Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T14:02:43.950Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal Control of Leukemic Cell Population Dynamics

Published online by Cambridge University Press:  07 February 2014

X. Dupuis
Get access

Abstract

We are interested in optimizing the co-administration of two drugs for some acute myeloid leukemias (AML), and we are looking for in vitro protocols as a first step. This issue can be formulated as an optimal control problem. The dynamics of leukemic cell populations in culture is given by age-structured partial differential equations, which can be reduced to a system of delay differential equations, and where the controls represent the action of the drugs. The objective function relies on eigenelements of the uncontrolled model and on general relative entropy, with the idea to maximize the efficiency of the protocols. The constraints take into account the toxicity of the drugs. We present in this paper the modeling aspects, as well as theoretical and numerical results on the optimal control problem that we get.

Keywords

acute myeloid leukemiaoptimal controldelay differential equationspopulation dynamicsgeneral relative entropyin vitro therapeutic optimization
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 1: Issue dedicated to Michael Mackey , 2014 , pp. 4 - 26
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adimy, M., Crauste, F.. Mathematical model of hematopoiesis dynamics with growth factor-dependent apoptosis and proliferation regulations. Math. Comput. Modelling, 49 (11-12) 2009, 2128–2137. CrossRefGoogle Scholar
Adimy, M., Crauste, F., El Abdllaoui, A.. Discrete maturity-structured model of cell differentiation with applications to acute myelogenous leukemia. J. Biol. Systems, 16(3) 2008, 395–424. CrossRefGoogle Scholar
J. L. Avila, C. Bonnet, J. Clairambault, H. Ozbay, S.-I. Niculescu, F. Merhi, R. Tang, J.-P. Marie. A new model of cell dynamics in Acute Myeloid Leukemia involving distributed delays. In 10th IFAC Workshop on Time Delay Systems, Boston, USA, 2012, 55–60.
A. Ballesta, F. Mehri, X. Dupuis, C. Bonnet, J.F. Bonnans, R. Tang, F. Fava, P. Hirsch, J.-P. Marie, J. Clairambault. In vitro dynamics of LAM patient blood sample cells and their therapeutic control by aracytine and an Flt3 inhibitor. In preparation.
Basdevant, C., Clairambault, J., Lévi, F.. Optimisation of time-scheduled regimen for anti-cancer drug infusion. M2AN Math. Model. Numer. Anal., 39(6) 2005, 1069–1086. CrossRefGoogle Scholar
Billy, F., Clairambault, J.. Designing proliferating cell population models with functional targets for control by anti-cancer drugs. Discrete Contin. Dyn. Syst. Ser. B, 18(4) 2013, 865–889. CrossRefGoogle Scholar
F. Billy, J. Clairambault, O. Fercoq, S. Gaubert, T. Lepoutre, T. Ouillon, S. Saito. Synchronisation and control of proliferation in cycling cell population models with age structure. Mathematics and Computers in Simulation, 2012.
J.F. Bonnans, X. Dupuis, L. Pfeiffer. Second-order necessary conditions in Pontryagin form for optimal control problems. Rapport de recherche RR-8306, INRIA, May 2013.
J.F. Bonnans, P. Martinon, V. Grélard. Bocop v1.0.3: A collection of examples. Url: www.bocop.org, June 2012.
Feichtinger, G., Tragler, G., Veliov, V.M.. Optimality conditions for age-structured control systems. J. Math. Anal. Appl., 288(1) 2003, 47–68. CrossRefGoogle Scholar
Gabriel, P., Garbett, S.P., Quaranta, V., Tyson, D.R., Webb, G.F.. The contribution of age structure to cell population responses to targeted therapeutics. J. Theoret. Biol., 311 2012, 19–27. CrossRefGoogle ScholarPubMed
Göllmann, L., Kern, D., Maurer, H.. Optimal control problems with delays in state and control variables subject to mixed control-state constraints. Optimal Control Appl. Methods, 30(4) 2009, 341–365. CrossRefGoogle Scholar
Guinn, T.. Reduction of delayed optimal control problems to nondelayed problems. J. Optimization Theory Appl., 18(3) 1976, 371–377. CrossRefGoogle Scholar
Halanay, A.. Optimal controls for systems with time lag. SIAM Journal on Control, 6(2) 1968, 215–234. CrossRefGoogle Scholar
Hinow, P., Wang, S., Arteaga, C., Webb, G.. A mathematical model separates quantitatively the cytostatic and cytotoxic effects of a her2 tyrosine kinase inhibitor. Theoretical Biology and Medical Modelling, 4(1) 2007, 14. CrossRefGoogle ScholarPubMed
Ledzewicz, U., Maurer, H., Schättler, H.. Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy. Math. Biosci. Eng., 8(2) 2011, 307–323. CrossRefGoogle ScholarPubMed
Ledzewicz, U., Schättler, H.. Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy. Math. Biosci., 206(2) 2007, 320–342. CrossRefGoogle ScholarPubMed
Mackey, M.C.. Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis. Blood, 51(5) 1978, 941–956. Google Scholar
Marquet, C., Adimy, M.. On the stability of hematopoietic model with feedback control. C. R. Math. Acad. Sci. Paris, 350(3-4) 2012, 173–176. CrossRefGoogle Scholar
Michel, P., Mischler, S., Perthame, B.. General relative entropy inequality: an illustration on growth models. J. Math. Pures Appl. (9), 84(9) 2005, 1235–1260. CrossRefGoogle Scholar
Özbay, H., Bonnet, C., Benjelloun, H., Clairambault, J.. Stability analysis of cell dynamics in leukemia. Math. Model. Nat. Phenom., 7(1) 2012, 203–234. CrossRefGoogle Scholar
Peixoto, D., Dingli, D., Pacheco, J.M.. Modelling hematopoiesis in health and disease. Mathematical and Computer Modelling, 53(7,8) 2011, 1546–1557. CrossRefGoogle Scholar
B. Perthame. Transport equations in biology. Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007.
Rowe, J.M.. Why is clinical progress in acute myelogenous leukemia so slow? Best Practice & Research Clinical Haematology, 21(1): 2008, 1–3. CrossRefGoogle Scholar
Stiehl, T., Marciniak-Czochra, A.. Mathematical modeling of leukemogenesis and cancer stem cell dynamics. Math. Model. Nat. Phenom., 7(1) 2012, 166–202. CrossRefGoogle Scholar