Finite element techniques for three-dimensional specific absorption rate (SAR) computation in anatomically based human models are presented. The formulations center on Helmholtz weak forms which have been shown to be numerically robust and to afford additional sparsity in the resulting system of algebraic equations. Practical solution of these equations depends critically on the realization of an effective sparse matrix solver. Experience with several conjugate gradient-type methods is reported. The findings show that convergence rate (and even convergence in some cases) degrades significantly with increasing matrix rank and decreasing electrical loss for mesh spacings which adequately resolve the physical wavelengths of the electromagnetic wave propagation. However, with proper choice of algorithm and preconditioning, reliable convergence has been achieved for matrix ranks exceeding 2 x 10(5) on domains having sizeable volumes of electrically lossless regions. An automatic grid generation scheme for constructing meshes which consist of variable element sizes that conform to a predefined set of boundaries is discussed. Example meshes of homogeneous and heterogeneous human anatomies, the boundaries of which have been derived from CT-scan information, are shown. These results highlight the fact that 3D finite element mesh generation remains a difficult problem, but usable meshes with this level of complexity can be generated. Integration of the finite element formulation, the sparse matrix solver, and the mesh generation scheme is shown to lead to algorithms that can be implemented on inexpensive reduced instruction set computer (RISC) workstations with run times on the order of hours. An example of hyperthermia device simulation is presented which suggests that the finite element method is a practical alternative that rivals the impressive finite-difference time-domain (FDTD) computations that have appeared.

An initial series of comparisons are made between finite element computations and laboratory measurements obtained during heterogeneous phantom heating with the Sigma 60 applicator. The phantom is a relatively complex, though still idealized, rendering of the pelvic area which has been used to study the deep heating characteristics of the Sigma 60 in this anatomy. Direct electric field measurements as well as inferred SAR through transient temperature analysis are plotted against computed results along 11 one-dimensional tracks through the phantom. Quantitative comparisons provided through the track-by-track analysis show generally good agreement between computation and measurement. The finite element method is found to predict well the jumps in the electric field when polarized perpendicularly to a muscle/fat interface. Visualizations of the complete three-dimensional distributions are also highlighted and correlate well with physical reasoning about the expected behaviour of the fields produced. Some discrepancies in the data persist and are discussed and analysed in depth. They underscore the difficulties that can arise in performing comparisons between measured and computed results and stress the need for careful and thorough investigations when attempting these types of model validation studies.

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