[DECRYPTAGE] Numerical simulation of 4D flow MRI

4D flow MRI, a tool of choice for improving patient care

Historically reserved for morphological studies, technological advances over the last few decades have made it possible to adapt Magnetic Resonance Imaging (MRI) to the exploration of blood flow, thanks to 3D phase contrast MRI (or 4D flow MRI) [1]. By providing access to the temporal evolution of the velocity field in all three spatial directions, in addition to the morphology of the arterial sector of interest, it is a tool of choice in clinical practice for the management and monitoring of patients with hemodynamically-related pathologies (aneurysms, stenoses, dissections, congenital or valvular heart disease). In addition to the simplicitý of data analysis, it offers the possibility of estimating new biomarkers derived from the velocity field and previously inaccessible using conventional imaging techniques, such as parietal shear stress [2], static pressure [3], or residence time.

Technological limitations and constraints

Despite the interest aroused by this modality, certain technological constraints (acquisition time, dependence on encoding speeds) still limit its use in clinical routine. What’s more, the intrinsic complexities of the acquisition process and the many user-dependent acquisition parameters mean that it is generally difficult to identify the sources of measurement error. Compromises such as reduced spatiotemporal resolution or the use of parallel imaging can reduce acquisition time, to the detriment of measurement accuracy. It therefore seems essential to quantify and characterize these measurement errors in order to avoid possible diagnostic errors and thus open up access to 4D flow MRI in clinical practice.

MRI simulation to optimize sequences and characterize measurement errors

One way of characterizing these image distortions is to digitally simulate the MRI acquisition process. The dynamics of nuclear magnetization described by Bloch’s equations [4] are at the heart of MRI acquisition, and are the physical phenomenon responsible for the contrast in an MRI image. With such simulations, it is then possible to reconstruct a synthetic image free from any experimental error specific to MRI measurement. Thus, by comparing experimentally acquired images with compatible synthetic images obtained by solving the Bloch equations, errors relating to the imaging technique (sequence, reconstruction) can be separated from errors of an experimental nature (antenna failure, non-linearity of magnetic fields, noise). Simulation can also be used to optimize the acquisition parameters of the sequences in question.

However, simulating 4D flow sequences is a major challenge, as it requires modeling of particle dynamics, which considerably increases computational costs. So, to take account of fluid displacement in simulations, the solution of Bloch’s equations can be coupled with Computational Fluid Dynamics (CFD). Since both phenomena involve very different physical time scales, it is nevertheless complex to simulate such a configuration effectively. What’s more, simulating 4D flow sequences lasting several minutes can be prohibitively expensive.

Work objectives

This work presents a new approach to simulating 4D flow MRI under realistic flow conditions. To this end, Bloch’s equations are numerically advanced on Lagrangian tracers transported by a simultaneously solved flow field [5]. In order to reduce computation time, a semi-analytical solution of the Bloch equations and a periodic particle seeding strategy are introduced. After validating each elementary step in the simulation process, the gain associated with this formulation is evaluated. Finally, a well-controlled test rig delivering a pulsed flow typical of aortic flows within a rigid phantom is designed [6]. Several 4D flow MRI acquisitions are performed and compared with compatible MRI simulations. Since fluid rheology and phantom morphology are well known, the classic sources of uncertainty encountered in-vivo, such as segmentation errors, wall movement or blood properties, are eliminated. This makes it possible to eliminate errors due to the patient and isolate errors induced by the MRI process.

Results and discussion
Performance evaluation shows that the proposed semi-analytical resolution method reduces computation time by a factor of 5, for a constant residual error. Likewise, periodic seeding of the particles guarantees a homogeneous particle distribution throughout the calculation, with less than 0.05% of cells having a particle density greater than 5. Analysis of velocity and acceleration errors in the in-vitro flow configuration suggests that the main distortions are linked to the presence of velocity displacement artifacts [7]. Although a proof-of-concept of the methodology is presented in this paper, certain physical effects (gradient non-linearity, chemical shift or magnetic susceptibility) remain to be taken into account in the simulation in order to establish a more detailed characterization of the errors. In addition, the simulation of more complex clinical sequences should make it possible to eliminate certain errors induced by the idealized sequence. Nevertheless, at this stage, the versatility of the tool developed enables us to simulate a multitude of morphological sequences and flows.


  1. Michael Markl, Frandics P. Chan, Marcus T. Alley, Kris L. Wedding, Mary T. Draney, Chris J. Elkins, David W. Parker, Ryan Wicker, Charles A. Taylor, Robert J. Herfkens, and Norbert J. Pelc. Time resolved three dimensional phase contrast MRI. Journal of Magnetic Resonance Imaging, 18(3):396-396, 2003.

  2. Wouter V. Potters, Pim Ooij, Henk Marquering, Ed vanBavel, and Aart J. Nederveen. Volumetric arterial wall shear stress calculation based on cine phase contrast MRI. Journal of Magnetic Resonance Imaging, 41(2):505-516, 2015.

  3. Pablo Lamata, Alex Pitcher, Sebastian Krittian, David Nordsletten, Malenka M. Bissell, Thomas Cassar, Alex J. Barker, Michael Markl, Stefan Neubauer, and Nicolas P. Smith. Aortic relative pressure components derived from four-dimensional flow cardiovascular magnetic resonance. Magnetic Resonance in Medicine72(4):1162-1169, March 2014.

  4. Bloch F. Nuclear Induction. Physical Review. 1946; 70:460-474.

  5. YALES2BIO. https://imag.umontpellier.fr/~yales2bio/

  6. Puiseux T, Sewonu A, Meyrignac O, et al. Reconciling PC-MRI and CFD: an in-vitro study. NMR in Biomedicine. 2019;e4063.

  7. Steinman DA, Ethier CR, Rutt BK. Combined analysis of spatial and velocity displacement artifacts in phase contrast measurements of complex flows. J Magn Reson Imaging. 1997 Mar-Apr;7(2):339-46.

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