Abstract

Motivated by a recent report by Dickey et al. [Phys. Med. Biol. 46, 2359 (2001)], who demonstrated optical property retrieval by using relative radiance measurements at a single position, we investigate the uniqueness of relative radiance measurements for quantifying the optical properties of turbid media by studying the solutions of the diffusion and P3 approximations of the Boltzmann transfer equation for a point source. Using the P3 approximation, we investigate the potential of radiance measurements for optical property recovery by examining the optical property response surface for point radiance information. We further derive first-order similarity relations for relative point radiance measurements and use these expressions to examine analytically the effects of noise on optical property retrieval over a wide range of optical properties typical of biological tissue. Finally, optimal experimental configurations are studied and explicit conditions for uniqueness derived that suggest potential strategies for improving optical property recovery. It is expected that point radiance measurements will prove valuable for both on-line treatment planning of minimally invasive laser therapies and optical characterization of tissues.

© 2006 Optical Society of America

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References

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    [PubMed]
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    [PubMed]
  3. T. C. Zhu, A. Dimofte, J. C. Finlay, D. Stripp, T. Busch, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732 nm measured in vivo during motexafin lutetium-mediated photodynamic therapy," Photochem. Photobiol. 81, 96-105 (2005).
    [PubMed]
  4. O. Bajaras, A. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, "Monte Carlo modeling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry," Phys. Med. Biol. 42, 1675-1687 (1997).
    [PubMed]
  5. L. C. L. Chin, B. C. Wilson, W. M. Whelan, and I. A. Vitkin, "Radiance-based monitoring of the coagulation boundary during laser interstitial thermal therapy," Opt. Lett. 29, 959-961 (2004).
    [PubMed]
  6. D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, "Light dosimetry using the P3 approximation," Phys. Med. Biol. 46, 2359-2370 (2001).
    [PubMed]
  7. L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
    [PubMed]
  8. J. C. Finlay and T. H. Foster, "Hemoglobin oxygen saturations in phantoms and in vivo from measurements of steady-state diffuse reflectance at a single, short source-detector separation," Med. Phys. 31, 1949-1959 (2004).
    [PubMed]
  9. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
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    [PubMed]
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    [PubMed]
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    [PubMed]
  13. J. P. Marijnissen and W. M. Star, "Calibration of isotropic light dosimetry probes based on scattering bulbs in clear media," Phys. Med. Biol. 41, 1191-1208 (1996).
    [PubMed]
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    [PubMed]
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    [PubMed]
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  18. W. W. Whelan, P. Chun, L. C. Chin, M. D. Sherar, and I. A. Vitkin, "Laser thermal therapy: utility of interstitial fluence monitoring for locating optical sensors," Phys. Med. Biol. 46, N91-N96 (2001).
    [PubMed]

Alianelli, L.

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation," Phys. Med. Biol. 45, 1359-1373 (2000).
[PubMed]

Bajaras, O.

O. Bajaras, A. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, "Monte Carlo modeling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry," Phys. Med. Biol. 42, 1675-1687 (1997).
[PubMed]

Ballangrud, A. M.

O. Bajaras, A. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, "Monte Carlo modeling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry," Phys. Med. Biol. 42, 1675-1687 (1997).
[PubMed]

Bassani, M.

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation," Phys. Med. Biol. 45, 1359-1373 (2000).
[PubMed]

Bevilacqua, F.

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 Pennylvania, 1996).
[PubMed]

Busch, T.

T. C. Zhu, A. Dimofte, J. C. Finlay, D. Stripp, T. Busch, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732 nm measured in vivo during motexafin lutetium-mediated photodynamic therapy," Photochem. Photobiol. 81, 96-105 (2005).
[PubMed]

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Chin, L.

L. Chin, M. Pop, W. Whelan, M. Sherar, and A. Vitkin, "Optical method using fluence or radiance measurements to monitor thermal therapy," Rev. Sci. Instrum. 74, 393-395 (2003).

Chin, L. C.

W. W. Whelan, P. Chun, L. C. Chin, M. D. Sherar, and I. A. Vitkin, "Laser thermal therapy: utility of interstitial fluence monitoring for locating optical sensors," Phys. Med. Biol. 46, N91-N96 (2001).
[PubMed]

Chin, L. C. L.

Chun, P.

W. W. Whelan, P. Chun, L. C. Chin, M. D. Sherar, and I. A. Vitkin, "Laser thermal therapy: utility of interstitial fluence monitoring for locating optical sensors," Phys. Med. Biol. 46, N91-N96 (2001).
[PubMed]

Depeursinge, C.

Dickey, D. J.

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, "Light dosimetry using the P3 approximation," Phys. Med. Biol. 46, 2359-2370 (2001).
[PubMed]

Dimofte, A.

T. C. Zhu, A. Dimofte, J. C. Finlay, D. Stripp, T. Busch, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732 nm measured in vivo during motexafin lutetium-mediated photodynamic therapy," Photochem. Photobiol. 81, 96-105 (2005).
[PubMed]

Douplik, A.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49, 3209-3225 (2004).
[PubMed]

Finlay, J. C.

T. C. Zhu, A. Dimofte, J. C. Finlay, D. Stripp, T. Busch, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732 nm measured in vivo during motexafin lutetium-mediated photodynamic therapy," Photochem. Photobiol. 81, 96-105 (2005).
[PubMed]

J. C. Finlay and T. H. Foster, "Hemoglobin oxygen saturations in phantoms and in vivo from measurements of steady-state diffuse reflectance at a single, short source-detector separation," Med. Phys. 31, 1949-1959 (2004).
[PubMed]

Foster, T. H.

J. C. Finlay and T. H. Foster, "Hemoglobin oxygen saturations in phantoms and in vivo from measurements of steady-state diffuse reflectance at a single, short source-detector separation," Med. Phys. 31, 1949-1959 (2004).
[PubMed]

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

Glatstein, E.

T. C. Zhu, A. Dimofte, J. C. Finlay, D. Stripp, T. Busch, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732 nm measured in vivo during motexafin lutetium-mediated photodynamic therapy," Photochem. Photobiol. 81, 96-105 (2005).
[PubMed]

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
[PubMed]

Hahn, S. M.

T. C. Zhu, A. Dimofte, J. C. Finlay, D. Stripp, T. Busch, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732 nm measured in vivo during motexafin lutetium-mediated photodynamic therapy," Photochem. Photobiol. 81, 96-105 (2005).
[PubMed]

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
[PubMed]

Hull, E. L.

Jacques, S. L.

L. H. Wang, S. L. Jacques, and L. Zheng, "MCML--Monte Carlo modeling of light transport in multilayered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[PubMed]

Jankun, J.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49, 3209-3225 (2004).
[PubMed]

Keck, R. W.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49, 3209-3225 (2004).
[PubMed]

Lilge, L.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49, 3209-3225 (2004).
[PubMed]

Malkowicz, S. B.

T. C. Zhu, A. Dimofte, J. C. Finlay, D. Stripp, T. Busch, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732 nm measured in vivo during motexafin lutetium-mediated photodynamic therapy," Photochem. Photobiol. 81, 96-105 (2005).
[PubMed]

Marijnissen, J. P.

J. P. Marijnissen and W. M. Star, "Performance of isotropic light dosimetry probes based on scattering bulbs in turbid media," Phys. Med. Biol. 47, 2049-2058 (2002).
[PubMed]

J. P. Marijnissen and W. M. Star, "Calibration of isotropic light dosimetry probes based on scattering bulbs in clear media," Phys. Med. Biol. 41, 1191-1208 (1996).
[PubMed]

Martelli, F.

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation," Phys. Med. Biol. 45, 1359-1373 (2000).
[PubMed]

Miles, J.

T. C. Zhu, A. Dimofte, J. C. Finlay, D. Stripp, T. Busch, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732 nm measured in vivo during motexafin lutetium-mediated photodynamic therapy," Photochem. Photobiol. 81, 96-105 (2005).
[PubMed]

Miller, G. G.

O. Bajaras, A. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, "Monte Carlo modeling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry," Phys. Med. Biol. 42, 1675-1687 (1997).
[PubMed]

Moore, R. B.

O. Bajaras, A. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, "Monte Carlo modeling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry," Phys. Med. Biol. 42, 1675-1687 (1997).
[PubMed]

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, "Light dosimetry using the P3 approximation," Phys. Med. Biol. 46, 2359-2370 (2001).
[PubMed]

Patterson, M. S.

Pestka, M.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49, 3209-3225 (2004).
[PubMed]

Pomerleau-Dalcourt, N.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49, 3209-3225 (2004).
[PubMed]

Pop, M.

L. Chin, M. Pop, W. Whelan, M. Sherar, and A. Vitkin, "Optical method using fluence or radiance measurements to monitor thermal therapy," Rev. Sci. Instrum. 74, 393-395 (2003).

Rayner, D. C.

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, "Light dosimetry using the P3 approximation," Phys. Med. Biol. 46, 2359-2370 (2001).
[PubMed]

Selman, S. H.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49, 3209-3225 (2004).
[PubMed]

Sherar, M.

L. Chin, M. Pop, W. Whelan, M. Sherar, and A. Vitkin, "Optical method using fluence or radiance measurements to monitor thermal therapy," Rev. Sci. Instrum. 74, 393-395 (2003).

Sherar, M. D.

W. W. Whelan, P. Chun, L. C. Chin, M. D. Sherar, and I. A. Vitkin, "Laser thermal therapy: utility of interstitial fluence monitoring for locating optical sensors," Phys. Med. Biol. 46, N91-N96 (2001).
[PubMed]

Star, W. M.

J. P. Marijnissen and W. M. Star, "Performance of isotropic light dosimetry probes based on scattering bulbs in turbid media," Phys. Med. Biol. 47, 2049-2058 (2002).
[PubMed]

J. P. Marijnissen and W. M. Star, "Calibration of isotropic light dosimetry probes based on scattering bulbs in clear media," Phys. Med. Biol. 41, 1191-1208 (1996).
[PubMed]

Stripp, D.

T. C. Zhu, A. Dimofte, J. C. Finlay, D. Stripp, T. Busch, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732 nm measured in vivo during motexafin lutetium-mediated photodynamic therapy," Photochem. Photobiol. 81, 96-105 (2005).
[PubMed]

Szkudlarek, M.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49, 3209-3225 (2004).
[PubMed]

Tochner, Z.

T. C. Zhu, A. Dimofte, J. C. Finlay, D. Stripp, T. Busch, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732 nm measured in vivo during motexafin lutetium-mediated photodynamic therapy," Photochem. Photobiol. 81, 96-105 (2005).
[PubMed]

Tulip, J.

O. Bajaras, A. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, "Monte Carlo modeling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry," Phys. Med. Biol. 42, 1675-1687 (1997).
[PubMed]

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, "Light dosimetry using the P3 approximation," Phys. Med. Biol. 46, 2359-2370 (2001).
[PubMed]

Vitkin, A.

L. Chin, M. Pop, W. Whelan, M. Sherar, and A. Vitkin, "Optical method using fluence or radiance measurements to monitor thermal therapy," Rev. Sci. Instrum. 74, 393-395 (2003).

Vitkin, I. A.

W. W. Whelan, P. Chun, L. C. Chin, M. D. Sherar, and I. A. Vitkin, "Laser thermal therapy: utility of interstitial fluence monitoring for locating optical sensors," Phys. Med. Biol. 46, N91-N96 (2001).
[PubMed]

L. C. L. Chin, B. C. Wilson, W. M. Whelan, and I. A. Vitkin, "Radiance-based monitoring of the coagulation boundary during laser interstitial thermal therapy," Opt. Lett. 29, 959-961 (2004).
[PubMed]

Wang, L. H.

L. H. Wang, S. L. Jacques, and L. Zheng, "MCML--Monte Carlo modeling of light transport in multilayered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[PubMed]

Whelan, W.

L. Chin, M. Pop, W. Whelan, M. Sherar, and A. Vitkin, "Optical method using fluence or radiance measurements to monitor thermal therapy," Rev. Sci. Instrum. 74, 393-395 (2003).

Whelan, W. M.

Whelan, W. W.

W. W. Whelan, P. Chun, L. C. Chin, M. D. Sherar, and I. A. Vitkin, "Laser thermal therapy: utility of interstitial fluence monitoring for locating optical sensors," Phys. Med. Biol. 46, N91-N96 (2001).
[PubMed]

Whittington, R.

T. C. Zhu, A. Dimofte, J. C. Finlay, D. Stripp, T. Busch, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732 nm measured in vivo during motexafin lutetium-mediated photodynamic therapy," Photochem. Photobiol. 81, 96-105 (2005).
[PubMed]

Wilson, B. C.

Wyman, D. R.

Zaccanti, G.

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation," Phys. Med. Biol. 45, 1359-1373 (2000).
[PubMed]

Zangheri, L.

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation," Phys. Med. Biol. 45, 1359-1373 (2000).
[PubMed]

Zheng, L.

L. H. Wang, S. L. Jacques, and L. Zheng, "MCML--Monte Carlo modeling of light transport in multilayered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[PubMed]

Zhu, T. C.

T. C. Zhu, A. Dimofte, J. C. Finlay, D. Stripp, T. Busch, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732 nm measured in vivo during motexafin lutetium-mediated photodynamic therapy," Photochem. Photobiol. 81, 96-105 (2005).
[PubMed]

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Appl. Opt. (1)

Comput. Methods Programs Biomed. (1)

L. H. Wang, S. L. Jacques, and L. Zheng, "MCML--Monte Carlo modeling of light transport in multilayered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[PubMed]

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

Med. Phys. (1)

J. C. Finlay and T. H. Foster, "Hemoglobin oxygen saturations in phantoms and in vivo from measurements of steady-state diffuse reflectance at a single, short source-detector separation," Med. Phys. 31, 1949-1959 (2004).
[PubMed]

Opt. Lett. (1)

Phys. Med. Biol. (7)

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, "Light dosimetry using the P3 approximation," Phys. Med. Biol. 46, 2359-2370 (2001).
[PubMed]

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49, 3209-3225 (2004).
[PubMed]

W. W. Whelan, P. Chun, L. C. Chin, M. D. Sherar, and I. A. Vitkin, "Laser thermal therapy: utility of interstitial fluence monitoring for locating optical sensors," Phys. Med. Biol. 46, N91-N96 (2001).
[PubMed]

O. Bajaras, A. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, "Monte Carlo modeling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry," Phys. Med. Biol. 42, 1675-1687 (1997).
[PubMed]

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation," Phys. Med. Biol. 45, 1359-1373 (2000).
[PubMed]

J. P. Marijnissen and W. M. Star, "Calibration of isotropic light dosimetry probes based on scattering bulbs in clear media," Phys. Med. Biol. 41, 1191-1208 (1996).
[PubMed]

J. P. Marijnissen and W. M. Star, "Performance of isotropic light dosimetry probes based on scattering bulbs in turbid media," Phys. Med. Biol. 47, 2049-2058 (2002).
[PubMed]

Rev. Sci. Instrum. (1)

L. Chin, M. Pop, W. Whelan, M. Sherar, and A. Vitkin, "Optical method using fluence or radiance measurements to monitor thermal therapy," Rev. Sci. Instrum. 74, 393-395 (2003).

Other (4)

T. C. Zhu, A. Dimofte, J. C. Finlay, D. Stripp, T. Busch, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732 nm measured in vivo during motexafin lutetium-mediated photodynamic therapy," Photochem. Photobiol. 81, 96-105 (2005).
[PubMed]

L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
[PubMed]

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

D. A. Boas, "Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications," Ph.D. dissertation (University of Pennylvania, 1996).
[PubMed]

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Figures (14)

Fig. 1
Fig. 1

Geometry for point radiance measurements.

Fig. 2
Fig. 2

(a) Monte Carlo simulations of fluence distribution normalized to r = 0.5 cm for three combinations of μ a and μ s and the same μeff. Beyond ∼0.2 cm the three profiles are virtually indistinguishable. (b) Monte Carlo simulations of the radiance distribution normalized to θ = 0.9° at r = 1 cm for the same optical property combinations as for (a). The three curves are well separated despite the equivalent fluence distributions.

Fig. 3
Fig. 3

Comparison of Monte Carlo (circles) generated relative radiance versus the diffusion (dotted curves) and P3 (solid curves) forward models for optical property sets of (μ a = 0.01 cm−1, μ s = 10 cm−1) and (μ a = 1.75 cm−1, μ s = 10 cm−1) at sensor positions of r = 0.5, 1.3 cm.

Fig. 4
Fig. 4

Chi-square surfaces generated with the P3 forward model for optical property sets of (μ a = 0.01 cm−1, μ s = 10 cm−1) and (μ a = 1.75 cm−1, μ s = 10 cm−1) at sensor positions of r = 0.5, 1.5 cm.

Fig. 5
Fig. 5

Optical property pairs that generate the same relative radiance distribution for two optical property sets at sensor positions of 0.5 and 1.5 cm.

Fig. 6
Fig. 6

χ2 difference between P3 radiance distributions calculated by use of the true properties of (μ a = 1.75 cm−1, μ s = 10 cm−1) and similar optical property pairs generated with Eq. (10) as a function of (a) μ s and (b) μ a at 1 cm. Dotted lines, the χ2 difference between radiance generated by the use of the true properties with and without noise. The intersection point of the χ2 distribution with the noise threshold provides an estimation of the minimum (μ a = 1.6 cm−1, μ s = 8.03 cm−1) and the maximum (μ a = 2.07 cm−1, μ s = 11.42 cm−1) deviations in optical properties owing to noise.

Fig. 7
Fig. 7

Relative radiance distribution for optical properties of μ a = 1.75 cm−1, μ s = 10 cm−1, and r = 1.0 cm (dashed black curve). Circles, the same radiance distribution with added noise. Also plotted are the radiance distributions for the minimum (μ a = 1.6 cm−1, μ s = 8.03 cm−1) and the maximum (μ a = 2.07 cm−1, μ s = 11.42 cm−1) deviations in optical properties that are due to noise (solid green curve). Within the noise level, the curves are virtually identical.

Fig. 8
Fig. 8

Contour plots of maximum uncertainty in recovered (a) μ a and (b) μ s owing to the presence of experimental noise at a sensor position of 1 cm.

Fig. 9
Fig. 9

Fractional uncertainty owing to noise in recovered (a) μ a and (b) μ s for three optical property sets as a function of increasing source–sensor separation.

Fig. 10
Fig. 10

Fractional uncertainty owing to noise in recovered (a) μ a and (b) μ s as a function of angular sampling interval for three optical property sets and a sensor position of 1 cm. All radiance data span an angular range of 0°–180°.

Fig. 11
Fig. 11

Fractional uncertainty owing to noise in recovered (a) μ a and (b) μ s as a function of maximum angle sampled for three optical property sets and a sensor position of 1 cm.

Fig. 12
Fig. 12

Contour plots demonstrating the dependence of εavg on both Δθ and θMAX for a fixed optical property set of μ a = 0.5 cm−1 and μ s = 10 cm−1 for both (a) μ a and (b) μ s .

Fig. 13
Fig. 13

χ2 space calculated for a (a) two sensor measurements at 0.5 and 1.5 cm and (b) a single-sensor measurement at 0.5 cm by use of a μeff constraint. The true optical properties are μ a = 0.01 cm−1 and μ s = 10 cm−1.

Fig. 14
Fig. 14

Fractional uncertainty in recovered (a) μ a and (b) μ s owing to noise for true optical properties of μ s = 10 cm−1 and μ a ranging from 0.001 to 5 cm−1. The cases illustrated are a single sensor with (dashed curve) and without a μeff constraint (circles), and two sensors (solid curve).

Equations (20)

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L ( r , θ ) = S o l = 0 3 2 l + 1 4 π [ C h l ( v ) Q l ( v r ) + D h l ( v + ) Q l ( v + r ) ] P l ( θ ) .
h o ( v ± ) = 1 ,        h 1 ( v ± ) = μ a v ± ,
h 2 ( v ± ) = [ 1 2 + 3 μ a μ t   (1) 2 v ± 2 ] , h 3 ( v ± ) = [ 9 μ a μ t   (1) 14μ t   (3) v ± + 3 v ± 14 μ t ( 3 ) ] ,
μ s ( 1 g 2 ) = 1.85 μ s ,      μ s ( 1 g 3 ) = 2.6 μ s .
C = ( v ) 2 2 π v 3 [ 3 μ a μ t ( 1 ) v + 2 ] [ 6 μ a 2 μ t ( 1 ) ( v 2 v + 2 ) ] ,
D = ( v ) 2 2 π v + 3 [ 3 μ a μ t ( 1 ) v 2 ] [ 6 μ a 2 μ t ( 1 ) ( v + 2 v 2 ) ] .
C 1 2 μ a μ eff - 2 ,     h 1 ( v ) μ a μ eff ,
h 2 ( v ) [ 1 2 + μ eff           2 2 μ eff           2 ] = 0 ,
h 3 ( v ) [ 3 μ eff           2 14 μ t ( 3 ) μ eff + 3 μ eff 14 μ t ( 3 ) ] = 0.
L ( r , θ ) = P o 4 π 1 4 π r D [ 1 + 3 ( D r + μ a D ) cos θ ] exp ( μ eff r ) ,
χ 2 ( μ a , μ s )
= i = l 1 N ( θ ) ( { L [ ( r , θ i ) exp L P 3 ( r , θ i , μ a , μ s ) P 3 ] } 2 δ exp             i 2 ) .
L rel ( r , θ ) = C [ 1 + 3 ( D r + μ a D ) cos θ ] .
K = 1 ( μ s + μ a ) { 1 / r + [ 3 μ a ( μ s + μ a ) ] 1 / 2 } .
A μ a 2 + B μ a + C = 0 ,
A = K 2 3 ,
B = 2 K 2 μ s 2 K r 3 μ s ,
C = ( K μ s ) 2 2 K μ s r + 1 r 2 .
ε MAX = | μ s , true μ s , MAX | μ s , true .
1 S = μ eff r + μ eff + 3 μ a 3 μ eff           2 .

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