Abstract

We present an investigation of the effect of a 3D non-scattering gap region on image reconstruction in diffuse optical tomography. The void gap is modelled by the Radiosity-Diffusion method and the inverse problem is solved using the adjoint field method. The case of a sphere with concentric spherical gap is used as an example.

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References

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  1. S. R. Arridge, "Optical Tomography in Medical Imaging," Inverse Problems 15, R41-R93 (1999).
    [CrossRef]
  2. S. R. Arridge and J. C. Hebden, "Optical Imaging in Medicine: II. Modelling and Reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
    [CrossRef] [PubMed]
  3. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Comparison of Finite-Difference Transport and Diffusion Calculations for Photon Migration in Homogeneous and Hetergeneous Tissue," Phys. Med. Biol. 43, 1285-1302 (1998).
    [CrossRef] [PubMed]
  4. O. Dorn, "A Transport-BackTransport Method for Optical Tomography," Inverse Problems 14, 1107-1130 (1998).
    [CrossRef]
  5. A. D. Klose, A. H. Hielscher, "Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer," Med. Phys. 26 1698-1707 (1999).
    [CrossRef] [PubMed]
  6. A. D. Klose, A. H. Hielscher, "Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer," Med. Phys. 26 1698-1707 (1999).
  7. O. Dorn, "Scattering and absorption transport sensitivity functions for optical tomography," Opt. Express 7, 492-506 (2000), http://www.opticsexpress.org/oearchive/source/26901.htm.
    [CrossRef] [PubMed]
  8. M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, "An investigation of light transport through scattering bodies with non-scattering regions," Phys. Med. Biol. 41, 767-783 (1996).
    [CrossRef] [PubMed]
  9. S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, "The Finite Element Model for the Propagation of Light in Scattering Media : A Direct Method for Domains with Non-Scattering Regions," Med. Phys. 27, 252-264 (2000).
    [CrossRef]
  10. J. Ripoll, S. R. Arridge, H. Dehghani, and M. Nieto-Vesperinas, "Boundary conditions for light propagation in diffusive media with nonscattering regions," J. Opt. Soc. Am. A 17, 1671-1681 (2000).
    [CrossRef]
  11. H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, "Optical Tomography in the Presence of Void Regions," J. Opt. Soc. Am. A 17, 1659-1670 (2000).
    [CrossRef]
  12. M. Schweiger and S. R. Arridge, "Comparison of 2D and 3D reconstruction algorithms in Optical Tomography," Appl. Opt. 37, 7419-7428 (1998).
  13. J. Ripoll, Ph.D. thesis, University Autonoma of Madrid, 2000.
    [CrossRef] [PubMed]
  14. S. R. Arridge and M. Schweiger, "The Finite Element Model for the Propagation of Light In Scattering Media: Boundary and Source Conditions," Med. Phys. 22, 1779-1792, (1995).
    [CrossRef]
  15. F. Natterer and F. Wubbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, 2001).
  16. J. Schoberl, "NetGen", http://www.sfb013.uni-linz.ac.at/ joachim/netgen/
  17. M. F. Cohen and J. R. Wallace, Radiosity and Realistic Image Synthesis (Academic, London, 1993).
  18. H. R. Zatz, Master's thesis, Cornell University, 1993.
    [CrossRef] [PubMed]
  19. S. R. Arridge and M. Schweiger, "Photon Measurement Density Functions. Part 2: Finite Element Calculations," Appl. Opt. 34, 8026-8037 (1995).
  20. V. Kolehmainen, M. Vaukhonen, J. P. Kaipio and S. R. Arridge, "Recovery of piecewise constant coefficients in optical diffusion tomography," Opt. Express 7, 468-480 (2000), http://www.opticsexpress.org/oearchive/source/24842.htm

Other

S. R. Arridge, "Optical Tomography in Medical Imaging," Inverse Problems 15, R41-R93 (1999).
[CrossRef]

S. R. Arridge and J. C. Hebden, "Optical Imaging in Medicine: II. Modelling and Reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Comparison of Finite-Difference Transport and Diffusion Calculations for Photon Migration in Homogeneous and Hetergeneous Tissue," Phys. Med. Biol. 43, 1285-1302 (1998).
[CrossRef] [PubMed]

O. Dorn, "A Transport-BackTransport Method for Optical Tomography," Inverse Problems 14, 1107-1130 (1998).
[CrossRef]

A. D. Klose, A. H. Hielscher, "Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer," Med. Phys. 26 1698-1707 (1999).
[CrossRef] [PubMed]

A. D. Klose, A. H. Hielscher, "Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer," Med. Phys. 26 1698-1707 (1999).

O. Dorn, "Scattering and absorption transport sensitivity functions for optical tomography," Opt. Express 7, 492-506 (2000), http://www.opticsexpress.org/oearchive/source/26901.htm.
[CrossRef] [PubMed]

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

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, "The Finite Element Model for the Propagation of Light in Scattering Media : A Direct Method for Domains with Non-Scattering Regions," Med. Phys. 27, 252-264 (2000).
[CrossRef]

J. Ripoll, S. R. Arridge, H. Dehghani, and M. Nieto-Vesperinas, "Boundary conditions for light propagation in diffusive media with nonscattering regions," J. Opt. Soc. Am. A 17, 1671-1681 (2000).
[CrossRef]

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, "Optical Tomography in the Presence of Void Regions," J. Opt. Soc. Am. A 17, 1659-1670 (2000).
[CrossRef]

M. Schweiger and S. R. Arridge, "Comparison of 2D and 3D reconstruction algorithms in Optical Tomography," Appl. Opt. 37, 7419-7428 (1998).

J. Ripoll, Ph.D. thesis, University Autonoma of Madrid, 2000.
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, "The Finite Element Model for the Propagation of Light In Scattering Media: Boundary and Source Conditions," Med. Phys. 22, 1779-1792, (1995).
[CrossRef]

F. Natterer and F. Wubbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, 2001).

J. Schoberl, "NetGen", http://www.sfb013.uni-linz.ac.at/ joachim/netgen/

M. F. Cohen and J. R. Wallace, Radiosity and Realistic Image Synthesis (Academic, London, 1993).

H. R. Zatz, Master's thesis, Cornell University, 1993.
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, "Photon Measurement Density Functions. Part 2: Finite Element Calculations," Appl. Opt. 34, 8026-8037 (1995).

V. Kolehmainen, M. Vaukhonen, J. P. Kaipio and S. R. Arridge, "Recovery of piecewise constant coefficients in optical diffusion tomography," Opt. Express 7, 468-480 (2000), http://www.opticsexpress.org/oearchive/source/24842.htm

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Figures (4)

Fig. 1.
Fig. 1.

Left : cutaway of spherical mesh. Right : location of sources and detectors on the sphere surface

Fig. 2.
Fig. 2.

distribution of photon density, photons/mm3 (top row) and mean time of photon flight, picoseconds (bottom row) over sphere surface. Left to right, solid sphere (no gap), gap widths 3mm, 4mm, 5mm.

Fig. 3.
Fig. 3.

Sensitivity functions for the 3mm gap case. Left intensity (photons/mm2), right mean time(picoseconds mm). The functions plotted are cross-sections through the equatorial plane of the sphere. Also available as a QuickTime movie, pmdf.mov. (3.8MB)

Fig. 4.
Fig. 4.

Target images (top row) and reconstructions (bottom row) for the 3mm gap case. The images are transverse, sagittal and coronal slices through the true centre of the blob, orientated according to the diagram in the top right panel. Bottom right shows a profile along the equatorial diameter through the blob centre. A movie showing a rotating orthographic view is attached rotating3d.mov (3.8MB).

Equations (14)

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· κ ( r ) Φ ( r ; ω ) + ( µ a ( r ) + i ω c ) Φ ( r ; ω ) = 0 r Ω d ( Ξ d Ω d )
Φ ( m ; ω ) + 2 A κ ( m ) Φ ( m ; ω ) ν = η ( m ; ω ) m Ω 1 +
Φ ( m ; ω ) + 2 A κ Φ ( m ; ω ) ν = 1 π Ξ cos θ cos θ Φ ( m ; ω ) 2 A h ( m , m ) ×
exp [ ( μ a + i ω c ) m m ] m m 2 d m m , m Ξ
cos θ = ν ̂ ( m ) · m m m m , cos θ = ν ̂ ( m ) · m m m m
y η ( m ; ω ) = 𝓟 η ( μ a κ ) κ ( m ) Φ η ( m ; ω ) ν , m Ω 1 +
C = 1 2 j = 1 s g j 𝓟 j μ a κ , g j 𝓟 j ( μ a κ ) L 2 ( Ω )
( α β ) = ( Re ( Φ j - Ψ j ) Re ( Φ j - · Ψ j ) )
· κ ( r ) Ψ j ( r ; ω ) + ( μ a ( r ) i ω c ) Ψ j ( r ; ω ) = 0 r Ω d \ ( Ξ d Ω d )
Ψ j ( m ; ω ) + 2 A κ ( m ) Ψ j ( m ; ω ) ν = g j ( m ; ω ) 𝓟 j ( μ a κ ) m Ω 1 +
Ψ j ( m ; ω ) + 2 A κ Ψ j ( m ; ω ) ν = 1 π Ξ cos θ cos θ ' Ψ j ( m ; ω ) 2 A h ( m , m ) x
exp [ ( μ a i ω c ) m m ] m m 2 d m m , m Ξ
f ( m , m ) = cos θ ( m ) cos θ ( m ) π m m 2
f τ α , τ α ( m , m ) = k = 1 N k = 1 N u n ( k ) ( m ) u n ( k ) ( m ) cos θ n ( k ) cos θ n ( k ) π N n ( k ) N n ( k ) 2

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