Page 38 - Mathematical Techniques of Fractional Order Systems

28 Mathematical Techniques of Fractional Order Systems

In an effort to reduce parameters in the original models of Ayatietal.
(2010), the action of tumor through coefficients r ij is replaced by the vari-
able order derivative αðtÞ applied in the operator D of CðtÞ and BðtÞ equa-
tions. However, since the resulting model’s steady state, from a
mathematical standpoint, is the same as the local healthy one, for all con-
templated cases here, and most parameters remain with the same value, the
associated activity of osteoclasts and osteoblasts in bone mass must differ
from the original case. As such, the bone resorption formation ratio, given
by Eq. (1.26), is determined between 0 and t, that corresponds to the com-
pletion time of a single cycle of CðtÞ in the new model. When treatments
are applied, the duration of an individual remodeling cycle is changed
whenever an introduction is done. Consequently, three resorption rates
must be determined: when the tumor begins, when treatment is applied, and
when the tumor is extinguished and the treatment is consequently stopped.
Each ratio is determined for the first complete cycle after the induced
change. Such is the same method as used originally in Ayatietal. (2010).
Bone resorption and formation activities are then given by κ C 5 rR and
κ B 5 r, respectively.

Ð t
max½0; CðtÞ 2 C ss Š
0 ð1:26Þ
t
R 5 Ð
max½0; BðtÞ 2 B ss Š
0
For the local and nonlocal tumor bone remodeling models, their simplifi-
cation is presented in Eqs. (1.27) and (1.28), respectively.
D αðtÞ CðtÞ 5 α C CðtÞ g CC BðtÞ g BC 2 β CðtÞ ð1:27aÞ
C
D αðtÞ BðtÞ 5 α B CðtÞ g CB BðtÞ g BB 2 β BðtÞ ð1:27bÞ
B
1
D zðtÞ 52 κ C max½0; CðtÞ 2 C SS Š 1 κ B max½0; BðtÞ 2 B SS Š ð1:27cÞ

1 L T
D TðtÞ 5 γ TðtÞlog ð1:27dÞ
T
TðtÞ
αðtÞ 5 1 2 θ 3 t 3 TðtÞ ð1:27eÞ
Fig. 1.11 compares the results for a bone with tumor obtained with this
simplified model with the results obtained with the model in Eq. (1.18).Itis
clear that the results are qualitatively similar, even though there are differ-
ences. Where the evolution of osteoclasts and osteoblasts is concerned, the
simplified model is even more similar to the model with no tumor. However,
it is not yet known if the tumorous-induced changes in the action of osteo-
blasts are the result of a reduction of their number or of a lessened capability