Page 27 - Mathematical Techniques of Fractional Order Systems

Variable Order Fractional Derivatives and Bone Remodeling Chapter | 1 17

Healthy bone dynamics & initial conditions
15
C0 = 11.16, B0 = 231.72
C0 = 11.16, B0 = 131.72
Osteoclasts [cells] 10 5 C0 = 1.271, B0 = 131.72
C0 = 1.271, B0 = 91.72

0
0 100 200 300 400 500 600 700 800 900 1000
900 C0 = 11.16, B0 = 231.72
800
Osteoblasts [cells] 700 C0 = 1.271, B0 = 91.72
C0 = 11.16, B0 = 131.72
600
C0 = 1.271, B0 = 131.72
500
400
300
200
100
0
0 100 200 300 400 500 600 700 800 900 1000
Time t [days]
FIGURE 1.5 Osteoclasts and Osteoblasts evolutions, respectively, for a different set of initial
conditions Cð0Þ 5 C 0 and Bð0Þ 5 B 0 , according to the model of Eq. (1.17). Remaining parameters
are the same as in Ayati et al. (2010) and presented in Table 1.2.
the activity of the bone cells. In Fig. 1.6, represented in full lines, a simula-
tion of the bone microenvironment for Eq. (1.18) can be compared to a
healthy case, represented in dashed lines.

TðtÞ TðtÞ
1 g CC 11r CC L T g BC 11r BC L T
D CðtÞ 5 α C CðtÞ BðtÞ 2 β CðtÞ ð1:18aÞ
C

g CB
TðtÞ
1 TðtÞ g BB 2r BB L T
D BðtÞ 5 α B CðtÞ 11r CB L T BðtÞ 2 β BðtÞ ð1:18bÞ
B

1
D TðtÞ 5 γ TðtÞlog L T ð1:18cÞ
T
TðtÞ
In Coelho et al. (2015), a PK/PD based treatment of osteolytic bone
metastases through anticancer and antiresorptive therapy is proposed, adapt-
ing the model of Ayati et al. (2010) as presented in Eq. (1.19). It corresponds
to the administration of anticancer treatment (chemotherapy—paclitaxel,
d 3 ðtÞ) and antiresorptive therapy (monoclonal anibodies—denosumab, d 1 ðtÞ;
or bisphosphonates—zoledronic acid, d 2 ðtÞ). Bisphosphonates (e.g., zoledro-
nic acid or alendronate) promote osteoclast apoptosis, and monoclonal anti-
bodies (e.g., denosumab) indirectly inhibit osteoclast formation by acting as
a decoy receptor for RANKL. Together with chemotherapy, the drug effect
of this treatment was included in the model through their PK/PD action. The
bone mass equation, zðtÞ, remains equal to Eq. (1.17c). Simulations can be
found in Fig. 1.7.