New Computational Tools for Modeling Chronic Myelogenous Leukemia

M. M. Peet; P. S. Kim; S.-I. Niculescu; D. Levy

Mathematical Modelling of Natural Phenomena (2009)

  • Volume: 4, Issue: 2, page 119-139
  • ISSN: 0973-5348

Abstract

top
In this paper, we consider a system of nonlinear delay-differential equations (DDEs) which models the dynamics of the interaction between chronic myelogenous leukemia (CML), imatinib, and the anti-leukemia immune response. Because of the chaotic nature of the dynamics and the sparse nature of experimental data, we look for ways to use computation to analyze the model without employing direct numerical simulation. In particular, we develop several tools using Lyapunov-Krasovskii analysis that allow us to test the robustness of the model with respect to uncertainty in patient parameters. The methods developed in this paper are applied to understanding which model parameters primarily affect the dynamics of the anti-leukemia immune response during imatinib treatment. The goal of this research is to aid the development of more efficient modeling approaches and more effective treatment strategies in cancer therapy.

How to cite

top

Peet, M. M., et al. "New Computational Tools for Modeling Chronic Myelogenous Leukemia." Mathematical Modelling of Natural Phenomena 4.2 (2009): 119-139. <http://eudml.org/doc/222231>.

@article{Peet2009,
abstract = { In this paper, we consider a system of nonlinear delay-differential equations (DDEs) which models the dynamics of the interaction between chronic myelogenous leukemia (CML), imatinib, and the anti-leukemia immune response. Because of the chaotic nature of the dynamics and the sparse nature of experimental data, we look for ways to use computation to analyze the model without employing direct numerical simulation. In particular, we develop several tools using Lyapunov-Krasovskii analysis that allow us to test the robustness of the model with respect to uncertainty in patient parameters. The methods developed in this paper are applied to understanding which model parameters primarily affect the dynamics of the anti-leukemia immune response during imatinib treatment. The goal of this research is to aid the development of more efficient modeling approaches and more effective treatment strategies in cancer therapy. },
author = {Peet, M. M., Kim, P. S., Niculescu, S.-I., Levy, D.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {delay-differential equations; model verification; optimization; polynomials; sum-ofsquares; stability; Lyapunov-Krasovskii; chronic myelogenous leukemia; imatinib; polynomials; sum-of squares; stability},
language = {eng},
month = {3},
number = {2},
pages = {119-139},
publisher = {EDP Sciences},
title = {New Computational Tools for Modeling Chronic Myelogenous Leukemia},
url = {http://eudml.org/doc/222231},
volume = {4},
year = {2009},
}

TY - JOUR
AU - Peet, M. M.
AU - Kim, P. S.
AU - Niculescu, S.-I.
AU - Levy, D.
TI - New Computational Tools for Modeling Chronic Myelogenous Leukemia
JO - Mathematical Modelling of Natural Phenomena
DA - 2009/3//
PB - EDP Sciences
VL - 4
IS - 2
SP - 119
EP - 139
AB - In this paper, we consider a system of nonlinear delay-differential equations (DDEs) which models the dynamics of the interaction between chronic myelogenous leukemia (CML), imatinib, and the anti-leukemia immune response. Because of the chaotic nature of the dynamics and the sparse nature of experimental data, we look for ways to use computation to analyze the model without employing direct numerical simulation. In particular, we develop several tools using Lyapunov-Krasovskii analysis that allow us to test the robustness of the model with respect to uncertainty in patient parameters. The methods developed in this paper are applied to understanding which model parameters primarily affect the dynamics of the anti-leukemia immune response during imatinib treatment. The goal of this research is to aid the development of more efficient modeling approaches and more effective treatment strategies in cancer therapy.
LA - eng
KW - delay-differential equations; model verification; optimization; polynomials; sum-ofsquares; stability; Lyapunov-Krasovskii; chronic myelogenous leukemia; imatinib; polynomials; sum-of squares; stability
UR - http://eudml.org/doc/222231
ER -

References

top
  1. 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.  
  2. E.P. Alyea, R.J. Soiffer, C. Canning, D. Neuberg, R. Schlossman, C. Pickett, H. Collins, Y. Wang, K.C. Anderson, J. Ritz. 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.  
  3. G.R. Angstreich, B.D. Smith, R.J. Jones. Treatment options for chronic myeloid leukemia: imatinib versus interferon versus allogeneic transplant. Curr. Opin. Oncol., 16 (2004), No. 2, 95–99.  
  4. 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.  
  5. A. Bagg. Chronic myeloid leukemia: a minimalistic view of post-therapeutic monitoring. J. Mol. Diagn., 4 (2002), No. 1, 1–10.  
  6. 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, benson/dsdp, (Submitted to ACM Transactions on Mathematical Software).  URIhttp://www.mcs.anl.gov/
  7. D.L. Chao, S. Forrest, M.P. Davenport, A.S. Perelson. Stochastic stage-structured modeling of the adaptive immune system. Proc. IEEE Comput. Soc. Bioinform. Conf., 2 (2003), 124–131.  
  8. C.I. Chen, H.T. Maecker, P.P. Lee. Development and dynamics of robust T-cell responses to CML under imatinib treatment. Blood, 111 (2008), No. 11, 5342-5349.  
  9. C. Colijn, M.C. Mackey. A mathematical model of hematopoiesis–I. Periodic chronic myelogenous leukemia. J. Theor. Biol., 237 (2005), No. 2, 117–132.  
  10. C. Colijn, M.C. Mackey. A mathematical model of hematopoiesis–II. Cyclical neutropenia. J. Theor. Biol., 237 (2005), No. 2, 133–146.  
  11. R.H. Collins, 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.  
  12. J. Cortes, M. Talpaz, S. O'Brien, D. Jones, R. Luthra, J. Shan, F. Giles, S. Faderl, S. Verstovsek, G. Garcia-Manero, M.B. Rios, H. Kantarjian. Molecular responses in patients with chronic myelogenous leukemia in chronic phase treated with imatinib mesylate. Clin. Cancer Res., 11 (2005), No. 9, 3425-3432.  
  13. S.M. Kaech, R. Ahmed. Memory CD8+ T cell differentiation: initial antigen encounter triggers a developmental program in naïve cells. Nat. Immunol., 2 (2001), No. 5, 415–422.  
  14. P.S. Kim. Mathematical Models of the Activation and Regulation of the Immune System. PhD thesis, Stanford University (2007).  
  15. T. Klingebiel, P.G. Schlegel. GVHD: overview on pathophysiology, incidence, clinical and biological features. Bone Marrow Transplant., 21 (1998), Suppl. 2, S45–S49.  
  16. 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.  
  17. N.L. Komarova, D. Wodarz. Drug resistance in cancer: Principles of emergence and prevention. Proc. Natl. Acad. Sci. USA, 102 (2005), No. 27, 9714–9719.  
  18. N.N. Krasovskii. Stability of Motion. Stanford University Press, 1963.  
  19. 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.  
  20. S.J Lee. Chronic myelogenous leukaemia. Br. J. Haematol., 111 (2000), No. 4, 993–1009.  
  21. T. Luzyanina, K. Engelborghs, S. Ehl, P. Klenerman, G. Bocharov. Low level viral persistence after infection with LCMV: a quantitative insight through numerical bifurcation analysis. Math. Biosci., 173 (2004), No. 1, 1–23.  
  22. W.A.E. Marijt, M.H.M. Heemskerk, F.M. Kloosterboer, E. Goulmy, M.G.D Kester, M.A.W.G. van der Hoorn, S.A.P. van Luxemburg-Heys, M. Hoogeboom, T. Mutis, J.W. Drijfhout, J.J. van Rood, R. Willemze, J.H.F. Falkenburg. 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.  
  23. 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).  
  24. R. Mercado, S. Vijh, S.E. Allen, K. Kerksiek, I.M. Pilip, E.G. Pamer. Early programming of T cell populations responding to bacterial infection. J. Immunol., 165 (2000), No. 12, 6833–6839.  
  25. F. Michor, T.P. Hughes, Y. Iwasa, S. Branford, N.P. Shah, C.L. Sawyers, M.A. Nowak. Dynamics of chronic myeloid leukaemia. Nature, 435 (2005), No. 7046, 1267–1270.  
  26. J.J. Molldrem, P.P. Lee, C. Wang, K. Felio, H.M. Kantarjian, R.E. Champlin, M.M. Davis. Evidence that specific T lymphocytes may participate in the elimination of chronic myelogenous leukemia. Nat. Med., 6 (2000), No. 8, 1018–1023.  
  27. H. Moore, N.K. Li. A mathematical model for chronic myelogenous leukemia (CML) and T cell interaction. J. Theor. Biol., 225 (2004), No. 4, 513–523.  
  28. K. Murali-Krishna, J.D. Altman, M. Suresh, D.J.D. Sourdive, D.J.D. Zajac, J.D. Miller, J. Slansky, R. Ahmed. Counting antigen-specific CD8+ T cells: a re-evaluation of bystander activation during viral infection. Immunity, 8 (1998), No. 2, 177–187.  
  29. B. Neiman. A mathematical model of chronic myelogenous leukaemia. Master's thesis University College, Oxford University, (2002).  
  30. P.W. Nelson, A.S. Perelson. Mathematical analysis of delay differential equation models of HIV-1 infection. Math. Biosci., 179 (2002), No. 1, 73–94.  
  31. 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).  
  32. 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.  
  33. 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.  
  34. M.M. Peet. Web site for Matthew M. Peet. mpeet, (2009).  URIhttp://mmae.iit.edu/
  35. M.M. Peet, A. Papachristodoulou, S. Lall. Positive forms and stability of linear time-delay systems. SIAM Journal on Control and Optimization, 47 (2009), No. 6, 3237–3258.  
  36. A.S. Perelson, G. Weisbuch. Immunology for Physicists Rev. Mod. Phys., 69 (1997), No. 4, 1219–1267.  
  37. L. Pujo-Menjouet, M.C. Mackey. Contribution to the study of periodic chronic myelogenous leukemia. Comptes Rendus Biologiques, 327 (2004), 235–244.  
  38. I. Roeder, M. Horn, I. Glauche, A. Hochhaus, M.C. Mueller, M. Loeffler. Dynamic modeling of imatinib-treated chronic myeloid leukemia: functional insights and clinical implications. Nat. Med., 12 (2006), No. 10, 1181–1184.  
  39. C.L. Sawyers. Chronic myeloid leukemia. New Engl. J. Med., 340 (1999), No. 17, 1330–1340.  
  40. 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.  
  41. C.A. Schiffer, R. Hehlmann, R. Larson. Perspectives on the treatment of chronic phase and advanced phase CML and Philadelphia chromosome positive ALL. Leukemia, 17 (2003), No. 4, 691–699.  
  42. G. Stengle. A nullstellensatz and a positivstellensatz in semialgebraic geometry. Mathematische Annalen, 207 (1973), 87–97.  
  43. 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.  
  44. S.F.T. Thijsen, G.J. Schuurhuis, J.W. van Oostveen, G.J. Ossenkoppele. Chronic mlyeloid leukemia from basics to bedside. Leukemia, 13 (1999), No. 11, 1646–1674.  
  45. M. Uzunel, J. Mattsson, M. Brune, J-E. Johansson, J. Aschan, O. Ringden. 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.  
  46. M.J. van Stipdonk, E.E. Lemmens, S.P. Schoenberger. Naïve CTLs require a single brief period of antigenic stimulation for clonal expansion and differentiation. Nat. Immunol., 2 (2001), No. 5, 423–429.  
  47. M. Villasana, A. Radunskaya. A delay differential equation model for tumor growth. J. Math. Biol., 47 (2003), No. 3, 270–294.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.