A mathematical model of the treatment and survival of patients with high-grade brain tumours

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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 2 Gy) 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.

Introduction

Primary malignant tumours of the brain and central nervous system (CNS) represent a major clinical problem. Primary tumours of the brain and CNS are dominated by high-grade gliomas, malignant tumours of glial cells, which support, nourish and facilitate the function of neurons in the brain. Of the high-grade gliomas, glioblastoma (GBM) is the most aggressive. Although relatively uncommon, accounting for approximately 2% of cancer cases, mortality rates from primary brain tumours are high. Considered from the perspective of an individual patient, the average years of life lost is higher than for any other adult solid tumour, at approximately 20 years per patient (Burnet et al., 2005).

Despite major developments in surgery, radiotherapy, imaging, and molecular biology, therapeutic results have changed little over the last century. High-grade gliomas are highly invasive and, although the tumour bulk can be effectively excised, residual tumour, which has infiltrated normal, functioning brain, is invariably left behind. Radiotherapy is effective at reducing the number of tumour cells present, but typically these tumours are not sterilized by standard doses of radiotherapy. In theory, higher doses of radiotherapy might improve outcome, but could be expected to increase damage to normal brain. Chemotherapy can produce responses in tumours, and lengthen the time to recurrence in patients (Stupp et al., 2005), but the cell killing from this modality is also insufficient to sterilize primary high-grade glial tumours. Imaging has been transformed by the introduction of computed tomography (CT) using X-rays and magnetic resonance scanning. Although, the use of these modalities has assisted diagnosis and overall management, it is generally accepted that they have not contributed to improved survival. Our understanding of the underlying genetic mutations associated with malignant brain tumours continues to advance, but as yet, an understanding of these features has not translated into new treatment strategies. This situation is in contrast to most other solid tumours where substantial improvements in survival have been achieved.

Treatment with radiotherapy is given with either radical intent, which is with the objective of curing the patient, or palliative intent, when the aim is to alleviate symptoms. Alternatively, patients who are already very ill, typically with substantial neurological deficits, may be offered supportive care without radiotherapy. Radical radiotherapy is intended to sterilize a tumour, whilst palliative treatment is intended to reduce its size. In both cases, treatment is intended to avoid causing any further damage to the normal brain cells, i.e. beyond that already caused by the tumour.

Factors, which govern the decision concerning the choice of treatment, include assessment of neurological function. For radical treatment patients are normally expected to have minimal or no neurological deficit, i.e. their performance status is excellent, because such treatment is arduous. Patients are assessed for performance status at presentation to the oncology department and are reassessed prior to commencing treatment. There is always an interval between the decision to treat and the commencement of radiotherapy, for the preparation and planning of treatment, and additional delay may be imposed because of limitations in treatment resources. Some patients become so poorly in this interval that radical treatment is no longer indicated, and the treatment intent must then be changed.

Against this background the development of computer models which could be used to evaluate new treatment strategies is an important objective (Murray, 2003). Such modelling might allow the execution of clinical trials “in silico”, in situations where therapeutic strategies would otherwise be impossible to perform (Burnet et al., 2006; Kirkby et al., 2002a, Kirkby et al., 2002b, Kirkby et al., 2005, UKRO3). An example of such a study is radiotherapy dose escalation. Models might also assist in the calculation of patient numbers required for randomized clinical trials of new treatments, where the relative efficacy of a new treatment is subject to uncertainty. Clinical problems such as the delay to start treatment (Burnet et al., 2006; Kirkby et al., 2005, UKRO3) and the value of the extent of surgical resection might be examined by such modelling. Newer imaging modalities may well produce biological information on individual patients which could be utilized to individualize treatment (Jena et al., 2005), and the incorporation of such information to computer models might facilitate this. Modelling of patient data might allow the extraction of biological information from population data and may indicate which parameters would be valuable to measure on an individual patient basis. In this way, efforts to develop clinical measurements could be focussed on those parameters, which would produce the greatest gain. In the translation of treatments based on molecular genetics from the bench to the bedside, mathematical models may contribute to the development of optimized treatment strategies.

We have developed a model to investigate some aspects of radiotherapy treatment for GBM, including the potential value of radiotherapy dose escalation and the adverse effects of delays to start treatment. Considerable effort has been put into modelling the development of solid tumours, but the consequences of these tumours for an individual patient have been largely ignored. Little effort has been directed at the effects of radiotherapy on tumours, although this modality is, after surgery, the most important modality for the curative treatment of cancer (SBU, 1996).

In what follows, we describe a model of an individual patient and from that the construction of a population of patients. The patient model includes the growth of the tumour, the effects of the tumour on the patient, and the effects of radiotherapy treatment on the tumour. In constructing a population of patients, it has proved necessary to include a representation of the clinical decision to offer radical radiotherapy treatment (i.e. treatment given with curative intent). Having constructed the model, we explore the parameter values by comparison of the population outcome with real clinical data.

Section snippets

Model of a patient

In the model, it is assumed there are two types of cell in the brain of each patient: normal cells (N(t)) and tumour cells (C(t)). Throughout this work, and consistent with clinical trials, t=0 represents the time of first presentation to a hospital oncology unit.

Parameter distributions

In order to turn the model of a single patient described above into a model of a population of patients, it is necessary to make assumptions about the distributions of certain parameters within the single patient model.

In this work, it is assumed that six parameters of the single patient model are distributed statistically. The following five parameters are assumed to be normally distributed

  • (1)

    The number of undamaged normal brain cells at presentation (N0).

  • (2)

    The doubling time of the tumour (tD).

  • (3)

    The

Fit to clinical data

The model can be successfully fitted to real clinical data, as shown in Fig. 2, with resulting parameter values as shown in Table 1. The closeness of the fit to the clinical data is excellent. There is a close fit at early times, which is largely controlled by the 2 clinical patient selection criteria (Section 3.2). There is an equally good fit at later times where the long-term survivors are represented.

There is no statistical method to compare the modelled population survival curve with the

Discussion

High-grade primary brain tumours, of which GBMs are the most malignant, represent a major clinical challenge. Sadly, a very small number of patients survive long-term, which demonstrates the need for new treatment strategies. Moreover, the potential benefit of such strategies may be difficult to prove because these tumours are comparatively rare. A mathematical model, which allows the investigation of, some determinants of patient outcome, and the evaluation of potential clinical trials would

Conclusions

The model achieves an excellent fit to the clinical data, including the very few long-term survivors. We hope that it will aid development of clinical studies using radiotherapy for patients with GBM, especially in guiding the size of study required. The model also suggests the presence of severe hypoxia, and emphasizes that strategies to address hypoxia are warranted. Since the model is built in a generic form, it could potentially be applied to other tumours. It could also be extended to

Acknowledgments

NFK wishes to thank the Life Sciences Interface of the Engineering and Physical Sciences Research Council for the Discipline Hopping Grant that made this work possible.

We are grateful to Dr. Karen Young for advice on the use of kernel density functions, and to Mr. John Gleave for designing the project, which contributed data on variation in the volume of the brain.

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