Improving Susceptibility Mapping Using a Threshold-Based K-space/Image Domain Iterative Reconstruction Approach

Abstract

To improve susceptibility quantification, a threshold-based k-space/image domain iterative approach that uses geometric information from the susceptibility map itself as a constraint to overcome the ill-posed nature of the inverse filter is introduced. Simulations were used to study the accuracy of the method and its robustness in the presence of noise. In vivo data was processed and analyzed using this method. Both simulations and in vivo results show that most streaking artifacts inside the susceptibility map caused by the ill-defined inverse filter were suppressed by the iterative approach. In simulated data, the bias toward lower mean susceptibility values inside vessels has been shown to decrease from around 10% to 2% when choosing an appropriate threshold value for the proposed iterative method. Typically, three iterations are sufficient for this approach to converge and this process takes less than 30 seconds to process a 512 × 512 × 256 dataset. This iterative method improves quantification of susceptibility inside vessels and reduces streaking artifacts throughout the brain for data collected from a single-orientation acquisition. This approach has been applied to vessels alone as well as to vessels and other structures with lower susceptibility to generate whole brain susceptibility maps with significantly reduced streaking artifacts.

INTRODUCTION

Susceptibility weighted imaging (SWI) using phase information has become an important clinical tool [1–3]. However, the use of phase information itself has stimulated great interest both as a source of contrast [4–6] and a source for producing susceptibility maps (SM) [7–24]. The impetus for solving the inverse problem from magnetic field perturbation came from the work described in Deville et al. [25]. This was noted by Marques and Bowtell in 2005 [26]. Salomir et al. [27] were the first group to utilize this concept in MRI. Unfortunately, this inverse process is ill-posed and requires a regularization procedure to estimate the susceptibility map. There are a variety of approaches to tackle this problem [7–24]. One unique method uses a multiple orientation data acquisition to remove the singularities [17]. Constrained regularizations [14,20,22,23] have shown good overall results, but they require longer reconstruction times and assumptions about the contrast in or near the object to be detected. Threshold-based, single-orientation regularization methods (TBSO) [11,15,18,24] provide the least acquisition time and the shortest computational time to calculate SM. However, their calculated SMs lead to underestimated susceptibility values (Δχ) and display severe streaking artifacts especially around structures with large Δχ, such as veins or parts of the basal ganglia.

Based on TBSO approaches, we propose an iterative method to overcome the singularities in the inverse filter and produce improved accuracy for susceptibility mapping. In this approach, we iteratively replace k-space values of the susceptibility maps, χ(k), near the singularities to obtain an almost artifact free SM, χ(r). The k-space values used for substitutions are estimated using structural information from the masked version of χ(r). Simulations using 2D cylinders and full brain 3D models were performed to examine the efficacy of this iterative approach. High resolution human data are also evaluated.

MATERIALS AND METHODS

Briefly, the expression for the susceptibility distribution [26,27] derived from the phase data can be written as (for a right handed system[28]):

$$\chi (\text{r})={\text{FT}}^{-1}\left[\frac{1}{\text{g}(\text{k})}•\text{FT}\left[\frac{\mathrm{\Phi }(\text{r})}{-\gamma {\text{B}}_0\text{TE}}\right]\right]$$ [1]

where,

$$\text{g}(\text{k})=\frac{1}{3}-\frac{{{\text{k}}_{\text{z}}}^2}{{{\text{k}}_{\text{x}}}^2+{{\text{k}}_{\text{y}}}^2+{{\text{k}}_{\text{z}}}^2}$$ [2]

and Φ(r) is the phase distribution, TE is the echo time, γ is the gyromagnetic ratio for hydrogen protons, B₀ is the main field strength, k_x, k_y and k_z are coordinates in k-space, and g(k) is the Green’s function or filter. Clearly, the analytic inverse filter g⁻¹(k) = 1/g(k), is ill-posed when g(k) is equal or close to zero, i.e., points on or near two conical surfaces in k-space at the magic angles of 54.7° and 125.3° from the B_o axis. This ill-posedness leads to severe artifacts (including severe streaking) in χ(k) and noise amplification [29]. Thus, for a proper pixel-by-pixel reconstruction of χ(r), recovering the correct values of χ(k) near the region of singularities is critical.

K-space Iterative Approach

If the shapes of the structures of interest are known, then one can use this information in the SM to create a more accurate k-space of said SM in the conical region. The structure of the vessels is obtained directly from the first pass susceptibility map χ_i=0(r). The detailed steps of the iterative method are discussed below and shown in Fig. 1.

Figure 1.

Illustration of the iterative reconstruction algorithm to obtain artifact free χ(r) maps.

• Step-1An initial estimate of the SM, χ_i=0(r), is obtained by applying a regularized version of the threshold-based inverse filter, g_reg⁻¹(k) [18], in Eq.[1] using the suggested threshold value, thr=0.1. The subscript “i” denotes the SM after the i^th iteration (“i=0” denotes the initial step before doing the iterative method and i=1 for the first iteration).
• Step-2The geometry of the structures of interest is extracted from χ_i=0(r) using a binary vessel mask, i.e. outside the veins, the signal in the mask is set to zero, and inside it is set to 1. Since streaking artifacts associated with veins in the SM are usually outside the vessels, after multiplying the χ_i(r) map by the mask, little streaking remains in the SM. This leads to χ_{vm, i}(r) as shown in part (b) of Fig.1. Vessel mask generation will be addressed in the next section.
• Step-3χ_{vm, i}(k) is obtained by Fourier transformation of χ_{vm, i}(r) (part (c) in Fig. 1).
• Step-4The pre-defined ill-posed region of k-space in χ_{vm, i}(k) is extracted (part (d) in Fig. 1). These extracted k-space data are denoted by χ_{vm, cone, i}(k). The size of χ_{vm, cone, i}(k) is decided by a threshold value, a, which is assigned to g(k). For the matrix size 512×512×512, the percentages of the cone region in k-space for a given a, are 2.4% (a=0.01), 24.1% (a=0.1), 47.1% (a=0.2) and 70.6% (a=0.3), respectively. When a increases, the size of χ_{vm, cone, i}(k) increases too and if a is increased too much then most of the original information will be lost.
• Step-5Data from χ_{vm, cone, i}(k) and χ_i=0(k) (part (e) in Fig. 1) are merged. This means part of χ_i=0(k) has been replaced by χ_{vm, cone, i}(k). The merged data are denoted by χ′_{merged, i}(k) (part (f) in Fig. 1).
• Step-6Inverse Fourier transformation of χ′_{merged, i}(k) gives the improved SM, χ_i+1(r) (part (g) in Fig. 1).
• Step-7
  χ_i(r) in step-1 is replaced by χ_i+1(r) from step-6 and the algorithm is repeated until

  $$\sqrt{\mathrm{\sum }\left[{({\chi }_i(r)-{\chi }_{i+1}(r))}^2\right]/N}<\epsilon$$ [3]

  where N is the number of pixels in χ_i(r) and ε is the tolerance value (chosen here to be 0.004ppm).

Binary Vessel Mask Generation

The binary vessel mask was generated using thresholding from the χ(r) map itself. The detailed steps are discussed below and shown in Fig. 2.

Figure 2.

Illustration of the binary vessel mask generation process.

• Step-1A threshold, th₁, is applied to χ_i=0(r) to create an initial binary vessel mask, M₀. The pixels whose susceptibility values are lower than th₁ will be set to zero while those greater than or equal to th₁ will be set to unity. In this study, a relatively low susceptibility of 0.07 ppm is used for th₁ to capture most vessels. However, this choice of threshold inevitably includes other brain structures in M₀ that have high susceptibility.
• Step-2A morphological operation i.e., a closing operation is performed to fill in holes in M₀ to generate an updated mask M₁.
• Step-3A median filter is applied to remove noise in M₁ and create M_2.
• Step-4False positive data points from M₂ are removed as follows: First, the χ(r) map is Mipped over 5 slices centered about the slice of interest to better obtain contiguous vessel information, as seen in χ_MIP(r). Second, another threshold, th₂ = 0.25ppm, is performed on χ_MIP(r) to create a new χ_{MIP_vm}(r) and binary mask M_P, which only contains predominantly vessels. Here, 0.25 ppm was chosen to isolate the major vessels in the MIP image. Third, each slice from M₂ is compared with M_P on a pixel by pixel basis to create M₃. If a data point from M₂ does not appear on M_P, this data point will be treated as a false positive and removed from M_2. This process can be equally well applied to extract other tissues by choosing appropriate values for th₁ and th₂.

2D Cylinder Simulations

Simulation of a two dimensional cylinder and its induced phase was first performed using a 8192 × 8192 matrix. A lower resolution complex image was then obtained by taking the Fourier transform of this matrix and applying an inverse Fourier transform of the central 512 × 512 matrix in k-space. This procedure is to simulate Gibbs ringing effects caused by finite sampling which we usually see in conventionally required MR data sets. Gibbs ringing comes from discontinuities in both the magnitude and phase images. To avoid Gibbs ringing from magnitude discontinuities, we used a magnitude image with a uniform signal of unity. A large sized object of 32 pixels was chosen to avoid major partial volume effects. All phase simulations were performed using a forward method [8,26,27,32] with B_o= 3T, Δχ =0.45 ppm in SI units, TE = 5 ms, and the cylinder perpendicular to the main magnetic field. The susceptibility value of 0.45ppm represents venous blood when the hematocrit (Hct) = 0.44, Δχ_do = 4π·0.27ppm [30] and the oxygen saturation level = 70%, where Δχ_do is the susceptibility difference between fully deoxygenated and fully oxygenated blood [31]. A relatively small echo time was chosen to avoid phase aliasing that can affect the estimated susceptibility values.

Selection of a TBSO Method to Generate the χ_i=0(r) Map

Threshold based single-orientation (TBSO) methods [11, 15, 18, 24] use a truncated g(k) to solve the singularity problem in the inverse filter g⁻¹(k) when g(k) is less than a predetermined threshold value, thr. When g(k) < thr, g⁻¹(k) is either set to zero [11, 24]; or to 1/thr [15]; or set to g⁻¹(k) = 1/thr first and then g⁻¹(k) is brought smoothly to zero as k approaches k_zo. This smoothing is accomplished by multiplying g⁻¹(k) by α²(k_z) with α(k_z) = (k_z−k_zo)/|k_zthr−k_zo | where k_z is the z component of that particular point in k-space, k_zo is the point at which the function g ⁻¹(k) becomes undefined, and k_zthr is the k_z coordinate value where |g(k)| = thr [18].

SMs using the methods in ref. [11,15,18] were calculated based on Eq. 1 using the 2D cylindrical model. Eq.1 is also valid to use for the simulated 2D cylinder phase to calculate SM since the 2D version is a special case of a 3D study with 1 slice and our actual study model is essentially an infinite cylinder perpendicular to the main field. Streaking artifacts are obvious in all three SMs (figures are not shown). The calculated mean susceptibility values inside the cylinder are around 0.40±0.01ppm for all SMs. The background noise levels, (i.e., standard deviation of the susceptibility values) measured from a region outside the streaking artifact in SM using ref. [18] are around 1/2 to 2/3 of the background noise levels in SMs using ref. [11,15] using thr = 0.06, 0.07 and 0.1, which are the optimal threshold values suggested in [11, 12, 18]. Given this result, the method [18] was chosen to generate a χ_i=0(r) map.

Finding an Optimal Threshold Value

To find the optimal threshold, a series of χ(r) maps were reconstructed by the iterative method using threshold values a of 0.01, 0.03, 0.07, 0.1, 0.15, 0.2, 0.25 and 0.3. The larger this threshold, the closer the final estimate for χ(r) will be to χ_vm(r). The optimal threshold value was found by comparing the accuracy of the estimated susceptibility values as well as the effects on reducing streaking artifacts in the reconstructed χ(r) maps. To study the effect of noise in χ(r) maps due to the noise in phase images, complex datasets for cylinders of diameter 2, 4, 8, 32 voxels, respectively, were simulated with Gaussian noise added to both real and imaginary channels. Noise was added in the complex images to simulate an SNR_magnitude of 40:1, 20:1, 10:1 and 5:1 in the magnitude images. Since σ_phase = 1/SNR_magnitude, this corresponds to σ_phase = 0.025, 0.05, 0.1 and 0.2 radians.

To estimate the improvement in the SM by the iterative method, we used a root mean squared error (RMSE) to measure streaking artifacts outside the cylinder. Background noise in the SM is measured in a region away from all major sources of streaking artifacts to compare to the noise measured in the phase image (i.e., so we can correlate noise in the phase with the expected noise enhancement from the inversion process).

Effect of High-Pass Filter

The effect of high-pass (HP) filtering the phase data on the χ(r) map generated by the iterative method was also studied. Phase images of a cylindrical geometry with diameters of 2, 4, 8, 16, 32, and 64 voxels were simulated. Homodyne HP filters [33] with a 2D hanning filter (full width at half-maximum, FWHM = 4, 8, 16, and 32 pixels) were applied on these phase images in both in-plane directions. SM reconstructions were stopped based on the criteria in step 7 of the iterative process.

Three Dimensional Brain Model Simulations

To address the potential of the iterative technique to improve the SM of general structures such as the basal ganglia, a 3D model of the brain was created including the: red nucleus (RN), substantia nigra (SN), crus cerebri (CC), thalamus (TH), caudate nucleus (CN), putamen (PUT), globus pallidus (GP), grey matter (GM), white matter (WM), cerebrospinal fluid (CSF) and the major vessels [34]. The structures in the 3D brain model were extracted from two human 3D T1 weighted and T2 weighted data sets. Basal ganglia and vessels are from one person; grey matter and white matter are from the other person’s data set. Since all structures are from in vivo human data sets, this brain model represents realistic shapes and positions of the structures in the brain. Susceptibility values in parts per million (ppm) for the structures SN, RN, PUT and GP, were taken from ref. [12] and others were from measuring the mean susceptibility value in a particular region from SMs using ref. [18] from in vivo human data: RN = 0.13, SN = 0.16, CC = −0.03, TH = 0.01, CN = 0.06, PUT = 0.09, GP = 0.18, vessels = 0.45, GM = 0.02, CSF = −0.014 and WM=0. All structures were set inside a 512×512×256 matrix of zeros. The phase of the 3D brain model was created by applying the forward method [8,26,27,32] to the 3D brain model with different susceptibility distributions using the imaging parameters: TE = 5ms and B₀ = 3T. A comparison between the phase maps from this brain model and a real data set is shown in Fig.3. To match the imaging parameters of the real data set, B₀=3T and TE=18ms were applied for the results presented in Fig.3. Except for Fig.3, all other figures in the paper associated with the 3D brain were simulated by using TE=5ms.

Figure 3.

a) A transverse view of the 3D brain model. b) The simulated phase map from the model using parameters: B_o=3T and TE=18ms which are consistent with imaging parameters in the real data c). Images b) and c) have the same window level setting.

In Vivo MR Data Collection and Processing

A standard high-resolution 3D gradient echo SWI sequence was used for data acquisition. A transverse 0.5 mm isotropic resolution brain dataset was collected at 3T from a 23-year-old healthy volunteer. The sequence parameters were: TR = 26 ms, flip angle = 15^o, read bandwidth = 121 Hz/pixel, TE = 14.3 ms, 192 slices, and a matrix size of 512 × 368. To reconstruct χ_i=0(r) with minimal artifacts, the following steps were carried out:

1. The unwanted background phase variations were removed using either: a) a homodyne HP filter (FWHM = 16 pixels) [33] or b) Prelude in FSL [35] to unwrap the phase, followed by the process of Sophisticated Harmonic Artifact Reduction for Phase data (SHARP) [36] with a filter radius of 6 pixels. To reduce artifacts in the calculated SMs, regions with the highest phase deviations due to air/tissue interfaces were removed manually from the HP filtered phase images and the phase in those regions were set to be zero.

2. A complex threshold approach [37] was used to separate the brain from the skull.

3. The phase image with an original matrix size of 512 × 368 × 192 was zero filled to 512 × 512 × 256 to increase the field-of-view and to avoid streaking artifacts caused by the edge of brain to alias back to the reconstructed SM.

4. The regularized inverse filter, g_reg⁻¹(k) [18] was applied to obtain χ_i=0(r), followed by the iterative process using a = 0.1. For in vivo data, the iterative program was terminated at the third iterative step.

RESULTS

Selection of threshold level based on simulations

To find the optimal threshold value, SMs were reconstructed using a= 0.01, 0.03, 0.07, 0.1, 0.15, 0.2, 0.25 and 0.3, respectively, with different noise levels (Fig. 4). The streaking artifacts shown in χ_i=0(r) (the first column in Fig. 4a) have been significantly reduced by the iterative method and fall below the noise level when a >= 0.1. Also, when a >= 0.1, the mean susceptibility value inside the cylinder was found to increase to 0.44ppm when the diameter of the cylinder was larger than 8 pixels (Fig. 4b) and this trend is independent of the object size and the noise in the phase image. The optimal result in terms of obtaining the true susceptibility value was with a threshold of 0.1. Fig. 4c shows a plot of RMSE of the susceptibility values from the whole region outside the 32-pixel cylinder using different a. The RMSE of the susceptibility values decreases as a increases. Therefore, for vessels, a value of 0.3 would be the optimal value. However, a large threshold value means replacing more original k-space with the k-space only consisting of vessel information which will reduce the signals from other brain structures and blur these structures. Since the SM using a = 0.1 already reveals the optimal susceptibility value for the vessels and an acceptable RMSE, it is appropriate to choose 0.1 for more general application to study the entire brain.

Figure 4.

Simulations showing the comparison of the calculated susceptibility distributions for a cylinder perpendicular to B_o at different threshold values (a) applied to g(k) as well as the initial χ_i=0(r) map. The direction of B_o is indicated by a black long arrow. The susceptibility, Δχ, inside the cylinder is 0.45 ppm. a) The comparison of the converged χ_i=b(r) map with the χ_i=0(r) map for a diameter of 32-pixel cylinder, where b is the iterative step required to reach convergence. In this data, b = 2 when a = 0.03, b = 3 when a = 0.1 and b = 4 when a = 0.2when σ_phase = 0. The top row of images shows simulations with no phase noise. The second and the third row show simulations with added phase noises σ_phase = 0.025 and 0.05 radians, respectively. The first column of images show initial χ_i=0(r) maps for reference. b) The variation of the mean calculated susceptibility inside the cylinder with different threshold value, a, for diameter (d) = 2, 4, 8 and 32 pixels cylinders. The mean susceptibility value is independent of the noise level; therefore, only mean values from σ_phase = 0 were provided. c) The variation of the RMSE of the susceptibility values outside the cylinder as a function of the threshold value, a, and the noise level. The d=32 pixels cylinder was used to generate (c).

Fig. 4a compares the converged χ_i=b(r) map with the χ_i=0(r) map, where b is the iterative step required to reach convergence. In this data, b = 2 when a = 0.01 and 0.03, b = 3 when a = 0.07, 0.1 and 0.15 and b = 4 when a = 0.2, 0.25 and 0.3 for σ_phase = 0. When σ_phase increases, more iterative steps were required to reach convergence. For instance, the maximum iterative step number is 9 when σ_phase = 0.2 radians. Using a noise level of 0.025 radians in the phase image as an example, g_reg⁻¹(k) [18] leads to a susceptibility noise of roughly 0.025 ppm in the χ_i=0(r) map. The iterative approach leads to a slight decrease in background noise, 0.021 ppm, in χ_i=3(r) map when a= 0.1. The background noise was measured in a region outside the streaking artifact indicated by the black circle in Fig. 4a. The overall decrease in RMSE in the background (Fig. 4c) is a consequence of both a decrease in streaking artiacts and a reduction in thermal noise contribution.

Selection of the Optimal Iterative Step

The inverse process [18] was applied to the dipole field in Fig. 5a to give the χ_i=0(r) map shown in Fig. 5b; prominent streaking artifacts are evident in this image. Streaking artifacts are significantly reduced at each step of the iterative method quickly reaching convergence (Figs. 5c to 5e). The largest improvement is seen in the first iterative step, which is verified by Fig. 5f, showing the difference between Fig. 5c (χ_i=1(r) map) and Fig. 5b (χ_i=0(r) map). After the second iteration, we can see some minor streaking reductions (Fig. 5g, the difference between the χ_i=1(r) map and χ_i=2(r) map). The mean susceptibility value approaches 0.44ppm in a single step. Similar results (not shown) are also obtained when the iterative method is run with different aspect ratios between the in-plane resolution and the through plane resolution (such as 1:2 and 1:4). The iterative results always lead to higher final susceptibility values compared to the initial value in χ_i=0(r). Finally, even when an HP filter is applied, up to a 10% increase in the susceptibility is realized (Fig. 5i). Large vessels benefit from a HP filter (FWMH = 4 pixels) and small vessels up to 8 pixels can still benefit from a HP filter (FWMH = 16 pixels).

Figure 5.

a) Phase images from a cylinder with a diameter of 32 pixels are simulated with: Δχ=0.45ppm, Bo=3T and TE=5ms. The cylinder is perpendicular to the main field. No thermal noise was added in these images. b) The initial χ_i=0(r) map. c) The SM from the first iteration, χ_i=1(r) map, d) χ_i=2(r) map and e) χ_i=3(r) map using threshold value a = 0.1. The SM has converged at χ_i=3(r) map. The streaking artifacts are reduced as the number of iterative steps increases. f) The difference image of c) subtracted from b) illustrates that the streaking artifacts were reduced by the iterative procedure and the largest improvement happens in this first iterative step. g) The difference image of the χ_i=1(r) map subtracted from the χ_i=2(r) map indicates that the streaking artifacts were further reduced by the second iterative step. h) The difference image of χ_i=2(r) map subtracted from χ_i=3(r) map shows much less improvement at the third iterative step. Thus it indicates a convergence of the iterative procedure. All images were set to the same window level setting for direct comparisons and for enhancing the presence of the streaking and the remnant error. i) The effect of the iterative approach on the changes in susceptibility values from HP filtered phase images. Differences between the values in iterative and non-iterative susceptibility map reconstruction (i.e. χ_converged(r) − χ_i=0(r)) from HP filtered phase images are plotted for different filter sizes. Results for four filter sizes (FWHM = 4, 8, 16 and 32 pixels) are shown here. Applying an HP filter leads to an underestimation of Δχ [18]. The iterative approach helps to improve the accuracy of the estimated susceptibility values.

Effect of the Iterative Approach on Surrounding Brain Tissues in the 3D Brain Model

a) SM Reconstruction using a vessel mask only

Figs. 6a and 6d represent χ_i=0(r), without noise and with 0.025 radians of noise in phase images. Fig. 6f is the vessel map. Streaking artifacts (delineated by the black arrows) are obvious in Figs. 6a and 6d and completely disappeared in the χ_i=3(r) maps (Figs. 6b and 6e) using a = 0.1. Fig. 6c is the χ_i=3(r) map using a = 0.2. As can be seen, when a increases, the iterative method still works for vessels, but brain tissues become more blurred. Fig. 7a plots the mean susceptibility values inside the vessel (vein of Galen), GP, SN, RN, PUT and CN from χ_i=3(r) maps generated by using a=0.1, 0.15, 0.2, 0.25 and 0.3, respectively. The susceptibility value in the brain model and χ_i=0(r) map are also provided in the plot as references. Generally, the susceptibility values of brain tissues except vessels decrease as a increases while, for vessels, the susceptibility value is 0.41ppm in the χ_i=0(r) map and is increased to 0.45ppm in the χ_i=3(r) maps.

Figure 6.

Results before and after the iterative method using a region of interest map which consists of either only vessels or specific brain structures (in this case the basal ganglia) plus vessels. a) The initial χ_i=0(r) map without noise added in the original simulated images. b) χ_i=3(r) map of (a) using threshold value a=0.1. c) Similar to (b), a=0.2. d) The initial χ_i=0(r) map with noise added in original images, resulting a standard deviation of 0.025 radian in phase images. e) χ_i=3(r) map of (d) using a=0.1. f) The associated vessel map. g) The χ_i=0(r) map in the coronal plane as a reference. The streaking artifacts are clearly shown in every structure. h) The χ_i=3(r) maps created by using a region of interest map which consists GP, SN, RN, PUT, CN and vessels. i) The difference image of (g) and (h). j)The initial χ_i=0(r) map in the transverse plane has “false” internal capsule (IC) (pointed by an arrow) around GP; k) The χ_i=3(r) map shows no “IC.” This matches the originally simulated model (l). No noises were added to images from (g) to (l).

Figure 7.

The plots of mean susceptibility values inside the vessel (vein of Galen), GP, SN, RN, PUT and CN from χ_i=3(r) maps. The first two data points of each curve is the value inside each structure from the brain model and the χ_i=0(r) map, respectively. a) χ_i=3(r) maps generated by applying a region of interest map which consists only vessels using a=0.1, 0.15, 0.2, 0.25 and 0.3, respectively. b) χ_i=3(r) maps generated by applying a region of interest map which consists the GP, SN, RN, PUT, CN and vessels using a=0.1.

b) SM Reconstruction using a mask including vessels and brain structures

The iterative method is not limited to improving SM from just vessels; it can also be applied to the entire brain. Figure 6g shows a coronal view of the χ_i=0(r) map for the brain model. The χ_i=3(r) map using a mask keeping all major structures (GP, SN, RN, PUT, CN) and vessels is shown in Fig. 6h. In practice, this is equivalent to setting thresholds in the χ_i=0(r) map to be greater than 0.009ppm to extract all these high susceptibility structures from the χ_i=0(r) map to create the mask. Fig. 6h reveals that streaking artifacts associated with veins as well as all major structures have been reduced. Fig. 6i shows the difference between Fig. 6g and 6h. In addition, streaking artifacts sometimes cause the appearance of “false” structures. For instance, there is no internal capsule (IC) included in the model (Fig. 6l), yet we see an IC like structure in the χ_i=0(r) map (Fig. 6j) (indicated by a dashed white arrow in Figs. 6j to 6l). The iterative method removes the streaking artifacts and the “false” IC (Fig. 6k). Fig. 7b shows susceptibility values in each structure in the brain model for χ_i=0(r) and χ_i=3(r) when the mask includes vessels and all major structures. The underestimated susceptibility values of all major structures and vessels in the χ_i=0(r) map have been recovered by the iterative method in the χ_i=3(r) map.

Effect of Errors in the Vessel Map

Accurately extracting vessels from χ_i=0(r) is critical for the iterative method. Figs. 8b–8d and the corresponding enlarged views (Figs. 8f–8h) from the rectangular region indicated in Fig. 8a, show the χ_i=3(r) maps using an accurate (Fig. 8j), a dilated (Fig. 8k) and an eroded (Fig. 8l) vessel map to show the effect of errors in the vessel mask on the χ_i=3(r) map. The dilated and eroded vessel maps were generated using Matlab functions based on a 3-by-3 square structuring element object. The susceptibility values measured from a vein indicated by the white arrow in Fig. 8e are 0.40±0.03ppm (Fig. 8e), 0.45±0.03ppm (Fig. 8f), 0.45±0.03ppm (Fig. 8g) and 0.40±0.07ppm (Fig. 8h), respectively. The iterative method still works if the vessel is slightly enlarged but does little to change the original χ_i=0(r) map if the vessels are too small or absent in the mask. As we just discussed, streaking artifacts produced “false” vessels indicated by the dashed black arrow in Fig. 8e since these vessels are not in the model (Fig. 8i). These false vessels disappeared in Fig. 8f.

Figure 8.

Comparison of the reconstructed χ_i=3(r) maps using j) accurate, k) dilated and l) eroded vessel maps. Their corresponding vessel maps and the enlarged views from the rectangular regions are provided in b) – d) and f) – h). a) and e) The initial χ_i=0(r) maps and i) the original brain model as references. The circle in the midbrain in the χ(r) maps represents the red nucleus (RN) and is indicated by a black arrow in i). Other hyper-intense regions in SMs are vessels.

Results from the In Vivo Dataset

In the in vivo example, we compare the differences between SHARP (Figs. 9a to 9d) and a homodyne HP filter (FWHM = 16 pixels) (Figs. 9e to 9h). Compared to the transverse view, streaking artifacts are more obvious in the sagittal or coronal view. Fig. 9a shows the χ_i=0(r) map with severe streaking artifacts. The streaking artifacts were significantly reduced in the χ_i=3(r) map (Fig. 9b) using a = 0.1. The streaking artifacts associated with the superior sagittal sinus vein (indicated by two black arrows in Fig. 9a) were significantly decreased in Figs. 9b and 9d. The subtracted image (Fig. 9c), Fig. 9a minus Fig. 9b, reveals the removed streaking artifacts. These streaking artifacts are one of the reasons why the χ_i=0(r) maps appear noisy. In the χ_i=3(r) map, the reduction in streaking artifacts from individual veins leads to a decrease of noise therefore an increased SNR of veins. If veins are the only interest, even a threshold of 0.2 can work reasonably well (Fig. 9d). Two relatively big veins, V1 and V2, indicated by the white dashed and white solid arrows, respectively, in Fig. 9b, were chosen to measure the susceptibility values. Results are provided in Table 1. The susceptibility values of these two veins have been improved by roughly 16% by the iterative method. The standard deviation of the susceptibility values measured from a uniform region inside the white matter decreased from 0.042 ppm in χ_i=0(r) map to 0.035ppm and 0.023ppm in the χ_i=3(r) map with a = 0.1 and 0.2, respectively. The baseline susceptibilities of the major structures are higher with SHARP than with the HP filter. The iterative method works for brain structures also when the structure is included in the mask. For instance, the mean susceptibility values of the GP and SN have been increased from 0.155±0.058 ppm and 0.162±0.067ppm in the χ_i=0(r) map to 0.163±0.070ppm and 0.186±0.083ppm in the χ_i=3(r) map, from the dataset processed using SHARP. The result after HP filtering (Fig. 9e) shows more edge artifacts indicated by the left arrow in Fig. 9e. Much of this error was reduced by the iterative method (Fig. 9f). It seems that the iterative method compensated for the worse first guess (Fig. 9e) and ended up with almost the same result (Figs. 9f and 9h) as having started with SHARP (Fig. 9b and 9d) from the image perspective. Since a small sized HP filter cannot remove rapid phase wrapping at air-tissue interfaces; we had to cut out the region near the sinuses in the phase images.

Figure 9.

Comparisons of SMs using SHARP or a HP filter (FWHM = 16 pixels) to remove the background field. The iterative method with a = 0.1 and 0.2 is applied after the background is removed. a – d) and e – h) are results after the application of SHARP and the HP filter, respectively. a) and e) the initial χ_i=0(r) maps. b) and f) The χ_i=3(r) maps generated from the iterative method with a = 0.1. c) and g) The differences of images between (a) and (b), and between (e) and (f), respectively. These two images show the successful reduction of the streaking artifacts. d) and h) The χ_i=3(r) maps generated from the iterative method with a = 0.2.

Table 1.

Δχ measured in vivo in two veins in χ_i=0(r) maps and χ_i=3(r) maps with different threshold values

	χi=0(r) map	χi=3(r) map/a=0.1	χi=3(r) map/a=0.2
V1 (SHARP)	0.32 ± 0.07	0.37 ± 0.08	0.38 ± 0.09
V1 (HP)	0.24 ± 0.05	0.28 ± 0.06	0.28 ± 0.06
V2 (SHARP)	0.35 ± 0.04	0.40 ± 0.05	0.41 ± 0.05
V2 (HP)	0.25 ± 0.05	0.31 ± 0.06	0.30 ± 0.06

Mean and standard deviation for the susceptibility values (in ppm) of two veins processed using SHARP and a HP filter (FWHM = 16 pixels), respectively, were chosen from the 0.5 mm isotropic resolution data. V1 and V2 are shown in Fig. 9. The susceptibility values of these two veins have been slightly increased by the iterative method. There is not much variation of the susceptibility value with different threshold values.

DISCUSSION

In this article, a threshold-based k-space/image domain iterative approach has been presented. Simulations and in vivo results show that the ill-posed problems of streaking artifacts and biases in the estimates of susceptibilities can be significantly reduced. The replacement of the χ(k) values near the singularities by χ_vm(k), which is obtained from the geometric information from the χ(r) map itself, obviates many of the current problems seen in the TBSO methods. Since χ_vm(r) contains little streaking artifacts itself, the values used inside the thresholded regions in χ(k) now contain no artifact either. In this sense, we obtain an almost perfect k-space without bad data points in the region of singularities. This explains why this method converges quickly and the major improvement is in the first iterative step (Fig. 5).

The proposed iterative approach is different from the other threshold based methods [11,15,18,19,24] which fill a pre-defined conical region using a constant, zero or 1/thr threshold [11, 15, 24] or the first-order derivative of g⁻¹(k) [19]. The iterative method uses full geometry information from the SM (vessels or pre-defined structures and not edge information) to iteratively change k-space values in the conical region using the forward model. This is also quite different than other currently proposed solutions [9, 12, 20, 22]. Even though spatial priors such as gradients of the magnitude are used [9, 12, 20, 22], in those methods, the meaningful values of the singularity regions in k-space are obtained through solving the complex cost function problem. However, the iterative method uses priors not from the magnitude image but from the SM. The missing data in the singularity regions are obtained through iterating back and forth between the SMs and their k-space. The advantage of cost function approaches is that they do not need to pre-define the singularity region in k-space which is solved by the optimization process automatically (although the optimization process itself is usually quite time-consuming). On the other hand, the iterative method is the most time-efficient. It is fast enough to reconstruct SMs for a 512 × 512 × 256 data set using an Intel Core i7 CPU 3.4GHz processor in less than 30 seconds, since in practice usually 3 iterations are good enough to generate decent results.

The threshold value also plays a key role. A threshold value of 0.1 is a reasonable choice since a lower threshold value leads to an increase in noise and a higher threshold value leads to a blurring of the object (Fig. 6c, Figs. 9d and 9h).

It is known that the ill-posedness of the inverse filter will increase the noise level from the phase to the SM. Based on both simulations and real data, we find that there is a factor of 4 increase in noise in the SM relative to the original phase data. This result and the fact that at Bo = 3T, TE = 5ms and σ_χi=0(r) = 0.025ppm, makes it possible to write the white noise error in χ_i=0(r) as 0.025·4· (3/B_o) · (5/TE) in ppm. White noise error in χ_i=3(r) will be less than this value since the iterative method will decrease noise in SM.

The iterative method can be used to remove streaking artifacts associated with not only vessels but also other brain structures as well. Fig. 6h shows a reduction in artifacts associated specifically with iron-rich regions such as the GP and CN.

Accurately extracting vessels from the χ_i=0(r) map is critical for the iterative method (Fig. 8). In this study, vessels were segmented directly from the SM (Fig. 2). It may also possible to segment veins from original magnitude images [9,12,20,22], phase images and/or SWI images. Extraction of accurate anatomic information from phase data sometimes is difficult since phase is orientation dependent and phase changes are generally nonlocal. SWI images work better for an anisotropic dataset rather than an isotropic dataset since phase cancellation is needed to highlight vessel information. Therefore, we may consider combining SMs with magnitude images, phase images and/or SWI images together to segment the veins, since different types of images can compensate for missing information.

The iterative method appears to help even in the presence of non-isotropic resolution with partial volume effects and to a minor degree when an HP filter is applied. A smaller sized HP filter would be better, since a larger HP filter will significantly underestimate the susceptibility value (Table 1). SHARP gave us better results compared with the HP filter (FWHM=16 pixels) (Fig. 9), but SHARP requires phase unwrapping which can be time consuming and is noise dependent [19]. From this perspective, an HP filter has the advantage since it does not need unwrapped phase. If the forward modeling approach of Neelavalli et al [38] can be used to reduce air/tissue interface fields, then it may be possible to use a small size HP filter (FWHM=8 pixels) which may provide similar results to SHARP.

Severe streaking artifacts associated with structures having high susceptibility values such as veins can lead to major changes in the appearance of the brain structures with low susceptibility. Practically, the susceptibility of the veins is a factor of 2.5 to 20 times higher than other structures in the brain. Therefore, even a 10% streaking effect can overwhelm the information in the rest of the brain and create false appearing structures as in (Fig. 6j) and in (Fig. 8e). The reduction of these artifacts makes a dramatic difference in the ability to properly extract the susceptibility of other tissues.

In conclusion, both simulations and human studies have demonstrated that the proposed iterative approach can dramatically reduce streaking artifacts and improve the accuracy of susceptibility quantification inside the structures of interest such as veins or other brain tissues. Given its fast processing time, it should be possible to expand its use into more daily clinical practice. With the improved accuracy of the susceptibility values inside veins, this method could be used potentially to improve quantification of venous oxygen saturation [18].