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

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

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.
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, (1995).

[2]

E. K. Afenya, Cancer treatment strategies and mathematical modeling,, in, (1998), 1.

[3]

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

[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.

[5]

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

[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.

[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. doi: 10.1016/j.nonrwa.2011.02.027.

[8]

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

[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.

[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. doi: 10.1016/j.mbs.2010.11.007.

[11]

A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides,", Springer, (1988).

[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. doi: 10.1137/S0036139902413489.

[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. doi: 10.1137/S0036139998338509.

[14]

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

[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. doi: 10.1016/S0022-5193(03)00221-2.

[16]

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

[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. doi: 10.2165/00003088-198106060-00002.

[18]

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

[19]

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

[20]

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

[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. doi: 10.1093/imammb/18.1.25.

[22]

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

[23]

L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules,, Cancer Treat Rep., 61 (1977), 1307.

[24]

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

[25]

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

[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.

[27]

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

[28]

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

[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.

show all references

References:
[1]

E. K. Afenya and D. E. Bentil, Models of acute myeloblastic leukemia and its chemotherapy,, in, (1995).

[2]

E. K. Afenya, Cancer treatment strategies and mathematical modeling,, in, (1998), 1.

[3]

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

[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.

[5]

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

[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.

[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. doi: 10.1016/j.nonrwa.2011.02.027.

[8]

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

[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.

[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. doi: 10.1016/j.mbs.2010.11.007.

[11]

A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides,", Springer, (1988).

[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. doi: 10.1137/S0036139902413489.

[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. doi: 10.1137/S0036139998338509.

[14]

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

[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. doi: 10.1016/S0022-5193(03)00221-2.

[16]

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

[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. doi: 10.2165/00003088-198106060-00002.

[18]

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

[19]

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

[20]

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

[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. doi: 10.1093/imammb/18.1.25.

[22]

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

[23]

L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules,, Cancer Treat Rep., 61 (1977), 1307.

[24]

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

[25]

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

[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.

[27]

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

[28]

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

[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.

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