2015, 12(1): 163-183. doi: 10.3934/mbe.2015.12.163

On viable therapy strategy for a mathematical spatial cancer model describing the dynamics of malignant and healthy cells

1. 

Moscow State University, GSP-1, Leninskie Gory, Moscow, Russian Federation

2. 

Federal Science and Clinical Center of the Federal Medical and Biological Agency, 28 Orehovuy boulevard, Moscow, 115682, Russian Federation

3. 

Mannheim University of Applied Sciences, Paul-Wittsack-Str. 10, 68163 Mannheim

Received  November 2013 Revised  October 2014 Published  December 2014

A mathematical spatial cancer model of the interaction between a drug and both malignant and healthy cells is considered. It is assumed that the drug influences negative malignant cells as well as healthy ones. The mathematical model considered consists of three nonlinear parabolic partial differential equations which describe spatial dynamics of malignant cells as well as healthy ones, and of the concentration of the drug. Additionally, we assume some phase constraints for the number of the malignant and the healthy cells and for the total dose of the drug during the whole treatment process.
    We search through all the courses of treatment switching between an application of the drug with the maximum intensity (intensive therapy phase) and discontinuing administering of the drug (relaxation phase) with the objective of achieving the maximum possible therapy (survival) time. We will call the therapy a viable treatment strategy.
Citation: Alexander S. Bratus, Svetlana Yu. Kovalenko, Elena Fimmel. On viable therapy strategy for a mathematical spatial cancer model describing the dynamics of malignant and healthy cells. Mathematical Biosciences & Engineering, 2015, 12 (1) : 163-183. doi: 10.3934/mbe.2015.12.163
References:
[1]

E. K. Afenya and C. P. Calderón, Growth kinetic of cancer cells prior to detection and treatment an alternative view,, Discrete and Continuous Dynamical Systems- Series B, 4 (2004), 25.   Google Scholar

[2]

E. K. Afenya and C. P. Calderón, Diverse ideas on the growth kinetics of disseminated cancer cells,, Bulletin of Mathematical Biology, 62 (2000), 527.   Google Scholar

[3]

E. C. Alvord and C. M. Show, Neoplasms affecting the nervous system in the elderly,, In S.Duckett, (1991), 210.   Google Scholar

[4]

A. V. Antipov and A. S. Bratus, Mathematical Model of Optimal Chemotherapy Strategy with Allowance for Cell Population Dynamics in a Heterogenous Tumour,, Comp. Math. and Math. Physics, 49 (2009), 1825.  doi: 10.1134/S0965542509110013.  Google Scholar

[5]

A. S. Bratus and E. S. Chumerina, Optimal Control in Therapy of Solid Tumour Grouth,, Comp. Math. and Math. Phys., 48 (2008), 892.  doi: 10.1134/S096554250806002X.  Google Scholar

[6]

A. S. Bratus and S. Yu. Zaichik, Smooth Solution of the Hamilton-Jacobi-Bellman Equation in Mathematical Model of Optimal Treatment of Viral Infection,, Diff. Equat., 46 (2010), 1571.  doi: 10.1134/S0012266110110054.  Google Scholar

[7]

A. S. Bratus, E. Fimmel and S. Kovalenko, On assessing quality of therapy in non-linear distributed mathematical models for brain tumor growth dynamics,, Mathematical Biosciences, 248 (2014), 88.  doi: 10.1016/j.mbs.2013.12.007.  Google Scholar

[8]

A. S. Bratus, Y. Todorov, I. Yegorov and D. Yurchenko, Solution of the Feedback Control Problem in a Mathematical Model of Leukaemia Therapy,, Journ. of Opt. Theor. Appl., 159 (2013), 590.  doi: 10.1007/s10957-013-0324-6.  Google Scholar

[9]

A. S. Bratus, E. Fimmel, Y. Todorov, Yu. S. Semenov and F. Nuernberg, On strategies on a mathematical model for leukemia therapy,, Nonlinear Analysis: Real World Applications, 13 (2012), 1044.  doi: 10.1016/j.nonrwa.2011.02.027.  Google Scholar

[10]

P. K. Burgess, P. M. Kulesa, J. D. Murray and Jr. E. C. Alvord, The interaction of growth rates and diffusion coefficients in a three-dimensional mathematical model of gliomas,, Journal of Neuropathology & Experimental Neurology, 56 (1997), 704.   Google Scholar

[11]

M. R. Chicoine and D. L. Silbergeld, Assessment of brain tumour cell motility in vivo and in vitro,, J. Neurosurgery, 82 (1995), 615.   Google Scholar

[12]

E. S. Chumerina, Choice of optimal strategy of tumour chemoterapy in gompertz model,, Chumerina, 48 (2009), 325.  doi: 10.1134/S1064230709020154.  Google Scholar

[13]

D. L. Collins, A. P. Zijdenbos, V. Kollokian, J. G. Sled, N. J. Kabani, C. J. Holmes and A. C. Evans, Design and construction of realistic digital brain phantom,, IEEE Trans. Medical Imaging, 17 (1998), 463.   Google Scholar

[14]

K. R. Fister and J. C. Panetta, Optimal Control Applied to Competing Chemotherapeutic Cell-Kill Strategies,, SIAM Journal on Applied Mathematics, 63 (2003), 1954.  doi: 10.1137/S0036139902413489.  Google Scholar

[15]

A. V. Fursikov, Optimal Control of Distributed Systems (Theory and Application),, Theory and applications. Translated from the 1999 Russian original by Tamara Rozhkovskaya. Translations of Mathematical Monographs, (1999).   Google Scholar

[16]

A. Giese and M. Westphal, Glioma invasion in the central nervous system,, J. Neurosurgery, 39 (1996), 235.   Google Scholar

[17]

D. Henri, Geometric Theory of Semilinear Parabolic Equations,, New York, (1981).   Google Scholar

[18]

T. Hines, Mathematically Modeling the Mass-Effect of invasive Brain Tumors,, preprint Arizona State University, (2010).   Google Scholar

[19]

M. Kimmel and J. Sverniak, Mathematical modeling as a tool for planning anticancer therapy,, European Journal of Pharmacology, 625 (2009), 108.   Google Scholar

[20]

U. Ledzewicz, M. Naghnaenian and H. Schattler, An optimal control, approach to cancer treatment under immunological activity,, Applicationes Mathematicae, 38 (2011), 17.  doi: 10.4064/am38-1-2.  Google Scholar

[21]

U. Ledzewicz and H. Schattler, The influence of PK/PD on the structure of optimal controls in cancer chemoterapy models,, Math. Biosc. Eng., 2 (2005), 561.  doi: 10.3934/mbe.2005.2.561.  Google Scholar

[22]

X. Li and J. Young, Optimal Control Theory for Unfinite Dimensional System,, Birkhaeuser, (1995).   Google Scholar

[23]

S. G. Mikhlin, Variational Methods in Mathematical Physics,, Pergamon Press, (1964).   Google Scholar

[24]

D. Mackenzie, Mathematical Modeling and Cancer,, SIAM News, (2004).   Google Scholar

[25]

Y. Matsukado, C. S. McCarthy and J. W. Kernohan, The Growth of glioblastoma multiforme (asytrocytomas, grades 3 and 4) in neurosurgical practice,, J. Neuwsurg., 18 (1961), 636.   Google Scholar

[26]

J. D. Murray, Mathematical Biology,, II. Spatial models and biomedical applications. Third edition. Interdisciplinary Applied Mathematics, (2003).   Google Scholar

[27]

J. D. Murray, Mathematical Biology,, Second edition. Biomathematics, (1993).  doi: 10.1007/b98869.  Google Scholar

[28]

P. Neittaanmaeki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems. Theory, Algorithms and Applications,, Marcel Dekker, (1994).   Google Scholar

[29]

G. Powathil, M. Kohandel, S. Sivaloganathan, A. Oza and M. Milosevic, Mathematical modeling of brain tumors: Effects of radiotherapy and chemotherapy,, Phys. Med. Biol., 52 (2007).   Google Scholar

[30]

D. L. Silbergeld and M. R. Chicoine, Isolation and Characterization of human malignant glioma cells from histologically normal brain,, J.Neurosurg., 86 (1997), 525.   Google Scholar

[31]

K. R. Swanson, Jr. E. C. Alvord and J. D. Murray, Virtual resection of gliomas: Effect of extent of resection on recurrence,, Mathematical and Computer Modelling, 37 (2003), 1177.  doi: 10.1016/S0895-7177(03)00129-8.  Google Scholar

[32]

K. R. Swanson, Mathematical Modeling of the Growth and Control of Tumours,, PhD thesis, (1999).   Google Scholar

[33]

K. R. Swanson, Jr. E. C. Alvord and J. D. Murray, A quantitative model for differential motility of gliomas in grey and white matter,, Cell Proliferation, 33 (2000), 317.   Google Scholar

[34]

Y. Todorov, E. Fimmel, A. S. Bratus, Yu. S. Semenov and F. Nuernberg, An optimal strategy for leukemia therapy: A multi-objective approach,, Russian Journal of Numerical Analysis and Mathematical Modelling, 26 (2011), 589.  doi: 10.1515/rjnamm.2011.035.  Google Scholar

show all references

References:
[1]

E. K. Afenya and C. P. Calderón, Growth kinetic of cancer cells prior to detection and treatment an alternative view,, Discrete and Continuous Dynamical Systems- Series B, 4 (2004), 25.   Google Scholar

[2]

E. K. Afenya and C. P. Calderón, Diverse ideas on the growth kinetics of disseminated cancer cells,, Bulletin of Mathematical Biology, 62 (2000), 527.   Google Scholar

[3]

E. C. Alvord and C. M. Show, Neoplasms affecting the nervous system in the elderly,, In S.Duckett, (1991), 210.   Google Scholar

[4]

A. V. Antipov and A. S. Bratus, Mathematical Model of Optimal Chemotherapy Strategy with Allowance for Cell Population Dynamics in a Heterogenous Tumour,, Comp. Math. and Math. Physics, 49 (2009), 1825.  doi: 10.1134/S0965542509110013.  Google Scholar

[5]

A. S. Bratus and E. S. Chumerina, Optimal Control in Therapy of Solid Tumour Grouth,, Comp. Math. and Math. Phys., 48 (2008), 892.  doi: 10.1134/S096554250806002X.  Google Scholar

[6]

A. S. Bratus and S. Yu. Zaichik, Smooth Solution of the Hamilton-Jacobi-Bellman Equation in Mathematical Model of Optimal Treatment of Viral Infection,, Diff. Equat., 46 (2010), 1571.  doi: 10.1134/S0012266110110054.  Google Scholar

[7]

A. S. Bratus, E. Fimmel and S. Kovalenko, On assessing quality of therapy in non-linear distributed mathematical models for brain tumor growth dynamics,, Mathematical Biosciences, 248 (2014), 88.  doi: 10.1016/j.mbs.2013.12.007.  Google Scholar

[8]

A. S. Bratus, Y. Todorov, I. Yegorov and D. Yurchenko, Solution of the Feedback Control Problem in a Mathematical Model of Leukaemia Therapy,, Journ. of Opt. Theor. Appl., 159 (2013), 590.  doi: 10.1007/s10957-013-0324-6.  Google Scholar

[9]

A. S. Bratus, E. Fimmel, Y. Todorov, Yu. S. Semenov and F. Nuernberg, On strategies on a mathematical model for leukemia therapy,, Nonlinear Analysis: Real World Applications, 13 (2012), 1044.  doi: 10.1016/j.nonrwa.2011.02.027.  Google Scholar

[10]

P. K. Burgess, P. M. Kulesa, J. D. Murray and Jr. E. C. Alvord, The interaction of growth rates and diffusion coefficients in a three-dimensional mathematical model of gliomas,, Journal of Neuropathology & Experimental Neurology, 56 (1997), 704.   Google Scholar

[11]

M. R. Chicoine and D. L. Silbergeld, Assessment of brain tumour cell motility in vivo and in vitro,, J. Neurosurgery, 82 (1995), 615.   Google Scholar

[12]

E. S. Chumerina, Choice of optimal strategy of tumour chemoterapy in gompertz model,, Chumerina, 48 (2009), 325.  doi: 10.1134/S1064230709020154.  Google Scholar

[13]

D. L. Collins, A. P. Zijdenbos, V. Kollokian, J. G. Sled, N. J. Kabani, C. J. Holmes and A. C. Evans, Design and construction of realistic digital brain phantom,, IEEE Trans. Medical Imaging, 17 (1998), 463.   Google Scholar

[14]

K. R. Fister and J. C. Panetta, Optimal Control Applied to Competing Chemotherapeutic Cell-Kill Strategies,, SIAM Journal on Applied Mathematics, 63 (2003), 1954.  doi: 10.1137/S0036139902413489.  Google Scholar

[15]

A. V. Fursikov, Optimal Control of Distributed Systems (Theory and Application),, Theory and applications. Translated from the 1999 Russian original by Tamara Rozhkovskaya. Translations of Mathematical Monographs, (1999).   Google Scholar

[16]

A. Giese and M. Westphal, Glioma invasion in the central nervous system,, J. Neurosurgery, 39 (1996), 235.   Google Scholar

[17]

D. Henri, Geometric Theory of Semilinear Parabolic Equations,, New York, (1981).   Google Scholar

[18]

T. Hines, Mathematically Modeling the Mass-Effect of invasive Brain Tumors,, preprint Arizona State University, (2010).   Google Scholar

[19]

M. Kimmel and J. Sverniak, Mathematical modeling as a tool for planning anticancer therapy,, European Journal of Pharmacology, 625 (2009), 108.   Google Scholar

[20]

U. Ledzewicz, M. Naghnaenian and H. Schattler, An optimal control, approach to cancer treatment under immunological activity,, Applicationes Mathematicae, 38 (2011), 17.  doi: 10.4064/am38-1-2.  Google Scholar

[21]

U. Ledzewicz and H. Schattler, The influence of PK/PD on the structure of optimal controls in cancer chemoterapy models,, Math. Biosc. Eng., 2 (2005), 561.  doi: 10.3934/mbe.2005.2.561.  Google Scholar

[22]

X. Li and J. Young, Optimal Control Theory for Unfinite Dimensional System,, Birkhaeuser, (1995).   Google Scholar

[23]

S. G. Mikhlin, Variational Methods in Mathematical Physics,, Pergamon Press, (1964).   Google Scholar

[24]

D. Mackenzie, Mathematical Modeling and Cancer,, SIAM News, (2004).   Google Scholar

[25]

Y. Matsukado, C. S. McCarthy and J. W. Kernohan, The Growth of glioblastoma multiforme (asytrocytomas, grades 3 and 4) in neurosurgical practice,, J. Neuwsurg., 18 (1961), 636.   Google Scholar

[26]

J. D. Murray, Mathematical Biology,, II. Spatial models and biomedical applications. Third edition. Interdisciplinary Applied Mathematics, (2003).   Google Scholar

[27]

J. D. Murray, Mathematical Biology,, Second edition. Biomathematics, (1993).  doi: 10.1007/b98869.  Google Scholar

[28]

P. Neittaanmaeki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems. Theory, Algorithms and Applications,, Marcel Dekker, (1994).   Google Scholar

[29]

G. Powathil, M. Kohandel, S. Sivaloganathan, A. Oza and M. Milosevic, Mathematical modeling of brain tumors: Effects of radiotherapy and chemotherapy,, Phys. Med. Biol., 52 (2007).   Google Scholar

[30]

D. L. Silbergeld and M. R. Chicoine, Isolation and Characterization of human malignant glioma cells from histologically normal brain,, J.Neurosurg., 86 (1997), 525.   Google Scholar

[31]

K. R. Swanson, Jr. E. C. Alvord and J. D. Murray, Virtual resection of gliomas: Effect of extent of resection on recurrence,, Mathematical and Computer Modelling, 37 (2003), 1177.  doi: 10.1016/S0895-7177(03)00129-8.  Google Scholar

[32]

K. R. Swanson, Mathematical Modeling of the Growth and Control of Tumours,, PhD thesis, (1999).   Google Scholar

[33]

K. R. Swanson, Jr. E. C. Alvord and J. D. Murray, A quantitative model for differential motility of gliomas in grey and white matter,, Cell Proliferation, 33 (2000), 317.   Google Scholar

[34]

Y. Todorov, E. Fimmel, A. S. Bratus, Yu. S. Semenov and F. Nuernberg, An optimal strategy for leukemia therapy: A multi-objective approach,, Russian Journal of Numerical Analysis and Mathematical Modelling, 26 (2011), 589.  doi: 10.1515/rjnamm.2011.035.  Google Scholar

[1]

Ben Sheller, Domenico D'Alessandro. Analysis of a cancer dormancy model and control of immuno-therapy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1037-1053. doi: 10.3934/mbe.2015.12.1037

[2]

Urszula Ledzewicz, Helen Moore. Optimal control applied to a generalized Michaelis-Menten model of CML therapy. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 331-346. doi: 10.3934/dcdsb.2018022

[3]

Shuo Wang, Heinz Schättler. Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1223-1240. doi: 10.3934/mbe.2016040

[4]

Jerzy Klamka, Helmut Maurer, Andrzej Swierniak. Local controllability and optimal control for\newline a model of combined anticancer therapy with control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 195-216. doi: 10.3934/mbe.2017013

[5]

Urszula Ledzewicz, Heinz Schättler. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 129-150. doi: 10.3934/dcdsb.2006.6.129

[6]

Urszula Ledzewicz, Heinz Schättler, Mostafa Reisi Gahrooi, Siamak Mahmoudian Dehkordi. On the MTD paradigm and optimal control for multi-drug cancer chemotherapy. Mathematical Biosciences & Engineering, 2013, 10 (3) : 803-819. doi: 10.3934/mbe.2013.10.803

[7]

Wei Feng, Shuhua Hu, Xin Lu. Optimal controls for a 3-compartment model for cancer chemotherapy with quadratic objective. Conference Publications, 2003, 2003 (Special) : 544-553. doi: 10.3934/proc.2003.2003.544

[8]

Harsh Vardhan Jain, Avner Friedman. Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 945-967. doi: 10.3934/dcdsb.2013.18.945

[9]

Avner Friedman, Xiulan Lai. Antagonism and negative side-effects in combination therapy for cancer. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2237-2250. doi: 10.3934/dcdsb.2019093

[10]

Avner Friedman, Xiulan Lai. Free boundary problems associated with cancer treatment by combination therapy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6825-6842. doi: 10.3934/dcds.2019233

[11]

Shuo Wang, Heinz Schättler. Optimal control for cancer chemotherapy under tumor heterogeneity with Michealis-Menten pharmacodynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2383-2405. doi: 10.3934/dcdsb.2019100

[12]

Urszula Ledzewicz, Heinz Schättler, Shuo Wang. On the role of tumor heterogeneity for optimal cancer chemotherapy. Networks & Heterogeneous Media, 2019, 14 (1) : 131-147. doi: 10.3934/nhm.2019007

[13]

Luis A. Fernández, Cecilia Pola. Catalog of the optimal controls in cancer chemotherapy for the Gompertz model depending on PK/PD and the integral constraint. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1563-1588. doi: 10.3934/dcdsb.2014.19.1563

[14]

Urszula Ledzewicz, Heinz Schättler. Controlling a model for bone marrow dynamics in cancer chemotherapy. Mathematical Biosciences & Engineering, 2004, 1 (1) : 95-110. doi: 10.3934/mbe.2004.1.95

[15]

Marzena Dolbniak, Malgorzata Kardynska, Jaroslaw Smieja. Sensitivity of combined chemo-and antiangiogenic therapy results in different models describing cancer growth. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 145-160. doi: 10.3934/dcdsb.2018009

[16]

Rachid Ouifki, Gareth Witten. A model of HIV-1 infection with HAART therapy and intracellular delays. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 229-240. doi: 10.3934/dcdsb.2007.8.229

[17]

Urszula Ledzewicz, Heinz Schättler. The Influence of PK/PD on the Structure of Optimal Controls in Cancer Chemotherapy Models. Mathematical Biosciences & Engineering, 2005, 2 (3) : 561-578. doi: 10.3934/mbe.2005.2.561

[18]

Cristian Morales-Rodrigo. A therapy inactivating the tumor angiogenic factors. Mathematical Biosciences & Engineering, 2013, 10 (1) : 185-198. doi: 10.3934/mbe.2013.10.185

[19]

Hsiu-Chuan Wei. Mathematical and numerical analysis of a mathematical model of mixed immunotherapy and chemotherapy of cancer. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1279-1295. doi: 10.3934/dcdsb.2016.21.1279

[20]

Craig Collins, K. Renee Fister, Bethany Key, Mary Williams. Blasting neuroblastoma using optimal control of chemotherapy. Mathematical Biosciences & Engineering, 2009, 6 (3) : 451-467. doi: 10.3934/mbe.2009.6.451

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

[Back to Top]