Asymptotic solution of electromagnetic heating of skin tissue with lateral heat conduction

Abstract We study the temperature evolution in the three-dimensional skin tissue exposed to an electromagnetic beam of millimeter wavelength. The skin absorption coefficient at the beam frequency determines how deep the electromagnetic energy penetrates into the skin tissue, which gives a sub-millimeter penetration depth for a 94 GHz wave. In contrast, in the lateral directions perpendicular to the depth, the beam size is usually much larger than the penetration depth. Based on this separation of length scales, we establish a formulation in which each term has separable dependences on the depth coordinate and on the lateral coordinates. We solve the equations in the formulation analytically to obtain a two-term asymptotic solution of the temperature distribution in the three-dimensional skin tissue. This closed-form analytical solution provides a practical and accurate way of predicting the temperature. It is useful when the beam size is moderately larger than the penetration depth. In that situation, the effect of lateral heat conduction is not negligible and is well captured in the asymptotic solution.