Abstract

We present an analytical solution for the scattering of diffuse photon density waves from an infinite circular, cylindrical inhomogeneity embedded in a homogeneous highly scattering turbid medium. The analytical solution, based on the diffusion approximation of the Boltzmann transport equation, represents the contribution of the cylindrical inhomogeneity as a series of modified Bessel functions integrated from zero to infinity and weighted by different angular dependencies. This series is truncated at the desired precision, similar to the Mie theory. We introduce new boundary conditions that account for specular reflections at the interface between the background medium and the cylindrical inhomogeneity. These new boundary conditions allow the separate recovery of the index of refraction of an object from its absorption and reduced scattering coefficients. The analytical solution is compared with data obtained experimentally to evaluate the predictive capability of the model. Optical properties of known cylindrical objects are recovered accurately. However, as the radius of the cylinder decreases, the required measurement signal-to-noise ratio rapidly increases. Because of the new boundary conditions, an upper limit can be placed on the recovered size of cylindrical objects with radii below 0.3 cm if they have a substantially different index of refraction from that of the background medium.

© 1998 Optical Society of America

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References

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    [CrossRef]
  3. D. A. Boas, M. A. Oleary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
    [CrossRef] [PubMed]
  4. G. Gratton, M. Fabiani, D. Friedman, M. A. Franceschini, S. Fantini, P. M. Corballis, E. Gratton, “Rapid changes of optical parameters in the human brain during a tapping task,” J. Cognitive Neurosci. 7, 446–456 (1994).
    [CrossRef]
  5. D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
    [CrossRef] [PubMed]
  6. 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]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  15. S. A. Walker, S. Fantini, E. Gratton, “Image reconstruction using back-projection from frequency domain optical measurements in highly scattering media,” Appl. Opt. 35, 170–179 (1996).
  16. S. Fantini, M. A. Franceschini, J. B. Fishkin, B. Barbieri, E. Gratton, “Quantitative determination of the absorption spectra of chromophores in strongly scattering media: a light-emitting-diode based technique,” Appl. Opt. 33, 5204–5213 (1994).
    [CrossRef] [PubMed]
  17. J. M. Beechem, E. Gratton, M. Ameloot, J. R. Knutson, L. Brand, “The global analysis of fluorescence intensity and anisotropy decay data: second generation theory and programs,” in Topics in Fluorescence Spectroscopy, II, J. R. Lakowicz, ed. (Plenum, New York, 1991), Chap. 5, pp. 241–305.

1997 (1)

1996 (1)

S. A. Walker, S. Fantini, E. Gratton, “Image reconstruction using back-projection from frequency domain optical measurements in highly scattering media,” Appl. Opt. 35, 170–179 (1996).

1995 (1)

1994 (5)

1993 (2)

1992 (1)

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]

1989 (2)

1987 (1)

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Alter, C. A.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Ameloot, M.

J. M. Beechem, E. Gratton, M. Ameloot, J. R. Knutson, L. Brand, “The global analysis of fluorescence intensity and anisotropy decay data: second generation theory and programs,” in Topics in Fluorescence Spectroscopy, II, J. R. Lakowicz, ed. (Plenum, New York, 1991), Chap. 5, pp. 241–305.

Aronson, R.

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]

Barbieri, B.

Beechem, J. M.

J. M. Beechem, E. Gratton, M. Ameloot, J. R. Knutson, L. Brand, “The global analysis of fluorescence intensity and anisotropy decay data: second generation theory and programs,” in Topics in Fluorescence Spectroscopy, II, J. R. Lakowicz, ed. (Plenum, New York, 1991), Chap. 5, pp. 241–305.

Benaron, D. A.

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

Boas, D. A.

D. A. Boas, M. A. Oleary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. Oleary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Bolin, F. P.

Brand, L.

J. M. Beechem, E. Gratton, M. Ameloot, J. R. Knutson, L. Brand, “The global analysis of fluorescence intensity and anisotropy decay data: second generation theory and programs,” in Topics in Fluorescence Spectroscopy, II, J. R. Lakowicz, ed. (Plenum, New York, 1991), Chap. 5, pp. 241–305.

Chance, B.

Cope, M.

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]

Corballis, P. M.

G. Gratton, M. Fabiani, D. Friedman, M. A. Franceschini, S. Fantini, P. M. Corballis, E. Gratton, “Rapid changes of optical parameters in the human brain during a tapping task,” J. Cognitive Neurosci. 7, 446–456 (1994).
[CrossRef]

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]

Fabiani, M.

G. Gratton, M. Fabiani, D. Friedman, M. A. Franceschini, S. Fantini, P. M. Corballis, E. Gratton, “Rapid changes of optical parameters in the human brain during a tapping task,” J. Cognitive Neurosci. 7, 446–456 (1994).
[CrossRef]

Fantini, S.

S. A. Walker, S. Fantini, E. Gratton, “Image reconstruction using back-projection from frequency domain optical measurements in highly scattering media,” Appl. Opt. 35, 170–179 (1996).

G. Gratton, M. Fabiani, D. Friedman, M. A. Franceschini, S. Fantini, P. M. Corballis, E. Gratton, “Rapid changes of optical parameters in the human brain during a tapping task,” J. Cognitive Neurosci. 7, 446–456 (1994).
[CrossRef]

S. Fantini, M. A. Franceschini, E. Gratton, “Semi-infinite-geometry boundary problem for light migration in highly scattering media: a frequency-domain study in the diffusion approximation,” J. Opt. Soc. Am. B 11, 2128–2138 (1994).
[CrossRef]

S. Fantini, M. A. Franceschini, J. B. Fishkin, B. Barbieri, E. Gratton, “Quantitative determination of the absorption spectra of chromophores in strongly scattering media: a light-emitting-diode based technique,” Appl. Opt. 33, 5204–5213 (1994).
[CrossRef] [PubMed]

Ference, R. J.

Fishkin, J. B.

Franceschini, M. A.

Friedman, D.

G. Gratton, M. Fabiani, D. Friedman, M. A. Franceschini, S. Fantini, P. M. Corballis, E. Gratton, “Rapid changes of optical parameters in the human brain during a tapping task,” J. Cognitive Neurosci. 7, 446–456 (1994).
[CrossRef]

Gratton, E.

S. A. Walker, S. Fantini, E. Gratton, “Image reconstruction using back-projection from frequency domain optical measurements in highly scattering media,” Appl. Opt. 35, 170–179 (1996).

G. Gratton, M. Fabiani, D. Friedman, M. A. Franceschini, S. Fantini, P. M. Corballis, E. Gratton, “Rapid changes of optical parameters in the human brain during a tapping task,” J. Cognitive Neurosci. 7, 446–456 (1994).
[CrossRef]

S. Fantini, M. A. Franceschini, E. Gratton, “Semi-infinite-geometry boundary problem for light migration in highly scattering media: a frequency-domain study in the diffusion approximation,” J. Opt. Soc. Am. B 11, 2128–2138 (1994).
[CrossRef]

S. Fantini, M. A. Franceschini, J. B. Fishkin, B. Barbieri, E. Gratton, “Quantitative determination of the absorption spectra of chromophores in strongly scattering media: a light-emitting-diode based technique,” Appl. Opt. 33, 5204–5213 (1994).
[CrossRef] [PubMed]

J. B. Fishkin, E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
[CrossRef] [PubMed]

J. M. Beechem, E. Gratton, M. Ameloot, J. R. Knutson, L. Brand, “The global analysis of fluorescence intensity and anisotropy decay data: second generation theory and programs,” in Topics in Fluorescence Spectroscopy, II, J. R. Lakowicz, ed. (Plenum, New York, 1991), Chap. 5, pp. 241–305.

J. S. Maier, E. Gratton, “Frequency-domain methods in optical tomography: detection of localized absorbers and a backscattering reconstruction scheme,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 440–451 (1993).
[CrossRef]

Gratton, G.

G. Gratton, M. Fabiani, D. Friedman, M. A. Franceschini, S. Fantini, P. M. Corballis, E. Gratton, “Rapid changes of optical parameters in the human brain during a tapping task,” J. Cognitive Neurosci. 7, 446–456 (1994).
[CrossRef]

Haskell, R. C.

Jacques, S. L.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Knutson, J. R.

J. M. Beechem, E. Gratton, M. Ameloot, J. R. Knutson, L. Brand, “The global analysis of fluorescence intensity and anisotropy decay data: second generation theory and programs,” in Topics in Fluorescence Spectroscopy, II, J. R. Lakowicz, ed. (Plenum, New York, 1991), Chap. 5, pp. 241–305.

Maier, J. S.

J. S. Maier, E. Gratton, “Frequency-domain methods in optical tomography: detection of localized absorbers and a backscattering reconstruction scheme,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 440–451 (1993).
[CrossRef]

Oleary, M. A.

D. A. Boas, M. A. Oleary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. Oleary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Patterson, M. S.

Prahl, S. A.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Preuss, L. E.

Stevenson, D. K.

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

Svaasand, L. O.

Taylor, R. C.

Tromberg, B. J.

Tsay, T.-T.

Walker, S. A.

S. A. Walker, S. Fantini, E. Gratton, “Image reconstruction using back-projection from frequency domain optical measurements in highly scattering media,” Appl. Opt. 35, 170–179 (1996).

Wilson, B. C.

Yodh, A. G.

D. A. Boas, M. A. Oleary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. Oleary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Appl. Opt. (5)

J. Cognitive Neurosci. (1)

G. Gratton, M. Fabiani, D. Friedman, M. A. Franceschini, S. Fantini, P. M. Corballis, E. Gratton, “Rapid changes of optical parameters in the human brain during a tapping task,” J. Cognitive Neurosci. 7, 446–456 (1994).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

Lasers Life Sci. (1)

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Phys. Med. Biol. (1)

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]

Proc. Natl. Acad. Sci. USA (1)

D. A. Boas, M. A. Oleary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Science (1)

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

Other (3)

J. S. Maier, E. Gratton, “Frequency-domain methods in optical tomography: detection of localized absorbers and a backscattering reconstruction scheme,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 440–451 (1993).
[CrossRef]

J. D. Jackson, Classical Electrodynamics (Wiley, New York), Chap. 16.

J. M. Beechem, E. Gratton, M. Ameloot, J. R. Knutson, L. Brand, “The global analysis of fluorescence intensity and anisotropy decay data: second generation theory and programs,” in Topics in Fluorescence Spectroscopy, II, J. R. Lakowicz, ed. (Plenum, New York, 1991), Chap. 5, pp. 241–305.

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Figures (7)

Fig. 1
Fig. 1

Helmholtz equation solved in cylindrical coordinates with the origin at the object center. The source is positioned on the x axis (ϕ s = 0, ρ s = r s , and z s = 0), and the axis of the infinite cylinder is along the z axis (coming out of the page). Sample contours of equal phase of the incoming spherical wave are shown before interaction with the cylinder. The radius of the cylinder is a and the vector pointing to the detector is r d d , ρ d , and z d ).

Fig. 2
Fig. 2

Experimental setup. (a) Side view of the 16-L glass container filled with Intralipid/India Ink mixture. Fiber optics carry light to the source and from the detector. The fiber tips are separated by 5 cm inside the tank and scanned by means of a mechanical XYZ scanner with a positioning accuracy of 10 μm. The object is suspended in the medium by a 3-mm-diameter glass rod. (b) Top view of the cylinder. The source and the detector make a one-dimensional scan perpendicular to the object across the volume of interest. Each scan consists of 41 measurements at 2-mm intervals for a total of 8 cm.

Fig. 3
Fig. 3

Sample fits to experimental data are presented for a 0.5-cm-radius cylinder of material type A- (Table 1). The open squares and the filled diamonds represent the ac and the phase measurements, respectively. The theory is represented by the solid and the dotted curves for ac and phase predictions. The experimental parameters are given in Table 1.

Fig. 4
Fig. 4

Comparison of the precision of optical property recovery for different sized cylinders. As the radius of each cylinder decreases, the object’s absorption coefficient cannot be determined independently from its radius. Open squares represent the best fit to experimental data by varying the absorption coefficient μ a at different assumed cylinder radii. Filled triangles represent the goodness of fit as measured by the reduced χ2 (ΔACmeas = 0.2%, ΔPhasemeas = 0.1°). (a) True cylinder radius, 0.75 cm. The dotted curve represents an approximation of the absorption versus the radius (∝r -1.67) normalized to the value of μ a at the true radius (r = 0.75 ± 0.02 cm). (b) True cylinder radius, 0.5 cm. The dotted curve represents an approximation of the absorption versus the radius (∝r -1.34) normalized to the value of μ a at the true radius (r = 0.5 ± 0.1 cm). (c) True cylinder radius, 0.25 cm. The dotted curve represents an approximation of the absorption versus the radius (∝r -0.97) normalized to the value of μ a at the true radius. Owing to the small radius of this cylinder, the true radius cannot be determined independently from the absorption coefficient.

Fig. 5
Fig. 5

Best fit of the analytical solution to measured data varying μ a , μ s ′, n, and radius. Filled squares represent the best-fit radius of each cylinder plotted versus the true radius. Recovered values of the cylinder optical properties can be compared with the values in Table 1. The values for the largest cylinders are recovered most accurately. As the cylinder radius decreases, the size and the optical properties of each cylinder are separated less accurately.

Fig. 6
Fig. 6

Effect of Fresnel reflections on measured ac and phase. The index of refraction mismatch between background and cylinder media is n background = 1.33 versus n object = 1.45. The infinite cylinder is centered between the source and the detector, which are separated by 5.0 cm. The background medium absorption and the scattering coefficients are μ a = 0.1 cm-1 and μ s ′ = 10 cm-1, whereas the absorption and the scattering coefficients of the cylinder vary (0.01 cm-1 < μ a < 0.2 cm-1, 5 cm-1 < μ s ′ < 20 cm-1). (a) Perturbation in measured ac intensity due to Fresnel reflections. As the cylinder becomes more (less) transparent than the background medium, the measured ac intensity is increased (decreased) owing to Fresnel reflections. (b) Difference in measured phase caused by Fresnel boundary conditions. This phase lag increases as the absorption coefficient of the cylinder decreases, reaching a maximum value of 0.5° when μ a = 0.01 cm-1.

Fig. 7
Fig. 7

Differences in experimental data generated by different-sized cylinders. Simulated experimental data are generated for the source–detector geometry shown in Fig. 2 with a cylinder of radius r true (background: μ a = 0.1 cm-1, μ s ′ = 10 cm-1, n background = 1.33; object: μ a = 0.2 cm-1, μ s ′ = 10 cm-1, n object = 1.33). These data are then fit with the theoretical model by our varying the absorption coefficient of a single cylinder in the same position with an assumed radius r fitted. We then make a simple evaluation of the differences between the theoretical model (assumed radius) and the simulated data (true radius) by calculating the reduced χ2. (a) Lines of reduced χ2 = 0.3, 3, and 30 for our experimental conditions. Here the size of cylinders with a radius larger than 0.4 cm can be determined with increasing accuracy as the radius r true increases. (b) Simulated data generated as in (a) by our adding an index of refraction mismatch between the background medium and the object (n background = 1.33; n object = 1.45). These data are then fitted as in (a). The effect of the object’s different index of refraction is a decrease in the required signal-to-noise ratio for accurate recovery of a cylinder’s size and optical properties as the true radius of the cylinder decreases to smaller than 0.4 cm. The index of refraction mismatch allows the placement of a limit on the recovered size and optical properties of cylinders with radii below 0.3 cm.

Tables (2)

Tables Icon

Table 1 Cylinder and Background Measured Optical Properties

Tables Icon

Table 2 Cylinder and Background Recovered Optical Properties

Equations (19)

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2 Φ r ,   t - μ a D   Φ r ,   t - 1 vD Φ r ,   t t = - 1 D   S r ,   t ,
Φ r ,   t = Φ dc r ,   t + Φ ac r ,   t exp - i ω t ,
2 + k 2 Φ ac r = - q r ,   ω D ,
k 2 = - v μ a + i ω vD ,
Φ inc r s ,   r d = S   exp ik out | r s - r d | 4 π D | r s - r d | ,
Φ inc r s ,   r d = S 2 π 2 D out n = 0 0 d p   cos n ϕ d × cos pz d I n x < K n x > .
Φ scatt r s ,   r d = n = 0 0 d p   cos n ϕ d cos pz d × A n p I n x + B n p K n x .
Φ in r s ,   r d = n = 0 0 d p   cos n ϕ d cos pz d × C n p I n y + D n p K n y ,
1 - R 21 Φ out = 1 - R 21 + 2 R 12 - R 21 D in ρ Φ in at   ρ = a .
  D out ρ   Φ out = D in ρ   Φ in at   ρ = a .
R 12 R 12 ϕ + R 12 j 2 - R 12 ϕ + R 12 j ,
R 12 ϕ 0 π / 2   2   sin θ cos θ R fr 12 θ d θ ,
R 12 j 0 π / 2   3   sin θ cos 2 θ R fr 12 θ d θ ,
R fr θ = 1 2 n   cos θ - n out cos θ n   cos θ + n out cos θ 2 + 1 2 n   cos θ - n out cos θ n   cos θ + n out cos θ 2 when   0 θ θ c = 1   when   θ c θ π / 2
n   sin θ = n out sin θ .
A n p = D n p = 0 ,
B n p = - S 2 π 2 D out   K n z b × D out x b I n x b I n y b X fr - D in y b I n x b I n y b D out x b K n x b I n y b X fr - D in y b K n x b I n y b ,
C n p = - S 2 π 2 D out   K n z b × D out x b K n x b I n x b - D out x b K n x b I n x b D out x b K n x b I n y b X fr - D in y b K n x b I n y b ,
X fr = 1 - R 12 1 - R 21 + 2 R 12 - R 21 1 - R 21   D in y b a I n y I n y ,

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