Biomedical Engineering Reference
In-Depth Information
B
[%]
F
E
A
B
0
= 100
D
C
B
-10
-8
-6 -4 -2 0
Exponent of Parameter Change
2
4
6
8
10
FIGURE 7.7
Schematic illustration of ideal response curve for combined changes of model parameters.
(Modified from Pivonka, P. et al.
Bone
43 (2): 249-263, 2008.)
In the work reported in Pivonka et al. [3], the “idealized” regulatory
response by functionally active BMUs is discussed. As was stated earlier,
the bone remodeling system starts from a steady state in which it can be
identified that Δ
BMC
= 0, Δ
BFE
= 0, and concentrations of various hormones,
growth factors that cause fluctuations of initial values of differentiation and
apoptosis rates in BMUs. In order to respond to minor changes in concentra-
tions, it is expected that BMUs should be rather insensitive to the fluctua-
tions mentioned.
From Figure 7.7, it can be recognized that point A is the threshold concen-
tration, which means that any change in a model parameter below A causes
no change in BMC (/BFE). Further, regions around the usual operation sta-
tus of BMUs should be found with relatively small gradients of change in
BMC (/BFE) in response to changes in differentiation rates (regions C-D and
D-E in Figure 7.7), and regions with larger gradients of change for larger
changes in differentiation rates (region E-F in Figure 7.7). It is expected that
this response in BMC (/BFE) changes will remain limited if the differentia-
tion rates increase significantly (region beyond point F in Figure 7.7) because
an unlimited increase of BMC (/BFE) is not physiologically realistic.
Conversely, it is expected that the rate of BMC (/BFE) change will also
decrease in a limited manner if the differentiation rates decrease signifi-
cantly. In fact, physiologically, it is reasonable for the BMC (/BFE) change
to be zero for extremely small differentiation rates. Additionally, it can be
observed from Figure 7.7 that point F marks the maximum change in BMC
(/BFE) (ΔB max). Since we have a point A that is the maximum concentra-
tion that does not lead to further modifications of BMC (/BFE), there must
be a transition region from point C to A that is characterized by point B, the
lowest value of BMC (/BFE).
