Angelo Sassaroli, Fabrizio Martelli, and Sergio Fantini, "Perturbation theory for the diffusion equation by use of the moments of the generalized temporal point-spread function. II. Continuous-wave results," J. Opt. Soc. Am. A 23, 2119-2131 (2006)

We investigate the performance of the method proposed in Part I of this paper in several situations of interest in diffuse optical imaging of biological tissues. Monte Carlo simulations were extensively used to validate the approximate scaling relationship between higher-order and first-order self moments of the generalized temporal point-spread function in semi-infinite and slab geometry. More specifically we found that in a wide range of cases the scaling parameters ${c}_{1}$, ${c}_{2}$, ${c}_{3}$ [see Eq. (36) of Part I] lie in the intervals (1.48, 1.58), (3.1, 3.7), and (8.5, 11.5), respectively. The scaling relationships between higher-order and first-order self moments are useful for the calculation of the perturbation of a single defect in a straightforward way. Although these relationships are more accurate for inclusions of linear size less than $\approx 6\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$, their performance is also studied for larger inclusions. A good agreement, to within $\approx 10\%$, was found between the perturbations of single and multiple defects calculated with the proposed method and those obtained by Monte Carlo simulations. We also provide formulas for the calculation of the moments up to the fourth order for which it is clear how lower-order moments can be used for the calculation of higher-order moments.

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Coordinates and Size^{
} of Inclusions: Slab Geometry

Sphere No.

x

y

z

r

1

8

$-5$

0

1

2

8

$-2$

0

1

3

8

0

0

1

4

8

2

0

1

5

8

4

0

1

6

11

$-10$

0

2

$\underset{\u0331}{\mathbf{7}}$

11

$-4$

0

2

8

11

0

0

2

$\underset{\u0331}{\mathbf{9}}$

11

4

0

2

10

11

8

0

2

11

16

$-14$

0

3

$\underset{\u0331}{12}$

16

$-6$

0

3

13

16

0

0

3

$\underset{\u0331}{14}$

16

6

0

3

15

16

12

0

3

$\underset{\u0331}{16}$

24

$-6$

0

3

$\underset{\u0331}{17}$

24

6

0

3

$\underset{\u0331}{\mathbf{18}}$

29

$-4$

0

2

$\underset{\u0331}{\mathbf{19}}$

29

4

0

2

$\mathbf{20}$

32

$-2$

0

1

$\mathbf{21}$

32

2

0

1

Coordinates of the centers and radius (in millimeters) of 21 spherical defects in the slab geometry. The spherical defects 2, 4, 20, 21 (bold), 7, 9, 18, 19 (underlined bold), and 12, 14, 16, 17 (underlined) are symmetric with respect to source and central detector for the spherical defects of radius $r=1$, 2, 3, respectively.

Table 2

Coordinates^{
} of $2\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ Inclusions: Semi-infinite Geometry

Sphere No.

x

y

z

1

8

5

0

2

8

15

0

3

8

25

0

4

13

5

0

5

13

15

0

6

13

25

0

7

18

5

0

8

18

15

0

9

18

25

0

10

8

5

6

11

8

15

6

12

8

25

6

13

13

5

6

14

13

15

6

15

13

25

6

16

18

5

6

17

18

15

6

18

18

25

6

19

8

5

12

20

8

15

12

21

8

25

12

22

13

5

12

23

13

15

12

24

13

25

12

25

18

5

12

26

18

15

12

27

18

25

12

Coordinates (in millimeters) of 27 spherical defects of radius $2\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$.

Table 3

Coordinates^{
} of $5\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ Inclusions: Slab Geometry

Sphere No.

x

y

z

1

12

$-10$

0

2

12

0

0

3

12

10

0

4

23

$-20$

0

5

23

$-10$

0

6

23

0

0

7

28

10

0

8

12

0

12

9

23

$-10$

12

10

23

0

15

11

23

0

$-23$

Coordinates (in millimeters) of 11 spherical defects of radius $5\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$. Sphere No. 1, 3, 7 (bold) are used for estimating the average value and variance of the parameters ${c}_{n}$ for the same position of the defect.

Table 4

Comparison of the Monte Carlo and Diffusion Equation Calculated Moments^{
a
}

In the table is reported comparisons of the moments calculated from MC data and those obtained from the theory (DE) for the situation shown in Fig. 8. The table is divided into two sections. In the upper section the first three rows correspond to the moments relative to the defects located at $({x}_{1},{y}_{1},{z}_{1})=(12,5,0)$, $({x}_{2},{y}_{2},{z}_{2})=(15,10,0)$, $({x}_{3},{y}_{3},{z}_{3})=(15,16,0)\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$, respectively, and detector located at $20\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ from the source; the other three rows correspond to the same defects (in the same order) and the detector located at $30\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ from the source. In the lower section the three defects (in the same order as before) are named 1, 2, 3 in the subscripts, respectively. The mixed moments, three for $\u27e8{l}_{i}{l}_{j}\u27e9$ and seven for $\u27e8{l}_{i}{l}_{j}{l}_{k}\u27e9$ are listed sequentially for the detectors located at 20 and $30\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ from the source, respectively.

Tables (4)

Table 1

Coordinates and Size^{
} of Inclusions: Slab Geometry

Sphere No.

x

y

z

r

1

8

$-5$

0

1

2

8

$-2$

0

1

3

8

0

0

1

4

8

2

0

1

5

8

4

0

1

6

11

$-10$

0

2

$\underset{\u0331}{\mathbf{7}}$

11

$-4$

0

2

8

11

0

0

2

$\underset{\u0331}{\mathbf{9}}$

11

4

0

2

10

11

8

0

2

11

16

$-14$

0

3

$\underset{\u0331}{12}$

16

$-6$

0

3

13

16

0

0

3

$\underset{\u0331}{14}$

16

6

0

3

15

16

12

0

3

$\underset{\u0331}{16}$

24

$-6$

0

3

$\underset{\u0331}{17}$

24

6

0

3

$\underset{\u0331}{\mathbf{18}}$

29

$-4$

0

2

$\underset{\u0331}{\mathbf{19}}$

29

4

0

2

$\mathbf{20}$

32

$-2$

0

1

$\mathbf{21}$

32

2

0

1

Coordinates of the centers and radius (in millimeters) of 21 spherical defects in the slab geometry. The spherical defects 2, 4, 20, 21 (bold), 7, 9, 18, 19 (underlined bold), and 12, 14, 16, 17 (underlined) are symmetric with respect to source and central detector for the spherical defects of radius $r=1$, 2, 3, respectively.

Table 2

Coordinates^{
} of $2\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ Inclusions: Semi-infinite Geometry

Sphere No.

x

y

z

1

8

5

0

2

8

15

0

3

8

25

0

4

13

5

0

5

13

15

0

6

13

25

0

7

18

5

0

8

18

15

0

9

18

25

0

10

8

5

6

11

8

15

6

12

8

25

6

13

13

5

6

14

13

15

6

15

13

25

6

16

18

5

6

17

18

15

6

18

18

25

6

19

8

5

12

20

8

15

12

21

8

25

12

22

13

5

12

23

13

15

12

24

13

25

12

25

18

5

12

26

18

15

12

27

18

25

12

Coordinates (in millimeters) of 27 spherical defects of radius $2\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$.

Table 3

Coordinates^{
} of $5\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ Inclusions: Slab Geometry

Sphere No.

x

y

z

1

12

$-10$

0

2

12

0

0

3

12

10

0

4

23

$-20$

0

5

23

$-10$

0

6

23

0

0

7

28

10

0

8

12

0

12

9

23

$-10$

12

10

23

0

15

11

23

0

$-23$

Coordinates (in millimeters) of 11 spherical defects of radius $5\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$. Sphere No. 1, 3, 7 (bold) are used for estimating the average value and variance of the parameters ${c}_{n}$ for the same position of the defect.

Table 4

Comparison of the Monte Carlo and Diffusion Equation Calculated Moments^{
a
}

In the table is reported comparisons of the moments calculated from MC data and those obtained from the theory (DE) for the situation shown in Fig. 8. The table is divided into two sections. In the upper section the first three rows correspond to the moments relative to the defects located at $({x}_{1},{y}_{1},{z}_{1})=(12,5,0)$, $({x}_{2},{y}_{2},{z}_{2})=(15,10,0)$, $({x}_{3},{y}_{3},{z}_{3})=(15,16,0)\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$, respectively, and detector located at $20\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ from the source; the other three rows correspond to the same defects (in the same order) and the detector located at $30\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ from the source. In the lower section the three defects (in the same order as before) are named 1, 2, 3 in the subscripts, respectively. The mixed moments, three for $\u27e8{l}_{i}{l}_{j}\u27e9$ and seven for $\u27e8{l}_{i}{l}_{j}{l}_{k}\u27e9$ are listed sequentially for the detectors located at 20 and $30\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ from the source, respectively.