Abstract

We evaluate the numerical accuracy of finite-difference time-domain (FDTD) analysis of optical transport in a three-dimensional scattering medium illuminated by an isotropic point source. This analysis employs novel boundary conditions for the diffusion equation. The power radiated from an isotropic point source located at a depth equal to the reciprocal of the reduced scattering coefficient (1/μs) below the surface at the irradiated position is introduced to the integral form of the diffusion equation. Finite- difference approximations of the diffusion equation for a surface cell are derived by utilizing new boundary conditions that include an isotropic source even in a surface cell. Steady-state and time-resolved reflectances are calculated by FDTD analysis for a semi-infinite uniform scattering medium illuminated by an isotropic point source. The numerical results agree reasonably with the analytical solutions for μs=13mm1 without resizing the mesh elements.

© 2011 Optical Society of America

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  1. M. Fabiani, D. D. Schmorrow, and G. Gratton, “Optical imaging of the intact human brain (guest editorial),” IEEE Eng. Med. Biol. Mag. 26 (4), 14–16 (2007).
    [CrossRef] [PubMed]
  2. T. O. McBride, B. W. Pogue, E. D. Gerety, S. B. Poplack, U. L. Osterberg, and K. D. Paulsen, “Spectroscopic diffuse optical tomography for the quantitative assessment of hemoglobin concentration and oxygen saturation in breast tissue,” Appl. Opt. 38, 5480–5490 (1999).
    [CrossRef]
  3. A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction schema for time-resolved optical tomography,” IEEE Trans. Med. Imag. 18, 262–271(1999).
    [CrossRef]
  4. F. G. Gao, H. Z. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41, 779–791 (2002).
    [CrossRef]
  5. R. D. Frostig, In Vivo Optical Imaging of Brain Function (CRC, 2002), Chap. 8.
    [CrossRef]
  6. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997), Chap. 9.
  7. T. Tanifuji and M. Hijikata, “Finite difference time domain (FDTD) analysis of optical pulse responses in biological tissues for spectroscopic diffused optical tomography,” IEEE Trans. Med. Imag. 21, 181–184 (2002).
    [CrossRef]
  8. M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
    [CrossRef] [PubMed]
  9. T. J. Farrell, M. S. Patterson, and B. 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]
  10. T. Tanifuji, “Alternative boundary conditions for solving optical diffusion equations by a finite difference time domain analysis in three-dimensional scattering medium,” Opt. Rev. 16, 283–289 (2009).
    [CrossRef]
  11. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical property,” Appl. Opt. 28, 2331–2336(1989).
    [CrossRef] [PubMed]
  12. T. Tanifuji and K. Ichitsubo, “Finite difference time domain analysis of diffusion equations with nonuniform grids for time-resolved reflectance of an optical pulse in three-dimensional scattering medium,” Opt. Rev. 12, 480–485 (2005).
    [CrossRef]
  13. M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delphy, “An investigation of light transport through scattering body with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
    [CrossRef] [PubMed]
  14. T. Tanifuji, “Extension of grid separation in the finite difference time domain analysis for predicting time-resolved reflectance of an optical pulse from scattering medium with non-scattering regions,” Opt. Rev. 16, 452–453 (2009).
    [CrossRef]

2009

T. Tanifuji, “Alternative boundary conditions for solving optical diffusion equations by a finite difference time domain analysis in three-dimensional scattering medium,” Opt. Rev. 16, 283–289 (2009).
[CrossRef]

T. Tanifuji, “Extension of grid separation in the finite difference time domain analysis for predicting time-resolved reflectance of an optical pulse from scattering medium with non-scattering regions,” Opt. Rev. 16, 452–453 (2009).
[CrossRef]

2007

M. Fabiani, D. D. Schmorrow, and G. Gratton, “Optical imaging of the intact human brain (guest editorial),” IEEE Eng. Med. Biol. Mag. 26 (4), 14–16 (2007).
[CrossRef] [PubMed]

2005

T. Tanifuji and K. Ichitsubo, “Finite difference time domain analysis of diffusion equations with nonuniform grids for time-resolved reflectance of an optical pulse in three-dimensional scattering medium,” Opt. Rev. 12, 480–485 (2005).
[CrossRef]

2002

F. G. Gao, H. Z. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41, 779–791 (2002).
[CrossRef]

T. Tanifuji and M. Hijikata, “Finite difference time domain (FDTD) analysis of optical pulse responses in biological tissues for spectroscopic diffused optical tomography,” IEEE Trans. Med. Imag. 21, 181–184 (2002).
[CrossRef]

1999

1996

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delphy, “An investigation of light transport through scattering body with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

1995

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

1992

T. J. Farrell, M. S. Patterson, and B. 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]

1989

Arridge, S. R.

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delphy, “An investigation of light transport through scattering body with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

Chance, B.

Delphy, D. T.

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delphy, “An investigation of light transport through scattering body with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Delpy, D. T.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

Fabiani, M.

M. Fabiani, D. D. Schmorrow, and G. Gratton, “Optical imaging of the intact human brain (guest editorial),” IEEE Eng. Med. Biol. Mag. 26 (4), 14–16 (2007).
[CrossRef] [PubMed]

Farrell, T. J.

T. J. Farrell, M. S. Patterson, and B. 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]

Firbank, M.

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delphy, “An investigation of light transport through scattering body with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Frostig, R. D.

R. D. Frostig, In Vivo Optical Imaging of Brain Function (CRC, 2002), Chap. 8.
[CrossRef]

Gao, F. G.

F. G. Gao, H. Z. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41, 779–791 (2002).
[CrossRef]

Gerety, E. D.

Gratton, G.

M. Fabiani, D. D. Schmorrow, and G. Gratton, “Optical imaging of the intact human brain (guest editorial),” IEEE Eng. Med. Biol. Mag. 26 (4), 14–16 (2007).
[CrossRef] [PubMed]

Hanson, K. M.

A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction schema for time-resolved optical tomography,” IEEE Trans. Med. Imag. 18, 262–271(1999).
[CrossRef]

Hielscher, A. H.

A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction schema for time-resolved optical tomography,” IEEE Trans. Med. Imag. 18, 262–271(1999).
[CrossRef]

Hijikata, M.

T. Tanifuji and M. Hijikata, “Finite difference time domain (FDTD) analysis of optical pulse responses in biological tissues for spectroscopic diffused optical tomography,” IEEE Trans. Med. Imag. 21, 181–184 (2002).
[CrossRef]

Hiraoka, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

Ichitsubo, K.

T. Tanifuji and K. Ichitsubo, “Finite difference time domain analysis of diffusion equations with nonuniform grids for time-resolved reflectance of an optical pulse in three-dimensional scattering medium,” Opt. Rev. 12, 480–485 (2005).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997), Chap. 9.

Klose, A. D.

A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction schema for time-resolved optical tomography,” IEEE Trans. Med. Imag. 18, 262–271(1999).
[CrossRef]

McBride, T. O.

Osterberg, U. L.

Patterson, M. S.

T. J. Farrell, M. S. Patterson, and B. 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, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical property,” Appl. Opt. 28, 2331–2336(1989).
[CrossRef] [PubMed]

Paulsen, K. D.

Pogue, B. W.

Poplack, S. B.

Schmorrow, D. D.

M. Fabiani, D. D. Schmorrow, and G. Gratton, “Optical imaging of the intact human brain (guest editorial),” IEEE Eng. Med. Biol. Mag. 26 (4), 14–16 (2007).
[CrossRef] [PubMed]

Schweiger, M.

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delphy, “An investigation of light transport through scattering body with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

Tanifuji, T.

T. Tanifuji, “Alternative boundary conditions for solving optical diffusion equations by a finite difference time domain analysis in three-dimensional scattering medium,” Opt. Rev. 16, 283–289 (2009).
[CrossRef]

T. Tanifuji, “Extension of grid separation in the finite difference time domain analysis for predicting time-resolved reflectance of an optical pulse from scattering medium with non-scattering regions,” Opt. Rev. 16, 452–453 (2009).
[CrossRef]

T. Tanifuji and K. Ichitsubo, “Finite difference time domain analysis of diffusion equations with nonuniform grids for time-resolved reflectance of an optical pulse in three-dimensional scattering medium,” Opt. Rev. 12, 480–485 (2005).
[CrossRef]

T. Tanifuji and M. Hijikata, “Finite difference time domain (FDTD) analysis of optical pulse responses in biological tissues for spectroscopic diffused optical tomography,” IEEE Trans. Med. Imag. 21, 181–184 (2002).
[CrossRef]

Wilson, B.

T. J. Farrell, M. S. Patterson, and B. 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]

Wilson, B. C.

Yamada, Y.

F. G. Gao, H. Z. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41, 779–791 (2002).
[CrossRef]

Zhao, H. Z.

F. G. Gao, H. Z. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41, 779–791 (2002).
[CrossRef]

Appl. Opt.

IEEE Eng. Med. Biol. Mag.

M. Fabiani, D. D. Schmorrow, and G. Gratton, “Optical imaging of the intact human brain (guest editorial),” IEEE Eng. Med. Biol. Mag. 26 (4), 14–16 (2007).
[CrossRef] [PubMed]

IEEE Trans. Med. Imag.

A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction schema for time-resolved optical tomography,” IEEE Trans. Med. Imag. 18, 262–271(1999).
[CrossRef]

T. Tanifuji and M. Hijikata, “Finite difference time domain (FDTD) analysis of optical pulse responses in biological tissues for spectroscopic diffused optical tomography,” IEEE Trans. Med. Imag. 21, 181–184 (2002).
[CrossRef]

Med. Phys.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, and B. 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]

Opt. Rev.

T. Tanifuji, “Alternative boundary conditions for solving optical diffusion equations by a finite difference time domain analysis in three-dimensional scattering medium,” Opt. Rev. 16, 283–289 (2009).
[CrossRef]

T. Tanifuji, “Extension of grid separation in the finite difference time domain analysis for predicting time-resolved reflectance of an optical pulse from scattering medium with non-scattering regions,” Opt. Rev. 16, 452–453 (2009).
[CrossRef]

T. Tanifuji and K. Ichitsubo, “Finite difference time domain analysis of diffusion equations with nonuniform grids for time-resolved reflectance of an optical pulse in three-dimensional scattering medium,” Opt. Rev. 12, 480–485 (2005).
[CrossRef]

Phys. Med. Biol.

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delphy, “An investigation of light transport through scattering body with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Other

R. D. Frostig, In Vivo Optical Imaging of Brain Function (CRC, 2002), Chap. 8.
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997), Chap. 9.

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Figures (12)

Fig. 1
Fig. 1

Positions of field parameters in the Yee grid. The accuracy of the finite difference is within O ( Δ t 2 ) + O ( Δ z 2 ) due to the central subtraction schema.

Fig. 2
Fig. 2

Schematic illustration showing the alternative boundary conditions in a surface cell of a 3D scattering medium. Undefined field parameters at k = 1 / 4 are interpolated, and Eq. (2) is integrated over a surface cell using the Gauss theorem.

Fig. 3
Fig. 3

Isotropic point source allocation to discrete points on the z axis in the Yee grid. The allocation has three cases depending on the position of the point source z 0 .

Fig. 4
Fig. 4

Lattice space increments ( Δ z ) extended by utilizing the proposed boundary conditions. (a)  Δ z must be refined in the conventional boundary conditions when z 0 is smaller than Δ z , (b) whereas no refinement is necessary with the proposed boundary conditions.

Fig. 5
Fig. 5

Steady-state reflectance calculated using step responses for seven different source–detector separations d.

Fig. 6
Fig. 6

Dependences of steady-state diffuse reflectance on source position calculated using conventional boundary conditions (BCs) and the proposed boundary conditions ABC-1 and ABC-2 for μ s = (a) 1, (b) 2, and (c)  3 mm 1 and (a), (b), (c)  μ a = 0.02 .

Fig. 7
Fig. 7

Dependences of steady-state diffuse reflectance on source-detector separation for three different reduced scattering coefficients μ s in the (a) horizontal and (b) diagonal directions.

Fig. 8
Fig. 8

Error in the steady-state diffuse reflectance calculated using conventional boundary conditions for three different reduced scattering coefficients μ s in the (a) horizontal and (b) diagonal directions.

Fig. 9
Fig. 9

Analytical and numerically calculated time-resolved diffuse reflectances for reduced scattering coefficients μ s of (a) 1, (b) 2, and (3)  3 mm 1 and for an absorption coefficient of μ a = 0.02 mm 1 .

Fig. 10
Fig. 10

Fluctuation amplitude dependence on source position k z and beam diameter in 1 / e maximum full width ( 1 / e MFW ).

Fig. 11
Fig. 11

Time-resolved reflectance for Δ z = 0.5 , 1, and 2 mm of (a) δ-shaped beam illumination and (b) Gaussian beam illumination.

Fig. 12
Fig. 12

Time-resolved reflectance for Δ z = 0.5 , 1, and 2 mm for μ s of (a) 1 and (b)  2.5 mm 1 .

Equations (26)

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1 c J ( r , t ) t + 1 3 ϕ ( r , t ) + μ tr J ( r , t ) = 4 π ε ( r , s ^ , t ) s ^ d ω ,
1 c ϕ ( r , t ) t + · J ( r , t ) + μ a ϕ ( r , t ) = 4 π ε ( r , s ^ , t ) d ω ,
ε ( r , s ^ , t ) = 1 4 π μ s μ a + μ s f ( t z c ) δ ( r r 0 ) , r = ( x 0 , y 0 , z 0 ) ,
4 π ε ( r , s ^ , t ) d ω = μ s μ a + μ s f ( t z c ) δ ( r r 0 ) ,
4 π ε ( r , s ^ , t ) ( s ^ · x ^ ) d ω = 4 π ε ( r , s ^ , t ) ( s ^ · y ^ ) d ω = 4 π ε ( r , s ^ , t ) ( s ^ · z ^ ) d ω = 0.
ϕ ( r , t ) + 2 A J n ( r , t ) = 0 ,
1 c ϕ ( r , t ) t = · [ D ( r ) ϕ ( r , t ) ] μ a ( r ) ϕ ( r , t ) · [ 1 μ tr ( r ) 4 π ε r i ( r , s ^ , t ) s ^ d ω ] + 4 π ε r i ( r , s ^ , t ) d ω ,
( 1 c t + μ tr ) ϕ ( r , t ) 2 A 3 ϕ ( r , t ) n = 0 ,
P ( ρ ) = 1 4 π [ z 0 ( μ eff + 1 r 1 ) e μ eff r 1 r 1 2 + ( z 0 + 2 z b ) ( μ eff + 1 r 2 ) e μ eff r 2 r 2 2 ] ,
r ( ρ , t ) = ( 4 π D c ) 3 / 2 e μ a c t [ z 0 e z 0 2 + ρ 2 4 D c t + ( z 0 + 2 z b ) e ( z 0 + 2 z b ) 2 + ρ 2 4 D c t ] .
R ( ρ , t ) = r ( ρ , t t ) f ( t ) d t .
J x n + 1 2 ( i + 1 2 , j , k ) = C 1 J x n 1 2 ( i + 1 2 , j , k ) + C 1 Δ x [ ϕ n ( i + 1 , j , k ) ϕ n ( i , j , k ) ] ,
J y n + 1 2 ( i , j + 1 2 , k ) = C 1 J y n 1 2 ( i , j + 1 2 , k ) + C 1 Δ y [ ϕ n ( i , j + 1 , k ) ϕ n ( i , j , k ) ] ,
J z n + 1 2 ( i , j , k + 1 2 ) = C 1 J z n 1 2 ( i , j , k + 1 2 ) + C 1 Δ z [ ϕ n ( i , j , k + 1 ) ϕ n ( i , j , k ) ] ,
ϕ n + 1 ( i , j , k ) = C 2 ϕ n ( i , j , k ) + C 2 Δ x [ J x n + 1 2 ( i + 1 2 , j , k ) J x n + 1 2 ( i 1 2 , j , k ) ] + C 2 Δ y [ J y n + 1 2 ( i , j + 1 2 , k ) J x n + 1 2 ( i , j 1 2 , k ) ] + C 2 Δ z [ J z n + 1 2 ( i , j , k + 1 2 ) J z n + 1 2 ( i , j , k 1 2 ) ] + C 2 S μ ¯ s ( r ) μ ¯ t ( r ) P 0 T s ( i i 0 , j j 0 ) × D ( k , z 0 ) Δ x Δ y Δ z .
C 1 ( r ) = 2 c μ ¯ tr ( r ) Δ t 2 + c μ ¯ tr ( r ) Δ t , C 1 δ ( r ) = 1 3 δ 2 c Δ t 2 + c Δ t μ ¯ tr ( r ) , C 1 S ( r ) = 2 c Δ t 2 + c Δ t μ ¯ tr ( r ) , δ = Δ x , Δ y , Δ z ,
C 2 ( r ) = 2 c μ ¯ a ( r ) Δ t 2 + c μ ¯ a ( r ) Δ t , C 2 δ ( r ) = 1 δ 2 c Δ t 2 + c Δ t μ ¯ a ( r ) , C 2 S ( r ) = 2 c Δ t 2 + c Δ t μ ¯ a ( r ) ,
T s ( i i 0 , j j 0 ) = { 1 ( i = i 0 j = j 0 ) 0 ( else ) .
D s ( k , z 0 ) = f [ ( n t + 1 ) Δ t k Δ z / c ] + f [ n Δ t k Δ z / c ] 2 ( k + 1 ) Δ z z 0 Δ z ,
D s ( k + 1 , z 0 ) = f [ ( n t + 1 ) Δ t ( k + 1 ) Δ z / c ] + f [ n Δ t ( k + 1 ) Δ z / c ] 2 z 0 k Δ z Δ z .
ϕ n ( i , j , 1 / 4 ) = 3 4 ϕ n ( i , j , 0 ) + 1 4 ϕ n ( i , j , 0 ) ,
J x n + 1 / 2 ( i ± 1 / 2 , j , 1 / 4 ) = 3 4 J x n + 1 / 2 ( i ± 1 / 2 , j , 0 ) + 1 4 J x n + 1 / 2 ( i ± 1 / 2 , j , 1 ) ,
J y n + 1 / 2 ( i , j ± 1 / 2 , 1 / 4 ) = 3 4 J y n + 1 / 2 ( i , j ± 1 / 2 , 0 ) + 1 4 J y n + 1 / 2 ( i , j ± 1 / 2 , 1 ) .
C 1 B ϕ n + 1 ( i , j , 0 ) = C 2 B ϕ n ( i , j , 0 ) [ ϕ n + 1 ( i , j , 1 ) ϕ n ( i , j , 1 ) ] / 4 / Δ z [ ϕ n + 1 ( i , j , 1 ) + ϕ n ( 1 , i , j ) ] μ a / 8 3 [ J x n + 1 / 2 ( i + 1 2 , j , 0 ) J x n + 1 / 2 ( i 1 2 , j , 0 ) ] / 4 / Δ x [ J x n + 1 / 2 ( i + 1 2 , j , 1 ) J x n + 1 / 2 ( i 1 2 , j , 1 ) ] / 4 / Δ x 3 [ J y n + 1 / 2 ( i , j + 1 2 , 0 ) J x n + 1 / 2 ( i , j 1 2 , 0 ) ] / 4 / Δ y [ J y n + 1 / 2 ( i , j + 1 2 , 1 ) J y n + 1 / 2 ( i , j 1 2 , 1 ) ] / 4 / Δ y [ J z n + 1 / 2 ( i , j , 3 2 ) + J z n + 1 / 2 ( i , j , 1 2 ) ] / 2 Δ z + μ ¯ s ' μ ¯ t P 0 T s ( i i 0 , j j 0 ) × D ( k , z 0 ) Δ x Δ y ( Δ z 2 ) ,
C 1 B = 3 4 c Δ t + 3 8 μ a ( 0 , i , j ) + 1 4 A Δ z , C 2 B = 3 4 c Δ t 3 8 μ a ( 0 , i , j ) 1 4 A Δ z .
J z n + 1 / 2 ( i , j , 1 / 2 ) J z n + 1 / 2 ( i , j , 0 ) Δ z / 2 { J z n + 1 / 2 ( i , j , 3 2 ) + J z n + 1 / 2 ( i , j , 1 2 ) } / 2 + { ϕ n + 1 ( i , j , 0 ) + ϕ n ( i , j , 0 ) } / 4 Δ z .

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