The Evolution of Tumour Composition During Fractionated Radiotherapy: Implications for Outcome

Bull Math Biol. 2018 May;80(5):1207-1235. doi: 10.1007/s11538-018-0391-9. Epub 2018 Feb 27.


Current protocols for delivering radiotherapy are based primarily on tumour stage and nodal and metastases status, even though it is well known that tumours and their microenvironments are highly heterogeneous. It is well established that the local oxygen tension plays an important role in radiation-induced cell death, with hypoxic tumour regions responding poorly to irradiation. Therefore, to improve radiation response, it is important to understand more fully the spatiotemporal distribution of oxygen within a growing tumour before and during fractionated radiation. To this end, we have extended a spatially resolved mathematical model of tumour growth, first proposed by Greenspan (Stud Appl Math 51:317-340, 1972), to investigate the effects of oxygen heterogeneity on radiation-induced cell death. In more detail, cell death due to radiation at each location in the tumour, as determined by the well-known linear-quadratic model, is assumed also to depend on the local oxygen concentration. The oxygen concentration is governed by a reaction-diffusion equation that is coupled to an integro-differential equation that determines the size of the assumed spherically symmetric tumour. We combine numerical and analytical techniques to investigate radiation response of tumours with different intratumoral oxygen distribution profiles. Model simulations reveal a rapid transient increase in hypoxia upon regrowth of the tumour spheroid post-irradiation. We investigate the response to different radiation fractionation schedules and identify a tumour-specific relationship between inter-fraction time and dose per fraction to achieve cure. The rich dynamics exhibited by the model suggest that spatial heterogeneity may be important for predicting tumour response to radiotherapy for clinical applications.

Keywords: Avascular tumours; Fractionated radiotherapy; Hypoxia; Moving boundaries.

Publication types

  • Research Support, Non-U.S. Gov't

MeSH terms

  • Cell Death / radiation effects
  • Computer Simulation
  • Dose Fractionation, Radiation
  • Humans
  • Linear Models
  • Mathematical Concepts
  • Models, Biological*
  • Neoplasms / metabolism
  • Neoplasms / pathology
  • Neoplasms / radiotherapy*
  • Oxygen / metabolism
  • Radiation Tolerance
  • Spheroids, Cellular / metabolism
  • Spheroids, Cellular / pathology
  • Spheroids, Cellular / radiation effects
  • Tumor Hypoxia / radiation effects


  • Oxygen