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Explaining current patterns on linear metallic implants during MRI exams using the transfer matrix.
Janot P. Tokaya1, Peter R.S. Stijnman1,2, Peter R. Luijten1, Cornelis A.T. Berg1, and Alexander J.E. Raaijmakers1,2

1Center for Image Sciences, UMC Utrecht, Utrecht, Netherlands, 2Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, Netherlands

Synopsis

Currents induced on relatively(<20cm) short linear metallic implants typically occur in certain patterns. Knowledge of these patterns can help current monitoring and identification of hazardous exposure conditions. The transfer matrix (TM) of an implant predicts what patterns will appear. The eigenvalues of the TM indicate which modes are induced naturally. The projection of the eigenvectors onto realistic incident electric fields obtained from electromagnetic simulations explain which modes are likely and effectively excited. Over 80% of electric field distributions excite the first eigenmode of the investigated structures more efficiently. Moreover, all severe currents follow the pattern of this first eigenmode.

Introduction

MRI examinations of patients with (elongated) metallic implants pose a safety risk. The RF fields induce currents on the implants that cause temperature elevations at the tips of such implants1–3. The tip heating can be calculated from the incident RF electric field with an implant characteristic called the transfer function(TF)4. Recently, the TF concept has been extended to the transfer matrix(TM)5 allowing MR-based TF determination. This work will show the eigenmode spectrum of the TM contains valuable information on the most likely occurring current distributions on the implant. This information can help identify hazardous situations and critical incident electric fields. Knowledge of likely current patterns could furthermore benefit MR-based current monitoring6–8.

A simulation study is performed to investigate the eigenmodes of various elongated implant structures. This, in combination with a realistic collection of incident electric field distributions provides a framework to investigate the likelihood and shape of induced patterns.

Methods

The TM,$$$\,M$$$, describes the one-dimensional current profile,$$$\,I_{ind}$$$, that will be induced on an implant exposed to an incident tangential one-dimensional electric field,$$$\,E_{inc}$$$, distribution through,$$$I_{ind}=ME_{inc}$$$. The TM is determined by FDTD electromagnetic simulations (Sim4Life, ZMT, Zurich, Switzerland). The implant is exposed to subsequently repositioned localized electric fields5 in a typical test medium9 ($$$\sigma=0.47$$$S/m,$$$\epsilon=78$$$). The TM is determined for the structures shown in figure 2.

Possible realistic 1D electric field distributions are determined from a FDTD simulation of “Duke” from the Virtual Family10 in a 1.5T birdcage coil (figure 3). From the electric field distribution inside Duke all possible $$$E_{z,inc}$$$ exposures of 10cm, 20cm, 30cm and 40cm length are extracted. The wires are assumed to be z-oriented.

The eigenvalue distributions indicate that the first eigenvector will be induced most strongly. An $$$E_{inc}$$$ resembling this mode would be worst case. Nevertheless, the current pattern may resemble the second eigenvector if $$$E_{inc}$$$ predominantly projects onto this mode. To investigate how often this occurs, the $$$E_{z,inc}$$$ distributions from Duke are projected onto the eigenvectors. This identifies eigenmodes that are ‘excited’ more heavily. The higher the projection value, $$$\alpha_i$$$=$$$\frac{v_i\cdot E_{z,inc}}{||E_{z,inc}||_2}$$$, the more $$$E_{z,inc}$$$ induces this eigenmode,$$$\,v_i$$$. Secondly, the total induced currents are computed through a multiplication of the $$$E_{z,inc}$$$ distributions with the TM. The largest possible currents dominated by the first and second eigenmode are compared.

Results

Figure 1 shows two examples of TMs with their first 4 eigenvectors that resemble sine waves with decreasing wavelength. Figure 2 shows that the 1st eigenvalue is generally much bigger than the 2nd for shorter implants. This indicates that the implants support the first mode better than the others.

In total 2.83 million $$$E_{z,inc}$$$ of 10cm, 2.27 million $$$E_{z,inc}$$$ of 20cm, 1.85 million $$$E_{z,inc}$$$ of 30cm and 1.53 million $$$E_{z,inc}$$$ distributions of 40cm are extracted from the electric field inside Duke. The projection of these onto the 1st and 2nd eigenvector of the TM are denoted by $$$\alpha_1$$$ and $$$\alpha_2$$$. Figure 4 shows the $$$\frac{\alpha_2}{\alpha_1}$$$ histogram for the extracted collection of incident electric fields. The first eigenvector is excited more strongly for over 80% of $$$E_{z,inc}$$$ distributions.

The contribution of each eigenvector to the currents computed from the $$$E_{z,inc}$$$ distributions by multiplication with the TM are calculated. In figure 5 the worst case currents dominated by the first and second eigenmode are displayed.

Discussion

The eigenvectors of the TM provide a basis set of modes to describe induced currents. Currents induced on an implant at 1.5T almost always appear in a pattern similar to the first dominant eigenvector. Which mode dominates depends on the length of the implant and the wavelength of the RF field in the medium, which is 44cm11.

Dominant eigenmodes have one antinode for implants investigated in this work (figure 2). Furthermore the first modes for all structures are excited more effectively by realistic $$$E_{z,inc}$$$ distributions. Figure 4 shows that over 80% of the potential $$$E_{z,inc}$$$ distributions induce the first eigenmode more effectively than the second.

Figure 5 shows that the worst case induced currents not dominated by the first eigenmode are much lower than the true maximal current. The contribution of each eigenvector to $$$I_{ind}$$$ is influenced by both the ability of the implant to support it and the effectiveness of $$$E_{z,inc}$$$ in exciting this specific mode.

Conclusion

The eigenvalues/vectors of the TM can be used to investigate the tendency for certain current patterns to appear. An analysis for four elongated implant structures of various lengths has shown that current patterns induced on smaller (<20cm) implants are dominated by a single eigenvector. To induce strong currents, this dominant eigenvector has to be excited more strongly than others, which happens for over 80% of realistic $$$E_{z,inc}$$$ distributions. The resultant current distributions, when significant, hence always resemble this dominant eigenvector.

Acknowledgements

This project is funded by The Netherlands Organisation for Scientific Research (NWO) under project number: 15739.

References

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5. Tokaya JP, Raaijmakers AJ, Luijten PR, van den Berg CA, Janot Tokaya CP. MRI-based, wireless determination of the transfer function of a linear implant: Introduction of the transfer matrix. 2018:1–14. doi: 10.1002/mrm.27218.

6. van den Bosch MR, Moerland M a, Lagendijk JJW, Bartels LW, van den Berg C a T. New method to monitor RF safety in MRI-guided interventions based on RF induced image artefacts. Med. Phys. 2010;37:814–821. doi: 10.1118/1.3298006.

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Figures

The TM of the bare (top) and insulated wire (bottom) of 20cm together with the first 4 eigenvectors. The eigenvectors are sine shaped modes with increasing number of nodes over the length of the implant. For the structures investigated in this work the first eigenvector always is the mode with least nodes. This potentially changes if the length of the implants exceeds the maximal length investigated here. It will result in a different ordering of the modes, but not in an alteration of their appearance. The electric field that would cause the higest current would follow the distribution of the dominant eigenvector.

The structures investigated in this work are bare and insulated wires of 10cm, 20cm, 30cm and 40cm, a 20cm half-capped wire and a 20cm bare wire attached to a conductive box that represents an implantable pulse generator (IPG). The first and second eigenvector of the TMs of these structures are shown as black and red lines respectively. The fraction between the corresponding 1st and 2nd eigenvalue, i.e. $$$\lambda_1$$$ and $$$\lambda_2$$$, is indicated and increases with increasing length. For the lengths displayed here the first eigenvector always has one antinode. The orange parts of the structures are copper and the green parts are insulation ($$$\sigma$$$=0 S/m,$$$\epsilon$$$=3) .

The simulation setup of Duke in the birdcage is shown on the left. The resulting electric field distributions (scaled to 30 μT peak B1+), show that the electric field inside Duke is predominantly z-oriented. From the electric field distribution inside Duke all 10cm, 20cm, 30cm and 40cm one-dimensional distributions in the z-direction are extracted. These distributions represent all possible electric field exposures that an implant of identical length directed along the z-axis can encounter.

The extracted electric field distributions of various lengths from various locations in the Duke model are projected onto the first two eigenvectors of the TM of the various implants with the same length. Here, the fraction between both projections is displayed for the distribution of incident electric fields. If $$$\frac{\alpha_2}{\alpha_1}$$$>0, the incident electric field has a larger projection onto the second eigenvector. The percentage of electric fields with $$$\frac{\alpha_2}{\alpha_1}$$$>0 is given for each distribution. For the wires shorter than 20cm more than 90% of the electric field distribution are more efficient in exciting the first eigenmode of the TM.

The worst case currents for the various structures are shown as solid lines in figures F1, F2, F3 and F4. The worst case currents with a larger projection onto the second eigenvector of the TM are shown as dashed lines. These currents are much lower as is emphasized by table 1. In this table the fraction between the maxima of dashed and solid lines in figures F1, F2, F3 and F4 are given. It shows that currents more similar to the second eigenmode are more than three times lower than worst case currents, which always resemble the first eigen mode.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)
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