Motion artifact correction in free-breathing abdominal MRI using overlapping partial samples to recover image deformations

Authors


Abstract

This article presents a method to reconstruct liver MRI data acquired continuously during free breathing, without any external sensor or navigator measurements. When the deformations associated with k-space data are known, generalized matrix inversion reconstruction has been shown to be effective in reducing the ghosting and blurring artifacts of motion. This article describes a novel method to obtain these nonrigid deformations. A breathing model is built from a fast dynamic series: low spatial resolution images are registered and their deformations parameterized by overall superior–inferior displacement. The correct deformation for each subset of the subsequent imaging data is then found by comparing a few lines of k-space with the equivalent lines from a deformed reference image while varying the deformation over the model parameter. This procedure is known as image deformation recovery using overlapping partial samples (iDROPS). Simulations using 10 rapid dynamic studies from volunteers showed the average error in iDROPS-derived deformations within the liver to be 1.43 mm. A further four volunteers were imaged at higher spatial resolution. The complete reconstruction process using data from throughout several breathing cycles was shown to reduce blurring and ghosting in the liver. Retrospective respiratory gating was also demonstrated using the iDROPS parameterization. Magn Reson Med, 2009. © 2009 Wiley-Liss, Inc.

MRI has increasing importantance in diagnostic radiology of the liver for a variety of conditions, including diffuse disease such as cirrhosis and hepatocellular carcinoma (1) as well as focal liver disease (2, 3). Respiratory motion within the acquisition of a single image can cause ghosting and blurring artifacts in the reconstructed data. These artifacts can be reduced by navigator and breath-holding techniques. Breath-holding can be problematic for sick patients, and although gating provides an alternative that improves image quality (4), both breath-holding and gating limit continuous acquisition and lengthen the scan time. Navigator tracking does allow for continuous acquisition, but only provides basic rigid correction of the imaged volume; a substantial proportion of liver motion is nonrigid (5).

Artifacts caused by motion between the acquisition of phase encodings can be reduced by generalized matrix inversion reconstruction (GMIR) (6). This involves constructing a matrix representation of the acquisition process, incorporating deformations during the scan, and applying a matrix inversion algorithm such as LSQR (7, 8) to recover an artifact-free image. This reconstruction procedure requires that the deformation fields associated with each sample of k-space (for example, with each shot) be known in advance. Deformations may also be found by optimizing some property of the final reconstructed images over possible sets of deformation fields (6). Individual reconstructions are computationally intensive, however, and this approach is therefore only feasible when the set of deformation fields varies with only a small number of parameters.

Respiratory motion causes large, nonrigid deformations of the abdominal organs. GMIR has been demonstrated in a simulated setting where the deformation fields are known in advance (9); however, determining these deformations during “real” acquisition is challenging. One approach applicable to respiratory motion, which is approximately periodic (10), is to construct a patient-specific model of breathing motion, relating the full nonrigid deformation to some surrogate breathing parameter. Estimating or measuring this parameter subsequently allows the deformation to be predicted (11, 12). Models of this type have been derived from MRI data that have been respiratory-gated and re-sorted by 1D or 2D respiratory phase for radiotherapy guidance in the liver (5, 13), and by registering rapidly acquired dynamic MRI data in the lung (14, 15), liver (16), and heart (17).

Such models have also been applied to provide deformation fields for GMIR using externally measured surrogate parameters (such as ECG, pulse oxymeter, spirometer, or chest bellows), either deriving a model from a distinct training scan (18) or using a “self-calibration” scheme based only on imaging data and parameter measurements (19). The present article describes a novel approach for image deformation recovery using overlapping partial samples (iDROPS), which requires a training acquisition to derive a model but does not require any external or internal measurements of breathing during imaging.

THEORY

The goal of iDROPS, combined with a motion model and GMIR, is to obtain a full-resolution unghosted image from data acquired continuously during free breathing. The method does not require recording of any surrogate breathing parameter during imaging. The two stages of acquisition and three stages of data processing are illustrated in Fig. 1.

Figure 1.

Overview of iDROPS as part of a complete free-breathing image reconstruction process, with deformation series ut and breathing parameter ρt.

The two acquisitions give low-resolution training and high-resolution imaging data, respectively. In the first processing step, the training data are nonrigidly registered to give deformation fields. The average superior–inferior displacement of each field is used as a surrogate breathing parameter, and a model is built to relate it to the deformation at each pixel. The second step, iDROPS, uses this model to estimate the deformation associated with each group of k-space lines acquired consecutively in time in the imaging data. Aliased images formed from subsets of k-space acquired during the imaging phase are compared with those from from the equivalent k-space lines of deformed low-resolution training data. The third step incorporates these deformations into a reconstruction formulation, GMIR, to reduce ghosting and blurring in the reconstructed imaging data.

Data Acquisition

The two acquisition phases needed are illustrated in Fig. 2. The goal is to reconstruct an image with N = Nx × Ny × Nz samples of k-space.

Figure 2.

The two acquisition phases with Ny = 32 and gimg = gtrain = 4. 2D k-space acquisition schemes are shown, with sampled areas shaded in gray.

Low-resolution training data are acquired sampling only the central gtrain-th part of k-space, giving a total of N/gtrain samples. Sampling reduction can be made in one or both phase-encoding directions: in this article we consider reduction only along Ny, so Ny/gtrain central lines are acquired. The factor gtrain is set to give each training frame an acquisition period of 1 s or less; that is, sufficiently short that several images are acquired in a typical breathing cycle [the average cycle duration measured in (10) was 3.6 s]. T frames are acquired, covering many breathing cycles (several minutes).

Imaging data are also acquired with the same field of view, using an interleaved “sliding window” sampling so each frame contains every gimg-th line of k-space: Ny/gimg lines spread over k-space. The offset of selected lines is incremented in successive frames so that a complete k-space is acquired after gimg frames. Again, gimg is selected to reduce motion within individual frames, so typically gimggtrain (although there is no methodological requirement for them to be identical). A total of Timg frames are acquired. Note that individual frames form aliased images.

Data are acquired in two or three dimensions but the linear operations of image reconstruction are conveniently represented as a series of matrix multiplications on image data written as one-dimensional vectors. Image vectors ct (training) and st (imaging) have length N = Nx × Ny × Nz, with all the image columns concatenated. These vectors can then be multiplied by matrix representations of linear operations: Fourier transformation, k-space subsampling, and nonrigid spatial deformation. The principal symbols used to denote images and these linear operations are shown in Table 1.

Table 1. The Principal Symbols Used to Denote Images and Linear Operations in the Modeling, iDROPS, and Reconstruction Methods
SymbolMeaning
N; Nx, Ny, NzTotal number of samples; number along each axis
T, TimgNumber of training frames, number of imaging frames
gtrain, gimgUndersampling factors for training and imaging
ct, stImages as 1D vectors (c is training, s is imaging)
c0, s0Reference exhale images; s0 is the true (unknown) image
cmath image, smath imageAliased images for iDROPS cost evaluation
equation imagek-space data; equation image includes all Timg imaging frames equation image
ut, vtDeformation fields (u is corrective, v is distorting)
ρt, ρtest;MBreathing parameter vectors with length M
α, β; C(r)Model coefficients for each element of u; coefficient matrix
equation imageMatrix applying non-rigid deformation u to image vector
Atrain, [Aimg]tk-space sampling matrices for training and imaging
F, F−1, FHForward, inverse, and Hermitian transpose FT matrices
PtMatrix to pack each imaging frame equation image into equation image

In addition to the requirement that motion should be minimal within individual training and imaging frames, the usable combinations of N, gimg, and gtrain are constrained by the requirements of the iDROPS step. iDROPS optimization uses only those k-space locations sampled during both training and an individual imaging frame: approximately Ny/(gimggtrain) k-space lines. A typical system with Ny = 256 and gimg = gtrain = 4 gives an overlap of 16 lines of k-space.

Building a Motion Model

Breathing motion is described by an examination-specific model built from the training data. Each training frame ct is reconstructed by Fourier transform after zero-filling to full k-space sample size N. A reference image c0 at the exhale position is manually selected from this training series, and an image-based registration algorithm [in this case a fluid registration (20)] is applied between each image ct and c0. This results in a series of three-dimensional nonrigid deformation fields ut, whose rows correspond to locations in the reference image c0; each has three columns giving the offset of corresponding tissue in ct along each axis. The row of ut corresponding to position r in c0 is written u(t,r) = [ux(t,r), uy(t,r), uz(t,r)].

A motion model relates the deformation field ut to a vector ρt of M surrogate breathing parameters. In this work, we use a scalar parameter (M = 1) derived directly from the deformation field. The parameter is the instantaneous mean displacement of the whole field-of-view in the superior–inferior direction x:

equation image(1)

We will describe this parameter as “liver height,” since it is dominated by the motion of the liver, the largest moving object within the field of view. Note that we do not require any segmentation to obtain this.

The model consists of a set of coefficients relating ρ to u at each image location and for each directional component of displacement. Each individual linear fit takes the form u = α + β · ρt, with separate coefficients α and β for each scalar element of ut:

u(t,r) = [1ρt] C(r), where

equation image(2)

Hence C(r) is also defined for each of the N spatial positions r in c0. It is derived at each point r by solving three systems in the least-squares sense, one for each displacement component. Using M = 1 in this article, and writing ρt for the parameter value measured at time t during training:

equation image(3)

with similar forms for y and z. Using M > 1 would add additional columns to the left-hand parameter matrix and additional rows of βm,x to the coefficient vector by which it is multiplied. Notice that this measured parameter matrix is invariant over r; so all C(r) can be found efficiently by combining one pseudo-inversion (here singular value decomposition) of the left-hand matrix with each possible right-hand vector formed from u{x,y,z}(1 … T, r).

With iDROPS, no physical measurements of the parameter are needed during the imaging acquisition [although a similar approach can be applied in the case of known parameters (18)]. Thus choosing appropriate parameters derived directly from the training deformations to which they are fitted, as above, gives the potential for very strong correlation and a well-fitted model.

Estimating Imaging Deformations

Parameter estimates ρt for each imaging frame st are obtained by iDROPS, performed separately for each frame. The iteration cycle is illustrated in Fig. 3. The fundamental comparison is between a few lines of k-space in equation image (where the tilde denotes k-space representation) and the equivalent lines from a deformed, Fourier-transformed training image. These “overlapping” lines will be sparse (because of the imaging sampling pattern) and spread over only the central region of k-space (because of the training sampling pattern).

Figure 3.

Illustration of a single iDROPS iteration with number of lines Ny = 32 and reduction factor gimg =. The cost function is repeatedly evaluated and maximized to find the parameter ρtest most closely reproducing the true deformation.

To evaluate a given parameter ρtest for accuracy in describing frame st, the breathing model Eq. [2] is first used to generate the corresponding deformation field utest. It is helpful here to distinguish clearly between deformation directions. The model-derived utest is a corrective deformation, transforming an image distorted by breathing into the exhale position. However, it is not possible to form an unaliased image st, because only sparsely sampled data k-space lines are available for each frame; so utest cannot be applied directly. Instead, iDROPS first calculates the inverse deformation field vtest = inv(utest)—a distorting deformation, because it transforms an exhale image to the distorted breathing position—and uses this to deform low-resolution reference training image c0.

We now wish to calculate (and optimize) a cost function between this deformed reference and the acquired imaging frame. The deformed c0 is Fourier-transformed back to k-space; then any area of the k-spaces of either image that does not appear in both is zeroed. The resulting sparse datasets are Fourier-transformed to image space for comparison with mutual information (MI).

This process is illustrated in Fig. 3. A single iteration, which can be implemented efficiently using nonrigid spatial resampling and the fast fourier transform, is equivalent to the following matrix formulation:

equation image(4)
equation image(5)

F and F−1 are matrices applying the forward and reverse Fourier transformations, respectively. equation image is the acquired k-space data corresponding to training reference image c0 (such that equation image), and similarly equation image is the acquired undersampled k-space data corresponding to the tth imaging frame (equation image). Zeroing of unsampled data is written as a multiplication of k-space by sampling matrices Atrain and [Aimg]t, both taking a value of one on leading diagonal elements corresponding to sampled positions in k-space and zero elsewhere. Spatial deformation is written as a linear operation equation image on the image vector, where the N × N transformation matrix equation image includes appropriate interpolation (trilinear interpolation was used in our implementation). Within the iDROPS loop, the approximation used was that v = −u: this is reasonable when the deformations are small and the deformation field spatially smooth, and it is extremely fast to calculate. A more sophisticated approach is discussed later for GMIR.

A typical smath image (where the * denotes an image formed only from “overlapping” samples) is shown in Fig. 4. The resulting pair of gimg-aliased, low-resolution images can be compared by evaluating their MI (21):

equation image(6)
Figure 4.

Typical smath image image, the aliased low-resolution data from which cost is evaluated in iDROPS optimization, with gimg = 4. The reduced phase-encoding is shown horizontally across the page.

As ρtest approaches the correct respiratory parameter for a given frame, Qt should become maximized. MI gives a degree of robustness to aliasing. Consider a change in deformation in which one feature becomes aligned between images, but a second does not, such that the alias of the second feature additively superimposes the first in only one of the two images. Mutual information will typically give an incremental increase in this case, while simpler measures (such as the sum of squared differences) will penalize the intensity difference caused by the additive aliasing.

Because smath image is constant with respect to ρtest, only cmath image needs re-evaluation for each iteration. Any suitable algorithm can be used to maximize Qttest) over ρtest. In our implementation, a coarse linear search was used to provide an initial guess for the global maximum, followed by Brent golden-section optimization (22, 23).

Generalized Matrix Inversion Reconstruction

Having estimated ρt for each imaging frame 1 ≤ tTimg, and derived (via the model) corresponding corrective deformations ut, we now perform GMIR. Batchelor et al. (6) described a method using the LSQR algorithm to invert the matrix; here we apply a variant where the data are evaluated in k-space (rather than using their Fourier transform). This allows direct handling of multiple samples of each k-space point without either averaging or grouping them into images prior to reconstruction.

Acquisition is described by a composition of spatial distortion and k-space sampling matrices. Here, a distorting transform is needed, and hence ut must again be inverted to give vt: a more precise estimate is obtained than used in the inner iDROPS loop by scattered data interpolation (SDI) (24). Like u, inv(u) should have a defined value at each grid point r, but it is actually precisely known at the set of scattered points r + u(r). In the SDI approach, inv(u(r)) at each grid point r is set to the inverse-square weighted sum of nearby points. For N pixels, this calculation has order O(N2). This can be improved by considering only those pixels within a search radius, initially quite small and expanded as necessary until sufficient points are found.

The acquisition process, from an unknown true image s0 of length N, is thus described by

equation image(7)

where equation image is the (known) acquired k-space data of length NTimg/gimg across all imaging frames, and Pt a packing matrix. Pt has size (NTimg/gimg) × N, and maps a k-space vector such as Fs0 into the longer k-space vector equation image which includes all k-space samples from all frames of an imaging series. It is thus dependent on frame number t; in implementation it can often be absorbed into Aimg, but it is conceptually distinct as a feature of the chosen row arrangement of the system, rather than a physical property of the acquisition1. All other symbols have the same meanings as in Eq. [4].

The LSQR algorithm (7, 8) is used to invert the linear system in Eq. [7] in the least-squares sense. The matrix B is large, but LSQR requires only the matrix-vector products Bx and BHy for arbitrary x and y. These can both be implemented efficiently using a non-rigid spatial resampling and a Fast Fourier Transform (6). The Hermitian transpose operation, on a packed k-space vector equation image (which includes all imaging time-points), is written as follows:

equation image(8)

We assume, as in (6), that equation image; that is, the Hermitian transpose of a transformation is approximately equal to the transformation produced by the inverse deformation field. Care must be taken with the 1/N factors when implementing F and FH using the forward and inverse FFT; depending on formulation, scaling may be needed between the Hermitian transpose and inverse.

In our implementation, complex data from separate abdominal array receiver elements were added before the start of reconstruction to form a single k-space (our acquisition protocol did not demonstrate any regions of signal cancellation from this assumption of minimal phase difference between elements). LSQR uses a numerically stable conjugate gradient scheme. Terminating after a small number of iterations provides effective regularization of the solution with these types of iterative inversion methods (9, 25, 26). Here we terminate after two iterations, chosen by visual observation of the images; further iterations do not improve image quality but do introduce additional noise-like features.

An alternative use of the iDROPS-derived parameter ρt is its use as a tool for binning images by respiratory phase. This allows reconstruction of stationary images from a subset of the imaging data for comparison with GMIR images from the full dataset, without the need for any physical sensor navigator measurements.

METHODS

This section presents two experimental methods. The first simulates iDROPS acquisitions to test the accuracy of estimated deformation fields, and the second demonstrates iDROPS as part of a complete reconstruction process on real data. All MRI images were obtained following written informed consent and under ethical and institutional scientific committee approved protocol.

iDROPS Deformation Accuracy

For testing iDROPS accuracy, both training and imaging series were derived by appropriately reducing the resolution of a single longer series, acquired using multi-shot EPI and SENSE on a 1.5 T Philips Intera MR system. Sixty consecutive 3D volumes of 128 × 128 × 45 voxels were acquired with a four-element abdominal array coil, covering the majority of the liver every 1.2 sec during free breathing, and reconstructed in the sagittal plane with spatial resolution 2.7 × 2.7 × 4.5 mm. Five volunteers (four male, one female, labeled 1–5) were each imaged supine on two occasions (labeled a, b) with this protocol. To obtain true deformation fields for each frame, all images were fully reconstructed, and the resulting series of volumes registered to an exhale reference image using an Eulerian fluid algorithm (20).

k-space lines were selected from Fourier transforms of these reconstructed images to simulate the two acquisition phases used for iDROPS. The training series consisted of 32 central lines of k-space from the first 30 frames (gtrain = 4). The imaging series was formed from the remaining 30 frames, each including every fourth line (gimg = 4) such that the whole of k-space was sampled every four frames. The model was built from the training series, and iDROPS used to estimate deformations in the imaging series, using the instantaneous mean superior–inferior displacement defined in Eq. [1] as the surrogate breathing parameter ρt.

The resulting deformation fields were compared with those derived from the original images. The liver was manually segmented in one sagittal slice (in the plane of the first visible major bifurcation of the right branch of the hepatic portal vein) for each study; the mean RMS difference between the “true” deformation and that estimated by iDROPS was calculated within this liver region for all imaging time-points.

Full Reconstruction: iDROPS with GMIR

Data were also acquired from four supine volunteers in free breathing, obtaining training and imaging data separately, with gtrain = gimg = 4. A 2D gradient echo sequence was used with a full in-plane acquisition matrix of 256 × 180 voxels (resolution 1.5 × 1.4 mm) in a single 8 mm slice. The training data comprised 80 frames, each of 45 central lines acquired in 1.1 sec per frame. The imaging dataset included a further 80 frames, each including every fourth line (i.e. 20 samples of each k-space point) and also acquired in 1.1 sec. These data were acquired using a four-element abdominal array coil and handled as complex-valued throughout. Studies were repeated with and without saturation slabs on either side of the imaging slice, which remove bright inflow artifacts from blood vessels.

iDROPS was applied to derive the 80 estimated imaging deformation fields, followed by GMIR to recreate an artifact-free image composed of data from all 80 imaging frames. The same parameter formulation (instantaneous mean superior–inferior displacement) and cost function (mutual information) were used as the iDROPS error measurement experiment.

The iDROPS-estimated parameters were also used to create a retrospectively gated image for comparison with the reconstruction. Imaging frames were divided, by this parameter, into six evenly populated bins over the breathing cycle. The image, or averaged pair of images, closest to exhale were chosen to give a clear reference. The images are not in general formed from the same number of averages of each k-space location, and the mean number of samples per k-space point is provided with the examples shown.

For further comparison, three full k-space acquisitions were also performed during a single exhale breath-hold, taking a total of 15 sec to acquire; the best (least artifacted) of these was chosen to provide a reference with which to compare the reconstructed image. This reference image therefore includes only a single measurement of each k-space sample, rather than 20 averages as in the iDROPS imaging acquisition; all other imaging parameters were the same.

Data Processing

The iDROPS and GMIR artifact correction methods described in the “Theory” section above were implemented in Objective Caml (27) and using the GNU Scientific Library (GSL) (23). The code was executed on a cluster of 64-bit Intel-based machines. The fluid registration implementation of Crum et al. (20) was used to register the training data, taking a few minutes per 3D training frame. Execution time for iDROPS deformation estimation was also on the order of a few minutes per imaging frame, with the majority of time spent computing Fourier transforms and deformations in evaluating Qttest). GMIR reconstruction of a full imaging dataset (combining all imaging frames) takes between a few minutes and a few tens of minutes depending on problem size. Complete processing of all frames from a typical acquisition thus requires a few CPU-hours. The R statistical computing environment (28) was used to analyze the results.

RESULTS

iDROPS Deformation Accuracy

A typical plot of mutual information Qttest) over ρtest is shown in Fig. 5. This curve, from the first imaging frame of Study 1a, is typical, with a sharp global maximum (in this case at about ρtest = −1, the correct value for this frame). The points shown are taken from the global search step of iDROPS, performed before optimization to ensure the global maximum is found.

Figure 5.

Optimization of mutual information over the breathing parameter for a typical imaging frame. The line is a cubic spline.

Figure 6 shows the estimated breathing parameters for two studies. For each the horizontal (time) axis is divided in half to show the simulated training and imaging data. The true parameters (from the original data) are shown as a line. Two examples are shown: the upper plot (Study 1a) demonstrates regular reproducible breathing across the whole period, while the lower plot (Study 2b) shows considerable variation in the frequency and depth of breathing throughout both acquisitions. In each case there is good agreement between measured and estimated values.

Figure 6.

Comparison of iDROPS estimated ρt from simulated imaging data with true values; two examples are shown. In each plot, the solid line shows the true (measured) values from the original data; circles show ρt calculated by simulating training and imaging data from the original images.

Errors in the estimates were quantified by comparing the iDROPS-predicted deformations ut to the actual deformations equation image measured (by fluid registration) for each corresponding imaging frame. The original magnitude of breathing motion averaged over all breathing frames equation image is mapped in Fig. 7 for one study (1a) with the corresponding error in iDROPS, equation image, shown adjacent. The bright area of high error in the lower-anterior liver corresponds to the location of the gall bladder.

Figure 7.

A single slice from the liver in volunteer 1a (left), showing maps of the average breathing motion magnitude in mm (center) and average magnitude of residual error between iDROPS-estimated deformation fields ut and true values equation image (right).

This error calculation was repeated for all imaging frames of all 10 volunteer studies: these data are shown in Fig. 8, averaged over the selected liver region. The overall mean magnitude of the error in iDROPS estimates, across all studies, was 1.43 mm (compared to an overall mean magnitude of breathing motion of 3.62 mm). A similar calculation was performed to estimate the contribution of errors in the model to this residual, by taking an average (within the liver region) of the magnitude of the residual from the model-fitting step in Eq. [3] at each point in the training data. This leads to an overall model residual of 1.24 mm across all 10 studies. If the model residual is mapped over the slice shown in Fig. 7, the results take a very similar form with an area of large error centred on the gall bladder, suggesting a local nonlinearity of motion with breathing in this region of the organ.

Figure 8.

Error in iDROPS-predicted deformations, averaged over the liver (equivalent to the region marked on Fig. 7). Solid dot shows median, box shows interquartile range, whiskers show extreme values (except those more than 2 SD from the mean which are plotted as outliers).

GMIR Artifact Correction

Figure 9 shows single-slice images from three of the four volunteer studies corrected with iDROPS and GMIR. We were unable to obtain a good training registration for the other volunteer due to fast, shallow breathing causing motion artifacts within the training frames themselves. Examples are shown both without (volunteers 1 and 2) and with (volunteer 3) saturation slabs, removing the bright inflow artifact from blood vessels.

Figure 9.

Single-slice images corrected with iDROPS from three volunteers. The bracketed values indicate the number of averages per k-space sample. First column (“reference”) shows a single breath-hold. Second column (“uncorrected”) shows 20 averages (80 frames) during free breathing. The third column (“gated”) shows a subset of these averages, gated to the exhale position using iDROPS estimates. The fourth column (“corrected”) shows all 20 averages using GMIR with iDROPS-derived deformations. Volunteer 3 shows data with saturation slabs on both sides of the slice, to remove bright inflow in blood vessels.

In all cases the visibility and definition of blood vessels is improved in the iDROPS-gated and iDROPS+GMIR corrected images compared to both the breath-hold reference and the uncorrected imaging data.

Blood vessel resolvability in the iDROPS-gated and iDROPS+GMIR corrected images is comparable. In volunteers 1 and 2 (both without saturation slabs), blood vessels appear slightly less blurred in the iDROPS-gated reconstruction compared to GMIR; in volunteer 3 the iDROPS+GMIR corrected image appears sharpest. The ghosting artifacts visible in the iDROPS-gated reconstruction in volunteer 1 is removed in the iDROPS+GMIR image; and in all cases the signal-to-noise ratio is considerably improved by iDROPS+GMIR, which incorporates all imaging data rather than just bins at one respiratory phase.

The breath-hold reference images show very low signal-to-noise ratios, consisting of only one k-space average. They do, however, all show gross blood-vessel features matching those in the reconstructions from free-breathing imaging data, ruling out any major artifacts in the reconstruction of vascular features.

Examination of iDROPS-gated images in all respiratory phases from volunteer 1, whose breathing depth was higher than the other volunteers, showed some vascular features moving in and out of plane in the superior-posterior part of the liver at inhale, accounting for the difference in reconstruction between the exhale iDROPS-gated image and the full iDROPS+GMIR corrected image which includes data from all phases.

DISCUSSION

The measurements presented of the errors in iDROPS-derived deformation fields demonstrate good accuracy, with the mean error across 10 studies of 1.43 mm, roughly half the voxel dimension of 2.7 mm along the primary direction of motion. iDROPS therefore appears to provide a useful approach to estimating deformation fields during imaging. It is encouraging that the recovered parameters seem relatively reliable even when the breathing reproducibility is poor both within the training acquisition and between the training and imaging acquisitions, as in Study 2b shown in Fig. 6. This is important because breathing irregularities and drifts often occur over the timescale of an MR examination (13). These data also suggest that the training acquisition should be long enough to measure several breathing cycles, to capture variability when it is present.

The measured error in the deformation fields arising from the motion model (about 1.24 mm) suggests that motion in some areas of the image is not linear with the chosen parameter (superior–inferior displacement). Although the 1.24 mm average error is still low compared to the image resolution, the error in specific local regions of tissue which are not moving linearly with breathing may be high: the example shown demonstrated this in the gall bladder (Fig. 7), which is anchored to the biliary system above and below the liver and may move with digestive motion.

Using a single breathing parameter, as in the examples shown in this article, does not allow for modeling hysteresis in the breathing cycle. Two or more parameter components would permit certain forms of variation and hysteresis to be captured by the model (13); it may also be worthwhile to include parameter components describing the time-derivative of the diaphragm displacement. The use of composite parameters (constructed, for example, by principal component analysis of the deformation field) may also assist in capturing the full variance of the breathing cycle. The generalization of Eq. [3] to model-building over multiple parameters is straightforward and has been discussed. Adding other parameters, perhaps derived from motion along different axes, might also allow applicability with slice geometries that do not give good coverage and resolution in the superior–inferior direction. Increasing the dimensionality of the parameter search space will lengthen computation time, though the degree of increase will depend on both the optimization algorithm and on the data itself.

The choice of image registration method used to derive the initial deformation fields ut between the training series may also influence the quality of the model. The fluid registration used here gives smooth deformation fields and hence generally a well-regularized motion model; however, with other methods explicit spatial regularization of the model (18) may be advantageous.

Other physical factors—such as age, gender, and illness, which have not been investigated in this study—may also affect the patient's ability to breathe with sufficient reproducibility for successful motion modeling and iDROPS. They may also influence the patient's breathing rate. If it is too high, the assumption that motion is minimal within individual training and undersampled imaging frames will be violated, leading to artifacts within the individual frames and possible failure of both the training registration and iDROPS steps (observed in the fourth volunteer for GMIR correction). As noted in the Theory section, the acquisition parameters (including undersampling factors and the choice of imaging sequence) must satisfy both the requirements of iDROPS and the need to obtain enough frames without internal motion artifacts in each breathing cycle.

The examples shown in Fig. 9 demonstrate that iDROPS is effective as part of a reconstruction process in free-breathing acquisition, either as a source of gating information or as an input to artifact correction with GMIR. Both the iDROPS-gated and iDROPS+GMIR images allow the resolution of blood vessel features not visible in either the breath-hold or uncorrected imaging data. As noted earlier, neither is unequivocally better in this set of examples; in two of the three examples, gating produces slightly sharper blood vessels, while GMIR removes ghosting artifacts and improves the signal-to-noise ratio. In practice, the relative preference of GMIR or gating will be determined by the requirements of continuous acquisition and signal-to-noise in the study. Depending on the hardware and acquisition, it may also be necessary to incorporate coil sensitivity profiles into the GMIR formulation in order to correctly combine coil array data containing phase differences. This type of approach has been previously been demonstrated for reconstructing diffusion images of the brain (29).

The data from volunteer 3 show extremely good recovery of detailed vascular features throughout the liver by GMIR. Conversely, the data from volunteer 1 show a small distortion of the posterior surface of the liver into the ribs; this is clearly anatomically incorrect and not present in the reference or gated images. Although further investigation is needed, this error seems to be the result of inaccuracies in the initial registration step for this case, probably caused by physical artifacts in the training images. Residual blurring of the abdominal organs inferior to the liver with GMIR is also to be expected, as motion of these organs more distant from the diaphragm will be less strongly correlated with breathing and also influenced by digestive activity.

The method has not been tested on tumours which may be encapsulated by the liver but have differing biomechanical properties. Note the deformation field can be nonrigid but the relationship to breathing parameters in the model should be linear.

Although the complete process is somewhat computationally intensive, all of the steps can be straightforwardly parallelized. The T − 1 training image registrations can be run simultaneously, as can the subsequent iDROPS deformation estimates for the Timg imaging frames. Registration time might also be decreased by using a Graphics Processing Unit (GPU) for some calculations (30). Within the iDROPS and GMIR procedures, the majority of time is spent on multidimensional FFT and non-rigid spatial resampling, both of which can also use multiple processors efficiently. Computation time depends on problem size, but with careful implementation using several multicore CPUs, final images within 10–20 min on typical-sized acquisitions should be realistic.

There are two particular areas where iDROPS may have useful impact on clinical scans. First, it may allow structural imaging of the liver or other abdominal organs where it would otherwise be difficult, when breath-holding is not well tolerated by the patient and navigator gating would result in an unacceptably long scan. Second, it may be applicable to reconstruction of dynamic contrast-enhanced data, where breath-holding cannot provide a sufficiently long imaging window and gating is disadvantageous due to the loss of imaging data during rapid contrast uptake. In these circumstances, training reference images (to use in generating cmath image for the iDROPS step) might be acquired both before and after the contrast injection, and a synthetic frame for comparison with similar tissue contrast generated by a time-varying interpolation of these two images (similar approaches have been suggested (31) to generate reference images for constrained reconstruction of dynamic series). iDROPS and GMIR would then be applied to groups of successive frames, generating a series with appropriate time resolution for analysis of dynamic contrast uptake but without the motion artifacts that would occur within these individual dynamic images had they been reconstructed conventionally.

CONCLUSION

The iDROPS approach, given suitable training and imaging data, allows deformations to be recovered with an error of a few mm in the region of the liver. These deformations can be used to remove ghosting and blurring from data acquired continuously during free breathing. Additionally, they may be used to provide retrospective respiratory gating. The method should be applicable to both dynamic and morphological MRI studies where acquisition times exceed the duration of practical breath-holds.

Acknowledgements

The authors are grateful to Philip Batchelor for several helpful discussions.

  • 1

    In the simplest case, equation image is is a simple concatenation of the Timg/gimg complete k-spaces sampled over the full imaging series; hence Pt can be a simple concatenation of one N × N identity matrix and (Timg/gimg) − 1 zero matrices of the same size, the location of the identity matrix varying with t.

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