title: A mathematical model of the treatment and survival of patients with high-grade brain tumours.
abstract: More years of life per patient are lost as the result of primary brain tumours than any other form of cancer. The most aggressive of these is known as glioblastoma (GBM). The median survival time of patients with GBM is under 10 months and the outlook has hardly improved over the past 20 years. Generally, these tumours are remarkably resistant to radiotherapy and yet about 2-3% of all GBMs appear to be cured. The objectives of this study were to formulate a mathematical and phenomenological model of tumour growth in a population of patients with GBM to predict survival, and to use the model to extract biological information from clinical data. The model describes the growth of the tumour and the resulting damage to the normal brain using simple concepts borrowed from chemical reaction engineering. Death is assumed to result when the amount of surviving normal brain falls to a critical level. Radiotherapy is assumed to destroy tumour but not healthy brain. Simple rules are included to represent approximately the clinician's decisions about what type of treatment to offer each patient. A population of patients is constructed by assuming that key parameters can be sampled from statistical distributions. Following Monte Carlo simulation, the model can be fitted to data from clinical trials. The model reproduces clinical data extremely accurately. This suggests that the long-term survivors are not a separate sub-population but are the 'lucky tail' of a unimodal distribution. The estimated values of radiation sensitivity (represented as SF2, the survival fraction after 2Gy) suggest the presence of severe hypoxia, which renders cells less sensitive to radiation. The model can predict the probable age distribution of tumours at presentation. The model shows the complicated effects of waiting times for treatment on the survival outcomes, and is used to predict the effects of escalation of radiotherapy dose. The model may aid the design of clinical trials using radiotherapy for patients with GBM, especially in helping to estimate the size of trial required. It is also designed in a generic form, and might be applicable to other tumour types.
authors.affiliation: Fluids & Systems Research Centre, School of Engineering (D2), University of Surrey, Guildford, Surrey GU2 7XH, UK. n.kirkby@surrey.ac.uk
authors.forename: Norman F
authors.initials: NF
authors.lastname: Kirkby
authors.forename: Sarah J
authors.initials: SJ
authors.lastname: Jefferies
authors.forename: Raj
authors.initials: R
authors.lastname: Jena
authors.forename: Neil G
authors.initials: NG
authors.lastname: Burnet
mesh_terms: D017677:Age Distribution; D001921:Brain / Q000503:physiopathology; D001932:Brain Neoplasms / Q000401:mortality* / Q000503:physiopathology / Q000532:radiotherapy; D005909:Glioblastoma / Q000401:mortality* / Q000503:physiopathology / Q000532:radiotherapy; D006801:Humans; D008433:Mathematics; D008954:Models, Biological; D009010:Monte Carlo Method; D016019:Survival Analysis; D013997:Time Factors