The success of laser hyperthermia techniques depends on precise prediction and control of temperature in the tissue. The study of time-domain analytical solutions can not only verify the results of the numerical models but also contribute to the implementation of technical solutions with few technical errors. Because the excitation time of a laser source is very short compared to tissue equilibrium time, non-Fourier models have become important in theoretical research. Currently, constant heat flux is regarded as a time-dependent physical quantity, which is inconsistent with experiment results in the literatures. In this study, based on modified constant heat flux, a corrected non-Fourier boundary condition is established, and the analytical solution is obtained by integration transformation. The temperature rise and distribution curves obtained using a modified model are consistent with the experimental results in the literatures. In addition, the time-independent non-Fourier boundary condition is inconsistent with the thermal equilibrium. Furthermore, the different heat transfer mechanisms of the single-phase lag model (SPLM) and double-phase lag model (DPLM) under modified non-Fourier boundary conditions are analyzed, and their differences from those of existing models are discussed. The result shows that the time factor should be considered in constant-heat-flux models of biological tissues, otherwise the predicted results will be inconsistent with the experiment results.
Based on the delayed non-Fourier law, which includes non-Fourier single- and two-phase lag equations, non-Fourier heat conduction equations of biological tissue in one-dimensional space were established, including an SPLM, a DPLM, and the classical Peens biological model (PBM). Considering the strong scattering biological surface, the time-dependent non-Fourier boundary condition of constant-heat-flux irradiation was established, and the theoretical solution was obtained via integration transformation under a quasi-static initial condition. Also, a theoretical solution of the time-independent constant-heat-flux irradiation problem was obtained.
Based on the obtained analytical solutions, the heat transfer mechanism among PBM, SPLM, and DPLM are discussed and compared to some results in the literatures. The obtained results are as follows:
1) When constant-heat-flux is treated as time-independent, the temperature distribution at any time does not obey the law of energy conservation, and among PBM, SPLM, and DPLM, the temperature distribution predicted by PBM is the highest at any time, whereas that of SPLM is the lowest. Also, the temperature variation of SPLM has no jump in wavefront at any position [Fig. 2(a)].
2) The temperature distribution of a closed solution with a corrected boundary condition at any time obeys the law of energy conservation, and among PBM, SPLM, and DPLM, the temperature distribution predicted by SPLM at any time is the highest, whereas that of PBM is the lowest, and that of DPLM prediction is between them near the surface. It is worth noticing that only SPLM predicts a sudden rise in wavefront, while DPLM and PBM do not. [Fig. 2(b)].
3) Under the corrected non-Fourier boundary condition, SPLM predicts a rapid jump in the temperature change at all positions, which is consistent with the experimental results in the literatures (Fig. 3). Under the wrong non-Fourier boundary condition of time-independent heat flux, PBM predicts a faster change in temperature than SPLM and DPLM, whereas the temperature is the highest, which is in contrast to the experimental results in the literatures.
4) Space-time temperature fields were compared. The corrected wavefront of SPLM is like a vertical cliff [Fig. 5(a)], which differs from the existing wavefront [Fig. 7(b)]. Besides, due to the wrong boundary conditions, the corresponding temperature predicted by PBM (Fig. 4) is higher than that of DPLM and SPLM (Fig. 7).
5) Under the corrected non-Fourier boundary condition, the bigger the heat-flux lagging time τq,the lower the thermal velocity, and the higher the rising amplitude when the lagging time of the temperature gradient is fixed (Figs. 8 and 9). At a constant lagging time of heat flux τq,with an increase in the temperature gradient time τT,the temperature predicted by DPLM decreases at all positions (Fig. 11), and the lower temperature at any time less than that of the SPLM near the surface (Fig. 10). These results are consistent with the experimental results in the literatures, but the existing constant-heat-flux boundary conditions are very different, and many conclusions are opposite.
Based on the results, we conclude that:
1) When constant-heat-flux is regarded as time-independent, the temperature distribution at any time is inconsistent with the law of energy conservation. Thus, time-independent boundary conditions cannot satisfy the heat balance equation on the boundary.
2) With a corrected boundary condition, the temperature distribution at any time obeys the law of energy conservation. Thus, the corrected boundary condition is energy conservation and satisfies the non-Fourier biological heat transfer equation.
