Abstract

Near-infrared (NIR) light propagations in strongly scattering tissue have been studied in the past few decades and diffusion approximations (DA) have been extensively used under the assumption that the refractive index is constant throughout the medium. When the index is varying, the discontinuity of the fluence rate arises at the index-mismatched interface. We introduce the finite element method (FEM) incorporating the refractive index mismatch at the interface between the diffusive media without any approximations. Intensity, mean time, and mean optical path length were computed by FEM and by Monte Carlo (MC) simulations for a two-layer slab model and a good agreement between the data from FEM and from MC was found. The absorption sensitivity of intensity and mean time measurements was also analyzed by FEM. We have shown that mean time and absorption sensitivity functions vary significantly as the refractive index mismatch develops at the interface between the two layers.

© 2004 Optical Society of America

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References

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Appl. Opt. (2)

Inverse Problems (1)

S. R. Arridge, �??Optical tomography in medical imaging,�?? Inverse Problems 15, R41-R93 (1999)
[CrossRef]

J. Opt. Soc. Am. A (4)

Phys. Med. Biol. (2)

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. Paulsen, �??The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,�?? Phys. Med. Biol. 48, 2713-2727 (2003)
[CrossRef] [PubMed]

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. Van Der Zee, and D. T. Delpy, �??A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,�?? Phys. Med. Biol. 38, 1859-1876 (1993)
[CrossRef] [PubMed]

Other (1)

A. Ishimaru, Wave propagation and scattering in random media (Academic, 1978), Chap 7-9.

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Figures (12)

Fig. 1.
Fig. 1.

Splitting of the domain Ω into Ω1 and Ω2 and its external boundary Γ into Γ1 and Γ2 at the index-mismatched interface S by duplications of nodes and facets in 3D (or edges in 2D) at S. As a result, Ω L has the boundary given by Γ LS for L=1, 2. The domains Ω1 and Ω2 are coupled by the discontinuity condition for the fluence rate on S.

Fig. 2.
Fig. 2.

Reproduction of boundary integral results [5] by FEM. The data show the normalized |R-R s |ϕ where the detector location is R=(0, 0, z), the source lacation is R s =(0, 0, 10) in mm, and ϕ is the fluence rate. S is the index-mismatched interface between medium 1 with refractive index n 1 and medium 2 with n 2. The scattering coefficient µs , the absorption coefficient µa , and the anisotropy factor g are the same as those given in [5]: µs ,1=7.5 mm-1, µa ,1=0.0035 mm-1, g 1=0.8 for medium 1 and µs ,2=5.0 mm-1, µa ,2=0.024 mm-1, g 2=0.8 for medium 2. In the figure, (a) is the result for n 2=1.0 and (b) is the result for n 2=2.0 with n 1=1.333 in both cases.

Fig. 3.
Fig. 3.

A planar two-layer slab model [7] and the generated FEM mesh for the truncated domain. In (b), the truncated domain has 160 mm width and 80 mm height. 22474 triangle elements and 11496 nodes were initially generated over the domain. For the index-mismatched case, 161 nodes on the interface were duplicated and total 11657 nodes were used. The source was applied to the centre of the surface of the layer 1. n 1 and n 2 are indices of refraction of the layer 1 and 2, respectively.

Fig. 4.
Fig. 4.

Plots of log10 [fluence rate (W/mm2)] for (a) n 1=1.33, n 2=1.33, (b) n 1=1.33, n 2=1.58, (c) n 1=1.58, n 2=1.33, and (d) n 1=1.58, n 2=1.58. n 1 and n 2 are indices of refraction of the layer 1 and 2, respectively. Maximum value of the contour is 0.0 and contour step is 0.5 in logarithmic scale.

Fig. 5.
Fig. 5.

Intensity and mean time (of flight) data computed by FEM and MC. n 1 and n 2 are refractive indices of the layer 1 and the layer 2 respectively.

Fig. 6.
Fig. 6.

Partial mean optical pathlength in the layer 1 and layer 2.

Fig. 7.
Fig. 7.

Absorption sensitivity in intensity measurements for (a) n 1=1.33, n 2=1.33, (b) n 1=1.33, n 2=1.58, (c) n 1=1.58, n 2=1.33, and (d) n 1=1.58, n 2=1.58. Source-detector separations are 10, 30, and 50 mm from the left column to the right.

Fig. 8.
Fig. 8.

Absorption sensitivity in mean time measurements for (a) n 1=1.33, n 2=1.33, (b) n 1=1.33, n 2=1.58, (c) n 1=1.58, n 2=1.33, and (d) n 1=1.58, n 2=1.58. Source-detector separations are 10, 30, and 50 mm from the left column to the right.

Fig. 9.
Fig. 9.

Cut-through view of the tetrahedral mesh and log10[fluence rate (W/mm2)] computed by FEM for the 3D two-layer slab model: (a) tetrahedral mesh, (b) fluence rate for (n 1, n 2)=(1.33, 1.58), and (c) fluence rate for (n 1, n 2)=(1.58, 1.33). n 1 and n 2 are refractive indices of the layer 1 and the layer 2, respectively. Maximum value of the contour is 0.0 and contour step is 0.5 in logarithmic scale.

Fig. 10.
Fig. 10.

Intensity, mean time (of flight), and partial mean optical path length data computed by FEM and MC in 3D. Symbols are results from MC simulations and solid lines are results from FEM. n 1 and n 2 are refractive indices of the layer 1 and the layer 2, respectively.

Fig. 11.
Fig. 11.

Absorption sensitivity in mean time measurements at the detection position located at 30 mm radial distance computed by 3D FEM for (a) (n 1, n 2)=(1.33, 1.58) and for (b) (n 1, n 2)=(1.58, 1.33). n 1 and n 2 are refractive indices of the layer 1 and the layer 2, respectively.

Fig. 12.
Fig. 12.

Scattered positions of (ϕ 1/3Jn , ϕ 2/3Jn ) on the locus imposed by the constraint and the spatial variation of the ratio ϕ 1/ϕ 2 along the interface. In the left figure, the pairs outside the range have not been shown. In the right figure, x is the distance from the centre of the interface.

Equations (32)

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· D ( r ) ϕ ( r ) + [ μ a ( r ) + i ( ω c ) ] ϕ ( r ) = q 0 ( r ) ,
J ( r ) = D ( r ) ϕ ( r ) ,
n ̂ · J = ζ ϕ
ζ = 1 2 ( 1 R ϕ 1 + R J )
R ϕ = 2 0 π 2 R p ( cos θ ) cos θ sin θ d θ ,
R J = 3 0 π 2 R p ( cos θ ) cos 2 θ sin θ d θ ,
n ̂ i · J = ζ ( i ) ϕ ( i ) ζ ( j ) ϕ ( j )
ζ ( i ) = 1 R ϕ ( i ) 2 ( R J ( i ) + R J ( j ) )
· D ( L ) ( r ) ϕ ( L ) ( r ) + [ μ a ( L ) ( r ) + i ( ω c ( L ) ) ] ϕ ( L ) ( r ) = q 0 ( L ) ( r )
[ K ( L ) + C ( L ) i ω B ( L ) + ζ ( L ) A ( L ) ] Φ ( L ) + Λ ( L ) [ n ̂ L · J ] = Q ( L )
Λ i ( L ) [ n ̂ L · J ] = s d S φ i ( L ) ( r ) n ̂ L ( r ) · J ( r ) .
[ F ( 1 ) + ζ ' ( 1 ) Σ ( 11 ) ζ ' ( 2 ) Σ ( 12 ) ζ ' ( 1 ) Σ ( 2 ) F ( 2 ) + ζ ' ( 2 ) Σ ( 22 ) ] [ Φ ( 1 ) Φ ( 2 ) ] = [ Q ( 1 ) Q ( 2 ) ]
K ij ( L ) = Ω L d Ω D ( L ) ( r ) φ i ( L ) ( r ) φ j ( L ) ( r ) ,
C ij ( L ) = Ω L d Ω μ a ( L ) ( r ) φ i ( L ) ( r ) φ j ( L ) ( r ) ,
B ij ( L ) = 1 c 0 Ω L d Ω n ( L ) ( r ) φ i ( L ) ( r ) φ j ( L ) ( r ) ,
A ij ( L ) = Γ L d Γ φ i ( L ) ( r ) φ j ( L ) ( r ) ,
ij ( LM ) = s d S φ i ( L ) ( r ) φ j ( M ) ( r ) .
Q i ( L ) = Ω L φ i ( L ) ( r ) q 0 ( L ) ( r )
[ K + C i ω B + A + Σ ] Φ = Q
A ( ζ ( 1 ) A ( 1 ) 0 0 ζ ( 2 ) A ( 2 ) )
Σ ( ζ ( 1 ) Σ ( 11 ) ζ ( 2 ) Σ ( 12 ) ζ ( 1 ) Σ ( 21 ) ζ ( 2 ) Σ ( 22 ) ) ,
[ K + C + A + Σ ] Φ 0 = Q .
[ K + C + A + Σ ] Φ 1 = B Φ 0 .
· D ( r ) ψ ( r ) + [ μ a ( r ) i ( ω c ) ] ψ ( r ) = 0 ,
ψ ( r ) + 2 AD ( r ) ψ ( r ) n ̂ = δ ( r m ) ,
[ K + C + i ω B + A + Σ ] Ψ = q
[ K + C + A + Σ ] Ψ 0 = q ,
[ K + C + A + Σ ] Ψ 1 = B Ψ 0 .
J ( E ) ( r ) = ψ 0 ( r ) ϕ 0 ( r )
J ( < t > ) ( r ) = [ J ( T ) ( r ) < t ( m ) > J ( E ) ( r ) ] E ( m )
J ( T ) ( r ) = ψ 1 ( r ) ϕ 0 ( r ) + ψ 0 ( r ) ϕ 1 ( r ) .
ϕ 1 J n = ( n 1 n 2 ) 2 ( ϕ 2 J n ) + C 1

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