Abstract
In this paper, the temperature distribution in a finite biological tissue in presence of metabolic and external heat source when the surface subjected to different type of boundary conditions is studied. Classical Fourier, single-phase-lag (SPL) and dual-phase-lag (DPL) models were developed for bio-heat transfer in biological tissues. The analytical solution obtained for all the three models using Laplace transform technique and results are compared. The effect of the variability of different parameters such as relaxation time, metabolic heat source, spatial heat source, different type boundary conditions on temperature distribution in different type of the tissues like muscle, tumor, fat, dermis and subcutaneous based on three models are analyzed and discussed in detail. The result obtained in three models is compared with experimental observation of Stolwijk and Hardy (Pflug Arch 291:129–162, 1966). It has been observe that the DPL bio-heat transfer model provides better result in comparison of other two models. The value of metabolic and spatial heat source in boundary condition of first, second and third kind for different type of thermal therapies are evaluated.
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Abbreviations
- T :
-
Local tissue temperature (\(^\circ \hbox {C}\))
- \(T_0\) :
-
Initial temperature of the body (\(^\circ \hbox {C}\))
- \(T_f\) :
-
Ambient temperature (\(^\circ \hbox {C}\))
- \(T_w\) :
-
Temperature on the surface (\(^\circ \hbox {C}\))
- x :
-
Distance of the tissue from the skin surface or space coordinate (m)
- t :
-
Time (s)
- L :
-
Depth of the biological tissue (m)
- k :
-
Thermal conductivity of tissue \((\text {W}\,\text {m}^{-1}\,\text {K}^{-1})\)
- \(\rho\) :
-
Tissue density \((\text {kg}\,\text {m}^{-3})\)
- c :
-
Specific heat of tissue \((\text {J}\,\text {kg}^{-1}\,\text {K}^{-1})\)
- \(c_b\) :
-
Specific heat of blood \((\text {J}\,\text {kg}^{-1}\text {K}^{-1})\)
- \(W_b\) :
-
Mass flow rate of the blood per unit volume of the tissue \((\text {kg}\,\text {m}^{-3}\,\text {s}^{-1})\)
- \(W_m\) :
-
Temperature dependent blood perfusion \((\text {s}^{-1})\)
- \(Q_{met}\) :
-
Heat source due to metabolic heat generation in the tissue \((\text {W}\,\text {m}^{-3})\)
- \(Q_{hs}\) :
-
External heat source \((\text {W}\,\text {m}^{-3})\)
- \(\tau _{q}\) :
-
Relaxation time or lagging time due to heat flux (s)
- \(\tau _{T}\) :
-
Relaxation time or lagging time due to temperature gradient (s)
- \(q_w\) :
-
Heat flux \((\text {W}\,\text {m}^{-2})\)
- h :
-
Heat transfer coefficient \((\text {W}\,\text {m}^{-2}\,\text {K}^{-1})\)
- X :
-
Space coordinate
- \(F_o\) :
-
Time
- \(\theta\) :
-
Local tissue temperature
- \(\theta _b\) :
-
Arterial blood temperature
- \(P_f\) :
-
Perfusion coefficient
- \(P_m\) :
-
Metabolic coefficient
- \(P_r\) :
-
External heat source coefficient
- \(K_i\) :
-
Kirchhoff number
- \(B_i\) :
-
Biot number
- \(\theta _w\) :
-
Temperature on surface
- \(F_{o_{q}}\) :
-
Relaxation time due to heat flux
- \(F_{o_{T}}\) :
-
Relaxation time due to temperature gradient
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Acknowledgments
The authors expresses their sincere thanks to reviewers for their valuable suggestions in the improvement of this article. Authors also would like to thanks Prof. Umesh Singh, Co-ordinator DST-Centre of Interdisciplinary Mathematical Sciences, Banaras Hindu University Varanasi, India, for providing necessary facilities.
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Kumar, D., Singh, S. & Rai, K.N. Analysis of classical Fourier, SPL and DPL heat transfer model in biological tissues in presence of metabolic and external heat source. Heat Mass Transfer 52, 1089–1107 (2016). https://doi.org/10.1007/s00231-015-1617-0
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DOI: https://doi.org/10.1007/s00231-015-1617-0