Changes in the retreatment radiation tolerance of the spinal cord with time after the initial treatment

Abstract

Purpose: To estimate, from experimental data, the retreatment radiation ‘tolerances’ of the spinal cord at different times after initial treatment.

Materials and methods: A model was developed to show the relationship between the biological effective doses (BEDs) for two separate courses of treatment with the BED of each course being expressed as a percentage of the designated ‘retreatment tolerance’ BED value, denoted and . The primate data of Ang et al. (2001) were used to determine the fitted parameters. However, based on rodent data, recovery was assumed to commence 70 days after the first course was complete, and with a non-linear relationship to the magnitude of the initial BED (BED_init).

Results: The model, taking into account the above processes, provides estimates of the retreatment tolerance dose after different times. Extrapolations from the experimental data can provide conservative estimates for the clinic, with a lower acceptable myelopathy incidence. Care must be taken to convert the predicted value into a formal BED value and then a practical dose fractionation schedule.

Conclusions: Used with caution, the proposed model allows estimations of retreatment doses with elapsed times ranging from 70 days up to three years after the initial course of treatment.

Keywords:

• Radiotherapy
• re-treatment
• CNS tolerance
• mathematical model
• myelopathy

Note

Acknowledgements

The authors remember the late Kian Ang whose unique study in the rhesus monkey provided the database for the present study. In addition, one of us (JWH) gratefully acknowledges personal discussions with him, which contributed to some of the thinking in the present manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Although the application of conservatism depends on a chosen risk level of myelopathy (with the illustrative risk level being taken to be 1%), this dependency has little influence on the BED calculation. In particular, for low incidences of risk of myelopathy and low levels of conservatism it makes very little difference at which level of risk conservatism is applied, see Appendix D.

Additional information

Funding

TEW would like to thank St John's College, Oxford and BBSRC grant [BKNXBKOO BK00.16] for their financial support. JBB and GFC gratefully acknowledge the Mathematical Institute (University of Oxford) for its hospitality and the José Castillejo Fellowship Program numbers [CAS14/00383 and CAS14/00363], respectively, funded by the Spanish Ministry of Education, Culture and Sports, for their research stay at the Mathematical Institute. They also thank funding from projects supported by Ministerio de Economía y Competitividad/FEDER, Spain [grant numbers: MTM2012-31073 and MTM2015-71200-R], Consejería de Educación, Cultura y Deporte from Junta de Comunidades de Castilla-La Mancha, Spain [grant number PEII-2014-031-P] and the James S. McDonnell Foundation 21st Century Science Initiative in Mathematical and Complex Systems Approaches for Brain Cancer, USA [Collaborative-Planning Grant 220020420 and Collaborative Award 220220450].

Notes on contributors

Thomas E. Woolley

Thomas E. Woolley studied mathematics at the University of Oxford between 2004 and 2017 and is now a Lecturer of Applied Mathematics at Cardiff University. His research focuses on applying mathematical techniques to biological problems in pattern formation, stem cell movement, oncology and neurobiology.

Juan Belmonte-Beitia

Juan Belmonte-Beitia graduated in Mathematics and Physics at Universidad Complutense de Madrid and obtained PhD in Mathematics (2008) at University of Castilla-La Mancha. His research interests are dynamical systems, differential equations and partial differential equations and their applications to Mathematical Biology, specifically the mathematical modeling of tumor growth.

Gabriel F. Calvo

Gabriel F. Calvo obtained PhD in Physics at Autonoma University, Madrid. Research activities have spanned from nonlinear optics to quantum information processing with photons, nanoparticles for biomedical applications and mathematical oncology. Currently, Associate Professor, Department of Mathematics and senior member at the MôLAB-Mathematical Oncology Laboratory (UCLM-University, Castilla-La Mancha).

John W. Hopewell

John W. Hopewell was the Director of Radiobiological Research, University of Oxford, between 1980 and 2001. His primary research interest has been the study of different aspects of normal tissue toxicity related to radiotherapy: pathogenesis, fractionation and dose-rate, volume effects and retreatment.

Eamonn A. Gaffney

Eamonn A. Gaffney is an Associate Professor at the Mathematical Institute, University of Oxford, with interests in mathematical and computational modeling for the life sciences.

Bleddyn Jones

Bleddyn Jones is a medically qualified radiobiologist and Professor of Clinical Radiobiology at The University of Oxford. He has made many contributions to explicit mathematical models of radiotherapy and chemotherapy, as well as their application in the clinic. He started re-treating brain and spinal tumor patients in 1991.