Free Access
Issue
Math. Model. Nat. Phenom.
Volume 4, Number 2, 2009
Delay equations in biology
Page(s) 119 - 139
DOI https://doi.org/10.1051/mmnp/20094206
Published online 26 March 2009
  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. [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [Google Scholar]
  5. A. Bagg. Chronic myeloid leukemia: a minimalistic view of post-therapeutic monitoring. J. Mol. Diagn., 4 (2002), No. 1, 1–10. [CrossRef] [PubMed] [Google Scholar]
  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, http://www.mcs.anl.gov/ benson/dsdp, (Submitted to ACM Transactions on Mathematical Software). [Google Scholar]
  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. [PubMed] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [Google Scholar]
  10. C. Colijn, M.C. Mackey. A mathematical model of hematopoiesis–II. Cyclical neutropenia. J. Theor. Biol., 237 (2005), No. 2, 133–146. [Google Scholar]
  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. [PubMed] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [PubMed] [Google Scholar]
  14. P.S. Kim. Mathematical Models of the Activation and Regulation of the Immune System. PhD thesis, Stanford University (2007). [Google Scholar]
  15. T. Klingebiel, P.G. Schlegel. GVHD: overview on pathophysiology, incidence, clinical and biological features. Bone Marrow Transplant., 21 (1998), Suppl. 2, S45–S49. [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  18. N.N. Krasovskii. Stability of Motion. Stanford University Press, 1963. [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  20. S.J Lee. Chronic myelogenous leukaemia. Br. J. Haematol., 111 (2000), No. 4, 993–1009. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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). [Google Scholar]
  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. [PubMed] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  29. B. Neiman. A mathematical model of chronic myelogenous leukaemia. Master's thesis University College, Oxford University, (2002). [Google Scholar]
  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. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  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). [Google Scholar]
  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. [Google Scholar]
  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. [Google Scholar]
  34. M.M. Peet. Web site for Matthew M. Peet. http://mmae.iit.edu/ mpeet, (2009). [Google Scholar]
  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. [CrossRef] [Google Scholar]
  36. A.S. Perelson, G. Weisbuch. Immunology for Physicists Rev. Mod. Phys., 69 (1997), No. 4, 1219–1267. [Google Scholar]
  37. L. Pujo-Menjouet, M.C. Mackey. Contribution to the study of periodic chronic myelogenous leukemia. Comptes Rendus Biologiques, 327 (2004), 235–244. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  39. C.L. Sawyers. Chronic myeloid leukemia. New Engl. J. Med., 340 (1999), No. 17, 1330–1340. [CrossRef] [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  42. G. Stengle. A nullstellensatz and a positivstellensatz in semialgebraic geometry. Mathematische Annalen, 207 (1973), 87–97. [CrossRef] [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [PubMed] [Google Scholar]
  47. M. Villasana, A. Radunskaya. A delay differential equation model for tumor growth. J. Math. Biol., 47 (2003), No. 3, 270–294. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]

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