Abstract

A novel method for the determination of the optical properties of tissue from time-domain measurements is presented. The data analysis is based on the evaluation of the first moment and the second centralized moment, i.e., the mean time of flight and the variance of the measured distribution of times of flight (DTOF) of photons injected by short (picosecond) laser pulses. Analytical expressions are derived for calculation of absorption and of reduced scattering coefficients from these moments by application of diffusion theory for infinite and semi-infinite homogeneous media. The proposed method was tested on experimental data obtained with phantoms, and results for absorption and reduced scattering coefficients obtained by the proposed method are compared with those obtained by fitting of the same data with analytical solutions of the diffusion equation. Furthermore, the accuracy of the moment analysis was investigated for a range of integration limits of the DTOF. The moment analysis may serve as a comparatively fast method for evaluating optical properties with sufficient accuracy and can be used, e.g., for on-line monitoring of optical properties of biological tissue.

© 2003 Optical Society of America

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  1. E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris, “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Anal. Biochem. 195, 330–351 (1991).
    [CrossRef] [PubMed]
  2. B. C. Wilson, M. S. Patterson, D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci. 1, 235–244 (1986).
    [CrossRef]
  3. V. Quaresima, R. Sfareni, A. Pizzi, M. Ferrari, “Measurement of the muscle optical properties on muscular dystrophy patients by a frequency-domain photometer,” in Biomedical Optical Spectroscopy and Diagnostics, E. Sevick-Muraca, D. Benaron, eds., Vol. 3 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 123–125.
  4. M. Kohl, M. Essenpreis, M. Cope, “The influence of glucose-concentration upon the transport of light in tissue-simulating phantoms,” Phys. Med. Biol. 40, 1267–1287 (1995).
    [CrossRef] [PubMed]
  5. M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  6. R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Compact tissue oximeter based on dual-wavelength multichannel time-resolved reflectance,” Appl. Opt. 38, 3670–3680 (1999).
    [CrossRef]
  7. S. J. Matcher, “Closed-form expressions for obtaining the absorption and scattering coefficients of a turbid medium with time-resolved spectroscopy,” Appl. Opt. 36, 8296–8302 (1997).
    [CrossRef]
  8. V. Kirchner, “Beugung von Photonendichtewellen an sphärischen und zylindrischen Inhomogenitäten,” Diploma thesis, Freie Universität Berlin, Berlin, 1996 (in German).
  9. S. R. Arridge, M. Cope, 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]
  10. E. W. Small, L. J. Libertini, D. W. Brown, J. R. Small, “Extensions of the method of moments for deconvolution of experimental data,” in Fluorescence Detection III, E. R. Menzel, ed., Proc. SPIE1054, 36–52 (1989).
    [CrossRef]
  11. A. Liebert, H. Wabnitz, D. Grosenick, R. Macdonald, “Fiber dispersion in time domain measurements compromising the accuracy of optical properties of strongly scattering media,” J. Biomed. Opt. 8, 512–516 (2003).
    [CrossRef] [PubMed]
  12. G. M. Hale, M. R. Querry, “Optical constants of water in the 200-nm to 200-μm wavelength region,” Appl. Opt. 12, 555–563 (1973).
    [CrossRef] [PubMed]
  13. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1994).

2003

A. Liebert, H. Wabnitz, D. Grosenick, R. Macdonald, “Fiber dispersion in time domain measurements compromising the accuracy of optical properties of strongly scattering media,” J. Biomed. Opt. 8, 512–516 (2003).
[CrossRef] [PubMed]

1999

1997

S. J. Matcher, “Closed-form expressions for obtaining the absorption and scattering coefficients of a turbid medium with time-resolved spectroscopy,” Appl. Opt. 36, 8296–8302 (1997).
[CrossRef]

1995

M. Kohl, M. Essenpreis, M. Cope, “The influence of glucose-concentration upon the transport of light in tissue-simulating phantoms,” Phys. Med. Biol. 40, 1267–1287 (1995).
[CrossRef] [PubMed]

1992

S. R. Arridge, M. Cope, 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]

1991

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris, “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

1989

1986

B. C. Wilson, M. S. Patterson, D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci. 1, 235–244 (1986).
[CrossRef]

1973

Arridge, S. R.

S. R. Arridge, M. Cope, 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]

Brown, D. W.

E. W. Small, L. J. Libertini, D. W. Brown, J. R. Small, “Extensions of the method of moments for deconvolution of experimental data,” in Fluorescence Detection III, E. R. Menzel, ed., Proc. SPIE1054, 36–52 (1989).
[CrossRef]

Burns, D. M.

B. C. Wilson, M. S. Patterson, D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci. 1, 235–244 (1986).
[CrossRef]

Chance, B.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris, “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

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

Cope, M.

M. Kohl, M. Essenpreis, M. Cope, “The influence of glucose-concentration upon the transport of light in tissue-simulating phantoms,” Phys. Med. Biol. 40, 1267–1287 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, 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]

Cubeddu, R.

Delpy, D. T.

S. R. Arridge, M. Cope, 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]

Essenpreis, M.

M. Kohl, M. Essenpreis, M. Cope, “The influence of glucose-concentration upon the transport of light in tissue-simulating phantoms,” Phys. Med. Biol. 40, 1267–1287 (1995).
[CrossRef] [PubMed]

Ferrari, M.

V. Quaresima, R. Sfareni, A. Pizzi, M. Ferrari, “Measurement of the muscle optical properties on muscular dystrophy patients by a frequency-domain photometer,” in Biomedical Optical Spectroscopy and Diagnostics, E. Sevick-Muraca, D. Benaron, eds., Vol. 3 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 123–125.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1994).

Grosenick, D.

A. Liebert, H. Wabnitz, D. Grosenick, R. Macdonald, “Fiber dispersion in time domain measurements compromising the accuracy of optical properties of strongly scattering media,” J. Biomed. Opt. 8, 512–516 (2003).
[CrossRef] [PubMed]

Hale, G. M.

Kirchner, V.

V. Kirchner, “Beugung von Photonendichtewellen an sphärischen und zylindrischen Inhomogenitäten,” Diploma thesis, Freie Universität Berlin, Berlin, 1996 (in German).

Kohl, M.

M. Kohl, M. Essenpreis, M. Cope, “The influence of glucose-concentration upon the transport of light in tissue-simulating phantoms,” Phys. Med. Biol. 40, 1267–1287 (1995).
[CrossRef] [PubMed]

Leigh, J.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris, “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

Libertini, L. J.

E. W. Small, L. J. Libertini, D. W. Brown, J. R. Small, “Extensions of the method of moments for deconvolution of experimental data,” in Fluorescence Detection III, E. R. Menzel, ed., Proc. SPIE1054, 36–52 (1989).
[CrossRef]

Liebert, A.

A. Liebert, H. Wabnitz, D. Grosenick, R. Macdonald, “Fiber dispersion in time domain measurements compromising the accuracy of optical properties of strongly scattering media,” J. Biomed. Opt. 8, 512–516 (2003).
[CrossRef] [PubMed]

Macdonald, R.

A. Liebert, H. Wabnitz, D. Grosenick, R. Macdonald, “Fiber dispersion in time domain measurements compromising the accuracy of optical properties of strongly scattering media,” J. Biomed. Opt. 8, 512–516 (2003).
[CrossRef] [PubMed]

Maris, M.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris, “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

Matcher, S. J.

S. J. Matcher, “Closed-form expressions for obtaining the absorption and scattering coefficients of a turbid medium with time-resolved spectroscopy,” Appl. Opt. 36, 8296–8302 (1997).
[CrossRef]

Nioka, S.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris, “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

Patterson, M. S.

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

B. C. Wilson, M. S. Patterson, D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci. 1, 235–244 (1986).
[CrossRef]

Pifferi, A.

Pizzi, A.

V. Quaresima, R. Sfareni, A. Pizzi, M. Ferrari, “Measurement of the muscle optical properties on muscular dystrophy patients by a frequency-domain photometer,” in Biomedical Optical Spectroscopy and Diagnostics, E. Sevick-Muraca, D. Benaron, eds., Vol. 3 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 123–125.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1994).

Quaresima, V.

V. Quaresima, R. Sfareni, A. Pizzi, M. Ferrari, “Measurement of the muscle optical properties on muscular dystrophy patients by a frequency-domain photometer,” in Biomedical Optical Spectroscopy and Diagnostics, E. Sevick-Muraca, D. Benaron, eds., Vol. 3 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 123–125.

Querry, M. R.

Sevick, E. M.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris, “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

Sfareni, R.

V. Quaresima, R. Sfareni, A. Pizzi, M. Ferrari, “Measurement of the muscle optical properties on muscular dystrophy patients by a frequency-domain photometer,” in Biomedical Optical Spectroscopy and Diagnostics, E. Sevick-Muraca, D. Benaron, eds., Vol. 3 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 123–125.

Small, E. W.

E. W. Small, L. J. Libertini, D. W. Brown, J. R. Small, “Extensions of the method of moments for deconvolution of experimental data,” in Fluorescence Detection III, E. R. Menzel, ed., Proc. SPIE1054, 36–52 (1989).
[CrossRef]

Small, J. R.

E. W. Small, L. J. Libertini, D. W. Brown, J. R. Small, “Extensions of the method of moments for deconvolution of experimental data,” in Fluorescence Detection III, E. R. Menzel, ed., Proc. SPIE1054, 36–52 (1989).
[CrossRef]

Taroni, P.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1994).

Torricelli, A.

Valentini, G.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1994).

Wabnitz, H.

A. Liebert, H. Wabnitz, D. Grosenick, R. Macdonald, “Fiber dispersion in time domain measurements compromising the accuracy of optical properties of strongly scattering media,” J. Biomed. Opt. 8, 512–516 (2003).
[CrossRef] [PubMed]

Wilson, B. C.

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

B. C. Wilson, M. S. Patterson, D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci. 1, 235–244 (1986).
[CrossRef]

Anal. Biochem.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris, “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

Appl. Opt.

J. Biomed. Opt.

A. Liebert, H. Wabnitz, D. Grosenick, R. Macdonald, “Fiber dispersion in time domain measurements compromising the accuracy of optical properties of strongly scattering media,” J. Biomed. Opt. 8, 512–516 (2003).
[CrossRef] [PubMed]

Lasers Med. Sci.

B. C. Wilson, M. S. Patterson, D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci. 1, 235–244 (1986).
[CrossRef]

Phys. Med. Biol.

S. R. Arridge, M. Cope, 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]

M. Kohl, M. Essenpreis, M. Cope, “The influence of glucose-concentration upon the transport of light in tissue-simulating phantoms,” Phys. Med. Biol. 40, 1267–1287 (1995).
[CrossRef] [PubMed]

Other

E. W. Small, L. J. Libertini, D. W. Brown, J. R. Small, “Extensions of the method of moments for deconvolution of experimental data,” in Fluorescence Detection III, E. R. Menzel, ed., Proc. SPIE1054, 36–52 (1989).
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1994).

V. Quaresima, R. Sfareni, A. Pizzi, M. Ferrari, “Measurement of the muscle optical properties on muscular dystrophy patients by a frequency-domain photometer,” in Biomedical Optical Spectroscopy and Diagnostics, E. Sevick-Muraca, D. Benaron, eds., Vol. 3 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 123–125.

V. Kirchner, “Beugung von Photonendichtewellen an sphärischen und zylindrischen Inhomogenitäten,” Diploma thesis, Freie Universität Berlin, Berlin, 1996 (in German).

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Figures (8)

Fig. 1
Fig. 1

Experimental setup: PMT, photomultiplier tube; HV, high-voltage supply; PDL800, semiconductor laser controller; SPC300, single photon counting electronics plug-in card; sync., synchronization signal; ρ, separation between transmitting and receiving optical fibers.

Fig. 2
Fig. 2

Analysis of deviations of moments (m 1, squares and V, circles) and optical properties (μ s ′, up triangles and μ a , down triangles) from their theoretical values versus (a) upper and (b) lower integration limits. The DTOF was computed for an infinite medium with μ s ′ = 10 cm-1 and μ a = 0.04 cm-1 for source-detector separation r sd = 3 cm according to Eq. (1). Integration was carried out up to those times of flight at which the number of counts of the DTOF had dropped below a certain fraction of its maximum value. All quantities were normalized to their theoretical values.

Fig. 3
Fig. 3

DTOFs measured for (a) the infinite-medium geometry and (b) the semi-infinite-medium geometry. Distance ρ between transmitting and receiving fibers (see Fig. 1) is indicated for each DTOF. The instrumental response function is given in both cases.

Fig. 4
Fig. 4

Top, mean time of flight 〈t〉 and bottom variance V calculated for DTOFs measured in the infinite and semi-infinite configurations versus fiber separation. The theoretical curves were computed for μ a = 0.0265 cm-1 and μ s ′ = 11 cm-1 as obtained from fitting of calculated photon fluence rates to the experimental DTOFs measured in infinite geometry (see Fig. 5).

Fig. 5
Fig. 5

Top, reduced scattering coefficient μ s ′ and bottom, absorption coefficient μ a versus separation of fibers. Values obtained by the method of moments (open symbols) are compared with results derived by fitting of DTOFs (filled symbols) for infinite-medium (squares) and semi-infinite-medium (circles) geometries. Horizontal lines, values used for calculating the theoretical curves in Fig. 4.

Fig. 6
Fig. 6

Comparison of (a) reduced scattering coefficient μ s ′ and (b) absorption coefficient μ a obtained by different methods of analysis (fit, moments) and for different geometries (infinite medium, semi-infinite medium) from measurements at various fiber separations ρ. Mean values (marked by filled squares) obtained by the method of moments are compared with mean values derived by fitting of DTOFs for infinite-medium and semi-infinite-medium geometries. The rectangles represent ±1 standard deviation obtained from measurements taken at various source-detector separations. The horizontal lines inside the rectangles mark median values; the vertical bars reflect the maximum-minimum ranges.

Fig. 7
Fig. 7

Optical properties (λ = 687 nm) of dilute milk (volume fraction, 20%) with various amounts of ink added, measured in infinite-medium (squares) and semi-infinite-medium (circles) geometries. Results of fitting of DTOFs (filled symbols) are compared with results obtained from the analysis of moments (open symbols).

Fig. 8
Fig. 8

Coefficients of variation (i.e., ratios of standard deviation and mean value) of (a) μ a and (b) μ s ′ as calculated by the fitting procedure (filled circles) and by the analysis of moments (open circles) plotted versus total photon count N tot of the DTOF. Measurements were performed on a solid phantom (μ a = 0.13 cm-1 and μ s ′ = 9.8 cm-1) in semi-infinite geometries at the interoptode distance ρ = 3 cm and wavelength λ = 687 nm. The solid curves refer to the analysis of moments and represent the theoretical uncertainties that are due to photon statistics.

Tables (1)

Tables Icon

Table 1 Relationship between Momentsa of the DTOF and Scattering and Absorption Coefficients for Infinite and Semi-Infinite Homogeneous Media According To Diffusion Theoryb

Equations (22)

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ϕrsd, t=cSp4πDct3/2exp-rsd24Dct-μact,
Rrsd, t=z0Sp4πDc3/2t5/2exp-rsd24Dct-μact,
mk=tk=- tkgtdt- gtdt.
m1F=m1E+m1f,
VF=VE+Vf.
σt=VNtot,
σV=m4c-V2Ntot,
km1NkVNk Nk=m3cNtot,
m1=rsd2cμaD
m1=rsd22cDrsdμa+D
V=rsd4c2μa3D
V=rsd34c2μaDrsdμa+D2
μa=m12cV
μa=m132cVm12+V
D=Vrsd22cm13
D=Vrsd22cm1m12+V
μs=2m13c3rsd2V
μs=2m1cm12+V3rsd2V
Vm12=Drsdμa
Vm12=Drsdμa
μeff=m12Vrsd
μeff=m12Vrsd

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