2. Towards Reconstructing Blood Velocity Profiles from Noisy and Sparse Time Resolved Phase Contrast Magnetic Resonance Flow Data

Main problem and introduction:

Phase-contrast MR imaging (4D PCMR) can provide time-resolved, three-dimensional velocity fields, thus making it possible to non-invasively measure blood flow in vivo. Velocities are encoded in the phase of the acquired MR signal through use of special RF-coil activation sequences. However there are serious limitations in both spatial and temporal resolution of the signals. Besides the resolution issues, data is also corrupted by noise introduced to the system by various elements (e.g. reading noise, magnetic field’s noise and etc). Therefore, the raw data thus obtained present difficulties for computing clinicaly relevant flow descriptors such as wall shear stresses, pressure gradients and flow residence time which depend on spacial derivatives of the velocity field (derivatives amplift noise in a signal).

Methodology:

Proper Orthogonal Decomposition (POD) is a method to find a set of ordered orthonormal basis vectors in a subspace where a random vector in the sample space can be expressed optimally using a linear combination of the selected first l basis vectors.POD has been typically used for dimension reduction in reduced order modeling of discretized partial differential equations.

One of the methods to calculate the POD is the snapshot method. Full order models (in this case computational fluid dynamics (CFD) model of patient specific geometry) are simulated for a range of boundary conditions (BCs). Since the exact BCs are not known, we run a series of simulation with boundary conditions “near” the noisy BC obtained from the PCMR data. The time resolved solutions x(k), k = 1… n are collected. These solutions are arranged in a matrix as:

We then compute the Singular Value Decomposition (SVD) of the matrix X as:

The set of left eigen vectors U span the subspace of the enumerated solutions. Any solution of the CFD model near the enumerated solutions can be written as a linear combination of the basis vectors . This potentially provides a way to remove noise and artifacts from the 4D PCMR data by simply projecting the noisy PCMR image v into the space of solutions as:

Finally, the noise free solution that is closest to is then reconstructed as:

This Procedure discussed above  is summarized in the figure shown below:

Results and Conclusions:

The results of this project demonstrate that the proposed method based on POD can provide very accurate velocity fields in patient-specific geometries by combining CFD simulations with in vivo MRI measurements.

In this project the aim was to develop an online service where clients
can submit their noise PC-MR data and receive noise and divergence free data. Where the service steps:
1- Submission 1: Client has to submit the noisy data.
2- Pre-process: Data will be pre-processed (Segmentation and
mesh generation).
3- Submission 2: Data submission into our software which is
running on a computational cloud.
4- Review: Reviewing processed result and write the report.
5- Report: Delivering noise free, high resolution data along with
requested report to the client.