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Constitutive Equations Developed for Modeling of Heat Conduction in Bio-tissues: A Review

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Abstract

In order to further study the mechanism of heat conduction in bio-tissues, so as to provide a solid theoretical basis for the research of related fields, this paper summarizes in detail the theoretical models used by scholars from various countries to analyze the heat conduction process in the photothermal effect of laser-bio-tissues for many years. The rationality of the theoretical model largely determines the accuracy of the response process description, the development of various models is reviewed longitudinally and the critical nodes in the development process are introduced, and the comparative analysis of the respective advantages and limitations of different models are made horizontally, focuses on the latest theoretical breakthroughs and cutting-edge applications in this field while summarizing the methods. Finally, based on the analysis of a large amount of literature, the paper refines several main problems faced in the research in this field and makes some prospects for the development direction in the future.

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References

  1. K.W. Guan, Y.Q. Jiang, C.S. Sun, H.J.O. Yu, Opt. Laser Technol. 43, 425 (2011)

    Article  ADS  Google Scholar 

  2. F. Zhang, Z. Han, W. Hao, J. Laser Med. Sci. 10, 10908 (2019)

    Google Scholar 

  3. E. Khalkhal, M. Rezaei-Tavirani, M.R. Zali, Z. Akbari, J. Laser Med. 10, 7–12 (2019)

    Article  Google Scholar 

  4. N. Houreld, Clin. Dermatol. 37, 358 (2019)

    Article  Google Scholar 

  5. M. Etehadtavakol, E. Ng, Therm. Sci. Eng. Prog. 100681 (2020)

  6. H. Pennes, J. Appl. Physiol. 1, 93 (1948)

    Article  ADS  Google Scholar 

  7. D. Maillet, Int. J. Therm. Sci. 139, 424 (2019)

    Article  Google Scholar 

  8. A. Hobiny, I. Abbas, J. Therm. Stresses 43, 503 (2020)

    Article  Google Scholar 

  9. Z. Yong, B. Chen, L. Dong, Int. J. Heat Mass Transf. 108, 1428 (2017)

    Article  Google Scholar 

  10. N. Afrin, J. Zhou, Y. Zhang, D.Y. Tzou, J.K. Chen, Numer. Heat Tranf. A 61, 483 (2012)

    Article  Google Scholar 

  11. R.C. Birkebak, Int. Rev. Gen. Exp. Zool 2, 269 (1966)

    Article  Google Scholar 

  12. H.G. Klinger, Bull. Math. Biol. 36, 403–415 (1974)

    MathSciNet  Google Scholar 

  13. D.Y. Tzou, J. Thermophys. Heat Transf. 9, 686 (1995)

    Article  Google Scholar 

  14. S. Thomsen, Photochem. Photobiol. 53, 825 (1991)

    Article  Google Scholar 

  15. I. Raouf, S. Khalid, A. Khan, J. Lee, H.S. Kim, M.H. Kim, J. Therm. Biol. 102644, 2 (2020)

    Google Scholar 

  16. A.R. Khaled, K. Vafai, Int. J. Heat Mass Transf. 46, 4989 (2003)

    Article  Google Scholar 

  17. C. Rossmann, D. Haemmerich, Crit. Rev. Biomed. 42, 6 (2014)

    Google Scholar 

  18. A. Bhowmik, R. Singh, R. Repaka, S.C. Mishra, J. Therm. Biol. 38, 107 (2013)

    Article  Google Scholar 

  19. M. Jaunich, S. Raje, K. Kim, K. Mitra, Z. Guo, Int. J. Heat Mass Transf. 51, 5511 (2008)

    Article  Google Scholar 

  20. N. Sahoo, S. Ghosh, A. Narasimhan, S.K. Das, Int. J. Therm. Sci. 76, 208 (2014)

    Article  Google Scholar 

  21. W. Sheng, S. He, W.J. Seare, A. Almutairi, J. Biomed. Opt. 22, 080901 (2017)

    Article  ADS  Google Scholar 

  22. A. Srivastava, Proc. Nat. Acad. Sci. India A. 88, 461 (2018)

    Google Scholar 

  23. M.R. Salimpour, E. Shirani, J. Therm. Biol 64, 7 (2017)

    Article  Google Scholar 

  24. H.M. Patil, R. Maniyeri, Sci. Eng. Prog. 10, 42 (2019)

    Google Scholar 

  25. H. Arkin, L. Xu, K. Holmes, IEEE Trans. Biomed. Eng. 41, 97 (1994)

    Article  Google Scholar 

  26. J. Fourier, Théorie analytique de la chaleur (1822)

  27. E. Majchrzak, B. Mochnacki, M. Jasiński, Com. Flu. Sol. Mech. 1766, 91 (2003)

    Google Scholar 

  28. M. Jasiński, J. Theor. Appl. Mech. 52, 825 (2014)

    Google Scholar 

  29. M. Ciesielski, B. Mochnacki, A. Mechanics, J. Theor. Appl. Mech. 52, 927 (2014)

    Article  Google Scholar 

  30. W. Wulff, IEEE Trans. Biomed. Eng. 6, 494 (1974)

    Article  Google Scholar 

  31. W.P. Hansen, S. Fine, Appl. Opt. 7, 155 (1968)

    Article  ADS  Google Scholar 

  32. W.J. Song, S. Weinbaum, L.M. Jiji, Biomech. Eng. 109, 72 (1987)

    Article  Google Scholar 

  33. M. Chen, K. Holmes, Ann. N. Y. Acad. Sci. 335, 137 (1980)

    Article  ADS  Google Scholar 

  34. S. Weinbaum, L. Jiji, D. Lemons, J. Biomech. Eng 106, 331 (1984)

    Article  Google Scholar 

  35. S. Gabay, I. Kremer, I. Bensira, G. Erez, Lasers Surg. Med. 8, 418 (1988)

    Article  Google Scholar 

  36. S. Weinbaum, L. Jiji, Biomech. Eng. 107, 131–139 (1985)

    Article  Google Scholar 

  37. C. Guiot, E. Madon, D. Allegro, P. Pianta, B. Baiotto, P. Gabriele, Phys. Med. Biol. 43, 2831 (1998)

    Article  Google Scholar 

  38. J. Baish, P. Ayyaswamy, K. Foster, J. Biomech. Eng. 108, 246 (1986)

    Article  Google Scholar 

  39. J.W. Baish, P.S. Ayyaswamy, K.R. Foster, J. Biomech. Eng. 108, 324 (1986)

    Article  Google Scholar 

  40. H.E. Wissler, J. Biomech. Eng. 109, 226 (1987)

    Article  Google Scholar 

  41. S. Weinbaum, L. Jiji, J. Biomech. Eng. 109, 234 (1987)

    Article  Google Scholar 

  42. S. Weinbaum, L. Jiji, J. Biomech. Eng. 111, 271 (1989)

    Article  Google Scholar 

  43. C. Charny, R. Levin, J. Biomech. Eng. 111, 263 (1989)

    Article  Google Scholar 

  44. J. Baish, J. Biomech. Eng. 116, 521 (1994)

    Article  Google Scholar 

  45. S. Weinbaum, L. Xu, L. Zhu, A. Ekpene, J. Biomech. Eng. 119, 278 (1997)

    Article  Google Scholar 

  46. R. Roemer, A. Dutton, J. Biomech. Eng. 120, 395 (1998)

    Article  Google Scholar 

  47. C. Li, J. Miao, K. Yang, X. Guo, J. Tu, P. Huang, D. Zhang, J. Appl. Phys. 123, 17 (2018)

    Google Scholar 

  48. A.T. Ramu, J.E. Bowers, Appl. Phys. Lett. 106, 26 (2015)

    Article  Google Scholar 

  49. J.L. Molina, J. Eng. Math. 123, 89 (2020)

    Article  Google Scholar 

  50. H. Xin, B. Chen, Z. Zhou, D. Li, J. Tian, Numer. Heat Tranf. A 77, 730 (2020)

    Article  Google Scholar 

  51. Z.K. Bojdi, A. Hemmat, A. Hemmat, Optik 130, 345 (2017)

    Article  ADS  Google Scholar 

  52. K. Grysa, A. Maciag, A. Maciag, Int. J. Numer. Method 1, 7336–7340 (2019)

    Google Scholar 

  53. A. Lopes, R. Gomes, M. Castiñeras, J.M. Coelho, J.P. Santos, Lasers Med. Sci. 35(43), 43 (2020)

    Article  Google Scholar 

  54. S. Lin, J. Mech. Med. Biol. 13, 1350063 (2013)

    Article  ADS  Google Scholar 

  55. M. Silva, B. Freitas, R. Andrade, J. Espregueira-Mendes, Ann. Biomed. Eng. 48, 1616 (2020)

    Article  Google Scholar 

  56. D. Yang, M. Converse, D. Mahvi, IEEE Trans. Biomed. Eng. 54, 1382 (2007)

    Article  Google Scholar 

  57. V. Saunders, Contemp. Phys. 1, 486 (1960)

    Article  ADS  Google Scholar 

  58. E.H. Wissler, J. Appl. Physiol. 16, 734 (1961)

    Article  Google Scholar 

  59. R.J. Crosbie, J.D. Hardy, IRE Trans. Bio-Med. Electron. 55, 627 (1963)

    Google Scholar 

  60. J. Stolwijk, J. Hardy, J. Appl. Physiol. 21, 967 (1966)

    Article  Google Scholar 

  61. B. Gao, S. Langer, P. Corry, Int. J. Hyperthermia 11, 267 (1995)

    Article  Google Scholar 

  62. A. Kumar, S. Kumar, V. Katiyar, S. Telles, J. Therm. Biol. 69, 228 (2017)

    Article  Google Scholar 

  63. E.H. Wissler, J. Appl. Physiol. 85, 35 (1998)

    Article  Google Scholar 

  64. K. Das, S.C. Mishra, J. Therm. Biol. 45, 103 (2014)

    Article  Google Scholar 

  65. G. Chenhua, C. Ruixian, Chin. Sci. Bull. 50, 2791 (2005)

    Article  Google Scholar 

  66. K. Khanafer, A. Alamiri, I. Flow and Heat Transfer in Biological Tissues (Springer, New York, 2008), pp. 237–259

    Google Scholar 

  67. C.K. Charny, S. Weinbaum, R.L. Levin, J. Biomech. Eng. 112, 80 (1990)

    Article  Google Scholar 

  68. D. Shrivastava, R.B. Roemer, Int. J. Heat Mass Transf. 47, 4293 (2004)

    Article  Google Scholar 

  69. R. Alagirusamy, A. Das, Int. J. Heat Mass Transf. 78, 1068 (2014)

    Article  Google Scholar 

  70. E. Majchrzak, B. Mochnacki, Int. J. Multiscale Comput. Eng. 12, 1 (2014)

    Article  Google Scholar 

  71. J.R. Ho, C.P. Kuo, W.S. Jiaung, Int. J. Heat Mass Transf. 46, 55 (2003)

    Article  Google Scholar 

  72. K. Kim, Z. Guo, Comput. Meth. Progr. Biomed. 86, 112 (2007)

    Article  Google Scholar 

  73. A. Vedavarz, S. Kumar, M.K. Moallemi, J. Heat. Trans. 116, 221 (1994)

    Article  Google Scholar 

  74. Z. Kang, P. Zhu, D. Gui, L. Wang, Int. J. Heat Mass Transf. 115, 250 (2017)

    Article  Google Scholar 

  75. W. Ulbrich, Phys. Rev. 123, 2001 (1961)

    Article  ADS  MathSciNet  Google Scholar 

  76. B. Mochnacki, W. Tuzikiewicz, J. Appl. Math. Comput. Mech. 15, 137 (2016)

    Article  MathSciNet  Google Scholar 

  77. M. Ciesielski, M. Duda, B. Mochnacki, J. Appl. Math. Comput. Mech. 15, 33 (2016)

    Article  MathSciNet  Google Scholar 

  78. C. Cattaneo, Comptes Rendus 247, 431 (1958)

    Google Scholar 

  79. P. Antaki, J. Heat. Trans. 127, 189–193 (2005)

    Article  Google Scholar 

  80. E. Majchrzak, Ł Turchan, J. Dziatkiewicz, Arch. Mech. 67, 6 (2015)

    Google Scholar 

  81. W. Kaminski, J. Heat. Trans. 112, 555–560 (1990)

    Article  Google Scholar 

  82. K. Mitra, S. Kumar, A. Vedevarz, M. Moallemi, J. Heat. Trans. 117, 568 (1995)

    Article  Google Scholar 

  83. A. Stratigos, J.S. Dover, J. Dover, Dermatol. Ther. 13, 2 (2000)

    Article  Google Scholar 

  84. M. Ciesielski, B. Mochnacki, Numer. Heat Tranf. A 74, 1685 (2018)

    Article  Google Scholar 

  85. F. Xu, K. Seffen, T. Lu, Int. J. Heat Mass Transf. 51, 2237 (2008)

    Article  Google Scholar 

  86. M.E. Gurtin, A.C. Pipkin, Arch. Rational Mech. Anal. 31, 113 (1968)

    Article  ADS  MathSciNet  Google Scholar 

  87. M. Ciesielski, M. Duda, B. Mochnacki, C. Mechanics, Appl. Math. Comput. Mech. 15, 33 (2016)

    Article  MathSciNet  Google Scholar 

  88. G. Xu, J. Wang, Appl. Math. Mech. 39, 1465 (2018)

    Article  Google Scholar 

  89. J. Aguilar, T. Fraga, J. Jiménez, R. Jiménez, V. Peregrino, G. Ramírez, Math (Probl, Eng, 2016).

    Google Scholar 

  90. B. Mochnacki, M. Paruch, Comp. Ass. Meth. Eng. Sci. 25, 240 (2018)

    Google Scholar 

  91. J.L. Auriault, Int. J. Eng. Sci. 101, 45 (2016)

    Article  Google Scholar 

  92. D.Y. Tzou, J. Heat. Trans. 117, 8 (1995)

    Article  Google Scholar 

  93. E. Majchrzak, Sci. Comput. Model. Eng. 69, 43 (2010)

    Google Scholar 

  94. E. Majchrzak, L. Turchan, Eng. Anal. Bound. Elem. 50, 76 (2015)

    Article  MathSciNet  Google Scholar 

  95. E. Majchrzak, Comput. Model. Eng. 69, 43 (2010)

    Google Scholar 

  96. E. Majchrzak, B. Mochnacki, Appl. Math. Comput. Mech. 17, 43 (2018)

    Article  MathSciNet  Google Scholar 

  97. M. Zisik, D. Tzou, J. Heat. Trans. 116, 526 (1994)

    Article  Google Scholar 

  98. J. Zhou, Y. Zhang, J. Chen, Numer. Heat Tranf. A 54, 1 (2008)

    Article  Google Scholar 

  99. S. Kumar, A. Srivastava, Appl. Math. Model. 52, 378 (2017)

    Article  MathSciNet  Google Scholar 

  100. K. Liu, J. Wang, Int. J. Heat Mass Transf. 70, 621 (2014)

    Article  Google Scholar 

  101. B. Mochnacki, E. Majchrzak, Int. J. Heat Mass Transf. 108, 1 (2017)

    Article  Google Scholar 

  102. J. Ma, X. Yang, S. Liu, Y. Sun, J. Yang, Int. J. Heat Mass Transf. 124, 1107 (2018)

    Article  Google Scholar 

  103. P. Biswas, S. Singh, A. Srivastava, Int. J. Therm. Sci. 147, 106139 (2020)

    Article  Google Scholar 

  104. J. Zhou, J. Chen, Y. Zhang, Comput. Biol. Med. 39, 286 (2009)

    Article  Google Scholar 

  105. K.C. Liu, Int. J. Therm. Sci. 47, 507 (2008)

    Article  Google Scholar 

  106. J. Zhou, Y. Zhang, J. Chen, Int. J. Therm. Sci. 48, 1477 (2009)

    Article  Google Scholar 

  107. K. Liu, H. Chen, Int. J. Heat Mass Transf. 52, 1185 (2009)

    Article  Google Scholar 

  108. K. Liu, Y. Chen, Int. J. Therm. Sci. 103, 1 (2016)

    Article  Google Scholar 

  109. D. Kumar, S. Singh, K. Rai, Heat Mass Transf. 52, 1089 (2016)

    Article  ADS  Google Scholar 

  110. F. Xu, T. Lu, K. Seffen, E. Ng, Appl. Mech. Rev. 62, 5 (2009)

    Article  Google Scholar 

  111. K.C. Liu, Y.N. Wang, Y.S. Chen, Int. J. Therm. Sci. 58, 29 (2012)

    Article  Google Scholar 

  112. X. Jiang, M. Xu, Phys. A 389, 3368 (2010)

    Article  MathSciNet  Google Scholar 

  113. Y.Z. Povstenko, J. Therm. Stresses 28, 83 (2004)

    Article  Google Scholar 

  114. X. Huan, J. Xiao, Chin. Phys. B. 24, 034401 (2015)

    Article  Google Scholar 

  115. A. Hobiny, I. Abbas, Mech. Based Des. Struct. Mech. 1, 1–18 (2019)

    Google Scholar 

  116. D. Kumar, K. Rai, J. Therm. Biol. 67, 49 (2017)

    Article  Google Scholar 

  117. A. Narasimhan, S. Sadasivam, Int. J. Heat Mass Transf. 60, 591 (2013)

    Article  Google Scholar 

  118. A. Phadnis, S. Kumar, A. Srivastava, J. Therm. Biol. 61, 16 (2016)

    Article  Google Scholar 

  119. E. Majchrzak, L. Turchan, M. Jasiński, Iran J. Sci. Technol. Trans Mech. Eng. 43, 539 (2018)

    Article  Google Scholar 

  120. M. Ewa, T. Lukasz, Iran J. Sci. Technol. Trans Mech. Eng. 43, 548 (2019)

    Google Scholar 

  121. Z. Yong, B. Chen, Int. J. Heat Mass Transf. 108, 1428 (2017)

    Article  Google Scholar 

  122. M. Ezzat, A. Elbary, N. Alsowayan, Anim. Sci. J. 87, 1304 (2016)

    Article  Google Scholar 

  123. X. Li, C. Li, Z. Xue, X. Tian, Int. J. Therm. Sci. 124, 459 (2018)

    Article  Google Scholar 

  124. F. Xu, T. Lu, K. Seffen, Acta Mech. Sin. 24, 1 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  125. Y. Zhang, Int. J. Heat Mass Transf. 52, 4829 (2009)

    Article  Google Scholar 

  126. K. Khanafer, K. Vafai, Heat Mass Transf. 42, 939 (2006)

    Article  ADS  Google Scholar 

  127. W. Minkowycz, A. Sheikh, K. Vafai, Int. J. Heat Mass Transf. 42, 3373 (1999)

    Article  Google Scholar 

  128. E. Majchrzak, L. Turchan, J. Dziatkiewicz, Arch. Mech. 67, 417 (2015)

    Google Scholar 

  129. M. Jasiński, E. Majchrzak, L. Turchan, Appl. Math. Model. 40, 750 (2016)

    Article  MathSciNet  Google Scholar 

  130. N. Afrin, Y. Zhang, J. Chen, Int. J. Heat Mass Transf. 54, 2419 (2011)

    Article  Google Scholar 

  131. A. Alla, S. Dahab, A. Kilany, Mech. Based Des. Struct. Mech. 1, 12 (2020)

    Google Scholar 

  132. P. Hooshmand, A. Moradi, B. Khezry, Int. J. Therm. Sci. 90, 214 (2015)

    Article  Google Scholar 

  133. M. Jasinski, E. Majchrzak, L. Turchan, Appl. Math. Model. 40, 750 (2016)

    Article  MathSciNet  Google Scholar 

  134. G. Kałuża, E. Majchrzak, Ł Turchan, Appl. Math. Model. 49, 498 (2017)

    Article  MathSciNet  Google Scholar 

  135. J. Liu, X. Zhang, C. Wang, Chin. Sci. Bull. 42, 289 (1997)

    Article  Google Scholar 

  136. J. Liu, X. Chen, L.X. Xu, IEEE Trans. Biomed. Eng. 46, 420 (1999)

    Article  Google Scholar 

  137. K.C. Liu, H.T. Chen, Int. J. Therm. Sci. 49, 1138 (2010)

    Article  Google Scholar 

  138. K.C. Liu, J. Phys. D 38, 3722 (2005)

    Article  ADS  Google Scholar 

  139. K. Liu, Comput. Phys. Commun. 177, 307 (2007)

    Article  ADS  Google Scholar 

  140. M.K. Zhang, B.Y. Cao, Y.C. Guo, J. Therm. Sci. 84, 9 (2014)

    Article  Google Scholar 

  141. D. Kumar, S. Singh, K.N. Rai, Heat Mass Transf. 52, 1089 (2015)

    Article  ADS  Google Scholar 

  142. D. Kumar, K. Rai, Mech. Based Des. Struct. Mech. 1, 15 (2020)

    Google Scholar 

  143. E. Majchrzak, L. Turchan, M. Jasiński, Iran J. Sci. Technol. Trans Mech. Eng. 43, 539 (2019)

    Article  Google Scholar 

  144. W. Kaminski, ASME J. Heat Transfer 112, 555 (1990)

    Article  Google Scholar 

  145. C.M. Pacella, L. Breschi, D. Bottacci, L. Masotti, Image-Guided Laser Ablation (Springer, New York, 2020).

    Book  Google Scholar 

  146. D. Zhu, Q. Luo, G. Zhu, W. Liu, Lasers Surg. Med. 31, 313 (2002)

    Article  Google Scholar 

  147. W. Roetzel, N. Putra, S.K. Das, Int. J. Therm. Sci. 42, 541 (2003)

    Article  Google Scholar 

  148. S. Jaunich, K. Kim, K. Mitra, Int. J. Heat Mass Transf. 51, 5511 (2008)

    Article  Google Scholar 

  149. X. Li, Z. Xue, X. Tian, Int. J. Therm. Sci. 132, 249 (2018)

    Article  Google Scholar 

  150. H. Ye, S. De, Burns 43, 909 (2017)

    Article  Google Scholar 

  151. M. Mobaraki, M. Ghaffari, A. Yazdanpanah, Y. Luo, D. Mills, Bioprinting 18, 80 (2020)

    Article  Google Scholar 

  152. M. Kojic, M. Milosevic, V. Simic, V. Geroski, Comput. Biol. Med. 108, 288 (2019)

    Article  Google Scholar 

  153. S. Haddad, A. Samani, Comput. Methods Biomech Biomed. Eng. 20, 215 (2017)

    Article  Google Scholar 

  154. S. Murphy, P. Coppi, A. Atala, Nat. Biomed. Eng. 1 (2019)

  155. A. Lee, A. Hudson, D. Shiwarski, J. Tashman, T. Hinton, S. Yerneni, J. Bliley, P. Campbell, A. Feinberg, Science 365, 482 (2019)

    Article  ADS  Google Scholar 

  156. I. Matai, G. Kaur, A. Seyedsalehi, A. McClinton, C.T. Laurencin, Biomaterials 226, 119536 (2020)

    Article  Google Scholar 

  157. K. Hynynen, C. Martin, D. Watmough, J. Mallard, Brit. J. Radiol. 56, 969 (1983)

    Article  Google Scholar 

  158. A. Martin, Ultrasonics 21, 85 (1983)

    Article  Google Scholar 

  159. A. Massalimova, A. Aimbetova, N. Aimakov, A. Abdullaev, Z. Utegulov, B. Ooi, D. Tosi, J. Sens. 17, 1–13 (2017)

    Article  Google Scholar 

  160. T.H. Nguyen, S. Park, K.K. Hlaing, H.W. Kang, Biomed Opt. Express 7, 1932 (2016)

    Article  Google Scholar 

  161. I. Krylov, R.A. Akasov, V. Rocheva, N. Sholina, D. Khochenkov, A. Nechaev, N. Melnikova, A. Dmitriev, A. Ivanov, A. Generalova, Front. Chem. 8, 295 (2020)

    Article  ADS  Google Scholar 

  162. J. Yang, Y. Liu, Y. Zhao, Z. Gong, M. Zhang, D. Yan, H. Zhu, C. Liu, C. Xu, Chem. Mat. 29, 8119 (2017)

    Article  Google Scholar 

  163. M. Andres, S. Blauth, C. Leithäuser, N. Siedow, J. Math. Ind. 10, 1 (2020)

    Article  Google Scholar 

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CL was the major contributor in writing the manuscript. RZ determined the topic and direction of the paper. All authors read and approved the final manuscript.

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Correspondence to Chen-yang Lyu.

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Lyu, Cy., Zhan, Rj. Constitutive Equations Developed for Modeling of Heat Conduction in Bio-tissues: A Review. Int J Thermophys 42, 27 (2021). https://doi.org/10.1007/s10765-020-02770-0

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