Heat Transfer In Vivo: Phenomena and Models

  • Alexander I. Zhmakin
Reference work entry


Physical phenomena encountered in heat transfer in living tissues and mathematical models used to simulate them are discussed. Effects of high or low temperature on the biological systems at the different levels – cell, tissue, organism – as well a role of the blood circulation and of the structure of the vascular network on heat transfer are considered. The classic Pennes bioheat equation, a number of non-Fourier heat transfer models (including single-phase-lag and dual-phase-lag models), porous media models, models based on the fractional differential equations, and discrete vascular models are analyzed. A few selected exact solutions are presented.



Concentration (m−3)


Specific heat (Jkg−1m−3)


Blood specific heat (Jkg−1 m−3)


Activation energy (J)


Multi-regional Green function


The Graetz number


The permeability of the porous medium (m2)


Thermal equilibration length (m)


The direction cosines of the vessels


The Nusselt number


The Pecle number


Heat source (Wm−3)


Heat flux (Wm−2)


Heat source due to blood convection (Wm−3)


Heat source due to blood perfusion (Wm−3)

\( {\dot{q}}_{\mathrm{met}} \)

Metabolic heat rate (Wm−3)


The Reynolds number


Temperature (K)


Temperature of the arterial blood (K)


Temperature of the vessel’s wall (K)


Ambient temperature (K)


Time (s)


Volume (m3)

\( \overline{u} \)

Mean blood velocity (ms−1)

Greek Symbols


Order of fractional derivative


Euler Gamma function

δ (s)

Dirac delta function


The angle between the direction of the blood vessels and the tissue temperature gradient




Thermal diffusivity (m2 s−1)


Thermal conductivity (Wm−1K−1)


Blood thermal conductivity (Wm−1K−1)


“Perfusional” thermal conductivity (Wm−1K−1)


Effective thermal conductivity (Wm−1K−1)


Dynamic viscosity (kgm−1 s−1)


Density (kgm−3),


Surface tension on the interface between phases i and j


Relaxation time (s)


Phase lag for the heat flux vector (s)


Phase lag temperature gradient (s)


Irreversible thermal ‘damage”


The blood perfusion rate (kgs−1m−3)

Mixed Symbols


Time interval (s)


Elementary volume (m3)


  1. Akbarzadeh AH, Pasini D (2014) Phase-lag heat conduction in multilayered cellular media with imperfect bonds. Int J Heat Mass Transf 75:656–667CrossRefGoogle Scholar
  2. Arkin H, Xu LX, Holmes KR (1994) Recent developments in modeling heat transfer in blood perfused tissues. IEEE Trans Biomed Eng 41:97–107CrossRefGoogle Scholar
  3. Ashwood-Smith MJ, Morris GJ, Fowler R, Appleton TC, Ashorn R (1988) Physical factors are involved in the destruction of embryos and oocytes during freezing and thawing procedures. Hum Reprod 3:795–802CrossRefGoogle Scholar
  4. Baish JW (1994) Formulation of a statistical model of heat transfer in perfused tissue. J Biomech Eng 116:521–527CrossRefGoogle Scholar
  5. Baish JW (2000) Microvascular heat transfer, ch. 98. In: Bonzano JD (ed) The biomedical engineering handbook, 2nd edn. CRC Press, Boca RatonGoogle Scholar
  6. Bald WB, Fraser J (1982) Cryogenic surgery. Rep Prog Phys 45:1381–1434CrossRefGoogle Scholar
  7. Baldwin RL (2005) Early days of studying the mechanism of protein folding. In: Buchner J, Kiefhaber T (eds) Protein folding handbook. Wiley-VCH, Weinheim, pp 3–21Google Scholar
  8. Bale JS (1996) Insect cold hardiness: a matter of life and death. Eur J Entomol 93:369–382Google Scholar
  9. Bali R, Sharma S (2011) A model for intra-articular heat exchange in a knee joint. Tribol Lett 41:379–386CrossRefGoogle Scholar
  10. Bank H, Mazur P (1973) Visualization of freezing damage. J Cell Biol 57:729–742CrossRefGoogle Scholar
  11. Baust JG, Gage AA (2005) The molecular basis of cryosurgery. BJU Int 95:1187–1191CrossRefGoogle Scholar
  12. Becker S, Kuznetsov A (eds) (2015) Heat transfer and fluid flow in biological processes. Academic, AmsterdamGoogle Scholar
  13. Behura AK, Prasad BN, Prasad L (2013) Burn depth prediction using analytical and numerical solution of Pennes bioheat equation. Int J Innov Appl Stud 3:215–220Google Scholar
  14. Bejan A (2001) The tree of convective heat streams: its thermal insulation function and the predicted 3/4-power relation between the body heat loss and body size. Int J Heat Mass Transf 44:699–704zbMATHCrossRefGoogle Scholar
  15. Benson EE, Lynch PT, Jones J (1992) Detection of lipid peroxidation products in cryoprotected and frozen rice cells: consequences for post-thaw survival. Plant Sci 85:107–114CrossRefGoogle Scholar
  16. Bhowmik A, Singh R, Repaka R, Mishra SC (2013) Conventional and newly developed bioheat transport models in vascularized tissues: a review. J Therm Biol 38:107–125CrossRefGoogle Scholar
  17. Bilgili M, Simsek E, Sahin A, Ozbek A (2015) Examination of human heat loss in five Mediterranean regions. Physiol Behav 149:61–68CrossRefGoogle Scholar
  18. Bingi VN, Savin AV (2003) Effects of weak magnetic fields on biological systems: physical aspects. Phys Uspekhi 46:259–292CrossRefGoogle Scholar
  19. Bischof J, He X (2005) Thermal stability of proteins. Ann N Y Acad Sci 1066:1–22CrossRefGoogle Scholar
  20. Bligh J, Johnson KG (1973) Glossary of terms for thermal physiology. J Appl Physiol 35:941–961CrossRefGoogle Scholar
  21. Breton G, Danyluk J, Ouellet F, Sarhan F (2000) Biotechnological applications of plant freezing associated proteins. Biotechnol Annu Rev 6:57–99Google Scholar
  22. Brey EM, King TW, Johnson C, McIntire LV, Reece GP, Patric CW (2002) A technique for quantitative three-dimensional analysis of microvascular structure. Microvasc Res 63:279–294CrossRefGoogle Scholar
  23. Brix G, Seebass M, Hellwig G, Griebel J (2002) Estimation of heat transfer and temperature rise in partial-body regions during MR procedures: an analytical approach with respect to safety considerations. Magn Reson Imag 20:65–76CrossRefGoogle Scholar
  24. Canney MS, Khokhlova VA, Bessonova OV, Bailey MR, Crum LA (2010) Shock-induced heating and millisecond boiling in gels and tissues due to high intensity focused ultrasound. Ultrasound Med Biol 36:250–267CrossRefGoogle Scholar
  25. Cebremedhin KG, Wu B, Perano K (2016) Modeling conductive cooling for thermally stressed dairy cows. J Therm Biol 56:91–99CrossRefGoogle Scholar
  26. Charny CK (1992) Mathematical models of bioheat transfer. Adv Heat Transf 22:19–155CrossRefGoogle Scholar
  27. Charny CK, Weinbaum S, Lewin RL (1990) An evaluation of the Weinbaum-Jiji bioheat equation for normal and hyperthermic conditions. J Biomech Eng 112:80–87CrossRefGoogle Scholar
  28. Chato JC (1980) Heat transfer to blood vessels. J Biomech Eng 102:110–118CrossRefGoogle Scholar
  29. Chato JC (1992) A view of the history of heat transfer in bioengineering. Adv Heat Transf 22:1–18CrossRefGoogle Scholar
  30. Chen MM, Holmes KR (1980) Microvascular contribution to tissue heat transfer. Ann N Y Acad Sci 335:137–150CrossRefGoogle Scholar
  31. Cheng H, Plewes D (2002) Tissue thermal conductivity by magnetic resonance thermometry and focused ultrasound heating. J Magn Reson Imag 16:598–609CrossRefGoogle Scholar
  32. Choi JH, Han B, Bishof JC (2004) Effects of a cryoprotective agent on thermal properties of solutions at subzero temperatures. ASME international mechanical engineering congress, Anaheim, 2004, pp 1–2Google Scholar
  33. Cholewska A, Stanek A, Kwiatek S, Cholewska A, Cieslar G, Straszak D, Gibinska J, Sieron-Stoltny K (2016) Proposal of thermal imaging application in photodynamic therapy – preliminary report. Photodiagn Photodyn Ther 14:34–39CrossRefGoogle Scholar
  34. Chua KJ (2013) Fundamental experiments and numerical investigation of cryofreezing incorporating vascular network with enhanced nano-freezing. Int J Therm Sci 70:17–31CrossRefGoogle Scholar
  35. Cicekli U (2003) Computational model for heat transfer in the human eye using the finite element method. MS thesis, Louisiana State UniversityGoogle Scholar
  36. Cohen ML (1977) Measurements of the thermal properties of human skin. A review. J Investig Dermatol 69:333–338CrossRefGoogle Scholar
  37. Cooper TE, Trezek GJ (1971) Correlation of thermal properties of some human tissue with water content. Aerosp Med 42:24–27Google Scholar
  38. Craciunescu O, Clegg TS (1997) Perturbations of large vessels on induced temperature distributions. Part a: three-dimensional simulation study. Adv Heat Mass Transf Biotechnol, HTD 335:193–198Google Scholar
  39. Craciunescu O, Das SK, Dewhirst MK (1999) Three-dimensional microvascular networks fractal structure: a potential for tissue characterization? Adv Heat Mass Transf Biotechnol, HTD 363:9–13Google Scholar
  40. Crezee J, Lagendjik JJW (1992) Temperature uniformity during hyperthermia: impact of large vessels. Phys Med Biol 37:1321–1337CrossRefGoogle Scholar
  41. Cui ZF, Barbene JC (1991) The influence of model parameter values on the prediction of skin surface temperature: II. Contact problems. Phys Med Biol 36:1607–1620CrossRefGoogle Scholar
  42. Cvetkovic M, Polyak D, Hirata A (2016) The electromagnetic-thermal dosimetry for the homogeneous human brain model. Eng Anal Bound Elem 63:61–73MathSciNetCrossRefzbMATHGoogle Scholar
  43. Dagan Z, Weinbaum S, Jiji LM (1986) Parametric study of the three layer microcirculatory model for surface tissue energy exchange. J Biomech Eng 108:89–96CrossRefGoogle Scholar
  44. Damor RS, Kumar S, Shukla AK (2013) Numerical solution of fractional bioheat equation with constant and sinusoidal heat flux condition on skin tissue. Am J Math Anal 1:20–24Google Scholar
  45. Damor RS, Kumar S, Shukla AK (2014) Numerical simulation of fractional bioheat equation in hyperthermia treatment. J Mech Med Biol 14:1450,018CrossRefGoogle Scholar
  46. Damor RS, Kumar S, Shukla AK (2015) Parametric study of fractional bioheat equation in skin tissue with sinusoidal heat flux. Fract Differ Calculus 5:43–53MathSciNetCrossRefGoogle Scholar
  47. Damor RS, Kumar S, Shukla AK (2016) Solution of fractional bioheat equation in terms of Fox’s H-function. SprinerPlus 5:111CrossRefGoogle Scholar
  48. Das K, Mishra SC (2014) Study of thermal behavior of a biological tissue: an equivalence of Pennes bioheat equation and Wulff continuum model. J Therm Biol 45:103–109CrossRefGoogle Scholar
  49. de Dear RJ, Arens E, Zhang H, Ouro M (1997) Convective and radiative heat transfer coefficients for individual human body segments. Int J Meteorol 40:141–156Google Scholar
  50. Debenetti PG (2003) Supercooled and glassy water. J Phys Condens Matter 15:R1669–R1726CrossRefGoogle Scholar
  51. Deng ZS, Liu J (2001) Blood perfusion-based model for characterizing the temperature fluctuations in living tissues. Physica A 300:521–530zbMATHCrossRefGoogle Scholar
  52. Deng ZS, Liu J (2004a) Mathematical modeling of temperature mapping over skin surface and its application in thermal disease diagnostics. Comput Biol Med 34:8CrossRefGoogle Scholar
  53. Deng ZS, Liu J (2004b) Monte Carlo simulation of the effects of large blood vessels during hyperthermia. Lect Notes Comput Sci 3314:437–442zbMATHCrossRefGoogle Scholar
  54. Deng ZS, Liu J (2006) Numerical study of the effects of large blood vessels on three-dimensional tissue temperature profiles during cryosurgery. Numer Heat Transfer, Part A 49:47–67CrossRefGoogle Scholar
  55. Devireddy RV, Leo PH, Lowengrub JS, Bischof JC (2002) Measurement and numerical analysis of freezing in solutions enclosed in a small container. Int J Heat Mass Transf 45:1915–1931CrossRefGoogle Scholar
  56. Dewhirst MW, Abraham J, Viglianti B (2015) Evolution of thermal dosimetry for application of hyperthermia to treat cancer. Adv Heat Tran 47:397–416CrossRefGoogle Scholar
  57. Diller KR (1992) Modeling of bioheat transfer processes at high and low temperatures. Adv Heat Tran 22:157–357CrossRefGoogle Scholar
  58. Dixit A, Gade U (2015) A case study on human bio-heat transfer and thermal comfort within CFD. Build Environ 94:122–130CrossRefGoogle Scholar
  59. Dumont F, Marechal PA, Gervais P (2006) Involvement of two specific causes of cell mortality in freeze-thaw cycles with freezing to −196 °C. Appl Environ Microbiol 72:21330–21335CrossRefGoogle Scholar
  60. Durkee JW, Antich PP (1991a) Characterization of bioheat transport using an exact solution to the cylindrical geometry, multi-region, time-dependent bioheat equation. Phys Med Biol 36:1377–1406CrossRefGoogle Scholar
  61. Durkee JW, Antich PP (1991b) Exact solutions to the multi-region time-dependent bioheat equation with transient heat sources and boundary conditions. Phys Med Biol 36:345–368CrossRefGoogle Scholar
  62. Durkee JW, Antich PP, Lee CE (1990) Exact solutions to the multiregion time-dependent bioheat equation. I: solution development. Phys Med Biol 35:847–867CrossRefGoogle Scholar
  63. Emery AF, Kramar PO, Guy AW, Lin JC (1975) Microwave induced temperature rises in rabbit eyes in cataract research. J Heat Transf 97:123–128CrossRefGoogle Scholar
  64. English MJM (2001) Physical principles of heat transfer. Curr Anesth Crit Care 12:66–71CrossRefGoogle Scholar
  65. Fan J, Wang L (2011a) Analytical theory of bioheat transport. J Appl Phys 109:104,202Google Scholar
  66. Fan J, Wang L (2011b) A general bioheat model at macroscale. Int J Heat Mass Transf 54:722–726zbMATHCrossRefGoogle Scholar
  67. Ferras LL, Ford NJ, Morgado ML, Nobrea JM, Rebelo MS (2015) Fractional Pennes bioheat equation: theoretical and numerical studies. Fract Calculus Appl Anal 18:1080–1106MathSciNetzbMATHGoogle Scholar
  68. Ferreira MS, Yanagihara JI (2009) A transient three-dimensional heat transfer model of the human body. Int Comm Heat Mass Transf 36:718–724CrossRefGoogle Scholar
  69. Fhuong NL, Yamashita M, Yoo S, Ito K (2016) Prediction of convective heat transfer coefficient of human upper and lower airway surfaces in steady and unsteady breathing conditions. Build Environ 100:172–185CrossRefGoogle Scholar
  70. Foster KR (2000) Thermal and nonthermal mechanisms of interaction of radiofrequency energy with biological systems. IEEE Trans Plasma Sci 28:15–23CrossRefGoogle Scholar
  71. Franks F (2003) Nucleation of ice and its management in ecosystems. Philos Trans R Soc Lond A 361:557–574CrossRefGoogle Scholar
  72. Fujikawa S (1981) The effect of various cooling rates on the membrane ultrastructure of frozen human erythrocites and its relation to the extent of haemolysis after thawing. J Cell Sci 49:369–382Google Scholar
  73. Gabris E, Rybaczuk M, Kedzia A (2005) Fractal models of circulatory system. Symmetrical and asymmetrical approach comparison. Chaos Solit Fractals 24:707–715zbMATHCrossRefGoogle Scholar
  74. Gabris E, Rybaczuk M, Kedzia A (2006) Blood flow simulation through fractal models of circulatory system. Chaos Solit Fractals 27:1–7zbMATHCrossRefGoogle Scholar
  75. Gage AA, Baust JG (2002) Cryosurgery – a review of recent advances and current issues. Cryo Letters 23:69–78Google Scholar
  76. de Gennes PJG (1985) Wetting: statics and dynamics. Rev Mod Phys 57:827–863CrossRefGoogle Scholar
  77. Goodarzi-Ardakani V, Taebi-Rahni M, Salimi MR, Ahmadi G (2016) Computational simulation of temperature and velocity distribution in human upper respiratory airway during inhalin of hot air. Respir Physiol Neurobiol 223:49–58CrossRefGoogle Scholar
  78. Gupta PK, Singh J, Rai KN (2010) Numerical simulation for heat transfer in tissues during thermal therapy. J Therm Biol 35:295–301CrossRefGoogle Scholar
  79. Gurtin ME, Pipkin AC (1969) A general theory of heat conduction with finite wave speed. Arch Rat Mech Anal 31:113–126MathSciNetzbMATHCrossRefGoogle Scholar
  80. Han B, Bishof JC (2004) Thermodynamic nonequilibrium phase change behavior and thermal properties of biological solutions for cryobiology applications. J Biomech Eng 126:196–203CrossRefGoogle Scholar
  81. Hansen E (2003) Modelling heat transfer in a bone-cement-prothesis system. J Biomech 36:787–795CrossRefGoogle Scholar
  82. Hassanpour S, Saboonchi A (2014) Interstitial hyperthermia treatment of countercurrent vascular tissue: a comparison of Pennes, WJ and porous media bioheat models. J Therm Biol 46:47–55CrossRefGoogle Scholar
  83. Hassanpour S, Saboonchi A (2015) The numerical assessment of volume averaging method in heat transfer modeling of tissue-like porous media. Int Comm Heat Mass Transf 63:41–48CrossRefGoogle Scholar
  84. Havlin S, Buldyrev SV, Goldberger AL, Mantegna RN, Ossadnik SM, Peng CK, Simons M, Stanley HE (1995) Fractals in biology and medicine. Chaos, Solitons Fractals 6:171–201MathSciNetzbMATHCrossRefGoogle Scholar
  85. Howells EB (2015) Measuring temperature. Anesth Intensive Care Med 16:358–362CrossRefGoogle Scholar
  86. Hubel A, Norman J, Darr TB (1999) Cryobiology characteristics of genetically modified hematpoietic progenitor cells. Cryobiology 38:140–153CrossRefGoogle Scholar
  87. Huttunen JMJ, Huttunen T, Malinen M, Kaipio JP (2006) Determination of heterogeneous thermal parameters using ultrasound induced heating and MR thermal mapping. Phys Med Biol 51:1011–1032CrossRefGoogle Scholar
  88. Izhar LI, Petrou M (2012) Thermal imaging in medicine. Adv Imag Electron Phys 171:41–114CrossRefGoogle Scholar
  89. Jenne JW, Preusser T, Gunther M (2012) High-intensity focused ultrasound: principles, therapy guidance, simulations and applications. Z Med Phys 22:311–322CrossRefGoogle Scholar
  90. Jiang SC, Ma N, Li HJ, Zhang XX (2002) Effects of thermal properties and geometrical dimensions on skin burn injuries. Burns 28:713–717CrossRefGoogle Scholar
  91. Joseph DD, Presiosi L (1989) Heat waves. Rev Mod Phys 61:41–73MathSciNetCrossRefGoogle Scholar
  92. Kandra D, Devireddy R (2008) Numerical simulation of local temperature distortions during ice nucleation of cells in suspension. Int J Heat Mass Transf 51:5655–5661CrossRefGoogle Scholar
  93. Karch R, Neumann F, Neumann M, Schreiner W (1999) A three-dimensional model for arterial tree representation, generated by constrained constructive optimization. Comput Biol Med 29:19–38CrossRefGoogle Scholar
  94. Karlsson JOM (2002) Cryopreservation: freezing and vitrification. Science 296:655–656CrossRefGoogle Scholar
  95. Kay JE, Tsemekhman V, Larson B, Baker M, Swanson B (2003) Comment on evidence for surface-initiated homogeneous nucleation. Atmos Chem Phys 3:1439–1443CrossRefGoogle Scholar
  96. Khaled ARA, Vafai K (2003) The role of porous media in modeling flow and heat transfer in biological tissues. Int J Heat Mass Transf 46:4989–5003zbMATHCrossRefGoogle Scholar
  97. Khan AA, Vincent JFV (1996) Mechanical damage induced by controlled freezing in apple and potato. J Texture Stud 27:143–157CrossRefGoogle Scholar
  98. Kilbas AA, Srivastave HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. North Holland, AmsterdamzbMATHCrossRefGoogle Scholar
  99. Kolios MC, Sherar MD, Hunt JW (1995) Large blood vessel cooling in heated tissue: a numerical study. Phys Med Biol 40:477–494CrossRefGoogle Scholar
  100. Kolios MC, Worthington AE, Sherar MD, Hunt JW (1998) Experimental evaluation of two simple thermal models using transient temperature analysis. Phys Med Biol 43:3325–3340CrossRefGoogle Scholar
  101. Kolios MC, Worthington AE, Holdsworth DW, Sherar MD, Hunt JW (1999) An investigation of the flow dependence of temperature gradients near large vessels during steady state and transient tissue heating. Phys Med Biol 44:1479–1497CrossRefGoogle Scholar
  102. Kondratiev TV, Wold R, Aasum E, Tveita T (2008) Myocardial mechanical dysfunction and calcium overload following rewarming from experimental hypothermia in vivo. Cryobiology 56:15–21CrossRefGoogle Scholar
  103. Konstas AA, Neimark MA, Laine AF, Pile-Spellman J (2007) A theoretical model of selective cooling using intracarotid cold saline infusion in the human brain. J Appl Physiol 102:1329–1340CrossRefGoogle Scholar
  104. Korpan NN (2001) Basics of cryosurgery. Springer, WienGoogle Scholar
  105. Kotte A, van Leeuwen G, de Bree J, van der Koijk J, Crezee H, Lagendjik J (1996) A description of discrete vessel segments in thermal modelling of tissues. Phys Med Biol 41:865–884CrossRefGoogle Scholar
  106. Kou HS, Shih TC, Lin WL (2003) Effect of the directional blood flow on thermal dose distribution during thermal therapy: an application of a Green function based on the porous model. Phys Med Biol 48:1577–1589CrossRefGoogle Scholar
  107. Kucsko G, Maurer PG, Yao NY, Kubo M, Noh H, Lo P, Park H, Lukin MD (2013) Nanometer scale thermometry in a living cell. Nature 500:54–58CrossRefGoogle Scholar
  108. Kumar P, Kumar D, Rai KN (2015) A mathematical model for hyperbolic space-fractional bioheat transfer during thermal therapy. Procedia Eng 127:56–62CrossRefGoogle Scholar
  109. Kwak HS, Im HG, Shim EB (2016) A model for allometric scaling of mammalian metabolism with ambient heat loss. Integr Med Res 5:30–36CrossRefGoogle Scholar
  110. Lagendijk JJW (1982) A mathematical model to calculate temperature distributions in human and rabbit eyes during hyperthermic treatment. Phys Med Biol 27Google Scholar
  111. Lahiri BB, Bagavathiappan S, Jayakumar T, Philip J (2012) Medical applications of infrared thermography: a review. Infrared Phys Technol 55:221–235CrossRefGoogle Scholar
  112. Lai D, Chen Q (2016) A two-dimensional model for calculating heat transfer in the human body in a transient and non-uniform thermal environment. Energ Buildings 118:114–122CrossRefGoogle Scholar
  113. Lakssass A, Kengne E, Semmaoui H (2010) Modified Pennes’ equation modelling bio-heat transfer in living tissues: analytical and numerical analysis. Nat Sci 2:1375–1385Google Scholar
  114. Leeuwen GMJV, Kotte ANT, de Bree J, der Koijk JFV, Crezee H, Lagendjik JJW (1997) Accuracy of geometrical modelling of heat transfer from tissue to blood vessels. Phys Med Biol 42:1451–1460CrossRefGoogle Scholar
  115. Li B, Wang J (2003) Anomalous heat conduction and anomalous diffusion in one-dimensional systems. Phys Rev Lett 91:044,301CrossRefGoogle Scholar
  116. Li L, Yu B, Liang M, Yang S, Zou M (2014) A comprehensive study of the effective thermal conductivity of living biological tissue with randomly distributed vascular trees. Int J Heat Mass Transf 72:616–621CrossRefGoogle Scholar
  117. Lifshits IM, Grossberg AY, Khokhlov AR (1979) Volume interactions in the statistical physics of a polymer macromolecule. Sov Phys Usp 22:123–142CrossRefGoogle Scholar
  118. Lillford PJ, Holt CB (2002) In vitro use of biological cryoprotectants. Phil Trans R Soc Lond B 357:945–951CrossRefGoogle Scholar
  119. Lin M, Xu F, Lu TJ, Bai BF (2010) A review of heat transfer 4 in human tooth – experimental characterization and mathematical modeling. Dent Mater 26:501–513CrossRefGoogle Scholar
  120. Liu J (2000) Preliminary survey on the mechanisms of the wave-like behaviors of heat transfer in living tissues. Forsch Ingenieur 66:1–10CrossRefGoogle Scholar
  121. Liu KC (2015) Analysis for high-order effects in thermal lagging to thermal responses in biological tissue. Int J Heat Mass Transf 81:347–354CrossRefGoogle Scholar
  122. Liu J, Xu LX (1999) Estimation of blood perfusion using phase shift in temperature response to sinusoidal heating the skin surface. IEEE Trans Biomed Eng 46:1037–1043CrossRefGoogle Scholar
  123. Loitsanskii LG (1970) Mechanics of gases and liquids (in Russian), 3rd edn. Nauka, MoscowGoogle Scholar
  124. Ma W, Liu W, Li M (2015) Modeling heat transfer from warm water to foot; analytical solution and experimental validation. Int J Therm Sci 98:364–371CrossRefGoogle Scholar
  125. Marciak-Kozlowska J, Kozlowski M (2010) Heat analysis of biological tissue exposed to laser pulses. Lasers Eng 20:279–295zbMATHGoogle Scholar
  126. Matzarakis A, Mayer H, Iziomon MG (1999) Application of a universal thermal index: physiological effective temperature. Int J Biometeorol 43:76–84CrossRefGoogle Scholar
  127. McKenzie JM, Voss IC, Siegel DI (2007) Groundwater flow with energy transport and water-ice phase change: numerical simulations, benchmarks, and application to freezing in peat bogs. Adv Wat Res 30:966–983CrossRefGoogle Scholar
  128. Moghadam MN, Abdel-Sayed P, Camine VM, Pioletti DP (2015) Impact of synovial fluid flow on temperature regulation in knee cartilage. J Biomech 48:370–374CrossRefGoogle Scholar
  129. Mohammed Y, Verhey JF (2005) A finite element method model to simulate laser interstitial thermotherapy in anatomical inhomogeneous regions. Biomed Eng Online 4:1–16CrossRefGoogle Scholar
  130. Mullen SF, Rosenbaum M, Critser JK (2007) The effect of osmotic stress on the cell volume, metaphase II spindle and developmental potential of in vitro matured porcine oocytes. Cryobiology 54:281–289CrossRefGoogle Scholar
  131. Nakayama A, Kuwahara F (2008) A general bioheat transfer model based on the theory of porous media. Int J Heat Mass Transf 51:3190–3199zbMATHCrossRefGoogle Scholar
  132. Narasimhan A, Jha KK, Gopal L (2010) Transient simulations of heat transfer in human eye undergoing laser surgery. Int J Heat Mass Transf 53:482–490zbMATHCrossRefGoogle Scholar
  133. Niemz MH (2007) Laser-tissue interactions, fundamentals and applications, 3rd edn. Springer, BerlinzbMATHCrossRefGoogle Scholar
  134. Oldham KB, Spanie J (1974) The fractional calculus. Academic, San DiegoGoogle Scholar
  135. Orel D, Rozman J (2003) A computer simulation of ultrasound thermal bio-effect in embryonic models. Radioengineer 12:26–30Google Scholar
  136. Pandey H, Gurung DB (2015) Analytical and numerical approximation solution of bio-heat equation. LAP Lambert Academic Publishing, SaarbrückenGoogle Scholar
  137. Parsons KC (1993) Human thermal environments. Taylor & Francis, LondonCrossRefGoogle Scholar
  138. Pearce RS (1999) Molecular analysis of acclimation to cold. Plant Growth Regul 29:47–76CrossRefGoogle Scholar
  139. Pegg DE (1966) Cryobiology. Phys Med Biol 11:209–224CrossRefGoogle Scholar
  140. Pegg DE, Wusteman MC, Boylan S (1996) Fractures in cryopreserved elastic arteries: mechanism and prevention. Cryobiology 33:658–659Google Scholar
  141. Peng T, O’Neill DP, Payne SJ (2011) A two-equation coupled system for determination of liver tissue temperature during thermal ablation. Int J Heat Mass Transf 54:2100–2109zbMATHCrossRefGoogle Scholar
  142. Pennes HH (1948) Analysis of tissue and arterial blood flow temperatures in the resting human forearm. J Appl Physiol 1:93–122; reprinted: Ibid, 1998, 85, 5–34Google Scholar
  143. Podlubny I (1998) Fractional differential equations. Academic, San DiegozbMATHGoogle Scholar
  144. Polge C, Smith AU, Parkers AS (1949) Revival of spermatozoa after vitrification and dehydration at low temperatures. Nature 164:666–676CrossRefGoogle Scholar
  145. Ponder E (1962) The coefficient of thermal conductivity of blood and of various tissues. The J Gener Physiol 45:545–551CrossRefGoogle Scholar
  146. Powell SL (2002) Contact cooling and it’s effects on manual dexterity. PhD thesis, Loughborough UniversityGoogle Scholar
  147. Preusser T, Weihusen A, Peitgen HO (2005) On the modelling of perfusion in the simulation of RF-ablation. In: Proceedings of simulation and visualization (SimVis), Magdeburg, pp 259–268Google Scholar
  148. Raabe D (2007) A texture-component avrami model for predicting recrystallization textures, kinetics and grain size. Modelling Simul Mater Sci Eng 15:39–63CrossRefGoogle Scholar
  149. Roemer RB (1999) Engineering aspects of hyperthermia therapy. Ann Rev Biomed Eng 1:347–376CrossRefGoogle Scholar
  150. Rubinsky B (2000) Cryosurgery. Annu Rev Biomed Eng 2:157–187CrossRefGoogle Scholar
  151. Rubinsky B (2003) Principles of low temperature cell preservation. Heart Fail Rev 8:277–284CrossRefGoogle Scholar
  152. Salloum M, Ghaddar N, Ghali K (2007) A new bioheat model of human body and its integration to clothing models. Int J Therm Sci 46:371–384CrossRefGoogle Scholar
  153. Sanz PD, de Elvira C, Martino M, Zaritzky N, Otero L, Carrasco JA (1999) Freezing rate simulation as an aid to reducing crystallization damage in foods. Meat Sci 52:275–278CrossRefGoogle Scholar
  154. Schäfer AT, Kauffmann JD (1999) What happens in freezing bodies? Experimental study of histological tissue change caused by freezing injuries. Forensic Sci Int 102:149–158CrossRefGoogle Scholar
  155. Schmidt JD, Doyle J, Larison S (1998) Prostate cryoablation: update 1998. CA Cancer J Clin 48:239–253CrossRefGoogle Scholar
  156. Scott JA (1988) A finite element model of heat transport in the human eye. Phys Med Biol 33:227–241CrossRefGoogle Scholar
  157. Shi J, Chen Z, Shi M (2009) Simulation of heat transfer of biological tissue during cryosurery based on vascular trees. Appl Therm Eng 29:1792–1798CrossRefGoogle Scholar
  158. Shih TC, Kou HS, Liauch CT, Lin WL (2002a) Thermal models of bioheat transfer equations in living tissue and thermal dose equivalence due to hypertermia. Biomed Eng – Appl Basis Comm 14:40–56CrossRefGoogle Scholar
  159. Shih TC, Kou HS, Lin WL (2002b) Effect of effective tissue conductivity onthermal dose distributions of living tissues with directional blood flow during thermal therapy. Int Comm Heat Mass Transf 29:115–126CrossRefGoogle Scholar
  160. Shih TC, Yuan P, Lin WL, Kou HS (2007) Analytical analysis of the Pennes bioheat transfer equation with sinusoidal heat flux condition on skin surface. Med Eng Phys 29:946–953CrossRefGoogle Scholar
  161. Shrivastava D, Roemer R (2005a) An analytical study of heat transfer in a finite region with two blood vessels and general dirichlet boundary conditions. Int J Heat Mass Transf 48:4090–4102zbMATHCrossRefGoogle Scholar
  162. Shrivastava D, Roemer R (2005b) A general analytical derivation of a new, source term dependent 2-D Poisson conduction shape factors. Int J Heat Mass Transf 47:4293–4300zbMATHCrossRefGoogle Scholar
  163. Shrivastava D, Roemer RB (2005c) An analytical study of “Poisson conduction shape factors” for two thermally significant vessels in a finite, heated tissue. Phys Med Biol 50:3627–3641CrossRefGoogle Scholar
  164. Sparks JP, Campbell GS, Black RA (2000) Liquid water content of wood tissue at temperatures below 0 °C. Can J For Res 30:624–630CrossRefGoogle Scholar
  165. Stańczyk M, Telega JJ (2002) Modelling of heat transfer in biomechanics – a review. Part I. Soft tissues. Acta Bioeng Biomech 4:31–61Google Scholar
  166. Stanley HE, Buldyrev SV, Goldberger AL, Goldberger ZD, Havlin S, Mantegna RN, Ossadnik SM, Peng CK, Simons M (1994) Statistical mechanics in biology: how ubiquitous are long-range correlations? Physica A 205:214–253CrossRefGoogle Scholar
  167. Steck LN, Sparrow EM, Abraham JP (2011) Non-invasive measurement of the human core temperature. Int J Heat Mass Transf 54:975–982zbMATHCrossRefGoogle Scholar
  168. Stranges DF, Khayat RE, Albaalbaki B (2013) Thermal convection in non-Fourier fluids. Linear stability. Int J Therm Sci 74:14–23CrossRefGoogle Scholar
  169. Sturesson C (1998) Medical laser-induced thermotherapy. Models and applications. Lund Report on Atomic Physics LRAP-235Google Scholar
  170. Sumida S (2006) Transfusion and transplantation of cryopreserved cells and tissues. Cell Tissue Bank 7:265–365CrossRefGoogle Scholar
  171. Tarasov VE (2016) Heat transfer in fractal materials. Int J Heat Mass Transf 93:427–430CrossRefGoogle Scholar
  172. Torvi CL, Dale JD (1994) A finite element model of skin subjected to a flash fire. ASME J Biomed Eng 116:250–255CrossRefGoogle Scholar
  173. Tsou DY (1996) Macro- to microscale heat transfer: the lagging behavior. Taylor & Francis, New YorkGoogle Scholar
  174. Tungjikusolmun S, Tyler ST, Haemmerich D, Tsai JZ, Cao H, Webster JG, Lee FT, Mahvi DM, Vorperian VR (2002) Three-dimensional finite-element analysis for radio-frequency hepatic tissue ablation. IEEE Trans Med Eng 49:3–8CrossRefGoogle Scholar
  175. Turk JR, Laughin MH (2004) Physical activity and atherosclerosis: which animal model? Can J Appl Physiol 29:657–683CrossRefGoogle Scholar
  176. Tzou DY (1993) An engineering assessment to the relaxation time in thermal wave propagation. Int J Heat Mass Transf 36:1845–1851zbMATHCrossRefGoogle Scholar
  177. Tzou DY, Dai W (2009) Thermal lagging in multi-carrier systems. Int J Heat Mass Transf 52:1206–1213zbMATHCrossRefGoogle Scholar
  178. Vafai K (ed) (2010) Porous media: applications in biological systems and biotechnology. CRC Press, Boca RatonzbMATHGoogle Scholar
  179. Vafai K (ed) (2015) Handbook of porous media, 3rd edn. CRC Press, Boca RatonzbMATHGoogle Scholar
  180. Vajda T (1999) Cryo-bioorganic chemistry: molecular interactions at low temperature. Cell Mol Life Sci 56:398–414CrossRefGoogle Scholar
  181. Vanne A, Hynynen K (2003) MRI feedback temperature control for focused ultrasound surgery. Phys Med Biol 48:31–43CrossRefGoogle Scholar
  182. Vyas DCM, Kumar S, Srivastava A (2016) Porous media based bio-heat transfer analysis on counter-current artery vein tissue phantoms: applications in photo thermal therapy. Int J Heat Mass Transf 99:122–140CrossRefGoogle Scholar
  183. Wang Z, Zhao G, Wang T, Yu Q, Su M, He X (2015) Three-dimensional numerical simulation of the effects of fractal vascular tree on tissue temperature and intracellular ice formation during combined cancer therapy of cryosurgery and hyperthermia. Appl Therm Eng 90:296–304CrossRefGoogle Scholar
  184. Wang G, Zhang L, Wang X, Tai BL (2016) An inverse method to reconstruct the heat flu produced by bone grinding tools. Int J Therm Sci 101:85–92CrossRefGoogle Scholar
  185. Weinbaum S, Jiji LM (1985) A new simplified bioheat equation for the effect of blood flow on average tissue temperature. J Biomech Eng 107:131–139CrossRefGoogle Scholar
  186. Weinbaum S, Jiji L, Lemons DE (1984) Theory and experiment for the effect of vascular temperature on surface tissue heat transfer–part 2: model formulation and solution. J Biomech Eng 106:331–341CrossRefGoogle Scholar
  187. Werner J, Buse M (1988) Temperature profiles with respect to inhomogeneity and geometry of human body. J Appl Physiol 65:1110–1118CrossRefGoogle Scholar
  188. Wessapan T, Rattanadecho P (2014) Influence of ambient temperature on heat transfer in the human eye during exposure to electromagnetic field at 900 MHz. Int J Heat Mass Transf 70:378–388CrossRefGoogle Scholar
  189. Wessapan T, Rattanadecho P (2016) Flow and heat transfer in biological tissue due to electromagnetic near-field exposure effects. Int J Heat Mass Transf 97:174–184CrossRefGoogle Scholar
  190. Wilson SB, Spence VA (1988) A tissue heat transfer model for relating dynamic skin temperature changes to physiological parameters. Phys Med Biol 33:895–912CrossRefGoogle Scholar
  191. Wissler EH (1998) Pennes’ 1948 paper revisited. J Appl Physiol 85:35–41CrossRefGoogle Scholar
  192. Wu YL, Weinbaum S, Jiji L (1993) A new analytic technique for 3-D heat transfer from a countercurrent blood vessels. Int J Heat Mass Transf 36:1073–1083zbMATHCrossRefGoogle Scholar
  193. Wu HL, Ma Y, Peng XF (2004) Freezing-thawing characteristics of botanical tissue and influence of water morphology. Chin Phys Lett 21:345–347CrossRefGoogle Scholar
  194. Wu C, Chen X, Zhou X (2016) Performance of novel solar assisted bian stone thermal therapy. Int J Heat Mass Transf 100:445–450CrossRefGoogle Scholar
  195. Xu F, Lu TJ (2009) Skin biothermomechanics: modeling and experimental characterization. Adv Appl Math 43:147–248Google Scholar
  196. Xu F, Lu TJ (2011) Introduction to skin biothermomechanics and thermal pain. Springer, BerlinCrossRefGoogle Scholar
  197. Xu F, Seffen KA, TJ L (2008) Non-Fourier analysis of skin biothermomechanics. Int J Heat Mass Transf 51:2237–2259zbMATHCrossRefGoogle Scholar
  198. Yang D, Converse M, Mahvi DM, Webster JG (2007) Expanding the bioheat equation to include tissue internal water evaporation during heating. IEEE Trans Biomed Eng 54:1382–1388CrossRefGoogle Scholar
  199. Yu B, Jiang X, Wan C (2016a) Numerical algorithms to estimate relaxation parameters and Caputo fractional derivative for a fractional thermal wave model in spherical composite medium. Appl Math Comput 274:106–118MathSciNetGoogle Scholar
  200. Yu Y, Xu D, Xu YS, Zhang Q (2016b) Variational formulation for a fractional heat transfer model in firefighter protective clothing. Appl Math Model 40:9675–9691MathSciNetCrossRefGoogle Scholar
  201. Yue K, Yu C, Lei Q, Luo Y, Zhang X (2014) Numerical simulation of effect of vessel bifurcation on heat transfer in the magnetic fluid hyperthermia. Appl Therm Eng 69:11–18CrossRefGoogle Scholar
  202. Zamir M (2001) Fractal dimensions and multifractility in vascular branching. J Theor Biol 212:183–190CrossRefGoogle Scholar
  203. Zhang YT, Liu J, Zhou YX (2002) Pilot study on cryogenic heat transfer in biological tissues embedded with large blood vessels. Forsch Ingenier 67:188–197CrossRefGoogle Scholar
  204. Zhmakin AI (2009) Fundamentals of cryobiology: physical phenomena and mathematical models, Biological and medical physics, biomedical engineering. Springer, BerlinzbMATHCrossRefGoogle Scholar
  205. Zhu L, Weinbaum S (1995) A model for heat transfer from embedded blood vessels in 2-D tissue preparations. J Biomech Eng 117:64–73CrossRefGoogle Scholar
  206. Zolfaghari A, Maerefat M (2011) Bioheat transfer. In: Dos Santos Bernardes MA (ed) Developments in heat transfer. InTech, Rijeka pp 153–170Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.SoftImpact Ltd.St. PetersburgRussia
  2. 2.Ioffe InstituteSt. PetersburgRussia

Section editors and affiliations

  • Ram Devireddy
    • 1
  1. 1.Department of Mechanical and Industrial EngineeringLouisiana State UniversityBaton RougeUSA

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