Journal of Thermal Science

, Volume 13, Issue 3, pp 255–258 | Cite as

An analytic solution of one-dimensional steady-state Pennes’ bioheat transfer equation in cylindrical coordinates

  • Kai Yue
  • Xinxin Zhang
  • Fan Yu


Based on the Pennes’ bioheat transfer equation, a simplified one-dimensional bioheat transfer model of the cylindrical living tissues in the steady state has been set up for application in limb and whole body heat transfer studies, and by using the Bessel’s equation, its corresponding analytic solution has been derived in this paper. With the obtained analytic solution, the effects of the thermal conductivity, the blood perfusion, the metabolic heat generation, and the coefficient of heat transfer on the temperature distribution in living tissues are analyzed. The results show that the derived analytic solution is useful to easily and accurately study the thermal behavior of the biological system, and can be extended to such applications as parameter measurement, temperature field reconstruction and clinical treatment.

Key words

bioheat transfer Pennes’ equation analytic solution Bessel functions 

CLC number



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Diller, K R, Valvano, J W, Pearce, J A. Bioheat Transfer. London, Springer: The CRC Handbook of Thermal Engineering 2000, 2000. 4-114–4-176Google Scholar
  2. [2]
    Chato, J C. Reflections on the History of Heat and Mass Transfer in Bioengineering. ASME Journal of Biomechanical Eng., 1981, 103: 97–101CrossRefGoogle Scholar
  3. [3]
    Shitzer, A, Chato, J C. Analytical Solutions to the Problem of Transient Heat Transfer in Living Tissue. ASME Journal of Biomechanical Eng., 1978, 100: 202–210Google Scholar
  4. [4]
    Bardati, F, Gerosa, G. On the Solution of the Non-linear Bio-heat Equation. Journal of Biomechanics, 1990, 23: 791–798CrossRefGoogle Scholar
  5. [5]
    Pennes, H H. Analysis of Tissue and Arterial Temperatures in the Resting Human Torearm. Journal of Applied Physiology, 1948, (1): 93–122Google Scholar
  6. [6]
    Diller, K R, Ryan, T P. Heat Transfer in Living System: Current Opportunities. Transactions of the ASME, 1998, 120: 810–829Google Scholar
  7. [7]
    Özisik, M N, Yu, C M. Heat Conduction. Beijing: Higher Education Press. 1983. 712–715Google Scholar
  8. [8]
    Wang, B X, Wang, Y M. Study on the Basic Equations of Biomedical Heat Transfer. Transport Phenomena Science and Technology. Beijing: Higher Education Press. 1992. 773–776Google Scholar
  9. [9]
    Werner, J, Buse, M. Three-dimensional Simulation of Cold and Warm Defence in Man. J. Appl. Physiol., 1988, 65(3): 1110–1118Google Scholar
  10. [10]
    Stolwijk, J A J. A Mathematical Model of Physiological Temperature Regulation in Man. NASA, 1971: CR-1855Google Scholar

Copyright information

© Science Press 2004

Authors and Affiliations

  • Kai Yue
    • 1
  • Xinxin Zhang
    • 1
  • Fan Yu
    • 1
  1. 1.School of Mechanical EngineeringUniversity of Science and Technology BeijingBeijingChina

Personalised recommendations