Abstract

Improved solutions of the diffusion equation for time-resolved and steady-state spatially resolved reflectance are investigated for the determination of the optical coefficients of semi-infinite turbid media such as tissue. These solutions are derived for different boundary conditions at the turbid-medium–air interface and are compared with Monte Carlo simulations. Relative reflectance data are fitted in the time domain, whereas relative and absolute reflectance are investigated in the steady-state domain. It is shown that the error in deriving the optical coefficients is, especially for steady-state spatially resolved reflectance, considerably smaller for the solutions under study than for the commonly used solutions. Analysis of experimental measurements of absolute steady-state spatially resolved reflectance confirms these results.

© 1997 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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1996

A. Knüttel, S. Koch, R. Schork, and D. Böcker, “Tissue characterization with optical coherence tomography (OCT),” Biomedical Sensing, Imaging, and Tracking Technologies I, R. A. Lieberman, H. Podbielska, and T. V. O-Dinh, eds., Proc. SPIE 2676, 54–64 (1996).

A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996).
[CrossRef] [PubMed]

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, and B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
[CrossRef] [PubMed]

1995

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
[CrossRef]

G. J. Tearney, M. E. Brezinski, J. F. Southern, B. E. Bouma, M. R. Hee, and J. G. Fujimoto, “Determination of the refractive index of highly scattering human tissue by optical coherence tomography,” Opt. Lett. 20, 2258–2260 (1995).
[CrossRef] [PubMed]

H. Liu, A. Hielscher, B. Chance, S. L. Jacques, and F. K. Tittel, “Influence of blood vessels on the measurement of hemoglobin oxygenation as determined by time-resolved reflectance spectroscopy,” Med. Phys. 22, 1209–1217 (1995).
[CrossRef] [PubMed]

A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissue,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

1994

1993

1992

T. J. Farrell, B. C. Wilson, and M. S. Patterson, “The use of a neural network to determine tissue optical properties from spatially resolved diffuse reflectance measurements,” Phys. Med. Biol. 37, 2281–2286 (1992).
[CrossRef] [PubMed]

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

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

1989

1988

1983

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distribution of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

1941

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Aarnoudse, J. G.

Adam, G.

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distribution of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Alfano, R. R.

Aronson, R.

Arridge, S. R.

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

Barbieri, B.

Böcker, D.

A. Knüttel, S. Koch, R. Schork, and D. Böcker, “Tissue characterization with optical coherence tomography (OCT),” Biomedical Sensing, Imaging, and Tracking Technologies I, R. A. Lieberman, H. Podbielska, and T. V. O-Dinh, eds., Proc. SPIE 2676, 54–64 (1996).

Bolin, F. P.

Bouma, B. E.

Brezinski, M. E.

Chance, B.

H. Liu, A. Hielscher, B. Chance, S. L. Jacques, and F. K. Tittel, “Influence of blood vessels on the measurement of hemoglobin oxygenation as determined by time-resolved reflectance spectroscopy,” Med. Phys. 22, 1209–1217 (1995).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and 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.

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

Dassel, A. C. M.

de Mul, F. F. M.

Delpy, D. T.

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

Fantini, S.

Farrell, T. J.

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

T. J. Farrell, B. C. Wilson, and M. S. Patterson, “The use of a neural network to determine tissue optical properties from spatially resolved diffuse reflectance measurements,” Phys. Med. Biol. 37, 2281–2286 (1992).
[CrossRef] [PubMed]

Feng, T. C.

Ference, R. J.

Fishkin, S.

Franceschini, M. A.

Fujimoto, J. G.

Graaff, R.

Gratton, E.

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Haskell, R. C.

Hee, M. R.

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hibst, R.

Hielscher, A.

H. Liu, A. Hielscher, B. Chance, S. L. Jacques, and F. K. Tittel, “Influence of blood vessels on the measurement of hemoglobin oxygenation as determined by time-resolved reflectance spectroscopy,” Med. Phys. 22, 1209–1217 (1995).
[CrossRef] [PubMed]

Hielscher, A. H.

A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissue,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Jacques, S. L.

A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissue,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

H. Liu, A. Hielscher, B. Chance, S. L. Jacques, and F. K. Tittel, “Influence of blood vessels on the measurement of hemoglobin oxygenation as determined by time-resolved reflectance spectroscopy,” Med. Phys. 22, 1209–1217 (1995).
[CrossRef] [PubMed]

Keijzer, M.

Kienle, A.

Knüttel, A.

A. Knüttel, S. Koch, R. Schork, and D. Böcker, “Tissue characterization with optical coherence tomography (OCT),” Biomedical Sensing, Imaging, and Tracking Technologies I, R. A. Lieberman, H. Podbielska, and T. V. O-Dinh, eds., Proc. SPIE 2676, 54–64 (1996).

Koch, S.

A. Knüttel, S. Koch, R. Schork, and D. Böcker, “Tissue characterization with optical coherence tomography (OCT),” Biomedical Sensing, Imaging, and Tracking Technologies I, R. A. Lieberman, H. Podbielska, and T. V. O-Dinh, eds., Proc. SPIE 2676, 54–64 (1996).

Koelink, M. H.

Lilge, L.

Liu, F.

Liu, H.

H. Liu, A. Hielscher, B. Chance, S. L. Jacques, and F. K. Tittel, “Influence of blood vessels on the measurement of hemoglobin oxygenation as determined by time-resolved reflectance spectroscopy,” Med. Phys. 22, 1209–1217 (1995).
[CrossRef] [PubMed]

McAdams, M.

Patterson, M. S.

A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996).
[CrossRef] [PubMed]

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, and B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
[CrossRef] [PubMed]

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

T. J. Farrell, B. C. Wilson, and M. S. Patterson, “The use of a neural network to determine tissue optical properties from spatially resolved diffuse reflectance measurements,” Phys. Med. Biol. 37, 2281–2286 (1992).
[CrossRef] [PubMed]

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

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

Pogue, B. W.

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

Preuss, L. E.

Schork, R.

A. Knüttel, S. Koch, R. Schork, and D. Böcker, “Tissue characterization with optical coherence tomography (OCT),” Biomedical Sensing, Imaging, and Tracking Technologies I, R. A. Lieberman, H. Podbielska, and T. V. O-Dinh, eds., Proc. SPIE 2676, 54–64 (1996).

Southern, J. F.

Star, W. M.

Steiner, R.

Storchi, P. R. M.

Svaasand, L. O.

Taylor, R. C.

Tearney, G. J.

Tittel, F. K.

H. Liu, A. Hielscher, B. Chance, S. L. Jacques, and F. K. Tittel, “Influence of blood vessels on the measurement of hemoglobin oxygenation as determined by time-resolved reflectance spectroscopy,” Med. Phys. 22, 1209–1217 (1995).
[CrossRef] [PubMed]

A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissue,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Tromberg, B. J.

Tsay, T. T.

Wang, L.

A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissue,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Wilson, B. C.

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, and B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
[CrossRef] [PubMed]

T. J. Farrell, B. C. Wilson, and M. S. Patterson, “The use of a neural network to determine tissue optical properties from spatially resolved diffuse reflectance measurements,” Phys. Med. Biol. 37, 2281–2286 (1992).
[CrossRef] [PubMed]

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

M. S. Patterson, B. Chance, and 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 and G. Adam, “A Monte Carlo model for the absorption and flux distribution of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Yoo, K. M.

Zijlstr, W. G.

Appl. Opt.

Astrophys. J.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

J. Opt. Soc. Am. A

Med. Phys.

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

H. Liu, A. Hielscher, B. Chance, S. L. Jacques, and F. K. Tittel, “Influence of blood vessels on the measurement of hemoglobin oxygenation as determined by time-resolved reflectance spectroscopy,” Med. Phys. 22, 1209–1217 (1995).
[CrossRef] [PubMed]

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distribution of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Med. Biol.

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

T. J. Farrell, B. C. Wilson, and M. S. Patterson, “The use of a neural network to determine tissue optical properties from spatially resolved diffuse reflectance measurements,” Phys. Med. Biol. 37, 2281–2286 (1992).
[CrossRef] [PubMed]

A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996).
[CrossRef] [PubMed]

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

A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissue,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Proc. SPIE

A. Knüttel, S. Koch, R. Schork, and D. Böcker, “Tissue characterization with optical coherence tomography (OCT),” Biomedical Sensing, Imaging, and Tracking Technologies I, R. A. Lieberman, H. Podbielska, and T. V. O-Dinh, eds., Proc. SPIE 2676, 54–64 (1996).

Other

J. D. Moulton, “Diffusion modelling of picosecond laser pulse propagation of turbid media,” master’s degree dissertation (McMaster University, Hamilton, Ontario, Canada, 1990).

G. H. Bryan, “An application of the method of images to the conduction of heat,” Proc. London Math. Soc. 22, 424–430 (1891).

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).

J. P. Vanhouton, D. A. Benaron, S. Spilman, and D. K. Stevenson, “Imaging brain injuries using time-resolved near-infrared light scanning,” Pediatr. Res. 39, 470–476 (1996).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Chaps. 7 and 9.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1983), Chap. 11.

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Figures (12)

Fig. 1
Fig. 1

Comparison of time-resolved diffuse reflectance, calculated by means of diffusion theory with different boundary conditions, with Monte Carlo simulations at ρ=4.75, 9.75 mm. The optical coefficients are μs=1mm-1, μa=0.02mm-1, g=0.8, and n=1.4.

Fig. 2
Fig. 2

Comparison of time-resolved diffuse reflectance, calculated by means of diffusion theory with different boundary conditions, with Monte Carlo simulations at ρ=4.75, 9.75, and 14.75 mm. The optical coefficients are μs=0.5mm-1, μa=0.02mm-1, g=0.8, and n=1.4.

Fig. 3
Fig. 3

Reduced scattering coefficients derived from nonlinear regressions of different solutions of the diffusion equation for time-resolved reflectance to a Monte Carlo simulation versus the start time of the fitting range. The optical coefficients of the Monte Carlo simulation are μs=1mm-1, μa=0.02mm-1, and n=1.4, and the distance from the source is ρ=9.75 mm.

Fig. 4
Fig. 4

Absorption coefficients derived from nonlinear regressions of different solutions of the diffusion equation for time-resolved reflectance to a Monte Carlo simulation versus the start time of the fitting range. The optical coefficients of the Monte Carlo simulation are μs=1mm-1, μa=0.02mm-1, and n =1.4, and the distance from the source is ρ=9.75 mm.

Fig. 5
Fig. 5

Reduced scattering coefficients derived from nonlinear regressions of the EBC solution for time-resolved reflectance to Monte Carlo simulations with different μsρ values versus the start time of the fitting range. The optical coefficients of the Monte Carlo simulation are μs=1mm-1, μa=0.02mm-1, and n=1.4 and the distances from the source, ρ, are shown.

Fig. 6
Fig. 6

Absorption coefficients derived from nonlinear regressions of the EBC solution for time-resolved reflectance to Monte Carlo simulations with different μsρ values versus the start time of the fitting range. The optical coefficients of the Monte Carlo simulation are μs=1mm-1, μa=0.02mm-1, and n =1.4, and the distances from the source, ρ, are shown.

Fig. 7
Fig. 7

Comparison of steady-state spatially resolved reflectance, calculated by means of diffusion theory with different boundary conditions (EBCF and EBC), with Monte Carlo simulations. The optical coefficients are μs=1mm-1, μa =0.01mm-1, g=0.9, and n=1.4.

Fig. 8
Fig. 8

Comparison of steady-state spatially resolved reflectance, calculated by means of diffusion theory with different boundary conditions (EBCF, EBC, and PBCC), with Monte Carlo simulations. The distance range is decreased compared with that shown in Fig. 7. The optical coefficients are μs=1mm-1, μa =0.01mm-1, g=0.9, and n=1.4.

Fig. 9
Fig. 9

Absolute values of the relative errors of the reduced scattering coefficients derived from nonlinear regressions of the steady-state EBC solution to Monte Carlo data of absolute spatially resolved reflectance. The start and the end distances of the fitting range are varied. The optical coefficients of the Monte Carlo simulations are μs=1mm-1, μa=0.01mm-1, g =0.9, and n=1.4.

Fig. 10
Fig. 10

Absolute values of the relative errors of the absorption coefficients derived from nonlinear regressions of the steady-state EBC solution to Monte Carlo data of absolute spatially resolved reflectance. The start and the end distances of the fitting range are varied. The optical coefficients of the Monte Carlo simulations are μs=1mm-1, μa=0.01mm-1, g=0.9, and n=1.4.

Fig. 11
Fig. 11

Absolute values of the relative errors of the reduced scattering coefficients derived from nonlinear regressions of the steady-state EBC solution to Monte Carlo data of relative spatially resolved reflectance. The start and the end distances of the fitting range are varied. The optical coefficients of the Monte Carlo simulations are μs=1mm-1, μa=0.01mm-1, g =0.9, and n=1.4.

Fig. 12
Fig. 12

Absolute values of the relative errors of the absorption coefficients derived from nonlinear regressions of the steady state EBC solution to Monte Carlo data of relative spatially resolved reflectance. The start and the end distances of the fitting range are varied. The optical coefficients of the Monte Carlo simulations are μs=1mm-1, μa=0.01mm-1, g=0.9, and n=1.4.

Tables (1)

Tables Icon

Table 1 Optical Properties Derived from Nonlinear Regressions to Spatially Resolved Absolute Reflectance Measurements on Phantoms by Use of the EBC Solution [Eq. (8)] and the EBCF Solution [Eq. (6)]a

Equations (11)

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

Φ(ρ, z, t)=c(4πDct)3/2exp(-μact)×exp-(z-z0)2+ρ24Dct-exp-(z+z0+2zb)2+ρ24Dct,
zb=1+Reff1-Reff2D
Φs(ρ, z)=14πDexp{-μeff[(z-z0)2+ρ2]1/2}[(z-z0)2+ρ2]1/2-exp{-μeff[(z+z0+2zb)2+ρ2]1/2}[(z+z0+2zb)2+ρ2]1/2,
Rf(ρ, t)=-DΦ(ρ, z, t)·(-z)|z=0.
Rf(ρ, t)=½(4πDc)-3/2t-5/2 exp(-μact)×z0 exp-r124Dct+(z0+2zb)×exp-r224Dct,
Rfs(ρ)=14πz0μeff+1r1 exp(-μeffr1)r12+(z0+2zb)μeff+1r2 exp(-μeffr2)r22.
R(s)(ρ, t)=2πdΩ[1-Rfres(θ)] 14πΦ(s)(ρ, z=0, t)+3D Φ(s)(ρ, z=0, t)zcos θcos θ,
R(s)(ρ, t)=0.118Φ(s)(ρ, z=0, t)+0.306Rf(s)(ρ, t).
Φρ, z, t=c4πDct3/2exp-μact×exp-z-z02+ρ24Dct+exp-z+z02+ρ24Dct-2zb0dl exp-l/zb×exp-z+z0+l2+ρ24Dct},
Φs(ρ, z)=14πDexp{-μeff[(z-z0)2+ρ2]1/2}[(z-z0)2+ρ2]1/2+exp{-μeff[(z+z0)2+ρ2]1/2}[(z+z0)2+ρ2]1/2-2zb0dl exp(-l/zb)×exp{-μeff[(z+z0+l)2+ρ2]1/2}[(z+z0+l)2+ρ2]1/2.
R(s)(ρ, t)=0.170Φ(s)(ρ, z=0, t).

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