Explicit formula of finite difference method to estimate human peripheral tissue temperatures during exposure to severe cold stress

doi:10.1016/j.jtherbio.2014.12.010

Journal of Thermal Biology

Volume 48, February 2015, Pages 51-55

https://doi.org/10.1016/j.jtherbio.2014.12.010 Get rights and content

Highlights

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A model based on bio-heat equation has been formulated for the estimation of temperature distribution in the dermal regions of human body.
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The explicit formula of FDM has been used to solve the model for reasonable and realistic results.
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Conditions for the prediction of frostbite and other cold injuries at the human peripheral tissues were discussed in the model.
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The study may be helpful to understand the thermal behaviour of tissues in laser therapies and other medical science problems.

Abstract

During cold exposure, peripheral tissues undergo vasoconstriction to minimize heat loss to preserve the maintenance of a normal core temperature. However, vasoconstricted tissues exposed to cold temperatures are susceptible to freezing and frostbite-related tissue damage. Therefore, it is imperative to establish a mathematical model for the estimation of tissue necrosis due to cold stress. To this end, an explicit formula of finite difference method has been used to obtain the solution of Pennes' bio-heat equation with appropriate boundary conditions to estimate the temperature profiles of dermal and subdermal layers when exposed to severe cold temperatures. The discrete values of nodal temperature were calculated at the interfaces of skin and subcutaneous tissues with respect to the atmospheric temperatures of 25 °C, 20 °C, 15 °C, 5 °C, −5 °C and −10 °C. The results obtained were used to identify the scenarios under which various degrees of frostbite occur on the surface of skin as well as the dermal and subdermal areas. The explicit formula of finite difference method proposed in this model provides more accurate predictions as compared to other numerical methods. This model of predicting tissue temperatures provides researchers with a more accurate prediction of peripheral tissue temperature and, hence, the susceptibility to frostbite during severe cold exposure.

Introduction

Skin consists of several layers and plays an active role in maintaining the thermal homeostasis of the body by many necessary structural and functional tissue responses, mainly by regulating the exchange of thermal energy to the environment (Khanday, 2013). The tissue necrosis at the skin and deep tissues occur depending upon the severity of ambient temperature. To visualize the structure and network of dermal and subdermal layers, the schematic diagram is given in Fig. 1 (Khanday, 2013). Several computer-simulated models of temperature distribution in human dermal system have been developed, which provide important insight into the concerned problem. Patterson (1978) has experimentally determined temperature profiles of the outer skin using radio-camera. Chao et al. (1973) and Cooper and Trezek (1972) considered two simple models and obtained temperature distribution curve in skin and subcutaneous tissue (SST) for certain fixed values of parameters. Agarwal et al. (2010) used finite element method to study the heat flow in dermal regions of elliptical and tapered shape human limbs. Khanday and Saxena (2009b) worked on the study of thermoregulation in human head at cold environmental conditions using variational finite element method. In addition, Khanday and Saxena (2009c) used an advanced model for the estimation of cold effect on human dermal parts in which one dimensional steady state case has been described over five layered skin and subcutaneous tissues. Recently, Khanday et al. (2014) formulated a mathematical model in which numerical estimation of the fluid distribution pattern in human dermal regions with heterogeneous metabolic fluid generation has been worked out. The significant work in this direction has been done by Diller and Hayes (1983), they formulated a mathematical model using finite element method to study the process of burn injury in blood-perfused skin. Ng and Chau (2000) have demonstrated their study on mesh independent prediction of skin burns injury. Stoll (1969) conducted detailed experiments on the interaction of burn time and temperature to cause specified injury levels in a human skin model. Henriques and Moritz (1947) devised a damage function which has subsequently been used quite widely. With this injury function, the cumulative damage incurred during a burn can be predicted.

No doubt a good number of researchers have estimated the body tissue temperature at different atmospheric conditions. But so far, no discrete values of temperature profiles were computed at the interfaces of the layered skin and subcutaneous tissues with respect to various adverse ambient conditions. The discrete values of temperature profiles at nodal points may be helpful in medical sciences and tissue engineering while monitoring the temperature fluctuations of tissues in plastic surgeries and other laser therapies. The skin grafting and other dermal treatments can be performed on the basis of the nodal tissue temperatures computed in the model.

The purpose of this study was to estimate the temperature profiles of human skin with a bio-heat transfer equation using a three layer model. Explicit formula of finite difference method has been developed to predict the tissue temperature at dermal and subdermal layers. The conditions under which hypothermia and various degrees of frostbite that occur to the biological tissues have been discussed.

Section snippets

Mathematical formulation of the model

The mathematical model for heat flow in biological tissues for the one dimensional case, initially proposed by Pennes' (1948), is given by $ρ c \frac{\partial T}{\partial t} = K \frac{\partial^{2} T}{\partial x^{2}} + ρ_{b} m_{b} c_{b} (T - T_{A}) + S$ where ρ, c, K and S respectively denote the density, specific heat, thermal conductivity and rate of metabolic heat generation of tissues. The other parameters m_b, c_b and T_A represent the blood mass flow rate, specific heat of the blood and arterial blood temperature respectively.

The outer surface of the skin is exposed to the

Solution of the model

The bio-heat equation has been solved by various methods including finite difference method, finite element method, boundary element method, but the explicit formula of finite difference method has not been used so far for the solution purpose. Also, in the above-mentioned methods, most of the authors have taken the values of physiological parameters only as position dependent or constants. The use of explicit formula of finite difference method and the heterogeneity of various parameters give

Discussion and conclusion

A mathematical model based on bio-heat equation has been established and solved with the help of explicit formula of finite difference method. The disturbance to the human thermoregulatory system with respect to various adverse cold conditions has been studied. The domain of skin and subcutaneous layers was discretized on the basis of their physiological and biophysical properties. Moreover, the nodal temperature at the interfaces of the skin and subcutaneous layers has been estimated with

Acknowledgement

The authors are highly thankful to the SERB-DST, New Delhi, Govt. of India (DST-SERB/F/3582/2013-14) for their financial support to carry this research work. Also we are indebted to the Co-Editor of this journal for his valuable and fruitful suggestions for the improvement of the paper.

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View full text

Explicit formula of finite difference method to estimate human peripheral tissue temperatures during exposure to severe cold stress

Highlights

Abstract

Introduction

Section snippets

Mathematical formulation of the model

Solution of the model

Discussion and conclusion

Acknowledgement

Bull. Math. Biol.

Three dimensional finite element method to study heat flow in dermal regions of elliptical and tapered shape human limbs

Appl. Math. Comput.

A probe technique for determining the thermal conductivity of tissue

J. Heat Transf.

A finite element model of burn injury in blood-perfused skin

J. Bio-Mech. Eng.

Text Book of Medical Physiology

Studies of thermal injury. I. The conduction of heat to and through skin and the temperature attained therein. A theoretical and an experimental investigation

Am. J. Pathol.

Mathematical and numerical estimation of tissue damage in human peripherals due to cold injuries

Int. J. Mech. Prod. Eng. Res. Dev.