A Nonextensive Model for Radiobiology | SpringerLink

A Nonextensive Model for Radiobiology O. Sotolongo-Grau1, D. Rodriguez-Perez2, Oscar Sotolongo-Costa3, and J.C. Antoranz2,3 1

Hospital General Gregorio Marañón, Laboratorio de Imagen Médica, 28007 Madrid, Spain 2 UNED, Departamento de Física Matemática y de Fluidos, 28040 Madrid, Spain 3 University of Havana, Cátedra de Sistemas Complejos Henri Poincaré, Havana 10400, Cuba

Abstract— A nonextensive survival fraction model is extended from single dose to fractionated radiotherapy treatments, ruled by an extensivity index and a critical annihilation dose. The expected effects of doses given simultaneously and separately in time, lead to the general expression for the survival fraction with both limits linked by a new parameter describing the evolution of tissue radioresistance. This parameter determines the critical dose per fraction needed to achieve total target annihilation. The conclusion is that as the extensivity index of the tissue characterises the primary response of the tissue to radiation, the success of fractionation depends only on this critical dose per fraction value. Keywords— Radiobiology, Survival fraction, Entropy.

I. INTRODUCTION One of the radiobiology main goals is to find the fraction of cells surviving a given dose of radiation. The well known linear quadratic (LQ) model [1-3] seems to answer this difficult problem and also provides simple isoeffect relationship for doses around 2 Gy/session. However the LQ model, and its associated family, has raised controversy [4] and has been both extensively criticized [5-7] and defended [8, 9]. Nevertheless it has become the standard radiobiology model, replacing the nominal standard dose (NSD) model [10] for isoeffect relationships, even when the community is well aware of its limitations and statistical flaws on its validation and fitting [11]. On the other hand there is not a suitable model for nonstandard treatments as in hypofractionated therapies. In this work we present, from a survival fraction general model using the framework of nonextensive thermodynamics [12, 13], a first approach to fractionated therapies and the physical meaning of the tissue recovering/repopulation processes according to this model.

II. STATISTICAL DESCRIPTION OF SURVIVAL FRACTION The survival fraction is a probabilistic approach to describe dose effects, thus it seems reasonable to look for it in terms of some maximum entropy principle [14]. First some

assumptions need to be made. The probability density of cell death per unit dose must be a function of a dimensionless radiation dose. Let the entropy be the BoltzmannnGibss’s entropy (BG). Then maximum entropy principle, with the usual normalization condition and mean value existence, leads to the exponential distribution. As far as the problem has been discussed as an extensive problem, the survival probability, or survival fraction, will fulfill the additive property. Under these conditions, the survival fraction takes the form of the linear model of radiobiology. However, this model fits the experimental data only for some tissues, under low radiation doses [1], so more complex empiric models have been introduced. Under the assumption that the cell death could be provoked by one or two lethal events, the linear quadratic (LQ) model [3] provides Fs = exp ª¬ −α D − β D 2 º¼ , where D is the radiation

dose. This model is in better agreement with the experimental data for moderate doses. 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. A main threat here is that the superposition principle is not fulfilled. Indeed, it is easy to show that if the survival fractions 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. This suggested that the radiobiological problem should be approached from a non extensive formulation [15]. 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 can not 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 complicated function. In order to cope with non-extensivity, we apply to the radiobiological problem the Tsallis entropy [15]. This condition is completed by two constraints on the probability density: the normalization condition and the finiteness of the “q-mean value” [16]. If the probability distribution support is Ω = [0; ∞ ) , then the solutions vanish subexponentially implying a value q ≥ 1 . However, if Ω is considered

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bounded from above, the solutions vanish superexponentially in Ω for q < 1 [17]. We require this latter property for Ω , because there is clinical evidence that for some tissues a finite threshold effect exists that completely removes all the cells. To apply the Tsallis maximum entropy principle to the problem we postulate the existence of some amount of absorbed radiation D0 < ∞ (or its equivalent “minimal annihilation effect”, E0 = α 0 D0 ) 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 Sq =

(

E0 1 1 − ³ p q ( E )dE 0 q −1

)

is subject to the normalization condition, and the finite q-mean value

³

E0

0

expression of ln ( Fs ) has been fitted for 23 experimental

data sets, corresponding to 5 different tissues under different radiation conditions (see [12] for details).

(1)

³

E0

0

p( E )dE = 1 ,

p q ( E ) EdE = E

q

δ c , and after enough fractions, Δ n > 1 , meaning that effective dose has reached the critical value and every single cell of tissue has been removed by the treatment. Then it is possible to find n0 , the threshold value of n , that kills every cell, for a given therapy protocol. The parameter ‹ is a cornerstone of isoeffect relationships. A fractionated therapy of fully independent fractions requires a greater radiation dose per fraction, or more fractions, in order to reach the same isoeffect as would a treatment with more correlated fractions. The ‹ coefficient acts here as a relaxation term. Immediately after radiation damage occurs ( ‹ = 1 ) tissue begins to recover, as ‹ decreases, until the tissue eventually reaches its initial radiation response capacity ( ‹ = 0 ). In other words, the formerly applied radiation results in a decrease of the annihilation dose (initially equal to D0 ) describing the effect of the next fraction. The more correlated a fraction is to the previous one, the larger the value of ‹ and, thus, the larger the effect on the critical dose will be. Notice that unlike γ , that characterizes the tissue primary response to radiation, ‹ characterizes the tissue trend to recover its previous radioresistance. As it was already shown in [13], nonextensivity properties of tissue response to radiation for single doses are more noticeable for higher doses than predicted by current models. On the contrary, nonextensive properties for fractionated therapies stand out at lower doses per fraction. Indeed,

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for high dosage a few fractions are applied in a treatment and different ‹ values would not require to change n . However, in the lower dosage case, more radiation fractions need to be applied and the ‹ parameter may become crucial. In this case n values move away from each other for isoeffect treatments with different ‹ . So, in order to achieve the desired therapy effects, fractionated radiotherapy must be planned for a tissue described by γ , varying δ according to ‹ . This radioresistance should be experimentally studied as its value tunes the annihilation dose along radiotherapy protocols.

IV. CONCLUSIONS A new model for radiobiology survival fractions has been developed taking into account only two hypotheses, the existence of a minimal annihilation dose and the maximum entropy principle. The model fits experimental data and predicts a universal behaviour for tissue reaction under radiation. Based on this model a first approach to fractionated therapies has been done introducing an interpolation coefficient that is interpreted as the tendency of the surviving tissue towards recovery, i.e. its resistance to new radiation doses. The existence of a critical dose per fraction has been found that divides therapies in two groups based on the possibility of achieving the total elimination of the tissue.

ACKNOWLEDGMENT Authors acknowledge the financial support from the Spanish Ministerio de Ciencia e Innovación under ITRENIO project (TEC2008-06715-C02-01).

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