Accelerated 4D‐flow MRI with 3‐point encoding enabled by machine learning

Abstract

Purpose

To investigate the acceleration of 4D‐flow MRI using a convolutional neural network (CNN) that produces three directional velocities from three flow encodings, without requiring a fourth reference scan measuring background phase.

Methods

A fully 3D CNN using a U‐net architecture was trained in a block‐wise fashion to take complex images from three flow encodings and to produce three real‐valued images for each velocity component. Using neurovascular 4D‐flow scans (n = 144), the CNN was trained to predict velocities computed from four flow encodings by standard reconstruction including correction for residual background phase offsets. Methods to optimize loss functions were investigated, including magnitude, complex difference, and uniform velocity weightings. Subsequently, 3‐point encoding was evaluated using cross validation of pixelwise correlation, flow measurements in major arteries, and in experiments with data at differing acceleration rates than the training data.

Results

The CNN‐produced 3‐point velocities showed excellent agreements with the 4‐point velocities, both qualitatively in velocity images, in flow rate measures, and quantitatively in regression analysis (slope = 0.96, R ² = 0.992). Optimizing the training to focus on vessel velocities rather than the global velocity error and improved the correlation of velocity within vessels themselves. The lowest error was observed when the loss function used uniform velocity weighting, in which the magnitude‐weighted inverse of the velocity frequency uniformly distributed weighting across all velocity ranges. When applied to highly accelerated data, the 3‐point network maintained a high correlation with ground truth data and demonstrated a denoising effect.

Conclusion

The 4D‐flow MRI can be accelerated using machine learning requiring only three flow encodings to produce three‐directional velocity maps with small errors.

Short abstract

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1. INTRODUCTION

Recent technical advances have enabled 4D‐flow MRI cardiovascular exams to provide a wealth of hemodynamic information. ¹ However, the long scan time of 4D‐flow MRI continues to be a major factor hindering its utilization, especially in emerging applications outside of aorta. ² To address this, a number of investigators have proposed the use of universal acceleration techniques including efficient trajectories ³ , ⁴ , ⁵ and temporal–spatial acceleration. ⁶ , ⁷ , ⁸ , ⁹ While certainly effective, the achievable acceleration provided by these techniques is limited by noise amplification and artifacts introduced at high accelerations. In cases using efficient trajectories, such as EPI, the flow field can be additionally subject to subtle artifacts due to higher order motion. ¹⁰ More recently, physics‐informed deep learning has been proposed as a method to both denoise the velocity field and improve the spatial resolution of the velocity field itself. ¹¹ , ¹² , ¹³ This approach uses flow‐specific information that can potentially allow accelerated acquisitions when applied as a post‐processing step to velocity fields.

As a complementary form of acceleration, alternative efforts have been proposed to apply acceleration based on the intrinsic encoding of the velocity into the phase of the magnetization. The phase of each image has spatially varying background offsets that result from field inhomogeneity, concomitant fields, and gradient eddy currents. ¹⁴ As these phase offsets must be determined to calculate the velocity components from the phase images, 4D‐flow MRI must inevitably solve for four unknowns and hence require a minimum of four flow encodings. While many variations of flow encoding exist, ¹⁵ , ¹⁶ , ¹⁷ , ¹⁸ the simplest approach uses a fourth “reference” scan dedicated to measuring the phase offset common to all velocity encodings, often called the “background phase.” This background phase measurement enabled the removal of the common phase from each flow encoding leaving the velocity induced phase and a smaller “residual background phase” from differences in eddy currents and concomitant fields in each of the flow encodings. This residual background phase is smoothly varying and usually removed using polynomial fitting to static background signal. As a consequence of having to encode velocities in three orthogonal directions and to perform a separate reference scan, the scan time of 4D‐flow MRI is at minimum four times longer than an equivalent non‐flow scan. Methods have been proposed using directionality constraints on the velocity field over time ¹⁹ , ²⁰ to reduce the number of velocity encodings; however, these methods can result in a reduction in the velocity‐to‐noise ratio (VNR) and make assumptions regarding the flow field itself.

As an alternative to methods attempting to impose constraints on the velocity field, it may be possible to target the substantially different background phase and acquire data without an additional reference encoding. The 4D‐flow MRI without a reference scan has been shown with phase contrast balanced SSFP (PC‐bSSFP). Due to the refocusing properties of bSSFP, the background phase can be estimated by polynomial fitting within segmented static tissue. ²¹ This assumes the background phase is smoothly varying, devoid of banding artifacts, and that there is sufficient surrounding static tissue. However, this approach is not directly applicable to most 4D‐flow MRI scans which, due to a spoiled gradient echo sequence, have a background phase which is not well fit with smooth polynomial functions. Furthermore, this method requires background tissue segmentation that can be challenging without an angiogram.

The hypothesis of this work is that reference free (3‐point) 4D‐flow MRI can be performed using deep learning. This is motivated by the fact that the background field will be governed by the physics of magnetic susceptibility (e.g., dipole kernel convolution) with additional smoothly varying offsets from eddy currents and concomitant gradients (e.g., low order spatial polynomials). The physical nature of these offsets is thus quite different from phase from velocity which will be governed by the physics of fluid flow (e.g., divergent free, Navier–Stokes, etc.). The use of deep learning is motivated by its performance in the realm of QSM. QSM requires the modeling of the background phase from spoiled gradient echo sequence, the same background phase present in the flow compensated reference scans in 4D‐flow MRI. Deep learning results have suggested that convolutional neural networks are highly capable of predicting and modeling complex phase maps from gradient echo images. ²² , ²³ , ²⁴ These prospects suggest that trained neural networks may be able to perform background phase removal from the velocity‐encoded phase maps so that three directional velocity components can be determined without a dedicated reference scan.

In this study, we investigated the feasibility and effectiveness of using convolutional neural networks to accurately produce three directional velocities without the use of the fourth reference encoding. Specifically, we use existing 4‐point 4D‐flow data to train a network to predict three directional velocities using only three of the acquired flow encodings. We quantitatively and qualitatively examine the agreement between the 3‐point and 4‐point velocities and investigated methods to improve the training.

2. METHODS

2.1. Subjects and image acquisition

This study was Health Insurance Portability and Accountability Act (HIPAA) compliant and approved by the local institutional review board (IRB). Data was included from 140 patients who underwent neurovascular 4D‐flow MRI scans in 3 T scanners (SIGNA Premier or MR750, GE Healthcare, Waukesha, Wisconsin, USA) using 48‐channel (for SIGNA Premier) or 32‐channel (for MR750) head coils. The 4D‐flow was performed using an undersampled 3D radial sequence ³ with 0.69 mm isotropic resolution, 22.0 × 22.0 × 14.0 cm³ FOV, 320 × 320 × 320 matrix size, TE/TR = 2.5 ms/7.7 ms, 5:39 scan time, 11 000 radial projections, 8° flip angle, 4‐point referenced flow encoding, and v _enc = 80 cm/s. A standard 4D‐flow reconstruction pipeline was used to produce magnitude and three directional velocity images (v _x, v _y , v _z ) computed from the four flow encodings (called “4‐point velocities”). This processing included the removal of residual background phase from concomitant fields using scanner geometries and calculated maxwell terms. This operation was performed in the complex domain to prevent errors from concomitant gradient included phase wrap. Following this, velocity images were computed from the phase difference images of the complex data. To correct for additional residual background phase, predominantly from eddy currents, the velocity images were corrected using second‐order polynomial fitting to segmented static background tissue. The segmentation of static background tissue was computed by thresholding the magnitude and complex difference angiogram images. In addition to this, raw complex images, prior to any correction for residual phase offsets from concomitant gradients or eddy currents, were saved from this process using low resolution images for coil combination. Of 140 total subjects, 112 subjects (4/5 of the total subjects) formed the training group, with the remaining 28 subjects (1/5 of the total subjects) formed the non‐training group. The non‐training group was equally divided into two groups forming validation and test sets (n = 14 each).

2.2. Training

A block‐wise fully three‐dimensional CNN was constructed in a U‐Net ²⁵ architecture as shown in Supporting Information Figure S1. It was constructed to take six channel images, corresponding to the real and imaginary channels from only three flow encodings without a reference encoding, as input and produce three real‐valued velocity images corresponding to v _x , v _y , and v _z as its output. Instead of taking the entire 3D volume of the source images, the network took a 3D sub‐volume, or block, of the source images as input and produced output images of the same size as the input block. The CNN was purely spatial and did not incorporate multiple time frames. This block‐wise training was required due to GPU memory limitations. The original data was collected with balanced reference encoding, from which we sub‐selected the flow encodings along each of x,y, and z: (−1,−1, +1), (−1, +1,−1), and (+1,−1,−1). The reference scan (−1,−1,−1) was not included as input to the CNN. The input images were retrospectively prepared from each subject's scan by taking those three flow encodings. The velocities produced by the CNN based on these three flow encodings are termed “3‐point velocities” here.

The network was trained to predict the 4‐point velocities from those three flow encodings by minimizing the following loss function ($f_{\text{loss}})$ that measured the discrepancy between the 3‐point (${\mathbf{v}}_{3\mathit{pt}}$) and 4‐point (${\mathbf{v}}_{4\mathit{pt}}$) velocities:

$$f_{\text{loss}}=\frac{1}{N}\sum _{N}W{‖{\mathbf{v}}_{4\mathit{pt}}-{\mathbf{v}}_{3\mathit{pt}}‖}^2,$$ (1)

which is the mean square error of the discrepancy between the two types of velocities, weighted by a weight image $W$, computed over the $N$ voxels within the patch. We examined several choices of the weight image $W$ described below to optimize the training to yield the best agreements between the two velocity types.

To reduce bottlenecks due to disk access, the network was trained using subject by subject training. First, a single 320 × 320 × 320 image was loaded to form a single case. Subsequently, a batch of eight randomly selected 64 × 64 × 64 blocks were extracted and used to update the CNN parameters using gradient descent optimization. This was repeated 15 times to roughly cover the entire 320 × 320 × 320 volume. Due to the random nature of the block selection, sampling did not include every pixel in the volume. Every 15 gradient descent steps/batches, a new case was loaded. Image augmentation was performed on a block level using random flip in each of three orthogonal directions prior to being used as an input. Once this process completed through all training subjects in one epoch, the (validation) loss was computed based on the validation subjects. The training continued until 400 epochs or until over‐fitting was observed in the validation loss. An Adam optimizer with a learning rate of 10⁻⁴ was used for all training.

2.3. Loss function weight

We examined several different choices for the weight image $W$ used in the loss function (see Eq. 1). The examined choices included the signal magnitude, complex difference (CD), and magnitude‐weighted inverse of velocity frequencies. In phase‐contrast MRI, the velocity noise is inversely proportional to the strength of the signal magnitude $M$. Thus, the use of magnitude image as weight $W_{\mathrm{mag}}=M$ effectively weighs the velocity discrepancies by the precision of the velocity measurements at each voxel location. This ensures that low weights are assigned where there is little MR signal (e.g., air) and high velocity noise. The complex difference angiogram is a high‐contrast, bright‐blood vessel image, routinely generated as flow‐derived angiogram in phase‐contrast MRI:

$$W_{\mathrm{CD}}=\left\{\begin{array}{cc}M\sin \frac{\pi \left|v\right|}{2v_{\mathrm{enc}}}& \left|v\right|<v_{\mathrm{enc}}\\ M& \text{otherwise}\end{array}\right.$$ (2)

It is an increasing function of velocity magnitude ($\left|v\right|)$ that is weighted by the signal magnitude $M$ (more specifically, the first quarter of sine function is used for velocity domains between 0 and v _enc). Thus, the complex difference image serves to selectively weigh velocities within vessels while suppressing the background where there is no flow. Lastly, we examined uniform velocity weighting, in which the inverse of velocity frequencies, weighted by the signal magnitude, uniformly distributed weights across all velocity ranges:

$$W_{\mathrm{vel}}=\frac{M}{p(v)}$$ (3)

where $p(v)$ is the estimated velocity distribution. To obtain the estimated velocity distribution, the frequency histogram of velocity magnitudes was computed over the entire pool of training subjects, using their 4‐point velocities. For binning, we used 100 bins between velocities of 0 and 0.3v _enc, followed by the last bin for velocities of 0.3v _enc or higher. 0.3v _enc was chosen for the last bin because the velocity distribution had negligible counts above this velocity. The cutoff velocity and the number of bins were chosen to ensure that every bin is represented with meaningful counts such that the histogram optimally reflects the distribution of relevant velocities in the training subjects. Velocity values in air were excluded from this histogram analysis using a manually selected threshold of the magnitude image.

2.4. Training Evaluation

Following training, the network was used to generate 3‐point velocities in each test subject using only three flow encodings without a reference scan. Like the training, the generation of 3‐point velocities were done in patches until the entire image volume was sequentially covered. To avoid possible artifacts at the edge of the patches, the 3‐point velocities were also generated with the patch location shifted by half of its width in all three axes. The final 3‐point velocities were taken to be as the average of the two results having shifted patch positions. Representative images were visualized as raw velocity images and also using streamlines calculated in Ensight (Ansys, Canonsburg, Pennsylvania, USA). The 3‐point and 4‐point velocities were quantitatively compared component‐wise within a mask of vessels. To obtain these masks, vessels were automatically segmented in each test subject's 4‐point complex difference image by applying a threshold at the highest 0.1% intensity (corresponding to approximately 50,000 voxels) and subsequently removing single isolated voxels. Scatter plots between 3‐point and 4‐point velocities were obtained for each velocity component to quantitatively examine their agreements in terms of correlation, slope, and mean squared error. This analysis was repeated with and without additional correction of 3‐point data for residual phase offsets using polynomial fitting to static tissue, with more details provided in Supporting Information Table S4. With the 140 total subjects divided into five groups, a five‐fold cross‐validation was performed to examine the generalizability of the proposed training. Each fold had one unique group designated to be its non‐training subjects while the rest subjects formed the training group. As before, the non‐training subjects were split equally between validation and test subjects. The network was trained with its own training subjects independently in these five folds, with the trained network tested with each fold's unique test subjects. RMS error (RMSE) was computed using velocity values normalized by v _enc.

2.5. Flow Measurements

For the 14 test subjects, flow measurements were made in the 3‐point and 4‐point images using a previously described tool. ²⁶ Measurements were made in the internal carotid arteries (ICA), middle cerebral arteries (MCA), basilar artery (BA), and superior sagittal sinus (SSS). Plane locations were perfectly matched given the co‐registered nature of the 3‐point and 4‐point data. For each vessel segment, the total flow, maximum velocity, and mean velocity were recorded of the blood in segmented vessel lumen. These measures were compared between the 3‐point and 4‐point velocities using a linear regression model generated and evaluated with MATLAB functions fitlm and coefCI (The MathWorks, Inc., Natick, Massachusetts, USA) to provide the significance of the correlation and the 95% confidence interval of the slope between 4‐point and 3‐point velocities.

2.6. Effect of Acceleration on Performance

To test the effect of acceleration on the performance of the 3‐point network, the trained network was evaluated with data applied with further retrospective undersampling. The original 4D‐Flow data was collected with 3D radial sampling, which allowed for retrospective subsampling to higher acceleration factors. The network was not retrained for this evaluation. This is a test of generalization of the network trained above to changes in the noise and undersampling, with the geometry and flow being otherwise similar. The original 3D radial data was retrospectively and randomly subsampled to additional acceleration factors from 1× to 6×. The 3‐point network trained on the original, unaccelerated data was used to infer the 3‐point velocities from these accelerated data. Additionally, accelerated 4‐point velocities were also computed from these accelerated data. Following this, the accelerated 3‐point and accelerated 4‐point velocities were compared against the original, unaccelerated 4‐point velocities within a vessel mask. The vessel mask was computed as above and used to estimate the slope, R ², and the RMSE.

3. RESULTS

Using only three flow encodings as input, the network produced three directional velocities that agreed with the 4‐point velocities, with the level of agreement depending on the variations of loss functions examined in this study. As the velocity images show in Figure 1, the CNN‐produced 3‐point velocities showed excellent resemblance to the 4‐point velocities but with different appearance depending on the training method. When trained using magnitude weighting, the network tended to reduce the velocity noise in the static tissue. For example, in Figure 1, the SD of the velocity distribution in the static tissue was lower in the magnitude weighted image than the ground truth 4‐point images. This static tissue velocity noise removal was reduced as the training weight changed from the magnitude image to complex difference and to magnitude‐weighted inverse of the velocity frequency. The sequential reduction in the denoising was most apparent within the static tissue where the velocity images show a higher level of velocity noise in the absence of flow. Despite the velocity images showing subtle differences, the complex difference PC‐MRA images were nearly identical across the training methods and showed high resemblance to the 4‐point angiogram (Figure 1, right).

FIGURE 1.

Velocity (shown only for L/R‐component in cm/s), difference from the 4‐point ground truth, and maximum intensity projection (MIP) of the PC‐MRA images of 4‐point velocity from standard reconstruction (4 pt) and 3‐point (3 pt) velocities produced by trained neural network using only three flow encodings. The 3‐point velocities were trained with the following loss function weights: magnitude image (3pt_mag), complex difference image (3pt_CD), and uniform velocity weighting (3pt_vel), in which the inverse of the velocity frequency was used to distribute the weight evenly across all velocities before magnitude weighting. The velocity error is shown as the difference from the 4‐point velocity. The velocity images qualitatively demonstrate the subtle denoising of the 3‐point velocities within the static tissue, which become reduced as the training improved (from top to bottom), while the PC‐MRA images show little changes. In all cases, there is excellent agreement between 4‐point and 3‐point images

The 3‐point and 4‐point velocities were plotted on scatter plots (Figure 2) to quantitatively examine their agreements and determine the slope of the fit, correlation (R ²), and the RMSE of the fit (Table 1). Consistent with the above observations that the noise characteristics were dependent on the training methods, the scatter plots also showed training‐dependent velocity underestimations that were reflected in the slope values. The lowest agreement was observed when the magnitude (W _mag) was used as the weight in the loss function, yielding 3‐point velocities being underestimated by 6.9% (when averaged over three velocity components) compared to the 4‐point velocities. When the complex difference angiogram (W _CD) was used for the weight, favoring stronger weighting within vessels, the slope improved to 0.960, yielding a 4.0% underestimation. Finally, when the magnitude‐weighted inverse of the velocity frequency (W _vel) uniformly distributed weights across all velocities, the training yielded the best agreement and the lowest velocity underestimation (3.1%) among the three variations examined. In all cases, the 3‐point velocities underestimated the 4‐point velocities, with the degree of underestimation varying with the specific loss function used in the training.

FIGURE 2.

Scatter plots between 4‐point velocities and 3‐point velocities, shown for different trainings in which the loss function was weighted by the magnitude image (A), complex difference image (B), and uniform velocity weighting (C). As the training increased weighting within vessels (from A to C), the scatter plots showed the slope reaching closer to 1 and increasingly correlated data points taking thinner distributions. The values of the slope, correlation, and root mean square errors are tabulated in Table 1

TABLE 1.

Statistical quantities for the regression lines in Figure 2 that measure the agreement between 3‐point and 4‐point velocities, shown for three variations of trainings. For each, the slope of the fit, correlation (R ²), and RMSE in units of v _enc are shown for x‐, y‐, and z‐components of the velocity. As the training improved (from top to bottom), the slope and correlation approached closer to 1, and RMSE decreased

Training weight	Slope	R 2	RMSE
Magnitude	0.934, 0.926, 0.932	0.984, 0.982, 0.989	0.0204, 0.0199, 0.0196
Complex difference	0.956, 0.944, 0.981	0.983, 0.980, 0.987	0.0216, 0.0217, 0.0222
Inverse of the velocity frequency, magnitude‐weighted	0.970, 0.970, 0.966	0.992, 0.991, 0.994	0.0150, 0.0151, 0.0149

The improved training explained above was accompanied not only by reduced velocity underestimations (slope) but also by higher correlations (R ²) and lower velocity errors (RMSE) (see Table 1). The improved accuracy (reflected by slopes) resulted in the distribution of data points tilting closer to the unity‐slope reference lines as the training improved (Figure 2, top to bottom). The improved precision (measured by R ² and RMSE) was represented by the data points being consistently closer to the reference line, making their distributions to be thin and have smaller widths (Figure 2C). This tighter distribution from the increased precision was most noticeable at high velocity magnitudes, where data points tended to be scattered in the other two trainings (Figure 2A,B). Consequently, the uniform velocity weighting (W _vel) produced highest correlation values and smallest RMSE of the fit (R ² = 0.992, RMSE = 0.0150), as opposed to the magnitude weighting (R ² = 0.985, RMSE = 0.0200) or complex difference weighting (R ² = 0.983, RMSE = 0.0218).

Having performed the training using the uniform velocity weighting (W _vel), the 3‐point and 4‐point agreement was five‐fold cross‐validated to examine the generality of the training method. Supporting Information Figure S2 and Table S1 show the results of this cross‐validation. The scatter plots shown in Supporting Information Figure S2 demonstrate the same qualitative features associated with improved accuracy and precision across all five folds, such as narrow distributions with tapered ends staying close to the unity‐slope reference line, which were observed in the previous uniform velocity weighted training (Figure 2C). The statistical measures across the five folds agreed with each other in their slope, correlation, and RMSE values, indicating that similar trainings were obtained independent of the particular choice of training subjects. When averaged over the five folds, the averaged slope (0.984 ± 0.011), correlations (0.992 ± 0.001), and RMSE (0.0154 ± 0.0011) agreed with the previous uniform velocity weighting results described above.

Supporting Information Table S2 demonstrates the effect of additional correction for residual background phase using second‐order polynomial fitting. Results demonstrate that the 3‐point velocity images are not significantly improved by background phase offset correction and that 3‐point velocity images have a mean correction of less than 1% of the v _enc.

Figure 3 shows the flow, maximum velocity, and average velocity scatter plots across the test subjects in the ICA, MCA, BA, and SSS, using the uniform velocity weighted training approach. All three derived parameters show good agreement between 3‐point and 4‐point measures; however, again there was an underestimation of the flow measures. Supporting Information Table S3 shows the 95% confidence intervals and statistical tests. All correlations were significant.

FIGURE 3.

Scatter plots comparing flow (left), maximum velocity (middle), and mean velocity (right) between 4‐point and 3‐point flow encodings. In all cases and in all vessels examined, there were excellent correlations between the two methods. Similar to the velocity correlations, there is a slight underestimation of flow parameters in 3‐point compared to 4‐point

Figure 4 shows representative streamline plots in the Circle of Willis comparing 4‐point and 3‐point with and without additional acceleration. The acceleration results are demonstrated in quantitative measures shown in Figure 5 and Supporting Information Table S4, and correlation plots shown in Supporting Information Figures S3–S5. Interestingly, when high levels of additional acceleration were used, the RMSE and correlations relative to the unaccelerated 4‐point ground truth were higher for 3‐point flow encoding than 4‐point flow encoding. As the 3‐point velocity images were derived from the same raw images, this does suggest a potentially denoising effect. This quantitative result is visualized in Figure 4, where the accelerated 4‐point data shows greater discrepancies from the ground truth compared to the accelerated 3‐point data.

FIGURE 4.

Representative streamlines comparing 4‐point (4 pt) and 3‐point (3 pt) with 6× additional acceleration and without additional acceleration (1×). The 3‐point flow encoded images are in good agreement with 6× and 1× additional acceleration as compared to the 4‐point without additional acceleration. The 4‐point with additional acceleration shows greater streamline and velocity heterogeneity and a greater disagreement with the unaccelerated reference

FIGURE 5.

RMSE, correlation analysis slope, and R ² as a function of additional acceleration for 4‐point and 3‐point velocity encoded images (all derived from the same data). All values were computed in segmented vessels. The 3‐point encoding is effective even at high accelerations and even produces lower RMSE and higher correlations compared to 4‐point at the same acceleration. The 3‐point flow encoding does produce a lower velocity value 4‐point and the underestimation increases with additional acceleration

4. DISCUSSION

In this study, we demonstrated the effectiveness of convolutional neural networks to produce three directional velocities using only three flow encodings of 4D‐flow MRI. A neural network trained using supervised learning produced 3‐point velocities in excellent agreement with the velocities derived from standard reconstruction using all four flow encodings. The discrepancy between the 3‐point and 4‐point velocities was minimal, with the slope indicating 3.0% systematic velocity underestimation, ∼1.5% errors in the velocity values from RMSE, and correlations as high as 0.992. The performance of the neural network depended on the loss function used for training, with improved performance when velocities across the spectrum were evenly weighted in the loss function, hence increasing the weight of vessels compared to static background tissue. We proposed a weighting using the inverse of the velocity distribution in the training data. With this weighting, the resulting velocity correlation with the reference image were improved. The performance of the velocity distribution weighted network was further demonstrated in flow measurements and in data with additional acceleration. In general, the 3‐point encoded network demonstrated good agreement with 4‐point velocities; however, the 3‐point setup did slightly underestimate velocities. Interestingly, the 3‐point network did outperform 4‐point velocities when high levels of additional acceleration were used.

The proposed 3‐point velocity encoding technique is a promising technique to provide additional acceleration to current 4D‐flow techniques used in clinical practice and developing research applications. By omitting a fourth reference scan dedicated to measuring the background phase, our proposed technique can produce 25% reduction in scan time at the same temporal resolution, or alternatively a 25% acceleration in temporal resolution while keeping the same scan time. These accelerations have important implications for clinical 4D‐flow MRI, as most 4D‐flow MRI scans require long scan times or are challenged by the low spatial and temporal resolution provided by clinically feasible protocols. Scan time remains a considerable challenge for applications outside of the aortic arch, including neurovascular, ² , ²⁷ abdominal and hepatic, ²⁸ , ²⁹ and whole‐heart imaging. ⁴ , ³⁰ 3‐point acceleration is also highly relevant to emerging dynamic applications including respiratory resolved 4D‐flow, ³¹ , ³² 4D‐flow during challenges (e.g., exercise ³³ ), and real‐time 4D‐flow. ³⁴ , ³⁵ These applications aim to collect transient dynamics which require high acceleration. Increasing scan time often does not improve these methods as they are inherently imaging transient effects. The provided acceleration from 3‐point imaging is highly complementary to other spatiotemporal and fluid dynamic constraints and thus has the potential, in aggregate, to provide the required acceleration to enable these methods in clinical populations.

A key to effective training of the CNN was to optimize the loss function to handle the relative distribution of velocities in the training dataset and the desired performance. In this work, training was performed on large‐field‐of‐view neurovascular datasets, with many small vessels occupying a small fraction of the imaging volume. When the magnitude weighted mean squared error was used, the vast majority of the loss is attributed to voxels in the static brain tissue, resulting in much heavier weighting on minimizing the error outside the vessels. As a result, we speculate that the neural network was biased to predict lower velocities from being overwhelmingly trained with voxels having no flow. The complex difference weighted mean squared error was heuristically chosen as a solution to this class imbalance between the vessels and non‐vessels. These angiographic images show flow with high intensities so that weighting can be significantly increased within vessels when used as the weight. Despite this improvement, due to velocity noise and much higher prevalence of voxels outside voxels, the collective weight from the voxels having near‐zero velocities still greatly exceeded those with meaningful velocities, resulting in 10 times greater weighting of non‐vessels. As a more rigorous, proposed solution, the last method aimed to distribute weights equally across all velocities by using the inverse of the velocity frequency, which placed heavier weights for less frequent velocities (higher velocities) and lighter weights for more frequent velocities (e.g., velocities close to zero). The actual loss function, however, was the measure weighted by the magnitude image, which ensured that the velocity precision is accounted for in the weighting. With this improved choice of weighting, we achieved the least velocity underestimation among the three training variations examined.

The 3‐point encoded methods is still likely limited by the distribution of velocities in the training dataset. Despite the large training set of 140 subjects, there are still few velocity values at or near the v _enc, due both to the sparsity of high flow vessels in the volume and due to the choice of v _enc to avoid phase aliasing. This forced us to use a threshold bin of 0.3v _enc but also more generally meant the CNN was likely skewed toward lower velocities. This is suggested by the fall off in linear behavior at high velocities in the scatter plots (e.g., Figure 2). It is possible that this could be improved using alternative training such as augmentation with synthetic data or using data collected with a lower v _enc. We have not investigated the effect of phase aliasing on 3‐point velocities, and future work is needed to investigate this.

It is worthwhile to note that the network was trained with 4‐point velocities that were corrected for the residual background phase caused by eddy currents and concomitant fields. This means that the network not only learned to predict velocity components from three flow encodings but also to correct for the residual phase offsets that are typically corrected in routine pre‐processing of 4D‐flow data in an additional step. An additional residual background phase correction performed on the 3‐point velocities showed that this additional step produced no difference as the neural network had successfully learned to correct for these phase errors from the training. Eddy current residual background phase correction has been applied to phase contrast MRI using deep learning ³⁶ ; however, these approaches were applied to velocity images whereas our methodology predicted background‐corrected velocities directly from the complex data.

While the velocity images showed varying degrees of bias that depended on the training, the appearance of vessels in the complex difference images remained unchanged. This is largely attributed to the fact that the complex difference angiograms not only reflect measured velocity but also incorporates the signal magnitudes, whose blurring is unaffected by the 3‐point velocity production.

While our proposed technique was demonstrated with balanced, referenced flow encoding, it remains open‐ended whether it can be applied to other types of 4‐point flow encodings used in 4D‐flow MRI, such as unbalanced reference flow encoding or Hadamard encodings. In referenced flow encoding, each of the velocity‐encoded phase images are simply the background phase plus the velocity‐dependent phase that is directly proportional to each velocity component. In Hadamard encoding schemes, however, each phase images have voxel intensities that are a linear combination of the background phase and all three velocity components. As the relation between the phase images and the velocity components is more complicated in these encoding schemes, application of this proposed technique may require more sophisticated training or network structure. It is also possible that the same principles may be used to reduce the number of encodings for higher order encoding schemes such as 5‐point, ¹⁶ 6‐point schemes, ³⁷ , ³⁸ and dual v _enc encoding. These approaches will receive proportionally lower acceleration gains and may also lose the ability to quantify turbulence.

Future work is also needed to investigate the effects of including temporal components in the 3‐point prediction. In the case of the brain, the background phase is highly stable across cardiac time frames and thus a 4D CNN may be more effective. This was not performed in this work due to the substantial increase in memory that would be associated with 4D training and the general lack of support for 4D in current machine learning frameworks.

With the network trained with exclusively cranial 4D‐flow scans in this study, the network trained with cranial examples may not directly apply to 4D‐flow scans of other body parts, where the field inhomogeneity may take different appearance. Furthermore, as a single‐institution study that included scans from two different scanners but with identical scan parameters and sequence, the performance of the trained network may not directly generalize to scans acquired with different scan parameters, degree of undersampling, or imaging sequence. Nevertheless, the consistency and accuracy of the demonstrated technique strongly suggest that current 4D‐flow scans can be accelerated by 25% without sacrifice with the help of machine learning.

5. CONCLUSIONS

Current 4D‐flow MRI scans can be accelerated without further undersampling by using a convolutional neural network to produce three directional velocities from three flow encodings. The training of the convolutional network is dependent on the loss function used, with compensation needed to account for distribution of velocities in the volume of interest.

Supporting information

Figure S1 The structure of the convolutional neural network used in this study. The network takes a 64×64×64 sub‐volume of the six 4D‐flow phase images from three flow encodings (real and complex, separately), and produces 3 real‐valued velocity images of the same dimension.

Figure S2. Scatter plots between the 3‐point and 4‐point velocities shown for the five folds in the cross‐validation, in which the loss function employed uniform velocity weighting. The similarity and agreement among the results obtained with different training subjects indicate the generality of the training method. The values of the slope, correlation, and root mean square errors in this cross‐validation study are tabulated in Table S1.

Figure S3. Scatter plots comparing the 3‐point with additional acceleration and 4‐point without additional acceleration. Tabulated results are shown in Table S3.

Figure S4. Scatter plots comparing the 3‐point with additional acceleration to 4‐point results compared to the same level of additional. Tabulated results are shown in Table S3.

Figure S5. Scatter plots comparing the 4‐point with additional acceleration to 4‐point without additional acceleration. Tabulated results are shown in Table S3.

Table S1. Statistical measures for the regression lines in Figure S2, which show the results of the 5‐fold cross‐validation. As in Table 1, the precision and accuracy of the 3‐point velocities are measured using slope of the fit, correlation (R ²), and root mean square of the error (RMSE) for x‐, y‐, and z‐components of the velocity. The consistency of these measures in the five folds reflect that the training was not dependent on the training subjects, indicating the generality of the training method.

Table S2. Statistical measures for flow, maximum velocity, and mean velocity measures extracted from vessel segments of internal cervical arteries (ICA), middle cerebral arteries (MCA), basilar artery (BA), superior sagittal sinus (SSS), and a pooled dataset of all segments. Metrics including correlation (R ²), significance of the correlation, slope of the fit, and 95% confidence interval (CI) are calculated from a linear regression model comparing the results from 3‐point flow encodings to the results from 4‐point imaging.

Table S3. Statistical measures for the regression analysis as a function of acceleration. 3‐point velocities were compared against full sampled 4‐point images and undersampled 4‐point images. Values are also provided for undersampled versus, fully sampled 4‐point velocities for reference. Both undersampling and the use of 3‐point imaging results in an underestimation of the velocity values. Interestingly, as the undersampling value increases, the undersampled 3‐point velocity values provide lower RMSE and higher correlation with fully sampled 4‐point as compared to undersampled 4‐point velocity values. This suggests a filtering effect from the CNN that is providing denoising in addition to the background phase effect. While the results do show generalization, the error in 3‐point velocity values increases when acceleration factors are used beyond that used for training.

Table S4. For the test data, the native 3‐point velocity images were corrected using 2nd order polynomial residual background phase correction using a background tissue mask segmented from the magnitude data and complex difference angiogram of the ground truth 4‐point velocities. The input complex data to the 3‐point network was not corrected for concomitant or residual background phase. Correlation analysis was performed in vessels against 4‐point velocities for 3‐point velocities with and without the additional background phase correction. In addition, the mean tissue correction and mean absolute tissue offset were calculated in segmented background tissue masks. There were no significant differences between any of the parameters. There were trends of reduced correlation coefficients and increase RMSE after residual background phase correction.

Kim D, Jen M‐L, Eisenmenger LB, Johnson KM. Accelerated 4D‐flow MRI with 3‐point encoding enabled by machine learning. Magn Reson Med. 2023;89:800‐811. doi: 10.1002/mrm.29469

DATA AVAILABILITY STATEMENT

Code and weights for this training are available: https://github.com/uwmri/ThreePoint4DFlow.