0
Research Papers

One-Dimensional Transient Heat Conduction in Composite Living Perfuse Tissue

[+] Author and Article Information
S. M. Becker

Department of Mechanical Engineering,
University of Canterbury,
Private Bag 4800,
Christchurch 8401, NZ
e-mail: sid.becker@canterbury.ac.nz

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received December 1, 2011; final manuscript received October 27, 2012; published online June 21, 2013. Assoc. Editor: Franz-Josef Kahlen.

J. Heat Transfer 135(7), 071002 (Jun 21, 2013) (11 pages) Paper No: HT-11-1542; doi: 10.1115/1.4024063 History: Received December 01, 2011; Revised October 27, 2012

Modeling the conduction of heat in living tissue requires the consideration of sudden spatial discontinuities in property values as well as the presence of the body's circulatory system. This paper presents a description of the separation of variables method that results in a remarkably simple solution of transient heat conduction in a perfuse composite slab for which at least one of the layers experiences a zero perfusion rate. The method uses the natural analytic approach and formats the description so that the constants of integration of each composite layer are expressed in terms of those of the previous layer's eigenfunctions. This allows the solution to be “built” in a very systematic and sequential manner. The method is presented in the context of the Pennes bioheat equation for which the solution is developed for a system composed of any number of N layers with arbitrary initial conditions.

FIGURES IN THIS ARTICLE
<>
Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.

References

Carslaw, H. S., and Jaeger, J. C., 1959, Conduction of Heat in Solids, Clarendon Press, London.
Özişik, M. N., 1993, Heat Conduction, John Wiley and Sons, New York.
Sun, Y. Z., and Wichman, I. S., 2004, “On Transient Heat Conduction in a One-Dimensional Composite Slab,” Int. J. Heat Mass Trans., 47(6–7), pp. 1555–1559. [CrossRef]
de Monte, F., 2000, “Transient Heat Conduction in One-Dimensional Composite Slab. A ‘Natural’ Analytic Approach,” Int. J. Heat Mass Trans., 43(19), pp. 3607–3619. [CrossRef]
de Monte, F., 2002, “An Analytic Approach to the Unsteady Heat Conduction Processes in One-Dimensional Composite Media,” Int. J. Heat Mass Trans., 45(6), pp. 1333–1343. [CrossRef]
de Monte, F., 2006, “Multi-Layer Transient Heat Conduction Using Transition Time Scales,” Int. J. Therm. Sci., 45(9), pp. 882–892. [CrossRef]
Mikhailov, M. D., Ozisik, M. N., and Vulchanov, N. L., 1983, “Diffusion in Composite Layers With Automatic Solution of the Eigenvalue Problem,” Int. J. Heat Mass Trans., 26(8), pp. 1131–1141. [CrossRef]
Özişik, M., 1968, Boundary Value Problems of Heat Conduction, International Textbook Company, Scranton, PA.
Hickson, R. I., Barry, S. I., and Mercer, G. N., 2009, “Critical Times in Multilayer Diffusion—Part 1: Exact Solutions,” Int. J. Heat Mass Trans., 52(25–26), pp. 5776–5783. [CrossRef]
Durkee, J. W., and Antich, P. P., 1991, “Exact-Solutions to the Multiregion Time-Dependent Bioheat Equation With Transient Heat-Sources and Boundary-Conditions,” Phys. Med. Biol., 36(3), pp. 345–368. [CrossRef]
Durkee, J. W., Antich, P. P., and Lee, C. E., 1990, “Exact-Solutions to the Multiregion Time-Dependent Bioheat Equation 1. Solution Development,” Phys. Med. Biol., 35(7), pp. 847–867. [CrossRef] [PubMed]
Durkee, J. W., Antich, P. P., and Lee, C. E., 1990, “Exact-Solutions to the Multiregion Time-Dependent Bioheat Equation 2. Numerical Evaluation of the Solutions,” Phys. Med. Biol., 35(7), pp. 869–889. [CrossRef] [PubMed]
Xu, F., Lu, T. J., Seffen, K. A., and Ng, E. Y. K., 2009, “Mathematical Modeling of Skin Bioheat Transfer,” Appl. Mech. Rev., 62(5), p. 050801. [CrossRef]
Pennes, H. H., 1948, “Analysis of Tissue and Arterial Blood Temperatures in the Resting Human Forearm,” J. Appl. Physiol., 1(2), pp. 93–122. [PubMed]
Weinbaum, S., and Jiji, L. M., 1985, “A New Simplified Bioheat Equation for the Effect of Blood-Flow on Local Average Tissue Temperature,” ASME J. Biomech. Eng., 107(2), pp. 131–139. [CrossRef]
Chato, J. C., 1980, “Heat-Transfer to Blood-Vessels,” ASME J. Biomech. Eng., 102(2), pp. 110–118. [CrossRef]
Hodson, D. A., Barbenel, J. C., and Eason, G., 1989, “Modeling Transient Heat-Transfer Through the Skin and a Contact Material,” Phys. Med. Biol., 34(10), pp. 1493–1507. [CrossRef] [PubMed]
Becker, S. M., 2012, “Analytic One Dimensional Transient Conduction Into a Living Perfuse/Non-Perfuse Two Layer Composite System,” Heat Mass Trans., 48(2), pp. 317–327. [CrossRef]
deMonte, F., Beck, J., and Amos, D., 2012, “Solving Two-Dimensional Cartesian Unsteady Heat Conduction Problems for Small Values of the Time,” Int. J. Therm. Sci., 60, pp. 106–113. [CrossRef]
de Monte, F., Beck, J. V., and Amos, D. E., 2008, “Diffusion of Thermal Disturbances in Two-Dimensional Cartesian Transient Heat Conduction,” Int. J. Heat Mass Trans., 51(25–26), pp. 5931–5941. [CrossRef]
Pontrelli, G., and de Monte, F., 2009, “Modeling of Mass Dynamics in Arterial Drug-Eluting Stents,” J. Porous Media, 12(1), pp. 19–28. [CrossRef]
Beck, J. V., Cole, K. D., Haji-Sheikh, A., and Litkouhi, B., 1992, Heat Conduction Using Green's Functions, Hemisphere Publishing Corporation, Washington, DC.

Figures

Grahic Jump Location
Fig. 1

Steady state two-layer system composite representation in which transport in layer i − 1 is governed by pure conduction and transport within layer i has both perfusion and conduction

Grahic Jump Location
Fig. 2

Composite system representation of the perfuse tissue slab

Grahic Jump Location
Fig. 3

Schematic of the three layered slab in which the outer layers are exposed to convection at the periphery, the first layer experiences no perfusion, and the third layer experiences volumetric energy generation

Grahic Jump Location
Fig. 4

Maximum error at small dimensionless times defined by Eq. (52): comparison between the test case solution and the semi-infinite Green's function solution of Eq. (51)

Grahic Jump Location
Fig. 5

Transient solution of the case study at different dimensionless times corresponding to test case parameter values: δ1 = 9/200, δ2 = 1/50, δ3 = 1/1000, HO = 1, HL = 10, m1 = 0, m2 = 1, m3 = 4, K2/K1 = 5/9, K3/K1 = 5/18, Δθ2 = 0, Δθ3 = −1, ΔθO = 1, ΔθL = 1, and ϕ3 = 5

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In