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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 18 — Sep. 1, 2008
  • pp: 13643–13650
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Fast time-domain diffuse optical tomography using pseudorandom bit sequences

Weirong Mo and Nanguang Chen  »View Author Affiliations


Optics Express, Vol. 16, Issue 18, pp. 13643-13650 (2008)
http://dx.doi.org/10.1364/OE.16.013643


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Abstract

We report a novel time-domain diffuse optical tomography to determine the optical properties in a faster speed than the conventional ones. Instead of using the ultrashort pulse laser, a 2.5 Gbps pseudorandom bit sequence is used to modulate the near-infrared light for tissue-like phantom illumination. The time-resolved signal can be retrieved very quickly by demodulation with the reference signal. The system impulse response has a full width at half maximum around 800 picoseconds and the 2-dimentional maps of optical properties can be obtained within a few seconds. The high signal-to-noise ratio and the environmental illumination insensitivity warrant a great potential for applications in clinical noninvasive breast cancer detection.

© 2008 Optical Society of America

1. Introduction

In this article, we report a novel time-domain DOT design which can achieve faster data acquisition speed than the conventional ones. The calibration results show a system impulse response with a full width at half maximum (FWHM) about 800 ps and a rapid data acquisition speed (<5 s) for 9 sources and 4 detectors. In addition, the design features a relative insensitivity to the environmental illumination as well as a high signal-to-noise ratio.

2. Method

2.1. Architecture

In this article, we report a novel sub-nanosecond PRBS-based design for the first time. The schematic is shown in Fig. 1. A PRBS analyzer transmitter (ME3620A, Anritsu) continuously generates a high bit rate (2488 Mbit/s) PRBS signal. The pattern length of 29-1 bits leads to a signal repetition time of 205.78 ns. A radio frequency (RF) power splitter (ZFSC-2-2500+, mini-circuits) evenly distributes this PRBS signal into branch A and branch B. Branch A is reserved as a reference signal for demodulation. Branch B is delayed in steps of 40 ps when going through a programmable digital delay line (PDDL5, GigaBaudics). A low power (5 mW) near-infrared (NIR) laser with a wavelength of 780 nm is used as the light source. An external intensity modulator (AZ-OK1-10-PFU-SFU-780-S, EOSPACE) modulates the intensity of this NIR light with the PRBS signal carried by branch B. The modulated light is multiplexed into 9 source fibers and guided to illuminate the tissue-like phantom from a handheld probe. Figure 2(a) shows the geometry of the handheld probe. The 9 fibers are switched on and off sequentially. The 4 fiber-bundles on the probe collect the light emitted from the phantom surface and are individually coupled to 4 high speed avalanche photodiodes (APDs, S2383-20, Hamamatsu). Each APD has a cut-off frequency around 650 MHz. The optoelectronic signal is amplified 42-dB by the RF amplifier before feeding into the mixer (SYM-2500, mini-circuits). The mixer demodulates this signal with respect to the reference signal (branch A). Since the mixer actually is not an ideal multiplier, the down-converted intermediate frequency (IF) signal contains not only the desired demodulation signals but also the undesired DC offset and the system noise. In order to get rid of the DC offset and reduce the noise level, a square wave with a frequency of 2.5 kHz is superimposed on the modulating PRBS signal and fed into the intensity modulator. Consequently, the demodulation results yield an IF signal of 2.5 kHz, which is further amplified 60-dB by a series of operational amplifiers. A personal computer acquires this signal via a data acquisition card (DAQ, PCI-6251, National Instruments). The sampling rate is 250 kS/s and the integration time is 4 ms for each temporal sampling point. Similarly, by correlating this signal with another 2.5 kHz square wave, the DC offset can be thoroughly eliminated and the system noise level is significantly reduced. The amplitude of the 2.5 kHz IF signal represents the detected light intensity at a specific time delay in response to a pulsed illumination. The entire temporal point spread function (TPSF) is obtained by scanning the time delay at a scan interval as small as 40 ps. In order to compromise the data length for the data acquisition speed, all TPSF curves were truncated at 5120 ps, or 128 temporal sampling points with a scan interval of 40 ps. The acquisition time for a TPSF of 128 points is 128×4 ms=512 ms. For 9 sources switched on sequentially, it needs a total of 4608 ms to acquire all the waveforms. The dead-time for 9 fibers switching is 20 ms×9=180 ms. Thus one complete scan can be finished in 5 s. The data acquisition time can be significantly reduced with a larger scan interval of, e.g., 200 ps. A scan interval of 40 ps is used in the current configuration as the acquisition speed is acceptable for our phantom experiments and the oversampled data set can be used explore different post-processing approaches such as deconvolution and Fourier analysis.

Fig. 1. Schematic of the fast time-domain DOT. The thick arrows represent the PRBS transmission path. The dashed arrows represent the optical signal path and the thin arrows represent the control signal.

It should be noted that the APDs are temperature sensitive components. The optoelectronic efficiency will degrade as the component’s temperature increases. In order to keep the photon sensitivity stable over a long running time, 4 temperature controllers (TEC) are used to limit the temperature fluctuation of the APDs within ±0.03 C°. In addition, because the performance of the intensity modulator is subject to its inherent photorefractive effect, the modulation depth and the amplitude of the measured TPSFs may not be repeatable. To keep the modulation depth constant, an automatic bias tracking and control module based on the addictive dither technique is implemented to set the working point (i.e. the positive quadratic point) of the intensity modulator ahead of each scanning.

2.2. Calibration

In order to characterize the system’s impulse response (SIR), we place a piece of diffusive white paper at an 18 cm distance from the handheld probe. Since the light beam from an optical fiber is diverging and the reflection from the white paper is diffusive, the 18 cm distance chosen for the SIR measurement provides appropriate attenuation of the light intensity arriving at the photodetectors. Figure 3(a) representatively plots one of 36 SIRs together with the theoretical prediction, i.e., the PRBS autocorrelation. The measured SIR has a FWHM approximately of 800 ps and the rise time (10%–90%) is about 600 ps. These values are slightly larger than the theoretical prediction with FWHM of 402 ps and the rise time of 360 ps. This signal integrity degradation is mainly caused by the junction capacitance (≈6 pF) of the APDs and the insufficient bandwidth of the components along the PRBS signal transmission path. If all components in the system have an adequate bandwidth, the shape of the measured SIRs should be approaching to the triangle shape of prediction. The error bars in Fig. 3(a) represent the standard deviation of the measured SIR among different channels, which suggests a very good uniformity.

Fig. 2. (a) Configuration of the reflection-mode handheld probe. The 4 big circles represent the detection fiber-bundles and the 9 small circles represent the source fiber. (b) Illustration of the experimental setup with the absorbing targets included. P1 and P2 represent the target positions. The coordinates of P1[x, y, z] and P2[x, y, z] are [0, 0, 2]cm and [1.5, 0, 2]cm, respectively.

3. Experiments and results

Fig. 3. (a) The measured SIRs (blue solid line) vs. the theoretical prediction (red dashed line). The measured SIR is slightly wider than the predicted one. The error bars represent the standard deviation of the measured SIR. (b) The homogeneous TPSF (blue solid line) obtained from the homogeneous Lipofundin solution and the heterogeneous TPSF (red dashed line) obtained from the heterogeneous solution for a channel where the source-to-detection distance is 3.3 cm. (c) The comparison of the TPSFs with the room light on (red circles) and off (blue line). The source-to-detector distance is 3.3 cm.

4. Image reconstruction

For the image reconstruction, we only concern the distribution of the absorption coefficient. As the handheld probe works in a reflection mode, a semi-infinite boundary condition is used together with the diffusion equation as the forward model for photon migration through the tissue-air interface. Born approximation based perturbation method is adopted to calculate the Jacobian Matrix that relates the measured perturbations in TPSF to local changes in the absorption coefficient linearly [23

23. 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, 2331–2337 (1996). [CrossRef]

, 24

24. D. A. Boas, T. Gaudette, and S. R. Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography,” Opt. Express 8, 263–270 (2001), www.opticsexpress.org/abstract.cfm?URI=OPEX-8-5-263. [CrossRef] [PubMed]

, 25

25. J. Wu, “Convolution picture of the boundary conditions in photon migration and its implications in time-resolved optical imaging of biological tissues,” J. Opt. Soc. Am. A 14, 280–287 (1997). [CrossRef]

]. To reconstruct the spatial variation of the optical properties, we applied a simultaneous iterative reconstruction technique (SIRT), a slightly modified algorithm of algebraic reconstruction technique (ART) to achieve a fast imaging speed. In addition,

Table 1. Analysis of the reconstructed absorption coefficient µa.

table-icon
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the positivity constraint is imposed on the absorption heterogeneity during iteration. Figure 4 illustrates the two-dimensional images of the absorption coefficient. Each figure covers a cross sectional area of 50 mm×25 mm.

Fig. 4. The reconstructed image of the absorption coefficient based on the SIRT. (a–c) represent the cross sectional view where y equals to 0, 0.25 and 0.5 cm, respectively. The target (absorber) is located at position P1 [0.0, 0.0, 2.0]. (d–f) represent the cross sectional view where y equals to 0, 0.25 and 0.5 cm, respectively. The target is shifted 1.5 cm along x-axis from P1 to P2 [1.5, 0.0, 2.0].

The target is placed at P1(0.0, 0.0, 2.0) and then P2(1.5, 0.0, 2.0) in the experiments. The depth remains the same at 2 cm. Figure 4(a–c) shows the case where the target is located at P1 and Fig. 4(d–f) shows the case where the target is moved to P2. From these figures, we can rather accurately determine the target’s position and size. For example, in the first case (Fig. 44 (a–c)), the recovered position of the absorber is ([x, y, z]=[0.013, 0, 1.918]cm), which is slightly different from the true position ([x, y, z]=[0, 0, 2]cm). In the second case (Fig. 44 (d–f)), the recovered position of the absorber is ([x, y, z]=[1.506, 0, 1.958] cm), which is slightly different from the true position ([x, y, z]=[1.5, 0, 2] cm). Table 1 provides the quantitative analysis of the reconstructed absorption coefficient. The mean and standard deviation are calculated with the reconstructed distribution within the true target region. It is evident that the reconstructed values are rather close to the true value. The total image reconstruction took about 3–5 s on a MATLAB platform including core 2 dual CPU at 1.8 GHz and 2 GB RAM. It is worthy of notice that our handheld probe is optimized for 2D cross-sectional imaging and 2.5D imaging in which a few slices in the Y-dimension can be obtained. For 3D imaging, we need to reconfigure the source and detector deployment. More sources and detectors may be necessary for high quality 3D imaging.

5. Summary

To summarize, we have developed a fast time-domain DOT prototype which features many advantages over the conventional time-domainDOT systems. The high speed on TPSF acquisition and system SNR, relatively simple system structure and low cost are also realized. To further strengthen these advantages, we expect to develop a real-time high resolution DOT system in the near future. We would like to thank the following for their funding support: Office of Life Science (R397-000-615-712), National University of Singapore and A*STAR/SERC (P-052 101 0098).

References and links

1.

D. R. Leff, O. Warren, L. C. Enfield, A. P. Gibson, T. Athanasiou, D. K. Pattern, J. C. Hebden, G. Z. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast - a systematic review,” Breast Cancer Res. Treat. 108, 9–22 (2008). [CrossRef]

2.

L. C. Enfield, A. P. Gibson, N. L. Everdell, D. T. Delpy, M. Schweiger, S. R. Arridge, C. Richardson, M. Keshtgar, M. Douek, and J. C. Hebden, “Three-dimensional time-resolved optical mammography of the uncompressed breast,” Appl. Opt. 46, 3628–3638 (2007). [CrossRef] [PubMed]

3.

A. Cerussi, N. Shah, D. Hsiang, A. Durkin, J. Butler, and B. J. Tromberg, “In vivo absorption, scattering, and physiologic properties of 58 malignant breast tumors determined by broadband diffuse optical spectroscopy,” J. Biomed. Opt. 11, 044005 (2006). [CrossRef] [PubMed]

4.

J. C. Hebden and T. Austin, “Optical tomography of the neonatal brain,” Euro. Rad. 17, 2926–2933 (2007). [CrossRef]

5.

T. Austin, J. C. Hebden, A.P. Gibson, G. Branco, R. Yusof, S. R. Arridge, J. H. Meek, D.T Delpy, and J. S. Wyatt, “Three-dimensional optical imaging of blood volume and oxygenation in the preterm brain,” Neuroimage , 31, 1426–1433 (2006). [CrossRef] [PubMed]

6.

M. A. Franceschini, D. K. Joseph, T. J. Huppert, S. G. Diamond, and D. A. Boas, “Diffuse optical imaging of the whole head,” J. Biomed. Opt. 11, 054007 (2006). [CrossRef] [PubMed]

7.

G. Gulsen, O. Birgul, B. Xiong, and O. Nalcioglu, “Design and implementation of a multi-Frequency diffuse optical tomography (MF-DOT) System,” J. Biomed. Opt. 11, 014020 (2006). [CrossRef] [PubMed]

8.

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh. “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235–247 (2003). [CrossRef] [PubMed]

9.

J. M. Yang, Y. H. Han, G. Yoon, B. S. Ahn, B. C. Lee, and K. S. Soh, “In vivo 783-channel diffuse reflectance imaging system and its application,” Appl. Opt. 46, 5991–6003 (2007). [CrossRef] [PubMed]

10.

M. Schweiger, A. Gibson, and S. R. Arridge, “Computational aspectcts of diffuse optical tomography,” Comput. Opt. 5, 33–41 (2001).

11.

F. Gao, H. J. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41, 778–791 (2002). [CrossRef] [PubMed]

12.

W. Becker, A. Bergmann, A. Gibson, N. Everdell, D. Jennions, M. Schweiger, A. R. Arridge, and J. C. Hebden, “Multi-dimensional time-correlated single photon counting applied to diffuse optical tomography,” Proc. SPIE 5693, 34–42 (2005). [CrossRef]

13.

F. Schmidt, M. Fry, E. M. C. Hillman, J. C. Hebden, and D. T. Delpy, “A 32-channel time-resolved instrument for medical optical tomography,” Rev. Sci. Instrum. 71, 256–265 (2000). [CrossRef]

14.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, and M. Takada, “Multi-channel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595–3602 (1999). [CrossRef]

15.

H. Zhao, F. Gao, Y. Tanikawa, K. Homma, and Y. Yamada, “Time-resolved diffuse optical tomographic imaging for the provision of both anatomical and functional information about biological tissue,” Appl. Opt. 44, 1905–1916 (2005). [CrossRef] [PubMed]

16.

C. V. Zint, W. Uhring, M. Torregrossa, B. Cunin, and P. Poulet, “Streak camera: a multidetector for diffuse optical tomography,” Appl. Opt. 42, 3313–3320 (2003). [CrossRef] [PubMed]

17.

J. C. Hebden, D. J. Hall, M. Firbank, and D. T. Delpy, “Time-resolved optical imaging of a solid tissue-equivalent phantom,” Appl. Opt. 34, 8038–8047 (1995). [CrossRef] [PubMed]

18.

N. G. Chen and Q. Zhu, “Time-resolved optical measurements with spread spectrum excitations,” Opt. Lett. 27, 1806–1808 (2002). [CrossRef]

19.

N. G. Chen and Q. Zhu, “Time-resolved diffusive optical imaging using pseudo-random bit sequences,” Opt. Express 11, 3445–3454 (2003), http://www.opticsexpress.org/abstract.cfm?uri=OE-11-25-3445. [CrossRef] [PubMed]

20.

W. F. Cheong, S. A. Prohl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990). [CrossRef]

21.

J. C. Hebden, B. D. Price, A. P. Gibson, and G. Royle, “A soft deformable tissue-equivalent phantom for diffuse optical tomography,” Phys. Med. Biol. 51, 5581–5590 (2006). [CrossRef] [PubMed]

22.

M. Firbank, M. Oda, and D. T. Delpy, “An improved design for a stable and reproducible phantom material for use in near-infrared spectroscopy and imaging.” Phys. Med. Biol. 40, 955–961 (1995). [CrossRef] [PubMed]

23.

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, 2331–2337 (1996). [CrossRef]

24.

D. A. Boas, T. Gaudette, and S. R. Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography,” Opt. Express 8, 263–270 (2001), www.opticsexpress.org/abstract.cfm?URI=OPEX-8-5-263. [CrossRef] [PubMed]

25.

J. Wu, “Convolution picture of the boundary conditions in photon migration and its implications in time-resolved optical imaging of biological tissues,” J. Opt. Soc. Am. A 14, 280–287 (1997). [CrossRef]

OCIS Codes
(170.3890) Medical optics and biotechnology : Medical optics instrumentation
(170.5280) Medical optics and biotechnology : Photon migration
(170.6920) Medical optics and biotechnology : Time-resolved imaging
(170.6960) Medical optics and biotechnology : Tomography

ToC Category:
Medical Optics and Biotechnology

History
Original Manuscript: June 23, 2008
Revised Manuscript: August 12, 2008
Manuscript Accepted: August 18, 2008
Published: August 20, 2008

Virtual Issues
Vol. 3, Iss. 11 Virtual Journal for Biomedical Optics

Citation
Weirong Mo and Nanguang Chen, "Fast time-domain diffuse optical tomography using pseudorandom bit sequences," Opt. Express 16, 13643-13650 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-13643


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References

  1. D. R. Leff, O. Warren, L. C. Enfield, A. P. Gibson, T. Athanasiou, D. K. Pattern, J. C. Hebden, G. Z. Yang, and A. Darzi, "Diffuse optical imaging of the healthy and diseased breast - a systematic review," Breast Cancer Res. Treat. 108, 9-22 (2008). [CrossRef]
  2. L. C. Enfield, A. P. Gibson, N. L. Everdell, D. T. Delpy, M. Schweiger, S. R. Arridge, C. Richardson, M. Keshtgar, M. Douek, and J. C. Hebden, "Three-dimensional time-resolved optical mammography of the uncompressed breast," Appl. Opt. 46, 3628-3638 (2007). [CrossRef] [PubMed]
  3. A. Cerussi, N. Shah, D. Hsiang, A. Durkin, J. Butler, and B. J. Tromberg, "In vivo absorption, scattering, and physiologic properties of 58 malignant breast tumors determined by broadband diffuse optical spectroscopy," J. Biomed. Opt. 11, 044005 (2006). [CrossRef] [PubMed]
  4. J. C. Hebden and T. Austin, "Optical tomography of the neonatal brain," Euro. Rad. 17, 2926-2933 (2007). [CrossRef]
  5. T. Austin, J. C. Hebden, A. P. Gibson, G. Branco, R. Yusof, S. R. Arridge, J. H. Meek, D. T. Delpy, and J. S. Wyatt, "Three-dimensional optical imaging of blood volume and oxygenation in the preterm brain," Neuroimage  31, 1426-1433 (2006). [CrossRef] [PubMed]
  6. M. A. Franceschini, D. K. Joseph, T. J. Huppert, S. G. Diamond, and D. A. Boas, "Diffuse optical imaging of the whole head," J. Biomed. Opt. 11, 054007 (2006). [CrossRef] [PubMed]
  7. G. Gulsen, O. Birgul, B. Xiong, and O. Nalcioglu, "Design and implementation of a multi-Frequency diffuse optical tomography (MF-DOT) System," J. Biomed. Opt. 11, 014020 (2006). [CrossRef] [PubMed]
  8. J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh. "Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging," Med. Phys. 30, 235-247 (2003). [CrossRef] [PubMed]
  9. J. M. Yang, Y. H. Han, G. Yoon, B. S. Ahn, B. C. Lee, and K. S. Soh, "In vivo 783-channel diffuse reflectance imaging system and its application," Appl. Opt. 46, 5991-6003 (2007). [CrossRef] [PubMed]
  10. M. Schweiger, A. Gibson, and S. R. Arridge, "Computational aspectcts of diffuse optical tomography," Comput. Opt. 5, 33-41 (2001).
  11. F. Gao, H. J. Zhao, and Y. Yamada, "Improvement of image quality in diffuse optical tomography by use of full time-resolved data," Appl. Opt. 41, 778-791 (2002). [CrossRef] [PubMed]
  12. W. Becker, A. Bergmann, A. Gibson, N. Everdell, D. Jennions, M. Schweiger, A. R. Arridge, and J. C. Hebden, "Multi-dimensional time-correlated single photon counting applied to diffuse optical tomography," Proc. SPIE 5693, 34-42 (2005). [CrossRef]
  13. F. Schmidt, M. Fry, E. M. C. Hillman, J. C. Hebden, and D. T. Delpy, "A 32-channel time-resolved instrument for medical optical tomography," Rev. Sci. Instrum. 71, 256-265 (2000). [CrossRef]
  14. H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, and M. Takada, "Multi-channel time-resolved optical tomographic imaging system," Rev. Sci. Instrum. 70, 3595-3602 (1999). [CrossRef]
  15. H. Zhao, F. Gao, Y. Tanikawa, K. Homma, and Y. Yamada, "Time-resolved diffuse optical tomographic imaging for the provision of both anatomical and functional information about biological tissue," Appl. Opt. 44, 1905-1916 (2005). [CrossRef] [PubMed]
  16. C. V. Zint,W. Uhring, M. Torregrossa, B. Cunin, and P. Poulet, "Streak camera: a multidetector for diffuse optical tomography," Appl. Opt. 42, 3313-3320 (2003). [CrossRef] [PubMed]
  17. J. C. Hebden, D. J. Hall, M. Firbank, and D. T. Delpy, "Time-resolved optical imaging of a solid tissue-equivalent phantom," Appl. Opt. 34, 8038-8047 (1995). [CrossRef] [PubMed]
  18. N. G. Chen and Q. Zhu, "Time-resolved optical measurements with spread spectrum excitations," Opt. Lett. 27, 1806-1808 (2002). [CrossRef]
  19. N. G. Chen and Q. Zhu, "Time-resolved diffusive optical imaging using pseudo-random bit sequences," Opt. Express 11, 3445-3454 (2003), http://www.opticsexpress.org/abstract.cfm?uri=OE-11-25-3445. [CrossRef] [PubMed]
  20. W. F. Cheong, S. A. Prohl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990). [CrossRef]
  21. J. C. Hebden, B. D. Price, A. P. Gibson, and G. Royle, "A soft deformable tissue-equivalent phantom for diffuse optical tomography," Phys. Med. Biol. 51, 5581-5590 (2006). [CrossRef] [PubMed]
  22. M. Firbank, M. Oda, and D. T. Delpy, "An improved design for a stable and reproducible phantom material for use in near-infrared spectroscopy and imaging." Phys. Med. Biol. 40, 955-961 (1995). [CrossRef] [PubMed]
  23. 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, 2331-2337 (1996). [CrossRef]
  24. D. A. Boas, T. Gaudette, and S. R. Arridge, "Simultaneous imaging and optode calibration with diffuse optical tomography," Opt. Express 8, 263-270 (2001), www.opticsexpress.org/abstract.cfm?URI=OPEX-8-5-263. [CrossRef] [PubMed]
  25. J. Wu, "Convolution picture of the boundary conditions in photon migration and its implications in time-resolved optical imaging of biological tissues," J. Opt. Soc. Am. A 14, 280-287 (1997). [CrossRef]

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