Abstract

We introduce a diffused optical detection system based on the administration of a fluorophore-antibody conjugate to diseased tissue. The conjugate interacts with the antigens expressed by the diseased tissue, resulting in fluorescent labeling of the antigen. By combining an optical detection system with a reconstruction algorithm developed on the basis of the random-walk model, we were able to determine the position of the fluorophore (and, thus, of the diseased cells) in the tissue. We present three-dimensional reconstructions of the location of a fluorophore (FITC-fluorescein isothiocyanate) in the tongues of mice. Measurements were performed with the fluorophore embedded at various simulated depths. The simulations were performed with agarose-based gel slabs applied to the tongue as tissuelike phantoms. Reconstructed fluorophore locations agree well with the actual values.

© 2003 Optical Society of America

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References

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2002

G. Gannot, I. Gannot, H. Vered, A. Buchner, Y. Keisari, “Increase in immune cell infiltration with progression of oral epithelium from hyperkeratosis to dysplasia and carcinoma,” Br. J. Cancer 86, 1444–1448 (2002).
[CrossRef] [PubMed]

2001

2000

D. J. Hawrysz, E. M. Sevick-Muraca, “Developments toward diagnostic breast cancer imaging using near-infrared optical measurements and fluorescent contrast agents,” Neoplasia 2, 388–417 (2000).
[CrossRef]

1999

V. Chernomordik, D. Hattery, I. Gannot, A. H. Gandjbakhche, “Inverse method 3D reconstruction of localized in vivo fluorescence—Application to Sjogren syndrome,” IEEE J. Sel. Top. Quantum Electron. 5, 930–935 (1999).
[CrossRef]

1998

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

G. A. Wagnières, W. M. Star, B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68, 603–632 (1998).
[PubMed]

1997

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

G. Wagnières, S. Cheng, M. Zellweger, N. Utke, D. Braichotte, J.-P. Ballini, H. van den Berg, “An optical phantom with tissue-like properties in the visible for use in PDT and fluorescence spectroscopy,” Phys. Med. Biol. 42, 1–12 (1997).
[CrossRef]

S. Andersson-Engels, C. Afklinteberg, K. Svanberg, S. Svanberg, “In vivo fluorescence imaging for tissue diagnostics,” Phys. Med. Biol. 42, 815–824 (1997).
[CrossRef] [PubMed]

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, G. H. Weiss, “Effects of multiple-passage probabilities on fluorescent signals from biological media,” Appl. Opt. 36, 4613–4619 (1997).
[CrossRef] [PubMed]

1996

1994

1993

1990

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of optical properties of biological tissue,” IEEE J. Quantum Electron. 2, 2166–2185 (1990).
[CrossRef]

Afklinteberg, C.

S. Andersson-Engels, C. Afklinteberg, K. Svanberg, S. Svanberg, “In vivo fluorescence imaging for tissue diagnostics,” Phys. Med. Biol. 42, 815–824 (1997).
[CrossRef] [PubMed]

Andersson-Engels, S.

S. Andersson-Engels, C. Afklinteberg, K. Svanberg, S. Svanberg, “In vivo fluorescence imaging for tissue diagnostics,” Phys. Med. Biol. 42, 815–824 (1997).
[CrossRef] [PubMed]

Aronson, R.

Arridge, S. R.

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Ballini, J.-P.

G. Wagnières, S. Cheng, M. Zellweger, N. Utke, D. Braichotte, J.-P. Ballini, H. van den Berg, “An optical phantom with tissue-like properties in the visible for use in PDT and fluorescence spectroscopy,” Phys. Med. Biol. 42, 1–12 (1997).
[CrossRef]

Barbour, R. L.

Bonner, R. F.

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

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, G. H. Weiss, “Effects of multiple-passage probabilities on fluorescent signals from biological media,” Appl. Opt. 36, 4613–4619 (1997).
[CrossRef] [PubMed]

Braichotte, D.

G. Wagnières, S. Cheng, M. Zellweger, N. Utke, D. Braichotte, J.-P. Ballini, H. van den Berg, “An optical phantom with tissue-like properties in the visible for use in PDT and fluorescence spectroscopy,” Phys. Med. Biol. 42, 1–12 (1997).
[CrossRef]

Buchner, A.

G. Gannot, I. Gannot, H. Vered, A. Buchner, Y. Keisari, “Increase in immune cell infiltration with progression of oral epithelium from hyperkeratosis to dysplasia and carcinoma,” Br. J. Cancer 86, 1444–1448 (2002).
[CrossRef] [PubMed]

Chang, J. H.

Cheng, S.

G. Wagnières, S. Cheng, M. Zellweger, N. Utke, D. Braichotte, J.-P. Ballini, H. van den Berg, “An optical phantom with tissue-like properties in the visible for use in PDT and fluorescence spectroscopy,” Phys. Med. Biol. 42, 1–12 (1997).
[CrossRef]

Cheong, W. F.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of optical properties of biological tissue,” IEEE J. Quantum Electron. 2, 2166–2185 (1990).
[CrossRef]

Chernomordik, V.

V. Chernomordik, D. Hattery, I. Gannot, A. H. Gandjbakhche, “Inverse method 3D reconstruction of localized in vivo fluorescence—Application to Sjogren syndrome,” IEEE J. Sel. Top. Quantum Electron. 5, 930–935 (1999).
[CrossRef]

Cotran, R. S.

R. S. Cotran, V. Kumar, S. L. Robbins, Robbins Pathological Basis of Disease, 5th ed. (Saunders, Philadelphia, 1994).

Delpy, D. T.

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

Eppstein, M. J.

Feld, M. S.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), p. 994.

Fox, P. C.

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

Gandjbakhche, A. H.

A. H. Gandjbakhche, “Diffused optical imaging and spectroscopy, in vivo,” C. R. Acad. Sci. Ser. IV 2, 1073–1079 (2001).

V. Chernomordik, D. Hattery, I. Gannot, A. H. Gandjbakhche, “Inverse method 3D reconstruction of localized in vivo fluorescence—Application to Sjogren syndrome,” IEEE J. Sel. Top. Quantum Electron. 5, 930–935 (1999).
[CrossRef]

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

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, G. H. Weiss, “Effects of multiple-passage probabilities on fluorescent signals from biological media,” Appl. Opt. 36, 4613–4619 (1997).
[CrossRef] [PubMed]

A. H. Gandjbakhche, I. Gannot, “Quantitative fluorescent imaging of specific markers of diseased tissue,” IEEE J. Sel. Top. Quantum Electron. 2, 914–921 (1996).
[CrossRef]

A. H. Gandjbakhche, G. H. Weiss, “Random walk and diffusion-like models of photon migration in turbid media,” in Progress in Optics XXXIV, E. Wolf, ed. (Elesevier, Amsterdam, 1995), pp. 335–402.

Gannot, G.

G. Gannot, I. Gannot, H. Vered, A. Buchner, Y. Keisari, “Increase in immune cell infiltration with progression of oral epithelium from hyperkeratosis to dysplasia and carcinoma,” Br. J. Cancer 86, 1444–1448 (2002).
[CrossRef] [PubMed]

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

Gannot, I.

G. Gannot, I. Gannot, H. Vered, A. Buchner, Y. Keisari, “Increase in immune cell infiltration with progression of oral epithelium from hyperkeratosis to dysplasia and carcinoma,” Br. J. Cancer 86, 1444–1448 (2002).
[CrossRef] [PubMed]

V. Chernomordik, D. Hattery, I. Gannot, A. H. Gandjbakhche, “Inverse method 3D reconstruction of localized in vivo fluorescence—Application to Sjogren syndrome,” IEEE J. Sel. Top. Quantum Electron. 5, 930–935 (1999).
[CrossRef]

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

A. H. Gandjbakhche, I. Gannot, “Quantitative fluorescent imaging of specific markers of diseased tissue,” IEEE J. Sel. Top. Quantum Electron. 2, 914–921 (1996).
[CrossRef]

Graber, H. L.

Hattery, D.

V. Chernomordik, D. Hattery, I. Gannot, A. H. Gandjbakhche, “Inverse method 3D reconstruction of localized in vivo fluorescence—Application to Sjogren syndrome,” IEEE J. Sel. Top. Quantum Electron. 5, 930–935 (1999).
[CrossRef]

Hawrysz, D. J.

D. J. Hawrysz, M. J. Eppstein, J. W. Lee, E. M. Sevick-Muraca, “Error consideration in contrast-enhanced three-dimensional optical tomography,” Opt. Lett. 26, 704–706 (2001).
[CrossRef]

D. J. Hawrysz, E. M. Sevick-Muraca, “Developments toward diagnostic breast cancer imaging using near-infrared optical measurements and fluorescent contrast agents,” Neoplasia 2, 388–417 (2000).
[CrossRef]

Hebden, J. C.

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

Keisari, Y.

G. Gannot, I. Gannot, H. Vered, A. Buchner, Y. Keisari, “Increase in immune cell infiltration with progression of oral epithelium from hyperkeratosis to dysplasia and carcinoma,” Br. J. Cancer 86, 1444–1448 (2002).
[CrossRef] [PubMed]

Kumar, V.

R. S. Cotran, V. Kumar, S. L. Robbins, Robbins Pathological Basis of Disease, 5th ed. (Saunders, Philadelphia, 1994).

Lee, J. W.

Nossal, R.

Ntziachristos, V.

Patterson, M. S.

Pogue, B. W.

Prahl, S. A.

S. A. Prahl, M. J. C. Van-Gemert, A. J. Welch, “Determining the optical properties of turbid media by using the adding doubling method,” Appl. Opt. 32, 559–568 (1993).
[CrossRef] [PubMed]

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of optical properties of biological tissue,” IEEE J. Quantum Electron. 2, 2166–2185 (1990).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), p. 994.

Rava, R. P.

Robbins, S. L.

R. S. Cotran, V. Kumar, S. L. Robbins, Robbins Pathological Basis of Disease, 5th ed. (Saunders, Philadelphia, 1994).

Roy, R.

Sevick-Muraca, E. M.

Smith, P. D.

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

Star, W. M.

G. A. Wagnières, W. M. Star, B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68, 603–632 (1998).
[PubMed]

Svanberg, K.

S. Andersson-Engels, C. Afklinteberg, K. Svanberg, S. Svanberg, “In vivo fluorescence imaging for tissue diagnostics,” Phys. Med. Biol. 42, 815–824 (1997).
[CrossRef] [PubMed]

Svanberg, S.

S. Andersson-Engels, C. Afklinteberg, K. Svanberg, S. Svanberg, “In vivo fluorescence imaging for tissue diagnostics,” Phys. Med. Biol. 42, 815–824 (1997).
[CrossRef] [PubMed]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), p. 994.

Utke, N.

G. Wagnières, S. Cheng, M. Zellweger, N. Utke, D. Braichotte, J.-P. Ballini, H. van den Berg, “An optical phantom with tissue-like properties in the visible for use in PDT and fluorescence spectroscopy,” Phys. Med. Biol. 42, 1–12 (1997).
[CrossRef]

van den Berg, H.

G. Wagnières, S. Cheng, M. Zellweger, N. Utke, D. Braichotte, J.-P. Ballini, H. van den Berg, “An optical phantom with tissue-like properties in the visible for use in PDT and fluorescence spectroscopy,” Phys. Med. Biol. 42, 1–12 (1997).
[CrossRef]

Van-Gemert, M. J. C.

Vered, H.

G. Gannot, I. Gannot, H. Vered, A. Buchner, Y. Keisari, “Increase in immune cell infiltration with progression of oral epithelium from hyperkeratosis to dysplasia and carcinoma,” Br. J. Cancer 86, 1444–1448 (2002).
[CrossRef] [PubMed]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), p. 994.

Wagnières, G.

G. Wagnières, S. Cheng, M. Zellweger, N. Utke, D. Braichotte, J.-P. Ballini, H. van den Berg, “An optical phantom with tissue-like properties in the visible for use in PDT and fluorescence spectroscopy,” Phys. Med. Biol. 42, 1–12 (1997).
[CrossRef]

Wagnières, G. A.

G. A. Wagnières, W. M. Star, B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68, 603–632 (1998).
[PubMed]

Weiss, G. H.

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, G. H. Weiss, “Effects of multiple-passage probabilities on fluorescent signals from biological media,” Appl. Opt. 36, 4613–4619 (1997).
[CrossRef] [PubMed]

A. H. Gandjbakhche, G. H. Weiss, “Random walk and diffusion-like models of photon migration in turbid media,” in Progress in Optics XXXIV, E. Wolf, ed. (Elesevier, Amsterdam, 1995), pp. 335–402.

Weissleder, R.

Welch, A. J.

S. A. Prahl, M. J. C. Van-Gemert, A. J. Welch, “Determining the optical properties of turbid media by using the adding doubling method,” Appl. Opt. 32, 559–568 (1993).
[CrossRef] [PubMed]

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of optical properties of biological tissue,” IEEE J. Quantum Electron. 2, 2166–2185 (1990).
[CrossRef]

Wilson, B. C.

G. A. Wagnières, W. M. Star, B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68, 603–632 (1998).
[PubMed]

Wu, J.

Zellweger, M.

G. Wagnières, S. Cheng, M. Zellweger, N. Utke, D. Braichotte, J.-P. Ballini, H. van den Berg, “An optical phantom with tissue-like properties in the visible for use in PDT and fluorescence spectroscopy,” Phys. Med. Biol. 42, 1–12 (1997).
[CrossRef]

Appl. Opt.

Br. J. Cancer

G. Gannot, I. Gannot, H. Vered, A. Buchner, Y. Keisari, “Increase in immune cell infiltration with progression of oral epithelium from hyperkeratosis to dysplasia and carcinoma,” Br. J. Cancer 86, 1444–1448 (2002).
[CrossRef] [PubMed]

C. R. Acad. Sci. Ser. IV

A. H. Gandjbakhche, “Diffused optical imaging and spectroscopy, in vivo,” C. R. Acad. Sci. Ser. IV 2, 1073–1079 (2001).

IEEE J. Quantum Electron.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of optical properties of biological tissue,” IEEE J. Quantum Electron. 2, 2166–2185 (1990).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

V. Chernomordik, D. Hattery, I. Gannot, A. H. Gandjbakhche, “Inverse method 3D reconstruction of localized in vivo fluorescence—Application to Sjogren syndrome,” IEEE J. Sel. Top. Quantum Electron. 5, 930–935 (1999).
[CrossRef]

A. H. Gandjbakhche, I. Gannot, “Quantitative fluorescent imaging of specific markers of diseased tissue,” IEEE J. Sel. Top. Quantum Electron. 2, 914–921 (1996).
[CrossRef]

Med. Phy.

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

Neoplasia

D. J. Hawrysz, E. M. Sevick-Muraca, “Developments toward diagnostic breast cancer imaging using near-infrared optical measurements and fluorescent contrast agents,” Neoplasia 2, 388–417 (2000).
[CrossRef]

Opt. Lett.

Photochem. Photobiol.

G. A. Wagnières, W. M. Star, B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68, 603–632 (1998).
[PubMed]

Phys. Med. Biol.

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

G. Wagnières, S. Cheng, M. Zellweger, N. Utke, D. Braichotte, J.-P. Ballini, H. van den Berg, “An optical phantom with tissue-like properties in the visible for use in PDT and fluorescence spectroscopy,” Phys. Med. Biol. 42, 1–12 (1997).
[CrossRef]

S. Andersson-Engels, C. Afklinteberg, K. Svanberg, S. Svanberg, “In vivo fluorescence imaging for tissue diagnostics,” Phys. Med. Biol. 42, 815–824 (1997).
[CrossRef] [PubMed]

Other

A. H. Gandjbakhche, G. H. Weiss, “Random walk and diffusion-like models of photon migration in turbid media,” in Progress in Optics XXXIV, E. Wolf, ed. (Elesevier, Amsterdam, 1995), pp. 335–402.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), p. 994.

R. R. Alfano, ed., Optical Biopsy IV, Proc. SPIE4613 (2002).

R. S. Cotran, V. Kumar, S. L. Robbins, Robbins Pathological Basis of Disease, 5th ed. (Saunders, Philadelphia, 1994).

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Figures (7)

Fig. 1
Fig. 1

H&E section of the tongue with SQCC tumor at day 1. A, Magnification ×2; B, higher magnification (×40) of the tumor section to show cell structure and organized structure.

Fig. 2
Fig. 2

H&E section of the tongue with SQCC tumor at day 20. A, Magnification ×2); B, Higher magnification (×40) of the tumor section to show dense and established tumor structure. The fluoresceinated antibodies bind to the infiltrating lymphocytes that appear around the tumor and infiltrate the tumor border.

Fig. 3
Fig. 3

Fluorescent imaging experimental setup.

Fig. 4
Fig. 4

A, Surface fluorescence light image, healthy mouse + CD19, 1-mm agarose slab; B, surface fluorescence light image, diseased mouse + CD19, 1.18-mm agarose slab (at day 2); C, surface fluorescence light image, diseased mouse + CD19, 1.95-mm agarose slab (20-day-old tumor).

Fig. 5
Fig. 5

Comparison of the detected intensity profiles with the profiles reconstructed from the theoretical model (taken at day 2). A, Healthy mouse, for z f = 1.0 mm nominal thickness of the turbid layer above the mouse tongue with antibody + fluorophore; reconstructed depth 0.98 mm. B, Healthy mouse, for z f = 1.0 mm nominal thickness of the turbid layer above the mouse tongue with antibody + fluorophore; reconstructed depth 1.0 mm.

Fig. 6
Fig. 6

Comparison of the detected intensity profiles with the profiles reconstructed from the theoretical model (taken at day 2). A, Sick mouse, for z f = 1.18 mm nominal thickness of the turbid layer above the mouse tongue with antibody + fluorophore; reconstructed depth 1.06 mm. B, Sick mouse, for z f = 1.18 mm nominal thickness of the turbid layer above the mouse tongue with antibody + fluorophore; reconstructed depth 1.09 mm.

Fig. 7
Fig. 7

Comparison of the detected intensity profiles with the profiles, reconstructed from the theoretical model (taken at day 20). A, Sick mouse, for z f = 1.95 mm nominal thickness of the turbid layer above the mouse tongue with antibody + fluorophore; reconstructed depth 1.78 mm. B, Sick mouse, for z f = 1.93 mm nominal thickness of the turbid layer above the mouse tongue with antibody + fluorophore; reconstructed depth 1.61 mm.

Equations (4)

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Γr, s=Φ μafμsfHα-, β--Hα-, β+-Hα+, β-+Hα+, β+exp-μaeμse1-μafμsf+μafμsf1+183π3/2m=11m3/2exp-2m μaiμsi,
Hα, β=1αβexp-2α μaiμsi+β μaeμse,
α±=34x¯f2+y¯f2+z¯f±2μsi2μsi2,
β±=34x¯f-x¯2+y¯f-y¯2+z¯f+2μse±2μse2μse2.

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