On the Controllability of a System Modeling Cell Dynamics Related to Leukemia

(original), ISI/JCR, paper

Abstract

In this paper, two control problems for a symmetric model of cell dynamics related to leukemia are considered. The first one, in connection with classical chemotherapy, is that the evolution of the disease under treatment should follow a prescribed trajectory assuming that the drug works by increasing the cell death rates of both malignant and normal cells. In the case of the second control problem, as for targeted therapies, the drug is assumed to work by decreasing the multiplication rate of leukemic cells only, and the control objective is that the disease state reaches a desired endpoint. The solvability of the two problems as well as their stability are proved by using a general method of analysis. Some numerical simulations are included to illustrate the theoretical results and prove their applicability. The results can possibly be used to design therapeutic scenarios such that an expected clinical evolution can be achieved.

Authors

Keywords

control problem; dynamic system; leukemia

References

[1] DeVita, V.T., Jr.; Chu, E. A,  History of cancer chemotherapy. Cancer Res. 2008, 68, 8643–8653. [CrossRef] [PubMed]
[2] Steensma, D.P.; Kyle, R.A., Hematopoietic stem cell discoverers. Mayo Clin. Proc. 2021, 96, 830–831. [CrossRef] [PubMed]
[3] Bonnet, D., Leukemic stem cells show the way. Folia Histochem. Cytobiol. 2005, 43, 183–186. [PubMed]
[4] Schättler, H.; Ledzewicz, U., Optimal Control for Mathematical Models of Cancer Therapies: An Application of Geometric Methods; Springer: New York, NY, USA, 2015.
[5] Friberg, L.E.; Henningsson, A.; Maas, H.; Nguyen, L.; Karlsson, M.O., Model of chemotherapy-induced myelosuppression with parameter consistency across drugs. J. Clin. Oncol. 2002, 20, 4713–4721. [CrossRef]
[6] Arimoto, M.K.; Nakamoto, Y.; Nakatani, K.; Ishimori, T.; Yamashita, K.; Takaori-Kondo, A.; Togashi, K., Increased bone marrow uptake of 18F-FDG in leukemia patients: preliminary findings. SpringerPlus 2015, 4, 521. [CrossRef]
[7] Ohanian, M.; Faderl, S.; Ravandi, F.; Pemmaraju, N.; Garcia-Manero, G.; Cortes, J.; Estrov, Z., Is acute myeloid leukemia a liquid tumor? Int. J. Cancer 2013, 133, 534. [CrossRef]
[8] Weaver, B.A. How, Taxol/Paclitaxel kills cancer cells. Mol. Biol. Cell 2014, 25, 2677–2681. [CrossRef]
[9] Engelhardt, D.; Michor, F. , A quantitative paradigm for decision-making in precision oncology. Trends Cancer 2021 , 7, 293–300.
[CrossRef]
[10] Afenya, E.K.,  Using mathematical modeling as a resource in clinical trials. Math. Biosci. Eng. 2005, 3, 421–436. [CrossRef]
[11] Afenya, E.K.; Bentil, D.E., Some perspectives on modeling leukemia. Math. Biosci. 1998, 150, 113–130. [CrossRef]
[12] Berezansky, L.; Bunimovich-Mendrazitsky, S.; Shklyar, B., Stability and controllability issues in mathematical modeling of the intensive treatment of leukemia. J. Optim. Theory Appl. 2015 , 167, 326–341. [CrossRef]
[13] Bratus, A.S.; Fimmel, E.; Todorov, Y.; Semenov, Y.S.; Nuernberg, F. On strategies on a mathematical model for leukemia therapy.
Nonlinear Anal. Real World Appl. 2012, 13, 1044–1059. [CrossRef]
[14] Crowell, H.L.; MacLean, A.L.; Stumpf, M.P.H., Feedback mechanisms control coexistence in a stem cell model of acute myeloid leukaemia. J. Theor. Biol. 2016, 401, 43–53. [CrossRef]
[15] Cucuianu, A.; Precup, R., A hypothetical-mathematical model of acute myeloid leukaemia pathogenesis. Comput. Math. Methods Med. 2010, 11, 49–65. [CrossRef]
[16] Dingli, D.; Michor, F., Successful therapy must eradicate cancer stem cells. Stem. Cells 2006, 24, 2603–2610. [CrossRef]
[17] Djulbegovic, B.; Svetina, S., Mathematical model of acute myeloblastic leukaemia: An investigation of the relevant kinetic parameters. Cell Prolif. 1985, 18, 307–319. [CrossRef]
[18] Foley, C.; Mackey, M.C., Dynamic hematological disease: A review. J. Math. Biol. 2009, 58, 285–322. [CrossRef]
[19] Kim, P.S.; Lee, P.P.; Levy, D., Modeling regulation mechanisms in the immune system. J. Theor. Biol. 2007, 246, 33–69. [CrossRef]
[20] Mac Lean, A.L.; Lo Celso, C.; Stumpf, M.P.H., Population dynamics of normal and leukaemia stem cells in the haematopoietic stem cell niche show distinct regimes where leukaemia will be controlled. J. R. Soc. Interface 2013, 10, 20120968. [CrossRef]
[21] Moore, H.; Li, N.K., A mathematical model for chronic myelogenous leukemia (CML) and T cell interaction. J. Theor. Biol. 2004, 227, 513–523. [CrossRef]
[22] Parajdi, L.G.; Precup, R.; Bonci, E.A.; Tomuleasa, C.,  A mathematical model of the transition from normal hematopoiesis to the chronic and accelerated-acute stages in myeloid leukemia. Mathematics 2020, 8, 376. [CrossRef]
[23] Precup, R., Mathematical understanding of the autologous stem cell transplantation. Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity 2012, 10, 155–167.
[24] Precup, R.; Serban, M.A.; Trif, D.; Cucuianu, A., A planning algorithm for correction therapies after allogeneic stem cell transplantation. J. Math. Model. Algorithms 2012, 11, 309–323. [CrossRef]
[25] Rubinow, S.I.; Lebowitz, J.L., A mathematical model of the acute myeloblastic leukemic state in man. Biophys. J. 1976, 16, 897–910. [CrossRef]
26] Sharp, J.A.; Browning, A.P.; Mapder, T.; Baker, C.M.; Burrage, K.; Simpson, M.J.,  Designing combination therapies using multiple optimal controls. J. Theor. Biol. 2020, 497, 110277. [CrossRef]
[27] Mackey, M.C.; Glass, L., Oscillation and chaos in physiological control systems. Science 1977, 197, 287–289. [CrossRef]
[28] Barbu, V.,  Mathematical Methods in Optimization of Differential Systems; Springer Science+Business Media: Dordrecht, The Netherlands, 1994.
[29] Becker, L.C.; Wheeler, M., Numerical and Graphical Solutions of Volterra Equations of the Second Kind; Maple Application Center: Waterloo, ON, Canada, 2005.
30] Burton, T.A., Volterra Integral and Differential Equation, 2nd ed.; Mathematics in Science & Engineering; Elsevier: Amsterdam, The Netherlands, 2005; Volume 202.
[31] Linz, P., Analytical and Numerical Methods for Volterra Equations; Studies in Applied Mathematics; SIAM: Philadelphia, PA, USA, 1985; Volume 7.

PDF

Scanned paper.

Latex version of the paper.

Cite this paper as:

I.Ş. Haplea, L.G. Parajdi, R. Precup, On the controllability of a system modeling cell dynamics related to leukemia,  Symmetry,  13 (2021) no. 10, 10.3390/sym13101867

About this paper

Journal

Symmetry

Publisher Name

MDPI

Print ISSN

Not available yet.

Online ISSN

2073-8994

Google Scholar Profile

Related Posts

Menu