Mathematical Modeling of Visual Acuity Loss in Chronic Retinal and Optic Nerve Diseases: A Differential-Equation Framework Linking Photoreceptor Apoptosis to LogMAR Visual Function in a Synthetic Cohort

Source abstract

This study presents a comprehensive quantitative framework for the kinetic mathematical modeling and prediction of Visual Acuity (VA) loss dynamics, measured on the LogMAR scale, in patients suffering from chronic retinal and optic nerve diseases such as Age-Related Macular Degeneration (AMD), Glaucoma, and Diabetic Retinopathy. The model is formulated and evaluated using real-world compliant virtual patient cohorts (N ≈ 1333 across four disease groups). These synthetic cohorts are generated using truncated multivariate normal distributions calibrated against published clinical population statistics (e.g., AREDS and Beaver Dam Eye Study) to ensure physiological and demographic fidelity. A system of non-linear kinetic ordinary differential equations is developed to couple cellular-level photoreceptor apoptosis and cumulative pathological stress with functional visual acuity via Fechner's psychophysical law. Model parameters (β, μ, δ, κ) were calibrated using Non-linear Least Squares (NLLS) optimization against simulated longitudinal clinical data (OCT-analogue photoreceptor density and ETDRS/LogMAR visual acuity), with numerical integration performed via the 4th-order Runge-Kutta (RK4) method; the baseline aging-apoptosis constant γ₀ was fixed from independent normative literature (Swanson & Horner) rather than jointly estimated, owing to its weak practical identifiability over a 5-year observation window. Simulations reveal a distinct cellular latency phase in which functional vision remains relatively stable — a direct consequence of the logarithmic nature of visual perception — followed by an abrupt, accelerating decline once a critical cell-loss threshold (30%–40%) is breached. The model differentiates the slow, near-linear progression of dry AMD from the rapidly accelerating trajectory of wet AMD and the intermediate-tempo course of diabetic retinopathy, and reproduces the LogMAR-flattening effect of simulated anti-VEGF therapy. Calibration against synthetic clinical data achieved R² values between 0.850 and 0.903 across all four disease groups (0.850 for dry AMD, 0.898 for wet AMD, 0.879 for glaucoma, and 0.903 for diabetic retinopathy; Table 1). Section 5.1 traces the gap between this range and an initially targeted R² > 0.93 benchmark to a structural identifiability issue between β and μ. The model offers a quantitative starting point for reasoning about therapeutic-window timing, though richer longitudinal data are needed before its parameter estimates can be used clinically.