2015, 12(5): 907-915. doi: 10.3934/mbe.2015.12.907

Thermal detection of a prevascular tumor embedded in breast tissue

1. 

School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 14623-5603, United States, United States

2. 

Department of Mathematics and Statistics, Rochester Institute of Technology, Rochester, New York 14623, United States

Received  January 2015 Revised  April 2015 Published  June 2015

This paper presents a mathematical model of heat transfer in a prevascular breast tumor. The model uses the steady state temperature of the breast at the skin surface to determine whether there is an underlying tumor and if so, verifies whether the tumor is growing or dormant. The model is governed by the Pennes equations and we present numerical simulations for versions of the model in two and three dimensions.
Citation: Ephraim Agyingi, Tamas Wiandt, Sophia A. Maggelakis. Thermal detection of a prevascular tumor embedded in breast tissue. Mathematical Biosciences & Engineering, 2015, 12 (5) : 907-915. doi: 10.3934/mbe.2015.12.907
References:
[1]

J. P. Agnelli, A. A. Barrea and C. V. Turner, Tumor location and parameter estimation by thermography,, Mathematical and Computer Modelling: An International Journal, 53 (2011), 1527.  doi: 10.1016/j.mcm.2010.04.003.  Google Scholar

[2]

N. Arora, D. Martins, D. Ruggerio, E. Tousimis, A. J. Swistel, M. P. Osborne and R. M. Simmons, Effectiveness of a noninvasive digital infrared thermal imaging system in the detection of breast cancer,, The American Journal of Surgery, 196 (2008), 523.  doi: 10.1016/j.amjsurg.2008.06.015.  Google Scholar

[3]

F. C. Baker, J. I. Waner, E. F. Vieira, S. R. Taylor, H. S. Driver and D. Mitchell, Sleep and 24 hour body temperatures: A comparison in young men, naturally cycling women and women taking hormonal contraceptives,, The Journal of Physiology, 530 (2001), 565.  doi: 10.1111/j.1469-7793.2001.0565k.x.  Google Scholar

[4]

G. F. Baronzio, A. Gramaglia, A. B. Baronzio and I. Freitas, Influence of Tumor Microenvironment on Thermoresponse: Biologic and Clinical Implications,, Madame Curie Bioscience Database . Austin (TX): Landes Bioscience, (2000).   Google Scholar

[5]

W. C. Black and H. G. Welch, Advances in diagnostic imaging and overestimations of disease prevalence and the benefits of therapy,, New England Journal of Medicine, 328 (1993), 1237.  doi: 10.1056/NEJM199304293281706.  Google Scholar

[6]

M. Brennan and N. Houssami, Thermography in breast cancer diagnosis, screening and risk assessment: systematic review,, Breast Cancer Management, 2 (2013), 163.  doi: 10.2217/bmt.13.4.  Google Scholar

[7]

J. D. Bronzino, Medical Devices and Systems. Boca Raton,, FL:CRC/Taylor & Francis; 2006., (2006).   Google Scholar

[8]

C. K. Charny, Mathematical models of bioheat transfer,, in Bioengineering Heat Transfer(ed. Y. I. Cho), (1992), 19.   Google Scholar

[9]

Z. Deng and J. Liu, Mathematical modeling of temperature mapping over skin surface and its implementation in thermal disease diagnostics,, Computers in Biology and Medicine, 34 (2004), 495.  doi: 10.1016/S0010-4825(03)00086-6.  Google Scholar

[10]

J. Folkman, Tumor angiogenesis: Therapeutic implications,, New England Journal of Medicine, 285 (1971), 1182.   Google Scholar

[11]

J. Folkman and R. Kalluri, Cancer without disease,, Nature, 427 (2004).  doi: 10.1038/427787a.  Google Scholar

[12]

F. J. González, Thermal simulation of breast tumors,, Revista Mexicana de Fisica, 53 (2007), 323.   Google Scholar

[13]

F. J. González, Non-invasive estimation of the metabolic heat production of breast tumors using digital infrared imaging,, QIRT Journal, 8 (2011), 139.   Google Scholar

[14]

H. P. Greenspan, Models for the growth of a solid tumor by diffusion,, Studies in Applied Mathematics, LI (1972), 317.   Google Scholar

[15]

D. R. Grimes, C. Kelly, K. Bloch and M. Partridge, A method for estimating the oxygen consumption rate in multicellular tumour spheroids,, Journal of the Royal Society Interface, 11 (2013).  doi: 10.1098/rsif.2013.1124.  Google Scholar

[16]

R. N. Lawson, Implications of surface temperatures in the diagnosis of breast cancer,, Canadian Medical Association Journal, 75 (1956), 309.   Google Scholar

[17]

R. N. Lawson and M. S. Chugtai, Breast cancer and body temperatures,, Canadian Medical Association Journal, 88 (1963), 68.   Google Scholar

[18]

Q. Y. Lin, H. Q. Yang, S. S. Xie, Y. H. Wang, Z. Ye and S. Q. Chen, Detecting early breast tumour by finite element thermal analysis,, Journal of Medical Engineering & Technology, 33 (2009), 274.  doi: 10.1080/03091900802106638.  Google Scholar

[19]

S. A. Maggelakis and A. E. Savakis, Heat transfer in tissue containing a prevascular tumor,, Applied Mathematics Letters, 8 (1995), 7.  doi: 10.1016/0893-9659(94)00101-H.  Google Scholar

[20]

M. Mital and E. P. Scott, Thermal Detection of Embedded Tumors Using Infrared Imaging,, Journal of Biomechanical Engineering, 129 (2006), 33.  doi: 10.1115/1.2401181.  Google Scholar

[21]

W. Mueller-Klieser, Method for the determination of oxygen consumption rates and diffusion coefficients in multicellular spheroids,, Biophysical Journal, 46 (1984), 343.  doi: 10.1016/S0006-3495(84)84030-8.  Google Scholar

[22]

M. Paruch and E. Majchrzak, Identification of tumor region parameters using evolutionary algorithm and multiple reciprocity boundary element method,, Engineering Applications of Artificial Intelligence, 20 (2007), 647.  doi: 10.1016/j.engappai.2006.11.003.  Google Scholar

[23]

H. H. Pennes, Analysis of tissue and arterial blood temperatures in the resting forearm,, J. Appl. Physiol, 1 (1948), 93.   Google Scholar

[24]

N. M. Sudharsan, E. Y. K. Ng and S. L. Teh, Surface Temperature Distribution of a Breast With and Without Tumour,, Computer Methods in Biomechanics and Biomedical Engineering, 2 (1999), 187.  doi: 10.1080/10255849908907987.  Google Scholar

show all references

References:
[1]

J. P. Agnelli, A. A. Barrea and C. V. Turner, Tumor location and parameter estimation by thermography,, Mathematical and Computer Modelling: An International Journal, 53 (2011), 1527.  doi: 10.1016/j.mcm.2010.04.003.  Google Scholar

[2]

N. Arora, D. Martins, D. Ruggerio, E. Tousimis, A. J. Swistel, M. P. Osborne and R. M. Simmons, Effectiveness of a noninvasive digital infrared thermal imaging system in the detection of breast cancer,, The American Journal of Surgery, 196 (2008), 523.  doi: 10.1016/j.amjsurg.2008.06.015.  Google Scholar

[3]

F. C. Baker, J. I. Waner, E. F. Vieira, S. R. Taylor, H. S. Driver and D. Mitchell, Sleep and 24 hour body temperatures: A comparison in young men, naturally cycling women and women taking hormonal contraceptives,, The Journal of Physiology, 530 (2001), 565.  doi: 10.1111/j.1469-7793.2001.0565k.x.  Google Scholar

[4]

G. F. Baronzio, A. Gramaglia, A. B. Baronzio and I. Freitas, Influence of Tumor Microenvironment on Thermoresponse: Biologic and Clinical Implications,, Madame Curie Bioscience Database . Austin (TX): Landes Bioscience, (2000).   Google Scholar

[5]

W. C. Black and H. G. Welch, Advances in diagnostic imaging and overestimations of disease prevalence and the benefits of therapy,, New England Journal of Medicine, 328 (1993), 1237.  doi: 10.1056/NEJM199304293281706.  Google Scholar

[6]

M. Brennan and N. Houssami, Thermography in breast cancer diagnosis, screening and risk assessment: systematic review,, Breast Cancer Management, 2 (2013), 163.  doi: 10.2217/bmt.13.4.  Google Scholar

[7]

J. D. Bronzino, Medical Devices and Systems. Boca Raton,, FL:CRC/Taylor & Francis; 2006., (2006).   Google Scholar

[8]

C. K. Charny, Mathematical models of bioheat transfer,, in Bioengineering Heat Transfer(ed. Y. I. Cho), (1992), 19.   Google Scholar

[9]

Z. Deng and J. Liu, Mathematical modeling of temperature mapping over skin surface and its implementation in thermal disease diagnostics,, Computers in Biology and Medicine, 34 (2004), 495.  doi: 10.1016/S0010-4825(03)00086-6.  Google Scholar

[10]

J. Folkman, Tumor angiogenesis: Therapeutic implications,, New England Journal of Medicine, 285 (1971), 1182.   Google Scholar

[11]

J. Folkman and R. Kalluri, Cancer without disease,, Nature, 427 (2004).  doi: 10.1038/427787a.  Google Scholar

[12]

F. J. González, Thermal simulation of breast tumors,, Revista Mexicana de Fisica, 53 (2007), 323.   Google Scholar

[13]

F. J. González, Non-invasive estimation of the metabolic heat production of breast tumors using digital infrared imaging,, QIRT Journal, 8 (2011), 139.   Google Scholar

[14]

H. P. Greenspan, Models for the growth of a solid tumor by diffusion,, Studies in Applied Mathematics, LI (1972), 317.   Google Scholar

[15]

D. R. Grimes, C. Kelly, K. Bloch and M. Partridge, A method for estimating the oxygen consumption rate in multicellular tumour spheroids,, Journal of the Royal Society Interface, 11 (2013).  doi: 10.1098/rsif.2013.1124.  Google Scholar

[16]

R. N. Lawson, Implications of surface temperatures in the diagnosis of breast cancer,, Canadian Medical Association Journal, 75 (1956), 309.   Google Scholar

[17]

R. N. Lawson and M. S. Chugtai, Breast cancer and body temperatures,, Canadian Medical Association Journal, 88 (1963), 68.   Google Scholar

[18]

Q. Y. Lin, H. Q. Yang, S. S. Xie, Y. H. Wang, Z. Ye and S. Q. Chen, Detecting early breast tumour by finite element thermal analysis,, Journal of Medical Engineering & Technology, 33 (2009), 274.  doi: 10.1080/03091900802106638.  Google Scholar

[19]

S. A. Maggelakis and A. E. Savakis, Heat transfer in tissue containing a prevascular tumor,, Applied Mathematics Letters, 8 (1995), 7.  doi: 10.1016/0893-9659(94)00101-H.  Google Scholar

[20]

M. Mital and E. P. Scott, Thermal Detection of Embedded Tumors Using Infrared Imaging,, Journal of Biomechanical Engineering, 129 (2006), 33.  doi: 10.1115/1.2401181.  Google Scholar

[21]

W. Mueller-Klieser, Method for the determination of oxygen consumption rates and diffusion coefficients in multicellular spheroids,, Biophysical Journal, 46 (1984), 343.  doi: 10.1016/S0006-3495(84)84030-8.  Google Scholar

[22]

M. Paruch and E. Majchrzak, Identification of tumor region parameters using evolutionary algorithm and multiple reciprocity boundary element method,, Engineering Applications of Artificial Intelligence, 20 (2007), 647.  doi: 10.1016/j.engappai.2006.11.003.  Google Scholar

[23]

H. H. Pennes, Analysis of tissue and arterial blood temperatures in the resting forearm,, J. Appl. Physiol, 1 (1948), 93.   Google Scholar

[24]

N. M. Sudharsan, E. Y. K. Ng and S. L. Teh, Surface Temperature Distribution of a Breast With and Without Tumour,, Computer Methods in Biomechanics and Biomedical Engineering, 2 (1999), 187.  doi: 10.1080/10255849908907987.  Google Scholar

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