Abstract

We present a novel procedure for localizing fluorescing-tagged objects embedded in turbid slab media from fluorescent intensity profiles acquired along a surface of interest. Using a numerical model based on a finite element code, we firstly develop a method devoted to lateral detection by varying the laser source position along one face of the tissue slab. Next, we mainly demonstrate the possibility to accurately assess the depth location by alternately changing the position of the source and the detector at the both sides of the slab. The dimensionless depth indicator derived from this procedure remains independent, over a wide range, on both the optical properties of the host tissue and the probe concentration. The overall findings validate the method in situations involving moderate size object-like tumors tagged with a new smart contrast agent (Cy 5.5) that offers high tumor-to-background contrast and great interest in early cancer diagnostic.

© 2006 Optical Society of America

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2006 (3)

B. Yuan and Q. Zhu, “Separately reconstructing the structural and functional parameters of a fluorescent inclusion embedded in a turbid medium,” Opt. Express 14, 7172–7187 (2006).
[Crossref]

X. Deulin and J. P. L’Huillier, “Finite element approach to photon propagation modelling in semi-infinite homogeneous and multilayered tissue structures,” Eur. Phys. J. Appl. Phys. 33, 133–146 (2006).
[Crossref]

A. M. Zysk, E. J. Chaney, and S. A. Boppart, “Refractive index of carcinogen-induced rat mammary tumours, Phys. Med. Biol. 51, 2165–2177 (2006).
[Crossref] [PubMed]

2005 (7)

2004 (5)

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[Crossref] [PubMed]

H. Quan and Z. Guo, “Fast 3-D optical imaging with transient fluorescence signals” Opt. Express 12, 449–457 (2004).
[Crossref] [PubMed]

S. Achilefu, “Lighting up tumors with receptor-specific optical molecular probes,” Technol. Cancer Res. Treat. 3, 393–409 (2004).
[PubMed]

L. Spinelli, A. Toricelli, A. Pifferi, P. Taroni, G. M. Danesini, and R. Cubeddu, “Bulk optical properties and tissue components in the female breast from multiwavelength time-resolved optical mammography,” J. Biomed. Opt. 9, 1137–1142 (2004).
[Crossref] [PubMed]

M. Pfister and B. Scholz, “Localization of fluorescent spots with space-space MUSIC for mammography-like measurements system,” J. Biomed. Opt. 9, 481–487 (2004).
[Crossref] [PubMed]

2003 (3)

I. Gannot, A. Garashi, G. Gannot, V. Chernomordik, and A. Gandjbakhche, “In vivo quantitative three dimensional localization of tumor labelled with exogenous specific fluorescence markers,” Appl. Opt. 42, 3073–3080 (2003).
[Crossref] [PubMed]

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thomson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
[Crossref] [PubMed]

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]

2002 (3)

I. Gannot, G. Gannot, A. Garashi, A. Gandjbakhche, A. Buchner, and Y. Keisari, “Laser activated fluorescence measurements and morphological features: an in vivo study of clearance time of fluorescein isothiocyanate tagged cell markers,” J. Biomed. Opt. 7, 14–19 (2002).
[Crossref] [PubMed]

V. Ntziachristos, J. Ripoll, and R. Weissleder, “Would near infrared fluorescence signals propagate through large human organs for clinical studies?” Opt. Lett. 27, 333–335 (2002).
[Crossref]

V. Ntziachristos, C. H. Tung, C. Bremer, and R. Weissleder, “Fluorescence mediated tomographic imaging system,” Nature Med. 8, 757–760 (2002).
[Crossref] [PubMed]

2001 (3)

1998 (3)

I. Gannot, R. F. Bonner, G. Gannot, P. C. Fox, P. D. Smith, and A. H. Gandjbakhche, “Optical simulations of a non-invasive technique for the diagnosis of diseased salivary glands in situ,” Med. Phys. 25, 1139–1144 (1998).
[Crossref] [PubMed]

K. A. Kang, D. F. Bruley, J. M. Londono, and B. Chance, “Localization of a fluorescent object in highly scattering media via frequency response analysis of near infrared-time resolved spectroscopy spectra,” Ann. Biomed. Eng. 26, 138–145 (1998).
[Crossref]

E. L. Hull, M. G. Nichols, and T. H. Foster, “Localization of luminescent inhomogeneities in turbid media with spatially resolved measurement of cw diffuse luminescence emittance,” Appl. Opt. 37, 2755–2765 (1998).
[Crossref]

1997 (1)

1996 (1)

H. Heusmann, J. Kölzer, and G. Mitic, “Characterization of female breasts in vivo by time resolved and spectroscopic measurements in near infrared spectroscopy,” J. Biomed. Opt. 1, 425–434 (1996).
[Crossref]

1995 (2)

M. Schweiger, S. R. Arridge, M. Hiroaka, and 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]

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Opt. 34, 3425–3430 (1995).
[Crossref] [PubMed]

1994 (2)

M. S. Patterson and B. W. Pogue, “Mathematical models for time-resolved and frequency domain fluorescence spectroscopy in biological tissues,” Appl. Opt. 33, 1963–1974 (1994).
[Crossref] [PubMed]

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A. 11, 2727–2741 (1994).
[Crossref]

1992 (1)

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

1989 (1)

Achilefu, S.

Andersson-Engels, S.

Arridge, S. R.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffusive optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
[Crossref] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiroaka, and 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]

Bengtsson, D.

Bloch, S. R.

Boas, D. A.

Bolin, F. P.

Bonner, R. F.

I. Gannot, R. F. Bonner, G. Gannot, P. C. Fox, P. D. Smith, and A. H. Gandjbakhche, “Optical simulations of a non-invasive technique for the diagnosis of diseased salivary glands in situ,” Med. Phys. 25, 1139–1144 (1998).
[Crossref] [PubMed]

Boppart, S. A.

A. M. Zysk, E. J. Chaney, and S. A. Boppart, “Refractive index of carcinogen-induced rat mammary tumours, Phys. Med. Biol. 51, 2165–2177 (2006).
[Crossref] [PubMed]

Born, M.

M. Born and E. Wolf, “Principles of Optics,” (MacMillan, N.Y., 1964).

Bouman, C. A.

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]

V. Ntziachristos, C. H. Tung, C. Bremer, and R. Weissleder, “Fluorescence mediated tomographic imaging system,” Nature Med. 8, 757–760 (2002).
[Crossref] [PubMed]

Bruley, D. F.

K. A. Kang, D. F. Bruley, J. M. Londono, and B. Chance, “Localization of a fluorescent object in highly scattering media via frequency response analysis of near infrared-time resolved spectroscopy spectra,” Ann. Biomed. Eng. 26, 138–145 (1998).
[Crossref]

Buchner, A.

I. Gannot, G. Gannot, A. Garashi, A. Gandjbakhche, A. Buchner, and Y. Keisari, “Laser activated fluorescence measurements and morphological features: an in vivo study of clearance time of fluorescein isothiocyanate tagged cell markers,” J. Biomed. Opt. 7, 14–19 (2002).
[Crossref] [PubMed]

Burnett, D. S.

D. S. Burnett, “Finite Element Analysis. from concepts to applications,” (Addison-Wesley, 1987).

Cerrusi, A. E.

A. E. Cerrusi, S. Fantini, J. S. Maier, W. W. Mantulin, and E. Gratton, “Chromophore detection by fluorescence spectroscopy in tissue-like phantoms,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model, and Human Studies II, B. Chance and R. R. Alfano, eds., Proc. SPIE2979, 139–150 (1997).
[Crossref]

Chance, B.

Chaney, E. J.

A. M. Zysk, E. J. Chaney, and S. A. Boppart, “Refractive index of carcinogen-induced rat mammary tumours, Phys. Med. Biol. 51, 2165–2177 (2006).
[Crossref] [PubMed]

Chernomordik, V.

Comelli, D.

C. D’Andrea, L. Spinelli, D. Comelli, G. Valentini, and R. Cubeddu, “Localization and quantification of fluorescent inclusions embedded in a turbid medium,” Phys. Med. Biol. 50, 2313–2327 (2005).
[Crossref] [PubMed]

Cubeddu, R.

C. D’Andrea, L. Spinelli, D. Comelli, G. Valentini, and R. Cubeddu, “Localization and quantification of fluorescent inclusions embedded in a turbid medium,” Phys. Med. Biol. 50, 2313–2327 (2005).
[Crossref] [PubMed]

L. Spinelli, A. Toricelli, A. Pifferi, P. Taroni, G. M. Danesini, and R. Cubeddu, “Bulk optical properties and tissue components in the female breast from multiwavelength time-resolved optical mammography,” J. Biomed. Opt. 9, 1137–1142 (2004).
[Crossref] [PubMed]

Culver, J. P.

D’Andrea, C.

C. D’Andrea, L. Spinelli, D. Comelli, G. Valentini, and R. Cubeddu, “Localization and quantification of fluorescent inclusions embedded in a turbid medium,” Phys. Med. Biol. 50, 2313–2327 (2005).
[Crossref] [PubMed]

Danesini, G. M.

L. Spinelli, A. Toricelli, A. Pifferi, P. Taroni, G. M. Danesini, and R. Cubeddu, “Bulk optical properties and tissue components in the female breast from multiwavelength time-resolved optical mammography,” J. Biomed. Opt. 9, 1137–1142 (2004).
[Crossref] [PubMed]

Dasari, R. R.

Delpy, D. T.

M. Schweiger, S. R. Arridge, M. Hiroaka, and 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]

Deulin, X.

X. Deulin and J. P. L’Huillier, “Finite element approach to photon propagation modelling in semi-infinite homogeneous and multilayered tissue structures,” Eur. Phys. J. Appl. Phys. 33, 133–146 (2006).
[Crossref]

Eppstein, M. J.

A. Godavarty, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Detection of single and multiple targets in tissue phantoms with fluorescence-enhanced optical imaging: feasibility study,” Radiology 235, 148–154 (2005).
[Crossref] [PubMed]

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[Crossref] [PubMed]

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thomson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
[Crossref] [PubMed]

Fantini, S.

A. E. Cerrusi, S. Fantini, J. S. Maier, W. W. Mantulin, and E. Gratton, “Chromophore detection by fluorescence spectroscopy in tissue-like phantoms,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model, and Human Studies II, B. Chance and R. R. Alfano, eds., Proc. SPIE2979, 139–150 (1997).
[Crossref]

Farrell, T. J.

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

Feld, M. S.

Feng, T. C.

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A. 11, 2727–2741 (1994).
[Crossref]

Ference, R. J.

Foster, T. H.

E. L. Hull, M. G. Nichols, and T. H. Foster, “Localization of luminescent inhomogeneities in turbid media with spatially resolved measurement of cw diffuse luminescence emittance,” Appl. Opt. 37, 2755–2765 (1998).
[Crossref]

T. H. Foster, E. L. Hull, M. G. Nichols, D. S. Rifkin, and N. Schwartz, “Two steady-state methods for localizing a fluorescent inhomogeneity in a turbid medium,” in Optical Tomography and spectroscopy of Tissue: Theory, Instrumentation, Model, and Human studies II, B. Chance and R. R. Alfano, eds., Proc. SPIE2979, 741–749 (1997).
[Crossref]

Fox, P. C.

I. Gannot, R. F. Bonner, G. Gannot, P. C. Fox, P. D. Smith, and A. H. Gandjbakhche, “Optical simulations of a non-invasive technique for the diagnosis of diseased salivary glands in situ,” Med. Phys. 25, 1139–1144 (1998).
[Crossref] [PubMed]

Gandjbakhche, A.

I. Gannot, A. Garashi, G. Gannot, V. Chernomordik, and A. Gandjbakhche, “In vivo quantitative three dimensional localization of tumor labelled with exogenous specific fluorescence markers,” Appl. Opt. 42, 3073–3080 (2003).
[Crossref] [PubMed]

I. Gannot, G. Gannot, A. Garashi, A. Gandjbakhche, A. Buchner, and Y. Keisari, “Laser activated fluorescence measurements and morphological features: an in vivo study of clearance time of fluorescein isothiocyanate tagged cell markers,” J. Biomed. Opt. 7, 14–19 (2002).
[Crossref] [PubMed]

Gandjbakhche, A. H.

I. Gannot, R. F. Bonner, G. Gannot, P. C. Fox, P. D. Smith, and A. H. Gandjbakhche, “Optical simulations of a non-invasive technique for the diagnosis of diseased salivary glands in situ,” Med. Phys. 25, 1139–1144 (1998).
[Crossref] [PubMed]

Gannot, G.

I. Gannot, A. Garashi, G. Gannot, V. Chernomordik, and A. Gandjbakhche, “In vivo quantitative three dimensional localization of tumor labelled with exogenous specific fluorescence markers,” Appl. Opt. 42, 3073–3080 (2003).
[Crossref] [PubMed]

I. Gannot, G. Gannot, A. Garashi, A. Gandjbakhche, A. Buchner, and Y. Keisari, “Laser activated fluorescence measurements and morphological features: an in vivo study of clearance time of fluorescein isothiocyanate tagged cell markers,” J. Biomed. Opt. 7, 14–19 (2002).
[Crossref] [PubMed]

I. Gannot, R. F. Bonner, G. Gannot, P. C. Fox, P. D. Smith, and A. H. Gandjbakhche, “Optical simulations of a non-invasive technique for the diagnosis of diseased salivary glands in situ,” Med. Phys. 25, 1139–1144 (1998).
[Crossref] [PubMed]

Gannot, I.

I. Gannot, A. Garashi, G. Gannot, V. Chernomordik, and A. Gandjbakhche, “In vivo quantitative three dimensional localization of tumor labelled with exogenous specific fluorescence markers,” Appl. Opt. 42, 3073–3080 (2003).
[Crossref] [PubMed]

I. Gannot, G. Gannot, A. Garashi, A. Gandjbakhche, A. Buchner, and Y. Keisari, “Laser activated fluorescence measurements and morphological features: an in vivo study of clearance time of fluorescein isothiocyanate tagged cell markers,” J. Biomed. Opt. 7, 14–19 (2002).
[Crossref] [PubMed]

I. Gannot, R. F. Bonner, G. Gannot, P. C. Fox, P. D. Smith, and A. H. Gandjbakhche, “Optical simulations of a non-invasive technique for the diagnosis of diseased salivary glands in situ,” Med. Phys. 25, 1139–1144 (1998).
[Crossref] [PubMed]

Garashi, A.

I. Gannot, A. Garashi, G. Gannot, V. Chernomordik, and A. Gandjbakhche, “In vivo quantitative three dimensional localization of tumor labelled with exogenous specific fluorescence markers,” Appl. Opt. 42, 3073–3080 (2003).
[Crossref] [PubMed]

I. Gannot, G. Gannot, A. Garashi, A. Gandjbakhche, A. Buchner, and Y. Keisari, “Laser activated fluorescence measurements and morphological features: an in vivo study of clearance time of fluorescein isothiocyanate tagged cell markers,” J. Biomed. Opt. 7, 14–19 (2002).
[Crossref] [PubMed]

Gibson, A. P.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffusive optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
[Crossref] [PubMed]

Godavarty, A.

A. Godavarty, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Detection of single and multiple targets in tissue phantoms with fluorescence-enhanced optical imaging: feasibility study,” Radiology 235, 148–154 (2005).
[Crossref] [PubMed]

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[Crossref] [PubMed]

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thomson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
[Crossref] [PubMed]

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thomson, and R. Roy, Near-infrared fluorescence imaging and spectroscopy in random media and tissues, in Biomedical photonics handbook, T. Vo Dinh ed., (CRC Press, 2003).

Gratton, E.

A. E. Cerrusi, S. Fantini, J. S. Maier, W. W. Mantulin, and E. Gratton, “Chromophore detection by fluorescence spectroscopy in tissue-like phantoms,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model, and Human Studies II, B. Chance and R. R. Alfano, eds., Proc. SPIE2979, 139–150 (1997).
[Crossref]

Guilbault, G. G.

G. G. Guilbault, “Practical Fluorescence,” (Marcel Dekker, Inc., New-York1973).

Guo, Z.

Gurfinkel, M.

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[Crossref] [PubMed]

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thomson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
[Crossref] [PubMed]

Haskell, R. C.

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A. 11, 2727–2741 (1994).
[Crossref]

Hebden, J. C.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffusive optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
[Crossref] [PubMed]

Heusmann, H.

H. Heusmann, J. Kölzer, and G. Mitic, “Characterization of female breasts in vivo by time resolved and spectroscopic measurements in near infrared spectroscopy,” J. Biomed. Opt. 1, 425–434 (1996).
[Crossref]

Hiroaka, M.

M. Schweiger, S. R. Arridge, M. Hiroaka, and 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]

Holboke, M. J.

Houston, J. P.

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thomson, and R. Roy, Near-infrared fluorescence imaging and spectroscopy in random media and tissues, in Biomedical photonics handbook, T. Vo Dinh ed., (CRC Press, 2003).

Hull, E. L.

E. L. Hull, M. G. Nichols, and T. H. Foster, “Localization of luminescent inhomogeneities in turbid media with spatially resolved measurement of cw diffuse luminescence emittance,” Appl. Opt. 37, 2755–2765 (1998).
[Crossref]

T. H. Foster, E. L. Hull, M. G. Nichols, D. S. Rifkin, and N. Schwartz, “Two steady-state methods for localizing a fluorescent inhomogeneity in a turbid medium,” in Optical Tomography and spectroscopy of Tissue: Theory, Instrumentation, Model, and Human studies II, B. Chance and R. R. Alfano, eds., Proc. SPIE2979, 741–749 (1997).
[Crossref]

Humeau, A.

J. P. L’Huillier and A. Humeau, “A computationally efficient model for simulating time-resolved fluorescence spectroscopy of thick biological tissues, in Photon Management, F. Wyrowski, ed., Proc. SPIE5456, 1–10 (2004).
[Crossref]

Intes, X.

Itzkan, I.

Kang, K. A.

K. A. Kang, D. F. Bruley, J. M. Londono, and B. Chance, “Localization of a fluorescent object in highly scattering media via frequency response analysis of near infrared-time resolved spectroscopy spectra,” Ann. Biomed. Eng. 26, 138–145 (1998).
[Crossref]

Keisari, Y.

I. Gannot, G. Gannot, A. Garashi, A. Gandjbakhche, A. Buchner, and Y. Keisari, “Laser activated fluorescence measurements and morphological features: an in vivo study of clearance time of fluorescein isothiocyanate tagged cell markers,” J. Biomed. Opt. 7, 14–19 (2002).
[Crossref] [PubMed]

Kennedy, M. D.

Kölzer, J.

H. Heusmann, J. Kölzer, and G. Mitic, “Characterization of female breasts in vivo by time resolved and spectroscopic measurements in near infrared spectroscopy,” J. Biomed. Opt. 1, 425–434 (1996).
[Crossref]

Kumar, S.

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time-resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[Crossref] [PubMed]

Kuwana, E.

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thomson, and R. Roy, Near-infrared fluorescence imaging and spectroscopy in random media and tissues, in Biomedical photonics handbook, T. Vo Dinh ed., (CRC Press, 2003).

L’Huillier, J. P.

X. Deulin and J. P. L’Huillier, “Finite element approach to photon propagation modelling in semi-infinite homogeneous and multilayered tissue structures,” Eur. Phys. J. Appl. Phys. 33, 133–146 (2006).
[Crossref]

J. P. L’Huillier and A. Humeau, “A computationally efficient model for simulating time-resolved fluorescence spectroscopy of thick biological tissues, in Photon Management, F. Wyrowski, ed., Proc. SPIE5456, 1–10 (2004).
[Crossref]

Londono, J. M.

K. A. Kang, D. F. Bruley, J. M. Londono, and B. Chance, “Localization of a fluorescent object in highly scattering media via frequency response analysis of near infrared-time resolved spectroscopy spectra,” Ann. Biomed. Eng. 26, 138–145 (1998).
[Crossref]

Low, P. S.

Maier, J. S.

A. E. Cerrusi, S. Fantini, J. S. Maier, W. W. Mantulin, and E. Gratton, “Chromophore detection by fluorescence spectroscopy in tissue-like phantoms,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model, and Human Studies II, B. Chance and R. R. Alfano, eds., Proc. SPIE2979, 139–150 (1997).
[Crossref]

Mantulin, W. W.

A. E. Cerrusi, S. Fantini, J. S. Maier, W. W. Mantulin, and E. Gratton, “Chromophore detection by fluorescence spectroscopy in tissue-like phantoms,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model, and Human Studies II, B. Chance and R. R. Alfano, eds., Proc. SPIE2979, 139–150 (1997).
[Crossref]

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R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A. 11, 2727–2741 (1994).
[Crossref]

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Mitic, G.

H. Heusmann, J. Kölzer, and G. Mitic, “Characterization of female breasts in vivo by time resolved and spectroscopic measurements in near infrared spectroscopy,” J. Biomed. Opt. 1, 425–434 (1996).
[Crossref]

Nichols, M. G.

E. L. Hull, M. G. Nichols, and T. H. Foster, “Localization of luminescent inhomogeneities in turbid media with spatially resolved measurement of cw diffuse luminescence emittance,” Appl. Opt. 37, 2755–2765 (1998).
[Crossref]

T. H. Foster, E. L. Hull, M. G. Nichols, D. S. Rifkin, and N. Schwartz, “Two steady-state methods for localizing a fluorescent inhomogeneity in a turbid medium,” in Optical Tomography and spectroscopy of Tissue: Theory, Instrumentation, Model, and Human studies II, B. Chance and R. R. Alfano, eds., Proc. SPIE2979, 741–749 (1997).
[Crossref]

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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]

V. Ntziachristos, J. Ripoll, and R. Weissleder, “Would near infrared fluorescence signals propagate through large human organs for clinical studies?” Opt. Lett. 27, 333–335 (2002).
[Crossref]

V. Ntziachristos, C. H. Tung, C. Bremer, and R. Weissleder, “Fluorescence mediated tomographic imaging system,” Nature Med. 8, 757–760 (2002).
[Crossref] [PubMed]

V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. 26, 893–895 (2001).
[Crossref]

O’Leary, M. A.

Patterson, M. S.

M. S. Patterson and B. W. Pogue, “Mathematical models for time-resolved and frequency domain fluorescence spectroscopy in biological tissues,” Appl. Opt. 33, 1963–1974 (1994).
[Crossref] [PubMed]

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

Patwardhan, S. V.

Perelman, L.

Pfister, M.

M. Pfister and B. Scholz, “Localization of fluorescent spots with space-space MUSIC for mammography-like measurements system,” J. Biomed. Opt. 9, 481–487 (2004).
[Crossref] [PubMed]

Pifferi, A.

L. Spinelli, A. Toricelli, A. Pifferi, P. Taroni, G. M. Danesini, and R. Cubeddu, “Bulk optical properties and tissue components in the female breast from multiwavelength time-resolved optical mammography,” J. Biomed. Opt. 9, 1137–1142 (2004).
[Crossref] [PubMed]

Pogue, B. W.

Preuss, L. E.

Quan, H.

Rifkin, D. S.

T. H. Foster, E. L. Hull, M. G. Nichols, D. S. Rifkin, and N. Schwartz, “Two steady-state methods for localizing a fluorescent inhomogeneity in a turbid medium,” in Optical Tomography and spectroscopy of Tissue: Theory, Instrumentation, Model, and Human studies II, B. Chance and R. R. Alfano, eds., Proc. SPIE2979, 741–749 (1997).
[Crossref]

Ripoll, J.

Riseborough, P.

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time-resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[Crossref] [PubMed]

Roy, R.

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[Crossref] [PubMed]

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thomson, and R. Roy, Near-infrared fluorescence imaging and spectroscopy in random media and tissues, in Biomedical photonics handbook, T. Vo Dinh ed., (CRC Press, 2003).

Sadoqi, M.

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time-resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[Crossref] [PubMed]

Scholz, B.

M. Pfister and B. Scholz, “Localization of fluorescent spots with space-space MUSIC for mammography-like measurements system,” J. Biomed. Opt. 9, 481–487 (2004).
[Crossref] [PubMed]

Schwartz, N.

T. H. Foster, E. L. Hull, M. G. Nichols, D. S. Rifkin, and N. Schwartz, “Two steady-state methods for localizing a fluorescent inhomogeneity in a turbid medium,” in Optical Tomography and spectroscopy of Tissue: Theory, Instrumentation, Model, and Human studies II, B. Chance and R. R. Alfano, eds., Proc. SPIE2979, 741–749 (1997).
[Crossref]

Schweiger, M.

M. Schweiger, S. R. Arridge, M. Hiroaka, and 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]

Sevick-Muraca, E. M.

A. Godavarty, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Detection of single and multiple targets in tissue phantoms with fluorescence-enhanced optical imaging: feasibility study,” Radiology 235, 148–154 (2005).
[Crossref] [PubMed]

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[Crossref] [PubMed]

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thomson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
[Crossref] [PubMed]

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thomson, and R. Roy, Near-infrared fluorescence imaging and spectroscopy in random media and tissues, in Biomedical photonics handbook, T. Vo Dinh ed., (CRC Press, 2003).

Smith, P. D.

I. Gannot, R. F. Bonner, G. Gannot, P. C. Fox, P. D. Smith, and A. H. Gandjbakhche, “Optical simulations of a non-invasive technique for the diagnosis of diseased salivary glands in situ,” Med. Phys. 25, 1139–1144 (1998).
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Spinelli, L.

C. D’Andrea, L. Spinelli, D. Comelli, G. Valentini, and R. Cubeddu, “Localization and quantification of fluorescent inclusions embedded in a turbid medium,” Phys. Med. Biol. 50, 2313–2327 (2005).
[Crossref] [PubMed]

L. Spinelli, A. Toricelli, A. Pifferi, P. Taroni, G. M. Danesini, and R. Cubeddu, “Bulk optical properties and tissue components in the female breast from multiwavelength time-resolved optical mammography,” J. Biomed. Opt. 9, 1137–1142 (2004).
[Crossref] [PubMed]

Svaasand, L. O.

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A. 11, 2727–2741 (1994).
[Crossref]

Svensson, J.

Swartling, J.

Taroni, P.

L. Spinelli, A. Toricelli, A. Pifferi, P. Taroni, G. M. Danesini, and R. Cubeddu, “Bulk optical properties and tissue components in the female breast from multiwavelength time-resolved optical mammography,” J. Biomed. Opt. 9, 1137–1142 (2004).
[Crossref] [PubMed]

Taylor, R. C.

Terike, K.

Theru, S.

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thomson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
[Crossref] [PubMed]

Thompson, A. B.

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[Crossref] [PubMed]

Thomson, A. B.

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thomson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
[Crossref] [PubMed]

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thomson, and R. Roy, Near-infrared fluorescence imaging and spectroscopy in random media and tissues, in Biomedical photonics handbook, T. Vo Dinh ed., (CRC Press, 2003).

Toricelli, A.

L. Spinelli, A. Toricelli, A. Pifferi, P. Taroni, G. M. Danesini, and R. Cubeddu, “Bulk optical properties and tissue components in the female breast from multiwavelength time-resolved optical mammography,” J. Biomed. Opt. 9, 1137–1142 (2004).
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Tromberg, B. J.

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A. 11, 2727–2741 (1994).
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R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A. 11, 2727–2741 (1994).
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Tung, C. H.

V. Ntziachristos, C. H. Tung, C. Bremer, and R. Weissleder, “Fluorescence mediated tomographic imaging system,” Nature Med. 8, 757–760 (2002).
[Crossref] [PubMed]

Valentini, G.

C. D’Andrea, L. Spinelli, D. Comelli, G. Valentini, and R. Cubeddu, “Localization and quantification of fluorescent inclusions embedded in a turbid medium,” Phys. Med. Biol. 50, 2313–2327 (2005).
[Crossref] [PubMed]

Wang, Y.

Webb, K. J.

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]

V. Ntziachristos, J. Ripoll, and R. Weissleder, “Would near infrared fluorescence signals propagate through large human organs for clinical studies?” Opt. Lett. 27, 333–335 (2002).
[Crossref]

V. Ntziachristos, C. H. Tung, C. Bremer, and R. Weissleder, “Fluorescence mediated tomographic imaging system,” Nature Med. 8, 757–760 (2002).
[Crossref] [PubMed]

V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. 26, 893–895 (2001).
[Crossref]

Wilson, B. C.

T. J. Farrell, M. S. Patterson, and B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the non-invasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
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M. Born and E. Wolf, “Principles of Optics,” (MacMillan, N.Y., 1964).

Wu, J.

Yodh, A. G.

Yuan, B.

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A. Godavarty, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Detection of single and multiple targets in tissue phantoms with fluorescence-enhanced optical imaging: feasibility study,” Radiology 235, 148–154 (2005).
[Crossref] [PubMed]

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[Crossref] [PubMed]

A. Godavarty, M. J. Eppstein, C. Zhang, S. Theru, A. B. Thomson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701–1720 (2003).
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Zysk, A. M.

A. M. Zysk, E. J. Chaney, and S. A. Boppart, “Refractive index of carcinogen-induced rat mammary tumours, Phys. Med. Biol. 51, 2165–2177 (2006).
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Ann. Biomed. Eng. (1)

K. A. Kang, D. F. Bruley, J. M. Londono, and B. Chance, “Localization of a fluorescent object in highly scattering media via frequency response analysis of near infrared-time resolved spectroscopy spectra,” Ann. Biomed. Eng. 26, 138–145 (1998).
[Crossref]

Appl. Opt. (8)

E. L. Hull, M. G. Nichols, and T. H. Foster, “Localization of luminescent inhomogeneities in turbid media with spatially resolved measurement of cw diffuse luminescence emittance,” Appl. Opt. 37, 2755–2765 (1998).
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D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
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I. Gannot, A. Garashi, G. Gannot, V. Chernomordik, and A. Gandjbakhche, “In vivo quantitative three dimensional localization of tumor labelled with exogenous specific fluorescence markers,” Appl. Opt. 42, 3073–3080 (2003).
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A. B. Milstein, M. D. Kennedy, P. S. Low, C. A. Bouman, and K. J. Webb, “Statistical approach for detection and localization of a fluorescing mouse tumor in intralipid,” Appl. Opt. 44, 2300–2310 (2005).
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Eur. Phys. J. Appl. Phys. (1)

X. Deulin and J. P. L’Huillier, “Finite element approach to photon propagation modelling in semi-infinite homogeneous and multilayered tissue structures,” Eur. Phys. J. Appl. Phys. 33, 133–146 (2006).
[Crossref]

Eur. Radiol. (1)

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]

J. Biomed. Opt. (5)

L. Spinelli, A. Toricelli, A. Pifferi, P. Taroni, G. M. Danesini, and R. Cubeddu, “Bulk optical properties and tissue components in the female breast from multiwavelength time-resolved optical mammography,” J. Biomed. Opt. 9, 1137–1142 (2004).
[Crossref] [PubMed]

H. Heusmann, J. Kölzer, and G. Mitic, “Characterization of female breasts in vivo by time resolved and spectroscopic measurements in near infrared spectroscopy,” J. Biomed. Opt. 1, 425–434 (1996).
[Crossref]

I. Gannot, G. Gannot, A. Garashi, A. Gandjbakhche, A. Buchner, and Y. Keisari, “Laser activated fluorescence measurements and morphological features: an in vivo study of clearance time of fluorescein isothiocyanate tagged cell markers,” J. Biomed. Opt. 7, 14–19 (2002).
[Crossref] [PubMed]

M. Pfister and B. Scholz, “Localization of fluorescent spots with space-space MUSIC for mammography-like measurements system,” J. Biomed. Opt. 9, 481–487 (2004).
[Crossref] [PubMed]

A. Godavarty, A. B. Thompson, R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488–496 (2004).
[Crossref] [PubMed]

J. Opt. Soc. Am. A. (1)

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A. 11, 2727–2741 (1994).
[Crossref]

Med. Phys. (3)

M. Schweiger, S. R. Arridge, M. Hiroaka, and 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]

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

I. Gannot, R. F. Bonner, G. Gannot, P. C. Fox, P. D. Smith, and A. H. Gandjbakhche, “Optical simulations of a non-invasive technique for the diagnosis of diseased salivary glands in situ,” Med. Phys. 25, 1139–1144 (1998).
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Figures (9)

Fig. 1.
Fig. 1.

Sketch of the photon propagation in a turbid slab medium with a fluorescent object and geometry of the system under investigation with L=100mm and d=40mm or 60mm. The fluorescing-tagged object was positioned at L/2, but displaced along z at different depths Zt. At each source location Y0, a fluorescence intensity profile can be computed and used to assess the localization of the object. The numbers 1, 1′, 2 and 2′ refer to the segments along which the boundary conditions were applied.

Fig. 2.
Fig. 2.

Principle of the lateral detection of a fluorophore probe (rt=3 mm, Zt=20 mm, C=1µM) by varying the position of the laser source over the top of a tissue slab surface, and probing. the fluorescence intensity profile at the opposite side. The optical parameters used for the simulations are the following: µax =µam= =0.015 mm-1, µ′sx =µ′sm =0.8 mm-1, µfx =µfm =0 for the scattering slab of thickness d=40 mm, and µfx =0.023 mm-1, µfm =0.0115 mm-1 for the inclusion (rt=3 mm)

Fig. 3.
Fig. 3.

Dependence of the on-axis maximum fluorescence signal detected at the surface of the slab on the fluorophore concentration for three different locations of the inclusion (Zt=5, 20 and 35 mm): (a) reflectance mode, (b) transmittance mode. The different simulations refer to µax =µam =0.003 mm-1, µ′sx =µ′sm =1 mm-1, µfx =µfm =0 for the scattering slab of thickness d=40 mm, while µfx =2 µfm was varied in the inclusion of radius rt=3 mm, from 0 (C=0 µM) to 0.23 mm-1 (C=10 µM).

Fig. 4.
Fig. 4.

Contour plots y-z of fluorescence photon flux density at emission wavelength. The computations refer to the same parameters as those used in Fig. 3 except that the fluorophore concentration in the inclusion was fixed at (a) 0.1 µM, (b) 1 µM, and (c) 10 µM, respectively.

Fig. 5.
Fig. 5.

Examples of simulated transmittance measurement of a fluorescing-tagged object of various radii rt and located at different depths Zt inside a turbid slab medium of thickness d=40 mm, (a) plot of normalized scan intensity profiles for a small object of radius rt=1 mm located at four different depths Zt=2, 20, 30 and 36 mm, (b) plot of normalized scan intensity profiles for five different sized objects rt=1, 3, 5, 6, and 10 mm located at Zt=20 mm. The simulations are based on the same optical parameters as those used in Fig. 2.

Fig. 6.
Fig. 6.

Contour plot y-z of fluorescence photon flux density at emission wavelength, for two different sized objects containing 1µM of ICG and embedded in the middle plane of a turbid slab medium of thickness d=40 mm. (a) radius rt=2 mm, (b) rt=6 mm.

Fig. 7.
Fig. 7.

(a). Plot of the depth indicator Fw (Zt) against depth location Zt, for a fluorescent object of radius 3 mm embedded in turbid slab media having different optical properties: -µax =µam =0.015 mm-1, µs x,m=0.8 mm-1 and 1.6 mm-1, -µax =µam =0.0015 mm-1, µs x,m=0.8 mm-1. (b) Plot of the depth indicator Fw (Zt) against depth location Zt, for three different sized objects (rt=1, 3 and 5mm) embedded in a turbid slab medium with µax =µam =0.015 mm-1, and µs x,m=0.8 mm-1. In both cases the fluorophore concentration inside the object is equal to 1µM.

Fig. 8.
Fig. 8.

Plot of the dimensionless indicator Fw(β) against the dimensionless depth β=Zt/d for a probe of radius 3mm containing various concentrations of markers and embedded inside a turbid slab of different thicknesses. (a) ICG/C=0.1µM, 1µM, and 3µM, Cy5.5/C=0.1µM, d=40mm. (b) Cy5.5/0.1µM, d=40mm and 60mm.

Fig. 9.
Fig. 9.

Plot of the dimensionless indicator Fw(β) against the dimensionless depth β=Zt/d for a probe of radius rt=3mm embedded inside a turbid slab of thickness d=40mm with three typical concentration contrasts equal to 1:0, 1:0.01 and 1:0.005.

Equations (10)

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. ( D x Φ x ( y , z ) ) + k x Φ x ( y , z ) = q x ( y , z )
. ( D m Φ m ( y , z ) ) + k m Φ m ( y , z ) = q m ( y , z )
q m ( y , z ) = ϕ μ fx Φ x ( y , z )
q x ( y 0 , z ) = μ sx L 0 e μ tx z ( 1 + g μ tx μ trx )
Φ m ( ξ ) + 2 A . n ̂ . D m . Φ m ( ξ ) = 0
A = 1 + R eff 1 R eff
Φ x ( ξ ) + 2 A . n ̂ . D x . Φ x ( ξ ) = g μ sx μ trx L 0
J m ( R ) ( y , z ) = D m Φ m ( y , z ) | z = 0
J m ( T ) ( y , z ) = D m Φ m ( y , z ) | z = d
F W ( Z t ) = FWHM ( Z t ) FWHM ( d Z t ) FWHM ( Z t ) + FWHM ( d Z t )

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