Page 19 - Mathematical Techniques of Fractional Order Systems

Variable Order Fractional Derivatives and Bone Remodeling Chapter | 1 9

Influence of accumulated values
0.55
T = ∞
T = 5000
0.5
T = 500
T = 100
0.45
0.4
0.35
0.3
0.25
0.2
0 5 10 15 20 25 30 35 40 45 50
Time t [days]
FIGURE 1.3 Testing the effects of accumulated values of order α. Obtained results for α 5 0:5
integrator output with initial conditions c 5 0:5 and different values of T (the length of initial
conditions function).

essential for the application of type-D variable order definition to the bone
remodeling models presented next.

1.3 BONE REMODELING

This section begins by introducing bone physiology concepts, in Section 1.3.1.
PK and PD models follow, in Section 1.3.2. Finally, a road-map through pub-
lished integer models, that reflect bone dynamic, is presented in Section 1.3.3.
All models presented here use dimensionless variables and parameters,
including the cell populations, except when explicitly said otherwise in
1
Table 1.1. D refers to the first order derivative in time, d , and D αðtÞ or
dt
D αðt;xÞ refers to the Gru ¨nwald Letnikoff type-D variable order derivative,
D αðtÞ or D αðt;xÞ , respectively.
2N D t 2N D t
1.3.1 Introducing Bone Physiology
The skeleton is an active metabolic tissue, besides providing support and
protection to the vital organs (Chen et al., 2010). It is not, however, static, as
it constantly undergoes remodeling. This process is spatially heterogeneous,
with regular but asynchronous cycles that can take place in 5% 25% of the
total bone surface available (Crockett et al., 2011). It is estimated that about
10% of the bone is renewed each year (Lerner, 2006). Cortical bone provides
strength and protection while trabecular is the most metabolically active.
Consequently, it is within the trabecular bone that most of bone turnover
occurs, under normal conditions and in diseases of bone loss or formation.
This reconstruction occurs within a Basic Multicellular Unit (BMU), a
temporary anatomical structure where bone is resorbed by cells termed
osteoclasts and sequentially formed due to cells called osteoblasts (Parfitt,
1994). This process is remarkably well balanced, as a tightly controlled