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Computational and Mathematical Methods in Medicine
Volume 2012, Article ID 960256, 16 pages
http://dx.doi.org/10.1155/2012/960256
Review Article

In Silico Modelling of Treatment-Induced Tumour Cell Kill: Developments and Advances

1Department of Medical Physics, Royal Adelaide Hospital, North Terrace, Adelaide, SA 5000, Australia
2School of Chemistry and Physics, University of Adelaide, Adelaide, SA 5000, Australia
3Faculty of Science, University of Oradea, 410087 Oradea, Romania

Received 14 February 2012; Revised 10 May 2012; Accepted 14 May 2012

Academic Editor: Scott Penfold

Copyright © 2012 Loredana G. Marcu and Wendy M. Harriss-Phillips. 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. C. E. Donaghey, “CELLSIM: cell cycle simulation made easy,” International Review of Cytology, vol. 66, pp. 171–210, 1980. View at Google Scholar · View at Scopus
  2. C. E. Donaghey, “CELLSIM and CELLGROW: tools for cell kinetic modeling,” ISA Transactions, vol. 22, no. 4, pp. 21–24, 1983. View at Google Scholar · View at Scopus
  3. J. A. O'Donoghue, “The response of tumours with Gompertzian growth characteristics to fractionated radiotherapy,” International Journal of Radiation Biology, vol. 72, no. 3, pp. 325–339, 1997. View at Publisher · View at Google Scholar · View at Scopus
  4. L. Marcu, T. van Doorn, and I. Olver, “Modelling of post-irradiation accelerated repopulation in squamous cell carcinomas,” Physics in Medicine and Biology, vol. 49, no. 16, pp. 3767–3779, 2004. View at Publisher · View at Google Scholar · View at Scopus
  5. L. Marcu, T. Van Doorn, S. Zavgorodni, and I. Olver, “Growth of a virtual tumour using probabilistic methods of cell generation,” Australasian Physical and Engineering Sciences in Medicine, vol. 25, no. 4, pp. 155–161, 2002. View at Google Scholar · View at Scopus
  6. I. F. Tannock, “Oxygen diffusion and the distribution of cellular radiosensitivity in tumours,” British Journal of Radiology, vol. 45, no. 535, pp. 515–524, 1972. View at Google Scholar · View at Scopus
  7. W. Duchting and T. Vogelsaenger, “Recent progress in modelling and simulation of three-dimensional tumor growth and treatment,” BioSystems, vol. 18, no. 1, pp. 79–91, 1985. View at Google Scholar · View at Scopus
  8. W. Duchting, T. Ginsberg, and W. Ulmer, “Modeling of radiogenic responses induced by fractionated irradiation in malignant and normal tissue,” Stem Cells, vol. 13, supplement 1, pp. 301–306, 1995. View at Google Scholar · View at Scopus
  9. W. Duchting, W. Ulmer, R. Lehrig, T. Ginsberg, and E. Dedeleit, “Computer simulation and modelling of tumor spheroid growth and their relevance for optimization of fractionated radiotherapy,” Strahlentherapie und Onkologie, vol. 168, no. 6, pp. 354–360, 1992. View at Google Scholar · View at Scopus
  10. W. Duechting and T. Vogelsaenger, “Three-dimensional pattern generation applied to spheroidal tumor growth in a nutrient medium,” International Journal of Bio-Medical Computing, vol. 12, no. 5, pp. 377–392, 1981. View at Google Scholar · View at Scopus
  11. M. Kocher, H. Treuer, and R. P. Müller, “Quantification of tumor reoxygenation during accelerated radiation therapy,” Radiology, vol. 205, no. 1, pp. 263–268, 1997. View at Google Scholar · View at Scopus
  12. M. Kocher, H. Treuer, J. Voges, M. Hoevels, V. Sturm, and R. P. Müller, “Computer simulation of cytotoxic and vascular effects of radiosurgery in solid and necrotic brain metastases,” Radiotherapy and Oncology, vol. 54, no. 2, pp. 149–156, 2000. View at Publisher · View at Google Scholar · View at Scopus
  13. B. G. Wouters and J. M. Brown, “Cells at intermediate oxygen levels can be more important than the “hypoxic fraction” in determining tumor response to fractionated radiotherapy,” Radiation Research, vol. 147, no. 5, pp. 541–550, 1997. View at Google Scholar · View at Scopus
  14. G. S. Stamatakos, E. A. Kolokotroni, D. D. Dionysiou, E. C. Georgiadi, and C. Desmedt, “An advanced discrete state-discrete event multiscale simulation model of the response of a solid tumor to chemotherapy: mimicking a clinical study,” Journal of Theoretical Biology, vol. 266, no. 1, pp. 124–139, 2010. View at Publisher · View at Google Scholar · View at Scopus
  15. G. S. Stamatakos, V. P. Antipas, N. K. Uzunoglu, and R. G. Dale, “A four-dimensional computer simulation model of the in vivo response to radiotherapy of glioblastoma multiforme: studies on the effect of clonogenic cell density,” British Journal of Radiology, vol. 79, no. 941, pp. 389–400, 2006. View at Publisher · View at Google Scholar · View at Scopus
  16. G. S. Stamatakos, E. I. Zacharaki, M. I. Makropoulou et al., “Modeling tumor growth and irradiation response in vitro—a combination of high-performance computing and web-based technologies including VRML visualization,” IEEE Transactions on Information Technology in Biomedicine, vol. 5, no. 4, pp. 279–289, 2001. View at Publisher · View at Google Scholar · View at Scopus
  17. D. D. Dionysiou, G. S. Stamatakos, D. Gintides, N. Uzunoglu, and K. Kyriaki, “Critical parameters determining standard radiotherapy treatment outcome for glioblastoma multiforme: a computer simulation,” The Open Biomedical Engineering Journal, vol. 2, pp. 43–51, 2008. View at Publisher · View at Google Scholar
  18. J. Nilsson, B. K. Lind, and A. Brahme, “Radiation response of hypoxic and generally heterogeneous tissues,” International Journal of Radiation Biology, vol. 78, no. 5, pp. 389–405, 2002. View at Publisher · View at Google Scholar · View at Scopus
  19. R. A. Popple, R. Ove, and S. Shen, “Tumor control probability for selective boosting of hypoxic subvolumes, including the effect of reoxygenation.,” International Journal of Radiation Oncology Biology Physics, vol. 54, no. 3, pp. 921–927, 2002. View at Google Scholar · View at Scopus
  20. K. Borkenstein, S. Levegrün, and P. Peschke, “Modeling and computer simulations of tumor growth and tumor response to radiotherapy,” Radiation Research, vol. 162, no. 1, pp. 71–83, 2004. View at Publisher · View at Google Scholar · View at Scopus
  21. C. Harting, P. Peschke, K. Borkenstein, and C. P. Karger, “Single-cell-based computer simulation of the oxygen-dependent tumour response to irradiation,” Physics in Medicine and Biology, vol. 52, no. 16, pp. 4775–4789, 2007. View at Publisher · View at Google Scholar · View at Scopus
  22. C. Harting, P. Peschke, and C. P. Karger, “Computer simulation of tumour control probabilities after irradiation for varying intrinsic radio-sensitivity using a single cell based model,” Acta Oncologica, vol. 49, no. 8, pp. 1354–1362, 2010. View at Publisher · View at Google Scholar · View at Scopus
  23. A. Dasu and J. Denekamp, “Superfractionation as a potential hypoxic cell radiosensitizer: prediction of an optimum dose per fraction,” International Journal of Radiation Oncology Biology Physics, vol. 43, no. 5, pp. 1083–1094, 1999. View at Publisher · View at Google Scholar · View at Scopus
  24. A. Daşu, I. Toma-Daşu, and M. Karlsson, “Theoretical simulation of tumour oxygenation and results from acute and chronic hypoxia,” Physics in Medicine and Biology, vol. 48, no. 17, pp. 2829–2842, 2003. View at Publisher · View at Google Scholar · View at Scopus
  25. A. Daşu, I. Toma-Daşu, and M. Karlsson, “The effects of hypoxia on the theoretical modelling of tumour control probability,” Acta Oncologica, vol. 44, no. 6, pp. 563–571, 2005. View at Publisher · View at Google Scholar · View at Scopus
  26. I. Toma-Daşu, A. Daşu, and A. Brahme, “Dose prescription and optimisation based on tumour hypoxia,” Acta Oncologica, vol. 48, no. 8, pp. 1181–1192, 2009. View at Publisher · View at Google Scholar · View at Scopus
  27. Å. Søvik, E. Malinen, Ø. S. Bruland, S. M. Bentzen, and D. R. Olsen, “Optimization of tumour control probability in hypoxic tumours by radiation dose redistribution: a modelling study,” Physics in Medicine and Biology, vol. 52, no. 2, pp. 499–513, 2007. View at Publisher · View at Google Scholar · View at Scopus
  28. B. Titz and R. Jeraj, “An imaging-based tumour growth and treatment response model: investigating the effect of tumour oxygenation on radiation therapy response,” Physics in Medicine and Biology, vol. 53, no. 17, pp. 4471–4488, 2008. View at Publisher · View at Google Scholar · View at Scopus
  29. W. M. Harriss-Phillips, E. Bezak, and E. K. Yeoh, “Monte Carlo radiotherapy simulations of accelerated repopulation and reoxygenation for hypoxic head and neck cancer,” British Journal of Radiology, vol. 84, no. 1006, pp. 903–918, 2011. View at Google Scholar
  30. W. Tuckwell, E. Bezak, E. Yeoh, and L. Marcu, “Efficient Monte Carlo modelling of individual tumour cell propagation for hypoxic head and neck cancer,” Physics in Medicine and Biology, vol. 53, no. 17, pp. 4489–4507, 2008. View at Publisher · View at Google Scholar · View at Scopus
  31. G. G. Steel, Ed., The Growth Kinetics of Tumours, Oxford Univeristy Press, Oxford, UK, 1977.
  32. M. W. Restsky, D. E. Swartzendruber, R. H. Wardwell, and P. D. Bame, “Is Gompertzian or exponential kinetics a valid description of individual human cancer growth?” Medical Hypotheses, vol. 33, no. 2, pp. 95–106, 1990. View at Publisher · View at Google Scholar · View at Scopus
  33. A. S. Qi, X. Zheng, C. Y. Du, and B. S. An, “A cellular automaton model of cancerous growth,” Journal of Theoretical Biology, vol. 161, no. 1, pp. 1–12, 1993. View at Publisher · View at Google Scholar · View at Scopus
  34. M. Gyllenberg and G. F. Webb, “Quiescence as an explanation of Gompertzian tumor growth,” Growth, Development and Aging, vol. 53, no. 1-2, pp. 25–33, 1989. View at Google Scholar · View at Scopus
  35. M. Gyllenberg and G. F. Webb, “A nonlinear structured population model of tumor growth with quiescence,” Journal of Mathematical Biology, vol. 28, no. 6, pp. 671–694, 1990. View at Publisher · View at Google Scholar · View at Scopus
  36. J. Smolle and H. Stettner, “Computer simulation of tumour cell invasion by a stochastic growth model,” Journal of Theoretical Biology, vol. 160, no. 1, pp. 63–72, 1993. View at Publisher · View at Google Scholar · View at Scopus
  37. J. Galle, M. Loeffler, and D. Drasdo, “Modeling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro,” Biophysical Journal, vol. 88, no. 1, pp. 62–75, 2005. View at Publisher · View at Google Scholar · View at Scopus
  38. M. Nordsmark, S. M. Bentzen, V. Rudat et al., “Prognostic value of tumor oxygenation in 397 head and neck tumors after primary radiation therapy. An international multi-center study,” Radiotherapy and Oncology, vol. 77, no. 1, pp. 18–24, 2005. View at Publisher · View at Google Scholar · View at Scopus
  39. L. H. Gray, A. D. Conger, M. Ebert, S. Hornsey, and O. C. Scott, “The concentration of oxygen dissolved in tissues at the time of irradiation as a factor in radiotherapy,” British Journal of Radiology, vol. 26, no. 312, pp. 638–648, 1953. View at Google Scholar
  40. S. E. Hill, “A simple visual method for demonstrating the diffusion of oxygen through rubber and various other substances,” Science, vol. 67, no. 1736, pp. 374–376, 1928. View at Google Scholar · View at Scopus
  41. E. A. Wright and P. Howard-Flanders, “The influence of oxygen on the radiosensitivity of mammalian tissues,” Acta Radiologica, vol. 4, no. 1, pp. 26–32, 1957. View at Google Scholar
  42. M. M. Elkind, R. W. Swain, T. Alescio, H. Sutton, and W. B. Moses, “Oxygen, nitrogen, recovery and radiation therapy,” in University of Texas, M.D. Anderson Hospital and Tumor Institute of Cellular Radiation Biology: A Symposium Considering Radiation Effects in the Cell and Possible Implications for Cancer Therapy, a Collection of Papers, pp. 442–461, Williams & Wilkins, Baltimore, Md, USA, 1965. View at Google Scholar
  43. I. F. Tannock and G. G. Steel, “Tumor growth and cell kinetics in chronically hypoxic animals.,” Journal of the National Cancer Institute, vol. 45, no. 1, pp. 123–133, 1970. View at Google Scholar · View at Scopus
  44. J. M. Brown, “Evidence for acutely hypoxic cells in mouse tumours, and a possible mechanism of reoxygenation,” British Journal of Radiology, vol. 52, no. 620, pp. 650–656, 1979. View at Google Scholar · View at Scopus
  45. D. L. S. McElwain, R. Callcott, and L. E. Morris, “A model of vascular compression in solid tumours,” Journal of Theoretical Biology, vol. 78, no. 3, pp. 405–415, 1979. View at Google Scholar · View at Scopus
  46. D. L. S. McElwain and G. J. Pettet, “Cell migration in multicell spheroids: swimming against the tide,” Bulletin of Mathematical Biology, vol. 55, no. 3, pp. 655–674, 1993. View at Publisher · View at Google Scholar · View at Scopus
  47. M. A. J. Chaplain and A. R. A. Anderson, “Mathematical modelling, simulation and prediction of tumour-induced angiogenesis,” Invasion and Metastasis, vol. 16, no. 4-5, pp. 222–234, 1996. View at Google Scholar · View at Scopus
  48. A. R. Kansal, S. Torquato, G. R. Harsh, E. A. Chiocca, and T. S. Deisboeck, “Simulated brain tumor growth dynamics using a three-dimensional cellular automaton,” Journal of Theoretical Biology, vol. 203, no. 4, pp. 367–382, 2000. View at Publisher · View at Google Scholar · View at Scopus
  49. K. Måseide and E. K. Rofstad, “Mathematical modeling of chronical hypoxia in tumors considering potential doubling time and hypoxic cell lifetime,” Radiotherapy and Oncology, vol. 54, no. 2, pp. 171–177, 2000. View at Publisher · View at Google Scholar · View at Scopus
  50. T. S. Deisboeck, M. E. Berens, A. R. Kansal, S. Torquato, A. O. Stemmer-Rachamimov, and E. A. Chiocca, “Pattern of self-organization in tumour systems: complex growth dynamics in a novel brain tumour spheroid model,” Cell Proliferation, vol. 34, no. 2, pp. 115–134, 2001. View at Publisher · View at Google Scholar · View at Scopus
  51. A. A. Patel, E. T. Gawlinski, S. K. Lemieux, and R. A. Gatenby, “A cellular automaton model of early tumor growth and invasion: the effects of native tissue vascularity and increased anaerobic tumor metabolism,” Journal of Theoretical Biology, vol. 213, no. 3, pp. 315–331, 2001. View at Publisher · View at Google Scholar · View at Scopus
  52. A. R. A. Anderson, “A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion,” Mathematical Medicine and Biology, vol. 22, no. 2, pp. 163–186, 2005. View at Publisher · View at Google Scholar · View at Scopus
  53. A. R. A. Anderson, M. A. J. Chaplain, C. García-Reimbert, and C. A. Vargas, “A gradient-driven mathematical model of antiangiogenesis,” Mathematical and Computer Modelling, vol. 32, no. 10, pp. 1141–1152, 2000. View at Publisher · View at Google Scholar · View at Scopus
  54. Y. Cai, S. Xu, J. Wu, and Q. Long, “Coupled modelling of tumour angiogenesis, tumour growth and blood perfusion,” Journal of Theoretical Biology, vol. 279, no. 1, pp. 90–101, 2011. View at Publisher · View at Google Scholar · View at Scopus
  55. L.G. Marcu and E. Bezak, “Neoadjuvant cisplatin for head and neck cancer: simulation of a novel schedule for improved therapeutic ratio,” Journal of Theoretical Biology, vol. 297, pp. 41–47, 2012. View at Google Scholar
  56. F. Ellis, “Dose, time and fractionation: a clinical hypothesis,” Clinical Radiology, vol. 20, no. 1, pp. 1–7, 1969. View at Google Scholar · View at Scopus
  57. L. Cohen, “A cell population kinetic model for fractionated radiation therapy. I. Normal tissues.,” Radiology, vol. 101, no. 2, pp. 419–427, 1971. View at Google Scholar · View at Scopus
  58. D. E. Lea and D. G. Catcheside, “The mechanism of the induction by radiation of chromosome aberrations in Tradescantia,” Journal of Genetics, vol. 44, no. 2-3, pp. 216–245, 1942. View at Publisher · View at Google Scholar · View at Scopus
  59. H. R. Whithers, J. M. G. Taylor, and B. Maciejewski, “The hazard of accelerated tumor clonogen repopulation during radiotherapy,” Acta Oncologica, vol. 27, no. 2, pp. 131–146, 1988. View at Google Scholar · View at Scopus
  60. T. Alper, Cellular Radiobiology, Cambridge University Press, Cambridge, UK, 1979.
  61. D. J. Brenner, L. R. Hlatky, P. J. Hahnfeldt, Y. Huang, and R. K. Sachs, “The linear-quadratic model and most other common radiobiological models result in similar predictions of time-dose relationships,” Radiation Research, vol. 150, no. 1, pp. 83–91, 1998. View at Publisher · View at Google Scholar · View at Scopus
  62. R. G. Dale and B. Jones, Radiobioloigcal Modelling in Radiation Oncology, The British Intistitue of Radiology, London, UK, 2007.
  63. J. F. Fowler, “The linear-quadratic formula and progress in fractionated radiotherapy,” British Journal of Radiology, vol. 62, no. 740, pp. 679–694, 1989. View at Google Scholar · View at Scopus
  64. J. F. Fowler, “21 years of biologically effective dose,” British Journal of Radiology, vol. 83, no. 991, pp. 554–568, 2010. View at Publisher · View at Google Scholar · View at Scopus
  65. S. M. Bentzen, “Quantitative clinical radiobiology,” Acta Oncologica, vol. 32, no. 3, pp. 259–275, 1993. View at Google Scholar · View at Scopus
  66. J. Dunst, P. Stadler, A. Becker et al., “Tumor volume and tumor hypoxia in head and neck cancers: the amount of the hypoxic volume is important,” Strahlentherapie und Onkologie, vol. 179, no. 8, pp. 521–526, 2003. View at Publisher · View at Google Scholar · View at Scopus
  67. M. Nordsmark and J. Overgaard, “Tumor hypoxia is independent of hemoglobin and prognostic for loco-regional tumor control after primary radiotherapy in advanced head and neck cancer,” Acta Oncologica, vol. 43, no. 4, pp. 396–403, 2004. View at Publisher · View at Google Scholar · View at Scopus
  68. D. Rischin, R. J. Hicks, R. Fisher et al., “Prognostic significance of [18F]-misonidazole positron emission tomography-detected tumor hypoxia in patients with advanced head and neck cancer randomly assigned to chemoradiation with or without tirapazamine: a substudy of Trans-Tasman Radiation Oncology Group study 98.02,” Journal of Clinical Oncology, vol. 24, no. 13, pp. 2098–2104, 2006. View at Publisher · View at Google Scholar · View at Scopus
  69. B. G. Wouters, S. A. Weppler, M. Koritzinsky et al., “Hypoxia as a target for combined modality treatments,” European Journal of Cancer, vol. 38, no. 2, pp. 240–257, 2002. View at Publisher · View at Google Scholar · View at Scopus
  70. L. Marcu, E. Bezak, and I. Olver, “Scheduling cisplatin and radiotherapy in the treatment of squamous cell carcinomas of the head and neck: a modelling approach,” Physics in Medicine and Biology, vol. 51, no. 15, pp. 3625–3637, 2006. View at Publisher · View at Google Scholar · View at Scopus
  71. I. Toma-Daşu, A. Daşu, and A. Brahme, “Quantifying tumour hypoxia by PET imaging—a theoretical analysis,” Advances in Experimental Medicine and Biology, vol. 645, pp. 267–272, 2009. View at Publisher · View at Google Scholar · View at Scopus
  72. I. Toma-Dasu, A. Dasu, and M. Karlsson, “Theoretical simulation of tumour hypoxia measurements,” Advances in Experimental Medicine and Biology, vol. 578, pp. 369–374, 2006. View at Google Scholar
  73. V. P. Antipas, G. S. Stamatakos, N. K. Uzunoglu, D. D. Dionysiou, and R. G. Dale, “A spatio-temporal simulation model of the response of solid tumours to radiotherapy in vivo: parametric validation concerning oxygen enhancement ratio and cell cycle duration,” Physics in Medicine and Biology, vol. 49, no. 8, pp. 1485–1504, 2004. View at Publisher · View at Google Scholar · View at Scopus
  74. D. D. Dionysiou and G. S. Stamatakos, “Applying a 4D multiscale in vivo tumor growth model to the exploration of radiotherapy scheduling: The effects of weekend treatment gaps and p53 gene status on the response of fast growing solid tumors,” Cancer Informatics, vol. 2, pp. 113–121, 2006. View at Google Scholar · View at Scopus
  75. G. S. Stamatakos, E. C. Georgiadi, N. Graf, E. A. Kolokotroni, and D. D. Dionysiou, “Exploiting clinical trial data drastically narrows the window of possible solutions to the problem of clinical adaptation of a multiscale cancer model,” PLoS ONE, vol. 6, no. 3, Article ID e17594, 2011. View at Publisher · View at Google Scholar · View at Scopus
  76. L. G. Hanin and M. Zaider, “Cell-survival probability at large doses: an alternative to the linear-quadratic model,” Physics in Medicine and Biology, vol. 55, no. 16, pp. 4687–4702, 2010. View at Publisher · View at Google Scholar · View at Scopus
  77. C. I. Armpilia, R. G. Dale, and B. Jones, “Determination of the optimum dose per fraction in fractionated radiotherapy when there is delayed onset of tumour repopulation during treatment,” British Journal of Radiology, vol. 77, no. 921, pp. 765–767, 2004. View at Publisher · View at Google Scholar · View at Scopus
  78. M. Carlone, D. Wilkins, B. Nyiri, and P. Raaphorst, “TCP isoeffect analysis using a heterogeneous distribution of radiosensitivity,” Medical Physics, vol. 31, no. 5, pp. 1176–1182, 2004. View at Publisher · View at Google Scholar · View at Scopus
  79. P. J. Keall and S. Webb, “Optimum parameters in a model for tumour control probability, including interpatient heterogeneity: evaluation of the log-normal distribution,” Physics in Medicine and Biology, vol. 52, no. 1, pp. 291–302, 2007. View at Publisher · View at Google Scholar · View at Scopus
  80. M. Stuschke and H. D. Thames, “Fractionation sensitivities and dose-control relations of head and neck carcinomas: analysis of the randomized hyperfractionation trials,” Radiotherapy and Oncology, vol. 51, no. 2, pp. 113–121, 1999. View at Publisher · View at Google Scholar · View at Scopus
  81. C. J. McGinn, P. M. Harari, J. F. Fowler, C. N. Ford, G. M. Pyle, and T. J. Kinsella, “Dose intensification in curative head and neck cancer radiotherapy-linear quadratic analysis and preliminary assessment of clinical results,” International Journal of Radiation Oncology Biology Physics, vol. 27, no. 2, pp. 363–369, 1993. View at Google Scholar · View at Scopus
  82. P. Barberet, F. Vianna, M. Karamitros et al., “Monte-Carlo dosimetry on a realistic cell monolayer geometry exposed to alpha particles,” Physics in Medicine and Biology, vol. 57, no. 8, pp. 2189–2207, 2012. View at Google Scholar
  83. S. Incerti, N. Gault, C. Habchi et al., “A comparison of cellular irradiation techniques with alpha particles using the Geant4 Monte Carlo simulation toolkit,” Radiation Protection Dosimetry, vol. 122, no. 1–4, pp. 327–329, 2006. View at Publisher · View at Google Scholar · View at Scopus
  84. J. Aroesty, T. Lincoln, N. Shapiro, and G. Boccia, “Tumor growth and chemotherapy: mathematical methods, computer simulations, and experimental foundations,” Mathematical Biosciences, vol. 17, no. 3-4, pp. 243–300, 1973. View at Publisher · View at Google Scholar · View at Scopus
  85. T. Weldon, Mathematical Models in Cancer Research, Institute of Physics, Bristol, UK, 1988.
  86. M. J. P. Welters, A. M. J. Fichtinger-Schepman, R. A. Baan et al., “Pharmacodynamics of cisplatin in human head and neck cancer: correlation between platinum content, DNA adduct levels and drug sensitivity in vitro and in vivo,” British Journal of Cancer, vol. 79, no. 1, pp. 82–88, 1999. View at Publisher · View at Google Scholar · View at Scopus
  87. C. M. Sorenson, M. A. Barry, and A. Eastman, “Analysis of events associated with cell cycle arrest at G2 phase and cell death induced by cisplatin,” Journal of the National Cancer Institute, vol. 82, no. 9, pp. 749–755, 1990. View at Google Scholar · View at Scopus
  88. A. V. Hill, “The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves,” The Journal of Physiology, vol. 40, supplement, pp. 4–7, 1910. View at Google Scholar
  89. S. N. Gardner, “A mechanistic, predictive model of dose-response curves for cell cycle phase-specific and—nonspecific drugs,” Cancer Research, vol. 60, no. 5, pp. 1417–1425, 2000. View at Google Scholar · View at Scopus
  90. I. F. Tannock and R. P. Hill, The Basic Science of Oncology, McGraw-Hill Professional, New York, NY, USA, 1998.
  91. A. W. El-Kareh and T. W. Secomb, “A mathematical model for cisplatin cellular pharmacodynamics,” Neoplasia, vol. 5, no. 2, pp. 161–169, 2003. View at Google Scholar · View at Scopus
  92. L. M. Levasseur, H. K. Slocum, Y. M. Rustum, and W. R. Greco, “Modeling of the time-dependency of in vitro drug cytotoxicity and resistance,” Cancer Research, vol. 58, no. 24, pp. 5749–5761, 1998. View at Google Scholar · View at Scopus
  93. J. H. M. Schellens, J. Ma, A. S. T. Planting et al., “Relationship between the exposure to cisplatin, DNA-adduct formation in leucocytes and tumour response in patients with solid tumours,” British Journal of Cancer, vol. 73, no. 12, pp. 1569–1575, 1996. View at Google Scholar · View at Scopus
  94. M. Kimmel and A. Swierniak, “An optimal control problem related to leukaemia chemotherapy,” Scientific Bulletins of the Silesian Technical University, vol. 65, pp. 120–113, 1983. View at Google Scholar
  95. A. Swierniak, A. Polanski, and M. Kimmel, “Optimal control problems arising in cell-cycle-specific cancer chemotherapy,” Cell Proliferation, vol. 29, no. 3, pp. 117–139, 1996. View at Google Scholar · View at Scopus
  96. U. Ledzewicz and H. Schättler, “Analysis of a cell-cycle specific model for cancer chemotherapy,” Journal of Biological Systems, vol. 10, no. 3, pp. 183–206, 2002. View at Publisher · View at Google Scholar · View at Scopus
  97. J. S. Au, S. H. Jang, J. Zheng et al., “Determinants of drug delivery and transport to solid tumors,” Journal of Controlled Release, vol. 74, no. 1–3, pp. 31–46, 2001. View at Publisher · View at Google Scholar · View at Scopus
  98. T. L. Jackson, “Intracellular accumulation and mechanism of action of doxorubicin in a spatio-temporal tumor model,” Journal of Theoretical Biology, vol. 220, no. 2, pp. 201–213, 2003. View at Publisher · View at Google Scholar · View at Scopus
  99. A. R. Tzafriri, E. I. Lerner, M. Flashner-Barak, M. Hinchcliffe, E. Ratner, and H. Parnas, “Mathematical modeling and optimization of drug delivery from intratumorally injected microspheres,” Clinical Cancer Research, vol. 11, no. 2, part 1, pp. 826–834, 2005. View at Google Scholar · View at Scopus
  100. R. Venkatasubramanian, M. A. Henson, and N. S. Forbes, “Integrating cell-cycle progression, drug penetration and energy metabolism to identify improved cancer therapeutic strategies,” Journal of Theoretical Biology, vol. 253, no. 1, pp. 98–117, 2008. View at Publisher · View at Google Scholar · View at Scopus
  101. B. Basse, B. C. Baguley, E. S. Marshall et al., “A mathematical model for analysis of the cell cycle in cell lines derived from human tumors,” Journal of Mathematical Biology, vol. 47, no. 4, pp. 295–312, 2003. View at Publisher · View at Google Scholar · View at Scopus
  102. J. H. Goldie and A. J. Coldman, “A mathematic model for relating the drug sensitivity of tumors to their spontaneous mutation rate,” Cancer Treatment Reports, vol. 63, no. 11-12, pp. 1727–1733, 1979. View at Google Scholar · View at Scopus
  103. B. G. Birkhead, W. M. Gregory, M. L. Slevin, and V. J. Harvey, “Evaluating and designing cancer chemotherapy treatment using mathematical models,” European Journal of Cancer and Clinical Oncology, vol. 22, no. 1, pp. 3–8, 1986. View at Google Scholar · View at Scopus
  104. M. Kimmel and D. E. Axelrod, “Mathematical models of gene amplification with applications to cellular drug resistance and tumorigenicity,” Genetics, vol. 125, no. 3, pp. 633–644, 1990. View at Google Scholar · View at Scopus
  105. N. Komarova, “Stochastic modeling of drug resistance in cancer,” Journal of Theoretical Biology, vol. 239, no. 3, pp. 351–366, 2006. View at Publisher · View at Google Scholar · View at Scopus
  106. N. L. Komarova and D. Wodarz, “Stochastic modeling of cellular colonies with quiescence: an application to drug resistance in cancer,” Theoretical Population Biology, vol. 72, no. 4, pp. 523–538, 2007. View at Publisher · View at Google Scholar · View at Scopus
  107. L. Marcu, E. Bezak, I. Olver, and T. van Doorn, “Tumour resistance to cisplatin: a modelling approach,” Physics in Medicine and Biology, vol. 50, no. 1, pp. 93–102, 2005. View at Publisher · View at Google Scholar · View at Scopus
  108. A. A. Katouli and N. L. Komarova, “The worst drug rule revisited: mathematical modeling of cyclic cancer treatments,” Bulletin of Mathematical Biology, vol. 73, no. 3, pp. 549–584, 2011. View at Publisher · View at Google Scholar · View at Scopus
  109. A. J. Davis and I. F. Tannock, “Repopulation of tumour cells between cycles of chemotherapy: a neglected factor,” Lancet Oncology, vol. 1, no. 2, pp. 86–93, 2000. View at Google Scholar · View at Scopus
  110. L. Marcu and E. Bezak, “Modelling of tumour repopulation after chemotherapy,” Australasian Physical and Engineering Sciences in Medicine, vol. 33, no. 3, pp. 265–270, 2010. View at Publisher · View at Google Scholar · View at Scopus
  111. S. N. Gardner, “Cell cycle phase-specific chemotherapy: computational methods for guiding treatment,” Cell Cycle, vol. 1, no. 6, pp. 369–374, 2002. View at Google Scholar · View at Scopus
  112. S. N. Gardner and M. Fernandes, “New tools for cancer chemotherapy: computational assistance for tailoring treatments,” Molecular Cancer Therapeutics, vol. 2, no. 10, pp. 1079–1084, 2003. View at Google Scholar · View at Scopus
  113. S. N. Gardner, “Modeling multi-drug chemotherapy: tailoring treatment to individuals,” Journal of Theoretical Biology, vol. 214, no. 2, pp. 181–207, 2002. View at Publisher · View at Google Scholar · View at Scopus
  114. J. H. Goldie, A. J. Coldman, V. Ng, H. A. Hopkins, and W. B. Looney, “A mathematical and computer-based model of alternating chemotherapy and radiation therapy in experimental neoplasms.,” Antibiotics and Chemotherapy, vol. 41, pp. 11–20, 1988. View at Google Scholar · View at Scopus
  115. W. Y. Hu, W. R. Zhong, F. H. Wang, L. Li, and Y. Z. Shao, “In silico synergism and antagonism of an anti-tumour system intervened by coupling immunotherapy and chemotherapy: a mathematical modelling approach,” Bulletin of Mathematical Biology, vol. 74, no. 2, pp. 434–452, 2012. View at Google Scholar