George Alexandrakis,
Thomas J. Farrell,
and Michael S. Patterson
The authors are with the Department of Medical Physics, Hamilton Regional Cancer Center and McMaster University, 699 Concession Street, Hamilton, Ontario, L8V 5C2 Canada.
George Alexandrakis, Thomas J. Farrell, and Michael S. Patterson, "Accuracy of the diffusion approximation in determining the optical properties of a two-layer turbid medium," Appl. Opt. 37, 7401-7409 (1998)
We have examined the possibility of determining the optical
properties of a two-layer medium by using a diffusion approximation
radiation transport model [Appl. Opt. 37, 779
(1998)]. Continuous-wave and frequency-domain (FD)
low-noise Monte Carlo (MC) data were fitted to the
model. Marquardt–Levenberg and a simulated annealing algorithm
were used and compared as optimization techniques. Our particular
choice of optical properties for the two-layer model was consistent
with skin and underlying fat in the presence of an exogenous
chromophore [Appl. Opt. 37, 1958 (1998)]. The
results are therefore specific to this set of optical
properties. It was found that the cw diffusion solution could never
be used to estimate all optical properties reliably. The combined
cw and FD solutions could not be used to estimate some of the top-layer
optical properties to an accuracy of better than 10%, although the
absorption and the transport scattering coefficients of the bottom
layer could be estimated to within 7% and 0.5%, respectively. No
improvement was found from simultaneously fitting MC data at three
different modulation frequencies. These results point to the need
for a more accurate radiation transfer model at small source–detector
separations.
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True Set of Optical Properties with which all MC Runs were
Performed and a Selected False Initial Guess of the Parameters Used as
Input to All Fitting Routinesa
Parameters
True
False
μa
1
/mm-1
7.75 × 10-2
1.75 × 10-2
μs
1
′/mm-1
1.49
1.0
μa
2
/mm-1
7.0 × 10-3
1.3 × 10-2
μs
2
′/mm-1
1.0
1.1
l/mm
1.5
3.5
Parameter definitions in text.
Table 2
Estimates of Optical Properties Obtained by Fitting MC
Data for Relative cw Reflectance Measurements by Marquardt–Levenberg
and a True Initial Guess (Mrqrt-t), a False Initial Guess (Mrqrt-f),
and Simulated Annealing with a False Initial Guess
(SA-f)a
Parameters
Mrqrt-t
Error (%)
Mrqrt-f
Error (%)
SA-f
Error (%)
μa
1
/mm-1
7.425 × 10-2
4.20
2.537 × 10-4
99.67
6.517 × 10-2
15.91
μs
1
′/mm-1
1.552
4.16
1.111
37.94
1.198
19.60
μa
2
/mm-1
8.674 × 10-3
-23.91
9.780 × 10-3
39.71
1.160 × 10-2
-65.67
μs
2
′/mm-1
0.8850
11.50
0.8753
12.48
0.6909
30.91
l/mm
1.476
1.61
2.327
-55.13
1.654
-10.29
qr/no units
0.9897
0.3547
0.7260
χmin2
2.30
2.42
2.29
Evaluations
660
2112
16,501
The χ2 of the true
parameters was χ2(t) =
21.6. Parameters are defined in the text.
Table 3
Estimates of Optical Properties Obtained by Fitting MC
Data for Absolute cw Reflectance Measurements
(qr = 1.0) by Mrqrt-t and SA-f where
χ2(t) =
23.3a
Parameters
Mrqrt-t
Error (%)
SA-f
Error (%)
μa
1
/mm-1
7.377 × 10-2
4.81
0.1680
-116.77
μs
1
′/mm-1
1.550
-4.05
0.7710
48.26
μa
2
/mm-1
8.805 × 10-3
-25.71
1.674 × 10-2
-139.14
μs
2
′/mm-1
0.8766
12.35
0.4600
54.00
l/mm
1.494
0.43
1.625
-8.33
χmin2
2.30
2.29
Evaluations
924
300,001
Mrqrt-f failed to converge.
Table 4
Estimates of Optical Properties Obtained by Fitting
MC Data for Relative Phase and cw Reflectance Measurements by Mrqrt-t,
SA-t, and
SA-fa
Parameters
Mrqrt-t
Error (%)
SA-t
Error (%)
SA-f
Error (%)
μa
1
/mm-1
6.764 × 10-2
12.72
6.621 × 10-2
14.57
6.447 × 10-2
16.82
μs
1
′/mm-1
1.686
-13.18
1.473
1.17
2.085
-39.94
μa
2
/mm-1
7.399 × 10
-3
-5.51
6.793 × 10
-3
2.96
7.932 × 10-3
-13.32
μs
2
′/mm-1
1.000
0.0
1.023
2.29
1.020
2.03
l/mm
1.439
4.07
0.8833
41.11
1.631
-8.73
qr/no units
0.9955
0.5343
1.595
qph/deg
-2.691 × 10
-3
-7.997 × 10
-3
1.821 × 10-2
χmin2
3.83
3.69
3.14
Evaluations
3642
109,001
194,001
The true parameters gave
χ2(t) = 24.3.
Table 5
Estimates of Optical Properties Obtained by Fitting MC
Data for Absolute Phase and cw Reflectance Measurements
(qr = 1.0, qph = 0.0)
by Mrqrt-t, SA-t, and SA-fa
Parameters
Mrqrt-t
Error (%)
SA-t
Error (%)
SA-f
Error (%)
μa1/mm-1
6.893 × 10-2
11.06
6.901 × 10-2
10.95
6.894 × 10-2
11.05
μs1′/mm-1
1.698
-13.93
1.696
-13.83
1.698
-13.95
μa2/mm-1
7.400 × 10-3
-5.71
7.393 × 10-3
-5.61
7.401 × 10-3
-5.72
μs2′/mm-1
1.003
-0.32
1.003
-0.32
1.003
-0.32
l/mm
1.419
5.42
1.419
5.39
1.418
5.44
χmin2
3.78
3.78
3.78
Evaluations
660
46,501
300,001
The true parameters gave
χ2(t) = 52.5.
Table 6
Estimates of Optical Properties Obtained by Fitting MC
Data for Absolute cw, ac Reflectance, and Phase Measurements
(qr = 1.0, qph = 0.0)
at a Modulation Frequency f = 195
MHza
Parameters
Mrqrt-t
Error (%)
SA-f
Error (%)
μa
1
/mm-1
6.660 × 10-2
14.07
6.663 × 10-2
14.03
μs
1
′/mm-1
1.698
-13.94
1.698
-13.94
μa
2
/mm-1
7.418 × 10-3
-5.97
7.412 × 10-3
-5.96
μs
2
′/mm-1
0.9985
0.16
0.9985
0.16
l/mm
1.446
3.58
1.446
3.60
χmin2
7.39
7.39
Evaluations
660
250,001
Mrqrt-t and SA-f converged to the
identical minimum. The true parameters gave
χ2(t) = 80.3.
Table 7
Estimates of Optical Properties Obtained by Fitting MC
Data for Absolute cw, ac Reflectance, and Phase Measurements
(qr = 1.0, qph = 0.0)
at a Modulation Frequency f = 195
MHza
Parameters
SA-t
Error (%)
μa
1
/mm-1
1.849 × 10-1
-138.50
μs
1
′/mm-1
1.478
0.83
μa
2
/mm-1
5.659 × 10-3
19.16
μs
2
′/mm-1
1.220
-21.95
l/mm
8.289 × 10-1
44.74
χmin2
2630
Evaluations
1.04 × 106
The number of reflectance measurements
included in the fits was the same as in all other cases (12), but
they were made at ρ = 1–6 (increment 1), 8–16 (increment
2), and 20 mm. Despite the large number of function evaluations,
SA-t gave poor quality parameter estimates. The true parameters gave
χ2(t) = 19,400.
Table 8
As in Table 6 but for a Modulation Frequency
f = 300 MHz [χ2(t) =
80.3]
Parameters
Mrqrt-t
Error (%)
SA-f
Error (%)
μa
1
/mm-1
6.843 × 10-2
11.70
6.999 × 10-2
9.69
μs
1
′/mm-1
1.691
-13.46
1.692
-13.57
μa
2
/mm-1
7.279 × 10-3
-3.98
7.256 × 10-3
-3.66
μs
2
′/mm-1
1.005
-0.46
1.006
-0.64
l/mm
1.433
4.47
1.414
5.73
χmin2
8.93
8.93
Evaluations
1584
200,001
Table 9
As in Table 6 but for a Modulation Frequency
f = 1 GHz and SA-t Shown Instead of SA-f
[χ2(t) = 1010]
Parameters
Mrqrt-t
Error (%)
SA-f
Error (%)
μa
1
/mm-1
6.859 × 10-2
11.49
6.867 × 10-2
11.39
μs
1
′/mm-1
1.695
-13.78
1.695
-13.76
μa
2
/mm-1
7.511 × 10-3
-7.30
7.509 × 10-3
-7.27
μs
2
′/mm-1
1.004
0.39
1.004
0.41
l/mm
1.415
5.64
1.415
5.69
χmin2
22.5
22.5
Evaluations
1320
58,001
Table 10
Continuous-Wave as Well as FD MC Data at Three Different
Modulation Frequencies (195 MHz, 300 MHz, 1 GHz) Simultaneously Fitted
as Absolute cw, ac Reflectance, and Phase Measurements
(qr = 1.0, qph =
0.0)a
Parameters
Mrqrt-t
Error (%)
SA-f
Error (%)
μa
1
/mm-1
6.767 × 10-2
14.78
6.853 × 10-2
11.57
μs
1
′/mm-1
1.680
-12.77
1.686
-13.15
μa
2
/mm-1
7.301 × 10-3
-4.29
7.344 × 10-3
-4.92
μs
2
′/mm-1
1.002
-0.23
1.004
0.37
l/mm
1.444
3.72
1.432
4.56
χmin2
54.6
42.5
Evaluations
1320
242,001
The true parameters gave
χ2(±) = 1200.
Tables (10)
Table 1
True Set of Optical Properties with which all MC Runs were
Performed and a Selected False Initial Guess of the Parameters Used as
Input to All Fitting Routinesa
Parameters
True
False
μa
1
/mm-1
7.75 × 10-2
1.75 × 10-2
μs
1
′/mm-1
1.49
1.0
μa
2
/mm-1
7.0 × 10-3
1.3 × 10-2
μs
2
′/mm-1
1.0
1.1
l/mm
1.5
3.5
Parameter definitions in text.
Table 2
Estimates of Optical Properties Obtained by Fitting MC
Data for Relative cw Reflectance Measurements by Marquardt–Levenberg
and a True Initial Guess (Mrqrt-t), a False Initial Guess (Mrqrt-f),
and Simulated Annealing with a False Initial Guess
(SA-f)a
Parameters
Mrqrt-t
Error (%)
Mrqrt-f
Error (%)
SA-f
Error (%)
μa
1
/mm-1
7.425 × 10-2
4.20
2.537 × 10-4
99.67
6.517 × 10-2
15.91
μs
1
′/mm-1
1.552
4.16
1.111
37.94
1.198
19.60
μa
2
/mm-1
8.674 × 10-3
-23.91
9.780 × 10-3
39.71
1.160 × 10-2
-65.67
μs
2
′/mm-1
0.8850
11.50
0.8753
12.48
0.6909
30.91
l/mm
1.476
1.61
2.327
-55.13
1.654
-10.29
qr/no units
0.9897
0.3547
0.7260
χmin2
2.30
2.42
2.29
Evaluations
660
2112
16,501
The χ2 of the true
parameters was χ2(t) =
21.6. Parameters are defined in the text.
Table 3
Estimates of Optical Properties Obtained by Fitting MC
Data for Absolute cw Reflectance Measurements
(qr = 1.0) by Mrqrt-t and SA-f where
χ2(t) =
23.3a
Parameters
Mrqrt-t
Error (%)
SA-f
Error (%)
μa
1
/mm-1
7.377 × 10-2
4.81
0.1680
-116.77
μs
1
′/mm-1
1.550
-4.05
0.7710
48.26
μa
2
/mm-1
8.805 × 10-3
-25.71
1.674 × 10-2
-139.14
μs
2
′/mm-1
0.8766
12.35
0.4600
54.00
l/mm
1.494
0.43
1.625
-8.33
χmin2
2.30
2.29
Evaluations
924
300,001
Mrqrt-f failed to converge.
Table 4
Estimates of Optical Properties Obtained by Fitting
MC Data for Relative Phase and cw Reflectance Measurements by Mrqrt-t,
SA-t, and
SA-fa
Parameters
Mrqrt-t
Error (%)
SA-t
Error (%)
SA-f
Error (%)
μa
1
/mm-1
6.764 × 10-2
12.72
6.621 × 10-2
14.57
6.447 × 10-2
16.82
μs
1
′/mm-1
1.686
-13.18
1.473
1.17
2.085
-39.94
μa
2
/mm-1
7.399 × 10
-3
-5.51
6.793 × 10
-3
2.96
7.932 × 10-3
-13.32
μs
2
′/mm-1
1.000
0.0
1.023
2.29
1.020
2.03
l/mm
1.439
4.07
0.8833
41.11
1.631
-8.73
qr/no units
0.9955
0.5343
1.595
qph/deg
-2.691 × 10
-3
-7.997 × 10
-3
1.821 × 10-2
χmin2
3.83
3.69
3.14
Evaluations
3642
109,001
194,001
The true parameters gave
χ2(t) = 24.3.
Table 5
Estimates of Optical Properties Obtained by Fitting MC
Data for Absolute Phase and cw Reflectance Measurements
(qr = 1.0, qph = 0.0)
by Mrqrt-t, SA-t, and SA-fa
Parameters
Mrqrt-t
Error (%)
SA-t
Error (%)
SA-f
Error (%)
μa1/mm-1
6.893 × 10-2
11.06
6.901 × 10-2
10.95
6.894 × 10-2
11.05
μs1′/mm-1
1.698
-13.93
1.696
-13.83
1.698
-13.95
μa2/mm-1
7.400 × 10-3
-5.71
7.393 × 10-3
-5.61
7.401 × 10-3
-5.72
μs2′/mm-1
1.003
-0.32
1.003
-0.32
1.003
-0.32
l/mm
1.419
5.42
1.419
5.39
1.418
5.44
χmin2
3.78
3.78
3.78
Evaluations
660
46,501
300,001
The true parameters gave
χ2(t) = 52.5.
Table 6
Estimates of Optical Properties Obtained by Fitting MC
Data for Absolute cw, ac Reflectance, and Phase Measurements
(qr = 1.0, qph = 0.0)
at a Modulation Frequency f = 195
MHza
Parameters
Mrqrt-t
Error (%)
SA-f
Error (%)
μa
1
/mm-1
6.660 × 10-2
14.07
6.663 × 10-2
14.03
μs
1
′/mm-1
1.698
-13.94
1.698
-13.94
μa
2
/mm-1
7.418 × 10-3
-5.97
7.412 × 10-3
-5.96
μs
2
′/mm-1
0.9985
0.16
0.9985
0.16
l/mm
1.446
3.58
1.446
3.60
χmin2
7.39
7.39
Evaluations
660
250,001
Mrqrt-t and SA-f converged to the
identical minimum. The true parameters gave
χ2(t) = 80.3.
Table 7
Estimates of Optical Properties Obtained by Fitting MC
Data for Absolute cw, ac Reflectance, and Phase Measurements
(qr = 1.0, qph = 0.0)
at a Modulation Frequency f = 195
MHza
Parameters
SA-t
Error (%)
μa
1
/mm-1
1.849 × 10-1
-138.50
μs
1
′/mm-1
1.478
0.83
μa
2
/mm-1
5.659 × 10-3
19.16
μs
2
′/mm-1
1.220
-21.95
l/mm
8.289 × 10-1
44.74
χmin2
2630
Evaluations
1.04 × 106
The number of reflectance measurements
included in the fits was the same as in all other cases (12), but
they were made at ρ = 1–6 (increment 1), 8–16 (increment
2), and 20 mm. Despite the large number of function evaluations,
SA-t gave poor quality parameter estimates. The true parameters gave
χ2(t) = 19,400.
Table 8
As in Table 6 but for a Modulation Frequency
f = 300 MHz [χ2(t) =
80.3]
Parameters
Mrqrt-t
Error (%)
SA-f
Error (%)
μa
1
/mm-1
6.843 × 10-2
11.70
6.999 × 10-2
9.69
μs
1
′/mm-1
1.691
-13.46
1.692
-13.57
μa
2
/mm-1
7.279 × 10-3
-3.98
7.256 × 10-3
-3.66
μs
2
′/mm-1
1.005
-0.46
1.006
-0.64
l/mm
1.433
4.47
1.414
5.73
χmin2
8.93
8.93
Evaluations
1584
200,001
Table 9
As in Table 6 but for a Modulation Frequency
f = 1 GHz and SA-t Shown Instead of SA-f
[χ2(t) = 1010]
Parameters
Mrqrt-t
Error (%)
SA-f
Error (%)
μa
1
/mm-1
6.859 × 10-2
11.49
6.867 × 10-2
11.39
μs
1
′/mm-1
1.695
-13.78
1.695
-13.76
μa
2
/mm-1
7.511 × 10-3
-7.30
7.509 × 10-3
-7.27
μs
2
′/mm-1
1.004
0.39
1.004
0.41
l/mm
1.415
5.64
1.415
5.69
χmin2
22.5
22.5
Evaluations
1320
58,001
Table 10
Continuous-Wave as Well as FD MC Data at Three Different
Modulation Frequencies (195 MHz, 300 MHz, 1 GHz) Simultaneously Fitted
as Absolute cw, ac Reflectance, and Phase Measurements
(qr = 1.0, qph =
0.0)a