Abstract
By means of a recently-proposed metric or structural derivative, called scale-q-derivative approach,we formulate differential equation that models the cell death by a radiation exposure in tumor treatments. The considered independent variable here is the absorbed radiation dose D instead of usual time. The survival factor, FS, for radiation damaged cell obtained here is in agreement with the literature on the maximum entropy principle, as it was recently shown and also exhibits an excellent agreement with the experimental data. Moreover, the well-known linear and quadratic models are obtained. With this approach, we give a step forward and suggest other expressions for survival factors that are dependent on the complex tumor structure.
Highlights
One of the authors (J.W.) demonstrated the existence of a possible relationship between q-deformed algebras in two different contexts of Non-extensive Statistical Mechanics (NESM), namely, the Tsallis' framework and the Kaniadakis' scenario, with local form of fractional-derivative operators denned in fractal media, the so-called Hausdorff derivatives, mapped into a continuous medium with a fractal measure [1].In addition, to describe complex systems, the q-calculus, in a NESM context, has its formal development based on the definition of deformed expressions for the logarithm and exponential [2], namely, the q-logarithm and the q-exponential
We introduced, for the first time, the mathematical tool of structural derivatives to model the survival factor for tissues irradiation
The models are analogous to the nuclear decay problem, that takes the time, t, as the independent variable; here, instead, our independent variable is the applied dose, x
Summary
One of the authors (J.W.) demonstrated the existence of a possible relationship between q-deformed algebras in two different contexts of Non-extensive Statistical Mechanics (NESM), namely, the Tsallis' framework and the Kaniadakis' scenario, with local form of fractional-derivative operators denned in fractal media, the so-called Hausdorff derivatives, mapped into a continuous medium with a fractal measure [1]. We set up a formalism that may yield an effective theory to model cell radiation absorption, without the use of statistical averages and without formally imposing any specific nonstandard statistics For this purpose, we apply the mathematical tool of structural derivatives [5] that includes q-derivatives-like, called scale-q- derivative [7] and may be thought with other kind of structural derivatives like Hausdorff derivatives [8, 9, 10] and conformable derivatives [11]. This energy exchange may be responsible for the resulting noninteger dimension of space-time, giving rise to an effective coarse-grained medium Notice that this kind of approach is suitable to treat systems with dissipative forces or non-holonomic systems since it include the scale in time letting to consider the effects of internal times of the systems. It is Interesting to observe that the referred author considered the internal energy of subsystems in such a way that they behave as particles with an internal structure
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