Abstract

Biomedical photoacoustic tomography (PAT) can provide qualitative images of biomedical soft tissue with high spatial resolution. However, whether it is possible to give accurate quantitative estimates of the spatially varying concentrations of the sources of photoacoustic contrast—endogenous or exogenous chromophores—remains an open question. Even if the chromophores’ absorption spectra are known, the problem is nonlinear and ill-posed. We describe a framework for obtaining such quantitative estimates. When the optical scattering distribution is known, adjoint and gradient-based optimization techniques can be used to recover the concentration distributions of the individual chromophores that contribute to the overall tissue absorption. When the scattering distribution is unknown, prior knowledge of the wavelength dependence of the scattering is shown to be sufficient to overcome the absorption-scattering nonuniqueness and allow both distributions of chromophore concentrations and scattering to be recovered from multiwavelength photoacoustic images.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. A. Kruger, P. Liu, Y. R. Fang, and C. R. Appledorn, “Photoacoustic ultrasound (PAUS)-reconstruction tomography,” Med. Phys. 22, 1605-1609 (1995).
    [CrossRef] [PubMed]
  2. C. G. A. Hoelen, F. F. M. de Mul, R. Pongers, and A. Dekker, “Three-dimensional photoacoustic imaging of blood vessels in tissue,” Opt. Lett. 23, 648-650 (1998).
    [CrossRef]
  3. 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, 803-806 (2003).
    [CrossRef] [PubMed]
  4. M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77, 041101 (2006).
    [CrossRef]
  5. E. Z. Zhang, J. G. Laufer, and P. C. Beard, “Backward-mode multiwavelength photoacoustic scanner using a planar Fabry-Perot polymer film ultrasound sensor for high-resolution three-dimensional imaging of biological tissues,” Appl. Opt. 47, 561-577 (2008).
    [CrossRef] [PubMed]
  6. J. G. 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]
  7. J. G. Laufer, C. Elwell, D. Delpy, and P. Beard, “Quantitative spatially resolved measurement of tissue chromophore concentrations using photoacoustic spectroscopy: application to the measurement of blood oxygenation and haemoglobin concentration,” Phys. Med. Biol. 52, 141-168 (2007).
    [CrossRef]
  8. M.-L. Li, J.-T. Oh, X. Xie, G. Ku, W. Wang, C. Li, G. Lungu, G. Stoica, and L. V. Wang, “Simultaneous molecular and hypoxia imaging of brain tumours in vivo using spectroscopic photoacoustic tomography,” Proc. IEEE 96, 481-489 (2008).
    [CrossRef]
  9. B. T. Cox, S. Arridge, K. Köstli, and P. Beard, “Quantitative photoacoustic imaging: fitting a model of light transport to the initial pressure distribution,” Proc. SPIE 5697, 49-55 (2005).
    [CrossRef]
  10. B. T. Cox, S. R. Arridge, K. Köstli, and P. C. Beard, “2D quantitative photoacoustic image reconstruction of absorption distributions in scattering media using a simple iterative method,” Appl. Opt. 45, 1866-1874 (2006).
    [CrossRef] [PubMed]
  11. K. Maslov, M. Sivaramakrishnan, H. F. Zhang, G. Stoica, and L. V. Wang, “Technical considerations in quantitative blood oxygenation measurement using photoacoustic microscopy in vivo,” Proc. SPIE 6086, 60860R (2006).
    [CrossRef]
  12. K. M. Stantz, B. Liu, M. Cao, D. Reinecke, K. Miller, and R. Kruger, “Photoacoustic spectroscopic imaging of intra-tumour heterogeneity and molecular identification,” Proc. SPIE 6086, 608605 (2006).
    [CrossRef]
  13. X. Wang, X. Xie, G. Ku, and L. V. Wang, “Noninvasive imaging of hemoglobin concentration and oxygenation in the rat brain using high-resolution photoacoustic imaging,” J. Biomed. Opt. 11, 024015 (2006).
    [CrossRef] [PubMed]
  14. K. Maslov, H. F. Zhang, and L. V. Wang, “Effects of wavelength-dependent fluence attenuation on the noninvasive photoacoustic imaging of hemoglobin oxygen saturation in subcutaneous vasculature in vivo,” Inverse Probl. 23, S113-S122 (2007).
    [CrossRef]
  15. M. Sivaramakrishnan, K. Maslov, H. F. Zhang, G. Stoica, and L. V. Wang, “Limitations of quantitative photoacoustic measurements of blood oxygenation in small vessels,” Phys. Med. Biol. 52, 1349-1361 (2008).
    [CrossRef]
  16. A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
    [CrossRef] [PubMed]
  17. B. T. Cox, S. R. Arridge, and P. C. Beard, “Simultaneous estimation of chromophore concentration and scattering distributions from multiwavelength photoacoustic images,” Proc. SPIE 6856, 68560Y (2008).
    [CrossRef]
  18. B. T. Cox, S. Kara, S. R. Arridge, and P. C. Beard, “K-space propagation models for acoustically heterogeneous media: application to biomedical photoacoustics,” J. Acoust. Soc. Am. 121, 3453-3464 (2007).
    [CrossRef] [PubMed]
  19. 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]
  20. M. Agranovsky, P. Kuchment, and L. Kunyansky, “On reconstruction formulas and algorithms for thermoacoustic and photoacoustic tomography,” arXiv:0706.1303v1 (2007).
  21. A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. Arridge, E. Hillman, and A. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44, 2082-2093 (2005).
    [CrossRef] [PubMed]
  22. A. J. Welch and M. van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Plenum, 1995).
  23. T. L. Troy and S. N. Thenadil, “Optical properties of human skin in the near infrared wavelength range of 1000to2200 nm,” J. Biomed. Opt. 6, 167-176 (2001).
    [CrossRef] [PubMed]
  24. A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400to2000 nm,” J. Phys. D: Appl. Phys. 38, 2543-2555 (2005).
    [CrossRef]
  25. S.-H. Tseng, A. Grant, and A. J. Durkin, “In vivo determination of skin near-infrared optical properties using diffuse optical spectroscopy,” J. Biomed. Opt. 13, 014016 (2008).
    [CrossRef] [PubMed]
  26. 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. (Plenum1995).
  27. L. Wang, S. L. Jacques, and L. Zheng, “MCML -Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
    [CrossRef] [PubMed]
  28. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
    [CrossRef]
  29. S. Arridge, M. Schweiger, M. Hiraoka, and D. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299-309 (1993).
    [CrossRef] [PubMed]
  30. M. Schweiger, S. Arridge, M. Hiraoka, and D. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779-1792 (1995).
    [CrossRef] [PubMed]
  31. Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography: Recovery of optical absorption coefficient maps of heterogeneous media,” Appl. Phys. Lett. 88, 231101 (2006).
    [CrossRef]
  32. B. Banerjee, S. Bagchi, R. M. Vasu, and D. Roy, “Quantitative photoacoustic tomography from boundary pressure measurements: noniterative recovery of optical absorption coefficient from the reconstructed absorbed energy map,” J. Opt. Soc. Am. A 25, 2347-2356 (2008).
    [CrossRef]
  33. J. Ripoll and V. Ntziachristos, “Quantitative point source photoacoustic inversion formulas for scattering and absorbing media,” Phys. Rev. E 71, 031912 (2005).
    [CrossRef]
  34. L. Yin, Q. Wang, Q. Zhang, and H. Jiang, “Tomographic imaging of absolute optical absorption coefficient in turbid media using combined photoacoustic and diffusing light measurements,” Opt. Lett. 32, 2556-2558 (2007).
    [CrossRef] [PubMed]
  35. 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]
  36. B. T. Cox, S. R. Arridge, and P. C. Beard, “Gradient-based quantitative photoacoustic image reconstruction for molecular imaging,” Proc. SPIE 6437, 64371T (2007).
    [CrossRef]
  37. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++, The Art of Scientific Computing (Cambridge U. Press, 2005).
  38. L. N. Trefethen and D. Bau, Numerical Linear Algebra (SIAM, 1997).
    [CrossRef]
  39. P. C. Hansen, “Deconvolution and regularization with Toeplitz matrices,” Numer. Algorithms 29, 323-378 (2002).
    [CrossRef]
  40. T. Spott and L. O. Svaasand, “Collimated light sources in the diffusion approximation,” Appl. Opt. 39, 6453-6465 (2000).
    [CrossRef]
  41. B. T. Cox, S. R. Arridge, and P. C. Beard, “Quantitative photoacoustic image reconstruction for molecular imaging,” Proc. SPIE 6086, 60861M (2006).
    [CrossRef]

2008

M.-L. Li, J.-T. Oh, X. Xie, G. Ku, W. Wang, C. Li, G. Lungu, G. Stoica, and L. V. Wang, “Simultaneous molecular and hypoxia imaging of brain tumours in vivo using spectroscopic photoacoustic tomography,” Proc. IEEE 96, 481-489 (2008).
[CrossRef]

M. Sivaramakrishnan, K. Maslov, H. F. Zhang, G. Stoica, and L. V. Wang, “Limitations of quantitative photoacoustic measurements of blood oxygenation in small vessels,” Phys. Med. Biol. 52, 1349-1361 (2008).
[CrossRef]

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Simultaneous estimation of chromophore concentration and scattering distributions from multiwavelength photoacoustic images,” Proc. SPIE 6856, 68560Y (2008).
[CrossRef]

S.-H. Tseng, A. Grant, and A. J. Durkin, “In vivo determination of skin near-infrared optical properties using diffuse optical spectroscopy,” J. Biomed. Opt. 13, 014016 (2008).
[CrossRef] [PubMed]

E. Z. Zhang, J. G. Laufer, and P. C. Beard, “Backward-mode multiwavelength photoacoustic scanner using a planar Fabry-Perot polymer film ultrasound sensor for high-resolution three-dimensional imaging of biological tissues,” Appl. Opt. 47, 561-577 (2008).
[CrossRef] [PubMed]

B. Banerjee, S. Bagchi, R. M. Vasu, and D. Roy, “Quantitative photoacoustic tomography from boundary pressure measurements: noniterative recovery of optical absorption coefficient from the reconstructed absorbed energy map,” J. Opt. Soc. Am. A 25, 2347-2356 (2008).
[CrossRef]

2007

L. Yin, Q. Wang, Q. Zhang, and H. Jiang, “Tomographic imaging of absolute optical absorption coefficient in turbid media using combined photoacoustic and diffusing light measurements,” Opt. Lett. 32, 2556-2558 (2007).
[CrossRef] [PubMed]

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]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Gradient-based quantitative photoacoustic image reconstruction for molecular imaging,” Proc. SPIE 6437, 64371T (2007).
[CrossRef]

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

K. Maslov, H. F. Zhang, and L. V. Wang, “Effects of wavelength-dependent fluence attenuation on the noninvasive photoacoustic imaging of hemoglobin oxygen saturation in subcutaneous vasculature in vivo,” Inverse Probl. 23, S113-S122 (2007).
[CrossRef]

B. T. Cox, S. Kara, S. R. Arridge, and P. C. Beard, “K-space propagation models for acoustically heterogeneous media: application to biomedical photoacoustics,” J. Acoust. Soc. Am. 121, 3453-3464 (2007).
[CrossRef] [PubMed]

2006

K. Maslov, M. Sivaramakrishnan, H. F. Zhang, G. Stoica, and L. V. Wang, “Technical considerations in quantitative blood oxygenation measurement using photoacoustic microscopy in vivo,” Proc. SPIE 6086, 60860R (2006).
[CrossRef]

K. M. Stantz, B. Liu, M. Cao, D. Reinecke, K. Miller, and R. Kruger, “Photoacoustic spectroscopic imaging of intra-tumour heterogeneity and molecular identification,” Proc. SPIE 6086, 608605 (2006).
[CrossRef]

X. Wang, X. Xie, G. Ku, and L. V. Wang, “Noninvasive imaging of hemoglobin concentration and oxygenation in the rat brain using high-resolution photoacoustic imaging,” J. Biomed. Opt. 11, 024015 (2006).
[CrossRef] [PubMed]

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

Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography: Recovery of optical absorption coefficient maps of heterogeneous media,” Appl. Phys. Lett. 88, 231101 (2006).
[CrossRef]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Quantitative photoacoustic image reconstruction for molecular imaging,” Proc. SPIE 6086, 60861M (2006).
[CrossRef]

B. T. Cox, S. R. Arridge, K. Köstli, and P. C. Beard, “2D quantitative photoacoustic image reconstruction of absorption distributions in scattering media using a simple iterative method,” Appl. Opt. 45, 1866-1874 (2006).
[CrossRef] [PubMed]

2005

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

J. Ripoll and V. Ntziachristos, “Quantitative point source photoacoustic inversion formulas for scattering and absorbing media,” Phys. Rev. E 71, 031912 (2005).
[CrossRef]

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400to2000 nm,” J. Phys. D: Appl. Phys. 38, 2543-2555 (2005).
[CrossRef]

J. G. 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]

B. T. Cox, S. Arridge, K. Köstli, and P. Beard, “Quantitative photoacoustic imaging: fitting a model of light transport to the initial pressure distribution,” Proc. SPIE 5697, 49-55 (2005).
[CrossRef]

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]

2003

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, 803-806 (2003).
[CrossRef] [PubMed]

2002

P. C. Hansen, “Deconvolution and regularization with Toeplitz matrices,” Numer. Algorithms 29, 323-378 (2002).
[CrossRef]

2001

T. L. Troy and S. N. Thenadil, “Optical properties of human skin in the near infrared wavelength range of 1000to2200 nm,” J. Biomed. Opt. 6, 167-176 (2001).
[CrossRef] [PubMed]

2000

1999

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

1998

1995

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

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

R. A. Kruger, P. Liu, Y. R. Fang, and C. R. Appledorn, “Photoacoustic ultrasound (PAUS)-reconstruction tomography,” Med. Phys. 22, 1605-1609 (1995).
[CrossRef] [PubMed]

1993

S. Arridge, M. Schweiger, M. Hiraoka, and D. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

Agranovsky, M.

M. Agranovsky, P. Kuchment, and L. Kunyansky, “On reconstruction formulas and algorithms for thermoacoustic and photoacoustic tomography,” arXiv:0706.1303v1 (2007).

Appledorn, C. R.

R. A. Kruger, P. Liu, Y. R. Fang, and C. R. Appledorn, “Photoacoustic ultrasound (PAUS)-reconstruction tomography,” Med. Phys. 22, 1605-1609 (1995).
[CrossRef] [PubMed]

Arridge, S.

B. T. Cox, S. Arridge, K. Köstli, and P. Beard, “Quantitative photoacoustic imaging: fitting a model of light transport to the initial pressure distribution,” Proc. SPIE 5697, 49-55 (2005).
[CrossRef]

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

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

S. Arridge, M. Schweiger, M. Hiraoka, and D. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

Arridge, S. R.

B. T. Cox, S. R. Arridge, and P. C. Beard, “Simultaneous estimation of chromophore concentration and scattering distributions from multiwavelength photoacoustic images,” Proc. SPIE 6856, 68560Y (2008).
[CrossRef]

B. T. Cox, S. Kara, S. R. Arridge, and P. C. Beard, “K-space propagation models for acoustically heterogeneous media: application to biomedical photoacoustics,” J. Acoust. Soc. Am. 121, 3453-3464 (2007).
[CrossRef] [PubMed]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Gradient-based quantitative photoacoustic image reconstruction for molecular imaging,” Proc. SPIE 6437, 64371T (2007).
[CrossRef]

B. T. Cox, S. R. Arridge, K. Köstli, and P. C. Beard, “2D quantitative photoacoustic image reconstruction of absorption distributions in scattering media using a simple iterative method,” Appl. Opt. 45, 1866-1874 (2006).
[CrossRef] [PubMed]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Quantitative photoacoustic image reconstruction for molecular imaging,” Proc. SPIE 6086, 60861M (2006).
[CrossRef]

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

Bagchi, S.

Banerjee, B.

Bashkatov, A. N.

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400to2000 nm,” J. Phys. D: Appl. Phys. 38, 2543-2555 (2005).
[CrossRef]

Bau, D.

L. N. Trefethen and D. Bau, Numerical Linear Algebra (SIAM, 1997).
[CrossRef]

Beard, P.

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

J. G. 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]

B. T. Cox, S. Arridge, K. Köstli, and P. Beard, “Quantitative photoacoustic imaging: fitting a model of light transport to the initial pressure distribution,” Proc. SPIE 5697, 49-55 (2005).
[CrossRef]

Beard, P. C.

E. Z. Zhang, J. G. Laufer, and P. C. Beard, “Backward-mode multiwavelength photoacoustic scanner using a planar Fabry-Perot polymer film ultrasound sensor for high-resolution three-dimensional imaging of biological tissues,” Appl. Opt. 47, 561-577 (2008).
[CrossRef] [PubMed]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Simultaneous estimation of chromophore concentration and scattering distributions from multiwavelength photoacoustic images,” Proc. SPIE 6856, 68560Y (2008).
[CrossRef]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Gradient-based quantitative photoacoustic image reconstruction for molecular imaging,” Proc. SPIE 6437, 64371T (2007).
[CrossRef]

B. T. Cox, S. Kara, S. R. Arridge, and P. C. Beard, “K-space propagation models for acoustically heterogeneous media: application to biomedical photoacoustics,” J. Acoust. Soc. Am. 121, 3453-3464 (2007).
[CrossRef] [PubMed]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Quantitative photoacoustic image reconstruction for molecular imaging,” Proc. SPIE 6086, 60861M (2006).
[CrossRef]

B. T. Cox, S. R. Arridge, K. Köstli, and P. C. Beard, “2D quantitative photoacoustic image reconstruction of absorption distributions in scattering media using a simple iterative method,” Appl. Opt. 45, 1866-1874 (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]

Bodapati, S.

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

Cao, M.

K. M. Stantz, B. Liu, M. Cao, D. Reinecke, K. Miller, and R. Kruger, “Photoacoustic spectroscopic imaging of intra-tumour heterogeneity and molecular identification,” Proc. SPIE 6086, 608605 (2006).
[CrossRef]

Chen, X.

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

Cheng, Z.

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

Choe, R.

Corlu, A.

Cox, B. T.

B. T. Cox, S. R. Arridge, and P. C. Beard, “Simultaneous estimation of chromophore concentration and scattering distributions from multiwavelength photoacoustic images,” Proc. SPIE 6856, 68560Y (2008).
[CrossRef]

B. T. Cox, S. Kara, S. R. Arridge, and P. C. Beard, “K-space propagation models for acoustically heterogeneous media: application to biomedical photoacoustics,” J. Acoust. Soc. Am. 121, 3453-3464 (2007).
[CrossRef] [PubMed]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Gradient-based quantitative photoacoustic image reconstruction for molecular imaging,” Proc. SPIE 6437, 64371T (2007).
[CrossRef]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Quantitative photoacoustic image reconstruction for molecular imaging,” Proc. SPIE 6086, 60861M (2006).
[CrossRef]

B. T. Cox, S. R. Arridge, K. Köstli, and P. C. Beard, “2D quantitative photoacoustic image reconstruction of absorption distributions in scattering media using a simple iterative method,” Appl. Opt. 45, 1866-1874 (2006).
[CrossRef] [PubMed]

B. T. Cox, S. Arridge, K. Köstli, and P. Beard, “Quantitative photoacoustic imaging: fitting a model of light transport to the initial pressure distribution,” Proc. SPIE 5697, 49-55 (2005).
[CrossRef]

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]

Dai, H.

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

de la Zerda, A.

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

de Mul, F. F. M.

Dekker, A.

Delpy, D.

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

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

S. Arridge, M. Schweiger, M. Hiraoka, and D. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

Durduran, T.

Durkin, A. J.

S.-H. Tseng, A. Grant, and A. J. Durkin, “In vivo determination of skin near-infrared optical properties using diffuse optical spectroscopy,” J. Biomed. Opt. 13, 014016 (2008).
[CrossRef] [PubMed]

Elwell, C.

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

J. G. 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]

Fang, Y. R.

R. A. Kruger, P. Liu, Y. R. Fang, and C. R. Appledorn, “Photoacoustic ultrasound (PAUS)-reconstruction tomography,” Med. Phys. 22, 1605-1609 (1995).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++, The Art of Scientific Computing (Cambridge U. Press, 2005).

Gambhir, S. S.

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

Genina, E. A.

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400to2000 nm,” J. Phys. D: Appl. Phys. 38, 2543-2555 (2005).
[CrossRef]

Grant, A.

S.-H. Tseng, A. Grant, and A. J. Durkin, “In vivo determination of skin near-infrared optical properties using diffuse optical spectroscopy,” J. Biomed. Opt. 13, 014016 (2008).
[CrossRef] [PubMed]

Hansen, P. C.

P. C. Hansen, “Deconvolution and regularization with Toeplitz matrices,” Numer. Algorithms 29, 323-378 (2002).
[CrossRef]

Hillman, E.

Hiraoka, M.

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

S. Arridge, M. Schweiger, M. Hiraoka, and D. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

Hoelen, C. G. A.

Jacques, S. L.

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

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. (Plenum1995).

Jiang, H.

Kara, S.

B. T. Cox, S. Kara, S. R. Arridge, and P. C. Beard, “K-space propagation models for acoustically heterogeneous media: application to biomedical photoacoustics,” J. Acoust. Soc. Am. 121, 3453-3464 (2007).
[CrossRef] [PubMed]

Kere, S.

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

Khuri-Yakub, B. T.

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

Kochubey, V. I.

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400to2000 nm,” J. Phys. D: Appl. Phys. 38, 2543-2555 (2005).
[CrossRef]

Köstli, K.

B. T. Cox, S. R. Arridge, K. Köstli, and P. C. Beard, “2D quantitative photoacoustic image reconstruction of absorption distributions in scattering media using a simple iterative method,” Appl. Opt. 45, 1866-1874 (2006).
[CrossRef] [PubMed]

B. T. Cox, S. Arridge, K. Köstli, and P. Beard, “Quantitative photoacoustic imaging: fitting a model of light transport to the initial pressure distribution,” Proc. SPIE 5697, 49-55 (2005).
[CrossRef]

Kruger, R.

K. M. Stantz, B. Liu, M. Cao, D. Reinecke, K. Miller, and R. Kruger, “Photoacoustic spectroscopic imaging of intra-tumour heterogeneity and molecular identification,” Proc. SPIE 6086, 608605 (2006).
[CrossRef]

Kruger, R. A.

R. A. Kruger, P. Liu, Y. R. Fang, and C. R. Appledorn, “Photoacoustic ultrasound (PAUS)-reconstruction tomography,” Med. Phys. 22, 1605-1609 (1995).
[CrossRef] [PubMed]

Ku, G.

M.-L. Li, J.-T. Oh, X. Xie, G. Ku, W. Wang, C. Li, G. Lungu, G. Stoica, and L. V. Wang, “Simultaneous molecular and hypoxia imaging of brain tumours in vivo using spectroscopic photoacoustic tomography,” Proc. IEEE 96, 481-489 (2008).
[CrossRef]

X. Wang, X. Xie, G. Ku, and L. V. Wang, “Noninvasive imaging of hemoglobin concentration and oxygenation in the rat brain using high-resolution photoacoustic imaging,” J. Biomed. Opt. 11, 024015 (2006).
[CrossRef] [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, 803-806 (2003).
[CrossRef] [PubMed]

Kuchment, P.

M. Agranovsky, P. Kuchment, and L. Kunyansky, “On reconstruction formulas and algorithms for thermoacoustic and photoacoustic tomography,” arXiv:0706.1303v1 (2007).

Kunyansky, L.

M. Agranovsky, P. Kuchment, and L. Kunyansky, “On reconstruction formulas and algorithms for thermoacoustic and photoacoustic tomography,” arXiv:0706.1303v1 (2007).

Laufer, J. G.

E. Z. Zhang, J. G. Laufer, and P. C. Beard, “Backward-mode multiwavelength photoacoustic scanner using a planar Fabry-Perot polymer film ultrasound sensor for high-resolution three-dimensional imaging of biological tissues,” Appl. Opt. 47, 561-577 (2008).
[CrossRef] [PubMed]

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

J. G. 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]

Lee, K.

Levi, J.

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

Li, C.

M.-L. Li, J.-T. Oh, X. Xie, G. Ku, W. Wang, C. Li, G. Lungu, G. Stoica, and L. V. Wang, “Simultaneous molecular and hypoxia imaging of brain tumours in vivo using spectroscopic photoacoustic tomography,” Proc. IEEE 96, 481-489 (2008).
[CrossRef]

Li, M.-L.

M.-L. Li, J.-T. Oh, X. Xie, G. Ku, W. Wang, C. Li, G. Lungu, G. Stoica, and L. V. Wang, “Simultaneous molecular and hypoxia imaging of brain tumours in vivo using spectroscopic photoacoustic tomography,” Proc. IEEE 96, 481-489 (2008).
[CrossRef]

Liu, B.

K. M. Stantz, B. Liu, M. Cao, D. Reinecke, K. Miller, and R. Kruger, “Photoacoustic spectroscopic imaging of intra-tumour heterogeneity and molecular identification,” Proc. SPIE 6086, 608605 (2006).
[CrossRef]

Liu, P.

R. A. Kruger, P. Liu, Y. R. Fang, and C. R. Appledorn, “Photoacoustic ultrasound (PAUS)-reconstruction tomography,” Med. Phys. 22, 1605-1609 (1995).
[CrossRef] [PubMed]

Liu, Z.

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

Lungu, G.

M.-L. Li, J.-T. Oh, X. Xie, G. Ku, W. Wang, C. Li, G. Lungu, G. Stoica, and L. V. Wang, “Simultaneous molecular and hypoxia imaging of brain tumours in vivo using spectroscopic photoacoustic tomography,” Proc. IEEE 96, 481-489 (2008).
[CrossRef]

Ma, T.-J.

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

Maslov, K.

M. Sivaramakrishnan, K. Maslov, H. F. Zhang, G. Stoica, and L. V. Wang, “Limitations of quantitative photoacoustic measurements of blood oxygenation in small vessels,” Phys. Med. Biol. 52, 1349-1361 (2008).
[CrossRef]

K. Maslov, H. F. Zhang, and L. V. Wang, “Effects of wavelength-dependent fluence attenuation on the noninvasive photoacoustic imaging of hemoglobin oxygen saturation in subcutaneous vasculature in vivo,” Inverse Probl. 23, S113-S122 (2007).
[CrossRef]

K. Maslov, M. Sivaramakrishnan, H. F. Zhang, G. Stoica, and L. V. Wang, “Technical considerations in quantitative blood oxygenation measurement using photoacoustic microscopy in vivo,” Proc. SPIE 6086, 60860R (2006).
[CrossRef]

Miller, K.

K. M. Stantz, B. Liu, M. Cao, D. Reinecke, K. Miller, and R. Kruger, “Photoacoustic spectroscopic imaging of intra-tumour heterogeneity and molecular identification,” Proc. SPIE 6086, 608605 (2006).
[CrossRef]

Ntziachristos, V.

J. Ripoll and V. Ntziachristos, “Quantitative point source photoacoustic inversion formulas for scattering and absorbing media,” Phys. Rev. E 71, 031912 (2005).
[CrossRef]

Oh, J.-T.

M.-L. Li, J.-T. Oh, X. Xie, G. Ku, W. Wang, C. Li, G. Lungu, G. Stoica, and L. V. Wang, “Simultaneous molecular and hypoxia imaging of brain tumours in vivo using spectroscopic photoacoustic tomography,” Proc. IEEE 96, 481-489 (2008).
[CrossRef]

Oralkan, O.

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

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, 803-806 (2003).
[CrossRef] [PubMed]

Pongers, R.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++, The Art of Scientific Computing (Cambridge U. Press, 2005).

Reinecke, D.

K. M. Stantz, B. Liu, M. Cao, D. Reinecke, K. Miller, and R. Kruger, “Photoacoustic spectroscopic imaging of intra-tumour heterogeneity and molecular identification,” Proc. SPIE 6086, 608605 (2006).
[CrossRef]

Ripoll, J.

J. Ripoll and V. Ntziachristos, “Quantitative point source photoacoustic inversion formulas for scattering and absorbing media,” Phys. Rev. E 71, 031912 (2005).
[CrossRef]

Roy, D.

Schweiger, M.

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

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

S. Arridge, M. Schweiger, M. Hiraoka, and D. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

Sivaramakrishnan, M.

M. Sivaramakrishnan, K. Maslov, H. F. Zhang, G. Stoica, and L. V. Wang, “Limitations of quantitative photoacoustic measurements of blood oxygenation in small vessels,” Phys. Med. Biol. 52, 1349-1361 (2008).
[CrossRef]

K. Maslov, M. Sivaramakrishnan, H. F. Zhang, G. Stoica, and L. V. Wang, “Technical considerations in quantitative blood oxygenation measurement using photoacoustic microscopy in vivo,” Proc. SPIE 6086, 60860R (2006).
[CrossRef]

Smith, B. R.

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

Spott, T.

Stantz, K. M.

K. M. Stantz, B. Liu, M. Cao, D. Reinecke, K. Miller, and R. Kruger, “Photoacoustic spectroscopic imaging of intra-tumour heterogeneity and molecular identification,” Proc. SPIE 6086, 608605 (2006).
[CrossRef]

Stoica, G.

M.-L. Li, J.-T. Oh, X. Xie, G. Ku, W. Wang, C. Li, G. Lungu, G. Stoica, and L. V. Wang, “Simultaneous molecular and hypoxia imaging of brain tumours in vivo using spectroscopic photoacoustic tomography,” Proc. IEEE 96, 481-489 (2008).
[CrossRef]

M. Sivaramakrishnan, K. Maslov, H. F. Zhang, G. Stoica, and L. V. Wang, “Limitations of quantitative photoacoustic measurements of blood oxygenation in small vessels,” Phys. Med. Biol. 52, 1349-1361 (2008).
[CrossRef]

K. Maslov, M. Sivaramakrishnan, H. F. Zhang, G. Stoica, and L. V. Wang, “Technical considerations in quantitative blood oxygenation measurement using photoacoustic microscopy in vivo,” Proc. SPIE 6086, 60860R (2006).
[CrossRef]

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, 803-806 (2003).
[CrossRef] [PubMed]

Svaasand, L. O.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++, The Art of Scientific Computing (Cambridge U. Press, 2005).

Thenadil, S. N.

T. L. Troy and S. N. Thenadil, “Optical properties of human skin in the near infrared wavelength range of 1000to2200 nm,” J. Biomed. Opt. 6, 167-176 (2001).
[CrossRef] [PubMed]

Trefethen, L. N.

L. N. Trefethen and D. Bau, Numerical Linear Algebra (SIAM, 1997).
[CrossRef]

Troy, T. L.

T. L. Troy and S. N. Thenadil, “Optical properties of human skin in the near infrared wavelength range of 1000to2200 nm,” J. Biomed. Opt. 6, 167-176 (2001).
[CrossRef] [PubMed]

Tseng, S.-H.

S.-H. Tseng, A. Grant, and A. J. Durkin, “In vivo determination of skin near-infrared optical properties using diffuse optical spectroscopy,” J. Biomed. Opt. 13, 014016 (2008).
[CrossRef] [PubMed]

Tuchin, V. V.

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400to2000 nm,” J. Phys. D: Appl. Phys. 38, 2543-2555 (2005).
[CrossRef]

Vaithilingam, S.

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

van Gemert, M.

A. J. Welch and M. van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Plenum, 1995).

Vasu, R. M.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++, The Art of Scientific Computing (Cambridge U. Press, 2005).

Wang, L.

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

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. (Plenum1995).

Wang, L. V.

M. Sivaramakrishnan, K. Maslov, H. F. Zhang, G. Stoica, and L. V. Wang, “Limitations of quantitative photoacoustic measurements of blood oxygenation in small vessels,” Phys. Med. Biol. 52, 1349-1361 (2008).
[CrossRef]

M.-L. Li, J.-T. Oh, X. Xie, G. Ku, W. Wang, C. Li, G. Lungu, G. Stoica, and L. V. Wang, “Simultaneous molecular and hypoxia imaging of brain tumours in vivo using spectroscopic photoacoustic tomography,” Proc. IEEE 96, 481-489 (2008).
[CrossRef]

K. Maslov, H. F. Zhang, and L. V. Wang, “Effects of wavelength-dependent fluence attenuation on the noninvasive photoacoustic imaging of hemoglobin oxygen saturation in subcutaneous vasculature in vivo,” Inverse Probl. 23, S113-S122 (2007).
[CrossRef]

X. Wang, X. Xie, G. Ku, and L. V. Wang, “Noninvasive imaging of hemoglobin concentration and oxygenation in the rat brain using high-resolution photoacoustic imaging,” J. Biomed. Opt. 11, 024015 (2006).
[CrossRef] [PubMed]

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

K. Maslov, M. Sivaramakrishnan, H. F. Zhang, G. Stoica, and L. V. Wang, “Technical considerations in quantitative blood oxygenation measurement using photoacoustic microscopy in vivo,” Proc. SPIE 6086, 60860R (2006).
[CrossRef]

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, 803-806 (2003).
[CrossRef] [PubMed]

Wang, Q.

Wang, W.

M.-L. Li, J.-T. Oh, X. Xie, G. Ku, W. Wang, C. Li, G. Lungu, G. Stoica, and L. V. Wang, “Simultaneous molecular and hypoxia imaging of brain tumours in vivo using spectroscopic photoacoustic tomography,” Proc. IEEE 96, 481-489 (2008).
[CrossRef]

Wang, X.

X. Wang, X. Xie, G. Ku, and L. V. Wang, “Noninvasive imaging of hemoglobin concentration and oxygenation in the rat brain using high-resolution photoacoustic imaging,” J. Biomed. Opt. 11, 024015 (2006).
[CrossRef] [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, 803-806 (2003).
[CrossRef] [PubMed]

Welch, A. J.

A. J. Welch and M. van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Plenum, 1995).

Xie, X.

M.-L. Li, J.-T. Oh, X. Xie, G. Ku, W. Wang, C. Li, G. Lungu, G. Stoica, and L. V. Wang, “Simultaneous molecular and hypoxia imaging of brain tumours in vivo using spectroscopic photoacoustic tomography,” Proc. IEEE 96, 481-489 (2008).
[CrossRef]

X. Wang, X. Xie, G. Ku, and L. V. Wang, “Noninvasive imaging of hemoglobin concentration and oxygenation in the rat brain using high-resolution photoacoustic imaging,” J. Biomed. Opt. 11, 024015 (2006).
[CrossRef] [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, 803-806 (2003).
[CrossRef] [PubMed]

Xu, M.

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

Yin, L.

Yodh, A.

Yuan, Z.

Zavaleta, C.

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

Zhang, E. Z.

Zhang, H. F.

M. Sivaramakrishnan, K. Maslov, H. F. Zhang, G. Stoica, and L. V. Wang, “Limitations of quantitative photoacoustic measurements of blood oxygenation in small vessels,” Phys. Med. Biol. 52, 1349-1361 (2008).
[CrossRef]

K. Maslov, H. F. Zhang, and L. V. Wang, “Effects of wavelength-dependent fluence attenuation on the noninvasive photoacoustic imaging of hemoglobin oxygen saturation in subcutaneous vasculature in vivo,” Inverse Probl. 23, S113-S122 (2007).
[CrossRef]

K. Maslov, M. Sivaramakrishnan, H. F. Zhang, G. Stoica, and L. V. Wang, “Technical considerations in quantitative blood oxygenation measurement using photoacoustic microscopy in vivo,” Proc. SPIE 6086, 60860R (2006).
[CrossRef]

Zhang, Q.

Zheng, L.

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

Appl. Opt.

Appl. Phys. Lett.

Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography: Recovery of optical absorption coefficient maps of heterogeneous media,” Appl. Phys. Lett. 88, 231101 (2006).
[CrossRef]

Comput. Methods Programs Biomed.

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

Inverse Probl.

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

K. Maslov, H. F. Zhang, and L. V. Wang, “Effects of wavelength-dependent fluence attenuation on the noninvasive photoacoustic imaging of hemoglobin oxygen saturation in subcutaneous vasculature in vivo,” Inverse Probl. 23, S113-S122 (2007).
[CrossRef]

J. Acoust. Soc. Am.

B. T. Cox, S. Kara, S. R. Arridge, and P. C. Beard, “K-space propagation models for acoustically heterogeneous media: application to biomedical photoacoustics,” J. Acoust. Soc. Am. 121, 3453-3464 (2007).
[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. Biomed. Opt.

T. L. Troy and S. N. Thenadil, “Optical properties of human skin in the near infrared wavelength range of 1000to2200 nm,” J. Biomed. Opt. 6, 167-176 (2001).
[CrossRef] [PubMed]

S.-H. Tseng, A. Grant, and A. J. Durkin, “In vivo determination of skin near-infrared optical properties using diffuse optical spectroscopy,” J. Biomed. Opt. 13, 014016 (2008).
[CrossRef] [PubMed]

X. Wang, X. Xie, G. Ku, and L. V. Wang, “Noninvasive imaging of hemoglobin concentration and oxygenation in the rat brain using high-resolution photoacoustic imaging,” J. Biomed. Opt. 11, 024015 (2006).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

J. Phys. D: Appl. Phys.

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400to2000 nm,” J. Phys. D: Appl. Phys. 38, 2543-2555 (2005).
[CrossRef]

Med. Phys.

R. A. Kruger, P. Liu, Y. R. Fang, and C. R. Appledorn, “Photoacoustic ultrasound (PAUS)-reconstruction tomography,” Med. Phys. 22, 1605-1609 (1995).
[CrossRef] [PubMed]

S. Arridge, M. Schweiger, M. Hiraoka, and D. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

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

Nat. Biotechnol.

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, 803-806 (2003).
[CrossRef] [PubMed]

Nat. Nanotechnol.

A. de la Zerda, C. Zavaleta, S. Kere, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T.-J. Ma, O. Oralkan, Z. Cheng, X. Chen, H. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotechnol. 3, 557-561 (2008).
[CrossRef] [PubMed]

Numer. Algorithms

P. C. Hansen, “Deconvolution and regularization with Toeplitz matrices,” Numer. Algorithms 29, 323-378 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

M. Sivaramakrishnan, K. Maslov, H. F. Zhang, G. Stoica, and L. V. Wang, “Limitations of quantitative photoacoustic measurements of blood oxygenation in small vessels,” Phys. Med. Biol. 52, 1349-1361 (2008).
[CrossRef]

J. G. 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]

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

Phys. Rev. E

J. Ripoll and V. Ntziachristos, “Quantitative point source photoacoustic inversion formulas for scattering and absorbing media,” Phys. Rev. E 71, 031912 (2005).
[CrossRef]

Proc. IEEE

M.-L. Li, J.-T. Oh, X. Xie, G. Ku, W. Wang, C. Li, G. Lungu, G. Stoica, and L. V. Wang, “Simultaneous molecular and hypoxia imaging of brain tumours in vivo using spectroscopic photoacoustic tomography,” Proc. IEEE 96, 481-489 (2008).
[CrossRef]

Proc. SPIE

B. T. Cox, S. Arridge, K. Köstli, and P. Beard, “Quantitative photoacoustic imaging: fitting a model of light transport to the initial pressure distribution,” Proc. SPIE 5697, 49-55 (2005).
[CrossRef]

K. Maslov, M. Sivaramakrishnan, H. F. Zhang, G. Stoica, and L. V. Wang, “Technical considerations in quantitative blood oxygenation measurement using photoacoustic microscopy in vivo,” Proc. SPIE 6086, 60860R (2006).
[CrossRef]

K. M. Stantz, B. Liu, M. Cao, D. Reinecke, K. Miller, and R. Kruger, “Photoacoustic spectroscopic imaging of intra-tumour heterogeneity and molecular identification,” Proc. SPIE 6086, 608605 (2006).
[CrossRef]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Simultaneous estimation of chromophore concentration and scattering distributions from multiwavelength photoacoustic images,” Proc. SPIE 6856, 68560Y (2008).
[CrossRef]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Quantitative photoacoustic image reconstruction for molecular imaging,” Proc. SPIE 6086, 60861M (2006).
[CrossRef]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Gradient-based quantitative photoacoustic image reconstruction for molecular imaging,” Proc. SPIE 6437, 64371T (2007).
[CrossRef]

Rev. Sci. Instrum.

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

Other

M. Agranovsky, P. Kuchment, and L. Kunyansky, “On reconstruction formulas and algorithms for thermoacoustic and photoacoustic tomography,” arXiv:0706.1303v1 (2007).

A. J. Welch and M. van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Plenum, 1995).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++, The Art of Scientific Computing (Cambridge U. Press, 2005).

L. N. Trefethen and D. Bau, Numerical Linear Algebra (SIAM, 1997).
[CrossRef]

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. (Plenum1995).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Images of absorbed optical energy density due to two chromophores with different absorpion spectra, shown at four wavelengths: (A) 650, (B) 750, (C) 850, (D) 950 nm . The image dimensions are 3.75 mm × 8 mm , a point source is positioned 0.25 mm beneath the upper surface, and the anisotropy factor is 0.9. Each image is normalized by its maximum value to optimize the visible dynamic range.

Fig. 2
Fig. 2

Specific absorption coefficients (mm/l/g) of oxyhemoglobin and deoxyhemoglobin, the two chromophores whose concentrations are shown in Figs. 3A and 3B.

Fig. 3
Fig. 3

True concentration distributions (g/l) of two chromophores (images A and B) and their estimates (images C and D) successfully obtained by minimizing Eq. (17) using a gradient-based algorithm (BFGS) with the multiwavelength absorbed energy images from Fig. 1 as input data. The functional gradients were calculated efficiently using an adjoint model, and the scattering was known a priori. The image dimensions are 3.75 mm × 8 mm .

Fig. 4
Fig. 4

Profiles through the chromophore concentrations (g/l) shown in Fig. 3. (A) Exact (solid) and estimated (dotted) profiles at 1.6 mm through Figs. 3A and 3C, respectively. (B) Profiles at 1.6 mm , exact (solid) and estimated (dotted), and at 2.3 mm , exact (dashed), estimated (dotted–dashed) through Figs. 3B and 3D, respectively.

Fig. 5
Fig. 5

Optical absorption-scattering nonuniqueness. The absorption and scattering coefficient distributions A and B give rise to the absorbed energy distribution (to which the photoacoustic image is proportional) shown in C. The absorption and scattering distributions D and E also give rise to the same absorbed energy distribution, which is shown in F. The differences between the absorption, scattering, and absorbed energy images are shown on the right in G, H, and J, respectively. The fact that these absorbed energy densities are indistinguishable (J is virtually zero everywhere) demonstrates the nonuniqueness of the relationship between a single-wavelength photoacoustic image and the underlying optical coefficients. The image dimensions are 4 mm × 8 mm , a point source is positioned 0.25 mm beneath the upper surface, and the anisotropy factor is 0.9.

Fig. 6
Fig. 6

Wavelength dependence of the chromophore absorption and scattering used in the example in Subsection 4E.

Fig. 7
Fig. 7

Singular value spectrum of the Hessian matrix when data at one, two, and four wavelengths are used in its construction. The nonuniqueness in the single wavelength case gives rise to a gap in the singular value spectrum of several orders of magnitude. The nonuniqueness, and therefore the gap in the spectrum, disappears when two or more wavelengths are used in the reconstruction. However, the condition number is still large due to a second type of ill-posedness caused by the diffusive nature of the light propagation. This can be treated using standard techniques such as Tikhonov regularization, as shown.

Fig. 8
Fig. 8

Results from Newton inversion using three iterations with Tikhonov regularization. (A) True chromophore concentration distribution c ( x ) in (g/l). (B) True scattering distribution a ( x ) . (C) Recovered chromophore concentration estimate. (D) Recovered scattering distribution estimate. The initial distributions were chosen to be uniform and equal to 5 for both the chromophore concentration and the scattering. The image dimensions are 3.6 mm × 7.5 mm . Profiles through these images are shown in Fig. 9. Both distributions have been recovered, without crosstalk between them, although the scattering is clearly more sensitive to the noise in this example.

Fig. 9
Fig. 9

Profiles for the multiwavelength inversion example described in Subsection 4E and Fig. 8. A central horizontal profile through the concentration distribution, and a central vertical profile through the scattering distribution show the true values (solid), initial guess (dashed), estimate after one Newton iteration (dotted–dashed) and after three iterations (dotted). The latter correspond to slices through Figs. 8C and 8D.

Tables (1)

Tables Icon

Table 1 Ranges of the Absorption and Reduced Scattering Coefficients Used in the Multiwavelength Inversion Example as a Function of Wavelength

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

H ( x , t ) = μ a ( x ) ϕ ( x ) δ ( t ) = h ( x ) δ ( t ) ,
T 0 = h ( ρ C v ) , p 0 = ( β v s 2 C p ) h = Γ h .
h = μ a ϕ ( μ a , μ s ) .
h ( x , λ ) = μ a ( x , λ ) ϕ [ x , λ , μ a ( x , λ ) , μ s ( x , λ ) ] .
μ a ( x , λ ) = k = 1 K c k ( x ) α k ( λ ) ,
μ s ( x , λ ) a ( x ) λ b ,
( μ a κ ) ϕ = q 0 ,
μ a ( n + 1 ) ( x ) = h ̂ ( x ) ( ϕ ( n ) ( x ) + ϵ ) ,
h 1 = μ a 1 ϕ 1 ,
h 2 = ( μ a 1 + μ a 2 ) ϕ 2 ,
h 2 h 1 = μ a 1 ( ϕ 2 ϕ 1 ) + μ a 2 ϕ 2 ,
argmin μ a ( x ) E μ a = 1 2 [ h ( μ a ) h ̂ ] 2 d Ω .
argmin μ s ( x ) E μ s = 1 2 [ h ( μ s ) h ̂ ] 2 d Ω .
( μ a κ ) ϕ * = μ a ( h h ̂ ) ,
E μ a μ a ( x ) = ϕ ( x ) [ h ( x ) h ̂ ( x ) ] ϕ * ( x ) ϕ ( x ) ,
E μ s μ s ( x ) = 3 κ ( x ) 2 ϕ * ( x ) ϕ ( x ) .
argmin c k ( x ) E c = 1 2 [ h ( c k ) h ̂ ] 2 d Ω d λ ,
E c c k = α k ( λ ) E μ a μ a ( λ ) d λ .
E nonunique = 1 2 [ h 2 ( μ a 2 , μ s 2 ) h 1 ( μ a 1 , μ s 1 ) ] 2 d Ω .
argmin c k ( x ) , a ( x ) E = 1 2 [ h ( c k , a ) h ̂ ] 2 d Ω d λ .
E c k = h c k [ h ( c k , a ) h ̂ ] d Ω d λ ,
E a = h a [ h ( c k , a ) h ̂ ] d Ω d λ .
h c k = α k h μ a ,
h a = λ b h μ s ,
h ( x ) μ a ( x ) = ϕ ( x ) δ ( x x ) + μ a ( x ) ϕ ( x ) μ a ( x ) ,
h ( x ) μ s ( x ) = μ a ( x ) ϕ ( x ) μ s ( x ) ,
( μ a κ ) ϕ ( x ) μ a ( x ) = ϕ ( x ) δ ( x x ) .
( μ a κ ) ϕ ( x ) κ ( x ) = [ δ ( x x ) ϕ ( x ) ] .
h = ( h λ 1 h λ L ) = ( h 1 λ 1 h M λ 1 h 1 λ L h M λ L ) , c = ( c 1 c K ) = ( c 11 c 1 N c K 1 c K N ) .
u = ( c a ) .
E ( u ) = 1 2 l = 1 L m = 1 M [ h m λ l ( u ) h ̂ m λ l ] 2 = 1 2 e T e ,
E ( u 0 + δ ) E ( u 0 ) + g T δ + 1 2 δ T H δ + ,
δ = H 1 g ( J T J ) 1 J e .
g = [ E c 11 , , E c K N ] [ E a 1 , , E a N ] T ,
H = [ 2 E c 11 2 2 E c 11 c K N 2 E c 11 a 1 2 E c 11 a N 2 E c K N c 11 2 E c K N 2 2 E c K N a 1 2 E c K N a N 2 E a 1 c 11 2 E a 1 c K N 2 E a 1 2 2 E a 1 a N 2 E a N c 11 2 E a N c K N 2 E a N a 1 2 E a N 2 ]
J c λ l = [ h 1 λ l c 1 h 1 λ l c K N h M λ l c 1 h M λ l c K N ] ,
J a λ l = [ h 1 λ l a 1 h 1 λ l a N h M λ l a 1 h M λ l a N ] .
J = [ J c λ 1 J a λ 1 J c λ 2 J a λ 2 J c λ L J a λ L ] .
δ = V Σ 1 U T g = i u i T g σ i v i .

Metrics