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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 4, Iss. 8 — Apr. 12, 1999
  • pp: 299–307
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Systematic diffuse optical image errors resulting from uncertainty in the background optical properties

Xuefeng Cheng and David A. Boas  »View Author Affiliations


Optics Express, Vol. 4, Issue 8, pp. 299-307 (1999)
http://dx.doi.org/10.1364/OE.4.000299


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Abstract

We investigated the diffuse optical image errors resulting from systematic errors in the background scattering and absorption coefficients, Gaussian noise in the measurements, and the depth at which the image is reconstructed when using a 2D linear reconstruction algorithm for a 3D object. The fourth Born perturbation approach was used to generate reflectance measurements and k-space tomography was used for the reconstruction. Our simulations using both single and dual wavelengths show large systematic errors in the absolute reconstructed absorption coefficients and corresponding hemoglobin concentrations, while the errors in the relative oxy- and deoxy- hemoglobin concentrations are acceptable. The greatest difference arises from a systematic error in the depth at which an image is reconstructed. While an absolute reconstruction of the hemoglobin concentrations can deviate by 100% for a depth error of ±1 mm, the error in the relative concentrations is less than 5%. These results demonstrate that while quantitative diffuse optical tomography is difficult, images of the relative concentrations of oxy- and deoxy-hemoglobin are accurate and robust. Other results, not presented, confirm that these findings hold for other linear reconstruction techniques (i.e. SVD and SIRT) as well as for transmission through slab geometries.

© Optical Society of America

1. Introduction

In this paper, we present absorption coefficient images of absorbing objects reconstructed from continuous-wave (CW) simulated reflectance data calculated for a single source and a two dimensional detector array with various errors in the background optical parameters as well as system noise. Then we apply two wavelengths to examine how these errors and noise affect the reconstruction of oxy- and deoxy-hemoglobin concentration images. We found that the absolute values of these parameters depend significantly on the systematic errors in the optical properties of the background, while the relative changes are in some instances independent of the background error. The forward algorithm we used for simulating the data is the nth order Born approximation. The inverse solution is based on the k-space reconstruction algorithm in reflectance mode [12

12. X. Cheng and D. A. Boas, “Diffuse optical reflectance tomography with continuous-wave illumination,” Opt. Express 3, 118–123 (1998)http://epubs.osa.org/oearchive/source/5663.htm. [CrossRef] [PubMed]

]. We have also performed tests using other linear reconstruction techniques (SVD [13

13. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambirdge Univ. Press, New York,1988) Ch2 p52.

] and SIRT [2

2. M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995). [CrossRef]

, 14

14. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York,1988).

]) in reflectance mode and transmission mode through a slab. The results (not shown) are consistent with those discussed for the k-space algorithm in reflectance mode. We thus draw the general conclusion that the results presented here have general applicability to all linear algorithms and geometries.

2. Forward algorithm and K-space theory

The diffusion of light through a highly scattering medium on a macroscopic level can be treated as a scalar wave, called a diffuse photon density wave (DPDW), whether or not the source of light is intensity modulated [15–17

15. M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992). [CrossRef]

]. In a medium with spatially uniform optical properties, this wave propagates unobstructed. However, the presence of spatial variations in the optical properties results in a perturbation of the DPDW, which is accurately modeled as a scattered DPDW [18

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

]. Therefore, light that diffuses through a heterogeneous medium can be described as a superposition of the incident intensity wave and those waves scattered from local variations in the optical properties.

The first Born approximation of the photon diffusion equation indicates that the wave scattered from spatial variations in the absorption coefficient is given by

ϕsc1(rd)=drϕinc(rs,r)vδμa(r)DG(r,rd).
(1)

ϕsc (r d) is the scattered photon fluence measured at the detector at position r d, such that what we measure at the detector is ϕ(r s ,r) = ϕinc (r s, r)+ϕsc (r s, r). ϕinc (r s, r) is the incident fluence at position r generated in the medium by a source at position r s. G(r, r d) is the Green’s function solution of the diffusion equation for the given medium. Both ϕinc (r s, r) and G(r, r d) are calculated given the spatially uniform, average background optical properties of the medium μs ′, the reduced scattering coefficient, and μa , the absorption coefficient. D = v/(3μs ′) is the photon diffusion coefficient [19

19. K. Furutsu, “Diffusion equation derived from the space-time transport equation in anisotropic random media,” J.Math.Phys. 24, 765–777 (1997).

, 20

20. T. Durduran, B. Chance, A. G. Yodh, and D. A. Boas, “Does the photon diffusion coefficient depend on absorption?,” J. Opt. Soc. Am. A 14, 3358–3365 (1997). [CrossRef]

], v is the speed of light in the medium, and δμa (r) is the spatial variation of the absorption coefficient from the average background value ma at position r. The second and nth order approximation are given by

ϕsc2(rd)=dr(ϕinc(rs,r)+ϕsc1(r))vδμa(r)DG(r,rd)
(2)
ϕscn(rd)=dr(ϕinc(rs,r)+ϕscn1(r))vδμa(r)DG(r,rd).
(3)

Diffuse optical tomography is essentially based on reconstructing an image of δμa (r) from measurements of ϕsc (r d) at multiple source, r s, and detector, r d, positions. Recognizing that eq. (1) resembles the Sommerfeld - Kirchoff scalar diffraction integral, we can use “diffraction tomography” to reconstruct an image of δμa (r) from measurements of ϕsc (r d) in a plane [21–23

21. C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Opt. Express 1, 6–11 (1997).http://epubs.osa.org/oearchive/source/1884.htm [CrossRef] [PubMed]

].

We start by noting that for imaging applications where measurements are made in reflection mode, we can usually approximate the highly scattering medium as semi-infinite. In this case, placing the free-space boundary in the xy-plane at z = 0, using a collimated pencil beam incident on the medium at x = y = z = 0, and making measurements in the same plane with the source fixed, eq. (1) can be re-written as

ϕ˜scn(ωx,ωy)=G˜(ωx,ωy,z)A(ωx,ωy,z)dz,
(4)

where we have taken the Fourier transform with respect to x, y, xd, and yd, and the object function A(ωx , ωy , z), which we wish to reconstruct an image of, is given by

A(ωx,ωy,z)=dxdyϕinc(rs,x,y,z)vδμa(x,y,z)Dexp(iωxx+iωyy)
(5)

An analytic inversion of eq. (1) can now be obtained by assuming that the object function is zero everywhere except in a slice of width h at position z. Thus, the analytic solution for the spatially varying absorption coefficient becomes

δμa(x,y,z)=Dhϕinc(x,y,z)FT1[ϕ˜scn(ωx,ωy)G˜(ωx,ωy,z)]
(6)

This k-space reconstruction method is described in more detail by others [21–23

21. C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Opt. Express 1, 6–11 (1997).http://epubs.osa.org/oearchive/source/1884.htm [CrossRef] [PubMed]

]. Figure 1 shows reconstructed absorption coefficient images of an absorbing object at various depths when there is no error in the background optical properties and no noise in the measurement. Simulation details are given below in the simulations section below. We see that the image is best focused at the exact depth position of the object. This suggests that the k-space algorithm is capable of pseudo-3D reconstruction, as described in more detail in [12

12. X. Cheng and D. A. Boas, “Diffuse optical reflectance tomography with continuous-wave illumination,” Opt. Express 3, 118–123 (1998)http://epubs.osa.org/oearchive/source/5663.htm. [CrossRef] [PubMed]

]. This feature, however, is not as clear when there is noise in the measurement and the background

Figure 1. Images reconstructed at various depths. Each image is 6 × 6 cm.

3. Spectroscopy for Oxy- and Deoxy-hemoglobin

μa(λ)=εHbo(λ)[HbO]+εHb(λ)[Hb]
(7)

where εHbO and εHb are the extinction coefficients of oxy- and deoxy-hemoglobin respectively and [HbO] and [Hb] are the corresponding concentrations. Below we represent the total hemoglobin concentration as [HbT] = [HbO] + [Hb] and the oxygen saturation as SO2 = [HbO] / [HbT]. The extinction spectra of hemoglobin are published in the literature [24–26

24. M. Cope, “The Development of a Near-Infrared Spectroscopy System and Its Application for Noninvasive Monitoring of Cerebral Blood and Tissue Oxygenation in the Newborn Infant,” University College London (1991).

] and are shown in figure 2. Measurements of the absorption coefficient at two or more wavelengths are necessary in order to solve for [HbO] and [Hb] as shown in eq. (8a) and eq. (8b).

Δ[Hb]=εHboλ2δμaλ1εHboλ1δμaλ2εHbλ1εHboλ2εHbλ2εHboλ1
(8a)
Δ[HbO]=εHboλ1δμaλ2εHboλ2δμaλ1εHbλ1εHboλ2εHbλ2εHboλ1
(8b)

where Δ[HbO] and Δ[Hb] refer to the change in oxy- and deoxy- hemoglobin concentration corresponding to the reconstructed absorption coefficient, δμaλ, at each wavelength. In this paper, we used wavelengths of 780 nm and 830 nm.

Figure 2. Absorption coefficients of hemoglobin (50 μmole/liter) and H2O.

4. Simulations

In our simulations, the background is a semi-infinite turbid media with a scattering coefficient of 10 cm-1 and an absorption coefficient of 0.05 cm-1. The object has dimensions of 0.5 by 0.5 by 0.2 cm located at a depth of 2 cm. It has the same scattering coefficient as the background but a different absorption coefficient. For our single wavelength simulation, the object has an absorption coefficient of 0.1cm-1. For the two-wavelength simulation, the background has an [HbT] of 100 μmole/liter and SO2 = 80%. For the object, [HbT] = 120 μmol/liter and SO2 = 60%. We used a wavelength of 780 nm and 830 nm in the simulation. From eq. (7), the background absorption coefficient is found to be 0.078 cm-1 and 0.092 cm-1 at 780 nm and 830 nm respectively and the absorption perturbation at the two wavelengths is 0.024 cm-1 and 0.012 cm-1 respectively. The measurement geometry consists of an array of 12 by 12 detectors uniformly distributed over an area of 6 by 6 cm. The light source is located at the center of the probe.

The procedure of the simulation is as follows: First, we calculate the diffuse reflectance using the fourth order Born approximation for which the object was divided into 5 by 5 by 3 voxels. Then we add Gaussian noise to this calculation to generate measurement noise. In reconstructing a 2D image using the k-space algorithm we calculate the Green’s function using systematically incorrect scattering and absorption coefficients for the background, as well as a systematically incorrect object depth. We then determine the image errors by comparing the reconstructed image value with that of the known object.

5. Results and discussions

In figure 3 we plot the systematic image errors as a function of the depth position at which the image is reconstructed. It shows that the image value is very sensitive to errors in the depth: 1 mm errors in the depth will cause between a 50 and 100% error in the reconstructed absorption coefficient. Underestimating the depth results in a smaller systematic error than overestimating the depth as the negative systematic errors cannot be greater than -100%. These large systematic errors originate from the exponentially decaying nature of the diffuse photon density wave inside the turbid media. This suggests that the systematic image errors will have a nonlinear relationship with the errors in the background absorption and scattering coefficient.

Figure 3. Systematic errors in the reconstructed image values of the object resulting from the errors in the depth at which the image is reconstructed.

Figure 4 shows the systematic image errors resulting from the errors in the background absorption and scattering coefficients. We can see that 20% errors in the background absorption results in an ~25% error in the image value while 20% errors in the background scattering coefficient causes an ~15% error in the image value. We also notice that an underestimation of the background optical properties gives a smaller error than the same overestimation for the same reason as underestimating the depth.

Figure 4. Systematic errors in the reconstructed image values of the object resulting from the errors in the background absorption (red line) and scattering coefficient (black line).

Figure 5 shows the uncertainties in the image value caused by measurement noise. In figure 5a, we plot the image uncertainties as a function of the Gaussian noise magnitude, which is defined as the ratio of the standard deviation of the noise with the measured total reflectance. It indicates that 10-5% noise will cause ~5% uncertainty in the image value. In figure 5b, the Gaussian noise is fixed at 10-5% and the object depth is varied. We see that the image uncertainty rises significantly as the depth of the object increases. This means that imaging a deep absorbing object is more prone to uncertainty than imaging a shallow object. Although this result is obvious, it does give us a method for designing the measurement geometry to optimize the depth sensitivity for a given application. Note that for noise levels as small as 10-5% we obtain image uncertainties greater than 10%. Noise levels in practice are expected to be on the order of 0.1%. This extreme sensitivity to noise in our example arises for two reasons: 1) the object contrast is small (<25%) and 2) the measurement geometry is not optimal.

Figure 5. Uncertainties in the image value caused by Gaussian measurement noise. In (a) the depth of the object is 2 cm and the noise is varied. In (b) the depth is varied while the noise is held constant at 3×10-5 percent.

The above results show that the reconstructed image value is sensitive to the errors in the background optical properties, the depth at which the 2D reconstruction is performed, and the measurement noise. These results confirm the difficulty of quantitative diffuse optical tomography. In the following, we demonstrate that it is possible to reduce the image errors by reconstructing the relative absorption changes. A practical example is the reconstruction of perturbations in the concentrations of oxy- and deoxy-hemoglobin.

Figure 6. Systematic errors in the reconstructed Δ[Hb], Δ[HbO], and Δ[Hb]/Δ[HbO] resulting from errors in the depth of the image reconstruction.

6. Summary

Table 1 and Table 2 summarize the numerical results of our one-wavelength and two-wavelength simulations respectively. These results demonstrate that quantitative image reconstruction using linear algorithms need accurate information about the background optical properties and especially the depth of the object. Small noise in the measurement can cause large uncertainties in the image values and these uncertainties increase significantly as the object depth increases. The reconstruction of the relative ratio Δ[Hb]/Δ[HbO] is much less sensitive to the depth of the object than the absolute measures of Δ[Hb] and Δ[HbO]. The ratio is also less sensitive to errors in the background optical properties. This distinction between absolute and relative imaging will play an important role in the application of diffuse optical tomography to studying hemodynamics. Ultimately, these results motivate the development of 3D reconstruction algorithms to better localize object position and to better determine background optical properties in order to minimize the systematic errors addressed here.

Table 1. Errors and uncertainties in the images values resulting from the systematic errors of background optical properties, the depth position of the image and random system noise in one-wavelength simulation.

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Table 2. Image errors in the absolute and ratio reconstruction resulting from the systematic errors of background optical properties, the depth position of the image in two-wavelength simulation

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7. Acknowledgements

The authors gratefully acknowledge funding from NIH R29NS37053 and from CIMIT, the Center for Innovative Minimally Invasive Therapies. This research was funded in part by the US Army, under Cooperative Agreement No. DAMD17-99-2-9001. This publication does not necessarily reflect theposition or the policy of the Government, and no official endorsement should be inferred. We thank Tom Gaudette for his assistance in preparing the manuscript.

References and links

1.

A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995). [CrossRef]

2.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995). [CrossRef]

3.

“Trends in Optics and Photonics Series,” R. Alfano ed., Advances in Optical Imaging and Photon Migration,Orlando, FLA, (OSA,1996).

4.

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

5.

R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, and J. G. Aarnoudse, “Condensed Monte Carlo simulations for the description of light transport,” Appl. Opt. 32, 426–434 (1993). [CrossRef] [PubMed]

6.

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]

7.

S. R. Arridge, J. P. Kaltenbach, R. L. Barbour, and G. Muller ed., Bellingham, Wa, (Proc. SPIE,1993).

8.

S. R. Arridge and J. C. Hebden, “Optical Imaging in Medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–854 (1997). [CrossRef] [PubMed]

9.

A. J. Devaney, “Reconstruction tomography with diffractive wave-fields,” Inverse Problems 2, 161–183 (1986). [CrossRef]

10.

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: Simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996). [CrossRef]

11.

A. D. Klose and A. H. Hielscher, “A transport-theory based reconstruction algorithm for optical tomography,” B. Chance, R. Alfano, and B. Tromberg ed., SPIE BiOS99, San Jose, CA, (SPIE,1999). [CrossRef]

12.

X. Cheng and D. A. Boas, “Diffuse optical reflectance tomography with continuous-wave illumination,” Opt. Express 3, 118–123 (1998)http://epubs.osa.org/oearchive/source/5663.htm. [CrossRef] [PubMed]

13.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambirdge Univ. Press, New York,1988) Ch2 p52.

14.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York,1988).

15.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992). [CrossRef]

16.

J. M. Schmitt, A. Knuttel, and J. R. Knutson, “Interference of diffusive light waves,” J.Opt.Soc.Am.A 9, 1832 (1992). [CrossRef] [PubMed]

17.

J. B. Fishkin and 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]

18.

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

19.

K. Furutsu, “Diffusion equation derived from the space-time transport equation in anisotropic random media,” J.Math.Phys. 24, 765–777 (1997).

20.

T. Durduran, B. Chance, A. G. Yodh, and D. A. Boas, “Does the photon diffusion coefficient depend on absorption?,” J. Opt. Soc. Am. A 14, 3358–3365 (1997). [CrossRef]

21.

C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Opt. Express 1, 6–11 (1997).http://epubs.osa.org/oearchive/source/1884.htm [CrossRef] [PubMed]

22.

X. D. Li, T. Durduran, A. G. Yodh, B. Chance, and D. N. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–575 (1997). [CrossRef] [PubMed]

23.

X. D. Li, T. Durduran, A. G. Yodh, B. Chance, and D. N. Pattanayak, “Diffraction tomography for biomedical imaging with diffuse photon density waves: errata,” Opt. Lett. 22, 1198 (1997). [CrossRef] [PubMed]

24.

M. Cope, “The Development of a Near-Infrared Spectroscopy System and Its Application for Noninvasive Monitoring of Cerebral Blood and Tissue Oxygenation in the Newborn Infant,” University College London (1991).

25.

S. Wray, M. Cope, and D. T. Delpy, “Characteristics of the near infrared absorption spectra of cytochrome aa3 and hemoglobin for the noninvasive monitoring of cerebral oxygenation.,” Biochim Biophys Acta 933, 184–192 (1988). [CrossRef] [PubMed]

26.

S. J. Matcher, C. E. Elwell, C. E. Cooper, M. Cope, and D. T. Delpy, “Performance comparison of several published tissue near-infrared spectroscopy algorithms,” Anal. Biochem. 227, 54–68 (1995). [CrossRef] [PubMed]

OCIS Codes
(170.5280) Medical optics and biotechnology : Photon migration
(170.6960) Medical optics and biotechnology : Tomography

ToC Category:
Focus Issue: Biomedical diffuse optical tomography

History
Original Manuscript: February 25, 1999
Published: April 12, 1999

Citation
Xuefeng Cheng and David Boas, "Systematic diffuse optical image errors resulting from uncertainty in the background optical properties," Opt. Express 4, 299-307 (1999)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-8-299


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References

  1. A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (1995). [CrossRef]
  2. M. A. OLeary, D. A. Boas, B. Chance and A. G. Yodh, "Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography," Opt. Lett. 20, 426-428 (1995). [CrossRef]
  3. "Trends in Optics and Photonics Series," R. Alfano ed., Advances in Optical Imaging and Photon Migration, Orlando, FLA, (OSA, 1996).
  4. S. R. Arridge, M. Schweiger, M. Hiraoka and D. T. Delpy, "A finite element approach for modeling photon transport in tissue," Med.Phys. 20, 299-309 (1993). [CrossRef] [PubMed]
  5. R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel and J. G. Aarnoudse, "Condensed Monte Carlo simulations for the description of light transport," Appl. Opt. 32, 426-434 (1993). [CrossRef] [PubMed]
  6. 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]
  7. S. R. Arridge, J. P. Kaltenbach and R. L. Barbour, G. Muller ed., Bellingham, Wa, (Proc. SPIE, 1993).
  8. S. R. Arridge and J. C. Hebden, "Optical Imaging in Medicine: II. Modelling and reconstruction," Phys. Med. Biol. 42, 841-854 (1997). [CrossRef] [PubMed]
  9. A. J. Devaney, "Reconstruction tomography with diffractive wave-fields," Inverse Problems 2, 161-183 (1986). [CrossRef]
  10. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue and M. S. Patterson, "Optical image reconstruction using frequency-domain data: Simulations and experiments," J. Opt. Soc. Am. A 13, 253-266 (1996). [CrossRef]
  11. A. D. Klose and A. H. Hielscher, "A transport-theory based reconstruction algorithm for optical tomography," B. Chance, R. Alfano and B. Tromberg ed., SPIE BiOS99, San Jose, CA, (SPIE, 1999). [CrossRef]
  12. X. Cheng and D. A. Boas, "Diffuse optical reflectance tomography with continuous-wave illumination," Opt. Express 3, 118-123 (1998). http://epubs.osa.org/oearchive/source/5663.htm [CrossRef] [PubMed]
  13. W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambirdge Univ. Press, New York, 1988) Ch2 p52.
  14. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).
  15. M. A. OLeary, D. A. Boas, B. Chance and A. G. Yodh, "Refraction of diffuse photon density waves," Phys. Rev. Lett. 69, 2658-2661 (1992). [CrossRef]
  16. J. M. Schmitt, A. Knuttel and J. R. Knutson, "Interference of diffusive light waves," J. Opt. Soc. Am. A 9, 1832 (1992). [CrossRef] [PubMed]
  17. J. B. Fishkin and 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]
  18. D. A. Boas, M. A. OLeary, B. Chance and A. G. Yodh, "Scattering of diffuse photon density waves by spherical inhomogeneties within turbid media: analytic solution and applications," Proc. Natl. Acad. Sci. USA 91, 4887- 4891 (1994). [CrossRef] [PubMed]
  19. K. Furutsu, "Diffusion equation derived from the space-time transport equation in anisotropic random media," J.Math.Phys. 24, 765-777 (1997).
  20. T. Durduran, B. Chance, A. G. Yodh and D. A. Boas, "Does the photon diffusion coefficient depend on absorption?," J. Opt. Soc. Am. A 14, 3358-3365 (1997). [CrossRef]
  21. C. L. Matson, "A diffraction tomographic model of the forward problem using diffuse photon density waves," Opt. Express 1, 6-11 (1997). http://epubs.osa.org/oearchive/source/1884.htm [CrossRef] [PubMed]
  22. X. D. Li, T. Durduran, A. G. Yodh, B. Chance and D. N. Pattanayak, "Diffraction tomography for biochemical imaging with diffuse-photon density waves," Opt. Lett. 22, 573-575 (1997). [CrossRef] [PubMed]
  23. X. D. Li, T. Durduran, A. G. Yodh, B. Chance and D. N. Pattanayak, "Diffraction tomography for biomedical imaging with diffuse photon density waves: errata," Opt. Lett. 22, 1198 (1997). [CrossRef] [PubMed]
  24. M. Cope, "The Development of a Near-Infrared Spectroscopy System and Its Application for Noninvasive Monitoring of Cerebral Blood and Tissue Oxygenation in the Newborn Infant," University College London (1991).
  25. S. Wray, M. Cope and D. T. Delpy, "Characteristics of the near infrared absorption spectra of cytochrome aa3 and hemoglobin for the noninvasive monitoring of cerebral oxygenation.," Biochim Biophys Acta 933, 184-192 (1988). [CrossRef] [PubMed]
  26. S. J. Matcher, C. E. Elwell, C. E. Cooper, M. Cope and D. T. Delpy, "Performance comparison of several published tissue near-infrared spectroscopy algorithms," Anal. Biochem. 227, 54-68 (1995). [CrossRef] [PubMed]

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