Truncated total least squares method with a practical truncation parameter choice scheme for bioluminescence tomography inverse problem

Gamma rays excited radioluminescence tomographic imaging

Background

Radionuclide-excited luminescence imaging is an optical radionuclide imaging strategy to reveal the distributions of radioluminescent nanophosphors (RLNPs) inside small animals, which uses radioluminescence emitted from RLNPs when excited by high energy rays such as gamma rays generated during the decay of radiotracers used in clinical nuclear medicine imaging. Currently, there is no report of tomographic imaging based on radioluminescence.

Methods

In this paper, we proposed a gamma rays excited radioluminescence tomography (GRLT) to reveal three-dimensional distributions of RLNPs inside a small animal using radioluminescence through image reconstruction from surface measurements of radioluminescent photons using an inverse algorithm. The diffusion equation was employed to model propagations of radioluminescent photons in biological tissues with highly scattering and low absorption characteristics.

Results

Phantom and artificial source-implanted mouse model experiments were employed to test the feasibility of GRLT, and the results demonstrated that the ability of GRLT to reveal the distribution of RLNPs such as Gd₂O₂S:Tb using the radioluminescent signals when excited by gamma rays produced from ^99mTc.

Conclusions

With the emerging of targeted RLNPs, GRLT can provide new possibilities for in vivo and noninvasive examination of biological processes at cellular levels. Especially, combining with Cerenkov luminescence imaging, GRLT can achieve dual molecular information of RLNPs and nuclides using single optical imaging technology.

High-Resolution Cortical Dipole Imaging Using Spatial Inverse Filter Based on Filtering Property

Cortical dipole imaging has been developed to visualize brain electrical activity in high spatial resolution. It is necessary to solve an inverse problem to estimate the cortical dipole distribution from the scalp potentials. In the present study, the accuracy of cortical dipole imaging was improved by focusing on filtering property of the spatial inverse filter. We proposed an inverse filter that optimizes filtering property using a sigmoid function. The ability of the proposed method was compared with the traditional inverse techniques, such as Tikhonov regularization, truncated singular value decomposition (TSVD), and truncated total least squares (TTLS), in a computer simulation. The proposed method was applied to human experimental data of visual evoked potentials. As a result, the estimation accuracy was improved and the localized dipole distribution was obtained with less noise.

Algorithm for localized adaptive diffuse optical tomography and its application in bioluminescence tomography

A reconstruction algorithm for diffuse optical tomography based on diffusion theory and finite element method is described. The algorithm reconstructs the optical properties in a permissible domain or region-of-interest to reduce the number of unknowns. The algorithm can be used to reconstruct optical properties for a segmented object (where a CT-scan or MRI is available) or a non-segmented object. For the latter, an adaptive segmentation algorithm merges contiguous regions with similar optical properties thereby reducing the number of unknowns. In calculating the Jacobian matrix the algorithm uses an efficient direct method so the required time is comparable to that needed for a single forward calculation. The reconstructed optical properties using segmented, non-segmented, and adaptively segmented 3D mouse anatomy (MOBY) are used to perform bioluminescence tomography (BLT) for two simulated internal sources. The BLT results suggest that the accuracy of reconstruction of total source power obtained without the segmentation provided by an auxiliary imaging method such as x-ray CT is comparable to that obtained when using perfect segmentation.

Cortical dipole imaging using truncated total least squares considering transfer matrix error

Cortical dipole imaging has been proposed as a method to visualize electroencephalogram in high spatial resolution. We investigated the inverse technique of cortical dipole imaging using a truncated total least squares (TTLS). The TTLS is a regularization technique to reduce the influence from both the measurement noise and the transfer matrix error caused by the head model distortion. The estimation of the regularization parameter was also investigated based on L-curve. The computer simulation suggested that the estimation accuracy was improved by the TTLS compared with Tikhonov regularization. The proposed method was applied to human experimental data of visual evoked potentials. We confirmed the TTLS provided the high spatial resolution of cortical dipole imaging.

Self-calibrated algorithms for diffuse optical tomography and bioluminescence tomography using relative transmission images

Reconstruction algorithms for diffuse optical tomography (DOT) and bioluminescence tomography (BLT) have been developed based on diffusion theory. The algorithms numerically solve the diffusion equation using the finite element method. The direct measurements of the uncalibrated light fluence rates by a camera are used for the reconstructions. The DOT is self-calibrated by using all possible pairs of transmission images obtained with external sources along with the relative values of the simulated data and the calculated Jacobian. The reconstruction is done in the relative domain with the cancelation of any geometrical or optical factors. The transmission measurements for the DOT are used for calibrating the bioluminescence measurements at each wavelength and then a normalized system of equations is built up which is self-calibrated for the BLT. The algorithms have been applied to a three dimensional model of the mouse (MOBY) segmented into tissue regions which are assumed to have uniform optical properties. The DOT uses the direct method for calculating the Jacobian. The BLT uses a reduced space of eigenvectors of the Green's function with iterative shrinking of the permissible source region. The reconstruction results of the DOT and BLT algorithms show good agreement with the actual values when using either absolute or relative data. Even a small calibration error causes significant degradation of the reconstructions based on absolute data.

Comparative studies of l(p)-regularization-based reconstruction algorithms for bioluminescence tomography

Inverse source reconstruction is the most challenging aspect of bioluminescence tomography (BLT) because of its ill-posedness. Although many efforts have been devoted to this problem, so far, there is no generally accepted method. Due to the ill-posedness property of the BLT inverse problem, the regularization method plays an important role in the inverse reconstruction. In this paper, six reconstruction algorithms based on l(p) regularization are surveyed. The effects of the permissible source region, measurement noise, optical properties, tissue specificity and source locations on the performance of the reconstruction algorithms are investigated using a series of single source experiments. In order to further inspect the performance of the reconstruction algorithms, we present the double sources and the in vivo mouse experiments to study their resolution ability and potential for a practical heterogeneous mouse experiment. It is hoped to provide useful guidance on algorithm development and application in the related fields.

Efficient sparse reconstruction algorithm for bioluminescence tomography based on duality and variable splitting

Bioluminescence tomography (BLT) can three-dimensionally and quantitatively resolve the molecular processes in small animals in vivo. In this paper, we propose a BLT reconstruction algorithm based on duality and variable splitting. By using duality and variable splitting to obtain a new equivalent constrained optimization problem and updating the primal variable as the Lagrangian multiplier in the dual augmented Lagrangian problem, the proposed method can obtain fast and stable source reconstruction even without the permissible source region and multispectral measurements. Numerical simulations on a mouse atlas and in vivo mouse experiments were conducted to validate the effectiveness and potential of the method.

Normalized Born Approximation-Based Two-Stage Reconstruction Algorithm for Quantitative Fluorescence Molecular Tomography

Fluorescence molecular tomography (FMT) is a promising technique forin vivosmall animal imaging. In this paper, a two-stage reconstruction method based on normalized Born approximation is developed for FMT, which includes two steps for quantitative reconstruction. First, the localization of fluorescent fluorophore is determined byl1-norm regularization method. Then, in the location region of fluorophore, which is provided by the first stage, algebraic reconstruction technique (ART) is utilized for the fluorophore concentration reconstruction. The validity of the two-stage quantitative reconstruction algorithm is testified by simulation experiments on a 3D digital mouse atlas and physical experiments on a phantom. The results suggest that we are able to recover the fluorophore location and concentration.

Bioluminescence tomography using eigenvectors expansion and iterative solution for the optimized permissible source region

A reconstruction algorithm for bioluminescence tomography (BLT) has been developed. The algorithm numerically calculates the Green's function at different wavelengths using the diffusion equation and finite element method. The optical properties used in calculating the Green's function are reconstructed using diffuse optical tomography (DOT) and assuming anatomical information is provided by x-ray computed tomography or other methods. A symmetric system of equations is formed using the Green's function and the measured light fluence rate and the resulting eigenvalue problem is solved to get the eigenvectors of this symmetric system of equations. A space can be formed from the eigenvectors obtained and the reconstructed source is written as an expansion of the eigenvectors corresponding to non-zero eigenvalues. The coefficients of the expansion are found to obtain the reconstructed BL source distribution. The problem is solved iteratively by using a permissible source region that is shrunk by removing nodes with low probability to contribute to the source. Throughout this process the permissible region shrinks from the entire object to just a few nodes. The best estimate of the reconstructed source is chosen that which minimizes the difference between the calculated and measured light fluence rates. 3D simulations presented here show that the reconstructed source is in good agreement with the actual source in terms of locations, magnitudes, sizes, and total powers for both localized multiple sources and large inhomogeneous source distributions.

Improved bioluminescence and fluorescence reconstruction algorithms using diffuse optical tomography, normalized data, and optimized selection of the permissible source region

Reconstruction algorithms are presented for two-step solutions of the bioluminescence tomography (BLT) and the fluorescence tomography (FT) problems. In the first step, a continuous wave (cw) diffuse optical tomography (DOT) algorithm is used to reconstruct the tissue optical properties assuming known anatomical information provided by x-ray computed tomography or other methods. Minimization problems are formed based on L1 norm objective functions, where normalized values for the light fluence rates and the corresponding Green's functions are used. Then an iterative minimization solution shrinks the permissible regions where the sources are allowed by selecting points with higher probability to contribute to the source distribution. Throughout this process the permissible region shrinks from the entire object to just a few points. The optimum reconstructed bioluminescence and fluorescence distributions are chosen to be the results of the iteration corresponding to the permissible region where the objective function has its global minimum This provides efficient BLT and FT reconstruction algorithms without the need for a priori information about the bioluminescence sources or the fluorophore concentration. Multiple small sources and large distributed sources can be reconstructed with good accuracy for the location and the total source power for BLT and the total number of fluorophore molecules for the FT. For non-uniform distributed sources, the size and magnitude become degenerate due to the degrees of freedom available for possible solutions. However, increasing the number of data points by increasing the number of excitation sources can improve the accuracy of reconstruction for non-uniform fluorophore distributions.

Sparse reconstruction for quantitative bioluminescence tomography based on the incomplete variables truncated conjugate gradient method

In this paper, we present an incomplete variables truncated conjugate gradient (IVTCG) method for bioluminescence tomography (BLT). Considering the sparse characteristic of the light source and insufficient surface measurement in the BLT scenarios, we combine a sparseness-inducing (ℓ1 norm) regularization term with a quadratic error term in the IVTCG-based framework for solving the inverse problem. By limiting the number of variables updated at each iterative and combining a variable splitting strategy to find the search direction more efficiently, it obtains fast and stable source reconstruction, even without a priori information of the permissible source region and multispectral measurements. Numerical experiments on a mouse atlas validate the effectiveness of the method. In vivo mouse experimental results further indicate its potential for a practical BLT system.

Sparse regularization-based reconstruction for bioluminescence tomography using a multilevel adaptive finite element method

Bioluminescence tomography (BLT) is a promising tool for studying physiological and pathological processes at cellular and molecular levels. In most clinical or preclinical practices, fine discretization is needed for recovering sources with acceptable resolution when solving BLT with finite element method (FEM). Nevertheless, uniformly fine meshes would cause large dataset and overfine meshes might aggravate the ill-posedness of BLT. Additionally, accurately quantitative information of density and power has not been simultaneously obtained so far. In this paper, we present a novel multilevel sparse reconstruction method based on adaptive FEM framework. In this method, permissible source region gradually reduces with adaptive local mesh refinement. By using sparse reconstruction with l(1) regularization on multilevel adaptive meshes, simultaneous recovery of density and power as well as accurate source location can be achieved. Experimental results for heterogeneous phantom and mouse atlas model demonstrate its effectiveness and potentiality in the application of quantitative BLT.