A theoretical framework of gradient coil designed to mitigate eddy currents for a permanent magnet MRI system

1.Introduction

For modern medical application, superconducting magnetic resonance imaging (MRI) system has been a powerful medicine device, especially in noninvasive imaging modality aspect compared with computed tomography (CT) and x-ray technology [1]. Gradient coils, as a vital component of an MRI system, are usually designed to provide a linear and orthogonal gradient field along the three axes to spatially encode the information of tissue [2, 3]. Gradient coil that using passive shield technology cannot prevent eddy currents on the surrounding conductors when the current switching quickly, which may lead to serious image distortion [4, 5], energy loss [6], mechanical vibration [7], and eddy artifact [8]. Comparatively, active masking technology has a better image and lower noise caused by eddy currents than passive shielded coils, which is widely used in the MRI systems.

However, high installation and operating cost have limited applications for many circumstances. In practice, primary and shielding coils cannot insert into the magnet pole simultaneously owing to deficient workspace for the planar permanent MRI systems [9]. In order to reduce the eddy current and improve magnetic field quality to get a better image result, more and more coil design methods were discussed in MRI systems. Generally, there are two different kinds of design philosophy. Pre-emphasis is frequently used method, e.g., gradient waveform alternation by providing extra current to compensate for the second magnetic field affected by eddy currents. Alternatively, different types of shielding method also can be used for preventing eddy currents. For the actively shielded method, the current direction reverses in the primary coil and shielding coil layers so that so that the magnetic flux beyond the gradient coil can be offset largely [10]. Passively shielded method is often implemented outside of the gradient coils by applying a certain thickness metallic cylinder or plate [11]. In order to reduce the eddy current and improve magnetic field quality, high permeability materials is used to insert between the magnet and coil layers to enhance the low-frequency shielding performance. However, the hysteresis characteristics is difficult to eliminate so that magnetic field propagates through the multi-cylinder cryostat vessel in the low frequency. For example, Moon proposed a minimum inductance coil design method around the gradient coil [12]. Zu et al. presented analytical expressions for the magnetic field by multilayer dielectric plane and line current model [13].

In this paper, at theoretical gradient coil design approach using mirror-image theory was proposed for planar permanent MRI system. The proposed gradient coils considered the mirror-current around the magnet pole and resist-eddy current plate, which offered sufficient magnet space for patient and MRI system components, such as cooling devices. In the gradient coil design procedure, the power dissipation, stored energy, and field linearity were optimized to achieve the designed gradient fields. For a better comparison, a set of counterpart coil with same dimension and parameter using actively shielded method were simulated.

2.Method

In this section, a mirror image concept was adopted to design the planar gradient coil [14]. A resistive eddy current plate model was incorporated to replace the shielding coils and cause a limitation on eddy currents in the permanent magnet MRI device. The gradient coil optimization was achieved using a stream function-combined finite difference method.

Figure 1.

The planar MRI configurations showing limited space is available for the gradient assembly.

2.2Mirror-image theory combined finite-difference method

In a vacuum space, the magnetic field generated by a current source could be calculated following Biot-Savart law. However, the effects of the magnetization current should be considered if there is a ferromagnetic medium nearby [15]. A whole resist-eddy current plate (thickness: d) can be divided into three regions, two boundaries are produced in the process, namely interface 1 and interface 2 (see the Fig. 2). As illustrated in Fig. 2, magnetic medium (relative permeability μ1) occupy the upper half-space of the reference plane, while lower space of the reference plane (interface 2) is filled with another magnetic materials (relative permeability μ3), see the Fig. 2a. The coil plane carrying the current goes parallel to the interface, and create a gap about a distance of a. As described mirror-image theory, the magnetic field is produced by the original line current I and mirror current I′ in the upper half-space, in which I′=k⁢I, k denotes the reflection coefficients.

Figure 2.

Three-layer region for the resist-eddy current plate (below coil plane z=-za) and positions of mirror-image current considering the second order mirror image for coil plane (z=-za). (a) Three-medium model; (b)mirror currents at different reflection coefficients in region 1; (c)mirror currents at different reflection coefficients in region 3. Dot lines represent the mirror-image of the coil layer.

In this work, a pair of resist eddy current plate was added to prevent the eddy current in the magnet pole. The coil layers located at z=±z⁢a, are parallel to the surface of magnetic materials layers (see the Fig. 1) with permeability μ2. Taken the below coil plane (z=-za) for granted, the distance the coil layer and interface 1 was a (as shown in Fig. 2). The locations of the mirror image (dot line) in the region 1 were shown in Fig. 2b, and those of the region 3 were given in Fig. 2c. This study assumed that the reflection coefficients could be adjusted by the relative permeability [13]:

(1)

k1=μ1-μ2μ1+μ2

k2=μ3-μ2μ3+μ2

Based on the mirror-image theory and Ampere loop theorem under medium/boundary condition, the locations and amplitudes of the mirror current were calculated as follows [13]:

(2)

{z10=d2-aI10=-k1⁢Iz1⁢n=d2-a-2⁢n⁢dI1⁢n=(k1⁢k2)n⁢(1-k1)⁢(1+k1)k1⁢I

(3)

{z2⁢n=-3⁢d2+a+2⁢n⁢dI2⁢n=(k1⁢k2)n⁢(1-k2)⁢(1+k1)k1⁢k2

where k1 and k2 denote reflection coefficients; d is the thickness of the resistive eddy current plate; a is the distance between the interface 1 and the coil plane. Since the distance between upper coil plane and lower coil plane is 2za, the transformed mirror current coordinates position z1⁢n′, z10′ (for the upper resist eddy current plate) for the coil plane (z=-za) can be obtained as well. The proposed coil plane is placed on the planes at z=±za , and mirror current is located on the planes at z=z1⁢n. The current density is defined as [12]:

(4)

J→=(Jrza⁢e⇀r+Jθza⁢e⇀θ)⁢δ⁢(z-za)+(Jr-za⁢e⇀r+Jr-za⁢e⇀θ)⁢δ⁢(z+za)+(Jrz10⁢e→r+Jrz10⁢e⇀θ)⁢δ⁢(z-z10)+(Jr-z10⁢e→r+Jr-z10⁢e⇀θ)⁢δ⁢(z+z10)+∑n∞(Jrz1⁢n⁢e→r+Jrz1⁢n⁢e⇀θ)⁢δ⁢(z-z1⁢n)+∑n∞(Jr-z1⁢n⁢e→r+Jr-z1⁢n⁢e⇀θ)⁢δ⁢(z+z1⁢n)+(Jrz10′⁢e→r+Jrz10′⁢e⇀θ)⁢δ⁢(z-z10′)+(Jrz10′⁢e→r+Jrz10′⁢e⇀θ)⁢δ⁢(z+z10′)+∑n∞(Jrz1⁢n′⁢e→r+Jrz1⁢n′⁢e⇀θ)⁢δ⁢(z-z1⁢n′)+∑n∞(Jrz1⁢n′⁢e→r+Jrz1⁢n′⁢e⇀θ)⁢δ⁢(z+z1⁢n′)

where the superscripts ±za denote the coil plane at z=±za; ±z10, ±z1⁢n, ±z10′ and ±z1⁢n′ are the positions of the mirror-image current density, which can be calculated by the Eqs (1–3).

A finite difference method (FDM) was employed to improve the gradient coil design [16, 17]. In the continuity equation of current density, J→ can be expressed as J→=Jr⁢e→r+Jθ⁢e→θ for the planar coil design [18]. By defining the potential function ψ, and it has following relation to the current density [19]:

(5)

Jr=∂⁡ψr⁢∂⁡θ

Jθ=-∂⁡ψ∂⁡r

In the Fig. 3, the biplanar gradient coils discretization have been divided two directions, radial and circumferential directions respectively (see the Fig. 3a). For the biplanar coil surface the current density components can be approximated as follows:

(6)

Jr=∂⁡ψr⁢∂⁡θ≈12⁢(ψA+ψB-ψC-ψD)r⋅Δ⁢θ

Jθ=-∂⁡ψ∂⁡r≈12⁢(ψB+ψC-ψA-ψD)Δ⁢r

According to the Biot-Savart law, the z-component of magnetic field in DSV region is defined as [20]:

(7)

Bz⁢(x,y,z)=μ04⁢π⁢∫V(Jr⁢s-Jθ⁢q)⁢r⁢d⁢r⁢d⁢θ⁢d⁢z(s2+q2+(z-z′)2)1.5

where μ0 denotes the magnetic permeability of free space; Jr and Jθ are current density components, respectively; s and q are the intermediate terms expressed in the following equations, respectively:

(8)

s=-x⁢sin⁡θ+y⁢cos⁡θ

(9)

q=x⁢cos⁡θ+y⁢sin⁡θ-r

Assuming that the contour level that is equally spaced is set as Nc, then we obtain ψ=ψmin+(i-0.5)⁢I, in which I=(ψmax-ψmin)/Nc. ψmin, ψmax shows the minimal and maximal stream function on the coil layer respectively. Once the stream function satisfy the Eq. (5), the stream function contours on the coil plane could reveal the coil patterns of optimized gradient coil [18].

Figure 3.

Biplanar coil surface’s discretization using FED mesh. (a) radial and circumferential components of current density, (b) the planar gradient coil mesh.

3.Results and discussion

In all different kinds of MRI, magnetic-field component is required to vary linearly along three directions in the z axis. Its linearity is a crucial indicator of the gradient coil quality defined as:

where Bz is the desired gradient strength. Therefore, for a linear gradient field in the simulation process, the gradient strength Gx, Gy, Gz should be uniform along each axis. To illustrate the basic theoretical approach, in this paper, x-axis gradient and z-axis gradient coil were considered separately. The weighting and parameters were significant for design target field, in the design simulation, the weighting factor λ1 and λ2) were 0.5, gradient field is 24 mT/s over DSV (diameter 0.4, 762 target points) srface, the maximum magnetic field error was 4%, and the maximum stray field intensity on the shielding region was 5 Gauss.

Figure 4.

Designed passively shielded and actively shielded x-axis coil winding: (a) and (b) are the plane views of different coils (μr= 400,1616) without considering pole piece effect, respectively; (c) and (d) are the plane views of different coils (μr= 400,1616) considering pole piece effect, respectively; (e) and (f) are the primary, shielding layers for x coil, respectively; (g) is the 3D coil winding of the actively shielded coil. The arrows indicates the currents direction.

Figure 4 shows the coil winding patterns with same function contour of x-transverse coil. To demonstrate the coil configuration clarify, different kinds of coil patterns were plotted, where Fig. 4a and b presents the coil patterns generated using mirror-image theory with different materials μr= 400 (Case I-1) and μr= 1616 (Case I-2), and Fig. 4c and d show coil patterns considering the pole-piece effect (as depicted in Case II-1 and Case II-2). Comparatively, an actively shielded coil design pattern was depicted in in Fig. 4e and f. Figure 4g presents the 3D geometry for the designed gradient coil. The actively-shielded x gradient primary (11 × 4 loops) and shielding coils (8 × 4 loops) had a radius of 0.43 m, 0.45 m, with z-coordinates of ± 0.25 m, ± 0.28 m, respectively. In this paper, the current directions in the primary and shielding coils were opposite in achieving the aim of the shielding. The proposed x-gradient coil performance using mirror-image theory design method and conventional unshielded/shielded coil are depicted in Table 2. During the coil designing stage, the figures of merit (FoM) η2/L, and η2/R were used to assess the coil performance; the minimum wire spacing of both coils were strictly controlled for the real engineering fabrication. The inductance and resistance of using mirror current theory designed coil have less than active shielding coil, the obvious drawback for active shielding coil is that it has higher operating currents and lower efficiency.

Figure 5.

Magnetic field distribution for the shielded and unshielded x-axis coils: (a) and (b) are the passively shielded coils (μr= 400) considering pole piece effect on the different plane (y= 0, z= 0); (c) and (d) are the passively shielded coils (μr= 1616) considering pole effect on the different cutting plane (y= 0, z= 0); (e) and (f ) are the unshielded coils on the different plane (y= 0, z= 0); (g) and (h) are the actively shielded coils on the different plane (y= 0, z= 0).

Figure 6.

A comparison of the magnetic field : (a) and (b) are the magnetic field for proposed coils (μr= 400,1616) without considering pole piece effect, respectively; (c) and (d) are the magnetic field for proposed coils (μr= 400,1616) considering pole piece effect, respectively; (e) and (f) are the magnetic field for unshielded coils with/without pole piece effect, respectively; (g) is the magnetic field for actively shielded coils.

Figure 7.

A comparison of z-coil winding: (a) is the plane view of designed coils with permeability μr= 440; (b) is coils with permeability μr= 1616; (c) is upper plane view for unshielded coil; (d) is the 3D coil winding patterns.

Figure 8.

Magnetic field distribution for the proposed z-coils using image current theory and unshielded z-coils; (a), (c) and (e) are the magnetic field distributions for the designed coils, μr= 440 (without considering pole piece effect), designed coils, μr= 440 (considering pole piece effect), unshielded coil, respectively; (b), (d) and (f) are the stray field related to (a), (c) and (e) on the sampling point, respectively.

The magnetic field error and linearity produced by the different cases (as depicted in Table 2) are listed in Fig. 5. The magnetic field distribution of both the shielded and unshielded coils in DSV region present a similar linear variation for the x axis because the same target field and dimensions constraints (magnetic field error: < 4%, depicted in red line, calculated by Eq. (2)). Figure 5a and b are the magnetic field distributions for passively shielded coils (μr= 40) considering pole piece effect on the different plane (y= 0, z= 0) over the DSV, respectively; Fig. 5c and d are the mangetic field distributions for the passively shielded coils (μr= 1616) considering pole effect on the different plane (y= 0, z= 0) over the DSV, respectively; moreover the magnetic field distributions of the actively shielded gradient coil were illustrated in Fig. 5g and h. By comparing Fig. 5a–d and e–f, the magnetic field distribution appears to have similar pattern owing to the same design parameters.

The magnetic field distributions for the shielding plate are illustrated in Fig. 6, however, the stray field distributions are quite different. For comparison, the magnetic field distributions on the sampling plane of unshielded and actively shielded coils are plotted, as depicted in Fig. 6e–g. The leaking magnetic field distributions of the coil patterns (μr= 400, 1616) considering the pole piece effect (Fig. 6c and d) are lower than that of unshielded coil patterns (Fig. 6f) owing to the resist-eddy current plate exists. Though the active x gradient coil had the same shielding effects as those of the proposed coil using image current theory (as shown in Fig. 6g), the operating currents, resistance and inductance were much higher than those of the proposed coils. The apparent advantage of the proposed coil in that it exhibits excellent magnetic field performance, e.g., the figure of merit. Moreover, the coil designed using higher permeability of the resist eddy plate (Fig. 6b and d) produced much lower stray field than that generated by coils designed with lower permeability (Fig. 6a and c).

Figure 7 displays the proposed coils for the different cases and unshielded z-gradient coil patterns. The unshielded coil plane (radius 0.43 m) was located on z=± 0.25 m, others design parameters were consistent with those of the x coils. Figure 7a and b show different design parameters coil winding patterns. The 3D coil winding patterns was plot in Fig. 7d, moreover, the upper plane view is illustrated in Fig. 7c. The magnetic field distribution for the proposed coil using the mirror-image theory is illustrated in Fig. 8. In Fig. 8a and c, the magnetic fields also present linear variation along the z-axis, in which magnetic gradient field error are less than 4% in all cases. The stray fields demonstrate the concentric circles format, as depicted in Fig. 8b and d. Owing to the stray field control parameters was strictly constrained in passively shielded coil by using mirror image theory, the magnetic leakage intensity is obviously lower than that of the unshielded magnetic leakage intensity. Comparatively, in the same simulation procedure and control parameters, the magnetic field distributions of using mirror current theory proposed coil considering pole effects have good shielding performance than that of proposed coil without considering pole effects and unshielded coil (as depieced in Fig. 8). The inductance and resistance designed by using mirror-image theory considering pole effects (μ= 440) in the z coil were 329.96 mΩ, 541.96 μH, respectively, which is less than that of unshielded coil (353.47 mΩ, 566.73 μH).

4.Conclusion

In this paper, a novel planar gradient coil design theoretical framework was proposed, which is beneficial to the development of permanent MRI device systems. The designed coil set can be used for planar MRI system by using resist-eddy current plate to eliminate the eddy currents, and combined the finite difference with mirror-image theory into the gradient coils design process, without using the actively shielded that usually requiring high currents. According to the results of the numerical calculation, for the x-gradient coil, the operating currents in the cases were reduced by 34.4% by replacing the active shielding using mirror-image theory, while a performance metric for the properties was maintained in this study. The performance of the coil will be re-verified evaluation in vivo experiments.