Study On The Influence Of Parameters In Time-fractional Cattaneo Heat Transfer Model On The Phenomenon Of Bio-heat Transfer

In recent years,the theory of fractional calculus have been widely used in vari-ous fields of science and engineering because of its non-local and historical memory characteristics,which provide a favorable tool for the description of many complex me-chanical and physical processes such as anomalous diffusion and constitutive relations of complex viscoelastic materials.Due to the complexity of biological tissue,the high energy and short-time thermal ablation process in the thermotherapy no longer meets the Fourier heat transfer law.The non-Fourier effect of the biological heat transfer is widely concerned.It is of great significance and prospect to study the mechanism and process of the biological heat transfer and establish the corresponding heat transfer mechanism and mathematical model for the development of the thermotherapy.Firstly,one-dimensional time-fractional Cattaneo heat transfer model(TFHE)is established for the heating process of tissue in vitro by using the fractional calculus theory.A convergent and absolutely stable finite difference scheme is established to solve the model numerically.At the same time,the classical one-dimensional Pennes model(Pennes)and Cattaneo-vernote model(CV)are solved by using finite difference method,and the difference schemes are convergent and absolutely stable.The heating process with constant temperature heat source and classical electromagnetic heat source was simulated by TFHE,Pennes and CV equations.The effects of the equation param-eters(relaxation time?_q,fractional order?)and energy parameters on the temperature of TFHE is analyzed and compared with Pennes and CV model.The results are as follows:1.When the relaxation time?_q>0,the temperature response is more delayed than Pennes,and with the increase of?_q,the delay is more obvious.In the heating stage,heating speed is more slower than Pennes,so the temperature of tissue is lower.2.With the increase of parameter fractional order?,the heating rate is higher,and then the internal temperature is higher at the same time.3.When the parameter?_q=0.01 s,?=1.99,the temperature curves simulated by TFHE coincide with those simulated by Pennes equation and CV equation respectively,So we can simulate more temperature changes by changing the parameters of TFHE.It is a generalized heat transfer model,which can be widely used in complex biological heat transfer process.4.When the pa-rameter?_q>0 and 1<?<2,the numerical results of two different heating processes show consistently.The temperature rise rate:Pennes>CV>TFHE,the tissue in-ternal temperature:Pennes>CV>TFHE,and the temperature differences simulated by the three models are greatly increased with the increase of heat source energy or the decrease of heating time.Secondly,in order to simulate the heating process of renal sympathetic dener-vation(RSD)in clinic,a two-dimensional time-fractional Cattaneo heat transfer model(TFHE)and two-dimensional Pennes model are established,in which the classical elec-tromagnetic heat source is used instead of the radio-frequency heat source and the blood flow heat transfer is considered.By using the alternating direction implicit method(A-DI),the difference schemes of two models which are convergent and absolutely stable are constructed for numerical solution.ADI method greatly simplifies the calcula-tion and storage.In the part of numerical simulation,different equation parameters(relaxation time?_q,fractional order?),energy parameters(energy flow density p₀,attenuation coefficient?)and blood flow velocity v_bare set.By analyzing the temper-ature curves at 0.4 cm from the heat source and the tissue temperature distribution at60 s under different parameter conditions,the influence of different parameters on the heat transfer phenomenon simulated by TFHE is obtained.The numerical results are compared with the temperature field simulated by the Pennes model.The numerical results show that:1.The relaxation time?_qdetermines the delay time of the tempera-ture response.The temperature rise curve and temperature distribution of TFHE with?_q=0.01 s are the same as those of the Pennes.Compared the TFHE simulation results of?_q=6 s and 15 s,the larger the?_qis,the slower the temperature rise speed is at 0.4cm,the time to reach the highest temperature is later,and the transverse diameter,lon-gitudinal diameter and area of the effective ablation area at 60 s are smaller.2.When the fractional order of the parameter?=1.5,1.7,1.99,the temperature change speed at 0.4 cm is greater with the increase of?,the effective ablation area increases at 60 s,and the increase of the transverse diameter is smaller than that of the longitudinal diam-eter.3.When the energy flow density P₀=12000 W/m³,compared with the numerical results of 8000 w/m³,the heating rate at 0.4 cm is faster,the tissue center temperature is higher and the effective ablation area is larger at 60 s.4.The size of parameter attenuation coefficient?determines the speed of energy attenuation,which is the main factor affecting the shape of effective ablation area.When?=300 m^-1,the heating rate at 0.4 cm is faster than that at?=200 m^-1,the central temperature is higher at60 s,and the aspect ratio(ratio of longitudinal diameter to transverse diameter)of the effective ablation area is smaller.5.The blood flow velocity v_bdetermines the size of blood flow heat transfer.Compared the numerical results of v_b=0 m/s,0.4 m/s and0.6 m/s,with the increase of flow rate,the temperature rise rate at 0.4 cm decreased,the central temperature of tissue decreased at 60 s,and the transverse,longitudinal and area of effective ablation area all decreased.In this study,we use the time-fractional Cattaneo heat transfer equation to simulate the one-dimensional and two-dimensional heat transfer process of biological tissues.The effects of various parameters on the heat transfer phenomenon simulated by time-fractional Cattaneo heat transfer model are preliminarily obtained.The advantages of TFHE compared with classical Pennes equation and CV equation in biological heat transfer are analyzed.It provides a reference for further application of time-fractional Cattaneo heat transfer equation in biological heat transfer simulation.