Effect of convective term on temperature distribution in biological tissue

Regular Article


We introduce a phase imprint into the order parameter describing the influence of blood flow on the temperature distribution in the tissue described by the one-dimensional Pennes equation and then engineer the imprinted phase suitably to generate a modified Pennes equation with a gradient term (known in the theory of biological systems as convective term) which is associated with the heat convected by the flowing blood. Using the derived model, we analytically investigate temperature distribution in biological tissues subject to two different spatial heating methods. The applicability of our results is illustrated by one of typical bio-heat transfer problems which is often encountered in therapeutic treatment, cancer hyperthermia, laser surgery, thermal injury evaluation, etc. Analyzing the effect of the convective term on temperature distribution, we found that an optimum heating of a biological system can be obtained through regulating the convective term.


Biological Tissue Skin Surface Convective Term Tissue Temperature Initial Temperature Distribution 
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  1. 1.
    F.C. Henriques, A.R. Moritz, Am. J. Pathol. 23, 431 (1947)Google Scholar
  2. 2.
    H.H. Pennes, J. Appl. Physiol. 1, 93 (1948)ADSGoogle Scholar
  3. 3.
    J.A.J. Stolwijk, J.D. Hardy, Pflugers Arch. 291, 129 (1966)CrossRefGoogle Scholar
  4. 4.
    E.H. Wissler, J. Physiol. (Paris) 63, 455 (1970)Google Scholar
  5. 5.
    A. Shitzer, R.C. Eberhart, Heat Transfer in Medicine and Biology (Plenum Press, New York, 1985)Google Scholar
  6. 6.
    M.J. Mantyla, J. Kuikka, A. Rekonnen, Brit. J. Radiol. 49, 335 (1976)CrossRefGoogle Scholar
  7. 7.
    J.A. Surrell, R.C. Alexander, S.D. Cohle Jr FRL, R.A. Wehrenberg, J. Trauma. 27, 935 (1987)CrossRefGoogle Scholar
  8. 8.
    W.P. Zhu, F.B. Tian, P. Ran, Int. J. Biomath. 5, 1250022 (2012)CrossRefMathSciNetGoogle Scholar
  9. 9.
    J. Liu, L. Zhu, X.L. Xu, ASME J. Biomech. Eng. 122, 372 (2000)CrossRefGoogle Scholar
  10. 10.
    G.T. Martin, H.F. Bowman, W.H. Newman, E.G. Cravalho, Adv. Biol. Heat Mass Transf. 18, 33 (1991)Google Scholar
  11. 11.
    A.M. Stoll, M.A. Chianta, J.R. Piergallini, Aviat. Space Env. Med. 50, 778 (1979)Google Scholar
  12. 12.
    R. Seip, E.S. Ebbini, IEEE Trans. Biomed. Eng. 42, 828 (1995)CrossRefGoogle Scholar
  13. 13.
    A.M. Stoll, J. Invest. Dermatol. 69, 328 (1977)CrossRefGoogle Scholar
  14. 14.
    E. Kengne, A. Lakhssassi, R. Vaillancourt, W.M. Liu, Eur. Phys. J. Plus 127, 89 (2012)CrossRefGoogle Scholar
  15. 15.
    E. Kengne, A. Lakhssassi, R. Vaillancourt, Appl. Math. 3, 217 (2012)CrossRefMathSciNetGoogle Scholar
  16. 16.
    A. Lakhssassi, E. Kengne, H. Semmaoui, Nat. Sci. 2, 131 (2010)Google Scholar
  17. 17.
    E.H. Wissler, J. Appl. Physiol. 85, 35 (1998)Google Scholar
  18. 18.
    R.B. Roemer, B.R. Paliwal, F.W. Hetzel, M.W. Dewhirst, Heat transfer in hyperthermia treatments: basic principles and applications, in Biological Physical and Clinical Aspects of Hyperthermia, edited by B.R. Paliwal, F.W. Hetzel, M.W. Dewhirst (AIP, New York, 1988) pp. 210-242Google Scholar
  19. 19.
    T.R. Gowrishankar, D.A. Stewart, G.T. Martin, J.C. Weaver, BioMed. Eng. On-Line 3, 1 (2004)Google Scholar
  20. 20.
    J. Erdmann, B. Lang, M. Seebass, IEEE Trans. Biomed. Eng. 46, 1129 (1999)CrossRefGoogle Scholar
  21. 21.
    W. Wulff, IEEE Trans. Biomed. Eng. 21, 494 (1974)CrossRefGoogle Scholar
  22. 22.
    H. Arkin, L.X. Xu, K.R. Holmes, IEEE Trans. Biomed. Eng. 41, 97 (1994)CrossRefGoogle Scholar
  23. 23.
    H.G. Klinger, Bull. Math. Biol. 36, 403 (1974)MATHMathSciNetGoogle Scholar
  24. 24.
    M.M. Chen, K.R. Holmes, Annu. NY Acad. Sci. 335, 137 (1980)ADSCrossRefGoogle Scholar
  25. 25.
    L.M. Jiji, S. Weinhaum, D.E. Lemons, ASME J. Biomech. Eng. 106, 331 (1984)CrossRefGoogle Scholar
  26. 26.
    S. Weinhaum, L.M. Jiji, ASME J. Biomech. Eng. 107, 131 (1985)CrossRefGoogle Scholar
  27. 27.
    J.W. Baish, ASME J. Biomech. Eng. 116, 521 (1994)CrossRefGoogle Scholar
  28. 28.
    E.H. Wissler, ASME J. Biomech. Eng. 109, 226 (1987)CrossRefGoogle Scholar
  29. 29.
    J. Lang, B. Erdmann, M. Seebass, IEEE Trans. Biomed. Eng. 46, 1129 (1999)CrossRefGoogle Scholar
  30. 30.
    K.R. Holmes, Biological Structures and Heat Transfer, in Allerton Workshop on the Future of Biothermal Engineering (1997)Google Scholar
  31. 31.
    J. Liu, L.X. Xu, Int. J. Heat Mass Transf. 43, 2827 (2000)CrossRefMATHGoogle Scholar
  32. 32.
    N.L. Markee, K.L. Hatch, H.I. Maibach, R.L. Barker, P. Radhakrishnaiah, S.S. Woo, Textile Res. J. 60, 561 (1990)CrossRefGoogle Scholar
  33. 33.
    J. Liu, L.X. Xu, IEEE Trans. Biomed. Eng. 46, 1037 (1999)CrossRefGoogle Scholar
  34. 34.
    S. Weinbaum, L.M. Jiji, D.E. Lemons, ASME J. Biomech. Eng. 106, 321 (1984)CrossRefGoogle Scholar
  35. 35.
    K.R. Diller, Adv. Heat Transfer 22, 157 (1992)CrossRefGoogle Scholar
  36. 36.
    J.H. Li, H. Liang, Laser Medicine - Applications of Laser in Biology and Medicine (Science Press, Beijing, 1989)Google Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Emmanuel Kengne
    • 1
  • Michel Saydé
    • 1
  • Ahmed Lakhssassi
    • 1
  1. 1.Département d’informatique et d’ingénierieUniversité du Québec en OutaouaisGatineau(PQ)Canada

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