A Python FDTD method algorithm for 1D planar acoustic wave propagation: Simulating high-frequency ultrasound in the brain and beyond

Abstract

Non-invasive techniques, such as high-frequency ultrasound, have emerged as promising therapeutic tools for neurological disorders, including Parkin-son’s disease and Alzheimer’s disease. By targeting specific brain regions, ultrasound stimulation modulates neural activity and induces beneficial physiological responses. However, simulating high-frequency acoustic wave propagation in biological tissues presents computational challenges due to the high spatial and temporal resolution required to satisfy the low Courant-Friedrichs-Lewy (CFL) condition for numerical stability and accuracy. This paper introduces a novel Python algorithm optimized for planar wave propa-gation, enabling efficient one-dimensional simulations of high-frequency ul-trasound. Utilizing the finite difference time domain (FDTD) method, the al-gorithm incorporates material-specific properties, including density, sound speed, and frequency-dependent attenuation, to model heterogeneous tissue structures such as skin, bone, cerebrospinal fluid, and brain tissue. The method accurately captures key acoustic phenomena, such as impedance mismatching and wave reflection, facilitating detailed analysis of energy transmission and absorption in complex biological interfaces. The algo-rithm’s performance is compared with COMSOL Multiphysics, which is in-herently limited to two and three-dimensional acoustic wave propagation. By reducing the problem to one dimension, the proposed method simplifies computational complexity while preserving key wave interactions, enabling early-stage analysis with lower computational costs. Beyond biomedical ap-plications, this approach is broadly applicable to any system governed by acoustic wave equations. By significantly reducing computational demands, it accelerates preliminary studies of wave propagation through multilayered media, contributing to the development of efficient ultrasound-based thera-peutic models and advancing acoustic research.