Improved acceleration of phase-contrast flow imaging with magnitude difference regularization

Abstract

Purpose:

To develop a regularized image reconstruction algorithm for improved scan acceleration of phase-contrast (PC) flow MRI.

Methods:

Based on the magnitude similarity between bipolar-encoded k-space data, magnitude-difference regularization was incorporated into the conventional compressed sensing (CS) reconstruction. The gradient of the magnitude regularization was derived so the reconstruction problem can be solved using non-linear conjugate gradient with backtracking line search. Phase contrast flow data obtained in the peripheral arteries of healthy and patient subjects were retrospectively undersampled for testing the proposed reconstruction method. Three-dimensional velocity-encoded PC flow MRI was performed with prospective 4-fold undersampling for measuring arotic flow velocity in a healthy volunteer.

Results:

In the femoral arteries of healthy volunteers, the root-mean-square (RMS) errors of mean velocities were 0.56±0.09 cm/s with CS-only reconstruction and 0.46±0.08 cm/s with addition of magnitude regularization for three-fold acceleration; 1.34±0.17 cm/s (CS only) and 1.08±0.15 cm/s (magnitude regularized) for four-fold acceleration. In the iliac arteries of the patient, the RMS errors of mean velocities were 0.72±0.12 cm/s and 0.56±0.10 for three-fold acceleration, and 1.75±0.21 and 1.24±0.19 cm/s for four-fold acceleration (in the order of CS-only and magnitude regularized reconstructions). In the popliteal arteries, the RMS errors were 0.61±0.10 cm/s and 0.42±0.11 for three-fold acceleration, and 1.41±0.19 and 1.12±0.17 cm/s for four-fold acceleration. The maximum through-plane mean flow velocities were measured as 63.2 cm/s and 84.5 cm/s in ascending and descending aortas, respectively.

Conclusion:

The addition of magnitude-difference regularization into conventional CS reconstruction improves the accuracy of image reconstruction using highly undersampled phase-contrast flow MR data.

1. Introduction

Phase contrast (PC) magnetic resonance imaging (MRI) has been established as a reliable tool for quantifying blood flow in the human vascular systems (1,2). Applicable vascular territories are extensive, including the heart and nearby great vessels, hepatic and portal veins, carotid and cerebral arteries, and peripheral arteries (3–6). PC flow MRI can be used not only for measuring and characterizing complex blood flow but also estimating wall shear stress which is known to be associated with the development of high-risk plaque (7,8). Owing to significant advances in pulse sequences, reconstruction algorithms and analysis techniques, PC MRI is now a necessary component of clinical protocols for patients with vascular disease.

The scan time of PC flow MRI is inevitably long due to the requirements of multiple acquisitions with varying bipolar gradients as well as high spatial and temporal resolution. Since the long scan time prevents more widespread use in clinical practice, various scan acceleration techniques have been applied in PC MRI. Non-2DFT readouts were the earliest approaches for accelerating PC MRI (9,10). Parallel imaging such as SENSE and GRAPPA is now the reference acceleration technique for PC flow imaging as in other MRI applications (11,12). Compressed sensing (CS), and low-rank-based reconstruction, as another general technique for accelerating MRI, can be used also for PC MRI, contributing to higher rate acceleration (13–18). More recently, PC-MRI-specific techniques were proposed, which reduce the total number of bipolar gradients for velocity-encoding and thus improve effective temporal resolution (19,20).

In this study, we propose another PC-MRI-specific technique which can improve image reconstruction from under-sampled PC data. The underlying rationale is that the magnitudes in reconstructed images barely vary over different bipolar gradients, which is embodied in the form of regularization of the cost function for image reconstruction. The feasibility of the proposed magnitude-regularized reconstruction is shown in healthy subjects and patients with peripheral artery disease (PAD).

2. Material and Methods

2.1. Magnitude regularized compressed sensing

Phase contrast flow MRI is based on the sensitivity of bipolar gradient to the phase of moving spin. By subtracting two phase images obtained from different bipolar encoded sequences, one can remove the background phase and obtain spin’s velocity value in the direction of the applied gradient. Since the reference and the bipolar encoded sequences have identical sequence parameters except the bipolar gradients, magnitude images, while not used for clinical diagnosis, should be similar to each other. This magnitude similarity can be utilized as a source of information redundancy and realized in the form of regularization.

Compressed sensing reconstruction with multi-channel receive coils is typically formulated as unconstrained minimization of the following cost function.

$$C^{(j)}\left({\mathbf{\text{m}}}^{(j)}\right)=\sum _{i=1}^{N_c}{\Vert {\mathbf{\text{y}}}_i^{(j)}-{\mathbf{\text{F}}}_u{\mathbf{\text{S}}}_i{\mathbf{\text{m}}}^{(j)}\Vert }_2^2+\alpha {\Vert \Psi {\mathbf{\text{m}}}^{\left(j\right)}\Vert }_{1}j=1,\cdots ,N_v$$ (1)

where m^(j) is a complex image obtained by applying j^th velocity-encoded bipolar gradient. The number of bipolar gradients N_v will be 1 + the number of velocity directions to be encoded where the addition of 1 is for the reference acquisition. ${\mathbf{\text{y}}}_i^{(j)}$ is the k-space data received through the i^th coil element when j^th bipolar gradient is applied, N_C is the number of coil elements, F_u indicates Fourier transform followed by undersampling, S_i is i^th coil sensitivity, and ψ is a sparsifying transform and set to the total variation operation in this study. Note that the minimization is performed independently for each of N_v data sets. The penalization of magnitude difference among different velocity-encoded data can be incorporated into the compressed sensing formulation to yield the following cost function

$$C(\mathbf{\text{m}})=\sum _{j=1}^{N_v}\left[\sum _{i=1}^{N_c}{\Vert {\mathbf{\text{y}}}_i^{(j)}-{\mathbf{\text{F}}}_u{\mathbf{\text{S}}}_i{\mathbf{\text{m}}}^{(j)}\Vert }_2^2+\alpha {\Vert \Psi {\mathbf{\text{m}}}^{\left(j\right)}\Vert }_1+\beta {\Vert \left|{\mathbf{\text{m}}}^{(j)}\right|-\left|{\mathbf{\text{m}}}^{(j+1)}\right|\Vert }_2^2\right]$$ (2)

The new regularization term enforces magnitude similarity across different velocity-encoded images in the same spatial pixel while CS regularization enforces the similarity across neighboring pixels in the same velocity encoding.

The proposed cost function C(m) can be minimized using a non-linear conjugate gradient (CG) method with backtracking line search (13) which requires the computation of the gradient of the cost function ∇_mC(m). The gradients of the data fildelity and l₁-penalty terms are written as

$$2\sum _{j=1}^{N_v}\sum _{i=1}^{N_c}{\mathbf{\text{S}}}_i^H{\mathbf{\text{F}}}_u^H\left({\mathbf{\text{F}}}_u{\mathbf{\text{S}}}_i{\mathbf{\text{m}}}^{(j)}-{\mathbf{\text{y}}}_i^{(j)}\right)+\alpha \sum _{j=1}^{N_v}{\Psi }^H\mathbf{\text{D}}\Psi {\mathbf{\text{m}}}^{(j)}$$ (3)

D is a diagonal matrix with the diagonal elements $d_i={\left({(\Psi m)}_i^{*}{(\Psi m)}_i+{\epsilon }^2\right)}^{-1/2}$ where ε² is a smoothing parmeter which makes the absolute value differentiable near the origin (13). The gradient of the last term can be written as (see the derivation details in Appendix)

$${\nabla }_{\mathbf{\text{m}}}\left(\sum _{j=1}^{N_v}{\Vert \left|{\mathbf{\text{m}}}^{(j)}\right|-\left|{\mathbf{\text{m}}}^{(j+1)}\right|\Vert }_2^2\right)=2{\Phi }^H{\text{T}}^H\text{T}\Phi \mathbf{\text{m}}$$ (4)

where the matrices Φ and T are defined in Eqs. A6 and A7, respectively in Appendix.

2.2. Retrospective in-vivo study

Healthy volunteer experiments

Phase contrast flow MRI was performed in seven healthy subjects for retrospective evaluation of the performance of the magnitude difference regularization. All subjects provided written consent forms approved by our institutional review board. The study was done on a Siemens 1.5T clinical MR scanner (Avanto; Siemens Medical Solutions, Erlangen, Germany) equipped with maximum gradient amplitude of 45 mT/m and maximum slew rate of 200 mT/m/s. One-dimensional through-plane flow velocity was measured with 2D axial scan planes placed on subjects’ thighs (i.e., N_v = 2). Imaging parameters included VENC = 80 cm/s, spatial resolution = 1.0×1.0 mm², FOV = 36×22 cm², slice thickness = 5 mm, view per segment = 2, and temporal resolution = 20.6 ms. For iterative reconstruction, the numbe of CG iterations was set to 100, and the initial guess was set to aliased images obtained from zero-filled under-sampled data. For data analysis, polygonal regions of interest (ROIs) were manually specified on the right and left femoral arteries. The mean velocity and peak velocity values were calculated within the ROIs for each cardiac phase. Root-mean-square (RMS) errors were calculated with the measurements from fully-sampled data as the reference.

$$E_{\text{RMS}}=\sqrt{\frac{{\sum }_{t=0}^{N_t}{\sum }_{i=0}^{N_{\mathit{\text{ROI}}}}{\left|v_{\text{rec},i}^{(t)}-v_{\text{ref},i}^{(t)}\right|}^2}{{\sum }_{t=0}^{N_t}{\sum }_{i=0}^{N_{\mathit{\text{ROI}}}}{\left|v_{\text{ref},i}^{(t)}\right|}^2}}$$ (5)

where $v_{\text{rec},i}^{(t)}$ and $v_{\text{ref},i}^{(t)}$ are reconstructed and reference velocity of i^th ROI at time frame t, respectively; N_t and N_ROI are the numbers of time frame and ROIs, respectively.

In two subjects, the effect of sampling pattern was investigated using the following protocol. The 1^st velocity-encoded k-space data set (out of N_v = 2 data sets) were retrospectively undersampled by selecting twenty phase encoding lines around the k-space center and randomly undersampling the peripheral regions. The resultant net acceleration factor was 3.0. The 2^nd velocity-encoded data were undersampled in a similar fashion while the peripheral regions were undersampled with three different strategies: i) The same peripheral locations were chosen as in the first velocity-encoded data; ii) The half of the peripheral samples was chosen from the sampled locations of the 1^st data, and the other half was chosen from the omitted locations of the 1^st undersampled data; iii) All peripheral samples were chosen from the omitted locations of the 1^st undersampled data. That is, the ratio of overlapped peripheral sample locations between the two data sets was 100%, 50% and 0%, respectively for the three cases. Image reconstruction was done iteratively using conjugate gradient algorithm (21) with the CS regularization coefficient α set to 10⁻³ (using normalized sensitivities and Fourier transform) (22) and the magnitude difference regularization coefficient β set to 10⁻⁴.

In five subjects, the addition of magnitude regularization was compared with CS only reconstruction by setting β to 0 (CS only) and 10⁻⁴ during the image reconstruction. Based on the finding of the first retrospective study, the two sampling patterns were chosen to be exclusive to each other in the peripheral k-space region. The net acceleration factors of 2,3 and 4 were implemented and tested.

Patient subject experiments

Phase contrast flow MRI was performed in a 52-year-old male patient with bilateral claudication (Rutherford class III) as diagnosed by prior digital subtraction angiography. One-dimensional through-plane velocity measurement was performed twice, one for imaging the iliac arteries and the other for popliteal arteries. Imaging parameters were VENC = 90 cm/s, spatial resolution = 1.2×1.7 mm² and FOV = 36×26 cm² for measuring iliac flow velocity, and VENC = 60 cm/s, spatial resolution = 1.2×1.2 mm² and FOV = 36×16 cm² for measuring popliteal flow velocity. Rate-3 and rate-4 scan accelerations were implemented by retrospective undersampling in a similar fashion to the healthy volunteer studies. The RMS errors of flow velocities were assessed for the reconstructions with and without the addition of magnitude regularization. For the same patient subject, non-gadolinium MR angiography images were also obtained using recently developed velocity-selective MRA technique (23,24).

2.3. Prospecitve in-vivo study

Three-dimensional velocity-encoded (thus N_v = 4) and 2D spatially encoded PC flow data were obtained in a healthy volunteer during breath-hold. The axially oriented scan plane included ascending and descending aorta and pulmonary arteries. Peripheral pulse gating was used. Imaging parameters included VENC = 150 cm/s, TR= 7.4 ms, spatial resolution = 1.4×1.8 mm², FOV = 26×26 cm², and temporal resolution = 29.6 ms. The undersampling factor was 4 which resulted in scan time of 36 heart heats.

3. Results

Figure 1 shows representative mean and peak velocities (a), and magnitude and phase images at the peak flow time (b) for the reference (full sampling) and three under-sampling schemes. Over 120 ROIs (resulting from 30 time frames and right and left arteries in two subjects), the RMS errors of mean and peak velocities were 0.38/0.92/1.26 cm/s and 1.59/ 3.04/ 7.11 cm/s, respectively for the undersamplings with no overlap, 50% overlap and 100% overlap between two bipolar-encoded data in the peripheral k-space region. With more overlaps in the peripheral k-space regions, the reconstruction error gets larger due to increased information redundancy among the data from different bipolar gradients. Base on this finding, all the subsequent studies used a set of sampling patterns that minimally share sample locations. The non-linear CG reconstruction was implemented in MATLAB on a desktop computer which included Intel i7–7700K processor (4.2 GHz), and DDR4 16G memory. With the number of CG iterations of 100 and the number of receive coil elements of 4–6, reconstruction of 1D flow image of thigh took approximately 6 min for each time frame.

Fig. 1.

a: Representative mean and peak velocities obtained from fully sampled data (reference) and three-fold undersampled data using undersampling schemes with varying overlaps between bipolar-encoded k-space sample locations. b: Raw magnitude and phase images obtained from fully sampled and undersampled data at the cardiac phase of peak arterial flow.

The comparisons between CS-only reconstruction and magnitude-regularized CS reconstruction are summarized in Figs. 2a and 2b. Over 360 ROIs (resulting from 30 time frames, and right and left arteries in five subjects), the RMS errors of mean velocities for the two reconstructions were 0.22±0.05 (CS only) / 0.24±0.06 (magnitude regularized) cm/s for two-fold acceleration (R=2), 0.56±0.09 / 0.46±0.08 cm/s for three-fold acceleration (R=3) and 1.34±0.17/ 1.08±0.15 cm/s for four-fold acceleration (R=4). The RMS errors of peak velocities were 0.65±0.23 (CS only) / 0.68±0.24 (magnitude regularized) cm/s for R=2, 2.11±0.42/ 1.69±0.36 cm/s for R=3, and 5.89±0.72/ 4.21±0.63 cm/s for R=4. In cases of acceleration factors of 3 and 4, the addition of magnitude regularization reduced the reconstruction errors by improving the condition of ill-posed encoding matrix resulting from high-rate undersampling. In case of acceleration factor of 2, the additional regularization did not help presumably because the sensitivity encoding and CS regularization were enough to estimate the omitted k-space samples. Figures 2c and 2d show representative mean and peak velocities obtained from the two reconstruction schemes for acceleration factor of 3.

Fig. 2:

Comparison of CS-only reconstruction and addition of magnitude regularization in RMS errors of mean velocity (a) and peak velocity (b) through femoral arteries in 5 healthy subjects. Representative mean velocity (c) and peak velocity (d) obtained from fully sampled data and three-fold accelerated data.

Figure 3 contains the result of patient studies, displaying the time-curves of the mean flow velocities of 4-fold accelerated data (Figs. 3a and 3d for popliteal and popliteal arteries respectively), and raw images reconstructed from different regularization and acceleration conditions (Figs. 3b and 3e). Over 60 ROIs on the iliac arteries, the RMS errors of mean velocities were 0.72±0.12 (CS only) / 0.56±0.10 cm/s (magnitude regularized) for three-fold acceleration, and 1.75±0.21/ 1.24±0.19 cm/s for four-fold acceleration. Over 60 ROIs on the popliteal arteries, the RMS errors of mean velocities were 0.61±0.10 (CS only) / 0.42±0.11 cm/s (magnitude regularized) for three-fold acceleration, and 1.41±0.19/ 1.12±0.17 cm/s for four-fold acceleration.

Fig. 3.

a: Representative mean and peak velocities through the iliac arteries of a patient subject obtained from fully sampled and four-fold accelerated data using CS only and additional magnitude regularization. b: Raw magnitude and phase images of reference and accelerated reconstructions. c: Independent MR angiography showing mild narrowings of iliac arteries. d: Representative mean and peak velocities through the popliteal arteries of the patient subject. e: Raw phase images of reference and accelerated reconstructions. f: Independent MR angiography showing severe narrowings and occlusions in the arteries

Three-dimensional velocity-encoded PC flow imaging could be completed in a single-breath-hold of around 30 sec with 4-fold acceleration. Figure 4 contains representative flow velocity maps in 3 directions at the time of peak systolic flow (Figs. 4a, 4b and 4c) and the time curve of through-plane velocity (v_z) averaged over ascending and descending aortic ROIs for each time frame (Fig. 4d). The through-plane mean flow velocity reached a peak value of 63.2 cm/s in ascending aorta, and the mean flow velocity reached a peak value of 84.5 cm/s in descending aorta, consistent with a previous study (25).

Fig. 4.

Representative 3D flow velocity maps at peak systole (a,b and c) and time curves of flow velocities averaged over ascending and descending aortic ROIs for each time frame (d). The maximum through-plane mean flow velocity was 63.2 cm/s in ascending aorta, and 84.5 cm/s in descending aorta.

4. Discussion

Regularization has long been used in various forms for image reconstruction from partially acquired data as it can improve the condition of corresponding ill-posed system matrix. While l2-norm of solution vector itself and l1-norm of finite difference are popular examples of regularization, the best type would depend on the nature of solution to be sought. PC flow MRI acquires multiple complex-valued data where phase values dominantly vary based on the size of the applied bipolar gradient but with little change in magnitude values. Therefore we proposed to impose penalty for the magnitude difference among the multiple data in a form of regularization. When added to compressed sensing reconstruction, the magnitude regularization turned out to improve the reconstruction accuracy in relatively high-rate acceleration.

The first retrospective test revealed that the location of acquired k-space samples has critical effect on the reconstruction fidelity. Except the fully sampled region near k-space origin, the remaining outer region needed to be sampled as exclusively as possible for different bipolar encodings. This implies that independent information from different sample locations is important for the magnitude regularization to be effective. Another related scan parameter would be the number of bipolar encodings. While the present study handles two velocity encodings for 1D velocity measurement (i.e. N_v = 2), larger N_v would enrich information redundancy, providing more potential for high-rate acceleration. However, the number of undersampling patterns needed also increases, which may make it difficult to achieve the mutual exclusiveness in sample locations.

The clinical feasibility of the proposed reconstruction scheme was shown on one PAD patient. Though the resultant RMS errors were comparable to those obtained in healthy volunteers, the effectiveness of magnitude regularization may be degraded depending on the flow pattern in the diseased vasculature. For instance, arterial blood spins in turbulent or vortex flow caused by stenosis or aneurysm may exhibit significant phase dispersion within a voxel. Then, the net magnitude of each voxel may significantly differ depending on the amount of dephasing as determined by the applied bipolar gradient. Eddy current is another potential source of inaccuracy of the proposed method (26). The distortion of bipolar gradient will induce phase errors and thus inaccuracies in the measured velocity values, but not the magnitude of spin, which may still support the key requirement of similar magnitudes among velocity-encoded data. However, as in cases of complex flow patterns, severe phase errors may result in intra-voxel dephasing and thus significant change in the net magnitude of each voxel especially for low spatial resolution. Further investigations on these effects would be warranted on a cohort of vascular patients.

Alternative acceleration techniques have been developed that utilizes the similarity between the reference and flow-encoded images. D. Kim et al applied compressed sensing for PC flow data in the domain of space and temporal principal component analysis (PCA). While doing so, they interleaved the reference and flow-encoded data in time so the concatenated signal becomes more sparse in the space-temporal PCA domain (15). Sun et al performed separate low-rank-based reconstruction for the reference image and the complex difference between the reference and flow-encoded data, after which the phase differences were extracted. (27). While these works utilize the similarity in complex domain which involves both magnitude and phase, the proposed method more specify the scope of the similarity in magnitude while not imposing constraints on phase. Previous works on separate magnitude and phase regularization, though not applied for phase contrast imaging, are also related to the present work (28,29). The additional separate regularization of phase may further improve the reconstruction fidelity of the proposed method.

5. Conclusions

We have developed a reconstruction algorithm for accelerating PC flow MRI, which combines the conventional compressed sensing reconstruction with magnitude regularization. The proposed magnitude regularization utilizes the magnitude similarity among different bipolar encoded data, and has shown to improve reconstruction accuracy as demonstrated by healthy and patient subject studies. The performance of the proposed technique in cases of arterial pathology needs to be further investigated.

Appendix

Let R_MD denotes the regularization of the magnitude difference among the images obtained using j^th bipolar gradient.

$$R_{MD}=\sum _{j=1}^{N_v}{\Vert \left|{\mathbf{\text{m}}}^{(j)}\right|-\left|{\mathbf{\text{m}}}^{(j+1)}\right|\Vert }_2^2$$ [A1]

The k^th pixel of m^(j) can be written as

$$m_k^{(j)}=\left|m_k^{(j)}\right|e^{j{\varphi }_k^{(j)}}$$ [A2]

where ${\varphi }_k^{(j)}$ is the phase of $m_k^{(j)}$ Then the partial derivative of R_MD with respective to $m_k^{(j)}$ is written as

$$\frac{\partial R_{MD}}{\partial m_k^{(j)}}=2\left(\left|m^{(j)}\right|-\left|m^{(j+1)}\right|\right)e^{j{\varphi }_k^{(j)}}$$ [A3]

When m denotes a column vector where all m^(j) are concatenated, i.e.

$$\mathbf{\text{m}}=\left[\begin{array}{c}{\mathbf{\text{m}}}^{(1)}\\ {\mathbf{\text{m}}}^{(2)}\\ ⋮\\ {\mathbf{\text{m}}}^{\left(N_v\right)}\end{array}\right]$$ [A4]

The gradient of R_MD with respect to m can be written as

$${\nabla }_{\mathbf{\text{m}}}{R}_{MD}=2{\Phi }^H{\text{T}}^H\text{T}\Phi \mathbf{\text{m}}$$ [A5]

Φ is a block diagonal matrix defined as

$$\Phi =\left[\begin{array}{cccc}{\Phi }^{(1)}& & & \\ & {\Phi }^{(2)}& & \\ & & \ddots & \\ & & & {\Phi }^{\left(N_v\right)}\end{array}\right]\text{\hspace{0.17em}\hspace{0.17em}where\hspace{0.17em}\hspace{0.17em}}{\Phi }^{(j)}=\left[\begin{array}{cccc}{e}^{j{\varphi }_1^{(j)}}& 0& \cdots & 0\\ 0& e^{j{\varphi }_2^{(j)}}& & 0\\ ⋮& & \ddots & \\ 0& 0& & e^{j{\varphi }_N^{(j)}}\end{array}\right]$$ [A6]

T is a block two-band matrix defined as

$$\text{T}=\left[\begin{array}{cccc}{\text{T}}_{-1}& {\text{T}}_1& & \\ & {\text{T}}_{-1}& {\text{T}}_1& \\ & & \ddots & \\ & & & \text{O}\end{array}\right]\text{\hspace{0.17em}\hspace{0.17em}where\hspace{0.17em}}{\text{T}}_{-1}=\left[\begin{array}{cccc}-1& 0& \cdots & 0\\ 0& -1& & 0\\ ⋮& & \ddots & \\ 0& 0& & -1\end{array}\right]\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}}{\text{T}}_1=\left[\begin{array}{cccc}1& 0& \cdots & 0\\ 0& 1& & 0\\ ⋮& & \ddots & \\ 0& 0& & 1\end{array}\right]$$ [A7]