Page 26 - Mathematical Techniques of Fractional Order Systems

16 Mathematical Techniques of Fractional Order Systems

1.3.3 Modeling Bone Remodeling Cycles—Integer Models
Models based on differential equations have been applied to analyze and
simulate biochemical and biomechanical interactions between the tumor
cells with the bone microenvironment. They can be divided into local or
nonlocal constructions (mainly one-dimensional models) and, within each
category, three stages of bone behavior are encompassed: healthy
bone microenvironment, tumor disrupted bone dynamics, and therapy
counteraction.
The simplest model for bone remodeling was proposed in Komarova
et al. (2003), taking an S-system form described by Eq. (1.17) (Savageau,
1988). Coupling of osteoclasts, CðtÞ, and osteoblasts, BðtÞ, behavior is done
through biochemical autocrine ðg CC ; g BB Þ and paracrine ðg BC ; g CB Þ factors
expressed implicitly in the system’s exponents. Bone mass density, zðtÞ,is
determined through the extent which values of CðtÞ and BðtÞ populations
exceed their nontrivial steady state, C SS and B SS , respectively. Consequently,
bone mass is but the reflection of the bone cells activities. Production and
death rate of the bone cells are encompassed in α C;B and β , respectively,
C;B
and constants κ C and κ B represent the bone resorption and formation activ-
ity, respectively.
1 g CC g BC
D CðtÞ 5 α C CðtÞ BðtÞ 2 β CðtÞ ð1:17aÞ
C
1 g CB g BB
D BðtÞ 5 α B CðtÞ BðtÞ 2 β BðtÞ ð1:17bÞ
B
1
D zðtÞ 52 κ C max½0; CðtÞ 2 C ss Š 1 κ B max½0; BðtÞ 2 B ss Š ð1:17cÞ
This model is capable of representing single or periodical remodeling
cycles by setting the autocrine and paracrine parameters to the appropriate
values, specially the osteoblast-derived osteoclast paracrine regulator g BC .
The RANK/RANKL/OPG pathway is also implicitly encoded in this parame-
ter. Response amplitude and frequency depend on the initial conditions, trig-
gered by a deviation from the steady state, as seen in Fig. 1.5 for periodical
cycles only. Parameter values are given in Table 1.2.
Disruptive pathologies to the bone microenvironment were also added. In
Ayati et al. (2010) the previous model was extended to incorporate the effect
of MM in the bone dynamics, as presented in Eq. (1.18). TðtÞ represents the
tumor cells density at time t, with a Gompertz form of constant growth
γ . 0, and acts through the autocrine and paracrine regulations pathways in
T
the form of r ij parameters. The tumor action is considered independent of the
bone mass, with a possible maximum tumor size of L T . Periodic remodeling
cycles are deregulated and bone mass density decreases. The bone mass
equation is the same as that of Eq. (1.17c), remaining as a consequence of