Overview of Adaptive and Low-Rank Approximation Algorithms for Modeling of The Influence of Electromagnetic Waves Generated by The Cell Phone Antenna on The Human Head

This paper presents an overview of formulations and algorithms dedicated to modeling the influence of electromagnetic waves on the human head. We start from the three-dimensional MRI scan of the human head. We approximate the MRI scan by the continuous approximation span over three-dimensional h adaptive mesh with quadratic polynomials. Next, we introduce time-harmonic Maxwell equations with a 1.8 GHz cell-phone antenna. We solve the problem of the propagation of electromagnetic waves on the human head. We compute the specific absorption rate used as the heat source for the Pennes bioheat equation. Finally, we introduce the Pennes bio-heat equation modeling the heat generated by the electromagnetic waves propagating through the skull, tissue, and air layers in the human head. We discuss the discretization and time-stepping algorithm for the Pennes equation’s solution over the human head. Namely, we focus on the Crank-Nicolson time integration scheme, to solve the bioheat transfer equations. We employ the hp finite elements with hierarchical shape functions and hp adaptive algorithm in three-dimensions. We propose an adaptive algorithm mixed with time-stepping iterations, where we simultaneously adapt the computational mesh, solve the Maxwell and Pennes equations, and we iterative with time steps. We employ the sparse Gaussian elimination algorithm with low-rank compression of the off-diagonal matrix blocks for the factorization of matrices. We conclude with the statement that 15 minutes of talk with a 1.8 GHz antenna of 1 Wat power results in increased brain tissue temperature up to 38.4 Celsius degree.