Table 1.
Review of the published works in flow MRI simulations.
Fig 1.
Illustration of the NMR experiment.
(a) When no magnetic field is applied, the spins are randomly oriented. When a B0 field is applied along the z-axis, all the spins precess around the z-axis and (b) an equilibrium magnetization arises, oriented along the same axis. The equilibrium magnetization is shifted towards the transverse xy-plane by the effects of an RF-pulse (B1) applied at resonance frequency (c). When the RF-excitation is released, the magnetization relaxes towards its equilibrium value (d) with a precession frequency that depends on the magnetic properties of the isochromat considered. Note that the magnetization shift due to the RF-pulse is around two or three orders of magnitude faster than the relaxation process.
Fig 2.
Diagram of a typical gradient echo pulse sequence with frequency and phase encoding gradients along x-axis and y-axis, and slice selection gradient along z-axis.
Each k-space line is filled during the readout event where the signal is measured by the receiver coil. This pulse sequence is repeated changing the phase encoding gradient amplitudes (Gy) after each repetition time TR to incrementally fill the k-space lines, as represented by the several amplitudes. For full Cartesian k-space sampling, the time between the RF-excitation and half the readout corresponds to the echo time and is denoted TE. It is a characteristic time of the evolution of the isochromats between the excitation by the RF and the signal measurement.
Fig 3.
Illustration of the effects of applying either flow compensating (left) or bipolar encoding velocity gradients (right) on the phase of the magnetization vector.
The static spins (blue) result in a zero accumulated phase at the end of the application, while the moving spins (red) result in a non-zero phase.
Fig 4.
Main steps of the CFD-MRI simulation procedure.
NSE: Navier-Stokes Equations. BC: Boundary conditions. The grey block corresponds to the simulation framework kernel, while the red/blue blocks are inputs/outputs to the simulation. tf is the final time of the simulation. The output “simulated images” correspond to three phase difference images and a magnitude image.
Fig 5.
Schematic illustration of the particles position at different instants between two consecutive RF excitations in a pipe flow with an aneurysm-like wall bulging.
The black spheres represent the classical particles injection strategy where particles are initially seeded within the whole domain and continuously injected from the inlet boundary surface. In contrast, in the proposed seeding strategy (blue spheres), the particles are periodically re-seeded at their initial location, and the steady-state magnetization is prescribed. In the classical strategy, zones of agglomerated and empty particles can be observed especially in the aneurysm sac [23], while the proposed approach allows to keep homogeneous the particle repartition.
Fig 6.
Evolution of the simulation time step over an arbitrary pulse sequence (RF and gradient) as a function of the magnetic event.
Within each CFD iteration of time ΔtCFD, the fluid velocity is kept constant. For indication, in a pipe of 5 cm radius and 10 cm length, with T2 = 10 ms, and with a gradient strength Gmax = 10 mT/m of 0.1 ms rise time, the order of magnitude for the minimum time steps would be: Δtstab ∼ 10−7 s, Δtmag ∼ 10−5 s (with bn = 1), Δtgrad ∼ 10−5 s, Δtseq ∼ 10−3 s and ΔtCFD ∼ 10−3 s.
Fig 7.
(a) Schematic illustration of the flow configuration used in Yuan et al. [31]. (b) 90° slice selective excitation sequence simulated. (c) Evolution along the centerline of the magnetization (Mx, My, Mz) for several velocities imposed to the particles. The dashed lines correspond to the YALES2BIO simulation results, while solid lines correspond to Yuan et al. data [31].
Fig 8.
Poiseuille flow test case simulated with a 2D PC-MRI sequence.
(a) Boundaries of the domain (flow in black and static tissues in gray) simulated. (b) Axial velocity profile along the x-axis reconstructed by MRI (dots) and compared with the imposed Poiseuille analytical solution (solid line). (c) Maximum and (d) mean errors as a function of the particle density.
Fig 9.
Computational cost for the MRI simulation of a Poiseuille flow with a 2D PC-MRI sequence, at different spin densities with an imaging matrix of size 16 × 16 and a FOV of 32 × 32 × 10 mm3.
The proposed semi-analytic formulation is compared to the full numerical integration method, where a fourth-order Runge-Kutta numerical scheme is adopted to discretize the Bloch equation. The computational cost is defined as the computational time divided by the lowest time.
Fig 10.
(a) Flow phantom schematic representation annotated with locations of surfaces of interest. (b) Flow rate waveform over a cycle at different locations of the phantom (inlet, bend, collateral).
Fig 11.
a) Particle density histogram with y axis expressed in logarithmic scale and b) particle density maps over the surface of the flow domain, after a simulation time of 8 s. Note that the particle density was mapped with a log scale to highlight both the dense and sparse regions. The ‘seeding’ strategy is the proposed particle reinitialization strategy while the ‘injection’ approach corresponds to the continuous injection from the inlet boundary surface. Note that only the external surface is shown as the largest zones of particles agglomeration and sparsity are located at the boundary cells.
Fig 12.
Comparison in the XZ-middle plane between (top) ||uSMRI|| and (bottom) ||uHR|| at four different phases during the cycle.
Fig 13.
(a). Evolution along a cycle of the Pearson’s correlation calculated between HR-CFD and SMRI, as well as between LR-CFD and SMRI. (b). Bland-Altman and (c). linear regression plots at phase t/Tc = 0.2 for the SMRI/LR-CFD comparison.
Fig 14.
Evolution of the (a) mean and (b) maximum mismatch between LR-CFD and MRI simulation for different spin densities and voxel sizes at t/Tc = 0.44.
Fig 15.
Map of the velocity mismatch at t/Tc = 0.44 between LR-CFD and SMRI at three different spatial resolutions.
Left 2 mm3, middle 3 mm3, and right 4 mm3. The error is calculated based on the LR-CFD, which is specific to each spatial resolution. In all the simulations, 48 particles/voxel were seeded in the domain.
Fig 16.
Largest patterns of phase-averaged CFD acceleration (top) and L2-norm of the SMRI/LR-CFD velocity mismatch (bottom) at two instants in the cycle.
The threshold values were set to (left) 25 and (right) 7% of the maximum CFD acceleration and (left) 14 and (right) 12% of the maximum mismatch at t/Tc = 0.44 − t/Tc = 0.91, respectively.
Fig 17.
Velocity L2-norm error in the XZ-middle plane of the (left) experimental and (right) simulated MRI at peak systole t/Tc = 0.44.
The L2-norm error is calculated based on the LR-CFD obtained from a 2 mm characteristic mesh size.