Janot P. Tokaya^{1}, Alexander J.E. Raaijmakers^{1,2}, Alessandro Sbrizzi^{1}, Peter R. Luijten^{1}, and Cornelis A.T. Berg^{1}

Two findings that are steps towards fully MR-based TF determination in realistic situations are presented. Firstly, the transfer matrix, that is an extension of the transfer function (TF), can describe the RF response of an implant in a human model. Secondly, a joint minimization of the incident field and the scattered field, determined with the transfer matrix, can describe measured and simulated complex B$$$_1^+$$$ distributions in a phantom with a linear implant present. The TM and TF result from this minimization, which enables their determination from a single MRI acquisition with the implant in place.

Purpose

For RF safety assessment of elongated implants the transfer function (TF) has been introducedTheory

The transfer matrix,
$$$M$$$, relates the induced current $$$I_{ind}$$$ in an implant to an incident
electric field distribution, $$$E_{inc}$$$, on an implant trough,
$$$I_{ind}=ME_{inc}$$$. Here $$$M$$$ can be parameterized with an attenuated
wave model as $$$M(c_k)$$$ with $$$c_k$$$ being the set of
three complex parameters, {$$$\sigma_{eff}$$$,$$$r$$$,$$$k$$$} characterizing
the implant^{5}.
The incident magnetic field can be
decomposed in SPherical And CYlindrical (SPACY) harmonics^{6},
$$$f_n^m$$$,$$B_{1,inc}^+(\alpha_{mn})\approx\sum_{n=1}^{N}\sum_{m=-n}^{n}\alpha_{mn}f_n^m.$$From
$$$B_{1,inc}^+$$$ the incident electric field can be calculated by a 1st order
differentiation^{7}. The scattered field due to the implant is
subsequently calculated from the induced current (Ampere’s law) that is modeled
by multiplying the incident E-field with the TM. For implants aligned along the
z-axis, the total magnetic field is given
by,$$\hspace{-50mm}B_{1,tot}^+(\alpha_{mn},c_k)=B_{1,inc}^+(\alpha_{mn})+B_{1,scatter}^+(\alpha_{mn},c_k)$$$$\hspace{-19mm}=B_{1,inc}^+(\alpha_{mn})+\frac{\mu_0I_{ind}(\alpha_{mn},c_k)\frac{e^{i\theta}}{2}}{2\pi\hspace{0.3mm}r},$$$$\hspace{-4mm}=B_{1,inc}^+(\alpha_{mn})+\frac{\mu_0M(c_k)E_{inc}(\alpha_{mn})\vert_{\vec{r}_{wire}}\frac{e^{i\theta}}{2}}{2\pi\hspace{0.3mm}r},$$$$\hspace{51mm}\approx\sum_{n=1}^{N}\sum_{m=-n}^n\alpha_{mn}f_n^m+\frac{\mu_0M(\alpha_{mn})\frac{1}{\sigma+i\omega\epsilon}(-2i\partial_x-2\partial_y)\alpha_{mn}f_n^m\vert_{\vec{r}_{wire}}\frac{e^{i\theta}}{2}}{2\pi\hspace{0.3mm}r},\hspace{8mm}(1)$$where
$$$\theta$$$ is the azimuthal angle around the wire and $$$r$$$ is the distance
to the wire. The differentiation provides the incident electric field from
incident B$$$^+_1$$$ SPACY expansion.

Methods

A) Numerical simulations (Sim4Life, ZMT,Zurich) were performed
to determine $$$M$$$ for a bare copper wire of 20cm in the aorta of
the virtual family model Duke. Two independent RF exposures from a high pass
birdcage coil at 1.5T (IQ and QI-feed) were used to verify that the TM
correctly predicts the current distribution for various incident E-field
distributions(these were determined seperately from simulations
without an implant). The transfer matrix parameters $$$c_k$$$ were estimated by minimization of^{5},$$\textrm{arg}\min_{c_k\in\mathbb{R}}\|I_{ind}-M(c_k)E_{inc}\|,$$for the IQ-feed simulations. Subsequently, this TM
was used to calculate the induced current for QI-feed and compared to the
actual current. B) The main drawback
of the previously presented method^{5} is the need for matching separate
simulations to determine the incident, tangential E-field. Here we present an
alternative. We estimate the TM and the incident E-field in a joint
minimization approach from a simulated or measured distribution combined with the model of equation 1.$$\hspace{55mm}\textrm{arg}\hspace{-5.3mm}\min_{c_k\in\mathbb{R},\alpha_{mn}\in\mathbb{C}}\|B_{1,meas/sim}^+-B_{1,tot}^+(\alpha_{mn},c_k)\|.\hspace{40mm}(2)$$The
parameters $$$c_k$$$ describing the TM of the implant follow
from fitting $$$B_{1,meas/sim}$$$. The
main advantage of this joint estimation is that no
separate simulations or measurements of the incident electric fields are needed.

The distribution was measured with a multi-flip
angle^{8} SPGR acquisition on a 1.5T MR system (Ingenia, Philips). The
$$$B_1^+\hspace{0.1mm}$$$phase was considered to be half the transceive phase that is acquired with two multi-echo
SPGR acquisitions with opposite gradient polarities^{9}. Simulations are
performed to validate the experiments. The measurement
and simulation setup are shown in figure 2.

Figure 1 shows the actual and predicted induced current in the implant. The agreement (Pearson correlation R=0.96) demonstrates that the TM can describe the implant response in a human subject reasonably well.

Figure 2 presents simulated and measured
$$$|B_1^+|$$$ and transceive phase distributions for an implant in an
elliptical phantom. From these distributions, with a region around the wire
removed, the incident $$$B_1^+$$$ distributions can be estimated with a
SPACY decomposition. (shown in figure 3). From the incident distribution
the incident E-field is calculated^{7}. The simulated E-field is shown for comparison. The coefficients from the
SPACY decomposition of the background field are provided as initial guess in
the joint minimization (equation 2) of the total field. The model is able to
accurately fit the total measured and simulated field as shown in figure
4. From this fit the parameters that describe the TM and TF follow (figure
5).

1. Park SM, Kamondetdacha R, Nyenhuis JA. Calculation of MRI-induced heating of an implanted medical lead wire with an electric field transfer function. J. Magn. Reson. Imaging 2007;26:1278–1285. doi: 10.1002/jmri.21159.

2.ISO/TS 10974:2012(en). Assessment of the safety of magnetic resonance imaging for patients with an active implantable medical device.

3. Feng S, Qiang R, Kainz W, Chen J. A Technique to Evaluate MRI-Induced Electric Fields at the Ends of Practical Implanted Lead. IEEE Trans. Microwve Theory Tech. 2015;63:305–313. doi: 10.1109/TMTT.2014.2376523.

4.** ** Tokaya JP, Raaijmakers AJE, Luijten PR, Bakker JF, van den Berg CAT. MRI-based transfer function determination for the assessment of implant safety. Magn Reson Med. 2017 Feb 5. doi: 10.1002/mrm.26613.

5. Tokaya JP, Raaijmakers AJE, Luijten PR, van den Berg CAT. MRI based RF safety characterization of implants using the implant response matrix: a simulation study. ISMRM proceedings 2017.

6. Sbrizzi A, Hoogduin H, Lagendijk JJ, Luijten P, van den Berg CAT. Robust reconstruction of B1+ maps by projection into a spherical functions space. Magn. Reson. Med. 2014;71:394–401. doi: 10.1002/mrm.24640.

7. Buchenau S, Haas M, Splitthoff DN, Hennig J, Zaitsev M. Iterative separation of transmit and receive phase contributions and B1+ -based estimation of the specific absorption rate for transmit arrays. Magn. Reson. Mater. Physics, Biol. Med. 2013;26:463–476. doi: 10.1007/s10334-013-0367-6.

8. van den Bosch MR, Moerland MA, Lagendijk JJW, Bartels LW, van den Berg CAT. 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.

9. van Lier ALHMW, Brunner DO, Pruessmann KP, Klomp DWJ, Luijten PR, Lagendijk JJW, van den Berg CAT. B1+ Phase mapping at 7 T and its application for in vivo electrical conductivity mapping. Magn. Reson. Med. 2012;67:552–561. doi: 10.1002/mrm.22995.

10. Yarnykh VL. Actual flip-angle imaging in the pulsed steady state: A method for rapid three-dimensional mapping of the transmitted radiofrequency field. Magn. Reson. Med. 2007;57:192–200. doi: 10.1002/mrm.21120.

Simulations were performed to test the TM concept in a
human model (Duke). Simulations of Duke in a high pass birdcage coil at 64MHz
were performed without an implant in IQ and QI-feed to determine the $$$E_{inc}$$$. These simulations were repeated with an implant
placed in Duke’s chest close to his heart. The resultant electric field is
shown on the left. $$$I_{ind}$$$ in IQ-feed is used to
determine the TM (shown on
the right). With this TM the expected current in QI-feed is calculated and compared to
the actually induced current. The comparison indicates the
TM can predict induced currents in human subjects reasonably well.

Simulated and measured $$$|B_1^+|$$$ and transceive phase distributions in the phantom with a wire. The $$$|B_1^+|$$$
distribution was experimentally determined by fitting a multi flip angle 3D
spoiled gradient echo acquisition with its signal equation on a voxel-by-voxel
basis. The transceive phase distributions were determined experimentally with
correction for eddy current contributions by acquiring two scans with opposite
gradient polarities^{9}. The combination of $$$|B_1^+|$$$
and the transceive phase are the $$$B_{1,meas/sim}^+$$$ distributions that are fitted with equation
1. A separate dual TR AFI $$$|B_1^+|$$$ map^{10} without the implant was acquired to investigate the performance of the SPACY decomposition in reconstructing the incident field.

The $$$B_1^+$$$ is determined from the $$$|B_1^+|$$$ field together with the transceive phase
distribution in the phantom with an implant. A mask of 4 cm radius around the
wire is removed from the data that is used to estimate the background field
with a SPACY decomposition. The true background field was determined in a
separate simulation and experiment (using a dual TR AFI acquisition^{10})
without a wire. These fields were used to confirm the agreement between
experiments and simulations and to check how well the SPACY decomposition
reconstructs the background field from the masked data. From the background distribution the electric field can
be estimated.

The measured and simulated $$$|B_1^+|$$$ distributions shown in the top row can be
accurately fit with equation 1.The corresponding fitted distributions are shown
in the bottom row, The fit was aided by an adequate initial guess for the complex
coefficients in the SPACY decomposition,$$$c_k$$$ , from the approximation of the background field as
shown in figure 3.

The transfer functions and transfer matrices computed with the $$$c_k$$$ parameters found with the minimization described in
equation 2 and the model of the $$$B_1^+$$$ distribution given in equation 1.
The black curves show the TM determined in simulation with a piecewise
excitation method^{5} (where the TM is determined with a sequentially
repositioned electric field excitation) that are considered the ground
truth. The red and green curve respectively show the TM and TF that follow the
simulated and measured $$$B_1^+$$$ distribution in the phantom with the wire
shown in figure 4.