Molina Juan A López, Rivera Maria J, Trujillo Macarena, Berjano Enrique J
Applied Mathematics Department, Instituto de Matemática Pura y Aplicada, Valencia Polytechnic University, Valencia, Spain.
Phys Med Biol. 2008 Mar 7;53(5):1447-62. doi: 10.1088/0031-9155/53/5/018. Epub 2008 Feb 19.
To date, all radiofrequency heating (RFH) theoretical models have employed Fourier's heat transfer equation (FHTE), which assumes infinite thermal energy propagation speed. Although this equation is probably suitable for modeling most RFH techniques, it may not be so for surgical procedures in which very short heating times are employed. In such cases, a non-Fourier model should be considered by using the hyperbolic heat transfer equation (HHTE). Our aim was to compare the temperature profiles obtained from the FHTE and HHTE for RFH modeling. We built a one-dimensional theoretical model based on a spherical electrode totally embedded and in close contact with biological tissue of infinite dimensions. We solved the electrical-thermal coupled problem analytically by including the power source in both equations. A comparison of the analytical solutions from the HHTE and FHTE showed that (1) for short times and locations close to the electrode surface, the HHTE produced temperatures higher than the FHTE, however, this trend became negligible for longer times, when both equations produced similar temperature profiles (HHTE always being higher than FHTE); (2) for points distant from the electrode surface and for very short times, the HHTE temperature was lower than the FHTE, however, after a delay time, this tendency inverted and the HHTE temperature increased to the maximum; (3) from a mathematical point of view, the HHTE solution showed cuspidal-type singularities, which were materialized as a temperature peak traveling through the medium at a finite speed. This peak rose at the electrode surface, and clearly reflected the wave nature of the thermal problem; (4) the differences between the FHTE and HHTE temperature profiles were smaller for the lower values of thermal relaxation time and locations further from the electrode surface.
迄今为止，所有射频加热（RFH）理论模型均采用傅里叶传热方程（FHTE），该方程假定热能传播速度为无限大。尽管此方程可能适用于大多数RFH技术的建模，但对于采用极短加热时间的外科手术而言，可能并非如此。在这种情况下，应考虑使用双曲传热方程（HHTE）的非傅里叶模型。我们的目的是比较从FHTE和HHTE获得的用于RFH建模的温度分布。我们基于一个完全嵌入并与无限大尺寸生物组织紧密接触的球形电极构建了一维理论模型。我们通过在两个方程中都包含电源来解析求解电热耦合问题。HHTE和FHTE的解析解比较表明：（1）在短时间且靠近电极表面的位置，HHTE产生的温度高于FHTE，然而，对于较长时间，这种趋势变得可以忽略不计，此时两个方程产生相似的温度分布（HHTE始终高于FHTE）；（2）对于远离电极表面的点且在极短时间内，HHTE温度低于FHTE，然而，经过延迟时间后，这种趋势反转，HHTE温度升至最高；（3）从数学角度来看，HHTE解显示出尖点型奇点，其表现为一个温度峰值以有限速度穿过介质。这个峰值在电极表面上升，清楚地反映了热问题的波动性质；（4）对于较低的热弛豫时间值和远离电极表面的位置，FHTE和HHTE温度分布之间的差异较小。
