Thermoacoustic Wave Propagation With Heat Diffusion In Biomedical Imaging By Using Lie Symmetry Theory

Abstract

Thermoacoustic imaging (TAI) is a hybrid imaging modality that integrates electromagnetic excitation with acoustic wave detection to achieve high-resolution, non-invasive imaging of biological tissues. The governing equations, namely the heat equation for temperature diffusion and the wave equation for acoustic propagation, are coupled partial differential equations (PDEs) with spatially varying coefficients in heterogeneous tissues. This paper employs Lie symmetry theory to analyse these PDEs, uncovering their symmetry groups, invariant solutions, and similarity reductions. We derive the Lie point symmetries for both the heat and wave equations in a two-dimensional heterogeneous medium, reducing them to simpler forms applicable to TAI. Five detailed examples—Three for the wave equation and two for the heat equation—illustrate the application of these solutions to model thermoacoustic wave propagation and heat diffusion. The results enhance the mathematical modelling of TAI, improve computational efficiency in image reconstruction, and offer insights into optimizing imaging algorithms for clinical applications such as cancer detection. This work underscores the potential of Lie symmetry theory in advancing biomedical imaging technologies.