Abstract

Currently, we are developing a computational optical biopsy technology for molecular sensing. We use the diffusion equation to model photon propagation but have a concern about the accuracy of diffusion approximation when the optical sensor is close to a bioluminescent source. We derive formulas to describe photon fluence for point and ball sources and measurement formulas for an idealized optical biopsy probe. Then, we numerically compare the diffusion approximation and the radiative transport as implemented by Monte Carlo simulation in the cases of point and ball sources. Our simulation results show that the diffusion approximation can be accurately applied if μsμa even if the sensor is very close to the source (>1mm). Furthermore, an approximate formula is given to describe the measurement of a cut-end fiber probe for a ball source.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
  4. G. Wang, Y. Li, and M. Jiang, "Uniqueness theorems in bioluminescence tomography," Med. Phys. 31, 2289-2299 (2004).
    [CrossRef] [PubMed]
  5. M. Jiang and G. Wang, "Image reconstruction for bioluminescence tomography," in Proc. SPIE 5535, 335-351 (2004).
    [CrossRef]
  6. V. Ntziachristos, C. Bremer, and R. Weissleder, "Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging," Eur. Radiol. 13, 195-208 (2003).
    [PubMed]
  7. G. Wang, Y. Li, and M. Jiang, "Computational optical biopsy methods, techniques and apparatus," patent disclosure filed with the University of Iowa Research Foundation in December 2003; provisional patent filed in 2004.
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    [CrossRef] [PubMed]
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  18. E. Aydin, C. Oliveira, and A. J. H. Goddard, "A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method," Med. Phys. 29, 2013-2023 (2002).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  27. S. Prahl, M. Keijzer, S. Jacques, and A. Welch, "A Monte Carlo model of light propagation in tissue," in Proc. SPIE 5, 102-111 (1989).
  28. S. Flock, B. Wilson, and M. Patterson, "Monte Carlo modeling of light propagation in highly scattering tissues--II: comparison with measurements in phantoms," IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
    [CrossRef] [PubMed]
  29. L. Wang, S. Jacques, and L. Zheng, "MCML--Monte Carlo modeling of photon transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
    [CrossRef] [PubMed]
  30. H. Li, J. Tian, F. Zhu, W. Cong, L. Wang, E. Hoffman, and G. Wang, "A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena with the Monte Carlo method," Acad. Radiol. 11, 1029-1038 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2005

2004

H. Li, J. Tian, F. Zhu, W. Cong, L. Wang, E. Hoffman, and G. Wang, "A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena with the Monte Carlo method," Acad. Radiol. 11, 1029-1038 (2004).
[CrossRef] [PubMed]

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, "A practical method to determine the light source distribution in bioluminescent imaging," in Proc. SPIE 5535, 679-686 (2004).
[CrossRef]

G. Wang, Y. Li, and M. Jiang, "Uniqueness theorems in bioluminescence tomography," Med. Phys. 31, 2289-2299 (2004).
[CrossRef] [PubMed]

M. Jiang and G. Wang, "Image reconstruction for bioluminescence tomography," in Proc. SPIE 5535, 335-351 (2004).
[CrossRef]

W. Cong, L. Wang, and G. Wang, "Formulation of photon diffusion from spherical bioluminescent sources in an infinite homogeneous medium," Biomed. Eng. Online 3:12 (2004).
[CrossRef] [PubMed]

2003

V. Ntziachristos, C. Bremer, and R. Weissleder, "Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging," Eur. Radiol. 13, 195-208 (2003).
[PubMed]

G. Wang, E. Hoffman, G. McLennan, L. Wang, M. Suter, and J. Meinel, "Development of the first bioluminescent CT scanner," Radiology 229, 566 (2003).

2002

E. Aydin, C. Oliveira, and A. J. H. Goddard, "A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method," Med. Phys. 29, 2013-2023 (2002).
[CrossRef] [PubMed]

2001

B. Rice, M. Cable, and M. Nelson, "In vivo imaging of light emitting probes," J. Biomed. Opt. 6, 432-440 (2001).
[CrossRef] [PubMed]

1998

S. Jacques, "Light distributions from point, line, and plane sources for photochemical reactions and fluorescence in turbid biological tissues," Photochem. Photobiol. 67, 23-32 (1998).
[CrossRef] [PubMed]

1995

S. Arridge, M. Hiraoka, and M. Schweiger, "Statistical basis for the determination of optical path length in tissue," Phys. Med. Biol. 40, 1539-1558 (1995).
[CrossRef] [PubMed]

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

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

1994

1990

W. Cheong, S. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2184 (1990).
[CrossRef]

1989

S. Flock, M. Patterson, B. Wilson, and D. Wyman, "Monte Carlo modeling of light propagation in highly scattering tissues-I: model predictions and comparison with diffusion theory," IEEE Trans. Biomed. Eng. 36, 1162-1168 (1989).
[CrossRef] [PubMed]

S. Prahl, M. Keijzer, S. Jacques, and A. Welch, "A Monte Carlo model of light propagation in tissue," in Proc. SPIE 5, 102-111 (1989).

S. Flock, B. Wilson, and M. Patterson, "Monte Carlo modeling of light propagation in highly scattering tissues--II: comparison with measurements in phantoms," IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

1983

1976

J. Joseph, W. Wiscombe, and J. Weinman, "The delta-Eddington approximation for radiative flux transfer," J. Atmos. Sci. 33, 2452-2459 (1976).
[CrossRef]

1970

E. Shettle and J. Weinman, "The transfer of solar irradiance through inhomogeneous turbid atmospheres evaluated by Eddington's approximation," J. Atmos. Sci. 27, 1048-1055 (1970).
[CrossRef]

Adam, G.

B. Wilson and G. Adam, "A Monte Carlo model for the absorption and flux distributions of light in tissue," Med. Phys. 10, 824-830 (1983).
[CrossRef] [PubMed]

Arridge, S.

S. Arridge, M. Hiraoka, and M. Schweiger, "Statistical basis for the determination of optical path length in tissue," Phys. Med. Biol. 40, 1539-1558 (1995).
[CrossRef] [PubMed]

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

Aydin, E.

E. Aydin, C. Oliveira, and A. J. H. Goddard, "A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method," Med. Phys. 29, 2013-2023 (2002).
[CrossRef] [PubMed]

Boas, D.

D. Boas, "Diffuse photon probes of structural and dynamical properties of turbid media," Ph.D. dissertation (University of Pennsylvania, 1996).

Bosch, J. T.

Bremer, C.

V. Ntziachristos, C. Bremer, and R. Weissleder, "Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging," Eur. Radiol. 13, 195-208 (2003).
[PubMed]

Cable, M.

B. Rice, M. Cable, and M. Nelson, "In vivo imaging of light emitting probes," J. Biomed. Opt. 6, 432-440 (2001).
[CrossRef] [PubMed]

Case, K.

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

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon, 1950).

Cheong, W.

W. Cheong, S. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2184 (1990).
[CrossRef]

Cong, A.

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, "A practical reconstruction method for bioluminescence tomography," Opt. Express 13, 6756-6771 (2005).
[CrossRef] [PubMed]

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, "A practical method to determine the light source distribution in bioluminescent imaging," in Proc. SPIE 5535, 679-686 (2004).
[CrossRef]

Cong, W.

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, "A practical reconstruction method for bioluminescence tomography," Opt. Express 13, 6756-6771 (2005).
[CrossRef] [PubMed]

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, "A practical method to determine the light source distribution in bioluminescent imaging," in Proc. SPIE 5535, 679-686 (2004).
[CrossRef]

W. Cong, L. Wang, and G. Wang, "Formulation of photon diffusion from spherical bioluminescent sources in an infinite homogeneous medium," Biomed. Eng. Online 3:12 (2004).
[CrossRef] [PubMed]

H. Li, J. Tian, F. Zhu, W. Cong, L. Wang, E. Hoffman, and G. Wang, "A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena with the Monte Carlo method," Acad. Radiol. 11, 1029-1038 (2004).
[CrossRef] [PubMed]

Delpy, D.

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

Feng, T.

Ferwada, H.

Flock, S.

S. Flock, B. Wilson, and M. Patterson, "Monte Carlo modeling of light propagation in highly scattering tissues--II: comparison with measurements in phantoms," IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

S. Flock, M. Patterson, B. Wilson, and D. Wyman, "Monte Carlo modeling of light propagation in highly scattering tissues-I: model predictions and comparison with diffusion theory," IEEE Trans. Biomed. Eng. 36, 1162-1168 (1989).
[CrossRef] [PubMed]

Goddard, A. J. H.

E. Aydin, C. Oliveira, and A. J. H. Goddard, "A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method," Med. Phys. 29, 2013-2023 (2002).
[CrossRef] [PubMed]

Groenhuis, R.

Haskell, R.

Hiraoka, M.

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

S. Arridge, M. Hiraoka, and M. Schweiger, "Statistical basis for the determination of optical path length in tissue," Phys. Med. Biol. 40, 1539-1558 (1995).
[CrossRef] [PubMed]

Hoffman, E.

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, "A practical reconstruction method for bioluminescence tomography," Opt. Express 13, 6756-6771 (2005).
[CrossRef] [PubMed]

H. Li, J. Tian, F. Zhu, W. Cong, L. Wang, E. Hoffman, and G. Wang, "A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena with the Monte Carlo method," Acad. Radiol. 11, 1029-1038 (2004).
[CrossRef] [PubMed]

G. Wang, E. Hoffman, G. McLennan, L. Wang, M. Suter, and J. Meinel, "Development of the first bioluminescent CT scanner," Radiology 229, 566 (2003).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Jacques, S.

S. Jacques, "Light distributions from point, line, and plane sources for photochemical reactions and fluorescence in turbid biological tissues," Photochem. Photobiol. 67, 23-32 (1998).
[CrossRef] [PubMed]

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

S. Prahl, M. Keijzer, S. Jacques, and A. Welch, "A Monte Carlo model of light propagation in tissue," in Proc. SPIE 5, 102-111 (1989).

Jiang, M.

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, "A practical reconstruction method for bioluminescence tomography," Opt. Express 13, 6756-6771 (2005).
[CrossRef] [PubMed]

Y. Li, M. Jiang, and G. Wang, "Computational optical biopsy," Biomed. Eng. Online 4:36 (2005).
[CrossRef] [PubMed]

M. Jiang and G. Wang, "Image reconstruction for bioluminescence tomography," in Proc. SPIE 5535, 335-351 (2004).
[CrossRef]

G. Wang, Y. Li, and M. Jiang, "Uniqueness theorems in bioluminescence tomography," Med. Phys. 31, 2289-2299 (2004).
[CrossRef] [PubMed]

G. Wang, Y. Li, and M. Jiang, "Computational optical biopsy methods, techniques and apparatus," patent disclosure filed with the University of Iowa Research Foundation in December 2003; provisional patent filed in 2004.

Joseph, J.

J. Joseph, W. Wiscombe, and J. Weinman, "The delta-Eddington approximation for radiative flux transfer," J. Atmos. Sci. 33, 2452-2459 (1976).
[CrossRef]

Keijzer, M.

S. Prahl, M. Keijzer, S. Jacques, and A. Welch, "A Monte Carlo model of light propagation in tissue," in Proc. SPIE 5, 102-111 (1989).

Kumar, D.

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, "A practical reconstruction method for bioluminescence tomography," Opt. Express 13, 6756-6771 (2005).
[CrossRef] [PubMed]

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, "A practical method to determine the light source distribution in bioluminescent imaging," in Proc. SPIE 5535, 679-686 (2004).
[CrossRef]

Li, H.

H. Li, J. Tian, F. Zhu, W. Cong, L. Wang, E. Hoffman, and G. Wang, "A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena with the Monte Carlo method," Acad. Radiol. 11, 1029-1038 (2004).
[CrossRef] [PubMed]

Li, Y.

Y. Li, M. Jiang, and G. Wang, "Computational optical biopsy," Biomed. Eng. Online 4:36 (2005).
[CrossRef] [PubMed]

G. Wang, Y. Li, and M. Jiang, "Uniqueness theorems in bioluminescence tomography," Med. Phys. 31, 2289-2299 (2004).
[CrossRef] [PubMed]

G. Wang, Y. Li, and M. Jiang, "Computational optical biopsy methods, techniques and apparatus," patent disclosure filed with the University of Iowa Research Foundation in December 2003; provisional patent filed in 2004.

Liu, Y.

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, "A practical reconstruction method for bioluminescence tomography," Opt. Express 13, 6756-6771 (2005).
[CrossRef] [PubMed]

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, "A practical method to determine the light source distribution in bioluminescent imaging," in Proc. SPIE 5535, 679-686 (2004).
[CrossRef]

Manno, I.

I. Manno, Introduction to the Monte Carlo Method (Akadémiai Kiadó, 1999).

McAdams, M.

McCray, P.

McLennan, G.

Meinel, J.

G. Wang, E. Hoffman, G. McLennan, L. Wang, M. Suter, and J. Meinel, "Development of the first bioluminescent CT scanner," Radiology 229, 566 (2003).

Nelson, M.

B. Rice, M. Cable, and M. Nelson, "In vivo imaging of light emitting probes," J. Biomed. Opt. 6, 432-440 (2001).
[CrossRef] [PubMed]

Ntziachristos, V.

V. Ntziachristos, C. Bremer, and R. Weissleder, "Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging," Eur. Radiol. 13, 195-208 (2003).
[PubMed]

Oliveira, C.

E. Aydin, C. Oliveira, and A. J. H. Goddard, "A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method," Med. Phys. 29, 2013-2023 (2002).
[CrossRef] [PubMed]

Patterson, M.

S. Flock, M. Patterson, B. Wilson, and D. Wyman, "Monte Carlo modeling of light propagation in highly scattering tissues-I: model predictions and comparison with diffusion theory," IEEE Trans. Biomed. Eng. 36, 1162-1168 (1989).
[CrossRef] [PubMed]

S. Flock, B. Wilson, and M. Patterson, "Monte Carlo modeling of light propagation in highly scattering tissues--II: comparison with measurements in phantoms," IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

Prahl, S.

W. Cheong, S. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2184 (1990).
[CrossRef]

S. Prahl, M. Keijzer, S. Jacques, and A. Welch, "A Monte Carlo model of light propagation in tissue," in Proc. SPIE 5, 102-111 (1989).

S. Prahl, "Light transport in tissue," Ph.D. dissertation (University of Texas at Austin, l988).

Rice, B.

B. Rice, M. Cable, and M. Nelson, "In vivo imaging of light emitting probes," J. Biomed. Opt. 6, 432-440 (2001).
[CrossRef] [PubMed]

Schweiger, M.

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

S. Arridge, M. Hiraoka, and M. Schweiger, "Statistical basis for the determination of optical path length in tissue," Phys. Med. Biol. 40, 1539-1558 (1995).
[CrossRef] [PubMed]

Shettle, E.

E. Shettle and J. Weinman, "The transfer of solar irradiance through inhomogeneous turbid atmospheres evaluated by Eddington's approximation," J. Atmos. Sci. 27, 1048-1055 (1970).
[CrossRef]

Suter, M.

G. Wang, E. Hoffman, G. McLennan, L. Wang, M. Suter, and J. Meinel, "Development of the first bioluminescent CT scanner," Radiology 229, 566 (2003).

Svaasand, L.

Tian, J.

H. Li, J. Tian, F. Zhu, W. Cong, L. Wang, E. Hoffman, and G. Wang, "A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena with the Monte Carlo method," Acad. Radiol. 11, 1029-1038 (2004).
[CrossRef] [PubMed]

Tormberg, B.

Tsay, T.

Tuchin, V.

V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis (SPIE, 2000).

Wang, G.

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, "A practical reconstruction method for bioluminescence tomography," Opt. Express 13, 6756-6771 (2005).
[CrossRef] [PubMed]

Y. Li, M. Jiang, and G. Wang, "Computational optical biopsy," Biomed. Eng. Online 4:36 (2005).
[CrossRef] [PubMed]

M. Jiang and G. Wang, "Image reconstruction for bioluminescence tomography," in Proc. SPIE 5535, 335-351 (2004).
[CrossRef]

G. Wang, Y. Li, and M. Jiang, "Uniqueness theorems in bioluminescence tomography," Med. Phys. 31, 2289-2299 (2004).
[CrossRef] [PubMed]

H. Li, J. Tian, F. Zhu, W. Cong, L. Wang, E. Hoffman, and G. Wang, "A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena with the Monte Carlo method," Acad. Radiol. 11, 1029-1038 (2004).
[CrossRef] [PubMed]

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, "A practical method to determine the light source distribution in bioluminescent imaging," in Proc. SPIE 5535, 679-686 (2004).
[CrossRef]

W. Cong, L. Wang, and G. Wang, "Formulation of photon diffusion from spherical bioluminescent sources in an infinite homogeneous medium," Biomed. Eng. Online 3:12 (2004).
[CrossRef] [PubMed]

G. Wang, E. Hoffman, G. McLennan, L. Wang, M. Suter, and J. Meinel, "Development of the first bioluminescent CT scanner," Radiology 229, 566 (2003).

G. Wang, Y. Li, and M. Jiang, "Computational optical biopsy methods, techniques and apparatus," patent disclosure filed with the University of Iowa Research Foundation in December 2003; provisional patent filed in 2004.

Wang, L.

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, "A practical reconstruction method for bioluminescence tomography," Opt. Express 13, 6756-6771 (2005).
[CrossRef] [PubMed]

W. Cong, L. Wang, and G. Wang, "Formulation of photon diffusion from spherical bioluminescent sources in an infinite homogeneous medium," Biomed. Eng. Online 3:12 (2004).
[CrossRef] [PubMed]

H. Li, J. Tian, F. Zhu, W. Cong, L. Wang, E. Hoffman, and G. Wang, "A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena with the Monte Carlo method," Acad. Radiol. 11, 1029-1038 (2004).
[CrossRef] [PubMed]

G. Wang, E. Hoffman, G. McLennan, L. Wang, M. Suter, and J. Meinel, "Development of the first bioluminescent CT scanner," Radiology 229, 566 (2003).

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

Weinman, J.

J. Joseph, W. Wiscombe, and J. Weinman, "The delta-Eddington approximation for radiative flux transfer," J. Atmos. Sci. 33, 2452-2459 (1976).
[CrossRef]

E. Shettle and J. Weinman, "The transfer of solar irradiance through inhomogeneous turbid atmospheres evaluated by Eddington's approximation," J. Atmos. Sci. 27, 1048-1055 (1970).
[CrossRef]

Weissleder, R.

V. Ntziachristos, C. Bremer, and R. Weissleder, "Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging," Eur. Radiol. 13, 195-208 (2003).
[PubMed]

Welch, A.

S. Prahl, M. Keijzer, S. Jacques, and A. Welch, "A Monte Carlo model of light propagation in tissue," in Proc. SPIE 5, 102-111 (1989).

Welch, A. J.

W. Cheong, S. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2184 (1990).
[CrossRef]

Wilson, B.

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Figures (7)

Fig. 1
Fig. 1

Illustration of the point and ball source models. r ̂ is the vector from the center of the detector of the source, u ̂ is the unit normal vector of the detector surface, ω is the half-angle of the cone that indicates the maximum incident angle of photons intercepted by the detector, σ is the angle between r ̂ and u ̂ , and R is the radius of the ball source. (a) Point source. (b) Solid ball source.

Fig. 2
Fig. 2

Illustration of the Monte Carlo simulation of the point and ball sources in an infinite medium.

Fig. 3
Fig. 3

Comparison of MC P F and ϕ P F [Eq. (1)] for a point source in an infinite medium with g = 0.9 and μ s = { 50,150,250 } ( cm 1 ) . (a) μ a = 0.2 ( cm 1 ) , (b) μ a = 0.5 ( cm 1 ) , (c) μ a = 1.0 ( cm 1 ) , (d) μ a = 2.0 ( cm 1 ) .

Fig. 4
Fig. 4

Comparison of MC P M and ϕ P M [Eq. (4)] for a point source in an infinite medium with g = 0.9 and μ s = { 50,150,250 } ( cm 1 ) . (a) μ a = 0.2 ( cm 1 ) , (b) μ a = 0.5 ( cm 1 ) , (c) μ a = 1.0 ( cm 1 ) , (d) μ a = 2.0 ( cm 1 ) .

Fig. 5
Fig. 5

(a) Comparison of MC B F and ϕ B F [Eq. (2)] and (b) comparison of MC B M and ϕ B M [Eq. (5)] with μ a = 0.5 , μ s = 150 , and g = 0.9 for different R = { 0.05 , 0.1 , 0.2 , 0.3 } cm.

Fig. 6
Fig. 6

(a) Comparison of MC B F and MC P F ; (b) comparison of MC B M and MC P M ; (c) comparison of MC B F MC P F and the diffusion prediction scale F ; (d) comparison of MC B M MC P M and the diffusion prediction scale M . μ a = 0.5 , μ s = 150 , and g = 0.9 for different R = { 0.05 , 0.1 , 0.2 , 0.3 } cm. (a) and (b) show that MC B F MC P F and MC B M MC P M are constant for r > R . (c) and (d) show that MC B F MC P F scale F and MC B M MC P M scale M for r > R .

Fig. 7
Fig. 7

Comparison of scale M and scale F with R = { 0.05 , 0.1 , 0.2 , 0.3 } cm, μ a = 0.5 , μ s = 150 , and g = 0.9 . This figure shows that if r > R , scale M scale F .

Equations (9)

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ϕ P F ( r ̂ ) = P e μ eff r 4 π D r ,
ϕ B , r R F ( r ̂ , R ) = ϕ P F ( r ̂ ) 3 [ ( e R μ eff e R μ eff ) + R μ eff ( e R μ eff + e R μ eff ) ] 2 R 3 μ eff 3 ,
ϕ B , r < R F ( r ̂ , R ) = ϕ P F ( r ̂ ) 3 e R μ eff { 1 e 2 r μ eff + [ 2 e ( r + R ) μ eff r + R e 2 r μ eff R ] μ eff } 2 R 3 μ eff 3 .
L ( r ̂ , s ̂ ) = 1 4 π [ ϕ P F ( r ̂ ) + 3 j ( r ̂ ) s ̂ ] ,
ϕ P M ( r ̂ , u ̂ , ω ) = ϕ P F ( r ̂ ) r sin ( ω ) 2 + 2 D cos ( σ ) [ 1 cos ( ω ) 3 ] ( 1 + r μ eff ) 4 r .
ϕ B M ( r ̂ , u ̂ , ω , R ) = Ω ϕ P M ( r ̂ , u ̂ , ω ) d Ω ,
ϕ Point M ( r ̂ , u ̂ , π 2 ) = ϕ Point F ( r ̂ ) r + 2 D ( 1 + r μ eff ) 4 r .
scale F = ϕ B F ϕ P F = { 3 [ ( e R μ eff e R μ eff ) + R μ eff ( e R μ eff + e R μ eff ) ] 2 R 3 μ eff 3 , r R 3 e R μ eff { 1 e 2 r μ eff + [ 2 e ( r + R ) μ eff r + R e 2 r μ eff R ] μ eff } 2 R 3 μ eff 3 , r < R .
ϕ B , r R M ( r ̂ , u ̂ , ω , R ) = ϕ P M ( r ̂ , u ̂ , ω ) 3 [ ( e R μ eff e R μ eff ) + R μ eff ( e R μ eff + e R μ eff ) ] 2 R 3 μ eff 3 .

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