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.

References

References
1.
Carslaw
,
H. S.
, and
Jaeger
,
J. C.
,
1959
,
Conduction of Heat in Solids
,
Clarendon Press
,
London
.
2.
Özişik
,
M. N.
,
1993
,
Heat Conduction
,
John Wiley and Sons
,
New York
.
3.
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
.10.1016/j.ijheatmasstransfer.2003.09.011
4.
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
.10.1016/S0017-9310(00)00008-9
5.
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
.10.1016/S0017-9310(01)00226-5
6.
de Monte
,
F.
,
2006
, “
Multi-Layer Transient Heat Conduction Using Transition Time Scales
,”
Int. J. Therm. Sci.
,
45
(
9
), pp.
882
892
.10.1016/j.ijthermalsci.2005.11.006
7.
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
.10.1016/S0017-9310(83)80167-7
8.
Özişik
,
M.
,
1968
,
Boundary Value Problems of Heat Conduction
,
International Textbook Company
,
Scranton, PA
.
9.
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
.10.1016/j.ijheatmasstransfer.2009.08.013
10.
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
.10.1088/0031-9155/36/3/004
11.
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
.10.1088/0031-9155/35/7/004
12.
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
.10.1088/0031-9155/35/7/005
13.
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
.10.1115/1.3124646
14.
Pennes
,
H. H.
,
1948
, “
Analysis of Tissue and Arterial Blood Temperatures in the Resting Human Forearm
,”
J. Appl. Physiol.
,
1
(
2
), pp.
93
122
.
15.
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
.10.1115/1.3138533
16.
Chato
,
J. C.
,
1980
, “
Heat-Transfer to Blood-Vessels
,”
ASME J. Biomech. Eng.
,
102
(
2
), pp.
110
118
.10.1115/1.3138205
17.
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
.10.1088/0031-9155/34/10/011
18.
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
.10.1007/s00231-011-0886-5
19.
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
.10.1016/j.ijthermalsci.2012.05.002
20.
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
.10.1016/j.ijheatmasstransfer.2008.05.015
21.
Pontrelli
,
G.
, and
de Monte
,
F.
,
2009
, “
Modeling of Mass Dynamics in Arterial Drug-Eluting Stents
,”
J. Porous Media
,
12
(
1
), pp.
19
28
.10.1615/JPorMedia.v12.i1.20
22.
Beck
,
J. V.
,
Cole
,
K. D.
,
Haji-Sheikh
,
A.
, and
Litkouhi
,
B.
,
1992
,
Heat Conduction Using Green's Functions
,
Hemisphere Publishing Corporation, Washington, DC
.
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