Dutta Jaideep, Kundu Balaram
Department of Mechanical Engineering, Jadavpur University, Raja S. C. Mallick Road, Kolkata 700032, West Bengal, India.
Department of Mechanical Engineering, Jadavpur University, Raja S. C. Mallick Road, Kolkata 700032, West Bengal, India.
J Therm Biol. 2018 Jan;71:41-51. doi: 10.1016/j.jtherbio.2017.10.012. Epub 2017 Nov 2.
This research article determines an exact analytical expression for 2-D thermal field in single layer living tissues under a therapeutic condition by means of Fourier and non-Fourier heat transfer approaches. An actual spatially dependent initial condition has been adopted to analyze the heat propagation in tissues. The exact analytical determination for this actual initial condition for temperature may be difficult. However, in this study, an approximate analytical method has newly been established for an appropriate initial condition. With this initial expression, an exact temperature distribution for 2-D heat conduction in plane co-ordinates has been investigated for the predefined therapeutic boundary condition to have knowledge for practical aspects of the thermal therapy. Laplace Transform Method (LTM) in conjunction with the Inversion Theorem is used for the analytical solution treatment. We have utilized both Pennes' bioheat equation (PBHE) and thermal wave model of bioheat equation (TWMBHE) for the analysis. The influence of thermo-biological behavior on 2-D heat conduction in tissues has been studied with the variation of several dependable parameters in relation to the Hyperthermia treatment protocol in a moderate temperature range (42-45°C). The result in the present study has been evidenced for the biological heat transfer for the enforcement of different circumstances and also has been validated with the published value where the maximum temperature deviation of 2.6% has been recorded. We conclude that the temperature curve for TWMBHE model shows a higher waveform nature for low thermal relaxation time and this wavy nature gradually diminishes with an increase in relaxation time. The maximum peak temperature attains 46.3°C for the relaxation time = 2s and with the increase in the relaxation time the peak temperature gradually falls. The impact of blood perfusion rate on the relaxation time has also been established in this paper.
本研究文章通过傅里叶和非傅里叶热传递方法，确定了治疗条件下单层活体组织中二维热场的精确解析表达式。采用了实际的空间相关初始条件来分析组织中的热传播。对于这种实际的温度初始条件进行精确解析确定可能很困难。然而，在本研究中，针对合适的初始条件新建立了一种近似解析方法。利用这个初始表达式，针对预定义的治疗边界条件，研究了平面坐标中二维热传导的精确温度分布，以了解热疗法实际应用方面的知识。拉普拉斯变换法（LTM）结合反演定理用于解析解处理。我们在分析中同时使用了彭尼斯生物热方程（PBHE）和生物热方程的热波模型（TWMBHE）。在中等温度范围（42 - 45°C）内，通过改变与热疗治疗方案相关的几个可靠参数，研究了热生物学行为对组织中二维热传导的影响。本研究结果证明了在不同情况下生物热传递的情况，并且与已发表的值进行了验证，记录到的最大温度偏差为2.6%。我们得出结论，TWMBHE模型的温度曲线在低热弛豫时间时显示出更高的波形特性，并且随着弛豫时间的增加，这种波动特性逐渐减弱。弛豫时间 = 2s时，最高峰值温度达到46.3°C，并且随着弛豫时间的增加，峰值温度逐渐下降。本文还确定了血液灌注率对弛豫时间的影响。
