MP-You A Web-based MPI Simulation Tool (2024)

The signal measured in an MPI setup can be modeled using:

with $\rho$ the particle distribution, $m$ is the magnetic moment of a single particle given by the product of the saturation magnetization $M_{s}$ in [A/m] and the volume of the core of a single particle, that itself depends on the particle’s diameter $d$, $B_{s}$ is the coil sensitivity given in [T/A], and $\mathcal{M}_{\textsf{eff}}$ is a function describing the magnetization response to a time-varying applied field $H_{\textsf{app}}$.

A commonly applied model for this magnetization behaviour is:

with the so-called Langevin function

and $\kappa=\frac{\mu_{0}m}{k_{B}T}$, $k_{B}$ is the Boltzmann constant and $T$ is the temperature in $K$. MPI utilizes the non-linear magnetization response by creating a field-free point (FFP) that is sensitive to high-frequency changes of the applied field. This is done by combining a gradient field with a gradient $G$ matrix, and a so-called drive field $H_{d}$, such that the position of the FFP is given by

In the present case, the drive field is defined via

which leads to a so-called Lissajous trajectory for the FFP.

The MPI system is a linear-shift invariant system [5]. Hence, the $n$-dimensional signal vector $y$ stemming from an arbitrary distribution of particles $\rho$ can be understood as a weighted sum of signals from $\delta$-samples placed at different positions within the FOV that has been discretized into $m$ voxels. The weights are proportional to the number of particles at that position, and can be determined from a simple matrix equation:

The matrix $S$ is called the system matrix, its columns correspond to the measured $n$-dimensional signal from a $\delta$-sample put at the $m$ different positions within the FOV, the term $\epsilon$ denotes possible noise in the measurement. In general $m<n$, which means we are faced with an overdetermined system, so the inverse of the matrix $S$ can not be defined in a straight-forward manner, and instead of solving for $\rho$ directly we need to consider the optimization problem [7]:

Since this system of linear equations is overdetermined, there can be more than one solution that fits the measurement data well. It is however reasonable to expect the solution to this equation to be somewhat regular or smooth, or in other words to have a small $l^{2}$-norm, leading us to add a so-called regularization term, given by $\lambda_{r}\|\rho\|$ to our cost function. This leads to the regularized version of our problem:

By varying the value of $\lambda_{r}$ we can balance between the goodness-of-fit to the signal and the desired smoothness of the solution.

Note, that the ground truth particle distribution is not affected by varying the experimental parameters, as are the $x$ and $y$ dimensions of the FOV. The M vs. H curve changes with $M_{s}$, $d$, and $g_{00}$ (see Table 1) such that an increase in $M_{s}$ or $d$ leads to a steeper shape, and a change in only $g_{00}$ only changes the range of $H$ over which we plot the M vs. H curve. It is important to note that the user only provides a single value $g_{00}$ for the gradient matrix G, from we which construct the gradient matrix as: