Abstract

Solutions of the diffusion approximation to the radiative transport equation are derived for a turbid (rectangular) parallelepiped using the method of image sources and applying extrapolated boundary conditions. The derived solutions are compared with Monte Carlo simulations in the steady-state and time domains. It is found that the diffusion theory is in good agreement with Monte Carlo simulations provided that the light is detected sufficiently far from the incident beam. Applications of the derived solutions, including the determination of the optical properties of the turbid parallelepiped, are discussed.

© 2005 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  2. M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  3. T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo ,” Med. Phys. 19, 879–888 (1992).
    [CrossRef] [PubMed]
  4. S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlength in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
    [CrossRef] [PubMed]
  5. H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, (Clarendon, 1959).
  6. F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E 67, 056623-1–14 (2003).
    [CrossRef]
  7. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  8. D. Contini, F. Martelli, G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36, 4587–4599 (1997).
    [CrossRef] [PubMed]
  9. A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
    [CrossRef]
  10. A. Kienle, R. Steiner, “Determination of the optical properties of tissue by spatially resolved transmission measurements and Monte Carlo simulations,” Proc. SPIE 2077, 142–152 (1993).
    [CrossRef]
  11. T. J. Pfefer, J. K. Barton, E. K. Chan, M. G. Ducros, B. S. Sorg, T. E. Milner, J. S. Nelson, A. J. Welch, “A three-dimensional modular adaptable grid numerical model for light propagation duing laser irradiation of skin tissue,” IEEE J. Sel. Top. Quantum Electron. 2, 934–942 (1996).
    [CrossRef]
  12. A. H. Barnett, “A fast numerical method for time-resolved photon diffusion in general stratified turbid media,” J. Comput. Phys. 201, 771–797 (2004).
    [CrossRef]

2004 (1)

A. H. Barnett, “A fast numerical method for time-resolved photon diffusion in general stratified turbid media,” J. Comput. Phys. 201, 771–797 (2004).
[CrossRef]

2003 (1)

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E 67, 056623-1–14 (2003).
[CrossRef]

1997 (2)

1996 (1)

T. J. Pfefer, J. K. Barton, E. K. Chan, M. G. Ducros, B. S. Sorg, T. E. Milner, J. S. Nelson, A. J. Welch, “A three-dimensional modular adaptable grid numerical model for light propagation duing laser irradiation of skin tissue,” IEEE J. Sel. Top. Quantum Electron. 2, 934–942 (1996).
[CrossRef]

1994 (1)

1993 (1)

A. Kienle, R. Steiner, “Determination of the optical properties of tissue by spatially resolved transmission measurements and Monte Carlo simulations,” Proc. SPIE 2077, 142–152 (1993).
[CrossRef]

1992 (2)

T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo ,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlength in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

1989 (1)

Arridge, S. R.

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlength in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Barnett, A. H.

A. H. Barnett, “A fast numerical method for time-resolved photon diffusion in general stratified turbid media,” J. Comput. Phys. 201, 771–797 (2004).
[CrossRef]

Barton, J. K.

T. J. Pfefer, J. K. Barton, E. K. Chan, M. G. Ducros, B. S. Sorg, T. E. Milner, J. S. Nelson, A. J. Welch, “A three-dimensional modular adaptable grid numerical model for light propagation duing laser irradiation of skin tissue,” IEEE J. Sel. Top. Quantum Electron. 2, 934–942 (1996).
[CrossRef]

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, (Clarendon, 1959).

Chan, E. K.

T. J. Pfefer, J. K. Barton, E. K. Chan, M. G. Ducros, B. S. Sorg, T. E. Milner, J. S. Nelson, A. J. Welch, “A three-dimensional modular adaptable grid numerical model for light propagation duing laser irradiation of skin tissue,” IEEE J. Sel. Top. Quantum Electron. 2, 934–942 (1996).
[CrossRef]

Chance, B.

Contini, D.

Cope, M.

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlength in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Del Bianco, S.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E 67, 056623-1–14 (2003).
[CrossRef]

Delpy, D. T.

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlength in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Ducros, M. G.

T. J. Pfefer, J. K. Barton, E. K. Chan, M. G. Ducros, B. S. Sorg, T. E. Milner, J. S. Nelson, A. J. Welch, “A three-dimensional modular adaptable grid numerical model for light propagation duing laser irradiation of skin tissue,” IEEE J. Sel. Top. Quantum Electron. 2, 934–942 (1996).
[CrossRef]

Farrell, T. J.

T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo ,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Feng, T. C.

Haskell, R. C.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, (Clarendon, 1959).

Kienle, A.

A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
[CrossRef]

A. Kienle, R. Steiner, “Determination of the optical properties of tissue by spatially resolved transmission measurements and Monte Carlo simulations,” Proc. SPIE 2077, 142–152 (1993).
[CrossRef]

Martelli, F.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E 67, 056623-1–14 (2003).
[CrossRef]

D. Contini, F. Martelli, G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36, 4587–4599 (1997).
[CrossRef] [PubMed]

McAdams, M.

Milner, T. E.

T. J. Pfefer, J. K. Barton, E. K. Chan, M. G. Ducros, B. S. Sorg, T. E. Milner, J. S. Nelson, A. J. Welch, “A three-dimensional modular adaptable grid numerical model for light propagation duing laser irradiation of skin tissue,” IEEE J. Sel. Top. Quantum Electron. 2, 934–942 (1996).
[CrossRef]

Nelson, J. S.

T. J. Pfefer, J. K. Barton, E. K. Chan, M. G. Ducros, B. S. Sorg, T. E. Milner, J. S. Nelson, A. J. Welch, “A three-dimensional modular adaptable grid numerical model for light propagation duing laser irradiation of skin tissue,” IEEE J. Sel. Top. Quantum Electron. 2, 934–942 (1996).
[CrossRef]

Patterson, M. S.

Pfefer, T. J.

T. J. Pfefer, J. K. Barton, E. K. Chan, M. G. Ducros, B. S. Sorg, T. E. Milner, J. S. Nelson, A. J. Welch, “A three-dimensional modular adaptable grid numerical model for light propagation duing laser irradiation of skin tissue,” IEEE J. Sel. Top. Quantum Electron. 2, 934–942 (1996).
[CrossRef]

Sassaroli, A.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E 67, 056623-1–14 (2003).
[CrossRef]

Sorg, B. S.

T. J. Pfefer, J. K. Barton, E. K. Chan, M. G. Ducros, B. S. Sorg, T. E. Milner, J. S. Nelson, A. J. Welch, “A three-dimensional modular adaptable grid numerical model for light propagation duing laser irradiation of skin tissue,” IEEE J. Sel. Top. Quantum Electron. 2, 934–942 (1996).
[CrossRef]

Steiner, R.

A. Kienle, R. Steiner, “Determination of the optical properties of tissue by spatially resolved transmission measurements and Monte Carlo simulations,” Proc. SPIE 2077, 142–152 (1993).
[CrossRef]

Svaasand, L. O.

Tromberg, B. J.

Tsay, T. T.

Welch, A. J.

T. J. Pfefer, J. K. Barton, E. K. Chan, M. G. Ducros, B. S. Sorg, T. E. Milner, J. S. Nelson, A. J. Welch, “A three-dimensional modular adaptable grid numerical model for light propagation duing laser irradiation of skin tissue,” IEEE J. Sel. Top. Quantum Electron. 2, 934–942 (1996).
[CrossRef]

Wilson, B. C.

T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo ,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

Yamada, Y.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E 67, 056623-1–14 (2003).
[CrossRef]

Zaccanti, G.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E 67, 056623-1–14 (2003).
[CrossRef]

D. Contini, F. Martelli, G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36, 4587–4599 (1997).
[CrossRef] [PubMed]

Appl. Opt. (2)

IEEE J. Sel. Top. Quantum Electron. (1)

T. J. Pfefer, J. K. Barton, E. K. Chan, M. G. Ducros, B. S. Sorg, T. E. Milner, J. S. Nelson, A. J. Welch, “A three-dimensional modular adaptable grid numerical model for light propagation duing laser irradiation of skin tissue,” IEEE J. Sel. Top. Quantum Electron. 2, 934–942 (1996).
[CrossRef]

J. Comput. Phys. (1)

A. H. Barnett, “A fast numerical method for time-resolved photon diffusion in general stratified turbid media,” J. Comput. Phys. 201, 771–797 (2004).
[CrossRef]

J. Opt. Soc. Am. A (2)

Med. Phys. (1)

T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo ,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Phys. Med. Biol. (1)

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlength in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (1)

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E 67, 056623-1–14 (2003).
[CrossRef]

Proc. SPIE (1)

A. Kienle, R. Steiner, “Determination of the optical properties of tissue by spatially resolved transmission measurements and Monte Carlo simulations,” Proc. SPIE 2077, 142–152 (1993).
[CrossRef]

Other (2)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, (Clarendon, 1959).

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Figures (6)

Fig. 1
Fig. 1

Geometry of the (rectangular) parallelepiped used for the calculations.

Fig. 2
Fig. 2

Two-dimensional arrangement of the positive (solid circles) and negative (open circles) sources to fulfill the extrapolated boundary conditions. The four sources shown at the top of the figure are those for l = 0 , n = 0 , and the four sources at the bottom are those for l = 0 , n = 1 ; see Eq. (5). The extrapolated boundaries are depicted by dashed lines. The arrows indicate that the point sources in each row are continued to infinity.

Fig. 3
Fig. 3

Comparison of the time-resolved transmittance from a cube ( l c = 19 mm ) and from a slab ( d = 19 mm ) calculated with the diffusion theory (dashed curves) with Monte Carlo simulations (solid curves).

Fig. 4
Fig. 4

Comparison of the time-resolved transmittance from a cube ( l c = 9 mm ) and from a slab ( d = 9 mm ) calculated with the diffusion theory (dashed curves) with Monte Carlo simulations (solid curves).

Fig. 5
Fig. 5

Comparison of the spatially resolved transmittance from the bottom and the lateral side of a cube ( l c = 9 mm ) calculated by the diffusion theory (solid curves) with Monte Carlo simulations (circles). The light beam is incident on the middle of the cube’s top side. The dashed lines in the inset give the location of the measurement of the transmittance from the lateral side, L s , and from the bottom side, T s .

Fig. 6
Fig. 6

Comparison of the spatially-resolved transmittance from the lateral side of a cube ( l c = 41 mm ) calculated by the diffusion theory (dashed curves) with Monte Carlo simulations (solid curves). The light beam is incident 1 mm and 3 mm away from the edge of the cube’s top side. The dashed line in the inset gives the location of the measurement of the transmittance, L s .

Equations (26)

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z b = 1 + R eff 1 R eff 2 D
1 c Φ ( r , t ) t D 2 Φ ( r , t ) + μ a Φ ( r , t ) = S ( r , t ) ,
Φ ( r , t ) = c ( 4 π D c t ) 3 2 exp ( μ a c t ) exp ( r 2 4 D c t ) ,
Φ ( r , t ) = c ( 4 π D c t ) 3 2 exp ( μ a c t ) l = m = n = [ exp ( ( x x 1 l ) 2 + ( y y 1 m ) 2 + ( z z 1 n ) 2 4 D c t ) exp ( ( x x 1 l ) 2 + ( y y 1 m ) 2 + ( z z 2 n ) 2 4 D c t ) exp ( ( x x 1 l ) 2 + ( y y 2 m ) 2 + ( z z 1 n ) 2 4 D c t ) + exp ( ( x x 1 l ) 2 + ( y y 2 m ) 2 + ( z z 2 n ) 2 4 D c t ) exp ( ( x x 2 l ) 2 + ( y y 1 m ) 2 + ( z z 1 n ) 2 4 D c t ) + exp ( ( x x 2 l ) 2 + ( y y 1 m ) 2 + ( z z 2 n ) 2 4 D c t ) + exp ( ( x x 2 l ) 2 + ( y y 2 m ) 2 + ( z z 1 n ) 2 4 D c t ) exp ( ( x x 2 l ) 2 + ( y y 2 m ) 2 + ( z z 2 n ) 2 4 D c t ) ] ,
x 1 l = 2 l l x + 4 l z b + x u ,
x 2 l = 2 l l x + ( 4 l 2 ) z b x u ,
y 1 m = 2 m l y + 4 m z b + y u ,
y 2 m = 2 m l y + ( 4 m 2 ) z b y u ,
z 1 n = 2 n l z + 4 n z b + z 0 ,
z 2 n = 2 n l z + ( 4 n 2 ) z b z 0 .
Φ ( r , t ) = c ( 4 π D c t ) 3 2 exp ( μ a c t ) × l = [ exp ( ( x x 1 l ) 2 4 D c t ) exp ( ( x x 2 l ) 2 4 D c t ) ) × m = [ exp ( ( y y 1 m ) 2 4 D c t ) exp ( ( y y 2 m ) 2 4 D c t ) ] × n = [ exp ( ( z z 1 n ) 2 4 D c t ) exp ( ( z z 2 n ) 2 4 D c t ) ] .
T ( x , y , t ) = D z Φ ( x , y , z , t ) z = l z ,
T ( x , y , t ) = 1 2 ( 4 π D c ) 3 2 t 5 2 exp ( μ a c t ) × l = [ exp ( ( x x 1 l ) 2 4 D c t ) exp ( ( x x 2 l ) 2 4 D c t ) ] × m = [ exp ( ( y y 1 m ) 2 4 D c t ) exp ( ( y y 2 m ) 2 4 D c t ) ] × n = [ ( l z z 1 n ) exp ( ( l z z 1 n ) 2 4 D c t ) ( l z z 2 n ) exp ( ( l z z 2 n ) 2 4 D c t ) ] .
D 2 Φ s ( r ) μ a Φ s ( r ) = S s ( r ) .
Φ s ( r ) = 1 4 π D exp ( μ eff r ) r ,
Φ s ( x , y , z ) = 1 4 π D l = m = n = ( exp ( μ eff r 1 ) r 1 exp ( μ eff r 2 ) r 2 exp ( μ eff r 3 ) r 3 + exp ( μ eff r 4 ) r 4 exp ( μ eff r 5 ) r 5 + exp ( μ eff r 6 ) r 6 + exp ( μ eff r 7 ) r 7 exp ( μ eff r 8 ) r 8 ) ,
r 1 = ( ( x x 1 l ) 2 + ( y y 1 m ) 2 + ( z z 1 n ) 2 ) 1 2 ,
r 2 = ( ( x x 1 l ) 2 + ( y y 1 m ) 2 + ( z z 2 n ) 2 ) 1 2 ,
r 3 = ( ( x x 1 l ) 2 + ( y y 2 m ) 2 + ( z z 1 n ) 2 ) 1 2 ,
r 4 = ( ( x x 1 l ) 2 + ( y y 2 m ) 2 + ( z z 2 n ) 2 ) 1 2 ,
r 5 = ( ( x x 2 l ) 2 + ( y y 1 m ) 2 + ( z z 1 n ) 2 ) 1 2 ,
r 6 = ( ( x x 2 l ) 2 + ( y y 1 m ) 2 + ( z z 2 n ) 2 ) 1 2 ,
r 7 = ( ( x x 2 l ) 2 + ( y y 2 m ) 2 + ( z z 1 n ) 2 ) 1 2 ,
r 8 = ( ( x x 2 l ) 2 + ( y y 2 m ) 2 + ( z z 2 n ) 2 ) 1 2 ,
T s ( x , y ) = D z Φ ( x , y , z ) z = l z ,
T s ( x , y ) = 1 4 π l = m = n = ( ( l z z 1 n ) ( μ eff + 1 r 1 ) exp ( μ eff r 1 ) r 1 2 ( l z z 2 n ) ( μ eff + 1 r 2 ) exp ( μ eff r 2 ) r 2 2 ( l z z 1 n ) ( μ eff + 1 r 3 ) exp ( μ eff r 3 ) r 3 2 + ( l z z 2 n ) ( μ eff + 1 r 4 ) exp ( μ eff r 4 ) r 4 2 ( l z z 1 n ) ( μ eff + 1 r 5 ) exp ( μ eff r 5 ) r 5 2 + ( l z z 2 n ) ( μ eff + 1 r 6 ) exp ( μ eff r 6 ) r 6 2 + ( l z z 1 n ) ( μ eff + 1 r 7 ) exp ( μ eff r 7 ) r 7 2 ( l z z 2 n ) ( μ eff + 1 r 8 ) exp ( μ eff r 8 ) r 8 2 ) ,

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