Page 31 - Mathematical Techniques of Fractional Order Systems

Variable Order Fractional Derivatives and Bone Remodeling Chapter | 1 21

Osteoclasts C(t) [cells] 3.5 4 3 2 1
2.5
1.5
0.5
0
0
0.1
0.2
0.3
0.4 1800 2000
0.5 1400 1600
0.6 1200
0.7 1000
0.8 600 800
0.9 200 400
1 0 Time - t [days]
Distance - x ∈ [0,1]
450
Osteoclasts B(t) [cells] 350
400
300
250
200
150
100
0
0.1
0.2
0.3
0.4 2000
0.5 1800
0.6 1400 1600
0.7 1000 1200
0.8 800
Distance - x ∈ [0,1] 0.9 200 400 600
1 0
Time - t [days]
115
110
Bone mass Z(t) [%] 105
100
95
90
90
85
0
0.1
0.2
0.3
0.4 2000
0.5 1800
0.6 1400 1600
0.7 1000 1200
0.8 600 800
0.9 400
Distance - x ∈ [0,1] 1 0 200
Time - t [days]
FIGURE 1.8 Nonlocal simulation of Osteoclasts, Osteoblasts and Bone Mass. Fisrt row, for
healthy remodeling cycles (Eq. 1.20). Second row, for a tumor disrupted bone microenviron-
ment (Eq. 1.21). Parameters, initial, and boundary conditions follow exactly what was presented
in Ayati et al. (2010), and can be found in Table 1.2. Untreated tumor evolution, for all metasta-
ses disrupted models, is presented in the second graphic of Fig. 1.10.