American Institute of Mathematical Sciences

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2013, 10(1): 151-165. doi: 10.3934/mbe.2013.10.151

On optimal and suboptimal treatment strategies for a mathematical model of leukemia

Elena Fimmel 1, , Yury S. Semenov 2, and  Alexander S. Bratus 2,

1. 

Mannheim University of Applied Sciences, Paul-Wittsack-Str. 10, 68163 Mannheim, Germany
2. 

Moscow State University of Railway Engineering, Obraztsova 15, Moscow, 127994, Russian Federation, Russian Federation

Received  March 2012 Revised  August 2012 Published  December 2012

In this work an optimization problem for a leukemia treatment model based on the Gompertzian law of cell growth is considered. The quantities of the leukemic and of the healthy cells at the end of the therapy are chosen as the criterion of the treatment quality. In the case where the number of healthy cells at the end of the therapy is higher than a chosen desired number, an analytical solution of the optimization problem for a wide class of therapy processes is given. If this is not the case, a control strategy called alternative is suggested.
Keywords: cancer model., Optimal therapy control, chemotherapy.
Mathematics Subject Classification: 49J1.
Citation: Elena Fimmel, Yury S. Semenov, Alexander S. Bratus. On optimal and suboptimal treatment strategies for a mathematical model of leukemia. Mathematical Biosciences & Engineering, 2013, 10 (1) : 151-165. doi: 10.3934/mbe.2013.10.151
References:
[1]

E. K. Afenya and D. E. Bentil, Models of acute myeloblastic leukemia and its chemotherapy, in "Computational Medicine, Public Health, and Biotechnology Part I. World Scientific" New Jersey, (1995), pp. 397. Google Scholar

[2]

E. K. Afenya, Cancer treatment strategies and mathematical modeling, in "Mathematical Models in Medical and Health Sciences" (eds. M. A. Horn, G. Simonett and G. F. Webb), Vanderbilt University. Nashville, (1998), 1-8. Google Scholar

[3]

E. K. Afenya and C. P. Calderón, Modeling disseminated cancers: A review of mathematical models, Comm. Theor. Biol., 8 (2003), 225-253. doi: 10.1080/08948550302449.  Google Scholar

[4]

E. K. Afenya and C. P. Calderón, A brief look at a normal cell decline and inhibition in acute leukemia, J. Can. Det. Prev.,, 20 (1996), 171-179. Google Scholar

[5]

E. K. Afenya, Acute leukemia and chemotherapy: a modeling viewpoint, Math. Biosci., 138 (1996), 79-100. doi: 10.1016/S0025-5564(96)00086-7.  Google Scholar

[6]

A. V. Antipov and A. S. Bratus', Mathematical model of optimal chemotherapy strategy with allowance for cell population dynamics in a heterogeneous tumor, Zh. Vychisl. Mat. Mat. Fiz., 49 (2009), 1907-1919  Google Scholar

[7]

A. S. Bratus, E. Fimmel, Y. Todorov, Y. S. Semenov and F. Nürnberg, On strategies on a mathematical model for leukemia therapy, Nonlinear Analysis: Real World Applications, 13 (2012), 1044-1059. doi: 10.1016/j.nonrwa.2011.02.027.  Google Scholar

[8]

B. D. Clarkson, Acute myelocytic leukemia in adults, Cancer, 30 (1972), 1572-1582. doi: 10.1002/1097-0142(197212)30:6<1572::AID-CNCR2820300624>3.0.CO;2-M.  Google Scholar

[9]

B. Djulbegovic and S. Svetina, Mathematical model of acute myeloblastic leukemia: an investigation of a relevant kinetic parameters, Cell Tissue Kinet., 18 (1985), 307-319. Google Scholar

[10]

M. Engelhart, D. Lebiedz and S. Sager, Optimal control for selected cancer chemotherapy ODE models: A view on the potential of optimal schedules and choice of objective function, Mathematical Biosciences, 229 (2011), 123-134. doi: 10.1016/j.mbs.2010.11.007.  Google Scholar

[11]

A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides," Springer, 1988.  Google Scholar

[12]

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. doi: 10.1137/S0036139902413489.  Google Scholar

[13]

K. R. Fister and J. C. Panetta, Optimal control applied to cell-cycle-specific cancer chemotherapy, SIAM Journal on Applied Mathematics, 60 (2000), 1059-1072. doi: 10.1137/S0036139998338509.  Google Scholar

[14]

C. L. Frenzen and J. D. Murray:, A cell kinetics justification for Gompertz equation, SIAM J. Appl. Math., 46 (1986), 614-624. doi: 10.1137/0146042.  Google Scholar

[15]

C. Guiot, P. G. Degiorgis, P. P. Delsanto, P. Gabriele and T. S. Deisboeck, Does tumour growth follow a universal law?, J. Theor. Biol., 225 (2003), 147-151. doi: 10.1016/S0022-5193(03)00221-2.  Google Scholar

[16]

"Handbook of Cancer Models with Applications," (W.-Y. Tan, L. Hanin Eds.), Ser. Math. Biology and Medicine; World Scientific. Vol. 9, 2008. Google Scholar

[17]

N. H. G. Holford and L. B. Sheiner, Understanding the dose-effect relationship-clinical application of pharmacokinetic-pharmacodynamic models, Clin. Pharmacokin, 6 (1981), 429-453. doi: 10.2165/00003088-198106060-00002.  Google Scholar

[18]

D. E. Kirk, "Optimal Contol Theory: An Introduction," Prentice-Hall, 1970 Google Scholar

[19]

U. Ledzewicz, A. d'Onofrio, H. Maurer and H. Schaettler, On optimal delivery of combination therapy for tumors, Mathematical Biosciences, 222 (2009), 13-26. doi: 10.1016/j.mbs.2009.08.004.  Google Scholar

[20]

U. Ledzewicz and H. Schaettler, Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy, Mathematical Biosciences, 206 (2007), 320-342. doi: 10.1016/j.mbs.2005.03.013.  Google Scholar

[21]

A. S. Matveev and A. V. Savkin, Optimal control regimens: influence of tumours on normal cells and several toxicity constraints, IMA J. Math. Appl. Med. Biol., 18 (2001), 25-40. doi: 10.1093/imammb/18.1.25.  Google Scholar

[22]

L. Norton and R. Simon, The Norton-Simon Hypothesis: designing more effective and less toxic chemotherapeutic regimens, Nature Clinical Practice, 3 Nr. 8, (2006). Google Scholar

[23]

L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treat Rep., 61(1977) Oct, 1307-1317. PubMed PMID: 589597. Google Scholar

[24]

J. C. Panetta, A mathematical model of breast and ovarian cancer treated with paclitaxel, Mathematical Biosciences, 146 (1997), 89-113. doi: 10.1016/S0025-5564(97)00077-1.  Google Scholar

[25]

S. I. Rubinow and J. L. Lebowitz, A mathematical model of the acute myeloblastic leukemic state in man, Biophys. J., 16 (1976), 897-910. doi: 10.1016/S0006-3495(76)85740-2.  Google Scholar

[26]

F. Schabel, Jr., H. Skipper and W. Wilcox, Experimental evaluation of potential anti-cancer agents. XIII. On the criteria and kinetics associated with curability of experimental leukemia, Cancer Chemo. Rep., 25 (1964), 1-111. Google Scholar

[27]

L. B. Sheiner and N. H. G. Holford, Determination of maximum effect, Clin. Pharmacology & Therapeutics, 71 (2002), pp.304. doi: 10.1067/mcp.2002.122277.  Google Scholar

[28]

G. W. Swan and T. L. Vincent, Optimal control analysis in the chemotherapy of IgG multiple myeloma, Bull. Math. Biol., 39 (1977), 317-337. Google Scholar

[29]

Y. Todorov, E. Fimmel, A. S. Bratus, Y. S. Semenov and F. Nürnberg, An optimal strategy for leukemia therapy: A multi-objective approach, Russian Journal of Numerical Analysis and Mathematical Modelling, 26 (2011), 589-604.  Google Scholar

show all references

References:
[1]

E. K. Afenya and D. E. Bentil, Models of acute myeloblastic leukemia and its chemotherapy, in "Computational Medicine, Public Health, and Biotechnology Part I. World Scientific" New Jersey, (1995), pp. 397. Google Scholar

[2]

E. K. Afenya, Cancer treatment strategies and mathematical modeling, in "Mathematical Models in Medical and Health Sciences" (eds. M. A. Horn, G. Simonett and G. F. Webb), Vanderbilt University. Nashville, (1998), 1-8. Google Scholar

[3]

E. K. Afenya and C. P. Calderón, Modeling disseminated cancers: A review of mathematical models, Comm. Theor. Biol., 8 (2003), 225-253. doi: 10.1080/08948550302449.  Google Scholar

[4]

E. K. Afenya and C. P. Calderón, A brief look at a normal cell decline and inhibition in acute leukemia, J. Can. Det. Prev.,, 20 (1996), 171-179. Google Scholar

[5]

E. K. Afenya, Acute leukemia and chemotherapy: a modeling viewpoint, Math. Biosci., 138 (1996), 79-100. doi: 10.1016/S0025-5564(96)00086-7.  Google Scholar

[6]

A. V. Antipov and A. S. Bratus', Mathematical model of optimal chemotherapy strategy with allowance for cell population dynamics in a heterogeneous tumor, Zh. Vychisl. Mat. Mat. Fiz., 49 (2009), 1907-1919  Google Scholar

[7]

A. S. Bratus, E. Fimmel, Y. Todorov, Y. S. Semenov and F. Nürnberg, On strategies on a mathematical model for leukemia therapy, Nonlinear Analysis: Real World Applications, 13 (2012), 1044-1059. doi: 10.1016/j.nonrwa.2011.02.027.  Google Scholar

[8]

B. D. Clarkson, Acute myelocytic leukemia in adults, Cancer, 30 (1972), 1572-1582. doi: 10.1002/1097-0142(197212)30:6<1572::AID-CNCR2820300624>3.0.CO;2-M.  Google Scholar

[9]

B. Djulbegovic and S. Svetina, Mathematical model of acute myeloblastic leukemia: an investigation of a relevant kinetic parameters, Cell Tissue Kinet., 18 (1985), 307-319. Google Scholar

[10]

M. Engelhart, D. Lebiedz and S. Sager, Optimal control for selected cancer chemotherapy ODE models: A view on the potential of optimal schedules and choice of objective function, Mathematical Biosciences, 229 (2011), 123-134. doi: 10.1016/j.mbs.2010.11.007.  Google Scholar

[11]

A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides," Springer, 1988.  Google Scholar

[12]

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. doi: 10.1137/S0036139902413489.  Google Scholar

[13]

K. R. Fister and J. C. Panetta, Optimal control applied to cell-cycle-specific cancer chemotherapy, SIAM Journal on Applied Mathematics, 60 (2000), 1059-1072. doi: 10.1137/S0036139998338509.  Google Scholar

[14]

C. L. Frenzen and J. D. Murray:, A cell kinetics justification for Gompertz equation, SIAM J. Appl. Math., 46 (1986), 614-624. doi: 10.1137/0146042.  Google Scholar

[15]

C. Guiot, P. G. Degiorgis, P. P. Delsanto, P. Gabriele and T. S. Deisboeck, Does tumour growth follow a universal law?, J. Theor. Biol., 225 (2003), 147-151. doi: 10.1016/S0022-5193(03)00221-2.  Google Scholar

[16]

"Handbook of Cancer Models with Applications," (W.-Y. Tan, L. Hanin Eds.), Ser. Math. Biology and Medicine; World Scientific. Vol. 9, 2008. Google Scholar

[17]

N. H. G. Holford and L. B. Sheiner, Understanding the dose-effect relationship-clinical application of pharmacokinetic-pharmacodynamic models, Clin. Pharmacokin, 6 (1981), 429-453. doi: 10.2165/00003088-198106060-00002.  Google Scholar

[18]

D. E. Kirk, "Optimal Contol Theory: An Introduction," Prentice-Hall, 1970 Google Scholar

[19]

U. Ledzewicz, A. d'Onofrio, H. Maurer and H. Schaettler, On optimal delivery of combination therapy for tumors, Mathematical Biosciences, 222 (2009), 13-26. doi: 10.1016/j.mbs.2009.08.004.  Google Scholar

[20]

U. Ledzewicz and H. Schaettler, Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy, Mathematical Biosciences, 206 (2007), 320-342. doi: 10.1016/j.mbs.2005.03.013.  Google Scholar

[21]

A. S. Matveev and A. V. Savkin, Optimal control regimens: influence of tumours on normal cells and several toxicity constraints, IMA J. Math. Appl. Med. Biol., 18 (2001), 25-40. doi: 10.1093/imammb/18.1.25.  Google Scholar

[22]

L. Norton and R. Simon, The Norton-Simon Hypothesis: designing more effective and less toxic chemotherapeutic regimens, Nature Clinical Practice, 3 Nr. 8, (2006). Google Scholar

[23]

L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treat Rep., 61(1977) Oct, 1307-1317. PubMed PMID: 589597. Google Scholar

[24]

J. C. Panetta, A mathematical model of breast and ovarian cancer treated with paclitaxel, Mathematical Biosciences, 146 (1997), 89-113. doi: 10.1016/S0025-5564(97)00077-1.  Google Scholar

[25]

S. I. Rubinow and J. L. Lebowitz, A mathematical model of the acute myeloblastic leukemic state in man, Biophys. J., 16 (1976), 897-910. doi: 10.1016/S0006-3495(76)85740-2.  Google Scholar

[26]

F. Schabel, Jr., H. Skipper and W. Wilcox, Experimental evaluation of potential anti-cancer agents. XIII. On the criteria and kinetics associated with curability of experimental leukemia, Cancer Chemo. Rep., 25 (1964), 1-111. Google Scholar

[27]

L. B. Sheiner and N. H. G. Holford, Determination of maximum effect, Clin. Pharmacology & Therapeutics, 71 (2002), pp.304. doi: 10.1067/mcp.2002.122277.  Google Scholar

[28]

G. W. Swan and T. L. Vincent, Optimal control analysis in the chemotherapy of IgG multiple myeloma, Bull. Math. Biol., 39 (1977), 317-337. Google Scholar

[29]

Y. Todorov, E. Fimmel, A. S. Bratus, Y. S. Semenov and F. Nürnberg, An optimal strategy for leukemia therapy: A multi-objective approach, Russian Journal of Numerical Analysis and Mathematical Modelling, 26 (2011), 589-604.  Google Scholar

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