Modelling Magnetic Resonance Imaging (MRI)
Introduction
EM simulation involves the numerical approximation of Maxwell’s equations to determine the EM behaviour of detectors and radiating structures. No physical assumptions are made regarding the fields when using fullwave techniques. This is opposed to static or quasistatic approaches in which time rates of change of magnetic and electric flux densities are neglected (assumed zero). In very highfield MRI (>3T) the EM field interactions with the conductive dielectric media is such that it does not allow accurate prediction of the EM behavior using static or quasistatic approximations.
The most common fullwave EM methods are the Method of Moments (MoM), the Finite Difference Time Domain (FDTD) method, and the Finite Element Method (FEM). The MoM computation is done by discretising the surface current in the frequency domain, the FEM computation involves the discretisation of the E field in the frequency domain and the FDTD computation is performed in the time domain, discretising both the E and H fields [1].
FEKO is a powerful MoM solver which incorporates piecewise triangular functions for metallic wires and surfaces. It employs Green’s functions as well as surface and volume equivalence principles (SEP and VEP) for dielectric volumes [2]. FEKO has been successfully used in a number of MRI radio frequency (RF) coil designs and magnetic resonance (MR) RF safety analyses [3,4,5,6,7]. Several MR applications will be discussed in further detail below.
RF Coil Design
RF coil design in FEKO allows for the accurate prediction of signaltonoise ratio (SNR), field homogeneity and coil safety. Accurate prediction of coil performance not only helps to optimise the coil design itself, but also reduces the time and cost involved in developing a superior RF coil for a specific application in MR. FEKO modelling and simulation greatly assist in the development of surface coils, volume coils and array coils.
Surface Coils:
Surface coils made of copper strips and wires can be modelled to analyse the SNR, currents and the specific absorption rate (SAR) for transmit coils. Fig. 1 shows surface coils of various shapes and sizes at different field strengths and for different applications.
Figure 1. (a) Twoelement rodent spine detector at 300 MHz.  (b) Figure8 loop plus circular loop in quadrature mode (top) and two circular loops (bottom) at 128 MHz [5]. 


(c) Overlapping loops for 31P cardiac imaging at 128 MHz.  (d) QuasiTEM mode strip element B field profile (top) and current distribution on strip element (bottom) at 128 MHz. 
Coil Tuning:
The model setup used for the tuning of a surface coil is shown in Fig. 2(a). A port is created in the loop and a voltage source is added in series with a capacitor. Additional ports with lumped capacitors are placed in the loop at the desired locations and the coil is loaded with a lossy medium. The coil element can be tuned by inspecting the source impedance plotted on a Cartesian graph or a Smith chart as shown in Fig. 2(b). The frequency sweep of 100 MHz to 150 MHz with impedance calculations at 50 points for the 4 cm loop shown in Fig. 2(a) completed within 12 seconds on an Intel quadcore workstation with 8GB RAM. For faster results, adaptive frequency sampling (AFS) may be used to calculate the source impedance over the frequency range of interest.


Figure 2. (a) A 4 cm diameter loop tuned to series resonance with 3 capacitors (59 pF) at 126 MHz.  (b) Cartesian plot (top) and Smith plot (bottom) of the source impedance of the coil tuned to series resonance at 126 MHz. 
SNR:
The signaltonoise ratio of RF coils can be calculated from the FEKO computation of the H field generated for unit current in the coil, using the reciprocity principle. The noise resistance is determined from the real part of the input impedance of a tuned coil source. The intrinsic SNR is given by the following formula, when all other conditions of the experimental setup (e.g. magnetisation, frequency, bandwidth) are kept constant [3]:
Here B_{t}^{} represents the leftcircularly polarised component of the transverse (x,y components in scanner coordinates) magnetic flux density and R_{noise} is the noise resistance which includes both coil noise and sample noise [4]. More information on the prediction of the exact SNR from a voxel volume at a particular MRI frequency may be found in Kumar et al. [5].
Coil Coupling:
Inductive coupling between adjacent coil elements can be minimised by overlapping the coils [8]. Looking at the S_{21} scattering parameter, the coil overlap is adjusted to minimise coupling. A simple example is shown in Fig. 3.
Figure 3. (a) Nonoverlapping coils.  (b) The S_{21} value at resonance is greater than 20 dB without overlap. 
(c) Two overlapping coils.  (d) The S_{21} value is less than 30 dB with overlap. 
Dielectric Medium:
The human body or the imaging sample volume in MR can be modelled as a conductive dielectric medium with known conductivity and permittivity. FEKO allows the modelling of dielectric bodies with arbitrary shape and homogeneous, layered or inhomogeneous dielectric properties. Fig. 4 below shows some dielectric bodies as examples.


Figure 4. (a) Hugo human inhomogeneous phantom.  (b) IEEE head (top) and custom breasttorso phantom (bottom). 
Volume Coils:
Volume coils such as birdcage coils and volume arrays can be modelled in FEKO with relative ease. The birdcage coil is one of the most commonly used types of transmit coil geometry in MR. A birdcage coil is typically excited at two ports which are driven in quadrature. The cylindrical surface currents generated on the birdcage rungs produce circularly polarised transverse magnetic fields within the birdcage coil. RF shields are used to stabilise birdcage resonance by shielding interference and spurious radiation. Fig. 5 shows some volume coils modelled with FEKO.


Figure 5. (a) RF shielded sixteen rung quadrature birdcage body coil (63 cm diameter) with ASTM body phantom.  (b) Dome shaped head volume coil. 


(c) Twelve rung head coil at 128 MHz.  (d) Helmholtz volume coil at 200 MHz. 
Array Coils:
Coil arrays are commonly used to improve SNR while scan time is reduced by employing parallel imaging techniques [9,10]. Coil arrays can be surface coils or volume coils depending on the particular application. An array geometry with minimal coil coupling for improved SNR can be designed using FEKO.
In the next example, FEKO was employed to evaluate array performance based on the Noise Figure (NF) [4]. Fig. 6 shows a comparison between the NF of flat washer type loops and wire loops as the number of loops in the array is increased. The numerical calculations were validated by experimental measurements. The experimental NFs of the individual component loops were ~0.6 dB and ~0.7 dB for the wire and flat strip coils, respectively. The loops were loaded with a physiologically analogous agar gel phantom (ε_{r} = 63.5, σ = 0.72 S/m).
SENSE Parallel Imaging Performance:
Arrays of simultaneously operated receiver coils are widely used in MR to reduce scan time. One of the general parallel imaging techniques is sensitivity encoding (SENSE) [10]. FEKO can be used to evaluate the performance of an array coil geometry for SENSE imaging with the following steps:
 Compute the B_{t}^{} profile of each coil element in the array in the desired field of view (FOV) for the coil sensitivity matrix.
 Compute the E fields in the volume of interest separately for each coil element to determine the noise correlation matrix of the array.
 Calculate the optimum SNR [8].
 Calculate the gfactor (the geometry factor that quantifies the spatial interaction between coils) [10].
 Calculate the parallel imaging SNR with the desired acceleration factor.
The parallel imaging performances of different array coil geometries can be compared to find an array that is suitable for a given application.
RF Safety:
RF safety in MR is strictly monitored to limit heating due to RF exposure in patients below 1 degree Celsius for any MR procedure. SAR is the parameter of importance that determines how much power is deposited in the conductive sample volume under RF exposure. FEKO calculates very reliably [6,7] volume averaged SAR in 1g, 10g and whole body of interest for a given power input to coil element/s. Fig. 7 illustrates the SAR analysis done for a breast MR transmit coil developed at 7T (300 MHz) [6].
Figure 7. Peak 10g SAR (red cube) and axial magnetic field profile shown for a breast MR transmit coil. The power input to the coil elements is 20W CW (continuous wave). 
RF Safety of Interventional Devices:
Potentially dangerous levels of currents may be induced on metallic implantable devices such as pacemaker leads due to RF exposure during MRI. Currently, MR examinations are contraindicated to patients with pacemakers due to potential danger of tissue damage. Development of MR safe interventional devices would allow patients with implanted devices to receive MRI scans when indicated. Such devices are also important to ensure the safety of medical practitioners and patients during interventional MR procedures.
FEKO enables accurate modelling and simulation of the MR environment with interventional devices such as pacemaker leads and can be used to evaluate the safety of a device [7]. SAR computation near the implanted lead and at the tip of the electrodes that are exposed to the conductive tissue, reveal the heating potential of the device under MR RF exposure. Fig. 8 shows the SAR levels near a lead placed in a cylindrical conductive phantom.
Figure 8. SAR (1g) levels near the lead (placed along the phantom length in the zdirection) and at the electrode. 
Fig. 9 shows the SAR at the electrode calculated by FEKO for pacemaker leads with various lengths, pitches and insulation thicknesses, as well as the validation with experimental temperature measurements at the electrode. Here SAR is assumed to be directly proportional to temperature rise, neglecting the heat loss to the environment by diffusion.
Figure 9. (a) FEKO computation of SAR (1g) at the electrode for leads with various lengths, pitches and insulation thicknesses. (b) Experimental validation by temperature measurements at the electrodes. 
Conclusion
EM simulation is a very powerful tool for the accurate prediction of MRI RF coil performance for rapid prototyping. The computationally fast MoM core of FEKO allows consideration of a wide range of design parameters to arrive at optimal solutions for RF coil designs. FEKO RF safety analysis involving SAR computation gives reliable results for MRI RF transmit coil safety and it greatly assists in the development of MR safe implantable devices.
Acknowledgement
The content of this article was provided by Ananda Kumar, Ph.D., Resonant Research LLC, Baltimore, MD, USA and Visiting Research Associate, Department of Radiology, Johns Hopkins University, Baltimore, MD, USA.
References
 D. B. Davidson, Computational Electromagnetics for RF and Microwave Engineering, 2^{nd} edition, Cambridge University Press, 2006.
 U. Jakobus, "Comparison of different techniques for the treatment of lossy dielectric/magnetic bodies within the method of moments formulations", AEU Int J Electron Commun, vol. 54, pp. 163173, 2000.
 A. Kumar and P. A. Bottomley, "Optimizing the intrinsic signaltonoise ratio of MRI strip detectors", Magnetic Resonance in Medicine, vol. 56, pp. 157166, July 2006.
 A. Kumar, W. A. Edelstein, P. A. Bottomley, “Noise Figure Limits for Circular Loop MR Coils”, Magnetic Resonance in Medicine, vol. 61, pp. 12011209, May 2009.
 A. Kumar and P. A. Bottomley, "Optimized Quadrature Surface Coil Designs", Magnetic Resonance Materials in Physics, Biology and Medicine, vol. 21, pp. 4152, March 2008.
 A. Kumar, L. Blawat, M. Schar, P. Barker, "Development of Quadrature Transmit Elements for Breast MRI/MRSI at 7T", 18th Annual ISMRM Meeting, Stockholm, Sweden, May 2010.
 P. A. Bottomley, A. Kumar, W. A. Edelstein, J. M. Allen, P. V. Karmarkar, "Designing passive MRIsafe implantable conducting leads with electrodes", Medical Physics, vol. 37, no. 7, July 2010.
 P. B. Roemer, W. A. Edelstein, C. E. Hayes, S. P. Souza, and O. M. Mueller, "The NMR phased array", Magnetic Resonance in Medicine, vol. 16, pp. 192225, Nov. 1990.
 D. K. Sodickson and W. J. Manning, "Simultaneous acquisition of spatial harmonics (SMASH): fast imaging with radiofrequency coil arrays", Magnetic Resonance in Medicine, vol. 38, pp. 591603, Oct. 1997.
 K. P. Pruessmann, M. Weiger, M. B. Scheidegger, and P. Boesiger, "SENSE: sensitivity encoding for fast MRI", Magnetic Resonance in Medicine, vol. 42, pp. 952962, Nov. 1999.