Abstract

We describe a noniterative method for recovering optical absorption coefficient distribution from the absorbed energy map reconstructed using simulated and noisy boundary pressure measurements. The source reconstruction problem is first solved for the absorbed energy map corresponding to single- and multiple-source illuminations from the side of the imaging plane. It is shown that the absorbed energy map and the absorption coefficient distribution, recovered from the single-source illumination with a large variation in photon flux distribution, have signal-to-noise ratios comparable to those of the reconstructed parameters from a more uniform photon density distribution corresponding to multiple-source illuminations. The absorbed energy map is input as absorption coefficient times photon flux in the time-independent diffusion equation (DE) governing photon transport to recover the photon flux in a single step. The recovered photon flux is used to compute the optical absorption coefficient distribution from the absorbed energy map. In the absence of experimental data, we obtain the boundary measurements through Monte Carlo simulations, and we attempt to address the possible limitations of the DE model in the overall reconstruction procedure.

© 2008 Optical Society of America

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2007 (3)

J. Laufer, D. Delpy, C. Elwell, and P. Beard, “Quantitative spatially resolved measurement of tissue chromophore concentrations using photoacoustic spectroscopy: application to the measurement of blood oxygenation and haemoglobin oncentration,” Phys. Med. Biol. 52, 141-168 (2007).
[CrossRef]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Gradient-based quantitative photoacoustic image reconstruction for molecular imaging,” Proc. SPIE 6437, 1T1-1T9 (2007).

Z. Yuan, Q. Wang, and H. Jiang, “Reconstruction of optical absorption coefficient maps of heterogeneous media by photoacoustic tomography coupled with diffusion equation based regularized Newton method,” Opt. Express 15, 18076-18081 (2007).
[CrossRef] [PubMed]

2006 (5)

2005 (4)

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

B. T. Cox and P. C. Beard, “Fast calculation of pulsed photoacoustic fields in fluids using k-space methods,” J. Acoust. Soc. Am. 117, 3616-3627 (2005).
[CrossRef] [PubMed]

J. Ripoll and V. Ntziachristos, “Quantitative photoacoustic inversion formulas for scattering and absorption media,” Phys. Rev. E 71, 031912(1-9) (2005).
[CrossRef]

J. Laufer, C. Elwell, D. Delpy, and P. Beard, “In vitro measurements of absolute blood oxygen saturation using pulsed near-infrared photoacoustic spectroscopy: accuracy and resolution,” Phys. Med. Biol. 50, 4409-4428 (2005).
[CrossRef] [PubMed]

2003 (2)

X. Wang, Y. Pang, G. Ku, X. Xie, G. Stoica, and L. V. Wang, “Noninvasive laser-induced photoacoustic tomography for structural and functional in vivo imaging of the brain,” Nat. Biotechnol. 21, 813-816 (2003).
[CrossRef]

S. Srinivasan, B. W. Pogue, S. D. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100, 12349-12354 (2003).
[CrossRef] [PubMed]

2002 (2)

X. Wang, Y. Xu, M. Xu, S. Yokoo, E. S. Fry, and L. V. Wang, “Photoacoustic tomography of biological tissues with high cross-section resolution: Reconstruction and experiments,” Med. Phys. 29, 2799-2805 (2002).
[CrossRef]

G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. (Washington, D.C.) 103, 487-518 (2002).
[CrossRef]

2000 (2)

1999 (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

1998 (1)

1997 (1)

C. Zhu, R. H. Byrd, and J. Nocedal, “L-BFGS-B: algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization,” ACM Trans. Math. Softw. 23, 550-560 (1997).
[CrossRef]

1995 (2)

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

Y. Yamada, “Light-tissue interaction and optical imaging in biomedicine,” Annu. Rev. Heat Transfer 6, 1-59 (1995).

1980 (1)

Arridge, S. R.

B. T. Cox, S. R. Arridge, and P. C. Beard, “Gradient-based quantitative photoacoustic image reconstruction for molecular imaging,” Proc. SPIE 6437, 1T1-1T9 (2007).

B. T. Cox, S. R. Arridge, K. P. Köstli, and P. C. Beard, “Two-dimensional quantitative photoacoustic image reconstruction of absorption distributions in scattering media by use of a simple iterative method,” Appl. Opt. 45, 1866-1875 (2006).
[CrossRef] [PubMed]

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

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

S. R. Arridge and M. Schweiger, “A gradient-based optimization scheme for optical tomography,” Opt. Express 6, 213-226 (1998).
[CrossRef]

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

Beard, P.

J. Laufer, D. Delpy, C. Elwell, and P. Beard, “Quantitative spatially resolved measurement of tissue chromophore concentrations using photoacoustic spectroscopy: application to the measurement of blood oxygenation and haemoglobin oncentration,” Phys. Med. Biol. 52, 141-168 (2007).
[CrossRef]

J. Laufer, C. Elwell, D. Delpy, and P. Beard, “In vitro measurements of absolute blood oxygen saturation using pulsed near-infrared photoacoustic spectroscopy: accuracy and resolution,” Phys. Med. Biol. 50, 4409-4428 (2005).
[CrossRef] [PubMed]

Beard, P. C.

B. T. Cox, S. R. Arridge, and P. C. Beard, “Gradient-based quantitative photoacoustic image reconstruction for molecular imaging,” Proc. SPIE 6437, 1T1-1T9 (2007).

B. T. Cox, S. R. Arridge, K. P. Köstli, and P. C. Beard, “Two-dimensional quantitative photoacoustic image reconstruction of absorption distributions in scattering media by use of a simple iterative method,” Appl. Opt. 45, 1866-1875 (2006).
[CrossRef] [PubMed]

B. T. Cox and P. C. Beard, “Fast calculation of pulsed photoacoustic fields in fluids using k-space methods,” J. Acoust. Soc. Am. 117, 3616-3627 (2005).
[CrossRef] [PubMed]

Byrd, R. H.

C. Zhu, R. H. Byrd, and J. Nocedal, “L-BFGS-B: algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization,” ACM Trans. Math. Softw. 23, 550-560 (1997).
[CrossRef]

Cox, B. T.

B. T. Cox, S. R. Arridge, and P. C. Beard, “Gradient-based quantitative photoacoustic image reconstruction for molecular imaging,” Proc. SPIE 6437, 1T1-1T9 (2007).

B. T. Cox, S. R. Arridge, K. P. Köstli, and P. C. Beard, “Two-dimensional quantitative photoacoustic image reconstruction of absorption distributions in scattering media by use of a simple iterative method,” Appl. Opt. 45, 1866-1875 (2006).
[CrossRef] [PubMed]

B. T. Cox and P. C. Beard, “Fast calculation of pulsed photoacoustic fields in fluids using k-space methods,” J. Acoust. Soc. Am. 117, 3616-3627 (2005).
[CrossRef] [PubMed]

Dehghani, H.

S. Srinivasan, B. W. Pogue, S. D. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100, 12349-12354 (2003).
[CrossRef] [PubMed]

Delpy, D.

J. Laufer, D. Delpy, C. Elwell, and P. Beard, “Quantitative spatially resolved measurement of tissue chromophore concentrations using photoacoustic spectroscopy: application to the measurement of blood oxygenation and haemoglobin oncentration,” Phys. Med. Biol. 52, 141-168 (2007).
[CrossRef]

J. Laufer, C. Elwell, D. Delpy, and P. Beard, “In vitro measurements of absolute blood oxygen saturation using pulsed near-infrared photoacoustic spectroscopy: accuracy and resolution,” Phys. Med. Biol. 50, 4409-4428 (2005).
[CrossRef] [PubMed]

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

deMul, F. F. M.

Dyer, P. E.

G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. (Washington, D.C.) 103, 487-518 (2002).
[CrossRef]

Elwell, C.

J. Laufer, D. Delpy, C. Elwell, and P. Beard, “Quantitative spatially resolved measurement of tissue chromophore concentrations using photoacoustic spectroscopy: application to the measurement of blood oxygenation and haemoglobin oncentration,” Phys. Med. Biol. 52, 141-168 (2007).
[CrossRef]

J. Laufer, C. Elwell, D. Delpy, and P. Beard, “In vitro measurements of absolute blood oxygen saturation using pulsed near-infrared photoacoustic spectroscopy: accuracy and resolution,” Phys. Med. Biol. 50, 4409-4428 (2005).
[CrossRef] [PubMed]

Fry, E. S.

X. Wang, Y. Xu, M. Xu, S. Yokoo, E. S. Fry, and L. V. Wang, “Photoacoustic tomography of biological tissues with high cross-section resolution: Reconstruction and experiments,” Med. Phys. 29, 2799-2805 (2002).
[CrossRef]

Furutsu, K.

Gibson, A. P.

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

Gibson, J. J.

S. Srinivasan, B. W. Pogue, S. D. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100, 12349-12354 (2003).
[CrossRef] [PubMed]

Gu, X.

Hebden, J. C.

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

Hiraoka, M.

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

Hoelen, C. G. A.

Jacques, S. L.

S. L. Jacques and L. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical-Thermal Response of Laser-Irradiated Tissue, A.J.Welch and M.J. C.van Gemert, eds. (Plenum, 1995).

Jiang, H.

Jiang, S. D.

S. Srinivasan, B. W. Pogue, S. D. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100, 12349-12354 (2003).
[CrossRef] [PubMed]

Kogel, C.

S. Srinivasan, B. W. Pogue, S. D. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100, 12349-12354 (2003).
[CrossRef] [PubMed]

Köstli, K. P.

Ku, G.

X. Wang, Y. Pang, G. Ku, X. Xie, G. Stoica, and L. V. Wang, “Noninvasive laser-induced photoacoustic tomography for structural and functional in vivo imaging of the brain,” Nat. Biotechnol. 21, 813-816 (2003).
[CrossRef]

Laufer, J.

J. Laufer, D. Delpy, C. Elwell, and P. Beard, “Quantitative spatially resolved measurement of tissue chromophore concentrations using photoacoustic spectroscopy: application to the measurement of blood oxygenation and haemoglobin oncentration,” Phys. Med. Biol. 52, 141-168 (2007).
[CrossRef]

J. Laufer, C. Elwell, D. Delpy, and P. Beard, “In vitro measurements of absolute blood oxygen saturation using pulsed near-infrared photoacoustic spectroscopy: accuracy and resolution,” Phys. Med. Biol. 50, 4409-4428 (2005).
[CrossRef] [PubMed]

Nocedal, J.

C. Zhu, R. H. Byrd, and J. Nocedal, “L-BFGS-B: algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization,” ACM Trans. Math. Softw. 23, 550-560 (1997).
[CrossRef]

Ntziachristos, V.

J. Ripoll and V. Ntziachristos, “Quantitative photoacoustic inversion formulas for scattering and absorption media,” Phys. Rev. E 71, 031912(1-9) (2005).
[CrossRef]

Paltauf, G.

G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. (Washington, D.C.) 103, 487-518 (2002).
[CrossRef]

Pang, Y.

X. Wang, Y. Pang, G. Ku, X. Xie, G. Stoica, and L. V. Wang, “Noninvasive laser-induced photoacoustic tomography for structural and functional in vivo imaging of the brain,” Nat. Biotechnol. 21, 813-816 (2003).
[CrossRef]

Paulsen, K. D.

S. Srinivasan, B. W. Pogue, S. D. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100, 12349-12354 (2003).
[CrossRef] [PubMed]

Pogue, B. W.

S. Srinivasan, B. W. Pogue, S. D. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100, 12349-12354 (2003).
[CrossRef] [PubMed]

Poplack, S. P.

S. Srinivasan, B. W. Pogue, S. D. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100, 12349-12354 (2003).
[CrossRef] [PubMed]

Ripoll, J.

J. Ripoll and V. Ntziachristos, “Quantitative photoacoustic inversion formulas for scattering and absorption media,” Phys. Rev. E 71, 031912(1-9) (2005).
[CrossRef]

Schweiger, M.

S. R. Arridge and M. Schweiger, “A gradient-based optimization scheme for optical tomography,” Opt. Express 6, 213-226 (1998).
[CrossRef]

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

Soho, S.

S. Srinivasan, B. W. Pogue, S. D. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100, 12349-12354 (2003).
[CrossRef] [PubMed]

Srinivasan, S.

S. Srinivasan, B. W. Pogue, S. D. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100, 12349-12354 (2003).
[CrossRef] [PubMed]

Stoica, G.

X. Wang, Y. Pang, G. Ku, X. Xie, G. Stoica, and L. V. Wang, “Noninvasive laser-induced photoacoustic tomography for structural and functional in vivo imaging of the brain,” Nat. Biotechnol. 21, 813-816 (2003).
[CrossRef]

Tosteson, T. D.

S. Srinivasan, B. W. Pogue, S. D. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100, 12349-12354 (2003).
[CrossRef] [PubMed]

Wang, L.

S. L. Jacques and L. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical-Thermal Response of Laser-Irradiated Tissue, A.J.Welch and M.J. C.van Gemert, eds. (Plenum, 1995).

Wang, L. V.

M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77, 041101-22 (2006).
[CrossRef]

L. V. Wang, “Ultrasound-modulated biophotonic imaging: A review of acousto-optical tomography and photoacoustic tomography,” Dis. Markers 19, 123-138 (2003, 2004).
[PubMed]

X. Wang, Y. Pang, G. Ku, X. Xie, G. Stoica, and L. V. Wang, “Noninvasive laser-induced photoacoustic tomography for structural and functional in vivo imaging of the brain,” Nat. Biotechnol. 21, 813-816 (2003).
[CrossRef]

X. Wang, Y. Xu, M. Xu, S. Yokoo, E. S. Fry, and L. V. Wang, “Photoacoustic tomography of biological tissues with high cross-section resolution: Reconstruction and experiments,” Med. Phys. 29, 2799-2805 (2002).
[CrossRef]

G. Yao and L. V. Wang, “Theoretical and experimental studies of ultrasound-modulated optical tomography in biological tissue,” Appl. Opt. 39, 659-664 (2000).
[CrossRef]

Wang, Q.

Wang, X.

X. Wang, Y. Pang, G. Ku, X. Xie, G. Stoica, and L. V. Wang, “Noninvasive laser-induced photoacoustic tomography for structural and functional in vivo imaging of the brain,” Nat. Biotechnol. 21, 813-816 (2003).
[CrossRef]

X. Wang, Y. Xu, M. Xu, S. Yokoo, E. S. Fry, and L. V. Wang, “Photoacoustic tomography of biological tissues with high cross-section resolution: Reconstruction and experiments,” Med. Phys. 29, 2799-2805 (2002).
[CrossRef]

Wu, C.

Xie, X.

X. Wang, Y. Pang, G. Ku, X. Xie, G. Stoica, and L. V. Wang, “Noninvasive laser-induced photoacoustic tomography for structural and functional in vivo imaging of the brain,” Nat. Biotechnol. 21, 813-816 (2003).
[CrossRef]

Xu, M.

M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77, 041101-22 (2006).
[CrossRef]

X. Wang, Y. Xu, M. Xu, S. Yokoo, E. S. Fry, and L. V. Wang, “Photoacoustic tomography of biological tissues with high cross-section resolution: Reconstruction and experiments,” Med. Phys. 29, 2799-2805 (2002).
[CrossRef]

Xu, Y.

X. Wang, Y. Xu, M. Xu, S. Yokoo, E. S. Fry, and L. V. Wang, “Photoacoustic tomography of biological tissues with high cross-section resolution: Reconstruction and experiments,” Med. Phys. 29, 2799-2805 (2002).
[CrossRef]

Yamada, Y.

Y. Yamada, “Light-tissue interaction and optical imaging in biomedicine,” Annu. Rev. Heat Transfer 6, 1-59 (1995).

Yao, G.

Yokoo, S.

X. Wang, Y. Xu, M. Xu, S. Yokoo, E. S. Fry, and L. V. Wang, “Photoacoustic tomography of biological tissues with high cross-section resolution: Reconstruction and experiments,” Med. Phys. 29, 2799-2805 (2002).
[CrossRef]

Yuan, Z.

Zhang, Q.

Zhao, H.

Zhu, C.

C. Zhu, R. H. Byrd, and J. Nocedal, “L-BFGS-B: algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization,” ACM Trans. Math. Softw. 23, 550-560 (1997).
[CrossRef]

ACM Trans. Math. Softw. (1)

C. Zhu, R. H. Byrd, and J. Nocedal, “L-BFGS-B: algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization,” ACM Trans. Math. Softw. 23, 550-560 (1997).
[CrossRef]

Annu. Rev. Heat Transfer (1)

Y. Yamada, “Light-tissue interaction and optical imaging in biomedicine,” Annu. Rev. Heat Transfer 6, 1-59 (1995).

Appl. Opt. (4)

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Figures (10)

Fig. 1
Fig. 1

Sample plot showing how the error functional decreases with increasing iteration number in the main inversion algorithm to recover H ( x ) .

Fig. 2
Fig. 2

Object cross section showing the three absorption inhomogeneities. The diameter of the outer circle is 30 mm , and that of the inclusions is 2 mm each. The optical properties are: for the background, μ a = 0.01 mm 1 ; μ s = 0.1 mm 1 ; and for the inclusions μ a = 0.2 mm 1 , 0.15 mm 1 and 0.1 mm 1 , for 1, 2, and 3, respectively, with μ s the same value as that of the background.

Fig. 3
Fig. 3

Various results obtained when diffusion equation is used in the data generation step. The illumination is from a single broad source. (a) Exact absorbed energy distributions H ( x ) . (b) Reconstructed H ( x ) from boundary pressure data containing 1% noise. (c) Exact fluence distribution φ 0 ( x ) obtained through solving the DE assuming the optical properties. (d) Recovered fluence φ 0 ( x ) using the recovered H ( x ) in the DE. (e) Exact absorption coefficient distribution μ a ( x ) . (f) Reconstructed μ a ( x ) distribution ( SNR = 8.37 db ) .

Fig. 4
Fig. 4

Results of reconstruction when MC simulation is used in the data generation step. The illumination is from a single broad source. (a) Exact absorbed energy distributions H ( x ) obtained from the φ 0 ( x ) by MC simulation. (b) Reconstructed H ( x ) from boundary pressure data with 1% noise. (c) Recovered fluence φ 0 ( x ) from the recovered H ( x ) . (d) Reconstructed μ a ( x ) distribution ( SNR = 7.11 db ) .

Fig. 5
Fig. 5

Results similar to those shown in Fig. 4 except that the illumination is from three broad sources. The SNR in the recovered μ a ( x ) distribution (d) is 8.35 db .

Fig. 6
Fig. 6

Three-dimensional profiles (a), (b) of the absorption coefficients shown in Figs. 3e, 3f, respectively.

Fig. 7
Fig. 7

Cross-sectional plots through (a) AA and (b) BB as shown in Fig. 2.

Fig. 8
Fig. 8

Results from an object with one of the inclusions near the boundary. (a) Absorbed energy map generated by the MC simulation. (b) The recovered absorption coefficient map using the DE.

Fig. 9
Fig. 9

Cross-sectional plots through the inhomogeneities of Fig. 8 and their corrected version compared with the original absorption coefficient distribution.

Fig. 10
Fig. 10

Cross-sectional plots through the reconstructed absorption inhomogeneities compared with their exact values. The data generation and recovery are done using DE and the H ( x ) used is reconstructed from boundary pressure measurements.

Equations (11)

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2 p ( x , t ) 1 c 2 2 p ( x , t ) t 2 = β C P H t .
H ( x , t ) = H ( x ) δ ( t ) .
H ( x ) = μ a ( x ) φ 0 ( x ) .
2 p ( x , k ) + k 2 p ( x , k ) = i k C P β c H ( x ) ,
p ( x , k ) = i = 1 N p i ( k ) B i ( x ) , H ( x ) = i = 1 N H i B i ( x ) .
W p = A H .
p n = ( i k χ 2 + χ 2 8 i k + χ 3 8 k 3 ) p + ( 1 2 i k + χ 2 k 2 ) 2 p s 2 .
[ κ ( x ) φ 0 ( x ) ] + μ a ( x ) φ 0 ( x ) = q 0 δ ( x x 0 ) ,
ϵ ( H ) = 1 2 k p c ( x , k ) p e ( x , k ) x Ω 2 = 1 2 k i = 1 N b [ p i c ( x i , k ) p i e ( x i , k ) ] 2 ,
k J T J [ Δ H ] = k J T Δ p .
[ κ ( x ) φ 0 ( x ) ] = q 0 δ ( x x 0 ) H ( x ) .

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