Page 43 - Mathematical Techniques of Fractional Order Systems

Variable Order Fractional Derivatives and Bone Remodeling Chapter | 1 33

Riemann Liouville, Caputo, Gru ¨nwald Letnikoff, etc., formulations of
fractional derivatives.
It is important to retain that the different definitions, corresponding to
different operators, all make sense. They can be implemented using elec-
tronic circuits, and the different memories of past values of the order corre-
sponding to different ways of switching between constant orders. In other
words, the way the order changes modifies the memory that remains of what
happened before with the order. Different memories correspond to different
phenomena found in applications.
An application to the modeling of the biochemical phenomena of bone
remodeling has been presented. This application requires a memory of past
values of the order; in particular, a type-D formulation from Sierociuk et al.
(2015a,b) was used. The models simulate what happens to healthy bone tis-
sue, to bone tissue with a tumor, and to bone tissue with a tumor subject to
cancer treatments. It has been shown that variable order derivatives lead to
models which are simpler and easier to understand. It is not difficult to see
why, in this application, the variable order derivatives must have a memory
of past values of the order. The memory is here reasonable, as the order
models the effects of cancer, and these take time both to become manifest
and to disappear even if the disease can be successfully treated. But there are
other applications in which a different behavior of the order may be needed.
That is why different definitions exist.
As to the models found, they can be used to develop personalized clinical
decision systems for bone pathologies, aiming at the administration of the
most efficient targeted therapies, adapted and tailored to each particular
patient. This is a step towards achieving the impact that computational analy-
sis of bone physiological models is expected to have on the development of
clinical decision support systems in the future.
As to future developments of variable order derivatives, and beyond a
further study of the operators, it is interesting to try to determine the condi-
tions in which different memories of past values may be needed in applica-
tions, so as to more easily use the appropriate definitions in practice.
As to future developments of the bone remodeling equations, diffusion
terms, and consequent boundary conditions, should be considered for the
PK/PD treatments (because drugs are applied in a specific site, and will dif-
fuse), and the biomechanical effects in the bone should be incorporated. This
means that the changes in dynamics caused by mechanical solicitations
should be added to the variable order differential equations (Belinha et al.,
2015; Capacete, 2016). But the greatest challenge will likely be to measure
coefficients from experimental data, possibly extrapolated from experiments
with animals (Bonucci and Ballanti, 2014). While parameters found in the
literature are not more than educated guesses by clinicians and oncobiolo-
gists, this approach may provide additional proof of the importance of vari-
able order derivatives in this field.