Page 39 - Mathematical Techniques of Fractional Order Systems

Variable Order Fractional Derivatives and Bone Remodeling Chapter | 1 29

Comparison between bone models
Osteoclasts C(t) [cells] 20 5 0 0 200 400 600 800 1000 1200
25
15
10

Osteoblasts B(t) [cells] 1000 0 0 200 400 600 800 1000 1200
800
600
400
200

Bone mass z(t) [%] 100 Existing healthy model
Existing tumorous model
80
Simplified tumorous model
60
40
20
0
0 200 400 600 800 1000 1200
Time t [days]
FIGURE 1.11 Osteoclasts, Osteoblasts, and Bone Mass evolutions, respectively. In full lines,
simulation of Eq. (1.17) for the existing model replicating a healthy bone microenvironment. In
dashed lines, simulation of Eq. (1.18) and (1.27), for a bone microenvironment disrupted by a
developing tumor. Black lines represent stationary states. Tumor evolution is represented, in a
full-blue line, on the left-side graphic of Fig. 1.10. Parameters were set according to Ayati et al.
(2010), and can be found in Table 1.2.

of the existing cells to form bone (Holen, 2012). Consequently, these differ-
ences are immaterial, at least in the current state of knowledge about the
phenomenon under study, even though future research may prove one of the
alternatives the most correct.
The system of Eq. (1.27) is very sensitive to changes in order, as can be
seen in Fig. 1.12. The value of this parameter must be set carefully to repli-
cate the qualitative behavior desired, as was done in all other simulations in
this chapter. On the other hand, this proves that a mere change in the differ-
entiation order is in fact able to replicate different situations, and different
evolutions of the bone mass, corresponding to different evolutions of a
tumor, and different effects of the treatments that may be applied.
The extension of the simplified model to include diffusion over one
dimension (as was done to the integer order model in its extension of
Eq. 1.20) is trivial, and is given by the system of partial differential equa-
tions in Eq. (1.28). Simulating this system is not so trivial, but can be
achieved simulating simultaneously several BMUs, and calculating the sec-
ond order derivatives in order to space using centered finite differences.