The Limitation of the Linear-Quadratic Model at Low Doses per Fraction

  • M. C. Joiner
  • B. Marples
  • H. Johns
Part of the Medical Radiology book series (MEDRAD)


The Linear-Quadratic (LQ) equation is used widely to model and predict the increase in total dose with decreasing dose per fraction needed for an isoeffective response to radiotherapy in normal tissues and tumours (reviewed by Joiner 1989). It is thought that this relationship reflects the gradual decrease in radiation effectiveness (“per unit dose”) with lowered doses due to these doses being further and further back “on the shoulder” of an underlying survival curve for the cells at risk in tissues. A considerable amount of work has been devoted to defining the operational limits of this model in a range of normal tissues and tumours during the last 10 years although these studies have been largely done on animal models for ethical reasons. What information there is on human dose-fractionation characteristics and repair kinetics does indicate that the animal systems predict the apparent situation in man reasonably accurately (Thames et al. 1990).


Unit Dose Neutron Dose Incomplete Repair Clinical Radiosensitivity Isoeffect Dose 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Beck-Bornholdt HP, Maurer T, Becker S, Omniczynski M, Vogler H, Wurschmidt F (1989) Radiotherapy of the rhabdomyosarcoma R1H of the rat: hyperfractionation— 126 fractions applied within 6 weeks. Int J Radiat Oncol Biol Phys 16: 701–705PubMedCrossRefGoogle Scholar
  2. Boothman DA, Bouvard I, Hughes EN (1989) Identification and characterization of x-ray-induced proteins in human cells. Cancer Res 49: 2871–2878PubMedGoogle Scholar
  3. Cleaver JE (1968) Defective repair replication of DNA in xeroderma pigmentosum. Nature 218: 652–656PubMedCrossRefGoogle Scholar
  4. Crompton NE A, Barth B, Kiefer J (1990) Inverse dose-rate effect for the induction of 6-thioguanine-resistant mutants in Chinese hamster V79-S cells by 60Co ƴ rays. Radiat Res 124: 300–308PubMedCrossRefGoogle Scholar
  5. Denekamp J (1973) Changes in the rate of repopulation during multifraction irradiation of mouse skin. Br J Radiol 46: 381–387PubMedCrossRefGoogle Scholar
  6. Douglas BG, Fowler JF (1976) The effect of multiple small doses of x-rays on skin reactions in the mouse and a basic interpretation. Radiat Res 66: 401–426PubMedCrossRefGoogle Scholar
  7. Folkard M (1986) Determination of d(3)-Be and d(4)-Be neutron spectra using a recoil proton spectrometer. Phys Med Biol 31: 135–144CrossRefGoogle Scholar
  8. Friedberg EC (1985) DNA repair. W.H. Freeman, New YorkGoogle Scholar
  9. Goodhead DT (1989) The initial physical damage produced by ionizing radiations. Int J Radiat Biol 56: 623–634PubMedCrossRefGoogle Scholar
  10. Hopewell JW, van den Aardweg GJMJ (1991) Studies of dose-fractionation on early and late responses in pig skin: a reappraisal of the importance of the overall treatment time and its effects on radiosensitization and incomplete repair. Int J Radiat Oncol Biol Phys 21: 1441–1450PubMedCrossRefGoogle Scholar
  11. Horiot JC, le Fur R, N’Guyen T et al. (1990) Hyperfractionated compared with conventional radiotherapy in oropharyngeal carcinoma: an EORTC randomized trial. Eur J Cancer 26: 779–780PubMedCrossRefGoogle Scholar
  12. Ikushima T (1987) Chromosomal responses to ionizing radiation reminiscent of an adaptive response in cultured Chinese hamster cells. Mutat Res 180: 215–221PubMedCrossRefGoogle Scholar
  13. Ikushima T (1989) Radio-adaptive response: characterization of a cytogenetic repair induced by low-level ionizing radiation in cultured Chinese hamster cells. Mutat Res 227: 241–246PubMedCrossRefGoogle Scholar
  14. Johns H, Joiner MC (1991) A simple method for fitting curves to dose-effect data for functional damage. Int J Radiat Biol 60: 533–541PubMedCrossRefGoogle Scholar
  15. Joiner MC (1987) The design and interpretation of top-up experiments to investigate the effects of low radiation doses. Int J Radiat Biol 51: 115–130CrossRefGoogle Scholar
  16. Joiner MC (1989) The dependence of radiation response on the dose per fraction. In: McNally NJ (ed) The scientific basis for modern radiotherapy (BIR Report 19 ). British Institute of Radiology, London, pp 20–26Google Scholar
  17. Joiner MC, Denekamp J (1986) Evidence for a constant repair capacity over 20 fractions of x-rays. Int J Radiat Biol 49: 143–150CrossRefGoogle Scholar
  18. Joiner MC, Johns H (1988) Renal damage in the mouse: the response to very small doses per fraction. Radiat Res 114: 385–398PubMedCrossRefGoogle Scholar
  19. Joiner MC, Denekamp J, Maughan RL (1986) The use of top-up experiments to investigate the effect of very small doses per fraction in mouse skin. Int J Radiat Biol 49: 565–580CrossRefGoogle Scholar
  20. Joiner MC, Rojas A, Johns H (1989) Does the repair capacity of skin change with repeated exposure to x-rays? Int J Radiat Biol 55: 993–1005PubMedCrossRefGoogle Scholar
  21. Joiner MC, Rojas A, Johns H (1992) A test of equal effect per fraction in the kidney of the mouse. Radiat Res 130: 227–235PubMedCrossRefGoogle Scholar
  22. Koval TM (1984) Multiphasic survival response of a radioresistant lepidopteran insect cell line. Radiat Res 98: 642–648PubMedCrossRefGoogle Scholar
  23. Koval TM (1986) Inducible repair of ionizing radiation damage in higher eukaryotic cells. Mutat Res 173: 291–293PubMedCrossRefGoogle Scholar
  24. Koval TM (1988) Enhanced recovery from ionizing radiation damage in a lepidopteran insect cell line. Radiat Res 115: 413–420PubMedCrossRefGoogle Scholar
  25. Lambin P, Marples B, Fertil B, Malaise EP, Joiner MC (1993) Hypersensitivity of a human tumour cell line to very low radiation doses. Int J Radiat Biol 63: 639–650PubMedCrossRefGoogle Scholar
  26. Liversage WE (1969) A general formula for equating protracted and acute regimes of radiation. Br J Radiol 42: 432–440PubMedCrossRefGoogle Scholar
  27. Marples B, Joiner MC (1993) The response of V79 cells to low radiation doses: evidence of enchanced sensitivity of the whole cell population. Radiat Res 133: 41–51PubMedCrossRefGoogle Scholar
  28. Palcic B, Jaggi B (1986) The use of solid state sensor technology to detect and characterise live mammalian cells growing in tissue culture. Int J Radiat Biol 50: 345–352CrossRefGoogle Scholar
  29. Papathanasiou MA, Kerr NCK, Robbins JH et al. (1991) Induction by ionizing radiation of the gadd45 gene in cultured human cells: lack of mediation by protein kinase C. Mol Cell Biol 11: 1009–1016PubMedGoogle Scholar
  30. Parkins CS, Fowler JF (1985) Repair in mouse lung of multifraction x-rays and neutrons: extension to 40 fractions. Br J Radiol 58: 225–241PubMedCrossRefGoogle Scholar
  31. Parkins CS, Fowler JF (1986) The linear quadratic fit for lung function after irradiation with x-rays at smaller doses per fraction than 2 Gy. Br J Cancer 53 [Suppl VII]: 320–323Google Scholar
  32. Parkins CS, Fowler JF, Maughan RL, Roper MJ (1985) Repair in mouse lung for up to 20 fractions of x-rays or neutrons. Br J Radiol 58: 225–241PubMedCrossRefGoogle Scholar
  33. Powell S, McMillan TJ (1990) DNA damage and repair- following treatment with ionizing radiation. Radiother Oncol 19: 95–108PubMedCrossRefGoogle Scholar
  34. Shadley JD, Wiencke JK (1989) Induction of the adaptive response by x-rays is dependent on radiation intensity. Int J Radiat Biol 56: 107–118PubMedCrossRefGoogle Scholar
  35. Shadley JD, Wolff S (1987) Very low doses of x-rays can cause human lymphocytes to become less susceptible to ionizing radiation. Mutagenesis 2: 95–96PubMedCrossRefGoogle Scholar
  36. Sinclair WK (1972) Cell-cycle dependence of the lethal radiation response in mammalian cells. Curr Top Radiat Res Q 7: 264–285Google Scholar
  37. Spadinger I, Palcic B (1992) The relative biological effectiveness of 60Co ƴ-rays, 55kVp x-rays, 250 kVp x-rays and 11 MeV electrons at low doses. Int J Radiat Biol 61: 345–353PubMedCrossRefGoogle Scholar
  38. Steel GG, Deacon JM, Duchesne GM, Horwich A, Kelland LR, Peacock JH (1987) The dose-rate effect in human tumour cells. Radiother Oncol 9: 299–310PubMedCrossRefGoogle Scholar
  39. Stevens G, Joiner MC, Joiner B, Johns H, Denekamp J (1991) Early detection of damage following bilateral renal irradiation in the mouse. Radiother Oncol 20: 124–131PubMedCrossRefGoogle Scholar
  40. Thames HD, Ang KK, Stewart FA, van der Schueren E (1988) Does incomplete repair explain the apparent failure of the basic LQ model to predict spinal cord and kidney responses to low doses per fraction? Int J Radiat Biol 54: 13–19PubMedCrossRefGoogle Scholar
  41. Thames HD, Bentzen SM, Turesson I, Overgaard M, van den Bogaert W (1990) Time-dose factors in radiotherapy: a review of the human data. Radiother Oncol 19: 219–235PubMedCrossRefGoogle Scholar
  42. Tuschl H, Altmann H, Kovac R, Topaloglou A, Egg D, Gunther R (1980) Effects of low-dose radiation on repair processes in human lymphocytes. Radiat Res 81: 1–9PubMedCrossRefGoogle Scholar
  43. Watts ME, Hodgkiss RJ, Jones NR, Fowler JF (1986) Radio- sensitisation of Chinese hamster cells by oxygen and mis-onidazole at low x-ray doses. Int J Radiat Biol 50: 1009–1021CrossRefGoogle Scholar
  44. Wilson GD, Soranson JA, Lewis AA (1987) Cell kinetics of mouse kidney using bromodeoxyuridine incorporation and flow cytometry: preparation and staining. Cell Tissue Kinet 20: 125–133PubMedGoogle Scholar
  45. Withers HR, Maciejewski B, Taylor JMG (1989) Biology of options in dose fractionation. In: McNally NJ (ed) The scientific basis for modern radiotherapy (BIR Report 19 ). British Institute of Radiology, London, pp 27–36Google Scholar
  46. Wolff S, Wiencke JK, Afzal V, Youngblom J, Cortes F (1989) The adaptive response of human lymphocytes to very low doses of ionizing radiation: a case of induced chromosomal repair with the induction of specific proteins. In: Baverstock KF, Stather JW (eds) Low dose radiation: biological bases of risk assessment. Taylor & Francis, London, pp 446–454Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • M. C. Joiner
    • 1
  • B. Marples
    • 2
  • H. Johns
    • 1
  1. 1.CRC Gray LaboratoryMount Vernon HospitalNorthwood, MiddlesexUK
  2. 2.B.C. Cancer Research CentreVancouverCanada

Personalised recommendations