In this study, an analytical solution to the heat transfer equation in biological tissues during laser heating is presented. The results were compared to recently published numerical simulations.

References

1.
Goldstein
,
R. J.
,
Ibele
,
W. E.
,
Patankar
,
S. V.
,
Simon
,
T. W.
,
Kuehn
,
T. H.
,
Strykowski
,
P. J.
,
Tamma
,
K. K.
,
Heberlein
,
J. V. R.
,
Davidson
,
J. H.
,
Bischof
,
J.
,
Kulacki
,
F. A.
,
Kortshagen
,
U.
,
Garrick
,
S.
,
Srinivasan
,
V.
,
Ghosh
,
K.
, and
Mittal
,
R.
, 2010, “
Heat Transfer—A Review of 2005 Literature
,”
Int. J. Heat Mass Transfer
,
53
, pp.
4397
4447
.
2.
Narasimhan
,
A.
,
Kumar Jha
,
K.
, and
Gopa
,
L.
, 2010, “
Transient Simulations of Heat Transfer in Human Eye Undergoing Laser Surgery
,”
Int. J. Heat Mass Transfer
,
53
, pp.
482
490
.
3.
Zhang
,
Y.
, 2009, “
Generalized Dual-Phase Lag Bioheat Equations Based on Nonequilibrium Heat Transfer in Living Biological Tissues
,”
Int. J. Heat Mass Transfer
,
52
, pp.
4829
4834
.
4.
Manns
,
F.
,
Borja
,
D.
,
Parel
,
J. M.
,
Smiddy
,
W.
, and
Culbertson
,
W.
, 2003, “
Semi-Analytical Thermal Model Tissue: Application to Laser Thermokeratoplasty
,”
J. Biomed. Opt.
,
8
(
2
), pp.
288
297
.
5.
Manns
,
F.
,
Milne
,
P .J.
,
Gonzalez-Cirre
,
X.
,
Denham
,
D. B.
,
Parel
,
J. M.
, and
Robinson
,
D. S.
, 1998, “
In Situ Temperature Measurements With Thermocouple Probes During Laser Interstitial Thermometry (LITT): Quantification and Correction of a Measurement Artifact
,”
Lasers Surg. Med. Lasers Surg. Med.
,
23
, pp.
94
103
.
6.
Panjehpour
,
M.
,
Wilke
,
A.
,
Frazier
,
D. L.
, and
Overholt
,
B. F.
, 1991,”
Proc. SPIE
,
1427
, pp.
307
315
.
7.
Amin
,
Z.
,
Donald
,
J. J.
,
Masters
,
A.
,
R.
Kant
,
W.
Lees
,
S.
Brown
, 1993, “
Interstitial Laser Photocoagulation Therapy for Liver Tumours: Clinical Results
,”
Proc. SPIE
,
1882
, pp.
202
209
.
8.
Fried
,
M. P.
, 1991, “
Lasers in Clinical Otolaryngology: Current Uses and Future Applications
,”
Ear Nose Throat J.
,
70
, pp.
843
847
.
9.
Vogl
,
T. J.
,
Mack
,
M. G.
,
Straub
,
R.
,
Roggan
,
A.
, and
Felix
,
R.
1997, “
Percutaneous MRI-Guided Laser-Induced Thermotherapy for Hepatic Metastases for Colorectal Cancer
,”
Lancet
,
350
, pp.
329
350
.
10.
Storm
,
F. K.
, 1983,
Hyperthermiain Cancer Therapy
,
GK Hall Medical Publishers
,
Boston
.
11.
Hahn
,
G. M.
, 1979, “
Potential for Therapy of Drugs and Hyperthermia
,”
Cancer Res.
,
39
, pp.
2264
2268
.
12.
Ono
,
A.
,
Fujii
,
T.
,
Saito
,
Y.
Matsuda
,
T.
,
Lee
,
DT.
,
Gotoda
,
T.
, and
Saito
,
D.
, 2003, “
Endoscopic Submucosal Resection of Rectal Carcinoid Tumors With a Ligation Device
,”
Gastrointest. Endosc
,
57
(
4
), pp.
583
587
.
13.
Belhadj
,
A.
,
Onyango
,
O.
, and
Rozibaeva
,
N.
, 2009, “
Boubaker Polynomials Expansion Scheme-Related Heat Transfer Investigation Inside Keyhole Model
,”
J. Thermophys. Heat Transfer
,
23
, pp.
639
640
.
14.
Chaouachi
,
A.
,
Boubaker
,
K.
,
Amlouk
,
M.
, and
Bouzouita
,
H.
, 2007, “
Enhancement of Pyrolysis Spray Disposal Performance Using Thermal Time-Response to Precursor Uniform Deposition
,”
Eur. Phys. J. Appl. Phys.
,
37
, pp.
105
109
.
15.
Fridjine
,
S.
, and
Amlouk
,
M.
, 2009, “
A New Parameter: An abacus for Optimizing Functional Materials Using the Boubaker Polynomials Expansion Scheme
,”
Mod. Phys. Lett. B
,
23
, pp.
2179
2182
.
16.
Fridjine
,
S.
,
Boubaker
,
K.
, and
Amlouk
,
M.
, 2009, “
Some Electron Probe X-Ray Microanalysis (EPMA)- and (BPES)-Related Physical Investigations on ZnSSe Thin-Films Growth Composition-Related Kinetics
,”
Can. J. Phys.
,
87
, pp.
653
657
.
17.
Guezmir
,
N.
,
Ben Nasrallah
,
T.
,
Boubaker
,
K.
,
Amlouk
,
M.
, and
Belgacem
,
S.
, 2009, “
Optical Modeling of Compound CuInS2 Using Relative Dielectric Function Approach and Boubaker Polynomials Expansion Scheme BPES
,”
J. Alloys Compd.
,
481
, pp.
543
548
.
18.
Ouni
,
B.
,
Boukhachem
,
A.
,
Dabbous
,
S.
,
Amlouk
,
A.
,
Boubaker
,
K.
, and
Amlouk
,
M.
, 2010, “
Some Transparent Semi-Conductor Metal Oxides: Comparative Investigations in Terms of Wemple-DiDomenico Parameters, Mechanical Performance and Amlouk-Boubaker Opto-Thermal Expansivity
,”
Mater. Sci. Semicond. Process.
,
13
, pp.
281
287
.
19.
Amlouk
,
A.
,
Boubaker
,
K.
,
El Mir
,
L.
, and
Amlouk
,
M.
, 2011, “
TiO2 Films Annealing Temperature-Dependent Properties in Terms of the Amlouk-Boubaker Opto-Thermal Expansivity ψAB
,”
Eur. Phys. J. Appl. Phys.
,
53
, p.
20502
.
20.
Ghanouchi
,
J.
,
Labiadh
,
H.
, and
Boubaker
,
K.
, 2008, “
An Attempt to Solve the Heat Transfer Equation in a Model of Pyrolysis Spray Using 4q-Order Boubaker Polynomials
,”
Int. J. Heat Technol.
,
26
, pp.
49
53
.
21.
Ghrib
,
T.
,
Boubaker
,
K.
, and
Bouhafs
,
M.
, 2008, “
Investigation of Thermal Diffusivity-Microhardness Correlation Extended to Surface-Nitrured Steel Using Boubaker Polynomials Expansion
,”
Mod. Phys. Lett. B
,
22
, pp.
2893
2907
.
22.
Labiadh
,
H.
, and
Boubaker
,
K.
, 2007, “
A Sturm-Liouville Shaped Characteristic Differential Equation as a Guide to Establish a Quasi-Polynomial Expression to the Boubaker Polynomials
,”
Differ. Eq. Control Process.
,
2
, pp.
117
133
.
23.
Oyodum
,
O. D.
,
Awojoyogbe
,
O. B.
,
Dada
,
M.
, and
Magnuson
,
J.
, 2009, “
On the Earliest Definition of the Boubaker Polynomials
,”
Eur. Phys. J.-Appl. Phys.
,
46
, pp.
21201
21203
.
24.
Slama
,
S.
,
Bouhafs
,
M.
, and
Ben Mahmoud
,
K. B.
, 2008,“
A Boubaker, 2008, Polynomials Solution to Heat Equation for Monitoring A3 Point Evolution During Resistance Spot Welding
,”
Int. J. Heat Technol.
,
26
, pp.
141
146
.
25.
Slama
,
S.
,
Bessrour
,
J.
,
Bouhafs
,
M.
, and
Ben Mahmoud
,
K. B.
, 2009, “
Numerical Distribution of Temperature as a Guide to Investigation of Melting Point Maximal Front Spatial Evolution During Resistance Spot Welding Using Boubaker Polynomials
,”
Numer. Heat Transfer, Part A
,
55
, pp.
401
408
.
26.
Slama
,
S.
,
Boubaker
,
K.
,
Bessrour
,
J.
, and
Bouhafs
,
M.
, 2009, “
Study of Temperature 3D Profile During Weld Heating Phase Using Boubaker Polynomials Expansion
,”
Thermochim. Acta
,
482
, pp.
8
11
.
27.
Tabatabaei
,
S.
,
Zhao
,
T.
,
Awojoyogbe
,
O.
, and
Moses
,
F.
, 2009, “
Cut-Off Cooling Velocity Profiling Inside a Keyhole Model Using the Boubaker Polynomials Expansion Scheme
,”
Heat Mass Transfer
,
45
, pp.
1247
1251
.
28.
Zhao
,
T. G.
,
Wang
,
Y. X.
, and
Ben Mahmoud
,
K. B.
, 2008, “
Limit and Uniqueness of the Boubaker-Zhao Polynomials Imaginary Root Sequence
,”
Int. J. Math. Comput.
,
1
, pp.
13
16
.
29.
Barry
,
P.
, and
Hennessy
,
A.
, 2010, “
Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, Section 6: The Boubaker Polynomials
,”
J. Integer Seq.
,
13
, pp.
1
34
.
30.
Agida
,
M.
, and
Kumar
,
A. S.
, 2010, “
A Boubaker Polynomials Expansion Scheme Solution to Random Love Equation in the Case of a Rational Kernel
,”
Electron. J. Theor. Phys.
,
7
, pp.
319
326
.
31.
Yıldırım
,
A.
, and
Mohyud-Din
,
S. T.
,
Zhang
,
D. H.
, 2010, “
Analytical Solutions to the Pulsed Klein-Gordon Equation Using Modified Variational Iteration Method (MVIM) and Boubaker Polynomials Expansion Scheme (BPES)
,”
Comput. Math. Appl.
,
59
(
8
), pp.
2473
2477
.
32.
Awojoyogbe
,
O. B.
, and
Boubaker
,
K.
, 2009, “
A Solution to Bloch NMR Flow Equations for the Analysis of Homodynamic Functions of Blood Flow Using m-Boubaker Polynomials
,”
Curr. Appl. Phys.
,
9
, pp.
278
288
.
33.
Belhadj
,
A.
,
Bessrour
,
J.
,
Bouhafs
,
M.
, and
Barrallier
,
L.
, 2009, “
Experimental and Theoretical Cooling Velocity Profile Inside Laser Welded Metals Using Keyhole Approximation and Boubaker Polynomials Expansion
,”
J. Thermal Anal. Calorim.
,
97
, pp.
911
920
.
34.
Kumar
,
A. S.
, 2010, “
An Analytical Solution to Applied Mathematics-Related Love’s Equation Using the Boubaker Polynomials Expansion Scheme
,”
J. Franklin Inst.
,
347
, pp.
1755
1761
.
35.
Milgram
,
A.
, 2011, “
The Stability of the Boubaker Polynomials Expansion Scheme (BPES)-Based Solution to Lotka-Volterra Problem
,”
J. Theoret. Biol.
,
271
, pp.
157
158
.
36.
Tung
,
M. M.
,
Trujillo
,
M.
,
López Molina
,
J. A.
,
Rivera
,
M. J.
, and
Berjano
,
E. J.
, 2009, “
Modeling the Heating of Biological Tissue Based on the Hyperbolic Heat Transfer Equation
,”
Math. Comput. Model.
,
50
, pp.
665
672
.
37.
Trujillo
,
M.
,
Rivera
,
M. J.
,
López Molina
,
J. A.
, and
Berjano
,
E. J.
, 2009, “
Effect of the Thermal Wave in Radiofrequency Ablation Modeling: An Analytical Study
,”
Math. Med. Biol.
,
26
(
3
), pp.
187
200
.
You do not currently have access to this content.