Computational and Mathematical Methods in Medicine

Volume 2018, Article ID 5284969, 5 pages

https://doi.org/10.1155/2018/5284969

## The Time-Domain Integration Method of Digital Subtraction Angiography Images

^{1}School of Biological Sciences and Medical Engineering, Southeast University, Nanjing 210096, China^{2}Shanghai United Imaging Healthcare Co., Ltd., Shanghai 201807, China^{3}Department of Radiology, The Affiliated Drum Tower Hospital of Nanjing University Medical School, Nanjing 210008, China^{4}Institute of Cancer and Genomic Science, University of Birmingham, Birmingham B15 2TT, UK

Correspondence should be addressed to Yu Sun; moc.qq@8145566451

Received 21 May 2018; Accepted 28 August 2018; Published 30 September 2018

Academic Editor: Dominique J. Monlezun

Copyright © 2018 Shuo Huang 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

The clarity improvement and the noise suppression of digital subtraction angiography (DSA) images are very important. However, the common methods are very complicated. An image time-domain integration method is proposed in this study, which is based on the blood flow periodicity. In this method, the images of the first cardiac cycle after the injection of the contrast agent are integrated to obtain the time-domain integration image. This method can be used independently or as a postprocessing method of the denoising method on the signal image. The experimental results on DSA data from an aortic dissection patient show that the image time-domain integration method is efficient in image denoising and enhancement, which also has a good real-time performance. This method can also be used to improve the denoising and image enhancement effect of some common models.

#### 1. Introduction

Circulatory system diseases, such as aortic dissection, have been the focus of medical research [1, 2] for their dangerousness and high incidence. Improving the clarity of the captured medical image helps us to diagnose more accurately, which is an important research field.

Digital subtraction angiography (DSA), as a real-time approach, is commonly employed in the clinical diagnosis of circulatory system disease [3, 4], especially in the real-time surgical monitoring and the medical examination among small branches of blood vessels which are difficult to be measured by other methods. In order to protect the patient, it is important to shorten the shooting time and to reduce the dosage of contrast media when capturing the images.

A lot of image denoising methods have been proposed in recent years, but they are all problematic when applied to the DSA images. For example, the image reconstruction method based on the level set theory [5], wavelet decomposition and reconstruction method [6, 7], Bayesian method [8], and image denoising method based on anisotropic diffusion [9] generally need a long operation time, which cannot meet the real-time requirements of DSA image processing. Moreover, images processed by these approaches are usually not clear enough to show the details such as edges and textures. In 2004, Candes et al. [10] proposed an image denoising method based on the sparse decomposition. On this basis, Needell and Vershynin [11] proposed the regularized orthogonal matching pursuit (ROMP) method; Scholefield and Dragotti [12] used a sparse quadtree decomposition representation to remove the noise in images; Adler et al. employed the shrinkage learning approach to acquire the high-resolution reconstruction image [13–17]. However, the operation of these approaches is also very complicated. Therefore, it is necessary to find an image processing method which is more suitable for the real-time analysis of DSA images.

In this work, an image time-domain integration method based on blood flow periodicity has been proposed. In this algorithm, the DSA images of the first cardiac cycle after the injection of the contrast agent are extracted denoised by the wavelet reconstruction method firstly, and then these images are integrated to obtain the time-domain integration image, which is named after the TDI image in this paper. This method contributes to the diagnosis of circulatory system diseases.

#### 2. Materials and Methods

##### 2.1. The Noise Model

The theoretical gray-scale at a certain pixel on the *j*-th frame of DSA images can be obtained from the following equation by the Lambert–Beer law [18, 19]:where and are the X-ray transmission amount before and after the addition of the contrast agent, respectively. is the volume of blood vessels at pixel . and are the number and amount of substance concentration of contrast agent particles at pixel , respectively. donates the absorption coefficient.

The image quality degrades in the original DSA image, as a result of the limitations on the imaging system’s resolution and the influence of additive noise such as Gaussian noise, which is donated by and can be expressed in as follows [6]:where and represent the point spread function and the additive noise, respectively. Operator “” is the convolution operator. Equation (1b) can be rewritten to matrix form using the block Toeplitz matrix , as shown in the following equation:

It is difficult to solve Equation (1c) when only is given. However, since the gray-scale level of a certain pixel is proportional to the number of contrast agent particles in that pixel, the number of contrast agent particles follows the motion pattern of blood. And as for the blood motion pattern, on consideration of the periodicity of human heartbeat, the blood flow rate in human body is also cyclical, which can be expressed in the following equation:where is the velocity field of blood at time . denotes the average velocity field of blood at time , which is the mean flow velocity at the same time in multiple cardiac cycles. is the length of the cardiac cycle. characterizes the changes in flow velocity owing to factors such as the instability of human blood pressure. can be regarded as a zero-mean-value distribution with a small variance, since patients are under the total anesthesia during the shooting process and their vital signs remain stable. According to the Wilke–Chang equation [20], the free diffusion rate of the contrast agent in the blood is much smaller than the blood flow rate, and thus the contrast agent obeys the same movement law as the blood. Therefore, the periodicity of the blood flow rate can be employed to improve the clarity of DSA images.

##### 2.2. Image Integration

In order to decrease the shoot time and the contrast agent’s injection quantity, images in the first cardiac cycle (donated by the cardiac cycle *S*) after the injection of the contrast agent are analyzed in this study. Firstly, the time at which the cardiac cycle begins is set to be . Subsequently, the velocity of the *i-*th contrast agent particle in this cardiac cycle is expressed as . Since the motion of the contrast agent particles is consistent with that of the blood, and on consideration of the velocity stability shown in Equation (2), the velocity of the particle at each position on its trajectory can be regarded as a sample of the blood flow field at that location. Therefore, once the substantial number of particles is extracted, the average velocity field of the contrast agent at the pixel in the entire cardiac cycle can be estimated by the mean velocity field value of particles which flowed through that pixel during the calculated cardiac cycle, as shown in the following equation:where represents the time when the *i*-th particle approached pixel .

The total time length that the *i*-th particle appears in pixel during one cardiac cycle satisfies Equation (3b), where represents the distance of the *i*-th particle in the range of pixel and is set to be the magnitude of velocity component which is parallel to the image plane in pixel during that cardiac cycle since the photographing each DSA image can be regarded as a sample of each particle’s location:

A variate is set to represent the absorption capacity of light in the unit time of a single contrast agent particle. After that, the time-weighted gray-scale value of the *i-*th particle at pixel , , can be expressed by the following equation:

On combination of Equations (3a)–(3c), can be characterized by the sum of the time integral intensities of particles which have appeared in pixel during the cardiac cycle, as shown in the following equation:where is the average moving distance of the contrast agent particles within that pixel. Since the size of a pixel is small, is approximately equal to the length of each pixel, . According to Equations (1a) and (3d), when the frame rate tends to infinity, the following equation can be obtained:

Equation (3e) demonstrates that the overall time-domain integration value of pixel , , can be expressed as the integral of each picture’s gray value at that position in the entire cardiac cycle. On consideration that the time step is short in the actual case, Equation (3e) can be employed in the calculation of the captured DSA images. Therefore, the relationship shown in Equation (3f) can be established. And the image is named after the time-domain integration image or the TDI image:

Furthermore, to strength the denoising effect, the images are denoised by the median filter before the time-domain integration since Ling’s work [21] shows that the noiseless image is usually insensitive to a median filter. Table 1 shows the specific steps of our method.