Quantitative Susceptibility Mapping of Small Objects using Volume Constraints

Abstract

Microbleeds have been implicated to play a role in many neurovascular and neurodegenerative diseases. The diameter of each microbleed has been used previously as a possible quantitative measure for grading microbleeds. We propose that magnetic susceptibility provides a new quantitative measure of extravasated blood. Recently, a Fourier based method has been used that allows susceptibility quantification from phase images for any arbitrarily shaped structures. However, when very small objects, such as microbleeds, are considered, the accuracy of this susceptibility mapping method still remains to be evaluated. In this paper, air-bubbles and glass beads are taken as microbleed surrogates to evaluate the quantitative accuracy of the susceptibility mapping method. We show that when an object occupies only a few voxels, an estimate of the true volume of the object is necessary for accurate susceptibility quantification. Remnant errors in the quantified susceptibilities and their sources are evaluated. We show that quantifying magnetic moment, rather than the susceptibility of these small structures, may be a better and more robust alternative.

Introduction

The measurement of magnetic susceptibility offers an entirely new form of contrast in magnetic resonance imaging (1–6). More specifically, susceptibility quantification has already found applications in mapping out iron in the form of ferritin in brain tissues such as the basal ganglia (1, 4, 6) and in the form of deoxyhemoglobin for measuring the oxygen saturation in veins (1, 4). This new form of imaging may provide a means for monitoring longitudinal changes in iron content in dementia, multiple sclerosis (MS), traumatic brain injury (TBI) and Parkinson’s disease (PD). It may also be used to monitor microbleeds which have been implicated in the progression of vascular dementia (7), Alzheimer’s and other neurovascular disorders (8, 9).

One of the most recent susceptibility mapping methods is a Fourier based method (2, 3, 5, 10) which utilizes phase images. The accuracy of such a method depends on the volume measurement of the object. For example, in order to quantify the susceptibility of a given microbleed, usually the center and the radius of the microbleed have to be determined (10–13). Alternate volume estimations of the microbleed from high resolution spin echo images may overcome these limitations. With a gradient echo sequence, the apparent volume of the object is increased due to what is commonly referred to as the “blooming” effect, a signal loss around the object caused by T2* dephasing. This increased apparent volume may be used to obtain an estimated susceptibility of the object while the product of the apparent volume and the estimated susceptibility is much more robust and should still provide a good estimate of the magnetic moment of the object.

The goal of this paper is to evaluate the quantitative accuracy of a Fourier based susceptibility mapping method when it is applied to small structures, and to show that: 1) an accurate estimate of the magnetic moment is possible using multi-echo gradient echo imaging; and 2) the accuracy of the effective susceptibility can be improved using the magnetic moment when an estimate of the true volume is available. For validation, we used a gel phantom with air bubbles and glass beads to mimic the clinical situation of microbleeds. The method illustrated here does not depend on the susceptibility value or the size of the object.

Theory

Current susceptibility mapping methods are based on the relation between the susceptibility distribution and the magnetic field variation in the Fourier domain (1–6; 10):

$$\mathrm{\Delta }B(k)=G(k)\Delta \chi (k)$$ [1]

where ΔB(k) is the Fourier transform of the magnetic field variation ΔB(r), Δχ(k) is the Fourier transform of the susceptibility distribution Δχ(r), and G(k) is the Green’s function

$$G(k)=1/3-{k_z}^2/({k_x}^2+{k_y}^2+{k_z}^2)$$ [2]

assuming that the main field direction is in the z direction. Susceptibility quantification is an ill-posed inverse problem, due to zeros in the Green’s function G(k) along the magic angle in the k-space domain. As a result, regularization is required, as G(k)⁻¹ is needed in the evaluation of susceptibility. In this study, we applied the regularization procedure described in a previous study (1) in which the intensity of G(k)⁻¹ is reasonably attenuated when the absolute value of G(k) is below a threshold value. The selection of this threshold is a trade-off between the susceptibility-to-noise ratio of the reconstructed susceptibility map and the accuracy in susceptibility quantification (1, 6). The threshold value was chosen to be 0.1 in this study.

Although ideally Δχ is the sought after parameter, when reduced resolution or T2* effects confound a clean measurement of the object’s volume, it is more appropriate to investigate the associated magnetic moment (or, equivalently, the total or integrated susceptibility weighted by the voxel volume (12)). To see why this is the case, consider a sphere with a susceptibility difference Δχ, the induced magnetic field at point P(r, θ) outside the sphere is given by (14):

$$\mathrm{\Delta }{B}_{\mathit{\text{out}}}(r)=\frac{\mathrm{\Delta }\chi {r_0}^3(3{\text{cos}}^2\theta -1)B_0}{3r^3}$$ [3]

where Δχ=χ_in-χ_out, χ_in is the susceptibility inside the object, χ_out is the susceptibility outside the object, r₀ is the radius of the sphere, r is the distance from the point P(r, θ) to the center of the sphere, and θ is the angle between the point P and the main field direction. For simplicity, the meaning of susceptibility in this paper will be taken to be Δχ rather than χ_in or χ_out. Eq. [3] also indicates that the product ΔχV is independent of TE, where V=4πr₀³/3 is the true volume of the sphere. Since the magnetic dipole moment of the spherical object is given as (14):

$$\mu =4\pi {r_0}^{3}M(\mathrm{\Delta }\chi )/3\approx 4\pi {r_0}^3\mathrm{\Delta }\chi B_0/(3{\mu }_0),$$ [4]

when Δχ is much smaller than 1. For simplicity we refer to the product ΔχV as the magnetic moment in this study. The phase value at a particular echo time (TE) is given in a right handed system by:

$$\mathrm{\Delta }\phi (r)=-\gamma \mathrm{\Delta }{B}_{\mathit{\text{out}}}(r)TE.$$ [5]

The susceptibility Δχ may be quantified using the phase information, if the true volume (V) of the object is known. Otherwise, the magnetic moment (ΔχV) may be found. Since gradient echo images lead to a dephasing artifact and the object appears larger than its actual size, we defined an apparent volume V′ and assuming that the susceptibility of this larger object can be accurately quantified, the magnetic moment could still be accurately calculated. Expressed symbolically, an estimated susceptibility value Δχ′ can be calculated from the Fourier based method using Eq. [1] to Eq. [5]. The quantity Δχ′V′ provides an estimate of the magnetic moment. Finally, the true susceptibility Δχ can be calculated using the following equation:

$$\mathrm{\Delta }\chi =\mathrm{\Delta }\chi \prime V\prime /V.$$ [6]

In this study, we use three volume definitions. The first one is the true volume V. The second one is the apparent volume V′, which is used in estimating the magnetic moment. The apparent volume is related to the signal loss due to T2* dephasing and is determined from gradient echo magnitude images, as described later. The last one is the spin echo volume V_se, which is measured from the spin echo images. This volume is used as an MR based estimate of the true volume of the air bubble or glass bead. We used simulations and multi-echo gradient echo images of a gel phantom containing air-bubbles and glass beads of varying sizes to test Eq. [6]. While glass beads can be considered as almost perfect spheres, air bubbles are closer to the clinical situation of variable shaped microbleeds.

Methods

Simulations

To evaluate validity of Eq. 6 for susceptibility calculation of small objects, we simulated magnitude and phase images of 4 spheres with different radii at 21 different TEs (from 0 ms to 20 ms, with a step size of 1 ms). In each simulation, the sphere was placed in the center of a 1024³ matrix with complex elements. The radii of four spheres tested, within this 1024³ matrix, were 32, 48, 64 and 96 pixels, respectively. The magnitude inside each sphere was set to 0 while the background magnitude was set to 300 to simulate intensities in the experimental data from the gel phantom. The phase images of the spheres were generated according to Eq. [3] and Eq. [5] with Δχ = 9.4 ppm. In order to simulate Gibbs ringing as well as partial volume effects seen in actual MR data, a process simulating the MR data sampling was used. Complex images generated in each 1024³ matrix were Fourier transformed into k-space. The central 32³ region was selected from k-space and was inverse Fourier transformed back to the imaging domain generating low resolution data containing both Gibbs ringing and partial voluming effects. The radii of the four spheres became 1, 1.5, 2 and 3 pixels respectively in this final 32³ volume. White Gaussian noise was then added to the real and imaginary channels of the complex data in the image domain such that the SNR in resultant magnitude images was 10:1. Susceptibility and the magnetic moment values were quantified for each of the spheres at all echo times and errors associated with these measurements were evaluated.

Phantom experiments

A gel phantom, containing 14 small air-bubbles and 9 glass beads of varying sizes, was imaged at 3T (Siemens VERIO, Erlangen, Germany) using a five-echo 3D gradient echo sequence. The echo times (TEs) were 3.93ms, 9.60ms, 15.27ms, 20.94ms and 26.61ms. Other imaging parameters for the gradient echo sequence were: repetition time (TR) 33ms, flip angle (FA) 11°, read bandwidth (BW) 465 Hz/pixel, voxel size 0.5×0.5×0.5mm³, and matrix size 512×304×176. A multi-slice 2D spin echo dataset was also collected with FA = 90°, TR = 5000ms and TE = 15ms and with the same field of view (FOV), BW, resolution and matrix size as in the gradient echo dataset. This is to maintain a one-to-one correspondence of the spin echo with the gradient echo images of the phantom. To ensure that the field perturbation measured in the phase images is the actual perturbation profile from the gel phantom, we first performed shimming using a spherical phantom immediately before performing the imaging experiment. Manual shimming was performed on the spherical phantom, to a spectral full width at half maximum of 13 Hz and the shim coefficients were noted. The same shim settings were used while imaging the gel phantom to ensure that field perturbation profile due to the presence of the phantom in the magnet is not influenced by any additional shimming.

For the construction of the phantom, an agarose gel solution was prepared with an 8% concentration by weight and poured into a cylindrical container. In the lower portion of the container, the gel was first filled to 1/3 the height of the cylinder and 9 glass beads of various sizes were embedded in the gel. The true diameter of the glass beads was roughly measured using calipers before the glass beads were put into the gel solution. Specifically, 4 glass beads were 2mm in diameter, 3 glass beads were 3mm in diameter, 1 glass bead was 5mm in diameter, and the largest glass bead was 6mm in diameter. The phantom was allowed to cool so that the gel solidified and properly engulfed the glass beads. Rest of the prepared gel solution was then poured into the cylindrical container and variable sized bubbles were injected by pumping various amount of air into the gel using an empty syringe (two smallest air bubbles were excluded from this study, due to the limitation in volume estimation of small objects. Details are provided in later sections). The theoretical susceptibility difference between air and water is known to be 9.4ppm and will be used to compare with the measurements from our method. For glass beads, the susceptibility values were measured independently in a former study to be −1.8 +/− 0.3ppm (15).

First, in order to identify air bubbles and glass beads in the collected MR data, binary masks from magnitude data were used. The intensity variation in the magnitude images caused by the RF field inhomogeneity was first removed using a 2D quadratic fitting, before the binary masks were created. A reasonably uniform magnitude intensity profile across the phantom was obtained after this intensity correction. The binary masks were created by local thresholding of the corrected magnitude images (11). First, a relatively strict threshold is used to pick only the voxels where the signal is less than 50% of the signal-to-noise ratio (SNR) in the gel away from the air bubbles or glass beads, since both air bubbles and glass beads have much lower intensities than the intensity of the surrounding gel. Next, the mean (α_mag-gel) and standard deviation (σ_mag-gel) were calculated for a cubic 21×21×21 VOI for each bubble or glass bead. A voxel roughly at the center of the bubble or glass bead was first chosen to center this 21³ voxel window. The voxels picked up in the first step were excluded in the mean and standard deviation calculation. If a neighboring voxel has intensity lower than α_mag-gel-β·σ_mag-gel, it was regarded as a voxel belonging to air or glass bead. For the high SNR data used here, β was empirically chosen to be 4 to separate air bubbles and glass beads from gel.

Susceptibility Quantification

In order to reduce the background field or phase variation, a forward modeling approach was used to estimate air/gel-phantom interface effects (16). The phase processing steps were as follows:

1. The original phase images were first unwrapped using the phase unwrapping tool, PRELUDE, in FMRIB Software Library (FSL) (17). With the geometry of the gel phantom extracted from the magnitude images at the shortest TE (3.93ms in this study), the background field effects were reduced by fitting the predicted phase to the unwrapped phase by a least squares method. An additional 2D quadratic fitting was added in order to remove the induced phase due to eddy currents.

2. The phase value inside a particular air bubble/glass bead (where the binary mask is 1) was set to the mean phase (essentially zero) from the local 9260 voxels. This is due to the fact that the phase inside a sphere is theoretically zero and the nonzero phase is induced by the remnant background field variation as well as Gibbs ringing. This step also determines the apparent volume (V′) from magnitude images.

3. At each echo time, a 160×160×87 voxel volume was cropped from the original phase images. This volume was selected because it covers most of the air bubbles and glass beads while voxels near the edge of the gel phantom were excluded. The selected volume was then zero-filled to a 512×512×256 matrix.

4. Susceptibility maps were generated using a threshold based approach described previously in (1). The mean (α_χ-air or α_χ-glass) and standard deviation (σ_χ-air or σ_χ-glass) of the susceptibility values of air bubble (or glass bead) were measured, taking into account the background susceptibility of the gel. Measurements were obtained in the following manner: the background mean (α_χ-gel) and standard deviation (σ_χ-gel) of the local gel susceptibility value around each bubble or glass bead was first calculated from the 21³ voxel region centered around each of the bubble/bead. Within this 21³ volume, the voxels belonging to the bubble or glass bead, as determined by the binary mask, were excluded for this background mean and standard deviation calculation. Once these measures were obtained, for susceptibility of air bubbles, only voxels with susceptibility values higher than α_χ-gel+3·σ_χ-gel were used for calculation purposes; while for glass beads, only voxels with a susceptibility value lower than α_χ-gel-3·σ_χ-gel were used. This process assumes that the noise in the susceptibility maps follows a Gaussian distribution, and the susceptibility of a voxel consisting of air or glass is statistically different from a voxel consisting of gel. The change in sign is due to the fact that the air bubbles are paramagnetic relative to the gel while glass beads are diamagnetic. To account for the baseline shift caused by remnant field variation, the susceptibility of the air bubble (or glass bead) was taken as α_χ-air- α_χ-gel (or α_χ-glass- α_χ-gel).

Volume Measurement

The apparent volume of the air bubble or glass bead was determined from the binary masks directly, i.e., by counting the number of voxels inside the air bubble or glass bead. On the other hand, the spin echo volume is measured utilizing the "object strength" notion proposed by Tofts et al (18), in which the total intensity is measured for a particular volume of interest (VOI). For a volume composed of two types of tissues, a and b, the total intensity can be expressed as

$$I=I_{\mathrm{a}}·n_{\mathrm{a}}+I_{\mathrm{b}}·(N-n_{\mathrm{a}})=(I_{\mathrm{a}}-I_{\mathrm{b}})·n_{\mathrm{a}}+I_{\mathrm{b}}·N$$ [7]

where I is the total intensity, “I_a” and “I_b” are the intensities of the voxels containing purely tissue “a” or tissue “b”, respectively. The total number of voxels in this volume of interest is denoted by “N”, and the number of voxels occupied by tissue “a” is denoted by “n_a”. Consequently, the number of voxels occupied by tissue “b” can be expressed as N-n_a.

By varying the size of the VOI, the total intensity is linearly dependent on the number of voxels in the VOI. While "I_b" can be determined as the slope in the fit to Eq. [7], n_a can be calculated from the intercept if “I_a” is given (n_a may not be an integer as partial volume is included). In this study, "I_b" corresponds to the intensity of a voxel composed purely of gel, while "I_a" corresponds to the intensity of a voxel composed purely of air or glass. For a relatively large air bubble or glass bead, "I_a" is dominated by the thermal noise, which can be approximated as 1.25×σ_mag-gel, where σ _mag-gel is the measured standard deviation of the gel region in the magnitude images (19). For an air bubble or glass bead with a radius generally less than 3 pixels, "I_a" is a combination of thermal noise and Gibbs ringing. To best account for these fluctuations, "I_a" is calculated from:

$$I_{\mathrm{a}}=\{\begin{array}{c}{w}_1·{\alpha }_{\text{mag-air}}+w_2·1.25·{\sigma }_{\text{mag-gel}},\text{for air bubble}\hfill \\ w_1·{\alpha }_{\text{mag-glass}}+w_2·1.25·{\alpha }_{\text{mag-gel}},\text{for glass bead}\hfill \end{array}$$ [8]

where α_mag-air and α_mag-glass are the measured mean values inside the bubble and glass bead, respectively; w₁ and w₂ are two weighting factors. Based on our simulations (explained below), w₁ and w₂ were empirically determined from simulations to be 0.4 and 0.6 respectively, to minimize the error in estimation of the true volume.

Error in Volume Measurement

Although a regression method is used to measure the spin echo volume, it is still affected by partial volume effects, Gibbs ringing as well as random noise. The simulated magnitude images at TE=0 were used to mimic spin echo magnitude images and to study the error in spin echo volume estimation. In addition, to examine the stability of this method due to thermal noise, the volume measurement evaluation was performed 10 times for each simulated sphere, with independently generated random noise for each of these simulations. The errors were determined by comparing the measured volume with the true volume. Note that, this error estimation does not apply for the apparent volume which is determined directly from the binary masks.

Results

Simulations

Magnetic moments for simulated spheres were calculated with the measured susceptibilities and the apparent volume for each sphere at a given echo time. The results across different TEs are shown in Figure 1. The measured volumes at different TEs were normalized to the volume at the longest TE, while the measured magnetic moments were normalized to the true magnetic moment, which is the product of input volume (i.e., the true volume) of the sphere and the input susceptibility (true susceptibility) 9.4ppm. The normalized magnetic moment is roughly a constant for all spheres. However, for the sphere with a radius less than 2 pixels, the magnetic moments measured in the short TE range have more fluctuations than those measured at longer TEs. In addition, the magnetic moments are under-estimated for all spheres. The mean normalized magnetic moments were measured as: 0.85±0.04 (radius=1pixel), 0.82±0.05 (radius=1.5pixels), 0.81±0.03 (radius=2pixels) and 0.81± 0.02 (radius=3pixels).

Figure 1.

Apparent volume normalized to the volume at TE = 20 ms (first column), measured susceptibility (second column), and normalized magnetic moments (third column) measured at different TEs of four different spheres. The dashed lines in the second column (b, e, h and k) indicate the true susceptibility 9.4 ppm. For each sphere, the effective magnetic moments were normalized to the true effective magnetic moment.

After the magnetic moments were obtained, the susceptibility values were corrected using the actual known volume (i.e., true volume) using Eq. [6]. Specifically, the corrected susceptibilities are: 7.95±0.38ppm (radius=1pixel), 7.70±0.43ppm (radius=1.5pixels), 7.62±0.27 (radius=2pixels), and 7.63±0.21 (radius=3pixels). There is still a 15% to 19% under-estimation in the averaged susceptibility after attempting to correct the volume of the sphere.

To evaluate the stability of the volume measuring method, we carried out 10 simulations for each sphere at TE=0. The means and standard deviations of the percentage errors relative to true volume for each sphere are: 18.02±27.26% (radius=1pixel), 1.89±12.18% (radius=1.5pixels), 3.67±8.91% (radius=2pixels) and 2.09±2.54% (radius=3pixels). The algorithm failed to quantify, in two of the 10 simulations for sphere with radius of 1pixel. Larger errors and more variations of the volume measurements were seen in spheres with radii less than 2 pixels. For the sphere with a radius of 3 pixels, the error in the volume estimation appears to be within 5% using the proposed method. As can be expected, when the object radius is only 1 pixel, the volume measurement becomes unstable.

Phantom experiments

A total of 14 air bubbles and 9 glass beads were examined in the phantom data. The measured spin echo volumes of the glass beads and air bubbles are shown in Table 1 and Table 2. In these two tables, the glass beads as well as air bubbles are sorted based on their spin echo volumes, from small to large objects. The diameters of these glass beads calculated from their spin echo volumes agree reasonably well with their physically measured diameters, as shown in Table 1. Also note that, the error in volume measurement is unreliable for spherical objects with radii less than 1.5 pixels (14.13 voxels for the volume). The error is generally larger than 20%, as shown in the simulations. Thus, the first two smallest air bubbles were excluded from the analysis.

Table 1.

Spin echo volume (in voxels) and the diameter (in mm) calculated from spin echo volume for each glass bead.

Bead	1	2	3	4	5	6	7	8	9
Measure
Spin Echo Volume	35.4	37.5	38.2	39.1	96.5	103.8	113.6	516.6	912.1
Spin Echo Diameter	2.0	2.1	2.1	2.1	2.9	2.9	3.0	5.0	6.0
Actual Diameter	2.0	2.0	2.0	2.0	3.0	3.0	3.0	5.0	6.0

Table 2.

Spin echo volume (in voxel) of the 14 air bubbles.

Bubble	1	2	3	4	5	6	7
Volume	3.3	15.1	28.7	42.2	43.9	82.8	87.7
Bubble	8	9	10	11	12	13	14
Volume	92.7	118.2	170.5	238.6	288.7	322.7	897.2

Figure 2 shows three orthogonal views of the susceptibility map of the largest glass bead for the shortest TE and the longest TE. Using Eq. [6], the measured susceptibilities can be corrected with the volume estimated from the spin echo images. These results are shown in Table 3. The mean of the corrected susceptibility values of the glass beads averaged over all the TEs is −1.82±0.17ppm, which is within the range of the measured values in the previous study (15). The mean of the corrected susceptibility values of the air bubbles is 6.66±0.85ppm. This is to be compared to the actual susceptibility of 9.4ppm.

Figure 2.

Axial, sagittal and coronal views of the susceptibility maps with TE=3.93ms (a, b and c) and TE=26.61ms (d, e and f). The main field direction is in “y” direction. Glass bead No. 9 in Table 1 is pointed by the white arrows. The air bubbles are pointed by the white dashed arrows.

Table 3.

Mean measured and corrected susceptibilities (in ppm) of the glass beads and air bubbles and different TEs.

TE/ms	Glass Bead		Air bubble
	Measured	Corrected	Measured	Corrected
	3.93	−1.50±0.07	−1.79±0.13	3.15±1.16	6.13±0.77
9.60	−1.07±0.32	−1.76±0.13	1.70±0.61	6.44±0.90
15.27	−0.86±0.32	−1.82±0.19	1.30±0.45	6.64±0.76
20.94	−0.68±0.28	−1.82±0.18	1.07±0.36	6.88±0.77
26.61	−0.59±0.25	−1.88±0.21	0.93±0.34	7.22±0.74

Discussion

The susceptibility mapping technique using the regularized Fourier based method has certain advantages over other methods, especially in terms of time-efficiency and simplicity. However, it suffers from problems caused by the intrinsic singularities in the inverse of the Green's function, as well as partial volume effects which disrupt the true phase behavior. For small objects, susceptibility quantification using the inverse method (1) yields a significant under-estimation of the susceptibility. The increased apparent volume at long TE can be utilized to create a larger virtual object, for which the true volume can be more accurately measured and thus the Fourier based susceptibility quantification gives a relatively smaller error for the magnetic moment. At this point, the susceptibility close to the actual value can be extracted from the estimated magnetic moment with an estimation of the true volume, either if it is known ahead of time, or it can be estimated from a high resolution spin echo dataset.

Based on the discussions above, the error δΔχ in the corrected susceptibility Δχ comes from the estimated magnetic moment μ_a = Δχ′V′ and estimated volume (V). Through error propagation, the error in the corrected susceptibility is given by:

$$\frac{\delta \mathrm{\Delta }\chi }{\left|\mathrm{\Delta }\chi \right|}=\sqrt{{(\frac{\delta {\mu }_a}{{\mu }_a})}^2+{(\frac{\delta V}{V})}^2}$$ [9]

As we can see, the smaller the error in the estimated volume, the smaller the error in the corrected susceptibility. This equation explains the error seen in the corrected susceptibility of the air bubbles as well as glass beads.

In simulations, where the true volume is known, the remnant under-estimation in the averaged corrected susceptibility ranges from 15% to 19%. Since there is no error in the true volume, this error must be due to the error in the apparent volume measurement and Δχ′ quantification due to the regularization process. The level of under-estimation is related to the threshold value in the inverse of the Green’s function. A smaller threshold leads to less under-estimation, but more streaking artifacts in the susceptibility maps. The regularized Fourier based method, with threshold value of 0.1, can lead to an under-estimation of around 13% for objects with radii larger than 3 pixels and even worse for smaller objects (1, 6). This can be viewed as a systematic error.

In phantom studies, after using the volume estimated from the spin echo data, the corrected susceptibility values of the air bubbles have a maximum underestimation close to 44%, compared to the theoretical value 9.4ppm. This is essentially a consequence of error in the spin echo volume measurement and the under-estimation of Δχ′ quantified using the Fourier based method. To overcome these limitations, one has to go to high resolution images that can minimize volume quantification error and to relatively longer echo times that can improve accuracy in the magnetic moment quantification. However, the decreased SNR in high resolution spin echo images may introduce additional variation/noise in the final volume results.

There are a number of limitations to this study. Even though a forward calculation was carried out to reduce the geometry induced field variation, remnant background field variation still exists. To best account for it, the phase inside the spherical objects was set to the local average phase. This also helps to reduce the large variation in susceptibility estimate induced by Gibbs ringing and thermal noise. However, this phase correction process is based on the assumption that the object of interest is a sphere. For non-spherical objects, this phase correction process may lead to variations of magnetic moment at different TEs. In addition, phase correction also creates a virtually larger object. It is possible that the center of the created object deviates from the true center of the original object of interest. This leads to additional errors even for spherical objects, as seen from simulations. Thus, the phase inside the spherical object has significant effects to this method. Theoretically, only when the center of a simulated large sphere coincides with the original center of the sphere, and when the background phase value is 0, can we obtain constant magnetic moment across different TEs. Hence, slight variation in object definition from binary mask, which is used for phase substitution, can introduce variations in magnetic moment values. This is the essential source of shape dependence of the proposed method.

Although the estimated magnetic moments of the glass beads are almost a constant over different TEs, as indicated by the corrected susceptibilities, the estimated magnetic moments of the air bubbles are usually larger at a longer TE than at a shorter TE. This can be understood by the fact that the air bubbles are not perfect spherical objects compared to the glass beads. In fact, most of the air bubbles have ellipsoidal shapes, and any attempt of phase correction inside the bubble based on the assumption of the spherical shape will cause errors in the susceptibility measurement and thus lead to errors in the measurement of the magnetic moments.

Generally speaking, for small objects which can be well approximated as spheres, the theoretically expected errors in the estimated magnetic moment measurements are within 20% of the expected values and can be further reduced by adjusting the regularization thresholds in the susceptibility mapping method. Practically, the errors might be larger due to the limited knowledge of the true volume. While most small microbleeds can be well approximated as spheres, the use of more accurate volume estimation methods has the potential to reduce the error in susceptibility quantification of microbleeds.

Conclusions

In conclusion, we have shown that for very small structures, obtaining accurate magnetic susceptibility values is limited by the errors in the volume estimations of these structures and in the Fourier based method itself. Despite this inability to estimate the actual volume of a small object accurately (whether it is an air bubble or microbleed), the estimated magnetic moment is almost a constant over different TEs. This demonstrates that it is possible to measure the magnetic moment at a longer TE when the apparent volume is increased due to T2* dephasing. By measuring or knowing a priori the actual volume of an object, it is possible to obtain a reasonable estimate of the susceptibility.