BioMed Research International

Volume 2016 (2016), Article ID 4504161, 15 pages

http://dx.doi.org/10.1155/2016/4504161

## A Sparsity-Constrained Preconditioned Kaczmarz Reconstruction Method for Fluorescence Molecular Tomography

Life Science and Technology, Xidian University, Xi’an, Shaanxi 710071, China

Received 1 July 2016; Accepted 10 October 2016

Academic Editor: Yudong Cai

Copyright © 2016 Duofan Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Fluorescence molecular tomography (FMT) is an imaging technique that can localize and quantify fluorescent markers to resolve biological processes at molecular and cellular levels. Owing to a limited number of measurements and a large number of unknowns as well as the diffusive transport of photons in biological tissues, the inverse problem in FMT is usually highly ill-posed. In this work, a sparsity-constrained preconditioned Kaczmarz (SCP-Kaczmarz) method is proposed to reconstruct the fluorescent target for FMT. The SCP-Kaczmarz method uses the preconditioning strategy to minimize the correlation between the rows of the forward matrix and constrains the Kaczmarz iteration results to be sparse. Numerical simulation and phantom and in vivo experiments were performed to test the efficiency of the proposed method. The results demonstrate that both the convergence and accuracy of the proposed method are improved compared with the classical memory-efficient low-cost Kaczmarz method.

#### 1. Introduction

Fluorescence molecular tomography (FMT) is an imaging modality that can localize and quantify fluorescent markers to resolve biological processes at molecular and cellular levels. Being minimally invasive, relatively inexpensive, and portable, FMT has been successfully applied in small animal research and preclinical diagnostics such as cancer diagnosis, drug discovery, and gene expression visualization [1–4].

Due to a large number of unknowns and a limited number of measurements as well as the diffusive transport of photons in biological tissues, FMT reconstruction is an ill-posed inverse problem [5–7]. To improve the FMT imaging quality, both the noncontact FMT technique [8, 9] and the strategy of multiple excitations can be used to obtain more measurements. Structural a priori information provided by CT or MRI can also be incorporated into FMT imaging [3, 10–12]. Moreover, reconstruction algorithms can resort to regularization strategies and find meaningful and numerically stable solutions. In [13, 14], the Tikhonov regularization, namely, norm regularization, is employed for solving the inverse problem. In [15–17], the sparsity regularization is utilized in the form of norm penalty function for FMT reconstruction. Joint and TV regularization for FMT reconstruction is presented in [18]. In [19], a hybrid regularization method incorporating and norm penalty is proposed to recover the 3D fluorophore distribution. In these techniques, optimal selection of the regularization parameter is needed to avoid over- or underregularization. Being a memory-efficient low-cost numerical solver that avoids bulky matrix computations in large-scale problems, Kaczmarz algorithm, also known as algebraic reconstruction technique (ART), iteratively updates the solution using only one equation at a time and has been applied in optical tomographic reconstruction [20–22]. During reconstruction, the Kaczmarz method may use the measurements in the order that they are collected, which is known as the sequential access order. To speed up the convergence rate of the Kaczmarz method and give better results in the first iteration relative to the sequential access scheme, different access orders have been proposed [23–25]. The idea of these different access orders is to minimize the correlation between measurements that are successively accessed by the iterative projection inversion method. In [20], the influence of the data access order is investigated when Kaczmarz method is used to perform diffuse optical tomography. The study shows that the convergence speed can be significantly improved by selecting proper projection access order.

In FMT, the forward matrix maps the fluorescent targets to the surface measurements. Generally, the rows of the forward matrix are correlated because of the correlations among source-detector maps from the same projection and the interrelations of different projections [26, 27]. In this work, we present a strategy which computes a preconditioning matrix to minimize the coherence of the preconditioned forward matrix. Then the Kaczmarz method which uses the sequential access order is adopted to solve the preconditioned FMT reconstruction problem. After preconditioning, the projections are close to perpendicularity and the convergence rate of the Kaczmarz method can be speeded up. As most optical fluorophores are designed to accumulate in relatively small, specific regions in tissues, such as tumors, and hence the fluorophore distributes sparsely in the imaging domain, we propose sparsity-constrained reconstruction method to perform FMT and the method is named as sparsity-constrained preconditioned Kaczmarz (SCP-Kaczmarz) method. Different from the existed norm regularization methods, this proposed SCP-Kaczmarz method adopts a thresholding step to the Kaczmarz results to satisfy a user-defined sparsity value.

The remaining of this paper is organized as follows. We first describe the mathematical forward model for FMT imaging, then the SCP-Kaczmarz method is presented for FMT reconstruction, then the numerical simulation and physical phantom and in vivo experiments are performed to evaluate the proposed method, and finally the discussion and conclusion are given.

#### 2. Methods

##### 2.1. Forward Model for FMT Imaging

When a CW point laser is used as excitation light, the diffusion of excitation and emission lights through biological tissues can be described by two coupled diffusion equations with the Robin-type boundary condition, and the coupled diffusion equations can be presented as follows [28]: where , being the domain under consideration. The subscripts and denote excitation light and emission light, respectively. is the diffusion coefficient with as the scattering coefficient, is the anisotropy parameter, and is the absorption coefficient. denotes the photon density. The fluorescent yield is the unknown parameter to be reconstructed, which is denoted as hereafter. By using the finite element method (FEM), the linear relationship between the boundary measurements and the desired unknown fluorophore distribution can be obtained from (1) and is described bywhere is the forward matrix, the sizes of , , and are , , and , respectively. is the number of surface measurements and is the number of unknowns needed to be determined inside the imaging domain. Usually , and this means that the number of measurements is smaller than that of the unknowns.

##### 2.2. Sparsity-Constrained Preconditioned Kaczmarz Method

It is known to us that the convergence of the Kaczmarz method is affected by the data access order. If the measurements are prearranged in a scheme that the projections are close to perpendicularity, the convergence of the Kaczmarz method will be speeded up. In this paper, rather than changing the sequential data access order, we design a preconditioner to minimize the correlation between the rows of the forward matrix of FMT and hence to make the Kaczmarz method converge quickly. Denote the preconditioning matrix as and the preconditioned forward matrix as , then we get , and can be obtained by solving the following optimization problem:where is the identity matrix and is the Frobenius norm.

Considering the singular value decomposition of which is described by , where is unitary matrix, is diagonal nonnegative matrix and is an unitary matrix. Letting , we can get Equation (4) means that the rows of the preconditioned forward matrix are orthogonal to one another. If the preconditioner is badly conditioned, we can use the diagonal loading strategy to mitigate the ill condition and is calculated by , where is a constant [27].

Figure 1 provides a geometric insight into the iterative progress of the Kaczmarz and the preconditioned Kaczmarz algorithms. Figure 1(a) presents a geometrical interpretation of Kaczmarz applied to a 2D problem. Here, each line represents a hyperplane in the solution space corresponding to one of the equations, and the solution is the intersection of the dashed lines. The progress of Kaczmarz is represented by dark blue dots and arrow lines. As depicted in Figure 1(a), the points with dots iteratively progress toward the solution (intersection of the two dashed lines) by orthogonal successive projections onto the two lines [22]. In Figure 1(b), the blue diamond and arrow demonstrate the convergence of preconditioned Kaczmarz algorithm toward the solution. Because the forward matrix has been preconditioned, the two green dashed lines which demonstrate the hyperplanes corresponding to the two equations are perpendicular. In theory, only one iteration is needed for the algorithm to converge to the solution. However, because of the ill condition of the forward matrix in FMT imaging and the presence of noise, the two lines are not completely perpendicular.