Physics-Informed Processing for Super-Resolution, Denoising, & Anti-Aliasing of 4D-Flow MRI
What is 4D-Flow MRI?
Time resolved three-dimensional phase contrast magnetic resonance imaging, also known as 4D-Flow MRI is a non-invasive technique to measure in-vivo blood velocities in the human vascular system. The technique takes advantage of the motion sensitivity of MRI to encode blood velocities in the phase of the MRI image. The technique uses bipolar velocity encoding gradients wherein flow-dependent phase changes are detected using two acquisitions with different velocity dependent signal phases but otherwise identical sequence parameters.
Figure 1. 4D-Flow MRI acquisition process.
It is impossible acquire all the data in a single cardiac cycle. Typically, the 'k-space' is scanned over several cardiac cycles. Acquisition is gated to the cardiac cycle using the ECG signal. Consequently, the 3-D data for a given cardiac phase is actually an 'average' over several cardiac cycles. Figure 1. illustrates the 4D-Flow MRI acquisition process.
Limitations of 4D-Flow MRI
There are several limitation in 4D-Flow MRI that has prevented its application in routine clinical practice. They include
- Low spatio-temporal resolution
- Acquisition noise
- Eddy current induced phase offsets
- Velocity aliasing
Low spatio-temporal resolution: The sheer amount of data that needs to be scanned limits the spatio-temporal resolution of 4D-Flow MRI. It is impacted by the time required to setup the requisite field gradients for each direction and scanning. One can compromise on spatial resolution and increase temporal resolution and vice-versa. Typically, current resolutions are in the range of 0.7mm-1.2mm isotropic spatial and ~40-80ms temporal. This typically results in scan times of 25 min. Recent advances in compressed sensing and parallel imaging has significantly reduced the scan times. However, the spatio-temporal resolution issues remain.
Some researchers have used Computational Fluid Dynamics (CFD) using geometry form angiography scan segmentation and boundary flow conditions (BFCs) obtained from 4D-Flow MRI to address the issue of spatio-temporal resolution. However, errors in segmentation and BFCs (due to acquisition noise) , errors in flow model parameters (dynamic viscosity) and assumption regarding the flow regime (Newtonian vs non-Newtonian, laminar vs turbulent) impact the fidelity of results. Consequently, CFD is not used in clinical decision making.
A recent trend in addressing some of the limitations of CFD is to use data-assimilation techniques to correct the CFD by using the rich, albeit low resolution and noisy 4D-Flow MRI data. While this approach can address some of the limitations of pure CFD to an extent, there is an additional issue of co-locating the CFD domain with the 4D-Flow MRI domain using image registration which can significantly impact the accuracy of the results.
Figure 2: CFD simulation of an intra-cranial aneurysm
Acquisition noise: As with any analog acquisition process, MRI acquisition is contaminated by noise. However, MRI acquisition occurs in spatial frequency of 'k-space'. In k-space, the acquisition noise is zero-mean Gaussian and is additive in both the real and imaginary channels. The Fourier-inverse operation used to compute the image in image-space does not alter the zero-mean Gaussian nature of noise since the Fourier-inverse operation is linear. Converting the so-called 'cartesian' image to the polar form (magnitude and phase images) changes the nature of the noise to Rician because of the nonlinear operations involved in the conversion. This sets up signal strength dependent bias in both the magnitude and phase images . In case of 4D-Flow MRI, researchers have used flow physics constraints along with typical filtering techniques to attenuate noise in the velocity encodings.
Eddy current induced phase offsets: Switching time-varying magnetic field gradients induces eddy currents in the conducting parts of the scanner. These eddy currents impact the strength and duration of field gradients and results in spatio-temporally varying phase offsets in 4D-Flow MRI scans. Although pre-emphasis system in modern scanners mitigate this effect to some extent, not all eddy current effects can be neutralized. Consequently, in the post-processing stage, the typical approach to eddy current phase offset correction is to fit the phase in static regions of the image using polynomials and then to subtract it from the phase signal in the flow regions. Identifying static and flow voxels accurately is therefore important in this process and is typically done by monitoring the variance of voxels values over time. The idea here is that a voxel whose variance is low over time can be assume to be in the static region of the image.
Velocity aliasing: Flow velocities are mapped to phases that range from -pi to pi. The velocity encoding parameter of 'VENC' determines the range of velocities that can handled. Therefore flow velocities ranging from -VENC to +VENC get mapped to -pi to +pi phase respectively. The setting of the 'VENC' parameter is based on the guess of the maximum velocity that will be encountered in the scan. Suppose the actual velocity is greater than VENC, then the phase wraps around itself and results in a scan velocity that is mismapped as shown in Figure 3. This phenomenon is called velocity aliasing. It manifests itself as reversal of color in the flow regions in the phase images as shown in Figure 3. As is the case with acquisition noise, this phenomenon is purely a artifact of representing the image-space data in the polar form.
Figure 3: Velocity aliasing
Physics-informed deep neural nets for processing 4D-Flow MRI
Physics-informed deep neural nets is a new technique for data assimilation for phenomenon whose physics is well known. The basic idea is to approximate the field of interest (for example, fluid velocity and pressure in fluid flow) as a neural net. Next, the training process uses point data acquired from sensors for data fidelity and imposes physics (i.e. the strong form of PDEs) on an arbitrary number of points at arbitrary location in the phenomenon domain. The advantage of neural-net representation of the field is that spatio-temporal derivatives of any order can be computed with accuracy only limited by machine precision using automatic differentiation.
There are a few challenges in applying the basic physics-informed deep neural net method for data assimilation of 4D-Flow MRI. They include
- 4D-Flow MRI data is not point data but is spatio-temporally averaged
- While the acquisition noise is additive zero-mean gaussian, transformation to polar form (magnitude and phase images) causes the noise to become Rician
- Phase images can be discontinuous because of velocity aliasing issues
The following are modifications to the basic algorithm to enable its use for data assimilation of 4D-Flow MRI. The neural-net is a continuous function of position and time and therefore generates point data. However, the observed data, in this case the values of velocity in a given voxel are volume averaged over the voxel. In our approach, we compute the volume averaged neural net prediction using Gaussian quadrature. A high order quadrature minimizes approximation errors. The issue of Rician noise and velocity aliasing is simply handled by computing the data fidelity in the cartesian representation. Assuming a five point 4D-Flow MRI acquisition protocol, the five associated cartesian complex images can be generated as $latex v_i$
The discontinuities due to velocity aliasing do not appear in the cartesian representation of the image. Furthermore, the noise is zero-mean Gaussian. The architecture of the neural net is shown in Figure 4. In our implementation, we use the tanh activation function in the nodes. Note that in addition the velocities, the neural net also predicts the pressure, dynamic viscosity, and the phase offset due to eddy currents.
Figure 4: Neural net architecture
Figure 5: Rough segmentation to determine in, out of the flow region
We have tested our method on both 2-D steady state flow as well as 2-D time varying flow. For 2-D steady state flow, we used CFD to generate a reference flow. This flow was then down sampled and degraded with zero-mean white noise in k-space. Furthermore, we set the VENC value while generating the synthetic 4D-Flow MRI in a manner that caused velocity aliasing. Figure 6. shows the results of our method. The arrows in the figure indicate regions where velocity aliasing occurs in the synthetic 4D-Flow MRI data. However, our algorithm is able to faithfully reproduce the reference CFD flow patterns in high resolution. In this particular case, we increased the spatial resolution by a factor of 10 x 10
Figure 6: Testing on synthetic 4D-Flow MRI for the 2D steady state case
In the second case, we tested our algorithm on time varying 2-D flows. The reference flow (the bottom row in the video above) is sampled from a DNS simulation of flow over a cylinder. This simulation generated vivid von Karman vortices. The reference flow was down sampled by averaging in space by a factor of 10 x 10 and in time by a factor of 5 (the top row in the video). We then added noise in k-space to generate the synthetic 4D-Flow (second row from the top). Clearly, it can be seen that our algorithm is able to recover the reference flow with a remarkable level of accuracy (it is obviously not exact).
Dr. Roshan M. D'Souza, PhD
Complex System Simulation Lab @ UWM
University of Wisconsin-Milwaukee