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**|**Biomed Opt Express**|**v.2(12); 2011 December 1**|**PMC3233252

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- Abstract
- 1. Introduction
- 2. Methods
- 3. Numerical experiments
- 4. Phantom experiment
- 5. Conclusion
- References and links

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Biomed Opt Express. 2011 December 1; 2(12): 3334–3348.

Published online 2011 November 21. doi: 10.1364/BOE.2.003334

PMCID: PMC3233252

Received 2011 September 23; Revised 2011 November 14; Accepted 2011 November 14.

Copyright © 2011 Optical Society of America

This is an open-access article distributed under the terms of the Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License, which permits download and redistribution, provided that the original work is properly cited. This license restricts the article from being modified or used commercially.

This article has been cited by other articles in PMC.

An *l _{p}* (0 <

Diffuse optical tomography (DOT) reconstructs the distributions of the optical properties, such as the scattering and absorption coefficients in the biological media. Near infrared (NIR) light incident on the surface of the biological medium propagates diffusively inside the biological medium, and the fractions of the NIR light are reemitted from and detected at the surface of the medium. By solving the inverse problem based on the photon diffusion equation with the measurement data of the detected light, DOT images are obtained noninvasively [1, 2]

The reconstructed optical properties contain the information about not only the structure but also the constituents in the measured object. By using the spectroscopic technique, the concentrations of hemoglobin, lipid etc. can be calculated from the optical properties. The abnormal absorption coefficients which can be caused by the tumor with angiogenesis are detected by DOT [3–7]. Therefore DOT attracts attentions as a new imaging modality for breast cancer diagnoses. DOT also can monitor the brain activities. The large change in the regional blood flow due to the higher brain activities is detectable by DOT [8, 9].

The advantage of the near infrared optical imaging modalities is that the measuring instruments are smaller in size and easier to use than the other modalities. Therefore, it is desirable that the number of the detectors is small. However, when the number of the detectors is limited, the inverse problem of DOT is highly ill-posed. The reconstructed optical properties usually have a low spatial resolution, and measurement noises significantly affect the reconstructed images. One of the solutions of these problems is to use a regularization method in the image reconstruction. So, recently the regularization techniques are actively studied for improvement of the DOT image quality.

The important approach is to incorporate the structural prior information. Pogue et al [10] uses a spatially variant Tikhonov regularization to reduce high frequency noises in the reconstructed images. Boverman et al employed prior segmentation of breast into glandular and adipose tissues [11]. Yalavarthy et al uses MRI-derived breast geometry to regularize the inverse solution [12]. Douiri et al applied anisotropic diffusion regularization with a priori edge information of the object to preserve the edge of the inner structure [13]. An algorithm assuming that the targets have blocky structures for regularization is introduced by Hiltunen et al [14]. Mutual information and joint entropy are used by Panagiotou et al to reflect the structure obtained from an alternative high resolution modality in the DOT images [15].

Another possible approach to improve the spatial resolution of the DOT image is to use a sparsity constraint. Cao et al implements *L*_{1} norm minimization by use of an expectation maximization algorithm for a linearized DOT inverse problem, and show that the reconstructed region with abnormal optical properties are localized more than the other methods they use [16].

Sparsity regularization with *L*_{1} norm and other methods to obtain sparse solution are used to several applications of inverse problems. Image restoration with sparsity constrained regularization is proposed by Shankar et al [17]. The *L*_{1} sparsity constraint is applied to fluorescence/bioluminescence diffuse optical tomography (F/BDOT) [18, 19]. Okawa et al used recursive spatial filtering for F/BDOT [20]. And other applications are found in the functional brain imaging such as electroencephalography [21, 22].

When the changes in the optical properties, which can be caused by the regional blood flow changes or the breast cancer in early stage, are expected to be localized, the sparsity regularization will provide a good reconstruction in DOT inverse problem.

In this paper, we apply the *l _{p}* (0 <

The changes in the absorption coefficients are described by a parameter to solve the difficulty in *l _{p}* minimization. And the DOT images are reconstructed by minimizing the residual error between the measurement and predicted data sets and the

Light propagation in biological media is a phenomenon of the transport of radiative energy. Therefore, the radiative transport equation (RTE) rigorously describes the phenomenon. In DOT, NIR light illuminates the surface of the medium. The biological tissues strongly scatter and weakly absorb NIR light while propagating. The fraction of the light is reemitted from and detected at the surface of the medium. Because RTE is an integro-differential equation, solving and using RTE directly for the inverse problem is not an easy task. So the photon diffusion equation (PDE) is usually used [1]. PDE is a partial differential equation approximating RTE, and this approximation holds when the media are thicker than several centimeters and time after the pulse light incidence is longer than several hundred picoseconds.

Under the diffusion approximation, and by use of the spherical harmonic expansion of the quantities in RTE, following PDE for the fluence rate of light at the position *r* and time *t*, Φ(*r*,*t*), is obtained,

$$\left\{-\nabla \cdot D\left(r\right)\nabla +{\mu}_{a}\left(r\right)+\frac{1}{c}\cdot \frac{\partial}{\partial t}\right\}\Phi \left(r,t\right)={q}_{0}\left(r,t\right),$$

(1)

where *D* = 1/(3*μ*′* _{s}*) represents the diffusion coefficient with the reduced scattering coefficient

The fluxes of the fluence rate, Γ(*r*,*t*) = −*n* · *D*Φ, are the quantities measured by the detectors located at various positions. The mean time of flight (MTF) is calculated from Γ as
$<t>={\int}_{0}^{\infty}t\cdot \Gamma dt/{\int}_{0}^{\infty}\Gamma dt$. In this study, a set of the measured MTFs, *M*, is used as the input data for reconstruction of the distribution of *μ _{a}*. An efficient method of solving the forward problem introduced by Schweiger et al [24] is also employed in this study to calculate MTFs using the moments of the time-resolved profiles.

Reconstruction of the distribution of *μ _{a}* is carried out by minimizing the residual error between the measured MTF data and the MTF data calculated by solving the forward problem. Reconstruction to estimate the

$$\underset{{\mu}_{a}}{\text{min}}{\Vert M-\widehat{M}\left({\mu}_{a}\right)\Vert}^{2}+\lambda \cdot f\left({\mu}_{a}\right),$$

(2)

where *M* and are the sets of the measured and the calculated MTFs, respectively. *λ* is a regularization parameter, and *f* is a regularization function depending on the regularization method such as Tikhonov regularization method, the total variation method, etc [25]. The nonlinear optimization method such as Gauss-Newton method etc. is used.

To calculate the gradient of the residual error in the first term in Eq. (2), Jacobian matrix which represents the sensitivity of the optical properties to the measured data, * _{j}*/

To reconstruct the changes in *μ _{a}* localized in the small regions, we apply an

$$\Delta {\mu}_{{a}_{i}}={\left|{z}_{i}\right|}^{2/p}\cdot \text{sgn}\left({z}_{i}\right).$$

(3)

Thus *μ*_{ai} is described as

$${\mu}_{{a}_{i}}={\overline{\mu}}_{a}+\Delta {\mu}_{{a}_{i}},$$

(4)

where * _{a}* is the constant baseline. Then the DOT reconstruction with the

$$\underset{z}{\text{min}}{\Vert M-\widehat{M}\left(z\right)\Vert}^{2}+\lambda \cdot \sum _{i=1}^{I}{\left|{z}_{i}\right|}^{2},$$

(5)

where *z* represents a vector consisting of *z _{i}*. By solving this optimization problem, a solution which selects the changes in

We use the nonlinear conjugate gradient method [26] for the optimization which requires the gradient of the cost function. Therefore, Jacobian matrix, *J _{ji}*(

Numerical experiments are conducted to investigate the effect of the *l _{p}* sparsity regularization with

A *μ _{a}* distribution in a 2D circular medium with a radius of 40 mm is reconstructed with the regularizations. The medium has strongly absorbing targets. The sizes and the number of the targets are varied in the simulation.

16 positions working for both light sources and detectors are located at the boundary of the medium with an equal spacing. An ultra-short pulse light illuminates one of the 16 positions and the reemitted light after propagating inside the medium is detected at the rest of the 16 positions. Each position works as the light source one by one. Thus we obtain 16×15 MTF data.

By use of the same forward solver for generating simulated data and for solving inverse problem, the reconstruction can be fairly successful [28]. To avoid this ‘inverse crime’ problem, the measured MTF data sets are generated by solving PDE using FEM with 12800 elements and 6561 nodes, and the inverse problem is solved using FEM with 3200 elements and 1681 nodes. Gaussian noises with the standard deviation of one percent of MTF are added to the measured data, and the noisy data are used for reconstruction.

The spatial resolution of the reconstructed images is investigated. Two circular targets are placed in the medium. The centers of the targets are at (*x*,*y*) = (20 mm,10 mm) and (20 mm,−10 mm) with the origin of the coordinate located at the center of the medium. The radius and *μ _{a}* of both of the targets are 5 mm and 0.014 mm

Figure 1 shows the L-curves with *p* = 2, 1, 1/2 and 1/4. Reconstructions are carried out with *λ* = 10^{−10}, 10^{−9}, 10^{−8},···,10^{−2} for each *p*, and the L-curves are plotted with *R* = log_{10} ||*M* – ||^{2} for the abscissa and *F* = log_{10}*f*(*z*) for the ordinate. The corner of the L-curve, in which both terms in Eq. (5) are minimized with a good balance, is visually judged from the plots. The optimum *λ* determined at the corner decreases with the decrease in *p*.

L-curves for the reconstructions with (a) *p* = 2, (b) *p* = 1, (c) *p* = 1/2, (d) *p* = 1/4 in the simulations in section 3.2.1. *R* = log_{10} ||*M–*||^{2} for the abscissa and *F* = log_{10}*f*(*z*) for the ordinate.

The reconstructed *μ _{a}* distributions are shown in Fig. 2 with the same color scale for all images using (a) Tikhonov regularization with

The *μ*_{a} distributions reconstructed using (a) Tikhonov regularization with *p* = 2 and using the *l*_{p} sparsity regularization with (b) *p* = 1, (c) *p* = 1/2 and (d) *p* = 1/4 in the simulations in section 3.2.1. The black open circles indicate the true **...**

The quantitative evaluations of the reconstructed images obtained by the simulations in section 3.2.1: (a) The profiles of Δ*μ*_{a} along the lines in the *y*-direction passing through the two peaks of *μ*_{a}, reconstructed with *p* = 2 (green **...**

Figure 3(b) plots *ζ* = Δ*μ*_{amin}/Δ*μ*_{amax} as a function of *p*, where Δ*μ*_{amin} is the minimum of Δ*μ _{a}* reconstructed between the peaks, and Δ

The average of the peak values of Δ*μ _{a}*, Δ

The area of the peak, *S*, is defined as the sum of the area of the FEM elements having Δ*μ _{a}* ≥ Δ

The *μ _{a}* distribution reconstructed using Tikhonov regularization (Fig. 2(a)) has small undulations around the targets, which are caused by the random noise, and the difference in the FEM meshing between that for generating the measurement data and that for solving the inverse problem.

On the other hand, undulations are hardly observed in the *μ _{a}* distributions reconstructed with the

In this subsection we investigate the sensitivity of reconstruction to small changes in the absorption coefficient by reconstructing *μ _{a}* of a medium having two targets with different

Reconstructions are carried out with the same manner as in the previous section. *λ* is selected by the L-curve method.

Figure 4 shows the reconstructed images. The *μ _{a}* image reconstructed using Tikhonov regularization shown in Fig. 4(a) reveals weak two peaks with Δ

Reconstructed *μ*_{a} distributions using (a) Tikhonov regularization with *p* = 2 and the *l*_{p} sparsity regularization with (b) *p* = 1, (c) *p* = 1/2 and (d) *p* = 1/4 in the simulations in section 3.2.2. The black open circles indicate the true positions **...**

When the *l _{p}* sparsity regularization with

Nevertheless the *l _{p}* sparsity regularization is not always successful. When

The *l _{p}* sparsity regularization is found to be effective for reconstructing localized targets as shown in the previous subsections. However, the targets are not always localized well in practical applications. We demonstrate reconstructions of broad targets in this subsection.

The circular target has a radius of 10 mm, a center at (*x*,*y*) = (20 mm, 0 mm) and *μ _{a}* of 0.014 mm

Figure 6 shows the reconstructed *μ _{a}* distributions using Tikhonov and the

Reconstructed *μ*_{a} distributions using (a) Tikhonov regularization with *p* = 2 and the *l*_{p} sparsity regularization with (b) *p* = 1, (c) *p* = 1/2 and (d) *p* = 1/4 in the simulations in section 3.2.3. The black open circle indicates the true position of **...**

On the other hand, the *l _{p}* sparsity regularizations remove the undulations. The shape and size of the target are well reconstructed when

The quantitative evaluations of the reconstructed images obtained by the simulations in section 3.2.3: (a) Reconstructed peaks of Δ*μ*_{a} with the true Δ*μ*_{a} = 0.0070 mm^{−1}, and (b) the area at half maximum of Δ **...**

From the results of Figs. 6 and and7,7, it can be said that the *l _{p}* sparsity regularization is effective to improve the quality of the reconstructed images of broad targets by reducing the influence of noise. However, the quality of the reconstructed images highly depends on the parameter

In the simulations above, we investigated the effects of the *p* value on the quality of the reconstructed images in the cases of various sizes and *μ _{a}* of the targets. According to the simulation results, the optimum

However, other simulation results indicate that smaller *p* does not always lead to better reconstruction. For the case of multiple targets with different *μ _{a}*, too small

A criterion will be required for determination of the optimum *p* value. Prior information provided by other imaging modalities may be useful for that purpose. Prior information of the target size may help determine the optimum *p* value, for example.

The effect of the *l _{p}* sparsity regularization is validated by a phantom experiment in this section. The time-resolved measurement system consisted of an ultra-short pulse laser operating at the wavelength of 759 nm, time-correlated single-photon counting units and 16 source/detector optical fiber bundles. The ultra-short pulse light with a duration of about 100 ps, a mean power of 0.25 mW and a repetition rate of 5 MHz illuminated the measured object. The details of the time-resolved measurement system is found in the literature [29].

One of the optical fiber bundles worked as a light source and the others detected light reemitted from the surface of the measured object. Therefore, we acquired 16×15 time-resolved data. The MTF dataset obtained from the measured time-resolved data at the wavelength of 759 nm is used as the input for reconstruction of the *μ _{a}* distribution.

In the phantom experiment, we reconstructed *μ _{a}* in a 2D-like tissue simulating phantom which was a cylinder made of polyacetal resin with a height of 240 mm and a radius of 40mm. The phantom had the background optical properties of

The optical fiber bundles were circularly attached on the surface of the phantom with an equal spacing in a 2D plane perpendicular to the cylinder axis. The *μ _{a}* distribution in the 2D plane was reconstructed with Tikhonov or the

Figure 8 shows the L-curves plotted with the evaluating points with *λ* = 10^{−10},10^{−9},10^{−8}, ,10^{−2}. *λ* values at the corners are determined as 10^{−2} and 10^{−5} for *p* =2 and 1, respectively. However, it was difficult to obtain typical L-curves for *p* = 1/2 and 1/4. The evaluating points for *p* = 1/2 and for 1/4 in Fig. 8(c) and (d) subtly form the L-curves, and *λ* = 10^{−7} is at the corners in both cases. More points on the L-curve for *p* = 1/4 were evaluated to confirm the corner as shown in Fig. 8(d). In the simulation sections and this phantom experiment, the regularized solution with small *p* tends to change drastically within a narrow range of *λ*. So it is better to make the L-curve with fine evaluating points of *λ* to find an appropriate corner of the L-curve.

L-curves for the reconstructions with (a) *p* = 2, (b) *p* = 1, (c) *p* = 1/2, (d) *p* = 1/4 in the phantom experiment. *R* = log_{10} ||*M* – ||^{2} for the abscissa and *F* = log_{10}*f*(*z*) for the ordinate

The reconstructed *μ _{a}* distributions are shown in Fig. 9. As expected from the results in the previous section, the smaller the value of

The *μ*_{a} distributions reconstructed with (a) Tikhonov regularization with *p* = 2 and the *l*_{p} sparsity regularization with (b) *p* = 1, (c) *p* = 1/2 and (d) *p* = 1/4 in the phantom experiment. The black open circle indicates the true position of the target. **...**

The results of the phantom experiment indicates that the *l _{p}* sparsity regularization localizes the change in the

The quantitative evaluations of the reconstructed images obtained by the phantom experiment: (a) Reconstructed peaks of Δ*μ*_{a} with the true Δ*μ*_{a} = 0.0020 mm^{−1}, and (b) the area at half maximum of Δ*μ* **...**

The maximum values of the reconstructed Δ*μ _{a}* are 0.0047 mm

To improve the spatial resolution and the robustness to noise, the *l _{p}* (0 <

Numerical experiments show that the *l _{p}* sparsity regularization improves the spatial resolution. Two localized targets with the absorption coefficients higher than that of the background are clearly separated in the images reconstructed with the

The difference in the absorption coefficient between targets at different positions is also recovered well by the *l _{p}* sparsity regularization. However, it is demonstrated that a target with a small differences in the absorption coefficient from the background may disappear due to the excessive effect of the

The *l _{p}* sparsity regularization is also effective for reconstructing broad targets. The influence of noise such as non-targeted small undulations in the reconstructed image is reduced. The size of the reconstructed target is reconstructed well when

The *l _{p}* sparsity regularization can be useful to reconstruct the localized changes in the absorption coefficient. The criterion to determine the optimum

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