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2015, 12(6): 1257-1275. doi: 10.3934/mbe.2015.12.1257

Dynamics and control of a mathematical model for metronomic chemotherapy

1. 

Dept. of Mathematics and Statistics, Southern Illinois University, Edwardsville, Il 62025, United States

2. 

Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo 63130

Received  November 2014 Revised  December 2014 Published  August 2015

A $3$-compartment model for metronomic chemotherapy that takes into account cancerous cells, the tumor vasculature and tumor immune-system interactions is considered as an optimal control problem. Metronomic chemo-therapy is the regular, almost continuous administration of chemotherapeutic agents at low dose, possibly with small interruptions to increase the efficacy of the drugs. There exists medical evidence that such administrations of specific cytotoxic agents (e.g., cyclophosphamide) have both antiangiogenic and immune stimulatory effects. A mathematical model for angiogenic signaling formulated by Hahnfeldt et al. is combined with the classical equations for tumor immune system interactions by Stepanova to form a minimally parameterized model to capture these effects of low dose chemotherapy. The model exhibits bistable behavior with the existence of both benign and malignant locally asymptotically stable equilibrium points. In this paper, the transfer of states from the malignant into the benign regions is used as a motivation for the construction of an objective functional that induces this process and the analysis of the corresponding optimal control problem is initiated.
Citation: Urszula Ledzewicz, Behrooz Amini, Heinz Schättler. Dynamics and control of a mathematical model for metronomic chemotherapy. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1257-1275. doi: 10.3934/mbe.2015.12.1257
References:
[1]

N. André, M. Carré and E. Pasquier, Metronomics: Towards personalized chemotherapy?,, Nature Reviews Clinical Oncology, 11 (2014), 413.   Google Scholar

[2]

N. André, L. Padovani and E. Pasquier, Metronomic scheduling of anticancer treatment: The next generation of multitarget therapy?, Future Oncology, 7 (2011), 385.   Google Scholar

[3]

S. Benzekry, N. André, A. Benabdallah, J. Ciccolini, C. Faivre, F. Hubert and D. Barbolosi, Modeling the impact of anticancer agents on metastatic spreading,, Mathematical Modeling of Natural Phenomena, 7 (2012), 306.  doi: 10.1051/mmnp/20127114.  Google Scholar

[4]

S. Benzekry and P. Hahnfeldt, Maximum tolerated dose versus metronomic scheduling in the treatment of metastatic cancers,, J. Theoretical Biology, 335 (2013), 235.  doi: 10.1016/j.jtbi.2013.06.036.  Google Scholar

[5]

G. Bocci, K. Nicolaou and R. S. Kerbel, Protracted low-dose effects on human endothelial cell proliferation and survival in vitro reveal a selective antiangiogenic window for various chemotherapeutic drugs,, Cancer Research, 62 (2002), 6938.   Google Scholar

[6]

B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory,, Springer Verlag, (2003).   Google Scholar

[7]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control,, American Institute of Mathematical Sciences, (2007).   Google Scholar

[8]

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

[9]

B. I. Camara, H. Mokrani and E. Afenya, Mathematical modeling of glioma therapy using oncolytic viruses,, Mathematical Biosciences and Engineering-MBE, 10 (2013), 565.  doi: 10.3934/mbe.2013.10.565.  Google Scholar

[10]

L. Cesari, Optimization - Theory and Applications,, Springer, (1983).  doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[11]

U. Forys, Y. Keifetz and Y. Kogan, Critical-point analysis for three-variable cancer angiogenesis models,, Mathematical Biosciences and Engineering, 2 (2005), 511.  doi: 10.3934/mbe.2005.2.511.  Google Scholar

[12]

U. Forys, J. Waniewski and P. Zhivkov, Anti-tumor immunity and tumor anti-immunity in a mathematical model of tumor immunotherapy,, J. of Biological Systems, 14 (2006), 13.   Google Scholar

[13]

R. A. Gatenby, A. S. Silva, R. J. Gillies and B. R. Frieden, Adaptive therapy,, Cancer Research, 69 (2009), 4894.  doi: 10.1158/0008-5472.CAN-08-3658.  Google Scholar

[14]

E. V. Grigorieva, E. N. Khailov, N. Bondarenko and A. Korobeinikov, Modeling and optimal control for antiretroviral therapy,, J. of Biological Systems, 22 (2014), 199.  doi: 10.1142/S0218339014400026.  Google Scholar

[15]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,, Springer Verlag, (1983).  doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[16]

P. Hahnfeldt, J. Folkman and L. Hlatky, Minimizing long-term burden: The logic for metronomic chemotherapeutic dosing and its angiogenic basis,, J. of Theoretical Biology, 220 (2003), 545.  doi: 10.1006/jtbi.2003.3162.  Google Scholar

[17]

P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy,, Cancer Research, 59 (1999), 4770.   Google Scholar

[18]

D. Hanahan, G. Bergers and E. Bergsland, Less is more, regularly: metronomic dosing of cytotoxic drugs can target tumor angiogenesis in mice,, J. Clinical Investigations, 105 (2000), 1045.  doi: 10.1172/JCI9872.  Google Scholar

[19]

Y. B. Hao, S. Y. Yi, J. Ruan, L. Zhao and K. J. Nan, New insights into metronomic chemotherapy-induced immunoregulation,, Cancer Letters, 354 (2014), 220.  doi: 10.1016/j.canlet.2014.08.028.  Google Scholar

[20]

B. Kamen, E. Rubin, J. Aisner and E. Glatstein, High-time chemotherapy or high time for low dose?,, J. Clinical Oncology, 18 (2000), 2935.   Google Scholar

[21]

Y. Kim and A. Friedman, Interaction of tumor with its microenvironment: A mathematical model,, Bulletin of Mathematical Biology, 72 (2010), 1029.  doi: 10.1007/s11538-009-9481-z.  Google Scholar

[22]

G. Klement, S. Baruchel, J. Rak, S. Man, K. Clark, D. J. Hicklin, P. Bohlen and R. S. Kerbel, Continuous low-dose therapy with vinblastine and VEGF receptor-2 antibody induces sustained tumor regression without overt toxicity,, J. Clinical Investigations, 105 (2000).   Google Scholar

[23]

V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis,, Bulletin of Mathematical Biology, 56 (1994), 295.   Google Scholar

[24]

U. Ledzewicz, M. Naghnaeian and H. Schättler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics,, J. of Mathematical Biology, 64 (2012), 557.  doi: 10.1007/s00285-011-0424-6.  Google Scholar

[25]

U. Ledzewicz, O. Olumoye and H. Schättler, On optimal chemotherapy with a stongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth,, Mathematical Biosciences and Engineering - MBE, 10 (2013), 787.  doi: 10.3934/mbe.2013.10.787.  Google Scholar

[26]

U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy,, J. of Optimization Theory and Applications - JOTA, 114 (2002), 609.  doi: 10.1023/A:1016027113579.  Google Scholar

[27]

U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy,, J. of Biological Systems, 10 (2002), 183.  doi: 10.1142/S0218339002000597.  Google Scholar

[28]

U. Ledzewicz and H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem,, Discrete and Continuous Dynamical Systems, 6 (2006), 129.   Google Scholar

[29]

U. Ledzewicz and H. Schättler, Anti-angiogenic therapy in cancer treatment as an optimal control problem,, SIAM J. on Control and Optimization, 46 (2007), 1052.  doi: 10.1137/060665294.  Google Scholar

[30]

U. Ledzewicz and H. Schättler, A review of optimal chemotherapy protocols: From MTD towards metronomic therapy,, Mathematical Modeling of Natural Phenomena, 9 (2014), 131.  doi: 10.1051/mmnp/20149409.  Google Scholar

[31]

U. Ledzewicz and H. Schättler, On optimal chemotherapy for heterogeneous tumors,, J. of Biological Systems, 22 (2014), 177.  doi: 10.1142/S0218339014400014.  Google Scholar

[32]

U. Ledzewicz and H. Schättler, Tumor microenvironment and anticancer therapies: An optimal control approach,, in Mathematical Oncology 2013 (eds. A. d'Onofrio and A. Gandolfi), (2014), 295.  doi: 10.1007/978-1-4939-0458-7_10.  Google Scholar

[33]

E. Pasquier, M. Kavallaris and N. André, Metronomic chemotherapy: new rationale for new directions,, Nature Reviews|Clinical Oncology, 7 (2010), 455.  doi: 10.1038/nrclinonc.2010.82.  Google Scholar

[34]

E. Pasquier and U. Ledzewicz, Perspective on "More is not necessarily better'': Metronomic Chemotherapy,, Newsletter of the Society for Mathematical Biology, 26 (2013), 9.   Google Scholar

[35]

K. Pietras and D. Hanahan, A multi-targeted, metronomic and maximum tolerated dose "chemo-switch" regimen is antiangiogenic, producing objective responses and survival benefit in a mouse model of cancer,, J. of Clinical Oncology, 23 (2005), 939.   Google Scholar

[36]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, MacMillan, (1964).   Google Scholar

[37]

R. Retsky, Metronomic Chemotherapy was originally designed and first used in 1994 for early stage cancer - why is it taking so long to proceed?,, Bioequivalence and Bioavailability, 3 (2011).  doi: 10.4172/jbb.100000e6.  Google Scholar

[38]

H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples,, Springer Verlag, (2012).  doi: 10.1007/978-1-4614-3834-2.  Google Scholar

[39]

H. Schättler, U. Ledzewicz and B. Amini, Dynamical properties of a minimally parameterized mathematical model for metronomic chemotherapy,, J. of Mathematical Biology, (2015).   Google Scholar

[40]

H. E. Skipper, On mathematical modeling of critical variables in cancer treatment (goals: better understanding of the past and better planning in the future),, Bulletin of Mathematical Biology, 48 (1986), 253.  doi: 10.1007/BF02459681.  Google Scholar

[41]

N. V. Stepanova, Course of the immune reaction during the development of a malignant tumour,, Biophysics, 24 (1980), 917.   Google Scholar

[42]

J. B. Swann and M. J. Smyth, Immune surveillance of tumors,, J. Clinical Investigations, 117 (2007), 1137.  doi: 10.1172/JCI31405.  Google Scholar

[43]

G. W. Swan, Role of optimal control in cancer chemotherapy,, Mathematical Biosciences, 101 (1990), 237.  doi: 10.1016/0025-5564(90)90021-P.  Google Scholar

[44]

A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle,, IMACS Ann. Comput. Appl. Math., 5 (1989), 51.   Google Scholar

[45]

A. Swierniak, Cell cycle as an object of control,, J. of Biological Systems, 3 (1995), 41.  doi: 10.1142/S0218339095000058.  Google Scholar

[46]

A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance,, Nonlinear Analysis, 47 (2001), 375.  doi: 10.1016/S0362-546X(01)00184-5.  Google Scholar

[47]

A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy,, Int. J. Applied Mathematics and Computer Science, 13 (2003), 357.   Google Scholar

[48]

H. P. de Vladar and J. A. González, Dynamic response of cancer under the influence of immunological activity and therapy,, J. of Theoretical Biology, 227 (2004), 335.  doi: 10.1016/j.jtbi.2003.11.012.  Google Scholar

[49]

S. D. Weitman, E. Glatstein and B. A. Kamen, Back to the basics: the importance of concentration $\times$ time in oncology,, J. of Clinical Oncology, 11 (1993), 820.   Google Scholar

[50]

T. E. Wheldon, Mathematical Models in Cancer Research,, Boston-Philadelphia: Hilger Publishing, (1988).   Google Scholar

show all references

References:
[1]

N. André, M. Carré and E. Pasquier, Metronomics: Towards personalized chemotherapy?,, Nature Reviews Clinical Oncology, 11 (2014), 413.   Google Scholar

[2]

N. André, L. Padovani and E. Pasquier, Metronomic scheduling of anticancer treatment: The next generation of multitarget therapy?, Future Oncology, 7 (2011), 385.   Google Scholar

[3]

S. Benzekry, N. André, A. Benabdallah, J. Ciccolini, C. Faivre, F. Hubert and D. Barbolosi, Modeling the impact of anticancer agents on metastatic spreading,, Mathematical Modeling of Natural Phenomena, 7 (2012), 306.  doi: 10.1051/mmnp/20127114.  Google Scholar

[4]

S. Benzekry and P. Hahnfeldt, Maximum tolerated dose versus metronomic scheduling in the treatment of metastatic cancers,, J. Theoretical Biology, 335 (2013), 235.  doi: 10.1016/j.jtbi.2013.06.036.  Google Scholar

[5]

G. Bocci, K. Nicolaou and R. S. Kerbel, Protracted low-dose effects on human endothelial cell proliferation and survival in vitro reveal a selective antiangiogenic window for various chemotherapeutic drugs,, Cancer Research, 62 (2002), 6938.   Google Scholar

[6]

B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory,, Springer Verlag, (2003).   Google Scholar

[7]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control,, American Institute of Mathematical Sciences, (2007).   Google Scholar

[8]

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

[9]

B. I. Camara, H. Mokrani and E. Afenya, Mathematical modeling of glioma therapy using oncolytic viruses,, Mathematical Biosciences and Engineering-MBE, 10 (2013), 565.  doi: 10.3934/mbe.2013.10.565.  Google Scholar

[10]

L. Cesari, Optimization - Theory and Applications,, Springer, (1983).  doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[11]

U. Forys, Y. Keifetz and Y. Kogan, Critical-point analysis for three-variable cancer angiogenesis models,, Mathematical Biosciences and Engineering, 2 (2005), 511.  doi: 10.3934/mbe.2005.2.511.  Google Scholar

[12]

U. Forys, J. Waniewski and P. Zhivkov, Anti-tumor immunity and tumor anti-immunity in a mathematical model of tumor immunotherapy,, J. of Biological Systems, 14 (2006), 13.   Google Scholar

[13]

R. A. Gatenby, A. S. Silva, R. J. Gillies and B. R. Frieden, Adaptive therapy,, Cancer Research, 69 (2009), 4894.  doi: 10.1158/0008-5472.CAN-08-3658.  Google Scholar

[14]

E. V. Grigorieva, E. N. Khailov, N. Bondarenko and A. Korobeinikov, Modeling and optimal control for antiretroviral therapy,, J. of Biological Systems, 22 (2014), 199.  doi: 10.1142/S0218339014400026.  Google Scholar

[15]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,, Springer Verlag, (1983).  doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[16]

P. Hahnfeldt, J. Folkman and L. Hlatky, Minimizing long-term burden: The logic for metronomic chemotherapeutic dosing and its angiogenic basis,, J. of Theoretical Biology, 220 (2003), 545.  doi: 10.1006/jtbi.2003.3162.  Google Scholar

[17]

P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy,, Cancer Research, 59 (1999), 4770.   Google Scholar

[18]

D. Hanahan, G. Bergers and E. Bergsland, Less is more, regularly: metronomic dosing of cytotoxic drugs can target tumor angiogenesis in mice,, J. Clinical Investigations, 105 (2000), 1045.  doi: 10.1172/JCI9872.  Google Scholar

[19]

Y. B. Hao, S. Y. Yi, J. Ruan, L. Zhao and K. J. Nan, New insights into metronomic chemotherapy-induced immunoregulation,, Cancer Letters, 354 (2014), 220.  doi: 10.1016/j.canlet.2014.08.028.  Google Scholar

[20]

B. Kamen, E. Rubin, J. Aisner and E. Glatstein, High-time chemotherapy or high time for low dose?,, J. Clinical Oncology, 18 (2000), 2935.   Google Scholar

[21]

Y. Kim and A. Friedman, Interaction of tumor with its microenvironment: A mathematical model,, Bulletin of Mathematical Biology, 72 (2010), 1029.  doi: 10.1007/s11538-009-9481-z.  Google Scholar

[22]

G. Klement, S. Baruchel, J. Rak, S. Man, K. Clark, D. J. Hicklin, P. Bohlen and R. S. Kerbel, Continuous low-dose therapy with vinblastine and VEGF receptor-2 antibody induces sustained tumor regression without overt toxicity,, J. Clinical Investigations, 105 (2000).   Google Scholar

[23]

V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis,, Bulletin of Mathematical Biology, 56 (1994), 295.   Google Scholar

[24]

U. Ledzewicz, M. Naghnaeian and H. Schättler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics,, J. of Mathematical Biology, 64 (2012), 557.  doi: 10.1007/s00285-011-0424-6.  Google Scholar

[25]

U. Ledzewicz, O. Olumoye and H. Schättler, On optimal chemotherapy with a stongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth,, Mathematical Biosciences and Engineering - MBE, 10 (2013), 787.  doi: 10.3934/mbe.2013.10.787.  Google Scholar

[26]

U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy,, J. of Optimization Theory and Applications - JOTA, 114 (2002), 609.  doi: 10.1023/A:1016027113579.  Google Scholar

[27]

U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy,, J. of Biological Systems, 10 (2002), 183.  doi: 10.1142/S0218339002000597.  Google Scholar

[28]

U. Ledzewicz and H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem,, Discrete and Continuous Dynamical Systems, 6 (2006), 129.   Google Scholar

[29]

U. Ledzewicz and H. Schättler, Anti-angiogenic therapy in cancer treatment as an optimal control problem,, SIAM J. on Control and Optimization, 46 (2007), 1052.  doi: 10.1137/060665294.  Google Scholar

[30]

U. Ledzewicz and H. Schättler, A review of optimal chemotherapy protocols: From MTD towards metronomic therapy,, Mathematical Modeling of Natural Phenomena, 9 (2014), 131.  doi: 10.1051/mmnp/20149409.  Google Scholar

[31]

U. Ledzewicz and H. Schättler, On optimal chemotherapy for heterogeneous tumors,, J. of Biological Systems, 22 (2014), 177.  doi: 10.1142/S0218339014400014.  Google Scholar

[32]

U. Ledzewicz and H. Schättler, Tumor microenvironment and anticancer therapies: An optimal control approach,, in Mathematical Oncology 2013 (eds. A. d'Onofrio and A. Gandolfi), (2014), 295.  doi: 10.1007/978-1-4939-0458-7_10.  Google Scholar

[33]

E. Pasquier, M. Kavallaris and N. André, Metronomic chemotherapy: new rationale for new directions,, Nature Reviews|Clinical Oncology, 7 (2010), 455.  doi: 10.1038/nrclinonc.2010.82.  Google Scholar

[34]

E. Pasquier and U. Ledzewicz, Perspective on "More is not necessarily better'': Metronomic Chemotherapy,, Newsletter of the Society for Mathematical Biology, 26 (2013), 9.   Google Scholar

[35]

K. Pietras and D. Hanahan, A multi-targeted, metronomic and maximum tolerated dose "chemo-switch" regimen is antiangiogenic, producing objective responses and survival benefit in a mouse model of cancer,, J. of Clinical Oncology, 23 (2005), 939.   Google Scholar

[36]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, MacMillan, (1964).   Google Scholar

[37]

R. Retsky, Metronomic Chemotherapy was originally designed and first used in 1994 for early stage cancer - why is it taking so long to proceed?,, Bioequivalence and Bioavailability, 3 (2011).  doi: 10.4172/jbb.100000e6.  Google Scholar

[38]

H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples,, Springer Verlag, (2012).  doi: 10.1007/978-1-4614-3834-2.  Google Scholar

[39]

H. Schättler, U. Ledzewicz and B. Amini, Dynamical properties of a minimally parameterized mathematical model for metronomic chemotherapy,, J. of Mathematical Biology, (2015).   Google Scholar

[40]

H. E. Skipper, On mathematical modeling of critical variables in cancer treatment (goals: better understanding of the past and better planning in the future),, Bulletin of Mathematical Biology, 48 (1986), 253.  doi: 10.1007/BF02459681.  Google Scholar

[41]

N. V. Stepanova, Course of the immune reaction during the development of a malignant tumour,, Biophysics, 24 (1980), 917.   Google Scholar

[42]

J. B. Swann and M. J. Smyth, Immune surveillance of tumors,, J. Clinical Investigations, 117 (2007), 1137.  doi: 10.1172/JCI31405.  Google Scholar

[43]

G. W. Swan, Role of optimal control in cancer chemotherapy,, Mathematical Biosciences, 101 (1990), 237.  doi: 10.1016/0025-5564(90)90021-P.  Google Scholar

[44]

A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle,, IMACS Ann. Comput. Appl. Math., 5 (1989), 51.   Google Scholar

[45]

A. Swierniak, Cell cycle as an object of control,, J. of Biological Systems, 3 (1995), 41.  doi: 10.1142/S0218339095000058.  Google Scholar

[46]

A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance,, Nonlinear Analysis, 47 (2001), 375.  doi: 10.1016/S0362-546X(01)00184-5.  Google Scholar

[47]

A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy,, Int. J. Applied Mathematics and Computer Science, 13 (2003), 357.   Google Scholar

[48]

H. P. de Vladar and J. A. González, Dynamic response of cancer under the influence of immunological activity and therapy,, J. of Theoretical Biology, 227 (2004), 335.  doi: 10.1016/j.jtbi.2003.11.012.  Google Scholar

[49]

S. D. Weitman, E. Glatstein and B. A. Kamen, Back to the basics: the importance of concentration $\times$ time in oncology,, J. of Clinical Oncology, 11 (1993), 820.   Google Scholar

[50]

T. E. Wheldon, Mathematical Models in Cancer Research,, Boston-Philadelphia: Hilger Publishing, (1988).   Google Scholar

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