Mathematical Modelling of Survival of Glioblastoma Patients Suggests a Role for Radiotherapy Dose Escalation and Predicts Poorer Outcome After Delay to Start Treatment

Abstract

The outcome of patients with glioblastoma (GBM) remains extremely poor. We have developed a mathematical model, using pathological and radiation biology concepts, to assess the detrimental effect of delay to start radiotherapy, the possible benefit from dose escalation, and to extract biological data from clinical data. Survival data were available for 154 adult patients with GBM treated in our centre with curative intent to a dose of 60 Gy in 30 fractions between 1996 and 2002. Survival data for 129 patients from the 60 Gy arm of the MRC BR02 randomised trial of radiotherapy dose were obtained for comparison. The model generates the equivalent of individual patients with a brain tumour, and produces an explicit outcome, either death or survival. The tumour, assumed to be growing exponentially, causes normal cell damage in the brain, and death occurs when the number of normal brain cells falls below a critical level. The outcome for an individual patient is determined by values of the variables assigned by the model. Parameters for the single patient include tumour doubling time, surviving fraction of tumour cells after each fraction of radiotherapy, and a waiting time from presentation to the start of radiotherapy. A surrogate for performance status is implemented, using a rule that rejects patients whose tumours are too advanced at presentation to be suitable for radical radiotherapy. Values for the parameters that determine individual patient outcome are randomly assigned from a set of probability distributions, using Monte Carlo simulation. The simulation constructs survival results for a population, typically 2000 individuals. The descriptors of the probability distributions that are used to determine the parameters that define the patient characteristics are adjusted to optimise the fit of the modelled population to real clinical data, using a combination of folding polygon and simulated annealing techniques. The model fits the clinical data well. The results suggest that the surviving fraction of tumour cells after a radiation dose of 2 Gy (SF2) does influence patient outcome. The mean in vivo SF2 for the Addenbrooke's data is 0.80, implying that hypoxia is a serious problem in radiotherapy for GBM. The Addenbrooke's data suggest a mean tumour doubling time of 24 days, so that a delay to start radiotherapy would be expected to have an adverse effect. Considering patients by treatment intent, median survival plummets as delay increases, and almost no patients survive long term after a 70-day delay. Radiotherapy dose escalation has an important predicted effect on survival. Assuming that the treatment could be delivered safely, a dose of 74 Gy, given at 2 Gy/fraction, would extend the survival of all patients. The proportion of long-term survivors would increase, from 2.4% with 60 Gy, to 6.4% with 74 Gy. The model can be used to derive gamma50, which has a value of 0.42, lower than the typical value of 1-2. Using the model, we have extracted biological information from clinical data. The model could be used to assess the potential benefit, or lack of benefit, from a proposed radiotherapy trial, and to estimate the necessary size. It shows that a single modality is unlikely to achieve a major improvement in long-term survival, although radiotherapy dose escalation should have a role, provided it can be given safely. The model could be extended to include chemotherapy, bio-reductive drugs, or gene therapy.