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Optimal radiation fractionation for low-grade gliomas:
Insights from a mathematical model
Tatiana Galochkina
Federal Science and Clinical Center of the Federal Medical and Biological Agency, 28
Orehovuy boulevard, 115682 Moscow, Russia (tat.galochkina@gmail.com).
Alexander Bratus
Lomonosov Moscow State University, Faculty of Computational Mathematics and
Cybernetics, GSP-1, 1/52, Leninskie Gory, 119991 Moscow, Russia
(applmath1miit@yandex.ru).
V´ıctor M. P´erez-Garc´ıa
Departamento de Matem´aticas, E. T. S. I. Industriales and Instituto de Matem´atica
Aplicada a la Ciencia y la Ingenier´ıa (IMACI), Universidad de Castilla-La Mancha, 13071
Ciudad Real, Spain (victor.perezgarcia@uclm.es).
Abstract
We study optimal radiotherapy fractionations for low-grade glioma using mathe-
matical models. Both space-independent and space-dependent models are stud-
ied. Two different optimization criteria have been developed, the first one ac-
counting for the global effect of the tumor mass on the disease symptoms and the
second one related to the delay of the malignant transformation of the tumor.
The models are studied theoretically and numerically using the method of
feasible directions. We have searched for optimal distributions of the daily
doses djin the standard protocol of 30 fractions using both models and the two
different optimization criteria. The optimal results found in all cases are mi-
nor deviations from the standard protocol and provide only marginal potential
gains. Thus, our results support the optimality of current radiation fraction-
ations over the standard 6 week treatment period. This is also in agreement
with the observation that minor variations of the fractionation have failed to
provide measurable gains in survival or progression free survival, pointing out
to a certain optimality of the current approach.
Keywords: Low-grade gliomas; Radiotherapy; Mathematical models of tumor
growth; Mathematical model of tumor response
Preprint submitted to Elsevier June 9, 2015
1. Introduction
Primary brain tumors arise in the brain due to mutations and abnormal
functioning in cells of the brain tissues. Gliomas are the most frequent subtype
accounting for 50% of all primary brain tumors. With the exception of the
unfrequent grade I pylocytic astrocytoma these tumors are very infiltrative and
remain a challenge for oncologists due to the low effectivity of current therapies.
Thus, patients diagnosed with gliomas typically die because of the complications
related to the tumor evolution.
The so-called low grade gliomas (LGG) describe WHO grade II gliomas
of astrocytic and/or oligodendroglial origin [28]. Despite their slow growth
these tumors are generally incurable, the median survival time being of about
5 years [45, 47]. While many patients present with easy to control seizures and
remain stable for years, others undergo the so called malignant transformation
and progress rapidly, with increasing neurological symptoms, to a higher-grade
tumor.
Management of LGG is controversial because these patients present few, if
any, neurological symptoms. Historically, when LGG was diagnosed in a young,
healthy adult, a commonly accepted strategy was a watch and wait approach
because of the indolent nature and variable behavior of these tumors. The
support for this practice came from several retrospective studies showing that,
when therapy was deferred, patients had no difference in outcome (survival,
quality of life) from time of radiologic diagnosis [37].
The decision as to whether a patient with LGG should receive resection, ra-
diation therapy (RT), or chemotherapy is based on a number of factors including
age, performance status, location of tumor, and patient preference [51, 47]. Since
LGGs are such a heterogeneous group of tumors with variable natural histories,
the risks and benefits of each of the three therapies must be carefully balanced
with the data available from limited prospective studies.
As to RT, the clinical trial by Garcia et al [17] showed the advantage of using
radiotherapy in addition to surgery. However, although immediate radiotherapy
after surgery increases the time of response (progression-free survival), it does
not improve overall survival while it induces serious neurological deficits as a
result to normal brain damage [62]. Thus, radiotherapy is usually offered only
for patients with several risk factors such as age, sub-total resection, or diffuse
astrocytoma pathology [23].
Radiotherapy planning has historically been based only on limited clinical
evidences. Mathematical modeling has the potential to help in selecting LGGs
patients that may benefit from radiotherapy and in developing specific optimal
fractionation schemes for selected patient subgroups. There is thus a need for
models accounting for the crucial features of low-grade glioma dynamics and
its response to radiation therapy without involving excessive details on the -
often unknown- specific processes but allowing for qualitative understanding of
the phenomena involved. The increasing availability of systematic and quan-
titative measurements of tumor growth rates provides key information for the
development and validation of such models (see e.g. Pallud et al. [39, 40] for
2
LGGs).
The optimization of tumor radiotherapy fractionation has four decades of
history (see e.g. the pioneering paper of Bahrami and Kim [1] and also the
book of Wheldon [2] for early survey). Most available papers on mathemati-
cal modeling of gliomas have considered specific aspects of radiation therapy
for high-grade gliomas [49, 8, 26, 25, 2]. As to LGGs there have been very
few relevant works using mathematical models to describe the response to RT.
Ribba et al [48] developed a ODE model of the response of low-grade glioma to
different therapies with a number of undetermined parameters that can be fit
to describe the individual patient’s response with a good qualitative agreement.
More recently P´erez-Garc´ıa et al [42] developed a simple spatial model able to
describe the known phenomenology of the response of LGGs to RT including
the observations from Pallud et al. [40].
In this paper we want to study the problem of therapy planning from a
mathematical point of view taking as a basis the model of P´erez-Garc´ıa et al
[42], P´erez-Romasanta et al [44]. To do so, we will pose a precise optimization
problem and study how should radiation doses be distributed to get the best
tumor control while keeping toxicity at the same levels as with the standard
treatment.
Our plan in this paper is as follows. First in Sec. 2 we will present our
model. This will include a description of the tumor cell dynamics (Sec. 2.1),
a description of the response of the tumor cells to radiation (Sec. 2.2) and
the specification of the objective function and restrictions for our problem (Sec.
2.3) together with the choice of the parameter values (Sec. 2.4). Next in Sec.
3 we provide our results for the dynamics of the model focusing only on the
proliferative aspects of the tumor. Sec. 4 will be devoted to the study of the
full mathematical problem including diffusion. Finally in Sec. 5 we summarize
our conclusions.
2. The model
In this paper we intend to find optimal radiotherapy therapeutical protocols
for low-grade glioma. To do so we need to fix: (i) the dynamics of the tumor
cells in the intervals between radiation doses, (ii) the response of tumor cells to
a dose of radiation, and (iii) the optimization criterion to be used.
2.1. Tumor cell dynamics
For the tumor cell dynamics we will use the model developed by P´erez-Garc´ıa
et al [42]. The model classifies tumor cells into two generic compartments. The
first one accounts for the density u(x, t) of tumor cells in units of a maximal
cell number that proliferate with a typical time 1/ρ and have a characteristic
mobility (diffusion) coefficient D. The second compartment accounts for the
density of cells v(x, t) that, after the action of radiation, are not able to repair
the DNA damage. These lethally damaged cells will die after a time related to
the typical proliferation time 1/ρ and the average number of mitosis cycles k
that damaged cells are able to complete before dying. The model reads
3
∂u
∂t =D∆u+ρ(1 −u−v)u, (1a)
∂v
∂t =D∆v−ρ
k(1 −u−v)v. (1b)
on a domain Ω of the brain supplemented with initial data
u(x, t0) = u0(x), u0(x)∈C2(¯
Ω),(1c)
v(x, t0) = 0.(1d)
and no-flux boundary conditions
∂u
∂¯n∂Ω
= 0,∂v
∂¯n∂Ω
= 0.(1e)
2.2. Response to RT
A radiotherapy session lasts about ten minutes and cell damage is produced
in this short time in relation with the typical cellular proliferation times. Thus
we can assume RT effect to be instantaneous in relation with our typical cellular
time.
Following the standard practice in radiotherapy we will assume the damaged
fraction of tumor cells to be given by the classical linear-quadratic (LQ) model
[63]. Thus for a radiation dose dj(Gy) given at a time tj, we will take the
survival fraction Sj, i.e. the fraction of cells that are not lethally damaged by a
dose djto be given by
Sj=e−αtdj−βtd2
j,(2)
where αt(Gy−1) and βt(Gy−2) are respectively the linear and quadratic coef-
ficients for tumor cell damage of the LQ model, to be fixed later.
The full treatment consists of a total dose Dsplit in a series of ndoses dj
(typically equal) given at times tj(typically equispaced except for the weekends
were the treatment is stopped).
Taking into account this information, and for a radiation fractionation de-
fined by the number of doses n, the irradiation times {tj}j=1,...,n and doses per
fraction {dj}j=1,...,n we get the following matching conditions for the tumor cell
densities u, v at the irradiations times.
u(x, t+
j) = Sju(x, t−
j),(3a)
v(x, t+
j) = v(x, t−
j) + [1 −Sj]u(x, t−
j),for j≥1.(3b)
Thus the action of radiotherapy leads to a discontinuous behavior in each
tumor subpopulation considered independently but the total tumor cell num-
ber u+vis a continuous function of time, a fact that cannot be reproduced
with simpler tumor radiotherapy models such as those based on single Fisher-
Kolmogorov equations.
4
2.3. Optimization problem
The main limitation of radiation therapy is the need to limit the damage
to the normal tissue that can be obtained using Eq. (2) but using instead
the parameters of the normal tissue αh, βhwith αh/βh≃2 [66]. Technical
improvements such as Intensity-Modulated Radiation Therapy (IMRT) allow to
deliver the radiation dose onto the tumor with pinpoint precision. The ability
to spare healthy surrounding tissue by using this technique is very valuable for
clinicians [6]. Other ways to deliver radiation with higher precision such as
charged particle therapy may allow for further improvements [27]. However,
with current technologies special care must be used to keep the damage to the
normal tissue limited, what can be quantified, given a certain set of doses (tj, dj)
as (see e.g. Van der Kogel & Joiner [63] pp. 114).
Eh=−log
n
j=1
Sj
=αh
D+1
αh/βh
n
j=1
d2
j
.(4)
In addition acute tissue reactions and other secondary effects depend: (i) on
the total volume irradiated (the so-called volume effect) and (ii) on the maximal
dose per fraction d∗used.
Mathematical optimization has the potential to help in finding optimal ther-
apeutic schedules (e.g. fractionation), spatial dose distributions, etc. To do so
once a particular tumor evolution model has been chosen and paired with a
model for the response to radiation we must choose a cost functional related to
the optimization objective.
In the case of high grade brain tumors, where there is no metastatic spreading
it has been hypothesized that death occurs in high grade tumors after the tumor
reaches a critical diameter of about 6-7 cm [58, 64]. However, high grade gliomas
are very aggressive tumors that due to the formation of necrotic areas result is
a complete loss of functionality in areas affected by macroscopic tumor [41].
In LGGs, however, the situation is different. Because of the low density of
the tumor the tissue affected is still partially functional and because of the slow
growth the brain is able to dynamically remap the lost functions into healthy
areas. The most harmful effect of the tumor is its potential to transform into a
high-grade tumor. So the optimal therapy would be the one delaying the most
the malignant transformation of the tumor.
The malignant transformation is a process that is not completely understood.
One may think naively that the larger the total tumor mass is, the higher would
be the possibility of a tumor cell accumulating a sufficient number of mutations
to become more malignant. In that case one would like to maximize the time
the total tumor mass is below a certain value M∗as
T:
Ω
u(t, x) + v(t, x)dx ≤M∗(5)
Alternatively, the role of the tumor microenvironment is being recognized [21] as
a driving force in tumor progression. In principle it may suffice that the tumor
5
clonal evolution leads to a continuous density increase that below a certain
limit would lead to vessel damage, generation of hypoxic foci, the stabilization
of hypoxia dependent signaling molecules such as HIF-1αand the increase of
genomic instability [33, 52, 46]. Hypoxic areas, increased genetic instability and
transition to a higher malignancy. In this case, the goal of the therapy would
be to make
T: max
Ωu(t, x) + v(t, x)≤R(6)
as large as possible.
Maximizing Tgiven either by Eq. (5) or Eq. (6) on the space of possible
radiation fractionations with the given constraints will be our goal in this paper.
Thus given a number of doses Nwe will try to find the set of irradiation times
{tj}N
j=1 and doses {dj}N
j=1 maximizing
τ= max
t,d T, Ω⊂R2(7)
under the constraints
dj≤d∗, k = 1, . . . , N. (8a)
Eh=αh
N
j=1
dj+1
αh/βh
N
j=1
d2
j
≤E∗,(8b)
So we can see that vector dmust be chosen from the intersection of the ball
(total damage of the healthy cells given by Eq. (8a) and the cube for each dose
of radiation given by Eq. (8b) in N-dimensional space.
2.4. Parameter estimation
Finally, to completely determine our problem we must find ranges of pa-
rameters describing the behavior of LGGs. In this paper we will focus on two-
dimensional scenarios to incorporate spatial variations in the quantities under
study without the computational complexity of fully 3D problems. The maxi-
mal tumor density can be easily found by taking the typical astrocyte size to
be about 10 µm and has been provided by many authors (see e.g. Swanson et
al. [58]). Proliferation rates for LGGs have been estimated to be around 0.003
day−1in Badoual et al. [19]. These rates give typical doubling times of about 1
year, in agreement with the slow growth speed of these tumors. The diffusion
coefficient may be estimated either directly from the cell motility [24] or from
the estimate for ρand the typical radiological growth speeds [57] and is about
0.01 mm2/day.
As to the response to RT the precise values of the parameters are not very
well known despite many studies, because what is clinically more relevant is
the ratio αt/βtthat for glioma cells is around 10. The total dose of radiation
delivered to LGGs typically is between 45 and 54 Gy, given in fractions of 1.8
Gy during 5 or 6 weeks of treatment, the patient being irradiated typically
6
5 days per week (Monday to Friday). Trials on hypofractionated schemes in
gliomas have explored doses per fraction of up to 3.2 Gy since higher doses
are considered to be harmful and lead to potentially lethal acute reactions [31].
Higher doses are sometimes used in a single fraction but targeting small volumes
in what it is called radiosurgery. The intention with such high doses is not to
perform a repetitive number of irradiations but instead to burn a piece of tissue
as an alternative to normal surgery. As to the time between doses a maximum
of three doses per day with spacing of 4 hours between doses of 1.5 Gy, has been
used in clinical trials on accelerated hypofractionation schemes [30, 36].
Getting estimates for the number of tumor cells surviving after a radiation
dose is difficult due to the tumor heterogeneity, the effect of the microenviron-
ment on the response, etc. P´erez-Garc´ıa et al [42] have estimated this number
from the radiological response fitting the typically observed dynamics of the
mean tumor diameter [39] obtaining values around Sf∼0.85. These numbers
allow us to get the effect of the isoeffect Eh/αhon the normal tissue of the
standard fractionation scheme for 30 doses of 1.8 Gy computed from Eq. (4) to
be about 102.6 Gy.
As to the critical tumor mass, tumors with a diameter of about 4 cm are
considered to be too large and some therapeutical action is typically taken,
either surgical or with chemotherapy.
All the biological and medical parameters used in this paper are summarized
in Table 2.4.
3. Model without diffusion
3.1. Model statement
To better understand the system dynamics we will first consider the concen-
trated model without diffusion. In addition to providing a first approximation to
the mathematical behavior of the full problem, the diffusionless system has been
shown to describe certain aspects of the tumor dynamics even for moderately
long times [42]. Thus the dynamics will be ruled by the equations
du
dt =ρu(1 −u−v),(9a)
dv
dt =−ρ
kv(1 −u−v).(9b)
Thus the dynamics in this limit is ruled by a simple autonomous system together
with the matching conditions at the treatment times tjdefined by
u(t+
j) = Sju(t−
j),(10a)
v(t+
j) = v(t−
j) + [1 −Sj]u(t−
j),for j≥1.(10b)
7
Variable Description Value (Units) References
C∗Maximum tumor 106cell/cm2Swanson et al. [58]
cell density (2D)
DDiffusion coefficient 0.01 mm2/day Jbadi et al. [24]
for tumor cells
ρProliferation ∼0.003 day−1Badoual et al. [19]
rates
djRadiation dose ≤3.2 Gy
in the j-th fraction 1.8 Gy
nTotal number 25-30
of doses
∆tmin Minimum interval 4 h Nieder et al. [36]
between doses
αt/βtAlpha-beta ratio 10 Gy Wigg [66]
of the LQ model
for LGG cells
αh/βhAlpha-beta ratio 2 Gy Wigg [66]
of the LQ model
for normal tissue
SF1.8Survival fraction ∼0.85 P´erez-Garc´ıa et al [42]
of the LQ model (1.8 Gy)
E∗/αtIsoeffect value 102.6 Gy Eq. (4)
kNumber of ∼1 P´erez-Garc´ıa et al [42]
mitosis before the
mitotic catastrophe
Table 1: Summary of typical parameter values of the biological parameters in the model of
LGG evolution.
3.2. Equilibria and explicit solutions
The equilibria of Eqs. (9) are
u= 0, v = 0 (11a)
{u+v= 1, u ∈[0,1], v ∈[0,1]}(11b)
To define the type of the equilibrium point (0,0) we use the standard Jacobian
method
J(u, v) = ρ(1 −2u−v)−ρu
ρ/ky −ρ/k(1 −u−2v)(12)
Since J(0,0) = ρ0
0−ρ/k , the eigenvalues are λ1=ρ > 0 and λ2=−ρ/k <
0, thus (u, v) = (0,0) is a saddle point.
Statement 1. The set P={u > 0, v > 0, u +v < 1}is an invariant set for
Eqs. (9).
8
Proof. The boundaries of Pare also the system trajectories:
u= 0
dv
dt =−ρ
kv(1 −v),du
dt =ρu(1 −u)
v= 0,u+v= 1
du
dt = 0,dv
dt = 0.
Since the trajectories of Eqs. (9) cannot intersect we obtain that no trajectory
can leave P.
□
The phase plot corresponding to Eqs. (9) is shown in Fig. 1 were it is clear
how all orbits starting in P, stay there for all times. Since all the considered
initial values in order to be biologically relevant (densities below the maximal
one) lie in P, in what follows we will consider the dynamics only in P.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u
v
Figure 1: Phase portrait corresponding to Eqs. (9) in P.
We can note another feature of our system: the existence of a first integral
Statement 2. On any trajectory of Eqs. (9)
I(u, v) = u(t)vk(t) = const .(13)
Proof. We can note that: V(t) = ln u+kln vis a first integral for our system:
˙
V=ρ(1 −u−v)−kρ
k(1 −u−v) = 0.
Therefore we have V(t) = ln u+kln v= const, and thus uvk= const.
The continuous equations (9) are integrable. Dividing Eqs. (9a) and (9b)
we get, as one would expect du/dv =−u/kv, thus uvk=u0vk
0, while for the
second equation we find
dv
dt =−ρ
kv1−u0vk
0v−k−v=ρ
kv2−v+u0vk
0v1−k.
9
To simplify the computations we assume k= 1 in what follows. Then dv/(v2−
v+u0v0) = ρdt or equivalently
dv−1
2
v−1
22+u0v0−1
4
=ρdt (14)
Statement 3. For all u, v in our model:
uv < 1/4.
Proof. First, since v0= 0 obviously u0v0= 0 <1/4. Let us show that the
effect of the therapy given by Eqs. (10) does not violate this restriction. To do
so let us assume the contrary:
¯u¯v=Sju((1 −Sj)u+v)≥1
4⇔S2
ju2−Sj(uv +u2) + 1
4≤0
We obtain an inequality for Sjwhich is a parabola with the discriminant:
∆ = (uv +u2)2−u2=u2(u+v)2−1≤0 (15)
We can note that ∆ = 0 only at the equilibrium points. As soon as we do not
leave Pand u+v < 1, the discriminant ∆<0and there are no Sjsatisfying
the inequality. So we have a contradiction.
This completes the proof since for the continuous system uv = const.
Now, from Eq. (14) and defining p0=1
4−u0v0we can obtain the final
formula for v(t):
1
2plnv−1
2−p0v0−1
2+p0
v−1
2+p0v0−1
2−p0
=ρ(t−t0),(16)
what leads to the final values for v(t) and also u(t) from Eq. (13)
v(t) = e2ρ(t−t0)p0p0−1
2v0−1
2−p0+p0+1
2v0−1
2+p0
v0−1
2+p0−e2ρ(t−t0)p0v0−1
2−p,(17a)
u(t) = u0v0
v(t).(17b)
3.3. Evolution under the action of radiotherapy
From the explicit form of the solutions (17) we can obtain the outcome of a
given radiotherapy scheme in a very simple way. To do so let us define the family
of functions uj(t) defined on the j-th time section [tj, tj+1] as uj(t) = u(t) for
t∈Ij≡(tj, tj+1) and
uj(tj)≡lim
t→t+
j
u(t),(18a)
uj(tj+1)≡lim
t→t−
j+1
u(t),(18b)
10
So, we can compute the successive values of uj(tj) as follows. First the irradia-
tion connects (uj(tj), vj(tj)) with (uj−1(tj), vj−1(tj)) as
uj(tj) = Sjuj−1(tj),(19a)
vj(tj) = vj−1(tj) + (1 −Sj)uj−1(tj).(19b)
The dynamics for times t∈Ikleads to [cf. Eqs. (17)]
vj(tj+1) = e2ρpj(tj+1 −tj)(pj−1
2)vj(tj)−1
2−pj+ (pj+1
2)vj(tj)−1
2+pj
vj(tj)−1
2+pj−vj(tj)−1
2−pje2ρpj(tj+1−tj),(19c)
uj(tj+1) = uj(tj)vj(tj)
vj(tj+1),(19d)
(19e)
where pj=1
4−uj(tj)vj(tj).
After the completion of the treatment for t=tNwe get the final values
for (u(tN), v(tN)) ≡(uN+1(tN), vN+1(tN)) to be denoted hereafter as (U, V ).
Defining P≡pN+1, the subsequent evolution is given for all times by
v(t) = e2ρ(t−tN)PP−1
2V−1
2−P+P+1
2V−1
2+P
V−1
2+P−e2ρ(t−tN)PV−1
2−P,(20a)
u(t) = U V
v(t).(20b)
for all t > tN.
3.4. Optimization problem
On the basis of the space-independent model given by Eqs. (9) the optimiza-
tion criteria given by Eqs. (5) and (6) are equivalent. In both cases for a given
fractionation {tj, dj}N
j=1 we will define the time to the malignant transformation
as
τ= max {t:u(t) + v(t)≤R}(21)
The invariance of Pimplies that biologically meaningful values of u+v < 1,
thus R < 1.
We can get the value of the time τfor a set of final values after the radio-
therapy (U, V ) and from the equation
EP−1
2s−+P+1
2s+
s+−Es−
+UV (s+−Es−)
Ep∗−1
2s−+P+1
2s+
=R, (22)
where E=e2ρτp,s−=V−1
2−P,s+=V−1
2+P.
After some conversions we get the quadratic equation aE2+bE +c= 0, where
a=1
2−p(1 −R)s2
−,b=R−41
2−P1
2+Ps−s+,c=1
2+P(1 −
R)s2
+what leads to the following two solutions to Eq. (22)
E1,2=s+
s−
−(R−4UV )±(R2−4U V ) (1 −4U V )
21
2−P(1 −R)(23)
11
We already know that p < 1
2by definition and since R < 1, then R2−4uv > 0.
Then there are two roots of the equation as uv = const. We can note that the
first root E1>0 since 4uv > 0, R < 1 and (R2−4U V ) (1 −4UV )>1.
Statement 4. The second root E2of Eq. (23) is negative.
Proof. It suffices to prove that:
−R+ 4UV −(R2−4U V ) (1 −4U V )≤0.
To do so, we rewrite the inequality as
R≥4uv −(R2−4UV ) (1 −4UV )
If the right side is negative then the statement is proven. Otherwise we can
square both sides of the equation and obtain:
8uv −R2−1≤2(R2−4UV ) (1 −4UV )
We can see that while the right side is positive, the left side is negative, thereby
the inequality is proved.
Thus the only relevant root from our analysis is E1and from there we get
finally the formula for the time of malignant transformation from the end of the
treatment
τ=
ln 2V−1 + √1−4UV
2V−1−√1−4UV ·−R+ 4U V +(R2−4U V ) (1 −4UV )
1−√1−4UV (1 −R)
ρ√1−4UV ,
(24)
and correspondingly the total time of malignant transformation from the initial
time
T=tN+τ(25)
In the case in which we are interested in this paper the total treatment time is
fixed, thus maximizing Tis equivalent to maximizing τ.
3.5. Results on dose distribution in time for the standard scheduling
In this paper we will consider the case of the standard fractionation in which
N= 30 and the times tjcorrespond to 30 irradiations during 6 weeks of treat-
ment (radiations are held 5 times a week from Monday to Friday). Although
some improvement is to be expected in fast-growing tumors by avoiding the
weekend treatment gaps [60, 61], LGGs grow so slowly that treatment gaps
have no measurable effect on the treatment outcome and thus we will consider
the most common fractionation.
We will look for the values of the doses djmaximizing Tgiven by Eq. (25)
with the restrictions on the total damage of normal tissue Eq. (8a) and on the
maximum dose of irradiation (8b) described at the first section. In our case we
12
R dw1dw2dw3dw4dw5dw6∆T
Gy/day Gy/day Gy/day Gy/day Gy/day Gy/day days
0.4 1.6551 1.7649 1.8183 1.8428 1.8540 1.8594 1
0.6 1.6527 1.7644 1.8186 1.8438 1.8548 1.8600 1
0.8 1.6510 1.7642 1.8193 1.8446 1.8549 1.8602 2
Table 2: Doses per day in Gy maximizing Eq. (25) for the concentrated model given by Eqs.
(9) and satisfying the restrictions (8a) and (8b). The last column (∆T) shows the days gained
with respect to the standard treatment with all doses equal to 1.8 Gy. The results are shown
for different values of the threshold density R
will fix the daily doses to be equal during each treatment week so that we have
only six degrees of freedom to be denoted hereafter as dw1, dw2, ..., dw6.
To obtain the results to be described in this section we have used the method
of feasible directions (see e.g. Sun and Yuan [55]). The idea of this method is
to chose a starting point satisfying the constraints and to find a direction such
that the following conditions are satisfied: (i) a small move in that direction
remains feasible, and (ii) the objective function improves. One then moves a
finite distance in the determined direction, obtaining a new and better point.
The process is repeated until no directions satisfying both (i) and (ii) can be
found. In general, the terminal point is a constrained local (but not necessarily
global) minimum of the problem. Typically many different randomly chosen
initial points for the algorithm are taken to avoid reaching only local minima.
The computations have shown that this algorithm finds only one maximum
of our functional (survival time) for all the taken initial points. Table 3.5 sum-
marizes our results for different threshold values.
We can see that within the framework of this model, the optimal doses found
are very close to those of standard scheduling and the potential improvement in
survival time is marginal.
A typical system trajectory under the action of the optimal therapy is shown
in Fig. 3.5.
4. Model with diffusion
Let us now consider the full spatial model with diffusion with dynamics given
by Eqs. (1) and response to the radiotherapy defined by Eqs. (3).
Through this paper we will assume u(t, x), v(t, x) be strongly positive con-
tinuous functions of xon the square Ω = {[0, r]×[0, r]}and ∀x∈Ω, u(t, x)∈
C1[0,+∞), v(t, x)∈C1[0,+∞).
4.1. Basic properties
Statement 5. Let u(t, x), v(t, x)be the solutions of Eqs. (1) with boundary
conditions (1e) and initial data (1c) on Ω = {[0, r]×[0, r]}. Then the function
V(t) =
Ω
(ln u+kln v)dx, (26)
13
0 0.1 0.2 0.3 0.4 0.5 0.6
0
0.1
0.2
0.3
0.4
0.5
0.6
u
v
Figure 2: Concentrated system trajectory. Red dots: System dynamics during the course of
irradiations. Blue line: Dynamics after treatment. Black line: Critical concentration value.
Initial concentrations: u0= 0.3, v0= 0, critical concentration value: R= 0.6. Parameters of
the model: ρ= 0.003; Sj= 0.85, j = 1,30
is non-decreasing.
Proof. First let us notice that
˙
V=
Ω∆u
u+k∆v
vdx.
Next, for each of the partial derivatives we get:
Ω
∂2u
∂x2
1
1
udx =
r
0
r
0
∂2u
∂x2
1
1
udx1dx2=
r
0
∂u
∂x1
1
u
r
0
+
r
0∂u
∂x121
u2dx1
dx2
=
r
0
r
0∂u
∂x1
1
u2
dx1dx2=
Ω∂ln u
∂x12
dx,
using the conditions on ∂Ω, what leads us to
˙
V=
Ω(∇ln u)2+k(∇ln v)2dx ≥0.
For further conclusions we need some definitions.
Definition 1. Let the class of functions S1
2(Ω) be the functions w(x, t)such
that:
∀tln w(x, t)∈W1
2(Ω) ⇔
Ω(ln w)2+ (∇ln w)2dx < ∞.
14
In these terms we can define the norm
∥w(x, t)∥2
S1
2=
Ω(ln w)2+ (∇ln w)2dx. (27)
Definition 2. ¯w(x)∈S1
2(Ω) is said to be a limit state of Eqs. (1) if for any
solution w(x, t):
∃t1, t2, . . . , tj, . . . , lim
k→∞
tk= +∞: lim
k→∞ ∥¯w(x)−w(x, tk)∥S1
2= 0.
Statement 6. Let u(t, x), v(t, x)∈S1
2be the solutions of Eqs. (1) with bound-
ary conditions (1e) and initial data (1c). Then if there are any limit states in
our model they all are constant.
Proof. Let us consider again the function V(t)defined by Eq. (26). Obviously
V(t)is a functional of u, v and
V(t) =
Ω
(ln u+kln v)dx ≤
Ω
(ln u)2dx
1/2
+
Ω
k(ln v)2dx
1/2
<∞.
As we have shown earlier ˙
V≥0. Using the La Salle invariance principle for
PDEs (see e.g. Appendix B of Capasso [9]) we can conclude that in our system
all the limit states (¯u(x),¯v(x)) must belong to the set {˙
V= 0}. Furthermore we
can see that:
˙
V= 0 ⇔∇ln u= 0
∇ln v= 0 ⇔ln u= const
ln v= const ⇔u= const
v= const
Thus if there are any limit states of our system then: ¯u(x) = const,¯v(x) =
const.
4.2. Results on dose distribution for the standard scheduling
4.2.1. Results on dose distribution with tumor mass criterion
As in Sec. 3.5 we consider the course of 30 irradiations during 6 weeks of
treatment. First we will consider the problem of choosing the dose values dj
maximizing Tas given by Eq. (5) satisfying the restrictions (8).
The optimization problem will be solved computationally using the method
of feasible directions. The total cell concentration is calculated with standard
integration methods. We will take the initial data as given by the equation
u0=rexp −(x1−¯x1)2+ (x2−¯x2)2
σ.(28)
15
M∗Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 ∆T
Gy/day Gy/day Gy/day Gy/day Gy/day Gy/day day
1.5 1.8123 1.8245 1.8265 1.8043 1.8120 1.8123 1
2 1.8156 1.7989 1.8269 1.8345 1.8127 1.8127 0
2.5 1.8200 1.8051 1.8151 1.8273 1.8689 1.8130 1
Table 3: Doses per day in Gy maximizing Eq. (25) (optimization criterion 1) for the extended
model given by Eqs. (9) and satisfying the restrictions (8a) and (8b). The last column (∆T)
shows the days gained with respect to the standard treatment with all doses equal to 1.8 Gy.
Parameter values are r= 0.1,σ = 4,Ω = [0,8] ×[0,8]. The results are shown for different
values of the threshold adimensional tumor mass M∗.
We will consider the initial value of the concentration to be r= 0.1 and the
initial squared tumor width σ= 4. The computational domain is 8 cm ×8
cm as soon as usually the LGGs could be diagnosed from the size of 4-6 cm in
diameter.
The method is seeded with initial values for radiation doses as in case of
the concentrated model, different, random and lying in the restriction area.
This leads to different local minima found by the algorithm. The precision of
computations is limited by the process of comparing the total density value
with the critical value on each step and by the lack of information about our
functional for which the explicit expression cannot be obtained or analyzed in
easy way. A large number of computational experiments was run with random
initial parameter values and the method of feasible conditions did find some
local minima we can make some conclusions.
It is clear that for all experiments the optimal parameters for most local
minima (the values for the doses of radiation each week) lie near the reference
values of 1.8 Gy and the days gained because of the therapy are marginal. Also
the optimal doses of irradiation do not depend too much on the chosen threshold
value. For substantially different levels of the critical value (M∗) the resulting
doses of radiation are always close to the 1.8 Gy reference values.
Finally, the dynamics of the integrated quantities Ωu(x, t) and Ωv(x, t) is
very similar to that of the concentrated model. An example of system trajectory
is shown in Fig. 3
4.2.2. Results on dose distribution with the optimization criterion on the max-
imum density of cancer cells
The second optimization criterion discussed in Sec. 2.3 is based on the
threshold value of the maximum cancer cells density, e.g. consider the maxi-
mization of Tas given by Eq.(6) under restrictions (8). The initial distribution
of cancer cells was chosen as previously, to be given by Eq. (28). We have held
an extensive number of computational experiments using the same computa-
tional method to obtain the values of doses maximizing the survival time. The
results for the different critical values of the maximum allowed density of cancer
cells are shown in table 4.
16
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
∫Ω uT dx
∫Ω vT dx
Figure 3: Phase-space like dynamics of the integral quantities ∫Ωu(x, t) and ∫Ωv(x, t). Red
dots: System dynamics during the course of irradiations. Blue line: Dynamics after treatment.
Black line: Critical threshold value M∗= 2, initial distribution (28) with r= 0.1, σ = 4,Ω =
[0.8] ×[0.8]. Parameters values are: ρ= 0.003; D= 0.0001; Sj= 0.85, j = 1,30
We can see that the optimal doses for this type of criterion are very close
to the standard scheduling and do not provide a significant improvement in the
time to the malignant transformation and correspondingly in patient’s survival.
The system trajectories are very similar to the ones obtained in the framework
of the ODE model as Fig. 4 shows.
5. Discussion and conclusions
In this paper we have studied optimal radiotherapy fractionations for low-
grade glioma using a mathematical model. The basic model reproduces most
of the observed phenomenology of the response of LGGs to radiation [42] and
R dw1dw2dw3dw4dw5dw6∆T
Gy/day Gy/day Gy/day Gy/day Gy/day Gy/day days
0.5 1.7998 1.8000 1.8101 1.8001 1.8010 1.8010 1
0.6 1.7899 1.7998 1.8001 1.8101 1.8001 1.8200 1
0.7 1.7999 1.8002 1.8000 1.8001 1.8002 1.8003 1
Table 4: Doses per day in Gy maximizing Eq. (6) (optimization criterion 2) for the extended
model given by Eqs. (9) and satisfying the restrictions (8a) and (8b). The last column (∆T)
shows the days gained with respect to the standard treatment with all doses equal to 1.8 Gy.
Parameter values are r= 0.1,σ = 4,Ω = [0,8] ×[0,8]. The results are shown for different
values of the maximum density of cancer cells R
17
0 0.1 0.2 0.3 0.4 0.5 0.6
0
0.1
0.2
0.3
0.4
0.5
0.6
max(u)
max(v)
Figure 4: Phase-space like dynamics of the maximum density quantities max(u) and max(v).
Red dots: System dynamics during the course of irradiations. Blue line: Dynamics after
treatment. Black line: Critical threshold value R= 0.6. The initial distribution is given by
Eq. (28) with r= 0.4, σ = 4,Ω = [0,8] ×[0,8]. Parameters of the model: ρ= 0.003;
D= 0.0001; Sj= 0.85, j = 1,30.
provides a reasonable basis for the optimization procedure used. Other models
described in the literature [4] share the same basic structure (Fisher-Kolmogorov
equations) and thus we believe that the applicability of our results is beyond
the specifics of the models.
Both space-independent and space-dependent models have been studied.
Two different optimization criteria have been developed, the first one account-
ing for the global effect of the tumor mass on the disease sympthoms and the
second one related to the delay of the malignant transformation of the tumor.
Using the method of feasible directions we have searched for optimal dis-
tributions of the daily doses djin the standard protocol of 30 fractions using
both models and the two different optimization criteria. The optimal results
found in all cases are minor deviations from the standard protocol and provide
only marginal potential gains that are not relevant in real-life scenarios. Thus,
our results support the optimality of current radiation fractionations over the
standard 6 week treatment period. This is also in agreement with the observa-
tion that minor variations of the fractionation have failed to provide measurable
gains in survival or progression free survival, pointing out to a certain optimality
of the current approach.
Thus one can expect no improvements of radiation therapy by redistributing
doses in a different way over 6 weeks if the toxicity levels are to be maintained.
Thus any improvement if possible should come from a radically new approach
such as the recently proposed extreme protraction [43]. We hope that this
study will contribute to the understanding of the response of low grade gliomas
18
to radiotherapy and the conceptual limitations and potentials of this useful
therapeutical tool.
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