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Development of a compensation algorithm for accurate depth localization in diffuse optical tomography Haijing Niu,† Fenghua Tian,† Zi-Jing Lin, and Hanli Liu* Department of Bioengineering, Joint Graduate Program between University of Texas at Arlington and University of Texas Southwestern Medical Center, University of Texas at Arlington, Arlington, Texas 76019, USA *Corresponding author:
[email protected] Received September 24, 2009; revised December 21, 2009; accepted December 23, 2009; posted January 6, 2010 (Doc. ID 117661); published January 29, 2010 Diffuse optical tomography endures poor depth localization, since its sensitivity decreases severely with increased depth. In this study, we demonstrate a depth compensation algorithm (DCA), which optimally counterbalances the decay nature of light propagation in tissue so as to accurately localize absorbers in deep tissue. The novelty of DCA is to directly modify the sensitivity matrix, rather than the penalty term of regularization. DCA is based on maximum singular values (MSVs) of layered measurement sensitivities; these MSVs are inversely utilized to create a balancing weight matrix for compensating the measurement sensitivity in increased depth. Both computer simulations and laboratory experiments were performed to validate DCA. These results demonstrate that one (or two) 3-cm-deep absorber(s) can be accurately located in both lateral plane and depth within the laboratorial position errors. © 2010 Optical Society of America OCIS codes: 170.5280, 170.6960, 170.3010.
In recent years, diffuse optical tomography (DOT) has increasingly become an important research tool for noninvasive functional brain imaging [1]. Since near-IR (NIR) light is highly diffused in biological tissues, however, the photon density drops off quickly with increased depth [2], making the measurement sensitivity of DOT in deep tissue significantly lower than that in superficial tissue. When being reconstructed, a deep object is often projected toward superficial layers, leading to a significant position error in object depth [3]. Spatial variant regularization (SVR) [2,3] is the current approach to enhance depth sensitivity of DOT by modifying the penalty term of regularization, and thus benefiting the image quality of DOT. While a variety of efforts have been made [2–4] to accurately image a deep object, a significant improvement in depth localization still remains a challenge. In this Letter, we report our recent development on a depth compensation algorithm (DCA) by introducing a pseudoexponential adjustment matrix to compensate severe sensitivity decreases with increased depth. The novelty of DCA is to directly compensate the sensitivity matrix, rather than modify the penalty term of regularization being used in SVR. We demonstrate that DCA affords great improvement in depth localization of DOT, up to a depth of 3 cm based on various laboratory phantom experiments. Following the conventional approach, we use the linear Rytov approximation [5] to the photon diffusion equation: the forward model is y = Ax, where y is the vector of measured changes in optical density from all the source-detector pairs; x is the vector of absorption changes in the image space. Matrix A describes the distribution of measurement sensitivity to changes in absorption in the medium. To reconstruct a DOT image, we first apply the Tikhonov regularization [6] method, written as xˆ = AT共AAT + ␣smaxI兲−1y, 0146-9592/10/030429-3/$15.00
where I is the identity matrix, smax is the maximal singular value of matrix A, and ␣ is the regularization parameter. The elements of A exhibit a fast decay with an increase in depth, leading to a lower measurement sensitivity for a deeper absorber, and thus resulting in poor depth localization that biases the reconstructed image toward the superficial layers [7]. To solve this problem, we propose to compose a weight matrix M, which has a pseudoexponential increase in magnitude with depth so as to counterbalance the loss of sensitivity of A in depth. Specifically, the M matrix is obtained as follows: (1) Assuming that the image to be reconstructed contains l layers, we can decompose matrix A into l submatrices, i.e., A1 , A2 , . . . , Al, counting from top to bottom, each of which reflects the measurement sensitivity within the respective layer in the medium. (2) We calculate the maximum singular values (MSVs) for each of the submatrices, expressed as M共A1兲 , M共A2兲 , . . . , M共Al-1兲 , M共Al兲. These MSVs exhibit a pseudoexponential decrease from the superficial to deep layers, resulting from the decrease in photon density with depth. (3) We form the M matrix as M = 兵diag关M共Al兲 , M共Al-1兲 . . . M共A2兲 , M共A1兲兴其␥, where ␥ is an adjustable power and varies between 0 and 3. These maximal singular values are inversely arranged in the order from bottom to top of the imaged medium, guaranteeing a balanced increase with depth in matrix M and thus, compensating the decay in measurement sensitivity in depth. (4) We multiply matrix A by M and define an adjusted matrix A# as A# = AM to be used in the inverse calculation. The adjustable power, ␥, controls the compensating weight in M, so the adjustable distribution of measurement sensitivity in A# can be ultimately optimized. A larger ␥ value in M will produce a larger © 2010 Optical Society of America
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weight toward deep layers; the corresponding elements in A# for deep layers will offer improved measurement sensitivities. On the contrary, lessening the ␥ value will reduce the power of compensation; correspondingly, the elements in A# for deep layers will receive a smaller weight for modified measurement sensitivities. Specifically, we propose DCA by replacing A with A# in the image reconstruction equation; namely, we would use xˆ = A#T共A#A#T + ␣smaxI兲−1y = MAT共AM2AT + ␣smaxI兲−1y in the actual image reconstruction. In order for DCA to work optimally, we need to further investigate how to correctly select the ␥ value for the new sensitivity matrix A#, since ␥ critically determines the depth sensitivity in reconstruction. To do so, we performed computer simulations to understand the dependence of quality of reconstructed images on ␥. First, we considered a homogeneous semi-infinite medium, whose imaging area on the surface was 8 ⫻ 8 cm2 with a depth of 5.5 cm, having background absorption and reduced scattering coefficients of a = 0.08 cm−1 and s⬘ = 10 cm−1. A square array [8] of 5 ⫻ 5 bifurcated optodes (with an interval of 1.5 cm) was arranged on the medium surface, producing 188 measurements when using the first to sixth nearest source-detector pairs. An absorber with 4 mm in diameter and a = 0.3 cm−1 was embedded right under the center of the optode array. Gaussian random noise (0.1%) was introduced in the simulation based on noise levels observed in our actual experiments. Second, we performed multiple simulations with the object being located from z = −1 to ⫺5 cm below the measurement surface (i.e., z = 0). At each object depth, the ␥ values were varied from 0 to 3 to investigate how ␥ affected the quality of reconstructed images. The simulation intervals in depth and ␥ value were ⫺0.1 cm and 0.1. For each depth and ␥ value, an image was reconstructed based on the adjusted matrix A#. The image quality was evaluated by (1) the contrast-to-noise ratio (CNR) [9] and (2) the positional error (PE). The CNR is defined as the difference between the averaged absorption coefficient within the region of interest (ROI, defined by the FWHM of a reconstructed image) and the region of background (ROB) divided by the weighted average of the standard deviations in the ROI and ROB. The PE is defined as the distance between the centers of real and reconstructed object. Larger CNR and smaller PE values indicate better quality of a reconstructed image. Figures 1(a) and 1(b) show computed CNR and PE results as a function of object depth, z, and adjustment power, ␥. These simulating results clearly demonstrate that with the depth compensation matrix, M, there exist a small range of ␥ values between 1.0 to 1.6 (as outlined by the dashed boxes in the figure), providing best CNR and PE outputs, while the object depth varies from z = −1 to ⫺5 cm. These figures imply that with an optimal selection of ␥, an embedded object in tissue can be reconstructed through DOT with accurate depth localization at both superficial
Fig. 1. (Color online) Dependence of (a) CNR and (b) PE of reconstructed images on object depth, z, and ␥ for an object 共d = 4 mm兲 located at center of x–y plane. They were generated when simulations moved the object along z axis (z = −1 to ⫺5 cm) below the measurement surface 共z = 0兲, while ␥ value increased from 0 to 3. The dashed rectangles outline the uniform values of CNR and PE.
(e.g., 1 cm) and deep (e.g., 3 cm or deeper) tissue locations. While the results from the simulations are promising, we wish to validate DCA by actual experiments using laboratory tissue phantoms. We utilized a multichannel, cw NIR tomographic imager (DYNOT, NIRx Medical Technologies [10]) to perform the experiments with a container of 15⫻ 10⫻ 10 cm3 filled with a 1% Intralipid solution as a homogeneous medium. The background medium had a = 0.08 cm−1 and s⬘ = 8.8 cm−1. The probe geometry was set to be the same as that given in the simulations. In experiment I, a 1 cm, spherical absorber 共a = 0.3 cm−1兲 was placed in the center of the x–y plane and 3 cm below the surface of the phantom (i.e., z = −3 cm). The measured DOT data with and without the absorber were respectively acquired to generate the changes in optical signals seen by all the channels. In experiment II, keeping the experimental setup and parameters unchanged, we placed two 1-cm-diameter absorbers at z = −3 cm with a center-to-center separation of 3 cm along x axis. Figure 2 shows the reconstructed images of one embedded object in both x–y and x–z planes using ␥ = 0 (equivalent to the conventional DOT reconstruction without any compensation) and ␥ = 1.6 (the optimal adjustment power). It clearly demonstrates that the reconstructed image in z direction is pulled toward the surface with a depth error of 1–1.5 cm when ␥ = 0 [Fig. 2(c)], while the lateral location of the absorber in x–y plane is relatively accurate [Fig. 2(a)]. After DCA is applied with ␥ = 1.6, the reconstructed object is located accurately at the expected depth [Fig. 2(d)]. Similarly, DCA with ␥ = 1.6 is able to offer a remarkable improvement in depth localization even when two absorbers are included (Fig. 3). It is noteworthy that DCA with an optimal ␥ 共␥ = 1.6兲 has greatly improved the lateral contrast and localization of the two imaged absorbers, as compared between Figs. 3(a) and 3(b). While DCA has a great potential to improve spatial localizations of imaged objects in DOT, the key is to optimally select ␥ for the depth compensation matrix M. Initially, we set ␥ values empirically within the range of 1–1.6 based on simulation results (Fig. 1). After investigations through laboratory experiments,
February 1, 2010 / Vol. 35, No. 3 / OPTICS LETTERS
Fig. 2. (Color online) Reconstructed DOT images of a single embedded object placed in the center of x–y plane and at z = −3 cm, as marked by the dashed circle. (a) and (c) are obtained with ␥ = 0 for the x–y and x–z plane; (b) and (d) are obtained with ␥ = 1.6 for the same respective planes. The color scale is normalized between 0 and 1.
we consider ␥ = 1.2– 1.6 to be more appropriate in practice so as to obtain high-quality DOT images from embedded objects in deep tissue. While the optimal ␥ value is determined empirically, it can be selected approximately through simulations and with a relatively narrow range, and thus easy to be implemented as well as more applicable to different research studies with different experimental setups. The physical meaning of optimal ␥ should represent an equilibrium point which balances the depth-
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dependent decay of sensitivity in matrix A and depth-dependent weight increase in matrix M during image reconstruction. We further confirm through simulations that the image quality and the appropriate ␥ range are not altered greatly when background s⬘ values vary from 5.0 to 15.0 cm−1 and background a values vary from 0.1 to 0.3 cm−1. Thus, we conclude that DCA is effective even for multispectral DOT and for tissues having different background optical properties. In this Letter, we have derived and presented the evidence that DCA with an optimal ␥ has ability to accurately localize absorbers in deep tissue 共⬃3 cm兲 in three dimension. It is further expected that an even deeper object can be imaged with accurate depth localization when DCA is used along with a DOT system that has a large dynamic range or a lower system noise. We believe that DCA can be also extended to process the data taken from frequency and time-domain systems in DOT. While our preliminary study based on computer simulations supports our expectation, further investigation is needed to optimize ␥ values when different types of data are used. In particular, the process to optimize ␥ may become more complex if an iterative algorithm is involved. It may be also possible that the principle of DCA can be translated to other optical imaging applications, such as for bioluminescence imaging and fluorescence molecular imaging. This work was supported in part by National Institutes of Health (NIH) funding from NINDS (4R33NS052850-03). The authors also acknowledge the PMI DOT imaging toolbox [11]. H. Niu acknowledges Prof. Tianzi Jiang at Institute of Automation, Chinese Academy of Sciences, for his support in the initial study. †
Both authors contributed equally to this paper.
References
Fig. 3. (Color online) Reconstructed images of two objects placed symmetrically around the center of x–y plane and z = −3 cm (dashed circles). (a) and (c) are with ␥ = 0, while (b) and (d) with ␥ = 1.6. The color scale is normalized between 0 and 1.
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