A three-dimensional (3D) simulation of bioheat transfer is crucial to analyze the physiological processes and evaluate many therapeutic/diagnostic practices spanning from high to low temperature medicine. In this paper we develop an efficient numerical scheme for solving 3D transient bioheat transfer equations based on the alternating direction implicit finite-difference method (ADI-FDM). An algorithm is proposed to deal with the boundary condition for irregular domain which could capture accurately the complex boundary and reduce considerably the staircase effects. Furthermore, the local adaptive mesh technology is introduced to improve the computational accuracy for irregular boundary and the domains with large temperature gradient. The detailed modification to ADI-FDM is given to accommodate such special grid structure, in particular. Combination of adaptive-mesh technology and ADI-FDM could significantly improve the computational accuracy and decrease the computational cost. Extensive results of numerical experiments demonstrate that the algorithm developed in the current work is very effective to predict the temperature distribution during hyperthermia and cryosurgery. This work may play an important role in developing a computational planning tool for hyperthermia and cryosurgery in the near future.

References

References
1.
Roemer
,
R. B.
,
1999
, “
Engineering Aspects of Hyperthermia Therapy
,”
Annu. Rev. Biomed. Eng.
,
1
, pp.
347
376
.10.1146/annurev.bioeng.1.1.347
2.
Rubinsky
,
B.
,
2000
, “
Cryosurgery
,”
Annu. Rev. Biomed. Eng.
,
2
, pp.
157
187
.10.1146/annurev.bioeng.2.1.157
3.
Bischof
,
J. C.
,
2000
, “
Quantitative Measurement and Prediction of Biophysical Response During Freezing in Tissues
,”
Annu. Rev. Biomed. Eng.
,
2
, pp.
257
288
.10.1146/annurev.bioeng.2.1.257
4.
Liu
,
J.
,
Lv
,
Y. G.
, and
Zhang
,
J.
,
2006
, “
Theoretical Evaluation of Burns to the Human Respiratory Tract Due to Inhalation of Hot Gas in the Early Stage of Fires
,”
Burns
,
32
, pp.
436
446
.10.1016/j.burns.2005.11.006
5.
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
, pp.
495
521
.10.1016/S0010-4825(03)00086-6
6.
Zhao
,
J. J.
,
Zhang
,
J.
,
Kang
,
N.
, and
Yang
,
F. Q.
,
2005
, “
A Two Level Finite Difference Scheme for One Dimensional Pennes' Bioheat Equation
,”
Appl. Math. Comput.
,
171
, pp.
320
331
.10.1016/j.amc.2005.01.052
7.
Karaa
,
S.
,
Zhang
,
J.
, and
Yang
,
F. Q.
,
2005
, “
A Numerical Study of a 3D Bioheat Transfer Problem With Different Spatial Heating
,”
Math. Comput. Simul.
,
68
, pp.
375
388
.10.1016/j.matcom.2005.02.032
8.
Rossi
,
M. R.
,
Tanaka
,
D.
,
Shimada
,
K.
, and
Rabin
,
Y.
,
2007
, “
An Efficient Numerical Technique for Bioheat Simulations and Its Application to Computerized Cryosurgery Planning
,”
Comput. Methods Programs Biomed.
,
85
, pp.
41
50
.10.1016/j.cmpb.2006.09.014
9.
Bertaccini
,
D.
, and
Calvetti
,
D.
,
2007
, “
Fast Simulation of Solid Tumors Thermal Ablation Treatments With a 3D Reaction Diffusion Model
,”
Comput. Biol. Med.
,
37
, pp.
1173
1182
.10.1016/j.compbiomed.2006.10.008
10.
Velez
,
F. F.
,
Romanov
,
O. G.
, and
Diego
,
J. L. A.
,
2009
, “
Efficient 3D Numerical Approach for Temperature Prediction in Laser Irradiated Biological Tissues
,”
Comput. Biol. Med.
,
39
, pp.
810
817
.10.1016/j.compbiomed.2009.06.009
11.
Pisa
,
S.
,
Cavagnaro
,
M.
,
Piuzzi
,
E.
,
Bernardi
,
P.
, and
Lin
,
J. C.
,
2003
, “
Power Density and Temperature Distributions Produced by Interstitial Arrays of Sleeved-Slot Antennas for Hyperthermic Cancer Therapy
,”
IEEE Trans. Microwave Theory Techniques
,
51
, pp.
2418
2426
.10.1109/TMTT.2003.819214
12.
Belhamadiaa
,
A. F. Y.
,
2005
, “
Numerical Prediction of Freezing Fronts in Cryosurgery: Comparison With Experimental Results
,”
Comput. Methods Biomech. Biomed. Eng.
,
8
, pp.
241
249
.
13.
Yang
,
B. H.
,
Wan
,
R. G.
,
Muldrew
,
K. B.
, and
Donnelly
,
J. B.
,
2008
, “
A Finite Element Model for Cryosurgery With Coupled Phase Change and Thermal Stress Aspects
,”
Finite Elements Anal. Design
,
44
, pp.
288
297
.10.1016/j.finel.2007.11.014
14.
Li
,
E.
,
Liu
,
G. R.
,
Tan
,
V.
, and
He
,
Z. C.
,
2010
, “
An Efficient Algorithm for Phase Change Problem in Tumor Treatment Using Alpha FEM
,”
Int. J. Thermal Sci.
,
49
, pp.
1954
1967
.10.1016/j.ijthermalsci.2010.06.003
15.
Haemmerich
,
D.
,
Tungjitkusolmun
,
S.
,
Staelin
,
S. T.
,
Lee
,
F. T.
,
Mahvi
,
D. M.
, and
Webster
,
J. G.
,
2002
, “
Finite-Element Analysis of Hepatic Multiple Probe Radio-Frequency Ablation
,”
IEEE Trans. Biomed. Eng.
,
49
, pp.
836
842
.10.1109/TBME.2002.800790
16.
Wang
,
H.
, and
Qin
,
Q. H.
,
2010
, “
Fe Approach With Green's Function As Internal Trial Function for Simulating Bioheat Transfer in the Human Eye
,”
Arch. Mech.
,
62
, pp.
493
510
.
17.
Lu
,
W. Q.
,
Liu
,
J.
, and
Zeng
,
Y. T.
,
1998
, “
Simulation of the Thermal Wave Propagation in Biological Tissues by the Dual Reciprocity Boundary Element Method
,”
Eng. Anal. Boundary Elements
,
22
, pp.
167
174
.10.1016/S0955-7997(98)00039-3
18.
Deng
,
Z. S.
, and
Liu
,
J.
,
2004
, “
Modeling of Multidimensional Freezing Problem During Cryosurgery by the Dual Reciprocity Boundary Element Method
,”
Eng. Anal. Boundary Elements
,
28
, pp.
97
108
.10.1016/S0955-7997(03)00128-0
19.
Zhou
,
J. H.
,
Zhang
,
Y. W.
, and
Chen
,
J. K.
,
2008
, “
A Dual Reciprocity Boundary Element Method for Photothermal Interactions in Laser-Induced Thermotherapy
,”
Int. J. Heat Mass Transfer
,
51
, pp.
3869
3881
.10.1016/j.ijheatmasstransfer.2008.01.009
20.
Ng
,
E. Y. K.
,
Tan
,
H. M.
, and
Ooi
,
E. H.
,
2009
, “
Boundary Element Method With Bioheat Equation for Skin Burn Injury
,”
Burns
,
35
, pp.
987
997
.10.1016/j.burns.2009.01.010
21.
Ooi
,
E. H.
,
Ang
,
W. T.
, and
Ng
,
E. Y. K.
,
2007
, “
Bioheat Transfer in the Human Eye: A Boundary Element Approach
,”
Eng. Anal. Boundary Elements
,
31
, pp.
494
500
.10.1016/j.enganabound.2006.09.011
22.
Chan
,
C. K.
,
1992
, “
Boundary Element Method Analysis for the Bioheat Transfer Equation
,”
ASME J. Biomech. Eng.
,
114
, pp.
358
365
.10.1115/1.2891396
23.
Bottauscio
,
O.
,
Chiampi
,
M.
, and
Zilberti
,
L.
,
2011
, “
A Boundary Element Approach to Relate Surface Fields With the Specific Absorption Rate (SAR) Induced in 3-D Human Phantoms
,”
Eng. Anal. Boundary Elements
,
35
, pp.
657
666
.10.1016/j.enganabound.2010.11.012
24.
Deng
,
Z. S.
, and
Liu.
,
J.
,
2002
, “
Monte Carlo Method to Solve Multidimensional Bioheat Transfer Problem
,”
Numer. Heat Transfer Part B
,
42
, pp.
543
567
.10.1080/10407790260444813
25.
Zhang
,
H. F.
,
2008
, “
Lattice Boltzmann Method for Solving the Bioheat Equation
,”
Phys. Med. Biol.
,
53
, pp.
15
23
.10.1088/0031-9155/53/3/N01
26.
Bellia
,
S. A.
,
Saidane
,
A.
,
Hamou
,
A.
,
Benzohra
,
M.
, and
Saite
,
J. M.
,
2008
, “
Transmission Line Matrix Modelling of Thermal Injuries to Skin
,”
Burns
,
34
, pp.
688
697
.10.1016/j.burns.2007.09.003
27.
Amri
,
A.
,
Saidane
,
A.
, and
Pulko
,
S.
,
2011
, “
Thermal Analysis of a Three-Dimensional Breast Model With Embedded Tumour Using the Transmission Line Matrix (TLM) Method
,”
Comput. Biol. Med.
,
41
, pp.
76
86
.10.1016/j.compbiomed.2010.12.002
28.
Cao
,
L. L.
,
Qin
,
Q. H.
, and
Zhao
,
N.
,
2010
, “
An RBF-MFS Model for Analysing Thermal Behaviour of Skin Tissues
,”
Int. J. Heat Mass Transfer
,
53
, pp.
1298
1307
.10.1016/j.ijheatmasstransfer.2009.12.036
29.
Dillenseger
,
J. L.
, and
Esneault
,
S.
,
2010
, “
Fast FFT-Based Bioheat Transfer Equation Computation
,”
Comput. Biol. Med.
,
40
, pp.
119
123
.10.1016/j.compbiomed.2009.11.008
30.
Shih
,
T. C.
,
Horng
,
T. L.
,
Lin
,
W. L.
,
Liauh
,
C. T.
, and
Shih
,
T. C.
,
2007
, “
Effects of Pulsatile Blood Flow in Large Vessels on Thermal Dose Distribution During Thermal Therapy
,”
Med. Phys.
,
34
, pp.
1312
1320
.10.1118/1.2712415
31.
Chai
,
J. C.
, and
Yap
,
Y. F.
,
2008
, “
A Distance-Function-Based Cartesian (DIFCA) Grid Method for Irregular Geometries
,”
Int. J. Heat Mass Transfer
,
51
, pp.
1691
1706
.10.1016/j.ijheatmasstransfer.2007.07.012
32.
Mittal
,
R.
, and
Iaccarino
,
G.
,
2005
, “
Immersed Boundary Methods
,”
Annu, Rev, Fluid Mech.
,
37
, pp.
239
261
.10.1146/annurev.fluid.37.061903.175743
33.
Ingram
,
D. M.
,
Causon
,
D. M.
, and
Mingham
,
C. G.
,
2003
, “
Developments in Cartesian Cut Cell Methods
,”
Math. Comput. Simul.
,
61
, pp.
561
572
.10.1016/S0378-4754(02)00107-6
34.
Samaras
,
T.
,
Christ
,
A.
, and
Kuster
,
N.
,
2006
, “
Effects of Geometry Discretization Aspects on the Numerical Solution of the Bioheat Transfer Equation With the FDTD Technique
,”
Phys. Med. Biol.
,
51
, pp.
221
229
.10.1088/0031-9155/51/11/N02
35.
Neufeld
,
E.
,
Chavannes
,
N.
,
Samaras
,
T.
, and
Kuster
,
N.
,
2007
, “
Novel Conformal Technique to Reduce Staircasing Artifacts at Material Boundaries for FDTD Modeling of the Bioheat Equation
,”
Phys. Med. Biol.
,
52
, pp.
4371
4381
.10.1088/0031-9155/52/15/001
36.
Douglas
,
J. J.
, and
Gunn
,
J. E.
,
1964
, “
A General Formulation of Alternating Direction Methods—Part I: Parabolic and Hyperbolic Problems
,”
Numer. Math.
,
6
, pp.
428
453
.10.1007/BF01386093
37.
Zeng
,
P.
,
Deng
,
Z. S.
, and
Liu
J.
,
2011
, “
Parallel Algorithms for Freezing Problems During Cryosurgery
,”
Int. J. Inform. Eng. Electron. Bus.
,
2
, pp.
11
19
.10.5815/ijieeb.2011.02.02
38.
Gibou
,
F.
,
Chen
,
H.
, and
Min
,
C. H.
,
2007
, “
A Supra-Convergent Finite Difference Scheme for the Poisson and Heat Equations on Irregular Domains and Non-Graded Adaptive Cartesian Grids
,”
J. Sci. Comput.
,
31
, pp.
19
60
.10.1007/s10915-006-9122-8
39.
Consiglieri
,
L.
,
Santos
,
I. D.
, and
Haemmerich
,
D.
,
2003
, “
Theoretical Analysis of the Heat Convection Coefficient in Large Vessels and the Significance for Thermal Ablative Therapies
,”
Phys. Med. Biol.
,
48
, pp.
4125
4134
.10.1088/0031-9155/48/24/010
40.
He
,
Z. Z.
, and
Liu
,
J.
,
2011
, “
The Effects of Blood Flow on the Iceball Evolution During a Multiple Probe Cryosurgery
,”
ASME 2011 International Mechanical Engineering Congress & Exposition
.
You do not currently have access to this content.