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2013, 10(1): 221-234. doi: 10.3934/mbe.2013.10.221

Mathematical analysis and simulations involving chemotherapy and surgery on large human tumours under a suitable cell-kill functional response

Diego Samuel Rodrigues 1, and  Paulo Fernando de Arruda Mancera 2,

1. 

Universidade de São Paulo, Depto de Matemática Aplicada e Estatística, ICMC, USP, 13560-970, São Carlos, Brazil
2. 

Universidade Estadual Paulista, Depto de Bioestatística, IBB, UNESP, 18618-970, Botucatu, Brazil

Received  April 2012 Revised  September 2012 Published  December 2012

Dosage and frequency of treatment schedules are important for successful chemotherapy. However, in this work we argue that cell-kill response and tumoral growth should not be seen as separate and therefore are essential in a mathematical cancer model. This paper presents a mathematical model for sequencing of cancer chemotherapy and surgery. Our purpose is to investigate treatments for large human tumours considering a suitable cell-kill dynamics. We use some biological and pharmacological data in a numerical approach, where drug administration occurs in cycles (periodic infusion) and surgery is performed instantaneously. Moreover, we also present an analysis of stability for a chemotherapeutic model with continuous drug administration. According to Norton & Simon [22], our results indicate that chemotherapy is less efficient in treating tumours that have reached a plateau level of growing and that a combination with surgical treatment can provide better outcomes.
Keywords: Norton-Simon hypothesis, chemotherapy, mathematical modelling., Tumour.
Mathematics Subject Classification: Primary: 92B05; Secondary: 37N2.
Citation: Diego Samuel Rodrigues, Paulo Fernando de Arruda Mancera. Mathematical analysis and simulations involving chemotherapy and surgery on large human tumours under a suitable cell-kill functional response. Mathematical Biosciences & Engineering, 2013, 10 (1) : 221-234. doi: 10.3934/mbe.2013.10.221
References:
[1]

R. E. Bellman., "Mathematical Methods in Medicine,'' World Scientific Publishing Co. Inc., River Edge, 1983.  Google Scholar

[2]

T. Browder, C. E. Butterfield, B. M. Kraling, B. Shi, B. Marshall, M. S. O'Reilly and J. Folkman, Antiangiogenic scheduling of chemotherapy improves efficacy against experimental drug-resistant cancer, Cancer Res., 60 (2000), 1878-1886. Google Scholar

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L. G. de Pillis and A. E. Radunskaya, A mathematical tumor model model with immune resistance and drug therapy: An optimal control approach, J. Theor. Med., 3 (2001), 79-100. doi: 10.1080/10273660108833067.  Google Scholar

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L. G. de Pillis, W. Gu, K. R. Fister, T. Head, K. Maples, A. Murugan, T. Neal and K. Yoshida, Chemotherapy for tumors: Analysis of the dynamics and a study of quadratic and linear optimal controls, Math. Biosc., 209 (2007), 292-315. doi: 10.1016/j.mbs.2006.05.003.  Google Scholar

[6]

A. D'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, An optimal delivery of combination therapy for tumors, Math. Biosc., 222 (2009), 13-26. doi: 10.1016/j.mbs.2009.08.004.  Google Scholar

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R. T. Dorr and D. D. Von Hoff, "Cancer Chemotherapy Handbook,'' McGraw-Hill, 1994. Google Scholar

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FEC100 chemotherapy for breast cancer (Written by Jeremy Braybrooke)., Document number: ASWCS09 BR006 [internet] accessed 27/07/2011, available from , ().   Google Scholar

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{N. Ferrara and H. P. Gerber}, The role of vascular endothelial growth factor in angiogenesis, Acta Haematol., 106 (2001), 148-156. doi: 10.1159/000046610.  Google Scholar

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K. R. Fister and J. C. Panetta, Optimal control applied to competing chemotherapeutic cell-kill strategies, SIAM J. Appl. Math., 63 (2003), 1954-1971. doi: 10.1137/S0036139902413489.  Google Scholar

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R. A. Gatenby, Application of competition theory to tumour growth: Implications for tumour biology and treatment, Eur. J. Cancer, 32A (1996), 722-726. doi: 10.1016/0959-8049(95)00658-3.  Google Scholar

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R. S. Kerbel, Tumour angiogenesis: Past, present and the near future, Carcinogenesis, 21 (2000), 505-515. doi: 10.1093/carcin/21.3.505.  Google Scholar

[13]

M. Kohandel, S. Sivaloganathan and A. Oza, Mathematical modeling of ovarian cancer treatments: Sequencing of surgery and chemotherapy, J. Theor. Biol., 242 (2006), 62-68. doi: 10.1016/j.jtbi.2006.02.001.  Google Scholar

[14]

L. G. Marcu and E. Bezak, Neoadjuvant cisplatin for head and neck cancer: simulation of a novel schedule for improved therapeutic ratio, J. Theor. Biol., 297 (2012), 41-47. doi: 10.1016/j.jtbi.2011.12.001.  Google Scholar

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R. B. Martin, M. E. Fisher, R. F. Minchin and K. L. Teo, Low-intensity combination chemotherapy maximizes host survival time for tumors containing drug-resistant cells, Math. Biosc., 110 (1992), 221-252. doi: 10.1016/0025-5564(92)90039-Y.  Google Scholar

[16]

R. B. Martin and K. L. Teo, "Optimal Control of Drug Administration in Cancer Chemotherapy,'' World Scientific, 1994. Google Scholar

[17]

R. B. Martin, Optimal control drug scheduling of cancer chemotherapy, Automatica 28 (1992), 113-1123. doi: 10.1016/0005-1098(92)90054-J.  Google Scholar

[18]

MeadJohnson Oncology Products [internet], http://patient.cancerconsultants.com/druginserts/Cyclophosphamide.pdf., accessed 29/02/2012., ().  http://patient.cancerconsultants.com/druginserts/Cyclophosphamide.pdf.+" target="_new" title="Go to article in Google Scholar"> Google Scholar

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S. Mukherjee, "The Emperor of All Maladies: A Biography of Cancer,'' Scribner, 2010. Google Scholar

[21]

F. Nani and H. I. Freedman, A mathematical model of cancer treatment by immunotherapy, Math. Biosc., 163 (2000), 159-199. doi: 10.1016/S0025-5564(99)00058-9.  Google Scholar

[22]

L. Norton and R. Simon, The Norton-Simon hypothesis revisited, Cancer Treat. Rep., 70 (1986), 163-169. Google Scholar

[23]

S. T. R. Pinho, H. I. Freedman and F. K. Nani, A chemotherapy model for the treatment of cancer with metastasis, Math. Comp. Model., 36 (2002), 773-803. doi: 10.1016/S0895-7177(02)00227-3.  Google Scholar

[24]

S. T. R. Pinho, F. S. Bacelar, R. F. S. Andrade and H. I. Freedman, A mathematical model for the effect of anti-angiogenic therapy in the treatment of cancer tumours by chemotherapy, Nonlin. Anal.: Real World Appl., 14 (2013), 815-828. doi: 10.1016/j.nonrwa.2012.07.034.  Google Scholar

[25]

D. S. Rodrigues, S. T. R. Pinho and P. F. A. Mancera, Um modelo matemático em quimio-terapia, TEMA, 13 (2012), 1-12 (in portuguese). doi: 10.5540/tema.2012.013.01.0001.  Google Scholar

[26]

D. S. Rodrigues, P. F. A. Mancera and S. T. R. Pinho, Accessing the effect of metronomic chemotherapy through a simple mathematical model,, 2012, ().   Google Scholar

[27]

F. M. Schaebel, Concepts for systematic treatment of micrometastases, Cancer, 35 (1975), 15-24. doi: 10.1002/1097-0142(197501)35:1<15::AID-CNCR2820350104>3.0.CO;2-W.  Google Scholar

[28]

H. E. Skipper, F. M. Schaebel-Jr. and W. S. Wilcox, Experimental evaluation of potential anticancer agents XIII: on the criteria and kinetics associated with curability of experimental leukemia, Cancer Chemother. Rep., 35 (1964), 1-111. Google Scholar

[29]

J. S. Spratt, J. S. Meyer and J. A. Spratt, Rates of growth of human neoplasms: part II, J. Surg. Oncol., 61 (1996), 68-73. doi: 10.1002/1096-9098(199601)61:1<68::AID-JSO2930610102>3.0.CO;2-E.  Google Scholar

[30]

G. S. Stamatakos, E. A. Kolokotroni, D. D. Dionysiou, E. C. Georgiadi and C. Desmedt, An advanced discrete state-discrete event multiscale simulation model of the response of a solid tumour to chemotherapy: mimicking a clinical study, J. Theor. Biol., 266 (2010), 124-139. doi: 10.1016/j.jtbi.2010.05.019.  Google Scholar

[31]

V. G. Vaidya and F. J. Alexandro Jr., Evaluation of some mathematical models for tumor growth, Int. J. Biom. Comput., 13 (1982), 19-35. doi: 10.1016/0020-7101(82)90048-4.  Google Scholar

[32]

World Health Organization, http://www.who.int/cancer/en/, accessed 02/03/2012., ().  http://www.who.int/cancer/en/+" target="_new" title="Go to article in Google Scholar"> Google Scholar

[33]

R. A. Weinberg, "The Biology of Cancer,'' Garland Science, 2008. Google Scholar

show all references

References:
[1]

R. E. Bellman., "Mathematical Methods in Medicine,'' World Scientific Publishing Co. Inc., River Edge, 1983.  Google Scholar

[2]

T. Browder, C. E. Butterfield, B. M. Kraling, B. Shi, B. Marshall, M. S. O'Reilly and J. Folkman, Antiangiogenic scheduling of chemotherapy improves efficacy against experimental drug-resistant cancer, Cancer Res., 60 (2000), 1878-1886. Google Scholar

[3]

R. N. Buick, Cellular basis of chemotherapy, in "Cancer Chemotherapy Handbook'' (Ed. R. T. Dorr and D. D. V. Hoff), Appleton & Lange, (1994), p.9. Google Scholar

[4]

L. G. de Pillis and A. E. Radunskaya, A mathematical tumor model model with immune resistance and drug therapy: An optimal control approach, J. Theor. Med., 3 (2001), 79-100. doi: 10.1080/10273660108833067.  Google Scholar

[5]

L. G. de Pillis, W. Gu, K. R. Fister, T. Head, K. Maples, A. Murugan, T. Neal and K. Yoshida, Chemotherapy for tumors: Analysis of the dynamics and a study of quadratic and linear optimal controls, Math. Biosc., 209 (2007), 292-315. doi: 10.1016/j.mbs.2006.05.003.  Google Scholar

[6]

A. D'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, An optimal delivery of combination therapy for tumors, Math. Biosc., 222 (2009), 13-26. doi: 10.1016/j.mbs.2009.08.004.  Google Scholar

[7]

R. T. Dorr and D. D. Von Hoff, "Cancer Chemotherapy Handbook,'' McGraw-Hill, 1994. Google Scholar

[8]

FEC100 chemotherapy for breast cancer (Written by Jeremy Braybrooke)., Document number: ASWCS09 BR006 [internet] accessed 27/07/2011, available from , ().   Google Scholar

[9]

{N. Ferrara and H. P. Gerber}, The role of vascular endothelial growth factor in angiogenesis, Acta Haematol., 106 (2001), 148-156. doi: 10.1159/000046610.  Google Scholar

[10]

K. R. Fister and J. C. Panetta, Optimal control applied to competing chemotherapeutic cell-kill strategies, SIAM J. Appl. Math., 63 (2003), 1954-1971. doi: 10.1137/S0036139902413489.  Google Scholar

[11]

R. A. Gatenby, Application of competition theory to tumour growth: Implications for tumour biology and treatment, Eur. J. Cancer, 32A (1996), 722-726. doi: 10.1016/0959-8049(95)00658-3.  Google Scholar

[12]

R. S. Kerbel, Tumour angiogenesis: Past, present and the near future, Carcinogenesis, 21 (2000), 505-515. doi: 10.1093/carcin/21.3.505.  Google Scholar

[13]

M. Kohandel, S. Sivaloganathan and A. Oza, Mathematical modeling of ovarian cancer treatments: Sequencing of surgery and chemotherapy, J. Theor. Biol., 242 (2006), 62-68. doi: 10.1016/j.jtbi.2006.02.001.  Google Scholar

[14]

L. G. Marcu and E. Bezak, Neoadjuvant cisplatin for head and neck cancer: simulation of a novel schedule for improved therapeutic ratio, J. Theor. Biol., 297 (2012), 41-47. doi: 10.1016/j.jtbi.2011.12.001.  Google Scholar

[15]

R. B. Martin, M. E. Fisher, R. F. Minchin and K. L. Teo, Low-intensity combination chemotherapy maximizes host survival time for tumors containing drug-resistant cells, Math. Biosc., 110 (1992), 221-252. doi: 10.1016/0025-5564(92)90039-Y.  Google Scholar

[16]

R. B. Martin and K. L. Teo, "Optimal Control of Drug Administration in Cancer Chemotherapy,'' World Scientific, 1994. Google Scholar

[17]

R. B. Martin, Optimal control drug scheduling of cancer chemotherapy, Automatica 28 (1992), 113-1123. doi: 10.1016/0005-1098(92)90054-J.  Google Scholar

[18]

MeadJohnson Oncology Products [internet], http://patient.cancerconsultants.com/druginserts/Cyclophosphamide.pdf., accessed 29/02/2012., ().  http://patient.cancerconsultants.com/druginserts/Cyclophosphamide.pdf.+" target="_new" title="Go to article in Google Scholar"> Google Scholar

[19]

R. D. Mosteller, Simplified calculation of body surface area, N. Engl. J. Med., (1987), 1098. Google Scholar

[20]

S. Mukherjee, "The Emperor of All Maladies: A Biography of Cancer,'' Scribner, 2010. Google Scholar

[21]

F. Nani and H. I. Freedman, A mathematical model of cancer treatment by immunotherapy, Math. Biosc., 163 (2000), 159-199. doi: 10.1016/S0025-5564(99)00058-9.  Google Scholar

[22]

L. Norton and R. Simon, The Norton-Simon hypothesis revisited, Cancer Treat. Rep., 70 (1986), 163-169. Google Scholar

[23]

S. T. R. Pinho, H. I. Freedman and F. K. Nani, A chemotherapy model for the treatment of cancer with metastasis, Math. Comp. Model., 36 (2002), 773-803. doi: 10.1016/S0895-7177(02)00227-3.  Google Scholar

[24]

S. T. R. Pinho, F. S. Bacelar, R. F. S. Andrade and H. I. Freedman, A mathematical model for the effect of anti-angiogenic therapy in the treatment of cancer tumours by chemotherapy, Nonlin. Anal.: Real World Appl., 14 (2013), 815-828. doi: 10.1016/j.nonrwa.2012.07.034.  Google Scholar

[25]

D. S. Rodrigues, S. T. R. Pinho and P. F. A. Mancera, Um modelo matemático em quimio-terapia, TEMA, 13 (2012), 1-12 (in portuguese). doi: 10.5540/tema.2012.013.01.0001.  Google Scholar

[26]

D. S. Rodrigues, P. F. A. Mancera and S. T. R. Pinho, Accessing the effect of metronomic chemotherapy through a simple mathematical model,, 2012, ().   Google Scholar

[27]

F. M. Schaebel, Concepts for systematic treatment of micrometastases, Cancer, 35 (1975), 15-24. doi: 10.1002/1097-0142(197501)35:1<15::AID-CNCR2820350104>3.0.CO;2-W.  Google Scholar

[28]

H. E. Skipper, F. M. Schaebel-Jr. and W. S. Wilcox, Experimental evaluation of potential anticancer agents XIII: on the criteria and kinetics associated with curability of experimental leukemia, Cancer Chemother. Rep., 35 (1964), 1-111. Google Scholar

[29]

J. S. Spratt, J. S. Meyer and J. A. Spratt, Rates of growth of human neoplasms: part II, J. Surg. Oncol., 61 (1996), 68-73. doi: 10.1002/1096-9098(199601)61:1<68::AID-JSO2930610102>3.0.CO;2-E.  Google Scholar

[30]

G. S. Stamatakos, E. A. Kolokotroni, D. D. Dionysiou, E. C. Georgiadi and C. Desmedt, An advanced discrete state-discrete event multiscale simulation model of the response of a solid tumour to chemotherapy: mimicking a clinical study, J. Theor. Biol., 266 (2010), 124-139. doi: 10.1016/j.jtbi.2010.05.019.  Google Scholar

[31]

V. G. Vaidya and F. J. Alexandro Jr., Evaluation of some mathematical models for tumor growth, Int. J. Biom. Comput., 13 (1982), 19-35. doi: 10.1016/0020-7101(82)90048-4.  Google Scholar

[32]

World Health Organization, http://www.who.int/cancer/en/, accessed 02/03/2012., ().  http://www.who.int/cancer/en/+" target="_new" title="Go to article in Google Scholar"> Google Scholar

[33]

R. A. Weinberg, "The Biology of Cancer,'' Garland Science, 2008. Google Scholar

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