Technical Note: Extended field‐of‐view (FOV) MRI distortion determination through multi‐positional phantom imaging

Abstract Comprehensive characterization of geometric distortions for MRI simulators and MRI‐guided treatment delivery systems is typically performed with large phantoms that are costly and unwieldy to handle. Here we propose an easily implementable methodology for MR distortion determination of the entire imaging space of the scanner through the use of a compact commercially available distortion phantom. The MagphanRT phantom was scanned at several locations within a MR scanner. From each scan, an approximate location of the phantom was determined from a subset of the fiducial spheres. The fiducial displacements were determined, and a displacement field was fitted to the displacement data using the entire multi‐scan data set. An orthogonal polynomial expansion fitting function was used that had been augmented to include independent rigid‐body transformations for each scan. The rigid‐body portions of the displacement field were thereafter discarded, and the resultant fit then represented the distortion field. Multi‐positional scans of the phantom were used successfully to determine the distortion field with extended coverage. A single scan of the phantom covered 20 cm in its smallest dimension. By stitching together overlapping scans we extended the distortion measurements to 30 cm. No information about the absolute location or orientation of each scan was required. The method, termed the Multi‐Scan Expansion (MSE) method, can be easily applied for larger field‐of‐views (FOVs) by using a combination of larger phantom displacements and more scans. The implementation of the MSE method allows for distortion determination beyond the physical limitations of the phantom. The method is scalable to the user’s needs and does not require any specialized equipment. This approach could open up for easier determination of the distortion magnitude at distances further from the scanner’s isocenter. This is especially important in the newly proposed methodologies of MR‐only simulation in RT and in adaptive replanning in MR linac systems.


| INTRODUCTION
MR images are increasingly used in radiation therapy (RT) due to their superior soft tissue contrast allowing for increased accuracy in tumor and normal tissue delineation as compared to segmentation performed on CT images. [1][2][3][4][5] The preferred method of incorporating the MRI into the clinical workflow has been through rigid registration to a CT image set. The MR-defined contours are thus mapped onto the CT and the CT is then used for dose calculations and evaluation. Current research is focused on the use of MR simulation alone without the use of an underlying CT scan for dose calculation. [6][7][8] This would negate the need for multiple scans, thus relieving stress on the patient and eliminating the CT imaging dose. This would also open up the field of adaptive replanning in integrated MR/linac systems. 9,10 However, a drawback with MRI is the inherent geometric image distortion due to imperfections in the hardware of the system (system dependent geometric distortion) or due to distortions induced by the patient (patient dependent geometric distortion). Patient dependent distortion arises from magnetic susceptibility and chemical shift effects and is more difficult to correct for than system dependent distortion. For this reason, the distortion correction efforts have mostly been concentrated on the latter. Patient dependent distortion is generally also smaller in magnitude. 1 System distortion arises due to inhomogeneities in the main magnetic field and due to nonlinearities in the induced gradient fields. 3 As such, the system distortion will be dependent on the pulse sequence type and imaging parameters used, with increased distortion as a function of distance from the isocenter and strength of the main magnetic field. 1,2 To account for this, vendors of MRI systems supply the user with distortion correction algorithms. However, despite vendor provided 3D distortion corrections, maximum distortion values of 2-3 mm are still routinely measured on these systems. 1 The clinical effect of this residual distortion is highly site and technique dependent. The consensus of leading experts considers a system distortion of <1 mm in the Stereotactic Radiosurgery (SRS) setting and <2 mm elsewhere acceptable. 1,4,11 However, a <1 mm residual distortion should also be considered in non-SRS settings to limit the detrimental effect on plan quality indices to below 5%. 1,5 Another issue arises with MR-only simulation in the context of stereotactic body radiation therapy (SBRT) treatments. During these treatments the use of non-coplanar beams are often utilized with beams entering and exiting the patients at much greater distances from the isocenter than what is seen in conventional treatments.
With increased distance from the isocenter the distortion increases and therefore a need for accurate knowledge about the distortion at these off-target locations is essential to properly account for and protect organs-at-risk.
Phantoms are commercially available to accurately quantify MRI distortions. One of the challenges that remains, however, is measuring distortion over the entire MR bore in a practical way. Liquidfilled phantoms large enough to cover torso-sized regions of the bore are heavy to handle and unwieldy in a clinical environment.
Nonliquid-filled phantoms can cover a larger region, but are often still unwieldy due to their weight, and perform only distortion measurements. It is desirable to have a phantom that is small enough to be handled easily in a clinical environment and can perform a multitude of measurements so that an overall quality control protocol is simplified.
Here we propose an easily executed methodology for enlarging the dimensions of the MR distortion map through the use of a compact commercially available distortion phantom. By importing several off-center locations into the distortion calculation algorithm, we can effectively emulate a whole-bore phantom for accurate, reproducible, and practical determination of distortion beyond the physical volume of the phantom and facilitate the increased use of MRI in radiation therapy.

2.A | Phantom and scanners
The MagPhanRT phantom (The Phantom Laboratory Inc, Salem, NY) was used in this study (Fig. 1). 12 The distortion component of the phantom consists of 505 solid plastic spheres of diameters 1 and 1.5 cm, surrounded by a background fluid with relaxation time approximately 350 ms at 1.5 Tesla. 12 The spheres appear dark in an MR image. The spheres within the phantom used for distortion mapping are placed with typical spacing of 3-4 cm. They do not strictly adhere to a regular lattice or pattern. However, this spacing is adequate to characterize the distortion field as distortions are generally slowly varying except near the very extremes of the field-of-view (FOV) of the scanner. 11,13,14 Because the phantom is a liquid-filled phantom, it can also perform other important measurements needed in a quality assurance protocol such as slice thickness, resolution, SNR, and uniformity. The multi-purpose capability is of practical importance in clinical environments because an efficient quality assurance protocol is desired due to the frequency of quality assurance tests, which are often done on a daily or weekly basis.
The phantom [ Fig. 1  dimensions. These locations then represent the best-known position of the spheres, but in an arbitrary coordinate system different from that of the MR scanner. This coordinate system will be referred to as the 'phantom' coordinate system. A rigid-body transformation (three translational and three rotational degrees of freedom) will be needed to know the actual location of the fiducials in the scanner.

2.B | Extended FOV distortion determination
The means to obtain this transformation are described subsequently.

2.B.2 | Localization
An approximate location and orientation of each phantom module within the scanner is determined by first locating three spheres of slightly larger diameter (1.5 cm) than the rest of the fiducial spheres.
The spheres are located near the central portion of the phantom, arranged at the vertices of a scalene triangle so that they can be uniquely identified. The Orthogonal Procrustes algorithm is then used to generate a rigid-body transformation that best transforms the locations of the three spheres in the phantom coordinate system to their locations in the coordinate system of the MR scanner.
Because the phantom is itself a rigid body, this transformation can then be applied to any sphere in the phantom to transform its coordinates from the phantom coordinate system to the coordinate system of the MR scanner. The location of each fiducial sphere arrived at in this manner will be referred to as the 'designed' location.
Because only three spheres are used, and because the sphere locations are subject to geometric distortion in the scanner, the rigidbody transformation computed thus far (and hence the 'designed' location) is still approximate. The residual error is in the form of a small rigid-body transformation that, as explained below, will be determined from the full distortion fit and does not propagate to the distortion measurements.

2.B.3 | Displacement field
From the segmented locations of each sphere determined from the MR image, a displacement vector at each location is calculated as the difference between the segmented location and the designed location. The displacements are fit to an expansion of Chebyshev polynomials: In these basis functions, U m is a Chebyshev polynomial of the second kind of order m, and A mnpi is the fitting coefficient to be determined for the Φ  The fiducial spheres are first segmented. Based on a subset of fiducials, a rigid-body transformation (RBT) is applied to each phantom scan to determine the phantom placement within the scanner. A displacement vector is calculated for each fiducial and the resultant displacement field is fit to a polynomial expansion. From the central phantom position, the misalignment between the laser alignment and the MR coordinate system is determined and applied. The rigid-body transformations associated with each of the phantom acquisitions are discarded from the resulting fit.
The appropriate order of the fit in Eq. (1) is governed by the structure and orientation of the phantom. The phantom used in this study has internal fiducials that are approximately on a three-dimensional grid. The order of the fit in each direction [the maximum values of m, n, p in Eq. (1)] is then taken as the number of fiducials along each direction, with a limit placed on the sum m + n + p of 1.5 times larger than the highest of the three integers m, n, and p.
This choice is not critical. The aim is to have more than enough terms initially to fit the distortion field, and then apply Tikhonov Regularization to reduce the effective order of the fit to prevent overfitting.
Among the basic functions described above are terms with no spatial dependence (m = 0, n = 0, p = 0), which represent simple translations by a distance δ along the x, y, or z axes: Also among the basis function set described above are functions where each component is proportional to only x, y, or z, where either m, n, or p is 1 and the other two are zero. The subspace spanned by these components can be described in linear combination of these vectors that include terms to represent distortions corresponding to small, linearized rotations and translations. Small rotations about the x, y, and z axes, respectively, are represented by the following distortion basis functions: Thus, by simple rearrangement of the basis functions described in Eq. (1), the translational and small rotational components can be broken out as follows: where the prime over the last summation term in Eq. (2) Among the translational and rotational terms in Eq. The procedure is as follows:   3. For each series, determine an approximate location of each phantom by any convenient method, such as using a small number of landmark fiducials that can be segmented easily.

4.
Based on the approximate location of the phantom, use the known design of the phantom fiducials and the measured location within the images to determine a displacement measurement d k for each fiducial k, which is found within the image at location

2.C | Method evaluation and validation
To evaluate the distortion correction accuracy, a high-quality CT of the phantom was acquired. This CT image was used as our "gold

3.A | Distortion determination
The number of fiducial locations was increased from 505 to 1515 positions through the inclusion of a AE5 cm translation in the Sup/Inf direction [ Fig. 3(a) and 3(d)]. The increase in sampling points density and the extended FOV is seen in Fig. 3(d). The maximum distortion determined with LFOV was 1.8 mm [ Fig. 3  To further validate the MSE method and to remove any assumption about the absolute position of the phantom, the internal distances between all fiducial spheres were determined for each image set for each phantom position (Fig. 7). Again, the CT image was considered the golden standard and the distances between the spheres measured within the CT scan were considered the true distances.

3.B | Method evaluation and validation
The MR images acquired with the EFOV with either vendor correc- F I G . 5. The dependence of absolute sphere position disagreement on radial distance from the MR isocenter for extended field-of-view (EFOV) vendor-corrected MR and EFOV Multi-Scan Expansion (MSE) corrected MR as compared to CT. The boxes represent the interquartile range (IQR) between the third (Q3) and the first (Q1) quartile. The upper and lower whiskers extend to Q3 + 1.5 * IQR and Q1 -1.5 * IQR, respectively. The outliers (black circles) are defined as data points that fall below Q1 -1.5 * IQR or above Q3 + 1.5 * IQR. The orange lines are the median value. The median positional disagreement increased with increased distance from the MR scanner isocenter for the vendor-corrected MR images. No significant change in positional disagreement with distance from the MR isocenter was found for the MSE corrected MR images. * IQR, respectively. The outliers (black circles) are defined as data points that fall below Q1 -1.5 * IQR or above Q3 + 1.5 * IQR. The orange lines are the median value. The median positional disagreement increased with increased distance from the MR scanner isocenter for the vendor-corrected MR images. No significant change in positional disagreement with distance from the MR isocenter was found for the MSE corrected MR images. magnitude of the residual distortion between using an entire-bore phantom and using the MSE methodology revealed similar results. 11,13,14 Although this multi-scan approach requires several acquisitions of the phantom, each taking several minutes, there are substantial practical benefits of enhancing the ability of a smaller phantom to cover larger FOV. Routine quality assurance protocols are often performed daily or weekly, and don't necessarily require large FOV for the distortion measurement. A smaller phantom that can be used to cover the larger FOV on a less frequent basis has clinical value in keeping the primary quality control protocol efficient to execute, and prevents the need for an entirely different phantom to cover large fields of view.
In our study we used distance measurements in 3D phantom images acquired at different offset from the isocenter as surrogate metrics for evaluating the accuracy of our propose method. An alternative, more direct analysis would involve a comparison between the distortion map determined through the proposed methodology and a distortion map determined through the use of a whole-bore phantom. Along these lines, the validity of the FOV extension was verified in the region of overlap between the LFOV and EFOV data sets, but no such verification could be performed in the parts that extended beyond this region. Furthermore, only one specific imaging protocol was used for MR image acquisition in the current study. Different imaging protocols can have different distortion patterns.
The largest difference between protocols arises from using 2D vs 3D acquisition techniques, particularly if the vendor distortion correction is not activated in the slice direction on a 2D acquisition.
The other main source of protocol-to-protocol variation depends upon the relative importance of gradient nonlinearity vs static magnetic field inhomogeneity. Distortion arising from gradient nonlinearity will be largely independent of protocol (other than the 2D vs 3D issue), whereas distortion arising from static magnetic field inhomogeneity will be sensitive to readout bandwidth as well as the orientation of the scan plane and readout axis direction. The methodology presented here can be repeated to characterize distortion on a protocol-specific basis.

| CONCLUSION
In summary, we have presented a methodology that could enable the user to use a standard distortion phantom to extend the map-