Field‐Based Spatial Self‐Registration of Multicoil Hardware for B₀ Field Control

ABSTRACT

Purpose

To develop and validate a magnetic field‐based method for self‐registration of multicoil (MC) shim hardware in MRI systems, enabling accurate B₀ field control despite nonreproducible hardware placement between calibration and experimental sessions.

Methods

An algorithm was designed to estimate rigid transformations between 3D field maps acquired during hardware calibration (Reference field) and subsequent experiments (Measured field). Hardware misplacement was determined by co‐registration of the two fields within a defined region of interest. Simulations using Biot–Savart–derived MC basis fields and experimental validation with a 48‐channel MC array at 3 T were used to quantify localization accuracy under varying signal‐to‐noise ratio (SNR) levels. A genetic algorithm optimized coil currents to generate a Reference field shape that yielded minimal self‐registration error.

Results

Simulations and measurements demonstrated that submillimeter (≤ 0.5 mm) and subdegree (≤ 0.5°) localization accuracy is achievable at practical SNR levels (≥ 5). Errors decreased with increasing SNR and field complexity. The optimized field produced by the genetic algorithm yielded the lowest mean translation error (0.20 mm) and rotation error (0.11°), consistent with theoretical predictions. Self‐registration computation required less than 10 s per case.

Conclusion

The proposed field‐based self‐registration method enables rapid, accurate localization of MC shim hardware using only MR‐acquired field maps, eliminating the need for external tracking hardware or repeated calibrations. This approach enhances reproducibility of MC field control and supports improved B₀ shimming performance in insert‐based or repositionable MR systems.

1. Introduction

Magnetic field homogeneity is a prerequisite for high quality magnetic resonance imaging (MRI) and spectroscopy (MRS) [1]. B₀ shimming—the process by which the main magnetic field in MR is homogenized—is particularly important in regions in which magnetic susceptibility differences between air and tissue create large field inhomogeneities [1, 2, 3]. B₀ shimming can be passive, using strategically placed magnetizable materials to cancel inhomogeneities, or active, using dedicated electrically driven shim coils to generate counter‐fields that offset the inhomogeneities [4]. Active shimming is typically accomplished using shim coils with magnetic field shapes that resemble the spherical harmonic (SH) functions, with one shim coil corresponding to one orthogonal SH shape. Although any 3D field can be modeled by a set of such SH functions, exact reconstruction of arbitrary fields would require an infinite number of coils [5, 6]. Practical concerns limit shim systems to low‐order SH shapes which struggle to correct localized distortions such as in the prefrontal cortex of the brain [7].

Multicoil (MC) shimming, in which individually driven, generic DC coils form a nonorthogonal basis for field modeling [7, 8], can improve upon available SH shimming in the rodent and human brain [9, 10, 11], especially when applied dynamically. A practical limitation of many MC implementations, however, is that most are designed as inserts for existing scanners [10, 12, 13, 14, 15, 16, 17, 18, 19] and therefore operate independently of fixed hardware such as the gradient coils that define the scanner's coordinate system. This independence can create integration challenges for MC systems—or for any shim system implemented as a scanner insert.

The shimming process requires a set of known basis functions, that is, a repertoire of coil‐specific field shapes that are produced per unit current. Shim challenges can, thus, be decomposed into weighted sums of these basis maps, with the weights representing the current through each shim coil. Basis maps can be obtained by a calibration in which a series of field maps are taken at varying shim settings. For each coil, the field in each pixel is calculated through linear correlation analysis of the observed field change versus applied current. High‐quality calibration of an MC shim system is lengthy due to a significant number of MC elements (typically 24–48) and the acquisition of field maps at various current settings (typically 6–8 settings) for every coil. The multiple calibration settings per coil are used to improve the accuracy of the voxel‐wise linear regression used to estimate basis fields. This procedure enables the robust prediction of the expected shim fields at any reasonable current setting [4, 10]. In principle, the calibration process needs to be performed only once, and the resulting calibration maps can then be used for all subsequent shimming experiments.

Because of the calibration procedure, optimal B₀ field control relies on knowledge of the hardware placement. If the hardware is physically placed in a different location from calibration, shim fields will be applied in the wrong place. At best, this reduces shim performance and at worst this can exacerbate inhomogeneities (Figure 1). Even misplacements as small as 1 mm can seriously degrade shim performance (Figures 2 and S1). Accurate positioning is inherent in coil systems that are permanently attached to the MR scanner, such as conventional SH systems, for which conditions can be expected to be reproducible between calibration and application. Coil inserts, however, are removed and repositioned between sessions [10, 12, 13, 14, 15, 21, 22]. Perfectly reproducible hardware placement of both SH and MC inserts is often impossible, resulting in inconsistencies between calibration and application. Re‐calibration of the hardware after each replacement would require several hours of scanner time, if done thoroughly, for acquisition and processing of the full set of basis maps and is therefore not a practical alternative for routine use. Marker‐based registration approaches could in principle be considered [10, 23]; however, they would require additional acquisition and co‐registration steps and would introduce permanent image‐visible structures, potentially necessitating larger fields of view and imposing constraints on imaging protocols.

FIGURE 1.

Exemplary B₀ field analysis in a human brain after dynamic MC shimming with optimal and suboptimal placement of established MC hardware [10]. Representative slices are shown at multiple superior–inferior positions spanning regions of both high (e.g., prefrontal cortex) and relatively low intrinsic field variation. The bulk of the B₀ field imperfections in the human brain at 7 T (first row) can be mitigated with DYNAMITE B₀ shimming (second row), with shim currents determined according to the constrained optimization described in Ref. [10] (i.e., not a free fit). Selected shifts and rotations of the hardware result in suboptimal shimming, including field asymmetries and large hotspots. Effects are more extreme in areas of intrinsic field inhomogeneity but can be seen throughout the brain.

FIGURE 2.

(A) Illustration of a hardware z‐shift. (B) Analysis of residual fields as a function of mis‐localization along the z‐axis (foot‐to‐head) of the established MC hardware [10] shows doubling of whole‐brain residuals after dynamic shimming at just 0.5 mm misplacement. In practice, translational and rotational errors can happen concomitantly, amplifying each other. Field degradations of this magnitude have been shown in prior work [7, 20] to be associated with pronounced imaging artifacts, including signal dropout and EPI distortion.

The positioning of an MC insert overall may differ from the calibration scenario due to mechanical tolerances, technical limits of the patient bed control, and user error. These positioning differences can be reduced by reasonable engineering measures but cannot be eliminated altogether. The position of each MC element relative to the other elements, however, remains unchanged [9, 10]. The individual basis field shapes and the relative constellation of coils can thus be assumed to remain constant. Knowledge of the spatial positioning of the overall MC hardware within the magnet, particularly relative to the calibration experiments, enables proactive consideration of such displacements in the shimming analysis. A technique is therefore needed to quantify differences in position between calibration and the experiment at hand.

In this paper, we systematically assess the relevance of hardware positioning for MC field control and show that even small spatial discrepancies have the potential to significantly impair the quality of MC shimming. We introduce a novel method for field map‐based hardware self‐registration in which a MC‐generated field map from misplaced hardware is co‐registered with the equivalent combination of basis to reliably detect hardware misplacements. This method can be applied each time the hardware is placed in the scanner, finding a 3D rigid transformation that represents the difference between the hardware at that moment compared with the hardware during calibration. The approach is purely MR‐based, without the requirement of additional hardware. Parts of this research have been previously presented in abstract form [24].

2. Methods

2.1. Hardware Self‐Registration With a Theoretical 48‐Channel MC Test Setup

This hardware localization method is inspired by brain registration in neuroanatomical atlas generation or longitudinal neuroscientific studies [25]. We can consider two B₀ field maps: one of a field pattern generated by one or more coils of a MC array, and the other acquired in the same scanner with the same shim settings but an overall difference in MC hardware position. The co‐registration of these two field maps in an overlapping region yields a rigid transformation that resembles the difference in hardware positioning between the two acquisitions.

A MATLAB‐based hardware self‐registration algorithm was developed to determine the underlying hardware misplacement based on the co‐registration of two 3D field maps (Figure 3): a Reference field, which is a field map acquired at the Day 0 (i.e., calibration) hardware position, and a Measured field, which is the same field shape measured at Day N (i.e., the day of the experiment at hand). The Reference field is mathematically created by a combination of the calibration maps, and the Measured field is measured after the hardware is replaced in the scanner before an experiment or at the beginning of a study protocol. This can be done in a phantom provided the hardware is not removed and replaced before the subject enters but is ideally performed in an anatomical region of the subject such as the brain. A spherical volume is used in this illustration and in most of the work presented, but the method is ROI‐agnostic and any other volumes of reasonable size can be used.

FIGURE 3.

Schematic of the self‐registration procedure for hardware localization. A Reference field synthesized from calibration maps at Day 0 is compared to a Measured field acquired at Day N after hardware placement to derive differences in position. Because Day 0 maps are derived from thorough calibration, they can be considered nearly noise‐free, whereas Day N maps contain measurement noise. The self‐registration process must therefore be robust to noise to ensure accurate localization.

A nonlinear least squares regression compares the two field maps to find the transformation that best aligns the two fields. This transformation is a measure of the distance between the hardware on Day 0 and Day N and can then be factored into the subsequent shimming with rigid transformation of the basis field shapes to assemble B₀ shim geometries on Day N. The self‐registration algorithm in its current form outputs the x, y, and z‐translation and z‐rotation parameters that align the two fields, and shim fields can be proactively updated. Notably, x‐ and y‐rotations are not considered in this first proof‐of‐principle work (see Discussion). Full implementation details are provided in Algorithms A1, A2.

To generate simulated basis maps with which to test the method, an MC B₀ test and validation setup with six rows of eight coils was designed for fabrication and integration with a 3 T clinical scanner. A simulated basis set (i.e., theoretical calibration maps for the proposed design) was generated with Biot‐Savart simulations of the hardware design in custom MATLAB (Mathworks Inc., Natick, MA) software [26]. The basis maps were simulated at 3.125 × 3.125 × 3 mm [3] resolution and were weighted by shim setting and summed to mathematically create the Reference fields. Because the alignment must be compared in a region contained in both the original and shifted hardware, the regression compares the fields in a 10‐cm sphere region of interest (ROI) to ensure overlap between the two measurements. In practice, purely morphological erosion of the ROI is also feasible, for instance, when considering a human brain.

Simulations were used both to validate and improve the co‐registration approach. Simulated basis maps were combined to create a Reference field, and hardware misplacement was represented by rigid transformations of this field. Monte Carlo simulations of 5000 rigid transformations of the field were performed and recovered using the self‐registration algorithm. Transformations were created with four randomly generated values representing x‐, y‐, and z‐translation and z‐rotation, and mathematically applied to the Reference field. Together, these rigid transformations represent a placement of the hardware some distance from the original calibration position. To tailor to clinically realistic scenarios, each translation value was limited to ±10 mm and z‐rotation was limited to 10 degrees. As a noise‐free field is unrealistic in practice, Gaussian noise was added to the transformed fields prior to the self‐registration recovery of transformation parameters. Given the tradeoff between SNR, achievable spatial resolution, and scan time, multiple SNR levels were investigated to determine SNR thresholds for acceptable self‐registration accuracy. Reasonable SNR and spatial resolution thresholds can be set to determine the minimum field map duration needed for the self‐registration process as higher SNR can be achieved by things such as multiple averages, which increase scan time. Simulations were performed for 10 different noise levels. SNR was calculated as the maximum value of the field map amplitude divided by the standard deviation of the noise (note that this is the SNR of a magnetic field distribution instead of a grayscale MRI). The Euclidian norm of translation error for each transformation was calculated, and norm error was averaged for each SNR level to create an overall translation error. Z‐rotation errors were also averaged at each SNR level. For the most promising field shapes, two sets of error curves were generated: one with the Reference field generated by the simulated basis set and one with the Reference field generated by the basis set from the hardware calibration.

Finally, additional Monte Carlo simulations of only the simulated basis set were performed to assess other factors contributing to self‐registration accuracy. The effect of field map resolution was examined by performing self‐registration on maps with coarser (5 mm) and finer (1 mm) isotropic voxels. To illustrate the importance of a spatially unique Reference field, self‐registration simulations were performed using an intentionally nonunique Reference field, namely a pure X‐gradient, to demonstrate the limitations of nonunique field patterns.

2.2. Theoretical Field Shape Selection and Optimization

To determine translation and z‐rotation parameters accurately, the Reference field must be unique in all directions. Performance of Reference field shapes was assessed empirically by evaluating transformation recovery accuracy across extensive Monte Carlo simulations, and field shapes were selected based on their resulting registration performance. Initially, the field from only one MC element (Single coil field shape) and from only one ring of eight coils (Four‐lobe field shape) were tested as Reference fields. The Four‐lobe field was designed heuristically to provide a simple, reproducible spatial pattern expected to yield good self‐registration performance. To maximize accuracy, an optimization of the current in all 48 coils was undertaken. Due to the large search space, a genetic algorithm (GA), an optimization technique inspired by the principles of natural selection and genetics, was selected as the optimization method. A GA was chosen because it enables direct optimization of registration performance without requiring assumptions about the structure of the objective function. The fitness metric in this case is based on Monte Carlo simulations of the full self‐registration procedure and is therefore computationally expensive, noisy, and highly nonconvex. For this type of objective function, the GA provides a practical approach for exploring the large parameter space. In principle, alternative nonlinear optimization methods could also be used; however, the GA was selected as a practical compromise between robustness to noise, global search capability, and implementation simplicity. GAs begin with an initial population and use operators to categorize and select the “fittest” individuals to bring to the next generation, as well as crossover operators to create offspring from highly rated solutions, and mutation operators to induce random changes [27]. The fitness function, which evaluates the quality of the potential solutions and is minimized in this optimization, consisted of 500 random rigid transformations. The number of transformations was chosen to ensure sufficiently extensive testing within reasonable computation limits.

The GA (Figure 4A) was used to find a 48‐element vector of shim settings that creates a Reference field with the lowest possible error of transformation recovery. The solution emerged through iterative evolution of an initial random population; while global optimality is not guaranteed, convergence was assessed by examining the similarity among high‐performing solutions. The GA Optimization field resulting from this GA and the Four‐lobe field were chosen as the two examples to validate the method in a scanner environment.

FIGURE 4.

Genetic algorithm (GA) architecture and results. (A) Block diagram of the genetic algorithm used to find a 48‐element vector of coil‐specific currents that creates the most robust Reference field. The GA used a random initial population of 200 initial conditions constrained between −0.5 and 0.5 A, representing the current through the corresponding coils. Stochastic uniform selection, Gaussian mutation, and scattered crossover with crossover fraction of 0.8 were used to refine the population in each subsequent generation. The function tolerance was 10⁻⁶, with an additional stopping criteria of 50 generations passed without improvement. (B) Results from Generation 1 to convergence at Generation 511. Three other top‐performing results are shown in the three rightmost columns. All successful solutions had an asymmetric bi‐lobe distribution with large gradients in the center, and the consistency of these independently evolved high‐performing solutions suggests convergence toward a near‐optimal Reference field configuration.

2.3. Experimental Validation

To enable comparison with simulations, the previously described 48‐channel MC setup was physically constructed for integration with a 3 T Siemens Prisma (Siemens Healthineers, Erlangen, Germany) and an existing TX/RX RF birdcage coil (Quality Electrodynamics, Mayfield, OH) (Figure 5). Although previous experiments showed that placing shim coils inside the RF coil resulted in some sensitivity loss that is unacceptable for imaging [10], this was accepted here for design simplicity in this proof‐of‐concept work, and the array was used inside the RF coil.

FIGURE 5.

CAD representation (A) and realization (B) of a 48‐channel MC hardware for B₀ field control, designed for integration with a Siemens Prisma 3 T scanner and Quality Electrodynamics TX/RX CP Head Coil in receive‐only mode. Plugs at the head end of the coil (C) connect to a custom amplifier system. The coils were arranged in six rows of eight elements with 100 turns each, center diameter 70 mm, resistance 718 mΩ, and inductance 880 μH. The coils were wound with 20‐gauge copper wire and roughly 1 cm thick. The array was driven by a multichannel amplifier system (Resonance Research Inc., Billerica, MA) with ±5A available per channel and a custom controller [28]. Current changes were signaled by transistor‐transistor logic (TTL) triggers integrated into pulse sequences.

The MC array was calibrated by a set of field maps. For each coil, six field maps were acquired, each at a different shim setting ranging ±5% of the dynamic range of an amplifier channel (absolute dynamic range: ±5 A). Each field map was acquired with a gradient‐echo sequence with six echo times (1.91, 2.11, 2.51, 3.21, 4.01, and 4.91 ms) for high‐quality field mapping at a 64 × 58 × 68 matrix size close to isotropic resolution (3.125 × 3.125 × 3 mm) [3]. The field maps were processed individually and combined to create one calibration map per coil, resulting in the calibrated basis set to be used for shimming. To compare simulation and measured fields, the calibrated basis maps were then used for additional Monte Carlo simulations of Four‐lobe and GA Optimization fields. For clarity, Table 1 summarizes the different data types used in this work, distinguishing between simulated, calibrated, and measured Reference field maps (Day 0) and corresponding transformations or field maps acquired on Day N.

TABLE 1.

Characteristics of simulated, calibrated, and measured data labels used in presentation of self‐registration analysis.

Data type	Reference field maps (Day 0)	Transformation (Day N)
Simulated	Generated from fully simulated basis fields of the hardware	Obtained by applying simulated rigid transformations to the simulated field maps
Calibrated	Generated from experimentally calibrated basis fields	Obtained by applying simulated rigid transformations to the calibrated field maps
Measured	Generated from experimentally calibrated basis fields	Acquired in the scanner after physically repositioning the hardware

The errors of the self‐registration procedure as a function of SNR were tested in the scanner environment by physically moving the MC hardware a well‐defined distance, acquiring field maps, and comparing them to the Reference field using the self‐registration algorithm. The shifts derived by the self‐registration algorithm were then compared with the ground truth (i.e., the known hardware shift) to assess accuracy. The x‐ and y‐ transformations were accomplished by shifting the hardware itself, for example, by adding 3D printed shims at the base to lift the hardware a known amount. To increase measurement precision of the hardware position, caliper measurements (with an estimated experimental precision of 0.1 mm) of distance from the hardware to a reference point on the bed were taken before and after shifting the hardware, and these measurements were used as the ground truth measurements. Spatial z‐transformations were acquired by moving the patient bed of the scanner, with caliper measurements of the bed position (rather than the console reading) serving as the ground truth. Due to the rigid construction of the hardware, isolated z‐rotations were not easily possible. Instead, z‐rotations were achieved by inserting lifts on only one side of the hardware at the expense of inducing minor concomitant x‐ and y‐ transformations. The exact translation and rotation components were estimated with basic trigonometry based on the known dimensions of the hardware geometry.

To validate the relationship between algorithm errors and SNR of the field maps, field maps for one transformation were acquired at seven different SNR levels and accuracy was compared. Because SNR in field maps is not uniquely defined, we define SNR here as the ratio of the absolute maximum field value to the standard deviation of the noise. To systematically vary SNR for curve‐fitting and performance evaluation, we reduced the amplitude of the field patterns, thereby decreasing the signal component rather than artificially increasing noise. The resulting field maps were then scaled to a common signal range, such that differences in SNR were driven solely by changes in noise level (representative examples of field maps at two different SNR levels are shown in Figure 6). Additionally, 10 different transformations at SNR 20 were tested for each Reference field and the self‐registration results were compared to the known transformation values. Measurements were taken in a tissue‐mimicking phantom [29]. A background field map was taken prior to the application of MC fields and was subtracted from subsequent MC field maps to achieve maps of purely MC‐generated fields.

FIGURE 6.

Performance of field‐based self‐registration algorithm as a function of SNR levels for simulated (blue) and calibrated (red) basis fields. The black crosses represent self‐registration error of individual measured field maps at the corresponding SNR levels. Error was lower at higher SNR and decreased with field shape complexity. The dashed black lines represent the cutoff of 1 and 0.5 mm error. Calibrated and measured results closely matched theoretical predictions for all shapes at all SNR levels, and achievable errors with the GA approach were within fractions of a millimeter and degree. The signal in SNR was calculated as the maximum field value in the ROI in all cases. Noise was calculated as the standard deviation of the residual after subtracting spherical harmonic terms up to 10th order (i.e., removing all major field shapes to leave only noise) for measured field maps and added as gaussian noise in simulated and calibrated field maps. In measured field maps, SNR was degraded by reducing amplitudes of applied field (i.e., reducing “signal” rather than adding noise) and rescaling, which preserved realistic spatial noise characteristics. Very low SNR values were included to probe failure modes and derive a functional dependence, rather than to reflect typical experimental operating conditions. Translation error was calculated using the Euclidian norm error of x‐ y‐ and z‐error. Note the differing y‐axes for the Single coil results.

3. Results

3.1. Field Shape Optimization With a Genetic Algorithm

The “fittest” results after selected generations of the GA are shown in Figure 4B. As the generations progressed, the algorithm selected solutions that had strong fields in the center of the ROI while maintaining asymmetry. The best solution, along with next‐best solutions before selection of the final shape, shared an asymmetrical bi‐lobe field distribution in multiple dimensions. The algorithm converged in 511 generations, after the best fitness value had not improved for 50 generations. To quantify the spatial encoding strength of the Reference fields, we analyzed their spatial gradient magnitudes in a post hoc comparison. The GA‐optimized field exhibited a substantially higher mean gradient magnitude (127.0 Hz/cm) compared with the Four‐lobe (56.0 Hz/cm) and Single‐coil (17.7 Hz/cm) fields, consistent with its improved registration performance (Figure S2).

3.2. Simulated Validation of Hardware Self‐Registration Approach

As expected, the average errors of the Monte Carlo simulations using the simulated basis set decreased with increasing SNR for all tested field shapes (Figure 6). The Single coil field had the highest error at every SNR level, with only SNR levels above 35.9 having translation error below 1 mm. At SNR levels 21.5 and above, rotation error was within 1 degree. The Four‐lobe field had an average translation error below 1 mm for SNR levels above 4.6 and errors below 0.5 mm for SNR levels above 12.9. The GA Optimization field had lower translation error: below 1 mm for SNR levels above 2.8 and 0.5 mm for levels above 7.7. For both the GA Optimization and Four‐lobe fields, the rotation error was below 1 degree for SNR levels above 2.8. The standard deviation of the errors, represented by the error bars, also decreased with increasing SNR. In the Single coil case, the standard deviation of the error was higher at every SNR level. Smaller ranges of standard deviation were observed in the other two cases, with the GA Optimization having the lowest standard deviation at each SNR level. Simulated self‐registration at larger voxel size showed the same relationship of decreasing error with increasing SNR for all Reference fields, but the X gradient field showed high error at every SNR level (Figure 7).

FIGURE 7.

Monte Carlo simulations measuring impact of Reference field and voxel size on self‐registration accuracy. Errors increase with voxel size, while the same overall trend of decreasing error with increasing SNR is preserved (A). Results in this figure are shown for 1 and 5 mm isotropic voxel sizes; corresponding results for 3 mm isotropic voxels are shown in Figure 6. To simulate coarser and finer resolution, Biot‐Savart simulations for the hardware were re‐run to generate basis maps with larger and smaller voxel sizes, and the new basis maps were combined as normal to create the Reference field. There was therefore no need for resampling of any field maps, preventing the introduction of associated effects and potential errors. As expected, accurate localization of 3D translations is not possible with a nonunique (in this case, X gradient) Reference field (B), highlighting the importance of Reference field design.

3.3. Theoretical Validation With Calibrated Basis Maps

The average errors of Monte Carlo simulations using the calibrated basis set (i.e., the basis set measured in hardware calibration) followed the same pattern of decreasing error with increasing SNR (Figure 6). Both Reference fields had an average translation error below 1 mm for SNR levels above 4.6, and an average rotation error below 1 degree for SNR levels above 2.8 and 4.6 for the GA Optimization and Four‐lobe, respectively. For all SNR levels, the calibrated basis maps had a somewhat higher error than simulated maps. At SNR levels above 2.8, the Four‐lobe field had less than 0.5 mm error discrepancy between simulated and calibrated fields. For all SNR levels, the GA Optimization had less than 0.25 mm discrepancy between simulated and calibrated fields.

3.4. Experimental Validation With 48‐Channel MC Hardware

Self‐registration error of field maps measured before and after physically moving the MC hardware also decreased with increasing SNR (Figure 6, black). For the Four‐lobe field, six out of seven translation errors fell within one standard deviation of the average error for the calibrated basis set. Four out of seven rotation errors were lower than one standard deviation from the average error. For the GA field, all translation errors and all but one rotation error fell within one standard deviation of the average error. This indicates overall a good agreement between simulation and measurements.

Both Reference fields were able to recover all 10 additional controlled hardware shifts with an average translation error of 0.30 and 0.20 mm for the Four‐lobe and GA Optimization fields, respectively. All rotation errors were recovered with an average of 0.13 degrees and 0.11 degrees for the Four‐lobe and GA Optimization, respectively. The maximum translation and rotation errors were 0.44 and 0.17 degrees, both using the Four‐lobe field. Figure 8 illustrates the residuals calculated for a sample transformation. Computation of the translation and rotation parameters took between 3 and 7 s for all tested cases on a standard desktop PC.

FIGURE 8.

An example of the difference in residual when subtracting the Measured (i.e., transformed) versus corrected field from the Reference field after self‐registration to recover a 2.3 mm z‐translation. The z‐translation was accomplished by physically moving the patient bed and measuring the resulting distance with calipers. The corrected field was generated by a forward transformation of the Reference field (a reverse transformation of the Measured field could accomplish the same effect). Notably, this is for illustration purposes only. In practice, the transformation result from the self‐registration algorithm will be used to proactively correct shim fields.

4. Discussion

In this work, we assessed the impact of hardware positioning on the B₀ field control performance of MC hardware inserts. We showed that small hardware misplacements can seriously degrade field control, reducing the shim performance of the hardware. Reductions in MC performance can be avoided if the hardware position is known and considered in computational field management. Here we introduced a purely MC field‐based method for the spatial self‐registration of MC hardware inserts. This method compares a Reference field created by calibrated basis maps to a Measured field acquired using the same constellation of coil fields in misplaced hardware in an overlapping ROI. The rigid transformation needed to align the fields is a measure of the hardware displacement from calibration. Notably, this is only relevant in applications in which the MC system is used as an insert and is not built into the magnet bore.

A field shape optimized with a GA allowed for an average localization accuracy of 0.2 mm and 0.11 degrees from a sub‐1‐min B₀ map and analysis. From the analysis of residuals after hardware misplacement (Figure 2), we aimed to recover misplacements with a precision of 0.5 mm norm translation error and 1 degree z‐rotation error. The necessary steps of hardware localization are (1) the acquisition of a B₀ map of the Reference field (< 1 min), (2) the reconstruction of the field map (20–30 s), and (3) the recovery of the transformation parameters with the self‐registration algorithm (< 10 s). The entire pipeline is expected to be realized in 1–2 min total once implemented in the console environment. However, such streamlining was not part of this proof‐of‐concept implementation and is still outstanding.

All measurements and self‐registration were performed within an ROI centered around isocenter, where gradient linearity is highest and gradient‐based image distortions are not a concern. Extreme gradient nonlinearity, for instance in the periphery of the scanner bore, could however introduce differential spatial distortions between Day 0 and Day N, thereby affecting registration accuracy. No significant eddy currents or amplifier nonlinearities were observed with the robust field‐mapping protocol (e.g., unipolar gradients, high bandwidth, and prioritizing data quality over speed) used in this work. Underlying field inhomogeneities, either in a phantom or in vivo, are accounted for through subtraction of a background field map acquired under identical conditions. Physiological processes such as breathing have been reported to produce low‐amplitude, low‐order temporal variations in the B₀ field [30], but because the field maps in this work are acquired over approximately 1 min, these fluctuations are expected to average over multiple respiratory cycles and are therefore not expected to substantially affect the estimated rigid transformation or localization accuracy. In principle, the stability of the estimated transformation could be further improved by repeating the Reference field acquisition and combining the resulting transformations, as averaging would reduce the influence of random noise. This would be expected to increase accuracy, although at the cost of longer acquisition time. This was not explored in the present work, as the achieved accuracy was already sufficient for the intended application.

The patient bed of the employed Siemens Prisma scanner has a reported position tolerance of ±1 mm which was insufficient for the analysis at hand. Additionally, the researcher places the isocenter of the region using laser crosshairs, and differences in this isocenter placement can lead to position differences between acquisitions. This was problematic for our experimental validation, which required exact knowledge of the position differences between acquisitions to evaluate accuracy of the self‐registration method. We therefore directly measured the bed position with a mechanical caliper before and after moving the bed to determine the precise shift. After initial placement of the hardware and isocenter, a marker was placed flush with the foot of the patient bed. After the bed was moved, the distance between the marker and foot of the bed was measured and taken as the ground truth of z‐translation.

There was excellent agreement between simulation and experimentation across a range of SNR levels. Measured shift experiments were all capable of achieving translation error below 0.5 mm and rotation error below 0.5 degrees. The GA Optimization outperformed both the Single coil and the Four‐lobe distributions for all translations at every SNR level for simulated basis maps, calibrated basis maps, and measured field maps. While the Four‐lobe field slightly outperformed the GA Optimization field in rotation error of measured field maps, the GA Optimization field was clearly the best‐performing Reference field shape considering the combined metrics of translation and rotation, as expected. Although the field maps were acquired at 3 mm isotropic resolution, subvoxel sensitivity arises from the collective information contained in the field pattern rather than from any single voxel alone. However, sensitivity decreases with increasing voxel size (Figure 7). The use of larger ROIs with more contributing voxels leads to higher self‐registration accuracy (Figure S3). Some spatial erosion of the ROI during the self‐registration procedure is advised to ensure overlap between the Reference and Measured fields. At minimum, the ROI erosion should be chosen to be equal to or larger than the maximum expected hardware displacement.

The GA converged to solutions that maximized field variation toward the center of the ROI while maintaining high amplitudes and asymmetry. This result is expected and reasonable given that the self‐registration algorithm minimizes summed field values in voxels of a residual field calculated by subtracting the Reference field from the Measured field (Figure 8) and can be used as a roadmap for improved heuristic design. The resulting field shape characteristics from the GA optimization can be taken as suggestions, but the truly optimal Reference field shape is specific to the capabilities of the hardware at hand. The self‐registration method itself is not limited only to MC B₀ hardware. SH coil inserts have also shown significant improvement in shimming performance [21], and this method could also be applied to self‐registration of such inserts.

In this work, only rotations about the z‐axis, that is, the bore or foot‐to‐head direction, were considered in addition to potential x‐, y‐ and z‐shifts. Generated coil fields remain invariant in these cases and can, thus, be modeled as rigid transformations given that the z‐axis is aligned with the direction of the main magnetic field. Coil rotations about the x‐axis (left‐to‐right) or the y‐axis (back‐to‐front) on the other hand result in qualitative alterations of the produced coil fields. This requires updated field shapes, for example, through re‐calibration of the coil field at a given rotation angle, as part of the self‐registration procedure. Plans for future research include the addition of x‐ and y‐rotations to the method. Besides the time‐consuming brute‐force re‐calculation of the coil field shapes, we plan to consider a library‐based system to match field shapes to a set of measured and interpolated basis shape rotations, use of symmetry considerations to simplify recalculation, or even a deep learning‐based approach in which a model is trained on many simulated basis‐shape rotations. Of the two dimensions, rotations about the x‐axis have more potential relevance in vivo, for instance, in case of a cantilevered patient bed [10].

All scanner measurements in this study were taken in a spherical, 10 cm‐diameter ROI, but simulations in a brain ROI showed excellent self‐registration capabilities (Figure 9). Experiments were performed with roughly 3 mm isotropic voxel size, but larger voxel sizes can be used with a small tradeoff in accuracy (Figure 7). The scan time of the B₀ map can therefore be reduced if needed, given the accuracy reduction if acceptable. This self‐registration procedure could also be effectively used for instances in which differences in hardware placement are a feature of the technology, such as a cardiac array that could be placed higher or lower depending on the reach of the patient's chest or allow for relaxation of the engineering and time requirements for reproducible hardware placement. This study is intended as proof‐of‐principle demonstration of the feasibility and accuracy of the proposed localization strategy. Evaluation of downstream improvements for in vivo imaging or spectroscopy applications represents an important next step but falls beyond the scope of the current methodological work.

FIGURE 9.

The self‐registration procedure is agnostic to the region of interest (ROI). Biot–Savart simulations were performed to generate basis maps for a scaled‐up version of the test setup, with an outer diameter (OD) of 23 cm to match existing hardware [9] used for brain imaging. This example serves only to illustrate the ROI‐agnostic nature of the approach; the hardware was uniformly scaled and not specifically optimized for brain shimming.

In this study we have shown the possibility of field‐based spatial self‐registration of MC inserts. This approach enables the benefits of MC field control to be more fully realized by accurately finding the difference in hardware position between calibration and the present experiment. This presented technique paves the way for optimal shim capabilities despite intentionally or unintentionally misplaced hardware.

Funding

This work was supported by the National Institutes of Health (R01‐EB030560).

Supporting information

Algorithm A1. MC hardware localization.

Algorithm A2. Loss function.

Figure S1. Degradation of shimming performance due to a simulated 1 mm hardware misplacement along the z‐direction. Whole‐brain residual B₀ maps are shown for five subjects after dynamic multicoil shimming with no shift (left column) and with a 1 mm z‐shift applied to the shim fields before application (right column), corresponding to applying the “correct” shim solution at an incorrect spatial location. Even this small displacement leads to a pronounced increase in residual field inhomogeneity, most notably in the prefrontal cortex. Quantitatively, the whole‐brain standard deviation of the residual field after dynamic shimming increased from an average of 10.1 Hz (no shift) to 28.7 Hz following a 1 mm shift, illustrating the sensitivity of shimming performance to submillimeter hardware positioning errors. The resulting local field gradients reach values sufficient to cause substantial intravoxel dephasing at typical echo times used in susceptibility‐sensitive imaging (e.g., gradient‐echo and EPI). For example, gradients on the order of 80 Hz/cm across a 3 mm voxel produce approximately π phase dispersion at echo times of ˜20 ms, which is expected to result in significant signal loss and geometric distortion.

Figure S2. X‐, Y‐, and Z‐gradient components of selected Reference fields. Spatial gradient maps (Hz/cm) are shown for the GA‐optimized, Four‐lobe, and Single coil Reference fields within the region of interest. The GA field exhibits stronger and more spatially distributed gradient components across all three directions compared with the Four‐lobe and Single coil fields, consistent with its improved transformation recovery performance. Color scale indicates gradient magnitude in Hz/cm.

Figure S3. Self‐registration accuracy as a function of ROI diameter. Monte Carlo simulations were performed to evaluate translation (left) and rotation (right) errors across varying SNR levels for spherical ROIs of 6, 8, and 14 cm diameter. Larger ROIs yielded lower registration errors across SNR levels, reflecting the increased spatial information available for alignment, with the caveat that the Day N ROI must not contain any voxels entirely outside of the Day 0 ROI. Representative axial slices of the corresponding Reference fields for each ROI size are shown in the translation panel.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.