About this Journal Submit a Manuscript Table of Contents
Computational and Mathematical Methods in Medicine
Volume 2013 (2013), Article ID 567213, 11 pages
http://dx.doi.org/10.1155/2013/567213
Research Article

Comparison of Simple Models of Periodic Protocols for Combined Anticancer Therapy

Department of Automatic Control, Silesian University of Technology, ul. Akademicka 2A, 44-100 Gliwice, Poland

Received 23 January 2013; Accepted 27 February 2013

Academic Editor: Enzo Grossi

Copyright © 2013 Marzena Dołbniak and Andrzej Świerniak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Jemal, F. Bray, M. M. Center, J. Ferlay, E. Ward, and D. Forman, “Global cancer statistics,” CA Cancer Journal for Clinicians, vol. 61, no. 2, pp. 69–90, 2011. View at Publisher · View at Google Scholar · View at Scopus
  2. C. D. Mathers and D. Loncar, “Projections of global mortality and burden of disease from 2002 to 2030,” PLoS Medicine, vol. 3, no. 11, pp. 2011–2030, 2006. View at Publisher · View at Google Scholar · View at Scopus
  3. D. Hanahan and R. A. Weinberg, “Hallmarks of cancer: the next generation,” Cell, vol. 144, no. 5, pp. 646–674, 2011. View at Publisher · View at Google Scholar · View at Scopus
  4. J. Folkman, “Tumor angiogenesis: therapeutic implications,” New England Journal of Medicine, vol. 285, no. 21, pp. 1182–1186, 1971. View at Scopus
  5. J. M. L. Ebos and R. S. Kerbel, “Antiangiogenic therapy: impact on invasion, disease progression, and metastasis,” Nature Reviews Clinical Oncology, vol. 8, no. 4, pp. 210–221, 2011. View at Publisher · View at Google Scholar · View at Scopus
  6. G. Bergers and D. Hanahan, “Modes of resistance to anti-angiogenic therapy,” Nature Reviews Cancer, vol. 8, no. 8, pp. 592–603, 2008. View at Publisher · View at Google Scholar · View at Scopus
  7. G. Gasparini, R. Longo, M. Fanelli, and B. A. Teicher, “Combination of antiangiogenic therapy with other anticancer therapies: results, challenges, and open questions,” Journal of Clinical Oncology, vol. 23, no. 6, pp. 1295–1311, 2005. View at Publisher · View at Google Scholar · View at Scopus
  8. L. S. Teng, K. T. Jin, K. F. He, H. H. Wang, J. Cao, and D. C. Yu, “Advances in combination of antiangiogenic agents targeting VEGF-binding and conventional chemotherapy and radiation for cancer treatment,” Journal of the Chinese Medical Association, vol. 73, no. 6, pp. 281–288, 2010. View at Publisher · View at Google Scholar · View at Scopus
  9. R. K. Jain, “Molecular regulation of vessel maturation,” Nature Medicine, vol. 9, no. 6, pp. 685–693, 2003. View at Publisher · View at Google Scholar · View at Scopus
  10. J. Ma and D. J. Waxman, “Combination of antiangiogenesis with chemotherapy for more effective cancer treatment,” Molecular Cancer Therapeutics, vol. 7, no. 12, pp. 3670–3684, 2008. View at Publisher · View at Google Scholar · View at Scopus
  11. US National Institutes of Health, Clinical Trials, 2012, http://www.clinicaltrials.gov/.
  12. P. Hahnfeldt, D. Panigrahy, J. Folkman, and L. Hlatky, “Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy,” Cancer Research, vol. 59, no. 19, pp. 4770–4775, 1999. View at Scopus
  13. S. Benzekry, G. Chapuisat, J. Ciccolini, A. Erlinger, and F. Hubert, “A new mathematical model for optimizing the combination between antiangiogenic and cytotoxic drugs in oncology,” Comptes Rendus De L Académie Des Sciences I, vol. 350, pp. 23–28, 2012.
  14. T. Jackson and X. Zheng, “A cell-based model of endothelial cell migration, proliferation and maturation during corneal angiogenesis,” Bulletin of Mathematical Biology, vol. 72, no. 4, pp. 830–868, 2010. View at Publisher · View at Google Scholar · View at Scopus
  15. S. R. McDougall, A. R. A. Anderson, M. A. J. Chaplain, and J. A. Sherratt, “Mathematical modelling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies,” Bulletin of Mathematical Biology, vol. 64, no. 4, pp. 673–702, 2002. View at Publisher · View at Google Scholar · View at Scopus
  16. B. D. Sleeman, M. Hubbard, and P. F. Jones, “The foundations of a unified approach to mathematical modelling of angiogenesis,” International Journal of Advances in Engineering Sciences and Applied Mathematics, vol. 1, pp. 43–52, 2009.
  17. C. L. Stokes and D. A. Lauffenburger, “Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis,” Journal of Theoretical Biology, vol. 152, no. 3, pp. 377–403, 1991. View at Publisher · View at Google Scholar · View at Scopus
  18. M. J. Plank and B. D. Sleeman, “A reinforced random walk model of tumour angiogenesis and anti-angiogenic strategies,” Mathematical Medicine and Biology, vol. 20, no. 2, pp. 135–181, 2003. View at Scopus
  19. T. Alarcon, H. Byrne, P. Maini, and J. Panovska, “Mathematical modelling of angiogenesis and vascular adaptation,” Studies in Multidisciplinarity, vol. 3, no. C, pp. 369–387, 2006. View at Publisher · View at Google Scholar · View at Scopus
  20. A. R. A. Anderson and M. A. J. Chaplain, “Continuous and discrete mathematical models of tumor-induced angiogenesis,” Bulletin of Mathematical Biology, vol. 60, no. 5, pp. 857–900, 1998. View at Publisher · View at Google Scholar · View at Scopus
  21. R. D. M. Travasso, E. Corvera Poiré, M. Castro, J. C. Rodríguez-Manzaneque, and A. Hernández-Machado, “Tumor angiogenesis and vascular patterning: a mathematical model,” PLoS ONE, vol. 6, no. 5, Article ID e19989, 2011. View at Publisher · View at Google Scholar
  22. L. Arakelyan, V. Vainstein, and Z. Agur, “A computer algorithm describing the process of vessel formation and maturation, and its use for predicting the effects of anti-angiogenic and anti-maturation therapy on vascular tumor growth,” Angiogenesis, vol. 5, no. 3, pp. 203–214, 2002. View at Publisher · View at Google Scholar · View at Scopus
  23. A. D'Onofrio and A. Gandolfi, “Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al. (1999),” Mathematical Biosciences, vol. 191, no. 2, pp. 159–184, 2004. View at Publisher · View at Google Scholar · View at Scopus
  24. A. Ergun, K. Camphausen, and L. M. Wein, “Optimal scheduling of radiotherapy and angiogenic inhibitors,” Bulletin of Mathematical Biology, vol. 65, no. 3, pp. 407–424, 2003. View at Publisher · View at Google Scholar · View at Scopus
  25. A. d'Onofrio and A. Gandolfi, “Chemotherapy of vascularised tumours: role of vessel density and the effect of vascular ‘pruning’,” Journal of Theoretical Biology, vol. 264, no. 2, pp. 253–265, 2010. View at Publisher · View at Google Scholar · View at Scopus
  26. A. D'Onofrio and A. Gandolfi, “A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy,” Mathematical Medicine and Biology, vol. 26, no. 1, pp. 63–95, 2009. View at Publisher · View at Google Scholar · View at Scopus
  27. M. J. Piotrowska and U. Foryś, “Analysis of the Hopf bifurcation for the family of angiogenesis models,” Journal of Mathematical Analysis and Applications, vol. 382, no. 1, pp. 180–203, 2011. View at Publisher · View at Google Scholar · View at Scopus
  28. A. Świerniak, “Control problems related to three compartmental model of combined anticancer therapy,” in Proceedings of the 20th Mediterranean Conference on Control & Automation (MED '12), pp. 1428–1433, 2012.
  29. S. T. R. Pinho, F. S. Bacelar, R. F. S. Andrade, and H. I. Freedman, “A mathematical model for the effect of anti-angiogenic therapy in the treatment of cancer tumours by chemotherapy,” Nonlinear Analysis: Real World Applications, vol. 14, no. 1, pp. 815–828, 2013.
  30. Z. Agur, L. Arakelyan, P. Daugulis, and Y. Ginosar, “HOPF point analysis for angiogenesis models,” Discrete and Continuous Dynamical Systems B, vol. 4, no. 1, pp. 29–38, 2004. View at Scopus
  31. U. Forys, Y. Kheifetz, and Y. Kogan, “Critical-point analysis for three-variable cancer angiogenesis models,” Mathematical Biosciences and Engineering, vol. 2, no. 3, pp. 511–525, 2005.
  32. A. Świerniak, “Comparison of six models of antiangiogenic therapy,” Applicationes Mathematicae, vol. 36, no. 3, pp. 333–348, 2009.
  33. U. Ledzewicz, H. Schättler, and A. d’Onofrio, “Optimal control for combination therapy in cancer,” in Proceedings of the 47th IEEE Conference on Decision and Control, pp. 1537–1542, 2008.
  34. U. Ledzewicz and H. Schattier, “Analysis of optimal controls for a mathematical model of tumour anti-angiogenesis,” Optimal Control Applications and Methods, vol. 29, no. 1, pp. 41–57, 2008. View at Publisher · View at Google Scholar · View at Scopus
  35. A. Swierniak, A. D'Onofrio, and A. Gandolfi, “Control problems related to tumor angiogenesis,” in Proceedings of the 32nd Annual Conference on IEEE Industrial Electronics (IECON '06), pp. 677–681, November 2006. View at Publisher · View at Google Scholar · View at Scopus
  36. U. Ledzewicz, H. Maurer, and H. Schättler, “Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy,” Mathematical Biosciences and Engineering, vol. 8, no. 2, pp. 307–323, 2011. View at Publisher · View at Google Scholar · View at Scopus
  37. N. Nath, T. Burg, D. M. Dawson, and E. Iyasere, “Optimizing antiangiogenic therapy for tumor minimization,” in Proceedings of the American Control Conference (ACC '10), pp. 1242–1247, July 2010. View at Scopus
  38. U. Ledzewicz, H. Maurer, and H. Schättler, “Minimizing tumor volume for a mathematical model of anti-angiogenesis with linear pharmacokinetics,” in Recent Advances in Optimization and its Applications in Engineering, pp. 267–276, Springer, 2010.
  39. U. Ledzewicz, J. Marriott, H. Maurer, and H. Schättler, “Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment,” Mathematical Medicine and Biology, vol. 27, no. 2, pp. 157–179, 2010. View at Publisher · View at Google Scholar · View at Scopus
  40. U. Ledzewicz and H. Schättler, “Antiangiogenic therapy in cancer treatment as an optimal control problem,” SIAM Journal on Control and Optimization, vol. 46, no. 3, pp. 1052–1079, 2007. View at Publisher · View at Google Scholar · View at Scopus
  41. M. Engelhart, D. Lebiedz, and S. Sager, “Optimal control for selected cancer chemotherapy ODE models: a view on the potential of optimal schedules and choice of objective function,” Mathematical Biosciences, vol. 229, no. 1, pp. 123–134, 2011. View at Publisher · View at Google Scholar · View at Scopus
  42. A. Świerniak, “Direct and indirect control of cancer populations,” Bulletin of the Polish Academy of Sciences: Technical Sciences, vol. 56, no. 4, pp. 367–378, 2008. View at Scopus
  43. R. K. Jain, “Normalization of tumor vasculature and microenvironment in antiangiogenic therapies,” ASCO Annual Meeting, pp. 412–417, 2007.
  44. S. Szala and M. Jarosz, “Tumor blood vessels,” Advances in Hygiene and Experimental Medicine, vol. 65, pp. 437–446, 2011 (Polish).
  45. S. Vinogradov and X. Wei, “Cancer stem cells and drug resistance: the potential of nanomedicine,” Nanomedicine, vol. 7, no. 4, pp. 597–615, 2012.
  46. M. Tafani and M. A. Russo, “Reprogramming Cancer Stem Cells,” Journal of Cancer Science & Therapy, vol. 4, pp. 25–26, 2012.
  47. P. M. Biava, M. Basevi, L. Biggiero, A. Borgonovo, E. Borgonovo, and F. Burigana, “Cancer cell reprogramming: stem cell differentiation stage factors and an agent based model to optimize cancer treatment,” Current Pharmaceutical Biotechnology, vol. 12, no. 2, pp. 231–242, 2011. View at Scopus