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The use of the linear quadratic model in radiotherapy: a review

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Abstract

To be able to predict the impact of any radiotherapy treatment the physics of radiation interactions and the expected biological effect for any radiotherapy treatment situation (dose, fractionation, modality) must be both understood and modelled. This review considers the current use and accuracy of the linear quadratic model which can be used to consider the variation in tissue response with fraction size. Cell kill following radiation damage results from damage to the DNA which can take a variety of forms. In many cases the linear quadratic model is used to estimate the relative impact for different situations especially clinical studies relating to fraction size. This is mainly undertaken using parameters derived from the linear quadratic model such as biological effective dose and standard effective dose. The model has also been adapted to consider the effect of overall treatment time, repair during treatment (as occurs for brachytherapy treatments) and other situations. There are some concerns over its use, mainly in the small dose ranges (both total low doses and low doses per fraction) where studies have shown its inaccuracy. In other situations however it does appear to provide a reasonable estimate of relative clinical effect. As with all models, however results should never be considered out of clinical context.

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References

  1. Freund, L.,Nachtrag zu den Artikel ‘Ein rnit Rontgen-Strahlen behandelter Fall von Naevus pigmentosus pilijkrus“ Wiener Medizinische Wochenschrift 47, 856–60, 1897.

    Google Scholar 

  2. Freund, L.,Ein rnit Rontgen-Strahlen behandelter Fall von Naevus pigmentosus piliferus Wiener Medizinische Wochenschrift 47, 428–34, 1897.

    Google Scholar 

  3. Milan, J., Bentley, R. E.,The storage and manipulation of radiation dose data in a small digital computer Br J Radiol 47, 115–21, 1974.

    Article  CAS  PubMed  Google Scholar 

  4. Mohan, R., Chui, C., Lidofsky, L.,Dfferential pencil beam dose computation model for photons Med Phys 13, 64–73, 1986.

    Article  CAS  PubMed  Google Scholar 

  5. Mackie, T. R., Scrimger, J. W., Battista, J. J.,A convolution method of calculating dose for 15-W X rays Med Phys 12, 188–96, 1985.

    Article  CAS  PubMed  Google Scholar 

  6. Ahnesjo, A.,Collapsed cone convolution of radiant energy for photon dose calculation in heterogeneous media Med Phys 16, 577–92, 1989.

    Article  CAS  PubMed  Google Scholar 

  7. Brown, M.,Tumor radiosensitivity: It’s the subpopulations that count. Int. J. Radiation 0ncolo & Biol. Phys. 47, 549–550, 2000.

    Article  CAS  Google Scholar 

  8. Adams, G. E., inThe biological Basis of Radiotherapy G. G. Steel, G. E. Adams, A. Horwich, Eds. (Elsevier, Amsterdam) pp. 1–13, 1989.

    Google Scholar 

  9. Hall, E. J.,Radiobiology for the radiologist (J. B. Lippincott Company, Philadelphia, ed. 4th), 1994.

    Google Scholar 

  10. Steel, G. G., Ed.,Basic Clinical Radiobiology (Amold, London), 1997.

    Google Scholar 

  11. Benjamin, I., Bruchak, K., Hampshire, M, Buhle, E. L.,(Universiiy of Pennsylvania Cancer Center, 1999), vol. 1999,

  12. Levine, A. J., Peny, M. E., Chang, A., Silver, A., Dittmer, D., Wu, M., Welsh, D.,The 1993 Palter Hubert Lecture: the role of the p53 tumour-suppressor gene in tumorigenesis Br J Cancer 69, 409–16, 1994.

    CAS  PubMed  Google Scholar 

  13. Puck, T. T., Marcus, P. I.,Action of X-rays on mammalian Cells J Exp Med 103, 653–666, 1956.

    Article  CAS  PubMed  Google Scholar 

  14. Thames, H. D., Hendry, J. H.,Fractionation in radiotherapy (Taylor and Francis Ltd., London, ed. 1st), 1987.

    Google Scholar 

  15. Ellis, F., inModem Trends in Radiotherapy Deeley, Wood, Eds. (Butterworth, London), vol. 1, pp. 34–51, 1967.

    Google Scholar 

  16. Orton, C. G., Ellis, F.,A simplification in the use of the NSD concept in practical radiotherapy Br J Radiol 46, 529–37, 1973.

    Article  CAS  PubMed  Google Scholar 

  17. Liversage, W. E.,A critical look at the ret Br J Radiol 44, 91–100, 1971.

    Article  CAS  PubMed  Google Scholar 

  18. Fowler, J. F.,Experimental animal results relating to time-dose relationships in radiotherapy and te “ret” concept Br J Radiol 44, 81–90, 1971.

    Article  CAS  PubMed  Google Scholar 

  19. Curtis, S. B.,Lethal andpotentially lethal lesions induced by radiation-a unz3ed repair model (erratum Radiat Res 1989 119:584) Radiat Res 106, 252–70, 1986.

    Article  CAS  PubMed  Google Scholar 

  20. Steel, G. G., McMillan, T. J., Peacock, J. H.,The 5 R’s of radiotherapy Int J Radiat Oncol Biol Phys 56, 1056–1048, 1989.

    Google Scholar 

  21. Lea, D. E., Catcheside, D. G.,The mechansim of the induction by radiation of chromosome aberrations in tradescantia. J. Genet. 44, 216–45, 1942.

    Article  Google Scholar 

  22. Brenner, D. J., Hall, E. J.,The origins and basis of the linearquadratic model (letter) Int J Radiat Oncol Biol Phys 23, 252–3, 1992.

    CAS  PubMed  Google Scholar 

  23. Gilbert, C. W., Hendry, J. H., Major, D.,The approximation in the formulation for surivial S= exp−(AD+BDA∧2) Int. J. Radiat. Biol. 37, 469–71, 1980.

    Article  CAS  Google Scholar 

  24. Barendsen, G. W.,Dosefractionation, dose rate and iso-effect relationships for normal tissue responses Int J Radiat Oncol Biol Phys 8, 1981–97, 1982.

    CAS  PubMed  Google Scholar 

  25. Fowler, J. F.,The linear-quadratic formula and progress in fractionated radiotherapy Br J Radiol 62, 679–94, 1989.

    Article  CAS  PubMed  Google Scholar 

  26. Dale, R. G.,The 1989 James Kirk memorial lecture. The potential for radiobiological modelling in radiotherapy treatment design Radiother Oncol 19, 245–55, 1990.

    Article  CAS  PubMed  Google Scholar 

  27. Brenner, D. J., Hlatky, L. R., Hahnfeldt, P. J., Huang, Y., Sachs, R. K.,The linear-quadratic model and most other common radiobiological models result in similar predictions of time-dose relationships Radiat Res 150, 83–91, 1998.

    Article  CAS  PubMed  Google Scholar 

  28. Brenner, D. J.,The use of the linear-quadratic model in clinical radiation oncology can be defended on the basis of empirical evidence and theoretical argument Med Phys 24, 1245–8, 1997.

    Article  CAS  PubMed  Google Scholar 

  29. Herbert, D. E.,The use of the linear-quadratic model in clinical radiation oncology can be defended on the basis of empirical evidence and theoretical argument Med Phys 24, 1245–8, 1997.

    Article  PubMed  Google Scholar 

  30. Scott, O. C.,Divergence of opinion over the linear-quadratic model (letter) Br J Radiol 60, 1042–3, 1987.

    Article  CAS  PubMed  Google Scholar 

  31. Wheldon, T. E., Amin, A. E.,The linear-quadratic model (letter) Br J Radiol 61, 700–2, 1988.

    Article  Google Scholar 

  32. O’Donoghue, J. A.,The linear-quadratic model (letter) Br J Radiol 61, 700, 1988.

    Article  Google Scholar 

  33. Yaes, R. J.,The linear-quadratic model (letter) Br J Radiol 61, 703–704, 1988.

    Article  Google Scholar 

  34. Withers, H. R.,The linear-quadratic model (letter) Br J Radiol 61, 705–707, 1988.

    Article  Google Scholar 

  35. Fowler, J. F.,The linear-quadratic model (letter) Br J Radiol 61, 704–705, 1988.

    Article  Google Scholar 

  36. Zaider, M.,There is no mechanistic basis for the use of the linear-quadratic expression in cellular survival analysis (letter) Med Phys 25, 791–2, 1998.

    Article  CAS  PubMed  Google Scholar 

  37. Sanchez-Reyes, A.,A simple model of radiation action in cells based on a repair saturation mechanism Radiat Res 130, 139–47, 1992.

    Article  CAS  PubMed  Google Scholar 

  38. Sachs, R. K., Brenner, D. J.,The mechanistic basis of the linear-quadratic formalism (letter) Med Phys 25, 2071–3, 1998.

    Article  CAS  PubMed  Google Scholar 

  39. Travis, E. L., Parkins, C., Down, J., Fowler, J. F., Maughan, R.,Is there a loss of repair capaciy in mouse lungs with increasing numbers of dose fractions? Int. J. radiat. Oncol. Biol. Phys, 9, 691–699, 1983.

    CAS  PubMed  Google Scholar 

  40. Haston, C. K., Hill, R. P., Newcomb, C. H., Van Dyk, J.,Radiation-induced lung damage in rats: the influence of fraction spacing on effect perfraction Int J Radiat Oncol Biol Phys 28, 633–40, 1994.

    CAS  PubMed  Google Scholar 

  41. Ang, K. K., van der Kogel, A. J., van der Schueren, E.,Lack of evidence for increased tolerance of rat spinal cord with decreasingfraction doses below 2 Gy Int J Radiat Oncol Biol Phys 11, 105–10, 1985.

    CAS  PubMed  Google Scholar 

  42. Van der Schueren, E., Landuyt, W., Ang, K. K., van der Kogel, A. J.,From 2 Gy to 2 Gyperfraction: sparing effect in rat spinal cord? Int J of RadiatOncol Biol Phys 14, 297–300, 1988.

    Article  Google Scholar 

  43. Thames, H. D., Ang, K. K., Stewart, F. A., van der Schueren, E.,Does incomplete repair explain the apparent failure of the basic LQ model to predict spinal cord and kidney responses to low dosesperfraction? Int J Radiat Biol 54, 13–9, 1988.

    Article  CAS  PubMed  Google Scholar 

  44. Wong, C. S., Minkin, S., Hill, R. P.,Linear-quadratic model underestimates sparing effect of small doses per fraction in rat spinal cord Radiother Oncol 23, 176–84, 1992.

    Article  CAS  PubMed  Google Scholar 

  45. Wong, C. S., Minkin, S., Hill, R. P.,Effect of small doses per fraction in rat spinal cord: influence of initial vs. final top-up doses Radiother Oncol 28, 52–6, 1993.

    Article  CAS  PubMed  Google Scholar 

  46. Parkins, C. S., Fowler, J. F.,The linear quadraticfit for lung function after irradiation with X-rays at smaller doses per jraction than 2 Gy Br J Cancer Suppl 7, 320–3, 1986.

    CAS  PubMed  Google Scholar 

  47. Hamilton, C. S., Denham, J. W., O’Brien, M., Ostwald, P., Kron, T., Wright, S., Drr, W.,Underprediction of human skin eythema at low doses per fraction by the linear quadratic model Radiother Oncol 40, 23–30, 1996.

    Article  CAS  PubMed  Google Scholar 

  48. Taylor, J. M., Withers, H. R., Hu, Z.,A comparison of mathematical models for regeneration in acutely responding tissues Int J Radiat Oncol Biol Phys 15, 1389–400, 1988.

    CAS  PubMed  Google Scholar 

  49. Box, G. P. E.,, B. R. Paliwal, Ed.,Quoted in 2nd International conference on dose, time and fractionation on radiation oncology. (American Insitute of Physics), 1984.

  50. van de Geijn, J.,Incorporating the time factor into the linearquadratic model (letter) Br J Radiol 62, 296–8, 1989.

    Article  PubMed  Google Scholar 

  51. Dale, R. G.,Time-dependent tumour repopulation factors in linear-quadratic equations—implications for treatment strategies Radiother Oncol 15, 371–81, 1989.

    Article  CAS  PubMed  Google Scholar 

  52. Lee, S. P., Leu, M. Y., Smathers, J. B., McBride, W. H., Parker, R. G., Withers, H. R.,Biologically effective dose distribution based on the linear quadratic model and its clinical relevance Int J Radiat Oncol Biol Phys 33, 375–89, 1995.

    CAS  PubMed  Google Scholar 

  53. Withers, H. R., Taylor, J. M., Maciejewski, B.,The hazard of accelerated tumor clonogen repopulation during radiotherapy Acta Oncol 27, 131–46, 1988.

    Article  CAS  PubMed  Google Scholar 

  54. O’Donoghue, J. A.,The response of tumours with Gompertzian growth characteristics to fiactionated radiotherapy Int J Radiat Biol 72, 325–39, 1997.

    Article  PubMed  Google Scholar 

  55. Newcomb, C. H., Van Dyk, J., Hill, R. P.,Evaluation of isoeffect formulae for predicting radiation-induced lung damage Radiother Oncol 26, 51–63, 1993.

    Article  CAS  PubMed  Google Scholar 

  56. Roberts, S. A., Hendry, J. H.,The delay before onset of accelerated tumour cell repopulation during radiotherapy: a direct maximum-likelihood analysis of a collection of worldwide tumour-control data Radiother Oncol 29, 69–74, 1993.

    Article  CAS  PubMed  Google Scholar 

  57. Kajanti, M., Kaleta, R., Kankaamanta, L., Muhonen, T., Holsti, L.,Effect of overall treatment time on local control in radical radiotherapy for squamous cell carcinoma of esophagus Int J Radiat Oncol Biol Phys 32, 1017–23, 1995.

    CAS  PubMed  Google Scholar 

  58. Peter, B., Van Waarde, M. A., Vissink, A., s-Gravenmade, E. J., Konings, A. W.,Radiation-induced cellproliferation in the parotid and submandibular glands of the rat Radiat Res 140, 257–65, 1994.

    Article  CAS  PubMed  Google Scholar 

  59. Tsunemoto, H., Ando, K., Koike, S., Urano, M.,Repopulation of tumour cells following irradiation with X-rays or low energy neutrons Int J Radiat Biol 65, 255–61, 1994.

    Article  CAS  PubMed  Google Scholar 

  60. Saunders, M. I., Dische, S., Fowler, J. F., Denekamp, J., Dunphy, E. P., Grosch, E., Fermont, D., Ashford, R., Maher, J., Des Rochers, C.,Radiotherapy employing three fractions on each of twelve consecutive days Acta Oncol 27, 163–7, 1988.

    Article  CAS  PubMed  Google Scholar 

  61. Horiot, J. C., Bontemps, P., van den Bogaert, W., Le Fur, R., van den Weijngaert, D., Bolla, M., Bernier, J., Lusinchi, A., Stuschke, M., Lopez-Torrecilla, J., Begg, A. C., Pierart, M., Collette, L.,Accelerated fvactionation (AF) compared to conventional fractionation (CF) improves loco-regional control in the radiotherapy of advanced head and neck cancers: results of the EORTC 22851 randornized trial Radiother Oncol 44, 111–21, 1997.

    Article  CAS  PubMed  Google Scholar 

  62. Mills, J. A., Wayte, S. C.,Use of the linear quadratic model in order to accommodate a small reduction in the number of fractions of a standard radiotherapy treatment regime Br J Radiol 66, 447–51, 1993.

    Article  CAS  PubMed  Google Scholar 

  63. McGinn, C. J., Harari, P. M., Fowler, J. F., Ford, C. N., Pyle, G. M., Kinsella, T. J.,Dose intensifcation in curative head and neck cancer radiotherapy-linear quadratic analysis and preliminary assessment of clinical resulfs Int J Radiat Oncol Biol Phys 27, 363–9, 1993.

    CAS  PubMed  Google Scholar 

  64. Yaes, R. J.,Linear-quadratic model isoeffect relations for proliferating tumor cells for treatment with multiple fractions per day Int J Radiat Oncol Biol Phys 17, 901–5, 1989.

    CAS  PubMed  Google Scholar 

  65. Wong, W. W., Mick, R., Haraf, D. J., Weichselbaum, R. R., Vokes, E. E.,Time-dose relationship for local tumor control following alternate week concomitant radiation and chemotherapy of advanced head and neck cancer Int J Radiat Oncol Biol Phys 29, 153–62, 1994.

    CAS  PubMed  Google Scholar 

  66. Robertson, A. G., Robertson, C., Perone, C., Clarke, K., Dewar, J., Elia, M. H., Hurman, D., MacDougall, R. H., Yosef, H. M.,Effect of gap length andposition on results of treatment of cancer of the larynx in Scotland by radiotherapy: a linear quadratic analysis Radiother Oncol 48, 165–73, 1998.

    Article  CAS  PubMed  Google Scholar 

  67. Hendry, J. H., Bentzen, S. M., Dale, R.G., Fowler, J. F., Wheldon, T. E., Jones, B., Munro, A. J., Slevin, N. J., Robertson, A. G.,A modelled comparison of the effects of using dyfferent ways to compensate for missed treatment days in radiotherapy Clin Oncol 8, 297–307, 1996.

    Article  CAS  Google Scholar 

  68. Jones, L., Metcalfe, P., Hoban, P.,Accounting for treatment delays when treating highly prolifeative tumours Phys Med Bio 144, 223–34, 1999.

    Article  Google Scholar 

  69. Dale, R. G.,The application of the linear-quadratic doseeffect equation tofractionated andprotracted radiotherapy Br J Radio1 58, 515–28, 1985.

    Article  Google Scholar 

  70. Dale, R. G.,The application of the linear-quadratic model to fractionated radiotherapy when there is incomplete normal tissue recovery between fractions, and possible implications for treatments involving multiple fractions per day ( erratum Br J Radiol 1986 59: 1151) Br J Radiol 59, 919–27, 1986.

    Article  CAS  PubMed  Google Scholar 

  71. Chen, J., van de Geijn, J., Goflinan, T.,Extra lethal damage due to residual incompletely repaired sublethal damage in hyperfractionated and continuous radiation treatment Med Phys 18, 488–96, 1991.

    Article  CAS  PubMed  Google Scholar 

  72. Thames, H. D.,An ‘incomplete-repair‘ model for survival after fractionated and continuous irradiations Int J Radiat Biol Relat Stud Phys Chem Med 47, 319–39, 1985.

    Article  CAS  PubMed  Google Scholar 

  73. Nilsson, P., Thames, H. D., Joiner, M. C.,A generalized formulation of the ‘incomplete-repair’ model for cell survival and tissue response to fractionated low dose-rate irradiation Int J Radiat Biol 57, 127–42, 1990.

    Article  CAS  PubMed  Google Scholar 

  74. Rutz, H. P., Coucke, P. A., Mirimanoff, R. O.,A linearquadratic model of cell survival considering both sublethal and potentially lethal radiation damage Radiother Oncol 21, 273–6, 1991.

    Article  CAS  PubMed  Google Scholar 

  75. Fowler, J. F.,‘Fading times’ required for apparently complete repair in irradiated tissues assuming the linear quadratic model of dose response Int J Radiat Biol Relat Stud Phys Chem Med 50, 601–7, 1986.

    Article  CAS  PubMed  Google Scholar 

  76. Fowler, J. F.,Intervals between multiple fractions per day. Differences between early and late radiation reactions Acta Oncol 27, 181–3, 1988.

    Article  CAS  PubMed  Google Scholar 

  77. Jones, B., Dale, R. G., Bleasdale, C., Tan, L. T., Davies, M.,A mathematical model of intraluminal and intracavitary brachytherapy Br J Radiol 67, 805–12, 1994.

    Article  CAS  PubMed  Google Scholar 

  78. Dale, R. G., Jones, B., Coles, I. P.,Effect of tumour shrinkage on the biological effectiveness of permanent brachytherapy implants Br J Radiol 67, 63945, 1994.

    Google Scholar 

  79. Jones, B., Bleasdale, C.,Effect of overall time when radiotherapy includes teletherapy and brachytherapy: a mathematical model Br J Radiol 67, 63–70, 1994.

    Article  CAS  PubMed  Google Scholar 

  80. Leborgne, F., Fowler, J. F., Leborgne, J. H., Zubizarreta, E., Chappell, R.,Fractionation in medium dose rate brachytherapy of cancer of the cervix Int J Radiat Oncol Biol Phys 35, 907–14, 1996.

    CAS  PubMed  Google Scholar 

  81. Millar, W. T., Hendry, J. H., Canney, P. A.,The influence of the number offractions and bi-exponential repair kinetics on biological equivalence in pulsed brachytherapy Br J Radiol 69, 457–68, 1996.

    Article  CAS  PubMed  Google Scholar 

  82. Visser, A. G., van den Aardweg, G. J., Levendag, P. C.,Pulsed dose rate and fractionated high dose rate brachytherapy: choice of brachytherapy schedules to replace low dose rate treatments Int J Radiat Oncol Biol Phys 34, 497–505, 1996.

    CAS  PubMed  Google Scholar 

  83. Sarkaria, J. N., Petereit, D. G., Stitt, J. A., Hartman, T., Chappell, R., Thomadsen, B. R., Buchler, D. A., Fowler, J. F., Kinsella, T. J.,A comparison of the efficacy and complication rates of low dose-rate versus high dose-rate brachytherapy in the treatment of uterine cervical carcinoma Int J Radiat Oncol Biol Phys 30, 75–82, 1994.

    CAS  PubMed  Google Scholar 

  84. Hamilton, C. S., Sirnpson, S. A., Ferguson, S., Ostwald, P., Hsu, W., O’Brien, M., Denham, J. W.,Low dose rate teletherapy and tumour response Australas Radiol 37, 210–2, 1993.

    Article  CAS  PubMed  Google Scholar 

  85. Jones, B., Dale, R. G.,Cell loss factors and the linearquadratic model Radiother Oncol 37, 136–9, 1995.

    Article  CAS  PubMed  Google Scholar 

  86. Fowler, J. F., Lindstrom, M. J.,Loss of local control with prolongation in radiotherapy Int J Radiat Oncol Biol Phys 23, 457–67, 1992.

    CAS  PubMed  Google Scholar 

  87. Hendry, J. H., Potten, C. S.,Intestinal cell radiosensitivity: A comparison for cell death assayed by apoptosis or by a loss of clonogenicity Int J Radiat Oncol Biol Phys 42, 621–628, 1982.

    CAS  Google Scholar 

  88. Ling, C. C., Chen, C. H., Fuks, Z.,An equation for the dose response of radiation-induced apoptosis: possible incorporation with the LQ model Radiother Oncol 33, 17–22, 1994.

    Article  CAS  PubMed  Google Scholar 

  89. Olsen, D. R.,Calculation of the biological effect of fractionated radiotherapy: the importance of radiationinduced apoptosis Br J Radiol 68, 1230–6, 1995.

    Article  CAS  PubMed  Google Scholar 

  90. Withers, H. R.,The four R’s of radiotherapy Adv m Radiat Bio 15, 241–7, 1975.

    Google Scholar 

  91. Brenner, D. J., Hlatky, L. R., Hahnfeldt, P. J., Hall, E. J., Sachs, R. K.,A convenient extension of the linear-quadratic model to include redistribution and reoygenation Int J Radiat Oncol Biol Phys 32, 379–90, 1995.

    Article  CAS  PubMed  Google Scholar 

  92. Turesson, I.,Individual variation and odse dependency in the progression rate of skin telegiectasia Int. J. Radiat. Oncol. Biol. Phys. 19, 1569–1574, 1990.

    CAS  PubMed  Google Scholar 

  93. Thames, H. D., Jr., Withers, H. R., Peters, L. J., Fletcher, G. H.,Changes in early and late radiation responses with altered dose fractionation: implications for dose-survival relationships Int J Radiat Oncol Biol Phys 8, 219–26, 1982.

    PubMed  Google Scholar 

  94. Fowler, J. F.,Review: total doses in fractionated radiotherapy—implications of new radiobiological data Int J Radiat Biol Relat Stud Phys Chem Med 46, 103–20, 1984.

    Article  CAS  PubMed  Google Scholar 

  95. Douglas,-B. G., Fowler, J. F.,Radiat Res 66, 401, 1976.

    Article  CAS  PubMed  Google Scholar 

  96. Williams, M. V., Denekamp, J., Fowler, J. F.,A review of alpha/beta ratios for experimental tumors: implications for clinical studies of altered fractionation Int J Radiat Oncol Biol Phys 11, 87–96, 1985.

    CAS  PubMed  Google Scholar 

  97. Tucker, S. L.,Tests for thefit of the linear-quadratic model to radiation isoefect data Int J Radiat Oncol Biol Phys 10, 1933–9, 1984.

    CAS  PubMed  Google Scholar 

  98. de Boer, R. W.,The use of the D versus dD plot to estimate the alpha/beta ratio from iso-effect radiation damage data Radiother Oncol 11, 361–7, 1988.

    Article  PubMed  Google Scholar 

  99. Chmelevsky, D., Guttenberger, R., Kummermehr, J., Kellerer, A. M.,A nonparametric method for the derivation of alpha/beta ratios from the effect of fractionated irradiations Int J Radiat Biol 59, 1253–68, 1991.

    Article  CAS  PubMed  Google Scholar 

  100. Taylor, J. M., Mason, K. A., Vegesna, V., Withers, H. R.,A non-parametric method of reconstructing single-dose survival curves from multi-fraction experiments Int J Radiat Biol 74, 583–93, 1998.

    Article  CAS  PubMed  Google Scholar 

  101. Chappell, R.,Presenting the coeffients of the linear quadratic formula for clinical use (erratum Int J Radiat Oncol Phys 1994 29:1209) Int J Radiat Oncol Biol Phys 29, 191–3, 1994.

    CAS  PubMed  Google Scholar 

  102. Chappell, R., Nondahl, D. M., Rezvani, M., Fowler, J. F.,Further analysis of radiobiological parameters from the First and Second British Institute of Radiology randomized studies of larynx/pharynx radiotherapy Int J Radiat Oncol Biol Phys 33, 509–18, 1995.

    CAS  PubMed  Google Scholar 

  103. Barton, M. B., Keane, T. J., Gadalla, T., Maki, E.,The effect of treatment time and treatment interruption on tumour control following radical radiotherapy of laryngeal cancer Radiother Oncol 23, 13743, 1992.

    Google Scholar 

  104. Roberts, S. A., Hendry, J. H., Brewster, A. E., Slevin, N. J.,The influence of radiotherapy treatment time on the control of laryngeal cancer: a direct analysis of data from two British Institute of Radiology trials to calculate the lug period and the time factor Br J Radiol 67, 7904, 1994.

    Google Scholar 

  105. Peacock, J. H., Eady, J. J., Edwards, S. M., McMillan, T. J., Steel, G. G.,The intrinsic alpha/beta ratio for human tumour cells: is it a constant? Int J Radiat Biol 61, 479–87, 1992.

    Article  CAS  PubMed  Google Scholar 

  106. Bentzen, S. M., Overgaard, J., Thames, H. D., Overgaard, M., Vejby Hansen, P., von der Maase, H., Meder, J.,Clinical radiobiology of malignant melanoma Radiother Oncol 16, 169–82, 1989.

    Article  CAS  PubMed  Google Scholar 

  107. Bentzen, S. M., Johansen, L. V., Overgaard, J., Thames, H. D.,Clinical radiobiology of squamous cell carcinoma of the oropharynx Int J Radiat Oncol Biol Phys 20, 1197–206, 1991.

    CAS  PubMed  Google Scholar 

  108. Dubray, B., Henry-Amar, M., Meerwaldt, J. H., Noordijk, E. M., Dixon, D. O., Cosset, J. M., Thames, H. D.,Radiationinduced lung damage after thoracic irradiation for Hodgkin’s disease: the role offractionation Radiother Oncol 36, 211–7, 1995.

    Article  CAS  PubMed  Google Scholar 

  109. Maciejewski, B., Withers, H. R., Taylor, J. M., Hliniak, A.,Dose fractionation and regeneration in radiotherapy for cancer of the oral cavity and oropharynx: tumor doseresponse and repopulation Int J Radiat Oncol Biol Phys 16, 831–43, 1989.

    CAS  PubMed  Google Scholar 

  110. Turesson, I., Thames, H. D.,Repair capacity and kinetics of human skin during fractionated radiotherapy: erythema, desquamation, and telangiectasia after 3 and 5 year’s followup Radiother Oncol 15, 169–88, 1989.

    Article  CAS  PubMed  Google Scholar 

  111. Ten Haken, R. K., Fraass, B. A., Lichter, A. S., Marsh, L. H., Radany, E. H., Sandler, H. M.,A brain tumor dose escalation protocol based on effective dose equivalence to prior experience Int J Radiat Oncol Biol Phys 42, 137–41, 1998.

    CAS  PubMed  Google Scholar 

  112. Kwa, S. L., Lebesque, J. V., Theuws, J. C., Marks, L. B., Munley, M. T., Bentel, G., Oetzel, D., Spahn, U., Graham, M. V., Drzymala, R. E., Purdy, J. A., Lichter, A. S., Martel, M. K., Ten Haken, R. K.,Radiationpneumonitis as afinction of mean lung dose: an analysis of pooled data of 540 patients Int J Radiat Oncol Biol Phys 42, 1–9, 1998.

    CAS  PubMed  Google Scholar 

  113. Roberts, S. A., Hendry, J. H., Slevin, N. J.,Modelling the optimal radiotherapy regime for the control of T2 laryngeal carcinoma using parameters derivedfrom several datasets Int J Radiat Oncol Biol Phys 39, 1173–82, 1997.

    CAS  PubMed  Google Scholar 

  114. Dale, R. G., Coles, I. P., Deehan, C., O’Donoghue, J. A.,Calculation of integrated biological response in brachytherapy Int J Radiat Oncol Biol Phys 38, 633–42, 1997.

    CAS  PubMed  Google Scholar 

  115. Dale, R. G.,A graphical method to simplijy the application of the linear-quadratic dose-effect equation to fractionated radiotherapy Br J Radiol 59, 1111–5, 1986.

    Article  CAS  PubMed  Google Scholar 

  116. Clark, B. G., Souhami, L., Pla, C., Al-Amro, A. S., Bahary, J. P., Villemure, J. G., Caron, J. L., Olivier, A., Podgorsak, E. B.,The integral biologically effective dose to predict brain stem toxicity of hypofractionated stereotactic radiotherapy Int J Radiat Oncol Biol Phys 40, 667–75, 1998.

    CAS  PubMed  Google Scholar 

  117. Yaes, R. J., Patel, P., Maruyama, Y.,On using the linearquadratic model in daily clinical practice Int J Radiat Oncol Biol Phys 20, 1353–62, 1991.

    CAS  PubMed  Google Scholar 

  118. Flickinger, J. C., Kalend, A.,Use of normalized total dose to represent the biological effect of fractionated radiotherapy Radiother Oncol 17, 339–47, 1990.

    Article  CAS  PubMed  Google Scholar 

  119. Jereczek-Fossa, B., Jassem, J., Nowak, R., Badzio, A.,Late complications after postoperative radiotherapy in endometrial cancer: analysis of 317 consecutive cases with application of linear-quadratic model Int J Radiat Oncol Biol Phys 41, 329–38, 1998.

    CAS  PubMed  Google Scholar 

  120. Lebesque, J. V., Keus, R. B.,The simultaneous boost technique: the concept of relative normalized total dose Radiother Oncol 22, 45–55, 1991.

    Article  CAS  PubMed  Google Scholar 

  121. Withers, H. R., Thames, H. D., Jr., Peters, L. J.,A new isoeffect curve for change in dose per fraction Radiother Oncol 1, 187–91, 1983.

    Article  CAS  PubMed  Google Scholar 

  122. Wang, C. C., Efirt J. T.,Does prolonged treatment course adversely affect local control of carcinoma of the larynx? Int J Radiat Oncol Biol Phys 29, 657–60, 1994.

    CAS  PubMed  Google Scholar 

  123. Dische, S., Saunders, M., Barrett, A., Harvey, A., Gibson, D., Parmar, M.,A randomised multicentre trial of CHART versus conventional radiotherapy in head and neck cancer Radiother Oncol 44, 123–36, 1997.

    Article  CAS  PubMed  Google Scholar 

  124. Horiot, J. C., Le Fur, R., N’Guyen, T., Chenal, C., Schraub, S., Alfonsi, S., Gardani, G., Van Den Bogaert, W., Danczak, S., Bolla, M., et al.,Hyperfractionation versus conventional fracfionation in oropharyngeal carcinoma: final analysis of a randomized trial of the EORTC cooperative group of radiotherapy Radiother Oncol 25, 231–41, 1992.

    Article  CAS  PubMed  Google Scholar 

  125. Fu, K. K., Cooper, J. S., Marcial, V. A., Laramore, G. E., Pajak, T. F., Jacobs, J., Al-Sarraf, M., Forastiere, A. A., Cox, J. D.,Evolution of the Radiation Therapy Oncology Group clinical trials for head and neck cancer Int J Radiat Oncol Biol Phys 35, 425–38, 1996.

    Article  CAS  PubMed  Google Scholar 

  126. Langer, M., Morrill, S. S., Lane, R.,A test of the claim that plan rankings are determined by relative cornplication and tumor-control probabilities Int J Radiat Oncol Biol Phys 41, 451–7, 1998.

    CAS  PubMed  Google Scholar 

  127. Deehan, C., O’Donoghue, J. A.,Biological equivalence between fractionated radiotherapy treatments using the linear-quadratic model Br J Radiol 61, 1187–8, 1988.

    Article  CAS  PubMed  Google Scholar 

  128. Yaes, R. J., Feola, J., Wienbicki, J., Urano, M., Maruyama, Y.,Biological equivalence between fractionated radiotherapy treatments using the linear-quadratic model (letter) Br J Radiol 62, 873–4, 1989.

    Article  CAS  PubMed  Google Scholar 

  129. Scalliet, P., Cosset, J. M., Wambersie, A.,Application of the LQ model to the interpretation of absorbed dose distribution in the daily practice of radiotherapy Radiother Oncol 22, 180–9, 1991.

    Article  CAS  PubMed  Google Scholar 

  130. Zackrisson, B., Franzen, L., Henriksson, R., Littbrand, B.,Tolerance to acceleratedfractionation in the head and neck region Acta Oncol 33, 391–6, 1994.

    Article  CAS  PubMed  Google Scholar 

  131. Horiot, J. C., van den Bogaert, W., Ang, K. K., al., e., inFrontiers of radiation therapy and oncology J. M. Vaeth, J. L. Meyer, Eds. (Basel, Karger), vol. 22, pp. 149-61, 1988.

  132. Singer, J. M., F’rice, P., Dale, R. G.,Radiobiological prediction of normal tissue toxicities and tumour response in the radiotherapy of advanced non-small-cell lung cancer Br J Cancer 78, 1629–33, 1998.

    CAS  PubMed  Google Scholar 

  133. Fowler, J. F., Ritter, M. A.,A rationale for fractionation for slowly proliferating tumors such as prostatic adenocarcinoma Int J Radiat Oncol Biol Phys 32, 521–9, 1995.

    Article  CAS  PubMed  Google Scholar 

  134. Hall, E. J., Brenner, D. J.,The radiobiology of radiosurgery: rationale for different treatment regimes for AVMs and malignancies Int J Radiat Oncol Biol Phys 25, 381–5, 1993.

    CAS  PubMed  Google Scholar 

  135. Wigg, D. R.,A radiobiological basis for the treatment of arteriovenous malformations Acta Oncol 38,3–299, 1999.

    PubMed  Google Scholar 

  136. Lo, Y. C., Ling, C. C., Larson, D. A.,The effect of setup uncertainties on the radiobiological advantage of fractionation in stereotaxic radiotherapy Int J Radiat Oncol Biol Phys 34, 1113–9, 1996.

    CAS  PubMed  Google Scholar 

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Authors and Affiliations

  1. Department of Radiation Oncology, Prince of Wales Hospital, Randwick, NSW

    P. Hoban

  2. Illawarra Cancer Therapy Centre, Wollongong, NSW

    P. Metcalfe

  3. Department of Radiation Oncology, Liverpool Hospital, Locked Bag 7103, 1871, Liverpool BC, NSW

    L. Jones

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  1. L. Jones
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  2. P. Hoban
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  3. P. Metcalfe
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Correspondence to L. Jones.

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Jones, L., Hoban, P. & Metcalfe, P. The use of the linear quadratic model in radiotherapy: a review. Australas. Phys. Eng. Sci. Med. 24, 132–146 (2001). https://doi.org/10.1007/BF03178355

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