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Linear 3D reconstruction of time-domain diffuse optical imaging

differential data: improved depth localization and lateral

resolution

Juliette Selb1,*, Anders M. Dale2, and David A. Boas1

1 Massachusetts General Hospital, Athinoula A. Martinos Center for Biomedical Imaging,

Charlestown, MA 02129

2 Department of Neurosciences, Department of Radiology, University of California, San Diego,

CA 92093

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.

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 non-

invasive cerebral imaging [1–3]. 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 [4,5] (both absorption coefficient μa and reduced

scattering coefficient μs′), depth resolution with single source-detector separation [6,7], and

better sensitivity to cortical activation [8,9]. 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 [10,11], or moments of these TPSFs [12–14] are fit non-

*Corresponding author: juliette@nmr.mgh.harvard.edu.

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Published in final edited form as:

Opt Express. 2007 December 10; 15(25): 16400–16412.

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linearly with a model of light propagation in a homogeneous medium [4,11], a two-layer

medium [15,16], a more realistic tissue-type segmentation of the head [17], or a complete

voxelized volume in the case of tomographic imaging [18,19].

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–

23], 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]. Hiraoka et al. extended the concept to inhomogeneous media [25], 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]. 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]. They showed depth-resolved time-course

of local perfusion after dye bolus injection on healthy subjects and stroke patients [27]. 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]. 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]. However,

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

In this paper, we describe an actual 3D linear reconstruction for differential imaging, based

on inversion of the forward sensitivity matrix calculated in different delay gates, in a similar

way that is sometimes implemented in CW DOT imaging [29]. We discuss the performance

and limitations of the reconstruction technique with simulated data. As already demonstrated

in previous papers, we show that TD offers the ability to localize the depth of absorption

contrast, which is not achievable with single distance CW data. Furthermore, we

demonstrate improved lateral resolution and localization for TD compared to CW. We then

apply the reconstruction technique to phantom data obtained with our time-gated system

[30].

2. Reconstruction principle

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

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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.

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], and extrapolated-boundary condition [31]. 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],

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 = A Δμa, 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 = nSD nGates 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]:

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 rS(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], 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.

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] for a

transmission geometry.

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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 x̂̂ is calculated by inversion of the sensitivity matrix: x̂̂ = pAinvy, where pAinv is the

pseudo-inverse of matrix A, computed with the following regularization:

(1)

where:

•

B = A L−1,

L = diag (diag (ATA + λ)) is used to scale the sensitivity matrix to act as a spatially

varying regularization [34], 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(σy2),

σy2 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 σy2 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.

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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]. 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.

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: rCOM

= (Σi, Vox≥80% Riri)/(Σi, Vox≥80% Ri), where ri 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 = (Σi, Vox≥80% Ri|ρi − ρCOM|)/(Σi, Vox≥80% Ri),

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.

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