Page 37 - Mathematical Techniques of Fractional Order Systems

Variable Order Fractional Derivatives and Bone Remodeling Chapter | 1 27

The order expression, αðtÞ or αðt; xÞ, acts only on the osteoclasts and
osteoblasts equations, as the tumor equations are said to be independent of
the bone microenvironment (Ayati et al., 2010), and that bone mass varia-
tions are but a reflection of osteoclasts and osteoblasts activity (Komarova
et al., 2003). The order is influenced by the tumor dynamics, according to
Eq. (1.23), where θ is a constant term experimentally determined, and time t
is related to the beginning of the tumor growth. It is now responsible for
inducing in the original healthy model (Eq. 1.20) the same response as the
tumor disrupted bone one (Eqs. 1.18 and 1.21).

αðtÞ 5 1 2 θ 3 t 3 TðtÞ ð1:23aÞ
αðt; xÞ 5 1 2 θ 3 t 3 Tðt; xÞ ð1:23bÞ
It is noted that, being the osteoclasts and osteoblasts formulated in differ-
ential equations, the order will be subjected to an integration process repre-
sented by the term 1 2 ::: in the following formulation and translated in the
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integrator D , as explained in Eq. (1.24) for a generic function fðtÞ when a
derivative’s order αðtÞ is applied.
1
D 12αðtÞ fðtÞ 5 γ½fðtފ3D D 2αðtÞ fðtÞ 5 γ½fðtފ3
1 α 21 α ð1:24Þ
3D fðtÞ 5 D ðtÞ½γ½fðtފŠ3fðtÞ 5 D ½D ðtÞ½γ½fðtފŠŠ
α
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The last step in Eq. (1.24) is given by fðtÞ 5 D ½D ðtÞ½...ŠŠ. According
to the law of exponents of operator D, presented in Vale ´rio and Sa ´ da Costa
(2013) even if the order is time-varying the referred law is verified as long
as the conditions of Eq. (1.25) are achieved. For such a case, coefficient cor-
respondence is β 52 1 and α 5 αðtÞ.
α β α1β
c D tc D fðtÞ 5 c D t fðtÞ for β # 0Xα 1 β # 0 ð1:25Þ
t
The numerical Type-D formulation behavior, when the tumor is being
extinguished, is said to provide inaccurate results by the authors (Sierociuk
et al., 2015b). To solve such numerical behavior, when the tumor’s mass is
approximately zero, a switching scheme is applied bypassing the variable
order Simulink block from the toolbox described. An acceptable value for the
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switching is for an achieved tumor density of 10 %. Not only that, due to
intrinsic characteristics of its differential equation, the tumor is constantly
growing until it reaches the maximum size of 100%. When treatment is
applied, the tumor’s density decreases to such small values it can be consid-
ered inexistent. However, when the referred therapy is halted, tumor density
tends to retake its Gompertz form and grow again until maximum capacity is
achieved. Physiologically speaking, such situation is in accordance to a
tumor relapse, which is not meant to be mathematically replicated here. So,
to solve such innate behavior when the tumor’s density is approximatively
null, after treatment, it should be maintained that way. Here, this was
achieved through a simple Simulink step function.