Enhancing the Velocity Data From 4D Flow MR Images by Reducing its Divergence

Physics-informed neural networks for parameter estimation in blood flow models

BACKGROUND

Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving inverse problems, especially in cases where no complete information about the system is known and scatter measurements are available. This is especially useful in hemodynamics since the boundary information is often difficult to model, and high-quality blood flow measurements are generally hard to obtain.

METHODS

In this work, we use the PINNs methodology for estimating reduced-order model parameters and the full velocity field from scatter 2D noisy measurements in the aorta. Two different flow regimes, stationary and transient were studied.

RESULTS

We show robust and relatively accurate parameter estimations when using the method with simulated data, while the velocity reconstruction accuracy shows dependence on the measurement quality and the flow pattern complexity. Comparison with a Kalman filter approach shows similar results when the number of parameters to be estimated is low to medium. For a higher number of parameters, only PINNs were capable of achieving good results.

CONCLUSION

The method opens a door to deep-learning-driven methods in the simulations of complex coupled physical systems.

Physics-Informed Compressed Sensing for PC-MRI: An Inverse Navier-Stokes Problem

We formulate a physics-informed compressed sensing (PICS) method for the reconstruction of velocity fields from noisy and sparse phase-contrast magnetic resonance signals. The method solves an inverse Navier-Stokes boundary value problem, which permits us to jointly reconstruct and segment the velocity field, and at the same time infer hidden quantities such as the hydrodynamic pressure and the wall shear stress. Using a Bayesian framework, we regularize the problem by introducing a priori information about the unknown parameters in the form of Gaussian random fields. This prior information is updated using the Navier-Stokes problem, an energy-based segmentation functional, and by requiring that the reconstruction is consistent with the $k$ -space signals. We create an algorithm that solves this inverse problem, and test it for noisy and sparse $k$ -space signals of the flow through a converging nozzle. We find that the method is capable of reconstructing and segmenting the velocity fields from sparsely-sampled (15% $k$ -space coverage), low ( $\sim 10$ ) signal-to-noise ratio (SNR) signals, and that the reconstructed velocity field compares well with that derived from fully-sampled (100% $k$ -space coverage) high ( $> 40$ ) SNR signals of the same flow.

SRflow: Deep learning based super-resolution of 4D-flow MRI data

Exploiting 4D-flow magnetic resonance imaging (MRI) data to quantify hemodynamics requires an adequate spatio-temporal vector field resolution at a low noise level. To address this challenge, we provide a learned solution to super-resolve in vivo 4D-flow MRI data at a post-processing level. We propose a deep convolutional neural network (CNN) that learns the inter-scale relationship of the velocity vector map and leverages an efficient residual learning scheme to make it computationally feasible. A novel, direction-sensitive, and robust loss function is crucial to learning vector-field data. We present a detailed comparative study between the proposed super-resolution and the conventional cubic B-spline based vector-field super-resolution. Our method improves the peak-velocity to noise ratio of the flow field by 10 and 30% for in vivo cardiovascular and cerebrovascular data, respectively, for 4 × super-resolution over the state-of-the-art cubic B-spline. Significantly, our method offers 10x faster inference over the cubic B-spline. The proposed approach for super-resolution of 4D-flow data would potentially improve the subsequent calculation of hemodynamic quantities.

Assessment of 4D flow MRI's quality by verifying its Navier-Stokes compatibility

4D Flow Magnetic Resonance Imaging (MRI) is the state-of-the-art technique to comprehensively measure the complex spatio-temporal and multidirectional patterns of blood flow. However, it is subject to artifacts such as noise and aliasing, which due to the 3D and dynamic structure is difficult to detect in clinical practice. In this work, a new mathematical and computational model to determine the quality of 4D Flow MRI is presented. The model is derived by assuming the true velocity satisfies the incompressible Navier-Stokes equations and that can be decomposed by the measurements u→meas plus an extra field w→ . Therefore, a non-linear problem with w→ as unknown arises, which serves as a measure of data quality. A stabilized finite element formulation tailored to this problem is proposed and analyzed. Then, extensive numerical examples-using synthetic 4D Flow MRI data as well as real measurements on experimental phantom and subjects-illustrate the ability to use w→ for assessing the quality of 4D Flow MRI measurements over space and time.

A comprehensive comparison between shortest-path HARP refinement, SinMod, and DENSEanalysis processing tools applied to CSPAMM and DENSE images

We addressed comprehensively the performance of Shortest-Path HARP Refinement (SP-HR), SinMod, and DENSEanalysis using 2D slices of synthetic CSPAMM and DENSE images with realistic contrasts obtained from 3D phantoms. The three motion estimation techniques were interrogated under ideal and no-ideal conditions (with MR induced artifacts, noise, and through-plane motion), considering several resolutions and noise levels. Under noisy conditions, and for isotropic pixel sizes of 1.5 mm and 3.0 mm in CSPAMM and DENSE images respectively, the nRMSE obtained for the circumferential and radial strain components were 10.7 ± 10.8% and 25.5 ± 14.8% using SP-HR, 11.9 ± 2.5% and 29.3 ± 6.5% using SinMod, and 6.4 ± 2.0% and 18.2 ± 4.6% using DENSEanalysis. Overall, the results showed that SP-HR tends to fail for large tissue motions, whereas SinMod and DENSEanalysis gave accurate displacement and strain field estimations, being the last which performed the best.

HARP-I: A Harmonic Phase Interpolation Method for the Estimation of Motion From Tagged MR Images

We proposed a novel method called HARP-I, which enhances the estimation of motion from tagged Magnetic Resonance Imaging (MRI). The harmonic phase of the images is unwrapped and treated as noisy measurements of reference coordinates on a deformed domain, obtaining motion with high accuracy using Radial Basis Functions interpolations. Results were compared against Shortest Path HARP Refinement (SP-HR) and Sine-wave Modeling (SinMod), two harmonic image-based techniques for motion estimation from tagged images. HARP-I showed a favorable similarity with both methods under noise-free conditions, whereas a more robust performance was found in the presence of noise. Cardiac strain was better estimated using HARP-I at almost any motion level, giving strain maps with less artifacts. Additionally, HARP-I showed better temporal consistency as a new method was developed to fix phase jumps between frames. In conclusion, HARP-I showed to be a robust method for the estimation of motion and strain under ideal and non-ideal conditions.

Super-resolution and denoising of 4D-Flow MRI using physics-Informed deep neural nets

BACKGROUND AND OBJECTIVE

Time resolved three-dimensional phase contrast magnetic resonance imaging (4D-Flow MRI) has been used to non-invasively measure blood velocities in the human vascular system. However, issues such as low spatio-temporal resolution, acquisition noise, velocity aliasing, and phase-offset artifacts have hampered its clinical application. In this research, we developed a purely data-driven method for super-resolution and denoising of 4D-Flow MRI.

METHODS

The flow velocities, pressure, and the MRI image magnitude are modeled as a patient-specific deep neural net (DNN). For training, 4D-Flow MRI images in the complex Cartesian space are used to impose data-fidelity. Physics of fluid flow is imposed through regularization. Creative loss function terms have been introduced to handle noise and super-resolution. The trained patient-specific DNN can be sampled to generate noise-free high-resolution flow images. The proposed method has been implemented using the TensorFlow DNN library and tested on numerical phantoms and validated in-vitro using high-resolution particle image velocitmetry (PIV) and 4D-Flow MRI experiments on transparent models subjected to pulsatile flow conditions.

RESULTS

In case of numerical phantoms, we were able to increase spatial resolution by a factor of 100 and temporal resolution by a factor of 5 compared to the simulated 4D-Flow MRI. There is an order of magnitude reduction of velocity normalized root mean square error (vNRMSE). In case of the in-vitro validation tests with PIV as reference, there is similar improvement in spatio-temporal resolution. Although the vNRMSE is reduced by 50%, the method is unable to negate a systematic bias with respect to the reference PIV that is introduced by the 4D-Flow MRI measurement.

CONCLUSIONS

This work has demonstrated the feasibility of using the readily available machinery of deep learning to enhance 4D-Flow MRI using a purely data-driven method. Unlike current state-of-the-art methods, the proposed method is agnostic to geometry and boundary conditions and therefore eliminates the need for tedious tasks such as accurate image segmentation for geometry, image registration, and estimation of boundary flow conditions. Arbitrary regions of interest can be selected for processing. This work will lead to user-friendly analysis tools that will enable quantitative hemodynamic analysis of vascular diseases in a clinical setting.