Effects of Memory Dependent Derivative of Bio-heat Model in Skin Tissue exposed to Laser Radiation

INTRODUCTION: Thermal processes are the essence of living organisms and are necessary for understanding life. The study of Transfer of heat in tissues is known as Bioheat transfer. Many techniques are developed for the thermal treatment of skin and other disease such as skin cancer, skin burns and injured skin tissue with laser. The tissue is inhomogeneous and at times anisotropic with complex thermal properties. Moreover, there may be skin tissue damage when irradiated with a laser beam. OBJECTIVES: In this research a novel one-dimensional (1-D) bioheat model has been used with memory-dependent derivative (MDD) in Pennes’ bioheat transfer equation due to laser radiation and the thermal damage in tissue caused due to laser heating has been examined. METHODS: Bioheat transfer model has been used with memory-dependent derivative (MDD) in Pennes’ bioheat transfer equation. The problem is solved using Laplace transform technique. RESULTS: The temperature and thermal damage in the skin exposed to heating with laser radiation is calculated and obtained in physical form. The thermal reaction of skin tissues during laser radiation is studied under memory-dependent derivative (MDD) in Pennes’ bioheat transfer equation. CONCLUSION: Analyzed a novel bioheat mathematical model on the basis of MDD involving time-delay parameter χχ for the Pennes’ bioheat transfer equation and applied to examine the thermal properties of the skin tissue for burns caused due to laser radiations. The thermal damages can be measured in a better way with the MDD model. The blood perfusion prevent the tissue damage by developing the cooling function. Effect of memory dependent derivatives and time delay parameter are represented graphically and analysed.


Introduction
Thermal processes are the essence of living organisms and are necessary for understanding life. Transfer of heat in tissues is complicated as it is affected by the flow of blood and asymmetrical structure of vascular tissues. The tissue is inhomogeneous and at times anisotropic with complex thermal properties. It also produces heat as part of the active metabolism. Maintenance of a fairly constant body temperature in a range of thermal environments implies that there is a continuous exchange of energy between deep, surface tissues and the environment.  [24], Marin [25,26], Lata and Kaur [27][28][29][30], Riazet al. [31]worked on different theory of thermoelasticity.
In this research a novel one-dimensional (1-D) bioheat model has been used with memory-dependent derivative (MDD) in Pennes' bioheat transfer equation due to laser radiation and the thermal damage in tissue caused due to laser heating has been examined. Laplace transform technique is used to solve the given problem. The temperature and thermal damage in the skin exposed to heating with laser radiation is calculated and obtained in physical form. Effect of memory dependent derivatives and time delay parameter are represented graphically and analyzed.

Basic Equations
For the differentiable function f(t), Wang and Li [32] introduced the first-order MDD with respect to the time delay > 0 for a fixed time t: The K(t−ξ) and the time delay parameter depends on the material properties. The kernel function K(t − ξ) is differentiable with respect to the variables t and ξ. Following Ezzat et al. [33][34][35] the kernel function K(t − ξ) is taken here in the form where a and b are constants. Following Sarkar [14] and Hobiny et al. [16]the memory dependent bio heat transfer equation with instant surface heating due to laser irradiation is given by EAI Endorsed Transactions on Pervasive Health and Technology 01 2020 -05 2020 | Volume 6 | Issue 22 | e2 Where, (6) Following Jacques [36] 1 , 2 , 1 and 2 are given by

Method and solution of the problem
The temperature distribution in a semi-infinite biological tissue with instantaneous surface heating and with the laser thermal source of 1-D model of Memory-Dependent Derivative Pennes bioheat transfer equation (MDDPBE) in a finite medium is considered. The 1-D form of Eq.
(3) by taking as constant, is written as: The initial conditions are: The skin tissue is exposed to instantaneous surface heating. We consider that heat flux →0 deep inside the tissue. Therefore, applicable boundary conditions are: For simplify the solution, following non-dimensional quantities are given by Using these non-dimensional quantities defined in (15), the governing Equation (12) and initial and boundary conditions (13) and (14) can be written as (by ignoring dashes) Laplace transforms is given by (1 − e −sχ ) � 2 χ 2 s 2 � + �1 − 2 χs � , = 1, = 1. (20) where Thus we get where = .
which can be further simplified as Where EAI Endorsed Transactions on Pervasive Health and Technology 01 2020 -05 2020 | Volume 6 | Issue 22 | e2 And the boundary conditions after the application of Laplace transform (17) takes the form By using the boundary conditions defined in Eq. (29) in Eq. (24), the exact solution is obtained as: Where The thermal damage i.e. evaluation of burn caused by laser radiation, following Jasiński [37], Askarizadeh & Ahmadikia [38] is given by, The results are simulated using MATLAB software and illustrated graphically. The impact of laser source on the skin surface was incorporated. The proposed mathematical models depends on the bio-heat transfer found and suitable boundary conditions. The conducting heat source, metabolic and perfusions are used in the formulations of the mathematical model. Numerical results are presented graphically in Figures 1-4 to study the influence of memory dependent derivative, kernel function K(t−ξ) , the laser exposure time , the thermal relaxation time 0 and the time delay parameter on the temperature and the thermal damages. The skin tissue is considered as .03 m thick and the reference temperature is taken equal to skin normal temperature, that is, 0 = = 37 0 . Figure 1 exhibits the deviation of temperature along the distance x by keeping the values of 0 = 5 = 10 with different values of memory dependent derivative, kernel function K(t−ξ). It is seen that the temperature start rising with the distance and decrease as the blood perfusions in skin increases.   It is seen that the temperature start from the utmost value and decrease rapidly.

Results and discussion
The main objective of this research work is to analyze a novel bioheat mathematical model on the basis of MDD involving time-delay parameter for the Pennes' bioheat transfer equation and applied to examine the thermal properties of the skin tissue for burns caused due to laser radiations. The thermal damages can be measured in a better way with the MDD model. The blood perfusion prevent the tissue damage by developing the cooling function. In this research, the memory-dependent derivative involving time-delay parameter becomes a new measure of efficiency for bioheat transfer in the skin tissues. These results may be beneficial in the study and further improvements in the applications of thermotherapy in skin tissues.