×10⁻¹¹ m²/s). Implications: The model provides a framework for in silico testing of injection parameters to optimize drug delivery, potentially reducing the need for costly experiments.

STUDY TYPE

other

SPECIALTY

ophthalmology

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AI pipeline

FIDELITY CHECK

100% · A

ORIGINAL PAPER

PMC10001150

Models and Algorithms for the Refinement of Therapeutic Approaches for Retinal Diseases

Diagnostics

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ABSTRACT

We are developing a Virtual Eye for in silico therapies to accelerate research and drug development. In this paper, we present a model for drug distribution in the vitreous body that enables personalized therapy in ophthalmology. The standard treatment for age-related macular degeneration is anti-vascular endothelial growth factor (VEGF) drugs administered by repeated injections. The treatment is risky, unpopular with patients, and some of them are unresponsive with no alternative treatment. Much attention is paid to the efficacy of these drugs, and many efforts are being made to improve them. We are designing a mathematical model and performing long-term three-dimensional Finite Element simulations for drug distribution in the human eye to gain new insights in the underlying processes using computational experiments. The underlying model consists of a time-dependent convection-diffusion equation for the drug coupled with a steady-state Darcy equation describing the flow of aqueous humor through the vitreous medium. The influence of collagen fibers in the vitreous on drug distribution is included by anisotropic diffusion and the gravity via an additional transport term. The resulting coupled model was solved in a decoupled way: first the Darcy equation with mixed finite elements, then the convection-diffusion equation with trilinear Lagrange elements. Krylov subspace methods are used to solve the resulting algebraic system. To cope with the large time steps resulting from the simulations over 30 days (operation time of 1 anti-VEGF injection), we apply the strong A-stable fractional step theta scheme. Using this strategy, we compute a good approximation to the solution that converges quadratically in both time and space. The developed simulations were used for the therapy optimization, for which specific output functionals are evaluated. We show that the effect of gravity on drug distribution is negligible, that the optimal pair of injection angles is ( 50 ∘ , 50 ∘ ) , that larger angles can result in 38% less drug at the macula, and that in the best case only 40% of the drug reaches the macula while the rest escapes, e.g., through the retina, that by using heavier drug molecules, more of the drug concentration reaches the macula in an average of 30 days. As a refined therapy, we have found that for longer-acting drugs, the injection should be made in the center of the vitreous, and for more intensive initial treatment, the drug should be injected even closer to the macula. In this way, we can perform accurate and efficient treatment testing, calculate the optimal injection position, perform drug comparison, and quantify the effectiveness of the therapy using the developed functionals. We describe the first steps towards virtual exploration and improvement of therapy for retinal diseases such as age-related macular degeneration.

CITATION EXPORT

Elfriede Friedmann, Simon Dörsam, Gerd U. Auffarth. Models and Algorithms for the Refinement of Therapeutic Approaches for Retinal Diseases. Diagnostics. (2023). https://doi.org/10.3390/diagnostics13050975

AUTHORS

3 · 2 CENTRES

Elfriede Friedmann, Simon Dörsam, Gerd U. Auffarth

1Institute of Mathematics, Kassel University, Heinrich-Plett-Straße 40, 34132 Kassel, Germany
2Department of Ophthalmology, David J. Apple International Laboratory for Ocular Pathology, Heidelberg University, 69120 Heidelberg, Germany

EDITOR / REVIEWER

Jae-Ho Han