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Study of tumor growth under hyperthermia condition. (English) Zbl 1401.92114

Summary: The new concept of keeping primary tumor under control in situ to suppress distant foci sheds light on the treatment of metastatic tumor. Hyperthermia is considered as one of the means for controlling tumor growth. To simulate the tumor growth, a continuum mathematical model has been introduced. The newest understanding of the Warburg effect on the cellular metabolism and diffusion of the nutrients in the tissue has been taken into consideration. The numerical results are compared with the in vivo experimental data by fitting the tumor cell doubling time/tumor cell growth rate under different thermal conditions. Both the tumor growth curve and corresponding average glucose concentration have been predicted. The numerical results have quantitatively illustrated the controlling effect on tumor growth under hyperthermia condition in the initial stage.

MSC:

92C50 Medical applications (general)
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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[1] A. Jemal, R. Siegel, E. Ward, Y. Hao, J. Xu, and M. J. Thun, “Cancer statistics, 2009,” CA Cancer Journal for Clinicians, vol. 59, no. 4, pp. 225-249, 2009. · doi:10.3322/caac.20006
[2] O. Abe, R. Abe, K. Enomoto, et al., “Effects of radiotherapy and of differences in the extent of surgery for early breast cancer on local recurrence and 15-year survival: an overview of the randomised trials,” Lancet, vol. 366, no. 9503, pp. 2087-2106, 2005.
[3] B. Fisher, N. Gunduz, and E. A. Saffer, “Influence of the interval between primary tumor removal and chemotherapy on kinetics and growth of metastases,” Cancer Research, vol. 43, no. 4, pp. 1488-1492, 1983.
[4] K. Camphausen, M. A. Moses, W. D. Beecken, M. K. Khan, J. Folkman, and M. S. O’Reilly, “Radiation therapy to a primary tumor accelerates metastatic growth in mice,” Cancer Research, vol. 61, no. 5, pp. 2207-2211, 2001.
[5] N. L. Komarova, “Mathematical modeling of tumorigenesis: mission possible,” Current Opinion in Oncology, vol. 17, no. 1, pp. 39-43, 2005. · doi:10.1097/01.cco.0000143681.37692.32
[6] T. W. Secomb, D. A. Beard, J. C. Frisbee, N. P. Smith, and A. R. Pries, “The role of theoretical modeling in microcirculation research,” Microcirculation, vol. 15, no. 8, pp. 693-698, 2008. · doi:10.1080/10739680802349734
[7] T. Roose, S. J. Chapman, and P. K. Maini, “Mathematical models of avascular tumor growth,” SIAM Review, vol. 49, no. 2, pp. 179-208, 2007. · Zbl 1117.93011 · doi:10.1137/S0036144504446291
[8] A. K. Laird, “Dynamics of tumor growth,” British journal of cancer, vol. 13, pp. 490-502, 1964.
[9] R. C. Hu and X. G. Ruan, “A simple cellular automaton model for tumor-immunity system,” in Proceedings of the IEEE International Conference on Robotics, Intelligent Systems and Signal Processing, vol. 1-2, pp. 1031-1035, 2003.
[10] H. P. Greenspan, “Models for growth of a solid tumor by diffusion,” Applied Mathematics, vol. 51, no. 4, pp. 317-340, 1972. · Zbl 0257.92001
[11] A. C. Burton, “Rate of growth of solid tumours as a problem of diffusion,” Growth, Development and Aging, vol. 30, no. 2, pp. 157-176, 1966.
[12] Y. Kim and A. Friedman, “Interaction of tumor with its micro-environment: a mathematical model,” Bulletin of Mathematical Biology, vol. 72, no. 5, pp. 1029-1068, 2010. · Zbl 1197.92027 · doi:10.1007/s11538-009-9481-z
[13] H. M. Byrne and M. A. Chaplain, “Growth of nonnecrotic tumors in the presence and absence of inhibitors,” Mathematical Biosciences, vol. 130, no. 2, pp. 151-181, 1995. · Zbl 0836.92011 · doi:10.1016/0025-5564(94)00117-3
[14] A. R. A. Anderson, “A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion,” Mathematical Medicine and Biology, vol. 22, no. 2, pp. 163-186, 2005. · Zbl 1073.92013 · doi:10.1093/imammb/dqi005
[15] L. Preziosi and A. Tosin, “Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications,” Journal of Mathematical Biology, vol. 58, no. 4-5, pp. 625-656, 2009. · Zbl 1311.92029 · doi:10.1007/s00285-008-0218-7
[16] A. R. A. Anderson and M. A. J. Chaplain, “Continuous and discrete mathematical models of tumor-induced angiogenesis,” Bulletin of Mathematical Biology, vol. 60, no. 5, pp. 857-899, 1998. · Zbl 0923.92011 · doi:10.1006/bulm.1998.0042
[17] P. Macklin, S. McDougall, A. R. A. Anderson, M. A. J. Chaplain, V. Cristini, and J. Lowengrub, “Multiscale modelling and nonlinear simulation of vascular tumour growth,” Journal of Mathematical Biology, vol. 58, no. 4-5, pp. 765-798, 2009. · Zbl 1311.92040 · doi:10.1007/s00285-008-0216-9
[18] J. A. Adam, “Mathematical models of prevascular spheroid development and catastrophe-theoretic description of rapid metastatic growth/tumor remission,” Invasion and Metastasis, vol. 16, no. 4-5, pp. 247-267, 1996.
[19] A. M. Stein, T. Demuth, D. Mobley, M. Berens, and L. M. Sander, “A mathematical model of glioblastoma tumor spheroid invasion in a three-dimensional in vitro experiment,” Biophysical Journal, vol. 92, no. 1, pp. 356-365, 2007. · doi:10.1529/biophysj.106.093468
[20] D. Grecu, A. S. Carstea, A. T. Grecu, et al., “Mathematical modelling of tumor growth,” Romanian Reports in Physics, vol. 59, no. 2, pp. 447-455, 2007.
[21] Y. Jiang, J. Pjesivac-Grbovic, C. Cantrell, and J. P. Freyer, “A multiscale model for avascular tumor growth,” Biophysical Journal, vol. 89, no. 6, pp. 3884-3894, 2005. · doi:10.1529/biophysj.105.060640
[22] J. A. Sherratt and M. A. J. Chaplain, “A new mathematical model for avascular tumour growth,” Journal of Mathematical Biology, vol. 43, no. 4, pp. 291-312, 2001. · Zbl 0990.92021 · doi:10.1007/s002850100088
[23] J. P. Ward and J. R. King, “Mathematical modelling of avascular-tumour growth,” IMA Journal of Mathemathics Applied in Medicine and Biology, vol. 14, no. 1, pp. 39-69, 1997. · Zbl 0866.92011
[24] J. P. Ward and J. R. King, “Mathematical modelling of avascular-tumour growth II: modelling growth saturation,” IMA Journal of Mathemathics Applied in Medicine and Biology, vol. 16, no. 2, pp. 171-211, 1999. · Zbl 0943.92019
[25] J. J. Casciari, S. V. Sotirchos, and R. M. Sutherland, “Mathematical modelling of microenvironment and growth in EMT6/Ro multicellular tumour spheroids,” Cell Proliferation, vol. 25, no. 1, pp. 1-22, 1992.
[26] M. A. J. Chaplain and B. D. Sleeman, “A mathematical model for the growth and classification of a solid tumor: a new approach via nonlinear elasticity theory using strain-energy functions,” Mathematical Biosciences, vol. 111, no. 2, pp. 169-215, 1992. · Zbl 0761.92018 · doi:10.1016/0025-5564(92)90070-D
[27] M. A. J. Chaplain, “Avascular growth, angiogenesis and vascular growth in solid tumours: the mathematical modelling of the stages of tumour development,” Mathematical and Computer Modelling, vol. 23, no. 6, pp. 47-87, 1996. · Zbl 0859.92012 · doi:10.1016/0895-7177(96)00019-2
[28] J. J. Casciari, S. V. Sotirchos, and R. M. Sutherland, “Glucose diffusivity in multicellular tumor spheroids,” Cancer Research, vol. 48, no. 14, pp. 3905-3909, 1988.
[29] R. Venkatasubramanian, M. A. Henson, and N. S. Forbes, “Incorporating energy metabolism into a growth model of multicellular tumor spheroids,” Journal of Theoretical Biology, vol. 242, no. 2, pp. 440-453, 2006. · doi:10.1016/j.jtbi.2006.03.011
[30] O. Waruburg, “On the origin of cancer cells,” Science, vol. 123, pp. 309-314, 1956.
[31] O. Waruburg, The Prime Cause and Prevention of Cancer, Konrad Triltsch, Würzburg, Germany, 1969.
[32] M. G. V. Heiden, L. C. Cantley, and C. B. Thompson, “Understanding the warburg effect: the metabolic requirements of cell proliferation,” Science, vol. 324, no. 5930, pp. 1029-1033, 2009. · doi:10.1126/science.1160809
[33] C. V. Dang and G. L. Semenza, “Oncogenic alterations of metabolism,” Trends in Biochemical Sciences, vol. 24, no. 2, pp. 68-72, 1999. · doi:10.1016/S0968-0004(98)01344-9
[34] L. M. Postovit, M. A. Adams, G. E. Lash, J. P. Heaton, and C. H. Graham, “Oxygen-mediated regulation of tumor cell invasiveness-involvement of a nitric oxide signaling pathway,” Journal of Biological Chemistry, vol. 277, no. 38, pp. 35730-35737, 2002. · doi:10.1074/jbc.M204529200
[35] J. Bussink, J. H. A. M. Kaanders, and A. J. Van Der Kogel, “Tumor hypoxia at the micro-regional level: clinical relevance and predictive value of exogenous and endogenous hypoxic cell markers,” Radiotherapy and Oncology, vol. 67, no. 1, pp. 3-15, 2003. · doi:10.1016/S0167-8140(03)00011-2
[36] R. A. Gatenby and R. J. Gillies, “Why do cancers have high aerobic glycolysis?” Nature Reviews Cancer, vol. 4, no. 11, pp. 891-899, 2004. · doi:10.1038/nrc1478
[37] M. C. Brahimi-Horn and J. Pouysségur, “Oxygen, a source of life and stress,” FEBS Letters, vol. 581, no. 19, pp. 3582-3591, 2007. · doi:10.1016/j.febslet.2007.06.018
[38] P. P. Hsu and D. M. Sabatini, “Cancer cell metabolism: warburg and beyond,” Cell, vol. 134, no. 5, pp. 703-707, 2008. · doi:10.1016/j.cell.2008.08.021
[39] B. S. Peskin and M. J. Carter, “Chronic cellular hypoxia as the prime cause of cancer: what is the de-oxygenating role of adulterated and improper ratios of polyunsaturated fatty acids when incorporated into cell membranes?” Medical Hypotheses, vol. 70, no. 2, pp. 298-304, 2008. · doi:10.1016/j.mehy.2007.05.033
[40] R. Moreno-Sánchez, S. Rodríguez-Enríquez, A. Marín-Hernández, and E. Saavedra, “Energy metabolism in tumor cells,” FEBS Journal, vol. 274, no. 6, pp. 1393-1418, 2007.
[41] R. J. DeBerardinis, J. J. Lum, G. Hatzivassiliou, and C. B. Thompson, “The biology of cancer: metabolic reprogramming fuels cell growth and proliferation,” Cell Metabolism, vol. 7, no. 1, pp. 11-20, 2008. · doi:10.1016/j.cmet.2007.10.002
[42] Y. Chen, R. Cairns, I. Papandreou, A. Koong, and N. C. Denko, “Oxygen consumption can regulate the growth of tumors, a new perspective on the Warburg effect,” PLoS One, vol. 4, no. 9, Article ID e7033, 2009. · doi:10.1371/journal.pone.0007033
[43] P. R. Stauffer and S. N. Goldberg, “Introduction: thermal ablation therapy,” International Journal of Hyperthermia, vol. 20, no. 7, pp. 671-677, 2004. · doi:10.1080/02656730400007220
[44] A. Chicheł, J. Skowronek, M. Kubaszewska, and M. Kanikowski, “Hyperthermia-Description of a method and a review of clinical applications,” Reports of Practical Oncology and Radiotherapy, vol. 12, no. 5, pp. 267-275, 2007.
[45] J. R. Oleson, M. W. Dewhirst, J. M. Harrelson, K. A. Leopold, T. V. Samulski, and C. Y. Tso, “Tumor temperature distributions predict hyperthermia effect,” International Journal of Radiation Oncology Biology Physics, vol. 16, no. 3, pp. 559-570, 1989.
[46] J. R. Lepock, “Cellular effects of hyperthermia: relevance to the minimum dose for thermal damage,” International Journal of Hyperthermia, vol. 19, no. 3, pp. 252-266, 2003. · doi:10.1080/0265673031000065042
[47] M. Na, C. Chao, Z. Ai-li, et al., “Thermal environmental effect on breast tumor growth,” Journal of Shanghai Jiaotong University, vol. 27, no. 5, pp. 501-505, 2009.
[48] J. L. R. Roti, H. H. Kampinga, R. S. Malyapa, et al., “Nuclear matrix as a target for hyperthermic killing of cancer cells,” Cell Stress and Chaperones, vol. 3, no. 4, pp. 245-255, 1998.
[49] J. Dong, P. Liu, and L. X. Xu, “Immunologic response induced by synergistic effect of alternating cooling and heating of breast cancer,” International Journal of Hyperthermia, vol. 25, no. 1, pp. 25-33, 2009. · doi:10.1080/02656730802279534
[50] C. W. Song, “Effect of local hyperthermia on blood flow and microenvironment: a review,” Cancer Research, vol. 44, no. 10, supplement, pp. S4721-S4730, 1984.
[51] C. W. Song, A. Lokshina, and J. G. Rhee, “Implication of blood flow in hyperthermic treatment of tumors,” IEEE Transactions on Biomedical Engineering, vol. 31, no. 1, pp. 9-16, 1984.
[52] H. Maeda, J. Wu, T. Sawa, Y. Matsumura, and K. Hori, “Tumor vascular permeability and the EPR effect in macromolecular therapeutics: a review,” Journal of Controlled Release, vol. 65, no. 1-2, pp. 271-284, 2000. · doi:10.1016/S0168-3659(99)00248-5
[53] D. T. Connolly, D. M. Heuvelman, R. Nelson et al., “Tumor vascular permeability factor stimulates endothelial cell growth and angiogenesis,” Journal of Clinical Investigation, vol. 84, no. 5, pp. 1470-1478, 1989.
[54] Y. Shen, A. Zhang, and L. X. Xu, “Mechanical study on tumor microvessel damage induced by alternate cooling and heating treatment,” in Proceedings of the 11th ASME Summer Bioengineering Conference (SBC’09), part A and B, pp. 593-594, June 2009.
[55] J. P. Freyer and R. M. Sutherland, “Regulation of growth saturation and development of necrosis in EMT6/Ro multicellular spheroids by the glucose and oxygen supply,” Cancer Research, vol. 46, no. 7, pp. 3504-3512, 1986.
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