Abstract

A hybrid radiative-transfer–diffusion model for optical tomography is proposed. The light propagation is modeled with the radiative-transfer equation in the vicinity of the laser sources, and the diffusion approximation is used elsewhere in the domain. The solution of the radiative-transfer equation is used to construct a Dirichlet boundary condition for the diffusion approximation on a fictitious interface within the object. This boundary condition constitutes an approximative distributed source model for the diffusion approximation in the remaining area. The results from the proposed approach are compared with finite-element solutions of the radiative-transfer equation and the diffusion approximation and Monte Carlo simulation. The results show that the method improves the accuracy of the forward model compared with the conventional diffusion model.

© 2005 Optical Society of America

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2003 (4)

S. Prince, V. Kolehmainen, J. Kaipio, M. Franceschini, D. Boas, S. Arridge, “Time-series estimation of biological factors in optical diffusion tomography,” Phys. Med. Biol. 48, 1491–1504 (2003).
[CrossRef] [PubMed]

J. Heino, S. R. Arridge, J. Sikora, E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E 68, 031908: 1–8, (2003).
[CrossRef]

V. Kolehmainen, S. Prince, S. Arridge, J. Kaipio, “State-estimation approach to the nonstationary optical tomography problem,” J. Opt. Soc. Am. A 20, 876–889 (2003).
[CrossRef]

T. Hayashi, Y. Kashio, E. Okada, “Hybrid Monte Carlo-diffusion method for light propagation in tissue with a low-scattering region,” Appl. Opt. 42, 2888–2896 (2003).
[CrossRef] [PubMed]

2002 (1)

J. Heino, E. Somersalo, “Estimation of optical absorption in anisotropic background,” Inverse Probl. 18, 559–573 (2002).
[CrossRef]

2000 (4)

1999 (2)

J. Tervo, P. Kolmonen, M. Vauhkonen, L. Heikkinen, J. Kaipio, “A finite-element model of electron transport in radiation therapy and a related inverse problem,” Inverse Probl. 15, 1345–1361 (1999).
[CrossRef]

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

1998 (3)

A. H. Hielscher, R. E. Alcouffe, R. L. Barbour, “Comparison of finite difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

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

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, P. M. Schlag, “Contrast features of breast cancer in frequency-domain laser scanning mammography,” J. Biomed. Opt. 3, 129–136 (1998).
[CrossRef] [PubMed]

1997 (2)

A. Villringer, B. Chance, “Noninvasive optical spectroscopy and imaging of human brain function,” Trends Neurosci. 20, 435–442 (1997).
[CrossRef] [PubMed]

K. R. Heereken, H. Obrig, R. Wenzel, K. Eberle, J. Ruben, K. Villringer, R. Kurth, A. Villringer, “Cerebral haemoglobin oxygenation during sustained visual stimulation—a near-infrared spectroscopy study,” Philos. Trans. R. Soc. London Ser. B 352, 743–750 (1997).
[CrossRef]

1996 (1)

J. P. van Houten, D. A. Benaron, S. Splilman, D. K. Stevenson, “Imaging brain injury using time-resolved near-infrared light scanning,” Pediatr. Res. 39, 470–476 (1996).
[CrossRef] [PubMed]

1995 (1)

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite-element model for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

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]

L. Wang, S. L. Jacques, “Hybrid model of Monte Carlo simulation diffusion theory for light reflectance by turbid media,” J. Opt. Soc. Am. A 10, 1746–1752 (1993).
[CrossRef]

1990 (1)

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near-infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).

1988 (1)

1983 (1)

1980 (1)

1941 (1)

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Alcouffe, R. E.

A. H. Hielscher, R. E. Alcouffe, R. L. Barbour, “Comparison of finite difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

Alexandrakis, G.

Amaldi, E.

E. Amaldi, “The production and slowing down of neutrons,” in Encyclopedia of Physics, S. Flügge, ed., Vol. 38/2Neutrons and Related Gamma Ray Problems (Springer-Verlag, Berlin, 1959), pp. 1–659.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, San Diego, 1985).

Arridge, S.

S. Prince, V. Kolehmainen, J. Kaipio, M. Franceschini, D. Boas, S. Arridge, “Time-series estimation of biological factors in optical diffusion tomography,” Phys. Med. Biol. 48, 1491–1504 (2003).
[CrossRef] [PubMed]

V. Kolehmainen, S. Prince, S. Arridge, J. Kaipio, “State-estimation approach to the nonstationary optical tomography problem,” J. Opt. Soc. Am. A 20, 876–889 (2003).
[CrossRef]

Arridge, S. R.

J. Heino, S. R. Arridge, J. Sikora, E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E 68, 031908: 1–8, (2003).
[CrossRef]

S. R. Arridge, H. Dehghani, M. Schweiger, E. Okada, “The finite-element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

H. Dehghani, S. R. Arridge, M. Schweiger, D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A 17, 1659–1667 (2000).
[CrossRef]

J. Riley, H. Dehghani, M. Schweiger, S. R. Arridge, J. Ripoll, M. Nieto-Vesperinas, “3D optical tomography in the presence of void regions,” Opt. Express 7, 462–467 (2000).
[CrossRef] [PubMed]

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

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite-element model for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

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, M. Schweiger, “A general framework for iterative reconstruction algorithms in optical tomography, using a finite-element method,” in Computational Radiology and Imaging: Therapy and Diagnosis, C. Borgers, F. Natterer, eds., IMA Volumes in Mathematics and its Applications, Vol. 110 (Springer-Verlag, Berlin, 1998), pp. 45–70.
[CrossRef]

Barbour, R. L.

A. H. Hielscher, R. E. Alcouffe, R. L. Barbour, “Comparison of finite difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

Benaron, D. A.

J. P. van Houten, D. A. Benaron, S. Splilman, D. K. Stevenson, “Imaging brain injury using time-resolved near-infrared light scanning,” Pediatr. Res. 39, 470–476 (1996).
[CrossRef] [PubMed]

Bernarding, J.

C. Hirth, H. Obrig, K. Villringer, A. Thiel, J. Bernarding, W. Muhhlnickel, H. Flor, U. Dirnagl, A. Villringer, “Noninvasive functional mapping of the human motor cortex using near-infrared spectroscopy,” NeuroReport7, 1977–1981 (1996).
[CrossRef] [PubMed]

Boas, D.

S. Prince, V. Kolehmainen, J. Kaipio, M. Franceschini, D. Boas, S. Arridge, “Time-series estimation of biological factors in optical diffusion tomography,” Phys. Med. Biol. 48, 1491–1504 (2003).
[CrossRef] [PubMed]

Boas, D. A.

D. A. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1996).

Bosch, J. J. T.

Case, M. C.

M. C. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Chance, B.

A. Villringer, B. Chance, “Noninvasive optical spectroscopy and imaging of human brain function,” Trends Neurosci. 20, 435–442 (1997).
[CrossRef] [PubMed]

Cope, M.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near-infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).

Dehghani, H.

Delpy, D. T.

H. Dehghani, S. R. Arridge, M. Schweiger, D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A 17, 1659–1667 (2000).
[CrossRef]

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite-element model for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

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]

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near-infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).

Dirnagl, U.

C. Hirth, H. Obrig, K. Villringer, A. Thiel, J. Bernarding, W. Muhhlnickel, H. Flor, U. Dirnagl, A. Villringer, “Noninvasive functional mapping of the human motor cortex using near-infrared spectroscopy,” NeuroReport7, 1977–1981 (1996).
[CrossRef] [PubMed]

Dorn, O.

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

Eberle, K.

K. R. Heereken, H. Obrig, R. Wenzel, K. Eberle, J. Ruben, K. Villringer, R. Kurth, A. Villringer, “Cerebral haemoglobin oxygenation during sustained visual stimulation—a near-infrared spectroscopy study,” Philos. Trans. R. Soc. London Ser. B 352, 743–750 (1997).
[CrossRef]

Edwards, A. D.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near-infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).

Erdl, H.

H. Jess, H. Erdl, K. T. Moesta, S. Fantini, M. A. Franceschini, E. Gratton, “Intensity modulated breast imaging: technology and clinical pilot study results,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 126–129.

Fantini, S.

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, P. M. Schlag, “Contrast features of breast cancer in frequency-domain laser scanning mammography,” J. Biomed. Opt. 3, 129–136 (1998).
[CrossRef] [PubMed]

H. Jess, H. Erdl, K. T. Moesta, S. Fantini, M. A. Franceschini, E. Gratton, “Intensity modulated breast imaging: technology and clinical pilot study results,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 126–129.

Farrell, T.

Ferwerda, H. A.

Flor, H.

C. Hirth, H. Obrig, K. Villringer, A. Thiel, J. Bernarding, W. Muhhlnickel, H. Flor, U. Dirnagl, A. Villringer, “Noninvasive functional mapping of the human motor cortex using near-infrared spectroscopy,” NeuroReport7, 1977–1981 (1996).
[CrossRef] [PubMed]

Franceschini, M.

S. Prince, V. Kolehmainen, J. Kaipio, M. Franceschini, D. Boas, S. Arridge, “Time-series estimation of biological factors in optical diffusion tomography,” Phys. Med. Biol. 48, 1491–1504 (2003).
[CrossRef] [PubMed]

Franceschini, M. A.

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, P. M. Schlag, “Contrast features of breast cancer in frequency-domain laser scanning mammography,” J. Biomed. Opt. 3, 129–136 (1998).
[CrossRef] [PubMed]

H. Jess, H. Erdl, K. T. Moesta, S. Fantini, M. A. Franceschini, E. Gratton, “Intensity modulated breast imaging: technology and clinical pilot study results,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 126–129.

Gratton, E.

H. Jess, H. Erdl, K. T. Moesta, S. Fantini, M. A. Franceschini, E. Gratton, “Intensity modulated breast imaging: technology and clinical pilot study results,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 126–129.

Greenstein, J. L.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Groenhuis, R. A. J.

Hayashi, T.

Heereken, K. R.

K. R. Heereken, H. Obrig, R. Wenzel, K. Eberle, J. Ruben, K. Villringer, R. Kurth, A. Villringer, “Cerebral haemoglobin oxygenation during sustained visual stimulation—a near-infrared spectroscopy study,” Philos. Trans. R. Soc. London Ser. B 352, 743–750 (1997).
[CrossRef]

Heikkinen, L.

J. Tervo, P. Kolmonen, M. Vauhkonen, L. Heikkinen, J. Kaipio, “A finite-element model of electron transport in radiation therapy and a related inverse problem,” Inverse Probl. 15, 1345–1361 (1999).
[CrossRef]

Heino, J.

J. Heino, S. R. Arridge, J. Sikora, E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E 68, 031908: 1–8, (2003).
[CrossRef]

J. Heino, E. Somersalo, “Estimation of optical absorption in anisotropic background,” Inverse Probl. 18, 559–573 (2002).
[CrossRef]

Henyey, L. G.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hielscher, A. H.

A. H. Hielscher, R. E. Alcouffe, R. L. Barbour, “Comparison of finite difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

Hiraoka, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite-element model for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

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]

Hirth, C.

C. Hirth, H. Obrig, K. Villringer, A. Thiel, J. Bernarding, W. Muhhlnickel, H. Flor, U. Dirnagl, A. Villringer, “Noninvasive functional mapping of the human motor cortex using near-infrared spectroscopy,” NeuroReport7, 1977–1981 (1996).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

Jacques, S. L.

Jess, H.

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, P. M. Schlag, “Contrast features of breast cancer in frequency-domain laser scanning mammography,” J. Biomed. Opt. 3, 129–136 (1998).
[CrossRef] [PubMed]

H. Jess, H. Erdl, K. T. Moesta, S. Fantini, M. A. Franceschini, E. Gratton, “Intensity modulated breast imaging: technology and clinical pilot study results,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 126–129.

Kaipio, J.

V. Kolehmainen, S. Prince, S. Arridge, J. Kaipio, “State-estimation approach to the nonstationary optical tomography problem,” J. Opt. Soc. Am. A 20, 876–889 (2003).
[CrossRef]

S. Prince, V. Kolehmainen, J. Kaipio, M. Franceschini, D. Boas, S. Arridge, “Time-series estimation of biological factors in optical diffusion tomography,” Phys. Med. Biol. 48, 1491–1504 (2003).
[CrossRef] [PubMed]

J. Tervo, P. Kolmonen, M. Vauhkonen, L. Heikkinen, J. Kaipio, “A finite-element model of electron transport in radiation therapy and a related inverse problem,” Inverse Probl. 15, 1345–1361 (1999).
[CrossRef]

Kalos, M.

M. Kalos, P. Whitlock, Monte Carlo Methods (Wiley, New York, 1986).
[CrossRef]

Kaltenbach, J. P.

J. P. Kaltenbach, M. Kaschke, “Frequency- and time-domain modeling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

Kaschke, M.

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, P. M. Schlag, “Contrast features of breast cancer in frequency-domain laser scanning mammography,” J. Biomed. Opt. 3, 129–136 (1998).
[CrossRef] [PubMed]

J. P. Kaltenbach, M. Kaschke, “Frequency- and time-domain modeling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

Kashio, Y.

Keijzer, M.

Kolehmainen, V.

V. Kolehmainen, S. Prince, S. Arridge, J. Kaipio, “State-estimation approach to the nonstationary optical tomography problem,” J. Opt. Soc. Am. A 20, 876–889 (2003).
[CrossRef]

S. Prince, V. Kolehmainen, J. Kaipio, M. Franceschini, D. Boas, S. Arridge, “Time-series estimation of biological factors in optical diffusion tomography,” Phys. Med. Biol. 48, 1491–1504 (2003).
[CrossRef] [PubMed]

V. Kolehmainen, “Novel approaches to image reconstruction in diffusion tomography,” Ph.D. dissertation (University of Kuopio, Kuopio, Finland, 2001).

Kolmonen, P.

J. Tervo, P. Kolmonen, M. Vauhkonen, L. Heikkinen, J. Kaipio, “A finite-element model of electron transport in radiation therapy and a related inverse problem,” Inverse Probl. 15, 1345–1361 (1999).
[CrossRef]

Kurth, R.

K. R. Heereken, H. Obrig, R. Wenzel, K. Eberle, J. Ruben, K. Villringer, R. Kurth, A. Villringer, “Cerebral haemoglobin oxygenation during sustained visual stimulation—a near-infrared spectroscopy study,” Philos. Trans. R. Soc. London Ser. B 352, 743–750 (1997).
[CrossRef]

Moesta, K. T.

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, P. M. Schlag, “Contrast features of breast cancer in frequency-domain laser scanning mammography,” J. Biomed. Opt. 3, 129–136 (1998).
[CrossRef] [PubMed]

H. Jess, H. Erdl, K. T. Moesta, S. Fantini, M. A. Franceschini, E. Gratton, “Intensity modulated breast imaging: technology and clinical pilot study results,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 126–129.

Muhhlnickel, W.

C. Hirth, H. Obrig, K. Villringer, A. Thiel, J. Bernarding, W. Muhhlnickel, H. Flor, U. Dirnagl, A. Villringer, “Noninvasive functional mapping of the human motor cortex using near-infrared spectroscopy,” NeuroReport7, 1977–1981 (1996).
[CrossRef] [PubMed]

Nieto-Vesperinas, M.

Obrig, H.

K. R. Heereken, H. Obrig, R. Wenzel, K. Eberle, J. Ruben, K. Villringer, R. Kurth, A. Villringer, “Cerebral haemoglobin oxygenation during sustained visual stimulation—a near-infrared spectroscopy study,” Philos. Trans. R. Soc. London Ser. B 352, 743–750 (1997).
[CrossRef]

C. Hirth, H. Obrig, K. Villringer, A. Thiel, J. Bernarding, W. Muhhlnickel, H. Flor, U. Dirnagl, A. Villringer, “Noninvasive functional mapping of the human motor cortex using near-infrared spectroscopy,” NeuroReport7, 1977–1981 (1996).
[CrossRef] [PubMed]

Okada, E.

T. Hayashi, Y. Kashio, E. Okada, “Hybrid Monte Carlo-diffusion method for light propagation in tissue with a low-scattering region,” Appl. Opt. 42, 2888–2896 (2003).
[CrossRef] [PubMed]

S. R. Arridge, H. Dehghani, M. Schweiger, E. Okada, “The finite-element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

Patterson, M.

Prince, S.

S. Prince, V. Kolehmainen, J. Kaipio, M. Franceschini, D. Boas, S. Arridge, “Time-series estimation of biological factors in optical diffusion tomography,” Phys. Med. Biol. 48, 1491–1504 (2003).
[CrossRef] [PubMed]

V. Kolehmainen, S. Prince, S. Arridge, J. Kaipio, “State-estimation approach to the nonstationary optical tomography problem,” J. Opt. Soc. Am. A 20, 876–889 (2003).
[CrossRef]

Reynolds, E. O. R.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near-infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).

Richardson, C. E.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near-infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).

Riley, J.

Ripoll, J.

Ruben, J.

K. R. Heereken, H. Obrig, R. Wenzel, K. Eberle, J. Ruben, K. Villringer, R. Kurth, A. Villringer, “Cerebral haemoglobin oxygenation during sustained visual stimulation—a near-infrared spectroscopy study,” Philos. Trans. R. Soc. London Ser. B 352, 743–750 (1997).
[CrossRef]

Schlag, P. M.

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, P. M. Schlag, “Contrast features of breast cancer in frequency-domain laser scanning mammography,” J. Biomed. Opt. 3, 129–136 (1998).
[CrossRef] [PubMed]

Schweiger, M.

J. Riley, H. Dehghani, M. Schweiger, S. R. Arridge, J. Ripoll, M. Nieto-Vesperinas, “3D optical tomography in the presence of void regions,” Opt. Express 7, 462–467 (2000).
[CrossRef] [PubMed]

H. Dehghani, S. R. Arridge, M. Schweiger, D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A 17, 1659–1667 (2000).
[CrossRef]

S. R. Arridge, H. Dehghani, M. Schweiger, E. Okada, “The finite-element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite-element model for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

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, M. Schweiger, “A general framework for iterative reconstruction algorithms in optical tomography, using a finite-element method,” in Computational Radiology and Imaging: Therapy and Diagnosis, C. Borgers, F. Natterer, eds., IMA Volumes in Mathematics and its Applications, Vol. 110 (Springer-Verlag, Berlin, 1998), pp. 45–70.
[CrossRef]

Sikora, J.

J. Heino, S. R. Arridge, J. Sikora, E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E 68, 031908: 1–8, (2003).
[CrossRef]

Somersalo, E.

J. Heino, S. R. Arridge, J. Sikora, E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E 68, 031908: 1–8, (2003).
[CrossRef]

J. Heino, E. Somersalo, “Estimation of optical absorption in anisotropic background,” Inverse Probl. 18, 559–573 (2002).
[CrossRef]

Splilman, S.

J. P. van Houten, D. A. Benaron, S. Splilman, D. K. Stevenson, “Imaging brain injury using time-resolved near-infrared light scanning,” Pediatr. Res. 39, 470–476 (1996).
[CrossRef] [PubMed]

Star, W. M.

Stevenson, D. K.

J. P. van Houten, D. A. Benaron, S. Splilman, D. K. Stevenson, “Imaging brain injury using time-resolved near-infrared light scanning,” Pediatr. Res. 39, 470–476 (1996).
[CrossRef] [PubMed]

Storchi, P. R. M.

Tam, W. G.

Tervo, J.

J. Tervo, P. Kolmonen, M. Vauhkonen, L. Heikkinen, J. Kaipio, “A finite-element model of electron transport in radiation therapy and a related inverse problem,” Inverse Probl. 15, 1345–1361 (1999).
[CrossRef]

Thiel, A.

C. Hirth, H. Obrig, K. Villringer, A. Thiel, J. Bernarding, W. Muhhlnickel, H. Flor, U. Dirnagl, A. Villringer, “Noninvasive functional mapping of the human motor cortex using near-infrared spectroscopy,” NeuroReport7, 1977–1981 (1996).
[CrossRef] [PubMed]

Totkas, S.

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, P. M. Schlag, “Contrast features of breast cancer in frequency-domain laser scanning mammography,” J. Biomed. Opt. 3, 129–136 (1998).
[CrossRef] [PubMed]

van Houten, J. P.

J. P. van Houten, D. A. Benaron, S. Splilman, D. K. Stevenson, “Imaging brain injury using time-resolved near-infrared light scanning,” Pediatr. Res. 39, 470–476 (1996).
[CrossRef] [PubMed]

Vauhkonen, M.

J. Tervo, P. Kolmonen, M. Vauhkonen, L. Heikkinen, J. Kaipio, “A finite-element model of electron transport in radiation therapy and a related inverse problem,” Inverse Probl. 15, 1345–1361 (1999).
[CrossRef]

Villringer, A.

K. R. Heereken, H. Obrig, R. Wenzel, K. Eberle, J. Ruben, K. Villringer, R. Kurth, A. Villringer, “Cerebral haemoglobin oxygenation during sustained visual stimulation—a near-infrared spectroscopy study,” Philos. Trans. R. Soc. London Ser. B 352, 743–750 (1997).
[CrossRef]

A. Villringer, B. Chance, “Noninvasive optical spectroscopy and imaging of human brain function,” Trends Neurosci. 20, 435–442 (1997).
[CrossRef] [PubMed]

C. Hirth, H. Obrig, K. Villringer, A. Thiel, J. Bernarding, W. Muhhlnickel, H. Flor, U. Dirnagl, A. Villringer, “Noninvasive functional mapping of the human motor cortex using near-infrared spectroscopy,” NeuroReport7, 1977–1981 (1996).
[CrossRef] [PubMed]

Villringer, K.

K. R. Heereken, H. Obrig, R. Wenzel, K. Eberle, J. Ruben, K. Villringer, R. Kurth, A. Villringer, “Cerebral haemoglobin oxygenation during sustained visual stimulation—a near-infrared spectroscopy study,” Philos. Trans. R. Soc. London Ser. B 352, 743–750 (1997).
[CrossRef]

C. Hirth, H. Obrig, K. Villringer, A. Thiel, J. Bernarding, W. Muhhlnickel, H. Flor, U. Dirnagl, A. Villringer, “Noninvasive functional mapping of the human motor cortex using near-infrared spectroscopy,” NeuroReport7, 1977–1981 (1996).
[CrossRef] [PubMed]

Wang, L.

Wenzel, R.

K. R. Heereken, H. Obrig, R. Wenzel, K. Eberle, J. Ruben, K. Villringer, R. Kurth, A. Villringer, “Cerebral haemoglobin oxygenation during sustained visual stimulation—a near-infrared spectroscopy study,” Philos. Trans. R. Soc. London Ser. B 352, 743–750 (1997).
[CrossRef]

Whitlock, P.

M. Kalos, P. Whitlock, Monte Carlo Methods (Wiley, New York, 1986).
[CrossRef]

Wray, S. C.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near-infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).

Wyatt, J. S.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near-infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).

Zardecki, A.

Zweifel, P. F.

M. C. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Appl. Opt. (5)

Astrophys. J. (1)

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Inverse Probl. (4)

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

J. Heino, E. Somersalo, “Estimation of optical absorption in anisotropic background,” Inverse Probl. 18, 559–573 (2002).
[CrossRef]

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

J. Tervo, P. Kolmonen, M. Vauhkonen, L. Heikkinen, J. Kaipio, “A finite-element model of electron transport in radiation therapy and a related inverse problem,” Inverse Probl. 15, 1345–1361 (1999).
[CrossRef]

J. Appl. Physiol. (1)

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near-infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).

J. Biomed. Opt. (1)

K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, P. M. Schlag, “Contrast features of breast cancer in frequency-domain laser scanning mammography,” J. Biomed. Opt. 3, 129–136 (1998).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (3)

Med. Phys. (3)

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite-element model for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

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, H. Dehghani, M. Schweiger, E. Okada, “The finite-element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

Opt. Express (1)

Pediatr. Res. (1)

J. P. van Houten, D. A. Benaron, S. Splilman, D. K. Stevenson, “Imaging brain injury using time-resolved near-infrared light scanning,” Pediatr. Res. 39, 470–476 (1996).
[CrossRef] [PubMed]

Philos. Trans. R. Soc. London Ser. B (1)

K. R. Heereken, H. Obrig, R. Wenzel, K. Eberle, J. Ruben, K. Villringer, R. Kurth, A. Villringer, “Cerebral haemoglobin oxygenation during sustained visual stimulation—a near-infrared spectroscopy study,” Philos. Trans. R. Soc. London Ser. B 352, 743–750 (1997).
[CrossRef]

Phys. Med. Biol. (2)

S. Prince, V. Kolehmainen, J. Kaipio, M. Franceschini, D. Boas, S. Arridge, “Time-series estimation of biological factors in optical diffusion tomography,” Phys. Med. Biol. 48, 1491–1504 (2003).
[CrossRef] [PubMed]

A. H. Hielscher, R. E. Alcouffe, R. L. Barbour, “Comparison of finite difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

Phys. Rev. E (1)

J. Heino, S. R. Arridge, J. Sikora, E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E 68, 031908: 1–8, (2003).
[CrossRef]

Trends Neurosci. (1)

A. Villringer, B. Chance, “Noninvasive optical spectroscopy and imaging of human brain function,” Trends Neurosci. 20, 435–442 (1997).
[CrossRef] [PubMed]

Other (12)

C. Hirth, H. Obrig, K. Villringer, A. Thiel, J. Bernarding, W. Muhhlnickel, H. Flor, U. Dirnagl, A. Villringer, “Noninvasive functional mapping of the human motor cortex using near-infrared spectroscopy,” NeuroReport7, 1977–1981 (1996).
[CrossRef] [PubMed]

H. Jess, H. Erdl, K. T. Moesta, S. Fantini, M. A. Franceschini, E. Gratton, “Intensity modulated breast imaging: technology and clinical pilot study results,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 126–129.

S. R. Arridge, M. Schweiger, “A general framework for iterative reconstruction algorithms in optical tomography, using a finite-element method,” in Computational Radiology and Imaging: Therapy and Diagnosis, C. Borgers, F. Natterer, eds., IMA Volumes in Mathematics and its Applications, Vol. 110 (Springer-Verlag, Berlin, 1998), pp. 45–70.
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R. Williams, M. Beck, eds., Process Tomography, Principles, Techniques and Applications, (Butterworth-Heinemann, Oxford, 1995).

D. A. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1996).

V. Kolehmainen, “Novel approaches to image reconstruction in diffusion tomography,” Ph.D. dissertation (University of Kuopio, Kuopio, Finland, 2001).

M. C. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

J. P. Kaltenbach, M. Kaschke, “Frequency- and time-domain modeling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

E. Amaldi, “The production and slowing down of neutrons,” in Encyclopedia of Physics, S. Flügge, ed., Vol. 38/2Neutrons and Related Gamma Ray Problems (Springer-Verlag, Berlin, 1959), pp. 1–659.

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[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Computational domain: solid lines, domain Ω with the source on the center of the upper edge; dashed lines, domain Ωbe in which the RTE is solved; white, domain Ωrte where the RTE is used as a forward model; gray, domain Ωda in which the DA is solved; dotted curve, interface boundary Γ where the solution of the RTE is used to construct a Dirichlet boundary condition for the DA.

Fig. 2
Fig. 2

Radiance and first-order approximation: solid curve, radiance; dashed curve, corresponding first-order approximation, on two points of interface Γ. The points are marked with asterisks in Fig. 3 in respective order from left to right.

Fig. 3
Fig. 3

Points, *, at which the radiances in Fig. 2 are calculated and the vertical and horizontal lines at which the photon densities in Figs. 5 and 6 are calculated (dashed lines).

Fig. 4
Fig. 4

Photon density derived from the solution of the RTE at interface boundary Γ.

Fig. 5
Fig. 5

Top, logarithms of the photon densities; bottom, absolute values of the relative errors with respect to the Monte Carlo simulation on the vertical line downward from the source (See Fig. 3, x position, 0 mm).

Fig. 6
Fig. 6

Left-hand side, photon densities; right-hand side, absolute values of relative errors with respect to Monte Carlo simulation on the horizontal lines within the domain (See Fig. 3, from top to bottom, y position, 5, 10, 15, and 20 mm).

Fig. 7
Fig. 7

Norm of the difference between the photon densities obtained with the RTE and with the hybrid model against the radius of the semicircular interface Γ: dashed line, radius of the interface boundary (r = 7 mm) used in the simulations.

Equations (48)

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

( i ω c + s ^ · + μ s + μ a ) ϕ ( r ,     s ^ ;     ω ) = μ s S n - 1 ϕ ( r ,     s ^ ;     ω ) Θ ( s ^ ,     s ^ ) d s ^ + q ( r ,     s ^ ;     ω ) ,
S n - 1 Φ ( s ^ ,     s ^ ) d s ^ = 1.
Θ ( s ^ ,     s ^ ) = Θ ( s ^ · s ^ ) .
Θ ( s ^ · s ^ ) = 1 4 π 1 - g 2 ( 1 + g 2 - 2 g cos γ ) 3 / 2 ,
Θ ( s ^ · s ^ ) = 1 2 π 1 - g 2 ( 1 + g 2 - 2 g cos γ ) .
ϕ ( r ,     s ^ ) = 0 ,             r Ω ,             s · n ^ < 0 ,
ϕ ( r ,     s ^ ) = { ϕ 0 ( r ,     s ^ ) , r i ɛ i s · n ^ < 0 , 0 , r Ω \ i ɛ i , s · n ^ < 0.
- · κ Φ ( r ;     ω ) + μ a Φ ( r ;     ω ) + i ω c Φ ( r ;     ω ) = q 0 ( r ;     ω ) ,
Φ ( r ) = S n - 1 ϕ ( r ,     s ^ ) d s ^ .
κ = 1 n ( μ a + μ s ) ,
μ s = ( 1 - g 1 ) μ s ,
g 1 = S n - 1 ( s ^ · s ^ ) Θ ( s ^ · s ^ ) d s ^ = g .
ϕ ( r ,     s ^ ) 1 S n - 1 Φ ( r ) + n S n - 1 s ^ · J ( r ) ,
J ( r ) = - κ Φ ( r )
s ^ · n ^ < 0 ϕ ( r ,     s ^ ) ( s ^ · n ^ ) d s ^ = 0 ,             r Ω .
Φ ( r ) + 1 2 γ n κ ϑ Φ ( r ) n ^ = 0 ,             r Ω ,
Φ ( r ) + 1 2 γ n κ ϑ Φ ( r ) n ^ = { - Γ s γ n , r i ɛ i 0 , r Ω \ i ɛ i .
Ω S n - 1 i ω c ϕ ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d r - Ω S n - 1 s ^ · v ( r ,     s ^ ) ϕ ( r ,     s ^ ) d s ^ d r + Ω S n - 1 ( s ^ · n ^ ) + ϕ ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d S + Ω S n - 1 ( μ s + μ a ) ϕ ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d r - Ω S n - 1 μ s S n - 1 Θ ( s ^ · s ^ ) ϕ ( r ,     s ^ ) d s ^ v ( r ,     s ^ ) d s ^ d r = Ω S n - 1 ( s ^ · n ^ ) - ϕ 0 ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d S + Ω S n - 1 q ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d r ,
ϕ h ( r ,     s ^ ) = i - 1 N n = 1 N a α i ψ i ( r ) ψ ( s ^ ) ,
( A 0 + A 1 + A 2 + A 3 + A 4 ) α = b 1 ψ 0 + b 2 ,
A 0 ( n ,     s ) = i ω c Ω ψ i ( r ) ψ j ( r ) d r 0 2 π ψ ( θ ) ψ m ( θ ) d θ ,
A 1 ( n ,     s ) = - [ Ω ψ i ( r ) ψ j ( r ) x d r × 0 2 π cos θ ψ ( θ ) ψ m ( θ ) d θ + Ω ψ i ( r ) ψ j ( r ) y d r × 0 2 π sin θ ψ ( θ ) ψ m ( θ ) d θ ] ,
A 2 ( n ,     s ) = Ω ψ i ( r ) ψ j ( r ) d S 0 2 π ( n ^ x cos θ + n ^ y × sin θ ) + ψ ( θ ) ψ m ( θ ) d θ ,
A 3 ( n ,     s ) = Ω ( μ s + μ a ) ψ i ( r ) ψ j ( r ) d r × 0 2 π ψ ( θ ) ψ m ( θ ) d θ ,
A 4 ( n ,     s ) = - Ω μ s ψ i ( r ) ψ j ( r ) d r 0 2 π 0 2 π Θ ( θ · θ ) × ψ ( θ ) d θ ψ m ( θ ) d θ ,
b 1 ( n ,     s ) = Ω ψ i ( r ) ψ j ( r ) d S 0 2 π ( n ^ x cos θ + n ^ y sin θ ) - ψ ( θ ) ψ m ( θ ) d θ ,
b 2 ( n ) = Ω 0 2 π q ( r ,     θ ) ψ m ( θ ) d θ ψ j ( r ) d r ,
Φ h ( r ) = k = 1 N a k φ k ( r ) ,
( K + C + R + i ω Z ) a = G + E ,
K ( p ,     k ) = Ω κ ( r ) φ k ( r ) · φ p ( r ) d r ,
C ( p ,     k ) = Ω μ a ( r ) φ k ( r ) φ p ( r ) d r ,
Z ( p ,     k ) = 1 c Ω φ k ( r ) φ p ( r ) d r ,
R ( p ,     k ) = Ω 2 γ n ϑ φ k ( r ) φ p ( r ) d S ,
G = 0 ,
E ( p ) = Ω q 0 φ p ( r ) d r .
G ( p ) = Ω - 2 Γ s ϑ φ p ( r ) d S ,
E = 0.
V i ω c ϕ ( r ,     s ^ ) v ( r ,     s ^ ) d V + V v ( r ,     s ^ ) s ^ · ϕ ( r ,     s ^ ) d V + V ( μ s + μ a ) ϕ ( r ,     s ^ ) v ( r ,     s ^ ) d V - V μ s S n - 1 Θ ( s ^ · s ^ ) ϕ ( r ,     s ^ ) d s ^ v ( r ,     s ^ ) d V = V q ( r ,     s ^ ) v ( r ,     s ^ ) d V .
Ω S n - 1 i ω c ϕ ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d r + Ω S n - 1 v ( r ,     s ^ ) s ^ · ϕ ( r ,     s ^ ) d s ^ d r + Ω S n - 1 ( μ s + μ a ) ϕ ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d r - Ω S n - 1 μ s S n - 1 Θ ( s ^ · s ^ ) ϕ ( r ,     s ^ ) d s ^ v ( r ,     s ^ ) d s ^ d r = Ω S n - 1 q ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d r .
Ω S n - 1 v ( r ,     s ^ ) s ^ · ϕ ( r ,     s ^ ) d s ^ d r = - Ω S n - 1 s ^ · v ( r ,     s ^ ) ϕ ( r ,     s ^ ) d s ^ d r + Ω S n - 1 ( s ^ · n ^ ) ϕ ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d S ,
Ω S n - 1 i ω c ϕ ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d r - Ω S n - 1 s ^ · v ( r ,     s ^ ) ϕ ( r ,     s ^ ) d s ^ d r + d Ω S n - 1 ( s ^ · n ^ ) ϕ ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d S + Ω S n - 1 ( μ s + μ a ) ϕ ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d r - Ω S n - 1 μ s S n - 1 Θ ( s ^ · s ^ ) ϕ ( r ,     s ^ ) d s ^ v ( r ,     s ^ ) d s ^ d r = Ω S n - 1 q ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d r .
d Ω S n - 1 ( s ^ · n ^ ) ϕ ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d S = d Ω S n - 1 [ ( s ^ · n ^ ) + - ( s ^ · n ^ ) - ] ϕ ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d S ,
d Ω S n - 1 [ ( s ^ · n ^ ) + - ( s ^ · n ^ ) - ] ϕ ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d S = d Ω S n - 1 ( s ^ · n ^ ) + ϕ ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d S - d Ω S n - 1 ( s ^ · n ^ ) - ϕ 0 ( r ,     s ^ ) v ( r ,     s ^ ) d s ^ d S .
Ω S n - 1 i ω c i = 1 N n = 1 N a α i ψ i ( r ) ψ ( s ^ ) ψ j ( r ) ψ m ( s ^ ) d s ^ d r - Ω S n - 1 s ^ · ( ψ j ( r ) ψ m ( s ^ ) ) × i = 1 N n = 1 N a α i ψ i ( r ) ψ ( s ^ ) d s ^ d r + d Ω S n - 1 ( s ^ · n ^ ) + i = 1 N n = 1 N a × α i ψ i ( r ) ψ ( s ^ ) ψ j ( r ) ψ m ( s ^ ) d s ^ d S + Ω S n - 1 ( μ s + μ a ) i = 1 N n = 1 N a α i ψ i ( r ) ψ ( s ^ ) ψ j ( r ) ψ m ( s ^ ) d s ^ d r - Ω S n - 1 μ s S n - 1 Θ ( s ^ · s ^ ) × i = 1 N n = 1 N a α i ψ i ( r ) ψ ( s ^ ) d s ^ ψ j ( r ) ψ m ( s ^ ) d s ^ d r = Ω S n - 1 ( s ^ · n ^ ) - ϕ 0 ( r ,     s ^ ) ψ j ( r ) ψ m ( s ^ ) d s ^ d S + Ω S n - 1 q ( r ,     s ^ ) ψ j ( r ) ψ m ( s ^ ) d s ^ d r .
ϕ 0 ( r ,     s ^ ) = i = 1 N n = 1 N a ψ i 0 ψ i ( r ) ψ ( s ^ ) ,
S 1 ψ ( s ^ ) d s ^
S 1 ψ ( s ^ ) d s ^ 0 2 π ( ψ h ) ( θ ) θ h d θ = 0 2 π ψ ( θ ) d θ .
i ω c i = 1 N n = 1 N a α i Ω ψ i ( r ) ψ j ( r ) d r 0 2 π ψ ( θ ) ψ m ( θ ) d θ - i = 1 N n = 1 N a α i [ Ω ψ i ( r ) ψ j ( r ) x d r × 0 2 π cos θ ψ ( θ ) ψ m ( θ ) d θ + Ω ψ i ( r ) ψ j ( r ) y d r 0 2 π sin θ ψ ( θ ) ψ m ( θ ) d θ ] + i = 1 N n = 1 N a α i d Ω ψ i ( r ) ψ j ( r ) d S 0 2 π ( n ^ x cos θ + n ^ y sin θ ) + ψ ( θ ) ψ m ( θ ) d θ + i = 1 N n = 1 N a α i Ω ( μ s + μ a ) ψ i ( r ) ψ j ( r ) d r 0 2 π ψ ( θ ) ψ m ( θ ) d θ - i = 1 N n = 1 N a α i Ω μ s ψ i ( r ) ψ j ( r ) d r × 0 2 π 0 2 π Θ ( θ · θ ) ψ ( θ ) d θ ψ m ( θ ) d θ = i = 1 N n = 1 N a ψ i 0 d Ω ψ i ( r ) ψ j ( r ) d S 0 2 π ( n ^ x cos θ + n ^ y sin θ ) - ψ ( θ ) ψ m ( θ ) d θ + Ω 0 2 π q ( r ,     θ ) ψ m ( θ ) d θ ψ j ( r ) d r .

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