Anatomically weighted second-order total variation reconstruction of 23Na MRI using prior information from 1H MRI

NeuroImage 105 (2015) 452–461

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Anatomically weighted second-order total variation reconstruction of 23 Na MRI using prior information from 1H MRI Christine Gnahm, Armin M. Nagel ⁎ German Cancer Research Center (DKFZ), Division of Medical Physics in Radiology, Im Neuenheimer Feld 280, 69120 Heidelberg, Germany

a r t i c l e

i n f o

Article history: Accepted 2 November 2014 Available online 8 November 2014 Keywords: Non-proton MRI Sodium MRI Iterative reconstruction Compressed sensing Anatomical prior knowledge Projection reconstruction Total variation

a b s t r a c t Sodium (23Na) MRI is a noninvasive tool to assess cell viability, which is linked to the total tissue sodium concentration (TSC). However, due to low in vivo concentrations, 23Na MRI suffers from low signal-to-noise ratio (SNR) and limited spatial resolution. As a result, image quality is compromised by Gibbs ringing artifacts and partial volume effects. An iterative reconstruction algorithm that incorporates prior information from 1H MRI is developed to reduce partial volume effects and to increase the SNR in non-proton MRI. Anatomically weighted second-order total variation (AnaWeTV) is proposed as a constraint for compressed sensing reconstruction of 3D projection reconstruction (3DPR) data. The method is evaluated in simulations and a MR measurement of a multiple sclerosis (MS) patient by comparing it to gridding and other reconstruction techniques. AnaWeTV increases resolution of known structures and reduces partial volume effects. In simulated MR brain data (nominal resolution Δx3 = 3 × 3 × 3 mm3), the intensity error of four small MS lesions was reduced from (6.9 ± 3.8)% (gridding) to (2.8 ± 1.4)% (AnaWeTV with T2-weighted reference images). Compared to gridding, a substantial SNR increase of 130% was found in the white matter of the MS patient. The algorithm is robust against misalignment of the prior information on the order of the 23Na image resolution. Features without prior information are still reconstructed with high contrast. AnaWeTV allows a more precise quantification of TSC in structures with prior knowledge. Thus, the AnaWeTV algorithm is in particular beneficial for the assessment of tissue structures that are visible in both 23Na and 1H MRI. © 2014 Elsevier Inc. All rights reserved.

Introduction Sodium (23Na) MRI provides a noninvasive measure of tissue viability. Elevated levels of total sodium concentration (TSC) can be associated with pathological changes in tissue for a number of diseases such as cancer (Nagel et al., 2011b; Ouwerkerk et al., 2003; Thulborn et al., 1999), stroke (Hilal et al., 1983; Hussain et al., 2009; Jones et al., 2006; Thulborn et al., 2005), muscular disease (Constantinides et al., 2000a; Nagel et al., 2011a; Weber et al., 2011) or cartilage degeneration (Schmitt et al., 2011; Wheaton et al., 2004). In multiple sclerosis (MS), recent studies indicate elevated 23Na concentrations even in normalappearing white and gray matter of the brain (Inglese et al., 2010; Maarouf et al., 2014; Paling et al., 2013; Zaaraoui et al., 2012). However, the quantification of TSC is challenging. The low nuclear magnetic resonance (NMR) sensitivity and low in vivo concentration of 23Na lead to low signal-to-noise ratios (SNR), thus limiting the achievable spatial resolution of the images. As a consequence, image quality is corrupted by partial volume effects and Gibbs ringing artifacts. Furthermore, the biexponential relaxation behavior of the slowly tumbling 23Na ions ⁎ Corresponding author. Fax: +49 6221 42 3058. E-mail addresses: [email protected] (C. Gnahm), [email protected] (A.M. Nagel).

http://dx.doi.org/10.1016/j.neuroimage.2014.11.006 1053-8119/© 2014 Elsevier Inc. All rights reserved.

has to be taken into account (Hubbard, 1970). Short T2⁎ relaxation times result in a further broadening of the point spread function (PSF) and require dedicated imaging sequences (Konstandin and Nagel, 2013a). Since 2007, Compressed Sensing (CS) (Candès et al., 2006; Donoho, 2006) and related iterative reconstruction algorithms have experienced an ever growing popularity for MRI reconstructions (Lustig et al., 2007). Even though these techniques are widely applied in proton imaging, they are still rarely used in non-proton MRI (Ajraoui et al., 2010; Behl et al., 2014; Hu et al., 2008; Kampf et al., 2010; Madelin et al., 2011). While these approaches exploit the sparsity of images in some transform domain, the incorporation of prior anatomical knowledge in the reconstruction provides further opportunities to improve image quality. Even the use of a support region that matches the object shape as the most basic anatomical information can improve image quality (Ajraoui et al., 2012; Gnahm et al., 2014). Images of different nuclei are generally highly correlated: in the case of 23Na MRI, anatomical structures such as the cerebrospinal fluid (CSF) are well visible, matching well with 1H MRI. Proton images are available with excellent SNR and high resolution within short measurement times. Algorithms that incorporate prior anatomical information from proton MRI for resolution enhancement have been used well before the onset of CS, mostly in

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MR spectroscopic imaging (Eslami and Jacob, 2010; Haldar et al., 2006; Hu et al., 1988; Liang and Lauterbur, 1991; Plevritis and Macovski, 1995). Constantinides et al. (2000b) proposed the first algorithm incorporating anatomical information for 23Na MRI. Common to all these methods is the need for segmentation of the proton reference image before reconstruction. It would be desirable to develop algorithms that do not rely on segmentation, which is an additional source of error. Haldar et al. (2008) proposed to use anatomical weighting factors with quadratic regularization to reconstruct low-SNR proton images. However, quadratic regularization does not harness image sparsity in the context of CS. Here, we propose an anatomically weighted second-order total variation (AnaWeTV) constraint that advances the idea of anatomical weighting factors to fulfill the requirements of CS. Prior information is obtained from registered proton images with higher resolution. The performance of the algorithm is demonstrated in data simulations and for in vivo 23Na MRI of a MS patient.

where A is the system matrix, which is composed of a Fourier transformation followed by Kaiser-Bessel gridding (Jackson et al., 1991) to the kspace trajectory. x is the image vector and y is the vector containing measured data. Ri are regularization terms with a constant weighting τi. AnaWeTV is always used in combination with a regularization of the support region, which is derived as a binary mask (BM) from the reference proton image (Gnahm et al., 2014). For simplicity, reconstructions with combined BM- and AnaWeTV-regularization will be denoted with AnaWeTV instead of AnaWeTV&BM throughout this paper. The objective function is minimized using a conjugate gradient algorithm (Zhang et al., 2006). The code was implemented in C++ using the FFTW3 library (Frigo and Johnson, 2005). The algorithm is stopped if the criterion

Methods

is fulfilled ten times in a row. AnaWeTV was compared to BM&TV2 as well as gridding. In 23Na MRI, it is common to apply a Hamming filter (Hamming, 1989; Konstandin and Nagel, 2013b; Stobbe and Beaulieu, 2008) to increase SNR and to reduce Gibbs ringing artifacts. Therefore, a Hamming filtered gridding reconstruction was performed as well. Additionally, AnaWeTV was compared to the originally proposed method of anatomically weighted quadratic regularization (AnaWeQR) (Haldar et al., 2008). Weighting factors in the iterative reconstruction were optimized as detailed in (Gnahm et al., 2014).

Image reconstruction The idea of anatomical weighting as suggested for quadratic regularization (Haldar et al., 2008) is refined to meet the requirement of CS in terms of l1-norm minimization. An anatomically weighted secondorder total variation (AnaWeTV) is proposed: RAnaWeTV ðxÞ ¼

    X     ð1Þ  ð2Þ  λW α Dα x þ ð1−λÞW α Dα x ;

α¼x;y;z

1

1

ð1Þ

  x −x  kþ1 k 2 b10−6 x 

ð6Þ

kþ1 2

Simulations where x is the image vector, Dα(1) denotes the first-order derivative (1)T (1) computing the finite differences in dimension α, and D(2) α = Dα Dα is the second-order derivative. The relative weighting of the first- and second-order derivatives is chosen by λ = 0.77 (Block et al., 2007; Geman and Yang, 1995). Wα is a diagonal matrix containing anatomical weighting factors taking values between 0 and 1. For a weighting factor of 1, Eq. (1) becomes the normal second-order total variation (TV2) (Block et al., 2007; Geman and Yang, 1995). For a small weighting factor (Wα)ii, intensity changes between voxel i and its neighboring voxels in direction α are less penalized. Thus, intensity changes in the reconstructed image are promoted at positions of known tissue boundaries. The anatomical weighting factors are calculated directly from a registered high-SNR, high-resolution 1H MR reference image. The confidence cα of a tissue boundary is defined as the first derivative of the reference image r that has been normalized to its maximum value:   ð1Þ cα;i ¼ Dα r :

ð2Þ

i

Wα is then calculated from the inverse wα of the confidence cα, ðW α Þii ¼

8 < :

0:1

wα;i − minðwα Þ wmax − minðwα Þ 1

f or

wα;i bwmax

f or

wα;i ¼ wmax

ð3Þ

with wα;i ¼ min

   −1 cα;i ; wmax

ð4Þ

The parameter wmax is used to control the amount of included prior information. For small values of wmax, only the strongest signal variations in the reference contribute. The image is reconstructed by minimizing the objective function f ðxÞ ¼

X 1 2 τ i Ri ; kAx−yk2 þ 2 i

ð5Þ

Radial k-space data were generated based on simulated T 2 weighted Spin Echo (T2w SE) 1H MR brain images from the BrainWeb database (Cocosco et al., 1997; Kwan et al., 1996) as described in Gnahm et al. (2014), since this contrast is similar to a 23 Na MR image. Datasets for a healthy brain as well as for the same brain with artificially inserted MS lesions were simulated with a nominal resolution Δx3 = 3 × 3 × 3 mm3 and 5000 projections, corresponding to 25% of the required Nyquist samples. The MS lesions were simulated to have 89% higher signal intensity than the surrounding white matter (WM). Another dataset was simulated for the healthy brain with reduced nominal resolution Δx3 = 6 × 6 × 6 mm3 and 5000 projections to fulfill the Nyquist criterion. Synthetic complex Gaussian noise was added to all datasets. The gridding reconstruction of the fully sampled dataset with Δx3 = 1.5 × 1.5 × 1.5 mm3 served as ground truth. AnaWeTV reconstructions were performed with weighting factors from three reference images with different contrasts: T1-weighted Magnetization Prepared Rapid Gradient Echo (T1w MPRAGE), T2weighted Fluid Attenuated Inversion Recovery (T2w FLAIR) and the T2w SE. The T2w FLAIR and T1w MPRAGE contrast were obtained as BrainWeb custom simulations (Kwan et al., 1996). The T2w SE was identical to the ground truth image and therefore contained the same contrast as the simulated radial datasets. For the AnaWeQR reconstruction, the T2w SE image was used as reference. Iterative reconstructions of the high-resolution (Δx3 = 3 × 3 ×3 mm3) dataset were performed with the following weighting factors: τBM = 10, τTV2 = 2.5 × 10−4 (BM&TV2), τAnaWeQR = 1 (AnaWeQR), τBM = 10, τAnaWeTV = 2.5 × 10−4 (AnaWeTV, T1w MPRAGE reference), τBM = 0, τAnaWeTV = 5 × 10−4 (AnaWeTV, T2w FLAIR reference) and τBM = 7, τAnaWeTV = 7.5 × 10−4 (AnaWeTV, T2w SE reference). For the lowresolution (Δx3 = 6 × 6 × 6 mm3) dataset, τBM = 4, τTV2 = 1 × 10−3 (BM&TV2) and τBM = 0.7, τAnaWeTV =2.5 × 10−3 (AnaWeTV, T2w SE reference) were used. The influence of registration errors of the prior information was tested on the 3 × 3 × 3 mm3 dataset by misaligning the T2w SE reference in the anterior-posterior direction by 1.5, 3, and 4.5 mm. Furthermore, the reconstruction of features without prior information as well as the

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Δx3 = 0.7 × 0.7 × 2.5 mm3; FOV = 320 × 240 mm2; 49 slices; echo-train-length = 15; TAQ = 2 min 22 s

consequence of anatomical information that was not contained in the original data was investigated using the T2w SE reference.

The images were registered to the gridding reconstruction of the Na data (which was interpolated to a voxel size of 1.5 × 1.5 × 1.5 mm3) using the FLIRT tool of the FSL library (Jenkinson and Smith, 2001). Iterative reconstruction was performed with weighting factors τBM = 1, τTV2 = 4 × 10− 4 (BM&TV2) and τBM = 1, τAnaWeTV = 7 × 10− 4 (AnaWeTV for all reference images). 23

MR measurements In vivo 23Na data of the brain of a MS patient (male, age: 43 years) were acquired on a 7 T whole-body system (Magnetom 7 T, Siemens Healthcare, Erlangen, Germany) using a double-resonant (1H: 297.2 MHz; 23Na: 78.6 MHz) quadrature birdcage coil (Rapid Biomed GmbH, Rimpar, Germany). A density-adapted 3D radial projection pulse sequence (Nagel et al., 2009) was applied using 25% of the required Nyquist projections (TE/TR = 0.3/120 ms; readout time = 10 ms; α = 90°; nominal resolution Δx3 = 3 × 3 × 3 mm3; projections =5000; TAQ = 10 min). The measurement was performed with approval of the Medical Ethics Committee (Faculty of Medicine, University of Heidelberg, Germany), and written informed consent was received from the patient before enrollment. Anatomical 1H images were acquired as part of a routine clinical protocol on a 3 T MR system (Magnetom Tim Trio, Siemens Healthcare, Erlangen, Germany). Three different contrasts were used to calculate anatomical weighting factors: • T1w 3D-MPRAGE, sagittal orientation: TE/TR = 3.42/1740 ms; TI = 1000 ms; BW = 180 Hz/px; α = 15°; resolution Δx 3 = 0.8 × 0.8 × 1.0 mm 3 ; field of view (FOV) = 235 × 260 mm 2 ; 160 slices; T AQ = 3 min 43 s • T2w 2D-FLAIR, sagittal orientation: TE/TR = 131/8500 ms; TI = 2400 ms; BW = 130 Hz/px; α = 180°; resolution Δx 3 = 0.9 × 0.9 × 3.0 mm3; FoV = 230 × 230 mm2; 25 slices; echo-train-length = 21; TAQ = 1 min 26 s • T2w 2D-Turbo Spin Echo (TSE), transversal orientation: TE/ TR = 100/5064 ms; BW = 161 Hz/px; α = 180°; resolution

Reference

wx

To evaluate image quality, several measures were used to quantify the effect of parameter variations on different image characteristics. Root-mean-square error (RMSE) and structural similarity (SSIM) (Wang et al., 2004) were calculated for simulated data with respect to the ground truth. SSIM was evaluated with a Gaussian kernel (5 × 5 × 5 voxels, standard deviation = 1.5 voxels). Only voxels within the BM were included in the calculation of RMSE and mean SSIM. Furthermore, four small MS lesions (indicated by arrows in Fig. 1) were used to calculate the intensity error of the reconstruction. For this, signal intensity within each lesion was evaluated in a suitable region of interest (ROI) and its deviation from the ground truth was calculated. The intensity error was quantified as the average deviation over all four lesions. For all reconstructions, SNR maps were calculated using a pseudo multiple replica approach (Robson et al., 2008) with 100 replica. Results Anatomical weighting factors wx, wy and wz calculated from three different image contrasts of simulated data of a brain with MS lesions are shown in Fig. 1 for one slice of the simulated 3D dataset. The contrast

wy

wz

T1w MPRAGE

a

Image evaluation

1.00 0.10

b T2w FLAIR

0.08 0.06 0.04 0.02 0

Ground truth T2w SE

c

y x z

Fig. 1. Reference images for simulated data of the brain with MS lesions and derived anatomical weighting factors wx, wy and wz in all three dimensions. a) T1w MPRAGE (wmax = 18), b) T2w FLAIR (wmax = 14) and c) T2w SE (wmax = 20). This image is also the ground truth for the simulated data reconstructions. Arrows mark four MS lesions used to calculate the intensity error. The visibility of the outline of the lesions in the anatomical weighting factors depends on the image contrast.

C. Gnahm, A.M. Nagel / NeuroImage 105 (2015) 452–461

of the reference image determines which structures are outlined in the anatomical weighting. In the T1w MPRAGE image, the intensity of the MS lesions is only 21% below the intensity of surrounding WM. Therefore, the outline of lesions is hardly visible in the weighting factors with the chosen controlling parameter wmax = 18. The T2w FLAIR image yields highest contrast of the MS lesions (intensity 134% above surrounding WM) and the outline of lesions is clearly discernible in the anatomical weighting (wmax = 14). At the same time, the outline of CSF structures is hardly visible. In the T2w SE image (wmax = 20), MS lesions and CSF structures have similar contrast and are both represented in the anatomical weighting. Fig. 2 shows the reconstruction results for a simulated dataset of a brain with artificial MS lesions (Δx3 = 3 × 3 × 3 mm3) using different techniques. For comparison, the ground truth (2a) is shown as well. Quality parameters for the images of Fig. 2 are listed in Table 1. By applying a Hamming filter (2c), noise and artifacts are reduced compared to gridding without filter (2b). At the same time, the suppression of high k-space frequencies by the filter leads to a loss of spatial resolution and an increase of partial volume effects. As a consequence,

Ground truth

a

b

AnaWeQR ref: T2w SE

e

Gridding

small lesions are not visible anymore. Compared to gridding, all parameters except SNR decline (Table 1). Most severe is the 150% increase in intensity error due to stronger partial volume effects. The SNR increase is approximately constant over the whole image (190–200%). Image quality is greatly improved over gridding by the iterative reconstruction techniques. Results with AnaWeQR (2e) and AnaWeTV (2f–h) are better than with BM&TV2 (2d), where some noise and artifacts remain in the image. However, all quality parameters are improved with BM&TV2 compared to gridding. SSIM, RMSE, and SNR are further improved by AnaWeQR, while the intensity error increases slightly. When AnaWeTV is used with a T1w MPRAGE reference (2f), the SNR of the image is comparable to the BM&TV2 result (Table 1), but CSF structures and MS lesions are reconstructed with better contrast: SSIM is increased and the intensity error is further reduced. AnaWeTV with anatomical weighting from a T2w FLAIR (2g) or a T2w SE reference image (2h) leads to the best reconstruction of small structures with prior information. The reduction in intensity error by 59% is considerable (Table 1). Even the smallest MS lesions (arrows in Fig. 2) are clearly

BM&TV2

Grid. & Hamming

c

AnaWeTV ref: T1w MPRAGE

f

455

d

AnaWeTV ref: T2w FLAIR

g

AnaWeTV ref: T2w SE

h

Fig. 2. Two selected slices of simulated MR brain data (Δx3 = 3 × 3 × 3 mm3) for different reconstructions. a) Ground truth, (Δx3 = 1.5 × 1.5 × 1.5 mm3), b) gridding, c) gridding with additional Hamming filter, d) BM&TV2, e) AnaWeQR with T2w SE reference, f) AnaWeTV with T1w MPRAGE reference, g) AnaWeTV with T2w FLAIR reference and h) AnaWeTV with T2w SE reference. The Hamming filter (c) increases the SNR of the image at the expense of reduced resolution and increased partial volume effects. Image quality is greatly improved by the iterative reconstructions. AnaWeQR (e) leads to a smoother image than BM&TV2 (d), where noise is visible. When the T1w MPRAGE image is used as reference for AnaWeTV (f), the noise level is comparable to BM&TV2, but the contrast of CSF structures and MS lesions is enhanced. The proposed AnaWeTV method yields the highest image quality when a T2w FLAIR (g) or SE image (h) is used as reference. Noise is reduced compared to BM&TV2 and AnaWeQR. Furthermore, the contrast of small structures is enhanced compared to all other methods. Small lesions are visible that are not distinguishable from noise in the gridding image.

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Table 1 Image quality measures for the different reconstructions of simulated MR brain data shown in Fig. 2. AnaWeTV with T2w reference images performs superiorly compared to the other image reconstruction techniques. SNR increases significantly and the intensity error is reduced by 59% compared to gridding. RMSE SSIM Intensity error (%) SNR (WM) SNR (CSF) Gridding Gridding & Hamming BM&TV2 AnaWeQR ref: T2w SE AnaWeTV ref: MPRAGE AnaWeTV ref: T2w FLAIR AnaWeTV ref: T2w SE

0.18 0.20 0.17 0.15

0.63 0.56 0.68 0.70

0.17

6.9 17.4 5.2 7.6

± ± ± ±

3.8 3.7 4.1 2.3

10 30 18 24

22 62 31 44

0.70

4.5 ± 3.5

18

32

0.17

0.67

2.8 ± 1.5

33

43

0.15

0.75

2.8 ± 1.4

39

46

depicted when T2w reference images are used with AnaWeTV, while they are not distinguishable from noise with gridding and are hardly visible with BM&TV2 or AnaWeQR. AnaWeTV with a T2w SE reference image leads to best values for SSIM and RMSE. Due to the edge preserving properties of the TV2 regularization, the SNR increase compared to gridding is structure dependent: it is less in small structures like CSF (105%) and higher in larger structures like white matter (WM, 280%). Therefore, SNR with AnaWeTV (T2w SE reference) is higher in WM when compared to a Hamming filter, but lower within CSF. Also, the loss in resolution with a Hamming filter is apparent in these structures. Fig. 3 shows the results for AnaWeTV with different misalignment of the anatomical weighting from a T2w SE image. The corresponding quality parameters are listed in Table 2. With increasing misalignment, the contrast enhancement through anatomical weighting is lost (details indicated by arrows). Dark rims can appear around bright CSF structures. However, for a misalignment of 1.5 mm, the intensity error is smaller than with BM&TV2. For a misalignment of 3 mm, the error is still better than with gridding. SSIM and RMSE are both improved compared to BM&TV2 even for a misalignment of up to 4.5 mm. The reconstruction of features that are contained in the data but not in the anatomical prior information is demonstrated in Fig. 4. The reconstruction of simulated data of a brain with MS lesions (Δx3 = 3 × 3 × 3 mm3) is compared for BM&TV2 (4a), AnaWeTV

0 mm

a

1.5 mm

b

Table 2 Image quality measures for AnaWeTV reconstructions with misaligned anatomical information (reference: T2w SE). For a misalignment of 1.5 mm, image quality is still better than for a BM&TV2 reconstruction without anatomical information (Table 1). Only for a misalignment of more than 3 mm is the intensity error increased over the value for gridding reconstruction. Misalignment

RMSE

SSIM

Intensity error (%)

0 mm 1.5 mm 3.0 mm 4.5 mm

0.15 0.16 0.16 0.17

0.75 0.72 0.71 0.69

2.8 4.6 5.8 8.6

1.4 2.5 2.9 4.6

with complete prior knowledge (4b), and AnaWeTV where the lesion information is missing in the anatomical prior (4c). Image quality of the latter two is comparable except for the region of missing prior information, which is evident in the enlarged part of the image. As expected, the lesions appear sharper and with higher contrast when prior information is available (4b). Without prior information (4c), lesion contrast is comparable to that in the BM&TV2 image (4a), but the intensity error is higher: (11.3 ± 5.5)%. In addition, the AnaWeQR reconstruction using the same prior information without lesions is shown (4d). Here, the lesions are strongly blurred compared to the other reconstructions with an intensity error of (17.0 ± 10.2)%. More critical is the case of anatomical information that was not contained in the original data, as demonstrated in Fig. 5 for the example of simulated data of a healthy brain. The results for gridding (5a) and AnaWeTV with correct anatomical information (5b) are shown in comparison to the AnaWeTV and AnaWeQR reconstruction where the outline of lesions that were not contained in the data is included in the anatomical weighting (5c and 5d). Arrows in the enlarged image parts indicate the position of these lesions. The intensity error at the position of the lesions for all reconstructions is given in Table 3. The anatomical information that was not contained in the data leads to artificial structures in the image. With AnaWeTV, however, the intensity of these artificial lesions is different from that of true lesions. The deviation from the background intensity corresponds to the signal variations due to noise in the gridding image. The intensity error increases slightly compared to gridding. With AnaWeQR, some artificial structures are strongly blurred, while others appear as small bright spots that could be misinterpreted as true lesions, resulting in a high intensity error (Table 3).

3.0 mm

c

± ± ± ±

no anatomical weighting

4.5 mm

d

e

Fig. 3. Effect of misregistered prior information on the AnaWeTV reconstruction (reference: T2w SE). Images reconstructed with anatomical information that was shifted in the anteriorposterior direction by 0 (a), 1.5 (b), 3.0 (c), and 4.5 mm (d). For comparison, an image reconstructed with BM&TV2 without anatomical weighting is shown (e). With increasing misregistration, the resolution enhancement of small structures is lost and dark rims can appear next to bright structures (arrows). However, image quality remains good compared to an image without anatomical weighting (e).

C. Gnahm, A.M. Nagel / NeuroImage 105 (2015) 452–461

AnaWeTV, prior with lesions

BM&TV2

a

b

457

AnaWeTV, prior without lesions

AnaWeQR, prior without lesions

c

d

Fig. 4. Effect of missing data in the prior information demonstrated on simulated data of a brain with MS lesions (Δx3 = 3 × 3 × 3 mm3). a) BM&TV2 reconstruction without anatomical weighting. b) AnaWeTV reconstruction; prior information contains MS lesions. c) AnaWeTV reconstruction; prior information does not contain MS lesions. d) AnaWeQR reconstruction; prior information does not contain MS lesions. Anatomical weighting factors were calculated from a T2w SE image (Fig. 1c). The contrast of the MS lesions in the AnaWeTV reconstruction without prior information (c) is comparable to the BM&TV2 reconstruction (a). Missing prior information in AnaWeQR leads to strong blurring of the unknown image features (d).

Fig. 6 shows the results for a simulated low-resolution dataset (Δx3 = 6 × 6 × 6 mm3) of a healthy brain. At this resolution, the smoothing effect of the Hamming filter (6b) is more pronounced than at higher resolution (cf. Fig. 2c), and image details are heavily blurred. RMSE increases by 12%, SSIM decreases by 24%. With BM&TV2 (6c), image quality is only slightly improved compared to gridding with a RMSE reduction of 1% and SSIM increase of 9%. Gibbs ringing artifacts are reduced while anatomical details keep their resolution. AnaWeTV with a T2w TSE reference (6d) yields greatly improved image quality, where resolution of anatomical details is markedly enhanced. No noise or Gibbs ringing artifacts are visible. RMSE is decreased by 8%; SSIM is increased by 27%.

AnaWeTV, correct prior

Gridding

a

b

AnaWeTV performs similarly when applied to measured 23Na MR data of a MS patient. Fig. 7 shows a sagittal view of 3D images for the different reconstruction techniques; Fig. 8 shows a transversal view of the same images. A Hamming filter (7b/8b) leads to highest SNR but reduced resolution. Both iterative techniques yield improved image quality compared to gridding (7a/8a). With AnaWeTV (7d–f/8d–f), image details appear sharper and with higher contrast than with BM&TV2 (7c/8c). The best contrast of MS lesions is achieved when T2-weighted reference images are used. As the T2w FLAIR reference image was acquired with sagittal orientation and high in-plane-resolution, the AnaWeTV reconstruction using this image lead to sharpest structures in the sagittal view (7h). In analogy, AnaWeTV with a T2w TSE reference

AnaWeQR, prior with lesions

AnaWeTV, prior with lesions

c

d

Fig. 5. Consequence of anatomical information that was not present in the data. Simulated dataset of a healthy brain (Δx3 = 3 × 3 × 3 mm3). a) Gridding reconstruction. b) AnaWeTV reconstruction with correct anatomical information. c) AnaWeTV and d) AnaWeQR reconstruction with prior information of MS lesions that are not contained in the data. At the positions of the lesions, artificial structures appear in the AnaWeTV image (c, arrows). The intensity of the artificial structures corresponds to the signal variations in the gridding image due to noise and is not as bright as real MS lesions. With AnaWeQR, some of these features are strongly blurred, while others appear as bright spots.

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Table 3 Intensity error at the position of lesions contained in the anatomical weighting for the reconstructions shown in Fig. 5. Compared to gridding, the error increases when anatomical information is present that is not contained in the data. The increase is small with AnaWeTV and larger with AnaWeQR. Intensity error [%] Gridding AnaWeTV, correct prior AnaWeTV, prior with lesions AnaWeQR, prior with lesions

15.5 7.0 17.8 24.9

± ± ± ±

12.3 8.8 14.3 15.2

image lead to best results for a transversal orientation (8i). In both orientations, intensity inhomogeneity within the large MS lesion in AnaWeTV reconstructions of 23Na data is independent of signal variations in the 1H reference images. The SNR gain in WM compared to gridding is 180% when applying a Hamming filter, 90% for BM&TV2, and 130% with AnaWeTV, independent of the used reference image. Discussion We presented an AnaWeTV constraint that uses high-resolution anatomical 1H information. The constraint exploits the fact that intensity variations mostly occur at tissue boundaries that are the same in 1H and 23Na images. Through anatomical weighting, intensity changes in the 23Na image are less penalized by the TV2 if they occur at the position of known tissue boundaries. Thus, steep intensity gradients are encouraged for structures with prior knowledge, which helps in extrapolating high k-space frequencies. Compared to BM&TV2, this leads to enhanced resolution resulting in higher contrast and reduced partial volume effects of these structures in the reconstructed images. Reduced partial volume effects can explain the remarkable reduction in intensity error by 59% in four small lesions in the simulated data. Whereas the previously presented BM&TV method (Gnahm et al., 2014) does only incorporate the most basic prior information—the shape of the object—and thus hardly can introduce false prior information from the reference image, anatomical weighting has a stronger influence on the resulting 23Na MR image. It is crucial that 1H image intensities do not falsify the information contained in the 23Na image. AnaWeTV minimizes a potentially negative influence on the 23Na image by two means: First, the anatomical weighting does not depend on absolute image intensities, but rather on relative intensity changes between neighboring voxels. As the weighting factors are positive, T1as well as T2-weighted images can be taken as reference. Second, the weighting factors merely modulate the contribution of image regions to the TV2-regularization. Therefore, image intensity cannot be transferred from the 1H image to the 23Na image. However, the contrast of the 1H image determines which tissue boundaries are contained in the anatomical weighting factors (Fig. 1). Therefore, the reference image has an influence on the sharpness of tissue structures in the reconstructed 23Na image. Best results are achieved for anatomical weighting factors that contain the outline of the most prominent image structures

Gridding

a

Grid. & Hamming

b

in the 23Na image. This is confirmed by our simulation results (Fig. 2 and Table 1). With AnaWeTV, quality parameters are always better than with BM&TV2 and image intensities do not change for different reference images. However, the best result is achieved for a T2w SE reference image, which is identical to the ground truth. The smallest improvement over BM&TV2 is achieved for the T1w MPRAGE reference image whose contrast is very different from the ground truth (Fig. 1a). In T1w MPRAGE, the signal intensity of MS lesions does not differ much from that of surrounding WM and only the most chronic lesions are visible. In contrast, T2w MRI depicts more lesions with high contrast. Therefore, T2w contrasts like T2w FLAIR or T2w TSE are more suited as reference images for MS patient data (Fig. 2, 7 and 8). As the contrast of T2w TSE 1H images is most similar to the 23Na contrast, this contrast leads to the best results when used as reference for AnaWeTV. For even better results, a T2w image with high slice resolution should be acquired in addition to the clinical protocol to serve as reference image for AnaWeTV. The starting point for the development of AnaWeTV was the AnaWeQR algorithm (Haldar et al., 2008). As quadratic regularization is a l2-norm minimization, it does not promote sparsity in the image and does not benefit from the advantages of CS (Candès et al., 2006; Donoho, 2006). Image details that are not contained in the anatomical prior knowledge are blurred and lose resolution with AnaWeQR compared to AnaWeTV (Fig. 4c and 4d). All image quality parameters are improved versus the AnaWeQR reconstruction when using AnaWeTV with the same anatomical reference image. The first-order TV constraint is known to produce undesirable staircase artifacts at the position of smooth intensity gradients and can create patchy images (Block et al., 2007; Knoll et al., 2011). This is because of the underlying assumption of piecewise constant images in first-order TV. By including second-order derivatives of the image in a second-order TV, these unwanted effects are avoided with the AnaWeTV algorithm. In contrast to other algorithms incorporating high-resolution prior information (Constantinides et al., 2000b; Eslami and Jacob, 2010; Hu et al., 1988; Liang and Lauterbur, 1991; Plevritis and Macovski, 1995), AnaWeTV does not depend on segmentation of the reference. The only preparation step necessary is registration of the 1H reference onto the gridding reconstruction of the 23Na data. We showed that the proposed algorithm is robust against registration errors of the reference image on the order of 23Na image resolution (3 mm for a 3 × 3 × 3 mm3 dataset). Registration errors of this magnitude are obvious when visually verifying the registration results. Therefore, we recommend manually checking the registration before AnaWeTV reconstruction. The spill-in of signal intensity from CSF into surrounding tissue due to partial volume effects is a large source of error for TSC quantification. As CSF has an approximately 4-fold higher 23Na content than brain tissue, it gives rise to a bright signal in 23Na MRI (Mirkes et al., 2014). Attempts to correct for partial volume effects have been adapted from positron emission tomography (Rousset et al., 1998; Soret et al., 2007) and rely on segmentation of high-resolution 1H MRI and estimation of

BM&TV2

c

AnaWeTV

d

Ground truth

e

Fig. 6. Selected slice of simulated data of a healthy brain reconstructed with different techniques (Δx3 = 6 × 6 × 6 mm3): a) Gridding, b) gridding with Hamming filter, c) BM&TV2, and d) AnaWeTV with T2w SE reference (Fig. 1c). e) Ground truth (Δx3 = 1.5 × 1.5 × 1.5 mm3). While anatomical details are strongly blurred with the Hamming filter, they are enhanced with AnaWeTV.

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Fig. 7. Sagittal view of in vivo MR images of a patient with MS lesions. The first two rows show different reconstructions of 23Na MRI data (Δx3 = 3 × 3 × 3mm3): a) Gridding, b) gridding with Hamming filter, c) BM&TV2, d) AnaWeTV with T1w MPRAGE reference, e) AnaWeTV with T2w FLAIR reference, f) AnaWeTV with T2w TSE reference. The last row shows registered 1 H images (Δx3 = 1.5 × 1.5 × 1.5 mm3) used for calculation of anatomical weighting factors: g) T1w MPRAGE (wmax = 30), h) T2w FLAIR (wmax = 20) and i) T2w TSE (wmax = 16). The enlarged section of the image shows two MS lesions. The Hamming filter leads to a smooth image with increased partial volume effects. Image quality is greatly improved by the iterative reconstructions, where small anatomical details are well visible. Structures appear sharper with AnaWeTV compared to BM&TV2. Highest contrast is achieved when a T2w FLAIR is used as reference image for AnaWeTV.

the local PSF (Hoffmann et al., 2014; Paling et al., 2013). Our results indicate that AnaWeTV could reduce partial volume effects without needing segmentation of the proton reference or knowledge of the PSF. However, detailed quantification studies are necessary to further investigate this point. The reduction of intensity error by AnaWeTV allows for a more precise determination of TSC. In studies with MS patients, this could help in determining whether changes in TSC are significant or not (Paling et al.,

2013). Furthermore, our results show that for quantification purposes, the use of a Hamming filter can be inadvisable, as the increase of partial volume effects can falsify measured intensity values. In low-resolution data, AnaWeTV can achieve considerable resolution enhancement of CSF structures. In 23Na MRI, low spatial resolution occurs when inversion recovery sequences (Nagel et al., 2011b; Stobbe and Beaulieu, 2005), triple quantum filtering (Jaccard et al., 1986; Pekar et al., 1987), or biexponentially weighted sequences (Benkhedah et al.,

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Fig. 8. Transversal view of in vivo MR images of a patient with MS lesions. The first two rows show different reconstructions of 23Na MRI data (Δx3 = 3 × 3 × 3mm3): a) Gridding, b) gridding with Hamming filter, c) BM&TV2, d) AnaWeTV with T1w MPRAGE reference, e) AnaWeTV with T2w FLAIR reference, f) AnaWeTV with T2w TSE reference. The last row shows registered 1H images (Δx3 = 1.5 × 1.5 × 1.5 mm3) used for calculation of anatomical weighting factors: g) T1w MPRAGE (wmax = 30), h) T2w FLAIR (wmax = 20) and i) T2w TSE (wmax = 16). In this orientation, AnaWeTV yields highest image contrast when a T2w TSE is used as reference image.

2013, 2014) are used. Furthermore, the proposed algorithm is not limited to 23Na MRI, but can also be applied to low-SNR proton MRI or to MRI of other nuclei with even lower MR sensitivity such as 35Cl or 39K. MRI of these nuclei has recently been demonstrated to be feasible in humans (Atkinson et al., 2014; Nagel et al., 2014; Umathum et al., 2013). The approach could also be adapted to other imaging modalities such as positron emission tomography (PET). Anatomical information from 1H images that is not contained in the 23 Na data can impose artificial image structures on the reconstructed image. Therefore, it is crucial that only reliable prior information be used in the reconstruction. In regions where the 23Na distribution could deviate from structures in the proton image, for instance in tumors, no anatomical weighting should be used. In this case, anatomical weighting could be set to 1 in the corresponding image parts, or a pure BM&TV2 reconstruction could be performed. Well-known structures like CSF, gray and white matter, or MS lesions can safely be included in the anatomical weighting.

Conclusion We propose an AnaWeTV constraint for iterative reconstruction of Na MRI that employs high-resolution anatomical information from 1 H MRI. The approach leads to significantly increased SNR and enhanced 23

resolution of known structures in the images. Through reduced partial volume effects, the intensity error is reduced in small structures with prior knowledge. Therefore, the AnaWeTV algorithm is in particular beneficial for the assessment of tissue structures that are visible in both 23Na and 1H MRI.

Acknowledgment This work was funded in part by the Helmholtz Alliance ICEMED— Imaging and Curing Environmental Metabolic Diseases, through the Initiative and Networking Fund of the Helmholtz Association. We thank Dr. Armin Biller for help with the MS patient measurement. We thank Prof. Dr. Peter Bachert for fruitful discussions and appreciate the careful manuscript review by Prof. Dr. Mark E. Ladd.

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