Abstract

This paper introduces an improved diffusion model which is accurate and fast in computation for the cases of μa/μ’s < 007 as good as the conventional diffusion model for the cases of μa/μ’s < 0.007 for surface measurement, hence more suitable than the conventional model to be the forward model used in the image reconstruction in the diffuse optical tomography. Deviation of the diffusion approximation (DA) on the medium surface is first studied in the Monte Carlo (MC) diffusion hybrid model for reflectance setup. A modification of DA and an improved MC diffusion hybrid model based on this modified DA are introduced. Numerical tests show that for media with relatively strong absorption the present modified diffusion approach can reduce the surface deviation significantly in both the hybrid and pure diffusion model, and consumes nearly no more computation time than the conventional diffusion approach.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

2006

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. Kaipio, "Finite element model for the coupled radiative transfer equation and diffusion approximation," Int. J. Numer. Methods Eng., 65,383-405 (2006).
[CrossRef]

2003

2002

2001

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Signal Process Mag.57-75 (2001).
[CrossRef]

B. Q. Chen, K. Stamnes, and J. J. Stamnes, "Validity of the diffusion approximation in bio-optical imaging", Appl. Opt. 40,6356-6366 (2001).
[CrossRef]

2000

1999

S. R. Arridge, "Optical tomography in medical imaging," Inv. Probl. 15,R41-R93 (1999).
[CrossRef]

1998

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Comparison of finite difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues," Phys. Med. Biol. 43, 1285-1302 (1998).
[CrossRef] [PubMed]

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, and P. M. Schlag, "Constrast features of breast cancer in frequency-domain laser scanning mammography," J. Biomed. Opt. 3,129-136 (1998)
[CrossRef]

1997

1995

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48,34-40 (1995).
[CrossRef]

L. Wang, S. L. Jacques, and L. Zhen, "MCML - Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47,131-146 (1995).
[CrossRef] [PubMed]

1994

1993

1988

1983

Adam, G.

B. C. Wilson and G. Adam, "A Monte Carlo model for the absorption and flux distributions of light in tissue," Med. Phys. 10, 824-830 (1983).
[CrossRef] [PubMed]

Alcouffe, R. E.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Comparison of finite difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues," Phys. Med. Biol. 43, 1285-1302 (1998).
[CrossRef] [PubMed]

Alexandrakis, G.

Arridge, S. R.

S. R. Arridge, "Optical tomography in medical imaging," Inv. Probl. 15,R41-R93 (1999).
[CrossRef]

Bal, G.

G. Bal and Y. Maday, "Coupling of transport and diffusion models in linear transport theory," Math. Model Numer. Anal. 36,69-86 (2002).
[CrossRef]

Barbour, R. L.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Comparison of finite difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues," Phys. Med. Biol. 43, 1285-1302 (1998).
[CrossRef] [PubMed]

Boas, D. A.

Brooks, D. H.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Signal Process Mag.57-75 (2001).
[CrossRef]

Chance, B.

A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain function," Trends Neurosci. 20,435-442 (1997).
[CrossRef] [PubMed]

T. Durduran, A. G. Yodh, B. Chance, and D. A. Boas, "Does the photon-diffusion coefficient depend on absorption," J. Opt. Soc. Am. A 14,3358-3365 (1997).
[CrossRef]

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48,34-40 (1995).
[CrossRef]

Chen, B. Q.

Contini, D.

Culver, J. P.

Delpy, D. T.

Dimarzio, C. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Signal Process Mag.57-75 (2001).
[CrossRef]

Dunn, A. K.

Durduran, T.

Fantini, S.

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, and P. M. Schlag, "Constrast features of breast cancer in frequency-domain laser scanning mammography," J. Biomed. Opt. 3,129-136 (1998)
[CrossRef]

Farrrell, T. J.

Feng, T. C.

Ferwerda, H. A.

Franceschini, M. A.

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, and P. M. Schlag, "Constrast features of breast cancer in frequency-domain laser scanning mammography," J. Biomed. Opt. 3,129-136 (1998)
[CrossRef]

Gaudette, R. J.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Signal Process Mag.57-75 (2001).
[CrossRef]

Groenhuis, R. A. J.

Haskell, R. C.

Hayashi, T.

Hielscher, A. H.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Comparison of finite difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues," Phys. Med. Biol. 43, 1285-1302 (1998).
[CrossRef] [PubMed]

Jacques, S. L.

L. Wang, S. L. Jacques, and L. Zhen, "MCML - Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47,131-146 (1995).
[CrossRef] [PubMed]

L. Wang and S. L. Jacques, "Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media," J. Opt. Soc. Am. A 10,1746-1752 (1993).
[CrossRef]

Jess, H.

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, and P. M. Schlag, "Constrast features of breast cancer in frequency-domain laser scanning mammography," J. Biomed. Opt. 3,129-136 (1998)
[CrossRef]

Kaipio, J.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. Kaipio, "Finite element model for the coupled radiative transfer equation and diffusion approximation," Int. J. Numer. Methods Eng., 65,383-405 (2006).
[CrossRef]

Kaschke, M.

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, and P. M. Schlag, "Constrast features of breast cancer in frequency-domain laser scanning mammography," J. Biomed. Opt. 3,129-136 (1998)
[CrossRef]

Kashio, Y.

Keijzer, M.

Kilmer, M.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Signal Process Mag.57-75 (2001).
[CrossRef]

Kolehmainen, V.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. Kaipio, "Finite element model for the coupled radiative transfer equation and diffusion approximation," Int. J. Numer. Methods Eng., 65,383-405 (2006).
[CrossRef]

Maday, Y.

G. Bal and Y. Maday, "Coupling of transport and diffusion models in linear transport theory," Math. Model Numer. Anal. 36,69-86 (2002).
[CrossRef]

Martelli, F.

McAdams, M. S.

Miller, E. L.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Signal Process Mag.57-75 (2001).
[CrossRef]

Moesta, K. T.

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, and P. M. Schlag, "Constrast features of breast cancer in frequency-domain laser scanning mammography," J. Biomed. Opt. 3,129-136 (1998)
[CrossRef]

Okada, E.

Patterson, M. S.

Schlag, P. M.

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, and P. M. Schlag, "Constrast features of breast cancer in frequency-domain laser scanning mammography," J. Biomed. Opt. 3,129-136 (1998)
[CrossRef]

Spott, T.

Stamnes, J. J.

Stamnes, K.

Star, W. M.

Storchi, P. R. M.

Stott, J. J.

Svaasand, L. O.

Tarvainen, T.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. Kaipio, "Finite element model for the coupled radiative transfer equation and diffusion approximation," Int. J. Numer. Methods Eng., 65,383-405 (2006).
[CrossRef]

Ten Bosch, J. J.

Totkas, S.

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, and P. M. Schlag, "Constrast features of breast cancer in frequency-domain laser scanning mammography," J. Biomed. Opt. 3,129-136 (1998)
[CrossRef]

Tromberg, B. J.

Tsay, T. T.

Vauhkonen, M.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. Kaipio, "Finite element model for the coupled radiative transfer equation and diffusion approximation," Int. J. Numer. Methods Eng., 65,383-405 (2006).
[CrossRef]

Villringer, A.

A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain function," Trends Neurosci. 20,435-442 (1997).
[CrossRef] [PubMed]

Wang, L.

L. Wang, S. L. Jacques, and L. Zhen, "MCML - Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47,131-146 (1995).
[CrossRef] [PubMed]

L. Wang and S. L. Jacques, "Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media," J. Opt. Soc. Am. A 10,1746-1752 (1993).
[CrossRef]

Wilson, B. C.

B. C. Wilson and G. Adam, "A Monte Carlo model for the absorption and flux distributions of light in tissue," Med. Phys. 10, 824-830 (1983).
[CrossRef] [PubMed]

Yodh, A.

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48,34-40 (1995).
[CrossRef]

Yodh, A. G.

Zaccanti, G.

Zhang, Q.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Signal Process Mag.57-75 (2001).
[CrossRef]

Zhen, L.

L. Wang, S. L. Jacques, and L. Zhen, "MCML - Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47,131-146 (1995).
[CrossRef] [PubMed]

Appl. Opt.

Comput. Methods Programs Biomed.

L. Wang, S. L. Jacques, and L. Zhen, "MCML - Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47,131-146 (1995).
[CrossRef] [PubMed]

IEEE Signal Process Mag.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Signal Process Mag.57-75 (2001).
[CrossRef]

Int. J. Numer. Methods Eng.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. Kaipio, "Finite element model for the coupled radiative transfer equation and diffusion approximation," Int. J. Numer. Methods Eng., 65,383-405 (2006).
[CrossRef]

Inv. Probl.

S. R. Arridge, "Optical tomography in medical imaging," Inv. Probl. 15,R41-R93 (1999).
[CrossRef]

J. Biomed. Opt.

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, and P. M. Schlag, "Constrast features of breast cancer in frequency-domain laser scanning mammography," J. Biomed. Opt. 3,129-136 (1998)
[CrossRef]

J. Opt. Soc. Am. A

Math. Model Numer. Anal.

G. Bal and Y. Maday, "Coupling of transport and diffusion models in linear transport theory," Math. Model Numer. Anal. 36,69-86 (2002).
[CrossRef]

Med. Phys.

B. C. Wilson and G. Adam, "A Monte Carlo model for the absorption and flux distributions of light in tissue," Med. Phys. 10, 824-830 (1983).
[CrossRef] [PubMed]

Opt. Express

Phys. Med. Biol.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Comparison of finite difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues," Phys. Med. Biol. 43, 1285-1302 (1998).
[CrossRef] [PubMed]

Phys. Today

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48,34-40 (1995).
[CrossRef]

Trends Neurosci.

A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain function," Trends Neurosci. 20,435-442 (1997).
[CrossRef] [PubMed]

Other

A. Ishimaru, Wave propagation and scattering in random media, (Academic Press, New York 1978).

S. C. Brenner and L. R. Scott, Mathematical Theory of Finite Element Methods, Texts in Appl. Math. 15, (New York, Springer-Verlag 1994).

P. G. Cialet, Finite Element Method for Elliptic Problems, (North-Holland 1978).

J. M. Jin, Finite Element Method in Electromagnetics, 2nd ed. (New York, John Wiley & Sons 2002).

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Figures (7)

Fig. 1.
Fig. 1.

A schematic diagram of the photon transportation in a turbid medium and the detection of the diffuse light on the surface.

Fig. 2.
Fig. 2.

The pure MC model and the conventional MC diffusion hybrid model are used to evaluate the diffuse intensity on the surface of a semi-infinite homogeneous medium, and the results by the two models are compared. This figure shows the relative deviation of the surface diffuse intensity by the conventional MC diffusion hybrid model (from the pure MC model). The geometry of the setup is shown in Fig. 1. The optical parameters of the medium are: n = 1.35, g = 0.8, μs = 49.4cm-1, μa = 0.6cm-1. In the hybrid model in this example, the surface diffuse intensity in the region ρ ≥ 0.45cm is evaluated with the finite element method based on the conventional diffusion equation.

Fig. 3.
Fig. 3.

A thin beam (e.g., a Gaussian beam) of light incidents orthogonally onto the surface (the plane marked with “ABCD”) of a semi-infinite turbid medium. V 1 is a cubic volume (whose surface is square ABCD) of the medium. V 2 is a small cube (whose surface is square EFGH) included in V 1. Square E’F’G’H’ which includes EFGH is slightly larger than EFGH. Square ABCD, EFGH and E’F’G’H’ are all centered at the incident point O. MC simulation is used to evaluate the diffuse intensity in V 2 and on square E’F’G’H’. Finite element method is used to solve the diffusion equation in the volume of V 1 with the small cube V 2 excavated.

Fig. 4.
Fig. 4.

The relative deviations (from the result of the pure MC simulation) of the diffuse intensity on the positive x-axis obtained by Conventional Hybrids I-III

Fig. 5.
Fig. 5.

Four MC diffusion hybrid models (Conventional Hybrids I-III and Present Hybrid) are used to evaluate the diffuse intensity on the surface of semi-infinite homogeneous media, and their results are compared with the pure MC simulation. The deviations are shown on the x axis in the region {(x, 0, 0) ∈ Γ1 ∪ Γ2}. Comparisons are done for four sets of medium parameters: (a) n = 1.35, g = 0.8, μs = 49.4cm-1, μa = 0.6cm-1; (b) n = 1.35, g = 0.9, μs = 99.4cm-1, μa = 0.6cm-1; (c) n = 1.35, g = 0.7, μs = 49.1cm-1, μa = 0.9cm-1; (d) n = 1.35, g = 0.8, μt = 99.9cm-1, μa = 0.1cm-1.

Fig. 6.
Fig. 6.

Five models (Conventional Pure, Present Pure, Conventional Hybrid III, Present Hybrid, and the pure MC) are used to calculate the diffuse intensity on the surface of a semi-infinite inhomogeneous medium with cylindrical symmetry, and the results of the first four models are compared with the fifth model (the pure MC simulation). In the two hybrid models, the surface diffuse distribution in the region x ≤ 0.4cm is evaluated with MC simulation.

Fig. 7.
Fig. 7.

(a). The deviations of the modified DAs in the pure diffusion model; (b). The deviations of the modified DAs in the MC diffusion hybrid model. The medium parameters are the same as used for Fig. 2, and the waist radius of the incident Gaussian beam is 0.03cm.

Tables (1)

Tables Icon

Table 1. Nomenclature (used in Figs. 4–7) for the diffusion models

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

[ ( 1 3 μ t ( r ) ) + μ a ( r ) ] U d ( r ) = 0 , r Ω
μ a ( r ) U d ( r ) + 1 4 π F d ( r ) = 0
1 4 μ μ t ( r ) F d ( r ) + 1 3 U d ( r ) = 0
U d ( r ) = A 2 π n F d ( r ) , r S
A = 1 + 3 π 0 π 2 R ( θ ) cos 2 ( θ ) sin ( θ ) d θ 1 2 0 π 2 R ( θ ) cos ( θ ) sin ( θ ) d θ
J d r s U d ( r ) + 3 4 π F d ( r ) s
{ Ω = V 1 V 2 , volume of V 1 with V 2 excavated Γ 0 = V 1 ABCD , boundary area of V 1 except ABCD Γ 1 = ABCD E F G H , area of ABCD with E F G H excavated Γ 2 = E F G H EFGH , area of E F G H with EFGH excavated Γ 3 = Ω ˉ V 2 = V 2 EFGH , boundary area of V 2 except EFGH
{ U d ( r ) = 0 , r Γ 0 1 3 μ t ( r ) n U d ( r ) + 1 2 A U d ( r ) = 0 , r Γ 1 Γ 2 1 3 μ t ( r ) n U d ( r ) = 1 4 π n F d ( r ) , r Γ 3
1 3 μ t ( r ) n U d ( r ) = 1 4 π n F d ( r ) , r Γ 2
U d ( r ) = α ( r ) , r Γ 2
{ square EFGH = { ( x , y , z ) x 0.4380 cm , y 0.4380 cm , z = 0 } , square E F G H = { ( x , y , z ) x 0.4517 cm , y 0.4517 cm , z = 0 }
DEV Hybrid I , II , III ( x ) = sfUd Hybrid I , II , III ( x ) sfUd MC ( x ) 1
1 4 π μ t ( r ) F d ( r ) + 1 + η 1 ( r ) 3 U d ( r ) = 0
U d ( r ) = [ 1 + η 2 ( r ) ] A 2 π n F d ( r ) , r S
{ η 1 ( r ) c 1 ( n ) μ a ( r ) μ t ( r ) exp [ 0 z μ t ( x , y , z ) d z ] , η 2 ( r ) c 2 ( n ) μ a ( r ) μ t ( r ) , r S
[ ( D ( r ) ) + μ a ( r ) ] U d ( r ) = 0 , r Ω
{ U d ( r ) = 0 , r Γ 0 D ( r ) n U d ( r ) + 1 2 [ 1 + η 2 ( r ) ] A U d ( r ) = 0 , r Γ 1 U d ( r ) = α ( r ) , r Γ 2 D ( r ) n U d ( r ) = 1 4 π n F d ( r ) , r Γ 3
D ( r ) = 1 + η 1 ( r ) 3 μ t ( r )
{ D ( r ) = { 1 + 3.14 μ a ( r ) μ t ( r ) exp [ 0 z ( x , y , z ) d z ] } [ 3 μ t ( r ) ] U d ( r ) = 1 2 π [ 1 + 2.2 μ a ( r ) μ t ( r ) ] A n F d ( r ) , r S
Ω [ D ( r ) ϕ ( r ) U d ( r ) + μ a ( r ) ϕ ( r ) U d ( r ) ] d V + Γ 1 1 2 [ 1 + η 2 ( r ) ] A ϕ ( r ) U d ( r ) d σ = Γ 3 ϕ ( r ) 1 4 π ( n ) . F d ( r ) d σ
{ D ( r ) = [ 3 μ s ( r ) ] 1 U d ( r ) = 1 2 π A n F d ( r ) , r S
{ D ( r ) = [ 3 μ s ( r ) ] 1 U d ( r ) = 1 2 π [ 1 + 2.2 μ a ( r ) μ t ( r ) ] A n F d ( r ) , r S
[ ( D ( r ) ) + μ a ( r ) ] U d ( r ) = S ( r ) , r V 1
{ U d ( r ) = 0 , r Γ 0 D ( r ) n U d ( r ) + 1 2 [ 1 + η 2 ( r ) ] A U d ( r ) = 0 , r s q u a r e ABCD
{ n 1.35 , g 0.8 , μ s 49.4 cm 1 , μ a ( x , y , z ) = { [ 0.6 + 0.2 cos ( ( r 0.4 ) π ) ] cm 1 , r < 2.0 cm 0.4 cm 1 , r 2.0 cm

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