Abstract

A theoretical model of photon propagation in a scattering medium is presented, from which algebraic formulas for the detector-reading perturbations (ΔR) produced by one or two localized perturbations in the macroscopic absorption cross section (Δμa) are derived. Examination of these shows that when Δμa is titrated from very small to large magnitudes in one voxel, the curve traced by the corresponding ΔR values is a rectangular hyperbola. Furthermore, while ΔRlimΔμa ΔR is dependent on the location of the detector with respect to the source and the voxel, the ratio ΔR/ΔR is independent of the detector location. We also find that when Δμa is varied in two voxels simultaneously, the quantity ΔR(Δμa,1Δμa,2) is a bilinear rational function of the Δμas. These results apply not only in the case of steady-state illumination and detection but to time-harmonic measurements as well. The validity of the theoretical formulas is demonstrated by applying them to the results of selected numerical diffusion computations. Potential applications of the derived expressions to image-reconstruction problems are discussed.

© 1998 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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  5. H. L. Graber, J. Chang, R. Aronson, R. L. Barbour, “A perturbation model for imaging in dense scattering media: derivation and evaluation of imaging operators,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institutes for Advanced Optical Technologies (SPIE Press, Bellingham, Wash., 1993), pp. 121–143.
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    [CrossRef]

1997 (1)

1996 (3)

1995 (1)

R. L. Barbour, H. L. Graber, J. Chang, S.-L. S. Barbour, P. C. Koo, R. Aronson, “MRI-guided optical tomography: prospects and computation for a new imaging method,” IEEE Computational Sci. Eng. 2, 63–77 (1995).
[CrossRef]

1994 (1)

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

1992 (1)

F. A. Grünbaum, “Diffuse tomography: the isotropic case,” Inverse Probl. 8, 409–419 (1992).
[CrossRef]

1990 (2)

A. Q. Howard, W. C. Chew, M. C. Moldoveanu, “A new correction to the Born approximation,” IEEE Trans. Geosci. Remote Sens. 28, 394–399 (1990).
[CrossRef]

W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distributions using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[CrossRef]

Acton, F. S.

F. S. Acton, Numerical Methods That Work, 2nd printing (Mathematical Association of America, Washington, D.C.1990), Chap. 8, pp. 211–220.

Alcouffe, R. E.

R. E. Alcouffe, Group XTM, Los Alamos National Laboratory, Los Alamos, N. M. 87545 (personal communication, 1995).

Aronson, R.

J. Chang, H. L. Graber, R. L. Barbour, R. Aronson, “Recovery of optical cross-section perturbations in dense-scattering media using transport-theory-based imaging operators and steady-state simulated data,” Appl. Opt. 35, 3963–3978 (1996).
[CrossRef] [PubMed]

R. L. Barbour, H. L. Graber, J. Chang, S.-L. S. Barbour, P. C. Koo, R. Aronson, “MRI-guided optical tomography: prospects and computation for a new imaging method,” IEEE Computational Sci. Eng. 2, 63–77 (1995).
[CrossRef]

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institutes for Advanced Optical Technologies (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

H. L. Graber, J. Chang, R. Aronson, R. L. Barbour, “A perturbation model for imaging in dense scattering media: derivation and evaluation of imaging operators,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institutes for Advanced Optical Technologies (SPIE Press, Bellingham, Wash., 1993), pp. 121–143.

H. L. Graber, R. L. Barbour, J. Chang, R. Aronson, “Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium,” in Proceedings of Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 669–681 (1995).
[CrossRef]

Arridge, S. R.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near infrared absorption and scatter imaging,” in Proceedings of Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, pp. 360–371 (1993).
[CrossRef]

S. R. Arridge, “The forward and inverse problems in time resolved infra-red imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institutes for Advanced Optical Technologies (SPIE Press, Bellingham, Wash., 1993), pp. 35–65.

Barbour, R. L.

Y. Yao, Y. Wang, Y. Pei, W. Zhu, R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering distributions by a Born iterative method,” J. Opt. Soc. Am. A 14, 325–342 (1997).
[CrossRef]

J. Chang, H. L. Graber, R. L. Barbour, R. Aronson, “Recovery of optical cross-section perturbations in dense-scattering media using transport-theory-based imaging operators and steady-state simulated data,” Appl. Opt. 35, 3963–3978 (1996).
[CrossRef] [PubMed]

R. L. Barbour, H. L. Graber, J. Chang, S.-L. S. Barbour, P. C. Koo, R. Aronson, “MRI-guided optical tomography: prospects and computation for a new imaging method,” IEEE Computational Sci. Eng. 2, 63–77 (1995).
[CrossRef]

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institutes for Advanced Optical Technologies (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

H. L. Graber, J. Chang, R. Aronson, R. L. Barbour, “A perturbation model for imaging in dense scattering media: derivation and evaluation of imaging operators,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institutes for Advanced Optical Technologies (SPIE Press, Bellingham, Wash., 1993), pp. 121–143.

H. L. Graber, R. L. Barbour, J. Chang, R. Aronson, “Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium,” in Proceedings of Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 669–681 (1995).
[CrossRef]

F. H. Schlereth, J. M. Fossaceca, A. D. Keckler, R. L. Barbour, “Imaging in diffusing media with a neural net formulation: A problem in large scale computation,” in Proceedings of Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, pp. 46–57 (1992).
[CrossRef]

Barbour, S.-L. S.

R. L. Barbour, H. L. Graber, J. Chang, S.-L. S. Barbour, P. C. Koo, R. Aronson, “MRI-guided optical tomography: prospects and computation for a new imaging method,” IEEE Computational Sci. Eng. 2, 63–77 (1995).
[CrossRef]

Beltrami, E.

E. Beltrami, Mathematical Models in the Social and Biological Sciences (Jones and Bartlett, Boston, Mass., 1993), Chap. 1, pp. 1–19.

Boas, D. A.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Images of inhomogeneous turbid media using diffuse photon density waves,” in OSA Proceedings on Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 106–115.

Bonner, R. F.

Chance, B.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Images of inhomogeneous turbid media using diffuse photon density waves,” in OSA Proceedings on Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 106–115.

Chang, J.

J. Chang, H. L. Graber, R. L. Barbour, R. Aronson, “Recovery of optical cross-section perturbations in dense-scattering media using transport-theory-based imaging operators and steady-state simulated data,” Appl. Opt. 35, 3963–3978 (1996).
[CrossRef] [PubMed]

R. L. Barbour, H. L. Graber, J. Chang, S.-L. S. Barbour, P. C. Koo, R. Aronson, “MRI-guided optical tomography: prospects and computation for a new imaging method,” IEEE Computational Sci. Eng. 2, 63–77 (1995).
[CrossRef]

H. L. Graber, J. Chang, R. Aronson, R. L. Barbour, “A perturbation model for imaging in dense scattering media: derivation and evaluation of imaging operators,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institutes for Advanced Optical Technologies (SPIE Press, Bellingham, Wash., 1993), pp. 121–143.

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institutes for Advanced Optical Technologies (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

H. L. Graber, R. L. Barbour, J. Chang, R. Aronson, “Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium,” in Proceedings of Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 669–681 (1995).
[CrossRef]

Chew, W. C.

A. Q. Howard, W. C. Chew, M. C. Moldoveanu, “A new correction to the Born approximation,” IEEE Trans. Geosci. Remote Sens. 28, 394–399 (1990).
[CrossRef]

W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distributions using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[CrossRef]

Delpy, D. T.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near infrared absorption and scatter imaging,” in Proceedings of Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, pp. 360–371 (1993).
[CrossRef]

Duderstadt, J. J.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979), Chap. 2, pp. 69–72.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, New York, 1992), Chap. 15, pp. 678–683.

Fossaceca, J. M.

F. H. Schlereth, J. M. Fossaceca, A. D. Keckler, R. L. Barbour, “Imaging in diffusing media with a neural net formulation: A problem in large scale computation,” in Proceedings of Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, pp. 46–57 (1992).
[CrossRef]

Furutsu, K.

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Gandjbakhche, A. H.

Graber, H. L.

J. Chang, H. L. Graber, R. L. Barbour, R. Aronson, “Recovery of optical cross-section perturbations in dense-scattering media using transport-theory-based imaging operators and steady-state simulated data,” Appl. Opt. 35, 3963–3978 (1996).
[CrossRef] [PubMed]

R. L. Barbour, H. L. Graber, J. Chang, S.-L. S. Barbour, P. C. Koo, R. Aronson, “MRI-guided optical tomography: prospects and computation for a new imaging method,” IEEE Computational Sci. Eng. 2, 63–77 (1995).
[CrossRef]

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institutes for Advanced Optical Technologies (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

H. L. Graber, J. Chang, R. Aronson, R. L. Barbour, “A perturbation model for imaging in dense scattering media: derivation and evaluation of imaging operators,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institutes for Advanced Optical Technologies (SPIE Press, Bellingham, Wash., 1993), pp. 121–143.

H. L. Graber, R. L. Barbour, J. Chang, R. Aronson, “Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium,” in Proceedings of Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 669–681 (1995).
[CrossRef]

Grünbaum, F. A.

F. A. Grünbaum, “Diffuse tomography: the isotropic case,” Inverse Probl. 8, 409–419 (1992).
[CrossRef]

Hiraoka, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near infrared absorption and scatter imaging,” in Proceedings of Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, pp. 360–371 (1993).
[CrossRef]

Howard, A. Q.

A. Q. Howard, W. C. Chew, M. C. Moldoveanu, “A new correction to the Born approximation,” IEEE Trans. Geosci. Remote Sens. 28, 394–399 (1990).
[CrossRef]

Jiang, H.

Keckler, A. D.

F. H. Schlereth, J. M. Fossaceca, A. D. Keckler, R. L. Barbour, “Imaging in diffusing media with a neural net formulation: A problem in large scale computation,” in Proceedings of Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, pp. 46–57 (1992).
[CrossRef]

Koo, P. C.

R. L. Barbour, H. L. Graber, J. Chang, S.-L. S. Barbour, P. C. Koo, R. Aronson, “MRI-guided optical tomography: prospects and computation for a new imaging method,” IEEE Computational Sci. Eng. 2, 63–77 (1995).
[CrossRef]

Martin, W. R.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979), Chap. 2, pp. 69–72.

Moldoveanu, M. C.

A. Q. Howard, W. C. Chew, M. C. Moldoveanu, “A new correction to the Born approximation,” IEEE Trans. Geosci. Remote Sens. 28, 394–399 (1990).
[CrossRef]

Nossal, R.

O’Leary, M. A.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Images of inhomogeneous turbid media using diffuse photon density waves,” in OSA Proceedings on Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 106–115.

Osterberg, U. L.

Patterson, M. S.

Paulsen, K. D.

Pei, Y.

Pogue, B. W.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, New York, 1992), Chap. 15, pp. 678–683.

Schlereth, F. H.

F. H. Schlereth, J. M. Fossaceca, A. D. Keckler, R. L. Barbour, “Imaging in diffusing media with a neural net formulation: A problem in large scale computation,” in Proceedings of Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, pp. 46–57 (1992).
[CrossRef]

Schweiger, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near infrared absorption and scatter imaging,” in Proceedings of Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, pp. 360–371 (1993).
[CrossRef]

Strang, G.

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovitch, San Diego, Calif., 1988), Chap. 5, pp. 269–272.

G. Strang, Linear Algebra and its Applications, 3rd ed. (Harcourt Brace Jovanovitch, San Diego, Calif., 1988), Chap. 7, pp. 370–372.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, New York, 1992), Chap. 15, pp. 678–683.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, New York, 1992), Chap. 15, pp. 678–683.

Wang, Y.

Y. Yao, Y. Wang, Y. Pei, W. Zhu, R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering distributions by a Born iterative method,” J. Opt. Soc. Am. A 14, 325–342 (1997).
[CrossRef]

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institutes for Advanced Optical Technologies (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

Wang, Y. M.

W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distributions using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[CrossRef]

Weiss, G. H.

Yamada, Y.

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Yao, Y.

Yodh, A. G.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Images of inhomogeneous turbid media using diffuse photon density waves,” in OSA Proceedings on Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 106–115.

Zhu, W.

Appl. Opt. (2)

IEEE Computational Sci. Eng. (1)

R. L. Barbour, H. L. Graber, J. Chang, S.-L. S. Barbour, P. C. Koo, R. Aronson, “MRI-guided optical tomography: prospects and computation for a new imaging method,” IEEE Computational Sci. Eng. 2, 63–77 (1995).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

A. Q. Howard, W. C. Chew, M. C. Moldoveanu, “A new correction to the Born approximation,” IEEE Trans. Geosci. Remote Sens. 28, 394–399 (1990).
[CrossRef]

IEEE Trans. Med. Imaging (1)

W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distributions using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[CrossRef]

Inverse Probl. (1)

F. A. Grünbaum, “Diffuse tomography: the isotropic case,” Inverse Probl. 8, 409–419 (1992).
[CrossRef]

J. Opt. Soc. Am. A (2)

Phys. Rev. E (1)

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Other (16)

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Images of inhomogeneous turbid media using diffuse photon density waves,” in OSA Proceedings on Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 106–115.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near infrared absorption and scatter imaging,” in Proceedings of Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, pp. 360–371 (1993).
[CrossRef]

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institutes for Advanced Optical Technologies (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

H. L. Graber, J. Chang, R. Aronson, R. L. Barbour, “A perturbation model for imaging in dense scattering media: derivation and evaluation of imaging operators,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institutes for Advanced Optical Technologies (SPIE Press, Bellingham, Wash., 1993), pp. 121–143.

S. R. Arridge, “The forward and inverse problems in time resolved infra-red imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institutes for Advanced Optical Technologies (SPIE Press, Bellingham, Wash., 1993), pp. 35–65.

H. L. Graber, R. L. Barbour, J. Chang, R. Aronson, “Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium,” in Proceedings of Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 669–681 (1995).
[CrossRef]

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovitch, San Diego, Calif., 1988), Chap. 5, pp. 269–272.

G. Strang, Linear Algebra and its Applications, 3rd ed. (Harcourt Brace Jovanovitch, San Diego, Calif., 1988), Chap. 7, pp. 370–372.

F. S. Acton, Numerical Methods That Work, 2nd printing (Mathematical Association of America, Washington, D.C.1990), Chap. 8, pp. 211–220.

F. H. Schlereth, J. M. Fossaceca, A. D. Keckler, R. L. Barbour, “Imaging in diffusing media with a neural net formulation: A problem in large scale computation,” in Proceedings of Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, pp. 46–57 (1992).
[CrossRef]

At each surface, the partial sums of Eq. (4.A.1) must be computed in a specific manner to ensure that the boundary condition is satisfied. The nth partial sum for a reflected flux consists of the i=0,±1,…,±n terms, while the nth partial sum for a transmitted flux is the sum of the i=0,±1,…,±(n-1),+n terms. Under this procedure, the calculated |J(x, 0, 0)| arising from the source at depth z0 is nearly equal to the |J(x, 0, Z)| arising from the source at Z-z0, as expected.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, New York, 1992), Chap. 15, pp. 678–683.

PSI-PlotTM Version 4 for WindowsTM User’s Handbook (Poly Software International, Salt Lake City, Utah, 1995), Chap. 5, p. 148.

R. E. Alcouffe, Group XTM, Los Alamos National Laboratory, Los Alamos, N. M. 87545 (personal communication, 1995).

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979), Chap. 2, pp. 69–72.

E. Beltrami, Mathematical Models in the Social and Biological Sciences (Jones and Bartlett, Boston, Mass., 1993), Chap. 1, pp. 1–19.

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Figures (4)

Fig. 1
Fig. 1

Detector readings, homogeneous reference medium. (a) Sketch of three-dimensional homogeneous, isotropically scattering medium modeled in computation of reference detector readings by the relaxation code. Units of the indicated dimensions are numbers of cells. Also shown are source and detector locations; the two detector arrays are in the same plane. (b) Plot of base-ten logarithms of the computed detector readings for reflected (circles) and transmitted (squares) light, as a function of distance along the surface between the incident beam and the detector. Plotted curves are analytic diffusion equation solutions for an infinite homogeneous, nonabsorbing, isotropically scattering slab that is 11 mfp thick, with the diffuse intensity going to zero at the physical boundaries (solid curves) or at extrapolated boundaries 0.7104 mfp outside the physical boundaries (dashed curves).

Fig. 2
Fig. 2

Effect of a single localized Δμa on computed detector readings. (a) Sketch of 2-D section through the slab medium, showing the location of the perturbed cell [cell (31, 31, 5)] and four detectors. Units of indicated distances are numbers of cells. (b) ΔR vs. Δμa (in units of percent increase in the perturbed voxel’s μt owing to the increased absorption), for the four detectors indicated in (a). Curves are best fits of Eq. (4.B.1) to the data points, using only the data points for finite values of μa. Data point–detector correspondences are as follows: filled circles, detector r1; filled triangles, detector t1; open circles, detector r2; open triangles, detector t2. (c) ΔR/ΔR versus Δμa for the same detectors as in (b); symbols also are the same as in (b).

Fig. 3
Fig. 3

Dependence of ΔR/ΔR on location of a single Δμa. (a) Sketch of 2-D section through the slab medium, showing three alternative locations for the perturbed cell [(31, 31, 1), (31, 31, 5), or (31, 31, 7)]. Units of indicated distances are numbers of cells. For each Δμa location and magnitude the ratio ΔR(Δμa)/ΔR was calculated for each of the 62 detectors, and the mean and standard deviation over the set of detectors were then computed. (b) Plots of ΔR(Δμa)/ΔR¯ versus Δμa, for Δμa in each of the three cells indicated in (a). Data point–perturbed cell correspondences are as follows: circles, cell (31, 31, 5); squares, cell (31, 31, 7); triangles, cell (31, 31, 1). (c) Plots of log10{σ[ΔR(Δμa)/ΔR]} versus Δμa. Units and symbols are the same as in (b).

Fig. 4
Fig. 4

Mutual coupling between two localized absorption perturbations. (a) Sketch of 2-D section through the slab medium, showing locations of the two perturbed cells [(31, 31, 5) and (31, 31, 7)] and two detectors. Units of indicated distances are numbers of cells. (b) Effect of two simultaneous localized Δμa’s, ΔRc vs. Δμa,5 versus Δμa,7, for detector r indicated in (a). (c) Absolute mutual coupling between the two cells indicated in (a), i.e., ΔRc-ΔRi. (d) Percent relative mutual coupling between the two cells indicated in (a), i.e., 100(1-ΔRc/ΔRi), for detector r. (e)–(g) show the same quantities as in (b)–(d), but for detector t.

Tables (4)

Tables Icon

Table 1 Coefficients of Rational Functions for the Absolute Mutual Coupling ΔRq+ΔRp-ΔRpq and the Relative Mutual Coupling 1-[ΔRpq/(ΔRq+ΔRp)] between Two Perturbed Voxels (See Subsection 3.D), in Terms of the Relative Absorption Perturbations z and ν

Tables Icon

Table 2 Coefficients of Best Fits of Eq. (4.B.1) to the Four Sets of Data Points Shown in Fig. 2(b). Shown Also Are the Detector Readings’ Perturbations Obtained from the Relaxation Computation When a Perfect Absorber Was Placed in the Cell Shaded in Fig. 2(a)

Tables Icon

Table 3 Coefficients of Best Fits of Eq. (4.B.2) to the Three Sets of Data Points Shown in Fig. 3(b) and for Two Other Locations of the Absorption Perturbation Not Shown in Fig. 3

Tables Icon

Table 4 Coefficients Obtained by Fitting the Coupled ΔR and Independent ΔR for Detectors r and t [See Fig. 4(a)] to Eqs. (4.B.3a) and (4.B.3b), Respectively

Equations (64)

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ΔRi=jwijΔμj,
ϕ=n=0Pnφ,
ϕ=(I-P)-1φTφ.
Pijn(t)=k=1I0tPik(t)Pkjn-1(t-t)dt.
Pn(ω)=P(ω)Pn-1(ω).
P(ω)=-P(t)exp(-iωt)dt=P exp(-iωtij).
T=T0+T0(I-ΔPT0)-1ΔPT0.
T=T0-T0(I+P0YT0)-1P0YT0=T0-T0P0Y(I+T0P0Y)-1T0.
T0-XT=YT0+(I-Y)T0P0Y(I+T0P0Y)-1T0.
R=i=1Iriϕi=rTϕ,
R0=r0TT0φ0,
μs,jμt,j=μs,jμt,j0μt,j0μt,j=μs,jμt,j0μt,j0μt,j0+Δμa,jμs,jμt,j0xj.
R=rTXTφ=rTX(I-P0X)-1φ.
xk=exp(-Δμa,klk),
P=X1/2P0X1/2,
xk11+Δμa,klk.
R=rTX1/2(I-X1/2P0X1/2)-1X1/2φ.
ΔR=r0T(T0-XT)φ0.
T0-XT=T0YT.
T0=VYT.
(T0-XT)ij=(T0)iq(T0)qjVqq,
ΔR=1Vqqi,jr0,i(T0)iq(T0)qjφj0=1Vqqir0,i(T0)iqj(T0)qjφj0=1Vqqir0,i(T0)iqϕq0=r0Ttqϕq0Vqq,
ΔR=r0Ttqϕq0yq1+(T0-I)qqyq,
ΔR=r0Ttqϕq0Δμa,qμt,q0+(T0)qqΔμa,q
ΔRr0Ttqϕq0Δμa,q1lq+(T0)qqΔμa,q
ΔRqlimΔμa,q ΔR=r0Ttqϕq0(T0)qq,
limΔμa,q0 ΔR=r0Ttqϕq0κqΔμa,q,
wij=r0,iTtjϕj0κj,
ΔRΔRq=Δμa,qκq(T0)qq+Δμa,q.
ΔRijwij=κj(T0)jj,
(T0)qj=Vqq(YT)qj+(T0)qp(YT)pj,
(T0)pj=Vpp(YT)pj+(T0)pq(YT)qj.
(YT)qj=1Δ[Vpp(T0)qj-(T0)qp(T0)pj],
(YT)pj=1Δ[Vqq(T0)pj-(T0)pq(T0)qj],
(T0-XT)ij=(T0)iq[Vpp(T0)qj-(T0)qp(T0)pj]+(T0)ip[Vqq(T0)pj-(T0)pq(T0)qj]Δ.
ΔR=r0T tqVppϕq0+tpVqqϕp0-tq(T0)qpϕp0-tp(T0)pqϕq0Δ.
ΔR=r0T tqϕq0yq+tpϕp0yp+Aq,pyqyp1+(T0-I)qqyq+(T0-I)ppyp+Bq,pyqyp,
Aq,p=tq(T0-I)ppϕq0+tp(T0-I)qqϕp0-tq(T0)qpϕp0-tp(T0)pqϕq0,
Bq,p=(T0-I)qq(T0-I)pp-(T0)qp(T0)pq.
ΔR=r0T tqϕq0z+tpϕp0ν+[tq(T0)ppϕq0+tp(T0)qqϕp0-tq(T0)qpϕp0-tp(T0)pqϕq0]zν1+(T0)qqz+(T0)ppν+[(T0)qq(T0)pp-(T0)qp(T0)pq]zν.
ΔR=r0T tqϕq0z+tpϕp0ν+[tq(T0)ppϕq0+tp(T0)qqϕp0]zν1+(T0)qqz+(T0)ppν+(T0)qq(T0)ppzν.
|J(x, 0, 0)|=14πi=-[αi(x2+αi2)-3/2-βi(x2+βi2)-3/2],
ΔR=ΔRΔμaK+Δμa
ΔRΔR=λ ΔμaK+Δμa.
ΔR=anΔμa,5+bnΔμa,7+cnΔμa,5Δμa,71+adΔμa,5+bdΔμa,7+cdΔμa,5Δμa,7
ΔR=an(Δμa,5+Δμa,7)+cnΔμa,5Δμa,71+ad(Δμa,5+Δμa,7)+cdΔμa,5Δμa,7
ΔRij=wijΔRijΔμa,jΔRij+wijΔμa,j.
(T0)ij=μt,j0ϕij+0
ΔRij=ϕj0k=1Ir0,i,k(T0)kjΔμa,jκj+(T0)jjΔμa,j=ϕj0k=1Ir0,i,kμt,k0ϕkj+0Δμa,jκj+(μt,j0ϕjj+0)Δμa,j=k=1Ir0,i,kμt,k0wkjΔμa,jκj+(μt,j0ϕjj+0)Δμa,j,
ΔRijj=k=1Ir0,i,k{wkjz+wkjν+Aj,j,kzν}1+ϕjj+0z+ϕjj+0ν+Bj,jzν,
ΔRi=j=1I wijΔμa,j1+κjΔμa,j=j=1Iwij1+κjΔμa,jΔμa,jj=1IwijΔμa,j
ΔRi=j=1IwijΔμa,j-1I-1j=1I-1j=j+1I Pi(Δμa,j, Δμa,j)Qi(Δμa,j, Δμa,j),
r0T[tq(T0)qpϕp0+tp(T0)pqϕq0]
4(T0)qq(T0)pp-(T0)qp(T0)pq
2r0T[tq(T0)ppϕq0+tp(T0)qqϕp0]
r0T[tq(T0)qq(T0)qpϕp0+tp(T0)pq(T0)qqϕq0-tq(T0)qp(T0)pqϕq0]
(T0)qq[2(T0)qq(T0)pp-(T0)qp(T0)pq]
r0T[2tq(T0)qq(T0)ppϕq0+tp(T0)qq2ϕp0-tq(T0)qp(T0)pqϕq0]
r0T[tp(T0)pp(T0)pqϕq0+tq(T0)qp(T0)ppϕp0-tp(T0)pq(T0)qpϕp0]
(T0)pp[2(T0)qq(T0)pp-(T0)qp(T0)pq]
r0T[2tp(T0)qq(T0)ppϕp0+tq(T0)pp2ϕp0-tp(T0)pq(T0)qpϕq0]
r0T{tq(T0)qp(T0)pp[(T0)qqϕp0-(T0)pqϕq0]+tp(T0)pq(T0)qq[(T0)ppϕq0-(T0)qpϕp0]}
(T0)qq(T0)pp[(T0)qq(T0)pp-(T0)qp(T0)pq]
r0T[tq(T0)ppϕq0+tp(T0)qqϕp0]×[(T0)qq(T0)pp-(T0)qp(T0)pq]

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