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Mathematics + cancer: an undergraduate “bridge” course in applied mathematics. (English) Zbl 1431.97002

Summary: Most undergraduates have limited experience with mathematical modeling. In an effort to respond to various initiatives, such as the recommendations outlined in [S. Garfunkel (ed.) and M. Montgomery (ed.), GAIMME: guidelines for assessment & instruction in mathematical modeling education. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (2016)], this paper describes a course on the mathematical models of cancer growth and treatment. Among its aims is to provide a template for a “bridge” course between the traditional calculus and differential equations sequence and more advanced courses in mathematics and statistics. Prerequisites include a course in ordinary differential equations. Linear algebra is a useful corequisite but no previous programming experience is required. The content includes classical models of tumor growth as well as models for the growth of specific cancer types. Relevant research articles are provided for further study. Material for student projects and effective communication is supplied, as well as suggestions for homework assignments and computer labs. This paper aims to assist instructors in developing their own “Mathematics + Cancer” course.

MSC:

97M10 Modeling and interdisciplinarity (aspects of mathematics education)
97D30 Objectives and goals of mathematics teaching
97M60 Biology, chemistry, medicine (aspects of mathematics education)
97I70 Functional equations (educational aspects)
97B40 Educational policy for higher education
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[1] K. Akakura, N. Bruchovsky, S. L. Goldenberg, P. S. Rennie, A. R. Buckley, and L. D. Sullivan, Effects of intermittent androgen suppression on androgen-dependent tumors, Cancer, 71 (1993), pp. 2782-2790, https://doi.org/10.1002/1097-0142(19930501)71:9
[2] M. Alley, The Craft of Scientific Presentations: Critical Steps to Succeed and Critical Errors to Avoid, 2nd ed., Springer, New York, 2013, https://doi.org/10.1007/978-1-4419-8279-7.
[3] American Cancer Society, http://cancerstatisticscenter.cancer.org/ (accessed 2019-10-10).
[4] American Cancer Society, Classics in oncology: Charles Brenton Huggins, CA Cancer J. Clin., 22 (1972), pp. 230-231, https://doi.org/10.3322/canjclin.22.4.230.
[5] American Cancer Society, Cancer Facts & Figures 2019, Atlanta, GA, 2019, https://www.cancer.org/research/cancer-facts-statistics.html.
[6] R. P. Araujo and D. L. S. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling, Bull. Math. Biol., 66 (2004), pp. 1039-1091, https://doi.org/10.1016/j.bulm.2003.11.002. · Zbl 1334.92187
[7] P. Armitage and R. Doll, The age distribution of cancer and a multi-stage theory of carcinogenesis, Brit. J. Cancer, 8 (1954), pp. 1-12.
[8] P. Armitage and R. Doll, A two-stage theory of carcinogenesis in relation to the age distribution of human cancer, Brit. J. Cancer, 11 (1957), pp. 161-169.
[9] J. Baez and Y. Kuang, Mathematical models of androgen resistance in prostate cancer patients under intermittent androgen suppression therapy, Appl. Sci., 6 (2016), art. 352, https://doi.org/10.3390/app6110352.
[10] J. C. Beier, J. L. Gevertz, and K. E. Howard, Building context with tumor growth modeling projects in differential equations, PRIMUS, 25 (2015), pp. 297-325, https://doi.org/10.1080/10511970.2014.975881.
[11] S. Benzekry, C. Lamont, A. Beheshti, A. Tracz, J. M. L. Ebos, L. Hlatky, and P. Hahnfeldt, Classical mathematical models for description and prediction of experimental tumor growth, PLoS Comput. Biol., 10 (2014), art. e1003800, https://doi.org/10.1371/journal.pcbi.1003800.
[12] M. Bjerknes, Expansion of mutant stem cell populations in the human colon, J. Theoret. Biol., 178 (1996), pp. 381-385, https://doi.org/10.1006/jtbi.1996.0034.
[13] K. M. Bliss, K. R. Fowler, and B. J. Galluzzo, Math Modeling: Getting Started & Getting Solutions, SIAM, Philadelphia, 2014, https://m3challenge.siam.org/resources/modeling-handbook.
[14] V. Brown and K. A. J. White, The HPV vaccination strategy: Could male vaccination have a significant impact?, Comput. Math. Methods Med., 11 (2010), pp. 223-237, https://doi.org/10.1080/17486700903486613. · Zbl 1202.92033
[15] S. Bunimovich-Mendrazitsky, J. C. Gluckman, and J. Chaskalovic, A mathematical model of combined bacillus Calmette-Guerin (BCG) and interleukin (IL)-2 immunotherapy of superficial bladder cancer, J. Theoret. Biol., 277 (2011), pp. 27-40, https://doi.org/10.1016/j.jtbi.2011.02.008. · Zbl 1397.92315
[16] S. Bunimovich-Mendrazitskya, H. Byrne, and L. Stone, Mathematical model of pulsed immunotherapy for superficial bladder cancer, Bull. Math. Biol., 70 (2008), pp. 2055-2076, https://doi.org/10.1007/s11538-008-9344-z. · Zbl 1147.92013
[17] H. M. Byrne, Dissecting cancer through mathematics: From the cell to the animal model, Nat. Rev. Cancer, 10 (2010), pp. 221-230, https://doi.org/10.1038/nrc2808.
[18] (CDC), Centers for Disease Control and Prevention, http://nccd.cdc.gov/uscs/ (accessed 2019-10-10).
[19] G. D. Clapp, T. Lepoutre, R. El Cheikh, S. Bernard, J. Ruby, H. Labussière-Wallet, F. E. Nicolini, and D. Levy, Implication of the autologous immune system in BCR\textendashABL transcript variations in chronic myelogenous leukemia patients treated with Imatinib, Cancer Res., 75 (2015), pp. 4053-4062, https://doi.org/10.1158/0008-5472.CAN-15-0611.
[20] J. Cohen, The Earth is round \((p < .05)\), Am. Psychol., 49 (1994), pp. 997-1003, https://doi.org/10.1037/0003-066X.49.12.997.
[21] J. Couzin, In their prime, and dying of cancer, Science, 317 (2007), pp. 1160-1162, https://doi.org/10.1126/science.317.5842.1160.
[22] CPython, CPython, https://github.com/python/cpython (accessed 2019-10-10).
[23] E. D. Crawford, C. S. Higano, N. D. Shore, M. Hussain, and D. P. Petrylak, Treating patients with metastatic castration resistant prostate cancer: A comprehensive review of available therapies, J. Urology, 194 (2015), pp. 1537-1547, https://doi.org/10.1016/j.juro.2015.06.106.
[24] J. M. Crowley and P. R. Turner, eds., Modeling Across the Curriculum, SIAM, Philadelphia, 2012, https://www.siam.org/reports/modeling_12.pdf.
[25] L. de Pillis, T. Caldwell, E. Sarapata, and H. Williams, Mathematical modeling of regulatory T cell effects on renal cell carcinoma treatment, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), pp. 915-943, https://doi.org/10.3934/dcdsb.2013.18.915. · Zbl 1301.92036
[26] L. G. de Pillis, W. Gu, and A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretations, J. Theoret. Biol., 238 (2006), pp. 841-862, https://doi.org/10.1016/j.jtbi.2005.06.037. · Zbl 1445.92135
[27] L. G. dePillis, H. Savage, and A. E. Radunskaya, Mathematical model of colorectal cancer with monoclonal antibody treatments, Brit. J. Med. Med. Res., 4 (2014), pp. 3101-3131, https://doi.org/10.9734/BJMMR/2014/8393.
[28] M. R. Droop, Vitamin \(B_{12}\) and marine ecology, IV. The kinetics of uptake, growth and inhibition in Monochrysis lutheri, J. Mar. Biol. Assoc. UK, 48 (1968), pp. 689-733, https://doi.org/10.1017/S0025315400019238.
[29] C. L. Dym, Principles of Mathematical Modeling, 2nd ed., Elsevier, New York, 2004. · Zbl 1057.00008
[30] B. Efron and R. Tibshirani, Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy, Statist. Sci., 1 (1986), pp. 54-77, https://doi.org/10.1214/ss/1177013815. · Zbl 0587.62082
[31] B. Efron and R. Tibshirani, An Introduction to the Bootstrap, Chapman & Hall, 1993. · Zbl 0835.62038
[32] A. Eladdadi and D. Isaacson, A mathematical model for the effects of HER2 overexpression on cell proliferation in breast cancer, Bull. Math. Biol., 70 (2008), pp. 1707-1729, https://doi.org/10.1007/s11538-008-9315-4. · Zbl 1166.92024
[33] A. Eladdadi and D. Isaacson, A mathematical model for the effects of HER2 over-expression on cell cycle progression in breast cancer, Bull. Math. Biol., 73 (2011), pp. 2865-2887, https://doi.org/10.1007/s11538-011-9663-3. · Zbl 1402.92149
[34] H. Enderling, M. A. J. Chaplain, A. R. A. Anderson, and J. S. Vaidya, A mathematical model of breast cancer development, local treatment and recurrence, J. Theoret. Biol., 246 (2007), pp. 245-259, https://doi.org/10.1016/j.jtbi.2006.12.010. · Zbl 1451.92091
[35] C. L. Epstein, Introduction to the Mathematics of Medical Imaging, 2nd ed., SIAM, Philadelphia, 2007, https://doi.org/10.1137/9780898717792. · Zbl 1305.92002
[36] M. S. Feizabadi and T. M. Witten, Modeling the effects of a simple immune system and immunodeficiency on the dynamics of conjointly growing tumor and normal cells, Int. J. Biol. Sci., 7 (2011), pp. 700-707, https://doi.org/10.7150/ijbs.7.700.
[37] B. J. Feldman and D. Feldman, The development of androgen-independent prostate cancer, Nat. Rev. Cancer, 1 (2001), pp. 34-45, https://doi.org/10.1038/35094009.
[38] J. Gallaher, L. M. Cook, S. Gupta, A. Araujo, J. Dhillon, J. Y. Park, J. G. Scott, J. Pow-Sang, D. Basanta, and C. C. Lynch, Improving treatment strategies for patients with metastatic castrate resistant prostate cancer through personalized computational modeling, Clin. Exp. Metastasis, 31 (2014), pp. 991-999, https://doi.org/10.1007/s10585-014-9674-1.
[39] S. Garfunkel and M. Montgomery, eds., GAIMME: Guidelines for Assessment & Instruction in Mathematical Modeling Education, SIAM, Philadelphia, 2016, http://www.siam.org/reports/gaimme.php.
[40] S. N. Gentry and T. L. Jackson, A mathematical model of cancer stem cell driven tumor initiation: Implications of niche size and loss of homeostatic regulatory mechanisms, PLoS ONE, 8 (2013), art. e71128, https://doi.org/10.1371/journal.pone.0071128.
[41] GNU Octave, https://www.gnu.org/software/octave (accessed 2019-10-10).
[42] B. Gompertz, On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies, Phil. Trans. Royal Soc. London, 115 (1825), pp. 513-583, https://doi.org/10.1098/rstl.1825.0026.
[43] S. Goodman, A dirty dozen: Twelve p-value misconceptions, Semin. Hematol., 45 (2008), pp. 135-140, https://doi.org/10.1053/j.seminhematol.2008.04.003.
[44] T. Greenhalgh, How to Read a Paper: The Basics of Evidence-Based Medicine, 5th ed., John Wiley & Sons, Hoboken, NJ, 2014.
[45] J. D. Hamilton, M. Rapp, T. Schneiderhan, M. Sabel, A. Hayman, et al., Glioblastoma multiforme metastasis outside the CNS: Three case reports and possible mechanisms of escape, J. Clin. Oncol., 32 (2014), pp. e80-e84, https://doi.org/10.1200/JCO.2013.48.7546.
[46] D. Hanahan and R. A. Weinberg, Hallmarks of cancer: The next generation, Cell, 144 (2011), pp. 646-674, https://doi.org/10.1016/j.cell.2011.02.013.
[47] E. B. Hedican, J. T. Kemper, and N. M. Lanie, Modeling biomarker dynamics with implications for the treatment of prostate cancer, Comput. Math. Methods Med., 8 (2007), pp. 77-92, https://doi.org/10.1080/17486700701349021. · Zbl 1121.92041
[48] N. J. Higham, Handbook of Writing for the Mathematical Sciences, 2nd ed., SIAM, Philadelphia, 1998, https://doi.org/10.1137/1.9780898719550. · Zbl 0973.00011
[49] N. H. Holford and L. B. Sheiner, Pharmacokinetic and pharmacodynamic modeling in vivo, Crit. Rev. Bioeng., 5 (1981), pp. 273-332.
[50] C. B. Huggins, The hormone-dependent cancer, Bull. N. Y. Acad. Med., 39 (1963), pp. 752-757.
[51] M. Hussain, C. M. Tangen, D. L. Berry, C. S. Higano, D. E. Crawford, G. Liu, et al., Intermittent versus continuous androgen deprivation in prostate cancer, New England J. Med., 368 (2013), pp. 1314-1325, https://doi.org/10.1056/NEJMoa1212299.
[52] H. V. Jain and A. Friedman, Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), pp. 945-967, https://doi.org/10.3934/dcdsb.2013.18.945. · Zbl 1277.92013
[53] H. V. Jain and T. Jackson, Mathematical modeling of cellular cross-talk between endothelial and tumor cells highlights counterintuitive effects of VEGF-targeted therapies, Bull. Math. Biol., 80 (2018), pp. 971-1016, https://doi.org/10.1007/s11538-017-0273-6. · Zbl 1394.92059
[54] M. D. Johnston, C. M. Edwards, W. F. Bodmer, P. K. Maini, and S. J. Chapman, Mathematical modeling of cell population dynamics in the colonic crypt and in colorectal cancer, Proc. Natl. Acad. Sci. USA, 104 (2007), pp. 4008-4013, https://doi.org/10.1073/pnas.0611179104.
[55] H.-W. Kang, M. Crawford, M. Fabbri, G. Nuovo, M. Garofalo, S. P. Nana-Sinkam, and A. Friedman, A mathematical model for MicroRNA in lung cancer, PLoS ONE, 8 (2013), pp. 1-19, https://doi.org/10.1371/journal.pone.0053663.
[56] A. G. Knudson, Mutation and cancer: Statistical study of retinoblastoma, Proc. Natl. Acad. Sci. USA, 68 (1971), pp. 820-823.
[57] Y. Kogan, Z. Agur, and M. Elishmereni, A mathematical model for the immunotherapeutic control of the Th1/Th2 imbalance in melanoma, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), pp. 1017-1030, https://doi.org/10.3934/dcdsb.2013.18.1017. · Zbl 1301.92033
[58] M. Kohandel, S. Sivaloganathan, and A. Oza, Mathematical modeling of ovarian cancer treatments: Sequencing of surgery and chemotherapy, J. Theoret. Biol., 242 (2006), pp. 62-68, https://doi.org/10.1016/j.jtbi.2006.02.001. · Zbl 1447.92198
[59] F. Kozusko, M. Bourdeau, Z. Bajzer, and D. Dingli, A microenvironment based model of antimitotic therapy of Gompertzian tumor growth, Bull. Math. Biol., 69 (2007), pp. 1691-1708, https://doi.org/10.1007/s11538-006-9186-5. · Zbl 1298.92051
[60] R. A. Ku-Carrillo, S. E. Delgadillo, and B. Chen-Charpentier, A mathematical model for the effect of obesity on cancer growth and on the immune system response, Appl. Math. Model., 40 (2016), pp. 4908-4920, https://doi.org/10.1016/j.apm.2015.12.018. · Zbl 1459.92042
[61] Y. Kuang, J. D. Nagy, and S. E. Eikenberry, Introduction to Mathematical Oncology, CRC Press, Boca Raton, FL, 2016. · Zbl 1341.92002
[62] A. K. Laird, Dynamics of tumor growth, Brit. J. Cancer, 18 (1964), pp. 490-502, https://doi.org/10.1038/bjc.1964.55.
[63] A. K. Laird, Dynamics of tumor growth: Comparison of growth rates and extrapolation of growth curve to one cell, Brit. J. Cancer, 19 (1965), pp. 278-291, https://doi.org/10.1038/bjc.1965.32.
[64] A. G. López, J. M. Seoane, and M. A. F. Sanjuán, A validated mathematical model of tumor growth including tumor-host interaction, cell-mediated immune response and chemotherapy, Bull. Math. Biol., 76 (2014), pp. 2884-2906, https://doi.org/10.1007/s11538-014-0037-5. · Zbl 1329.92075
[65] Y. Louzoun, C. Xue, G. B. Lesinski, and A. Friedman, A mathematical model for pancreatic cancer growth and treatments, J. Theoret. Biol., 351 (2014), pp. 74-82, https://doi.org/10.1016/j.jtbi.2014.02.028. · Zbl 1412.92154
[66] N. V. Mantzaris, S. Webb, and H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis, J. Math. Biol., 49 (2004), pp. 111-187, https://doi.org/10.1007/s00285-003-0262-2. · Zbl 1109.92020
[67] Maplesoft, https://www.maplesoft.com/products/maple (accessed 2019-10-10).
[68] A. Marciniak-Czochra, T. Stiehl, A. D. Ho, W. Jäger, and W. Wagner, Modeling of asymmetric cell division in hematopoietic stem cells-Regulation of self-renewal is essential for efficient repopulation, Stem Cells Dev., 18 (2008), pp. 377-386, https://doi.org/10.1089/scd.2008.0143.
[69] F. Michor, T. P. Hughes, Y. Iwasa, S. Branford, N. P. Shah, C. L. Sawyers, and M. A. Nowak, Dynamics of chronic myeloid leukemia, Nature, 435 (2005), pp. 1267-1270, https://doi.org/10.1038/nature03669.
[70] A. B. Miller, C. Wall, C. J. Baines, P. Sun, T. To, and S. A. Narod, Twenty five year follow-up for breast cancer incidence and mortality of the Canadian National Breast Screening Study: Randomised screening trial, BMJ, 348 (2014), art. g366, https://doi.org/10.1136/bmj.g366.
[71] H. Moore and N. K. Li, A mathematical model for chronic myelogenous leukemia (CML) and T cell interaction, J. Theoret. Biol., 227 (2004), pp. 513-523, https://doi.org/10.1016/j.jtbi.2003.11.024. · Zbl 1439.92068
[72] P. Morreale, M. Burnett, A. Gates, J. Cossa, and N. Amato, REU-in-a-Box: Expanding the pool of Computing Researchers, National Center for Women & Information Technology, 2011, http://www.ncwit.org/reubox.
[73] C. Mufudza, W. Sorofa, and E. T. Chiyaka, Assessing the effects of estrogen on the dynamics of breast cancer, Comput. Math. Methods Med., 2012 (2012), art. 473572, https://doi.org/10.1155/2012/473572. · Zbl 1254.92048
[74] H. Murphy, H. Jaafari, and H. M. Dobrovolny, Differences in predictions of ODE models of tumor growth: A cautionary example, BMC Cancer, 16 (2016), art. 163, https://doi.org/10.1186/s12885-016-2164-x.
[75] J. D. Nagy, The ecology and evolutionary biology of cancer: A review of mathematical models of necrosis and tumor cell diversity, Math. Biosci. Eng., 2 (2005), pp. 381-418, https://doi.org/10.3934/mbe.2005.2.381. · Zbl 1070.92026
[76] S. Nanda, L. de Pillis, and A. Radunskaya, B cell chronic lymphocytic leukemia: A model with immune response, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), pp. 1053-1076, https://doi.org/10.3934/dcdsb.2013.18.1053. · Zbl 1277.92017
[77] National Science Foundation, Research Experiences for Undergraduates (REU), https://www.nsf.gov/funding/pgm_summ.jsp?pims_id=5517 (accessed 2019-10-10).
[78] A. D. Norden and P. Y. Wen, Glioma therapy in adults, Neurologist, 12 (2006), pp. 279-292, https://doi.org/10.1097/01.nrl.0000250928.26044.47.
[79] C. O. Nordling, A new theory on the cancer-inducing mechanism, Brit. J. Cancer, 7 (1953), pp. 68-72.
[80] L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treat. Rep., 61 (1977), pp. 1307-1317.
[81] L. Norton and R. Simon, The Norton-Simon hypothesis revisited, Cancer Treat. Rep., 70 (1986), pp. 163-169.
[82] D. Paquin, P. S. Kim, P. P. Lee, and D. Levy, Strategic treatment interruptions during Imatinib treatment of chronic myelogenous leukemia, Bull. Math. Biol., 73 (2011), pp. 1082-1100, https://doi.org/10.1007/s11538-010-9553-0. · Zbl 1215.92035
[83] T. Portz, Y. Kuang, and J. D. Nagy, A clinical data validated mathematical model of prostate cancer growth under intermittent androgen suppression therapy, AIP Adv., 2 (2012), art. 011002, https://doi.org/10.1063/1.3697848.
[84] A. Rhodes and T. Hillen, Mathematical modeling of the role of Survivin on dedifferentiation and radioresistance in cancer, Bull. Math. Biol., 78 (2016), pp. 1162-1188, https://doi.org/10.1007/s11538-016-0177-x. · Zbl 1348.92086
[85] R. Roe-Dale, D. Isaacson, and M. Kupferschmid, A mathematical model of breast cancer treatment with CMF and doxorubicin, Bull. Math. Biol., 73 (2011), pp. 585-608, https://doi.org/10.1007/s11538-010-9549-9. · Zbl 1226.92042
[86] K. Roesch, D. Hasenclever, and M. Scholz, Modeling lymphoma therapy and outcome, Bull. Math. Biol., 76 (2014), pp. 401-430, https://doi.org/10.1007/s11538-013-9925-3. · Zbl 1297.92044
[87] T. Roose, S. J. Chapman, and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Rev., 49 (2007), pp. 179-208, https://doi.org/10.1137/S0036144504446291. · Zbl 1117.93011
[88] R. K. Sachs, L. R. Hlatky, and P. Hahnfeldt, Simple ODE models of tumor growth and anti-angiogenic or radiation treatment, Math. Comput. Model., 33 (2001), pp. 1297-1305, https://doi.org/10.1016/S0895-7177(00)00316-2. · Zbl 1004.92023
[89] E. A. Sarapata and L. G. de Pillis, A comparison and catalog of intrinsic tumor growth models, Bull. Math. Biol., 76 (2014), pp. 2010-2024, https://doi.org/10.1007/s11538-014-9986-y. · Zbl 1300.92042
[90] C. S. Schumacher and M. J. Siegel, 2015 CUPM Curriculum Guide to Majors in the Mathematical Sciences, Mathematical Association of America, Washington, DC, 2015.
[91] H. E. Skipper, F. M. Schabel Jr., and W. S. Willcox, Experimental evaluation of potential anticancer agents XIII: On the criteria and kinetics associated with “curability” of experimental leukemia, Cancer Chemother. Rep., 35 (1964), pp. 1-111.
[92] T. Stiehl, C. Lutz, and A. Marciniak-Czochra, Emergence of heterogeneity in acute leukemias, Biology Direct, 11 (2016), art. 51, https://doi.org/10.1186/s13062-016-0154-1.
[93] M. Sturrock, W. Hao, J. Schwartzbaum, and G. A. Rempala, A mathematical model of pre-diagnostic glioma growth, J. Theoret. Biol., 380 (2015), pp. 299-308, https://doi.org/10.1016/j.jtbi.2015.06.003. · Zbl 1343.92247
[94] M. Sun, T. K. Choueiri, O.-P. R. Hamnvik, et al., Comparison of gonadotropin-releasing hormone agonists and orchiectomy: Effects of androgen-deprivation therapy, JAMA Oncology, 2 (2016), pp. 500-507, https://doi.org/10.1001/jamaoncol.2015.4917.
[95] Surveillance, Epidemiology, and End Results (SEER) Program of the National Cancer Institute (NCI), http://seer.cancer.gov/ (accessed 2019-10-10).
[96] K. R. Swanson, C. Bridge, J. D. Murray, and E. C. Alvord Jr., Virtual and real brain tumors: Using mathematical modeling to quantify glioma growth and invasion, J. Neurol. Sci., 216 (2003), pp. 1-10, https://doi.org/10.1016/j.jns.2003.06.001.
[97] K. R. Swanson, R. C. Rostomily, and E. C. Alvord Jr., A mathematical modeling tool for predicting survival of individual patients following resection of glioblastoma: A proof of principle, Brit. J. Cancer, 98 (2008), pp. 113-119, https://doi.org/10.1038/sj.bjc.6604125.
[98] A. Talkington, C. Dantoin, and R. Durrett, Ordinary differential equation models for adoptive immunotherapy, Bull. Math. Biol., 80 (2018), pp. 1059-1083, https://doi.org/10.1007/s11538-017-0263-8. · Zbl 1394.92066
[99] A. Talkington and R. Durrett, Estimating tumor growth rates in vivo, Bull. Math. Biol., 77 (2015), pp. 1934-1954, https://doi.org/10.1007/s11538-015-0110-8. · Zbl 1339.92035
[100] The MathWorks, The MathWorks, https://www.mathworks.com (accessed 2019-10-10).
[101] H. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, 2003. · Zbl 1054.92042
[102] C. Tomasetti, L. Li, and B. Vogelstein, Stem cell divisions, somatic mutations, cancer etiology, and cancer prevention, Science, 355 (2017), pp. 1330-1334, https://doi.org/10.1126/science.aaf9011.
[103] C. Tomasetti, L. Marchionni, M. A. Nowak, G. Parmigiani, and B. Vogelstein, Only three driver gene mutations are required for the development of lung and colorectal cancers, Proc. Natl. Acad. Sci. USA, 112 (2015), pp. 118-123, https://doi.org/10.1073/pnas.1421839112.
[104] C. Tomasetti and B. Vogelstein, Variation in cancer risk among tissues can be explained by the number of stem cell divisions, Science, 347 (2015), pp. 78-81, https://doi.org/10.1126/science.1260825.
[105] J. J. Tosoian, M. Mamawala, J. I. Epstein, P. Landis, S. Wolf, B. J. Trock, and H. B. Carter, Intermediate and longer-term outcomes from a prospective active-surveillance program for favorable-risk prostate cancer, J. Clin. Oncol., 33 (2015), pp. 3379-3385, https://doi.org/10.1200/JCO.2015.62.5764.
[106] I. Vergote, C. G. Tropé, F. Amant, G. B. Kristensen, T. Ehlen, N. Johnson, R. H. Verheijen, M. E. van der Burg, A. J. Lacave, P. B. Panici, G. G. Kenter, A. Casado, C. Mendiola, C. Coens, L. Verleye, G. C. Stuart, S. Pecorelli, and N. S. Reed, Neoadjuvant chemotherapy or primary surgery in stage IIIC or IV ovarian cancer, New England J. Med., 363 (2010), pp. 943-953, https://doi.org/10.1056/NEJMoa0908806.
[107] L. von Bertalanffy, Quantitative laws in metabolism and growth, Quart. Rev. Biol., 32 (1957), pp. 217-231, https://doi.org/10.1086/401873.
[108] S. Wilson and D. Levy, A mathematical model of the enhancement of tumor vaccine efficacy by immunotherapy, Bull. Math. Biol., 74 (2012), pp. 1485-1500, https://doi.org/10.1007/s11538-012-9722-4. · Zbl 1251.92023
[109] D. Wodarz and N. L. Komarova, Dynamics of Cancer: Mathematical Foundations of Oncology, World Scientific, Hackensack, NJ, 2014. · Zbl 1318.92001
[110] Wolfram Research, Wolfram Research, https://www.wolfram.com/mathematica (accessed 2019-10-10).
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