The Effect of Relaxation Time on the Heat Transfer and Temperature Distribution in Tissues

ABSTRACT

A normal mode analysis for characterizing the temperature fluctuation in tissues was proposed based on the Penne’s bio-heat transfer equation. Closed-form analytical solutions to obtain the heating pattern due to the propagation of ultrasonic waves in tissue system are presented. The evaluation of temporal and spatial distributions of temperature is investigated with the effect of relaxation time. The derived method is evaluated with numerical simulations in 2D which are applied to tissue medium in simplified geometry.

A normal mode analysis for characterizing the temperature fluctuation in tissues was proposed based on the Penne’s bio-heat transfer equation. Closed-form analytical solutions to obtain the heating pattern due to the propagation of ultrasonic waves in tissue system are presented. The evaluation of temporal and spatial distributions of temperature is investigated with the effect of relaxation time. The derived method is evaluated with numerical simulations in 2D which are applied to tissue medium in simplified geometry.

Cite this paper

nullM. Othman, M. Ali and R. Farouk, "The Effect of Relaxation Time on the Heat Transfer and Temperature Distribution in Tissues,"*World Journal of Mechanics*, Vol. 1 No. 6, 2011, pp. 283-287. doi: 10.4236/wjm.2011.16035.

nullM. Othman, M. Ali and R. Farouk, "The Effect of Relaxation Time on the Heat Transfer and Temperature Distribution in Tissues,"

References

[1] R. Seip and E. S. Ebbini, “Studies on the Three-Dimen- sional Temperature Response to Heating Fields Using Diagnostic Ultrasound,” IEEE Transactions on Biomedi- cal Engineering, Vol. 42, No. 8, 1995, pp. 828-839. doi:10.1109/10.398644

[2] 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. doi:10.1111/1523-1747.ep12507865

[3] A. M. Stoll and M. A. Chianta, J. R. Piergallini, “Thermal Conduction Effects in Human-Skin,” Aviation, Space, and Environmental Medicine, Vol. 50, 1979, pp. 778-787.

[4] C. J. Diederich and K. Hynynen, “Ultrasound Technology for Hyperthermia,” Ultrasound in Medicine and Biology, Vol. 25, No. 6, 1999, pp. 871-887. doi:10.1016/S0301-5629(99)00048-4

[5] H. S. Carslaw and J. C. Jaeger, “Conduction of Heat in Solids,” 2nd Edition, Oxford University Press, Oxford, 1959.

[6] J. H. Lienhard, “A Heat Transfer Textbook,” Prentice- Hall, Englewood Cliffs, 1987.

[7] P. J. Riu, K. R. Foster, D. W. Blick and E. R. Adair, “A Thermal Model for Human Thresholds of Microwave- evoked Warmth Sensations,” Bioelectromagnetics, Vol. 18, No. 8, 1997, pp. 578-583. doi:10.1002/(SICI)1521-186X(1997)18:8<578::AID-BEM6>3.0.CO;2-#

[8] H. F. Bowman, E. G. Cravalho and M. Woods, “Theory, Measurement, and Application of Thermal Properties of Biomaterials,” Annual Review of Biophysics and Bioen- gineering, Vol. 4, 1975, pp. 43-80. doi:10.1146/annurev.bb.04.060175.000355

[9] G. T. Martin, H. F. Bowman and W. H. Newman, “Basic Element Method for Computing the Temperature Field during Hyperthermia Therapy Planning,” Advanced Bio Heat Mass Transfer, Vol. 231, 1992, pp. 75-80.

[10] C. R. Davies, G. M. Saidel and H. Harasaki, “Sensitivity Analysis of One-Dimensional Heat Transfer in Tissue with Temperature-Dependent Perfusion,” Journal of Biome- chanical Engineering, Vol. 119, No. 1, 1997, pp. 77-80. doi:10.1115/1.2796068

[11] H. Arkin and K. R. Holmes, “Recent Developments in Modeling Heat Transfer in Blood Perfused Tissues,” IEEE Transactions on Biomedical Engineering, Vol. 41, No. 2, 1994, pp. 97-107. doi:10.1109/10.284920

[12] B. Erdmann, J. Lang and M. Seebass, “Optimization of Temperature Distributions for Regional Hyperthermia Based on a Nonlinear Heat Transfer Model,” Annals of the New York Academy of Sciences, Vol. 858, No. 1, 1998, pp. 36-46. doi:10.1111/j.1749-6632.1998.tb10138.x

[13] P. D. Yréus and J. Diederich, “Theoretical Model of In- ternally Cooled Interstitial Ultrasound Applicators for Thermal Therapy,” Physics in Medicine & Biology, Vol. 47, No. 7, 2002, pp. 1073-1089. doi:10.1088/0031-9155/47/7/306

[14] K. R. Diller, “Development and Solution of Finite-dif- ference Equations for Burn Injury with Spreadsheet Soft- ware”, The Journal of Burn Care & Rehabilitation, Vol. 20, No. 1, 1999, pp. 25-32.

[15] K. R. Diller, “Modeling Thermal Skin Burns on a Per- sonal Computer,” The Journal of Burn Care & Rehabili- ntation, Vol. 19, No. 5, 1998, pp. 420-429. doi:10.1097/00004630-199809000-00012

[16] S. C. Jiang, N. Ma, H. J. Li and X. Zhang, “Effects of Thermal Properties and Geometrical Dimensions on Skin Burn Injuries,” Burns, Vol. 28, No. 8, 2002, pp. 713-717. doi:10.1016/S0305-4179(02)00104-3

[17] C. L. Chan, “Boundary Element Method Analysis for the Bioheat Transfer Equation” Journal of Biomechanical En- gineering, Vol. 114, No. 3, 1992, pp. 358-365. doi:10.1115/1.2891396

[18] B. Mochnacki and E. Majchrzak, “Sensitivity of the Skin Tissue on the Activity of External Heat Sources,” Com- puter Modeling in Engineering & Sciences, Vol. 4, No. 4, 2003, pp. 431-438.

[19] M. I. A. Othman, “Effect of Rotation in Case of 2-D Pro- blems of Generalized Thermoelasticity with Thermal Re- laxation,” Mechanics & Mechanical Engineering, Vol. 8, No. 2, 2005, pp. 111-126.

[20] M. I. A. Othman, “Effect of Rotation on Plane Waves in Generalized Thermo-Elasticity with Two Relaxation Times,” International Journal of Solids and Structures, Vol. 41, No. 11-12, 2004, pp. 2939-2956. doi:10.1016/j.ijsolstr.2004.01.009

[21] M. I. A. Othman, “Lord-Shulman Theory under the De- pendence of the Modulus of Elasticity on the Reference Temperature in Two-dimensional Generalized Thermo-elas- ticity,” Journal of Thermal Sresses, Vol. 25, No. 11, 2002, pp. 1027-1045. doi:10.1080/01495730290074621

[22] J. N. Sharma, R. Chand and M. I. A. Othman, “On the Propagation of Lamb Waves in Visco-Thermoelastic Plates under Fluid Loadings,” International Journal of Engineer- ing Science, Vol. 47, No. 3, 2009, pp. 391-404. doi:10.1016/j.ijengsci.2008.10.008

[23] M. I. A. Othman and R. Kumar, “Reflection of Magneto- Thermoelastic Waves with Temperature Dependent Prop- erties in Generalized Thermoelasticity,” International Com- munications in Heat and Mass Transfer, Vol. 36, No. 5, 2009, pp. 513-520. doi:10.1016/j.icheatmasstransfer.2009.02.002

[24] M. I. A. Othman and B. Singh, “The Effect of Rotation on Generalized Micropolar Thermoelasticity for a Half- space under Five Theories”, International Journal of Sol- ids and Structures, Vol. 44, No. 9, 2007, pp. 2748-2762. doi:10.1016/j.ijsolstr.2006.08.016

[25] M. I. A. Othman, Kh. Lotfy and R. M. Farouk, “General- ized Thermo-Microstretch Elastic Medium with Tempe- rature Dependent Properties for Different Theories,” En- gineering Analysis with Boundary Elements, Vol. 34, No. 3, 2010, pp. 229-237. doi:10.1016/j.enganabound.2009.10.003

[26] M. I. A. Othman, “Electrohydrodynamic Stability in a Ho- rizontal Viscoelastic Fluid Layer in the Presence of a Ver- tical Temperature Gradient,” International Journal of En- gineering Science, Vol. 39, No. 11, 2001, pp. 1217-1232. doi:10.1016/S0020-7225(00)00092-6

[27] H. Pauly and H. P. Schwan, “Mechanism of Absorption of Ultrasound in Liver Tissue,” Journal of the Acoustical Society of America, Vol. 50, No. 2B, 1971, pp. 692-698. doi:10.1121/1.1912685

[1] R. Seip and E. S. Ebbini, “Studies on the Three-Dimen- sional Temperature Response to Heating Fields Using Diagnostic Ultrasound,” IEEE Transactions on Biomedi- cal Engineering, Vol. 42, No. 8, 1995, pp. 828-839. doi:10.1109/10.398644

[2] 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. doi:10.1111/1523-1747.ep12507865

[3] A. M. Stoll and M. A. Chianta, J. R. Piergallini, “Thermal Conduction Effects in Human-Skin,” Aviation, Space, and Environmental Medicine, Vol. 50, 1979, pp. 778-787.

[4] C. J. Diederich and K. Hynynen, “Ultrasound Technology for Hyperthermia,” Ultrasound in Medicine and Biology, Vol. 25, No. 6, 1999, pp. 871-887. doi:10.1016/S0301-5629(99)00048-4

[5] H. S. Carslaw and J. C. Jaeger, “Conduction of Heat in Solids,” 2nd Edition, Oxford University Press, Oxford, 1959.

[6] J. H. Lienhard, “A Heat Transfer Textbook,” Prentice- Hall, Englewood Cliffs, 1987.

[7] P. J. Riu, K. R. Foster, D. W. Blick and E. R. Adair, “A Thermal Model for Human Thresholds of Microwave- evoked Warmth Sensations,” Bioelectromagnetics, Vol. 18, No. 8, 1997, pp. 578-583. doi:10.1002/(SICI)1521-186X(1997)18:8<578::AID-BEM6>3.0.CO;2-#

[8] H. F. Bowman, E. G. Cravalho and M. Woods, “Theory, Measurement, and Application of Thermal Properties of Biomaterials,” Annual Review of Biophysics and Bioen- gineering, Vol. 4, 1975, pp. 43-80. doi:10.1146/annurev.bb.04.060175.000355

[9] G. T. Martin, H. F. Bowman and W. H. Newman, “Basic Element Method for Computing the Temperature Field during Hyperthermia Therapy Planning,” Advanced Bio Heat Mass Transfer, Vol. 231, 1992, pp. 75-80.

[10] C. R. Davies, G. M. Saidel and H. Harasaki, “Sensitivity Analysis of One-Dimensional Heat Transfer in Tissue with Temperature-Dependent Perfusion,” Journal of Biome- chanical Engineering, Vol. 119, No. 1, 1997, pp. 77-80. doi:10.1115/1.2796068

[11] H. Arkin and K. R. Holmes, “Recent Developments in Modeling Heat Transfer in Blood Perfused Tissues,” IEEE Transactions on Biomedical Engineering, Vol. 41, No. 2, 1994, pp. 97-107. doi:10.1109/10.284920

[12] B. Erdmann, J. Lang and M. Seebass, “Optimization of Temperature Distributions for Regional Hyperthermia Based on a Nonlinear Heat Transfer Model,” Annals of the New York Academy of Sciences, Vol. 858, No. 1, 1998, pp. 36-46. doi:10.1111/j.1749-6632.1998.tb10138.x

[13] P. D. Yréus and J. Diederich, “Theoretical Model of In- ternally Cooled Interstitial Ultrasound Applicators for Thermal Therapy,” Physics in Medicine & Biology, Vol. 47, No. 7, 2002, pp. 1073-1089. doi:10.1088/0031-9155/47/7/306

[14] K. R. Diller, “Development and Solution of Finite-dif- ference Equations for Burn Injury with Spreadsheet Soft- ware”, The Journal of Burn Care & Rehabilitation, Vol. 20, No. 1, 1999, pp. 25-32.

[15] K. R. Diller, “Modeling Thermal Skin Burns on a Per- sonal Computer,” The Journal of Burn Care & Rehabili- ntation, Vol. 19, No. 5, 1998, pp. 420-429. doi:10.1097/00004630-199809000-00012

[16] S. C. Jiang, N. Ma, H. J. Li and X. Zhang, “Effects of Thermal Properties and Geometrical Dimensions on Skin Burn Injuries,” Burns, Vol. 28, No. 8, 2002, pp. 713-717. doi:10.1016/S0305-4179(02)00104-3

[17] C. L. Chan, “Boundary Element Method Analysis for the Bioheat Transfer Equation” Journal of Biomechanical En- gineering, Vol. 114, No. 3, 1992, pp. 358-365. doi:10.1115/1.2891396

[18] B. Mochnacki and E. Majchrzak, “Sensitivity of the Skin Tissue on the Activity of External Heat Sources,” Com- puter Modeling in Engineering & Sciences, Vol. 4, No. 4, 2003, pp. 431-438.

[19] M. I. A. Othman, “Effect of Rotation in Case of 2-D Pro- blems of Generalized Thermoelasticity with Thermal Re- laxation,” Mechanics & Mechanical Engineering, Vol. 8, No. 2, 2005, pp. 111-126.

[20] M. I. A. Othman, “Effect of Rotation on Plane Waves in Generalized Thermo-Elasticity with Two Relaxation Times,” International Journal of Solids and Structures, Vol. 41, No. 11-12, 2004, pp. 2939-2956. doi:10.1016/j.ijsolstr.2004.01.009

[21] M. I. A. Othman, “Lord-Shulman Theory under the De- pendence of the Modulus of Elasticity on the Reference Temperature in Two-dimensional Generalized Thermo-elas- ticity,” Journal of Thermal Sresses, Vol. 25, No. 11, 2002, pp. 1027-1045. doi:10.1080/01495730290074621

[22] J. N. Sharma, R. Chand and M. I. A. Othman, “On the Propagation of Lamb Waves in Visco-Thermoelastic Plates under Fluid Loadings,” International Journal of Engineer- ing Science, Vol. 47, No. 3, 2009, pp. 391-404. doi:10.1016/j.ijengsci.2008.10.008

[23] M. I. A. Othman and R. Kumar, “Reflection of Magneto- Thermoelastic Waves with Temperature Dependent Prop- erties in Generalized Thermoelasticity,” International Com- munications in Heat and Mass Transfer, Vol. 36, No. 5, 2009, pp. 513-520. doi:10.1016/j.icheatmasstransfer.2009.02.002

[24] M. I. A. Othman and B. Singh, “The Effect of Rotation on Generalized Micropolar Thermoelasticity for a Half- space under Five Theories”, International Journal of Sol- ids and Structures, Vol. 44, No. 9, 2007, pp. 2748-2762. doi:10.1016/j.ijsolstr.2006.08.016

[25] M. I. A. Othman, Kh. Lotfy and R. M. Farouk, “General- ized Thermo-Microstretch Elastic Medium with Tempe- rature Dependent Properties for Different Theories,” En- gineering Analysis with Boundary Elements, Vol. 34, No. 3, 2010, pp. 229-237. doi:10.1016/j.enganabound.2009.10.003

[26] M. I. A. Othman, “Electrohydrodynamic Stability in a Ho- rizontal Viscoelastic Fluid Layer in the Presence of a Ver- tical Temperature Gradient,” International Journal of En- gineering Science, Vol. 39, No. 11, 2001, pp. 1217-1232. doi:10.1016/S0020-7225(00)00092-6

[27] H. Pauly and H. P. Schwan, “Mechanism of Absorption of Ultrasound in Liver Tissue,” Journal of the Acoustical Society of America, Vol. 50, No. 2B, 1971, pp. 692-698. doi:10.1121/1.1912685