Abstract

This article proposes the closed-form solution of the generalized non-Fourier model-based bioheat transfer equation (BHTE) in Cylindrical coordinates to understand the thermal behavior of living tissue heated by a pulsed laser. The axisymmetric living tissue exposed to the non-Gaussian temporal profile of laser heating has been considered to investigate the non-Fourier bioheat transfer phenomena. The closed-form solution of the generalized non-Fourier model-based BHTE with time-dependent thermal energy generation has been obtained through the finite integral transform (FIT) technique. The analytical solution was juxtaposed to the corresponding numerical solution in order to determine its reliability. The numerical solution of the aforementioned governing equation has been obtained by the finite volume method (FVM). The results of both analytical and numerical solutions have been verified using results given in published literature. Subsequently, the dual-phase-lag (DPL) model's findings were juxtaposed to those obtained using the hyperbolic and traditional Fourier models. The effect of different parameters like relaxation times corresponding to the temperature gradient and heat flux, metabolic energy generation, and blood perfusion on the resultant temperature distribution inside the axisymmetric living tissue exposed to pulsed laser heating has been discussed. The importance of this study might be found in various applications such as laser-based-photothermal therapy, melting of the surface of metal and alloys by laser heating.

References

1.
Bischof
,
J. C.
,
2000
, “
Quantitative Measurement and Prediction of Biophysical Response During Freezing in Tissues
,”
Annu. Rev. Biomed. Eng.
,
2
(
1
), pp.
257
288
.10.1146/annurev.bioeng.2.1.257
2.
Roemer
,
R. B.
,
1999
, “
Engineering Aspects of Hyperthermia Therapy
,”
Annu. Rev. Biomed. Eng.
,
1
(
1
), pp.
347
376
.10.1146/annurev.bioeng.1.1.347
3.
Ng
,
E. Y. K.
,
Tan
,
H. M.
, and
Ooi
,
E. H.
,
2009
, “
Boundary Element Method With Bioheat Equation for Skin Burn Injury
,”
Burns
,
35
(
7
), pp.
987
997
.10.1016/j.burns.2009.01.010
4.
Deng
,
Z. S.
, and
Liu
,
J.
,
2004
, “
Mathematical Modeling of Temperature Mapping Over Skin Surface and Its Implementation in Thermal Disease Diagnostics
,”
Comput. Biol. Med.
,
34
(
6
), pp.
495
521
.10.1016/S0010-4825(03)00086-6
5.
Pennes
,
H. H.
,
1948
, “
Analysis of Tissue and Arterial Blood Temperatures
,”
J. Appl. Physiol.
,
1
(
2
), pp.
93
122
.10.1152/jappl.1948.1.2.93
6.
Wuff
,
W.
,
1974
, “
The EnergyConservation Equation for Living Tissues
,”
IEEE Trans. Biomed. Eng.
,
BME-21
(
6
), pp.
494
495
.10.1109/TBME.1974.324342
7.
Jiji
,
L. M.
,
Weinbaum
,
S.
, and
Lemons
,
D. E.
,
1984
, “
Theory and Experiment for the Effect of Vascular Microstrucutre on Surface Tissue Heat Transfer—Part II: Model Formulation and Solution
,”
ASME J. Biomech. Eng.
,
106
(
4
), pp.
331
341
.10.1115/1.3138502
8.
Charny
,
C. K.
,
1992
, “
Mathematical Models of Bio-Heat Transfer
,”
Adv. Heat Transfer
,
22
, pp.
19
155
.10.1016/S0065-2717(08)70344-7
9.
Torvi
,
D. A.
, and
Dale
,
D. J.
,
1994
, “
A Finite Element Model of Skin Subjected to a Flash Fire
,”
ASME J. Biomech. Eng.
,
116
(
3
), pp.
250
255
.10.1115/1.2895727
10.
Lee
,
H. L.
,
Chen
,
W. L.
,
Chang
,
W. J.
,
Char
,
M. I.
, and
Yang
,
Y. C.
,
2016
, “
Numerical Analysis of Dual-Phase-Lag Heat Transfer for a Moving Finite Medium Subjected to Laser Heat Source
,”
Appl. Math. Model.
,
40
(
7–8
), pp.
4700
4711
.10.1016/j.apm.2015.12.005
11.
Kumar
,
S.
, and
Srivastava
,
A.
,
2014
, “
Numerical Investigation of Thermal Response of Laser Irradiated Tissue Phantoms Embedded With Optical Inhomogeneities
,”
Int. J. Heat Mass Transfer
,
77
, pp.
262
277
.10.1016/j.ijheatmasstransfer.2014.05.012
12.
Maurer
,
M. J.
, and
Thompson
,
H. A.
,
1973
, “
Non-Fourier Effects at High Heat Flux
,”
Trans. ASME J. Heat Transfer
,
95
(
2
), pp.
284
286
.10.1115/1.3450051
13.
Chester
,
M.
,
1963
, “
Second Sound in Solids
,”
Phys. Rev.
,
131
(
5
), pp.
2013
2015
.10.1103/PhysRev.131.2013
14.
Kazimi
,
M. S.
, and
Erdman
,
C. A.
,
1975
, “
On the Interface Temperature of Two Suddenly Contacting Materials
,”
Trans. ASME J. Heat Transfer
,
97
(
4
), pp.
615
617
.10.1115/1.3450441
15.
Mitra
,
K.
,
Kumar
,
S.
,
Vedevarz
,
A.
, and
Moallemi
,
M. K.
,
1995
, “
Experimental Evidence of Hyperbolic Heat Conduction in Processed Meat
,”
Trans. ASME J. Heat Transfer
,
117
(
3
), pp.
568
573
.10.1115/1.2822615
16.
Cattaneo
,
C.
,
1958
, “
A Form of Heat Conduction Equation Which Eliminates the Paradox of Instantaneous Propagation
,”
C. R.
,
247
, pp.
431
433
.
17.
Vernotte
,
P.
,
1961
, “
Some Possible Complications in the Phenomena of Thermal Conduction
,”
C. R.
,
252
, pp.
2190
2191
.
18.
Kumar
,
S.
, and
Srivastava
,
A.
,
2015
, “
Thermal Analysis of Laser-Irradiated Tissue Phantoms Using Dual Phase Lag Model Coupled With Transient Radiative Transfer Equation
,”
Int. J. Heat Mass Transfer
,
90
, pp.
466
479
.10.1016/j.ijheatmasstransfer.2015.06.077
19.
Tzou
,
D. Y.
,
1995
, “
A Unified Field Approach for Heat Conduction From Macro-to-Micro Scales
,”
Trans. ASME J. Heat Transfer
,
117
(
1
), pp.
8
16
.10.1115/1.2822329
20.
Narasimhan
,
A.
, and
Sadasivam
,
S.
,
2013
, “
Non-Fourier Bio Heat Transfer Modelling of Thermal Damage During Retinal Laser Irradiation
,”
Int. J. Heat Mass Transfer
,
60
, pp.
591
597
.10.1016/j.ijheatmasstransfer.2013.01.010
21.
Afrin
,
N.
,
Zhou
,
J.
,
Zhang
,
Y.
,
Tzou
,
D. Y.
, and
Chen
,
J. K.
,
2012
, “
Numerical Simulation of Thermal Damage to Living Biological Tissues Induced by Laser Irradiation Based on a Generalized Dual Phase Lag Model
,”
Numer. Heat Transfer, A
,
61
(
7
), pp.
483
501
.10.1080/10407782.2012.667648
22.
Mukherjee
,
A.
,
Lahiri
,
A.
, and
Mishra
,
S. C.
,
2016
, “
Analyses of Dual-Phase Lag Heat Conduction in 1-D Cylindrical and Spherical Geometry- an Application of the Lattice Boltzmann Method
,”
Int. J. Heat Mass Transfer
,
96
, pp.
627
642
.10.1016/j.ijheatmasstransfer.2016.01.048
23.
Singh
,
S.
, and
Kumar
,
S.
,
2014
, “
Numerical Study on Triple Layer Skin Tissue Freezing Using Dual Phase Lag Bio-Heat Model
,”
Int. J. Therm. Sci.
,
86
, pp.
12
20
.10.1016/j.ijthermalsci.2014.06.027
24.
Patidar
,
S.
,
Kumar
,
S.
,
Srivastava
,
A.
, and
Singh
,
S.
,
2016
, “
Dual Phase Lag Model-Based Thermal Analysis of Tissue Phantoms Using Lattice Boltzmann Method
,”
Int. J. Therm. Sci.
,
103
, pp.
41
56
.10.1016/j.ijthermalsci.2015.12.011
25.
Jasiński
,
M.
,
Majchrzak
,
E.
, and
Turchan
,
L.
,
2016
, “
Numerical Analysis of the Interactions Between Laser and Soft Tissues Using the Generalized Dual-Phase Lag Equation
,”
Appl. Math. Model.
,
40
(
2
), pp.
750
762
.10.1016/j.apm.2015.10.025
26.
Majchrzak
,
E.
, and
Turchan
,
L.
,
2015
, “
The General Boundary Element Method for 3D Dual-Phase Lag Model of Bioheat Transfer
,”
Eng. Anal. Bound. Elem.
,
50
, pp.
76
82
.10.1016/j.enganabound.2014.07.012
27.
Mochnacki
,
B.
, and
Majchrzak
,
E.
,
2017
, “
Numerical Model of Thermal Interactions Between Cylindrical Cryoprobe and Biological Tissue Using the Dual-Phase Lag Equation
,”
Int. J. Heat Mass Transfer
,
108
, pp.
1
10
.10.1016/j.ijheatmasstransfer.2016.11.103
28.
Zhou
,
J.
,
Zhang
,
Y.
, and
Chen
,
J. K.
,
2009
, “
An Axisymmetric Dual-Phase-Lag Bioheat Model for Laser Heating of Living Tissues
,”
Int. J. Therm. Sci.
,
48
(
8
), pp.
1477
1485
.10.1016/j.ijthermalsci.2008.12.012
29.
Lee
,
H. L.
,
Chen
,
W. L.
,
Chang
,
W. J.
,
Wei
,
E. J.
, and
Yang
,
Y. C.
,
2013
, “
Analysis of Dual-Phase-Lag Heat Conduction in Short-Pulse Laser Heating of Metals With Hybrid Method
,”
Appl. Therm. Eng.
,
52
(
2
), pp.
275
283
.10.1016/j.applthermaleng.2012.12.019
30.
Liu
,
K. C.
, and
Chen
,
H. T.
,
2015
, “
Analysis of the Bioheat Transfer Problem With Pulse Boundary Heat Flux Using a Generalized Dual-Phase-Lag Model
,”
Int. Commun. Heat Mass Transfer
,
65
, pp.
31
36
.10.1016/j.icheatmasstransfer.2015.04.004
31.
Liu
,
K. C.
,
2007
, “
Numerical Analysis of Dual-Phase-Lag Heat Transfer in a Layered Cylinder With Nonlinear Interface Boundary Conditions
,”
Comput. Phys. Commun.
,
177
(
3
), pp.
307
314
.10.1016/j.cpc.2007.02.110
32.
Liu
,
K. C.
, and
Chen
,
H. T.
,
2009
, “
Analysis for the Dual-Phase-Lag Bio-Heat Transfer During Magnetic Hyperthermia Treatment
,”
Int. J. Heat Mass Transfer
,
52
(
5–6
), pp.
1185
1192
.10.1016/j.ijheatmasstransfer.2008.08.025
33.
Wu
,
T. S.
,
Lee
,
H. L.
,
Chang
,
W. J.
, and
Yang
,
Y. C.
,
2015
, “
An Inverse Hyperbolic Heat Conduction Problem in Estimating Pulse Heat Flux With a Dual-Phase-Lag Model
,”
Int. Commun. Heat Mass Transfer
,
60
, pp.
1
8
.10.1016/j.icheatmasstransfer.2014.11.002
34.
Shih
,
T. C.
,
Yuan
,
P.
,
Lin
,
W. L.
, and
Kou
,
H.-S.
,
2007
, “
Analytical Analysis of the Pennes Bioheat Transfer Equation With Sinusoidal Heat Flux Condition on Skin Surface
,”
Med. Eng. Phys.
,
29
(
9
), pp.
946
953
.10.1016/j.medengphy.2006.10.008
35.
Deng
,
Z. S.
, and
Liu
,
J.
,
2002
, “
Analytical Study on Bioheat Transfer Problems With Spatial or Transient Heating on Skin Surface or Inside Biological Bodies
,”
ASME J. Biomech. Eng.
,
124
(
6
), pp.
638
649
.10.1115/1.1516810
36.
Dutta
,
J.
, and
Kundu
,
B.
,
2017
, “
A Revised Approach for an Exact Analytical Solution for Thermal Response in Biological Tissues Significant in Therapeutic Treatments
,”
J. Therm. Biol.
,
66
, pp.
33
48
.10.1016/j.jtherbio.2017.03.015
37.
Askarizadeh
,
H.
, and
Ahmadikia
,
H.
,
2014
, “
Analytical Analysis of the Dual-Phase-Lag Model of Bioheat Transfer Equation During Transient Heating of Skin Tissue
,”
Heat Mass Transfer
,
50
(
12
), pp.
1673
1684
.10.1007/s00231-014-1373-6
38.
Hooshmand
,
P.
,
Moradi
,
A.
, and
Khezry
,
B.
,
2015
, “
Bioheat Transfer Analysis of Biological Tissues Induced by Laser Irradiation
,”
Int. J. Therm. Sci.
,
90
, pp.
214
223
.10.1016/j.ijthermalsci.2014.12.004
39.
Torabi
,
M.
, and
Zhang
,
K.
,
2014
, “
Multi-Dimensional Dual-Phase-Lag Heat Conduction in Cylindrical Coordinates: Analytical and Numerical Solutions
,”
Int. J. Heat Mass Transfer
,
78
, pp.
960
966
.10.1016/j.ijheatmasstransfer.2014.07.038
40.
Askarizadeh
,
H.
, and
Ahmadikia
,
H.
,
2015
, “
Analytical Study on the Transient Heating of a Two-Dimensional Skin Tissue Using Parabolic and Hyperbolic Bioheat Transfer Equations
,”
Appl. Math. Model.
,
39
(
13
), pp.
3704
3720
.10.1016/j.apm.2014.12.003
41.
Kishore
,
P.
, and
Kumar
,
S.
,
2019
, “
Analysis of non-Fourier Heat Conduction Model Based Bio-Heat Transfer Equation in Cylindrical Coordinates Using the Method of Separation of Variables
,” Proceedings of the 25th National and Third International ISHMT-ASTFE Heat and Mass Transfer Conference (
IHMTC-2019
), Roorkee, India, Dec. 28–31, pp.
26
31
.10.1615/IHMTC-2019.50
42.
Abdel-Hamid
,
B.
,
1999
, “
Modelling Non-Fourier Heat Conduction With Periodic Thermal Oscillation Using the Finite Integral Transform
,”
Appl. Math. Model.
,
23
(
12
), pp.
899
914
.10.1016/S0307-904X(99)00017-7
43.
Cotta
,
R. M.
,
Cotta
,
B. P.
,
Naveira-Cotta
,
C. P.
, and
Cotta-Pereira
,
G.
,
2010
, “
Hybrid Integral Transform Analysis of the Bioheat Equation With Variable Properties
,”
Int. J. Therm. Sci.
,
49
(
9
), pp.
1510
1516
.10.1016/j.ijthermalsci.2010.04.019
44.
Kumar
,
S.
, and
Srivastava
,
A.
,
2017
, “
Finite Integral Transform-Based Analytical Solutions of Dual Phase Lag Bio-Heat Transfer Equation
,”
Appl. Math. Model.
,
52
, pp.
378
403
.10.1016/j.apm.2017.05.041
45.
Tang
,
D. W.
, and
Araki
,
N.
,
2000
, “
Non-Fourier Heat Conduction Behavior in Finite Mediums Under Pulse Surface Heating
,”
Mater. Sci. Eng., A
,
292
(
2
), pp.
173
178
.10.1016/S0921-5093(00)01000-5
46.
Verma
,
R.
, and
Kumar
,
S.
,
2020
, “
Computational Study on Constant and Sinusoidal Heating of Skin Tissue Using Radial Basis Functions
,”
Comput. Biol. Med.
,
121
, p.
103808
.10.1016/j.compbiomed.2020.103808
47.
Hahn
,
D. W.
, and
Ozisik
,
M. N.
,
2012
,
Heat Conduction
, 3rd ed.,
Wiley
, Hoboken,
NJ
.
48.
Patankar
,
S. V.
,
1980
,
Numerical Heat Transfer and Fluid Flow
,
Hemisphere Publishing Corporation
,
New York
.
49.
Antaki
,
P. J.
,
2005
, “
New Interpretation of Non-Fourier Heat Conduction in Processed Meat
,”
Trans. ASME J. Heat Transfer
,
127
(
2
), pp.
189
193
.10.1115/1.1844540
50.
Sahoo
,
N.
,
Narasimhan
,
A.
,
Dhar
,
P.
, and
Das
,
S. K.
,
2018
, “
Non-Fourier Thermal Transport Induced Structural Hierarchy and Damage to Collagen Ultrastructure Subjected to Laser Irradiation
,”
Int. J. Hyperthermia
,
34
(
3
), pp.
229
242
.10.1080/02656736.2017.1342873
51.
Torabi
,
M.
, and
Saedodin
,
S.
,
2011
, “
Analytical and Numerical Solutions of Hyperbolic Heat Conduction in Cylindrical Coordinates
,”
J. Thermophys. Heat Transfer
,
25
(
2
), pp.
239
253
.10.2514/1.51395
52.
Lin
,
C.-K.
,
Hwang
,
C.-C.
, and
Chuag
,
Y.-P.
,
1997
, “
The Unsteady Solutions of a Unified Heat Conduction Equation
,”
Int. J. Heat Transfer
,
40
(
7
), pp.
1716
1719
.10.1016/S0017-9310(96)00220-7
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