A revised approach for an exact analytical solution for thermal response in biological tissues significant in therapeutic treatments

https://doi.org/10.1016/j.jtherbio.2017.03.015Get rights and content

Highlights

  • A correct analysis for temperature response in living tissue.

  • A spatial dependency of initial condition for living tissues is for actual study.

  • Comparison analysis is done between present and published works.

  • Difference in results is dependent on Fourier and non-Fourier approaches.

  • Maximum difference is for non-Fourier heat transfer.

Abstract

The genesis of the present research paper is to develop a revised exact analytical solution of thermal profile of 1-D Pennes’ bioheat equation (PBHE) for living tissues influenced in thermal therapeutic treatments. In order to illustrate the temperature distribution in living tissue both Fourier and non-Fourier model of 1-D PBHE has been solved by ‘Separation of variables’ technique. Till date most of the research works have been carried out with the constant initial steady temperature of tissue which is not at all relevant for the biological body due to its nonhomogeneous living cells. There should be a temperature variation in the body before the therapeutic treatment. Therefore, a coupled heat transfer in skin surface before therapeutic heating must be taken account for establishment of exact temperature propagation. This approach has not yet been considered in any research work. In this work, an initial condition for solving governing differential equation of heat conduction in biological tissues has been represented as a function of spatial coordinate. In a few research work, initial temperature distribution with PBHE has been coupled in such a way that it eliminates metabolic heat generation. The study has been devoted to establish the comparison of thermal profile between present approach and published theoretical approach for particular initial and boundary conditions inflicted in this investigation. It has been studied that maximum temperature difference of existing approach for Fourier temperature distribution is 19.6% while in case of non-Fourier, it is 52.8%. We have validated our present analysis with experimental results and it has been observed that the temperature response based on the spatial dependent variable initial condition matches more accurately than other approaches.

Introduction

The latest amelioration in laser, microwave, ultrasonic technologies have conveyed the phenomena of thermal treatment of diseased organs, injured tissue such as skin burns or skin cancer. The accurate estimation of temperature response within living tissue is very major research interest to provide complete knowledge of design parameters of therapeutic surgical instrument (Alexander and Griffiths, 1993, Field and Bleehen, 1979). For example, the sole objective of hyperthermia is to raise the temperature of diseased tissue up to a standard temperature of 41–56 °C (therapeutic value) by external heat treatment and then thermal destruction takes place without any thermal disturbance of the body (Lagendijk, 2000).

The Pennes’ mathematical model (Pennes, 1948) of bioheat transfer is well established due to its simplicity. But this model is based on Classical Fourier's law of heat conduction which states that thermal wave propagates in any domain with an infinite speed. In actual case due to non-uniform and non-homogeneous inner structure of tissues, the propagation of thermal disturbance between blood and tissues always takes place at finite speed. Cattaneo (1958) and Vernotte (1958) concurrently developed C–V model (alternatively single phase lag model) with modification of classical Fourier's hypothesis:q(r,t+τq)=kT(r,t)

Eq. (1) demonstrates a time lag between application of heat flux vector and establishment of temperature gradient. The time delay (lag) between heat flux and temperature gradient is defined as thermal relaxation time τq which generally (theoretical time scale) ranges from 10−4 to 10−8 s. The conventional classical Fourier's model of heat conduction covers the macroscopic effects of the domain whereas in practical case after implementation of heat flux, the wave propagation requires certain fraction of time to generate energy and to transform to the nearby element. This is microscopic approach of energy transit in non-homogeneous structure (Tzou, 1996). However, Vedavarz et al. (1994) suggested that phase lag of few biological tissues lies in the range of 1–100 s at room temperature. Kaminski (1990) proposed the relaxation time of 25–30 s for meat products approximately by conducting experiment work.

At very earlier stage, on the basis of PBHE, establishment of analytical models of living tissues by considering the energy interaction and conservation between capillaries and blood vessels (Wulff, 1974, Chen et al., 1981, Weinbaum and Jiji, 1985, Song et al., 1987). In 1990s several authors have applied Green's function method to solve 1-D PBHE problems (Durkee et al., 1990, Durkee and Antich, 1991a, Durkee and Antich, 1991b, Vyas and Rustgi, 1992, Gao et al., 1995, Mitra et al., 1995, Weinbaum et al., 1997, Liu and Xu, 1999, Rai and Rai, 1999, Deng and Liu, 2002).

Shih et al. (2007) investigated thermal response of semi-infinite biological tissues with oscillatory boundary conditions in 1-D form of PBHE omitting metabolic heat generation. Tung et al. (2009) proposed 1-D ‘Hyperbolic heat transfer equation’ in skin tissue in comparison with conventional Fourier models. Liu (2008) investigated thermal behavior of living tissue subjected to different heating conditions in 1-D manner by implementing Laplace transform technique. Cotta et al. (2010) suggested ‘Generalized integral transform technique’ for solving 1-D PBHE with linearly variable thermophysical properties of tissue and blood perfusion term in a heterogeneous media. Liu and Chen (2009) presented an elaborated study of mathematical modelling of skin tissue in both Fourier and non-Fourier 1-D models in relation with coupled biological-thermal response under thermal agitation. Gupta et al. (2010) numerically studied heat transfer in biological tissues during thermal therapy by the influence of electromagnetic radiation. Askarizadeh and Ahmadikia (2014) solved 1-D dual phase lag model of bioheat transfer in skin tissue with the help of Laplace transform technique and developed a thorough study to portray the impact of thermal damage. Liu and Chen (2015) illustrated thermal analysis of highly absorbed laser irradiation in living tissues in 1-D dual phase lag form by a hybrid implementation of Laplace transform and discretization method. Kumar et al., 2016a, Kumar et al., 2016b demonstrated temperature distribution on the basis of Laplace transform in different tissues such as muscle, tumor, fat and dermis. A non-linear dual-phase-lag (DPL) bio-heat transfer model based on temperature dependent metabolic heat generation rate is established by Kumar et al., 2016a, Kumar et al., 2016b to analyze the heat transfer phenomena in biological tissues during thermal ablation treatment. Lin and Li (2016) portrayed analytical solution of Pennes', C-V model and Dual phase lag model with the help of ‘Shifting of variables’ method. Kumar et al. (2015) developed numerical solution of 1-D DPL model of bioheat equation by considering initial condition as arterial temperature of the tissue. Recently Kundu (2016) explored 1-D Fourier and non-Fourier model of biological heat transfer for different surface conditions.

From the exclusive literature survey as mentioned above we have attempted to figure out the best possible research work reported in different reputed international journals. We have found a major drawback behind the selection of initial condition for development of analytical solution of heat conduction in living tissue.

As per our vision on the basis of literature review we figured out that:

  • a.

    The research papers (Song et al., 1987, Durkee et al., 1990, Durkee and Antich, 1991a, Durkee and Antich, 1991b, Vyas and Rustgi, 1992, Gao et al., 1995, Mitra et al., 1995, Weinbaum et al., 1997, Liu and Xu, 1999, Rai and Rai, 1999, Deng and Liu, 2002, Shih et al., 2007, Tung et al., 2009, Liu, 2008 Cotta et al., 2010; Xu et al., 2009; Gupta et al., 2010; Askarizadeh and Ahmadikia, 2014; Liu and Chen, 2015; Kumar et al., 2016a, 2016b; Lin and Li, 2016; Alkhwaji et al., 2012; Liu et al., 1999; Kumar et al., 2015) deals with the establishment of analytical mathematical model of biological heat transfer have considered constant initial temperature Ti(S,0) for the solution. But unlike other engineering problem initial temperature of living tissue can’t be assumed as constant. As skin is exposed to environment, temperature gradient always changes due to coupled (conductive-convective) heat transfer. Though the research work (Alkhwaji et al., 2012) depicted the initial condition of skin tissue has not been considered as constant but metabolic heat generation has been omitted. The initial steady state temperature distribution of the skin tissue is obviously independent on time but always dependent on spatial coordinate. The impact of heat transfer coefficient in surrounding fluid near the skin surface should not be negligible while considering steady state temperature distribution. Thus it signifies a serious setback for development of exact temperature profile in living tissue under therapeutic heat treatment operation.

  • b.

    Deng and Liu (2002) first proposed spatial boundary condition while the inner core temperature of the tissue has kept constant. From mathematical point of view temperature of the inner core can be considered as constant for special case. But from biological aspect due to change in blood flow and numerous microscopic energy exchange in non-homogeneous tissue structures, constant temperature is mere an impossible phenomena.

  • c.

    The research paper published by Deng and Liu (2002), Liu et al. (1999), Liu (2008) and Kundu (2016) have solved Fourier and non-Fourier models of living tissue with different analytical methods. But the way initial steady state temperature distribution represented eliminates the metabolic heat generation permanently. Basically metabolic heat generation is considered as constant for a particular body. But eliminating this term completely from the basic governing equation may lead to development of incorrect thermal profile and it will surely affect the design variables of therapeutic instruments.

On the basis of research gap as mentioned above present work has been motivated to investigate the thermal profile of 1-D PBHE model. Both Classical Fourier and non-Fourier heat conduction has been considered. ‘Separation of Variables’ technique has been employed to construct the analytical temperature field for a predefined surface condition. The inner core conditions of the tissue have been satisfied with either maximum or minimum temperature. The initial condition in present research work has been reformed and to find out steady state temperature distribution we have implemented conductive-convective (coupled) boundary condition to justify the practical approach of skin heat transfer. A comparison analysis of tissue temperature of both Fourier and non-Fourier models has been established to display as a function of different non-dimensional parameters. Also the temperature difference between published theoretical approach (Deng and Liu, 2002, Liu et al., 1999, Liu, 2008, Kundu, 2016) and present approach has been established and justified with biomedical aspect of therapeutic treatment for same initial and boundary conditions imposed on skin tissue.

Section snippets

Mathematical formulation and physical modelling

In mathematical physics Fourier's law of heat conduction portrays one of the best understood logical models and it also possesses a number of postulates (Fourier, 1878). The problems dealt with transient heat transfer for extremely short period of time, highly induced heat flux, non-homogeneous medium, Fourier's law literally fails to deliver the accurate temperature field for instant.

Results and discussion

Uttermost biological materials consist of non-homogeneous cells (e.g. honeycomb pattern) soft tissues, capillary tubular structures (venule-arteriole) with variable cross-sections ranging from 55000μm diameter (Jiji, 2009). A very essential observation from literature survey which depicts that for human skin tissue no information regarding thermal relaxation time τ is available till now. But during therapeutic heat treatment, knowledge about this ‘characteristic time’ is indispensable to

Conclusions

From the detailed research work epitomized in this article, the concluding remarks can be summarized as follows:

  • a.

    A refined exact analytical approach of thermal response in 1-D PBHE model in both Fourier and non-Fourier form has been proposed in this research work with inclusion of metabolic heat generation.

  • b.

    The existing theoretical approach has been highlighted and we have suggested our mathematical approach with proper initial and boundary conditions on the basis of practical aspects (real-time

Jaideep Dutta was born in Kharagpur, West Bengal, India on 12th December 1990. He received B.Tech. in Mechanical Engineering from Dr. B.C. Roy Engineering College, Durgapur, India in July 2012; M.Tech. (Half Time Teaching Assistant) in Thermal Engineering specialization from National Institute of Technology Karnataka, Surathkal, India in May 2015. Currently he is pursuing Ph.D. in Heat Power Engineering at Jadavpur University, India. He was Assistant Professor in Manipal University Jaipur for

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    Jaideep Dutta was born in Kharagpur, West Bengal, India on 12th December 1990. He received B.Tech. in Mechanical Engineering from Dr. B.C. Roy Engineering College, Durgapur, India in July 2012; M.Tech. (Half Time Teaching Assistant) in Thermal Engineering specialization from National Institute of Technology Karnataka, Surathkal, India in May 2015. Currently he is pursuing Ph.D. in Heat Power Engineering at Jadavpur University, India. He was Assistant Professor in Manipal University Jaipur for one year. His present research interests include: Biological heat transfer in living tissues influenced in thermal therapies, Analytical methods of heat and mass transfer and heat transfer in moving point heat source.

    Dr Balaram Kundu is an Associate Professor in the Dept. of Mechanical Engineering at Jadavpur University. He received his Ph.D. Degree in Mechanical Engineering from IIT Kharagpur in 2000. He worked as a Research Professor in the School of Mechanical Engineering at Hanyang University, Seoul, South Korea for two years and nine months. He graduated in Mechanical Engineering from Regional Engineering College Durgapur (Presently NIT Durgapur) in 1993 and post graduated in Thermal Engineering from Bengal Engineering College (Deemed University) (at present IIESTS) in 1995. His Area of Expertise includes: Analytical Heat Transfer, Solar Collector, Thermal Insulation, Extended Surface Heat Transfer, Fin-and-Tube Heat Exchanger, Biological Heat Transfer, etc. He is the author of more than eighty technical papers published in reputed international journals. He is recipient of: Institutional Medal for the best paper published in the Journal of the Institution of Engineers (India), Outstanding Faculty Award – the VIFFA 2015, CEE Teacher's Excellence Award 2015, etc. He is Editorial Board Member of more than thirteen international journals, viz., Journal of Thermal Engineering, American Journal of Heat and Mass Transfer, Thermal and Mass Transport, Journal of Mechanical Engineering and Biomechanics, American Journal of Scientific Research and Essays, International Journal of Thermal Energy and Applications, Scientific Research and Essays, etc.

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