Some comments on radiobiological models and the consistent Taylor series model.

Auteurs

  • Gustavo Benitez Alvarez Universidade Federal Fluminense
  • Diomar Cesar Lobão Universidade Federal Fluminense

DOI :

https://doi.org/10.56814/pecen.v7i1.1987

Résumé

Here it is mathematically shown that the Linear Quadratic model is insuffcient to adequately describe the survival curve of some cell lines, and that the β parameter of this model is dependent on the dose range used for curve fitting. Therefore, higher-order polynomials are needed to have a single formula that describes the survival of all cell lines at all dose ranges. Based on the Taylor series and two mathematical hypotheses, it is possible to show that the free parameters are dependent on each other. A new approach is proposed to eliminate this interdependence between the free parameters of the Taylor series expansion. The available experimental data on survival curves indicate that there are at least three different behaviors. The theoretical analysis is tested for these three different behaviors, including also five known models not based on Taylor series. Based on experimental cell survival data it is possible to generate charts on the isoeffective total dose in fractionation. A comparative study on the performance of each model in different fractionation schemes is carried out. Experimental data show that fractionation in the low and medium dose ranges can present a non-monotonic behavior, and that most models not based on the Taylor series are unable to reproduce this behavior. Furthermore, to reproduce this nonmonotonic behavior it is necessary to use polynomials of order greater than five. Finally, it is shown that for some cell lines, hyperfractionation presents a considerable therapeutic gain when compared to conventional fractionation, since there are cases in which the isoeffective total dose in hyperfractionation is much lower than the total dose in conventional fractionation.

Biographie de l'auteur

  • Gustavo Benitez Alvarez, Universidade Federal Fluminense
    Universidade Federal Fluminense Centro Tecnológico Escola de Engenharia Industrial Metalúrgica de Volta Redonda Avenida dos Trabalhadores 420, Vila Santa Cecilia, Volta Redonda, Rio de Janeiro, Brasil. CEP 27255-250. Áreas: Física/Matemática/Engenharia Currículo Lattes

Références

Andisheh B., Edgren M., Belkić D., Mavroidis P., Brahme A. & Lind B.K. (2013) A comparative analysis of radiobiological models for cell surviving fractions at high doses. Technol. Cancer Res. Treat., 12: 183–192. doi:10.7785/tcrt.2012.500306.

Apostol T.M. (1967) Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. New York, NY: John WiJey & Sons.

Belkić D. & Belkić K. (2011) Padé–Froissart exact signal-noise separation in nuclear magnetic resonance spectroscopy. J. Phys. B: At. Mol. Opt. Phys., 44: 125003. doi: 10.1088/0953-4075/44/12/125003.

Bender M.A. & Gooch P.C. (1962) The kinetics of X-ray survival of mammalian cells in vitro. Int. J. Radiat. Biol., 5: 133–145. doi:10.1080/09553006214550651.

Burden R.L., Faires J.D. & Burden A.M. (2016) Numerical Analysis. Boston, MA: Cengage Learning. Carmichael J., Degraff W.G., Gamson J., Russo D., Gazdar A.F., Levitt M.L., Minna J.D. & Mitchell J.B. (1989) Radiation sensitivity of human lung cancer cell lines. Eur J Cancer Clin Oncol., 25: 527–534. doi:10.1016/0277-5379(89)90266-6.

Ekstrand K.E. (2010) The Hug–Kellerer equation as the universal cell survival curve. Phys. Med. Biol., 55: 267–273. doi:10.1088/0031-9155/55/10/n01.

Fernandez-Palomo C., Seymour C. & Mothersill C. (2016) Inter-Relationship between Low-Dose Hyper-Radiosensitivity and Radiation-Induced Bystander Effects in the Human T98G Glioma and the Epithelial HaCaT Cell Line. Radiat Res., 185: 124–133. doi:10.1667/RR14208.1.

Garcia L.M., Leblanc J., Wilkins D. & Raaphorst G.P. (2006) Fitting the linear–quadratic model to detailed data sets for different dose ranges. Phys. Med. Biol., 51: 2813–2823. doi:10.1088/0031-9155/51/11/009.

Garcia L.M., Wilkins D.E. & Raaphorst G.P. (2007) α/β ratio: A dose range dependence study. Int. J. Radiat. Oncol. Biol. Phys., 67: 587–593. doi:10.1016/j.ijrobp.2006.10.017.

Guerrero M. & Li X.A. (2004) Extending the linear–quadratic model for large fraction doses pertinent to stereotactic radiotherapy. Phys. Med. Biol., 49: 4825–4835. doi: 10.1088/0031-9155/49/20/012.

Hug O. & Kellerer A.M. (1963) Zur interpretation der dosiswirkungsbeziehungen in der strahlenbiologie. Biophysik, 1: 20–32.

Joiner M.C. & van der Kogel A.J. (2018) Basic Clinical Radiobiology. Boca Raton, FL: CRC Press.

Kavanagh B.D. & Newman F. (2008) Toward a unified survival curve: in regard to Park et al. (Int J Radiat Oncol Biol Phys 2008;70:847–852) and Krueger et al. (Int J Radiat Oncol Biol Phys 2007;69:1262–1271). Int. J. Radiat. Oncol. Biol. Phys., 71: 958–959. doi:10.1016/j.ijrobp.2008.03.016.

Krueger S.A., Collis S.J., Joiner M.C., Wilson G.D. & Marples B. (2007) Transition in survival from low-dose hyper-radiosensitivity to increased radioresistance is independent of activation of ATM Ser1981 activity. Int. J. Radiat. Oncol. Biol. Phys., 69: 1262–1271. doi:10.1016/j.ijrobp.2007.08.012.

Lind B.K., Persson L.M., Edgren M.R., Hedlöf I. & Brahme A. (2003) Repairable–conditionally repairable damage model based on dual Poisson processes. Radiat. Res., 160: 366–275. doi:10.1667/0033-7587(2003)160[0366:RRDMBO]2.0.CO;2.

McKenna F. & Ahmad S. (2009) Toward a unified survival curve: in regard to Kavanagh and Newman (Int J Radiat Oncol Biol Phys 2008;71:958–959) and Park et al. (Int J Radiat Oncol Biol Phys 2008;70:847–852). Int. J. Radiat. Oncol. Biol. Phys., 73: 640. doi:10.1016/j.ijrobp.2008.08.063.

McMahon S.J. (2019) The linear quadratic model: usage, interpretation and challenges. Phys. Med. Biol., 64: 01TR01. doi:10.1088/1361-6560/aaf26a.

Park C., Papiez L., Zhang S., Story M. & Timmerman R.D. (2008) Universal survival curve and single fraction equivalent dose: useful tools in understanding potency of ablative radiotherapy. Int. J. Radiat. Oncol. Biol. Phys., 70: 847–852. doi:10.1016/j.ijrobp.2007.10.059.

Scholz M. & Kraft G. (1994) Calculation of heavy ion inactivation probabilities based on track structure, X ray sensitivity and target size. Radiat. Prot. Dosim., 52: 29–33. doi:10.1093/oxfordjournals.rpd.a082156.

Shuryak I. & Cornforth M.N. (2021) Accounting for overdispersion of lethal lesions in the linear quadratic model improves performance at both high and low radiation doses. Int. J. Radiat. Biol., 97: 50–59. doi:10.1080/09553002.2020.1784489.

Steel G.G., Deacon J.M., Duchesne G.M., Horwich A., Kelland L.R. & Peacock J.H. (1987) The dose–rate effect in human tumour cells. Radiother Oncol., 9: 299–310. doi:10.1016/s0167-8140(87)80151-2.

Wang J.Z., Huang Z., Simon S.L., Yuh W.T.C. & Mayr N.A. (2010) A generalized linearquadratic model for radiosurgery, stereotactic body radiation therapy, and high–dose rate brachytherapy. Sci. Transl. Med., 2: 39ra48. doi:10.1126/scitranslmed.3000864.

Téléchargements

Publiée

2023-05-31

Numéro

Vol. 7 No 1 (2023): Pesquisa e Ensino em Ciências Exatas e da Natureza

Rubrique

CIÊNCIAS DA ENGENHARIA / ENGINEERING SCIENCES

Licence

Autores que publicam nesta revista concordam com os seguintes termos / Authors who publish in this journal agree to the following terms:

A) Autor(es) e o periódico mantêm os direitos da publicação; os autores concedem ao periódico o direito de primeira publicação, sendo vedada sua reprodução total ou parcial sob qualquer caráter de ineditismo; qualquer utilização subsequente de trechos de manuscritos (e.g., figuras, tabelas, gráficos etc.) publicados na Pesquisa e Ensino em Ciências Exatas e da Natureza deve reconhecer a autoria e publicação original / Author (s) and the journal maintain the rights of publication; the authors grant the journal the right of first publication, being prohibited their total or partial reproduction in any character of novelty; any subsequent use of excerpts from manuscripts (e.g., figures, tables, graphics, etc.) published in Research and Teaching in Exact and Natural Sciences must acknowledge the original authorship and publication.

B) Autores tem o direito de disseminar a própria publicação, podendo, inclusive disponibilizar o PDF final do trabalho em qualquer site institucional ou particular, bem como depositar a impressão da publicação em bibliotecas nacionais e internacionais / Authors have the right to disseminate the publication itself, and may also make available the final PDF of the work in any institutional or private site, as well as depositing the printing of the publication in national and international libraries.