International Journal of Biomedical Imaging

Volume 2016 (2016), Article ID 6120713, 11 pages

http://dx.doi.org/10.1155/2016/6120713

## An Analytical Approach for Fast Recovery of the LSI Properties in Magnetic Particle Imaging

^{1}Robots & Intelligent Systems Lab, Gyeongsang National University, Jinju, Republic of Korea^{2}School of Mechanical & Aerospace Engineering & ReCAPT, Gyeongsang National University, Jinju, Republic of Korea

Received 2 May 2016; Revised 18 August 2016; Accepted 22 September 2016

Academic Editor: D. L. Wilson

Copyright © 2016 Hamed Jabbari Asl and Jungwon Yoon. 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

Linearity and shift invariance (LSI) characteristics of magnetic particle imaging (MPI) are important properties for quantitative medical diagnosis applications. The MPI image equations have been theoretically shown to exhibit LSI; however, in practice, the necessary filtering action removes the first harmonic information, which destroys the LSI characteristics. This lost information can be constant in the -space reconstruction method. Available recovery algorithms, which are based on signal matching of multiple partial field of views (pFOVs), require much processing time and* a priori* information at the start of imaging. In this paper, a fast analytical recovery algorithm is proposed to restore the LSI properties of the -space MPI images, representable as an image of discrete concentrations of magnetic material. The method utilizes the one-dimensional (1D) -space imaging kernel and properties of the image and lost image equations. The approach does not require overlapping of pFOVs, and its complexity depends only on a small-sized system of linear equations; therefore, it can reduce the processing time. Moreover, the algorithm only needs* a priori* information which can be obtained at one imaging process. Considering different particle distributions, several simulations are conducted, and results of 1D and 2D imaging demonstrate the effectiveness of the proposed approach.

#### 1. Introduction

Magnetic particle imaging (MPI) is a new method for imaging the spatial distribution of magnetic nanoparticles, as tracers, with high resolution. The method was proposed by Gleich and Weizenecker [1] and exploits the nonlinear magnetization response of the nanoparticles to a time-variable magnetic field and allows for fast image acquisition. MPI has many applications, especially in medical diagnosis such as blood flow visualization for coronary artery diseases, cancer detection [2, 3], stem cell tracking [4], and molecular imaging [5].

MPI uses an oscillating drive field (excitation field) of sufficient amplitude to change the magnetization of the nanoparticles, which induces a voltage signal in the receive coils. To enable spatial encoding of the information, a static magnetic gradient field, also known as the selection field, is utilized in MPI. This field contains a spatial location named a field-free point (FFP) that has zero field magnitude, and only the particles located at the FFP induce the MPI signal in the receive coils [6–8].

Image reconstruction in MPI includes two main approaches [9]. The first approach makes use of the system matrix. This method, also known as frequency space reconstruction, employs a large system matrix and requires its inversion and some postprocessing to deal with poor conditioning. The approach has high computational load and requires the system matrix to be estimated before imaging [10–12]. The second approach is the -space method, which was introduced by Goodwill et al. [13–15]. This approach does not require matrix inversion or precharacterization and hence provides a robust reconstruction algorithm with a potential for real-time imaging.

Linearity and shift invariance (LSI) are important characteristics of most medical diagnostic systems. For example, since X-ray computed tomography (CT) is LSI, the CT images of tissue attenuation coefficient maps provide quantitative lumen diameters for cardiovascular diagnosis [16]. It has been theoretically proven that the MPI is an LSI system [17]. However, in practice, to detect MPI signal, it is necessary to utilize a filter to remove the drive field signal at its fundamental excitation frequency, which is an unavoidable phenomenon in current MPI techniques. This filtering action also removes the induced particle signal at the excitation frequency and hence the complete MPI signal is not available in practice which destroys the LSI properties of the MPI image.

To recover the complete MPI signal to facilitate the use of the fast -space imaging method in qualitative medical diagnosis, an algorithm is presented in [16] and is optimized in [18]. The approaches use the partial field of views (pFOVs), which are primarily generated for enlarging the FOV. It is shown that the lost information can be recovered by matching the two successive overlapped pFOVs, provided the particle concentration at one reference location is known. Although the methods provide satisfactory results, the computational load is reported in [19] to be relatively high, which increases the imaging time and so this cannot be used for ultrafast imaging.

In this paper, the effect of the filtered MPI image is analyzed and it is shown that its value is constant provided that the trajectory of the FFP is generated by a harmonic function, a property that has to be satisfied in both one-dimensional (1D) and multidimensional imaging. Assuming a constant image loss, a fast analytical algorithm is proposed to restore the lost information. The approach can be real time since its complexity depends on solution of a small-sized system of linear equations and does not require overlapping of pFOVs. The method utilizes the derivative of the magnetization curve of the particles and a mathematical model of the -space image. It also requires* a priori* information regarding the system, which can be obtained experimentally or estimated theoretically prior to imaging. This is another advantage of the proposed approach in comparison to previous works, which require a boundary condition to be obtained at the start of imaging. Several 1D and 2D imaging simulations are presented to demonstrate the effectiveness of the proposed algorithm.

#### 2. Methods

The developed recovery algorithm is based on the mathematical model of the MPI signal. In this section, the signal model of the -space method is presented and a model of the term removed from the signal as a result of filtering the fundamental frequency of the excitation signal is derived. Then, a new algorithm is proposed to recover the lost image.

##### 2.1. Mathematical Model of a 1D MPI Signal

In this section, the mathematical model of the 1D MPI signal is derived which is used in Section 2.2 to model the -space reconstruction method and to analyze the LSI properties of the -space.

When the excitation field in MPI is periodic, the voltage signal , induced by the particles’ magnetization, is periodic as well, and therefore it can be expanded into a Fourier series as follows:where are the Fourier coefficients and with as the frequency of the excitation field. The coefficients can be written aswhere .

Assuming a 1D distribution of the particles in the -direction and a constant sensitivity for the receive coil in this direction, the induced voltage by the time-varying magnetization of the particles can be written as follows [17]:where denotes the permeability of free space and is the particle magnetization, which depends on the magnetic field . The particle magnetization can be modeled using the Langevin function as follows:where is the particle magnetic moment, is a property of the magnetic particle, and is the particle concentration that has to be measured in an MPI image. According to (3), the time derivative of magnetization is of interest and can be written asOn the other hand, the magnetic field that particles experience in MPI is as follows:where is the selection field and is the excitation or drive field. Therefore, using (6), (5) can be written aswhere . Now, using (3) and (7), (2) can be written as follows:By defining the function(8) can be written as are the so-called* system function*, which in fact are the Fourier coefficients of an induced voltage by a point-like distribution of particles at position .

*Remark 1. *The region of integration in (3) is “object” if the particles are ideal (i.e., have a step-like magnetization cure). When the particle magnetization is modeled by the Langevin function, the field of view is slightly different from the object to be imaged. Therefore, in the remaining equations, the region of integration is replaced by “FOV.”

For a homogeneous harmonic drive field with cosine waveform asand a linear selection field , where is the gradient of the selection field in the -direction, it has been shown in [20] that can be written by a convolution aswhereis the Chebyshev polynomial of order . Now, substituting (12) in (10), one haswhere represent the system function for the ideal particles, which are defined asNow, one can rewrite (14) as follows:whereFrom (15) and (16), one can conclude that correspond to coefficients of a Chebyshev series (see, e.g., [21]), and therefore can be written based on the Chebyshev series as follows:

This equation is used in the following subsection to describe the 1D -space reconstruction method.

##### 2.2. -Space Imaging and LSI Characteristics

In the -space reconstruction method, the image is defined to be at position , where denotes the position of the FFP. can be measured from the induced voltage through the following relation [13]:where is the time derivative of , that is, the velocity of the FFP. The formulation of by a convolution in (17) demonstrates that the -space MPI image is a linear and shift-invariant image, properties which are characteristic of most clinical imaging systems, for example, ultrasound, CT, and magnetic resonance imaging (MRI). However, to acquire an MPI image while retaining its LSI properties, one needs the measurement of the true value of , which is not available in practice. In fact, a band-stop filter is used in practice to segregate from the signal induced by the drive field (11) in the receive coil. The filter suppresses the first harmonic of the signal at the frequency of the drive field and leaves the signal at higher harmonics.

According to (13) and (18), the filtered first harmonic can be written as [16]which is a constant value in the image. It should be noted that since is the Fourier coefficient of an odd real function (as a result of the sinusoidal drive field, is an odd function), its value is imaginary and therefore, according to (20), the lost image is a real constant value. Using (15) and (16), one can write (20) as follows:which shows that the lost value varies when is shifted due to the velocity term , and hence its envelope at different positions resembles the sinusoidal excitation pattern (note that ). This means that the lost value is maximal at the center of the FOV and minimal at its edges. In the next subsections, the effect of this constant loss is analyzed and a compensation algorithm is proposed.

##### 2.3. LSI Analysis

According to the definition, the MPI image is linear when the image intensity is linearly proportional to the concentration of the particles and is shift-invariant when the image intensity is independent of the location of the particles. Considering the expression of the lost image in (21) and the fact that the image loss depends on the location of the particles, it can be verified that the lost information destroys the LSI properties of the MPI image [16].

To visualize the abovementioned phenomenon, consider Figure 1 which shows the -space MPI images of a set of magnetic nanoparticles located at different positions of the FOV with concentration . As demonstrated, there is a difference in the maximum value of the acquired image intensity for the same particles at different locations. Also, it is clear from this figure that the lost constant value is larger when the particles are located at the center of the FOV, since the velocity of the FFP is maximum at the center.