Abstract

This paper presents a method to calculate photon-measurement density functions (PMDF’s), which were introduced in Part 1 [Appl. Opt. 34, 7395–7409 (1995), for near-infrared imaging and spectroscopy in complex and inhomogeneous objects through the use of a finite-element model. PMDF’s map the sensitivity of a measurement on the surface of an object to the perturbations of the optical parameters within the object. Data are presented for homogeneous and layered circular objects and for a complex two-dimensional model of a head. In particular the influence of the optical parameters on the shape of the PMDF and the distortions caused by boundary layers and complex inhomogeneties are investigated.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. R. Arridge, “Photon-measurement density functions. Part 1: analytical forms,” Appl. Opt. 34, 7395–7409 (1995).
    [CrossRef] [PubMed]
  2. S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite-element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
    [CrossRef] [PubMed]
  3. M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite-element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. (to be published).
  4. G. Strang, G. J. Fix, An Analysis of the Finite Element Method (Prentice-Hall, N.J., 1973).
  5. S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204–215 (1991).
  6. M. Schweiger, S. R. Arridge, D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imag. Vision 3, 263–283 (1993).
    [CrossRef]
  7. R. L. Barbour, H. L. Graber, J. Lubowsky, R. Aronson, “Model for 3 D optical imaging of tissue,” in Proceedings of the Tenth Annual IEEE International Geoscience and Remote Sensing Symposium (IGARSS), J. Ormsby, ed. (IEEE/Geoscience and Remote Sensing Society, 1990), Vol 2, pp. 1395–1399.
    [CrossRef]
  8. H. R. Gordon, “Equivalence of the point- and beam-spread functions of scattering media: a formal demonstration,” Appl. Opt. 33, 1120–1122 (1994).
    [CrossRef] [PubMed]
  9. J. Peraire, M. Vahdati, K. Morgan, O. C. Zienkiewicz, “Adaptive remeshing for compressible flow computations,” J. Comput. Phys. 72, 449–466 (1987).
    [CrossRef]
  10. P. van der Zee, “Measurement and modelling of the optical properties of human tissue in the near infrared,” Ph.D. dissertation (Department of Medical Physics and Bioengineering, University College London, London, 1993).
  11. M. Firbank, “The design, calibration and usage of a solid scattering and absorbing phantom for near infra red spectroscopy,” Ph.D. dissertation (Department of Medical Physics and Bioengineering, University College London, London, 1994).
  12. W.-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” J. Quantum Electron. 26, 2166–2185 (1990).
    [CrossRef]

1995 (1)

1994 (1)

1993 (2)

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite-element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imag. Vision 3, 263–283 (1993).
[CrossRef]

1990 (1)

W.-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

1987 (1)

J. Peraire, M. Vahdati, K. Morgan, O. C. Zienkiewicz, “Adaptive remeshing for compressible flow computations,” J. Comput. Phys. 72, 449–466 (1987).
[CrossRef]

Aronson, R.

R. L. Barbour, H. L. Graber, J. Lubowsky, R. Aronson, “Model for 3 D optical imaging of tissue,” in Proceedings of the Tenth Annual IEEE International Geoscience and Remote Sensing Symposium (IGARSS), J. Ormsby, ed. (IEEE/Geoscience and Remote Sensing Society, 1990), Vol 2, pp. 1395–1399.
[CrossRef]

Arridge, S. R.

S. R. Arridge, “Photon-measurement density functions. Part 1: analytical forms,” Appl. Opt. 34, 7395–7409 (1995).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imag. Vision 3, 263–283 (1993).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite-element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204–215 (1991).

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite-element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. (to be published).

Barbour, R. L.

R. L. Barbour, H. L. Graber, J. Lubowsky, R. Aronson, “Model for 3 D optical imaging of tissue,” in Proceedings of the Tenth Annual IEEE International Geoscience and Remote Sensing Symposium (IGARSS), J. Ormsby, ed. (IEEE/Geoscience and Remote Sensing Society, 1990), Vol 2, pp. 1395–1399.
[CrossRef]

Cheong, W.-F.

W.-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Cope, M.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204–215 (1991).

Delpy, D. T.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite-element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imag. Vision 3, 263–283 (1993).
[CrossRef]

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204–215 (1991).

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite-element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. (to be published).

Firbank, M.

M. Firbank, “The design, calibration and usage of a solid scattering and absorbing phantom for near infra red spectroscopy,” Ph.D. dissertation (Department of Medical Physics and Bioengineering, University College London, London, 1994).

Fix, G. J.

G. Strang, G. J. Fix, An Analysis of the Finite Element Method (Prentice-Hall, N.J., 1973).

Gordon, H. R.

Graber, H. L.

R. L. Barbour, H. L. Graber, J. Lubowsky, R. Aronson, “Model for 3 D optical imaging of tissue,” in Proceedings of the Tenth Annual IEEE International Geoscience and Remote Sensing Symposium (IGARSS), J. Ormsby, ed. (IEEE/Geoscience and Remote Sensing Society, 1990), Vol 2, pp. 1395–1399.
[CrossRef]

Hiraoka, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite-element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite-element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. (to be published).

Lubowsky, J.

R. L. Barbour, H. L. Graber, J. Lubowsky, R. Aronson, “Model for 3 D optical imaging of tissue,” in Proceedings of the Tenth Annual IEEE International Geoscience and Remote Sensing Symposium (IGARSS), J. Ormsby, ed. (IEEE/Geoscience and Remote Sensing Society, 1990), Vol 2, pp. 1395–1399.
[CrossRef]

Morgan, K.

J. Peraire, M. Vahdati, K. Morgan, O. C. Zienkiewicz, “Adaptive remeshing for compressible flow computations,” J. Comput. Phys. 72, 449–466 (1987).
[CrossRef]

Peraire, J.

J. Peraire, M. Vahdati, K. Morgan, O. C. Zienkiewicz, “Adaptive remeshing for compressible flow computations,” J. Comput. Phys. 72, 449–466 (1987).
[CrossRef]

Prahl, S. A.

W.-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Schweiger, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite-element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imag. Vision 3, 263–283 (1993).
[CrossRef]

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite-element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. (to be published).

Strang, G.

G. Strang, G. J. Fix, An Analysis of the Finite Element Method (Prentice-Hall, N.J., 1973).

Vahdati, M.

J. Peraire, M. Vahdati, K. Morgan, O. C. Zienkiewicz, “Adaptive remeshing for compressible flow computations,” J. Comput. Phys. 72, 449–466 (1987).
[CrossRef]

van der Zee, P.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204–215 (1991).

P. van der Zee, “Measurement and modelling of the optical properties of human tissue in the near infrared,” Ph.D. dissertation (Department of Medical Physics and Bioengineering, University College London, London, 1993).

Welch, A. J.

W.-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Zienkiewicz, O. C.

J. Peraire, M. Vahdati, K. Morgan, O. C. Zienkiewicz, “Adaptive remeshing for compressible flow computations,” J. Comput. Phys. 72, 449–466 (1987).
[CrossRef]

Appl. Opt. (2)

J. Comput. Phys. (1)

J. Peraire, M. Vahdati, K. Morgan, O. C. Zienkiewicz, “Adaptive remeshing for compressible flow computations,” J. Comput. Phys. 72, 449–466 (1987).
[CrossRef]

J. Math. Imag. Vision (1)

M. Schweiger, S. R. Arridge, D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imag. Vision 3, 263–283 (1993).
[CrossRef]

J. Quantum Electron. (1)

W.-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Med. Phys. (1)

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite-element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

Other (6)

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite-element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. (to be published).

G. Strang, G. J. Fix, An Analysis of the Finite Element Method (Prentice-Hall, N.J., 1973).

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204–215 (1991).

R. L. Barbour, H. L. Graber, J. Lubowsky, R. Aronson, “Model for 3 D optical imaging of tissue,” in Proceedings of the Tenth Annual IEEE International Geoscience and Remote Sensing Symposium (IGARSS), J. Ormsby, ed. (IEEE/Geoscience and Remote Sensing Society, 1990), Vol 2, pp. 1395–1399.
[CrossRef]

P. van der Zee, “Measurement and modelling of the optical properties of human tissue in the near infrared,” Ph.D. dissertation (Department of Medical Physics and Bioengineering, University College London, London, 1993).

M. Firbank, “The design, calibration and usage of a solid scattering and absorbing phantom for near infra red spectroscopy,” Ph.D. dissertation (Department of Medical Physics and Bioengineering, University College London, London, 1994).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

Contribution of the nodes to (a) the source determination, and (b) the source measurement: a, b, c, a′, b′, and c′ represent the nodes.

Fig. 2
Fig. 2

Flow chart of the steps in the calculation of the Jacobian by explicit perturbation of the mesh elements.

Fig. 3
Fig. 3

Geometry for a circular FEM mesh.

Fig. 4
Fig. 4

FEM mesh generated from an MRI scan. The gray scales (shadings) indicate (a) the absorption coefficient μ a and (b) the scattering coefficient μ s .

Fig. 5
Fig. 5

J α for a homogeneous circle (see Table 1 for parameters).

Fig. 6
Fig. 6

J ν for a homogeneous circle (see Table 1 for parameters).

Fig. 7
Fig. 7

Cross sections through normalized PMDF’s for a homogeneous circle along the 45° radial at an optode spacing of 90° as functions of μ a : (a) J α ( E ) , (b) J α ( t ) , (c) J ν ( E ) , (d) J ν ( t ) , (e) mean penetration depth, and (f) width of the PMDF as a function of μ a .

Fig. 8
Fig. 8

Cross sections through normalized PMDF’s as functions of μ s . The plots are analogous to those in Fig. 7.

Fig. 9
Fig. 9

J α for a layered circle (see Table 2 for parameters).

Fig. 10
Fig. 10

J ν for a layered circle (see Table 2 for parameters).

Fig. 11
Fig. 11

Cross sections through normalized PMDF’s for a layered circle along the 45° radial at an optode spacing of 90°. The radial distances are in fractions of the appropriate circle diameter (see Table 3 for parameters).

Fig. 12
Fig. 12

J α for the head mesh. The columns, from left to right, represent J α ( E ) , J α ( t ) , and J α ( t 4 ) . The rows, from top to bottom, are as follows: Row 1: an adult parameter set with a small optode spacing; row 2: a neonatal parameter set with a small optode spacing; row 3: an adult parameter set with a large optode spacing; and row 4: a neonatal parameter set with a large optode spacing. See Table 3 for the optical parameters.

Fig. 13
Fig. 13

J ν for the head mesh. The measurement types and parameter sets are represents as in Fig. 12, with the subscript ν substituted for α.

Tables (4)

Tables Icon

Table 1 Parameters Used for Fig. 4 (J α) and Fig. 5 (J ν)a

Tables Icon

Table 2 Optical Parameters for a Layered Circle

Tables Icon

Table 3 Parameters for the Images in Figs. 9 and 10

Tables Icon

Table 4 Optical Parameters Used for the MRI-Based Head Meshes

Equations (31)

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

Φ h ( r , t ) = j = 1 D Φ j ( t ) u j ( r )             h ,
[ K ( κ ) + C ( γ ) ] Φ + B Φ t = Q ,
[ K ( κ ) + C ( γ ) + i ω B ] Φ ^ ( ω ) = Q ^ ( ω ) ,
K i j = Ω κ u j ( r ) · u i ( r ) d Ω , C i j = Ω γ u j ( r ) u i ( r ) d Ω , B i j = Ω u j ( r ) u i ( r ) d Ω , Q j ( t ) = Ω u j ( r ) q 0 ( r , t ) d Ω , Φ = [ Φ 1 ( t ) , Φ 2 ( t ) , , Φ D ( t ) ] T
[ Q ] = [ Z ] [ w ] ,
[ Γ ] = [ P ] [ Φ ] ,
P i j = - κ n ^ ( ξ i ) · u j , if N j τ ( ξ i ) , = 0 , otherwise .
[ θ K ( κ ) + θ C ( γ ) + 1 Δ t B ] Φ n + 1 + [ ( 1 - θ ) K ( κ ) + ( 1 - θ ) C ( γ ) - 1 Δ t B ] Φ n = θ Q n + 1 + ( 1 - θ ) Q n ,
{ · κ - γ } Φ ( r ) = - q 0 r .
[ K ( κ ) + C ( γ ) ] Φ = Q ,
[ K ( κ ) + C ( γ ) + i ω B ] n Φ ^ ( ω ) ω n + i n B n - 1 Φ ^ ( ω ) ω n - 1 = n Φ ^ ( ω ) ω n .
[ K ( κ ) + C ( γ ) ] n Φ ^ ( ω ) ω n | ω = 0 = - i n B n - 1 Φ ^ ( ω ) ω n - 1 | ω = 0 ,
Φ ( T n ) = n [ K ( κ ) + C ( γ ) ] - 1 B Φ ( T n - 1 ) .
[ K ( κ + ν ) + C ( γ + α ) + i ω B ] [ Φ ^ ( ω ) + η ^ ( ω ) ] = Q ^ ( ω ) .
[ K ( κ ) + C ( γ ) + i ω B ] η ^ - [ K ( ν ) + C ( α ) ] Φ ^ ,
[ K ( κ ) + C ( γ ) η + B η t - [ K ( ν ) + C ( α ) ] Φ .
[ Φ int ] = [ G int ] [ Q int ] ,
[ Γ ] = [ P ] [ G int ] [ Q int ] ,
{ P 1 T P 2 T P j T P M T } .
Γ j ( i ) = P j T · G i
[ Φ int ] - [ G int ] [ Z ] [ Γ ] .
Φ i ( j ) = G i T · Z j .
Φ i ( j ) = Z j T · G i .
J ^ α ( Γ ) ( ξ j , ζ i , ω ; r ) = [ k N k τ ( r ) Φ ^ k ( i ) ( ω ) u k ( r ) ] × [ k N k τ ( r ) Φ ^ Adj , k ( j ) ( ω ) u k ( r ) ] .
( Φ ^ ( i ) × Φ ^ ( j ) ) ( ω , r ) [ k N k τ ( r ) Φ ^ k ( i ) ( ω ) u k ( r ) ] × [ k N k τ ( r ) Φ ^ k ( j ) ( ω ) u k ( r ) ] .
J ^ ν ( Γ ) ( ξ j , ζ i , ω ; r ) = k , m N k N m τ ( r ) Φ ^ k ( i ) ( ω ) Φ ^ Adj , m ( j ) ( ω ) r u k ( r ) · r u m ( r ) = ( Φ ^ ( i ) × Φ ^ Adj ( j ) ) ( ω , r ) .
J α ( E ) ( ξ j , ζ i ; r ) = ( Φ ( i ) × Φ Adj ( j ) ) ( r ) , J ν ( E ) ( ξ j , ζ i ; r ) = ( Φ ( i ) × Φ Adj ( j ) ) ( r ) .
J α ( T n ) ( ξ j , ζ i ; r ) = [ Φ ( i ) × Φ Adj ( j ) ] ( T n ) ( r ) = r = 0 n ( n r ) Φ ( i ) ( T r ) × Φ Adj ( j ) ( T n - r ) ,
J ν ( T n ) ( ξ j , ζ i ; r ) = Φ ( i ) × Φ Adj ( j ) ) ( T n ) ( r ) = r = 0 n ( n r ) Φ ( i ) ( T r ) × Φ Adj ( j ) ( T n - r ) .
J α ( Ψ ) ( ξ j , ζ i ; r ) = Im [ ( Φ ^ ( i ) × Φ ^ Adj ( j ) ) ( ω , r ) Γ ^ ( ξ j , ζ j , ω ) ] ,
J ν ( Ψ ) ( ξ j , ζ i ; r ) = Im [ ( Φ ^ ( i ) × Φ ^ Adj ( j ) ) ( ω , r ) Γ ^ ( ξ j , ζ j , ω ) ] .

Metrics