Biomedical Engineering Reference
In-Depth Information
TABLE 9.3
Fitting Parameters
Fitting Parameters
Sample no.
β
K
τ
d
(ms)
τ
c
(ms)
1
0.299
-0.272
11.79
94
2
0.302
-0.295
12.74
93
3
0.305
-0.212
10.98
60
4
0.276
-0.190
11.44
115
5
0.286
-0.214
9.30
81
6
0.291
-0.213
11.42
88
Mean ± StDev
0.293 ± 0.01
-0.233 ± 0.038
11.28 ± 1.037
88.5 ± 16.4
of the fitting function is characterized by three parameters: the proportional
coefficient
K,
stretching exponent β, and time constant τ
d
.
The second part
(typical exponential term) is characterized by only one parameter: the time
constant τ
c
,
which is easily determined. These parameters are independent
of the holding load
F
o
and loading time
t
o
.
To determine the three parameters
in the first part, at least five piezovoltage time curves were measured for each
sample corresponding to different holding loads
F
o
and loading times
t
o
,
as
shown in Figure 9.12. Hou et al. obtained the three parameters by fitting any
three of the five curves to Equation (9.1) using a trial-and-error method on
a computer. Then the other two curves were fitted by the three determined
parameters, with corresponding values of
F
o
and
t
o
,
to check the adequacy
of fitting. Hou et al. found that once the three parameters were determined
by any three curves, they were available for the remaining two curves. This
proved the uniqueness of the fitting. In other words, the stretched expo-
nential behavior merely represents the bone's inherent property. The fitting
parameters of the six samples are listed in Table 9.3.
The time constant τ
d
in the loading process was about one order of mag-
nitude smaller than the τ
c
in the loading hold process. The significance of
the stretched exponent β(0.276 < β < 0.305) is that the relaxation mechanism
in the loading process was different from that in the load holding process.
The fitting functions (9.5) and (9.6) and the measured piezovoltage curves in
Figures 9.12 and 9.13 show that the peaks of piezovoltage are proportional
to the maximum load
F
o
and inversely proportional to loading interval
t
o
.
Making use of the mean value of the fitting perameters listed in Table 9.3, the
fitting function (9.1) of the bone can be further written as
0.293
t
−
t
t
11.28
(
tt
≤
)
−
0.233
F
e
o
o
()
=
o
Vt
(9.7)
0.293
t
o
−
t
−
11.28
88.5
−
0.233
Fe
⋅
e
(
t
>
t
)
o
o
