Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-28T00:07:19.274Z Has data issue: false hasContentIssue false
  • English
  • Français

New Computational Tools for ModelingChronic Myelogenous Leukemia

Published online by Cambridge University Press:  26 March 2009

M. M. Peet ,
P. S. Kim ,
S.-I. Niculescu  and
D. Levy
M. M. Peet*
Affiliation:
Illinois Institute of Technology, Chicago, USA
P. S. Kim
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City 84102, USA
S.-I. Niculescu
Affiliation:
Laboratoire des Signaux et Systèmes, CNRS-Supélec, 91192 Gif-sur-Yvette, France
D. Levy
Affiliation:
Department of Mathematics and Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park 20742, USA
*
[email protected]
Get access

Abstract

In this paper, we consider a system of nonlinear delay-differential equations(DDEs) which models the dynamics of the interaction between chronic myelogenousleukemia (CML), imatinib, and the anti-leukemia immune response. Because of thechaotic nature of the dynamics and the sparse nature of experimental data, welook for ways to use computation to analyze the model without employing directnumerical simulation. In particular, we develop several tools usingLyapunov-Krasovskii analysis that allow us to test the robustness of the modelwith respect to uncertainty in patient parameters. The methods developed in thispaper are applied to understanding which model parameters primarily affect thedynamics of the anti-leukemia immune response during imatinib treatment. Thegoal of this research is to aid the development of more efficient modelingapproaches and more effective treatment strategies in cancer therapy.

Keywords

delay-differential equationsmodel verificationoptimizationpolynomialssum-ofsquaresstabilityLyapunov-Krasovskiichronic myelogenous leukemiaimatinib
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 2: Delay equations in biology , 2009 , pp. 119 - 139
Copyright
© EDP Sciences, 2009

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

M. Adimy, L. Pujo-Menjouet. A mathematical model describing cellular division with a proliferating phase duration depending on the maturity of cells. Electronic Journal of Differential Equations, (2003) No. 107, 1–14.
Alyea, E.P., Soiffer, R.J., Canning, C., Neuberg, D., Schlossman, R., Pickett, C., Collins, H., Wang, Y., Anderson, K.C., Ritz, J.. Toxicity and efficacy of defined doses of CD4+ donor lymphocytes for treatment of relapse after allogeneic bone marrow transplant. Blood, 19 (1998), No. 10, 3671–3680.
Angstreich, G.R., Smith, B.D., Jones, R.J.. Treatment options for chronic myeloid leukemia: imatinib versus interferon versus allogeneic transplant. Curr. Opin. Oncol., 16 (2004), No. 2, 95–99. CrossRef
R. Antia, C.T. Bergstrom, S.S. Pilyugin, S.M. Kaech, R. Ahmed. Models of CD8+ responses: 1. What is the antigen-independent proliferation program. J. Theor. Biol., 221 (2003), No. 4, 585–598.
Bagg, A.. Chronic myeloid leukemia: a minimalistic view of post-therapeutic monitoring. J. Mol. Diagn., 4 (2002), No. 1, 1–10. CrossRef
S.J. Benson, Y. Ye", DSDP5: Software For semidefinite programming. (Sept. 2005) Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, ANL/MCS-P1289-0905, http://www.mcs.anl.gov/ benson/dsdp, (Submitted to ACM Transactions on Mathematical Software).
Chao, D.L., Forrest, S., Davenport, M.P., Perelson, A.S.. Stochastic stage-structured modeling of the adaptive immune system. Proc. IEEE Comput. Soc. Bioinform. Conf., 2 (2003), 124–131.
Chen, C.I., Maecker, H.T., Lee, P.P.. Development and dynamics of robust T-cell responses to CML under imatinib treatment. Blood, 111 (2008), No. 11, 5342-5349. CrossRef
C. Colijn, M.C. Mackey. A mathematical model of hematopoiesis–I. Periodic chronic myelogenous leukemia. J. Theor. Biol., 237 (2005), No. 2, 117–132.
C. Colijn, M.C. Mackey. A mathematical model of hematopoiesis–II. Cyclical neutropenia. J. Theor. Biol., 237 (2005), No. 2, 133–146.
Collins, R.H., Jr., O. Shpilberg, W.R. Drobyski, D.L. Porter, S. Giralt, R. Champlin, S.A. Goodman, S.N. Wolff, W. Hu, C. Verfaillie, A. List, W. Dalton, N. Ognoskie, A. Chetrit, J.H. Antin, J. Nemunaitis. Donor leukocyte infusions in 140 patients with relapsed malignancy after allogeneic bone marrow transplantation. J. Clin. Oncol., 15 (1997), No. 2, 433–444. CrossRef
Cortes, J., Talpaz, M., O'Brien, S., Jones, D., Luthra, R., Shan, J., Giles, F., Faderl, S., Verstovsek, S., Garcia-Manero, G., Rios, M.B., Kantarjian, H.. Molecular responses in patients with chronic myelogenous leukemia in chronic phase treated with imatinib mesylate. Clin. Cancer Res., 11 (2005), No. 9, 3425-3432. CrossRef
Kaech, S.M., Ahmed, R.. Memory CD8+ T cell differentiation: initial antigen encounter triggers a developmental program in naïve cells. Nat. Immunol., 2 (2001), No. 5, 415–422.
P.S. Kim. Mathematical Models of the Activation and Regulation of the Immune System. PhD thesis, Stanford University (2007).
T. Klingebiel, P.G. Schlegel. GVHD: overview on pathophysiology, incidence, clinical and biological features. Bone Marrow Transplant., 21 (1998), Suppl. 2, S45–S49.
H.J. Kolb, A. Schattenberg, J.M. Goldman, B. Hertenstein, N. Jacobsen, W. Arcese, P. Ljungman, A. Ferrant, L. Verdonck, D. Niederwieser, et al. Graft-versus-leukemia effect of donor lymphocyte transfusions in marrow grafted patients. European Group for Blood and Marrow Transplantation Working Party Chronic Leukemia". Blood, 86 (1995), No. 5, 2041–2050.
Komarova, N.L., Wodarz, D.. Drug resistance in cancer: Principles of emergence and prevention. Proc. Natl. Acad. Sci. USA, 102 (2005), No. 27, 9714–9719. CrossRef
N.N. Krasovskii. Stability of Motion. Stanford University Press, 1963.
K-A. Kreuzer, C.A. Schmidt, J. Schetelig, T.K. Held, C. Thiede, G. Ehninger, W. Siegert. Kinetics of stem cell engraftment and clearance of leukaemia cells after allogeneic stem cell transplantation with reduced intensity conditioning in chronic myeloid leukaemia. Eur. J. Haematol., 69 (2002), No. 1, 7–10. CrossRef
Lee, S.J. Chronic myelogenous leukaemia. Br. J. Haematol., 111 (2000), No. 4, 993–1009. CrossRef
Luzyanina, T., Engelborghs, K., Ehl, S., Klenerman, P., Bocharov, G.. Low level viral persistence after infection with LCMV: a quantitative insight through numerical bifurcation analysis. Math. Biosci., 173 (2004), No. 1, 1–23. CrossRef
Marijt, W.A.E., Heemskerk, M.H.M., Kloosterboer, F.M., Goulmy, E., Kester, M.G.D, van der Hoorn, M.A.W.G., van Luxemburg-Heys, S.A.P., Hoogeboom, M., Mutis, T., Drijfhout, J.W., van Rood, J.J., Willemze, R., Falkenburg, J.H.F.. Hematopoiesis-restricted minor histocompatibility antigens HA-1- or HA-2-specific T cells can induce complete remissions of relapsed leukemia. Proc. Natl. Acad. Sci. USA, 100 (2003), No. 5, 2742–2747. CrossRef
F. Mazenc, P.S. Kim, S.-I. Niculescu. Stability of a combined Gleevec and immune model involving delays: linear and global analysis. Proceedings of the 47th IEEE Conference on Decision and Control (2008).
Mercado, R., Vijh, S., Allen, S.E., Kerksiek, K., Pilip, I.M., Pamer, E.G.. Early programming of T cell populations responding to bacterial infection. J. Immunol., 165 (2000), No. 12, 6833–6839. CrossRef
Michor, F., Hughes, T.P., Iwasa, Y., Branford, S., Shah, N.P., Sawyers, C.L., Nowak, M.A.. Dynamics of chronic myeloid leukaemia. Nature, 435 (2005), No. 7046, 1267–1270. CrossRef
Molldrem, J.J., Lee, P.P., Wang, C., Felio, K., Kantarjian, H.M., Champlin, R.E., Davis, M.M.. Evidence that specific T lymphocytes may participate in the elimination of chronic myelogenous leukemia. Nat. Med., 6 (2000), No. 8, 1018–1023. CrossRef
Moore, H., Li, N.K.. A mathematical model for chronic myelogenous leukemia (CML) and T cell interaction. J. Theor. Biol., 225 (2004), No. 4, 513–523. CrossRef
Murali-Krishna, K., Altman, J.D., Suresh, M., Sourdive, D.J.D., Zajac, D.J.D., Miller, J.D., Slansky, J., Ahmed, R.. Counting antigen-specific CD8+ T cells: a re-evaluation of bystander activation during viral infection. Immunity, 8 (1998), No. 2, 177–187. CrossRef
B. Neiman. A mathematical model of chronic myelogenous leukaemia. Master's thesis University College, Oxford University, (2002).
Nelson, P.W., Perelson, A.S.. Mathematical analysis of delay differential equation models of HIV-1 infection. Math. Biosci., 179 (2002), No. 1, 73–94. CrossRef
S. Niculescu, P.S. Kim, D. Levy, P.P. Lee. On stability of a combined Gleevec and immune model of chronic myelogenous leukemia: exploiting delay system structure. Proceedings of 2007 IFAC Symposium on Nonlinear Control (2007).
A. Papachristodoulou, M.M. Peet, S. Lall. Stability Analysis of Nonlinear Time-Delay Systems. IEEE Transactions on Automatic Control (Special Issue on Positive Polynomials in Control), 2009.
P. Paschka, M.C. Muller, K. Merx, S. Kreil, C. Schoch, T. Lahaye, A. Weisser, A. Petzold, H. Konig, U. Berger, H. Gschaidmeier, R. Hehlmann, A. Hochhaus. Molecular monitoring of response to imatinib (Glivec) in CML patients pretreated with interferon alpha. Low levels of residual disease are associated with continuous remission. Leukemia, 17 (2003), No. 9, 1687–1694.
M.M. Peet. Web site for Matthew M. Peet. http://mmae.iit.edu/ mpeet, (2009).
Peet, M.M., Papachristodoulou, A., Lall, S.. Positive forms and stability of linear time-delay systems. SIAM Journal on Control and Optimization, 47 (2009), No. 6, 3237–3258. CrossRef
A.S. Perelson, G. Weisbuch. Immunology for Physicists Rev. Mod. Phys., 69 (1997), No. 4, 1219–1267.
Pujo-Menjouet, L., Mackey, M.C.. Contribution to the study of periodic chronic myelogenous leukemia. Comptes Rendus Biologiques, 327 (2004), 235–244. CrossRef
Roeder, I., Horn, M., Glauche, I., Hochhaus, A., Mueller, M.C., Loeffler, M.. Dynamic modeling of imatinib-treated chronic myeloid leukemia: functional insights and clinical implications. Nat. Med., 12 (2006), No. 10, 1181–1184. CrossRef
Sawyers, C.L.. Chronic myeloid leukemia. New Engl. J. Med., 340 (1999), No. 17, 1330–1340. CrossRef
C.L. Sawyers, A. Hochhaus, E. Feldman, J.M. Goldman, C.B. Miller, O.G. Ottmann, C.A. Schiffer, M. Talpaz, F. Guilhot, M.W. Deininger, T. Fischer, S.G. O'Brien, R.M. Stone, C.B. Gambacorti-Passerini, N.H. Russell, J.J. Reiffers, T.C. Shea, B. Chapuis, S. Coutre, S. Tura, E. Morra, R.A. Larson, A. Saven, C. Peschel, A. Gratwohl, F. Mandelli, M. Ben-Am, I. Gathmann, R. Capdeville, R.L. Paquette, B.J. Druker", Imatinib induces hematologic and cytogenetic responses in patients with chronic myelogenous leukemia in myeloid blast crisis: results of a phase II study. Blood, 99 (2002), No. 10, 3530–3539.
Schiffer, C.A., Hehlmann, R., Larson, R.. Perspectives on the treatment of chronic phase and advanced phase CML and Philadelphia chromosome positive ALL. Leukemia, 17 (2003), No. 4, 691–699. CrossRef
Stengle, G.. A nullstellensatz and a positivstellensatz in semialgebraic geometry. Mathematische Annalen, 207 (1973), 87–97. CrossRef
J.F. Sturm. Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optimization Methods and Software, (1999), vol. 11–12, 625-653, Version 1.05 available at http://fewcal.kub.nl/sturm/software/sedumi.html.
Thijsen, S.F.T., Schuurhuis, G.J., van Oostveen, J.W., Ossenkoppele, G.J.. Chronic mlyeloid leukemia from basics to bedside. Leukemia, 13 (1999), No. 11, 1646–1674. CrossRefPubMed
Uzunel, M., Mattsson, J., Brune, M., Johansson, J-E., Aschan, J., Ringden, O.. Kinetics of minimal residual disease and chimerism in patients with chronic myeloid leukemia after nonmyeloablative conditioning and allogeneic stem cell transplantation. Blood, 101 (2003), No. 2, 469–472. CrossRef
van Stipdonk, M.J., Lemmens, E.E., Schoenberger, S.P.. Naïve CTLs require a single brief period of antigenic stimulation for clonal expansion and differentiation. Nat. Immunol., 2 (2001), No. 5, 423–429.
Villasana, M., Radunskaya, A.. A delay differential equation model for tumor growth. J. Math. Biol., 47 (2003), No. 3, 270–294. CrossRef