Abstract

Bioluminescence imaging has been a popular tool in small animal imaging. During the last decade, the efforts have focused on the development of tomographic systems. However, due to the difficulties in the nature of inverse source problem, multi-modal systems have been the center of attention for the last couple of years. These systems provide complementary information such that the difficulties of the inverse source problem could be overcome using the a priori information obtained. Motivated by these advances in multi-modal systems, this work presents a novel analytical reconstruction of the bioluminescent source. It is shown that if source strength is known a priori then source position could be calculated or vice versa, if source location is known a priori, source strength could be calculated as well as the photon fluence rate. The determination of the source location can be achieved by another imaging system such as X-ray computed tomography. Therefore, in bioluminescence tomography together with an imaging system can be utilized as a multi-modal system. In this work, conventional finite element based simulations are also performed and the numerical results are compared with the analytical ones. It turns out to be that the analytical results are in a good accordance with the numerical results.

© 2014 Optical Society of America

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2014 (2)

2013 (2)

H. Erkol and M. B. Unlu, “Virtual source method for diffuse optical imaging,” Appl. Opt. 52, 4933–4970 (2013).
[CrossRef] [PubMed]

J. A. Guggenheim, H. R. A. Basevi, J. Frampton, I. B. Styles, and H. Dehghani, “Multi-modal molecular diffuse optical tomography system for small animal imaging,” Meas. Sci. Technol. 24, 105405 (2013).
[CrossRef] [PubMed]

2012 (1)

H. Yan, Y. Lin, W. C. Barber, M. B. Unlu, and G. Gulsen, “A gantry-based tri-modality system for bioluminescence tomography,” Rev. Sci. Instrum. 83, 043708 (2012).
[CrossRef] [PubMed]

2011 (1)

W. Han, J. A. Eichholz, and G. Wang, “On a family of differential approximations of the radiative transfer equation,” J. Math. Chem. 50, 689–702 (2011).
[CrossRef]

2010 (6)

2008 (2)

Q. Zhang, L. Yin, Y. Tan, Z. Yuan, and H. Jiang, “Quantitative bioluminescence tomography guided by diffuse optical tomography,” Opt. Express 16, 1481–1486 (2008).
[CrossRef] [PubMed]

H. Dehghani and B. W. Pogue, “Spectrally resolved bioluminescence optical tomography using the reciprocity approach,” Med. Phys. 35, 4863–4871 (2008).
[CrossRef] [PubMed]

2007 (2)

F. Martelli, A. Sassaroli, S. D. Bianco, and G. Zaccanti, “Solution of the time-dependent diffusion equation for a three-layer medium: application to study photon migration through a simplified adult head model,” Phys. Med. Biol. 52, 2827–2843 (2007).
[CrossRef] [PubMed]

M. A. Anastasio, J. Zhang, D. Modgil, and P. J. L. Rivière, “Application of inverse source concepts to photoacoustic tomography,” Inverse Problems 23, 21–35 (2007).
[CrossRef]

2006 (5)

2005 (3)

2004 (3)

M. Shendeleva, “Radiative transfer in a turbid medium with a varying refractive index: comment,” J. Opt. Soc. Am. A 21, 2464–2467 (2004).
[CrossRef]

W. Cong, L. V. Wang, and G. Wang, “Formulation of photon diffusion from spherical bioluminescent sources in an infinite homogeneous medium,” Biomed. Eng. Online 3, 1–6 (2004).
[CrossRef]

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. 31(8), 2289–2299 (2004).
[CrossRef] [PubMed]

2003 (1)

E. Demiralp and H. Beker, “Properties of bound states of the Schrödinger equation with attractive Dirac delta potentials,” J. Phys. A 36, 7449–7459 (2003).
[CrossRef]

2002 (2)

F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, “Analytical approximate solutions of the timedomain diffusion equation in layered slabs,” J. Opt. Soc. Am. A. 19, 71–80 (2002).
[CrossRef]

G. W. Faris, “Diffusion equation boundary conditions for the interface between turbid media: a comment,” J. Opt. Soc. Am. A 19, 519–520 (2002).
[CrossRef]

2001 (3)

T. J. Farrel and M. S. Patterson, “Experimental verification of the effect of refractive index mismatch on the light fluence in a turbid medium,” J. Biomed. Opt. 6, 468–473 (2001).
[CrossRef]

T. Farrel and M. S. Patterson, “Experimental verification of the effect of refractive index mismatch on the light fluence in a turbid medium,” J. Biomed. Opt. 6, 468–473 (2001).
[CrossRef]

B. W. Pogue, S. Geimer, T. O. McBride, S. Jiang, U. L. Österberg, and K. D. Paulsen, “Three-dimensional simulation of near-infrared diffusion in tissue: boundary condition and geometry analysis for finite-element image reconstruction,” Appl. Opt. 40, 588–600 (2001).
[CrossRef]

2000 (2)

1999 (5)

B. W. Pogue, T. O. McBride, U. L. Osterberg, and K. D. Paulsen, “Comparison of imaging geometries for diffuse optical tomography of tissue,” Opt. Express 4, 270–286 (1999).
[CrossRef] [PubMed]

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, and M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

A. Kienle and T. Glanzmann, “In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model,” Phys. Med. Biol. 44, 2689–2702 (1999).
[CrossRef] [PubMed]

M. Schweiger and S. R. Arridge, “Optical tomographic reconstructon in a complex head model using a priori region boundary condition,” Phys. Med. Biol. 44, 2703–2721 (1999).
[CrossRef] [PubMed]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, 41–93 (1999).
[CrossRef]

1998 (1)

1997 (4)

1995 (1)

1994 (2)

B. W. Pogue and M. S. Patterson, “Frequency-domain optical absorption spectroscopy of finite tissue volumes using diffusion theory,” Phys. Med. Biol. 39, 1157–1180 (1994).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’leary, B. Chances, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: Analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91, 4887–4891 (1994).
[CrossRef] [PubMed]

1993 (2)

S. R. Arridge, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite element method for the forward and inverse model in optical tomography,” J. Mat. Imaging Vis. 3, 263–283 (1993).
[CrossRef]

1992 (3)

I. Dayan, S. Havlin, and G. H. Weiss, “Photon migration in a two-layer turbid medium A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

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

1991 (1)

M. S. Patterson and S. J. Madsen, “Diffusion equation representation of photon migration in tissue,” IEEE MTT-S Digest 2, 905–908 (1991).

1990 (1)

1989 (1)

1979 (1)

S. Takatani and M. D. Graham, “Theoretical analysis of diffuse reflectance from a two-layer tissue model,” IEEE Trans. Biomed. Eng. 26, 656–664 (1979).
[CrossRef] [PubMed]

Anastasio, M. A.

M. A. Anastasio, J. Zhang, D. Modgil, and P. J. L. Rivière, “Application of inverse source concepts to photoacoustic tomography,” Inverse Problems 23, 21–35 (2007).
[CrossRef]

Aronson, R.

Arridge, S. R.

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. R. Arridge, and J. Ripoll, “Diffuse photon propagation in multilayered geometries,” Phys. Med. Biol. 51, 497–516 (2006).
[CrossRef] [PubMed]

A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. M. C. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44(11), 2082–2093 (2005).
[CrossRef] [PubMed]

M. Schweiger and S. R. Arridge, “Optical tomographic reconstructon in a complex head model using a priori region boundary condition,” Phys. Med. Biol. 44, 2703–2721 (1999).
[CrossRef] [PubMed]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, 41–93 (1999).
[CrossRef]

E. Okada, M. Firbank, M. Schwiger, S. R. Arridge, M. Cope, and D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. 36, 21–31 (1997).
[CrossRef] [PubMed]

M. Schweiger and S. R. Arridge, “The finite-element method for the propagation of light in scattering media: frequency domain case,” Med. Phys. 24, 895–902 (1997).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite element method for the forward and inverse model in optical tomography,” J. Mat. Imaging Vis. 3, 263–283 (1993).
[CrossRef]

S. R. Arridge, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Avrillier, S.

Barber, W. C.

H. Yan, Y. Lin, W. C. Barber, M. B. Unlu, and G. Gulsen, “A gantry-based tri-modality system for bioluminescence tomography,” Rev. Sci. Instrum. 83, 043708 (2012).
[CrossRef] [PubMed]

Basevi, H. R. A.

J. A. Guggenheim, H. R. A. Basevi, J. Frampton, I. B. Styles, and H. Dehghani, “Multi-modal molecular diffuse optical tomography system for small animal imaging,” Meas. Sci. Technol. 24, 105405 (2013).
[CrossRef] [PubMed]

Beker, H.

E. Demiralp and H. Beker, “Properties of bound states of the Schrödinger equation with attractive Dirac delta potentials,” J. Phys. A 36, 7449–7459 (2003).
[CrossRef]

Berns, M. W.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, and M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Bianco, S. D.

F. Martelli, A. Sassaroli, S. D. Bianco, and G. Zaccanti, “Solution of the time-dependent diffusion equation for a three-layer medium: application to study photon migration through a simplified adult head model,” Phys. Med. Biol. 52, 2827–2843 (2007).
[CrossRef] [PubMed]

Birgul, O.

M. B. Unlu, O. Birgul, R. Shafiha, G. Gulsen, and O. Nalcıoğlu, “Diffuse optical tomographic reconstruction using multifrequency data,” J. Biomed. Opt. 11, 054008 (2006).
[CrossRef]

Boas, D. A.

S. A. Walker, D. A. Boas, and E. Gratton, “Photon density waves scattered from cylindrical inhomogeneities: theory and experiments,” Appl. Opt. 37, 1935–1944 (1998).
[CrossRef]

D. A. Boas, M. A. O’leary, B. Chances, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: Analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Brooksby, B. A.

Bunting, C. F.

Chances, B.

D. A. Boas, M. A. O’leary, B. Chances, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: Analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Chen, D.

Choe, R.

Cong, W.

Contini, D.

Cope, M.

E. Okada, M. Firbank, M. Schwiger, S. R. Arridge, M. Cope, and D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. 36, 21–31 (1997).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Corlu, A.

Darne, C.

C. Darne, Y. Lu, and E. M. Sevick-Muraca, “Small animal fluorescence and bioluminescence tomography: a review of approaches, algorithms and technology update,” Phys. Med. Biol. 59, R1–R64 (2014).
[CrossRef]

Davis, S. C.

Dayan, I.

I. Dayan, S. Havlin, and G. H. Weiss, “Photon migration in a two-layer turbid medium A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

Dehghani, H.

Delpy, D. T.

H. Dehghani and D. T. Delpy, “Near-infrared spectroscopy of the adult head: effect of scattering and absorbing obstructions in the cerebrospinal fluid layer on light distribution in the tissue,” Appl. Opt. 39, 4721–4729 (2000).
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M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite element method for the forward and inverse model in optical tomography,” J. Mat. Imaging Vis. 3, 263–283 (1993).
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S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
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Durduran, T.

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W. Han, J. A. Eichholz, and G. Wang, “On a family of differential approximations of the radiative transfer equation,” J. Math. Chem. 50, 689–702 (2011).
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Erkol, H.

Faris, G. W.

Farrel, T.

T. Farrel and M. S. Patterson, “Experimental verification of the effect of refractive index mismatch on the light fluence in a turbid medium,” J. Biomed. Opt. 6, 468–473 (2001).
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Farrel, T. J.

T. J. Farrel and M. S. Patterson, “Experimental verification of the effect of refractive index mismatch on the light fluence in a turbid medium,” J. Biomed. Opt. 6, 468–473 (2001).
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T. J. Farrel, M. S. Patterson, and B. C. Wilson, “A diffusion theory model of specially resolved steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef]

Firbank, M.

Fishkin, J. B.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, and M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
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J. A. Guggenheim, H. R. A. Basevi, J. Frampton, I. B. Styles, and H. Dehghani, “Multi-modal molecular diffuse optical tomography system for small animal imaging,” Meas. Sci. Technol. 24, 105405 (2013).
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Guggenheim, J. A.

J. A. Guggenheim, H. R. A. Basevi, J. Frampton, I. B. Styles, and H. Dehghani, “Multi-modal molecular diffuse optical tomography system for small animal imaging,” Meas. Sci. Technol. 24, 105405 (2013).
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Gulsen, G.

H. Yan, Y. Lin, W. C. Barber, M. B. Unlu, and G. Gulsen, “A gantry-based tri-modality system for bioluminescence tomography,” Rev. Sci. Instrum. 83, 043708 (2012).
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M. B. Unlu, O. Birgul, R. Shafiha, G. Gulsen, and O. Nalcıoğlu, “Diffuse optical tomographic reconstruction using multifrequency data,” J. Biomed. Opt. 11, 054008 (2006).
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W. Han, J. A. Eichholz, and G. Wang, “On a family of differential approximations of the radiative transfer equation,” J. Math. Chem. 50, 689–702 (2011).
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I. Dayan, S. Havlin, and G. H. Weiss, “Photon migration in a two-layer turbid medium A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
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Hoffman, E.

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J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. R. Arridge, and J. Ripoll, “Diffuse photon propagation in multilayered geometries,” Phys. Med. Biol. 51, 497–516 (2006).
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Jiang, S.

Jin, Z.

Kienle, A.

Lee, K.

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G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. 31(8), 2289–2299 (2004).
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Liemert, A.

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H. Yan, Y. Lin, W. C. Barber, M. B. Unlu, and G. Gulsen, “A gantry-based tri-modality system for bioluminescence tomography,” Rev. Sci. Instrum. 83, 043708 (2012).
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M. S. Patterson and S. J. Madsen, “Diffusion equation representation of photon migration in tissue,” IEEE MTT-S Digest 2, 905–908 (1991).

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F. Martelli, A. Sassaroli, S. D. Bianco, and G. Zaccanti, “Solution of the time-dependent diffusion equation for a three-layer medium: application to study photon migration through a simplified adult head model,” Phys. Med. Biol. 52, 2827–2843 (2007).
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F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, “Analytical approximate solutions of the timedomain diffusion equation in layered slabs,” J. Opt. Soc. Am. A. 19, 71–80 (2002).
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Mclennan, G.

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M. B. Unlu, O. Birgul, R. Shafiha, G. Gulsen, and O. Nalcıoğlu, “Diffuse optical tomographic reconstruction using multifrequency data,” J. Biomed. Opt. 11, 054008 (2006).
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O’leary, M. A.

D. A. Boas, M. A. O’leary, B. Chances, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: Analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91, 4887–4891 (1994).
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M. S. Patterson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
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Paulsen, K. D.

Pham, T.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, and M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
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H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, “Spectrally resolved bioluminescence optical tomography,” Opt. Lett. 31, 365–367 (2006).
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J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. R. Arridge, and J. Ripoll, “Diffuse photon propagation in multilayered geometries,” Phys. Med. Biol. 51, 497–516 (2006).
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Rivière, P. J. L.

M. A. Anastasio, J. Zhang, D. Modgil, and P. J. L. Rivière, “Application of inverse source concepts to photoacoustic tomography,” Inverse Problems 23, 21–35 (2007).
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Sassaroli, A.

F. Martelli, A. Sassaroli, S. D. Bianco, and G. Zaccanti, “Solution of the time-dependent diffusion equation for a three-layer medium: application to study photon migration through a simplified adult head model,” Phys. Med. Biol. 52, 2827–2843 (2007).
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F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, “Analytical approximate solutions of the timedomain diffusion equation in layered slabs,” J. Opt. Soc. Am. A. 19, 71–80 (2002).
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Schweiger, M.

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. R. Arridge, and J. Ripoll, “Diffuse photon propagation in multilayered geometries,” Phys. Med. Biol. 51, 497–516 (2006).
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A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. M. C. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44(11), 2082–2093 (2005).
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Schwiger, M.

Sevick-Muraca, E. M.

C. Darne, Y. Lu, and E. M. Sevick-Muraca, “Small animal fluorescence and bioluminescence tomography: a review of approaches, algorithms and technology update,” Phys. Med. Biol. 59, R1–R64 (2014).
[CrossRef]

Shafiha, R.

M. B. Unlu, O. Birgul, R. Shafiha, G. Gulsen, and O. Nalcıoğlu, “Diffuse optical tomographic reconstruction using multifrequency data,” J. Biomed. Opt. 11, 054008 (2006).
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Sikora, J.

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. R. Arridge, and J. Ripoll, “Diffuse photon propagation in multilayered geometries,” Phys. Med. Biol. 51, 497–516 (2006).
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Sinn, P.

Spott, T.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, and M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Styles, I. B.

J. A. Guggenheim, H. R. A. Basevi, J. Frampton, I. B. Styles, and H. Dehghani, “Multi-modal molecular diffuse optical tomography system for small animal imaging,” Meas. Sci. Technol. 24, 105405 (2013).
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Svaasand, L. O.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, and M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
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S. Takatani and M. D. Graham, “Theoretical analysis of diffuse reflectance from a two-layer tissue model,” IEEE Trans. Biomed. Eng. 26, 656–664 (1979).
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Tian, J.

Tinet, E.

Tromberg, B. J.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, and M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Tualle, J. M.

Unlu, M. B.

H. Erkol and M. B. Unlu, “Virtual source method for diffuse optical imaging,” Appl. Opt. 52, 4933–4970 (2013).
[CrossRef] [PubMed]

H. Yan, Y. Lin, W. C. Barber, M. B. Unlu, and G. Gulsen, “A gantry-based tri-modality system for bioluminescence tomography,” Rev. Sci. Instrum. 83, 043708 (2012).
[CrossRef] [PubMed]

M. B. Unlu, O. Birgul, R. Shafiha, G. Gulsen, and O. Nalcıoğlu, “Diffuse optical tomographic reconstruction using multifrequency data,” J. Biomed. Opt. 11, 054008 (2006).
[CrossRef]

Walker, E. C.

Walker, S. A.

Wall, R. T.

Wang, G.

W. Han, J. A. Eichholz, and G. Wang, “On a family of differential approximations of the radiative transfer equation,” J. Math. Chem. 50, 689–702 (2011).
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H. Gao, H. Zhao, W. Cong, and G. Wang, “Bioluminescence tomography in Gaussian prior,” Biomed. Opt. Express 1, 1259–1277 (2010).
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G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. Mclennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express 14, 7801–7809 (2006).
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W. Cong, L. V. Wang, and G. Wang, “Formulation of photon diffusion from spherical bioluminescent sources in an infinite homogeneous medium,” Biomed. Eng. Online 3, 1–6 (2004).
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G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. 31(8), 2289–2299 (2004).
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L. H. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging Hoboken (Wiley, 2007).

Wang, L. V.

W. Cong, L. V. Wang, and G. Wang, “Formulation of photon diffusion from spherical bioluminescent sources in an infinite homogeneous medium,” Biomed. Eng. Online 3, 1–6 (2004).
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Wang, Q.

Weiss, G. H.

I. Dayan, S. Havlin, and G. H. Weiss, “Photon migration in a two-layer turbid medium A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

Wilson, B. C.

T. J. Farrel, M. S. Patterson, and B. C. Wilson, “A diffusion theory model of specially resolved steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
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L. H. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging Hoboken (Wiley, 2007).

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Yalavarthy, P. K.

Yamada, Y.

F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, “Analytical approximate solutions of the timedomain diffusion equation in layered slabs,” J. Opt. Soc. Am. A. 19, 71–80 (2002).
[CrossRef]

Yan, H.

H. Yan, Y. Lin, W. C. Barber, M. B. Unlu, and G. Gulsen, “A gantry-based tri-modality system for bioluminescence tomography,” Rev. Sci. Instrum. 83, 043708 (2012).
[CrossRef] [PubMed]

Yin, L.

Yodh, A. G.

A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. M. C. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44(11), 2082–2093 (2005).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’leary, B. Chances, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: Analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Yuan, Z.

Zaccanti, G.

F. Martelli, A. Sassaroli, S. D. Bianco, and G. Zaccanti, “Solution of the time-dependent diffusion equation for a three-layer medium: application to study photon migration through a simplified adult head model,” Phys. Med. Biol. 52, 2827–2843 (2007).
[CrossRef] [PubMed]

F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, “Analytical approximate solutions of the timedomain diffusion equation in layered slabs,” J. Opt. Soc. Am. A. 19, 71–80 (2002).
[CrossRef]

D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation I. Theory,” Appl. Opt. 36, 4587–4599 (1997).
[CrossRef] [PubMed]

Zacharopoulos, A.

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. R. Arridge, and J. Ripoll, “Diffuse photon propagation in multilayered geometries,” Phys. Med. Biol. 51, 497–516 (2006).
[CrossRef] [PubMed]

Zhang, A.

Zhang, J.

Zhang, Q.

Zhao, H.

Zhou, G. X.

Appl. Opt. (9)

M. S. Patterson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation I. Theory,” Appl. Opt. 36, 4587–4599 (1997).
[CrossRef] [PubMed]

E. Okada, M. Firbank, M. Schwiger, S. R. Arridge, M. Cope, and D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. 36, 21–31 (1997).
[CrossRef] [PubMed]

S. A. Walker, D. A. Boas, and E. Gratton, “Photon density waves scattered from cylindrical inhomogeneities: theory and experiments,” Appl. Opt. 37, 1935–1944 (1998).
[CrossRef]

H. Dehghani and D. T. Delpy, “Near-infrared spectroscopy of the adult head: effect of scattering and absorbing obstructions in the cerebrospinal fluid layer on light distribution in the tissue,” Appl. Opt. 39, 4721–4729 (2000).
[CrossRef]

B. W. Pogue, S. Geimer, T. O. McBride, S. Jiang, U. L. Österberg, and K. D. Paulsen, “Three-dimensional simulation of near-infrared diffusion in tissue: boundary condition and geometry analysis for finite-element image reconstruction,” Appl. Opt. 40, 588–600 (2001).
[CrossRef]

H. Dehghani, B. A. Brooksby, P. W. Pogue, and K. D. Paulsen, “Effects of refractive index on near-infrared tomography of the breast,” Appl. Opt. 44, 1870–1878 (2005).
[CrossRef] [PubMed]

A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. M. C. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44(11), 2082–2093 (2005).
[CrossRef] [PubMed]

H. Erkol and M. B. Unlu, “Virtual source method for diffuse optical imaging,” Appl. Opt. 52, 4933–4970 (2013).
[CrossRef] [PubMed]

Biomed. Eng. Online (1)

W. Cong, L. V. Wang, and G. Wang, “Formulation of photon diffusion from spherical bioluminescent sources in an infinite homogeneous medium,” Biomed. Eng. Online 3, 1–6 (2004).
[CrossRef]

Biomed. Opt. Express (3)

IEEE MTT-S Digest (1)

M. S. Patterson and S. J. Madsen, “Diffusion equation representation of photon migration in tissue,” IEEE MTT-S Digest 2, 905–908 (1991).

IEEE Trans. Biomed. Eng. (1)

S. Takatani and M. D. Graham, “Theoretical analysis of diffuse reflectance from a two-layer tissue model,” IEEE Trans. Biomed. Eng. 26, 656–664 (1979).
[CrossRef] [PubMed]

Inverse Problems (2)

M. A. Anastasio, J. Zhang, D. Modgil, and P. J. L. Rivière, “Application of inverse source concepts to photoacoustic tomography,” Inverse Problems 23, 21–35 (2007).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, 41–93 (1999).
[CrossRef]

J. Biomed. Opt. (4)

M. B. Unlu, O. Birgul, R. Shafiha, G. Gulsen, and O. Nalcıoğlu, “Diffuse optical tomographic reconstruction using multifrequency data,” J. Biomed. Opt. 11, 054008 (2006).
[CrossRef]

T. J. Farrel and M. S. Patterson, “Experimental verification of the effect of refractive index mismatch on the light fluence in a turbid medium,” J. Biomed. Opt. 6, 468–473 (2001).
[CrossRef]

T. Farrel and M. S. Patterson, “Experimental verification of the effect of refractive index mismatch on the light fluence in a turbid medium,” J. Biomed. Opt. 6, 468–473 (2001).
[CrossRef]

A. Liemert and A. Kienle, “Light diffusion in N-layered turbid media: frequency and time domains,” J. Biomed. Opt. 15, 025002 (2010).
[CrossRef] [PubMed]

J. Mat. Imaging Vis. (1)

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite element method for the forward and inverse model in optical tomography,” J. Mat. Imaging Vis. 3, 263–283 (1993).
[CrossRef]

J. Math. Chem. (1)

W. Han, J. A. Eichholz, and G. Wang, “On a family of differential approximations of the radiative transfer equation,” J. Math. Chem. 50, 689–702 (2011).
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J. Mod. Opt. (1)

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Figures (9)

Fig. 1
Fig. 1

The circular region with an inclusion represented by a point source located at ρi.

Fig. 2
Fig. 2

Concentric two-layered domain.

Fig. 3
Fig. 3

Finite element mesh used in the simulations.

Fig. 4
Fig. 4

Photon fluence rate (number of photons per mm−2) vs. number of measurement for a point source located x = 10 mm and y = 0 where the radius of the circular domain R=20 mm.

Fig. 5
Fig. 5

Source strength α vs. source position ρ where ⋄ and + represent the calculated value and the true value, respectively; the radius of the circular domain, R = 20 mm and the true value of the source strength, αtrue = 1mm−1.

Fig. 6
Fig. 6

Position of the source ρ vs. strength of the source α for the radius of the circular domain, R = 20 mm where + and ⋄ represent the calculated and the true value, respectively.

Fig. 7
Fig. 7

Photon fluence rate (number of photons per mm−2) vs. number of measurement for a point source located x = 10 mm and y = 0 where the radii of the outer and inner circular domain R=20, 15mm, respectively.

Fig. 8
Fig. 8

Source strength α vs. source position ρ for the layered domain. Here, ⋄ and + represent the calculated values and the true value, respectively; the radii of the outer and inner domain R = 20, 15mm and the true value of the source strength αtrue = 1mm−1.

Fig. 9
Fig. 9

Source position ρ vs. source strength α for the layered circular domain. Here, the outer and inner radii R = 20, 15 mm where ⋄ and + represent the calculated and the true value, respectively.

Equations (39)

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Φ ( r , t ) c t + μ a Φ ( r , t ) [ D Φ ( r , t ) ] = S ( r , t )
( i ω c + μ a D 2 ) Φ ( r , ω ) = S ( r , ω )
2 Φ ( r , ω ) ( i ω c + μ a ) 1 D Φ ( r , ω ) = 1 D α i δ ( r r i )
2 Φ ( r ) μ a D Φ ( r ) = 1 D α i δ ( r r i ) .
2 Φ ( ρ , ϕ ) μ a D Φ ( ρ , ϕ ) = 1 D α i δ ( ρ ρ i , ϕ ϕ i )
Φ ( ρ , ϕ ) = m = 0 ( A m J m ( k ρ ) + B m Y m ( k ρ ) ) cos ( m ϕ )
Φ 1 ( k R , ϕ ) = m = 0 ( A m J m ( k R ) + B m Y m ( k R ) ) cos ( m ϕ ) .
A m J m ( k R ) + B m Y m ( k R ) = 1 π 0 2 π Φ 1 ( R , ϕ ) cos ( m ϕ ) d ϕ
0 2 π cos ( m ϕ ) cos ( m ϕ ) d ϕ = π δ m , m
Φ 1 ( R ) + 2 ζ D Φ 1 ( ρ ) ρ | ρ = R = 0
A m = R 2 I m ( Y m ( k R ) 2 ζ D + Y m ( k R ) )
B m = R 2 I m ( J m ( k R ) 2 ζ D + J m ( k R ) )
I m = 0 2 π Φ 1 ( R , ϕ ) cos ( m ϕ ) d ϕ ,
Φ 2 ( ρ , ϕ , k ) = m = 0 ( C m J m ( k ρ ) + D m Y m ( k ρ ) ) cos ( m ϕ )
Φ 2 ( ρ , ϕ , k ) = m = 0 C m J m ( k ρ ) cos ( m ϕ ) .
Φ 2 ( ρ , ϕ , k ) | ρ = ρ i = Φ 1 ( ρ , ϕ , k ) | ρ = ρ i .
m = 0 ( 1 ρ d d ρ ( ρ d d ρ g m ( k ρ ) ) cos ( m ϕ ) + 1 ρ 2 d 2 d ϕ 2 ( cos ( m ϕ ) ) g m ( k ρ ) μ a D Φ ( k ρ , ϕ ) ) = α i D δ ( ρ , ρ i )
m = 0 ( ρ i ε ρ i + ε ( d d ρ ( ρ d g m ( k ρ ) d ρ ) ) d ρ 0 2 π cos ( m ϕ ) cos ( m ϕ ) d ϕ = 1 D α i ρ i ε ρ i + ε 0 2 π cos ( m ϕ ) δ ( ρ , ρ i ) ρ d ρ d ϕ
δ ( ρ , ρ i ) = δ ( ρ ρ i ) δ ( ϕ ϕ i ) ρ
0 2 π cos ( m ϕ ) cos ( m ϕ ) d ϕ = π δ m m
d g m ( k ρ ) d ρ | ρ = ρ i + ε d g m ( k ρ ) d ρ | ρ = ρ i ε = α i π D ρ i cos ( m ϕ i ) .
A m J m ( k ρ i ) + B m Y m ( k ρ i ) = C m J m ( k ρ i )
A m J m ( k ρ i ) + B m Y m ( k ρ i ) = C m J m ( k ρ i ) α i π D ρ i cos ( m ϕ i ) .
C m = A m + B m Y m ( k ρ i ) J m ( k ρ i )
α i = DRI m J m ( k ρ i ) cos ( m ϕ i ) ( J m ( k R ) 2 ζ D + J m ( k R ) ) ,
Φ out ( ρ , ϕ ) = m = 0 ( A m J m ( k out ρ ) + B m Y m ( k out ρ ) ) cos ( m ϕ )
A m = R out 2 I m ( Y m ( k out R out ) 2 ζ t a D out + Y m ( k out R out ) )
B m = R out 2 I m ( J m ( k out R out ) 2 ζ t a D out + J m ( k out R out ) )
I m = 0 2 π Φ ( R out , ϕ ) cos ( m ϕ ) d ϕ ,
Φ in ( ρ , ϕ ) = m = 0 ( C m J m ( k in ρ ) + D m Y m ( k in ρ ) ) cos ( m ϕ )
( n 2 n 1 ) 2 Φ in ( k in ρ ) | ρ = R in Φ out ( k out ρ ) | ρ = R in = c ( n 2 n 1 ) ( D . Φ ) | ρ = R in
D in Φ in ρ | ρ = R in = D out Φ out ρ | ρ = R in ,
Φ in ( k in ρ ) | ρ = R in = Φ out ( k out ρ ) | ρ = R in .
C m = π R in 2 { A m [ J m ( k out R in ) Y m ( k in R in ) D out D in J m ( k out R in ) Y m ( k in R in ) ] + B m [ Y m ( k out R in ) Y m ( k in R in ) D out D in Y m ( k out R in ) Y m ( k in R in ) ] }
D m = π R in 2 { A m [ J m ( k out R in ) J m ( k in R in ) D out D in J m ( k out R in ) J m ( k in R in ) ] + B m [ Y m ( k out R in ) J m ( k in R in ) D out D in Y m ( k out R in ) J m ( k in R in ) ] } ,
α i = 2 D in J m ( k in ρ i ) cos ( m ϕ i ) D m .
I m = 0 2 π Φ 1 ( R , ϕ ) cos ( m ϕ ) d ϕ
I m = 0 2 π Φ ( R , ϕ ) cos ( m ϕ ) d ϕ = i = 1 128 Φ ( R , ϕ i ) cos ( m ϕ i ) .
α m = DRI m J m ( k ρ i ) cos ( m ϕ i ) [ J m ( k R ) 2 ζ D + J m ( k R ) ]

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