Radiology Research and Practice

Volume 2018, Article ID 6709525, 13 pages

https://doi.org/10.1155/2018/6709525

## MRI-Based Quantification of Magnetic Susceptibility in Gel Phantoms: Assessment of Measurement and Calculation Accuracy

Department of Medical Radiation Physics, Lund University, Skåne University Hospital Lund, 22185 Lund, Sweden

Correspondence should be addressed to Ronnie Wirestam; es.ul.dem@matseriw.einnor

Received 23 May 2018; Accepted 5 July 2018; Published 30 July 2018

Academic Editor: Paul Sijens

Copyright © 2018 Emma Olsson 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 local magnetic field inside and around an object in a magnetic resonance imaging unit depends on the magnetic susceptibility of the object being magnetized, in combination with its geometry/orientation. Magnetic susceptibility can thus be exploited as a source of tissue contrast, and susceptibility imaging may also become a useful tool in contrast agent quantification and for assessment of venous oxygen saturation levels. In this study, the accuracy of an established procedure for quantitative susceptibility mapping (QSM) was investigated. Three gel phantoms were constructed with cylinders of varying susceptibility and geometry. Experimental results were compared with simulated and analytically calculated data. An expected linear relationship between estimated susceptibility and concentration of contrast agent was observed. Less accurate QSM-based susceptibility values were observed for cylindrical objects at angles, relative to the main magnetic field, that were close to or larger than the magic angle. Results generally improved for large objects/high spatial resolution and large volume coverage. For simulated phase maps, accurate susceptibility quantification by QSM was achieved also for more challenging geometries. The investigated QSM algorithm was generally robust to changes in measurement and calculation parameters, but experimental phase data of sufficient quality may be difficult to obtain in certain geometries.

#### 1. Introduction

An object in an external magnetic field will become magnetized to a degree that is determined by the magnetic susceptibility of the object. The local magnetic field inside and in the surroundings of the object will thus depend on the magnetic susceptibility in combination with the geometry/orientation of the object being magnetized. In susceptibility-weighted magnetic resonance imaging (SWI), the local magnetic field distribution is explored to enhance image contrast and to improve the visibility of various structures on the basis of their magnetic susceptibility [1]. Additionally, quantitative susceptibility mapping (QSM) has developed into a promising method for calculating arbitrary magnetic susceptibility distributions from measured magnetic resonance imaging (MRI) phase data [2–4]. A more complete understanding of the phase behaviour in vivo may also require biophysical considerations of microstructural tissue anisotropy and magnetic susceptibility anisotropy [5].

In vivo, the magnetic susceptibility differs among tissue types and tissue regions, and it can thus be exploited as a source of contrast in MRI. For example, in reference to the cerebrospinal fluid (CSF), i.e., when the CSF magnetic susceptibility is set to 0 ppm, the globus pallidus, which is part of the basal ganglia, has a magnetic susceptibility of 0.105 ppm while white matter has a lower susceptibility of -0.030 ppm [6]. In addition to the effects of normal ageing [7], a number of conditions and processes can alter the magnetic susceptibility of tissue. For example, iron accumulation in inflamed myelin cells, as in a multiple sclerosis (MS) plaque, increases the susceptibility of the myelin [8]. Iron accumulation is also seen in other neurodegenerative diseases, for example, Alzheimer’s and Parkinson’s diseases [9]. The magnetic susceptibility is also dependent on the oxygen saturation level of blood, and the susceptibility increases with increasing levels of deoxyhemoglobin. Hence, venous blood will show a higher susceptibility than arterial blood, and quantification of magnetic susceptibility can thus be useful in estimations of oxygen extraction fraction (OEF) [10] and cerebral metabolic rate of oxygen (CMRO_{2}) [11]. The change in susceptibility with deoxygenation can also be manifested, for example, in extravasated blood from an intracranial haemorrhage [12]. Quantitative measurements of the susceptibility could also be potentially useful to determine the concentration of an external MRI contrast agent (CA). Relaxivity-based CA quantification, which is the currently most common approach, is associated with several methodological complications in, for example, perfusion and permeability measurements using dynamic contrast-enhanced MRI (DCE-MRI) and dynamic susceptibility contrast MRI (DSC-MRI) [13]. Hence, more accurate CA concentration quantification in vivo would indeed be beneficial, and a few examples of dynamic contrast-enhanced QSM studies have been presented [14–16].

In QSM applications, quantification of magnetic susceptibility in absolute terms is becoming increasingly important, and extensive validation is thus warranted. A number of QSM reconstruction tools exist, and the process of systematically characterizing differences in accuracy between algorithms has recently been initiated by other groups [17–19]. In experimental evaluations, phantoms have the advantage of offering well-defined contents and geometries and constitute an important, though not complete, part of the validation process, and the present investigation serves as a supplement to previous investigations related to the accuracy of phase and susceptibility quantification [e.g., [20–25]]. In the present study, previously described in preliminary terms by Olsson (unpublished report) [26], the QSM approach was evaluated in gel phantoms with inserted cylinders containing known concentrations of gadolinium CA, to establish whether the QSM method can deliver accurate results with respect to quantitative magnetic susceptibility values in absolute terms. Measured phase values and the corresponding magnetic susceptibility estimates, calculated by an established QSM algorithm, were compared to values based on theoretical relationships. Various phantom designs and simulated susceptibility distributions, as well as different parameters and settings in the measurements and in the susceptibility calculation, were investigated in order to establish optimal settings and important sources of error in the attempts to produce accurate magnetic susceptibility maps.

#### 2. Theory

##### 2.1. The Dipole Field and the Magic Angle

A magnetic moment with magnitude , pointing in the direction, produces a magnetic flux density component :where* d* is the distance from* m* and is the angle relative to the z-axis. The angle at which the factor equals zero is called the magic angle (i.e., approximately ±54.7° or 180°±54.7°). At the magic angle positions, the magnetic flux density component will be zero independently of the magnitude of the magnetic moment.

##### 2.2. Phase Shift and Magnetic Field

Variations in the local magnetic field with position* r* lead to differences in MRI resonance frequency and to subsequent phase-shift variations, and the MRI phase evolution* ϕ(r)* is given bywhere TE is the echo time. In order for the measured phase images to be useful, unwrapping and filtering of background field variations are required. The unwrapping can be accomplished by a region growing algorithm which identifies phase gradients that correspond to a difference by a multiple of 2 and subsequent addition or subtraction of 2

*π*[27]. Filtering is needed because the unwrapped image usually contains a remaining background phase gradient over the entire image. This phase does not arise from the susceptibility distribution inside the object but from, for example, imperfect shimming or susceptibility sources outside the imaging volume.

*Projection onto Dipole Fields*(PDF) [28, 29] is one method for background field removal that compares magnetic fields generated from magnetic dipoles inside and outside a region of interest. Other examples of filtering methods are

*Laplacian Boundary Value*(

*LBV*) [30] and

*Regularization Enabled Sophisticated Harmonic Artefact Reduction for Phase data*(

*RESHARP*) [31].

##### 2.3. Cylindrical Objects

The internal (in) and external (ex) magnetic field alterations B, caused by an infinitely long cylinder, are given by the following analytical expressions:where is the difference in susceptibility between the inside and the outside of the cylinder, is the radius of the cylinder, is the angle between the direction of the B_{0} field and the cylinder axis, and and are the cylindrical coordinates describing a point at distance and at an angle relative to a point at the centre of the cylinder.

##### 2.4. Magnetic Susceptibility and Magnetic Field

For more complicated geometries or shapes, the local field change caused by the introduction of an object in the external magnetic field can be described more generally [32, 33] and is often formulated as a convolution (denoted “”) of the arbitrary susceptibility distribution with a dipole field kernel; i.e., the corresponding phase is given bywhere and are spherical coordinates and denotes the magnetic susceptibility. The main idea of QSM is to extract the susceptibility distribution according to (5), using the information of the local magnetic field from the measured phase images. However, problems arise because the dipole kernel is zero at the magic angle. A convolution in real space represents a multiplication in k-space, and extracting the susceptibility distribution from (5) by deconvolution would therefore imply a division by zero at some coordinates in k-space which would, in principle, affect every point of the *(r)* solution in real space.

*Morphology Enabled Dipole Inversion *(MEDI) [29, 34–36] is a QSM reconstruction method, designed to solve the ill-posed inverse problem of resolving* χ(r)* according to (5). In the MEDI approach, the problem is formulated so that the difference between an estimated field map and the measured field map should be of the order of the noise level

*. This can be written aswhere is a weighting matrix, is the measured field, and*

*ε**D*is the representation of the dipole field in k-space. “

*FT*” and “

*FT*

^{−1}” denote the forward and inverse Fourier transform, respectively. Additionally, MEDI uses the fact that changes in susceptibility follow the morphological boundaries and that the susceptibility map therefore should have gradients in the same locations as the magnitude image [35].

In brief, the inverse problem is solved through an iterative process. An initial guess is made for the susceptibility distribution. Convolving this with the dipole kernel gives an estimated field map. The estimated field map is compared to the measured field map, i.e., the phase image, and the difference, the error, is used to update the initial guess. The updated susceptibility distribution is then used as input when this procedure is repeated. Iterations are made until the result fulfils the requirements. A regularization parameter determines how much magnitude versus phase image information is prioritized. The Lagrange multiplier method is used to reformulate the problem in (6) as a minimization of a cost function [35].

#### 3. Materials and Methods

##### 3.1. Phantom Design

In order to evaluate the QSM method with respect to phase measurement as well as mathematical reconstruction, three different phantoms were constructed. Thin-walled plastic cylinders were filled with a paramagnetic gadolinium (Gd) contrast agent solution (Dotarem, Guerbet, France). The employed plastic material and the low thickness of the cylinder walls (of the order of 100 *μ*m) imply that the susceptibility effects created by the cylinders should be negligible. The cylinders were sealed and glued onto the inside of a larger container. The container with cylinders was subsequently filled with agarose gel doped with a small amount of nickel in the form of nickel(II)nitrate hexahydrate, Ni(NO_{3})_{2}·6H_{2}O. The gel was designed according to a locally developed preparation routine using, in this study, 1% agarose and 0.24 mM Ni^{2+} [37]. The susceptibility of the gel was calculated using Wiedemann’s additivity law for the susceptibility of mixtures, i.e., *, *where is the concentration of substance [38].

The purpose of the contrast agent was to obtain a controlled increase of the susceptibility inside the cylinders to achieve a difference in susceptibility between the cylinders and the background, resembling different compartments in the human body (including cases of injected external contrast agent, for example, for the purpose of perfusion imaging). Table 1 shows the theoretical absolute values used for the susceptibility of water and nickel, as well as the most commonly reported value of the molar susceptibility for gadolinium (used as a reference value in this study [39]). The calculated susceptibility values for the gel and the 0.5 mM gadolinium solution are also included.(i)In the first phantom design, cylinders with 5 mm diameter, filled with 0.5 mM Gd solution, were positioned at five different angles relative to the main magnetic field (approximately 0, 30, 55, 75, and 90°). The actual angles were measured in the resulting images.(ii)The second phantom design consisted of 5 mm diameter cylinders in parallel, with varying concentrations of Gd contrast agent, i.e., 0, 0.2, 0.4, 0.6, 0.8, 1, 2, 4, 6, 8, 10 mM.(iii)In the third phantom, cylinders in parallel, containing 0.5 mM Gd solution with diameters of 2, 2.6, 4.7, 5, 7.4, 9, 10.8 mm, were used.