a mathematical model of the treatment and survival of patients with
TRANSCRIPT
10.1016/j.jtbi.2006.09.007www.elsevier.com/locate/yjtbi
A mathematical model of the treatment and survival of patients with high-grade brain tumours
Norman F. Kirkbya,, Sarah J. Jefferiesb, Raj Jenab,c, Neil G. Burnetb,c
aFluids & Systems Research Centre, School of Engineering (D2), University of Surrey, Guildford, Surrey GU2 7XH, UK bOncology Centre, Addenbrooke’s Hospital, P.O. Box 193, Cambridge CB2 2QQ, UK
cDepartment of Oncology, University of Cambridge, Oncology Centre, Addenbrooke’s Hospital, P.O. Box 193, Cambridge CB2 2QQ, UK
Received 13 April 2006; received in revised form 23 August 2006; accepted 6 September 2006
Available online 16 September 2006
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.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Brain cancer; Radiotherapy; Glioblastoma; Patient survival; Tumour growth
1. 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
e front matter r 2006 Elsevier Ltd. All rights reserved.
i.2006.09.007
esses: [email protected] (N.F. Kirkby),
brookes.nhs.uk (R. Jena), [email protected]
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, mortal- ity 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
Nomenclature
b bandwidth for approximation of distribution of tumour age at presentation (days)
C(t) number of cancer cells in the brain (cells) C0 number of cancer cells in brain at presentation
(cells) j number of fractions of radiotherapy (dimen-
sionless) kc rate constant for cancer cell growth (days1) kn rate constant for normal cell damage
(cells1 days1) ks normalization constant used in pdf for survival
fraction (Eq. (17)) (dimensionless) m exponent used in pdf for survival fraction (Eq.
(17)) (dimensionless) n exponent used in pdf for survival fraction (Eq.
(17)) (dimensionless) N(t) number of normal brain cells (cells) NBrain number of normal cells in a healthy brain (cells) Ncrit critical number of normal brain cells required
for patient to remain alive (cells) Np number of patients in virtual clinical trial
(dimensionless)
N0 number of normal brain cells left at presenta- tion (cells)
rc number growth rate of cancer cells (cells day1) rn number rate of damage to normal cells
(cells day1) t time (days) tage tumour age at presentation (days) tage,i tumour age at presentation in the ith patient
(days) tD tumour doubling time (days) tdelay time to commence treatment from presentation
(days) tp patient survival time from presentation if left
untreated (days) ts patient survival time from presentation (days) tt patient survival time from treatment if left
untreated (days) xs cancer cell survival fraction in response to one
fraction of radiotherapy (dimensionless)
a constant in pdf for survival fraction (Eq. (17)) (dimensionless)
N.F. Kirkby et al. / Journal of Theoretical Biology 245 (2007) 112–124 113
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 radiother-
apy 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 perfor- mance status is excellent, because such treatment is arduous. Patients are assessed for performance status at presentation to the oncology department and are reas- sessed prior to commencing treatment. There is always an interval between the decision to treat and the commence- ment 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, b, 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
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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.
2. 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 ¼ 0 represents the time of first presentation to a hospital oncology unit.
2.1. Patient death
A patient dies when the number of undamaged normal cells drops below a threshold value. Let N(ts) represent the number of normal cells in the brain. The patient dies when
NðtsÞ ¼ Ncrit, (1)
where ts is the survival time and Ncrit is the number of normal brain cells that constitute the minimum, critical
threshold. We currently assume that Ncrit is not affected by any treatment modality including radiotherapy.
2.2. Interaction between tumour and normal brain
The model assumes that the processes of damage to the normal cells due to the presence of the tumour can be represented as an irreversible elementary chemical reaction as follows:
N þ C! kn
C. (2)
Normal cells are assumed to be destroyed in a process related to the size of the tumour. This damage process is modelled as follows: let rn be the number rate of damage of normal cells. This is assumed to be related to the number of normal cells (N) and to the number of cancer cells (C) as follows:
rn ¼ knNC. (3)
2.3. Growth of tumour
If it is assumed that the growth of the tumour mass is a simple, first-order process, the number rate of growth is given by
rc ¼ kcC. (4)
In order to develop a solution to this model, it is assumed that the brain and the tumour are both closed systems with respect to cells, so that the number balances may written as follows:
dN
and
dC
2.4. General solution
Given the initial condition for cancer cells at presenta- tion,
Cðt ¼ 0Þ ¼ C0. (7)
Eq. (6) has the conventional, simple solution
CðtÞ ¼ C0 expðkctÞ, (8)
which relates to the tumour doubling time, tD, by
tD ¼ ln 2
The solution to Eq. (5) can now be developed as
dN
and on integration
ðexpðkctÞ 1Þ, (11)
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where N0 is the number of normal brain cells left at presentation (t ¼ 0). Eq. (11) can be simplified by back substitution of the number of cancer cells to give
ln N
ðC C0Þ. (12)
When required, Eq. (12), the state-space equation, is used to calculate the size of the tumour from the number of normal brain cells remaining.
Another application of Eq. (10) is that it allows the calculation of the age of the tumour at presentation, provided we assume that the tumour starts with a single cancer cell, with the rest of the brain completely intact (i.e. N ¼ NBrain). Therefore the tumour age is
tage ¼ 1
NBrain
N0
. (13)
The above model describes the growth of the tumour and the resulting damage done to the normal brain. Once the tumour is growing, a variety of symptoms may result in the patient seeking medical advice and, in time, a hospital appointment. On rare occasions this delay may result in death before presentation to hospital. Once in hospital it is assumed that diagnosis is made and the patient can be referred for treatment.
2.5. Model of treatment
In this model, the time delay between the patient presenting to the oncology department and the commence- ment of radiotherapy is a variable represented as tdelay. The delay to start treatment was included in the model, despite there being specific clinical data available, in order to be able to examine the effects of alteration in this delay.
Radical radiotherapy is normally given as a ‘‘fractio- nated’’ course, where the total dose is divided into multiple, equal, small-dose exposures, typically given once a day for 5 days a week. This ‘‘fractionation’’ has the effect of sparing normal tissue, in this case normal brain, relative to tumour. In the radical treatment of gliomas, 30 exposures are normally administered. Of those patients that are treated, it is assumed in this model that the radiotherapy is applied instantaneously, and that all of the j exposures are delivered at the same instant, and each results in the same fraction of tumour cells surviving, denoted here as xs. Hence,
C tþdelay
s, (14)
where tþdelay represents a time just after the radiation treatment, and tdelay a time immediately before the exposure. This expression was used in order to avoid introducing further variables describing the exact pattern of delivery and its effect on tumour growth (Kirkby et al., 2002a, b).
It is also assumed that all the normal brain cells survive the treatment, so, for normal brain cells,
N tþdelay
. (15)
On the basis of an individual patient, the tumour is assumed to be sterilized if
C tþdelay
o1. (16)
If the tumour is not sterilized, Eqs. (14) and (15) form the initial conditions for the regrowth of the tumour, and it is assumed that the parameters of the regrowth are the same as those that applied before treatment.
3. Model of a population of patients
3.1. 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)
(3)
(4)
(5)
The critical size of undamaged brain (Ncrit).
The remaining parameter to be treated statistically is the survival fraction of the cancer cells in response to a single exposure of radiation, xs. It is not appropriate to regard this parameter as being normally distributed since, for instance, the value must lie in the interval (0,1). Hence, the following probability density function was used:
pðxsÞ ¼ ksx n s ð1 xsÞ
meaxs . (17)
This distribution is not easily integrated in order to determine the normalization constant ks, the mean or the variance as functions of the three parameters n, m and a. There is also the need to generate a variate with this distribution. For the normal variates, the method of Box–Muller was used, but for the survival fraction, xs, a simple acceptance/rejection test was used. The value of the constant, ks, was determined by
trapezoidal integration of the denominator of the following equation required for normalization:
ks ¼ 1R 1
meaxs dxs
. (18)
In the course of the numerical integration of the denominator in Eq. (18) the maximum height of the
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function can be recorded. This was used to maintain a reasonable efficiency for the acceptance/rejection test, which otherwise used two uniformly distributed random variates, one of which was scaled to this maximum height of the function. Fig. 1 shows an example of Eq. (17) and a reconstructed distribution from the sampling method used.
Although there are four parameters required to describe the distribution of xs, the survival fraction, this represents only three degrees of freedom once the normalization procedure is completed. Each parameter, which is assumed to have a normal distribution, has two degrees of freedom. Therefore, there are a total of 13 adjustable parameters in this model.
3.2. Model of patient selection
In an attempt to mimic real clinical practice, it is assumed that clinicians are able to calculate exactly how long each patient will survive if left untreated and that two decisions that patients are suitable for radical treatment are made for each patient: the first at presentation and the second at the start of treatment. When first seen, if a patient will survive untreated longer than a specified time, tp, the patient is deemed sufficiently fit to be accepted for treatment. Typically, this is assumed to be of the order of 8 weeks. The criterion for selection is explicitly as follows:
tpp 1
Ncrit
N0
. (19)
Following the delay to treatment, each patient is reassessed just before the start of treatment, exactly as would happen clinically, and if the patient will survive untreated for longer than a specified time, tt (measured from the start of treatment), the patient is treated. This is a surrogate for excluding patients whose condition has deteriorated and who would not benefit from this treatment.
0
0.5
1
1.5
2
2.5
3
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Survival Fraction
P ro
b ab
ili ty
D en
si ty
As sampled
As required
Fig. 1. Theoretical distribution of survival fraction after one dose of
radiotherapy, showing both the required shape and the actual sampled
distribution, to demonstate that the acceptance/rejection method func-
tions correctly. This is not the fitted distribution, which is shown below
(Fig. 5).
Therefore, patients are selected if the following condition is also met:
tt þ tdelayp 1
Ncrit
N0
. (20)
In reality, clinical decisions are imperfect, but these rules are intended to represent a simplified version of real clinical practice. In this way, only patients who are considered fit enough are accepted for treatment. The two rules are applied independently which implies, for instance, that the consultant does not know what the delay to treatment will be at the time the decision is made to offer radical treatment. These two rules are a surrogate for good clinical performance status, that is, excellent health with minimal or no neurological deficit, but these rules are a very simple representation of a rather complicated situation.
3.3. Simulation of patient population
Monte Carlo simulation was used to generate a population of patients. In this case, it was generally found that a population of 2000 patients had to be generated so that the resulting patient population survival curve was independent of any further increase in the number of patients simulated. A larger number of patients must initially be generated, to allow for exclusion of those that ‘fail’ the selection criteria for patient fitness described above. The results of the simulation are presented as a
Kaplan–Meier survival curve in Fig. 2, i.e. a graph of the proportion of patients surviving versus time (Kaplan and Meier, 1958). This method allows for censorship of patients who are surviving at the time of interest. This is the standard method for presenting clinical survival data, though, in the case of the model, this censorship is redundant.
Time (days)
C um
ul at
iv e
S ur
vi va
l
0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Addenbrooke’s patient survival
Fig. 2. Kaplan–Meier survival curve for the Addenbrooke’s patient
population, and the fitted model.
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Table 1
Detail of values of the variables from the fit to the Addenbrooke’s patient
data
Factor
SF2a
Tumour doubling
31.5 daysc 183 days2c
aThese parameters are used in the probability density function for SF2.
These values give a mean SF2 ¼ 0.80, and median ¼ 0.83. bParameters not adjusted for this fitting. cParameters from clinical series.
N.F. Kirkby et al. / Journal of Theoretical Biology 245 (2007) 112–124 117
For each patient a tumour age is calculated and the probability density function for these ages is approximated using a Gaussian kernel density function (Silverman, 1998), as follows:
pðtageÞ ¼ 1
2 . (21)
The value of the bandwidth, b, was selected by trial and error. For a virtual trial of 2000 patients it was found acceptable to relate b to the standard deviation of the raw data for the tumour age distribution as follows:
b ¼ 0:2SDðtage;iÞ. (22)
3.4. Clinical data
Clinical data were available for a set of 154 adult patients with GBM who were given radical treatment in our unit between 1996 and 2002. Patients are only accepted for this treatment programme if they are in excellent health with minimal or no neurological deficit, in other words with normal performance status. Kaplan–Meier survival analysis was available (Kaplan and Meier, 1958). All patients were treated in 30 exposures, delivered daily for 5 days per week, delivering a total dose of 60Gy (Burnet et al., 2006). No adjuvant chemotherapy was given during this period. Data were available for the delay to start radiotherapy: mean 32 days, variance 183 days2 (SD 13.5 days).
3.5. Determination of the parameters of the model
The model is fitted to clinical survival data by minimiz- ing the weighted sum of squares of errors between the simulated and real Kaplan–Meier survival curves. The algorithm used for this minimization was the simulated annealing of folding polygons, as described by Press et al. (1996). The weighting was adjusted heuristically to guide the optimization to fit correctly the tail of the curve representing the long-term survivors.
Some of the parameters of the model can be calculated a priori, because they are contained within the clinical data, specifically the mean and variance of the delay to start treatment, noted above. Three further parameters have been estimated, namely the mean and variance of the normal brain cell number at presentation, and the mean number of normal brain cells remaining at the time of death. The estimates of mean and variance of the normal brain cell number at presentation are consistent with brain volume calculations for a data set of 100 patients, performed in relation to an anatomical study of the brain (not published).
The parameters of the distribution of xs are impossible to measure in vivo in patients, although some laboratory
data exist which suggest approximate values, taking into account the typically hypoxic nature of GBM. However, there are some parameters whose values are entirely unknown, without fitting. In particular, the mean and variance of the interaction constant, kn, are untestable. Typically, eight parameters were determined by fitting,
namely:
at the time of death,
mean and variance of the tumour doubling time tD the
three parameters determining xs, i.e. m, n and a.
4. Results
4.1. 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 real clinical data, because standard techniques such as logrank testing depend upon a specific size of the study population. Since the modelled population can be of any size, and there are
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M ea
n D
el ay
to S
ta rt
T re
at m
en t (
da ys
30.00
30.50
31.00
31.50
32.00
32.50
9.10E+11 9.20E+11 9.30E+11 9.40E+11 9.50E+11 9.60E+11 9.70E+11 9.80E+11 9.90E+11 1.00E+12
Fig. 3. Plane of contours of sum of squares of errors for two representative variances, mean delay to start treatment in days and mean number of normal
cells remaining at patient death, showing the covariance between them.
N.F. Kirkby et al. / Journal of Theoretical Biology 245 (2007) 112–124118
advantages in having a very large study population, such methods are invalidated. For further analysis see Burnet et al. (2006).
In the course of generating this fit, it was noted that several of the parameters are strongly covariant; to a large extent the variation in the population can be achieved by altering any of the variances in the main distributions. Fig. 3 shows a plane of contours of sum of squares of errors for 2 representative variances, showing this relation- ship. Encouragingly, many other parameters of the model are fitted without this covariance. An example is shown in Fig. 4. It is clear from Figs. 3 and 4 that the sum of squares of errors in the parameter space is complex and con- voluted. Therefore it is not possible to guarantee that the best possible fit has been discovered.
4.2. Distribution of radiation sensitivity and survival fraction
The shape of this survival fraction distribution is substantially skewed, as shown in Fig. 5. This distribution has a mean survival fraction of 0.80, and a median of 0.83. As discussed in Burnet et al. (2006), the results are consistent with those found from in-vitro experiments, especially allowing for the effects of hypoxia in increasing radio-resistance.
Since each radiation exposure was given in a dose of 2Gy, this figure shows the distribution of SF2, the survival
fraction after 2Gy. Two gray is a commonly used dose per exposure in clinical practice, and SF2 has been used extensively as a descriptor of cellular radiation sensitivity (Deacon et al., 1984).
4.3. Age distribution of tumours at presentation
The model predicts the age distribution of tumours at the time of presentation, when they contain of the order of 1011
cells. The distribution is shown in Fig. 6. The mean age at presentation is 1034 days (median 1029) and variance 8.6 104 days2. The distribution is slightly skewed (skew- ness is 0.25), but otherwise almost normal (kurtosis 3.08). However, it should be noted that the variance in tumour age results mainly from the variances of tumour doubling time and number of normal brain cells at presentation. These variances are co-variant with the variances of some of the other parameters in the model, and none has been estimated well.
4.4. Sensitivity analysis
A sensitivity analysis was conducted by varying each parameter separately by 710% from its best-fit value. The resulting sum of squares of errors was compared to the sum of squares of errors at the best fit, and the results are presented in Fig. 7. From this it is clear that the model
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V ar
ia nc
e of
D el
ay to
S ta
rt T
re at
m en
t ( da
ys 2 )
170.00
172.00
174.00
176.00
178.00
180.00
182.00
184.00
186.00
9.10E+11 9.20E+11 9.30E+11 9.40E+11 9.50E+11 9.60E+11 9.70E+11 9.80E+11 9.90E+11 1.00E+12
Fig. 4. Plane of contours of sum of squares of errors for two parameters well fitted in the model, variance of delay to start treatment (in days2) and mean
number of normal cells remaining at patient death, showing absence of covariance. This contrasts with Fig. 3.
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Survival Fraction
P ro
b ab
ili ty
D en
si ty
o f
S u
rv iv
al F
ra ct
io n
Fig. 5. Distribution of survival fraction after one dose of radiotherapy, xs,
resulting from fitting to the Addenbrooke’s data. The mean value of SF2 is
0.80, and the median is 0.83.
Tumour Age (days)
(1 /d
ay s)
0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400
0
0.0002
0.0004
0.0006
0.0008
0.0010
0.0012
0.0014
Fig. 6. Probability distribution of the age of tumours at presentation. The
mean is 1034 days, median 1029, and variance 86436 days2 (SD 294 days);
the skewness is 0.25, and kurtosis 3.08.
N.F. Kirkby et al. / Journal of Theoretical Biology 245 (2007) 112–124 119
is vastly more sensitive to some parameters than others, and not necessarily symmetrical even in the vicinity of the best fit.
In general terms, the sum of squares of errors is more sensitive to variation in the means of the model parameters than the variances. The mean of the interaction constant is the exception. The sum of squares of errors is also sensitive to the survival fraction parameters.
4.5. Clinical selection criteria
Within the model, a substantial number of patients are rejected from possible treatment. The majority are rejected because the model generates patients with implausible characteristics, such as tumours larger than the brain at presentation. This is the result of using independent normal distributions for the parameters of the model. In addition,
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Fig. 7. Sensitivity analysis of the main parameters in the model, ranked in order of sensitivity. Note that numbers given as 0 are below 0.5%, not actually
zero.
N.F. Kirkby et al. / Journal of Theoretical Biology 245 (2007) 112–124120
patients are rejected whose tumours render them too poorly to be accepted into the radical treatment pro- gramme. After presentation, and before treatment com- mences, some patients become too unwell and some even die; these are rejected by the relevant clinical criteria. For 1000 patients treated, 171 deteriorate and 47 die after acceptance but before treatment commences.
5. 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 be of sub- stantial value. We have sought to develop such a model to address this serious clinical problem. The model was designed to describe growth of a tumour and consequent patient death. Some of the parameter values were determined by fitting the model to real clinical data, and appear to be biologically plausible.
The model demonstrates that a delay to start radio- therapy has an obvious, predictable, deleterious effect of patient survival. This aspect of our results has been discussed elsewhere (Burnet et al., 2006).
5.1. Fit to clinical data
The population survival curve for patients with GBM is rather unusual in having such a small number of long-term survivors; most cancers have a rather better cure rate. It is tempting to ignore this group because it is so small, but they are important. To the patients themselves, long-term survival is enormously significant. Developments in treat- ment, which might improve outcome, should include increasing the proportion of patients within this group. Thus, the model must predict the existence of this group if it is to function as an effective tool to evaluate new treatment strategies. The fit to the clinical data achieved by the model is
excellent (Fig. 2). The model successfully fits the tail of long-term survivors, as well as the bulk of the patient population. For statistical considerations see Burnet et al. (2006).
5.2. Clinical selection criteria
It is important to note that the model has been fitted to data for patients who have actually completed radical treatment. Patients who deteriorate whilst waiting to start treatment are actively switched out of the radical treatment programme in order to deliver a more appropriate palliative or supportive care package. Real clinical decision-making regarding a patient’s
treatment is clearly not as simple as has been assumed in
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the model. Nevertheless, there are real decision points in a patient’s treatment pathway, the first at presentation to oncology and the second at the start of treatment. If the patient’s condition has deteriorated, then radical treatment is abandoned and an alternative, palliative strategy initiated. Decision points have been included in the model, at these times, to mimic this clinical situation. In the model, the clinician ‘‘knows’’ how long the patient will survive if untreated, but in reality only the current and previous performance status is known.
Within the model, a substantial number of patients are rejected from possible treatment at the first decision point because the model generates patients with implausible characteristics. There is no clinical equivalence of this group. At the second decision point, at the time of starting treatment, for each 1000 patients treated in the model, a further 171 are rejected because they deteriorate before treatment, and 47 more die after acceptance but before treatment commences, who are also not treated. This amounts to 18% of patients ‘‘accepted’’ for radical treatment in the model. In reality, 3% (4 of 123 with definite treatment data) of patients deteriorate whilst waiting to start treatment, and do not proceed with radical treatment.
These figures appear to suggest that the clinicians are better predictors of outcome than the model, despite it containing explicit information about patient survival. However, it is more likely that the model acceptance criteria are less strict than those used clinically. The model acceptance criteria are not distributed, and their values are not derived from fitting. This represents an area for future refinement of the model.
Performance status is not explicitly represented in the model, because all patients accepted for treatment with curative intent are required to be of a uniform excellent performance status. Therefore, it is not possible at present to model the outcome of those patients who deteriorate before treatment, or indeed those who are less well at presentation, who are initially offered palliative treatment. The inclusion of patients undergoing palliative radio- therapy is also an area for future work.
5.3. Radiation sensitivity
The results of the fitting, with such close agreement between the real clinical data and the modelled population, indicate that a second, separate population is not biologically necessary (Fig. 2). More specifically, the distribution of xs, the fraction of tumour cells surviving each radiation exposure, is unimodal. This distribution is skewed, indicating that a much smaller proportion of patients have sensitive tumours than resistant ones. Since the parameters of the distribution are derived from fitting to the real clinical data, this is a plausible result. It is also consistent with published data of intrinsic tumour cell radiosensitivity (West et al., 1993; Bjork-Eriksson et al., 1998), and normal fibroblast sensitivity (Peacock et al.,
2000). Although these distributions are often considered to be Gaussian, measurements indicate that in reality they are typically slightly skewed. The absolute values of estimates of xs, or SF2, from the
model represent in vivo estimates, and are consistent with the presence of severe hypoxia, which renders cells less sensitive to radiation. In patients, GBMs are known to contain severely hypoxic areas (Rampling et al., 1994). The results are also consistent with in vitro data, where hypoxic conditions have been used (Taghian et al., 1995); see Burnet et al. (2006) for further details. The model also predicts dependence of patient outcome on SF2, which has not been uniformly reported in the oncology literature (Taghian et al., 1993). Our work suggests that treatment strategies to address radiation resistance in general, and hypoxia in particular, are warranted. The model can be used to evaluate the effect of
radiotherapy dose escalation on the patient population. Our evaluation of this (Burnet et al., 2006) suggests that an escalation from 60Gy to 74Gy would increase the survival time of all patients treated, and also improve the proportion of patients achieving long-term survival, albeit modestly, from 2.4% to 6.4%. This would require that the treatment can be delivered safely, but imaging techniques which allow the individualization of the target volume might achieve this (Jena et al., 2005).
5.4. Tumour growth and patient performance status
Within the model the most central assumption is of exponential tumour growth. Therefore, decline in the number of remaining normal brain cells invariably accel- erates with tumour age. This would have the effect of producing a rapid and accelerating decline in performance status in model patients. Clinical experience suggests that this is realistic for many patients; others have a more uniform decline before death, which may be accounted for by a rather slower growth rate. An obvious and important area for future work is to
model the kinetics of tumour growth in more detail. The ability to model a tumour whose doubling time increases with tumour age would be helpful. This may of itself predict that some patients do not suffer an accelerating decline in normal brain cell number, and hence account for a slower, steady decline in performance status. More realistic tumour growth kinetics are an area for future development. However, we feel justified in starting where we have, because so much mathematical effort is currently being directed at this feature of tumour growth.
5.5. Age distribution of tumours at presentation
The model can predict the age distribution of tumours at the time of presentation (Fig. 6). The distribution is slightly skewed. This may be due to the exclusion of some young, rapidly growing tumours, which may cause early deteriora- tion in the patient’s clinical condition, leading to their
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exclusion from the radical treatment programme. The presence of a relatively wide range in tumour age at presentation, and the relatively high mean value, is clinically plausible, in our view. Patients present with a wide range of symptoms, but only once the tumour has reached a substantial size. Some of the common symptoms of high-grade brain tumour, such as headache, are also non-specific, which causes difficulty in rapid diagnosis. Moreover, tumours are known to have a wide variation in clinical growth rates. Therefore, the predicted range of age at presentation is credible, though it is impossible to validate clinically.
Our predictions of tumour age are dependent on the underlying assumption of exponential tumour cell growth. Whilst this is most likely to be a realistic representation of growth in the early phase, it is not true of older tumours in which nutrient and oxygen limitations occur through inefficient angiogenesis and neovascularization. Thus, most tumours have been growing faster in the past, so that the predictions of tumour age are likely to be over-estimates.
5.6. Parameter values and sensitivity analysis
The most striking feature of the sensitivity analysis (Fig. 7) is that the mean number of normal cells at presentation (N0) is the parameter, which shows the largest effect on the sum of squares of errors. Furthermore, the difference between increasing and decreasing this para- meter is a strong indication of the nonlinearity of the parameter space. Thus, the estimate for N0 is likely to be our most accurate and could in principle be measured clinically.
The second most influential parameter in the sensitivity analysis is the mean doubling time (Fig. 7). The value of 24.1 days indicates that these tumours may progress very rapidly, leading to deterioration in the patient’s clinical condition. In patients with GBM, this decline is typically not reversible with treatment. The fact that this parameter has been estimated with reasonable certainty is helpful, particularly in promoting the importance of a rapid patient journey from diagnosis to definitive treatment for patients with GBM (Burnet et al., 2006).
In the sensitivity analysis, the sum of squares of errors is generally more sensitive to variation in the means of the model parameters than the variances. The mean of the interaction constant is an exception; this may be reasonable because there are several mechanisms of interaction between tumour and normal brain tissue. For example, generalized pressure effects from a large tumour in a non- eloquent part of the brain may produce death through a different mechanism than direct damage to a particular region of normal brain which is critical to life, such as the brain stem. Thus, each separate interaction mechanism may be characterized by very different values for the respective interaction constants. Hence, lumping these phenomena together may have resulted in the relatively poor determination of both mean and variance of the
interaction constant. This is envisaged as an area for development of the model. However, detailed analysis of the mechanism of death in individual patients is not clinically available. Some indication of the sensitivity of the system is
provided in Fig. 7. In general, the sum of squares of errors is more sensitive to variation in the means of the model parameters than their variances. This is in part to be expected because the means relate strongly to the bulk of the cases in the distributions of the data, while the variances affect the relatively unusual cases.
5.7. Other assumptions in the model
A number of assumptions are required within the model. We have made no attempt to link the number of normal cells remaining to the severity of neurological deficit. However, subjectively, clinical observations suggest a relationship of this general type. More detailed examina- tion of this interaction would be of interest. It must, however, take into account differences in the clinical effect produced by tumours in different locations. For example, a small tumour in an eloquent area of brain such as the motor cortex can produce a substantial neurological deficit whilst one within a ‘‘silent’’ area of non-dominant poster- ior parietal lobe might well be substantial larger before causing any clinical effect. We have not included patient age as a specific parameter
affecting survival although it is known to be highly prognostic. Because of the excellent fit to the real clinical data, it seems possible that there is covariance between age and another variable in the model, such as the critical number of normal cells at death, or tumour doubling time. However, this is a weakness of the current model and is
an area for future work. In the model it is assumed that the normal brain and
tumour are a closed system, with respect to cells. Gliomas do not cause clinical metastases outside the cranial cavity, except with extreme rarity (Waite et al., 1999). This suggests that tumour cells do not leave the cranium, so treating this as a closed system is a reasonable assumption. There are suspicions that tumour cells may escape into the peripheral circulation, based on the observation that recipients of organ donation from patients with GBM can develop systemic GBM (Frank et al., 1998; Old et al., 1998). However, it seems likely that the number of cells escaping the cranium must be small. In the model, we have deliberately assumed that the
radiotherapy is given as an instantaneous treatment. This means that patients cannot deteriorate during the course. In practice, this problem affects very few patients, probably because of the stringency of patient selection, at both presentation and start of treatment. The instantaneous treatment also means that the model at present cannot examine the effects of interruptions in treatment, such as Christmas holidays. It also prevents direct comparison with palliative radiotherapy series for patients with GBM
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(typically given with large fractions over 2 weeks), or with clinical trials using altered fractionation for other tumours (such as Continuous Hyperfractionated Accelerated Radiotherapy (CHART) for lung cancer) (Saunders et al., 1997). The incorporation into the model of real fractionation patterns will require a modest but significant increase in computational effort. We intend to develop the model further to include treatments with different overall treatment times, and altered fractionation.
5.8. Clinical applications of the model
The model has the potential to aid the design of clinical trials, particularly by suggesting the magnitude of the difference that might be seen between standard and experimental treatment arms. This is needed in order to calculate the size of a trial required to achieve the necessary statistical power. The size consideration should also indicate whether a trial would be possible in patients with a relatively rare tumour. The model is built in a generic form, so that it could be applied to other tumours, provided the patient population data is available for the necessary fitting.
The model has allowed us to explore the potential role of increase in radiotherapy dose in patients with GBM. This appears to warrant further investigation, provided that such dose escalation can be delivered safely. Such a study might not have been considered, based on existing small- scale clinical data.
5.9. Development of other models
The model described in this paper is complementary to, rather than in competition with, work on more detailed aspects and features of this problem. For example, Swanson et al. (2002) have attempted to model tumour growth and cell migration in gliomas. This model was able to describe invasion and also predict a role for imaging techniques with better detection thresholds. Other model- ling, with more structural features, includes the work by Chaplain, Byrne and co-workers (Byrne and Chaplain, 1995; Chaplain, 2000; Levine et al., 2001) which describes a plausible route to angiogenic processes.
Some work has described the effect of radiotherapy on tumours. For instance, it is known that radiosensitivity is distributed around the cell cycle but few models have addressed this issue. Cell population modelling, with the incorporation of cell cycle-dependent radiation sensitivity, is at an early stage of development (Alarcon et al., 2003; Alarcon et al., 2005, Kirkby et al., 2002a) but shows considerable promise. The development of multi-scale models, incorporating cellular information, structural information and clinical population data may enhance our understanding of current treatment protocols and potential future treatment strategies (Antipas et al., 2004, Dionysiou et al., 2004).
6. 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 include other treatment modalities, such as chemotherapy. A number of other developments are suggested by the results so far, which we hope to explore further.
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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Introduction
Growth of tumour
Parameter distributions
Results
Age distribution of tumours at presentation
Sensitivity analysis
Parameter values and sensitivity analysis
Other assumptions in the model
Clinical applications of the model
Development of other models
A mathematical model of the treatment and survival of patients with high-grade brain tumours
Norman F. Kirkbya,, Sarah J. Jefferiesb, Raj Jenab,c, Neil G. Burnetb,c
aFluids & Systems Research Centre, School of Engineering (D2), University of Surrey, Guildford, Surrey GU2 7XH, UK bOncology Centre, Addenbrooke’s Hospital, P.O. Box 193, Cambridge CB2 2QQ, UK
cDepartment of Oncology, University of Cambridge, Oncology Centre, Addenbrooke’s Hospital, P.O. Box 193, Cambridge CB2 2QQ, UK
Received 13 April 2006; received in revised form 23 August 2006; accepted 6 September 2006
Available online 16 September 2006
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.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Brain cancer; Radiotherapy; Glioblastoma; Patient survival; Tumour growth
1. 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
e front matter r 2006 Elsevier Ltd. All rights reserved.
i.2006.09.007
esses: [email protected] (N.F. Kirkby),
brookes.nhs.uk (R. Jena), [email protected]
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, mortal- ity 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
Nomenclature
b bandwidth for approximation of distribution of tumour age at presentation (days)
C(t) number of cancer cells in the brain (cells) C0 number of cancer cells in brain at presentation
(cells) j number of fractions of radiotherapy (dimen-
sionless) kc rate constant for cancer cell growth (days1) kn rate constant for normal cell damage
(cells1 days1) ks normalization constant used in pdf for survival
fraction (Eq. (17)) (dimensionless) m exponent used in pdf for survival fraction (Eq.
(17)) (dimensionless) n exponent used in pdf for survival fraction (Eq.
(17)) (dimensionless) N(t) number of normal brain cells (cells) NBrain number of normal cells in a healthy brain (cells) Ncrit critical number of normal brain cells required
for patient to remain alive (cells) Np number of patients in virtual clinical trial
(dimensionless)
N0 number of normal brain cells left at presenta- tion (cells)
rc number growth rate of cancer cells (cells day1) rn number rate of damage to normal cells
(cells day1) t time (days) tage tumour age at presentation (days) tage,i tumour age at presentation in the ith patient
(days) tD tumour doubling time (days) tdelay time to commence treatment from presentation
(days) tp patient survival time from presentation if left
untreated (days) ts patient survival time from presentation (days) tt patient survival time from treatment if left
untreated (days) xs cancer cell survival fraction in response to one
fraction of radiotherapy (dimensionless)
a constant in pdf for survival fraction (Eq. (17)) (dimensionless)
N.F. Kirkby et al. / Journal of Theoretical Biology 245 (2007) 112–124 113
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 radiother-
apy 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 perfor- mance status is excellent, because such treatment is arduous. Patients are assessed for performance status at presentation to the oncology department and are reas- sessed prior to commencing treatment. There is always an interval between the decision to treat and the commence- ment 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, b, 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
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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.
2. 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 ¼ 0 represents the time of first presentation to a hospital oncology unit.
2.1. Patient death
A patient dies when the number of undamaged normal cells drops below a threshold value. Let N(ts) represent the number of normal cells in the brain. The patient dies when
NðtsÞ ¼ Ncrit, (1)
where ts is the survival time and Ncrit is the number of normal brain cells that constitute the minimum, critical
threshold. We currently assume that Ncrit is not affected by any treatment modality including radiotherapy.
2.2. Interaction between tumour and normal brain
The model assumes that the processes of damage to the normal cells due to the presence of the tumour can be represented as an irreversible elementary chemical reaction as follows:
N þ C! kn
C. (2)
Normal cells are assumed to be destroyed in a process related to the size of the tumour. This damage process is modelled as follows: let rn be the number rate of damage of normal cells. This is assumed to be related to the number of normal cells (N) and to the number of cancer cells (C) as follows:
rn ¼ knNC. (3)
2.3. Growth of tumour
If it is assumed that the growth of the tumour mass is a simple, first-order process, the number rate of growth is given by
rc ¼ kcC. (4)
In order to develop a solution to this model, it is assumed that the brain and the tumour are both closed systems with respect to cells, so that the number balances may written as follows:
dN
and
dC
2.4. General solution
Given the initial condition for cancer cells at presenta- tion,
Cðt ¼ 0Þ ¼ C0. (7)
Eq. (6) has the conventional, simple solution
CðtÞ ¼ C0 expðkctÞ, (8)
which relates to the tumour doubling time, tD, by
tD ¼ ln 2
The solution to Eq. (5) can now be developed as
dN
and on integration
ðexpðkctÞ 1Þ, (11)
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where N0 is the number of normal brain cells left at presentation (t ¼ 0). Eq. (11) can be simplified by back substitution of the number of cancer cells to give
ln N
ðC C0Þ. (12)
When required, Eq. (12), the state-space equation, is used to calculate the size of the tumour from the number of normal brain cells remaining.
Another application of Eq. (10) is that it allows the calculation of the age of the tumour at presentation, provided we assume that the tumour starts with a single cancer cell, with the rest of the brain completely intact (i.e. N ¼ NBrain). Therefore the tumour age is
tage ¼ 1
NBrain
N0
. (13)
The above model describes the growth of the tumour and the resulting damage done to the normal brain. Once the tumour is growing, a variety of symptoms may result in the patient seeking medical advice and, in time, a hospital appointment. On rare occasions this delay may result in death before presentation to hospital. Once in hospital it is assumed that diagnosis is made and the patient can be referred for treatment.
2.5. Model of treatment
In this model, the time delay between the patient presenting to the oncology department and the commence- ment of radiotherapy is a variable represented as tdelay. The delay to start treatment was included in the model, despite there being specific clinical data available, in order to be able to examine the effects of alteration in this delay.
Radical radiotherapy is normally given as a ‘‘fractio- nated’’ course, where the total dose is divided into multiple, equal, small-dose exposures, typically given once a day for 5 days a week. This ‘‘fractionation’’ has the effect of sparing normal tissue, in this case normal brain, relative to tumour. In the radical treatment of gliomas, 30 exposures are normally administered. Of those patients that are treated, it is assumed in this model that the radiotherapy is applied instantaneously, and that all of the j exposures are delivered at the same instant, and each results in the same fraction of tumour cells surviving, denoted here as xs. Hence,
C tþdelay
s, (14)
where tþdelay represents a time just after the radiation treatment, and tdelay a time immediately before the exposure. This expression was used in order to avoid introducing further variables describing the exact pattern of delivery and its effect on tumour growth (Kirkby et al., 2002a, b).
It is also assumed that all the normal brain cells survive the treatment, so, for normal brain cells,
N tþdelay
. (15)
On the basis of an individual patient, the tumour is assumed to be sterilized if
C tþdelay
o1. (16)
If the tumour is not sterilized, Eqs. (14) and (15) form the initial conditions for the regrowth of the tumour, and it is assumed that the parameters of the regrowth are the same as those that applied before treatment.
3. Model of a population of patients
3.1. 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)
(3)
(4)
(5)
The critical size of undamaged brain (Ncrit).
The remaining parameter to be treated statistically is the survival fraction of the cancer cells in response to a single exposure of radiation, xs. It is not appropriate to regard this parameter as being normally distributed since, for instance, the value must lie in the interval (0,1). Hence, the following probability density function was used:
pðxsÞ ¼ ksx n s ð1 xsÞ
meaxs . (17)
This distribution is not easily integrated in order to determine the normalization constant ks, the mean or the variance as functions of the three parameters n, m and a. There is also the need to generate a variate with this distribution. For the normal variates, the method of Box–Muller was used, but for the survival fraction, xs, a simple acceptance/rejection test was used. The value of the constant, ks, was determined by
trapezoidal integration of the denominator of the following equation required for normalization:
ks ¼ 1R 1
meaxs dxs
. (18)
In the course of the numerical integration of the denominator in Eq. (18) the maximum height of the
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function can be recorded. This was used to maintain a reasonable efficiency for the acceptance/rejection test, which otherwise used two uniformly distributed random variates, one of which was scaled to this maximum height of the function. Fig. 1 shows an example of Eq. (17) and a reconstructed distribution from the sampling method used.
Although there are four parameters required to describe the distribution of xs, the survival fraction, this represents only three degrees of freedom once the normalization procedure is completed. Each parameter, which is assumed to have a normal distribution, has two degrees of freedom. Therefore, there are a total of 13 adjustable parameters in this model.
3.2. Model of patient selection
In an attempt to mimic real clinical practice, it is assumed that clinicians are able to calculate exactly how long each patient will survive if left untreated and that two decisions that patients are suitable for radical treatment are made for each patient: the first at presentation and the second at the start of treatment. When first seen, if a patient will survive untreated longer than a specified time, tp, the patient is deemed sufficiently fit to be accepted for treatment. Typically, this is assumed to be of the order of 8 weeks. The criterion for selection is explicitly as follows:
tpp 1
Ncrit
N0
. (19)
Following the delay to treatment, each patient is reassessed just before the start of treatment, exactly as would happen clinically, and if the patient will survive untreated for longer than a specified time, tt (measured from the start of treatment), the patient is treated. This is a surrogate for excluding patients whose condition has deteriorated and who would not benefit from this treatment.
0
0.5
1
1.5
2
2.5
3
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Survival Fraction
P ro
b ab
ili ty
D en
si ty
As sampled
As required
Fig. 1. Theoretical distribution of survival fraction after one dose of
radiotherapy, showing both the required shape and the actual sampled
distribution, to demonstate that the acceptance/rejection method func-
tions correctly. This is not the fitted distribution, which is shown below
(Fig. 5).
Therefore, patients are selected if the following condition is also met:
tt þ tdelayp 1
Ncrit
N0
. (20)
In reality, clinical decisions are imperfect, but these rules are intended to represent a simplified version of real clinical practice. In this way, only patients who are considered fit enough are accepted for treatment. The two rules are applied independently which implies, for instance, that the consultant does not know what the delay to treatment will be at the time the decision is made to offer radical treatment. These two rules are a surrogate for good clinical performance status, that is, excellent health with minimal or no neurological deficit, but these rules are a very simple representation of a rather complicated situation.
3.3. Simulation of patient population
Monte Carlo simulation was used to generate a population of patients. In this case, it was generally found that a population of 2000 patients had to be generated so that the resulting patient population survival curve was independent of any further increase in the number of patients simulated. A larger number of patients must initially be generated, to allow for exclusion of those that ‘fail’ the selection criteria for patient fitness described above. The results of the simulation are presented as a
Kaplan–Meier survival curve in Fig. 2, i.e. a graph of the proportion of patients surviving versus time (Kaplan and Meier, 1958). This method allows for censorship of patients who are surviving at the time of interest. This is the standard method for presenting clinical survival data, though, in the case of the model, this censorship is redundant.
Time (days)
C um
ul at
iv e
S ur
vi va
l
0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Addenbrooke’s patient survival
Fig. 2. Kaplan–Meier survival curve for the Addenbrooke’s patient
population, and the fitted model.
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Table 1
Detail of values of the variables from the fit to the Addenbrooke’s patient
data
Factor
SF2a
Tumour doubling
31.5 daysc 183 days2c
aThese parameters are used in the probability density function for SF2.
These values give a mean SF2 ¼ 0.80, and median ¼ 0.83. bParameters not adjusted for this fitting. cParameters from clinical series.
N.F. Kirkby et al. / Journal of Theoretical Biology 245 (2007) 112–124 117
For each patient a tumour age is calculated and the probability density function for these ages is approximated using a Gaussian kernel density function (Silverman, 1998), as follows:
pðtageÞ ¼ 1
2 . (21)
The value of the bandwidth, b, was selected by trial and error. For a virtual trial of 2000 patients it was found acceptable to relate b to the standard deviation of the raw data for the tumour age distribution as follows:
b ¼ 0:2SDðtage;iÞ. (22)
3.4. Clinical data
Clinical data were available for a set of 154 adult patients with GBM who were given radical treatment in our unit between 1996 and 2002. Patients are only accepted for this treatment programme if they are in excellent health with minimal or no neurological deficit, in other words with normal performance status. Kaplan–Meier survival analysis was available (Kaplan and Meier, 1958). All patients were treated in 30 exposures, delivered daily for 5 days per week, delivering a total dose of 60Gy (Burnet et al., 2006). No adjuvant chemotherapy was given during this period. Data were available for the delay to start radiotherapy: mean 32 days, variance 183 days2 (SD 13.5 days).
3.5. Determination of the parameters of the model
The model is fitted to clinical survival data by minimiz- ing the weighted sum of squares of errors between the simulated and real Kaplan–Meier survival curves. The algorithm used for this minimization was the simulated annealing of folding polygons, as described by Press et al. (1996). The weighting was adjusted heuristically to guide the optimization to fit correctly the tail of the curve representing the long-term survivors.
Some of the parameters of the model can be calculated a priori, because they are contained within the clinical data, specifically the mean and variance of the delay to start treatment, noted above. Three further parameters have been estimated, namely the mean and variance of the normal brain cell number at presentation, and the mean number of normal brain cells remaining at the time of death. The estimates of mean and variance of the normal brain cell number at presentation are consistent with brain volume calculations for a data set of 100 patients, performed in relation to an anatomical study of the brain (not published).
The parameters of the distribution of xs are impossible to measure in vivo in patients, although some laboratory
data exist which suggest approximate values, taking into account the typically hypoxic nature of GBM. However, there are some parameters whose values are entirely unknown, without fitting. In particular, the mean and variance of the interaction constant, kn, are untestable. Typically, eight parameters were determined by fitting,
namely:
at the time of death,
mean and variance of the tumour doubling time tD the
three parameters determining xs, i.e. m, n and a.
4. Results
4.1. 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 real clinical data, because standard techniques such as logrank testing depend upon a specific size of the study population. Since the modelled population can be of any size, and there are
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M ea
n D
el ay
to S
ta rt
T re
at m
en t (
da ys
30.00
30.50
31.00
31.50
32.00
32.50
9.10E+11 9.20E+11 9.30E+11 9.40E+11 9.50E+11 9.60E+11 9.70E+11 9.80E+11 9.90E+11 1.00E+12
Fig. 3. Plane of contours of sum of squares of errors for two representative variances, mean delay to start treatment in days and mean number of normal
cells remaining at patient death, showing the covariance between them.
N.F. Kirkby et al. / Journal of Theoretical Biology 245 (2007) 112–124118
advantages in having a very large study population, such methods are invalidated. For further analysis see Burnet et al. (2006).
In the course of generating this fit, it was noted that several of the parameters are strongly covariant; to a large extent the variation in the population can be achieved by altering any of the variances in the main distributions. Fig. 3 shows a plane of contours of sum of squares of errors for 2 representative variances, showing this relation- ship. Encouragingly, many other parameters of the model are fitted without this covariance. An example is shown in Fig. 4. It is clear from Figs. 3 and 4 that the sum of squares of errors in the parameter space is complex and con- voluted. Therefore it is not possible to guarantee that the best possible fit has been discovered.
4.2. Distribution of radiation sensitivity and survival fraction
The shape of this survival fraction distribution is substantially skewed, as shown in Fig. 5. This distribution has a mean survival fraction of 0.80, and a median of 0.83. As discussed in Burnet et al. (2006), the results are consistent with those found from in-vitro experiments, especially allowing for the effects of hypoxia in increasing radio-resistance.
Since each radiation exposure was given in a dose of 2Gy, this figure shows the distribution of SF2, the survival
fraction after 2Gy. Two gray is a commonly used dose per exposure in clinical practice, and SF2 has been used extensively as a descriptor of cellular radiation sensitivity (Deacon et al., 1984).
4.3. Age distribution of tumours at presentation
The model predicts the age distribution of tumours at the time of presentation, when they contain of the order of 1011
cells. The distribution is shown in Fig. 6. The mean age at presentation is 1034 days (median 1029) and variance 8.6 104 days2. The distribution is slightly skewed (skew- ness is 0.25), but otherwise almost normal (kurtosis 3.08). However, it should be noted that the variance in tumour age results mainly from the variances of tumour doubling time and number of normal brain cells at presentation. These variances are co-variant with the variances of some of the other parameters in the model, and none has been estimated well.
4.4. Sensitivity analysis
A sensitivity analysis was conducted by varying each parameter separately by 710% from its best-fit value. The resulting sum of squares of errors was compared to the sum of squares of errors at the best fit, and the results are presented in Fig. 7. From this it is clear that the model
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V ar
ia nc
e of
D el
ay to
S ta
rt T
re at
m en
t ( da
ys 2 )
170.00
172.00
174.00
176.00
178.00
180.00
182.00
184.00
186.00
9.10E+11 9.20E+11 9.30E+11 9.40E+11 9.50E+11 9.60E+11 9.70E+11 9.80E+11 9.90E+11 1.00E+12
Fig. 4. Plane of contours of sum of squares of errors for two parameters well fitted in the model, variance of delay to start treatment (in days2) and mean
number of normal cells remaining at patient death, showing absence of covariance. This contrasts with Fig. 3.
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Survival Fraction
P ro
b ab
ili ty
D en
si ty
o f
S u
rv iv
al F
ra ct
io n
Fig. 5. Distribution of survival fraction after one dose of radiotherapy, xs,
resulting from fitting to the Addenbrooke’s data. The mean value of SF2 is
0.80, and the median is 0.83.
Tumour Age (days)
(1 /d
ay s)
0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400
0
0.0002
0.0004
0.0006
0.0008
0.0010
0.0012
0.0014
Fig. 6. Probability distribution of the age of tumours at presentation. The
mean is 1034 days, median 1029, and variance 86436 days2 (SD 294 days);
the skewness is 0.25, and kurtosis 3.08.
N.F. Kirkby et al. / Journal of Theoretical Biology 245 (2007) 112–124 119
is vastly more sensitive to some parameters than others, and not necessarily symmetrical even in the vicinity of the best fit.
In general terms, the sum of squares of errors is more sensitive to variation in the means of the model parameters than the variances. The mean of the interaction constant is the exception. The sum of squares of errors is also sensitive to the survival fraction parameters.
4.5. Clinical selection criteria
Within the model, a substantial number of patients are rejected from possible treatment. The majority are rejected because the model generates patients with implausible characteristics, such as tumours larger than the brain at presentation. This is the result of using independent normal distributions for the parameters of the model. In addition,
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Fig. 7. Sensitivity analysis of the main parameters in the model, ranked in order of sensitivity. Note that numbers given as 0 are below 0.5%, not actually
zero.
N.F. Kirkby et al. / Journal of Theoretical Biology 245 (2007) 112–124120
patients are rejected whose tumours render them too poorly to be accepted into the radical treatment pro- gramme. After presentation, and before treatment com- mences, some patients become too unwell and some even die; these are rejected by the relevant clinical criteria. For 1000 patients treated, 171 deteriorate and 47 die after acceptance but before treatment commences.
5. 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 be of sub- stantial value. We have sought to develop such a model to address this serious clinical problem. The model was designed to describe growth of a tumour and consequent patient death. Some of the parameter values were determined by fitting the model to real clinical data, and appear to be biologically plausible.
The model demonstrates that a delay to start radio- therapy has an obvious, predictable, deleterious effect of patient survival. This aspect of our results has been discussed elsewhere (Burnet et al., 2006).
5.1. Fit to clinical data
The population survival curve for patients with GBM is rather unusual in having such a small number of long-term survivors; most cancers have a rather better cure rate. It is tempting to ignore this group because it is so small, but they are important. To the patients themselves, long-term survival is enormously significant. Developments in treat- ment, which might improve outcome, should include increasing the proportion of patients within this group. Thus, the model must predict the existence of this group if it is to function as an effective tool to evaluate new treatment strategies. The fit to the clinical data achieved by the model is
excellent (Fig. 2). The model successfully fits the tail of long-term survivors, as well as the bulk of the patient population. For statistical considerations see Burnet et al. (2006).
5.2. Clinical selection criteria
It is important to note that the model has been fitted to data for patients who have actually completed radical treatment. Patients who deteriorate whilst waiting to start treatment are actively switched out of the radical treatment programme in order to deliver a more appropriate palliative or supportive care package. Real clinical decision-making regarding a patient’s
treatment is clearly not as simple as has been assumed in
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the model. Nevertheless, there are real decision points in a patient’s treatment pathway, the first at presentation to oncology and the second at the start of treatment. If the patient’s condition has deteriorated, then radical treatment is abandoned and an alternative, palliative strategy initiated. Decision points have been included in the model, at these times, to mimic this clinical situation. In the model, the clinician ‘‘knows’’ how long the patient will survive if untreated, but in reality only the current and previous performance status is known.
Within the model, a substantial number of patients are rejected from possible treatment at the first decision point because the model generates patients with implausible characteristics. There is no clinical equivalence of this group. At the second decision point, at the time of starting treatment, for each 1000 patients treated in the model, a further 171 are rejected because they deteriorate before treatment, and 47 more die after acceptance but before treatment commences, who are also not treated. This amounts to 18% of patients ‘‘accepted’’ for radical treatment in the model. In reality, 3% (4 of 123 with definite treatment data) of patients deteriorate whilst waiting to start treatment, and do not proceed with radical treatment.
These figures appear to suggest that the clinicians are better predictors of outcome than the model, despite it containing explicit information about patient survival. However, it is more likely that the model acceptance criteria are less strict than those used clinically. The model acceptance criteria are not distributed, and their values are not derived from fitting. This represents an area for future refinement of the model.
Performance status is not explicitly represented in the model, because all patients accepted for treatment with curative intent are required to be of a uniform excellent performance status. Therefore, it is not possible at present to model the outcome of those patients who deteriorate before treatment, or indeed those who are less well at presentation, who are initially offered palliative treatment. The inclusion of patients undergoing palliative radio- therapy is also an area for future work.
5.3. Radiation sensitivity
The results of the fitting, with such close agreement between the real clinical data and the modelled population, indicate that a second, separate population is not biologically necessary (Fig. 2). More specifically, the distribution of xs, the fraction of tumour cells surviving each radiation exposure, is unimodal. This distribution is skewed, indicating that a much smaller proportion of patients have sensitive tumours than resistant ones. Since the parameters of the distribution are derived from fitting to the real clinical data, this is a plausible result. It is also consistent with published data of intrinsic tumour cell radiosensitivity (West et al., 1993; Bjork-Eriksson et al., 1998), and normal fibroblast sensitivity (Peacock et al.,
2000). Although these distributions are often considered to be Gaussian, measurements indicate that in reality they are typically slightly skewed. The absolute values of estimates of xs, or SF2, from the
model represent in vivo estimates, and are consistent with the presence of severe hypoxia, which renders cells less sensitive to radiation. In patients, GBMs are known to contain severely hypoxic areas (Rampling et al., 1994). The results are also consistent with in vitro data, where hypoxic conditions have been used (Taghian et al., 1995); see Burnet et al. (2006) for further details. The model also predicts dependence of patient outcome on SF2, which has not been uniformly reported in the oncology literature (Taghian et al., 1993). Our work suggests that treatment strategies to address radiation resistance in general, and hypoxia in particular, are warranted. The model can be used to evaluate the effect of
radiotherapy dose escalation on the patient population. Our evaluation of this (Burnet et al., 2006) suggests that an escalation from 60Gy to 74Gy would increase the survival time of all patients treated, and also improve the proportion of patients achieving long-term survival, albeit modestly, from 2.4% to 6.4%. This would require that the treatment can be delivered safely, but imaging techniques which allow the individualization of the target volume might achieve this (Jena et al., 2005).
5.4. Tumour growth and patient performance status
Within the model the most central assumption is of exponential tumour growth. Therefore, decline in the number of remaining normal brain cells invariably accel- erates with tumour age. This would have the effect of producing a rapid and accelerating decline in performance status in model patients. Clinical experience suggests that this is realistic for many patients; others have a more uniform decline before death, which may be accounted for by a rather slower growth rate. An obvious and important area for future work is to
model the kinetics of tumour growth in more detail. The ability to model a tumour whose doubling time increases with tumour age would be helpful. This may of itself predict that some patients do not suffer an accelerating decline in normal brain cell number, and hence account for a slower, steady decline in performance status. More realistic tumour growth kinetics are an area for future development. However, we feel justified in starting where we have, because so much mathematical effort is currently being directed at this feature of tumour growth.
5.5. Age distribution of tumours at presentation
The model can predict the age distribution of tumours at the time of presentation (Fig. 6). The distribution is slightly skewed. This may be due to the exclusion of some young, rapidly growing tumours, which may cause early deteriora- tion in the patient’s clinical condition, leading to their
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exclusion from the radical treatment programme. The presence of a relatively wide range in tumour age at presentation, and the relatively high mean value, is clinically plausible, in our view. Patients present with a wide range of symptoms, but only once the tumour has reached a substantial size. Some of the common symptoms of high-grade brain tumour, such as headache, are also non-specific, which causes difficulty in rapid diagnosis. Moreover, tumours are known to have a wide variation in clinical growth rates. Therefore, the predicted range of age at presentation is credible, though it is impossible to validate clinically.
Our predictions of tumour age are dependent on the underlying assumption of exponential tumour cell growth. Whilst this is most likely to be a realistic representation of growth in the early phase, it is not true of older tumours in which nutrient and oxygen limitations occur through inefficient angiogenesis and neovascularization. Thus, most tumours have been growing faster in the past, so that the predictions of tumour age are likely to be over-estimates.
5.6. Parameter values and sensitivity analysis
The most striking feature of the sensitivity analysis (Fig. 7) is that the mean number of normal cells at presentation (N0) is the parameter, which shows the largest effect on the sum of squares of errors. Furthermore, the difference between increasing and decreasing this para- meter is a strong indication of the nonlinearity of the parameter space. Thus, the estimate for N0 is likely to be our most accurate and could in principle be measured clinically.
The second most influential parameter in the sensitivity analysis is the mean doubling time (Fig. 7). The value of 24.1 days indicates that these tumours may progress very rapidly, leading to deterioration in the patient’s clinical condition. In patients with GBM, this decline is typically not reversible with treatment. The fact that this parameter has been estimated with reasonable certainty is helpful, particularly in promoting the importance of a rapid patient journey from diagnosis to definitive treatment for patients with GBM (Burnet et al., 2006).
In the sensitivity analysis, the sum of squares of errors is generally more sensitive to variation in the means of the model parameters than the variances. The mean of the interaction constant is an exception; this may be reasonable because there are several mechanisms of interaction between tumour and normal brain tissue. For example, generalized pressure effects from a large tumour in a non- eloquent part of the brain may produce death through a different mechanism than direct damage to a particular region of normal brain which is critical to life, such as the brain stem. Thus, each separate interaction mechanism may be characterized by very different values for the respective interaction constants. Hence, lumping these phenomena together may have resulted in the relatively poor determination of both mean and variance of the
interaction constant. This is envisaged as an area for development of the model. However, detailed analysis of the mechanism of death in individual patients is not clinically available. Some indication of the sensitivity of the system is
provided in Fig. 7. In general, the sum of squares of errors is more sensitive to variation in the means of the model parameters than their variances. This is in part to be expected because the means relate strongly to the bulk of the cases in the distributions of the data, while the variances affect the relatively unusual cases.
5.7. Other assumptions in the model
A number of assumptions are required within the model. We have made no attempt to link the number of normal cells remaining to the severity of neurological deficit. However, subjectively, clinical observations suggest a relationship of this general type. More detailed examina- tion of this interaction would be of interest. It must, however, take into account differences in the clinical effect produced by tumours in different locations. For example, a small tumour in an eloquent area of brain such as the motor cortex can produce a substantial neurological deficit whilst one within a ‘‘silent’’ area of non-dominant poster- ior parietal lobe might well be substantial larger before causing any clinical effect. We have not included patient age as a specific parameter
affecting survival although it is known to be highly prognostic. Because of the excellent fit to the real clinical data, it seems possible that there is covariance between age and another variable in the model, such as the critical number of normal cells at death, or tumour doubling time. However, this is a weakness of the current model and is
an area for future work. In the model it is assumed that the normal brain and
tumour are a closed system, with respect to cells. Gliomas do not cause clinical metastases outside the cranial cavity, except with extreme rarity (Waite et al., 1999). This suggests that tumour cells do not leave the cranium, so treating this as a closed system is a reasonable assumption. There are suspicions that tumour cells may escape into the peripheral circulation, based on the observation that recipients of organ donation from patients with GBM can develop systemic GBM (Frank et al., 1998; Old et al., 1998). However, it seems likely that the number of cells escaping the cranium must be small. In the model, we have deliberately assumed that the
radiotherapy is given as an instantaneous treatment. This means that patients cannot deteriorate during the course. In practice, this problem affects very few patients, probably because of the stringency of patient selection, at both presentation and start of treatment. The instantaneous treatment also means that the model at present cannot examine the effects of interruptions in treatment, such as Christmas holidays. It also prevents direct comparison with palliative radiotherapy series for patients with GBM
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(typically given with large fractions over 2 weeks), or with clinical trials using altered fractionation for other tumours (such as Continuous Hyperfractionated Accelerated Radiotherapy (CHART) for lung cancer) (Saunders et al., 1997). The incorporation into the model of real fractionation patterns will require a modest but significant increase in computational effort. We intend to develop the model further to include treatments with different overall treatment times, and altered fractionation.
5.8. Clinical applications of the model
The model has the potential to aid the design of clinical trials, particularly by suggesting the magnitude of the difference that might be seen between standard and experimental treatment arms. This is needed in order to calculate the size of a trial required to achieve the necessary statistical power. The size consideration should also indicate whether a trial would be possible in patients with a relatively rare tumour. The model is built in a generic form, so that it could be applied to other tumours, provided the patient population data is available for the necessary fitting.
The model has allowed us to explore the potential role of increase in radiotherapy dose in patients with GBM. This appears to warrant further investigation, provided that such dose escalation can be delivered safely. Such a study might not have been considered, based on existing small- scale clinical data.
5.9. Development of other models
The model described in this paper is complementary to, rather than in competition with, work on more detailed aspects and features of this problem. For example, Swanson et al. (2002) have attempted to model tumour growth and cell migration in gliomas. This model was able to describe invasion and also predict a role for imaging techniques with better detection thresholds. Other model- ling, with more structural features, includes the work by Chaplain, Byrne and co-workers (Byrne and Chaplain, 1995; Chaplain, 2000; Levine et al., 2001) which describes a plausible route to angiogenic processes.
Some work has described the effect of radiotherapy on tumours. For instance, it is known that radiosensitivity is distributed around the cell cycle but few models have addressed this issue. Cell population modelling, with the incorporation of cell cycle-dependent radiation sensitivity, is at an early stage of development (Alarcon et al., 2003; Alarcon et al., 2005, Kirkby et al., 2002a) but shows considerable promise. The development of multi-scale models, incorporating cellular information, structural information and clinical population data may enhance our understanding of current treatment protocols and potential future treatment strategies (Antipas et al., 2004, Dionysiou et al., 2004).
6. 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 include other treatment modalities, such as chemotherapy. A number of other developments are suggested by the results so far, which we hope to explore further.
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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Introduction
Growth of tumour
Parameter distributions
Results
Age distribution of tumours at presentation
Sensitivity analysis
Parameter values and sensitivity analysis
Other assumptions in the model
Clinical applications of the model
Development of other models