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.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A finite element approach to modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

R. Graaf and K. Rinzema, "Practical improvements on photon diffusion theory: application to isotropic scattering," Phys. Med. Biol. 46, 3043-3050 (2001).
[CrossRef]

A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, "Study on the propagation of ultrashort pulse light in cylindrical optical phantoms," Phys. Med. Biol. 44, 2747-2763 (1999).
[CrossRef] [PubMed]

R. Graaf and K. Rinzema, "Practical improvements on photon diffusion theory: application to isotropic scattering," Phys. Med. Biol. 46, 3043-3050 (2001).
[CrossRef]

1999 (1)

A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, "Study on the propagation of ultrashort pulse light in cylindrical optical phantoms," Phys. Med. Biol. 44, 2747-2763 (1999).
[CrossRef] [PubMed]

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995), p. 934.

1993 (1)

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A finite element approach to modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995), p. 934.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A finite element approach to modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A finite element approach to modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

R. Graaf and K. Rinzema, "Practical improvements on photon diffusion theory: application to isotropic scattering," Phys. Med. Biol. 46, 3043-3050 (2001).
[CrossRef]

Hiraoka, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A finite element approach to modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

Imai, D.

A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, "Study on the propagation of ultrashort pulse light in cylindrical optical phantoms," Phys. Med. Biol. 44, 2747-2763 (1999).
[CrossRef] [PubMed]

A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, "Study on the propagation of ultrashort pulse light in cylindrical optical phantoms," Phys. Med. Biol. 44, 2747-2763 (1999).
[CrossRef] [PubMed]

R. Graaf and K. Rinzema, "Practical improvements on photon diffusion theory: application to isotropic scattering," Phys. Med. Biol. 46, 3043-3050 (2001).
[CrossRef]

A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, "Study on the propagation of ultrashort pulse light in cylindrical optical phantoms," Phys. Med. Biol. 44, 2747-2763 (1999).
[CrossRef] [PubMed]

Schweiger, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A finite element approach to modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995), p. 934.

Yamada, Y.

A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, "Study on the propagation of ultrashort pulse light in cylindrical optical phantoms," Phys. Med. Biol. 44, 2747-2763 (1999).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A finite element approach to modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

R. Graaf and K. Rinzema, "Practical improvements on photon diffusion theory: application to isotropic scattering," Phys. Med. Biol. 46, 3043-3050 (2001).
[CrossRef]

A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, "Study on the propagation of ultrashort pulse light in cylindrical optical phantoms," Phys. Med. Biol. 44, 2747-2763 (1999).
[CrossRef] [PubMed]

Other (1)

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995), p. 934.

Schematic of the slab and the semi-infinite geometries used for MC simulations. A collimated laser beam is incident on the scattering medium and three small detectors are placed at the distances $d=10,20$, $30\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ from the x axis in the semi-infinite geometry (right panel), and $d=0$, 10, $20\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ from the x axis in transmittance of the slab geometry (left panel). The thickness of the slab is $40\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$.

Estimates of the parameters ${c}_{n}$$(n=1,2,3)$ for the case of spherical inclusions in the slab geometry. The reduced scattering coefficient $\left({\mu}_{s}^{\prime}\right)$ of the background medium was $1.5\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$ and the absorption coefficient was ${\mu}_{a}=0.005\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$ and $0.02\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$, left and right panel, respectively. The parameters ${c}_{1}$, ${c}_{2}$, ${c}_{3}$ are plotted in the top, medium, and lower sections, respectively. Three different symbols, diamonds (position number $1\u201314$), triangles (position number $15\u201331$) and squares (position number $32\u201349$) refer to the defects having radius $r=1$, 2, $3\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$, respectively. The error bar is calculated by considering three independent MC simulations.

Estimates of the parameters ${c}_{n}$$(n=1,2,3)$ for the case of spherical inclusions having radius $r=2\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ in the semi-infinite geometry. The reduced scattering coefficient $\left({\mu}_{s}^{\prime}\right)$ of the background medium was $0.8\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$ and the absorption coefficient was ${\mu}_{a}=0.005\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$ and $0.03\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$, left and right panel, respectively. The parameters ${c}_{1}$, ${c}_{2}$, ${c}_{3}$ are plotted in the top, medium and lower sections, respectively. The error bar is calculated by considering three independent MC simulations.

Comparison of the parameters ${c}_{n}$$(n=1,2,3)$ for the case of spherical inclusions with radius $r=2\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ in the slab (left panel) and semi-infinite geometry (right panel). The reduced scattering coefficient $\left({\mu}_{s}^{\prime}\right)$ of the medium was $0.8\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$. The filled and empty symbols are defined in the text. The parameters ${c}_{1}$, ${c}_{2}$, ${c}_{3}$ are plotted as functions of the absorption coefficient of the medium in the first, second, and third rows, respectively.

Estimates of the parameters ${c}_{n}(n=1,2,3)$ for the case of spherical inclusions with radius $r=5\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ in the slab geometry. The reduced scattering coefficient $\left({\mu}_{s}^{\prime}\right)$ of the background medium was 0.8 (left panel) and $1.5\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$ (right panel). The filled and empty symbols are defined as in Fig. 4. The parameters ${c}_{1}$, ${c}_{2}$, ${c}_{3}$ are plotted as functions of the absorption coefficient of the medium in the first, second, and third rows, respectively.

Effect of a cubic defect having sides of $4\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ and placed with the center at $(x,y,z)=(15,16,0)\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ (see Fig. 1) in a semi-infinite medium with absorption and reduced scattering coefficient of $0.005\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$ and $0.8\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$, respectively. The relative change of CW with respect to the initial value (background value) is plotted against the absorption contrast between the defect and the medium for the source–detector distance of 20 and $30\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ in the left and right panels, respectively. The numbers 1, 2, 3, 4 indicate the first-, second-, third-, and fourth-order approximations, respectively, while MC indicates the CW change calculated from the MC data.

The reconstruction of Fig. 6 at the source-detector distance of $30\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ is recalculated by assigning to the scaling parameters ${c}_{n}$ the value of 1.

Effect of three cubic defects placed with centers at $({x}_{1},{y}_{1},{z}_{1})=(12,5,0),({x}_{2},{y}_{2},{z}_{2})=(15,10,0),({x}_{3},{y}_{3},{z}_{3})=(15,16,0)\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ (see Fig. 1) in a semi-infinite medium with absorption and reduced scattering coefficients of $0.005\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$ and $0.8\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$, respectively. The relative change of CW is plotted against the absorption contrast at the source–detector distances of 20 and $30\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$, left and right panel, respectively. The numbers 1, 2, 3, 4 indicate the first-, second-, third-, and fourth-order approximation, respectively, while MC indicates the CW change calculated from the MC data. The sides of the cubic defects are ${s}_{1}=2$, ${s}_{2}=3$, ${s}_{3}=4\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$, respectively.

Reconstruction of CW change of Fig. 8 by using only the mixed moments $\u27e8{l}_{i}{l}_{j}\u27e9$, $\u27e8{l}_{i}^{2}{l}_{j}\u27e9$, $\u27e8{l}_{i}{l}_{j}{l}_{k}\u27e9$, etc., (left panel) and by using only the self moments $\u27e8{l}_{i}^{n}\u27e9$ (right panel).

Effect of three adjacent cubic defects having sides of $4\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ placed with the center at $({x}_{1},{y}_{1},{z}_{1})=(13,17,0),({x}_{2},{y}_{2},{z}_{2})=(13,21,0),({x}_{3},{y}_{3},{z}_{3})=(17,19,0)\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ in a semi-infinite medium with absorption and reduced scattering coefficient of $0.005\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$ and $0.8\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$, respectively. The relative change of CW is plotted against the absorption contrast at the source–detector distances of 20 and $30\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$, left and right panels, respectively. The numbers 1, 2, 3, 4 indicate the approximation orders, while MC indicates the CW change calculated from the MC simulation.

Effect of a cubic defect having sides of $10\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ placed with the center at $(x,y,z)=(15,18,0)\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ in a semi-infinite medium with reduced scattering coefficient of $0.8\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$ and absorption coefficient of $0.005\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$ and $0.02\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{-1}$, top and bottom figures, respectively. The relative change of CW is plotted against the absorption contrast at the source-detector distance of $30\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$. The numbers 1, 2, 3, 4 in the figures on the left side indicate approximation orders, while MC indicates the CW change calculated from the MC simulation. In the figures on the right side are compared the fourth-order curves obtained by dividing the defect into 1, 8, and 64 voxels.

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.