 Technical notes
 Open Access
 Published:
Realtime phasecontrast flow cardiovascular magnetic resonance with lowrank modeling and parallel imaging
Journal of Cardiovascular Magnetic Resonance volume 19, Article number: 19 (2017)
Abstract
Background
Conventional phasecontrast cardiovascular magnetic resonance (PCCMR) employs cinebased acquisitions to assess blood flow condition, in which electrocardiogram (ECG) gating and respiration control are generally required. This often results in lower acquisition efficiency, and limited utility in the presence of cardiovascular pathology (e.g., cardiac arrhythmia). Realtime PCCMR, without ECG gating and respiration control, is a promising alternative that could overcome limitations of the conventional approach. But realtime PCCMR involves image reconstruction from highly undersampled (k, t)space data, which is very challenging. In this study, we present a novel modelbased imaging method to enable highresolution realtime PCCMR with sparse sampling.
Methods
The proposed method captures spatiotemporal correlation among flowcompensated and flowencoded image sequences with a novel lowrank model. The image reconstruction problem is then formulated as a lowrank matrix recovery problem. With proper temporal subspace modeling, it results in a convex optimization formulation. We further integrate this formulation with the SENSEbased parallel imaging model to handle multichannel acquisitions. The performance of the proposed method was systematically evaluated in 2D realtime PCCMR with flow phantom experiments and in vivo experiments (with healthy subjects). Additionally, we performed a feasibility study of the proposed method on patients with cardiac arrhythmia.
Results
The proposed method achieves a spatial resolution of 1.8 mm and a temporal resolution of 18 ms for 2D realtime PCCMR with one directional flow encoding. For the flow phantom experiments, both regular and irregular flow patterns were accurately captured. For the in vivo experiments with healthy subjects, flow dynamics obtained from the proposed method correlated well with those from the cinebased acquisitions. For the experiments with the arrhythmic patients, the proposed method demonstrated excellent capability of resolving the beatbybeat flow variations, which cannot be obtained from the conventional cinebased method.
Conclusion
The proposed method enables highresolution realtime PCCMR at 2D without ECG gating and respiration control. It accurately resolves beatbybeat flow variations, which holds great promise for studying patients with irregular heartbeats.
Background
Over the past few decades, phasecontrast cardiovascular magnetic resonance (PCCMR) has been developed into a powerful tool for quantification and visualization of blood flow dynamics in the heart and large vessels [1–5]. It has advanced the understanding and diagnosis of various cardiovascular diseases, such as atherosclerosis [6], aneurysms [7], and arteriovenous malformation [8]. Conventional PCCMR [9, 10] employs electrocardiogram (ECG) synchronized cine acquisitions with respiration control to acquire data from multiple cardiac cycles, from which averaged velocity maps are obtained. Although this approach has been widely used in biomedical research and clinical practice, it suffers from a number of wellknown limitations. For example, it often requires periodic or quasiperiodic cardiac motion to ensure efficient data acquisition; rejection of data caused by irregular cardiac motion often leads to prolonged acquisition time. Additionally, due to its underlying assumption, this approach only obtains averaged flow information over multiple cardiac cycles, failing to resolve beatbybeat flow variations associated with irregular cardiac motion (e.g., cardiac arrhythmia). Capturing physiological and/or pathological flow variabilities has long been an important goal of PCCMR research [11–14].
Realtime PCCMR [15, 16] without ECG gating and respiration control is a promising direction to address these limitations; however, it requires a much higher imaging speed, posing significant challenges for both data acquisition and image reconstruction. A number of techniques have been developed to advance realtime PCCMR. For example, advanced acquisition methods, such as echoplanar [17, 18], radial [19, 20], and spiral [21–24] acquisition schemes, have been employed for realtime PCCMR. In addition, realtime PCCMR also benefits from accelerated data acquisitions. For example, with the emergence of parallel imaging, sensitivity encoding (SENSE) [25] and generalized autocalibrating partially parallel acquisitions (GRAPPA) [26] have been applied to realtime PCCMR [27–32]. More recently, modelbased reconstruction methods [33, 34] using regularized nonlinear inversion [35] have been developed, achieving 2D realtime flow imaging with a spatial resolution of 1.5 mm and a temporal resolution of 25.6 ms by jointly reconstructing a proton density map, a phase map, and a set of coil sensitivities.
In this work, we present a new modelbased method for realtime PCCMR with sparse sampling. It is based on the integration of a novel lowrank model with parallel imaging. With temporal subspace modeling, the proposed method yields a convex optimization problem, thereby enabling efficient computation. The proposed method achieves realtime PCCMR without ECG gating and respiration control, and well resolves the beatbybeat flow variations that cannot be obtained from the conventional cine method. Compared with stateoftheart realtime PCCMR techniques, it provides higher temporal resolution. The effectiveness of the proposed method has been systematically evaluated in 2D realtime PCCMR using both phantom experiments and in vivo experiments. A preliminary account of this work was presented in [36, 37].
Theory
Ignoring flow during readout time, the imaging equation for realtime PCCMR can be modeled as follows:
where ρ _{ v }(r, t) denotes the dynamic image associated with either the flowcompensated (i. e., v = 1) or flowencoded image sequence (i. e., v = 2, ⋯, N _{ v }), S _{ i }(r) the sensitivity map for the i th receiver coil (i = 1, 2, ⋯, N _{ c }), and d _{ v,i }(k, t) and η _{ v,i }(k, t) respectively the (k, t)space measured data and measurement noise. Here, the goal is to reconstruct ρ _{ v }(r, t) from the undersampled data {d _{ v,i }(k, t)}, and then calculate the velocity maps as \( \mathrm{V}\left(\mathbf{r}, t\right)=\frac{\varDelta \phi \left(\mathbf{r}, t\right)}{\uppi}\cdot \)VENC, where Δϕ(r, t) = ∠ ρ _{ v }(r, t) − ∠ ρ _{1}(r, t) denotes the phase difference between the flowencoded and flowcompensated image sequences, and VENC the prespecified encoding velocity. Since in realtime PCCMR, there is no data sharing with ECG gating, (k, t)space data is often highly undersampled. Direct inversion of {d _{ v,i }(k, t)} can incur significant aliasing artifacts and lead to inaccurate velocity measurements.
Here we introduce a lowrank modelbased reconstruction method with parallel imaging to address the problem. For convenience, we consider a discrete image model, in which each flow image sequence can be represented as a spatiotemporal Casorati matrix [38], i.e.,
Similar to cardiac imaging applications [39–41], each C _{ v } admits a lowrank approximation due to strong spatiotemporal correlation of timeseries images. Moreover, due to the nature of flow encoding, there is also strong spatial and temporal correlation among different flow image sequences. To exploit such correlation, the following joint Casorati matrix is introduced:
on which we enforce the lowrank structure, i.e., rank(C) ≤ L. There are a number of ways of imposing lowrank constraints [38, 40, 42, 43]. Here, we use an explicit rank constraint via matrix factorization, i.e., C = UV, where U ∈ ℂ ^{N × L} and V ∈ ℂ ^{L × M}. In this lowrank representation, the columns of U and rows of V respectively span the spatial subspace and temporal subspace of C.
Next, we formulate the lowrank constrained reconstruction problem. First, note that with matrixvector notation, Eq. (1) can be written as:
where d _{ i } denotes the measured data, Ω the sparse sampling operator, F _{s} the spatial Fourier transform matrix, and S _{ i } and n _{ i } respectively the sensitivity map and measurement noise. Imposing the lowrank constraint, the image reconstruction problem can be formulated as
This problem is a nonconvex optimization problem, for which a number of algorithms can be applied (e.g., [44, 45]).
The image reconstruction problem can be further simplified. Extending the early work in cardiac imaging [38, 40, 41, 46], we can preestimate the temporal subspace V by acquiring training data with a specialized data acquisition scheme. Specifically, as shown in Fig. 1, we design an interleaved sampling pattern, in which both training data and imaging data are collected. Here, the training data are sampled from the central kspace, while the imaging data are acquired from the remaining (k, t)space region with a random sampling scheme. With this sampling scheme, the two sets of data provide the complementary information for the lowrank model: the training data have high temporal resolution, while the imaging data have high spatial resolution. From the training data, we estimate the temporal subspace using the principal component analysis [38, 47]. With the imaging data, we estimate the spatial subspace U. To match the timing between the two sets of data, a proper temporal interpolation is performed, which interpolates the training data into those at the same time instants as the imaging data. Note that with such a scheme, the temporal resolution for the proposed method is 2 × N _{ v } × TR. Moreover, note that the coil sensitivities S _{ i } can be estimated from temporal averaged (k, t)space data from the flowcompensated image sequence.
With \( \widehat{\mathbf{V}} \), we can determine U by solving the following convex optimization problem:
Due to the temporal subspace estimation, the lowrank matrix recovery problem has been reduced to a simple leastsquares problem. By solving Û, the joint Casorati matrix can be reconstructed as \( \widehat{\mathbf{C}}=\widehat{\mathbf{U}}\widehat{\mathbf{V}} \), from which we can obtain each flow image sequence and estimate the flow velocities. A diagram summarizing the proposed method is shown in Fig. 2.
Methods
We performed both phantom and in vivo studies to evaluate the performance of the proposed method for 2D realtime PCCMR. The experiments were conducted on a 3.0 T whole body MR scanner (Achieva, Philips Medical System, Best, The Netherlands), equipped with a 32channel cardiovascular coil. A gradientecho (GRE) based pulse sequence was adapted to implement the proposed realtime acquisition scheme as shown in Fig. 1. Here neither ECG gating nor respiration control was used to aid data acquisition. Additionally, we performed conventional cine PCCMR using a vendorprovided GREbased pulse sequence, in which retrospective ECG gating was used.
First, flow phantom experiments were performed to evaluate the capability of the proposed method in resolving various flow dynamics. Specifically, a 15mmdiameter plastic tube simulating large vessel in the aorta was filled with bloodmimicking fluid [48], and plugged into a container (filled with water and positioned in the magnetic isocenter along the zdirection). The tube was further connected with a computerprogrammable pump (CompuFlow 5000 MR, Toronto, Canada) [49], with which we can set up different flow waveforms for the phantom experiments. Here, the two flow waveforms were used: flow waveform (I), as shown in Fig. 3a, repeating at a 2 s period within which a 1 s bellshape flow is followed by a 1 s constant flow; and flow waveform (II), as shown in Fig. 3d, repeating at a 4 s period within which two different 1 s bellshape flows are separated by a constant flow. To obtain flow measurements, we performed a onedirectional velocity encoding along the foothead (FH) direction for both cine and realtime experiments. For cine flow imaging, we assumed that the heart beat period is 2 s for ECG gating. Under this assumption, the waveform (I) represents a periodic flow, whereas the waveform (II) represents aperiodic flow. For both cine and realtime flow experiments, we used the following imaging parameters: field of view (FOV) = 220 mm × 120 mm, matrix size = 182 × 100, spatial resolution = 1.20 mm × 1.20 mm, slice thickness = 5 mm, repetition time (TR) = 5.0 ms, echo time (TE) = 3.0 ms, flip angle = 10°, and VENC = 100 cm/s. Notice that the temporal resolution for the realtime acquisition is 4 × TR = 20 ms, while, for the cine acquisition, the temporal resolution is 56 ms (with 36 cardiac phases). The total acquisition time was around 42 s for both experiments.
Second, in vivo experiments were performed to evaluate the proposed method. Ten healthy volunteers (7 males, age: 22–29 years, median: 25 years), who had no symptoms of cardiovascular diseases, were recruited. In addition, we performed a feasibility study of applying the proposed method for arrhythmia detection, and recruited two patients (2 males, age: 23year old and 72year old). This study was approved by the Institutional Review Board at Tsinghua University, and all the subjects gave written informed consent. Both the cine and realtime flow experiments were performed on the planes perpendicular to the ascending aorta (AAo) and descending aorta (DAo) during free breathing, and with one directional velocity encoding along the FH direction. For the cine acquisition, the retrospective ECG gating was set according to an estimate of each subject’s heartbeat period, and three averages were performed to mitigate respiratory motion artifacts. For both the cine and realtime imaging experiments, the following imaging parameters were used: FOV = 240 mm × 225 mm, matrix size = 132 × 124, spatial resolution = 1.80 mm × 1.80 mm, slice thickness = 5 mm, TR/TE = 4.5/2.8 ms, flip angle = 10°, and VENC = 200 cm/s. For the realtime flow imaging, the temporal resolution is 4 × TR = 18 ms, whereas for the cine imaging, the temporal resolution is around 36 ms (with 28 cardiac phases). The total acquisition time was around 94 s for both experiments.
For cine flow imaging, the flowcompensated and flowencoded images were simply reconstructed from the fullysampled data. For the proposed realtime flow imaging, we followed the procedure illustrated in Fig. 2. Specifically, we first performed the temporal interpolation and estimated the temporal subspace V from the training data. We then estimated the coil sensitivity maps S _{ i } from the temporally averaged (k, t)space measurements. We further determined the spatial subspace U by solving Eq. (6), followed by forming the timeseries images for flowcompensated and flowencoded images. To improve the computational efficiency, proper coil compression (e.g., [50]) can be adopted. After image reconstruction, phase correction [51] was performed to correct the phase offsets caused by eddy currents. The velocity maps were then extracted for quantitative flow analysis.
We analyzed the results of the phantom and in vivo experiments. For the phantom experiments, the flow waveforms obtained from the cine and realtime flow imaging methods were analyzed for both periodic and aperiodic flow patterns. For the in vivo experiments with healthy subjects, we evaluated the degree of agreement between the flow measurements from the cine method and those from the proposed method. Specifically, we performed a BlandAltman analysis, as well as a paired Student’s ttest, on the peak velocities and stroke volumes obtained from the two methods. Here the peak velocity is defined as the maximum velocity within one cardiac cycle, and the stroke volume is the integral of the flow velocity over one cardiac cycle within the ascending aorta. For the experiments with arrhythmic patients, we evaluated the flow variabilities captured by the proposed method with reference to an external ECG recording of cardiac motion.
To evaluate the effectiveness of imposing a lowrank constraint on the joint Casorati matrix \( \mathbf{C}=\left[\begin{array}{cc}\hfill {\mathbf{C}}_1\hfill & \hfill {\mathbf{C}}_2\hfill \end{array}\right] \), we performed a comparison with an alternative formulation, in which the lowrank constraint is enforced for each individual flow image sequence. The signaltonoise (SNR) and velocitytonoise (VNR) were calculated for the magnitude images and velocity maps, respectively. Here SNR was calculated as a ratio between the mean signal intensity over a region of interest (ROI) and the standard deviation of the background, whereas VNR was calculated as a ratio between the mean velocity for the same ROI and the standard deviation for a region in the stationary tissue [52].
Results
Representative results are shown to illustrate the performance of the proposed method. Figure 3 shows the flow waveforms for the phantom experiments obtained from the conventional cine method and the proposed realtime imaging method. Here the input flow waveforms for the pump were also shown. As can be seen, for the flow waveform (I) (i.e., periodic flow), both the cine and realtime imaging methods can capture the flow dynamics. In particular, the peak flows obtained from the two methods were accurate. However, for the flow waveform (II) (i.e., aperiodic flow), only the proposed method resolves the significant flow variations. The conventional cine method, which integrates data into a single cardiac cycle, fails to reconstruct the aperiodic flow dynamics (e.g., erroneous peak flows).
Figure 4 shows the in vivo results for two healthy subjects. Here, we show the reconstructed magnitude images and velocity maps corresponding to a systolic cardiac phase and a diastolic cardiac phase. As can be seen, the proposed method provides at least comparable reconstruction quality to the cine method. Although both methods can resolve the vessel structure, the realtime imaging method is more motionrobust than the cine method. To better illustrate the proposed method, a reconstruction video for one healthy subject was included (see Additional file 1).
In addition, we analyzed the mean flow velocities associated with two ROIs in AAo and DAo. Figure 5a and b respectively show the velocity waveforms over 10 consecutive cardiac cycles for a healthy subject. Clearly, the proposed method well resolves beatbybeat variations. We further evaluated how the velocity waveforms from the realtime imaging are related to those from the conventional cine method. We averaged the velocity waveforms over 30 consecutive cardiac cycles from the proposed method into one velocity waveform associated with a synthetic cardiac cycle, and then compared it with that from the cine method. From Fig. 5c and d, it is evident that the averaged velocity waveforms for AAo and DAo correlate well with those from the conventional cine method. In particular, both methods yield very similar peak velocities for the AAo and DAo.
We also performed a statistical analysis of the results from the two methods for all ten healthy subjects. Figure 6a and b respectively show the BlandAltman plots of peak velocities and stroke volumes that compare the two methods. As can be seen, the results from the proposed method are in excellent agreement with those from the conventional cine method. In addition, we performed the paired Student’s ttest analysis on the two methods, and the correlation coefficients for peak velocities and stroke volumes are 0.94 (P < 0.0001) and 0.90 (P = 0.0002), respectively. This further confirms strong correlation between the two methods.
Figure 7 shows the reconstruction results for the 23yearold patient (with mild cardiac arrhythmia). As expected, the proposed method is able to reconstruct flow variations over different cardiac cycles. In particular, as shown in Fig. 7b, the proposed method nicely captures a sudden flow velocity drop occurring in an arrhythmic period. Note that this type of flow dynamics cannot be obtained from the conventional cine method. Further, it is worth noting that the flow velocity variations correlate well with the ECG signal recorded during the acquisition. Besides, we show three snapshot images from the proposed method. Clearly, the velocity maps confirm the dramatic flow variations within the arrhythmic period.
Figure 8 shows the reconstruction results for the 72yearold patient (with severe cardiac arrhythmia). The velocity waveforms associated with the AAo and DAo from the proposed method are shown in Fig. 8a. Again, the proposed method well captures irregular flow variations, which are more significant than the ones from the previous patient. Moreover, we show the reconstructed magnitude images and velocity maps in Fig. 8b, and include the corresponding reconstruction video in Additional file 2.
Figure 9 compares the magnitude images and velocity maps from the proposed method using the joint lowrank constraint with that using the separate lowrank reconstruction. Here the two methods reconstructed the same data set (i.e., a 40 s realtime PCCMR acquisition), and used the same rank value L = 20. As can be seen, the proposed method reconstructs the spatial images and velocity maps with improved quality over the alternative formulation. This illustrates the benefits of imposing the lowrank constraint on the joint Casorati matrix.
Discussion
In this work, we introduced a new realtime flow imaging method and systematically demonstrated its effectiveness with both flow phantom experiments and in vivo experiments. Here, it is worth reiterating the key characteristics of the proposed method. First, the proposed method can be used as a viable alternative to the conventional cine flow imaging method in that it provides comparable (if not superior) image quality and flow information for healthy subjects. Second, the proposed method is able to resolve beatbybeat physiological and/or pathological flow variations, which cannot be obtained from the conventional cine method based on ECG gating and respiration control. Such information is often clinically important (e.g., for assessing cardiac arrhythmia).
As with other modelbased methods, the proposed method involves model selection (i.e., selection of the rank L). Generally, the selection of L needs to balance the model representational power, the number of measurements (i.e., acquisition time), and signaltonoise ratio [40]. In this work, we manually selected L to trade off the above factors, and it consistently yielded good reconstruction performance, although it is worthwhile to investigate other principled model selection methods (e.g., [53, 54]) in future research.
The proposed formulation results in a convex optimization problem, which enables efficient computation. For example, the runtime for reconstructing an in vivo dataset (from 94 s realtime acquisition) takes around 10 min on a workstation with 64 GB RAM and 3.47 GHz CPU. The computational efficiency may be further improved by an implementation on graphical processing units. Such an investigation is beyond the scope of this paper, but is worthwhile to explore for future research.
In addition to rank constraint, sparsity constraint can also be incorporated to accelerate PCCMR. It has been demonstrated in [40, 43, 55] that joint lowrank and sparsity constrained reconstruction leads to improved performance for dynamic MRI. Along this line, we can extend the proposed realtime flow imaging method by exploiting our early work [56] in cine flow imaging, although such an extension will come with additional computational cost.
The flowcompensated and flowencoded images share similar magnitude but different phase differences. We can extend the proposed method to exploit such information and impose a stronger constraint in the modelbased reconstruction. However, the resulting formulation can involve a joint reconstruction of magnitude and phase images, which generally leads to a nonconvex optimization problem. To solve such a problem, specialized algorithms and proper initialization are often needed. In contrast, the proposed method here employs a lowrank model to exploit the spatiotemporal correlation between flow images, which leads to a simple convex problem formulation and efficient computation. Given that the two models may have different tradeoffs, comprehensively evaluating their advantages and drawbacks is a very interesting open problem to be explored in future work.
In this work, we demonstrate the performance of the proposed method for 2D realtime flow imaging, in which throughplane flow was imaged. Considering the complex flow patterns and blood vessel geometry, it is highly desirable to perform 3D realtime flow imaging. However, 3D realtime flow imaging generally involves a more challenging tradeoff between spatial resolution, temporal resolution, and imaging time, and a significantly more challenging computational problem. We are investigating an extension of the proposed method to 3D realtime flow imaging, and the results will be reported in future work.
This paper is focused on the development of a novel realtime flow imaging technique, which should serve as a foundation for our subsequent clinical studies. Given that the proposed method well resolves beatbybeat flow variations, it can provide more information on hemodynamics for patients with significant irregular heartbeats. In the future work, we plan to conduct systematic study of the proposed method for various potential clinical applications (e.g., atrial fibrillation, premature atrial contraction or congenital heart disease).
It is also worthwhile to remark on the potential limitations of the proposed method. First, note that the aforementioned spatial and temporal resolution both refer to nominal resolution. For a linear shiftinvariant reconstruction method (e.g., conventional Fourier reconstruction), the resolution can be characterized through the point spread function. However, for a nonlinear reconstruction method (e.g., sparsity [57] or lowrank constrained reconstruction [38, 40]), rigorously characterizing the resolution has been a longstanding open problem. In this work, we turn to reporting the nominal spatial and temporal resolution, although it is worthwhile to perform an indepth study of resolution characterization for these advanced image reconstruction methods in future research.
Second, it is useful to create gold standard data sets for studying realtime flow imaging. Due to the undersampling nature of realtime imaging experiments, it is often difficult to generate an ideal reference for systematic quantitative evaluation. For example, in the phantom experiments, the input flow waveforms for the pump deviate from the flow measurements during the constant flow due to the phantom response to the flow/pressure in the tubing system. In the future, we hope to build a more advanced flow imaging phantom, in which better reference data sets can be generated.
Conclusions
A new modelbased method was introduced for highresolution realtime PCCMR without ECG gating and respiration control. It integrates the novel lowrank model with parallel imaging, which enables highquality image reconstruction from highly undersampled (k, t)space data for realtime PCCMR. The effectiveness and utilities of the proposed method have been demonstrated for 2D realtime PCCMR with both phantom experiments and in vivo experiments. We expect that the proposed method will enhance the practical utility of realtime PCCMR for various clinical applications.
Abbreviations
 AAo:

Ascending aorta
 DAo:

Descending aorta
 ECG:

Electrocardiogram
 FOV:

Field of view
 GRAPPA:

Generalized autocalibrating partially parallel acquisitions
 PCCMR:

Phasecontrast cardiovascular magnetic resonance
 SENSE:

Sensitivity encoding
 TE:

Echo time
 TR:

Repetition time
 VENC:

Encoding velocity
References
 1.
van Dijk P. Direct cardiac NMR imaging of heart wall and blood flow velocity. J Comput Assist Tomogr. 1984;8(3):429–36.
 2.
Nayler G, Firmin D, Longmore D. Blood flow imaging by cine magnetic resonance. J Comput Assist Tomogr. 1986;10(5):715–22.
 3.
Gatehouse PD, Keegan J, Crowe LA, Masood S, Mohiaddin RH, Kreitner KF, Firmin DN. Applications of phasecontrast flow and velocity imaging in cardiovascular MRI. Eur Radiol. 2005;15(10):2172–84.
 4.
Markl M, Kilner PJ, Ebbers T. Comprehensive 4D velocity mapping of the heart and great vessels by cardiovascular magnetic resonance. J Cardiovasc Magn Reson. 2011;13(7):1–22.
 5.
Nayak KS, Nielsen JF, Bernstein MA, Markl M, Gatehouse PD, Botnar RM, Saloner D, Lorenz C, Wen H, Hu BS, Epstein FH, Oshinski JN, Raman SV. Cardiovascular magnetic resonance phase contrast imaging. J Cardiovasc Magn Reson. 2015;17(1):1.
 6.
Markl M, Frydrychowicz A, Kozerke S, Hope M, Wieben O. 4D flow MRI. J Magn Reson Imaging. 2012;36(5):1015–36.
 7.
Hope TA, Hope MD, Purcell DD, von Morze C, Vigneron DB, Alley MT, Dillon WP. Evaluation of intracranial stenoses and aneurysms with accelerated 4D flow. Magn Reson Imaging. 2010;28(1):41–6.
 8.
Ansari S, Schnell S, Carroll T, Vakil P, Hurley M, Wu C, Carr J, Bendok B, Batjer H, Markl M. Intracranial 4D flow MRI: Toward individualized assessment of arteriovenous malformation hemodynamics and treatmentinduced changes. Am J Neuroradiol. 2013;34(10):1922–8.
 9.
Lenz GW, Haacke EM, White RD. Retrospective cardiac gating: A review of technical aspects and future directions. Magn Reson Imaging. 1989;7(5):445–55.
 10.
Dyverfeldt P, Bissell M, Barker AJ, Bolger AF, Carlhäll CJ, Ebbers T, Francios CJ, Frydrychowicz A, Geiger J, Giese D, Hope MD, Kilner PJ, Kozerke S, Myerson S, Neubauer S, Wieben O, Markl M. 4D flow cardiovascular magnetic resonance consensus statement. J Cardiovasc Magn Reson. 2015;17(1):1–19.
 11.
Finn JP, Nael K, Deshpande V, Ratib O, Laub G. Cardiac MR imaging: State of the technology. Radiology. 2006;241(2):338–54.
 12.
Thavendiranathan P, Verhaert D, Walls MC, Bender JA, Rajagopalan S, Chung YC, Simonetti OP, Raman SV. Simultaneous right and left heart realtime, freebreathing CMR flow quantification identifies constrictive physiology. J Am Coll Cardiol Cardiovasc Imaging. 2012;5(1):15–24.
 13.
Kowallick JT, Joseph AA, UnterbergBuchwald C, Fasshauer M, van Wijk K, Merboldt KD, Voit D, Frahm J, Lotz J, Sohns JM. Realtime phasecontrast flow MRI of the ascending aorta and superior vena cava as a function of intrathoracic pressure (Valsalva manoeuvre). Br J Radiol. 2014;87(1042):20140401.
 14.
Körperich H, Barth P, Gieseke J, Müller K, Burchert W, Esdorn H, Kececioglu D, Beerbaum P, Laser KT. Impact of respiration on stroke volumes in paediatric controls and in patients after Fontan procedure assessed by MR realtime phasevelocity mapping. Eur Heart J Cardiovasc Imaging. 2015;16(2):198–209.
 15.
Riederer SJ, Wright RC, Ehman RL, Rossman PJ, HolsingerBampton AE, Hangiandreou NJ, Grimm RC. Realtime interactive color flow MR imaging. Radiology. 1991;181(1):33–9.
 16.
Liu CY, Varadarajan P, Pohost GM, Nayak KS. Realtime colorflow MRI at 3 T using variabledensity spiral phase contrast. Magn Reson Imaging. 2008;26(5):661–6.
 17.
Eichenberger AC, Schwitter J, McKinnon GC, Debatin JF, von Schulthess GK. Phasecontrast echoplanar MR imaging: Realtime quantification of flow and velocity patterns in the thoracic vessels induced by Valsalva’s maneuver. J Magn Reson Imaging. 1995;5(6):648–55.
 18.
Mohiaddin RH, Gatehouse PD, Moon JC, Youssuffidin M, Yang GZ, Firmin DN, Pennell DJ. Assessment of reactive hyperaemia using real time zonal echoplanar flow imaging. J Cardiovasc Magn Reson. 2002;4(2):283–7.
 19.
Shankaranarayanan A, Simonetti OP, Laub G, Lewin JS, Duerk JL. Segmented kspace and realtime cardiac cine MR imaging with radial tajectories. Radiology. 2001;221(3):827–36.
 20.
Joseph AA, Merboldt KD, Voit D, Zhang S, Uecker M, Lotz J, Frahm J. Realtime phasecontrast MRI of cardiovascular blood flow using undersampled radial fast lowangle shot and nonlinear inverse reconstruction. NMR Biomed. 2012;25(7):917–24.
 21.
Gatehouse PD, Firmin DN, Collins S, Longmore DB. Real time blood flow imaging by spiral scan phase velocity mapping. Magn Reson Med. 1994;31(5):504–12.
 22.
Nayak KS, Pauly JM, Kerr AB, Hu BS, Nishimura DG. Realtime color flow MRI. Magn Reson Med. 2000;43(2):251–8.
 23.
Steeden JA, Atkinson D, Taylor AM, Muthurangu V. Assessing vascular response to exercise using a combination of realtime spiral phase contrast MR and noninvasive blood pressure measurements. J Magn Reson Imaging. 2010;31(4):997–1003.
 24.
Kowalik GT, Steeden JA, Pandya B, Odille F, Atkinson D, Taylor A, Muthurangu V. Realtime flow with fast GPU reconstruction for continuous assessment of cardiac output. J Magn Reson Imaging. 2012;36(6):1477–82.
 25.
Pruessmann KP, Weiger M, Scheidegger MB, Boesiger P. SENSE: Sensitivity encoding for fast MRI. Magn Reson Med. 1999;42(5):952–62.
 26.
Griswold MA, Jakob PM, Heidemann RM, Nittka M, Jellus V, Wang J, Kiefer B, Haase A. Generalized autocalibrating partially parallel acquisitions (GRAPPA). Magn Reson Med. 2002;47(6):1202–10.
 27.
Weiger M, Pruessmann KP, Boesiger P. Cardiac realtime imaging using SENSE. Magn Reson Med. 2000;43(2):177–84.
 28.
Hoogeveen R, Leone B, van der Brink J. Realtime quantitative flow using EPI and SENSE. In: Proceedings of the 9th Annual Meeting of the International Society for Magnetic Resonance in Medicine: 2127 April 2001. Glasgow, vol. 9; 2001. p. 114.
 29.
Körperich H, Gieseke J, Barth P, Hoogeveen R, Esdorn H, Peterschröder A, Meyer H, Beerbaum P. Flow volume and shunt quantification in pediatric congenital heart disease by realtime magnetic resonance velocity mapping: a validation study. Circulation. 2004;109(16):1987–93.
 30.
Wintersperger BJ, Nikolaou K, Dietrich O, Rieber J, Nittka M, Reiser MF, Schoenberg SO. Single breathhold realtime cine MR imaging: Improved temporal resolution using generalized autocalibrating partially parallel acquisition (GRAPPA) algorithm. Eur Radiol. 2003;13(8):1931–6.
 31.
Jones A, Steeden JA, Pruessner JC, Deanfield JE, Taylor AM, Muthurangu V. Detailed assessment of the hemodynamic response to psychosocial stress using realtime MRI. J Magn Reson Imaging. 2011;33(2):448–54.
 32.
Kowalik GT, Knight DS, Steeden JA, Tann O, Odille F, Atkinson D, Taylor A, Muthurangu V. Assessment of cardiac time intervals using high temporal resolution realtime spiral phase contrast with UNFOLDedSENSE. Magn Reson Med. 2015;73(2):749–56.
 33.
Joseph A, Kowallick JT, Merboldt KD, Voit D, Schaetz S, Zhang S, Sohns JM, Lotz J, Frahm J. Realtime flow MRI of the aorta at a resolution of 40 msec. J Magn Reson Imaging. 2014;40(1):206–13.
 34.
Tan Z, Roeloffs V, Voit D, Joseph AA, Untenberger M, Merboldt KD, Frahm J. Modelbased reconstruction for realtime phasecontrast flow MRI: Improved spatiotemporal accuracy. Magn Reson Med. 2016. doi:10.1002/mrm.26192.
 35.
Uecker M, Hohage T, Block KT, Frahm J. Image reconstruction by regularized nonlinear inversionjoint estimation of coil sensitivities and image content. Magn Reson Med. 2008;60(3):674–82.
 36.
Zhao B, Sun A, Ma K, Li R, Christodoulou AG, Yuan C, Liang ZP. Realtime phase contrast cardiovascular flow imaging with joint lowrank and sparsity constraints. In: Proceedings of the 23rd Annual Meeting of the International Society for Magnetic Resonance in Medicine: 1016 May 2014; Milan; 2014. p. 743.
 37.
Sun A, Zhao B, Li Y, He Q, Zhou Z, Chen S, Li R, Yuan, C. A validation study of realtime phase contrast MRI with lowrank modeling. In: Proceedings of the 24th Annual Meeting of the International Society for Magnetic Resonance in Medicine: 713 May 2016; Singapore; 2016. p. 2702.
 38.
Liang ZP. Spatiotemporal imagingwith partially separable functions. In: Proceedings of IEEE International Symposium on Biomedical Imaging: April 2007; Washington, DC; 2007. pp. 988–91.
 39.
Zhao B, Haldar JP, Brinegar C, Liang ZP. Low rank matrix recovery for realtime cardiac MRI. In: Proceedings of IEEE International Symposium on Biomedical Imaging: April 2010; Rotterdam; 2010. pp. 996–9.
 40.
Zhao B, Haldar JP, Christodoulou AG, Liang ZP. Image reconstruction from highly undersampledspace data with joint partial separability and sparsity constraints. IEEE Trans Med Imaging. 2012;31(9):1809–20.
 41.
Christodoulou AG, Zhang H, Zhao B, Hitchens TK, Ho C, Liang ZP. Highresolution cardiovascular MRI by integrating parallel imaging with lowrankand sparse modeling. IEEE Trans Biomed Eng. 2013;60(11):3083–92.
 42.
Haldar JP, Liang ZP. Spatiotemporal imaging with partially separable functions: A matrix recovery approach. In: Proceedings of IEEE International Symposium on Biomedical Imaging: April 2010; Rotterdam; 2010. pp. 716–9.
 43.
Lingala SG, Hu Y, DiBella E, Jacob M. Accelerated dynamic MRI exploiting sparsity and lowrank structure: kt SLR. IEEE Trans Med Imaging. 2011;30(5):1042–54.
 44.
Haldar JP, Hernando D. Rankconstrained solutions to linear matrix equations using power factorization. IEEE Signal Proc Let. 2009;16(7):584–7.
 45.
Shen Y, Wen Z, Zhang Y. Augmented lagrangian alternating direction method for matrix separation based on lowrank factorization. Optim Method Softw. 2014;29(2):239–63.
 46.
Christodoulou AG, Zhao B, Zhang H, Ho C, Liang ZP. Fourdimensional MR cardiovascular imaging: Method and applications. In: Proceedings of IEEE Engineering in Medicine and Biology Society: Aug 2011; Boston; 2011. pp. 3732–5.
 47.
Gupta AS, Liang ZP. Dynamic imaging by temporal modeling with principal component analysis. In: Proceedings of the 9th Annual Meeting of the International Society for Magnetic Resonance in Medicine: 2127 April 2001; Glasgow; 2001. p. 10.
 48.
Traber J, Wurche L, Dieringer MA, Utz W, von KnobelsdorffBrenkenhoff F, Greiser A, Jin N, SchulzMenger J. Realtime phase contrast magnetic resonance imaging for assessment of haemodynamics: From phantom to patients. Eur Radiol. 2016;26(4):986–96.
 49.
Holdsworth D, Rickey D, Drangova M, Miller D, Fenster A. Computercontrolled positive displacement pump for physiological flow simulation. Med Biol Eng Comput. 1991;29(6):565–70.
 50.
Zhang T, Pauly JM, Vasanawala SS, Lustig M. Coil compression for accelerated imaging with cartesian sampling. Magn Reson Med. 2013;69(2):571–82.
 51.
Walker PG, Cranney GB, Scheidegger MB, Waseleski G, Pohost GM, Yoganathan AP. Semiautomated method for noise reduction and background phase error correction in MR phase velocity data. J Magn Reson Imaging. 1993;3(3):521–30.
 52.
Ringgaard S, Oyre SA, Pedersen EM. Arterial MR imaging phasecontrast flow measurement:improvements with varying velocity sensitivity during cardiac cycle. Radiology. 2004;232(1):289–94.
 53.
Stoica P, Selen Y. Modelorder selection: a review of information criterion rules. IEEE Signal Process Mag. 2004;21(4):36–47.
 54.
Haldar JP. Constrained imaging: denoising and sparse sampling. PhD thesis. University of Illinois at UrbanaChampaign, Electrical & Computer Engineering Department; 2011.
 55.
Zhao B, Haldar JP, Christodoulou AG, Liang ZP. Further development of image reconstruction from highly undersampled (k, t)space data with joint partial separability and sparsity constraints. In: Proceedings of IEEE International Symposium on Biomedical Imaging: April 2011; Chicago; 2011. pp. 1593–6.
 56.
Sun A, Zhao B, Ma K, Zhou Z, He L, Li R, Yuan C. Accelerated phase contrast flow imaging with direct complex difference reconstruction. Magn Reson Med. 2016. doi:10.1002/mrm.26184.
 57.
Lustig M, Donoho D, Pauly JM. Sparse MRI: The application of compressed sensing for rapid MR imaging. Magn Reson Med. 2007;58(6):1182–95.
Acknowledgements
The authors thank Prof. ZhiPei Liang at the University of Illinois at UrbanaChampaign for insightful discussions. The authors also thank Dr. Zechen Zhou and Shuo Chen at Tsinghua University for the help with data acquisition.
Funding
This work was partially supported by the National Key R&D Program during the “13th FiveYear Plan” (2016YFC1301601), and National Institute of Health (NIHRO1EB013695).
Availability of supporting data
Not applicable.
Author’s contributions
AS conceived the study, developed the imaging technique, performed data collection and analysis, and drafted the manuscript. BZ conceived the study, developed the imaging technique, assisted in the interpretation of the results, and revised the manuscript. YL and QH assisted with the experimental study. RL participated in the design of experimental study, assisted in the interpretation of the results, and revised the manuscript. CY supervised the project and revised the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Written informed consent was obtained from all the subjects for publication of their individual details and accompanying images in this manuscript.
Ethical approval and consent to participate
This study was approved by the Institutional Review Board at Tsinghua University, and all the subjects gave written informed consent.
Authors’ information
The authors have no additional information to report.
Author information
Additional files
Additional file 1:
Realtime PCCMR of a healthy subject. This video includes the reconstructed magnitude images and velocity maps by the proposed method for a healthy subject. (GIF 3557 kb)
Additional file 2:
Realtime PCCMR of an arrhythmic patient. This video includes the reconstructed magnitude images and velocity maps by the proposed method for an arrhythmic patient. (GIF 4693 kb)
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Sun, A., Zhao, B., Li, Y. et al. Realtime phasecontrast flow cardiovascular magnetic resonance with lowrank modeling and parallel imaging. J Cardiovasc Magn Reson 19, 19 (2017). https://doi.org/10.1186/s1296801703301
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Keywords
 Cardiovascular imaging
 Phasecontrast CMR
 Cine
 Realtime flow imaging
 Modelbased reconstruction
 Lowrank modeling
 Parallel imaging