Share This Article:

Extended Generalized Riccati Equation Mapping for Thermal Traveling-Wave Distribution in Biological Tissues through a Bio-Heat Transfer Model with Linear/Quadratic Temperature-Dependent Blood Perfusion

Abstract Full-Text HTML Download Download as PDF (Size:1767KB) PP. 1471-1484
DOI: 10.4236/am.2013.410199    3,888 Downloads   5,444 Views   Citations

ABSTRACT

Analytical thermal traveling-wave distribution in biological tissues through a bio-heat transfer (BHT) model with linear/quadratic temperature-dependent blood perfusion is discussed in this paper. Using the extended generalized Riccati equation mapping method, we find analytical traveling wave solutions of the considered BHT equation. All the travelling wave solutions obtained have been used to explicitly investigate the effect of linear and quadratic coefficients of temperature dependence on temperature distribution in tissues. We found that the parameter of the nonlinear superposition formula for Riccati can be used to control the temperature of living tissues. Our results prove that the extended generalized Riccati equation mapping method is a powerful tool for investigating thermal traveling-wave distribution in biological tissues.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Kengne, E. , Hamouda, F. and Lakhssassi, A. (2013) Extended Generalized Riccati Equation Mapping for Thermal Traveling-Wave Distribution in Biological Tissues through a Bio-Heat Transfer Model with Linear/Quadratic Temperature-Dependent Blood Perfusion. Applied Mathematics, 4, 1471-1484. doi: 10.4236/am.2013.410199.

References

[1] J. Liu, L. Zhu and X. L. Xu, “Studies on the Three-Dimensional Temperature Transients in the Canine Prostate during Transurethal Micorwave Thermal Therapy,” Journal of Biomechanical Engineering, Vol. 122, No. 4, 2000, pp. 372-379. http://dx.doi.org/10.1115/1.1288208
[2] G. T. Martin, H. F. Bowman, W. H. Newman and E. G. Cravalho, “Thermal Model with Temperature Dependent Perfusion for the Hyperthermia Treatment of Benign Prostatic Hyperplasia,” Advanced Heat and Mass Transfer, Vol. 18, 1991, pp. 33-37.
[3] R. Seip and E. S. Ebbini, “Studies on the Three-Dimensional Temperature Response to Heating Fields Using Diagnostic Ultrasound,” IEEE Transactions on Biomedical Engineering, Vol. 42, No. 8, 1995, pp. 828-839.
http://dx.doi.org/10.1109/10.398644
[4] S. Puccini, N. K. Bar, M. Bublat, T. Kahn and H. Busse, “Simulations of Thermal Tissue Coagulation and Their Value for the Planning and Monitoring of Laser-Induced Interstitial Thermotherapy (LITT),” Magnetic Resonance in Medicine, Vol. 49, No. 2, 2003, pp. 351-362.
http://dx.doi.org/10.1002/mrm.10357
[5] A. M. Stoll, “Thermal-Properties of Human-Skin Related to Nondestructive Measurement of Epidermal Thickness,” Journal of Investigative Dermatology, Vol. 69, 1977, pp. 328-332.
http://dx.doi.org/10.1111/1523-1747.ep12507865
[6] A. M. Stoll, M. A. Chianta and J. R. Piergallini, “Thermal Conduction Effects in Human-Skin,” Aviation, Space, and Environmental Medicine, Vol. 50, No. 8, 1979, pp. 778-787.
[7] E. Kengne, A. Lakhssassi, R. Vaillancourt and W. M. Liu, “Monitoring of Temperature Distribution in Living Biological Tissues via Blood Perfusion,” The European Physical Journal Plus, Vol. 127, No. 89, 2012, 15p.
[8] A. Lakhssassi, E. Kengne and H. Semmaoui, “Investigation of Nonlinear Temperature Distribution in Biological Tissues by Using Bioheat Transfer Equation of Pennes’ Type,” Natural Science, Vol. 2, No. 3, 2010, pp. 131-138.
http://dx.doi.org/10.4236/ns.2010.23022
[9] E. Kengne, A. Lakhssassi and R. Vaillancourt, “Temperature Distributions for Regional Hypothermia Based on Nonlinear Bioheat Equation of Pennes Type: Dermis and Subcutaneous Tissues,” Applied Mathematics, Vol. 3, No. 3, 2012, pp. 217-224.
http://dx.doi.org/10.4236/am.2012.33035
[10] F. C. Henriques and A. R. Moritz, “Studies of Thermal Injury, 1. The Conduction of Heat to and through Skin and the Temperatures Attained Therein. A Theoretical and Experimental Investigation,” American Journal of Pathology, Vol. 23, No. 4, 1947, pp. 431-549.
[11] H. H. Pennes, “Analysis of Tissue and Arterial Blood Temperature in the Resting Human Forearm,” Journal of Applied Physiology, Vol. 1, No. 2, 1948, pp. 93-122.
[12] J. A. J. Stolwijk and J. D. Hardy, “Temperature Regulation in Man-A Theoretical Study,” Pflugers Archives, Vol. 291, No. 2, 1966, pp. 129-162.
http://dx.doi.org/10.1007/BF00412787
[13] E. H. Wissler, “Comparison of Results Obtained from Two Mathematical Models—A Simple 14-Node Model and a Complex 250-Node Model,” Journal of Physiology (Paris), Vol. 63, No. 3, 1970, pp. 455-458.
[14] A. Shitzer and R. C. Eberhart, “Heat Transfer in Medicine and Biology,” Plenum Press, New York, 1985.
http://dx.doi.org/10.1007/978-1-4684-8285-0
[15] M. J. Mantyla, J. Kuikka and A. Rekonnen, “A Regional Blood Flow in Human Tumors with Special Reference to the Effect of Radiotherapy,” British Journal of Radiology, Vol. 49, 1976, pp. 335-338.
http://dx.doi.org/10.1259/0007-1285-49-580-335
[16] J. A. Surrell, R. C. Alexander, S. D. Cohle Jr., F. R. Lovell and R. A. Wehrenberg, “Effects of Microwave Radiation on Living Tissues,” Journal of Trauma-Injury Infection & Critical Care, Vol. 27, No. 8, 1987, pp. 935-939.
http://dx.doi.org/10.1097/00005373-198708000-00014
[17] Z.-S. Deng and J. Liu, “Parametric Studies on the Phase Shift Method to Measure the Blood Perfusion of Biological Bodies,” Medical Engineering & Physics, Vol. 22, No. 10, 2000, pp. 693-702.
http://dx.doi.org/10.1016/S1350-4533(01)00015-7
[18] Z.-S. Deng and J. Liu, “Mathematical Modelling of Temperature Mapping over Skin Surface and Its Implementation in Thermal Disease Diagnostics,” Computers in Biology and Medicine, Vol. 34, No. 6, 2004, pp. 495-521.
http://dx.doi.org/10.1016/S0010-4825(03)00086-6
[19] Z.-S. Deng and J. Liu, “Modelling of Multidimensional Freezing Problem during Cryosurgery by the Dual Reciprocity Boundary Element Method,” Engineering Analysis with Boundary Elements, Vol. 28, No. 2, 2004, pp. 97-108.
http://dx.doi.org/10.1016/S0955-7997(03)00128-0
[20] J. Liu and L. S. Xu, “Boundary Information Based Diagnostics on the Thermal States of Biological Bodies,” International Journal of Heat and Mass Transfer, Vol. 43, No. 16, 2000, pp. 2827-2839.
http://dx.doi.org/10.1016/S0017-9310(99)00367-1
[21] E. H. Wissler, “Pennes’ 1948 Paper Revisited,” Journal of Applied Physiology, Vol. 85, No. 1, 1998, pp. 35-41.
[22] R. B. Roemer, B. R. Paliwal, F. W. Hetzel and M. W. Dewhirst, “Heat Transfer in Hyperthermia Treatments: Basic Principles and Applications,” In: B. R. Paliwal, F. W. Hetzel and M. W. Dewhirst, Eds., Biological Physical and Clinical Aspects of Hyperthermia, American Institute in Physics, New York, 1988, pp. 210-242.
[23] S. Acharya, D. B. Gurung and V. P. Saxena, “Effect of Metabolic Reactions on Thermoregulation in Human Males and Females Body,” Applied Mathematics, Vol. 4, No. 5A, 2013, pp. 39-48.
http://dx.doi.org/10.4236/am.2013.45A005
[24] D. E. Haines and D. D. Watson, “Tissue Heating during Radiofrequency Catheter Ablation: A Thermodynamic Model and Observations in Isolated Perfused and Superfused Canine Right Ventricular Free Wall,” Pacing and Clinical Electrophysiology, Vol. 12, No. 6, 1989, pp. 962976.
http://dx.doi.org/10.1111/j.1540-8159.1989.tb05034.x
[25] K. R. Diller, “Modeling of Bioheat Transfer Processes at High and Low Temperatures,” Advances in Heat Transfer, Vol. 22, 1992, pp. 157-167.
http://dx.doi.org/10.1016/S0065-2717(08)70345-9
[26] R. W. Y. Habash, R. Bansal, D. Krewski and H. T. Alhafid, “Thermal Therapy, Part IV: Electromagnetic and Thermal Dosimetry,” Critical Reviews in Biomedical Engineering, Vol. 35, No. 1-2, 2007, pp. 123-182.
http://dx.doi.org/10.1615/CritRevBiomedEng.v35.i1-2.30
[27] E. Erdmann, J. Lang and M. Seebass, “Optimization of Temperature Distributions for Regional Hyperthermia Based on a Nonlinear Heat Equation,” In: K. Diller, Ed., Biotransport: Heat and Mass Transfer in Living Systems, 1998, pp. 36-46.
[28] K. R. Foster, A. Lozano-Nieto, P. J. Riu and T. S. Ely, “Heating of Tissues by Microwaves: A Model Analysis,” Bioelectromagnetics, Vol. 19, No. 7, 1998, pp. 420-428.
http://dx.doi.org/10.1002/(SICI)1521-186X(1998)19:7<420::AID-BEM3>3.0.CO;2-3
[29] P. J. Riu, K. R. Foster, D. W. Blick and E. R. Adair, “A Thermal Model for Human Thresholds of MicrowaveEvoked Warmth Sensations,” Bioelectromagnetics, Vol. 18, No. 8, 1997, pp. 578-583.
http://dx.doi.org/10.1002/(SICI)1521-186X(1997)18:8<578::AID-BEM6>3.0.CO;2-#
[30] H. F. Bowman, E. G. Cravalho and M. Woods, “Theory, Measurement, and Application of Thermal Properties of Biomaterials,” Annual Review of Biophysics and Bioengineering, Vol. 4, 1975, pp. 43-80.
http://dx.doi.org/10.1146/annurev.bb.04.060175.000355
[31] G. T. Martin, H. F. Bowman and W. H. Newman, “Basic Element Method for Computing the Temperature Field during Hyperthermia Therapy Planning,” Adv Bio Heat Mass Transf, Vol. 231, 1992, pp. 75-80.
[32] C. W. Song, A. Lokshina, J. G. Rhee, M. Patten and S. H. Levitt, “Implication of Blood Flow in Hyperthermic Treatment of Tumors,” IEEE Transactions on Biomedical Engineering, Vol. 31, No. 1, 1984, pp. 9-16.
http://dx.doi.org/10.1109/TBME.1984.325364
[33] J. Lang, B. Erdmann and M. Seebass, “Impact of Nonlinear Heat Transfer on Temperature Control in Regional Hyperthermia,” IEEE Transactions on Biomedical Engineering, Vol. 46, No. 9, 1999, pp. 1129-1138.
http://dx.doi.org/10.1109/10.784145
[34] C. R. Davies, G. M. Saidel and H. Harasaki, “Sensitivity Analysis of One-Dimensional Heat Transfer in Tissue with Temperature-Dependent Perfusion,” Journal of Biomechanical Engineering, Vol. 119, No. 1, 1997, pp. 7780. http://dx.doi.org/10.1115/1.2796068
[35] W. P. Partridge and L. C. Wrobel, “A Coupled Dual Reciprocity BEM/Genetic Algorithm for Identification of Blood Perfusion Parameters,” International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 19, No. 1, 2009, pp. 25-38.
http://dx.doi.org/10.1108/09615530910922134
[36] T. R. Gowrishankar, D. A. Stewart, G. T. Martin and J. C. Weaver, “Transport Lattice Models of Heat Transport in Skin with Spatially Heterogeneous, Temperature-Dependent Perfusion,” BioMedical Engineering on Line, Vol. 3, 2004, pp. 1-17.
[37] S. Lie and G. Scheffers, “Vorlesungenüber Continuierliche Gruppen mil Geometrischen und Anderen Anwend,” Chelsea Publishing Company, New York, 1967.
[38] J. D. Cole, “On a Quasilinear Parabolic Equation Occurring in Aerodynamics,” Quarterly of Applied Mathematics, Vol. 9, 1951, pp. 225-236.
[39] E. Hopf, “The Partial Differential Equation ut + uux = uxx,” Communications on Pure and Applied Mathematics, Vol. 3, No. 3, 1950, pp. 201-230.
http://dx.doi.org/10.1002/cpa.3160030302
[40] E. L. Ince, “Ordinary Differential Equations,” Dover Publ., New York, 1956.
[41] Ph. Hartman, “Ordinary Differential Equations,” John Wiley, New York, 1964.
[42] S. Zhu, “The Generalizing Riccati Equation Mapping Method In Non-Linear Evolution Equation: Application To (2+1)-Dimensional Boiti-Leon-Pempinelle Equation,” Chaos, Solitons and Fractals, Vol. 37, No. 5, 2008, pp. 1335-1342. http://dx.doi.org/10.1016/j.chaos.2006.10.015
[43] H. S. Carslaw and J. C. Jaeger, “Conduction of Heat in Solids,” Second Edition, Oxford University Press, Oxford, 1959.
[44] J. Liu and L. X. Xu, “Estimation of Blood Perfusion Using Phase Shift in Temperature Response to Sinusoidal Heating at the Skin Surface,” IEEE Transactions on Biomedical Engineering, Vol. 46, No. 9, 1999, pp. 10371043. http://dx.doi.org/10.1109/10.784134
[45] S. Weinbaum, L. M. Jiji and D. E. Lemons, “Theory and Experiment for the Effect of Vascular Microstructure on Surface Tissue Heat Transfer-Part I: Anatomical Foundation and Model Conceptualization,” Journal of Biomechanical Engineering, Vol. 106, No. 4, 1984, pp. 321-330.
http://dx.doi.org/10.1115/1.3138501
[46] F. Rossi, R. Pini and L. Menabuoni, “3D Simulation and Experimental Comparison of Temperature Dynamics in Laser Welded Cornea,” Proceedings of the COMSOL Users Conference, Milano, 2006.
[47] M. J. Rivera, J. A. Lopez Molina, M. Trujillo, V. Romero-Garcia and E. J. Berjano, “Analytical Validation of Comsol Multiphysics for Theoretical Models of Radiofrequency Ablation Including the Hyperbolic Bioheat Transfer Equation,” Engineering in Medicine and Biology Society (EMBC), Annual International Conference of the IEEE, 2010, pp. 3214-3217.

  
comments powered by Disqus

Copyright © 2018 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.