A hybrid equation for simulation of perfused tissue during thermal treatment

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.