Mathematical Problems in Engineering

Volume 2019, Article ID 2351878, 7 pages

https://doi.org/10.1155/2019/2351878

## An Analytical Method for Reducing Metal Artifacts in X-Ray CT Images

^{1}College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China^{2}Department of Radiology, Traditional Chinese Medicine Hospital of Huangdao District of Qingdao City, Qingdao 266500, China

Correspondence should be addressed to Ming Chen; moc.361@gnag_nehcgnim

Received 7 November 2018; Accepted 20 December 2018; Published 13 January 2019

Academic Editor: Haipeng Peng

Copyright © 2019 Ming 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

Medical CT imaging often encounters metallic implants or some metal interventional therapy apparatus. These metallic objects can produce metal artifacts in reconstruction images, which severely degrade image quality. In this paper, we analyze the difference between polychromatic projection data and Radon transform data and develop an analytical method to reduce metal artifacts. Approximate features of metal artifacts can be obtained by a simplified energy spectrum function of x-ray beam. The developed method can reduce most artifacts, and preserve more original details. It does not require prior knowledge of x-ray energy spectrum and original projection data, avoiding iterative calculation and saving reconstruction time. Simulation experimental results show that the method can greatly remove metal artifacts.

#### 1. Introduction

X-ray CT imaging plays an indispensable role in clinical diagnosis and interventional therapy. However, due to beam-hardening caused by some metallic implants in patients or some metallic interventional therapy apparatus, such as dental fillings, orthopaedic implants, or microwave ablation needle, there are strong streak or star-shape artifacts in reconstruction images [1–3], which are usually called metal artifacts. They seriously degrade image quality and bring trouble to clinical applications. Although many metal artifact reduction (MAR) methods were developed, their applications in clinical settings are not totally successful because of the complexity of their forming cause and characteristics. Currently there is no standard solution [4, 5]. Therefore, how to reduce metal artifacts still remain a challenging problem in x-ray medical CT imaging.

The effects of x-ray beam-hardening, the photon starvation, and the partial volume can all result in metal artifacts [6]. In the past few decades, a large number of MAR algorithms have been proposed to correct or reduce metal artifacts. Interpolation was widely used for data completion [7–10], where missing projection data are approximated by an interpolation technique. But due to the inaccuracy of data interpolation, additional fringe artifacts and other deformations were introduced in a new reconstruction image [10, 11]. The missing data became more accurate by use of forward projections of a prior image [12, 13]. A combination method of normalization and interpolation was proposed to remove most of the artifacts [11]. However, since artifacts are very strong for some cases, some pixels are often classified into wrong types, leading to unsatisfactory results. In order to correct beam-hardening, some iterative algorithms were proposed, which reconstructed images from some processed projections [14–16]. They can suppress some artifacts, but there is still no satisfactory result for all images. Recently, there were some researches about the deep learning strategy to reduce metal artifacts [17, 18]. One of their drawbacks is that there is no common CT image database for model training, and another is that some mild artifacts typically still remain.

The forming cause of metal artifacts is mainly the high attenuation of metallic objects, which leads to x-ray beam-hardening and aggravates the scattering phenomenon for the polychromatic x-ray beam spectrum. For low-atom number metals, satisfactory results were achieved by correcting beam-hardening [19–25]. Some dual-energy correction [21] and statistical iterative correction [23, 24] were proposed to reduce beam-hardening effects. The former required longer postprocessing and higher doses of radiation, while the latter needed more prior information about the energy spectrum of the incident x-ray and the energy-dependent attenuation coefficient of the materials. Park et al. [25] put forward a MAR method by giving the approximate expression of metal artifacts, where choosing the approximate alternative of energy spectrum is very critical to obtain more accurate geometrical characterizations of artifacts.

For a polychromatic x-ray spectrum CT imaging, this paper analyzes the geometrical features of metal artifacts in a two-dimensional fan-beam system, constructs an approximate energy spectrum function, and gives an approximate expression of metal artifacts by using an excess photon energy relation. The artifacts can be analytically expressed as the approximate mathematical equation in this method.

The rest of this paper is organized as follows. In Section 2, we summarize some background knowledge about Radon transform and the projection expression for a polychromatic x-ray spectrum. In Section 3, an analytical MAR method is developed by looking for geometrical characterizations of metal artifacts. In Section 4, we give numerical simulations to verify the effectiveness of the developed method. Finally, conclusions will be made and some related issues will be discussed.

#### 2. Background Knowledge

In this section, we give Radon transform of a two-dimensional (2D) function and projection expressions for a polychromatic x-ray spectrum. This section will give theoretical basis for finding the relations between single-energy projection data and polychromatic projection data in next section.

Mathematically, 2D Radon transforms can be seen as a line integral process from one 2D function to another. Let denote Radon transform. For and a line , can be expressed as follows:orwhere is a reconstructed point, is a arc length on , is a projection angle, and is a sampling variable along a detector direction. In fact, (2) is also called ray sum, line integral, or the projection under single energy [26].

When is fixed and is in (2), is a set of parallel projections shown in Figure 1. For all , is usually called parallel-beam projections in CT imaging. The expression of a fan-beam projection for single energy is similar to (2) by converting the parameters of parallel-beam scan mode to those of fan-beam scan mode.