American Institute of Mathematical Sciences

Advanced
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, (1995).   Google Scholar

[2]

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

[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.  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.   Google Scholar

[5]

E. K. Afenya, Acute leukemia and chemotherapy: a modeling viewpoint,, Math. Biosci., 138 (1996), 79.  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.   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.  doi: 10.1016/j.nonrwa.2011.02.027.  Google Scholar

[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.  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.   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.  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.  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.  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.  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.  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.  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.  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.  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.  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 (2006).   Google Scholar

[23]

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

[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.  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.  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.   Google Scholar

[27]

L. B. Sheiner and N. H. G. Holford, Determination of maximum effect,, Clin. Pharmacology & Therapeutics, 71 (2002).  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.   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.   Google Scholar

show all references

References:
[1]

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

[2]

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

[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.  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.   Google Scholar

[5]

E. K. Afenya, Acute leukemia and chemotherapy: a modeling viewpoint,, Math. Biosci., 138 (1996), 79.  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.   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.  doi: 10.1016/j.nonrwa.2011.02.027.  Google Scholar

[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.  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.   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.  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.  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.  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.  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.  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.  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.  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.  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.  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 (2006).   Google Scholar

[23]

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

[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.  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.  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.   Google Scholar

[27]

L. B. Sheiner and N. H. G. Holford, Determination of maximum effect,, Clin. Pharmacology & Therapeutics, 71 (2002).  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.   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.   Google Scholar

[1]

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
[2]

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
[3]

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
[4]

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
[5]

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
[6]

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
[7]

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
[8]

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
[9]

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
[10]

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
[11]

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
[12]

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
[13]

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
[14]

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
[15]

Clara Rojas, Juan Belmonte-Beitia, Víctor M. Pérez-García, Helmut Maurer. Dynamics and optimal control of chemotherapy for low grade gliomas: Insights from a mathematical model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1895-1915. doi: 10.3934/dcdsb.2016028
[16]

Arturo Alvarez-Arenas, Konstantin E. Starkov, Gabriel F. Calvo, Juan Belmonte-Beitia. Ultimate dynamics and optimal control of a multi-compartment model of tumor resistance to chemotherapy. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2017-2038. doi: 10.3934/dcdsb.2019082
[17]

Luis A. Fernández, Cecilia Pola. Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2577-2612. doi: 10.3934/dcdsb.2018266
[18]

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
[19]

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
[20]

Urszula Ledzewicz, Shuo Wang, Heinz Schättler, Nicolas André, Marie Amélie Heng, Eddy Pasquier. On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach. Mathematical Biosciences & Engineering, 2017, 14 (1) : 217-235. doi: 10.3934/mbe.2017014

2018 Impact Factor: 1.313

Tools

Metrics

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

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences