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Non-extensive radiobiology
O. Sotolongo-Grau∗, D. Rodriguez-Perez∗, J. C. Antoranz∗,† and O.
Sotolongo-Costa†
∗UNED, Departamento de Física Matemática y de Fluidos
†UH, Cátedra de Sistemas Complejos Henri Poincaré
Abstract. The expression of survival factors for radiation damaged cells is 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. A generalization of the exponential, the
logarithm and the product to a non-extensive framework, provides a simple formula for the survival
fraction corresponding to the application of several radiation doses on a living tissue. The obtained
expression shows a remarkable agreement with the experimental data found in the literature, also
providing a new interpretation of some of the parameters introduced anew. It is also shown how
the presented formalism may has direct application in radiotherapy treatment optimization through
the definition of the potential effect difference, simply calculated between the tumour and the
surrounding tissue.
Keywords: Radiobiology, Survival fraction, Entropy
PACS: 05.20.-y, 87.10.-e, 87.53.Ay, 87.55.dh
INTRODUCTION
One of the main concerns of a radiation oncologist is to find a treatment which, maxi-
mizing the damage over the tumor, minimizes it over the surrounding healthy tissue. In
order to reach a suitable treatment the radiobiologists have developed some empirical
models describing the interaction between radiation and living tissues (see [1] for a re-
view of radiobiology models) capable of finding the survival fraction, Fs, of cells under a
radiation dose, D. These models applicability limits are not clear so multiple corrections
have been developed in order to fit the experimental data [2].
The concept of tissue effect, E, raised from some of these models [3] is used to com-
pare different treatments each other. Usually expressed as E =− ln(Fs) is a dimension-
less magnitude that gathers several models of interaction between cells and ionizing
radiation.
The simplest radiobiology model is the linear one. Here the tissue effect is consid-
ered linear to the radiation dose, E = αD, and the survival fraction, Fs = exp(−αD),
is viewed as the cumulative survival probability of a cell under any dose below D. This
probability fullfils the additive property meaning that the effects of radiation are cumu-
lative following an additive model and the survival fraction for two doses could be found
as Fs [D1 +D2] = Fs [D1] Fs [D2].
However, this model only fits the experimental data for some tissues, under low
radiation doses [1], so the tissue effect must be corrected to E = αD+βD2, called the

linear quadratic (LQ) model. But then the survival fraction loses the additive property,
Fs [D1 +D2] < Fs [D1] Fs [D2] , and the tissue effect becomes a supperadditive quantity,
E [D1 +D2]> E [D1]+E [D2].
As a result of the nonlinear nature of E in this case, the superposition principle is not
fulfilled. However any model of interaction between radiation and living tissues must
allow to divide a continuous radiation in finite intervals and the resultant tissue effect
must be the same.
Indeed, it is easy to show that, under the LQ viewpoint, if the tissue effect were
additive for different radiation sessions, then the additivity of the dose would not hold.
Conversely, assuming that the dose is additive then the tissue effect is not equivalent to
the sum of the effects for different doses. This result suggests that the radiobiological
problem must be approached from a non extensive formulation [4].
In this work we use at the first stage the Boltzmann-Gibbs (BG) entropy in order to
find the expression of the tissue effect as a function of the absorbed dose. Later, along
with the Tsallis entropy [5] definition, it is assumed that a critical value of the radiation
dose kills every single cell and a general expression for survival fraction is found. This
survival fraction expression fits the experimental data even where previous empirical
models fail. Using the q-deformed functions [6, 7] a new expression to find the survival
fraction of a whole treatment is found allowing to show hints to find the best treatment.
FIRST STEP: THE CLASSICAL APPROACH
First we study the extensive problem applying the BG entropy (in units of the Boltzmann
constant),
S =−
∫
∞
0
ln[p(E)]p(E)dE, (1)
where in this case E is, as before, the tissue effect and p(E) is the cell killing probability
density.
According to the maximum entropy principle, if p(E) satisfies the normalization
condition and a finite mean value of the tissue effect does exist, then the problem of
finding the p(E) that extremizes the BG entropy under the above conditions can be
posed. It is well known that among all continuous probability distributions for a positive
continuous variable with a fixed mean value, the exponential distribution has the largest
entropy [8]. So,
p(E) = 1〈E〉e
− E〈E〉 , (2)
and the survival probability of a single cell will be
Fs =
∫
∞
E
p(x)dx = e−
E
〈E〉 . (3)
The survival probability here must fulfill the dose additivity property. This can be
achieved if, following the discussion in the previous section, the tissue effect is propor-
tional to the absorbed dose:
E = α0D, (4)

where α0 is chosen as a constant that makes E adimensional.
It must be noted that (3) is the experimentally proved and currently used expression
for the survival fraction as a function of tissue effect and justified in the literature only
through empirical arguments [1]. We can take α = α0/〈E〉= 1/〈D〉 and the expression
(3) becomes expressed in the known standard radiobiology form of the linear model.
Even when the BG treatment of the problem does not cover the available data, it shows
that the tissue effect must be defined as proportional to the absorbed dose of radiation.
However the empiric expressions already known show, as has been discussed in the
introduction, that the survival probability of a cell does not fulfill the additive property.
Since this is usually associated to non extensive problems the solution must be searched
using a non extensive definition of entropy. On the other hand, the Tsallis formulation
of the entropy has been proved its helpfulness when applied to problems of this nature.
ONE STEP FURTHER: THE GENERALIZED APPROACH
To apply the maximum entropy principle, in the Tsallis version, to the problem of finding
the survival fraction of a living tissue [9] that receives a radiation, we postulate the
existence of some amount of absorbed radiation D0 < ∞ (or its equivalent “minimal
annihilation effect”, E0 = α0D0) after which no cell survives. The application of the
maximum entropy principle performs like the usual one but with a few modifications.
The Tsallis entropy becomes
Sq =
1
q−1
(
1−
∫ E0
0
pq(E)dE
)
, (5)
the normalization condition is in this case
∫ E0
0 p(E)dE = 1 and the q-mean value be-
comes
∫ E0
0 p
q(E)EdE = 〈E〉q < ∞. With this definition, all properties of the tissue and
its characteristics of the interaction with radiation become included in 〈E〉q and therefore
in E0. This is the only parameter (besides q) entering in our description. It is clear that
the determination of 〈E〉q 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 (5) under the above conditions the well known method
of Lagrange multipliers [7] is applied, obtaining
E0 =
2−q
1−q
( 〈E〉q
2−q
)
1
2−q
, (6)
and
p(E) =
(
2−q
〈E〉q
)
1
2−q

1− 1−q
2−q
(
2−q
〈E〉q
)
1
2−q
E


1
1−q
. (7)

Then the survival factor is
Fs(E) =
∫ E0
E
p(x)dx =
(
1− E
E0
)
2−q
1−q
, (8)
with q < 1 for E < E0 and zero otherwise. It is not hard to see that when q → 1 then
E0 → ∞ and 〈E〉q → 〈E〉.
Equation (8) can be written
Fs(D) =
{(
1− DD0
)γ
∀D < D0
0 ∀D > D0
, (9)
where we introduced E =α0D, γ = 2−q1−q and D0 =E0/α0. Finally, the LQ model is easily
recovered from (9) in the limit q → 1 up to order two in a Taylor series expansion [10].
Tsallis based Survival fraction properties
The linear model for the tissue effect [1] implies that if the dose is additive the
corresponding survival fraction is multiplicative. Though this property belongs only to
the linear model and not to more general descriptions like the LQ model [1] and others,
we think it is worth to find a link between the additivity property of the dose and the
probabilistic properties of the cell survival fraction.
Let us define the expγ(x) function
expγ(x) =
[
1+ xγ
]γ
, (10)
and the lnγ(x) its inverse function
lnγ(expγ(x)) = x. (11)
Then, let us introduce the γ-product of two numbers x and y as
x⊗γ y = expγ
[
lnγ(x)+ lnγ(y)
]
=
[
x
1
γ + y
1
γ −1
]γ
. (12)
Note that definitions (10) and (11) are not essentially different from the q-exponential
and q-logarithm presented in [6]. We are just introducing these definitions to simplify
the calculations.
Let us now define the “generalized tissue effect” as E = −E0γ lnγ(Fs). We demand
this effect to satisfy the additive property. Then the survival fraction expressed as Fs =
expγ(−γ EE0 ) = expγ(−γ
D
D0 ), becomes γ-multiplicative. This implies that the statistical
independence of the survival fractions is only possible when γ → ∞ (q → 1).
The survival fraction for the sum of the effects after N doses becomes

Fs(NE) =
[
1−
N
∑
i=1
Ei
E0
]γ
= expγ
[
−
N
∑
i=1
γ Ei
E0
]
=
[
N
⊗
i=1
]
γ
Fs(Ei), (13)
where
[
⊗N
i=1
]
γ denotes the iterated application of the γ-product.
EXPERIMENTAL AGREEMENT
Equation (9) represents the survival fraction in terms of the measurable quantities D (ra-
diation 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 where
the survival fraction Fs is represented as a function of D for different radiation condi-
tions. However, if D is rescaled as 1−D/D0, as usual in phase transition phenomena,
all curves corresponding to the same tissue collapse to the same straight line in a log-log
plot.
The expression of ln [Fs] has been fitted for 23 experimental data sets, corresponding
to 5 different tissues, in terms of the rescaled variable ln [1−D/D0], minimizing the
appropriate least squares functional using the steepest descent method [11]. The slope
of these lines are the values of γ meaning that D0 is the natural unit of D.
Figure 1 shows, in a log-log plot, the comparison of our model with all these data
sets. In order to represent all data sets in the same plot the survival fraction is shown
normalized by γ as (Fs)1/γ .
All the information about the kind of radiation, radiation rate, etc. is contained in
the phenomenological term D0, whereas tissues are characterized by γ . This makes
(9) a very general expression with universal characteristics since the phase transition
described by (9) is homomorphic with the phase transition of ferromagnets near the
Curie point [17]. The exponent γ in this case, as in ferromagnetic phase transitions,
determines the universality class. Then γ in our case deals only with the kind of tissue
that interacts with radiation [10].
IS IT POSSIBLE TO DESIGN THE BEST PROTOCOL?
The expression for the survival fraction after N doses of radiation must allow to compare
the damage provoked on the tumor and surrounding tissue cells if the magnitudes γ and
D0 are known for both, tumor and tissue. Let us assume we know the recommended
radiation dose D+ per session for a treatment with N sessions at several values of N. We
will define the potential effect as
χ =− ln [Fs] , (14)
and then the recommended potential effect per dose over the tumor will be,
χ+ =−γ(t) ln
[
1− D+
D(t)0
]
, (15)

0.4
0.5
0.6
0.7
0.8
0.9
1
0.4 0.5 0.6 0.7 0.8 0.9 1
Fs
1/γ
1−D/D0
150 cGy/min (1)
7.6 cGy/min (1)
1.6 cGy/min (1)
Neutrons (2)
e− High (2)
e− Low (2)
e− Hypoxic (2)
Oxic (3)
Hypoxic (3)
N2 (4)
N2 + 1mM (4)
N2 + 10mM (4)
Air (4)
Air + 1mM (4)
Air + 10mM (4)
250 kVp x−rays (5)
14.9 MeV deuterons (5)
3 MeV deuterons (5)
26 MeV α−particles (5)
8.3 MeV α−particles (5)
5.1 MeV α−particles (5)
4 MeV α−particles (5)
2.5 MeV α−particles (5)
FIGURE 1. Normalized survival fractions, (Fs)1/γ , as a function of the rescaled radiation dose, 1−
D/D0. The straight line shown is y = x. Values for γ and D0 are detailed in table 1.
where γ(t) and D(t)0 are the characteristic radiation coefficients for the tumor. If we have
already characterized the healthy tissue around the tumor with γ(h) and D(h)0 then we can
find the survival fraction of cells after the treatment for the tumor,
Θ(t) = expγ(t)
[
N lnγ(t) [exp(−χ+)]
]
, (16)
and the healthy tissue,
Θ(h) = expγ(h)
[
γ(h)D(t)0
γ(t)D(h)0
N lnγ(t) [exp(−χ+)]
]
. (17)
In order to find the best treatment all we need is to calculate the difference of potentials
for the treatment,
∆χ = χ(t)−χ(h) = ln
[
Θ(h)
Θ(t)
]
, (18)
and guarantee that it will be positive. This could be achieved taking advantage of the
different responses to radiation of tumor and normal tissues. Also, if this response is too
close for an specific kind of radiation or a given dose rate, those conditions could be
changed looking for a higher potential difference.

CONCLUSIONS
A new theoretical expression for the survival fraction of cells under radiation has been
found, using the Tsallis formulation of entropy. The existence of a critical value for the
absorbed radiation dose under which no cells survive is introduced in the formulation
in order to get a proper expression. The new expression depends of two coefficients
that characterize the tissue behaviour under radiation (γ) and the specifics conditions in
which the radiation is applied (D0).
The Tsallis mathematical formalism allows to redefine the multiplication operation
giving a way to find the survival fraction after several radiation sessions. If the charac-
teristic coefficients are known for the tumor and the surrounding tissue then some hints
can be given to choose the less harmful, albeit most efficient, treatment to apply.
ACKNOWLEDGMENTS
The authors wish to thank Prof. Juan Antonio Santos, MD, for fruitful discussions.
They also acknowledge the Spanish Ministerio de Industria for its support through the
Proyecto CD-TEAM, CENIT.
REFERENCES
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4. C. Tsallis, Brazilian Journal of Physics 29, 1–35 (1999), URL arXiv:cond-mat/
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(2009), URL arXiv:0907.5551v2[physics.med-ph].
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TABLE 1. Values for γ and D0 obtained from the fitting with experimental data.
(1) Human melanoma (γ = 14.0± 0.9) irradiated at different dose rates. Data extracted
from [12].
150 cGy/min D0 = 27± 2 Gy
7.6 cGy/min D0 = 38± 4 Gy
1.6 cGy/min D0 = 46± 5 Gy
(2) Intestinal stem-cells (γ = 30.5± 0.4) irradiated with different particles an conditions.
Data extracted from [13].
Neutrons D0 = 36.0± 0.7 Gy
Electrons (high dose rate) D0 = 62.2± 1.2 Gy
Electrons (low dose rate) D0 = 68.8± 1.6 Gy
Electrons (hypoxic conditions) D0 = 162± 3 Gy
(3) Cultured mammalian cells (γ = 8.9± 0.6) exposed to x-rays under oxic and hypoxic
conditions. Data extracted from [14].
Oxic D0 = 21.7± 1.4 Gy
Hypoxic D0 = 61± 4 Gy
(4) Chinese hamster cells (γ = 14.2± 0.6) irradiated in the presence or absence of mis-
onidazole. Data extracted from [15].
Hypoxic D0 = 85± 7 Gy
Hypoxic, 1mM D0 = 46± 3 Gy
Hypoxic, 10mM D0 = 33± 2 Gy
Aerated D0 = 29± 3 Gy
Aerated, 1mM D0 = 28± 3 Gy
Aerated, 10mM D0 = 32± 4 Gy
(5) Human kidney cells (γ = 8.8± 0.3) exposed in vitro to radiations of different energies.
Data extracted from [16].
250kVp x-rays D0 = 22.4± 0.8 Gy
14.9MeV deuterons D0 = 22± 3 Gy
3MeV deuterons D0 = 17± 2 Gy
26MeV α-particles D0 = 15± 2 Gy
8.3MeV α-particles D0 = 9.1± 1.2 Gy
5.1MeV α-particles D0 = 7.5± 0.8 Gy
4MeV α-particles D0 = 6.9± 0.6 Gy
2.5MeV α-particles D0 = 9.2± 0.7 Gy