Abstract

We present a computational study of diffuse optical tomography using the one-way radiative transfer equation. The one-way radiative transfer is a simplification of the radiative transfer equation to approximate the transmission of light through tissues. The major simplification of this approximation is that the intensity satisfies an initial value problem rather than a boundary value problem. Consequently, the inverse problem to reconstruct the absorption and scattering coefficients from transmission measurements of scattered light is simplified. Using the initial value problem for the one-way radiative transfer equation to compute the forward model, we are able to quantitatively reconstruct the absorption and scattering coefficients efficiently and effectively for simple problems and obtain reasonable results for complicated problems.

© 2015 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1996).
  2. L. V. Wang and H.-I. Wu, Biomedical Optics (Wiley, 2007).
  3. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
    [Crossref]
  4. S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
    [Crossref]
  5. P. González-Rodríguez, A. Kim, and M. Moscoso, “Robust depth selectivity in mesoscopic scattering regimes using angle-resolved measurements,” Opt. Lett. 38, 787–789 (2013).
    [Crossref] [PubMed]
  6. O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998).
    [Crossref]
  7. A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698–1707 (1999).
    [Crossref] [PubMed]
  8. K. Ren, G. Bal, and A. H. Hielscher, “Frequency Domain Optical Tomography Based on the Equation of Radiative Transfer,” SIAM J. Sci. Comput. 28, 1463–1489 (2006).
    [Crossref]
  9. G. Bal, “Inverse transport theory and applications,” Inverse Probl. 25, 053001 (2009).
    [Crossref]
  10. H. K. Kim and A. H. Hielscher, “A diffusion-transport hybrid method for accelerating optical tomography,” J. Innov. Opt. Health Sci. 3, 1–13 (2010).
    [Crossref]
  11. T. Tarvainen, V. Kolehmainen, S. R. Arrdige, and J. P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model,” J. Quant. Spectrosc. Radiat. Transf. 112, 2600–2608 (2011).
    [Crossref]
  12. P. González-Rodríguez, B. Ilan, and A. D. Kim, “The one-way radiative transfer equation,” submitted for publication.
  13. P. González-Rodríguez and A. Kim, “Comparison of light scattering models for diffuse optical tomography,” Opt. Express 17, 8756–8774 (2009).
    [Crossref] [PubMed]
  14. P. González-Rodríguez, O. Dorn, M. Kindelan, and M. Moscoso, “History matching problem in reservoir engineering using the propagation-backpropagation method,” Inverse Probl. 21, 565–590 (2005).
    [Crossref]
  15. E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, La Grange Park, IL, 1993)
  16. H. Gao and H. Zhao, “A fast forward solver of radiative transfer equation,” Transport Theory Statist. Phys. 38, 149–192 (2009).
    [Crossref]

2013 (1)

2011 (1)

T. Tarvainen, V. Kolehmainen, S. R. Arrdige, and J. P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model,” J. Quant. Spectrosc. Radiat. Transf. 112, 2600–2608 (2011).
[Crossref]

2010 (1)

H. K. Kim and A. H. Hielscher, “A diffusion-transport hybrid method for accelerating optical tomography,” J. Innov. Opt. Health Sci. 3, 1–13 (2010).
[Crossref]

2009 (4)

G. Bal, “Inverse transport theory and applications,” Inverse Probl. 25, 053001 (2009).
[Crossref]

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[Crossref]

P. González-Rodríguez and A. Kim, “Comparison of light scattering models for diffuse optical tomography,” Opt. Express 17, 8756–8774 (2009).
[Crossref] [PubMed]

H. Gao and H. Zhao, “A fast forward solver of radiative transfer equation,” Transport Theory Statist. Phys. 38, 149–192 (2009).
[Crossref]

2006 (1)

K. Ren, G. Bal, and A. H. Hielscher, “Frequency Domain Optical Tomography Based on the Equation of Radiative Transfer,” SIAM J. Sci. Comput. 28, 1463–1489 (2006).
[Crossref]

2005 (1)

P. González-Rodríguez, O. Dorn, M. Kindelan, and M. Moscoso, “History matching problem in reservoir engineering using the propagation-backpropagation method,” Inverse Probl. 21, 565–590 (2005).
[Crossref]

1999 (2)

A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698–1707 (1999).
[Crossref] [PubMed]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[Crossref]

1998 (1)

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998).
[Crossref]

Arrdige, S. R.

T. Tarvainen, V. Kolehmainen, S. R. Arrdige, and J. P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model,” J. Quant. Spectrosc. Radiat. Transf. 112, 2600–2608 (2011).
[Crossref]

Arridge, S. R.

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[Crossref]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[Crossref]

Bal, G.

G. Bal, “Inverse transport theory and applications,” Inverse Probl. 25, 053001 (2009).
[Crossref]

K. Ren, G. Bal, and A. H. Hielscher, “Frequency Domain Optical Tomography Based on the Equation of Radiative Transfer,” SIAM J. Sci. Comput. 28, 1463–1489 (2006).
[Crossref]

Dorn, O.

P. González-Rodríguez, O. Dorn, M. Kindelan, and M. Moscoso, “History matching problem in reservoir engineering using the propagation-backpropagation method,” Inverse Probl. 21, 565–590 (2005).
[Crossref]

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998).
[Crossref]

Gao, H.

H. Gao and H. Zhao, “A fast forward solver of radiative transfer equation,” Transport Theory Statist. Phys. 38, 149–192 (2009).
[Crossref]

González-Rodríguez, P.

P. González-Rodríguez, A. Kim, and M. Moscoso, “Robust depth selectivity in mesoscopic scattering regimes using angle-resolved measurements,” Opt. Lett. 38, 787–789 (2013).
[Crossref] [PubMed]

P. González-Rodríguez and A. Kim, “Comparison of light scattering models for diffuse optical tomography,” Opt. Express 17, 8756–8774 (2009).
[Crossref] [PubMed]

P. González-Rodríguez, O. Dorn, M. Kindelan, and M. Moscoso, “History matching problem in reservoir engineering using the propagation-backpropagation method,” Inverse Probl. 21, 565–590 (2005).
[Crossref]

P. González-Rodríguez, B. Ilan, and A. D. Kim, “The one-way radiative transfer equation,” submitted for publication.

Hielscher, A. H.

H. K. Kim and A. H. Hielscher, “A diffusion-transport hybrid method for accelerating optical tomography,” J. Innov. Opt. Health Sci. 3, 1–13 (2010).
[Crossref]

K. Ren, G. Bal, and A. H. Hielscher, “Frequency Domain Optical Tomography Based on the Equation of Radiative Transfer,” SIAM J. Sci. Comput. 28, 1463–1489 (2006).
[Crossref]

A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698–1707 (1999).
[Crossref] [PubMed]

Ilan, B.

P. González-Rodríguez, B. Ilan, and A. D. Kim, “The one-way radiative transfer equation,” submitted for publication.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1996).

Kaipio, J. P.

T. Tarvainen, V. Kolehmainen, S. R. Arrdige, and J. P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model,” J. Quant. Spectrosc. Radiat. Transf. 112, 2600–2608 (2011).
[Crossref]

Kim, A.

Kim, A. D.

P. González-Rodríguez, B. Ilan, and A. D. Kim, “The one-way radiative transfer equation,” submitted for publication.

Kim, H. K.

H. K. Kim and A. H. Hielscher, “A diffusion-transport hybrid method for accelerating optical tomography,” J. Innov. Opt. Health Sci. 3, 1–13 (2010).
[Crossref]

Kindelan, M.

P. González-Rodríguez, O. Dorn, M. Kindelan, and M. Moscoso, “History matching problem in reservoir engineering using the propagation-backpropagation method,” Inverse Probl. 21, 565–590 (2005).
[Crossref]

Klose, A. D.

A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698–1707 (1999).
[Crossref] [PubMed]

Kolehmainen, V.

T. Tarvainen, V. Kolehmainen, S. R. Arrdige, and J. P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model,” J. Quant. Spectrosc. Radiat. Transf. 112, 2600–2608 (2011).
[Crossref]

Lewis, E. E.

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, La Grange Park, IL, 1993)

Miller, W. F.

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, La Grange Park, IL, 1993)

Moscoso, M.

P. González-Rodríguez, A. Kim, and M. Moscoso, “Robust depth selectivity in mesoscopic scattering regimes using angle-resolved measurements,” Opt. Lett. 38, 787–789 (2013).
[Crossref] [PubMed]

P. González-Rodríguez, O. Dorn, M. Kindelan, and M. Moscoso, “History matching problem in reservoir engineering using the propagation-backpropagation method,” Inverse Probl. 21, 565–590 (2005).
[Crossref]

Ren, K.

K. Ren, G. Bal, and A. H. Hielscher, “Frequency Domain Optical Tomography Based on the Equation of Radiative Transfer,” SIAM J. Sci. Comput. 28, 1463–1489 (2006).
[Crossref]

Schotland, J. C.

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[Crossref]

Tarvainen, T.

T. Tarvainen, V. Kolehmainen, S. R. Arrdige, and J. P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model,” J. Quant. Spectrosc. Radiat. Transf. 112, 2600–2608 (2011).
[Crossref]

Wang, L. V.

L. V. Wang and H.-I. Wu, Biomedical Optics (Wiley, 2007).

Wu, H.-I.

L. V. Wang and H.-I. Wu, Biomedical Optics (Wiley, 2007).

Zhao, H.

H. Gao and H. Zhao, “A fast forward solver of radiative transfer equation,” Transport Theory Statist. Phys. 38, 149–192 (2009).
[Crossref]

Inverse Probl. (5)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[Crossref]

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[Crossref]

G. Bal, “Inverse transport theory and applications,” Inverse Probl. 25, 053001 (2009).
[Crossref]

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998).
[Crossref]

P. González-Rodríguez, O. Dorn, M. Kindelan, and M. Moscoso, “History matching problem in reservoir engineering using the propagation-backpropagation method,” Inverse Probl. 21, 565–590 (2005).
[Crossref]

J. Innov. Opt. Health Sci. (1)

H. K. Kim and A. H. Hielscher, “A diffusion-transport hybrid method for accelerating optical tomography,” J. Innov. Opt. Health Sci. 3, 1–13 (2010).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (1)

T. Tarvainen, V. Kolehmainen, S. R. Arrdige, and J. P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model,” J. Quant. Spectrosc. Radiat. Transf. 112, 2600–2608 (2011).
[Crossref]

Med. Phys. (1)

A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698–1707 (1999).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

SIAM J. Sci. Comput. (1)

K. Ren, G. Bal, and A. H. Hielscher, “Frequency Domain Optical Tomography Based on the Equation of Radiative Transfer,” SIAM J. Sci. Comput. 28, 1463–1489 (2006).
[Crossref]

Transport Theory Statist. Phys. (1)

H. Gao and H. Zhao, “A fast forward solver of radiative transfer equation,” Transport Theory Statist. Phys. 38, 149–192 (2009).
[Crossref]

Other (4)

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, La Grange Park, IL, 1993)

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1996).

L. V. Wang and H.-I. Wu, Biomedical Optics (Wiley, 2007).

P. González-Rodríguez, B. Ilan, and A. D. Kim, “The one-way radiative transfer equation,” submitted for publication.

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Figures (6)

Fig. 1
Fig. 1 The two-dimensional imaging system we are considering here. An intensity modulated plane wave is incident on the source plane, y = 0. We measure the transmittance on the detector plane, y = y0, due to a slab containing a disk, each of which is composed of a uniform absorbing and scattering medium except for a scattering obstacle and an absorbing obstacle in the disk. We measure multiple transmittance measurements by rotating the disk and seek to reconstruct quantitative images of the scatterer and absorber using those measurements.
Fig. 2
Fig. 2 The “mass,” Mf (θ), defined in Eq. (4.6), for the Henyey-Greenstein scattering phase function defined in (3.2) computed with respect to θ = π/2.
Fig. 3
Fig. 3 Reconstructions of a sample containing one absorber and one scatterer.
Fig. 4
Fig. 4 Reconstructions of a sample containing two absorbers and one scatterer situated in distinct regions in the sample.
Fig. 5
Fig. 5 Reconstructions of a sample containing two absorbers and one scatterer situated with overlapping regions in the sample.
Fig. 6
Fig. 6 Reconstructions using the heat kernel for regularization of the sample shown in Fig. 5.

Equations (62)

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sin θ I y + cos θ I x + i ω c I + μ t I μ s 0 2 π p ( θ θ ) I ( θ , x , y ) d θ = 0 in 0 < y < y 0 ,
I ( θ , x , 0 ) = I 0 δ ( θ π / 2 ) on 0 < θ < π ,
I ( θ , x , y 0 ) = 0 on π < θ < 2 π .
p ( θ θ ) = 1 2 π 1 g 2 1 + g 2 2 g cos ( θ θ ) .
0 2 π p ( θ θ ) d θ = 1.
T ( x ) = N A I ( θ , x , y 0 ) sin θ d θ ,
I + ( θ , x , y ) = I ( θ , x , y ) , 0 < θ < π ,
I ( θ , x , y ) = I ( θ + π , x , y ) , 0 < θ < π .
sin θ I + y + cos θ I + x + i ω c I + + μ t I + μ s P | f I + = μ s P b I ,
sin θ I y cos θ I x + i ω c I + μ t I μ s P f I = μ s P b I + ,
I + ( θ , x , 0 ) = I 0 δ ( θ π / 2 ) ,
I ( θ , x , y 0 ) = 0.
P f [ ] = 0 π p ( θ θ ) [ ] d θ ,
P f [ ] = 0 π p ( θ θ ) [ ] d θ ,
T ( x ) = N A I + ( θ , x , y 0 ) sin θ d θ .
M f ( θ ) = θ π / 2 θ + π / 2 p ( θ θ ) d θ .
P f I ± P b I .
sin θ I + y + cos θ I + x + ( μ t + i ω c ) I + μ s P f I + = 0.
sin θ I ˜ + y cos θ I ˜ + x + ( μ t + i ω c ) I ˜ + μ s P f I ˜ + = 0 ,
I ˜ + ( θ , x , y 0 ) = I ˜ 0 ( θ , x ) .
sin θ I y + cos θ I x + ( μ t + i ω c ) I μ s P f I = q in 0 < y y 0 ,
I = f ( θ , x ) on y = 0 ,
sin θ I ˜ y cos θ I ˜ x + ( μ t + i ω c ) I ˜ μ s P f I ˜ = q ˜ in 0 < y y 0 ,
I ˜ = f ˜ ( θ , x ) on y = y 0 .
1 1 f ( x ) d x m = 1 2 M + 1 f ( x m ) W m ,
P f 2 M + 1 I ( θ , x , y ) = π 2 m = 1 2 M + 1 p ( θ θ m ) I ( θ m , x , y ) W m ,
sin θ m I m y + cos θ m I m x + ( μ t + i ω c ) I m μ s P f 2 M + 1 I m = q m ,
I m ( x , 0 ) = f ( θ m , x ) ,
sin θ m I ˜ m y cos θ m I ˜ m x + ( μ t + i ω c ) I ˜ m μ s P f 2 M + 1 I ˜ m = q ˜ m ,
I ˜ m ( x , y 0 ) = f ˜ ( θ m , x ) ,
sin θ m I m , i y + cos θ m ( I m , i + 1 I m , i 1 2 h ) + ( μ t + i ω c ) I m , i μ s , i P f 2 M + 1 I m , i = q m , i ,
I m , i ( 0 ) = f ( θ m , x i ) ,
sin θ m I ˜ m , i y + cos θ m ( I ˜ m , i + 1 I ˜ m , i 1 2 h ) + ( μ t + i ω c ) I ˜ m , i μ s , i P f 2 M + 1 I ˜ m , i = q ˜ m , i ,
I ˜ m , i ( y 0 ) = f ˜ ( θ m , x i ) ,
D x N = 1 2 h [ 0 1 1 1 0 1 1 0 1 1 1 0 ] ,
T c = diag [ cos θ 1 , cos θ 2 , , cos θ 2 M + 1 ] ,
T s = diag [ sin θ 1 , sin θ 2 , , sin θ 2 M + 1 ] .
A ( y ) = diag [ μ a ( x 1 , y ) , μ a ( x 2 , y ) , , μ a ( x N , y ) ] ,
S ( y ) = diag [ μ s ( x 1 , y ) , μ s ( x 2 , y ) , , μ s ( x N , y ) ] .
B d I ( y ) d y + C ( y ) I = q ( y ) , I ( 0 ) = f .
B = I N T s ,
C ( y ) = D x N T c + [ ( A ( y ) + S ( y ) + i ω c ) I 2 M + 1 ] + S ( y ) P f 2 M + 1 .
B d I ˜ ( y ) d y + C ˜ ( y ) I ˜ = q ˜ ( y ) , I ( y 0 ) = f ˜ ,
C ˜ ( y ) = D x N T c + [ ( A ( y ) + S ( y ) i ω c ) I 2 M + 1 ] + S ( y ) P f 2 M + 1 .
[ B + Δ y 2 C ( y j + 1 ) ] I j + 1 = [ B Δ y 2 C ( y j ) ] I j + Δ y 2 [ q ( y j + 1 ) + q ( y j ) ] ,
[ B Δ y 2 C ˜ ( y j ) ] I ˜ j = [ B + Δ y 2 C ˜ ( y j + 1 ) ] I ˜ j + 1 + Δ y 2 [ q ˜ ( y j + 1 ) + q ( y ˜ j ) ] ,
sin θ m I ( k ) + y + cos θ I ( k ) + x + ( μ t + i ω c ) I ( k ) + μ s P f I ( k ) + = μ s P b I ( k 1 ) , I ( k ) + ( θ , x , 0 ) = I 0 δ ( θ π / 2 ) ,
sin θ m I ( k ) y cos θ I ( k ) x + ( μ t + i ω c ) I ( k ) μ s P f I ( k ) = μ s P b I ( k ) + , I ( k ) ( θ , x , y 0 ) = 0 ,
T ( k ) ( x i ) θ m N A I m , i ( k ) + ( y 0 ) sin θ m w m , i = 1 , , N .
K ( μ a , μ s ) = 1 2 R ( μ a , μ s ) L 2 2 ,
R ( μ a , μ s ) = m ( μ a , μ s ) d
μ a = μ ¯ a + δ μ a ,
μ s = μ ¯ s + δ μ s .
R ( μ a , μ s ) = R ( μ ¯ a , μ ¯ s ) + R ( μ ¯ a , μ ¯ s ) ( δ μ ¯ a , μ ¯ s ) ,
( δ μ a , δ μ s ) = R ˜ ( μ ¯ a , μ ¯ s ) [ R ( μ ¯ a , μ ¯ s ) R ˜ ( μ ¯ a , μ ¯ s ) ] 1 R ( μ ¯ a , μ ¯ s ) .
( δ μ a , δ μ s ) β R ˜ ( μ ¯ a , μ ¯ s ) R ( μ ¯ a , μ ¯ s ) ,
δ μ a ( x , y ) = β 0 2 π I ˜ + ( θ , x , y ) I + ( θ , x , y ) d θ ,
δ μ s ( x , y ) = β 0 2 π I ˜ + ( θ , x , y ) [ I + ( θ , x , y ) P f I + ( θ , x , y ) ] d θ .
sin θ I + y + cos θ I + x + ( μ ¯ a + μ ¯ s + i ω c ) I + μ ¯ s P f I + = 0 , in 0 < y y 0 I + = ( θ , x , 0 ) = I 0 δ ( θ π / 2 ) 0 < θ < π ,
ρ i = T one-way ( x i ; φ j ) T RTE ( x i ; φ j ) , i = 1 , , N s .
sin θ I ˜ + y cos θ I ˜ + y + ( μ ¯ a + μ ¯ s + i ω c ) I ˜ + μ ¯ s P f I ˜ + = 0 , in 0 y < y 0 I + ( θ , x , y 0 ) = { ρ * / Δ θ on x 0 / 2 x x 0 / 2 , a n d ( π Δ θ ) / 2 θ ( π + Δ θ ) / 2 , 0 otherwise ,
μ ¯ a μ ¯ a + δ μ a , μ ¯ s μ ¯ s + δ μ s .

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