Constructal Theory Applied to Vascular Countercurrent Networks

  • Weizhong DaiEmail author
Part of the Understanding Complex Systems book series (UCS)


Heat transfer within the human skin is a complicated process involving metabolic heat generation, heat conduction and blood perfusion in tissue, convection and perfusion of the arterial-venous blood through the capillary, and interaction with the environment. Modeling of heat-related phenomena such as bioheat transfer is important in the development of biological and biomedical technologies, such as thermotherapy of skin cancer and the design of heating or cooling garments, as well as protecting human life in cases of accidental or natural disasters [1, 2]. In this chapter, the constructal theory of multi-scale tree-shaped heat exchangers is applied to the vascular countercurrent network embedded in a three-dimensional triple layered skin structure. Based on the designed vascular countercurrent network, we present our mathematical models and numerical results for predicting skin burn injury induced by intense radiation heating and for optimizing skin temperature induced by laser or electromagnetic radiations related to hyperthermia cancer treatments.


Skin Surface Constructal Theory Skin Structure Cellular Tissue Blood Temperature 
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  1. 1.
    Borkebak RC. Heat transfer in biological systems. Int Rev Gen Exper Zool. 1966;2:269–344.Google Scholar
  2. 2.
    Bowman HF, Cravalho EG, Woods M. Theory, measurement and application of properties of biomaterials. Annu Rev Biophys Bioeng. 1975;4:43–80.CrossRefGoogle Scholar
  3. 3.
    Bergman TL, Faghri A, Viskanta R. Frontiers in transport phenomena research and education: energy systems, biological systems, security, information technology and nanotechnology. Int J Heat Mass Tran. 2008;53:4599–613.CrossRefGoogle Scholar
  4. 4.
    Lubashevsky IV, Gafiychuk V. Mathematical description of heat transfer in living tissue. Unpublished bookGoogle Scholar
  5. 5.
    Chato JC. Fundamentals of bioheat transfer. In: Gautherie M, editor. Clinical thermology: thermal dosimetry and treatment planning. Berlin: Springer; 1990.Google Scholar
  6. 6.
    Gartner LP. Color atlas of histology. 3rd ed. Philadelphia: Lippincott Williams and Wilkins; 2000.Google Scholar
  7. 7.
    Bejan A. Shape and structure, from engineering to nature. Cambridge: Cambridge University Press; 2000.zbMATHGoogle Scholar
  8. 8.
    Bejan A. The tree of convective heat streams: its thermal insulation function and the predicted 3/4-power relation between body heat loss and body size. Int J Heat Mass Tran. 2001;44:699–704.zbMATHCrossRefGoogle Scholar
  9. 9.
    Bejan A, Lorente S. Constructal theory of generation of configuration in nature and engineering. J Appl Phys. 2006;100:041301.CrossRefGoogle Scholar
  10. 10.
    Bejan A, Lorente S. Design with constructal theory. Hoboken: Wiley; 2008.CrossRefGoogle Scholar
  11. 11.
    da Silva AK, Lorente S, Bejan A. Constructal multi-scale tree-shaped heat exchangers. J Appl Phys. 2004;96:1709–18.CrossRefGoogle Scholar
  12. 12.
    Dai W, Bejan A, Tang X, Zhang L, Nassar R. Optimal temperature distribution in a 3D triple layered skin structure with embedded vasculature. J Appl Phys. 2006;99:104702.CrossRefGoogle Scholar
  13. 13.
    Dai W, Wang H, Jordan PM, Mickens RE, Bejan A. A mathematical model for skin burn injury induced by radiation heating. Int J Heat Mass Tran. 2008;51:5497–510.zbMATHCrossRefGoogle Scholar
  14. 14.
    Tang X, Dai W, Nassar R, Bejan A. Optimal temperature distribution in a 3D triple layered skin structure embedded with artery and vein vasculature. Numer Heat Transfer A. 2006;50:809–43.CrossRefGoogle Scholar
  15. 15.
    Wang H, Dai W, Bejan A. Optimal temperature distribution in a 3D triple layered skin structure embedded with artery and vein vasculature and induced by electromagnetic radiation. Int J Heat Mass Tran. 2007;50:1843–54.zbMATHCrossRefGoogle Scholar
  16. 16.
    Zeng X, Dai W, Bejan A. Vascular countercurrent network for 3D triple-layered skin structure with radiation heating. Numer Heat Transfer A. 2010;57:369–91.CrossRefGoogle Scholar
  17. 17.
    Huang H, Chen ZP, Roemer R. A counter current vascular network model of heat transfer in tissues. J Biomech Eng. 1996;118:120–9.CrossRefGoogle Scholar
  18. 18.
    Majchrzak E, Mochnacki B. Numerical model of heat transfer between blood vessel and biological tissue. Comput Assist Mech Eng Sci. 1999;6:439–47.zbMATHGoogle Scholar
  19. 19.
    Liu J. Preliminary survey on the mechanisms of the wave-like behaviors of heat transfer in living tissues. Forschung im Ingenieurwesen. 2000;66:1–10.CrossRefGoogle Scholar
  20. 20.
    Liu J, Chen X, Xu LX. New thermal wave aspects on burn evaluation of skin subjected to instantaneous heating. IEEE Trans Biomed Eng. 1999;46:420–8.CrossRefGoogle Scholar
  21. 21.
    Tzou DY. Macro-to-microscale heat transfer: the lagging behavior. Washington: Taylor and Francis; 1996.Google Scholar
  22. 22.
    Mitra K, Kumar S, Vedavarz A, Moallemi MK. Experimental evidence of hyperbolic heat conduction in processed meat. Trans ASME J Heat Transfer. 1995;117:568–73.CrossRefGoogle Scholar
  23. 23.
    Sturesson C, Andersson-Engels A. A mathematical model for predicting the temperature distribution in laser-induced hyperthermia: experimental evaluation and applications. Phys Med Biol. 1995;40:2037–52.CrossRefGoogle Scholar
  24. 24.
    Henriques FC, Mortiz AR. Studies of thermal injury in the conduction of heat to and through skin and the temperature attained therein: a theoretical and experimental investigation. Am J Pathol. 1947;23:531–49.Google Scholar
  25. 25.
    Diller KR. Modeling of bioheat transfer processes at high and low temperature. In: Cho YI, Hartnett JP, Irvine Jr TF, editors. Advanced in heat transfer, 22. New York: Academic; 1992. p. 157–357.Google Scholar
  26. 26.
    Jaesung H, Klavs FJ. Combined experimental and modeling studies of laser-assisted chemical vapor deposition of copper from copper (I)-hexafluoroacetylacetonate-trimethylviny-lsilane. J Appl Phys. 1994;75:2240–50.CrossRefGoogle Scholar
  27. 27.
    Cole KS, Cole RH. Dispersion and absorption in dielectrics I. alternating current characteristics. J Chem Phys. 1941;9:341–51.CrossRefGoogle Scholar
  28. 28.
    Gabriel S, Lau RW, Gabriel C. The dielectric properties of biological tissues: III. parametric models for the dielectric spectrum of tissues. Phys Med Biol. 1996;41:2271–93.CrossRefGoogle Scholar
  29. 29.
    Dincov DD, Parrott KA, Pericleous KA. A new computational approach to microwave heating of two-phase porous materials. Int J Numer Method Heat & Fluid Flow. 2004;14:783–802.zbMATHCrossRefGoogle Scholar
  30. 30.
    Sullivan DM. Electromagnetic simulation using the FDTD method. New York: IEEE; 1999.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Mathematics and Statistics, College of Engineering and ScienceLouisiana Tech UniversityRustonUSA

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