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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 3, Iss. 1 — Jan. 29, 2008
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Linear 3D reconstruction of time-domain diffuse optical imaging differential data: improved depth localization and lateral resolution

Juliette Selb, Anders M. Dale, and David A. Boas  »View Author Affiliations


Optics Express, Vol. 15, Issue 25, pp. 16400-16412 (2007)
http://dx.doi.org/10.1364/OE.15.016400


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Abstract

We present 3D linear reconstructions of time-domain (TD) diffuse optical imaging differential data. We first compute the sensitivity matrix at different delay gates within the diffusion approximation for a homogeneous semi-infinite medium. The matrix is then inverted using spatially varying regularization. The performances of the method and the influence of a number of parameters are evaluated with simulated data and compared to continuous-wave (CW) imaging. In addition to the expected depth resolution provided by TD, we show improved lateral resolution and localization. The method is then applied to reconstructing phantom data consisting of an absorbing inclusion located at different depths within a scattering medium.

© 2007 Optical Society of America

1. Introduction

In complement of traditional techniques such as functional Magnetic Resonance Imaging, Diffuse Optical Tomography (DOT) is emerging as a low-cost and portable method for noninvasive cerebral imaging [1

1. H. Obrig and A. Villringer, “Beyond the visible — imaging the human brain with light,” J. Cereb. Blood Flow Metab. 23, 1–18 (2002). [PubMed]

3

3. Special section on Optics in Neuroscience, J. Biomed.Opt.10 (2005).

]. Based on the measurement of diffuse near-infrared light attenuation through the scalp, skull and brain, DOT assesses local optical absorption due essentially to the concentrations in oxy- and deoxy-hemoglobin. Modeling of light propagation through the head and solving of the inverse problem enable imaging of total hemoglobin concentration and oxygenation in the brain.

Most DOT instruments used in neuroscience are continuous wave (CW) systems, but time-domain (TD) technology is a recent promising alternative with advantages compensating for its increased cost and difficulty of implementation. These advantages include absolute characterization of tissue optical properties [4

4. 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, 2331–36 (1989). [CrossRef] [PubMed]

,5

5. V. Ntziachristos, XH. Ma, A.G. Yodh, and B. Chance, “Multichannel photon counting instrument for spatially resolved near infrared spectroscopy,” Rev. Sci. Instrum. 70, 193–201 (1999). [CrossRef]

] (both absorption coefficient µa and reduced scattering coefficient µs’), depth resolution with single source-detector separation [6

6. J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, “Determining changes in NIR absorption using a layered model of the human head,” Phys. Med. Biol. 46, 879–896 (2001). [CrossRef] [PubMed]

,7

7. A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, “Timeresolved reflectance at null source-detector separation: improving contrast and resolution in diffuse optical imaging,” Phys. Rev. Lett. 95 (078101), 1–4, (2005).

], and better sensitivity to cortical activation [8

8. J. Selb, J.J. Stott, M.A. Franceschini, A.G. Sorenson, and D.A. Boas, “Improved sensitivity to cerebral dynamics during brain activation with a time-gated optical system: analytical model and experimental validation,” J. Biomed. Opt. 10, 011013 (2005). [CrossRef]

,9

9. B. Montcel, R. Chabrier, and P. Poulet, “Detection of cortical activation with time-resolved diffuse optical methods,” Appl. Opt. 44, 1942–1947 (2005). [CrossRef] [PubMed]

]. TD systems introduce short pulses of light into the tissue (up to tens to hundreds of picoseconds) and measure the temporal point spread function (TPSF) of photons exiting after propagation through the tissue.

One application of TD system is the determination of the absolute optical properties µa and µs’ of a tissue. One to several source-detector pairs are placed on the surface of the studied organ. The measured TPSFs [10

10. H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595–3602 (1999). [CrossRef]

,11

11. J. Swartling, J.S. Dam, and S. Andersson-Engels, “Comparison of spatially and temporally resolved diffuse-reflectance measurements systems for determination of biomedical optical properties,” Appl. Opt. 42, 4612–4620 (2003). [CrossRef] [PubMed]

], or moments of these TPSFs [12

12. M. Schweiger and S.R. Arridge, “Application of temporal filters to time resolved data in optical tomography,” Phys. Med. Biol. 44, 1699–1717 (1999). [CrossRef] [PubMed]

14

14. A. Liebert, H. Wabnitz, D. Grosenick, M. Möller, R. Macdonald, and H. Rinneberg, “Evaluation of optical properties of highly scattering media by moments of distributions of times of flight of photons,” Appl. Opt. 42, 5785–5790 (2003). [CrossRef] [PubMed]

] are fit nonlinearly with a model of light propagation in a homogeneous medium [4

4. 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, 2331–36 (1989). [CrossRef] [PubMed]

,11

11. J. Swartling, J.S. Dam, and S. Andersson-Engels, “Comparison of spatially and temporally resolved diffuse-reflectance measurements systems for determination of biomedical optical properties,” Appl. Opt. 42, 4612–4620 (2003). [CrossRef] [PubMed]

], a two-layer medium [15

15. A. Kienle, T. Glanzmann, G. Wagnières, and H. van den Bergh, “Investigation of two-layered turbid media with time-resolved reflectance,” Appl. Opt. 37, 6852–6862 (1998). [CrossRef]

,16

16. F. Martelli, S. Del Bianco, G. Zaccanti, A. Pifferi, A. Toricelli, A. Bassi, P. Taroni, and R. Cubeddu, “Phantom validation and in vivo application of an inversion procedure for retrieving the optical properties of diffusive layered media from time-resolved reflectance measurements,” Opt. Lett. 29, (2004). [CrossRef] [PubMed]

], a more realistic tissue-type segmentation of the head [17

17. A. H. Barnett, J.P. Culver, A.G. Sorensen, A. Dale, and D.A. Boas, “Robust inference of baseline optical properties of the human head with three-dimensional segmentation from magnetic resonance imaging,” Appl. Opt. 42, 3095–3108, (2003). [CrossRef]

], or a complete voxelized volume in the case of tomographic imaging [18

18. S.R. Hintz, D.A. Benaron, J.P. van Houten, J.L. Duckworth, F.W.H. Liu, S.D. Spilman, D.K. Stevenson, and W.-F. Cheong, “Stationary headband for clinical time-of-flight optical imaging at the bedside,” Photochem. Photobiol. 68, 361–369 (1999). [CrossRef]

,19

19. J.C. Hebden, A. Gibson, R. Yusof, N. Everdell, E.M.C. Hillman, D.T. Delpy, S.R. Arridge, T. Austin, J.H. Meek, and J.S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002) [CrossRef] [PubMed]

]. However, for functional studies, differential imaging between a baseline state and an activated state is sufficient, and a linear relation between changes in absorption and changes in intensity is generally assumed valid for the small absorption changes associated with brain function. In the backprojection method widely used for CW data reconstruction [20

20. A. Maki, Y. Yamashita, Y. Ito, E. Watanabe, Y. Mayanagi, and H. Koizumi, “Spatial and temporal analysis of human motor activity using noninvasive NIR topography,” Med. Phys. 22, 1997–2005 (1995). [CrossRef] [PubMed]

23

23. D.A. Boas, T.J. Gaudette, G. Strangman, X. Cheng, J.J.A. Marota, and J.B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76–90 (2001). [CrossRef] [PubMed]

], the change in intensity between each source and detector is translated into a local change in absorption. A two-dimensional image is obtained by interpolation of all local absorption changes. The method is based on a modified Beer-Lambert law where a differential path factor (DPF) accounts for larger propagation length than source-detector separation [24

24. D.T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, and J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988). [CrossRef] [PubMed]

]. Hiraoka et al. extended the concept to inhomogeneous media [25

25. M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S.R. Arridge, P. van der Zee, and D.T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993). [CrossRef] [PubMed]

], by introducing partial DPFs describing the optical pathlengths in different tissue types, hence taking into account the contribution from different layers to the change in attenuation. However, no depth information is available with single distance CW data, and both cerebral and superficial activations are projected on a single imaging plane.

This limitation of NIRS imaging in terms of depth resolution can be overcome with TD data, where depth information is contained in the photons’ time of flights. Steinbrink et al. applied the concept of partial DPFs to the time domain by introducing time-dependant mean partial pathlengths (TMPP) [6

6. J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, “Determining changes in NIR absorption using a layered model of the human head,” Phys. Med. Biol. 46, 879–896 (2001). [CrossRef] [PubMed]

]. They used a model of fifteen 2 mm thick layers, and performed Monte Carlo simulations to calculate the time-dependant sensitivity to each layer. They thus obtained a sensitivity matrix which they inverted by singular value decomposition. They could distinguish between extra and intra-cerebral signals during brain activation, with a single source-detector pair measurement. Liebert et al. used a similar method but computed the sensitivity of three moments of the TPSF — integrated intensity, mean time of flight, variance — to each layer [26

26. A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Möller, R. Macdonald, A. Villringer, and H. Rinneberg, “Time-resolved multidistance near-infrared spectroscopy of the adult head: intracerebral and extracerebral absorption changes from moments of distribution of times of flight of photons,” Appl. Opt. 43, 3037–3047 (2004). [CrossRef] [PubMed]

]. They showed depth-resolved time-course of local perfusion after dye bolus injection on healthy subjects and stroke patients [27

27. A. Liebert, H. Wabnitz, J. Steinbrink, M. Möller, R. Macdonald, H Rinneberg, A. Villringer, and H. Obrig, “Bed-side assessment of cerebral perfusion in stroke patients based on optical monitoring of a dye-bolus by time-resolved diffuse reflectance,” Neuroimage 24, 426–435 (2005). [CrossRef] [PubMed]

]. We used a simplified 3-layer model — scalp, skull, and brain — and showed that we could experimentally distinguish between superficial systemic signals and cerebral activation signals during a motor stimulus on a single source-detector pair [8

8. J. Selb, J.J. Stott, M.A. Franceschini, A.G. Sorenson, and D.A. Boas, “Improved sensitivity to cerebral dynamics during brain activation with a time-gated optical system: analytical model and experimental validation,” J. Biomed. Opt. 10, 011013 (2005). [CrossRef]

]. In all these studies, depth resolution has been shown, but no 3D imaging was implemented, since a single source-detector pair was used. The technique can be extended to 2D imaging with depth resolution by simple interpolation of all source-detector pair’s measurements [28

28. M. Kacprzak, A. Liebert, P. Sawosz, N. Zolek, and R. Maniewski, “Time-resolved optical imager for assessment of cerebral oxygenation”, J. Biomed. Opt. 12, 034019 (2007). [CrossRef] [PubMed]

]. However, this approach limits the lateral resolution to approximately the source-detector separation.

2. Reconstruction principle

In this section, we describe the formalism implemented for the image reconstructions.

2.1 Medium and probe geometry

For computational simplicity, the studied medium is modeled as a volume of 10 cm by 10 cm laterally, and 3 cm in depth, divided into nvox voxels of 2.5×2.5×2.5 mm3 (see Fig. 1(a)). The background optical properties were set to µs’=10 cm-1 and µa=0.1 cm-1, unless otherwise stated. The probe set on the surface of the medium has a square geometry of 4×4 sources and 3×3 detectors (source-detector separation 2.5 cm) as shown in Figs. 1(a) and 1(b). Only nearest neighbor source-detector pairs are taken into account for the reconstructions.

Fig. 1. (a) 3D representation of the modeled medium and probe. (b) Probe geometry. Three lateral locations of the inclusion commonly used in the following simulations are also displayed (c) Simulated TPSF for one source-detector pair, with the delay gates used in the reconstruction.

2.2 Forward problem

We model light propagation in the medium with the analytical solution of the diffusion equation for a semi-infinite homogeneous medium, calculated with the image source technique [4

4. 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, 2331–36 (1989). [CrossRef] [PubMed]

], and extrapolated-boundary condition [31

31. R.C. Haskell, L.O. Svaasand, T.T. Tsay, T.C. Feng, M.S. McAdams, and B.J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994). [CrossRef]

]. Figure 1(c) shows the resulting TPSF obtained for one source-detector pair, all source-detector pairs yielding identical TPSF’s since the medium is assumed semi-infinite and homogeneous. Since our experimental TD device described in Ref. 30 is based on a time-gated detection of the TPSF, we use in our simulations gated detection. The measurements consist in the intensity at nGates delay gates, integrated over the gate width wGate. The gates considered in our simulations are presented in Fig. 1(c). To comply with our experimental parameters [29

29. D.A. Boas, K. Chen K, D. Grebert, and M.A. Franceschini, “Improving the diffuse optical imaging spatial resolution of the cerebral hemodynamic response to brain activation in humans,” Opt. Lett. 29, 1506–1508 (2004). [CrossRef] [PubMed]

], each gate width was set to 300 ps and the delay between two gates to 500 ps.

Differential images are obtained using the normalized Born approximation for a change in absorption: ΔI/I0=AΔµa, where the sensitivity matrix A linearly relates the changes in the absorption coefficient to the changes in intensity. Δµa is the vector of absorption changes at each voxel, of length nvox, and ΔI/I0 is the vector of changes in the normalized measured intensity, of length the number of measurements nMeas=nSDnGates where nSD is the number of source-detector pairs and nGates the number of delay gates for each pair. In the time domain, the terms of the sensitivity matrix A can be computed by convolution of the direct and adjoint Green’s functions [32

32. S.R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. 34,7395–7409 (1995). [CrossRef] [PubMed]

]:

Ai,j=1G(rS(i),rD(i),τi)+G(rD(i),rj,τ)G(rj,rS(i),τiτ)dτ

where G(rS, rD, τ) is the time domain Green’s function (GF) solution of the diffusion equation at delay τ for a source at position rS and a detector at position rD, rj is the position of the jth voxel, and r S(i), rD(i) and τi are respectively the source position, detector position and delay of the ith measurement.

In practice, we first calculated the GFs solutions of the diffusion equation at each voxel of the medium in the frequency domain [32

32. S.R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. 34,7395–7409 (1995). [CrossRef] [PubMed]

], for each optode (source or detector), at 101 frequencies (0 to 20 GHz, frequency step 200 MHz). The frequency-domain solutions were then Fourier-transformed to yield the GFs in the time domain with a 25 ps time step. To obtain the sensitivity matrix for each source-detector pair at a specific delay τ, the source forward GF at time τ’ and the detector adjoint GF at time τ-τ’ were multiplied, the result summed for τ’ varying between 0 and τ, and then integrated over the width wGate of the detection gate.

Fig. 2. Sensitivity profiles of a single source-detector pair for the 8 delay gates presented in Fig. 1(c) in (a) a vertical plane along the source and detector and (b) a (x,y) plane located at depth 0.75 cm. The sensitivity is given per unit volume (cm3) and unit change in absorption (cm-1).

Note that alternative methods to compute the sensitivity matrix could be used, in particular analytical solutions in the time-domain for a perturbation, like described in Ref. [33

33. S. Carraresi, T.S.M. Shatir, F. Martelli, and G. Zaccanti, “Accuracy of a perturbation model to predict the effect of scattering and absorbing inhomogeneities on photon migration,” Appl. Opt. 40, 4622–4632 (2001). [CrossRef]

] for a transmission geometry.

Figure 2 shows profiles of the obtained sensitivity matrix for the 8 delay gates depicted in Fig. 1(c) (delays between 0.1 and 3.6 ns with 500 ps between two gates; wGate=300 ps). Figure 2(a) shows profiles of the sensitivity matrix along a source-detector vertical plane, and Fig.2(b) shows (x,y) profiles of the matrix at depth z=0.75 cm. A logarithmic grayscale was used to allow visualization of the large dynamic range of sensitivities of all 8 gates. As expected, sensitivities at longer delays go deeper inside the medium, and also probe a wider lateral region. From this additional information relative to traditional CW sensitivity profiles, we can expect both depth resolution and improved lateral resolution and localization.

2.3 Inverse Problem

From the measurements y=ΔI/I0 and the computed sensitivity matrix A, the reconstructed image x̂ is calculated by inversion of the sensitivity matrix: x̂=pAinvy, where pAinv is the pseudo-inverse of matrix A, computed with the following regularization:

pAinv=L1BT(BBT+αsmaxσy2)1,
(1)

where:

B=A L-1,

L=diag(diag (AT A+λ)) is used to scale the sensitivity matrix to act as a spatially varying regularization [34

34. J.P. Culver, A.M. Siegel, J.J. Stott, and D.A. Boas, “Volumetric diffuse optical tomography of brain activity,” Opt. Lett. 28, 2061–2063 (2003). [CrossRef] [PubMed]

], giving higher weight to voxels with lower sensitivity. It is a diagonal matrix of size nvox×nvox where each diagonal element is the aggregate squared sensitivity to the corresponding voxel. The coefficient λ=max(diag(ATA))/β enables a thresholding of the matrix in order not to give large weighting to voxels with very small sensitivity. The choice of factor β will be discussed in section 3.4 below.

α is a regularization parameter, set to 10-3 in the simulations unless stated otherwise.

smax=max[diag (BBT)]/max(σ2y),

σ2y is the measurement covariance matrix, assumed to be diagonal. In our simulations, we assumed a square root dependence of the noise on I with the intensity, and thus σ2y is inversely proportional to intensity. This regularization acts to penalize noisier measurements.

2.4. CW reconstructions

CW sensitivity matrix and data were simulated with the same model, by integration of TD data over all time steps. CW and TD reconstructions will be compared in Section 3 to assess the improvement offered by TD imaging.

3. Simulations: optimization, performance and comparison with CW

In this section, we assume a point-like change in absorption, simulate the corresponding measurement vector, and reconstruct the 3D map of the absorption changes. This gives us the imaging point spread function of our reconstruction algorithm and enables us to assess the performance of the method and compare it to a CW reconstruction scheme.

3.1 Reconstruction examples

Figure 3 shows examples of reconstructions for a point-like inclusion located at the lateral position 1, between source and detector, as defined in Fig. 1(a), and with a depth z varying between 0.5 cm and 3 cm by steps of 0.5 cm. Both CW and TD reconstructions are presented. The rendered volume shows the contour of 80% of the maximum in the reconstructed absorption change. The following parameters were used: 5 gates starting at delays 0.6, 1.1, 1.6, 2.1, and 2.6 ns with a width wGate=300 ps (gates 2 to 6 in Fig. 1(c)); β=20; α=10-3; signal to noise ratio =100 at the peak of the TPSF.

The following general observations can be made: the CW data always reconstruct the inclusion at the same depth. If a traditional Tikhonov regularization is used instead (i.e. if A is used in place of B in Eq.1), the CW reconstruction is pulled towards the surface (data not shown), where the sensitivity is maximum [35

35. D.A. Boas and A.M. Dale, “Simulation study of magnetic resonance imaging-guided cortically constrained diffuse optical tomography of the human brain function,” Appl. Opt. 44, 1957–1968 (2005). [CrossRef] [PubMed]

]. The effect of the spatially varying regularization matrix L is to force the reconstruction deeper under the surface. However, the reconstructed depth is unchanged with different actual depths for the CW data.

On the contrary, the TD method reconstructs the inclusion deeper as its actual depth increases. For the true inclusion at a depth of 1.5 and 2 cm, the reconstructed depth is actually slightly over-estimated, which results from the effect of the regularization matrix L. As the true inclusion gets deeper, the reconstructed depth becomes under-estimated (see inclusion at true depth 3 cm).

We also observe that the reconstructed volume at 80% of the maximum contrast is smaller for TD than for CW reconstructions, showing improved lateral resolution for this location of the inclusion.

More quantitative and systematic assessment of the effect of different parameters and of the improvement of TD over CW will be investigated in the following paragraphs.

Fig. 3. CW and TD reconstructions of a point-like inclusion located at position 1 and at a depth z varying between 0.5 cm and 3 cm. The volumes show the contour of 80% of the maximum reconstructed change in absorption.

3.2 Performance assessment

The reconstruction performances were assessed by a number of parameters. The location of the center of mass (COM) of the reconstructed inclusion was calculated by taking into account all voxels with a contrast above 80% of the maximum contrast in absorption: r COM=(∑i,Vox≥80% Riri)/(∑i,Vox≥80% Ri), where r i and Ri are respectively the position and absorption contrast of the ith voxel. We define the localization error as the distance between the true inclusion and the COM, both in depth and laterally. We call lateral resolution the contrast-weighted sum of the lateral distances to the COM, over all voxels with a contrast above 80% of the maximum contrast: Res(∑i,Vox≥80% Ri|ρi-ρCOM|)/(∑i,Vox≥80% RiRi), where ρi and ρCOM are the lateral positions of the ith voxel and the COM respectively.

3.3 Optimal number of gates

We studied the influence of the number of gates included in the reconstruction. Figure 4(a) shows the evolution of the reconstructed depth (depth of the COM) as a function of the true depth of an inclusion located at lateral position 1, for different number of gates included (starting from gate 1 on Fig. 1(c)). The reconstruction improves, more strikingly for deep inclusions, as more late gates are included, up to 6 gates, after which the reconstruction does not improve anymore as we include more noisy data. The first gate only brings minor improvement for superficial inclusion (data not shown), and does not contribute to the data reconstruction for deep inclusions. We tried other combinations (data not shown), and found that the best one with our parameters was 5 gates every 500 ps from 0.6 ns to 2.6 ns.

With these parameters, the inclusion can be reconstructed with a depth error under 15 % down to approximately 2.5 cm. The reconstructed depths of deeper inclusions continue to increase, but with larger errors, e.g. an inclusion at 3 cm is reconstructed at 2.3 cm (an underestimation of almost 25%).

Fig. 4. Reconstructed depth of the COM as a function of the inclusion true depth, for a pointlike inclusion in position 1, for (a) different delay gate combinations (β=20), and (b) different threshold coefficients in the spatially varying regularization matrix L (gates 2 to 6 in Fig. 1(c)).

3.4 Influence of the spatially varying regularization

By penalizing more the voxels with higher sensitivity, the L matrix enables us to reconstruct the inclusion better than a simple Tikhonov regularization [34

34. J.P. Culver, A.M. Siegel, J.J. Stott, and D.A. Boas, “Volumetric diffuse optical tomography of brain activity,” Opt. Lett. 28, 2061–2063 (2003). [CrossRef] [PubMed]

]. However this matrix has to be thresholded so that regions far from a source-detector pair do not inappropriately receive larger weighting. The effect of this threshold is presented in Fig. 4(b), where the depth of the reconstructed COM is plotted vs. the true depth of the inclusion, for CW data and TD data with the 5-gate combination discussed above, and for a β factor of 10, 20, 50 and 100. With CW data, the absorption is reconstructed at a constant depth whatever its true depth, and this reconstructed depth simply increases as β increases. For TD data, a smaller threshold (large β) enables better reconstructions for deep inclusions (z>2.5 cm), but also leads to an overestimated reconstructed depth for medium-deep inclusions (z around 1.5 to 2.5 cm). Moreover, for inclusions very close to the surface (under 1 cm) located just underneath a source or detector as in position 3 from Fig. 1(b), a large β also leads to strong artifacts (data not shown). A β=20 was used in the following simulations, enabling a good compromise for reconstruction of different depths between 1 and 3 cm.

3.5 Influence of background optical properties

The evaluation of the baseline optical properties of a medium is subject to uncertainty [36

36. D. Comelli, A. assi, A. Pifferi, P. Taroni, A. Torricelli, R. Cubeddu, F. Martelli, and G. Zaccanti, “In vivo time-resolved reflectance spectroscopy of the human forehead,” Appl. Opt. 46, 1717–1725 (2007). [CrossRef] [PubMed]

]. Therefore, we tested the influence of an error in the background optical properties on the reconstruction. We do not present data for this study, but enunciate general results we observed. An overestimation of the scattering coefficient (µ’s,recon>µ’s,true) by 20% led to an inclusion being reconstructed closer to the surface by approximately 1 to 3 mm, and vice versa for an underestimation of the scattering coefficient. We did not observe an effect of an over- or under-estimated absorption coefficient (by up to 50%) on the reconstructed depth. For these results, the true depth of the inclusion was varied between 0.5 cm and 3 cm.

3.6 Variation of depth localization error with lateral position

One major advantage of TD data is the depth information provided by the time of flight of photons. In Fig. 4(a), we showed the evolution of the reconstructed depth as a function of the true depth for one particular lateral position. In Fig. 5, we present the evolution of the reconstructed depth as a function of the lateral position of the inclusion, for a 1 cm and 2 cm deep inclusion. In both cases, the depth error is small (within 20% at 1 cm, and about 5% away from the edges of the medium at 2 cm).

Fig. 5. Depth of the COM for (a) a 1 cm deep and (b) a 2 cm deep inclusion reconstructed with TD data (5 gates) as a function of the inclusion lateral position.

For the inclusion located at 2cm below the surface, it also worth noting that the reconstructed depth varies very little with the lateral position of the inclusion. This means that the reconstruction method has no “blind zone”, and the depth reconstruction performance remains good for any position of the inclusion. Note that we do not compare the depth error with that obtained by CW measurements as no depth information is provided without overlapping or multi-distance CW measurement.

3.7 Lateral localization and resolution

In this section, we study the performance of the reconstruction method in terms of lateral localization and lateral resolution, and compare it with a CW reconstruction method. Figures 6 and 7 show the evolution of, respectively, the lateral localization error and the lateral resolution, for CW (left) and TD (right) reconstructions, at two different depths of the inclusion (1 cm, top, and 2 cm, bottom) as a function of the inclusion lateral position.

Fig. 6. Lateral error (cm) for a 1 cm deep inclusion (top) and a 2 cm deep inclusion (bottom) reconstructed with CW (left) and TD (right) data, as a function of the inclusion lateral position.

For both depths, TD shows reduced error in the lateral localization and better lateral resolution. Importantly we also note that both the lateral resolution and the lateral localization for TD reconstructions are more uniform with the lateral position of the inclusion relative to the probe than with a CW reconstruction.

These improvements of TD over CW for the lateral localization and resolution can be explained intuitively based on the sensitivity profiles presented in Fig. 2. For later delay gates, the sensitivity profile of a given source-detector probes deeper into the medium giving us depth resolution, but also probes a larger region laterally providing additional information to improve lateral localization and resolution.

Fig. 7. Lateral resolution (cm) for a 1cm deep inclusion (top) and a 2 cm deep inclusion (bottom) reconstructed with CW (left) and TD (right) data, as a function of the inclusion lateral position.

3.8 Contrast to noise ratio improvement

In this section, we varied the regularization parameter α between 10-4 and 10, both for CW and TD reconstructions, and studied the evolution of CNR and lateral resolution. Figure 8 is a parametric curve of the image CNR as a function of the lateral resolution. The true inclusion is located at position 1, and at a depth z=1.5 cm. We compute the CNR as the average contrast to noise ratio for all voxels of contrast above 80% of the maximum contrast. The image noise was obtained by propagation of the measurement noise by: σ2x=pAinvσ2y pATinv, where σ2x is the image covariance matrix.

The plot illustrates the trade-off between CNR and resolution: as the regularization is increased, the CNR of the reconstructed inclusion increases, but is counter-balanced by a worsening of the lateral resolution. We observe that for an identical CNR of the image, the lateral resolution is improved by TD reconstructions compared to CW. Similarly, at identical lateral resolution, TD reconstruction enables a much higher CNR of the image.

Fig. 8. CNR versus lateral resolution for a regularization parameter α varying between 10-4 and 10. CNR is given per unit volume (cm3) and unit change in absorption (cm-1) of the inclusion. Regularization parameters of 1, 10-2 and 10-4 are indicated on the plot.

4. Phantom reconstructions

4.1 TD instrument

4.2 Phantom experiment

The system was tested on a liquid phantom containing a spherical absorbing inclusion. One half of the probe was set over a tank (19×19×9 cm3) filled with a solution of intralipid and ink (estimated optical properties: µa=0.14 cm-1, µs’=10 cm-1), containing a hollow glass sphere of diameter 15 mm. The sphere was filled with the same intralipid solution with 70 times higher ink concentration. Images were acquired for 30 seconds with the sphere located at position 2 (as defined on Fig. 1(b)), followed by 30 seconds of acquisition with the sphere outside of the field of view of the probe. This procedure was repeated 5 times to increase signal to noise ratio. We repeated the experiment for three different depths of the inclusion: top of the sphere at a depth of 9.5 mm, 18.5 mm and 32.5 mm below the surface.

4.3 Phantom reconstructions

We calculated the contrast of the image as the percentage change in intensity between the “off” state when the sphere is outside of the field of view, and the “on” state when it is under the probe. We averaged the contrasts obtained from the 5 successive on/off states and used these data to reconstruct an image. We used a regularization parameter α=10-2 for the first two depths of the sphere. However, for the deepest inclusion, the contrast to noise ratio was very low, and we had to change the regularization parameter to 10-1 to obtain a reasonable reconstruction. Figure 9 shows the reconstructions for all 3 depths of the inclusion, as well as the true position of the top of the absorbing sphere.

Fig. 9. Phantom reconstruction for three different depths of the glass sphere. The rendered volumes show the contours at 80% of the maximum absorption contrast. The small circles represent the top of the 7.5 mm radius sphere.

The superficial inclusion is reconstructed at a depth of 7.5 mm, the second at a depth of 12.5 mm, and the deepest at 22.5 mm. In all cases the reconstruction is closer to the surface than the true inclusion, even for medium-deep inclusions, which differs from our simulations. Note that our simulations used a point absorption at a particular depth, while the experimental setup consists of a 15 mm diameter inclusion. Moreover our hypothesis of small perturbation breaks down as the solution in the glass sphere is strongly absorbing, hence the linear assumption is probably not valid anymore. In addition, we had to use a larger regularization parameter α than in the simulations because of the increased noise level of the experimental data. The effect of a higher regularization parameter is also to pull the reconstruction towards the surface (simulation data not shown). However, the formalism developed here enables reconstruction of experimental differential data at three different depths down to 3 cm of the true inclusion.

5. Conclusion

We presented a 3D reconstruction technique of differential TD gated data, and showed with simulated data that the method allows both depth resolution and improved lateral resolution and lateral localization compared to traditional CW reconstruction. With our typical experimental parameters, we also showed that only a limited number of gates is useful for optimal image reconstruction. The technique was used to reconstructed dynamic phantom images, where a spherical absorbing inclusion was embedded at various depths within a scattering intralipid solution. The technique presented here for gated measurements would be easily adaptable to moments of TPSF, provided that careful treatment of the covariance matrix is used in the reconstruction regularization.

Acknowledgments

We thank Anand Kumar for useful discussions on the spatially varying regularization. This work was supported by the NIH grants P41-RR14075, R01-EB002482 and NIBIB-EB00790.

References and links

1.

H. Obrig and A. Villringer, “Beyond the visible — imaging the human brain with light,” J. Cereb. Blood Flow Metab. 23, 1–18 (2002). [PubMed]

2.

A.P. Gibson, J.C. Hebden, and S.R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005). [CrossRef] [PubMed]

3.

Special section on Optics in Neuroscience, J. Biomed.Opt.10 (2005).

4.

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, 2331–36 (1989). [CrossRef] [PubMed]

5.

V. Ntziachristos, XH. Ma, A.G. Yodh, and B. Chance, “Multichannel photon counting instrument for spatially resolved near infrared spectroscopy,” Rev. Sci. Instrum. 70, 193–201 (1999). [CrossRef]

6.

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, “Determining changes in NIR absorption using a layered model of the human head,” Phys. Med. Biol. 46, 879–896 (2001). [CrossRef] [PubMed]

7.

A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, “Timeresolved reflectance at null source-detector separation: improving contrast and resolution in diffuse optical imaging,” Phys. Rev. Lett. 95 (078101), 1–4, (2005).

8.

J. Selb, J.J. Stott, M.A. Franceschini, A.G. Sorenson, and D.A. Boas, “Improved sensitivity to cerebral dynamics during brain activation with a time-gated optical system: analytical model and experimental validation,” J. Biomed. Opt. 10, 011013 (2005). [CrossRef]

9.

B. Montcel, R. Chabrier, and P. Poulet, “Detection of cortical activation with time-resolved diffuse optical methods,” Appl. Opt. 44, 1942–1947 (2005). [CrossRef] [PubMed]

10.

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

11.

J. Swartling, J.S. Dam, and S. Andersson-Engels, “Comparison of spatially and temporally resolved diffuse-reflectance measurements systems for determination of biomedical optical properties,” Appl. Opt. 42, 4612–4620 (2003). [CrossRef] [PubMed]

12.

M. Schweiger and S.R. Arridge, “Application of temporal filters to time resolved data in optical tomography,” Phys. Med. Biol. 44, 1699–1717 (1999). [CrossRef] [PubMed]

13.

F. Gao, Y. Tanikawa, H. Zhao, and Y. Yamada, “Semi-three-dimensional algorithm for time-resolved diffuse optical tomography by use of the generalized pulse spectrum technique,” Appl. Opt. 41, 7346–7358 (2002). [CrossRef] [PubMed]

14.

A. Liebert, H. Wabnitz, D. Grosenick, M. Möller, R. Macdonald, and H. Rinneberg, “Evaluation of optical properties of highly scattering media by moments of distributions of times of flight of photons,” Appl. Opt. 42, 5785–5790 (2003). [CrossRef] [PubMed]

15.

A. Kienle, T. Glanzmann, G. Wagnières, and H. van den Bergh, “Investigation of two-layered turbid media with time-resolved reflectance,” Appl. Opt. 37, 6852–6862 (1998). [CrossRef]

16.

F. Martelli, S. Del Bianco, G. Zaccanti, A. Pifferi, A. Toricelli, A. Bassi, P. Taroni, and R. Cubeddu, “Phantom validation and in vivo application of an inversion procedure for retrieving the optical properties of diffusive layered media from time-resolved reflectance measurements,” Opt. Lett. 29, (2004). [CrossRef] [PubMed]

17.

A. H. Barnett, J.P. Culver, A.G. Sorensen, A. Dale, and D.A. Boas, “Robust inference of baseline optical properties of the human head with three-dimensional segmentation from magnetic resonance imaging,” Appl. Opt. 42, 3095–3108, (2003). [CrossRef]

18.

S.R. Hintz, D.A. Benaron, J.P. van Houten, J.L. Duckworth, F.W.H. Liu, S.D. Spilman, D.K. Stevenson, and W.-F. Cheong, “Stationary headband for clinical time-of-flight optical imaging at the bedside,” Photochem. Photobiol. 68, 361–369 (1999). [CrossRef]

19.

J.C. Hebden, A. Gibson, R. Yusof, N. Everdell, E.M.C. Hillman, D.T. Delpy, S.R. Arridge, T. Austin, J.H. Meek, and J.S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002) [CrossRef] [PubMed]

20.

A. Maki, Y. Yamashita, Y. Ito, E. Watanabe, Y. Mayanagi, and H. Koizumi, “Spatial and temporal analysis of human motor activity using noninvasive NIR topography,” Med. Phys. 22, 1997–2005 (1995). [CrossRef] [PubMed]

21.

G. Gratton, P.M. Corballis, E. Cho, M. Fabiani, and D.C. Hood, “Shades of gray matter: noninvasive optical images of human brain responses during visual stimulation,” Psychophysiology 32, 505–509 (1995). [CrossRef] [PubMed]

22.

M.A. Franceschini, V. Toronov, M. Filiaci, E. Gratton, and S. Fantini, “On-line optical imaging of the human brain with 160 ms temporal resolution,” Opt. Express 6, 49–57 (2000). [CrossRef] [PubMed]

23.

D.A. Boas, T.J. Gaudette, G. Strangman, X. Cheng, J.J.A. Marota, and J.B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76–90 (2001). [CrossRef] [PubMed]

24.

D.T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, and J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988). [CrossRef] [PubMed]

25.

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S.R. Arridge, P. van der Zee, and D.T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993). [CrossRef] [PubMed]

26.

A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Möller, R. Macdonald, A. Villringer, and H. Rinneberg, “Time-resolved multidistance near-infrared spectroscopy of the adult head: intracerebral and extracerebral absorption changes from moments of distribution of times of flight of photons,” Appl. Opt. 43, 3037–3047 (2004). [CrossRef] [PubMed]

27.

A. Liebert, H. Wabnitz, J. Steinbrink, M. Möller, R. Macdonald, H Rinneberg, A. Villringer, and H. Obrig, “Bed-side assessment of cerebral perfusion in stroke patients based on optical monitoring of a dye-bolus by time-resolved diffuse reflectance,” Neuroimage 24, 426–435 (2005). [CrossRef] [PubMed]

28.

M. Kacprzak, A. Liebert, P. Sawosz, N. Zolek, and R. Maniewski, “Time-resolved optical imager for assessment of cerebral oxygenation”, J. Biomed. Opt. 12, 034019 (2007). [CrossRef] [PubMed]

29.

D.A. Boas, K. Chen K, D. Grebert, and M.A. Franceschini, “Improving the diffuse optical imaging spatial resolution of the cerebral hemodynamic response to brain activation in humans,” Opt. Lett. 29, 1506–1508 (2004). [CrossRef] [PubMed]

30.

J. Selb, D.K. Joseph, and D.A. Boas, “Time-gated optical system for depth-resolved functional brain imaging,” J. Biomed. Opt. 11, 044008 (2006). [CrossRef] [PubMed]

31.

R.C. Haskell, L.O. Svaasand, T.T. Tsay, T.C. Feng, M.S. McAdams, and B.J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994). [CrossRef]

32.

S.R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. 34,7395–7409 (1995). [CrossRef] [PubMed]

33.

S. Carraresi, T.S.M. Shatir, F. Martelli, and G. Zaccanti, “Accuracy of a perturbation model to predict the effect of scattering and absorbing inhomogeneities on photon migration,” Appl. Opt. 40, 4622–4632 (2001). [CrossRef]

34.

J.P. Culver, A.M. Siegel, J.J. Stott, and D.A. Boas, “Volumetric diffuse optical tomography of brain activity,” Opt. Lett. 28, 2061–2063 (2003). [CrossRef] [PubMed]

35.

D.A. Boas and A.M. Dale, “Simulation study of magnetic resonance imaging-guided cortically constrained diffuse optical tomography of the human brain function,” Appl. Opt. 44, 1957–1968 (2005). [CrossRef] [PubMed]

36.

D. Comelli, A. assi, A. Pifferi, P. Taroni, A. Torricelli, R. Cubeddu, F. Martelli, and G. Zaccanti, “In vivo time-resolved reflectance spectroscopy of the human forehead,” Appl. Opt. 46, 1717–1725 (2007). [CrossRef] [PubMed]

37.

F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by means of time-domain detection: methodology and phantom validation,” Phys. Med. Biol. 49, 1055–1078 (2004). [CrossRef] [PubMed]

OCIS Codes
(100.3190) Image processing : Inverse problems
(170.3010) Medical optics and biotechnology : Image reconstruction techniques
(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:
Image Processing

History
Original Manuscript: July 31, 2007
Revised Manuscript: September 21, 2007
Manuscript Accepted: September 25, 2007
Published: November 26, 2007

Virtual Issues
Vol. 3, Iss. 1 Virtual Journal for Biomedical Optics

Citation
Juliette Selb, Anders M. Dale, and David A. Boas, "Linear 3D reconstruction of time-domain diffuse optical imaging differential data: improved depth localization and lateral resolution," Opt. Express 15, 16400-16412 (2007)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-25-16400


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References

  1. H. Obrig and A. Villringer, "Beyond the visible - imaging the human brain with light," J. Cereb. Blood Flow Metab. 23,1-18 (2002). [PubMed]
  2. A.P. Gibson, J.C. Hebden, and S.R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50,R1-R43 (2005). [CrossRef] [PubMed]
  3. Special section on Optics in Neuroscience, J. Biomed. Opt. 10 (2005).
  4. M.S. Patterson, B. Chance, and B.C. Wilson, "Time-resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties," Appl. Opt. 28, 2331-36 (1989). [CrossRef] [PubMed]
  5. V. Ntziachristos, XH. Ma, A.G. Yodh, and B. Chance, "Multichannel photon counting instrument for spatially resolved near infrared spectroscopy," Rev. Sci. Instrum. 70, 193-201 (1999). [CrossRef]
  6. J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, H. Rinneberg, "Determining changes in NIR absorption using a layered model of the human head," Phys. Med. Biol. 46, 879-896 (2001). [CrossRef] [PubMed]
  7. A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, "Time-resolved reflectance at null source-detector separation: improving contrast and resolution in diffuse optical imaging," Phys. Rev. Lett. 95 (078101), 1-4, (2005).
  8. J. Selb, J.J. Stott, M.A. Franceschini, A.G. Sorenson and D.A. Boas, "Improved sensitivity to cerebral dynamics during brain activation with a time-gated optical system: analytical model and experimental validation," J. Biomed. Opt. 10, 011013 (2005). [CrossRef]
  9. B. Montcel, R. Chabrier, P. Poulet, "Detection of cortical activation with time-resolved diffuse optical methods," Appl. Opt. 44, 1942-1947 (2005). [CrossRef] [PubMed]
  10. H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, "Multichannel time-resolved optical tomographic imaging system," Rev. Sci. Instrum. 70, 3595-3602 (1999). [CrossRef]
  11. J. Swartling, J.S. Dam, and S. Andersson-Engels, "Comparison of spatially and temporally resolved diffuse-reflectance measurements systems for determination of biomedical optical properties," Appl. Opt. 42, 4612-4620 (2003). [CrossRef] [PubMed]
  12. M. Schweiger and S.R. Arridge, "Application of temporal filters to time resolved data in optical tomography," Phys. Med. Biol. 44, 1699-1717 (1999). [CrossRef] [PubMed]
  13. F. Gao, Y. Tanikawa, H. Zhao, and Y. Yamada, "Semi-three-dimensional algorithm for time-resolved diffuse optical tomography by use of the generalized pulse spectrum technique," Appl. Opt. 41, 7346-7358 (2002). [CrossRef] [PubMed]
  14. A. Liebert, H. Wabnitz, D. Grosenick, M. Möller, R. Macdonald, and H. Rinneberg, "Evaluation of optical properties of highly scattering media by moments of distributions of times of flight of photons," Appl. Opt. 42, 5785-5790 (2003). [CrossRef] [PubMed]
  15. A. Kienle, T. Glanzmann, G. Wagnières, and H. van den Bergh, "Investigation of two-layered turbid media with time-resolved reflectance," Appl. Opt. 37, 6852-6862 (1998). [CrossRef]
  16. F. Martelli, S. Del Bianco, G. Zaccanti, A. Pifferi, A. Toricelli, A. Bassi, P. Taroni, and R. Cubeddu, "Phantom validation and in vivo application of an inversion procedure for retrieving the optical properties of diffusive layered media from time-resolved reflectance measurements," Opt. Lett. 29, (2004). [CrossRef] [PubMed]
  17. A. H. Barnett, J.P. Culver, A.G. Sorensen, A. Dale, and D.A. Boas, "Robust inference of baseline optical properties of the human head with three-dimensional segmentation from magnetic resonance imaging," Appl. Opt. 42, 3095-3108, (2003). [CrossRef]
  18. S.R. Hintz, D.A. Benaron, J.P. van Houten, J.L. Duckworth, F.W.H. Liu, S.D. Spilman, D.K. Stevenson, W.-F. Cheong, "Stationary headband for clinical time-of-flight optical imaging at the bedside," Photochem. Photobiol. 68, 361-369 (1999). [CrossRef]
  19. J.C. Hebden, A. Gibson, R. Yusof, N. Everdell, E.M.C. Hillman, D.T. Delpy, S.R. Arridge, T. Austin, J.H. Meek, J.S. Wyatt, "Three-dimensional optical tomography of the premature infant brain," Phys. Med. Biol. 47, 4155-4166 (2002) [CrossRef] [PubMed]
  20. A. Maki, Y. Yamashita, Y. Ito, E. Watanabe, Y. Mayanagi, and H. Koizumi, "Spatial and temporal analysis of human motor activity using noninvasive NIR topography," Med. Phys. 22, 1997-2005 (1995). [CrossRef] [PubMed]
  21. G. Gratton, P.M. Corballis, E. Cho, M. Fabiani, and D.C. Hood, "Shades of gray matter: noninvasive optical images of human brain responses during visual stimulation," Psychophysiology 32, 505-509 (1995). [CrossRef] [PubMed]
  22. M.A. Franceschini, V. Toronov, M. Filiaci, E. Gratton, and S. Fantini, "On-line optical imaging of the human brain with 160 ms temporal resolution," Opt. Express 6, 49-57 (2000). [CrossRef] [PubMed]
  23. D.A. Boas, T.J. Gaudette, G. Strangman, X. Cheng, J.J.A. Marota, and J.B. Mandeville, "The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics," Neuroimage 13, 76-90 (2001). [CrossRef] [PubMed]
  24. D.T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, and J. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1988). [CrossRef] [PubMed]
  25. M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S.R. Arridge, P. van der Zee, and D.T. Delpy, "A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy," Phys. Med. Biol. 38, 1859-1876 (1993). [CrossRef] [PubMed]
  26. A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Möller, R. Macdonald, A. Villringer, and H. Rinneberg, "Time-resolved multidistance near-infrared spectroscopy of the adult head: intracerebral and extracerebral absorption changes from moments of distribution of times of flight of photons," Appl. Opt. 43, 3037-3047 (2004). [CrossRef] [PubMed]
  27. A. Liebert, H. Wabnitz, J. Steinbrink, M. Möller, R. Macdonald, H Rinneberg, A. Villringer, and H. Obrig, "Bed-side assessment of cerebral perfusion in stroke patients based on optical monitoring of a dye-bolus by time-resolved diffuse reflectance," Neuroimage 24, 426-435 (2005). [CrossRef] [PubMed]
  28. M. Kacprzak, A. Liebert, P. Sawosz, N. Zolek, and R. Maniewski, "Time-resolved optical imager for assessment of cerebral oxygenation", J. Biomed. Opt. 12, 034019 (2007). [CrossRef] [PubMed]
  29. D.A. Boas, K. Chen K, D. Grebert, and M.A. Franceschini, "Improving the diffuse optical imaging spatial resolution of the cerebral hemodynamic response to brain activation in humans," Opt. Lett. 29, 1506-1508 (2004). [CrossRef] [PubMed]
  30. J. Selb, D.K. Joseph, and D.A. Boas, "Time-gated optical system for depth-resolved functional brain imaging," J. Biomed. Opt. 11, 044008 (2006). [CrossRef] [PubMed]
  31. R.C. Haskell, L.O. Svaasand, T.T. Tsay, T.C. Feng, M.S. McAdams, B.J. Tromberg, "Boundary conditions for the diffusion equation in radiative transfer," J. Opt. Soc. Am. A 11, 2727-2741 (1994). [CrossRef]
  32. S.R. Arridge, "Photon-measurement density functions. Part I: Analytical forms," Appl. Opt. 34,7395-7409 (1995). [CrossRef] [PubMed]
  33. S. Carraresi, T.S.M. Shatir, F. Martelli, and G. Zaccanti, "Accuracy of a perturbation model to predict the effect of scattering and absorbing inhomogeneities on photon migration," Appl. Opt. 40, 4622-4632 (2001). [CrossRef]
  34. J.P. Culver, A.M. Siegel, J.J. Stott, and D.A. Boas, "Volumetric diffuse optical tomography of brain activity," Opt. Lett. 28, 2061-2063 (2003). [CrossRef] [PubMed]
  35. D.A. Boas, and A.M. Dale, "Simulation study of magnetic resonance imaging-guided cortically constrained diffuse optical tomography of the human brain function," Appl. Opt. 44, 1957-1968 (2005). [CrossRef] [PubMed]
  36. D. Comelli, A. assi, A. Pifferi, P. Taroni, A. Torricelli, R. Cubeddu, F. Martelli, and G. Zaccanti, "In vivo time-resolved reflectance spectroscopy of the human forehead," Appl. Opt. 46, 1717-1725 (2007). [CrossRef] [PubMed]
  37. F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, "Optical tomographic mapping of cerebral haemodynamics by means of time-domain detection: methodology and phantom validation," Phys. Med. Biol. 49, 1055-1078 (2004). [CrossRef] [PubMed]

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