Math. Model. Nat. Phenom.
Volume 7, Number 1, 2012Cancer modeling
|Page(s)||203 - 234|
|Published online||25 January 2012|
- H. T. Alaoui, R. Yafia. Stability and Hopf bifurcation in an approachable haematopoietic stem cells model. Mathematical Biosciences, 206 (2007), 176–184. [CrossRef] [MathSciNet] [PubMed]
- M. Adimy, F. Crauste. Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay. Discrete and Continuous Dynamical Systems - Series B, 8 (2007), No. 1, 19–38. [CrossRef] [MathSciNet]
- M. Adimy, F. Crauste. Mathematical model of hematopoiesis dynamics with growth factor-dependent apoptosis and proliferation regulations. Mathematical and Computer Modelling, 49 (2009), No. 11–12, 2128–2137. [CrossRef] [MathSciNet]
- M. Adimy, F. Crauste, A. El Abdllaoui. Asymptotic behavior of a discrete maturity structured system of hematopoietic stem cell dynamics with several delays. Mathematical Modelling of Natural Phenomena, 1, (2006), No. 2, 1-19. [CrossRef] [EDP Sciences]
- M. Adimy, F. Crauste, A. El Abdllaoui. Discrete maturity-structured model of cell differentiation with applications to acute myelogenous leukemia. J. Biological Systems, 16 (2008), No. 3, 395–424. [CrossRef]
- M. Adimy, F. Crauste, A. El Abdllaoui. Boundedness and Lyapunov function for a nonlinear system of hematopoietic stem cell dynamics C. R. Acad. Sci. Paris, Ser. I, 348 (2010), No. 7-8, 373–377. [CrossRef]
- M. Adimy, F. Crauste, C. Marquet. Asymptotic behavior and stability switch for a mature-immature model of cell differentiation. Nonlinear Analysis : Real World Applications, 11 (2010), No. 4, 2913–2929. [CrossRef] [MathSciNet]
- M. Adimy, F. Crauste, S. Ruan. A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia. SIAM J. Appl. Math., 65 (2005), No. 4, 1328–1352. [CrossRef] [MathSciNet]
- M. Adimy, F. Crauste, S. Ruan. Modelling hematopoiesis mediated by growth factors with applications to periodic hematological diseases. Bulletin of Mathematical Biology, 68 (2006), No. 8, 2321–2351. [CrossRef] [MathSciNet] [PubMed]
- P-A. Bliman. Extension of Popov absolute stability criterion to nonautonomous systems with delays. INRIA Technical Report 3625, February 1999.
- P-A. Bliman. Robust absolute stability of delay systems. Nonlinear Control in the Year 2000, A. Isidori, F. Lamnabhi-Lagarrigue, W. Respondek, eds., LNCIS vol. 258, Springer Verlag, 2000, 207–237.
- B. M. Boman, M. S. Wicha. Cancer stem cells : a step toward the cure. J. Clinical Oncology, 26 (2008), No. 17, 2795–2799. [CrossRef]
- D. Bonnet, J. E. Dick. Human acute myeloid leukemia is organized as a hierarchy that originates from a primitive hematopoietic cell. Nature Medicine, 3 (1997), No. 7, 730–737. [CrossRef] [PubMed]
- N. G. Čebotarev, N. N. Meĭman. The Routh-Hurwitz problem for polynomials and entire functions. (in Russian) Trudy Mat. Inst. Steklov., 26 (1949), 3-331.
- C. Colijn, M. C. Mackey. A mathematical model of hematopoiesis : I. Periodic chronic myelogenous leukemia. J. Theoretical Biology, 237 (2005), No. 2, 117–132. [CrossRef] [PubMed]
- C. Corduneanu. Integral equations and stability of feedback systems. Academic Press, New York, 1973.
- R. F. Curtain, H. Logemann, O. Staffans. Stability results of Popov-type for infinite-dimensional systems with applications to integral control. Proc. London Math. Soc., 86 (2003), No. 3, 779–816. [CrossRef] [MathSciNet]
- C. A. Desoer, M. Vidyasagar. Feedback systems : input-output properties. SIAM Classics in Applied Mathematics, 55, SIAM, 2009.
- D. Dingli, F. Michor. Successful therapy must eradicate cancer stem cells. Stem Cells, 24 (2006), No. 12, 2603–2610. [CrossRef] [PubMed]
- D. Dingli, J. M. Pacheco. Modeling the architecture and dynamics of hematopoiesis. Wiley Interdisciplinary Reviews : Systems Biology and Medicine, 2 (2010), No. 2, 235–244. [CrossRef]
- R. C. Dorf, R. H. Bishop. Modern Control Systems (12th Edition). Pearson Educatiýn Inc., New Jersey, 2011.
- J. Dyson, R. Villella-Bressan, G. F. Webb. A singular transport equation modelling a proliferating maturity structured cell population. Canadian Applied Mathematics Quarterly, 4 (1996), No. 1, 65–95. [MathSciNet]
- P. Fortin, M. C. Mackey. Periodic chronic myelogenous leukemia : spectral analysis of blood counts and aetilogical implications. British Journal of Haematology, 104 (1999), No. 2, 336–345. [CrossRef] [PubMed]
- C. Foley, M.C. Mackey. Dynamic hematological disease : a review. J. Mathematical Biology, 58 (2009), No. 1-2, 285–322. [CrossRef] [MathSciNet] [PubMed]
- L. Grüne. Input-to-state dynamical stability and its Lyapunov function characterization. IEEE Trans. on Automatic Control, 47 (2002), No. 9, 1499–1504. [CrossRef]
- K. Gu. An improved stability criterion for systems with distributed delays. Int. J. Robust and Nonlinear Control, 13 (2003), No. 9, 819–831. [CrossRef]
- P. B. Gupta, C. L. Chaffer, R.A. Weinberg. Cancer stem cells : mirage or reality ? Nature Medicine, 15 (2009), No. 9, 1010–1012. [CrossRef] [PubMed]
- T. Haferlach. Molecular genetic pathways as therapeutic targets in acute myeloid leukemia. Hematology, American Society of Hematology Educational Program, (2008), No. 1, 400–411. [CrossRef]
- A. Halanay. Differential Equations, Stability, Oscillations, Time Lags. (in Romanian), Editura Academiei R.P.R., Bucharest, 1963, English version by Academic Press, 1966.
- C. Haurie, D. C. Dale, M. C. Mackey. Cyclical neutropenia and other periodic hematological diseases : A review of mechanisms and mathematical models. Blood, 92 (1998), No. 8, 2629–2640. [PubMed]
- K. J. Hope, L. Jin, J. E. Dick. Human acute myeloid leukemia stem cells. Archives of Medical Research, vol.34 (2003), No. 6, 507–514. [CrossRef] [PubMed]
- B. J. P. Huntly, D. G. Gilliland. Leukemia stem cells and the evolution of cancer-stem-cell research. Nature Reviews : Cancer, 5 (April 2005), 311–321. [CrossRef] [PubMed]
- M. E. King, J. Rowe. Recent developments in acute myelogenous leukemia therapy. The Oncologist, 12 (2007), Suppl. 2, 14–21. [PubMed]
- L. Kold-Andersen, M. C. Mackey. Resonance in periodic chemotherapy : A case study of acute myelogenous leukemia. J. Theoretical Biology, 209 (2001), No. 1, 113–130. [CrossRef] [PubMed]
- X. Lai, S. Nikolov, O. Wolkenhauer, J. Vera. A multi-level model accounting for the effects of JAK2-STAT5 signal modulation in erythropoiesis. Computational Biology and Chemistry, 33 (2009), No. 4, 312–324. [CrossRef] [PubMed]
- Z. Ling, Z. Lin. Traveling wavefront in a hematopoiesis model with time delay. Applied Mathematics Letters, 23 (2010), No. 4, 426–431. [CrossRef]
- M. C. Mackey, L. Glass. Oscillation and chaos in physiological control systems. Science, 197 (1977), No. 4300, 287–289. [CrossRef] [PubMed]
- M. C. Mackey. Unified hypothesis for the origin of aplastic anaemia and periodic hematopoiesis. Blood, 51 (1978), No. 5, 941–956. [PubMed]
- M. C. Mackey, C. Ou, L. Pujo-Menjouet, J. Wu. Periodic oscillations of blood cell populations in chronic myelogenous leukemia. SIAM J. Math. Anal., 38 (2006), No. 1, 166–187. [CrossRef] [MathSciNet]
- J. P. Marie. Private communication. Hôpital St. Antoine, Paris, France, July 2010.
- J. A. Martinez-Climent, L. Fontan, R. D. Gascoyne, R. Siebert, F. Prosper. Lymphoma stem cells : enough evidence to support their existence ? Haematologica, 95 (2010), No. 2, 293–302. [CrossRef] [PubMed]
- A. G. McKendrick. Applications of mathematics to medical problems. Proc. Edinburgh Math. Soc., 44 (1925), 98–130, DOI : 10.1017/S0013091500034428. [CrossRef]
- W. Michiels, S. Mondie, D. Roose, M. Dambrine. The effect of approximating distributed delay control laws on stability. Advances in time-delay systems, S-I. Niculescu, K. Gu, Eds., Springer-Verlag, LNCSE 38, 2004, 207–222.
- F. Michor, T. P. Hughes, Y. Iwasa, S. Branford, N. P. Shah, C. L. Sawyers, M. Nowak. Dynamics of chronic myeloid leukaemia. Nature, 435 (30 June 2005), 1267–1270. [CrossRef] [PubMed]
- C-I. Morarescu, S-I. Niculescu, W. Michiels. Asymptotic stability of some distributed delay systems : an algebraic approach. International Journal of Tomography & Statistics, 7 (2007), No. F07, 128–133. [MathSciNet]
- U. Münz, J. M. Rieber, F. Allgöwer. Robust stabilization and control of uncertain distributed delay systems. Topics in Time Delay Systems, J. J. Loiseau et al. Eds., LNCIS 388 (2009), 221–231.
- S. L. Noble, E. Sherer, R. E. Hannemann, D. Ramkrishna, T. Vik, A. E. Rundell. Using adaptive model predictive control to customize maintenance therapy chemotherapeutic dosing for childhood acute lymphoblastic leukemia. Journal of Theoretical Biology, 264 (2010), No. 3, 990–1002. [CrossRef] [PubMed]
- S-I. Niculescu, V. Ionescu, D. Ivanescu, L. Dugard, J-M. Dion. On generalized Popov theory for delay systems. Kybernetica, 36 (2000), No. 1, 2–20.
- S-I. Niculescu, P. S. Kim, K. Gu, P. P. Lee, D. Levy. Stability crossing boundaries of delay systems modeling immune dynamics in leukemia. Discrete and Continuous Dynamical Systems Series B, 13, (2010), No. 1, 129–156. [CrossRef]
- A. V. Oppenheim, A. S. Willsky, H. Nawab. Signals & Systems 2nd ed., Prentice Hall, New Jersey, 1997.
- H. Özbay. Introduction to feedback control theory. CRC Press LLC, Boca Raton FL, 2000.
- H. Özbay, C. Bonnet, J. Clairambault. Stability analysis of systems with distributed delays and application to hematopoietic cell maturation dynamics. Proc. of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, December 2008, 2050–2055.
- H. Özbay, H. Benjelloun, C. Bonnet, J. Clairambault. Stability conditions for a system modeling cell dynamics in leukemia. Preprints of IFAC Workshop on Time Delay Systems, TDS2010, Prague, Czech Republic, June 2010.
- D. Peixoto, D. Dingli, J.M. Pacheco. Modelling hematopoiesis in health and disease. Mathematical and Computer Modelling, 53 (2011), 7-8, 1546–1557. [CrossRef]
- B. Perthame. Transport equations in biology. Frontiers in Mathematics, Birkhäuser Verlag, 2007.
- V. M. Popov, A. Halanay. On stability for nonlinear systems with time delay. (in Russian), Automat. i Telemekhanika, 23 (1962), No. 7, 849–851.
- Y. Qu, J. Wei, S. Ruan. Stability and bifurcation analysis in hematopoietic stem cell dynamics with multiple delays. Physica D : Nonlinear Phenomena, 239 (2010), No. 20–22, 2011–2024. [CrossRef] [MathSciNet]
- V. Răsvan. Absolute stability of time lag control systems. (in Romanian) Editura Academiei, R.S.R., Bucharest, 1975 (Russian version by Nauka, Moscow, 1983).
- V. Răsvan. “Lost” cases in theory of stability for linear time-delay systems. Mathematical Reports, 9 (2007), No. 1, 99–110.
- J. Rowe. Why is clinical progress in acute myelogenous leukemia so slow ? Best Practice & Research Clinical Haematology., vol. 21 (2008), No. 1, 1–3. [CrossRef] [MathSciNet] [PubMed]
- N. J. Savill, W. Chadwick, S. E. Reece. Quantitative analysis of mechanisms that govern red blood cell age structure and dynamics during anaemia. PLoS Computational Biology, 5 (2009), doi :10.1371/journal.pcbi.1000416
- E. D. Sontag. The ISS philosophy as a unifying framework for stability-like behavior. Nonlinear Control in the Year 2000, vol.2, LNCIS 259 (2001), 443-467, DOI : 10.1007/BFb0110320.
- W. R. Sperr, A. W. Hauswirth, S. Florian, L. Öhler, K. Geissler, P. Valent. Human leukaemic stem cells : a novel target of therapy. European Journal of Clinical Investigation, 34, Suppl.2, (August 2004), 31–40. [CrossRef] [PubMed]
- E. I. Verriest. Stability of systems with distributed delays. Preprints of the IFAC Conference on System, Structure and Control, Nantes, France, July 1995, 294–299.
- E. I. Verriest. Linear Systems with Rational Distributed Delay : Reduction and Stability. Proc. of the 1999 European Control Conference, DA-12, Karlsruhe, Germany, September 1999.
- M. Vidyasagar. Nonlinear system analysis, 2nd Ed., SIAM Classics in Applied Mathematics, vol. 42, SIAM, Philadelphia 2002.
- M. Ważewska-Czyżewska, A. Lasota. Mathematical problems of the dynamics of a system of red blood cells. (in Polish) Matematyka Stosowana, 6 (1976), 23–40.
- A. Wan, J. Wei. Bifurcation analysis in an approachable haematopoietic stem cells model. J. Math. Anal. Appl., 345 (2008), No. 1, 276–285. [CrossRef]
- G-M. Zou. Cancer stem cells in leukemia, recent advances. Journal of Cellular Physiology, 213 (2007), No. 2, 440–444. [CrossRef] [PubMed]
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