Michael Eijbersen^{1,2}, Bart Steensma^{1,2}, Cornelis A.T. van den Berg^{1,2}, and Alexander J.E. Raaijmakers^{1,2,3}

^{1}Radiotherapy, UMC Utrecht, Utrecht, Netherlands, ^{2}Computational Imaging Group, UMC Utrecht, Utrecht, Netherlands, ^{3}Biomedical Engineering, Eindhoven University of Technology, Eindhoven, Netherlands

A new computational method is presented for determining a transfer function of implants using a MRI only paradigm. By relaxing the transciever phase approximation better results are obtained for longer implants, however an optimization for both B_{1}^{+} and B_{1}^{-} based on magnitude only maps is required.

$$I=ME_{inc}$$.

A parameterized version of the TM can be determined if incident E

$$ E_z=\frac{1}{\mu_0\sigma+i\omega\mu_r\mu_0\epsilon_r\epsilon_0}\left(-i\frac{\partial B^{1+}}{\partial x}-\frac{\partial B^{1+}}{\partial y}+i\frac{\partial B^{1-}}{\partial x}-\frac{\partial B^{1-}}{\partial y}\right)$$

The transceive phase approximation assumes $$$\frac{\partial Bz}{dz}=0$$$ in which case the following relation holds:

$$\frac{\partial B_1^{+}}{\partial x}+i\frac{\partial B_1^{+}}{\partial y}=-\frac{\partial B_1^{-}}{\partial x}+i\frac{\partial B_1^{-}}{\partial y}$$

This allows the incident electric field to be determined by only B

$$B_{1,bg}^{\pm}=\sum_{n,l,m} j_l(kr)Y^l_m(\theta,\phi)$$ $$B_{1,sc}^\pm(\vec{r})\exp(i\omega t)=\frac{4\pi}{\mu_0}\int_{\text{wire}}I^\pm(\vec{r}')\exp\left(t-\frac{|\vec{r}'-\vec{r}|}{c_m}\right)\left(\frac{1}{|\vec{r}'-\vec{r}|^3}+\frac{i\omega}{c_m|\vec{r}'-\vec{r}|^2}\right)dr'\times (\vec{r}'-\vec{r})$$

The combined scattered and background field (d) in the data may be written as a matrix vector product:

$$\vec{d}=A\vec{x}+J\vec{y}$$

Here the matrix A and J contain discretized versions of the background and scattered field terms respectively. Since only magnitudes of the data points are available in experiments, the problem is reformulated in a Magnitude Squared Least Squares (MSLS) setting. This allows for working with inner products which are linear and allows us to rewrite the problem in terms of z, which is the concatenation of x and y with their real and imaginary part splitted. In this formulation we may derive equations for the cost function, gradient and hessian in order to use a Trust Region Newton algorithm to find an acceptable local minimum. A schematic representation of the derivation is provided in figure 2.

The proposed method was tested using simulated |B

- Implants ISA. Iso/ts 10974:2012(en). Assessment of the safety of magnetic resonance imaging for patients with an active implantable medical device. 2012
- Tokaya, Janot P., et al. MRI‐based transfer function determination through the transfer matrix by jointly fitting the incident and scattered field. Magnetic resonance in medicine, 2020, 83.3: 1081-1095.
- van den Bosch, Michiel R., et al. New method to monitor RF safety in MRI‐guided interventions based on RF induced image artefacts. Medical physics, 2010, 37.2: 814-821.

Figure 1: The Sim4Life model from three perspectives.

Figure 2: The Mathematics behind the MSLS formulation of the problem. By taking the magnitude squared it is possible to use the linearity of the inner product and separation of real and imaginary parts to derive the optimization equations. Note that $$$A_i \equiv a_i^H$$$ and $$$A_i \equiv a_i^H$$$ and $$$J_i \equiv j_i^H$$$ to make vectors lower case and non-hermitean vectors columns.

Figure 3: The results of the optimization compared with the simulations. The images show excellent agreement for both B_{1}^{+} and B1^{-}.

Figure 4: Comparison between the simulated induced current and incident electric fields. Note that the calculated incident electric field based on the transciever phase approximation yields an unacceptable estimate.

Figure 5: A comparison of the Transfer Matrix and Transfer functions between the simulation and the model reconstructions. The transfer function shows good agreement.

DOI: https://doi.org/10.58530/2022/0584