Page 33 - Mathematical Techniques of Fractional Order Systems

Variable Order Fractional Derivatives and Bone Remodeling Chapter | 1 23

@ 2
1
D Cðt; xÞ 5 σ C 2 Cðt; xÞ 2 β Cðt; xÞ 1
C
@x
ð1:21aÞ

TðtÞ TðtÞ
g CC 11r CC g BC 11r BC
L T L T
1 α C Cðt; xÞ Bðt; xÞ
@ 2
1
D Bðt; xÞ 5 σ B 2 Bðt; xÞ 2 β Bðt; xÞ 1
B
@x
ð1:21bÞ

g CB
TðtÞ TðtÞ
11r CB g BB 2r BB
L T L T
1 α B Cðt; xÞ Bðt; xÞ
@ 2
1 L T
D Tðt; xÞ 5 σ T 2 Tðt; xÞ 1 γ Tðt; xÞlog ð1:21cÞ
T
@x Tðt; xÞ
In Christ et al. (2018), the model of Ayati et al. (2010) adapted in Coelho
et al. (2015), is further extended to include the PK/PD action of anticancer
and antiresorptive therapy in the one-dimensional model of Eq. (1.22).
Simulation results are presented in Fig. 1.9 for a two-case scenario.
@ 2
1
d
C
D Cðt; xÞ 5 σ C 2 Cðt; xÞ 2 ð1 1 K s 2 2 ðtÞÞβ Cðt; xÞ 1
@x
ð1:22aÞ

TðtÞ TðtÞ
d
g CC 11r CC g BC 11r BC ð 11K s 1 1 ðtÞÞ
L T L T
1 α C Cðt; xÞ Bðt; xÞ
@ 2
1
D Bðt; xÞ 5 σ B Bðt; xÞ 2 β Bðt; xÞ 1
B
@x 2
ð1:22bÞ

g CB
TðtÞ TðtÞ
11r CB g BB 2r BB
L T L T
1 α B Cðt; xÞ Bðt; xÞ
@ 2
1 L T
d ðtÞ γ Tðt; xÞlog
D Tðt; xÞ 5 σ T 2 Tðt; xÞ 1 1 2 K i 34 c 34 T ð1:22cÞ
@x Tðt; xÞ
Involved variables and parameters are summarized in Tables 1.1 and 1.2.
1.4 VARIABLE ORDER MODELS—CREATING COMPACT
BIOCHEMICAL BONE REMODELING MODELS
Fractional and variable order derivatives have been already successfully used
in modeling the dynamics of bone remodeling. Christ et al. (2018) imple-
mented fractional derivatives in the differential equations of bone remodeling