Wren J, Karlsson M, Loyd D
Department of Mechanical Engineering, Linköping University, Sweden.
Int J Hyperthermia. 2001 Nov-Dec;17(6):483-98. doi: 10.1080/02656730110081794.
Bio-heat equations (BHEs) are necessary for predicting tissue temperature during thermal treatment. For some applications, however, existing BHEs describe the convective heat transfer by the blood perfusion in an unsatisfactory way. The two most frequently used equations, the BHE of Pennes and the k(eff) equation, use for instance either a heat sink or an increased thermal conductivity in order to account for the blood perfusion. Both these methods introduce modelling inaccuracies when applied to an ordinary tissue continuum with a variety of vessel sizes. In this study, a hybrid equation that includes both an increased thermal conductivity and a heat sink is proposed. The equation relies on the different thermal characteristics associated with small, intermediate and large sized vessels together with the possibilities of modelling these vessels using an effective thermal conductivity in combination with a heat sink. The relative importance of these two terms is accounted for by a coefficient beta. For beta = 0 and beta = 1, the hybrid equation coincides with the BHE of Pennes and the k(eff) equation, respectively. The hybrid equation is used here in order to simulate temperature fields for two tissue models. The temperature field is greatly affected by beta, and the effect is dependent on, e.g. the boundary conditions and the power supply. Since the BHE of Pennes and the k(eff) equation are included in the hybrid equation, this equation can also be useful for evaluation of the included equations. Both these heat transfer modes are included in the proposed equation, which enables implementation in standard thermal simulation programmes.
生物热方程（BHEs）对于预测热处理过程中的组织温度是必不可少的。然而，对于某些应用，现有的生物热方程对血液灌注引起的对流热传递的描述并不令人满意。例如，最常用的两个方程，彭尼斯生物热方程和有效热导率方程（k(eff)方程），要么使用热沉，要么使用增加的热导率来考虑血液灌注。当应用于具有各种血管大小的普通组织连续体时，这两种方法都会引入建模误差。在本研究中，提出了一个既包含增加的热导率又包含热沉的混合方程。该方程基于与小、中、大尺寸血管相关的不同热特性，以及使用有效热导率结合热沉对这些血管进行建模的可能性。这两个项的相对重要性由系数β来体现。当β = 0和β = 1时，混合方程分别与彭尼斯生物热方程和有效热导率方程一致。这里使用混合方程来模拟两种组织模型的温度场。温度场受β的影响很大，并且这种影响取决于例如边界条件和电源。由于彭尼斯生物热方程和有效热导率方程都包含在混合方程中，所以该方程也可用于评估所包含的方程。所提出的方程包含了这两种热传递模式，这使得它能够在标准热模拟程序中实现。
