Linear quadratic and tumour control probability modelling in external beam radiotherapy

Journal of Mathematical Biology

Volumn 58, Issue 4-5, 2009, Pages 799-817

  O'Rourke, S F C ^a   McAneney, H ^a   Hillen, T ^b  

^a QUEEN'S UNIVERSITY BELFAST   (United Kingdom)

^b UNIVERSITY OF ALBERTA   (Canada)

Author keywords

BED; Fractionation; Linear quadratic; Radiotherapy; Repopulation

Indexed keywords

BIOLOGICAL MODEL; CELL CYCLE; COMPUTER ASSISTED RADIOTHERAPY; COMPUTER SIMULATION; HUMAN; MATHEMATICAL PHENOMENA; NEOPLASM; PATHOLOGY; RADIOBIOLOGY; REVIEW; STATISTICAL MODEL; STATISTICS;

CELL CYCLE; COMPUTER SIMULATION; HUMANS; LINEAR MODELS; MATHEMATICAL CONCEPTS; MODELS, BIOLOGICAL; MODELS, STATISTICAL; NEOPLASMS; RADIOBIOLOGY; RADIOTHERAPY PLANNING, COMPUTER-ASSISTED;

EID: 58349095345     PISSN: 03036812     EISSN: 14321416     Source Type: Journal    
DOI: 10.1007/s00285-008-0222-y     Document Type: Review

References (104)

1. VP Antipas GS Stamatakos NK Uzunoglu DD Dionysiou RG Dale 2004 A spatio-temporal simulation model of the response of solid tumours to radiotherapy in vivo: parametric validation concerning oxygen enhancement ratio and cell cycle Phys Med Biol 49 1485 1504
2. CI Armpilia RG Dale B Jones 2004 Determination of the optimum dose per fraction in fractionated radiotherapy when there is delayed onset of tumour repopulation during treatment Br J Radio 77 765 767
3. GW Barendsen 1982 Dose fractionation, dose rate and iso-effect relationships for normal tissue responses Int J Radiat Oncol Bio Phys 8 1981 1997
4. JS Bedford M N.Cornforth 1987 Relationship between recovery from sublethal x-ray damage and the rejoining of chromosome breaks in normal human fibroblasts Radiat Res 111 406 423
5. SM Bentzen MI Saunders S Dische 1999 Repair halftimes estimated from observations of treatment-related morbidity after chart or conventional radiotherapy in head and neck cancer Radiother Oncol 53 219 226
6. A Brahme 1984 Dosimetric precision requirements in radiation therapy Acta Radiol Oncol 23 379 391
7. D Brenner LR Hltaky PJ Hahnfeldt Y Huang RK Sachs 1998 The linear quadratic and most other common radiobiogical models Radiat Res 150 83 88
8. DJ Brenner EJ Hall 1999 Fractionation and protraction for radiotherapy of prostate carcinoma Int J Radiat Oncol Biol Phys 43 1095 1101
9. DJ Brenner LR Hlatky PJ Hahnfeldt EJ Hall RK Sachs 1995 A convenient extension of the linear-quadratic model to include redistribution and reoxygenation Int J Radiat Oncol Biol Phys 32 379 390
10. JD Brenner 1993 Dose, volume and tumour control predictions in radiotherapy Int J Radiat Oncol Biol Phys 26 171 179
11. FM Buffa C West K Byrne JV Moore AE Nahum 2001 Radiation response and cure rate of human colon adenocarcinoma spheroids of different size: the significance of hypoxia on tumour control modelling Int J Radiat Oncol Biol Phys 49 1109 1118
12. EB Butler BS Teh WH Grant BM Uhl RB Kuppersmith JK Chiu DT Donovan SY Woo 1999 Smart (simultaneous modulated accelerated radiation therapy) boost: a new accelerated fractionation schedule for the treatment of head and neck cancer with intensity modulated radiotherapy Int J Radiat Oncol Biol Phys 45 21 32
13. DJ Carlson RD Stewart XA Li K Jennings JZ Wang M Guerro 2004 Comparison of in vitro and in vivo α/β ratios for prostate cancer Phys Med Biol 49 4477 4491
14. KH Chadwick HP Leenhouts 1973 A molecular theory of cell survival Phys Med Biol 18 78 87
15. Chadwick KH and Leenhouts HP (1981). The molecualr theory of radiation biology. Springer, Berlin
16. PL Chen DJ Brenner RK Sachs 1995 Ionizing radiation damage to cells: effects of cell cycle resdistribution Math Biosci 126 147 170
17. SB Curtis 1986 Lethal and potentially lethal lesions induced by radiation-a unified repair model Radiat Res 106 252 270
18. RG Dale 1985 The application of the linear-quadratic dose-effect equation to fractionated and protracted radiotherapy Br J Rad 58 515 528
19. RG Dale 1989 Time-dependent tumour repopulation factors in linear quadratic equations-implications for treatment strategies Radiother Oncol 15 371 382
20. RG Dale 1996 Dose-rate effects in targeted radiotherapy Phys Med Biol 41 1871 1884
21. A Dawson T Hillen 2006 Derivation of the tumour control probability (tcp) from a cell cycle model Comput Math Meth Med 7 121 142
22. 2 block, and cell division Radiat Res 124 201 207
23. DD Dionysiou GS Stamatakos 2006 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 tumours Cancer Inform 2 113 121
24. DD Dionysiou GS Stamatakos NK Uzunoglu KS Nikita A Marioli 2004 A four-dimensional simulation model of tumour response to radiotherapy in vivo: parametric validation considering radiosensitivity, genetic profile and fractionation J Theor Biol 230 1 20
25. GM Duchesne LJ Peters 1999 What is the α/β ratio for prostate cancer? rationale for hypofractionated high-dose-rate brachytherapy Int J Radiat Oncol Biol Phys 44 747 748
26. JD Fenwick 1998 Predicting the radiation control probability of heterogeneous tumour ensembles: data analysis and parameter estimation using a closed form expression Phys Med Biol 43 2159 2178
27. JF Fowler 1989 The linear-quadratic formula and progress in fractionated radiotherapy Br J Radiol 62 679 694
28. JF Fowler 1991 The phantom of tumor treatment-continually rapid proliferation unmasked Radiother Oncol 22 156 158
29. JF Fowler 2001 Biological factors influencing optimum fractionation in radiotherapy Acta Oncol 40 712 717
30. JF Fowler 2003 What hypofractionated protocols should be tested for prostate cancer? Int J Radiat Oncol Biol Phys 56 1093 1104
31. JF Fowler 2006 Development of radiobiology for oncology-a personal view Phys Med Biol 51 R263 R286
32. JF Fowler R Chappell 2000 Non-small cell lung tumours repopulate rapidly during radiation therapy Int J Radiat Oncol Biol Phys 46 516 517
33. JF Fowler BE Stern 1960 Dose-rate effects: some theoretical and practical considerations Br J Radiol 33 389 395
34. LH Gray AD Conger M Ebert S Hornsey OC Scott 1953 The concentration of oxygen dissolved in tissues at the time of irradiation as a factor in radiotherapy Br J Radiol 26 638 648
35. M Guerrero RD Stewart J Wang XA Li 2002 Equivalence of the linear-quadratic and two-lesion kinetic models Phys Med Biol 47 3197 3209
36. P Hahnfeldt L Hlatky 1996 Resensitization due to redistribution of cells in the phases of the cell cycle during arbitrary radiation protocols Radiat Res 145 134 143
37. Hall EJ and Giaccia AJ (2006). Radiobiology for the radiologist, 6th edn. JB Lippincott, Philadelphia
38. Harder D (1988). The pairwise lesion interaction model. In: Kiefer, J (eds) Quantitative mathematical models in radiative biology, pp 159-170. Springer, Berlin
39. LR Hlatky P Hahnfeldt RK Sachs 1994 Influence of time dependent stochastic heterogeneity on the radiation response of a cell population Math Biosci 122 201 220
40. JA Horas OR Olguin MG Rizzotto 2005 On the surviving fraction in irradiated multicellular tumour spheroids: calculation of overall radiosensitivity parameters, influence of hypoxia and volume effects Phys Med Biol 50 1689 1701
41. J Horiot RL Fur T N'Guyen C Chenal S Schraub S Alfonsi G Gardani WVD Bogaert S Danczak M Bolla 1992 Hyperfractionation versus conventional fractionation in oropharyngeal carcinoma: Final analysis of a randomized trial of the eortc cooperative group of radiotherapy Radiother Oncol 25 231 241
42. B Jones RG Dale 1995 Cell loss factors and the linear-quadratic model Radiother Oncol 37 136 139
43. B Jones RG Dale 1999 Mathematical models of tumour and normal tissue response Acta Oncol 38 883 893
44. AM Kellerer HH Rossi 1972 The theory of dual radiation action Curr Top Radiat Res Q 8 85 158
45. WS Kendal 1998 A closed form description of tumour control with fractionated radiotherapy and repopulation Radiat Biol 73 207 210
46. JJ Kim IF Tannock 2005 Repopulation of cancer cells during therapy: an important cause of treatment failure Nat Rev: Cancer 5 516 525
47. CR King TA DiPetrillo DE Wazer 2000 Optimal radiotherapy for prostate cancer: predictions for conventional external beam, imrt, and brachytherapy from radiobiologic models Int J Radiat Oncol Biol Phys 46 165 172
48. J Kirk WM Gray ER Watson 1971 Cumulative radiation effect. part 1: Fractionated treatment regimes Clin Radiol 22 145 155
49. JP Kirkpatrick LB Marks 2004 Modelling killing and repopulation kinetics of subclinical cancer: direct calculations from clinical data Int J Radiat Oncol Biol Phys 58 641 654
50. GJ Kutcher C Burman L Brewster M Goitein R Mohan 1991 Histogram reduction method for calculating complication probabilities for three dimensional treatment planning evaluations Int J Radiat Oncol Biol Phys 21 137 146
51. Lea DE (1946). Actions of radiations on living cells. Cambridge University Press, London
52. KA Lindsay EG Wheldon C Deehan TE Wheldon 2001 Radiation carcinogenesis modelling for risk of treatment-related second tumours following radiotherapy Br J Radiol 74 529 536
53. B Maciejewski S Majewski 1991 Dose fractionation and tumour repopulation in radiotherapy for bladder cancer Radiother Oncol 21 163 170
54. B Maciejewski K Skladowskia B Pileckia JMG Taylor RH Withersd L Miszczyka A Zajusza R Suwinskia 1996 Randomized clinical trial on accelerated 7 days per week fractionation in radiotherapy for head and neck cancer. preliminary report on acute toxicity Radiother Oncol 40 137 145
55. JH Mao KA Lindsay RJ Mairs TE Wheldon 2001 The effect of tissue-specific growth patterns of target stem cells on the spectrum of tumours resulting from multistage tumorigenesis J Theor Biol 210 93 100
56. H McAneney SFC O'Rourke 2007 Investigation of various growth mechanisms of solid tumour growth within the linear quadratic model for radiotherapy Phys Med Biol 52 1039 1054
57. R Mohan GS Mageras B Baldwin LJ Brewster GJ Kutcher 1992 Clinically relevant optimisation of 3d conformal treatments Med Phys 19 933 944
58. TR Munro CW Gilbert 1961 The relation between tumor lethal doses and the radiosensitivity of tumor cells Br J Radiol 34 246 251
59. Nahum AE and Tait DM (1992). Maximising control by customized dose prescription for pelvic tumours. In: Breit, A (eds) Advanced radiation therapy: tumour response monitoring and treatment planning, pp 425-431. Springer, Heidelberg
60. A Niemierko M Goitein 1991 Calculation of normal tissue complication probability and dose-volume histogram reduction schemes for tissues with a critical element architecture Radiother Oncol 20 166 176
61. A Niemierko M Goitein 1993 Implementation of a model for estimating tumour control probability for an inhomogeneously irradiated tumor Radiother Oncol 29 140 147
62. JA O'Donoghue 1997 The response of tumours with gompertzian growth characteristics to fractionated radiotherapy Int J Radiat Biol 72 325 339
63. JM O'Sullivan DP Hollywood N Cody J Dillon S Buckney MJ Moriarty JG Armstrong 2002 Accelerated radiation therapy, seven fractions per week, for advanced head and neck cancer-a feasibility study Clin Oncol (R Coll Radiol) 14 236 240
64. J Overaaard HS Hansen W Sapru M Overgaard C Grau K Jorgensen L Bastholt O Hansen L Specht A Berthelsen M Pedersen 1996 Conventional radiotherapy as the primary treatment of squamous cell carcinoma of the head and neck. a randomized multicentre study of 5 versus 6 fractions per week-preliminary report from the dahanca 6 and 7 trial Radiother Oncol 40 S31
65. L Peters KK Ang HD Thames 1988 Accelerated fractionation in the radiation treatment of head and neck cancer: a critical comparison of different strategies Acta Oncol 27 185 194
66. Porter EH (1980) The statistics of dose-cure relationships for irradiated tumors. Part I and II. Br J Radiol 53:210-227, 336-345
67. B Ribba T Cloin S Schnell 2006 A multiscale model of cancer, and its use in analyzing irradiation therapies Theor Biol Med Mod 3 1 19
68. RK Sachs DJ Brenner 1998 The mechanistic basis of the linear-quadratic model Med Phys 25 2071 2073
69. RK Sachs P Hahnfeld DJ Brenner 1997 The link between low-let dose-response relations and the underlying kinetics of damage production/repair/misrepair Int J Radiat Biol 72 351 374
70. RK Sachs LR Hlatky P Hahnfeldt 2001 Simple ode models of tumour growth and anti-angiogenic or radiation treatment Mathl Comput Model 33 1297 1305
71. MI Saunders AD Barrett 1997 Continuous hyperfractionated accelerated radiotherapy (chart) versus conventional radiotherapy in non-small cell lung cancer: a randomized multicenter trial Lancet 350 161 165
72. MI Saunders AD Barrett MK Pamar A Harvey A Gibson 1996 Randomized multicentre trials of chart v conventional radiotherapy in head and neck and non-small cell lung cancer Br J Cancer 73 1455 1462
73. E Sham RE Durand 1998 Cell kinetics and repopulation during multifraction irradiation of spheriods Radiother Oncol 46 201 207
74. WK Sinclair 1966 The shape of radiation survival curves of mammalain cells cultured in vitro Biophys Asp Radiat Qual Int At Energy Agency Tech Rep Ser 58 21 43
75. JA Spratt DDvon Fournier JS Spratt EE Weber 1993 Decelerating growth and human breast cancer Cancer 71 2013 2019
76. P Stavrev MWB Warkentin N Stavreva BG Fallone 2003 Radiation damage, repopulation and cell recovery analysis of in vitro tumour cell megacolony culture data using a non-poissonian cell repopulation tcp model Phys Med Biol 50 3053 3061
77. Steel GG (1977). Growth kinetics of tumours. Clarendon Press, Oxford
78. Steel GG (ed) (2002). Basic Clinical Radiobiology, 3rd edn. Arnold, London
79. GG Steel TJ McMillan JH Peacock 1989 The 5rs of radiobiology Int J Radiat Biol 56 1045 1048
80. A Swierniak A Polanski M Kimmel 1996 Optimal control problems arising in cell-cycle-specific cancer chemotherapy Cell Prolif 29 117 139
81. HD Thames SM Bentzen I Turesson M Overgaard WV den Bogaert 1990 Time-dose factors in radiotherapy: a review of the human data Radiother Oncol 19 219 235
82. Thames HD and Hendry JH (1987). Fractionation in radiotherapy. Taylor and Francis, London, 279
83. CA Tobias 1985 The repair-misrepair model in radiobiology, comparison to other models Radiat Res 8 S77 S95
84. WA Tomé J Fowler 2003 On the inclusion of proliferation in tumour control probability calculations for inhomogeneously irradiated tumours Phys Med Biol 48 N261 N268
85. EL Travis SL Tucker 1987 Isoeffect models and fractionated radiation therapy Int J Radiat Biol 13 283 287
86. SL Tucker JMG Taylor 1996 Improved models of tumor cure Int J Radiol Biol 70 539 553
87. SL Tucker HD Thames JMG Taylor 1990 How well is the probability of tumor cure after fractionated irradiation described by poisson statistics? Radiat Res 124 273 282
88. JR Usher 1980 Mathematical derivation of optimal uniform treatment schedule for the fractionated irradiation of human tumours Math Biosc 49 157 184
89. CC Wang 1988 Local control of oropharyngeal carcinoma after two accelerated hyperfractionated radiation therapy schemes Int J Radiat Oncol Biol Phys 14 1143 1146
90. S Webb 1994 Optimum parameters in a model for tumour control probability including interpatient heterogeneity Phys Med Biol 39 1895 1914
91. S Webb AE Nahum 1993 A model for calculating tumour control probability in radiotherapy including the effects of inhomogeneous distributions of dose and clonogenic cell density Phys Med Biol 38 653 666
92. LM Wein JE Cohen JT Wu 2000 Dynamic optimization of a linear-quadratic model with incomplete repair and volume-dependent sensitivity and repopulation Int J Radiat Biol Phys 47 1073 1083
93. EG Wheldon KA Lindsay TE Wheldon 2000 The dose-response relationship for cancer incidence in a two-stage radiation carcinogenesis model incorporating cellular repopulation Int J Radiat Biol 76 699 710
94. TE Wheldon AE Amin 1988 The linear-quadratic model Br J Radiol 61 700 702
95. TE Wheldon C Deehan EG Wheldon A Barrett 1998 The linear quadratic transformation of dose-volume histograms in fractionated radiotherapy Radiother Oncol 46 285 295
96. TE Wheldon J Kirk JS Orr 1977 Optimal radiotherapy of tumour cells following exponential-quadratic survival curves and exponential repopulation kinetics Br J Radiol 50 681 682
97. HR Withers 1975 The four r's of radiotherapy Adv Radiat Biol 5 241 247
98. Withers HR (1988). Some changes in concepts of dose fractionation over 20 years. In: Vaeth, JM and Meyer, J (eds) Time, dose and Fractionation in the Radiation Therapy of Cancer Frontiers of Radiation Therapy and Oncology, vol. 22, pp 1-13. Karger, Basel
99. HR Withers HD Thames LJ Peters 1983 A new isoeffect curve for change in dose per fraction Radiother Oncol 1 187 191
100. BG Woulters JM Brown 1997 Cells at intermediate oxygen levels can be more important than the hypoxic fraction in determining tumour response to fractionated radiotherapy Radiat Res 147 541 550
101. CR Wratten MG Poulsen S Williamson Tripcony J Keller G Dickie 2002 Effect of surgery on normal tissue toxicity in patients treated with accelerated radiotherapy Acta Oncol 41 56 62
102. AY Yakovlev 1993 Comments on the distribution of clonogens in irradiated tumors Radiat Res 134 117 120
103. M Zaider GN Minerbo 2000 Tumour control probability: a formulation applicable to any temporal protocol of dose delivery Phys Med Biol 45 279 293
104. M Zaider CS Wuu GN Minerbo 1996 The combined effects of sublethal damage repair, cellular repopulation and redistribution in the mitotic cycle. i survival probabilities after exposure to radiation Radiat Res 145 457 466