Tissue Radiation Response with Maximum Tsallis Entropy
O. Sotolongo-Grau* and D. Rodrı´guez-Pe´rez
UNED, Departamento de Fı´sica Matema´tica y de Fluidos, 28040 Madrid, Spain
J. C. Antoranz
UNED, Departamento de Fı´sica Matema´tica y de Fluidos, 28040 Madrid, Spain,
and University of Havana, Ca´tedra de Sistemas Complejos Henri Poincare´, Havana 10400, Cuba
Oscar Sotolongo-Costa
University of Havana, Ca´tedra de Sistemas Complejos Henri Poincare´, Havana 10400, Cuba
(Received 22 June 2010; published 7 October 2010)
The expression of survival factors for radiation damaged cells is currently based on probabilistic
assumptions and experimentally fitted for each tumor, radiation, and conditions. Here, we show how the
simplest of these radiobiological models can be derived from the maximum entropy principle of the
classical Boltzmann-Gibbs expression. We extend this derivation using the Tsallis entropy and a cutoff
hypothesis, motivated by clinical observations. The obtained expression shows a remarkable agreement
with the experimental data found in the literature.
DOI: 10.1103/PhysRevLett.105.158105 PACS numbers: 87.10.
e, 05.20.
y, 87.53.Ay, 87.55.dh
One of the radiobiology main goals is to find the fraction
of cells surviving under a given dose of radiation [1]. This
fractionwill depend on the radiation deposited on the tissue,
which is on its own a very difficult transport [2] and timing
[3] optimization problem, not to mention the multiple
scales involved in this problem, as pointed out in [4].
Assuming the dose D of absorbed radiation is known, a
mean-field model of the survival fraction Fs can be devel-
oped [1]. Current models are mainly based in what is called
‘‘target theory,’’ proposing some targets inside the cell that
can be damaged by the radiation in a lethal event, provok-
ing cell death.
The simplest radiobiology model is the linear one [1]
where the survival fraction is expressed asFs ¼ expð

DÞ
arguing that the cell death under radiation follows a Poisson
process. Here the survival fraction is viewed as the cumu-
lative probability of cell survival (or complementary cumu-
lative probability of cell death) under any dose below D.
This probability fulfills the additive property meaning that
the effects of radiation are cumulative following an additive
model. So, the survival fraction for two doses can be found
as Fs½D1 þD2 ¼ Fs½D1Fs½D2.
This probabilistic framework calls to mind statistical
mechanics and one of its cornerstones, the notion of en-
tropy and the maximum entropy principle [5]. Statistical
mechanics is perhaps the most general theoretical frame-
work of physics. The general statistical approach provided
by Boltzmann and Gibbs [6] allows us to apply the notion
of entropy in diverse fields, including the theory of fluctu-
ations [6], information theory [7], and many others.
If Fs is viewed as a probability function, then the prob-
lem could be posed in terms of the maximum entropy
principle [8]. Let us denote by pðEÞ the probability density
of cell death and by EðDÞ a dimensionless form of the
radiation dose. This magnitude, usually expressed as E ¼

lnðFsÞ, is known as the tissue effect [9]. Then, under the
assumption that the entropy is the Boltzmann-Gibbs (BG)
entropy,
S ¼
Z


pðEÞ ln½pðEÞdE; (1)
where
represents all the states of E, when the maxi-
mum entropy principle is demanded, the normalization
condition,
R

pðEÞdE ¼ 1, and the mean value existence
R

pðEÞEdE ¼ hEi must be also fulfilled since pðEÞ is a
probability density.
It is known that the exponential distribution maxi-
mizes the entropy under these conditions. Then pðEÞ ¼
1
hEi
e
ðE=hEiÞ and Fs ¼
R
1
E pðxÞdx ¼ e

ðE=hEiÞ.
Until here the problem has been discussed as an exten-
sive problem. So, the survival probability, or survival
fraction, must fulfill the additive property and E must be
proportional to D as E ¼
D, where
is a constant that
makes E adimensional. The survival fraction takes then the
form
Fs ¼ e

ðD=hDiÞ; (2)
i.e., the linear model of radiobiology.
However, this model fits the experimental data only for
some tissues, under low radiation doses [1], so more com-
plex empirical models have been built. Under the assump-
tion that the cell death could be provoked by one or two
lethal events, the linear quadratic (LQ) model [10] is ex-
pressed as Fs ¼ exp½

D
D2. This model fits better
with the experimental data for moderate radiation doses.
PRL 105, 158105 (2010) P HY S I CA L R EV I EW LE T T E R S
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Some conclusions can be obtained from those models
and the relation between them. First, in the transition from
the linear to the LQ model, the survival fraction loses
the additive property, since Fs½D1 þD2<Fs½D1Fs½D2.
A main threat here is that the superposition principle is
not fulfilled. Indeed, it is easy to show that if the survival
fraction were multiplicative for different radiation sessions,
then the additivity of the dose would not hold. Conversely,
assuming that the dose is additive then the survival fraction
would not equal the product of survival fractions for
different doses. It suggests that the radiobiological problem
must be approached from a nonextensive formulation [11].
Second, in order to fit the experimental data a second order
term had to be added to the linear model, and since the
effect for high doses cannot be explained by any of those
models, the conclusion is that the LQ model resembles a
series approximation up to second order of a more compli-
cated function.
In 1988 Tsallis introduced a generalization of the BG
entropy function, called q entropy [12], which has been
shown to give surprisingly good results when applied to all
kinds of natural systems formed by many interacting ele-
ments (see [11] and references therein). In particular, it has
successfully described the course of damage caused by
large energy inputs leading to fractures [13] or earthquakes
[14]. Hence, to cope with nonextensivity, and motivated by
the analogy with those damage models, we will apply to
the radiobiological problem the Tsallis entropy [11],
Sq ¼
1
q
1
½1

Z


pqðEÞdE; (3)
where q is the index of nonextensivity. This condition is
completed by the constraints that the probability density
satisfies the normalization condition and the finiteness of
the q-mean value
R

p
q
ðEÞEdE ¼ hEiq <1 [15].
If the probability distribution support is
¼ ½0;1Þ,
then the solutions vanish subexponentially implying a
value q  1. However, if
is considered bounded from
above, the solutions vanish superexponentially in
for
q < 1 [16]. We will request this latter property for
,
because there is clinical evidence that for some tumors a
finite threshold effect exists enough to completely remove
them, and also the data support this superexponential decay
with the dose.
To apply the maximum entropy principle in its Tsallis
version to the problem of finding the survival fraction of an
irradiated living tissue [17], we postulate the existence of
some amount of absorbed radiation D0 <1 (or its equiva-
lent ‘‘minimal annihilation effect,’’ E0 ¼
0D0) after
which no cell survives. The application of the maximum
entropy principle is performed in the usual way but for a
few modifications.
The Tsallis entropy becomes
Sq ¼
1
q
1


1

Z E0
0
pqðEÞdE

; (4)
the normalization condition is in this case
RE0
0 pðEÞdE ¼ 1
and theq-meanvaluebecomes
RE0
0 p
q
ðEÞEdE ¼ hEiq <1.
With this definition, all properties of the tissue and its
characteristics of the interaction with radiation become
included in hEiq and therefore in E0. This is the only
parameter (besides q) entering in our description. It is clear
that the determination of hEiq for the different tissues under
different conditions of radiation would give the necessary
information for the characterization of the survival factor.
To calculate the maximum of Sq under the above
conditions the well-known method of Lagrange multipliers
is applied, obtaining
E0 ¼
2
q
1
q

hEiq
2
q

1=2
q
; (5)
and
pðEÞ ¼

2
q
hEiq

1=2
q


1

1
q
2
q

2
q
hEiq

1=2
q
E

1=1
q
:
(6)
Then the survival factor is
FsðEÞ ¼
Z E0
E
pðxÞdx ¼

1

E
E0

2
q=1
q
; (7)
with q < 1 for E < E0 and zero otherwise. It is not hard to
see that when q ! 1 then E0 ! 1 and hEiq ! hEi.
Equation (7) can be written
FsðDÞ ¼
(
ð1
D
D0
Þ

8 D < D0;
0 8 D  D0;
(8)
where we introduced E ¼
D,  ¼ 2
q
1
q
, and D0 ¼ E0=
.
Finally, the LQ model is easily recovered in the limit
q ! 1 up to order two in a Taylor series expansion. This
is a restatement of our conjecture that the LQ model comes
from an approximation of a more general model.
Equation (8) represents the survival fraction in terms
of the measurable quantities D (radiation dose) and D0
(minimal annihilation dose).
In order to compare our model with the experimental
data we have selected some survival curves from the
literature [17–21] where the survival fraction Fs is repre-
sented as a function of D for different radiation conditions.
Following the usual method of phase transition theory,
rescaling D as 1
D=D0, all curves corresponding to the
same tissue collapse to the same straight line in a log-log
plot as in Fig. 1. The expression of lnðFsÞ has been fitted for
23 experimental data sets, corresponding to 5 different
tissues under different radiation conditions. For each tissue
the steepest descent method [22] is used to minimize the
least squares functional,
2 ¼
X
i
X
ni
j¼1


 ln

1

Dj
D0;i


lnðFs;jÞ

2
(9)
where i denotes the tissue experimental set (different irra-
diation conditions), j runs along the ni experimental points
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