Abstract

The recent developments in light generation and detection techniques have opened new possibilities for optical medical imaging, tomography, and diagnosis at tissue penetration depths of ∼10 cm. However, because light scattering and diffusion in biological tissue are rather strong, the reconstruction of object images from optical projections needs special attention. We describe a simple reconstruction method for diffuse optical imaging, based on a modified backprojection approach for medical tomography. Specifically, we have modified the standard backprojection method commonly used in x-ray tomographic imaging to include the effects of both the diffusion and the scattering of light and the associated nonlinearities in projection image formation. These modifications are based primarily on the deconvolution of the broadened image by a spatially variant point-spread function that is dependent on the scattering of light in tissue. The spatial dependence of the deconvolution and nonlinearity corrections for the curved propagating ray paths in heterogeneous tissue are handled semiempirically by coordinate transformations. We have applied this method to both theoretical and experimental projections taken by parallel- and fan-beam tomography geometries. The experimental objects were biomedical phantoms with multiple objects, including in vitro animal tissue. The overall results presented demonstrate that image-resolution improvements by nearly an order of magnitude can be obtained. We believe that the tomographic method presented here can provide a basis for rapid, real-time medical monitoring by the use of optical projections. It is expected that such optical tomography techniques can be combined with the optical tissue diagnosis methods based on spectroscopic molecular signatures to result in a versatile optical diagnosis and imaging technology.

© 1997 Optical Society of America

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  51. M. B. van der Mark, G. W. ’t Hooft, D. G. Papaioannou, S. B. Colak, N. A. A. J. van Asten, J. C. J. Paasschens, “Boundary effects in optical tomography,” presented at Biomedical Optics Society Europe ’96, Vienna, 7–11 September 1996.

1996 (2)

X. Wang, C. J. Ritchie, Y. Kim, “Elevation direction deconvolution in three dimensional ultrasound imaging,” IEEE Trans. Med. Imag. 15, 389–394 (1996).
[CrossRef]

C. Chinnock, “Medical diagnostics: laser mammography continues development,” Laser Focus World 32 (2), 38–39 (1996).

1995 (2)

1994 (2)

S. J. Glick, B. C. Penney, M. A. King, C. L. Byrne, “Noniterative compensation for the distance dependent detector response and photon attenuation in SPECT imaging,” IEEE Trans. Med. Imag. 13, 363–374 (1994).
[CrossRef]

D. H. Burns, “Optical tomography for three-dimensional spectroscopy,” Appl. Spectrosc. 48, 12A–19A (1994).
[CrossRef]

1993 (3)

B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
[CrossRef] [PubMed]

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

B. B. Das, K. M. Yoo, R. R. Alfano, “Ultrafast time-gated imaging in thick tissues: a step toward optical mammography,” Opt. Lett. 18, 1092–1094 (1993).
[CrossRef] [PubMed]

1992 (1)

S. Rathee, Z. J. Koles, T. R. Overton, “Image restoration in computed tomography: restoration of experimental CT images,” IEEE Trans. Med. Imag. 11, 546–553 (1992).
[CrossRef]

1988 (1)

R. R. Alfano, G. C. Tang, A. Pradhan, M. Bleich, D. S. J. Choy, E. Opher, “Steady state and time resolved laser fluorescence from normal and tumor lung and breast tissues,” J. Tumor Marker Oncol. 3, 165–172 (1988).

1987 (3)

T. J. Yorkey, J. G. Webster, “A comparison of impedance tomographic reconstruction algorithms,” Clin. Phys. Physiol. Meas. 8, 55–62 (1987).
[CrossRef] [PubMed]

B. Brown, D. Barber, L. Tarassenko, eds., Special Issue on Electrical Impedance Tomography—Applied Potential Tomography, Clin. Phys. Physiol. Meas. 8, (1987).

D. C. Barber, A. D. Seager, “Fast reconstruction of resistance images,” Clin. Phys. Physiol. Meas. 8, 47–54 (1987).
[CrossRef] [PubMed]

1974 (1)

1929 (1)

M. Cutler, “Transillumination as an aid in the diagnosis of breast lesions,” Surg. Gynecol. Obstet. 48, 721–730 (1929).

’t Hooft, G. W.

S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, “Optical image reconstruction with deconvolutions in light diffusing media,” in Photon Propagation in Tissues, B. Chance, D. T. Delpy, G. J. Mueller, A. Katzir, eds., Proc. SPIE2626, 306–315 (1995).

G. W. ’t Hooft, D. G. Papaioannou, J. J. M. Baselmans, M. J. C. van Gemert, “Dependence of image quality on optical parameters in time-resolved transillumination experiments,” in Laser Interaction with Hard and Soft Tissue, M. J. C. van Gemert, R. W. Steiner, L. O. Svaasand, H. Albrecht, eds., Proc. SPIE2077, 153–158 (1993).
[CrossRef]

D. G. Papaioannou, S. B. Colak, G. W. ’t Hooft, “Resolution and sensitivity limits of optical imaging in highly scattering media,” in Photon Propagation in Tissues, B. Chance, D. T. Delpy, G. J. Mueller, A. Katzir, eds., Proc. SPIE2626, 218–227 (1995).
[CrossRef]

M. B. van der Mark, G. W. ’t Hooft, D. G. Papaioannou, S. B. Colak, N. A. A. J. van Asten, J. C. J. Paasschens, “Boundary effects in optical tomography,” presented at Biomedical Optics Society Europe ’96, Vienna, 7–11 September 1996.

Alfano, R. R.

B. B. Das, K. M. Yoo, R. R. Alfano, “Ultrafast time-gated imaging in thick tissues: a step toward optical mammography,” Opt. Lett. 18, 1092–1094 (1993).
[CrossRef] [PubMed]

R. R. Alfano, G. C. Tang, A. Pradhan, M. Bleich, D. S. J. Choy, E. Opher, “Steady state and time resolved laser fluorescence from normal and tumor lung and breast tissues,” J. Tumor Marker Oncol. 3, 165–172 (1988).

Arridge, S. R.

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

S. R. Arridge, “The forward and inverse problems in time resolved infrared imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, 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 Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 35–64.

Barber, D. C.

D. C. Barber, A. D. Seager, “Fast reconstruction of resistance images,” Clin. Phys. Physiol. Meas. 8, 47–54 (1987).
[CrossRef] [PubMed]

Barbour, R. L.

J. Chang, H. L. Graber, R. L. Barbour, “Image reconstruction of targets in random media from continuous wave laser measurements and simulated data,” in 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. 193–201.

Baselmans, J. J. M.

G. W. ’t Hooft, D. G. Papaioannou, J. J. M. Baselmans, M. J. C. van Gemert, “Dependence of image quality on optical parameters in time-resolved transillumination experiments,” in Laser Interaction with Hard and Soft Tissue, M. J. C. van Gemert, R. W. Steiner, L. O. Svaasand, H. Albrecht, eds., Proc. SPIE2077, 153–158 (1993).
[CrossRef]

Benaron, D.

D. Benaron, G. Muller, B. Chance, “Introduction: a medical perspective at the threshold of clinical optical tomography,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, 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 Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 3–9.

Benaron, D. A.

D. A. Benaron, J. P. van Houten, W.-F. Cheong, E. L. Kermit, R. A. King, “Early clinical results of time of flight optical tomography in a neonatal intensive care unit,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, A. Katzir, eds. Proc. SPIE2389, 582–596 (1995).
[CrossRef]

Bleich, M.

R. R. Alfano, G. C. Tang, A. Pradhan, M. Bleich, D. S. J. Choy, E. Opher, “Steady state and time resolved laser fluorescence from normal and tumor lung and breast tissues,” J. Tumor Marker Oncol. 3, 165–172 (1988).

Boas, D. A.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Images of inhomogeneous turbid media using diffuse photon density waves,” in 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.

Brooksby, J. W.

F. A. Marks, H. W. Tomlinson, J. W. Brooksby, “A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, A. Katzir, eds., Proc. SPIE1888, 500–510 (1993).
[CrossRef]

Burns, D. H.

Byrne, C. L.

S. J. Glick, B. C. Penney, M. A. King, C. L. Byrne, “Noniterative compensation for the distance dependent detector response and photon attenuation in SPECT imaging,” IEEE Trans. Med. Imag. 13, 363–374 (1994).
[CrossRef]

Cerussi, A. E.

S. A. Walker, A. E. Cerussi, E. Gratton, “Back-projection image reconstruction using photon density waves in tissues,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, A. Katzir, eds., Proc. SPIE2389, 350–357 (1995).
[CrossRef]

Chance, B.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
[CrossRef] [PubMed]

B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
[CrossRef] [PubMed]

S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, A. Katzir, eds., Proc. SPIE1888, 78–89 (1993).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Images of inhomogeneous turbid media using diffuse photon density waves,” in 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.

D. Benaron, G. Muller, B. Chance, “Introduction: a medical perspective at the threshold of clinical optical tomography,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, 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 Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 3–9.

K. A. Kang, B. Chance, S. Zhao, S. Srinivasan, E. Patterson, R. Troupin, “Breast tumor characterization using near infrared spectroscopy,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, A. Katzir, eds., Proc. SPIE1888, 487–499 (1993).
[CrossRef]

Chang, J.

J. Chang, H. L. Graber, R. L. Barbour, “Image reconstruction of targets in random media from continuous wave laser measurements and simulated data,” in 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. 193–201.

Cheong, W.-F.

D. A. Benaron, J. P. van Houten, W.-F. Cheong, E. L. Kermit, R. A. King, “Early clinical results of time of flight optical tomography in a neonatal intensive care unit,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, A. Katzir, eds. Proc. SPIE2389, 582–596 (1995).
[CrossRef]

Chinnock, C.

C. Chinnock, “Medical diagnostics: laser mammography continues development,” Laser Focus World 32 (2), 38–39 (1996).

Choy, D. S. J.

R. R. Alfano, G. C. Tang, A. Pradhan, M. Bleich, D. S. J. Choy, E. Opher, “Steady state and time resolved laser fluorescence from normal and tumor lung and breast tissues,” J. Tumor Marker Oncol. 3, 165–172 (1988).

Colak, S. B.

S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, “Optical image reconstruction with deconvolutions in light diffusing media,” in Photon Propagation in Tissues, B. Chance, D. T. Delpy, G. J. Mueller, A. Katzir, eds., Proc. SPIE2626, 306–315 (1995).

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B. Chance, K. Kang, L. He, J. Weng, E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
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Srinivasan, S.

K. A. Kang, B. Chance, S. Zhao, S. Srinivasan, E. Patterson, R. Troupin, “Breast tumor characterization using near infrared spectroscopy,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, A. Katzir, eds., Proc. SPIE1888, 487–499 (1993).
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R. R. Alfano, G. C. Tang, A. Pradhan, M. Bleich, D. S. J. Choy, E. Opher, “Steady state and time resolved laser fluorescence from normal and tumor lung and breast tissues,” J. Tumor Marker Oncol. 3, 165–172 (1988).

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F. A. Marks, H. W. Tomlinson, J. W. Brooksby, “A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, A. Katzir, eds., Proc. SPIE1888, 500–510 (1993).
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K. A. Kang, B. Chance, S. Zhao, S. Srinivasan, E. Patterson, R. Troupin, “Breast tumor characterization using near infrared spectroscopy,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, A. Katzir, eds., Proc. SPIE1888, 487–499 (1993).
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M. B. van der Mark, G. W. ’t Hooft, D. G. Papaioannou, S. B. Colak, N. A. A. J. van Asten, J. C. J. Paasschens, “Boundary effects in optical tomography,” presented at Biomedical Optics Society Europe ’96, Vienna, 7–11 September 1996.

S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, “Optical image reconstruction with deconvolutions in light diffusing media,” in Photon Propagation in Tissues, B. Chance, D. T. Delpy, G. J. Mueller, A. Katzir, eds., Proc. SPIE2626, 306–315 (1995).

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G. W. ’t Hooft, D. G. Papaioannou, J. J. M. Baselmans, M. J. C. van Gemert, “Dependence of image quality on optical parameters in time-resolved transillumination experiments,” in Laser Interaction with Hard and Soft Tissue, M. J. C. van Gemert, R. W. Steiner, L. O. Svaasand, H. Albrecht, eds., Proc. SPIE2077, 153–158 (1993).
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S. A. Walker, A. E. Cerussi, E. Gratton, “Back-projection image reconstruction using photon density waves in tissues,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, A. Katzir, eds., Proc. SPIE2389, 350–357 (1995).
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M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
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S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, A. Katzir, eds., Proc. SPIE1888, 78–89 (1993).
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Other (35)

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Y. Yamashita, M. Kaneko, “Visible and infrared diaphanoscopy for medical diagnostics,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, 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 Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 283–316.

J. Chang, H. L. Graber, R. L. Barbour, “Image reconstruction of targets in random media from continuous wave laser measurements and simulated data,” in 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. 193–201.

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S. R. Arridge, “The forward and inverse problems in time resolved infrared imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, 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 Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 35–64.

J. Kolzer, G. Mitic, J. Otto, W. Zinth, “Measurement of the optical properties of breast tissue using time-resolved transillumination,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Preizzhev, V. V. Tuchin, A. Katzir, eds., Proc. SPIE2326, 143–152 (1994).
[CrossRef]

M. Kashke, H. Jess, G. Gaida, J.-M. Kaltenbach, W. Wrobel, “Transillumination imaging of tissue by phase modulation,” in 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. 88–92.

H. Jess, K. T. Moesta, G. Gaida, W. Walch, H. Erdl, P. M. Schlag, M. Kaschke, “Optical mammography at Carl Zeiss,” presented at Biomedical Optics Society Europe ’95, Barcelona, Spain, 12–16 September 1995.

S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, “Optical image reconstruction with deconvolutions in light diffusing media,” in Photon Propagation in Tissues, B. Chance, D. T. Delpy, G. J. Mueller, A. Katzir, eds., Proc. SPIE2626, 306–315 (1995).

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H. Schomberg, “Nonlinear image reconstruction from projections of ultrasonic travel times and electric current densities,” in Mathematical Aspects of Computerized Tomography, G. T. Herman, F. Natterer, eds., Vol. 8 of Springer Lecture Notes in Medical Informatics Series (Springer, Berlin, 1981), pp. 270–291.

D. A. Benaron, J. P. van Houten, W.-F. Cheong, E. L. Kermit, R. A. King, “Early clinical results of time of flight optical tomography in a neonatal intensive care unit,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, A. Katzir, eds. Proc. SPIE2389, 582–596 (1995).
[CrossRef]

F. A. Marks, H. W. Tomlinson, J. W. Brooksby, “A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, A. Katzir, eds., Proc. SPIE1888, 500–510 (1993).
[CrossRef]

D. Benaron, G. Muller, B. Chance, “Introduction: a medical perspective at the threshold of clinical optical tomography,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, 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 Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 3–9.

K. A. Kang, B. Chance, S. Zhao, S. Srinivasan, E. Patterson, R. Troupin, “Breast tumor characterization using near infrared spectroscopy,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, A. Katzir, eds., Proc. SPIE1888, 487–499 (1993).
[CrossRef]

D. G. Papaioannou, S. B. Colak, G. W. ’t Hooft, “Resolution and sensitivity limits of optical imaging in highly scattering media,” in Photon Propagation in Tissues, B. Chance, D. T. Delpy, G. J. Mueller, A. Katzir, eds., Proc. SPIE2626, 218–227 (1995).
[CrossRef]

See, for example, papers in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, 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 Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993).

D. Boas, M. O’Leary, A. Yodh, University of Pennsylvania, Photon Migration Imaging (pmi) Software Package; User’s information is available on the Internet. The source code is available by ftp at sol1.lrsm.upenn.edu in /pub/pmi.

S. L. Jacques, M. R. Ostermeyer, L. Wang, A. H. Hielscher, “Effects of sources, boundaries, and heterogeneities on photon migration,” in 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. 83–87.

M. R. Ostermeyer, S. L. Jacques, A. H. Hielscher, L. Wang, “Accelerated modeling of light transport in heterogeneous tissue using superposition of virtual sources,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Priezzhev, V. V. Tuchin, A. Katzir, eds., Proc. SPIE2326, 56–64 (1995).
[CrossRef]

M. R. Ostermeyer, S. L. Jacques, “Perturbation theory for optical diffusion theory: a general approach for absorbing and scattering objects in tissue,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, A. Katzir, eds., Proc. SPIE2389, 98–102 (1995).
[CrossRef]

J.-M. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Ref. 34, pp. 65–86.

S. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, A. Katzir, eds., Proc. SPIE1888, 78–89 (1993).
[CrossRef]

E. M. Sevick, C. L. Burch, J. K. Frisoli, M. L. Johnson, K. Nowaczyk, H. Szmacinski, J. R. Lakowicz, “The physical basis of biomedical optical imaging using time-dependent measurements of photon migration in the frequency domain,” in Ref. 34, pp. 485–512.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Images of inhomogeneous turbid media using diffuse photon density waves,” in 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.

G. W. ’t Hooft, D. G. Papaioannou, J. J. M. Baselmans, M. J. C. van Gemert, “Dependence of image quality on optical parameters in time-resolved transillumination experiments,” in Laser Interaction with Hard and Soft Tissue, M. J. C. van Gemert, R. W. Steiner, L. O. Svaasand, H. Albrecht, eds., Proc. SPIE2077, 153–158 (1993).
[CrossRef]

L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. Dasari, M. S. Feld, “Photon paths in turbid media: theory and experimental observation,” in 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. 153–155.

A. Hielscher, F. K. Tittel, “Photon density wave diffraction tomography,” in 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. 78–82.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986), Chap. V, pp. 128–137.

D. C. Youla, “Mathematical theory of image restoration by the method of convex projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), Chap. 2.

See R. R. Alfano, ed., Advances in Optical Imaging and Photon Migration, Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994).

See papers in Tissue Optics—Applications in Medical Diagnostics and Therapy, V. V. Tuchin, ed., Vol. MS102 of SPIE Milestone Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1994).

S. W. Rowland, “Computer implementation of image reconstruction formulas,” in Image Reconstruction from Projections, G. T. Herman, ed. (Springer-Verlag, Berlin, 1979), pp. 9–79.
[CrossRef]

S. A. Walker, A. E. Cerussi, E. Gratton, “Back-projection image reconstruction using photon density waves in tissues,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, A. Katzir, eds., Proc. SPIE2389, 350–357 (1995).
[CrossRef]

M. B. van der Mark, G. W. ’t Hooft, D. G. Papaioannou, S. B. Colak, N. A. A. J. van Asten, J. C. J. Paasschens, “Boundary effects in optical tomography,” presented at Biomedical Optics Society Europe ’96, Vienna, 7–11 September 1996.

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Figures (60)

Fig. 1
Fig. 1

Image of the acronym PMS (for Philips Medical Systems) immersed in the center of a 10-mm-thick solution of 10% intralipid30s′ = 1/mm). The letter height is 6 mm. NIR laser pulses at the 780-nm wavelength are used for imaging. Here all the photons transmitted through the cell are integrated in time, giving an image that would approximate cw light projection. The transmitted pulse had a total width of ∼1 ns.

Fig. 2
Fig. 2

Image of the same phantom as in Fig. 1, with the acronym PMS, imaged only by the first-arriving ballistic photons. Photons within the first 12 ps of the transmitted pulse were used to form the image. Note the improvement in image sharpness compared with that of Fig. 1 with cw (integrated) light.

Fig. 3
Fig. 3

Spatial dependence of the perturbation function P(r) on a small point absorber is shown in the (x/a, y/a, 0) plane for rS = (-a, 0, 0), rD = (a, 0, 0).

Fig. 4
Fig. 4

Spatial dependence of the weight function W(r), which represents the inverse assignment of a perturbation strength that causes an observed measured transmitted intensity difference at a detector position rD. The plot is in the (x/a, y/a, 0) plane. The source and detector locations are rS = (- a, 0, 0) and rD = (a, 0, 0), respectively.

Fig. 5
Fig. 5

Perturbed part of the photon density computed within the diffusion theory for (a) a small absorber, (b) a small scatterer, with a source located at position 0.0 on the y axis for each case. The results are plotted only to demonstrate the chargelike and dipolelike effects of these perturbations of absorbing and scattering objects. The quantitative parameters are not relevant for the present discussion.

Fig. 6
Fig. 6

Spatial dependence of the banana function B(r) that represents the amount of blurring introduced in images in diffuse media. The plot is in the (x/a, y/a, 0) plane. The source and the detector locations are rS = (-a, 0, 0) and rD = (a, 0, 0), respectively.

Fig. 7
Fig. 7

Density profile of B(r) as a function of transverse dimension x of the contour picture shown in Fig. 6 for different distances from the source. These profiles are given approximately at distances of 3(rS - rD)/6 (center of banana), 2(rS - rD)/6, and (rS - rD)/6 (closest to the source–detector) for the outermost, middle, and inner curves, respectively, where (rS - rD) = 100 mm. Note that the width of the banana does not change rapidly unless the area of interest is very close to a source or a detector.

Fig. 8
Fig. 8

Parallel-beam projection geometry for obtaining the optical image of an object. The plane in the figure shows the (x, y) plane of a Cartesian coordinate system. The object is an infinite cylinder centered at the origin, r = {0, 0, 0}, and its cross section is indicated as the darker circle in the middle. The object is characterized with optical parameters μs′, μa, and n. This object is embedded in an infinite medium with optical parameters μs0′, μa0, and n0.

Fig. 9
Fig. 9

Example of intensity projection in transparent media, TST. This projection data is for a theoretical phantom with a more absorbing cylindrical object of 5-mm radius embedded in a homogeneous nonabsorbing medium. The numbers on the right axis give the maximum and the minimum quantitative values for this graph.

Fig. 10
Fig. 10

Light-projection data for a theoretical phantom with a scattering cylindrical object, OBJ, embedded in a scattering medium. The optical parameters of this system are given in Table 1. Parallel-beam projection geometry is used for obtaining the optical image of an object. The object is an infinite cylinder centered at the origin r = {0, 0, 0}. The numbers on the right axis give the maximum and the minimum quantitative photon density values for this graph.

Fig. 11
Fig. 11

Comparison of the light-projection data with a large cylinder from Fig. 10 (dotted curve) with those of a much smaller cylinder (solid curve) that represent the PSF of the background scattering medium. The optical parameters are given in Table 1. The numbers on the right axis give the maximum and the minimum quantitative values for this graph (which apply only to the solid curve). The dotted curve is redrawn from Fig. 10 for comparison purposes, and therefore the quantitative values do not apply on that projection.

Fig. 12
Fig. 12

Light projections that show the effect of object (cylinder) position and the strength of perturbation on the intensity projection shape. These demonstrate the effects of the spatially dependent blurring and the nonlinear image formation. The projections for the same object as shown in Figs. 10 and 11, OBJ, are drawn as dotted curves, the one on the right (circle labeled C) is for a centered object, and the one on the left (circle labeled OC) is for an off-centered object that is closer to a source line than to the detector line. The solid curves represent corresponding projections for another cylinder with the same parameters, except that its absorption coefficient is 50 times higher than that of the previous cylinder. In summary, a is the reference centered cylinder with weak absorption. Solid curve b shows what happens to the shape of the projection if the strength of the absorption is increased by a factor of 50. Dotted curve c and solid curve d show the shape of the projection if the weakly and the strongly absorbing cylinders, respectively, are moved off center to a position 10 mm away from the source line. The strongly absorbing cylinders distort the projection rays around themselves and result in the indicated difference as compared with a weak cylinder. This is a direct demonstration of the nonlinearities in the projection image formation.

Fig. 13
Fig. 13

X-ray normalized attenuation coefficient K projection that corresponds to the intensity projection in Fig. 9. The numbers on the right axis give quantitative values. These values are used in the text to compute contrast in the images.

Fig. 14
Fig. 14

Normalized attenuation coefficient K projection that corresponds to the diffuse intensity projection in Fig. 10. The numbers on the right axis give quantitative values.

Fig. 15
Fig. 15

Absolute values of the attenuation coefficient projections for cylinders of 5-mm radius in the same medium as in the previous figures. The background attenuation coefficient (0.173205/mm) has been subtracted from these projections. These correspond to the intensity projections given in Fig. 12. The labeling is as follows: centered cylinder with weak absorption (curve a, 1% perturbation), centered strong absorption (curve b, ×50 perturbation), and an off-centered weak absorption (curve c, 1% perturbation, ∼10 mm away from the source line).

Fig. 16
Fig. 16

Normalized values of the attenuation coefficient projections for cylinders of 5-mm radius as in Fig. 15. Dotted curve a is the reference centered cylinder with weak absorption. Solid curve b shows what happens to the shape of the projection if the strength of the absorption is increased by a factor of 50. Solid curve c shows the shape of the projection if the weakly absorbing centered cylinder is moved off center, as in Fig. 12.

Fig. 17
Fig. 17

Fan-beam projection geometry that shows the arrangement of sources and detectors schematically. The source and the detector locations are equally spaced on the periphery of the circular area. In the numerical examples and the experiments that are described below, we take 32 equally spaced source positions and 32 equally spaced detector positions. The source and the detector locations are interlaced with respect to each other.

Fig. 18
Fig. 18

Fan-beam light-projection data in (ϕS, ϕD) = (iS, jD) space for a theoretical phantom with a scattering cylindrical object, OBJ, embedded in scattering media. The optical parameters of this system are given in Table 2.

Fig. 19
Fig. 19

Normalized fan-beam attenuation projection data K in (ϕS, ϕD) = (iS, jD) space for a theoretical phantom with a cylindrical object, OBJ, embedded in scattering media. This figure is obtained from the intensity projection data in Fig. 18 by the use of the procedure described in the text for the attenuation constant calculations in fan-beam geometry.

Fig. 20
Fig. 20

Filtered backprojection tomographic image of the x-ray projection discussed in Fig. 13 for a cylinder of 5-mm diameter in transparent (TST) media.

Fig. 21
Fig. 21

Cross-sectional view of the tomographic image in Fig. 20. The numbers on the right axis give the maximum and the minimum quantitative values for this graph.

Fig. 22
Fig. 22

Results of filtered backprojection tomography when effects coming from light diffusion in the attenuation coefficient projection data of Fig. 14 are ignored. The FWHM of the peak in the image is ∼28 mm, and its peak is κmax = κ0 + 1.1 × 10-4 (1/mm), where κ0 = 0.174069/mm is the background attenuation corresponding to the base at K = 0.

Fig. 23
Fig. 23

Cross-sectional view of the tomographic image shown in Fig. 22. The half-width of the image is 28 mm (FWHM), which is much larger than the input object diameter, which is 10 mm. The numbers on the right axis give the maximum and the minimum quantitative values for this graph. These values are used in the text to compute contrast in the images.

Fig. 24
Fig. 24

Results of filtered backprojection tomography with a deconvolution (DeConv) step for correcting light diffusion in the image of Fig. 22. The FWHM of the peak in the deconvolved image is ∼12 mm and its peak is κmax = κ0 + 0.7 × 10-3 (1/mm), where κ0 = 0.174069/mm is the background attenuation coefficient that corresponds to the base of the figure at K = 0. These values agree well with the object parameters.

Fig. 25
Fig. 25

Cross-sectional view of the tomographic image shown in Fig. 24. The half-width of the image is 12 mm (FWHM), which agrees well with the input object diameter, which is 10 mm. The numbers on the right axis give the maximum and the minimum quantitative values for this graph.

Fig. 26
Fig. 26

Schematic illustration on the spatial dependence of the blurring function in the object space.

Fig. 27
Fig. 27

Schematic illustration of the projection geometry for the example that demonstrates the spatially dependent deblurring process. A thin offset cylinder is placed in a turbid medium and imaged by two orthogonal projections. One of the projections (0° angle) is shown in the figure. The other is orthogonal to that at 90°.

Fig. 28
Fig. 28

Normalized attenuation coefficient K projections that correspond to the thin offset cylinder geometry described in the diffuse intensity projection in Fig. 27. The optical parameters are given in Table 1. Parallel-beam geometry is used for obtaining the optical projection image of the object. The object is an infinite cylinder offset from the origin r = {0, 45, 0} mm. The dotted curves correspond to the two banana functions calculated by assuming an object–source (or –detector) distance found from the peak of the first-order image, which is discussed below.

Fig. 29
Fig. 29

Contour and surface plots of the tomographic first-order (no deblurring) image of the offset cylinder with the projections shown in Fig. 28. Only two projections are used.

Fig. 30
Fig. 30

Contour and surface plots of the tomographic first order (no deblurring) image of a centered cylinder with identical properties as those discussed for the offset cylinder. Only two projections are used.

Fig. 31
Fig. 31

FFT’s of the attenuation coefficient K projections given in Fig. 28. The dotted curves correspond to the FFT’s of the two banana functions shown in the same figure. We calculated these banana functions by taking the spatial dependence of the broadening.

Fig. 32
Fig. 32

Contour and surface plots of the tomographic second-order deblurred image of the offset cylinder with the projections shown in Fig. 28. As can be seen from the comparison of this image with the one shown in Fig. 29, the resolution has improved dramatically.

Fig. 33
Fig. 33

Contour and surface plots of the tomographic second-order deblurred image of the offset cylinder with the projections shown in Fig. 28. The resolution has improved dramatically in comparison with the one shown in Fig. 30.

Fig. 34
Fig. 34

Projection ray paths probing a media with a weakly absorbing cylinder (left) and a strongly absorbing cylinder (right). These are calculated semiempirically by the replacement of the refractive-index inhomogeneity term in the geometric ray equation with the attenuation constant changes. The distortions that are due to the strength of the object on the right are clearly visible, and these affect the shape of the bananas that are used to deblur the image.

Fig. 35
Fig. 35

Coordinate transformations used to compensate for the effects of ray-path bending demonstrated in Fig. 33.

Fig. 36
Fig. 36

Strongly absorbing cylinders distort the projection rays around themselves and result in the indicated difference compared with a weak cylinder. This curve gives the difference between curves a and b in Fig. 16.

Fig. 37
Fig. 37

Results of filtered backprojection tomography for a strong object with an absorption coefficient 50 times higher than that of the background medium. This object is identical to the one discussed in Fig. 24, except that the absorption coefficient of the object in Fig. 24 is only 1% higher than that of the background medium. Despite this difference in the strengths of these objects, the reconstructed image above, which applies deblurring and coordinate transformations to compensate for diffusion and imaging nonlinearities, respectively, is of comparable resolution with that of the image of the weak object shown in Fig. 24. The FWHM of the peak in the deconvolved image above is still ∼12 mm but its shape is rounded because of the additional filtering needed to decrease the effects of noise in the projections processed further for coordinate transformations.

Fig. 38
Fig. 38

Central cross section of the reconstructed image shown in Fig. 37. The profile of the image without deblurring is also indicated by the dotted curve for comparison.

Fig. 39
Fig. 39

Radial dependence of the width of the blurring function along the radial direction in the fan-beam tomography geometry. This is calculated from the dependence of the width of the banana function as a function of the source–detector distance for typical diffuse medium parameters.

Fig. 40
Fig. 40

Coordinate transformation for radial dependence of the deblurring. The radial expansion Δrn′, which is necessary at different normalized radial positions rn for equalizing the blurring process, is shown.

Fig. 41
Fig. 41

Tomographic image of the cylindrical object with the x-ray backprojection tomography in the (x, y) Cartesian coordinate system. This is obtained by transformation from Fig. 42.

Fig. 42
Fig. 42

First-order tomographic image of the cylindrical object with the x-ray backprojection tomography (r, ϕ) in the polar coordinate system.

Fig. 43
Fig. 43

Deblurring function used to deconvolve the first-order image along the ϕ direction. As discussed in the text, this is only an approximation to the real PSF because it ignores many of the additional corrections necessary in the fan-beam geometry.

Fig. 44
Fig. 44

Deblurring function used to deconvolve the first-order image along the r direction. Because of the radial coordinate transformations discussed in Fig. 40, a PSF with a single width is sufficient in the radial direction.

Fig. 45
Fig. 45

Second-order deblurred tomographic image of the cylindrical object with the x-ray backprojection tomography in the (r, ϕ) polar coordinate system.

Fig. 46
Fig. 46

Second-order deblurred tomographic image of the cylindrical object with the x-ray backprojection tomography in the (x, y) Cartesian coordinate system. This is obtained by transformation from Fig. 45. A comparison of the contours of the different gray levels in this figure for the deblurred image and the image in Fig. 41 before deblurring demonstrates the improvement in resolution in this geometry. The same color coding is used in both figures.

Fig. 47
Fig. 47

Geometry of the parallel-beam projection experiments with Delrin cylinders in an intralipid solution.

Fig. 48
Fig. 48

Optical projection images of two Delrin cylinders with 6-mm- and 10-mm-diameters embedded in an intralipid solution. The real profiles and positions of the objects are indicated schematically as shaded areas. The projections are taken from four angles corresponding to the views from 0°, 90°, 180°, and 270°.

Fig. 49
Fig. 49

2D surface plots for the reconstructed slice image of the double object sample with 10-mm- and 6-mm-diameter Delrin cylinders embedded in the center of a 1% intralipid solution tank. The projections of the object are given in Fig. 48. The original image is shown on the top, and the deblurred image is shown on the bottom of the figure. The deblurring process resolves the two objects clearly. The profiles of these objects, however, are highly smoothed because of the large amount of noise present in the experimental data.

Fig. 50
Fig. 50

Projections of a complex object composed of four cylinders with different optical properties. This composite object consists of two strongly absorbing cylinders (blackened metallic wires of ∼1-mm diameter) and two weakly scattering larger diameter cylinders (Delrin cylinders of 6-mm and 10-mm diameters). Note that the projections do not reveal any internal structure of this complex object. The dotted and solid curves correspond to projections taken from opposite sides.

Fig. 51
Fig. 51

Surface plot of the deblurred reconstructed image from the projections shown in Fig. 50. Deblurring helps to identify the two peaks coming from the two blackened metallic wires. However, the two Delrin cylinders are not resolved because of the screening effects of the high absorption contrast of the metallic wires. No corrections for imaging nonlinearities were applied.

Fig. 52
Fig. 52

Two orthogonal projections of the deblurred image given in Fig. 51. As discussed above, the oscillations around the objects are due to the crude 1/r filter with a sharp cutoff used in our calculations.

Fig. 53
Fig. 53

(a) Picture, (b) reconstructed image of an irregular 3D object in an intralipid. As seen in (a) the object consists of an irregularly shaped Delrin material attached to a black metal holder, just above the Delrin object. The reconstructed image, shown as an isosurface plot, reproduces the shape of the object well with realistic aspect ratios. Although the effects of nonlinearities were taken into account in the image reconstruction, its distortions are still evident, with the large unrealistic boundary close to the top of the image near the metallic object.

Fig. 54
Fig. 54

Schematic system layout of the optical tomography experiments in the fan-beam geometry. A/D, analog-to-digital; H.V., high voltage; PMT, photomultiplier tube.

Fig. 55
Fig. 55

Picture of the fan-beam optical tomography experimental setup. 32 equally spaced detector fibers are placed in the periphery of the ring around the cylindrical phantom. There is a single source fiber placed in the middle of two detector fiber positions. The cylindrical phantom is rotated around its axis to 32 equally spaced angles, effectively giving 32 equally spaced source positions interlaced between the detector positions.

Fig. 56
Fig. 56

Gray-level 2D tomographic image of a 12-mm-diameter cylindrical inhomogeneity probed in the fan-beam setup shown in Fig. 55. These images illustrate the effects of deblurring and provide a comparison of the original (left) and deblurred (right) images. The same gray-level coding has been applied to both images in order to provide a correct comparison to determine resolution improvements.

Fig. 57
Fig. 57

3D isosurface plot of the reconstructed image of a 5 mm × 5 mm × 5 mm absorptive cylindrical object in a nylon cylinder probed in the fan-beam geometry.

Fig. 58
Fig. 58

Cross-sectional contour plot of the image from Fig. 57. This image provides a more realistic deblurred profile of the inhomogeneity that cannot be judged from the previous 3D isosurface plot.

Fig. 59
Fig. 59

Cross-sectional contour plot of a phantom containing almost fully chicken breast tissue in the fan-beam projection setup. The gaps in the bucket containing this animal tissue were filled with intralipid at 1.2% concentration. There was also a single piece of chicken leg with bone included in the phantom. The high-intensity area in the image is the location of this piece with the bone. The images are deblurred and represent measurements at two different wavelengths as indicated. The difference in the images at these two wavelengths is not strong.

Fig. 60
Fig. 60

Cross-sectional color-coded contour plots of a phantom containing a single piece of chicken breast meat in the fan-beam projection setup. The rest of the bucket was filled with intralipid at various concentrations. The images are not deblurred. The measurements with 1.2% intralipid were repeated at two different wavelengths, as indicated. As discussed in the text, the difference in the images reflects a differing object contrast, which changes between a scattering and an absorptive property, depending on the wavelength and the intralipid concentration. The purpose of this figure is to emphasize the differences in reconstructed images for objects that have the same physical shape but different optical parameters. With further developments in optical tomography, these differences are expected to facilitate the realization of separate images for absorption and scattering parameters, leading to a better optical diagnostics potential.

Tables (2)

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Table 1 Optical and Geometrical Parameters of the Object and the Phantom

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Table 2 Optical and Geometrical Parameters of the Object and the Phantom

Equations (37)

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S=Soδr-rS=Ioδr-rS.
Δφr, t-cμaDφr, t-1Dtφr, t=-1DSr, t,
φ0r=So4πDexp-κr-rSr-rS,
ψ0r=So4πDexp-κr-rSr-rS.
Ioutr=A0ψ0r=SoA04πDexp-κr-rSr-rS
φr=φ0r+-qα1r, r1,
α1r, r1=So4πDexp-κr1-rSr1-rSexp-κr1-rr1-r.
φr=φ0r1-qα1r, r1φ0r=φ0r1-qPr, r1,
Pr, r1=exp-κr1-rS+r1-r-r-rSr1-rSr1-r/r-rS
qr=1PrD, r1-φrDφ0rD
qr=1PrD, rφ0rD-φrDφ0rD=WrD, rΔφrDφ0rD,
ΔφrD=φ0rD-φrD,
WrD, r=1PrD, r=r1-rSr1-rD/rD-rSexp-κr1-rS+r1-rD-rD-rS.
α1rS, rD, r=-So4πDpr-rS·rD-rrD-r+q×exp-κr-rSr-rSexp-κr-rDr-rD,
Br=PrD, r/Pa, 0, 0; x, 0, 0,
Br=exp-κr--axˆr--axˆexp-κr-axˆr-axˆexp-κx--ax--aexp-κx-ax-a.
s=x cos θ+y sin θ.
Is, θ=I0·expLμax, y, E, tdl,
Ins, θ=Is, θI0=exp--ss-ssμax, y×δx cos θ+y sin θ-sdxdy.
IoutrD=Iin4πD0exp-κ0rD-rSrD-rS×1-1VnVolqrPrD, rdr
IoutrD=IinC01-1VnVolqrPrD, rdr,
Ins, θ=Iouts, θC0Iin=1-qx, yPnrD, x, ydxdy,
κxrays, θ=IPs, θL=lnI0/Is, θL,
K=κs, θ-κminκmax-κmin,
κ0|21=1r2-r1lnI1r1I2r2=lnI2r2-lnI1r1r2-r1
κs, θ=κij-00=κ0+1rSDlnI00Iij=κ0+1rSDlnI00Is, θ,
κ0|10=1r1-r0lnI0r0I1r1=lnI1r1-lnI0r0r1-r0.
Δκij=κij-κ0,ave=lnI1r1-lnI00r00r1-r00-κ0,ave.
Mfy=VolKx, y; fxfxdx,
A·u=b,
Au·u=bu
κx, y=0π-F-1HfsFκs, θ×δx cos θ+y sin θ-sdsdθ,
κx, y=0πF-1HfsFκs, θdθ,
Hfs=fsRectfs2fs0,
Kx, y=0πF-1HfsFκs, θFPSFs, θdθ,
κx, y=κx, θ00κy, θ90,
x=r+Ceddxφ1x,

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