Skip to main content
SpringerLink
Log in

Optimization of Cytostatic Leukemia Therapy in an Advection–Reaction–Diffusion Model

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this work, we study an advection–reaction–diffusion system involving normal and cancer hematopoietic cells (HC) and modeling the development of leukemia in the bone marrow. The effects of a targeted cytostatic therapy, which inhibits the regeneration of cancer HC, are studied through an optimal control system. We prove the existence of a positive bounded solution and the existence of an optimal controls by semigroup techniques. Conditions of optimality are in terms of the dual system. The effects of targeted cytostatic therapy, upon the dynamics of normal and cancer HC, are discussed through numerical simulations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Coebergh, J.W.W., Reedijk, A.M.J., de Vries, E., Martos, C., Jakab, Z., Steliarova-Foucher, E., Kamps, W.A.: Leukaemia incidence and survival in children and adolescents in Europe during 1978–1997. Report from the Automated Childhood Cancer Information System Project. Eur. J. Cancer 42, 2019–2036 (2006)

    Article  Google Scholar 

  2. Segel, G.B., Lichtman, M.A.: Familial (inherited) leukemia, lymphoma, and myeloma: an overview. Blood Cells Mol. Dis. 32, 246–261 (2004)

    Article  Google Scholar 

  3. Aïnseba, B., Benosman, C.: Global dynamics of hematopoietic stem cells and differentiated cells in a chronic myeloid leukemia model. J. Math. Biol. 62, 975–997 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Reikvam, H., Hatfield, K.J., Kittang, A.O., Hovland, R.: Bruserud, Ø: acute myeloid leukemia with the t(8;21) translocation: clinical consequences and biological implications. J. Biomed. Biotechnol. 2011, 1–23 (2011)

    Article  Google Scholar 

  5. Bernt, K.M., Armstrong, S.A.: Leukemia stem cells and human acute lymphoblastic leukemia. Semin. Hematol. 46, 33–38 (2009)

    Article  Google Scholar 

  6. Lane, S.W., Gilliland, D.G.: Leukemia stem cells. Semin. Cancer Biol. 20, 71–76 (2010)

    Article  Google Scholar 

  7. Marchesi, F., Annibali, O., Cerchiara, E., Tirindelli, M.C., Avvisati, G.: Cytogenetic abnormalities in adult non-promyelocytic acute myeloid leukemia: a concise review. Crit. Rev. Oncol. Hematol. 80, 331–346 (2011)

    Article  Google Scholar 

  8. Yamazaki, H., Nishida, H., Iwata, S., Dang, N.H., Morimoto, C.: CD90 and CD110 correlate with cancer stem cell potentials in human T-acute lymphoblastic leukemia cells. Biochem. Biophys. Res. Commun. 383, 172–177 (2009)

    Article  Google Scholar 

  9. Prenkert, M.: On mechanisms of drug resistance in acute myeloid leukemia. PÖrebro University, thesis. ISBN 978-91-7668-729-1 (2010)

  10. Dietz, A.B., Souan, L., Knutson, G.J., Bulur, P.A., Litzow, M.R., Vuk-Pavlović, S.: Imatinib mesylate inhibits T-cell proliferation in vitro and delayed-type hypersensitivity in vivo. Blood 104, 1094–1099 (2004)

    Article  Google Scholar 

  11. Savage, D.G., Antman, K.H.: Imatinib mesylate: a new oral targeted therapy. Drug therapy. N. Engl. J. Med. 346, 683–693 (2002)

    Article  Google Scholar 

  12. Paganin, M., Ferrando, A.: Molecular pathogenesis and targeted therapies for NOTCH1-induced T-cell acute lymphoblastic leukemia. Blood Rev. 25, 83–90 (2011)

    Article  Google Scholar 

  13. Stone, R.M.: Targeted agents in AML: much more to do. Best Pract. Res. Clin. Haematol. 20, 39–48 (2007)

    Article  Google Scholar 

  14. Ma, W., Gutierrez, A., Goff, D.J., Geron, I., Sadarangani, A., Jamieson, C.A.M., Court, A.C., Shih, A.Y., Jiang, Q., Wu, C.C., Li, K., Smith, K.M., Crews, L.A., Gibson, N.W., Deichaite, I., Morris, S.R., Wei, P., Carson, D.A., Look, A.T., Jamieson, C.H.M.: Notch1 signaling promotes human T-cell acute lymphoblastic leukemia initiating cell regeneration in supportive niches. Plos One 7, 1–14 (2012)

    Google Scholar 

  15. Kettemann, A., Neuss-Radu, M.: Derivation and analysis of a system modeling the chemotactic movement of hematopoietic stem cells. J. Math. Biol. 56, 579–610 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Andasari, V., Gerisch, A., Lolas, G., South, A.P., Chaplain, M.J.: Mathematical modeling of cancer cell invasion of tissue: biological insight from mathematical analysis and computational simulation. J. Math. Biol. 63, 141–171 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Volpert, V., Petrovskii, S.: Reaction–diffusion waves in biology. Phys. Life Rev. 6, 267–310 (2009)

    Article  Google Scholar 

  18. Clairambault, J.: Modelling physiological and pharmacological control on cell proliferation to optimise cancer treatments. Math. Model. Nat. Phenom. 4, 12–67 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ducrot, A., Volpert, V.: On a model of leukemia development with a spatial cell distribution. Math. Model. Nat. Phenom. 2, 101–120 (2007)

    Article  MathSciNet  Google Scholar 

  20. Mughal, T.I., Schrieber, A.: Principal long-term adverse effects of imatinib in patients with chronic myeloid leukemia in chronic phase. Biologics 4, 315–323 (2010)

    Google Scholar 

  21. Harousseau, J.L., Lancet, J.E., Reiffers, J., Lowenberg, B., Thomas, X., Huguet, F., Fenaux, P., Zhang, S., Rackoff, W., De Porre, P., Stone, R.: A phase 2 study of the oral farnesyltransferase inhibitor tipifarnib in patients with refractory or relapsed acute myeloid leukemia. Blood 109, 5151–5156 (2007)

    Article  Google Scholar 

  22. Abbott, L.H., Michor, F.: Mathematical models of targeted cancer therapy. Br. J. Cancer 95, 1136–1141 (2006)

    Article  Google Scholar 

  23. Chotinantakul, K., Leeanansaksiri, W.: Hematopoietic stem cell development, niches, and signaling pathways. Bone Marrow Res. 2012, 1–16 (2012)

    Article  Google Scholar 

  24. Tajbakhsh, S.: Stem cell: what’s in a name? Nat. Rep. Stem Cells (2009). doi:10.1038/stemcells.2009.90

  25. Andreeff, M., Goodrich, D.W., Pardee, A.B.: Cell Proliferation, Differentiation, and Apoptosis. Holland-Frei Cancer Medicine, 5th edn. BC Decker, Hamilton (2000)

    Google Scholar 

  26. Greenspan, H.P.: Models for the growth of a solid tumor by diffusion. Stud. Appl. Math. 52, 317–340 (1972)

    Google Scholar 

  27. Ledzewicz, U., Schättler, H., Friedman, A., Kashdan, E.: Mathematical Methods and Models in Biomedicine. Springer, Heidelberg (2013)

    Book  MATH  Google Scholar 

  28. Demin, I., Ducrot, A., Volpert, V.: Spatial distribution of cell populations in the process of erythropoiesis. Int. Electron. J. Pure Appl. Math. 01, 143–161 (2010)

    Google Scholar 

  29. Cheng, K., Shen, D., Xie, Y., Cingolani, E., Malliaras, K., Marbàn, E.: Brief report: mechanism of extravasation of infused stem cells. Stem Cells 12, 2835–2842 (2012)

    Article  Google Scholar 

  30. Reya, T., Morrison, S.J., Clarke, M.F., Weissman, I.L.: Stem cells, cancer, and cancer stem cells. Nature 414, 105–111 (2001)

    Article  Google Scholar 

  31. Wirtz, D., Konstantopoulos, K., Searson, P.C.: The physics of cancer: the role of physical interactions and mechanical forces in metastasis. Nat. Rev. Cancer 11, 512–522 (2011)

    Article  Google Scholar 

  32. Ferraresi, V., Catricalà, C., Ciccarese, M., Ferrari, A., Zeuli, M., Cognetti, F.: Severe skin reaction in a patient with gastrointestinal stromal tumor treated with imatinib mesylate. Anticancer Res. 26, 4771–4774 (2006)

    Google Scholar 

  33. Van Oosterom, A.T., Judson, I.R., Verweij, J., Stroobants, S., Dumez, H., Donato di Paola, E., Sciot, R., Van Glabbeke, M., Dimitrijevic, S., Nielsen, O.S.: Update of phase I study of imatinib (STI571) in advanced soft tissue sarcomas and gastrointestinal stromal tumors: a report of the EORTC Soft Tissue and Bone Sarcoma Group. Eur. J. Cancer 38, S83–S87 (2002)

    Article  Google Scholar 

  34. Fimmel, E., Semenov, Y.S., Bratus, A.S.: On optimal and suboptimal treatment strategies for a mathematical model of leukemia. Math. Biosci. Eng. 10, 151–165 (2013)

    MATH  MathSciNet  Google Scholar 

  35. Fister, K.R., Panetta, J.C.: Optimal control applied to cell-cycle-specific cancer chemotherapy. SIAM J. Appl. Math. 60, 1059–1072 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  36. Amann, H.: Abstract Linear Theory, Linear and Quasilinear Parabolic Problems. Birkhauser, Basel (1995)

    Book  Google Scholar 

  37. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    Book  MATH  Google Scholar 

  38. Hieber, H., Prüss, J.: Heat kernels and maximal \(L^p-L^q\) estimates for parabolic evolution equations. Commun. Partial Differ. Equ. 22, 1647–1669 (1997)

    Article  MATH  Google Scholar 

  39. Prüss, J.: Evolutionary Integral Equations and Applications. Birkhauser, Basel (1993)

    Book  MATH  Google Scholar 

  40. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics No. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics No. 840. Springer, Berlin (1981)

    Google Scholar 

  41. Barbu, V., Iannelli, M.: Optimal control of population dynamics. J. Optim. Theory Appl. 102, 1–14 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  42. Aïnseba, B., Bendahmane, M., Ruiz-Baier, R.: Analysis of an optimal control problem for the tridomain model in cardiac electrophysiology. J. Math. Anal. Appl. 388, 231–247 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  43. Picart, D., Aïnseba, B., Milner, F.: Optimal control problem on insect pest populations. Appl. Math. Lett. 24, 1160–1164 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  44. Rodriguez-Brenes, I.A., Wodarz, D., Komarova, N.L.: Stem cell control, oscillations, and tissue regeneration in spatial and non-spatial models. Front. Oncol. 3, 1–10 (2013)

    Article  Google Scholar 

  45. Moore, S., Lyle, S.: Quiescent, slow-cycling stem cell populations in cancer: a review of the evidence and discussion of significance. J. Oncol. 2011, 1–11 (2011)

    Article  Google Scholar 

  46. Sehl, M., Zhou, H., Sinsheimer, J.S., Lange, K.L.: Extinction models for cancer stem cell therapy. Math. Biosci. 234, 132–146 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  47. Takeishi, S., Matsumoto, A., Onoyama, I., Naka, K., Hirao, A., Nakayama, K.I.: Ablation of Fbxw7 eliminates leukemia-initiating cells by preventing quiescence. Cancer Cell 23, 347–361 (2013)

    Article  Google Scholar 

  48. Calmelet, C., Prokop, A., Mensah, J., McCawley, L.J., Crooke, P.S.: Modeling the cancer stem cell hypothesis. Math. Model. Nat. Phenom. 5, 40–62 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors thank the editors and anonymous reviewers for their constructive comments that improved the manuscript.

Author information

Authors and Affiliations

  1. Département de Mathématiques, Université de Tlemcen, Tlemcen, Algeria

    Chahrazed Benosman

  2. IMB UMR CNRS 5251, Université de Bordeaux, Bordeaux, France

    Bedr’Eddine Aïnseba & Arnaud Ducrot

Authors
  1. Chahrazed Benosman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bedr’Eddine Aïnseba
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Arnaud Ducrot
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chahrazed Benosman.

Additional information

Communicated by Mimmo Iannelli.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Benosman, C., Aïnseba, B. & Ducrot, A. Optimization of Cytostatic Leukemia Therapy in an Advection–Reaction–Diffusion Model. J Optim Theory Appl 167, 296–325 (2015). https://doi.org/10.1007/s10957-014-0667-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-014-0667-7

Keywords

Mathematics Subject Classification

Search

Navigation