Predicting regrowth of low-grade gliomas after radiotherapy

Stéphane Plaszczynski 1,2*, Basile Grammaticos 1,2, Johan Pallud 3, 4, 5,
Jean-Eric Campagne 1,2, and Mathilde Badoual 1,2

1 Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France;

2 Université Paris-Cité, IJCLab, 91405 Orsay, France;

3 Department of Neurosurgery, GHU Paris Sainte-Anne Hospital, 75014 Paris,
France;

4 Université de Paris, Sorbonne Paris Cité, Paris, France;

5 Inserm, U1266, IMA-Brain, Institut de Psychiatrie et Neurosciences de Paris,
France.

*<EMAIL_ADDRESS>
## Abstract

Diffuse low grade gliomas are invasive and incurable brain tumours that
inevitably transform into higher grade ones. A classical treatment to delay
this transition is radiotherapy (RT). Following RT, the tumour gradually
shrinks during a period of typically 6 months to 4 years before regrowing. To
improve the patient’s health-related quality of life and help clinicians build
personalised follow-ups, one would benefit from predictions of the time during
which the tumour is expected to decrease. The challenge is to provide a
reliable estimate of this regrowth time shortly after RT (i.e. with few data),
although patients react differently to the treatment. To this end, we analyse
the tumour size dynamics from a batch of 20 high-quality longitudinal data,
and propose a simple and robust analytical model, with just 4 parameters. From
the study of their correlations, we build a statistical constraint that helps
determine the regrowth time even for patients for which we have only a few
measurements of the tumour size. We validate the procedure on the data and
predict the regrowth time at the moment of the first MRI after RT, with
precision of, typically, 6 months. Using virtual patients, we study whether
some forecast is still possible just three months after RT. We obtain some
reliable estimates of the regrowth time in 75% of the cases, in particular for
all “fast-responders”. The remaining 25% represent cases where the actual
regrowth time is large and can be safely estimated with another measurement a
year later. These results show the feasibility of making personalised
predictions of the tumour regrowth time shortly after RT.

## Introduction

Diffuse gliomas are primary brain tumours originating from glial cells
(oligodendrocytes and/or astrocytomas). In its 2016 classification, The World
Health Organization defines four grades [1]: while the first grade gliomas are
benign, second grade gliomas (or low grade gliomas, LGG) are invasive, growing
at a rate of $2$ to $8$ mm/year [2] in diameter, but without involving
metastasis or necrosis. Unfortunately, they cannot be cured by oncological
treatments [3] so one needs to contain their growth as long as possible,
before they transform into grade III and IV (glioblastomas) with a
dramatically low survival rate. LGG are detected with magnetic resonance
imaging (MRI) scans under a T2-FLAIR sequence. Since they are diffuse tumours
that extend beyond the observed boundaries [4, 5], the uncertainty on their
size is irreducible. Classical treatments include resection (when possible),
chemotherapy and radiotherapy (RT) [6].

Standard conformational radiotherapy for LGG is generally performed during 6
weeks (5 days a week) and the classical dose is around 50 Gy. Irradiation of
gliomas involves a large number of physical processes [7] and its effect
varies across patients. However, some general features emerge: the tumour
shrinks during a period that varies between a few months and several years,
before regrowing at a rate similar to the one observed before radiotherapy.

Mathematical modelling of natural and under treatment tumour growth has a long
and rich history (in particular for gliomas, one can refer to the recent
review [8]). For invasive tumour such as gliomas that cannot be removed by
surgery, one aspect that is of special interest for clinicians is the response
of tumour to treatments and in particular, radiotherapy [9, 10, 11, 12, 13].
Its primary goal is to optimize treatments “virtually”: for example, choosing
the optimal radiation fractionation of doses [14], finding the best way to
combine it to chemotherapy [15] or studying its interplay with the immune
system [16]. Beyond describing qualitatively the different processes at stake,
the real usefulness of a model would be to predict the response of individual
patients to a treatment, even before the end of the treatment. Such
predictions would allow the clinician to personalise the follow-up (and the
treatment) for each patient. There has been some attempts to predict tumour
growth and the effects of treatments on individual patients. If purely
statistical or image-based models can be used to predict glioma growth [17],
mechanistic models are usually used for instance to predict the metastatic
relapse in breast cancer [18], tumour growth in leukaemia and ovarian cancer
[19], response of high grade gliomas to chemoradiation [20], or the patient-
specific evolution of resistance in the context of prostate cancer [21].

For low-grade gliomas, individualized predictions from the tumour size
dynamics and genetic characteristics, have been made for the response to a
chemotherapy treatment [22]. To our knowledge, such individual predictions do
not exist in the case of low-grade glioma and RT. In this article, we show
that it is possible to predict the evolution of LGGs under RT, for individual
patients, with an approach based on a practical mechanistic model, even in the
case where the number of patients is not sufficient to apply standard machine-
learning techniques.

In order to be used for predictions, a model should have a limited number of
parameters. Models that are too detailed are useful to describe qualitatively
the tumour evolution but usually involve too many unknown parameters [23].
Given the scarcity of clinical data, only a small number of parameters can be
deduced. The goal here is to keep as few as possible parameters but still to
capture the essential dynamics of the tumour.

In a previous work [13], we analysed a large number (43) of LGG radial
evolutions under RT and proposed a physically motivated model, with 4
parameters, that fitted well all the profiles of tumour evolution during
patient’s follow-up. Following that work, we now try to make predictions using
that model. This is challenging given the variety of possible in-vivo
reactions to radiation. We have chosen to focus on the moment when the tumour
stops shrinking and starts to regrow, what we call in the following the
“regrowth time”. This is an essential feature of the tumour dynamics for two
reasons. First, the patients often ask their clinician when the tumour will
regrow in order to plan some major life projects (as having a child,
travelling, retiring, etc.). This would be a valuable information to improve
their life-quality. Second, the the dates for the next MRIs are currently
fixed and not optimal on an individual basis. By making predictions we may
adjust them more precisely for personalised follow-ups.

##  1 Materials and Methods

###  1.1 Standard protocol approvals, registrations and patient consents

The study received required authorizations (IRB#1: 2021/20) from the human
research institutional review board (IRB00011687). The requirement to obtain
informed consent was waived according to French legislation (observational
retrospective study).

###  1.2 The patients

We had at our disposal a set of 43 patients with LGGs who were diagnosed at
the Sainte-Anne Hospital (Paris, France) from 1989 to 2000. These patients
were selected according to precise criteria that are detailed elsewhere [24].
In short, only adults with typical LGGs (that is, no angiogenesis and, thus,
no contrast enhancement on gadolinium-T1 images), available clinical and
imaging follow-ups before, during, and after RT, and RT as their first
oncological treatment except for stereotactic biopsies were eligible. The
external conformational RT was given using the same methodology (total dose,
50.4–54 Gy; 6-week period) at 2 outside institutions. The patients had an MRI
follow-up before, during, and after RT. Three tumour diameters in the axial,
coronal, and sagittal planes on each MRI image with T2-weighted and FLAIR
sequences were measured manually. The mean radiological tumour radius was
defined as half the geometric mean of these three diameters and was measured
as a function of time. The error bars for the measured mean radii were
estimated by clinicians and were set to $\pm 1$ mm. From this cohort, we
discarded the patients that did not have any sign of tumour regrowth at the
last time point or those that had fewer than five time points in their follow-
up.

###  1.3 The model

A biologically motivated model with the effect of RT on LGG has been presented
in [13] and validated by the fits on 43 patient follow-ups. It is based on a
standard diffusion-proliferation equation [25] and RT is modelled with a time-
dependent death rate ($\kappa_{D}(t)$). The evolution of the glioma cell
density then follows the equation

$\displaystyle\dfrac{\partial\rho}{\partial
t}=D\Delta\rho+[\kappa-\kappa_{D}(t)]\rho(1-\rho),$ (1)

where $\rho(r,t)$ is a function of the radius $r$ (assuming a spherical
symmetry) and time $t$ (conventionally set to zero at the beginning of RT),
$D$ is the diffusion coefficient and $\kappa$ the proliferation rate. In its
most simple (thus predictive) form, the death-term is characterized by an
amplitude and a characteristic time

$\displaystyle\kappa_{D}(t)=\kappa_{d}e^{-t/\tau_{d}}\quad\textrm{for}~{}t\geq
0,$ (2)

and is considered as null before RT.

Assuming that the tumour growth-rate when patients consult is already in the
asymptotic state, i.e. that it evolves linearly with a speed
$v=\sqrt{2D\kappa}$, and neglecting diffusion after RT, the radius evolution
can be approximated by [13]

$\displaystyle
R(t)=R_{0}+vt-v\tau_{d}\dfrac{\kappa_{d}}{\kappa}(1-e^{-t/\tau_{d}}).$ (3)

Inspired by this formula, we simplify the model eq. 1 by proposing a purely
geometrical one in the form

$\displaystyle R(t)=R_{0}+vt-k(1-e^{-t/\tau}).$ (4)

which has 4 free parameters: $R_{0},v,k,\tau$. We emphasize that eq. 4 should
be considered as an ad-hoc model that cannot be related to the one obtained by
solving numerically eq. 1 since eq. 3 neglects diffusion. This simple
geometrical model has the considerable advantage of being analytical. The role
of each terms is clear and sketched in Fig 1. It captures the 3 phases of the
evolution: first the linear growth, then the exponential decay of a fraction
of the tumour and therefore of its radius, third, the regrowth with the same
velocity as before RT.

Fig 1:  Illustration of the analytical model describing the tumour radial
evolution. Before RT ($t<0$), the radius evolves linearly with the asymptotic
speed $v$ and reaches $R_{0}$ at $t=0$. Then it becomes the sum of an
exponential decay of amplitude $R_{0}-k$ and characteristic time $\tau$ (death
term) and a linear $vt$ regrowth.

To test whether this model does fit our data appropriately, we construct a
classical objective function as the mean squared error from the set of
measured values $\\{t_{i},R_{i}\\}$

$\displaystyle\chi^{2}(R_{0},v,k,\tau)=\sum_{i=1}^{N}[R_{i}-R(t_{i};R_{0},v,k,\tau)]^{2}/\sigma_{i}^{2}$
(5)

where $\sigma_{i}=1$ mm. This is a 4-parameter real valued function that we
minimize easily with a standard optimization algorithm [26] since the model is
analytical. To obtain physical results, we impose as limits that all
parameters be positive and that the radial asymptotic speed lie in the range
$0.5\leq v\leq 4$ mm/yr [2]. We then obtain the best-fit parameter values
given in Table 1 and show, in Fig 2, the comparison between the data and the
fitted model on a set of 20 patients who possess at least 9 data points. The
agreement is excellent. Although the results are quite similar to the ones
obtained in [13], we have considerably simplified the model and reduced
drastically the run-time which will be useful later in making predictions.

id | $R_{0}$ | $v$ | $k$ | $\tau$ | $t_{min}$
---|---|---|---|---|---
(0) | 17.95 | 0.50 | 7.29 | 1.98 | 3.95
(1) | 29.03 | 1.33 | 10.49 | 0.19 | 0.72
(2) | 24.56 | 3.24 | 17.69 | 1.40 | 1.91
(3) | 16.86 | 0.96 | 13.28 | 1.57 | 3.42
(4) | 25.40 | 1.11 | 5.20 | 0.10 | 0.39
(5) | 28.72 | 1.37 | 10.78 | 0.47 | 1.33
(6) | 27.00 | 1.99 | 17.00 | 2.39 | 3.04
(7) | 23.26 | 1.23 | 5.92 | 1.18 | 1.66
(8) | 15.83 | 2.45 | 6.84 | 0.45 | 0.82
(9) | 31.60 | 1.13 | 8.35 | 0.38 | 1.12
(10) | 26.45 | 4.00 | 16.15 | 0.45 | 0.98
(11) | 14.67 | 1.54 | 4.66 | 0.32 | 0.72
(12) | 41.20 | 4.00 | 30.74 | 1.83 | 2.63
(13) | 16.64 | 3.59 | 13.83 | 1.06 | 1.37
(14) | 20.21 | 1.30 | 14.84 | 3.64 | 4.16
(15) | 19.33 | 0.70 | 7.80 | 0.81 | 2.11
(16) | 23.61 | 3.37 | 23.15 | 2.28 | 2.51
(17) | 32.68 | 0.72 | 7.30 | 0.43 | 1.36
(18) | 35.02 | 2.19 | 15.32 | 0.99 | 1.93
(19) | 28.06 | 0.52 | 10.16 | 1.29 | 3.52
Table 1: Parameters of the least-square solutions of eq. 5 corresponding to
the fits shown on Fig 2. The first columns represents the patients’ ID,
($R_{0},v,k,\tau$) are the estimated parameters of the model, and the regrowth
time ($t_{min}$) is derived from them. Lengths ($R_{0},k$) are expressed in mm
and times ($\tau,t_{min}$) in years. Fig 2: Comparison between the measured
values of the tumour radius and the bestfit model for 20 patients. The points
represent the measured values and the red line our model obtained by
minimizing eq. 5. The abscissa represent time in years (with the origin set at
RT) and the ordinate the tumour radius (in mm). The error bars on the
measurements are of 1 mm. The dashed vertical red line shows the model
minimum, i.e. the moment regrowth starts.

###  1.4 Constraining the parameters space

We now study whether some common features appear in our best-fit parameters.
Fig 3 shows the histograms for each parameter on the 20 patients.

Fig 3: Histograms of the bestfit parameters

No parameter displays a clearly peaked distribution. For a given patient, the
expected parameters are random variables and a priori unpredictable, although
within some bounds.

We now consider the correlation between the variables by computing their
Pearson coefficients and show the results in Table 2.

| $R_{0}$ | $v$ | $k$ | $\tau$
---|---|---|---|---
$R_{0}$ | 1 | 0.17 | 0.46 | -0.09
$v$ |  | 1 | 0.73 | 0.10
$k$ |  |  | 1 | 0.52
$\tau$ |  |  |  | 1
Table 2: Correlation coefficients measured between the 20 bestfit parameters
of our model. Since the matrix is symmetric with ones on the diagonal we only
show its upper half.

The structure is far from being diagonal, indicating non-trivial correlations
among most pairs of variables. Of particular interest is the large ($k,v$)
correlation since it relates a quantity defined before RT ($v$) to a one after
RT ($k$).

To make use of the information in the most efficient way, we first decorrelate
the variables. This is performed by diagonalizing the covariance matrix
111which is always possible since the covariance matrix is by construction
always positive-definite.. From the eigenvectors, we build the transformation
matrix $\bm{T}$ that projects our parameters $\bm{p}^{T}=(R_{0},v,k,\tau)$
onto an orthogonal basis where the new variables
$\bm{X}^{T}=(x_{1},x_{2},x_{3},x_{4})$ are uncorrelated. From our data we
measure the following projection matrix:

$\displaystyle\bm{T}=$ $\displaystyle\begin{pmatrix}0.75&0.07&0.65&0.02\\\
0.65&-0.13&-0.74&-0.11\\\ -0.06&-0.64&0.17&-0.74\\\
0.01&-0.75&0.05&0.66\end{pmatrix}$ (6)

and the linear change of variables is then simply

$\displaystyle X=\bm{T}p$ (7)

Considering the important terms in the matrix, we see that the first 2 lines
link essentially the size of the tumour ($R_{0}$) to the amplitude of the RT
reaction ($k$). The next two ones relate in a non-trivial way, the growth
speed ($v$) to the RT effect ($k,\tau)$.

We now consider the distribution of these new $\\{x_{i=1,\cdots,4}\\}$
variables that which, we recall, are mutually uncorrelated by construction.
Their histograms are shown on Fig 4.

Fig 4: Histograms of the transformed decorrelated variables The new variables
are linear combinations of the fitted ($R_{0},v,k,\tau$) parameters as
described in the text. They are normalized to unit area and the result of the
Gaussian fit is shown in black.

The nice feature now is that, unlike the original variables (Fig 3), the
distributions are now approximately Gaussian 222although it is a slightly
questionable assertion for $x_{2}$, the standard deviation of the fit is large
enough to capture reasonably all the points.. For each variable we fit the
mean ($\mu_{i}$) and standard-deviation $\sigma_{i}$.

We can now build a term that contains the extra-information about the
correlations among the variables in the form

$\displaystyle\chi^{2}_{cons}(R_{0},v,k,\tau)=\sum_{i=1}^{4}\left[\dfrac{(x_{i}(R_{0},v,k,\tau)-\mu_{i})}{\sigma_{i}}\right]^{2},$
(8)

where the $x_{i}$’s are computed according to eq. 7, and
($\mu_{i},\sigma_{i}$) are the parameters of the Gaussian fits shown on Fig 4.

We then add this term to the original $\chi^{2}$ function (eq. 5)

$\displaystyle\chi^{2}_{TOT}(R_{0},v,k,\tau)=\chi^{2}(R_{0},v,k,\tau)+\chi^{2}_{cons}(R_{0},v,k,\tau)$
(9)

and perform the minimization. The constraint acts as a Bayesian prior, i.e. it
includes all the a priori information we have between the parameters. It has
no sense to use it on the previous fits (Fig 2) since it was derived from
them. But we have checked that the best-fits obtained using $\chi^{2}_{TOT}$
are exactly the same as the ones with only $\chi^{2}$, meaning that we are not
over-constraining the parameters with the $\chi^{2}_{cons}$ term.

So why add such a term? Suppose we have few data, for instance 2 measurements
before RT and one after, then we have only 3 points to determine 4 parameters.
Using eq. 8 we introduce some extra equations and the problem becomes at least
technically solvable.

In the following we focus on the regrowth time, which according to our model
(eq. 4) is

$\displaystyle t_{min}=\tau\ln\left(\dfrac{k/\tau}{v}\right).$ (10)

It depends mostly on $\tau$ and logarithmically on the relative speed between
the shrinkage due to RT ($k/\tau=v_{d}$) and the intrinsic tumour growth
($v$). The $\chi^{2}_{TOT}(R_{0},v,k,\tau)$ minimization leads to the
($\hat{R_{0}},\hat{v},\hat{k},\hat{\tau}$) estimates and we use those values
in eq. 10 to estimate the regrowth time.

##  2 Results

###  2.1 Data validation

We first validate the procedure on our dataset by assessing the performances
of our predictions with a single point after RT.

Among our patients, we choose 6 follow-ups, with at least two points before RT
and enough subsequent points for the minimum of the fit to be robust (see Fig
2). We then take the points before RT and the first one just after it, and
perform the constrained minimization (eq. 9). We obtain an estimate of
$t_{min}$ and compare it to the one from the full fit. In order to avoid
mixing the training and test samples, for each patient we rebuild the
constraint on the lines of section 1.4, removing each time the patient’s data
from the datasets. The results are shown in Fig 5.

Fig 5: Predictions for the regrowth time on real data with the first point
after RT. The black dashed curve shows the best fit model using all the data
points and the vertical black line shows its minimum (same as Fig 2) . We then
only consider the red circled points consisting of all the points before RT
and the first one just after, and perform the constrained minimization
described in the text. The result for the model is the dashed red curve with
the estimated regrowth time shown as the vertical red line.

The $t_{min}$ predictions for most of the patients are quite precise; they lie
within a few months of the value determined with all data. For patient (13) it
is slightly larger (10 months). This is an interesting case, since the point
after RT is above the one before. This can be due to statistical fluctuations
or to the fact that RT produces sometimes an oedema that can be misidentified
as the tumour radius. However even in this case, we obtain a reasonable
estimate. This shows that, at the date of the first MRI after RT, we could
have guessed in most cases efficiently the regrowth time of the tumour and
plan more efficiently the dates of the next MRIs.

###  2.2 Predictions

We now evaluate on virtual patients a strategy to estimate as soon as possible
the tumour regrowth time. To this aim, we must first fix the times of the MRI
measurements which are constrained in the following way.

  1. 1.

Although we showed results with many points before RT (Fig 5), today’s
clinical paradigm is to reduce the tumour as soon as possible. We thus
consider the case where the radiotherapy sessions are planned immediately
after the first MRI within typically 6 months.

  2. 2.

Since this is a central point, a second MRI should be performed around the RT
date.

  3. 3.

Fast-responders reacting within a few months, we propose to perform an MRI
measurement 3 months after RT.

We will then consider the cases where the measurement times are located at
$t_{\textrm{mes}}=[-6,0,+3]$ months and test if we can still make some
predictions for the regrowth time. This is a very challenging situation since
we only have 3 nearby points with important relative errors. To assess
statistically the performances of the prediction, we adopt a Monte-Carlo
approach. For a given set of “true” parameters ($R_{0},v,k,\tau$), we first
compute the tumour radius at $t_{\textrm{mes}}$. We then add to each point a
random Gaussian noise with a $\sigma=1$ mm standard deviation, and from these
virtual measurements, estimate the regrowth time. We repeat the procedure 1000
times and consider the mean of the predictions and the 95% confidence-level
interval (obtained from the $[0.05,0.95]$ percentiles) that we compare to the
true $t_{min}$ value. This procedure is illustrated in Fig 6.

Fig 6: Characterization of the regrowth time estimation with a Monte-Carlo
method. The black curve represents a model (which is here the bestfit of
patient (18)) with its minimum shown as the vertical black line. One draws
some Gaussian noise of $\sigma=1$ mm at the measurement times
$t_{\textrm{mes}}=[-6,0,3]$ months, and performs the $t_{min}$ estimation
described in the text. This is repeated 1000 times which allows to construct
the red histogram of all the $t_{min}$ estimates. The red vertical dashed line
shows its mean value and the horizontal one the [0.05,0.95] percentile region.

We use our 20 best-fits as a representative set of “true models”. We perform
the Monte-Carlo study described previously for each set of parameters and
compare the mean and 95% confidence-level interval of our estimated regrowth
times to the true value on Fig 7.

Fig 7: Performances of the regrowth time estimates with 3 measurements at
$t_{\textrm{mes}}=[-6,0,3]$ months for a set of true parameters corresponding
to the bestfits of our 20 patients. Black points represent the mean of the
estimates and the bars the 95% confidence-level interval. The red point is the
true value associated to each best-fit for the patients labelled on the
vertical axis (corresponding to the dashed lines in Fig 2). Note that only the
best-fit parameters of each patient are being used here.

First, we notice that 15 predictions out of 20 (75%) are good, the mean value
being typically within 6 months of the true. In these cases, the guess follows
roughly the true values which confirms that the method is not only driven by
the constraint (which would lead always to the same interval) but also
incorporates the information of the 3 measurements. Fast-responders (patients
(1),(4),(8) and (11)) are correctly predicted and tend to lead to predictions
under 1 year which could be the threshold to plan a next MRI rapidly (possibly
3 months later).

There are also 5 outliers out of 20 (25%) corresponding to the cases where the
true regrowth times are the largest ($3-4$ years), i.e. to the slowest
responders. A point at 3 months for them is much too soon to infer any
information about the curvature, so that the prediction is only driven by the
constraint and goes to its mean value of about 2 years. More precisely, by
Taylor-expanding our model near $t=0^{+}$

$\displaystyle R(t)=R_{0}+(v-v_{d})t+\epsilon t^{2}/2+\mathcal{O}(t^{3})$ (11)

where $v_{d}\equiv k/\tau$ is the speed of the collapse and the curvature term
is $\epsilon=v_{d}/\tau$. For slow-responders, there is almost no curvature at
3 months, $\epsilon\to 0$ and $\tau=v_{d}/\epsilon$ diverges leading to a very
broad (and even sometimes bi-modal) $t_{min}$ distribution. In this case, the
prediction is only driven by the constraint.

Although pessimistic for the patient, the predicted value is still large
(around 2 years, see Fig 7). Thus we can safely plan a next MRI 1 year after
RT. We consider the case where the times for the radial measurement are at
$t_{\textrm{mes}}=[-6,0,3,12]$ months and perform the prediction again. The
result is shown in black in Fig 8.

Fig 8: Monte-Carlo estimates of the regrowth time with an extra point at 1
year for the 5 outliers of Fig 7 ($t_{min}>3$ years). Black bars corresponds
to the 95% confidence-level intervals obtained from the standard constraint
(eq. 8) and the blue ones with the loose constraint described in the text. The
black/blue points corresponds to the mean values and the red point is the true
value of each model.

Unfortunately, the constraint eq. 8 is still pulling $t_{min}$ to too low
values. We need to switch to a looser constraint. As is clear from eq. 11, the
linear term, that is the best constrained, is related to the slopes measured
before ($v$) and after ($v_{d}$) RT. In the absence of good knowledge of the
curvature, we may try to relate these slopes to the regrowth time. Indeed, on
our dataset, we observe a strong correlation between $v_{d}$ and $t_{min}$
(Fig 9) that we fit to a power-law

$\displaystyle t_{min}=13/v_{d}^{0.78}.$ (12)

Fig 9: Correlation between $v_{d}$ and $t_{min}$ measured on our set of 20
patients. The red line shows the power-law fit.

The origin of this heuristic constraint remains to be understood, but we can
use it to build a new estimator for $t_{min}$: we fit on the data only the
linear terms in eq. 11 in order to get $v_{d}$ which we transform according to
eq. 12. Since this method uses a single correlation, we call it the loose
constraint. We show the result of applying this procedure on the outliers in
blue on Fig 8. The distributions are much better centred on the true values.
One may ask why not always use this constraint. As clear in the figure, the
uncertainty is larger with the loose-constraint method. This is the classical
bias/variance trade-off of any estimator. Although it gives indeed less biased
results for outliers, the method would miss fast-responders at 3 months, since
the slopes determination is then extremely noisy. On the contrary, with a
point at 1 year, there is enough lever-arm to determine the slope quite
precisely and take advantage from the correlation to let the data “speak for
themselves”.

We point out that the loose-constraint method is very simple and may be used
by any clinician without even a computer. First measure the slope before RT to
obtain $v$, then the slope after RT ($v-v_{d}$) to obtain $v_{d}$, and finally
use eq. 12 to predict the regrowth-time.

##  3 Discussion

We have proposed a new simple model to describe the evolution of diffuse low-
grade gliomas before and after radiotherapy. It is analytical and describes in
a satisfactory way the follow-ups of 20 patients with measured tumour radii
before and after RT. This model has 4 free parameters, 2 before RT and 2
after, that vary for each patient. From the study of the correlation between
all the parameters we proposed a way to include a prior information to any
follow-up, which allows to perform predictions for the regrowth-time of the
tumour rapidly after RT. From the data we had at our disposal, we showed that
including this information allows to predict the regrowth time of the tumour
at the very first MRI measurement after RT typically within 6 months. Using
virtual patients, we have shown that is is possible to predict reasonably well
the regrowth time with only one point 6 months before RT, one around RT and
one 3 months after, in 75% of the cases. The remaining 25% for which our
prediction is pessimistic, have all large regrowth-time ($\simeq$ 4 years) and
may draw benefit from another measurement 1 year after RT, leading to more
correct estimates.

These results assume that our database is representative of all LGG evolution
and would profit from incorporating more patients’ data. Similar profiles are
obtained for chemotherapy treatments [22, 27] and it would be interesting to
redo the analysis in this case.

This work is based on a 4-parameters model which is a simplified version of a
biologically motivated model. This choice can be challenged; why not use some
non-parametric method that are often efficient? First, the low dataset (43
patients but in practice 20 with a sufficient number of points to inform our
model) precludes the possibility of using general purpose Machine Learning
techniques like Deep Neural Networks, Random Forests, Boosted Decision Trees
(as described for instance in this recent review [28]), as well as Recurrent
Networks dedicated to Time Series (e.g. [29]). Second, we could think of using
Gaussian Processes (GP) method (e.g. [30]) that can work on small samples with
some optimized kernel. We have tried it, with a squared exponential kernel and
a white noise. However, by construction, outside the data input region the
naive “vanilla” model converges to a constant and cannot describe the regrowth
phase. To overcome this failure, one is forced to use a time dependent
function of the mean which is exactly the meaning of the 4-parameters model
developed in this article. This clarifies why modelling, especially based on
physical arguments, is superior to all purely statistical methods. This was
the key to the success of making predictions from a restricted dataset and
with very few data points.

Here, we have varied the patient’s population and shown that the method has
the potential to make some predictions among various patients profile. The
problem is different for a personalised follow-up (which is the practical
clinical case) since the prediction depends on the details of the measurements
(times and values). Using a Monte-Carlo Markov Chain technique, one can obtain
an individualised probability distribution of the regrowth-time that can help
clinicians adapt their treatment and the dates of the next MRIs. We plan to
provide such a tool that will be publicly available online.

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