Abstract

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 c1, c2, c3 [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 6mm, their performance is also studied for larger inclusions. A good agreement, to within 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.

© 2006 Optical Society of America

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References

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2006

2001

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

1999

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]

1997

1996

1995

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

1993

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.

Aronson, R.

Arridge, S. R.

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]

Boas, D. A.

Contini, D.

Delpy, D. T.

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]

Fantini, S.

Graaf, R.

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]

Jacques, S. L.

Jiang, H.

Martelli, F.

Osterberg, U. L.

Ostermeyer, M. R.

Patterson, M. S.

Paulsen, K. D.

Pogue, B. W.

Rinzema, K.

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

Sassaroli, A.

A. Sassaroli, F. Martelli, and S. Fantini, "Perturbation theory for the diffusion equation by use of the moments of the generalized point-spread function. I. Theory," J. Opt. Soc. Am. A 23, 2105-2118 (2006).
[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]

Zaccanti, G.

Appl. Opt.

J. Opt. Soc. Am. A

Med. Phys.

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]

Opt. Express

Phys. Med. Biol.

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

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

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Figures (12)

Fig. 1
Fig. 1

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 mm from the x axis in the semi-infinite geometry (right panel), and d = 0 , 10, 20 mm from the x axis in transmittance of the slab geometry (left panel). The thickness of the slab is 40 mm .

Fig. 2
Fig. 2

Estimates of the parameters c n ( n = 1 , 2 , 3 ) for the case of spherical inclusions in the slab geometry. The reduced scattering coefficient ( μ s ) of the background medium was 1.5 mm 1 and the absorption coefficient was μ a = 0.005 mm 1 and 0.02 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 14 ), triangles (position number 15 31 ) and squares (position number 32 49 ) refer to the defects having radius r = 1 , 2, 3 mm , respectively. The error bar is calculated by considering three independent MC simulations.

Fig. 3
Fig. 3

Estimates of the parameters c n ( n = 1 , 2 , 3 ) for the case of spherical inclusions having radius r = 2 mm in the semi-infinite geometry. The reduced scattering coefficient ( μ s ) of the background medium was 0.8 mm 1 and the absorption coefficient was μ a = 0.005 mm 1 and 0.03 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.

Fig. 4
Fig. 4

Comparison of the parameters c n ( n = 1 , 2 , 3 ) for the case of spherical inclusions with radius r = 2 mm in the slab (left panel) and semi-infinite geometry (right panel). The reduced scattering coefficient ( μ s ) of the medium was 0.8 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.

Fig. 5
Fig. 5

Estimates of the parameters c n ( n = 1 , 2 , 3 ) for the case of spherical inclusions with radius r = 5 mm in the slab geometry. The reduced scattering coefficient ( μ s ) of the background medium was 0.8 (left panel) and 1.5 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.

Fig. 6
Fig. 6

Effect of a cubic defect having sides of 4 mm and placed with the center at ( x , y , z ) = ( 15 , 16 , 0 ) mm (see Fig. 1) in a semi-infinite medium with absorption and reduced scattering coefficient of 0.005 mm 1 and 0.8 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 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.

Fig. 7
Fig. 7

The reconstruction of Fig. 6 at the source-detector distance of 30 mm is recalculated by assigning to the scaling parameters c n the value of 1.

Fig. 8
Fig. 8

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 ) mm (see Fig. 1) in a semi-infinite medium with absorption and reduced scattering coefficients of 0.005 mm 1 and 0.8 mm 1 , respectively. The relative change of CW is plotted against the absorption contrast at the source–detector distances of 20 and 30 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 mm , respectively.

Fig. 9
Fig. 9

Reconstruction of CW change of Fig. 8 by using only the mixed moments l i l j , l i 2 l j , l i l j l k , etc., (left panel) and by using only the self moments l i n (right panel).

Fig. 10
Fig. 10

Effect of three adjacent cubic defects having sides of 4 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 ) mm in a semi-infinite medium with absorption and reduced scattering coefficient of 0.005 mm 1 and 0.8 mm 1 , respectively. The relative change of CW is plotted against the absorption contrast at the source–detector distances of 20 and 30 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.

Fig. 11
Fig. 11

Effect of a cubic defect having sides of 10 mm placed with the center at ( x , y , z ) = ( 15 , 18 , 0 ) mm in a semi-infinite medium with reduced scattering coefficient of 0.8 mm 1 and absorption coefficient of 0.005 mm 1 and 0.02 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 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.

Fig. 12
Fig. 12

Same as Fig. 11 but with reduced scattering coefficient of 1.5 mm 1 .

Tables (4)

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Table 1 Coordinates and Size of Inclusions: Slab Geometry

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Table 2 Coordinates of 2 mm Inclusions: Semi-infinite Geometry

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Table 3 Coordinates of 5 mm Inclusions: Slab Geometry

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Table 4 Comparison of the Monte Carlo and Diffusion Equation Calculated Moments a

Equations (19)

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l i n c n 1 l i ( V i ϕ 0 ( r r i ) d r ) n 1 n 1 , c 0 = 1 ,
c n 1 = l i n l i ( V i ϕ 0 ( r r i ) d r ) n 1 n 1 , c 0 = 1
l i n = j = 1 M l i j n exp ( μ a i l i j ) exp ( μ a 0 l 0 j ) j = 1 M exp ( μ a i l i j ) exp ( μ a 0 l 0 j ) ,
ϕ 0 ( r i ) = 1 4 π D [ exp ( μ eff r 1 ) r 1 exp ( μ eff r 2 ) r 2 ] ,
Δ CW CW = l Δ μ a + 1 2 ! l 2 Δ μ a 2 1 3 ! l 3 Δ μ a 3 + 1 4 ! l 4 Δ μ a 4 .
Δ CW CW 0 = i = 1 3 l i Δ μ a ( r i ) + ( 1 ) k 1 + k 2 + k 3 k 1 + k 2 + k 3 = 2 4 Δ μ a 1 k 1 Δ μ a 2 k 2 Δ μ a 3 k 3 k 1 ! k 2 ! k 3 ! l 1 k l l 2 k 2 l 3 k 3 ,
l i 2 l k = j = 1 M l i j 2 l k j exp ( μ a i l i j ) exp ( μ a k l k j ) exp ( μ a 0 l 0 j ) j = 1 M exp ( μ a i l i j ) exp ( μ a k l k j ) exp ( μ a 0 l 0 j ) ,
l i = A 0 ( r b , r i ) ϕ 0 ( r i ) V i A 0 ( r b ) .
l i l j = P ( i , j ) A 0 ( r b , r i ) ϕ 0 ( r i , r j ) ϕ 0 ( r j ) V i V j A 0 ( r b ) i j ,
l i 2 = c 1 l i V i ϕ 0 ( r r i ) d r .
l i l j l k = P ( i , j , k ) A 0 ( r b , r i ) ϕ 0 ( r i , r j ) ϕ 0 ( r j , r k ) V i V j V k A 0 ( r b ) ,
i j k ,
l i 2 l k = c 1 l i l k V i ϕ 0 ( r r i ) d r + 2 l i [ ϕ 0 ( r i , r k ) ] 2 V i V k ,
l i 3 = c 2 l i ( V i ϕ 0 ( r r i ) d r ) 2 .
l i l j l k l m = 1 A 0 ( r b ) ( P ( i , j , k , m ) A 0 ( r b , r i ) ϕ 0 ( r i , r j ) ϕ 0 ( r j , r k ) × ϕ 0 ( r k , r m ) ϕ 0 ( r m ) V i V j V k V m ) i j k m ,
l i 2 l k 2 = c 1 2 l i l k V i ϕ 0 ( r r i ) d r V k ϕ 0 ( r r k ) d r + V i V k [ ϕ 0 ( r i , r k ) ] 2 { 4 l i l k + c 1 [ 2 l k V i ϕ 0 ( r r i ) d r + 2 l i V k ϕ 0 ( r r k ) d r ] } ,
l i 2 l j l k = c 1 l i l j l k V i ϕ 0 ( r r i ) d r + 2 l i l k [ ϕ 0 ( r i , r j ) ] 2 V i V j + 2 l i l j [ ϕ 0 ( r i , r k ) ] 2 V i V k + 2 l i [ ϕ 0 ( r i , r j ) ϕ 0 ( r j , r k ) ϕ 0 ( r k , r i ) ] V i V j V k ,
l i 3 l j = c 2 l i l j ( V i ϕ 0 ( r r i ) d r ) 2 + 6 c 1 V i ϕ 0 ( r r i ) d r [ ϕ 0 ( r i , r j ) ] 2 l i V i V j ,
l i 4 = c 3 l i ( V i ϕ 0 ( r r i ) d r ) 3 .

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