10.3 The Eight-Layer Inversion Algorithm
257
Table 10.5 Results of
application of eight-layer
inversion algorithm using
Fig. 10.7. The
Levenberg–Marquardt
parameter is 0.0001
Lesion No.
Layer Upper boundary (mm) Original
1
−0.10625
0.31122
2
−0.17965
0.21456
3
−0.25305
0.26258
4
−0.32645
0.32472
5
−0.39985
0.37471
6
−0.47325
0.40568
7
−0.54665
0.41624
8
−0.62005
0.40678
Table 10.6 Results of
application of eight-layer
inversion algorithm using a
Levenberg–Marquardt
parameter 5 × 10 −5 . The
unknown parameters occupy
the region between 0 mm and
−0.39985 mm
Lesion no.
Layer Upper boundary (mm) Original
1
0
0.084
2
−0.05
0.083
3
−0.10
0.184
4
−0.15
0.246
5
−0.20
0.279
6
−0.25
0.290
7
−0.30
0.282
8
−0.35
0.258
with 0.10 mm in the original. Using these two results, therefore, we define the new
grid, shown in Fig. 10.7, which will be used to generate the second-iteration results.
The results for the second iteration are obtained using a value of the Levenberg–
Marquardt parameter, LM = 0.0001, and are shown in Table 10.5.
There is a slight oscillation in the results over the last four layers, but the
average value of the conductivities is 0.4009, which clearly identifies these layers
as belonging to the smooth muscle cap. The average of the first four layers is
0.278, which means that we associate these layers with the fibrous tissue. Hence, we
estimate that the smooth muscle cap extends from −0.39985 mm to −0.69345 mm,
making it 0.2936 mm long, and that the fibrous tissue extends from −0.10625 mm to
−0.39985 mm, making it also 0.2936 mm. The true length of each of these materials
is 0.3 mm, so we are in the ballpark.
As one final iteration, however, we put the eight layers extending from 0 mm to
−0.39985, and then fixed the smooth muscle cap from −0.39985 to −0.69345 mm,
leaving the remainder of the trial lesion as in Fig. 10.7. Now the results, as shown in
Table 10.6, are strikingly improved. We used a Levenberg–Marquardt parameter of
5 × 10 −5 .
The calcium layer now clearly occupies 0.1 mm, and the average of the remaining
six conductivities is 0.2565, which is a better fit to 0.24 than before. Hence, we
conclude that the fibrous tissue occupies the next 0.29985 mm, after the calcium.
The final results for the original lesion are summarized in Table 10.7.
257
Table 10.5 Results of
application of eight-layer
inversion algorithm using
Fig. 10.7. The
Levenberg–Marquardt
parameter is 0.0001
Lesion No.
Layer Upper boundary (mm) Original
1
−0.10625
0.31122
2
−0.17965
0.21456
3
−0.25305
0.26258
4
−0.32645
0.32472
5
−0.39985
0.37471
6
−0.47325
0.40568
7
−0.54665
0.41624
8
−0.62005
0.40678
Table 10.6 Results of
application of eight-layer
inversion algorithm using a
Levenberg–Marquardt
parameter 5 × 10 −5 . The
unknown parameters occupy
the region between 0 mm and
−0.39985 mm
Lesion no.
Layer Upper boundary (mm) Original
1
0
0.084
2
−0.05
0.083
3
−0.10
0.184
4
−0.15
0.246
5
−0.20
0.279
6
−0.25
0.290
7
−0.30
0.282
8
−0.35
0.258
with 0.10 mm in the original. Using these two results, therefore, we define the new
grid, shown in Fig. 10.7, which will be used to generate the second-iteration results.
The results for the second iteration are obtained using a value of the Levenberg–
Marquardt parameter, LM = 0.0001, and are shown in Table 10.5.
There is a slight oscillation in the results over the last four layers, but the
average value of the conductivities is 0.4009, which clearly identifies these layers
as belonging to the smooth muscle cap. The average of the first four layers is
0.278, which means that we associate these layers with the fibrous tissue. Hence, we
estimate that the smooth muscle cap extends from −0.39985 mm to −0.69345 mm,
making it 0.2936 mm long, and that the fibrous tissue extends from −0.10625 mm to
−0.39985 mm, making it also 0.2936 mm. The true length of each of these materials
is 0.3 mm, so we are in the ballpark.
As one final iteration, however, we put the eight layers extending from 0 mm to
−0.39985, and then fixed the smooth muscle cap from −0.39985 to −0.69345 mm,
leaving the remainder of the trial lesion as in Fig. 10.7. Now the results, as shown in
Table 10.6, are strikingly improved. We used a Levenberg–Marquardt parameter of
5 × 10 −5 .
The calcium layer now clearly occupies 0.1 mm, and the average of the remaining
six conductivities is 0.2565, which is a better fit to 0.24 than before. Hence, we
conclude that the fibrous tissue occupies the next 0.29985 mm, after the calcium.
The final results for the original lesion are summarized in Table 10.7.