Citation preview
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: n.kirkby@surrey.ac.uk (N.F. Kirkby),
brookes.nhs.uk (R. Jena), ngb21@cam.ac.uk
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
ARTICLE IN PRESS N.F. Kirkby et al. / Journal of Theoretical
Biology 245 (2007) 112–124114
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)
ARTICLE IN PRESS N.F. Kirkby et al. / Journal of Theoretical
Biology 245 (2007) 112–124 115
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
ARTICLE IN PRESS N.F. Kirkby et al. / Journal of Theoretical
Biology 245 (2007) 112–124116
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.
ARTICLE IN PRESS
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
ARTICLE IN PRESS
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
ARTICLE IN PRESS
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,
ARTICLE IN PRESS
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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Biology 245 (2007) 112–124 121
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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Biology 245 (2007) 112–124122
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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Biology 245 (2007) 112–124 123
(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