BioMed Research International

Volume 2017, Article ID 2010512, 18 pages

https://doi.org/10.1155/2017/2010512

## Image Restoration for Fluorescence Planar Imaging with Diffusion Model

Engineering Research Center of Molecular and Neuro Imaging of the Ministry of Education & School of Life Science and Technology, Xidian University, Xi’an, Shaanxi 710071, China

Correspondence should be addressed to Shouping Zhu; nc.ude.naidix@uhzps

Received 6 June 2017; Accepted 5 November 2017; Published 27 November 2017

Academic Editor: Toshiyuki Sawaguchi

Copyright © 2017 Xuanxuan Zhang 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 planar imaging (FPI) is failure to capture high resolution images of deep fluorochromes due to photon diffusion. This paper presents an image restoration method to deal with this kind of blurring. The scheme of this method is conceived based on a reconstruction method in fluorescence molecular tomography (FMT) with diffusion model. A new unknown parameter is defined through introducing the first mean value theorem for definite integrals. System matrix converting this unknown parameter to the blurry image is constructed with the elements of depth conversion matrices related to a chosen plane named focal plane. Results of phantom and mouse experiments show that the proposed method is capable of reducing the blurring of FPI image caused by photon diffusion when the depth of focal plane is chosen within a proper interval around the true depth of fluorochrome. This method will be helpful to the estimation of the size of deep fluorochrome.

#### 1. Introduction

Fluorescence imaging techniques have become indispensable tools for numerous biomedical applications attributing to the everlasting development of fluorescent probes [1]. With the help of various fluorescent probes and fluorescence reporter techniques [2, 3], fluorescence imaging techniques are capable of tracing biomedical processes at cellular and subcellular levels* in vivo* and noninvasively in wide applications such as gene expression, protein function, and cell therapy [4–8].

Up to the present, a number of fluorescence imaging techniques have been developed [9–15]. Microscopic fluorescence imaging techniques provide high spatial resolutions but suffer from small fields of vision. On the contrary, macroscopic fluorescence imaging techniques can capture whole-body images for small animals but with a limited spatial resolution. Fluorescence planar imaging (FPI) [16–20] is the most widely used macroscopic fluorescence imaging technique, which directly detects the fluorescence photons on the surface of an imaged small animal using camera. According to the locations of excitation light source and camera, FPI can be formed in two different modes [1, 21]: epi-illumination mode and transillumination mode. Epi-illumination mode places excitation source and camera at the same side of the imaged small animal, which collects fluorescent photons in the same direction of the reflected excitation lights; thus it is also called fluorescence reflectance imaging (FRI). The defect of this mode is the difficulty of the excitation of deep fluorochromes. As an alternative, transillumination mode places the imaged small animal between excitation light source and camera. This mode can easily excite the fluorochromes far away from camera but the images are more heavily contaminated by excitation lights than epi-illumination mode although the excitation lights are attenuated by filters.

Whichever mode is applied, FPI is incapable of imaging deep fluorochromes with high spatial resolution. It is well-known that the penetration depth of near-infrared light in tissues is several centimeters [22]. Nevertheless, due to the elastic scattering, near-infrared photons are diffused after several millimeters of propagation in tissues [23]. So the fluorescent images acquired with camera are blurred. The deeper the fluorochromes are, the more strongly the fluorescent photons are diffused and the more blurry the images are. This restricts the applications of FPI in many cases. For instance, when imaging deep tumors, it is difficult to estimate the sizes of tumors through FPI images because they are strongly blurred.

Image restoration techniques aim to eliminate or reduce the impact of image degradation such as blurring. The causations of blurring can be classified into three types [24]: medium-induced, optical, and mechanical. The blurring derived from photon diffusion belongs to the first and second types due to the elastic scattering in medium. Blurring can be described with linear or nonlinear models, which depends on the specific problem. The general linear model can be summarized as [24–26], where denotes noise, is the system matrix, and and are the blurry and expected images, respectively. The key of deblurring is the construction of , which is known as the point spread function (PSF) in many applications. Because the linear model is usually formed with a convolution like [24–26], deblurring is also called deconvolution. During the last two decades, the researches of deblurring in fluorescence imaging focused on microscopic fluorescence imaging techniques [26–33] which are known as techniques with almost no photon diffusion. In these investigations, researchers implemented deconvolution methods to deal with the blurring derived from imaging system, that is, the mechanical type of blurring through PSFs of imaging system.

In this paper, we aim to build a method to reduce the impact of the blurring derived from photon diffusion in FPI. This will be helpful to the estimation of the size of deep fluorochrome. The scheme of the proposed image restoration method is conceived based on a reconstruction scheme in fluorescence molecular tomography (FMT) [34–37], in which the diffusion model [38–40] is used to describe the photon propagation in tissues, the Born approximation [41–43] is applied to solve the diffusion equation, and the Kirchhoff approximation [44, 45] is implemented to obtain Green’s function. Different from the blurring in fluorescence microscopic imaging, the blurring in FPI is not caused by imaging system. Consequently, the construction method of system matrix in fluorescence microscopic imaging is not applicable to the deblurring in FPI. The primary contribution of this work is the construction of the system matrix for FPI. Through introducing the first mean value theorem for definite integrals, we define a new unknown parameter as the restoration target rather than the fluorescent yield. The new unknown parameter is a weighted average of the voxel values of fluorescent yield along detection direction. To construct the system matrix that converts this parameter to the blurry image, depth conversion matrix is defined, which consists of the weights of the voxels with different depths related to the same pixel of the expected image. Subsequently, the elements of depth conversion matrices related to a chosen plane named focal plane are selected to construct the system matrix according to a proportional relationship. Finally, the Levenberg-Marquardt method [46, 47] is applied to solve the system equation and acquire the restored image. Phantom and mouse experiments are carried out to validate the proposed method.

#### 2. Methods

The generation of fluorescence consists of two processes: excitation and emission. In the excitation process, photons from excitation source propagate to fluorochromes. Subsequently, fluorescent photons emitted from fluorochromes propagate to detectors in the emission process. Each process can be modeled by the diffusion equation with Robin-type boundary condition as follows [35–39]:where denotes the position, is the photon density, is the source term, and and are the absorption and diffusion coefficients, respectively. The diffusion coefficient is defined as , where is the reduced scattering coefficient. Ω is the domain of the object and *∂*Ω is the corresponding boundary. denotes the outward normal vector and is a coefficient related to the reflective index mismatch at boundary [40]. For the excitation process, the source term is determined by the location of excitation source and commonly approximated as an isotropy point source located one scattering length below the surface when a collimated source is used [39, 40]. As for the emission process, the source term is determined by the distribution of the photon density for excitation as well as the fluorescent yield of fluorochrome.

In this paper, the Born approximation [41–43] is used to solve the diffusion equations as follows:where is Green’s function solution to the diffusion equation and is the fluorescent yield of fluorochrome. denotes the photon density for emission at the position of detector when a point source is located at position . denotes the photon density for excitation at position when the source is located at . If the excitation source is a point source, is also a Green’s function solution to the diffusion equation; otherwise, is the convolution of Green’s function and the distribution function of the source. is the fluorescent photon density for a pair of source and detector. The analytic formula of Green’s function solution to the diffusion equation can be achieved only for infinite space, semi-infinite space, and several simple geometries. To obtain Green’s function in geometries with arbitrary boundaries, the Kirchhoff approximation is implemented [44, 45].

Let us consider an imaging situation with transillumination mode as shown in Figure 1(a). A collimated source is used to excite fluorochrome and a planar detector is applied to capture fluorescent images. The imaged object is assumed to be a cube with a spherical fluorescent target located at the center. An illustration of the image restoration problem is shown in Figure 1(b). The fluorescent image acquired with the detector should be a blurry image due to the photon diffusion. The image we expect to achieve through the image restoration (hereinafter abbreviated as expected image) should be a projection along the detection direction, that is, -axis.