Elsevier

Optics Communications

Volume 284, Issue 24, 1 December 2011, Pages 5871-5876
Optics Communications

Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography

https://doi.org/10.1016/j.optcom.2011.07.071Get rights and content

Abstract

In this paper, an efficient l1-regularized reconstruction method named the primal-dual interior-point (PDIP) method is presented for three-dimensional bioluminescence tomography (BLT) based on the adaptive finite element framework. Taking into account the sparse characteristic of the bioluminescent source, the BLT inverse problem is considered to be a linear programming problem and is represented by its primal and dual form. The source localization and quantification are obtained by solving the primal-dual Newton equation system. The comparisons between PDIP and the classical conjugate gradient least square (CGLS) algorithm are implemented to validate our method. Results from numerical simulation and an in vivo mouse experiment demonstrate the credibility and the potential of the proposed method in practical BLT reconstruction.

Introduction

As a promising optical molecular imaging technique, bioluminescence tomography (BLT) suggests enormous potential in drug development and preclinical oncological investigation due to its significant advantages in specificity, sensitivity, safety and cost-effectiveness [1], [2], [3], [4]. By integrating surface measured light flux distribution, geometrical structures and tissue optical properties, the goal of BLT is to reconstruct the distribution of bioluminescent probes inside a small living animal, achieving accurate tomographic reconstruction and visualization in three-dimensional (3D) [5], [6], [7]. However, the reconstruction problem remains a challenging issue because of the inherent ill-posedness nature of the inverse problem.

Recently, intensive interests have been given to the reconstruction algorithms and many feasible approaches have been proposed to handle the inverse source problem [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. To alleviate the ill-posedness, some forms of a priori information have been employed to remarkably improve the source reconstruction, including permissible source region strategy that restricts the source in a specific area [8] and a spectrally-resolved approach that attains the reconstruction using multispectral data [9], [17]. No matter what kind of a priori information used, the source reconstruction problem is regularized to a least squares optimization problem. Many numerical methods have been applied to solve this optimization problem, such as the conjugate gradient [13], level set strategy [7], [14], trust region method [15] and a differential evolution approach [6]. Most of these methods are based on the l2 norm regularization, which tries to solve the problem by combining a quadratic error term and an l2 norm term. However, l2 norm regularization has been proven to smooth solutions and bring multi-pseudo sources to surround the true source. Because of the sparsity of bioluminescent source distribution [16], l1 regularization has been applied in BLT and became a mainstream trend [17], [18].

In this paper, considering the sparse characteristic of the bioluminescent source, a primal-dual interior-point (PDIP) method was proposed for BLT reconstruction, which successfully integrates the adaptive finite element (AFE) framework and the l1 norm regularization strategy. During the last twenty years, interior-point methods have been proven to be highly efficient in both theory and practice, which are robust for numerically solving optimization problems, such as linear, quadratic, second-order cone, geometric, and semidefinite programming [19], [20], [21], [22], [23], [24]. The interior-point method was first proposed for solving linear programming problems by Karmarkar in 1984 [25]. By the early 1990s, a subclass of the interior-point methods distinguished itself as the most efficient practical approach, and turned out to be a strong competitor for large-scale problems. As the first approaches developed for solving sparse problems via convex optimization, interior-point methods have been widely applied in sparse signal reconstruction and processing, statistics, and related fields over the past few years [26]. As one branch of the interior-point methods, PDIP inherits the highly efficient and numerical robustness of the interior-point methods and it poses good convergence since it only requires a total of Ο(n) iterations. As one of the best and most perfect polynomial algorithms of linear programming in theory, the PDIP combines the penalty function method and Newton method and can be used to resolve the sparse problem with l1 regularization. Therefore, the PDIP method can be utilized to solve the source reconstruction problem in BLT.

In the proposed PDIP method, diffusion approximation (DA) was employed to describe light propagation in biological tissues. Based on the AFE framework, the DA is formalized as a linear matrix equation between the unknown source variables and the surface measurement. Because of the sparsity of the bioluminescent source distribution and the insufficiency of the surface measurement, the exact solution of the inverse source problem is the l0 regularizer, which is one of the hardest combinatory optimization problems. In order to solve such a complicated problem, we replaced the l0 regularizer with the l1 norm for simplicity. By introducing the logarithmic barrier term, the PDIP method takes the l1 norm problem as a minimization problem of linear programming. In order to obtain the optimal solution of the minimization problem, the Karush–Kuhn–Tucker (KKT) conditions were used to restrain the solving process. Using the Newton method, we obtained the optimal solution to the primal-dual equation. Reconstructed results on numerical simulation and an in vivo mouse experiment validated the ability of the proposed method for locating and quantifying the bioluminescent source.

Section snippets

Forward formula of BLT

It is well acknowledged that the radiative transfer equation (RTE) can be used as the most accurate model to describe the photon transport in biological tissues [10], [27]. However, due to the implementation complexity in a numerical setting and the difficulty in the direct calculation, the application of RTE for BLT is difficult [28]. Because photon transport in biological tissues exhibits high scattering and weak absorption, DA and its Robin boundary condition have been well accepted [8]:Dr

Experiments and results

In this section, a series of verification experiments were designed to evaluate our proposed reconstruction algorithm. In order to analyze the results quantitatively, we defined the Location Error of the distance between the actual and reconstructed source and the Relative Error of the source density between the actual and reconstructed source as follows:LocationError=(xx0)2+(yy0)2+(zz0)2RelativeError=SreconSreal/Srealwhere (x, y, z) is the coordinate of the reconstructed source with the

Conclusion and discussion

Because of the ill-posedness of the BLT inverse problem, multiple solutions and aberrant source reconstructions are frequent problems. Therefore, regularization methods are generally adopted to ease ill-posedness in the inverse problem. Among these methods, l1 regularization has become a powerful tool for solving the underdetermined inverse problem because of the inherent sparse distribution characteristics of the bioluminescence source and the insufficient measurement data in BLT. In this

Acknowledgments

This work is supported by the Program of the National Basic Research and Development Program of China (973) under Grant Nos. 2011CB707702, the National Natural Science Foundation of China under Grant Nos. 81090272, 81000632, and 30900334, the Shaanxi Provincial Natural Science Foundation Research Project under Grant No. 2009JQ8018, and the Fundamental Research Funds for the Central Universities.

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