Analytical solutions for light propagation of LED

Analytical solutions of diffusion theory for light propagation in turbid media are essential for optical diagnostics and therapeutic applications, including cerebral oximetry, hemodynamic monitoring, and photostimulation. While existing solutions work reasonably well for collimated light sources-lasers and optical fibers-analytical solutions for LEDs remain missing, despite the growing use of LEDs in wearable and implantable bioelectronics. We present a method to solve the diffusion theory and derive analytical solutions for two biomedically relevant configurations: 1) surface-mounted LEDs on semi-infinite media (e.g., wearable devices) and 2) embedded LEDs in infinite media (e.g., implantable devices). Beyond a distance of 4 times the scattering length of the medium to the LED source, our analytical solutions are reasonably accurate, within 6% error for 1) and 3% for 2). This represents significant improvements over existing analytical solutions, characterized by 26% and 15% error, respectively. Using our analytical solutions, we derive tissue optical properties ([Formula: see text] and [Formula: see text]) from diffuse reflectance results with <7% error, and we determine the irradiance threshold for photostimulation, aligned with experimental optogenetic activation data. Our analytical solutions are readily adaptable to various biomedical applications, offering a rigorous theoretical foundation for next-generation LED-based bioelectronics, to enable more accurate optical diagnostics and therapies in clinical applications.