Abstract

We introduce a robust method to recover optical absorption, reduced scattering, and single-scattering asymmetry coefficients (μa, μs, g 1) of infinite turbid media over a range of (μsa) spanning 3 orders of magnitude. This is accomplished through the spatially resolved measurement of irradiance at source-detector separations spanning 0.25–8 transport mean free paths (l*). These measurements are rapidly processed by a multistaged nonlinear optimization algorithm in which the measured irradiances are compared with predictions given by the δ-P 1 variant of the diffusion approximation to the Boltzmann transport equation. The ability of the δ-P 1 model to accurately describe radiative transport within media of arbitrary albedo and on spatial scales comparable to l* is the key element enabling the separation of g 1 from μs.

© 2004 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  21. J. S. You, V. Venugopalan are preparing a manuscript to be titled, “Delta-P1 approximation to radiative transport in the frequency domain.”

2004

S. A. Carp, S. A. Prahl, V. Venugopalan, “Radiative transport in the delta-P 1 approximation: accuracy of fluence rate and optical penetration depth predictions in turbid semi-infinite media,” J. Biomed. Opt. 9, 632–647 (2004).
[CrossRef] [PubMed]

2003

T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, M. N. Ediger, “Reflectance-based determination of optical properties in highly attenuating tissue,” J. Biomed. Opt. 8, 206–215 (2003).
[CrossRef] [PubMed]

2002

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. L. Österberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

I. Charvet, P. Theuler, B. Vermeulen, M. Saint-Ghislain, C. Biton, J. Jacquet, F. Bevilacqua, C. Depeursinge, P. Meda, “A new optical method for the non-invasive detection of minimal tissue alterations,” Phys. Med. Biol. 47, 2095–2108 (2002).
[CrossRef] [PubMed]

2001

E. L. Hull, T. H. Foster, “Steady-state reflectance spectroscopy in the P3 approximation,” J. Opt. Soc. Am. A 18, 584–599 (2001).
[CrossRef]

T. P. Moffitt, S. A. Prahl, “Sized-fiber reflectometry for measuring local optical properties,” IEEE J. Sel. Top. Quantum Electron. 7, 952–958 (2001).
[CrossRef]

A. Kienle, F. K. Forster, R. Hibst, “Influence of the phase function on determination of the optical properties of biological tissue by spatially resolved reflectance,” Opt. Lett. 26, 1571–1573 (2001).
[CrossRef]

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

2000

1999

F. Bevilacqua, D. Piguet, P. Marquet, J. D. Gross, B. J. Tromberg, C. Depeursinge, “In vivo local determination of tissue optical properties: applications to human brain,” Appl. Opt. 38, 4939–4950 (1999).
[CrossRef]

F. Bevilacqua, C. Depeursinge, “Monte Carlo study of diffuse reflectance at source-detector separations close to one transport mean free path,” J. Opt. Soc. Am. A 16, 1935–1945 (1999).
[CrossRef]

1998

M. R. Jones, Y. Yamada, “Determination of the asymmetry parameter and scattering coefficient of turbid media from spatially resolved reflectance measurements,” Opt. Rev. 5, 72–76 (1998).
[CrossRef]

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

1997

1996

1994

1992

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

1989

1976

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Bennett, C. L.

T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, M. N. Ediger, “Reflectance-based determination of optical properties in highly attenuating tissue,” J. Biomed. Opt. 8, 206–215 (2003).
[CrossRef] [PubMed]

Berger, A. J.

Bevilacqua, F.

I. Charvet, P. Theuler, B. Vermeulen, M. Saint-Ghislain, C. Biton, J. Jacquet, F. Bevilacqua, C. Depeursinge, P. Meda, “A new optical method for the non-invasive detection of minimal tissue alterations,” Phys. Med. Biol. 47, 2095–2108 (2002).
[CrossRef] [PubMed]

F. Bevilacqua, C. Depeursinge, “Monte Carlo study of diffuse reflectance at source-detector separations close to one transport mean free path,” J. Opt. Soc. Am. A 16, 1935–1945 (1999).
[CrossRef]

F. Bevilacqua, D. Piguet, P. Marquet, J. D. Gross, B. J. Tromberg, C. Depeursinge, “In vivo local determination of tissue optical properties: applications to human brain,” Appl. Opt. 38, 4939–4950 (1999).
[CrossRef]

Bigio, I. J.

Biton, C.

I. Charvet, P. Theuler, B. Vermeulen, M. Saint-Ghislain, C. Biton, J. Jacquet, F. Bevilacqua, C. Depeursinge, P. Meda, “A new optical method for the non-invasive detection of minimal tissue alterations,” Phys. Med. Biol. 47, 2095–2108 (2002).
[CrossRef] [PubMed]

Boas, D. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Boyer, J.

Brooks, D. H.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Carp, S. A.

S. A. Carp, S. A. Prahl, V. Venugopalan, “Radiative transport in the delta-P 1 approximation: accuracy of fluence rate and optical penetration depth predictions in turbid semi-infinite media,” J. Biomed. Opt. 9, 632–647 (2004).
[CrossRef] [PubMed]

Charvet, I.

I. Charvet, P. Theuler, B. Vermeulen, M. Saint-Ghislain, C. Biton, J. Jacquet, F. Bevilacqua, C. Depeursinge, P. Meda, “A new optical method for the non-invasive detection of minimal tissue alterations,” Phys. Med. Biol. 47, 2095–2108 (2002).
[CrossRef] [PubMed]

Depeursinge, C.

I. Charvet, P. Theuler, B. Vermeulen, M. Saint-Ghislain, C. Biton, J. Jacquet, F. Bevilacqua, C. Depeursinge, P. Meda, “A new optical method for the non-invasive detection of minimal tissue alterations,” Phys. Med. Biol. 47, 2095–2108 (2002).
[CrossRef] [PubMed]

F. Bevilacqua, C. Depeursinge, “Monte Carlo study of diffuse reflectance at source-detector separations close to one transport mean free path,” J. Opt. Soc. Am. A 16, 1935–1945 (1999).
[CrossRef]

F. Bevilacqua, D. Piguet, P. Marquet, J. D. Gross, B. J. Tromberg, C. Depeursinge, “In vivo local determination of tissue optical properties: applications to human brain,” Appl. Opt. 38, 4939–4950 (1999).
[CrossRef]

DiMarzio, C. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Durkin, A. J.

T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, M. N. Ediger, “Reflectance-based determination of optical properties in highly attenuating tissue,” J. Biomed. Opt. 8, 206–215 (2003).
[CrossRef] [PubMed]

A. J. Berger, V. Venugopalan, A. J. Durkin, T. Pham, B. J. Tromberg, “Chemometric analysis of frequency-domain photon migration data: quantitative measurements of optical properties and chromophore concentrations in multicomponent turbid media,” Appl. Opt. 39, 1659–1667 (2000).
[CrossRef]

Ediger, M. N.

T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, M. N. Ediger, “Reflectance-based determination of optical properties in highly attenuating tissue,” J. Biomed. Opt. 8, 206–215 (2003).
[CrossRef] [PubMed]

Fantini, S.

Farrell, T. J.

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Forster, F. K.

Foster, T. H.

Franceschini, M. A.

Gall, J. A.

T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, M. N. Ediger, “Reflectance-based determination of optical properties in highly attenuating tissue,” J. Biomed. Opt. 8, 206–215 (2003).
[CrossRef] [PubMed]

Gaudette, R. J.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Gratton, E.

Gross, J. D.

Hibst, R.

Hielscher, A. H.

Hull, E. L.

Jacquet, J.

I. Charvet, P. Theuler, B. Vermeulen, M. Saint-Ghislain, C. Biton, J. Jacquet, F. Bevilacqua, C. Depeursinge, P. Meda, “A new optical method for the non-invasive detection of minimal tissue alterations,” Phys. Med. Biol. 47, 2095–2108 (2002).
[CrossRef] [PubMed]

Jiang, S.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. L. Österberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

Jones, M. R.

M. R. Jones, Y. Yamada, “Determination of the asymmetry parameter and scattering coefficient of turbid media from spatially resolved reflectance measurements,” Opt. Rev. 5, 72–76 (1998).
[CrossRef]

Joseph, J. H.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Kienle, A.

Kilmer, M.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Lilge, L.

Marquet, P.

Matchette, L. S.

T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, M. N. Ediger, “Reflectance-based determination of optical properties in highly attenuating tissue,” J. Biomed. Opt. 8, 206–215 (2003).
[CrossRef] [PubMed]

McBride, T. O.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. L. Österberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

Meda, P.

I. Charvet, P. Theuler, B. Vermeulen, M. Saint-Ghislain, C. Biton, J. Jacquet, F. Bevilacqua, C. Depeursinge, P. Meda, “A new optical method for the non-invasive detection of minimal tissue alterations,” Phys. Med. Biol. 47, 2095–2108 (2002).
[CrossRef] [PubMed]

Miller, E. L.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Moffitt, T. P.

T. P. Moffitt, S. A. Prahl, “Sized-fiber reflectometry for measuring local optical properties,” IEEE J. Sel. Top. Quantum Electron. 7, 952–958 (2001).
[CrossRef]

Mourant, J. R.

Österberg, U. L.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. L. Österberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

Patterson, M. S.

Paulsen, K. D.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. L. Österberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

Pfefer, T. J.

T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, M. N. Ediger, “Reflectance-based determination of optical properties in highly attenuating tissue,” J. Biomed. Opt. 8, 206–215 (2003).
[CrossRef] [PubMed]

Pham, T.

Piguet, D.

Pogue, B. W.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. L. Österberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

Poplack, S.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. L. Österberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

Prahl, S. A.

S. A. Carp, S. A. Prahl, V. Venugopalan, “Radiative transport in the delta-P 1 approximation: accuracy of fluence rate and optical penetration depth predictions in turbid semi-infinite media,” J. Biomed. Opt. 9, 632–647 (2004).
[CrossRef] [PubMed]

T. P. Moffitt, S. A. Prahl, “Sized-fiber reflectometry for measuring local optical properties,” IEEE J. Sel. Top. Quantum Electron. 7, 952–958 (2001).
[CrossRef]

Saint-Ghislain, M.

I. Charvet, P. Theuler, B. Vermeulen, M. Saint-Ghislain, C. Biton, J. Jacquet, F. Bevilacqua, C. Depeursinge, P. Meda, “A new optical method for the non-invasive detection of minimal tissue alterations,” Phys. Med. Biol. 47, 2095–2108 (2002).
[CrossRef] [PubMed]

Soho, S.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. L. Österberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

Steiner, R.

Theuler, P.

I. Charvet, P. Theuler, B. Vermeulen, M. Saint-Ghislain, C. Biton, J. Jacquet, F. Bevilacqua, C. Depeursinge, P. Meda, “A new optical method for the non-invasive detection of minimal tissue alterations,” Phys. Med. Biol. 47, 2095–2108 (2002).
[CrossRef] [PubMed]

Tromberg, B. J.

Venugopalan, V.

S. A. Carp, S. A. Prahl, V. Venugopalan, “Radiative transport in the delta-P 1 approximation: accuracy of fluence rate and optical penetration depth predictions in turbid semi-infinite media,” J. Biomed. Opt. 9, 632–647 (2004).
[CrossRef] [PubMed]

A. J. Berger, V. Venugopalan, A. J. Durkin, T. Pham, B. J. Tromberg, “Chemometric analysis of frequency-domain photon migration data: quantitative measurements of optical properties and chromophore concentrations in multicomponent turbid media,” Appl. Opt. 39, 1659–1667 (2000).
[CrossRef]

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

J. S. You, V. Venugopalan are preparing a manuscript to be titled, “Delta-P1 approximation to radiative transport in the frequency domain.”

Vermeulen, B.

I. Charvet, P. Theuler, B. Vermeulen, M. Saint-Ghislain, C. Biton, J. Jacquet, F. Bevilacqua, C. Depeursinge, P. Meda, “A new optical method for the non-invasive detection of minimal tissue alterations,” Phys. Med. Biol. 47, 2095–2108 (2002).
[CrossRef] [PubMed]

Weinman, J. A.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Wells, W. A.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. L. Österberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

Wilke, J. N.

T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, M. N. Ediger, “Reflectance-based determination of optical properties in highly attenuating tissue,” J. Biomed. Opt. 8, 206–215 (2003).
[CrossRef] [PubMed]

Wilson, B.

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Wilson, B. C.

Wiscombe, W. J.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Wyman, D. R.

Yamada, Y.

M. R. Jones, Y. Yamada, “Determination of the asymmetry parameter and scattering coefficient of turbid media from spatially resolved reflectance measurements,” Opt. Rev. 5, 72–76 (1998).
[CrossRef]

You, J. S.

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

J. S. You, V. Venugopalan are preparing a manuscript to be titled, “Delta-P1 approximation to radiative transport in the frequency domain.”

Zhang, Q.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Appl. Opt.

IEEE J. Sel. Top. Quantum Electron.

T. P. Moffitt, S. A. Prahl, “Sized-fiber reflectometry for measuring local optical properties,” IEEE J. Sel. Top. Quantum Electron. 7, 952–958 (2001).
[CrossRef]

IEEE Signal Process. Mag.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

J. Atmos. Sci.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

J. Biomed. Opt.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. L. Österberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, M. N. Ediger, “Reflectance-based determination of optical properties in highly attenuating tissue,” J. Biomed. Opt. 8, 206–215 (2003).
[CrossRef] [PubMed]

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J. S. You, V. Venugopalan are preparing a manuscript to be titled, “Delta-P1 approximation to radiative transport in the frequency domain.”

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Figures (5)

Fig. 1
Fig. 1

Experiment setup for radial measurement of light transport within turbid phantoms.

Fig. 2
Fig. 2

Schematic of multistaged optimization algorithm.

Fig. 3
Fig. 3

Normalized irradiance versus sd separation for samples with (μs a ) = 290 and 3.1. Symbols represent measured data (N = 0.37), whereas curves represent δ-P 1 model prediction corresponding to the recovered set of optical properties. The normalized irradiance Ī = 4πI r /P0μt*2 2, P 0 being the power of the source.

Fig. 4
Fig. 4

Errors in the recovered optical coefficients for each phantom as a function of the number of measurements utilized for l* (○), μ a (◇), μ s (△), and g* (▽).

Fig. 5
Fig. 5

Route to the converged optical properties in Stage 1 for the case (μs a ) = 14 with an initial guess of [μ a , μs*, g*] = [0.0004, 0.02, 0.43] when every measurement (●) is used and when every eighth measurement (□). The symbols denote the optical properties found after every iteration. A total of 22 iterations were necessary when every measurement was used, whereas 159 iterations were necessary when only every eighth measurement was used.

Tables (3)

Tables Icon

Table 1 Optical Properties and Data Range and Number of the Five Turbid Media Tested

Tables Icon

Table 2 Estimated Optical Properties after Stage 1 of the Proposed Inversion Algorithma

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Table 3 Final Optical Property Values Obtained from the Optimization Algorithma

Equations (13)

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ω · Lr, ω=-μtLr, ω+μs4π Lr, ωpωωdω+Sr, ω,
pδ-P1ωω=14π1-f1+3g*ω · ω+2fδ1-ω · ω,
f=g2=g12, g*=g1g1+1.
ω · Lr, ω=-μt*Lr, ω+μs*4π Lr, ωp*ωωdω+Sr, ω,
Lr, ω14π ϕdr+34πjr · ω+12π Pr, ωδ1-ω · ω0,
ϕdr=3μs*μt*+g*μaP0 expμt*r08πμeffr×E1μt*r0-μeffr0-E1μt*r0+μeffr0-2g* sinhμeffr0r0μt*+g*μaexpμt*r0-E1μt*r-μeffrexp-μeffr+E1μt*r+μeffrexpμeffr,
jr=13μtrFGμeffr+1r2exp-μeffr-FHμeffr-1r2expμeffr+3g*μs*P0×exp-μt*r-r04πr2,
F=3μs*μt*+g*μaP0 expμt*r08πμeff,
G=E1μt*r0-μeffr0-E1μt*r0+μeffr0-2g* sinhμeffr0r0μt*+g*μaexpμt*r0-E1μt*r-μeffr,
H=E1μt*r+μeffr.
Irr, θ=ϕcr+14sin2 θϕdr+21-cos3 θjr,
ϕcr=P0 exp-μt*r-r04πr2.
χ2μa, μs*, g*=i=1MImri-Ipri, μa, μs*, g*σi2,

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