Lupine Publishers | Open Access Journal of Oncology and Medicine
Introduction
The
study of complex systems and investigation of their structural and dynamical
properties have attracted considerable interests among scientists in general
and physicists, biologists and medical researchers in particular. Complex
systems can be found almost everywhere however the highest level of
complexities is related to living and biological organisms and systems. Due to
the lack of a reliable and effective tool to investigate such systems, we have
not reached to the complete understanding and comprehensive pictures of the
phenomena and processes which occur in these systems. Of course a comprehensive
knowledge of biological and biomedical complex phenomena will be achieved when
we employ simultaneously different field of science and engineering including:
biology, chemistry, physics, mathematics, mechanical engineering and so on.
Fortunately
in recent year's powerful tool of fractional calculus has been proposed for
study of complex and nonlinear phenomena. It is in fact very useful tool for
describing the behavior of nonlinear systems which are characterized by:
special kind of non-locality, long-term memory and fractal properties. There
exist many biological objects and systems with memory, nonlocal effects and
nonlinear behaviors and such these non-localities and memory effects in
biological objects and systems mean that the next state of the organism or
system relies not only on its present state but also upon all of its previous
states. As a result, the concept of fractional dynamics and in fact adopting
fractional calculus can play an important role in the study of dynamical
biological systems. Up to now few number of important issues such as: protein
folding phenomena and mechanics of cancer cells (for more details see the
references which have investigated physics of protein and physics of cancer in
detail) have been investigated using the framework of fractional dynamics [1].
However
many other important issues still remain as open issues, such as: modeling of
interactions between light (laser) and biological tissue and modeling of
intracellular (and intercellular) interactions in the framework of fractional
dynamics. As a physicist or biologists and even medical researchers, we always
are able to model natural phenomena for instance modeling of tumor growth using
systems of differential equations and nowadays it is well know that the
fractional-order ones are more comprehensive and also incorporate memory effect
and the concept of non-locality in the model.
Mathematically
the idea is in fact, to rewrite the ordinary governing differential equations
in the fractional form by replacing the standard derivative with a fractional
derivative of arbitrary order which is defined in the Caputo sense as follows:
where Γ denotes the Gamma function and , . And its Laplace transform can be given by:
where Γ denotes the Gamma function and , . And its Laplace transform can be given by:
Where, F(s) is the
Laplace transform of f (t). Solutions of
fractional differential equations generally will be expressed using a
generalized special function named as Mittag-Leffler function. This function
can be considered as a generalized exponential function and has several
different forms. For instance the one-parameter Mittag-Leffler function is
defined by the series expansion as:
Where C is the set of complex numbers? It is worth mentioning that the exponential function is just a special case of α = l Mittag-Leffler function, for example for the special case of , the Mittag-Leffler function Eq. (3) reduces to the exponential function E1(z) = ez . This point is very important because of that the natural exponential function has been considered as a fundamental function of natural science and in particular biology up to now, so that many phenomena could be described using it and now scientist are able to think that with such this new framework (i.e. fractional differential equations and their solutions in terms of Mittag-Leffler functions) they can find many new results and information about biological and biomedical phenomena [2,3].
Where C is the set of complex numbers? It is worth mentioning that the exponential function is just a special case of α = l Mittag-Leffler function, for example for the special case of , the Mittag-Leffler function Eq. (3) reduces to the exponential function E1(z) = ez . This point is very important because of that the natural exponential function has been considered as a fundamental function of natural science and in particular biology up to now, so that many phenomena could be described using it and now scientist are able to think that with such this new framework (i.e. fractional differential equations and their solutions in terms of Mittag-Leffler functions) they can find many new results and information about biological and biomedical phenomena [2,3].
Finally,
based on all above mentioned reasons, as a conclusion we should say that we
believe that the powerful tool of fractional calculus and in fact the frame
work of fractional dynamics can give.com new insights in understanding and
modeling of nonlinear complex phenomena in various living cellular structures
and their interactions and we invite all biologist and medical researchers to
consider this new powerful approach for their future studies.
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