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Mathematical Biosciences and Engineering (MBE)
 

On viable therapy strategy for a mathematical spatial cancer model describing the dynamics of malignant and healthy cells
Pages: 163 - 183, Issue 1, February 2015

doi:10.3934/mbe.2015.12.163      Abstract        References        Full text (777.9K)           Related Articles

Alexander S. Bratus - Moscow State University, GSP-1, Leninskie Gory, Moscow, Russian Federation (email)
Svetlana Yu. Kovalenko - Federal Science and Clinical Center of the Federal Medical and Biological Agency, 28 Orehovuy boulevard, Moscow, 115682, Russian Federation (email)
Elena Fimmel - Mannheim University of Applied Sciences, Paul-Wittsack-Str. 10, 68163 Mannheim, Germany (email)

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), 25-28.       
2 E. K. Afenya and C. P. Calderón, Diverse ideas on the growth kinetics of disseminated cancer cells, Bulletin of Mathematical Biology, 62 (2000), 527-542.
3 E. C. Alvord and C. M. Show, Neoplasms affecting the nervous system in the elderly, In S.Duckett, editor, the Pathology of the Aging Human Nervous System, Lea and Febiger, Philadelphia, (1991), 210-281.
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), 1825-1836.       
5 A. S. Bratus and E. S. Chumerina, Optimal Control in Therapy of Solid Tumour Grouth, Comp. Math. and Math. Phys., 48 (2008), 892-911.       
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), 1571-1583.       
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), 88-96.       
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), 590-605.       
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), 1044-1059.       
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), 704-713.
11 M. R. Chicoine and D. L. Silbergeld, Assessment of brain tumour cell motility in vivo and in vitro, J. Neurosurgery, 82 (1995), 615-622, Bulletin of Mathematical Biology, 62 (2000), 527-542.
12 E. S. Chumerina, Choice of optimal strategy of tumour chemoterapy in gompertz model, , J. Comp. Syste. Sci. Int., 48 (2009), 325-331.       
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), 463-468.
14 K. R. Fister and J. C. Panetta, Optimal Control Applied to Competing Chemotherapeutic Cell-Kill Strategies, SIAM Journal on Applied Mathematics, 63 (2003), 1954-1971.       
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, 187. American Mathematical Society, Providence, RI, 2000.       
16 A. Giese and M. Westphal, Glioma invasion in the central nervous system, J. Neurosurgery, 39 (1996), 235-252.
17 D. Henri, Geometric Theory of Semilinear Parabolic Equations, New York, Springer Verlag, 1981.       
18 T. Hines, Mathematically Modeling the Mass-Effect of invasive Brain Tumors, preprint Arizona State University, July 28, 2010.
19 M. Kimmel and J. Sverniak, Mathematical modeling as a tool for planning anticancer therapy, European Journal of Pharmacology, 625 (2009), 108-121.
20 U. Ledzewicz, M. Naghnaenian and H. Schattler, An optimal control, approach to cancer treatment under immunological activity, Applicationes Mathematicae, 38 (2011), 17-31.       
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), 561-578.       
22 X. Li and J. Young, Optimal Control Theory for Unfinite Dimensional System, Birkhaeuser, Boston, 1995.
23 S. G. Mikhlin, Variational Methods in Mathematical Physics, Pergamon Press, 1964.       
24 D. Mackenzie, Mathematical Modeling and Cancer, SIAM News, Vol 37, No.1, 2004.
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), 636-644.
26 J. D. Murray, Mathematical Biology, II. Spatial models and biomedical applications. Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.       
27 J. D. Murray, Mathematical Biology, Second edition. Biomathematics, 19. Springer-Verlag, Berlin, 1993.       
28 P. Neittaanmaeki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems. Theory, Algorithms and Applications, Marcel Dekker, New York, 1994.       
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), p3291.
30 D. L. Silbergeld and M. R. Chicoine, Isolation and Characterization of human malignant glioma cells from histologically normal brain, J.Neurosurg., 86 (1997), 525-531.
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), 1177-1190.
32 K. R. Swanson, Mathematical Modeling of the Growth and Control of Tumours, PhD thesis, University of Washington, Seattle, WA, 1999.       
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), 317-329.
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), 589-604.       

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