In order to capture dephasing due to (nonlinear) imaging gradients or B₀ inhomogeneities in simulations, the object has to be modelled with a very large number of spins. This can be prohibitive in terms of computing speed and memory. In this work, we improve and accelerate simulations by explicitly modelling intra-voxel dephasing. The model captures dephasing within a voxel and allows a reduction of the number of spins, hence speeding up simulation substantially. The same model has previously been used to improve the consistency between measurement and reconstruction for encoding with nonlinear gradients¹ and for trajectory optimization.²

The signal equation for an RF channel $$$l$$$ is given in Eq. (1),$$s_l(t_i)=\int m(\mathbf{x})c_l(\mathbf{x})e^{j\mathbf{k}^T(t_i)\mathbf{\psi}(\mathbf{x})}d\mathbf{x}\,\,\,(1)$$where $$$m$$$ is the magnetization and $$$c_l$$$ is the coil sensitivity of the $$$l$$$th receive channel. B₀ inhomogeneities can be treated like an additional nonlinear gradient, hence $$$\mathbf{\psi}$$$ is a matrix containing encoding fields and B₀ inhomogeneities and $$$\mathbf{k}$$$ is the corresponding trajectory. To discretize Eq. (1), instead of modelling voxels with a delta function, we assume box functions with voxel dimensions $$$W_x$$$, $$$W_y$$$ and $$$W_z$$$ to model intra-voxel dephasing. The signal equation reduces to sepearate integrals representing the Fourier transform of a box function in each spatial dimension, which yields,^1,2

$$s_l(t_i)=\sum_pm_pc_l(\mathbf{x}_p)\mathrm{sinc}\left(W_x\mathbf{k}^T(t_i)\mathbf{g}_{p,x}\right)\mathrm{sinc}\left(W_y\mathbf{k}^T(t_i)\mathbf{g}_{p,y}\right)\mathrm{sinc}\left(W_z\mathbf{k}^T(t_i)\mathbf{g}_{p,z}\right)e^{j\mathbf{k}^T(t_i)\mathbf{\psi}(\mathbf{x}_p)}\,\,\,(2)$$

where $$$\mathbf{g}_{p,x}$$$, $$$\mathbf{g}_{p,y}$$$ and $$$\mathbf{g}_{p,z}$$$ are the derivatives of both the encoding fields and the inhomogeneities in $$$x$$$, $$$y$$$, $$$z$$$ direction at position $$$\mathbf{x}_p$$$. These derivates can be obtained analytically (if possible) or via finite differences.

We evaluate the capabilities of the model by simulating artifacts induced by an MR-safe guidewire (0.36mm diameter) consisting of a glass-fiber reinforced structure with embedded iron microparticles with a mean radius of $$$8\mu m$$$ (MaRVis Medical, Germany). The iron particles are modelled as spheres and are randomly distributed across the volume of the wire, each creating a dipole field.^3,4 The dipole fields and their derivatives are obtained analytically. The guidewire is located in the x-z plane and the simulation parameters are FOV=64×64mm², slice thickness of 5mm and a readout BW=465Hz/px. The object is encoded at 128×128 points with a Cartesian trajectory and a single ideal receive channel. Spins not excited by the slice-selection pulse due to the guidewire-induced field inhomogeneities are masked out so that they do not contribute to the signal. A simulation with a large number of spins ($$$x×y×z=4096\times103\times4096\approx1.7\cdot10^9$$$) is used as ground truth. In other simulations, the number of spins in-slice and through-slice direction is varied independently. Since more accurate signal modelling improves image quality, the signals are reconstructed by 2DFFT. The resulting images are compared to the ground truth via the Structural Similarity Index Measure (SSIM)⁵, which captures visual appearance better than l2-norm based error measures. Results with a SSIM >0.997 compared to the ground truth are accepted. For comparison, 2D GRE images of the guidewire were acquired at 1.5T (Tim Symphony, Siemens, Erlangen, Germany) with a FOV of 256×256mm², TE=5ms, TR=25ms, a flip angle of 15°, and a 512×512 Cartesian trajectory to match the resolution of the simulations.

Figs. 1(a,f) show the measured data and a corresponding subsection, which we compare to simulations (Figs. 1(b-e,g-j)). The simulations do not exactly match the measurement, which is probably due to imperfections in the wire model. The result obtained using the intra-voxel dephasing model (Figs. 1(c,h), $$$spins=512\times9\times512=2.3\cdot10^{6}$$$) closely resembles the ground truth (Figs. 1(b,g)) whereas the same number of spins without modelling dephasing leads to a noisy result (Figs. 1(d,i)). Comparable image quality (in terms of SSIM) is achived by increasing the number of spins to $$$896\times41\times896=12.6\cdot10^6$$$ (Figs. 1(e,j)). However, the image still appears noisier than the result including intra-voxel dephasing. As seen, a comparable image quality can be achieved with a reduction factor of ~5.4 in the number of spins if the dephasing model is used. In terms of memory requirements, the reduction is from 3100 to 574 Gigabytes for the encoding matrix in double precision without degrading image quality. Fig. 2 shows the SSIM of results with different numbers of spins compared to the ground truth. For a low number of spins, the model consistently performs better, however, with large numbers of spins, the influence of the model diminishes.

We show that a substantial reduction in memory requirements is possible. The overhead of including intra-voxel dephasing into the simulation is minimal compared to the other operations and hence negligible. Dephasing due to iron particles is quite pronounced, therefore it might be possible to reduce the number of spins even further for other applications. This model can also be beneficial in other applications such as imaging near metallic implants.