Investigation of Magnetic Field Gradient Waveforms in the Presence of ...

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Investigation of Magnetic Field Gradient Waveforms in the Presence of a Metallic Vessel in Magnetic Resonance Imaging Through Simulation Frédéric G. Goora

, Hui Han , Matthew Ouellette

, Bruce G. Colpitts , and Bruce J. Balcom

Department of Electrical and Computer Engineering, University of New Brunswick, Fredericton, NB E3B 5A3, Canada MRI Centre, Department of Physics, University of New Brunswick, Fredericton, NB E3B 5A3, Canada Department of Mechanical Engineering, University of New Brunswick, Fredericton, NB E3B 5A3, Canada We have previously established the ability to characterize eddy currents and their impact on the magnetic fields in metal vessel magnetic resonance imaging applications through comparison with measured data. We now use simulation to investigate the spatiotemporal characteristics of eddy currents and their magnetic fields encountered in metal vessel magnetic resonance imaging that may not easily be experimentally verified. Simulations using CST EM Studio have revealed the impact of an offset in the positioning of the metal vessel in a superconducting magnet bore as well as the consequences of the different amplitude magnetic gradients employed during imaging experiments. Furthermore, we investigate the response of the metal vessel to magnetic field gradients generated from nonplanar and planar gradient coil geometries (encountered in superconducting and permanent magnet-based systems). Establishing the basic electromagnetic properties of the metal vessel in a magnetic resonance sense through simulation permits us to replace our simple metal vessel with a more sophisticated rock core holder design and investigate its properties. Index Terms—Eddy currents, electromagnetic transients, gradient coil, magnetic resonance imaging, simulation.

I. INTRODUCTION

T

HIS work is part of a large scale investigation of the use of simulation to investigate the electromagnetic impact of metal vessels when used in magnetic resonance imaging (MRI) applications. The ability to characterize eddy currents and their impact on the magnetic fields in metal vessel MRI [1]–[3] applications through comparison with measured data was established in our previous work [4]. Correspondingly, the electromagnetic impact of a variety of scenarios that one anticipates encountering in metal vessel MRI that are not easily experimentally verified can now be considered through simulation. These scenarios include the impact of offsets in the positioning of the metal vessel within the bore of a superconducting magnet as its insertion is subject to an offset between the vessel and gradient coil isocenter. The effects of the variable amplitude magnetic field gradients used during imaging experiments are of interest and will also be investigated within this paper. The temporal evolution and spatial distribution of the magnetic field within the metallic vessel upon the application of a magnetic field gradient during an imaging experiment will also be investigated. The temporal evolution of the magnetic field reveals the suitability of metal vessels for MRI applications. The spatial distribution of eddy currents provides a visualization of the eddy current circulation patterns which aid in the determination of the suitability of more sophisticated vessel designs. The nonlinearity of the magnetic field as it temporally evolves is also an important consideration. A magnetic field evolving linearly with time permits an imaging experiment to be completed even when the amplitude of the eddy currents are sufficient to cause Manuscript received October 16, 2012; revised November 30, 2012; accepted December 10, 2012. Date of publication December 20, 2012; date of current version May 30, 2013. Corresponding author: B. J. Balcom (e-mail: bjb@unb. ca). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2012.2234758

a significant distortion to the magnetic field as experienced by the sample. The orientation of the metal vessel in [4] was suitable for superconducting magnet-based applications. However, the compatibility of the metal vessel in permanent magnet-based systems, such that the static magnetic field is laterally incident on the metal vessel and not longitudinal as in superconducting magnet-based systems, is of equal importance and must be considered. Therefore, simulation results from the application of magnetic field gradients that are induced through planar gradient coils are presented. This type of coil geometry is commonly used in permanent magnet-based systems as they do not obstruct access to the static magnetic field and permit insertion of the RF probe and sample into the region of the static magnetic field (denoted ) necessary for magnetic resonance. The gradient coils are typically mounted on or near the magnet pole pieces. The following sections characterize the metal vessel, the magnetic field gradient coils, and the electromagnetic response of the vessel to the switched magnetic field gradients used in MRI.

II. METAL VESSEL PROPERTIES Simulations were employed to extract the magnetic field gradient waveforms for vessels constructed of Nitronic 60 stainless steel. The electrical conductivity and relative magnetic permeability of this metal were 1.02 10 S/m and 1.003, respectively [5]. The modeled vessel length is 15.2 cm (6.0 in) with outer and inner diameters of 7.6 cm (3.0 in) and 5.1 cm (2.0 in), respectively. The vessel is positioned such that the origin of the cylinder is coincident with the gradient isocenter. Any deviations from this basic vessel geometry and position will be clearly indicated in the relevant sections.

0018-9464/$31.00 © 2012 IEEE

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TABLE I SUMMARY OF PLANAR

AND

COIL DIMENSIONS

A. Nonplanar Magnetic Field Gradients The nonplanar magnetic field gradient coils, referred to as a Maxwell pair [6]–[8] and Golay coils [6]–[8], are employed in superconducting magnet structures. They are identical to those discussed in [4]. Note that the static magnetic field is collinear with the long axis of the metal vessel. B. Planar Magnetic Field Gradients

Fig. 1. Simulated planar magnetic field gradient coils with a metallic vessel in a geometry, permanent magnet-based system: (a) Maxwell pair geometry, (b) geometry. and (c)

III. SIMULATION PARAMETERS The simulation parameters outlined in [4] were preserved for the data presented here. Additional parameters specific to this work are outlined below.

Planar gradient coil geometries are typically used in permanent magnet-based systems. Note that the static magnetic field is transverse to the long axis of the metal vessel. The following subsections outline the geometries and dimensions of each coil design. As was the case in [4], simplified gradient coil geometries were selected to reduce the overall simulation complexity. We discuss the planar gradient coil geometries in greater detail as follows. 1) Maxwell Pair—Planar Geometry: The ratio of the dimensions that provide the optimal homogeneity of the magnetic field gradient for a planar Maxwell pair is such that the separation between the coils, , is equal to the lengths of the square coil [9]. The coil separation was set to be 10.16 cm which is similar to the distance between the pole pieces of a permanent magnet available within the UNB MRI Centre. Therefore, the dimensions of the square coil were also set to 10.16 cm. The current was set to 30 A with 50 turns in each coil. The planar Maxwell pair with the metallic vessel present is shown in Fig. 1(a). 2) Transverse Gradients—Planar Geometry: The transverse gradient geometry and dimensional relationships which produce an optimal magnetic field gradient from [9] have been summarized in Table I. Since the coil separation was set to 10.16 cm (refer to Section III-B1), the remaining dimensions were also calculated and are also summarized in Table I. The coil length l is not explicitly defined in [9]; a longer coil length results in the reduction of undesired gradients inherently produced by the wires in the coil geometry [9]. However, increasing the length also increases the inductance and resistance of the coil. Therefore, length l was chosen such that it was twice the Golay coil width which equals 15.74 cm. For the transverse gradient coil simulations, the current was set to 30 A with 50 turns in each coil. The transverse coils which produce an x- and y-directed magnetic field gradient are referred to as and coils, respectively. The and coils with the metallic vessel present are shown in Fig. 1(b) and (c), respectively. Note that the gradient coil was accomplished by rotating the metallic cylinder within the simulation space. IV. SIMULATION RESULTS As in [4], the data was simulated using the low frequency time domain solver of CST EM Studio.

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A. Effect of Metal Vessel Offset in Gradient Set The insertion of the metal vessel within the bore of a superconducting magnet is unlikely in practice to result in a vessel exactly centered in the gradient coil. A z-directed offset in a superconducting magnet case (Section III-A) is considered in order to simplify the investigation of the effects of the vessel offset in the gradient coil. The following summary of the spatial dependence of eddy currents is based on the development of the theory found in [10]. The excitation current waveform applied to the gradient coils produces a pulsed magnetic field which results in the induction of eddy currents on the metallic vessel whose magnetic field opposes the applied field. This field can be denoted as . The Taylor series expansion of this eddy current induced magnetic field is (1) The first expansion term, , is referred to as the eddy current [10]. The second expansion term, , is referred to as the linear eddy current [10]. The three Cartesian components of , , , and , represent the magnetic field gradient induced along the three physical gradient axes , , and , respectively, due to eddy currents [10]. Higher order terms are not usually considered and do not have standardized names [10]. 1) Temporal Evolution of Magnetic Field: The temporal evolution of the magnetic field for discrete times of 0.1, 0.2, 0.3, 0.4, and 0.8 ms are shown in Fig. 2(a). The solid lines represent the simulation results for a centered vessel whereas the dashed lines represent a vessel that has been translated along the z-axis by 1 cm. A temporal dependence of the magnetic field is expected as outlined in (1) [10]. Furthermore, the shifting of the metallic vessel results in an asymmetry of the vessels exposure to the magnetic field gradient imposed due to the Maxwell pair. This asymmetry results in asymmetric eddy currents being generated which would result in an asymmetry in the resulting magnetic field. Also, the corresponding diffusion of eddy currents [4], [11]–[15] results in a spatiotemporal shift of eddy currents which results in a spatiotemporal shift of the magnetic field gradient. The positional shift of the magnetic field from 0.1, 0.2, 0.3, 0.4, and 0.8 ms was 0.924, 0.916, 0.831, 0.674, and 0.151 mm, respectively [shown in Fig. 2(b)]. The magnetic field offsets temporally evolve with time as eddy currents diffuse and the system approaches a quasi-steady state response. At the quasi-steady state, the amplitude of the circulating eddy currents results in a negligible impact on the magnetic field within the metal vessel. This will be referred to as the quasi-steady state of the system for the remainder of this paper. Note that the positional shift of the field will ultimately be 0 at the quasi-steady state. It is observed that the translation of the metallic vessel within the gradient coils results in eddy currents that produce magnetic fields. These fields are similar to those in the centered vessel case but with the addition of a constant magnetic field offset term and edge effects due to the asymmetric exposure of the metal vessel to the applied magnetic field gradients. The field

Fig. 2. Magnetic field within the metallic vessel due to eddy currents upon application of a nonplanar Maxwell pair induced magnetic field gradient for (a) both a centered (solid line) and translated (dashed line) metallic vessels and (b) a translated metallic vessel showing the magnetic field offset. Indicated times are those following the application of a magnetic field gradient.

offset varies with time and is due to the diffusion of eddy currents as their spatial distribution within the metallic vessel temporally evolves [4], [11]–[15]. Although the diffusion of eddy currents results in a temporally dependent shift in the magnetic field stated in (1), the magnitude of the overall effect is insignificant due to the concurrent predominately single exponential decay in eddy current amplitude [1], [4], [15]. 2) Spatial Distribution and Temporal Evolution of Eddy Currents—Nonplanar Maxwell Pair: The observation of the spatial distribution of eddy currents on the metal vessel provides additional insight and intuition regarding the temporal evolution of the magnetic field [4]. We consider the eddy current response of the metal vessel due to a switched magnetic field from a nonplanar Maxwell pair (Section III-A). The vessel in these simulations was translated by 1 cm along the -axis. Correspondingly, the edge of the vessel along is closer to the Maxwell pair gradient coil. The spatial distribution of eddy currents 0.1 and 0.8 ms after the application of the magnetic field gradient are shown in Fig. 3.

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Fig. 3. Spatial distribution of eddy currents on a translated metallic vessel shown at (a) 0.1 ms and (b) 0.8 ms following viewed in the -plane the application of a magnetic field gradient from a nonplanar Maxwell pair magnetic field gradient coil.

Diffusion of eddy current [4], [11]–[15] is evident in Fig. 3. Diffusion of eddy currents from transient resistance effects is observed from time 0.1 to 0.2 ms after the application of the magnetic field (refer to [4]). From 0.2 ms through to 0.8 ms after the application of the magnetic field, diffusion of eddy currents affects the current null (which is initially located away from gradient isocenter). This time varying spatial distribution of eddy currents causes the time varying shift (positional offset) that was observed in the zero-crossing of the magnetic fields due to the eddy currents. In the event that the metallic cylinder is not positioned at the isocenter of the gradient coils, the desired magnetic field gradient will still be applied to the sample with the addition of another linear term. Overall, the effects of eddy current diffusion are reduced since the eddy current amplitude is exponentially decaying concurrently with the diffusion process. Therefore, the impact of small vessel translations will be negligible to the final measurement. B. Effect of Varying Gradient Amplitude Levels The strength of the magnetic field gradients applied during imaging experiments vary greatly. The resulting electromagnetic effects will be discussed in the following subsections. A nonplanar Maxwell pair gradient coil (Section III-A) was used as the magnetic field gradient coil. 1) Simulation Results: The magnetic field gradient waveform (MFG waveform) was simulated for the Nitronic 60 stainless steel vessel. The current flowing through the nonplanar Maxwell pair gradient coil was varied in three separate simulations. The excitation current was applied with discrete current values of 15, 30, and 60 A. The simulated MFG waveforms are shown in Fig. 4(a).

Fig. 4. (a) Magnetic field gradient waveform and (b) normalized magnetic field gradient waveform simulations at position (1 cm, 1 cm, 1 cm) inside a Nitronic 60 stainless steel vessel at 15, 30, and 60 A of excitation current through a nonplanar Maxwell pair gradient coil.

2) Discussion of Results: The corresponding MFG waveform rise and fall time constants are not dependent on the applied gradient strength as clearly shown in the plots of the normalized waveforms in Fig. 4(b). Varying the magnetic field gradient strength results in consistent magnetic field gradient waveforms with identical rise and fall time constants. C. Nonplanar Magnetic Field Gradient—Maxwell Pair The metal vessel described in Section II is positioned within the nonplanar Maxwell pair gradient coil isocenter (Section III-A). 1) Temporal Evolution of Magnetic Field: The temporal evolution of the magnetic field within the centered metal vessel at discrete times of 0.1, 0.2, 0.3, 0.4, and 0.8 ms after the application of current to the magnetic field gradient coils are shown in Fig. 2(a). The quasi-steady state magnetic field within the metallic vessel has resulted 0.8 ms after the field is applied, which is in agreement with the single exponential time constant calculated in [4] of 172 s. Note that the behavior of the eddy currents following the removal of the magnetic field will be identical and opposite to the behavior observed as the field is applied. The application of current to the Maxwell pair gradient coils induces a magnetic field gradient across the metallic vessel. In accordance with Faraday’s law of electromagnetic induction [4],

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TABLE II REGIONS OF 2 AND 5% MAGNETIC FIELD NONLINEARITY

[16], the resultant electromotive force on the electrically conductive metallic vessel induces eddy currents such that they oppose the changing magnetic field. It is this opposition to the changing magnetic field that results in a nonlinear magnetic field resulting instantaneously after the magnetic field is applied. The severity of this nonlinearity at an instantaneous time is related to the conductivity of the metal, the metal vessel geometry, and the field incident on the metal vessel. The eddy currents follow a predominately single exponential decay [1], [4], [15] after initial gradient turn on. The differences in the magnetic field between 0.4 and 0.8 ms are small; the inclusion of magnetic fields beyond 0.8 ms are unnecessary and would not provide any additional information regarding the system dynamics. Note that this is the case for all following simulation results and will not be reiterated in subsequent sections. 2) Magnetic Field Nonlinearity: MRI requires the application of orthogonal magnetic fields which vary linearly along the z-component of the magnetic field [8]. The nonlinearity of a magnetic field gradient can be defined as Nonlinearity

gradient-gradient of best fit line gradient of best fit line

(2) The best fit line is based on the region within the metal vessel that is 2.0 cm in length (for -directed magnetic field gradients). The length of the regions with nonlinearities of less than 2% and 5% are summarized in Table II. The nonlinearity of the system at the quasi-steady state is a function of the gradient coils only. Note this applies to all of the gradient coil configurations and will not be reiterated in the following sections. 3) Spatial Distribution and Temporal Evolution of Eddy Currents—Nonplanar Maxwell Pair: The spatial distribution and temporal evolution of eddy currents are shown in Fig. 5. Note that similar data was presented in [4] in order to highlight the effects of eddy current diffusion [11], [12] and transient resistance [13], [14]. Nevertheless, the figures shown illustrate current diffusion effects; transient resistance effects are observed immediately following the application of the magnetic field gradient (shown in [4]). Note that the relative scales in Fig. 5(a), (b), and (c) were required to be adjusted as the amplitude of the eddy currents decays exponentially; comparison of the scales between Fig. 5(c) and (a) reveal that the scale in (c) has been reduced by a factor of 160. D. Nonplanar Magnetic Field Gradient—Golay Coils Transverse magnetic field gradients used in superconducting magnets are generated from nearly identical gradient coils positioned orthogonally to each other. The inherent symmetry of

Fig. 5. Spatial distribution of eddy currents on a centered metallic vessel at (a) 0.1 ms, (b) 0.2 ms, and (c) 0.8 ms after viewed in the -plane the application of a magnetic field from a nonplanar Maxwell pair gradient coil.

these transverse magnetic field gradients (i.e., - and -directed magnetic field gradients) and the metal vessel result in the requirement to only discuss the magnetic fields resulting from a single transverse gradient coil. The results for the orthogonal gradient coil will be identical due to superposition. However, an inherent asymmetry between the gradient coil geometry and the metal vessel now exists. The impact of this asymmetry will be investigated in the following sections. 1) Temporal Evolution of Magnetic Field: The following simulation results summarize the temporal evolution of the magnetic field within the metal vessel at discrete times of 0.1 ms, 0.2 ms, 0.3 ms, 0.4 ms, and 0.8 ms. The resulting magnetic fields for the centered vessel at the discrete times stated above are shown in Fig. 6. Recall that the inner diameter of the vessel is 5.1 cm (refer to Section II). Correspondingly, the application of the magnetic field gradient induces eddy currents within the conductive vessel walls that oppose the applied magnetic field; these

GOORA et al.: INVESTIGATION OF MAGNETIC FIELD GRADIENT WAVEFORMS IN THE PRESENCE OF A METALLIC VESSEL

Fig. 6. Temporal evolution of the magnetic field following the application of an -directed magnetic field gradient from a nonplanar Golay coil for a centered metallic vessel. Indicated times are those following the application of a magnetic field gradient.

TABLE III REGIONS OF 2 AND 5% MAGNETIC FIELD NONLINEARITY

effects are distinctly observed in Fig. 6. The resulting field distortions decay rapidly and are negligible after 0.8 ms. 2) Magnetic Field Nonlinearity—Nonplanar Golay Coils: The magnetic field nonlinearity of the transverse magnetic field gradient is calculated using (1). The best fit-line is based on a region within the metal vessel that is 2.0 cm in length (for and/or -directed magnetic field gradients). The length of the regions with nonlinearities of less than 2% and 5% are summarized in Table III. Note that the 2% nonlinearity region within the vessel at time 0.1 ms is greater than that at quasi-steady state. This result states that the presence of the metal vessel improves the linearity of the magnetic field gradient within the vessel. Nevertheless, the magnetic field gradient has not achieved the desired amplitude for imaging experiments at these early time intervals. Therefore, the amplitude of the required magnetic field gradient for the imaging experiment and that experienced by the sample must be carefully considered during experiments conducted with encoding times prior to achieving a quasi-steady state. 3) Spatial Distribution and Temporal Evolution of Eddy Currents—Nonplanar Golay Coils: The spatial distribution and temporal evolution of eddy currents resulting from a transverse magnetic field gradient is more difficult to depict graphically (as compared to Section IV-C3). Two views of the vessel are required for a given time in order to fully depict the spatial pattern of the eddy current. These two views of the vessel 0.1 ms after the magnetic field has been applied are shown in Fig. 7. As observed in [4], the observed eddy current patterns in Fig. 7 are similar to the current distribution on the adjacent Golay coil windings. The observed distribution is also very similar to the coil windings found in sophisticated gradient coils

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Fig. 7. (a) Side and (b) top view of the spatial distribution of current flow on a metallic vessel 0.1 ms after the application of a magnetic field from a nonplanar Golay coil.

which use distributed windings [6], [7]. The presence of mirror image symmetry is expected due to the mirror image symmetry present in the Golay coil geometry [4]. The induced current flow opposes the current flow in the gradient coil and is a direct result Faraday’s law of electromagnetic induction and Lenz’s law [4], [16]. Similarly, two planes must be selected in order to sufficiently depict the temporal evolution of the spatial distribution of eddy currents. One plane corresponds to the (or -plane) and the other plane corresponds to the (or -plane). The -plane permits the representation of eddy currents that flow in the -direction at the edges of the metal vessel in a cross-section of the vessel [as shown in Fig. 7(a)]. The -plane shows eddy currents that flow in the -direction at the edges of the metal vessel in an orthogonal cross-section of the vessel [as shown in Fig. 7(b)]. The spatial distribution of eddy currents in the slice of the -plane is shown in Fig. 8. This orientation shows the magnitude of the -component of eddy currents only. The -component of the eddy current is nil in this slice. A small -component of the eddy current is present which is the result of the circulation path of eddy currents on the cylindrical wall of the vessel. Note that the scales of each consecutive image are modified such that the magnitude of current illustrates the spatial distribution of the eddy currents flowing on the vessel at the selected time. Fig. 8 reveals that the spatial distribution of the -component of current does not change significantly as the eddy current amplitude decays exponentially. Note that diffusion effects occur and are most significant from 0.1 to 0.2 ms. The spatial distribution of eddy currents of the slice in the -plane is shown in Fig. 9. This figure shows the magnitude of the -component of eddy currents only; the x-component is nil in this slice. An -component of the eddy current is present which is the result of the circulation path of eddy currents on the cylindrical wall of the vessel. The spatial distribution of the -component of current does not change significantly as the eddy current amplitude decays

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Fig. 10. Magnetic field evolution following the application of a -directed magnetic field gradient for a centered metallic vessel from a planar Maxwell pair gradient coil. Indicated times are those following the application of a magnetic field gradient.

TABLE IV REGIONS OF 2 AND 5% MAGNETIC FIELD NONLINEARITY

Fig. 8. Spatial distribution of eddy currents on a centered metallic vessel (a) 0.1 ms and (b) 0.8 ms after the application viewed in the -plane of a magnetic field from a nonplanar Golay gradient coil.

Fig. 9. Spatial distribution of eddy currents on a centered metallic vessel -plane shown 0.1 ms after the application of a viewed in the magnetic field from a nonplanar Golay gradient coil.

exponentially. Diffusion effects are present and are most significant from 0.1 to 0.2 ms. Although the combination of nonplanar Golay coils and metallic vessels initially appear to be unfavorable for MRI applications, it is observed that the flow of eddy currents follow mirror image symmetry and their spatiotemporal properties do not vary significantly with time. E. Planar Magnetic Field Gradient—Maxwell Pair The following subsections outline the simulation results from a metal vessel applicable for use in permanent magnet-based

MRI application. MRI studies of rock cores are typically conducted at lower fields which often employ permanent magnets as the source of the static magnetic field. Correspondingly, the electromagnetic characterization of the metallic vessel response to planar magnetic field gradients is important. The planar Maxwell pair (Section III-B1) is the magnetic field gradient source and its geometry is shown in Fig. 1(a). The orientation of the metallic vessel is such that the static magnetic field is laterally incident to the vessel. The static field orientation is -directed as per convention. The vertical axis is specified to be -directed in this configuration and coordinate system. 1) Temporal Evolution of the Magnetic Field: The following simulation results summarize the temporal evolution of the magnetic field within the metal vessel at discrete times of 0.1, 0.2, 0.3, 0.4, and 0.8 ms. The resulting magnetic fields for the centered vessel at the discrete times stated previously are shown in Fig. 10. The inner diameter of the vessel is 5.1 cm (refer to Section II) so the region of interest of the magnetic field is 2.5 cm. 2) Magnetic Field Nonlinearity: The magnetic field nonlinearity of the transverse magnetic field gradient is calculated using (1). The best fit-line is based on a region within the metal vessel that is 1.5 cm in length. The length of the regions with nonlinearities of less than 2% and 5% are summarized in Table IV. 3) Spatial Distribution and Temporal Evolution of Eddy Currents: The front and side views of the spatial representation spatial distribution of eddy currents on the metallic vessel resulting from a planar Maxwell pair gradient coils are shown in Fig. 11.

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Fig. 11. (a) Front and (b) side view of the spatial representation of current flow on a metallic vessel 0.1 ms after the application of a magnetic field from a planar Maxwell pair gradient coil.

Fig. 12. Spatial distribution of eddy currents on a centered metallic vessel -plane (a) 0.1 ms and (b) 0.8 ms after the applicaviewed in the tion of a magnetic field from a planar Maxwell pair gradient coil.

Due to the symmetry of the system and the anti-symmetry with current flow in the individual coils of the Maxwell pair, anti-symmetric spatial patterns of eddy currents result. In order to characterize these patterns, the - and -planes are selected in order to characterize to the temporal evolution of the spatial distribution of the eddy currents. The spatial distribution of the temporal evolution of eddy currents in the -plane from 0.1 to 0.8 ms after the application of the magnetic field gradient is shown in Fig. 12.

Diffusion effects occur between times 0.1 and 0.2 ms and are not observed beyond 0.2 ms. Note that the scale of the eddy currents as shown has been adjusted such that the spatial representation of the eddy currents is clearly observed. The spatial distribution of the temporal evolution of eddy currents in the -plane is similar to that of the -plane. This is due to the symmetry of the metal vessel and the resulting circulation pattern of the eddy currents within the vessel.

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Fig. 13. Magnetic field evolution following the application of an -directed gramagnetic field gradient for a centered metallic vessel from a planar dient coil. Indicated times are those following the application of a magnetic field gradient.

TABLE V REGIONS OF 2 AND 5% MAGNETIC FIELD NONLINEARITY

F. Planar Magnetic Field Gradient—

Coils

coil) geometry is The planar magnetic field gradient coil ( shown in Fig. 1(b). 1) Temporal Evolution of Magnetic Field: The following simulation results summarize the temporal evolution of the magnetic field within the metal vessel at discrete times of 0.1, 0.2, 0.3, 0.4, and 0.8 ms after the application of the magnetic field gradient. The resulting magnetic fields for the centered vessel at the discrete times stated previously are shown in Fig. 13. The inner diameter of the vessel is 5.1 cm (refer to Section II) so the region of interest of the magnetic field is 2.5 cm. 2) Magnetic Field Nonlinearity: The magnetic field nonlinearity of the transverse magnetic field gradient is calculated using (1). The best fit-line is based on a region within the metal vessel that is 1.5 cm in length. The length of the regions with nonlinearities of less than 2% and 5% are summarized in Table V. 3) Spatial Distribution and Temporal Evolution of Eddy Currents—Planar Coils: The front and side views of the spatial distribution of eddy currents on the metal vessel resulting from a planar Golay pair gradient coil 0.1 ms after the application of a magnetic field gradient are shown in Fig. 14. coils and the anti-symmetry of curThe symmetry of the rent flow in the individual coils result in anti-symmetric spatial patterns of eddy currents. In order to characterize these patterns, the - and -planes are selected in order to characterize to the temporal evolution of the spatial distribution of the eddy currents.

Fig. 14. (a) Front and (b) side view of the spatial distribution of current flow on a metallic vessel 0.1 ms after the application of a magnetic field from a planar gradient coil.

The spatial distributions of the temporal evolution of eddy currents in the - and -plane 0.1 ms after the application of a magnetic field gradient are shown in Fig. 15. The symmetry between gradient coils and the positioning of the vessel result in negligible current diffusion following the application of the magnetic field gradient as the system reaches the quasi-steady state. Correspondingly, only the spatial distribution of eddy currents induced 0.1 ms after the application of the magnetic field gradient is shown. G. Planar Magnetic Field Gradient—

Coils

coil) geometry is The planar magnetic field gradient coil ( shown in Fig. 1(c). 1) Temporal Evolution of the Magnetic Field: The following simulation results summarize the temporal evolution of the magnetic field within the metal vessel at discrete times of 0.1,

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Fig. 16. Magnetic field evolution following the application of a -directed gradient magnetic field gradient for a centered metallic vessel from a planar coil. Indicated times are those following the application of a magnetic field gradient.

TABLE VI REGIONS OF 2 AND 5% MAGNETIC FIELD GRADIENT LINEARITY

Fig. 15. Spatial distribution of eddy currents on a centered metallic vessel shown at (a) 0.1 ms and (b) 0.8 ms following viewed in the -plane gradient coil. the application of a magnetic field from a planar

0.2, 0.3, 0.4, and 0.8 ms after the application of a magnetic field gradient. The resulting magnetic fields for the centered vessel at the discrete times stated previously are shown in Fig. 16. 2) Magnetic Field Nonlinearity: The magnetic field nonlinearity of the transverse magnetic field gradient is calculated using (1). The best fit-line is based on a region within the metal vessel that is 1.5 cm in length. The length of the regions with nonlinearities of less than 2% and 5% are summarized in Table VI. 3) Spatial Distribution and Temporal Evolution of Eddy Currents: The front and side view of the spatial distribution of eddy currents on the metallic vessel 0.1 ms after the application of magnetic field gradient from a planar gradient coil are shown in Fig. 17. Note that the orientation of the metallic vessel has been modified such that the static magnetic field is laterally incident to the vessel. Therefore, the static field orientation is main-

tained to be -directed as per convention. The vertical axis is specified to be -directed in this configuration and coordinate system. Again, due to the symmetry of the system and the anti-symmetry with current flow in the individual coils of the gradient coils, anti-symmetric spatial patterns of eddy currents result. In order to characterize these patterns, the - and -planes are selected in order to characterize to the temporal evolution of the spatial distribution of the eddy currents. The spatial distribution of the temporal evolution of eddy currents in the - and -plane is shown in Fig. 18. The symmetry between the gradient coils and the positioning of the vessel result in negligible current diffusion from 0.1 to 0.8 ms (quasisteady state). Correspondingly, only the spatial distribution of eddy currents induced 0.1 ms after the application of the magnetic field gradient is shown. H. Realistic Metal Vessel Design and Simulation Establishing that the basic properties of a metallic vessel exposed to a switched magnetic field could be accurately simulated within CST EM Studio has permitted more complicated vessel designs to be investigated through simulation. A detailed mechanical engineering analysis was performed prior to vessel construction to ensure the physical properties of the vessel. Modifications of the basic vessel design to ensure strength and accessibility to the interior of the vessel for RF and fluid connections would also affect the electrical performance of the vessel. A vessel meeting the required physical properties was designed and is reported on in [17]. This design was imported into CST and a cutaway view is shown in Fig. 19.

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Fig. 17. (a) Front and (b) side view of the spatial representation of current flow on a metallic vessel 0.1 ms after the application of a magnetic field from a planar gradient coils.

Fig. 18. Spatial distribution of eddy currents on a centered metallic vessel -plane and (b) -plane 0.1 ms after viewed in the (a) gradient coil. the application of a magnetic field from a planar

1) Temporal Evolution of Magnetic Field: The following simulation results summarize the temporal evolution of the magnetic field within the metal vessel at discrete times of 0.1, 0.2, 0.3, 0.4, and 0.8 ms. As in Section IV-C1, the steady-state magnetic field within the metallic vessel has effectively reached a steady state at 0.8 ms. The time steps represent discrete times following the turn on of the magnetic field gradient (refer to

Section IV-C1 for details). The resulting magnetic fields for the centered vessel at the discrete times stated previously are shown in Fig. 20. It is observed that the presence of the realistic metal vessel design does not result in a degradation of the switched magnetic field that a sample enclosed by the vessel would experience during MRI applications. The general eddy current behavior

GOORA et al.: INVESTIGATION OF MAGNETIC FIELD GRADIENT WAVEFORMS IN THE PRESENCE OF A METALLIC VESSEL

Fig. 19. Cut perspective view of a realistic metal pressure vessel/rock core holder imported into CST EM Studio. Note the hole locations required for RF connections, fluid input/output connections, reinforcing screw locations, as well as grooves for O-rings on the end caps.

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field nonlinearities of less than 2% exceeded 10.0 cm during the temporal evolution of the field. A prototype high-pressure rock core holder based on this design has been fabricated and direct measurement using the magnetic field gradient waveform monitor (MFGM) [1], [18], [19] method as well as imaging experiments [17] have verified that the electromagnet characteristics of the metal vessel based on the properties reported here are accurate. 2) Spatial Distribution and Temporal Evolution of Eddy Currents: The spatial distribution of currents induced on the metallic vessel 0.1 ms after the application of a magnetic field gradient is shown in Fig. 21. The increase in current density near the discontinuities within the vessel do not cause any significant impact on the magnetic field as experienced by the sample enclosed by the metal vessel. We also investigated the impact of having end caps both electrically connected and electrically isolated from the cylindrical metal vessel. We observed that the impact of having the end caps electrically connected or isolated resulted in a negligible impact on the electromagnetic properties of the vessel in MRI applications. As observed above, diffusion of eddy currents was observed to be minimal for the magnetic field gradient geometries used in both superconducting and permanent magnet-based systems. Correspondingly, electrically connected or isolated end caps will not cause a significant difference in the spatial pattern of eddy currents that are established following the application (and removal) of magnetic field gradients used in MRI applications. The temporal evolution of linear magnetic fields within the metal vessel result, which are desired for MRI applications. V. CONCLUSION

Fig. 20. Magnetic field evolution following the application of a -directed magnetic field gradient for a centered realistic metal vessel (rock core holder) from a nonplanar Maxwell pair gradient coil. Indicated times are those following the application of a magnetic field gradient.

Fig. 21. Spatial distribution of eddy currents on the metal pressure vessel 0.1 ms after the application of a magnetic field gradient induced from a nonplanar Maxwell pair gradient coil.

outlined in this paper has been observed and permits the prediction of the spatial distribution of eddy currents on the metal vessel as well as the temporal evolution of the magnetic field enclosed by the cylinder. The length of the regions with magnetic

Based on the conclusions reported in [4] along with the results reported in Section IV, we have characterized the electromagnetic response of a metal vessel used in a MRI application. The introduction of a vessel offset when positioned in a superconducting magnet bore will introduce a eddy current that causes a shift in the field as well as linear eddy currents that diffuse following the application of a magnetic field gradient. Correspondingly, the vessel should be inserted within the magnet bore with care to ensure that these effects are minimized. The amplitude of the applied magnetic field gradients was confirmed not to cause any difference in eddy current behavior and time constant associated with the rising and falling edge of the transient response. The nonlinearity of the temporal evolution of the magnetic field generated by nonplanar as well as planar magnetic field gradient coils was investigated. Fundamentally, the quasi-steady state nonlinearity of the system is governed by the nonlinearity of the gradient coils used (which is based on their geometry and design [6]–[9]). The nonlinearity of the magnetic field gradient is impacted by the presence of the metal vessel. However, the overall impact on the MRI experiment can be minimized through the selection of appropriate timings in the experiment. The properties investigated using a simple metal vessel design permitted the intuitive understanding and visualization of the spatiotemporal distribution of eddy currents and the resulting magnetic field impact. These concepts have been further

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 6, JUNE 2013

developed and observed through the simulation results from a realistic metal vessel [17]. These results, the properties reported in [4], and the measurements reported in [1] and [17] confirm that metal vessel MRI is a viable method for the investigation of material properties under high pressure and temperature in both superconducting and permanent magnet-based systems. Furthermore, a fundamental understanding of the electromagnetic response of the metal vessel to magnetic field gradients permits the development of magnetic field gradient compensation strategies (in addition to [20]–[30]) that are compatible with metallic vessels in superconducting and permanent magnet-based systems. ACKNOWLEDGMENT The work of B. J. Balcom was supported by a Research Chair in MRI of Materials (2009-2016) from the Canada Chairs Program and by a Discovery grant from NSERC of Canada. REFERENCES [1] H. Han and B. Balcom, “Magnetic resonance imaging inside metallic vessels,” Meas. Sci. Technol., vol. 21, 2010. [2] H. Han, D. Green, M. Ouellette, R. MacGregor, and B. Balcom, “NonCartesian sampled centric scan SPRITE imaging with magnetic field gradient and B0(t) field measurements for MRI in the vicinity of metal structures,” J. Magn. Reson., vol. 206, pp. 97–104, 2010. [3] H. Han, M. Ouellette, B. MacMillan, F. Goora, R. MacGregor, D. Green, and B. Balcom, “High pressure magnetic resonance imaging with metallic vessels,” J. Magn. Reson., vol. 213, no. 1, pp. 90–97, Dec. 2011. [4] F. Goora, H. Han, B. Colpitts, and B. Balcom, “Simulation and verification of magnetic field gradient waveforms in the presence of a metallic vessel in magnetic resonance imaging,” IEEE Trans. Magn., vol. 48, no. 9, pp. 2440–2448, Sep. 2012. [5] Armco, “Nitronic 60 stainless steel bar and wire UNS-S21800 product data bulletin,” Nitronic 60 Stainless Steel Data Sheet 1990. [6] R. Turner, “Gradient coil design: A review of methods,” Magn. Reson. Imag., vol. 11, pp. 903–920, 1993. [7] E. M. Haacke, R. W. Brown, M. R. Thompson, and R. Venkatasen, Magnetic Resonance Imaging Physical Principles and Sequence Design. Toronto, Canada: Wiley-Liss, 1999. [8] S. Hidalgo-Tobon, “Theory of gradient coil design methods for magnetic resonance imaging,” Concepts in Magn. Resonance Part A, vol. 36, no. 4, pp. 223–242, 2010. [9] Y. Xia, K. R. Jeffrey, and P. T. Callaghan, “Purpose-designed probes and their applications for dynamic NMR microscopy in an electromagnet,” Magn. Reson. Imaging, vol. 10, no. 3, pp. 411–426, 1992. [10] M. A. Bernstein, K. F. King, and X. J. Zhou, Handbook of MRI Pulse Sequences. New York, NY, USA: Elsevier, 2004. [11] K. W. Miller, “Diffusion of electric current into rods, tubes, and flat surfaces,” AIEE Trans., vol. 66, pp. 1496–1502, 1947.

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