Generalized Reconstruction by Inversion of Coupled Systems (GRICS) applied to free-breathing MRI

Authors

  • Freddy Odille,

    1. Imagerie Adaptative Diagnostique et Interventionnelle, Nancy University, Nancy, France
    2. Inserm, ERI13, Nancy, France
    3. Pôle Imagerie, University Hospital Nancy, Nancy, France
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  • Pierre-André Vuissoz,

    1. Imagerie Adaptative Diagnostique et Interventionnelle, Nancy University, Nancy, France
    2. Inserm, ERI13, Nancy, France
    3. Pôle Imagerie, University Hospital Nancy, Nancy, France
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  • Pierre-Yves Marie,

    1. Department of Nuclear Medicine, University Hospital Nancy, Nancy, France
    2. Inserm, U684, Nancy, France
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  • Jacques Felblinger

    Corresponding author
    1. Imagerie Adaptative Diagnostique et Interventionnelle, Nancy University, Nancy, France
    2. Inserm, ERI13, Nancy, France
    3. Pôle Imagerie, University Hospital Nancy, Nancy, France
    • INSERM (ERI 13), Tour Drouet 4, CHU de Nancy Brabois, Rue du Morvan, Vandoeuvre-lès-Nancy, 54511 France
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Abstract

A reconstruction strategy is proposed for physiological motion correction, which overcomes many limitations of existing techniques. The method is based on a general framework allowing correction for arbitrary motion–nonrigid or affine, making it suitable for cardiac or abdominal imaging, in the context of multiple coil, arbitrarily sampled acquisition. A model is required to predict motion in the field of view at each sample time point, based on prior knowledge provided by external sensors. A theoretical study is carried out to analyze the influence of motion prediction errors. Small errors are shown to propagate linearly in that reconstruction algorithm, and thus induce a reconstruction residue that is bounded (stability). Furthermore, optimization of the motion model is proposed in order to minimize this residue. This leads to reformulating reconstruction as two inverse problems which are coupled: motion-compensated reconstruction (known motion) and model optimization (known image). A fixed-point multiresolution scheme is described for inverting these two coupled systems. This framework is shown to allow fully autocalibrated reconstructions, as coil sensitivities and motion model coefficients are determined directly from the corrupted raw data. The theory is validated with real cardiac and abdominal data from healthy volunteers, acquired in free-breathing. Magn Reson Med 60:146–157, 2008. © 2008 Wiley-Liss, Inc.

INTRODUCTION

Patient motion is a major impediment to the acquisition of high-quality images. This is especially true in thoracic and abdominal imaging, as organs move during breathing. Motion is likely to induce several consequences on MR signal formation. Intraview and interview motion have to be distinguished between: motion is intraview when occurring during individual MR experiments (between RF excitation and echo formation), whereas motion is interview when occurring between individual MR experiments. Whenever the period of motion is slow compared to the period of MR acquisition, as defined by the repetition time TR, the assumption can be made that motion is interview. This is often a reasonable assumption when considering pseudo-periodic motion induced by respiration, and also possibly by cardiac contraction, which are the two principal sources of motion in cardiac and abdominal imaging (typically, the adult respiratory period is about 4–5 s, and TR ⋍ 10 ms for fast imaging).

Interview motion results in spatial encoding inconsistencies, and hence in image deterioration that can take complex forms (blurring/ghosting artifacts) (1) as acquisition is performed in a Fourier space. Several strategies can be employed in order to handle patient motion better. Patient cooperation is the most commonly used method. However, breath-holds cannot last much longer than 20 s and physiological drifts are likely to occur before the scan is complete (2). This leads to limitation on the time-period of signal recording and thus, signal-to-noise ratio (SNR). Moreover, the position of organs in successive breath-holds may not be reproducible. Synchronization techniques are well-established and systematically used in clinical protocols, but they require a high-level of motion reproducibility. This is often a limiting factor considering heart rate variability (whether in free breathing or during a breath-hold), and respiratory variability in terms of amplitude and frequency. This is why alternative techniques have been put forward with the aim of inverting the process of spatial encoding of moving structures that underlies artifacts.

With the technique proposed in (3), it is possible to correct for motion prospectively, by modulating the magnetic field gradients and RF fields in order to cancel the effect of motion in the Bloch equations. The main drawback of the method is that it is limited to correcting, at best, affine motion, due to magnetic field gradient systems being linear. Motion can also be compensated for in reconstruction. But all existing methods have been restricted to rigid or affine motion, either in theory (4, 5) or in practice (6, 7), due either to computational issues or to the difficulty of modeling complex deformation fields.

This article starts by presenting a reconstruction framework also descibed in (8), generalizing methods in (7) and (9), in order to correct for artifacts caused by elastic motion (nonrigid or affine) in the context of multiple coil acquisition. Practical implementation requires knowledge of complete displacement fields at each acquisition time. To overcome this problem, a model of motion in the field of view is introduced. This model is driven by a reduced number of input signals (e.g. external sensors, navigator echoes…), and described by certain coefficient maps. Assuming the model has been established, the problem amounts to solving a linear system of equations described by a simple matrix equation.

Analysis of the influence of motion prediction errors in this generalized reconstruction is then proposed. Under the assumption that the MR signal is conserved during the acquisition process, it is shown that small motion prediction errors propagate linearly in the reconstruction algorithm. This raises two important conclusions: the first one is related to stability with respect to prediction errors, and the second one is the possibility of optimizing the description of motion, and hence improving reconstruction further.

From the latter observations, it is shown that the problem can be reformulated as two linear optimization problems, which are coupled: the generalized reconstruction—which requires an estimate of motion, and the construction of an optimal motion-model—which, itself, requires an estimate of the solution image. The coupled system is solved by means of a multiresolution iterative method. The proposed resolution scheme is totally autocalibrated, which means that the high-resolution scan of interest can be autocalibrated in terms of parallel imaging (determination of coil sensitivity maps) and in terms of motion correction (determination of model coefficients). Hence, an additional scan is no longer required. Experimental results are presented in cardiac and abdominal image reconstructions, with only a few respiratory monitoring signals being needed as inputs of the motion model, provided by external sensors (respiratory bellows).

THEORY

Generalized Reconstruction Framework for Arbitrary Motion Compensation

Correction for arbitrary motion, in reconstruction, has been demonstrated in (7). This work can be extended to multiple coil acquisition (8), by combination with the general parallel imaging method proposed in (9). The resulting framework involves modeling the acquisition pipeline as shown in Fig. 1. We consider acquisition of a volume of size Nx × Ny × Nz, using a coil array of Nγ receivers. Thus, the k-space signal vector s is related to the 3D image at a certain reference time t0, denoted ρ0, via a generalized encoding operator E:

equation image(1)
Figure 1.

Description of MR acquisition pipeline by linear operators applied successively. If spatial transformation operators are known, this representation allows motion-compensated reconstruction by inversion of one single linear system.

E is comprised of the different operators applied in cascade that are represented in Fig. 1:

equation image(2)

where we used the following notations: ξmath image is the sampling operator at time tn, of size Nmath image × NxNyNz (sparse matrix, with only 0 or 1 values in case of Cartesian sampling), F is the 3D Fourier transform operator, of size NxNyNz × NxNyNz, σγ is the γth coil sensitivity weighting operator, of size NxNyNz × NxNyNz (diagonal matrix), Tmath image is the spatial transformation operator at time tn, of size NxNyNz × NxNyNz (sparse matrix), representing motion between t0 and tn.

This formulation is very general because it allows arbitrary motion to be compensated for, and it includes the parallel imaging theory with arbitrary k-space sampling, as it also generalizes (9).

Instead of solving Eq. [1], a more practical form is found with the Hermitian symmetric system defined by:

equation image(3)

with

equation image(4)

Equation [3] can be solved by means of an iterative system solver that does not require explicit knowledge of all elements in operator EH E, such as conjugate gradient methods, or the Generalized Minimum RESidual (GMRES). Such iterative methods only require explicit knowledge of the function equation image which can be computed by applying the different operators in the sum [4] successively. In practice, this enables the demand in RAM to be kept relatively low, and also to make use of fast Fourier transform algorithms. The spatial transformations Tmath image are interpolation operators, so they can be represented by sparse matrices. The number of nonzero elements depends on the interpolation basis that is chosen (linear, cubic, sinc with Kaiser-Bessel window…).

Conditionning of the encoding operator can be increased by simply repeating the scan several times (number of excitations NEX > 1), which amounts to overdetermining the problem. In a conventional approach, data would be averaged prior to reconstructing, which would result in blurring, with a convolution kernel width on the order of motion amplitude, as shown in (10). Here, data from each excitation are treated independently, as they have been acquired in a different spatial configuration. This allows new linearly independent equations to be added to the system. Tikhonov regularization may also be employed if E remains not well conditioned for numerical resolution, which is done, in practice, by replacing EHE by EHE + λ Id, with Id being the identity matrix and λ a small tuning parameter.

Propagation of Motion Prediction Errors - Stability Analysis

Assuming that only an estimate T̂math image of each spatial transformation operator Tmath image is available, Eq. [1] can be rewritten as above:

equation image(5)

ε represents a residue term expressed as:

equation image(6)

where equation image and equation image denote the real image at time tn and its estimate, respectively. To begin with, we consider this estimation error as small, which can be thought of as a small perturbation of the system. Furthermore, we assume that the MR signal is conserved during the acquisition process. We propose to express the latter hypothesis locally by the optical flow equation. This equation has been widely used in image processing applications (11, 12), including recent research work (13). The optical flow equation is applied here to linearization of the small difference between real image at time tn and its estimate. Introducing, for each acquisition time tn, the displacement error vector field equation image, made on each displacement field estimate, it reads:

equation image(7)

As a result, the residue can be expressed as a function of the error made on displacement fields, instead of the error made on image estimates:

equation image(8)

Equation [8] clearly shows how motion prediction errors are propagated in the reconstruction algorithm. They result in a reconstruction residue that is related to the prediction error equation image by a linear operator. As a result, the reconstruction defined by inversion of Eq. [1] is stable with respect to motion prediction errors, since small prediction errors induce bounded reconstruction errors. Moreover, a regularized solution to Eq. [1] is known to ensure existence and uniqueness of the solution. So we can conclude that, under the signal conservation assumption, the motion-compensated reconstruction is well-posed in Hadamard's sense (existence, uniqueness, and stability).

It also appears that, due to the spatial gradient term in Eq. [8], prediction errors have more weight when located in image areas with large intensity variations. On the contrary, the reconstruction is not very sensitive to prediction errors in homogeneous areas. Moreover, since the terms equation image are weighted by image gradients, they contain high spatial frequencies that will enter the Fourier operator in Eq. [8]. Hence peripheral k-space data will be affected in particular. Therefore, with nonrigid, local deformations, accurate (high resolution) displacement fields are required to cancel all artifacts.

From a computational point of view, Eq. [8] is very interesting due to the linear relationship between prediction errors and reconstruction errors. Indeed an optimization scheme can be implemented in order to find better displacement field estimates, minimizing the reconstruction residue.

Use of a Motion Model

Optimizing Eq. [8] with respect to motion is not easy, as it still contains a high number of unknowns (displacement error fields at each acquisition time tn). This may be a major obstacle to practical implementation as the problem may be highly underdetermined. One way to overcome this difficulty is to introduce a model for patient motion that drives the motion estimation process. Thus the number of parameters describing motion can be reduced significantly.

Rigid or affine models have already been proposed to this end (3–6). These models allow global translational or rotational motion to be described, which makes them well-suited for correction of head motion. Scaling and shear terms can also be added, but they are of limited relevance. Indeed, although these kinds of displacements are able to describe nonrigid motion locally, they are unlikely to describe motion on a global scale in the field of view. Affine motion requires displacements to be linearly dependent on the spatial coordinate, so the representation depends on the chosen coordinate system, which is a major limitation (in particular, the rotation center must be the same as the homothety center).

Given the insufficient number of degrees of freedom allowed by affine models, it is proposed to describe motion by locally free deformations, while constraining the temporal dimension. This is done by forcing the time evolution of each voxel to be expressed as a linear combination of certain input signals, reasonably correlated with true motion. This kind of motion model, driven by a few input signals, has similarities with existing ones proposed in prospective motion correction problems (3, 14) and in postprocessing motion analysis (15–17). Example signals that may be used as model inputs include navigator echoes and external sensors (respiratory belts or signals derived from the ECG). Thus, denoting S(t) = [1(t) … SK(t)[T the model input signals, and u(r,t) the displacement fields at position (r) = [x,y,z[T and time t, the proposed model allows elastic displacements to be described by the following linear combinations:

equation image(9)

The coefficients maps equation image describe the motion model (there is one coefficient map per input signal and per spatial dimension).

Using the proposed representation of motion, prediction errors in [8] can be rewritten as a function of the model parameters α, instead of the displacements u:

equation image(10)

yielding an expression of the form:

equation image(11)

The linear operator R involved in the residue expression depends on the image estimates that are already available, given the solution image ρ0 and an initial motion model, described by certain coefficient maps α. δα represents the modification that should be applied to the coefficient maps in order to cancel the reconstruction residue ε.

Generalized Reconstruction by Inversion of Coupled Systems (GRICS)

We propose to merge the two inverse problems defined by Eqs. [1] and [11]: generalized reconstruction, using motion predicted by a given model, and model optimization, minimizing the reconstruction residue. The solution image to the generalized reconstruction depends on the model parameters; however, the optimal model depends on the solution image. So the two inverse problems are coupled, as described below:

equation image(12)

METHODS

Proposed Multiresolution Fixed-Point Scheme

The graphical representation in Fig. 2 describes the coupled inverse problem and the following resolution scheme. We propose to solve system [11] using a fixed-point method. Starting from an initial model estimate α(0), the first optimization problem is solved (reconstruction), with the model being fixed. A first solution image ρmath image is found. The reconstruction residue is computed and is nonzero in the presence of residual motion. Then, by fixing the solution image, the model optimization step is performed, yielding a model modification δα(0). Afterwards, the model is updated: equation image. Then, all the process is repeated, with a new reconstruction based on the updated model. Iterations are performed until a stopping condition is reached, e.g. when the residue stops decreasing.

Figure 2.

Schematic diagram representing the GRICS framework († denotes a pseudo-inverse). The reconstruction step (inversion of the encoding operator E described by Fig. 1) depends on motion, which is represented by a model comprised of certain parameters, α. The model can be optimized by minimizing the reconstruction residue with respect to model coefficients. This minimization itself, depends on the solution image. So, the two inverse problems (motion-compensated reconstruction and model optimization) are coupled. The solutions to the coupled problem are a solution image and an optimal motion model.

The optical flow equation in [7], and hence the whole rational, are valid only for small values of the displacement errors δu. As initialization, it is possible to use a model built during a calibration step, played in free breathing before the high-resolution scan of interest, in a similar way to methods described in (14–16). When no prior knowledge is available, it is still possible to handle large displacements by implementing the fixed-point algorithm inside a multiresolution scheme. Indeed, there always exists a resolution at which motion can be considered as small. So it is possible to use an initial model with null coefficients (i.e. equation image). The fixed-point iterations are performed at the starting (low) resolution, using only the most central k-space data. Thus, a low-resolution image is reconstructed, and low-resolution coefficient maps are determined. Once the fixed-point iteration has been completed, coefficient maps are interpolated to the next resolution level, and serve as initialization for the new resolution level. The resulting multiresolution, fixed-point algorithm is summarized in Appendix A.

Generalized Reconstruction Step

The generalized reconstruction step also requires knowledge of the coil sensitivity maps. Here we propose an autocalibrated method in which k-space acquisition is fully sampled. Sensitivity maps are built by extracting the central k-space data. This amounts to considering that the influence of motion on coil sensitivities is negligible at the resolution chosen for sensitivity map computation. This may not always be exact, however this is not a limitation, since sensitivity maps may easily be reevaluated after each iteration. Thus possible errors in their determination, due to motion artifacts, would be compensated for.

Additionally, it is important to understand that coil sensitivity variations seen by voxels which are moving are actually taken into account in the formulation [1], because spatial transformations are applied before coil weighting operators (this point is also described and discussed in (5)). Only explicit time dependency of coil sensitivities is not taken into account, that is, changes due to non fixed coil arrays, or due to variations in coil loading. These effects are considered to be negligible here.

Model Optimization Step

The model optimization step has similarities with optical flow motion detection methods used in video processing. The optical flow equation leading to an underdetermined system (1 equation, 3 unknowns for each voxel in 3D), means that it is necessary to introduce an additional constraint based on geometry. This problem is slightly different because the source data (the coil k-spaces) are in the MRI encoding space, rather than in the spatial domain. Here, with the chosen motion model, we use several NEX, so the problem is mathematically less underdetermined than the classical optical flow formulation that can be found in (11) (Horn and Schunk formulation). By analogy with the Horn and Schunk method, we propose solving the problem by using an additional constraint based on geometry, ensuring smooth spatial variations in the solution. The model modification at iteration k is found by:

equation image(13)

So Eq. [12] involves minimizing a functional comprised of two terms: a quadratic error term derived from Eq. [10], and the squared norm of spatial gradients in the coefficient maps. equation image denotes the Euclidean norm. Further implementation details are given in Appendix B.

Convergence

We have not proved mathematically that the fixed-point multiresolution scheme converges. However, we propose an experimental study of convergence by monitoring the evolution of the residue ε over iterations (see Results section). Fixed-point iterations were performed until ε stopped decreasing, with a maximum number of iterations, at each resolution level, set to 8. If the residue is found to increase after a certain iteration, the result corresponding to the best residue is kept for the next resolution level. This strategy ensures a robust implementation because, in practice, GRICS reconstruction cannot be worse than standard reconstruction (equivalent to that obtained with a model α≡0) in terms of minimal reconstruction residue.

MRI Experiments

MR experiments were performed using a 1.5 T GE Signa Excite HDX MR system (General Electric, Milwaukee, WI). 2D and 3D MRI data were acquired in both phantom experiments and healthy volunteers, in free breathing. Pulse sequence details are given in Table 1. Cardiac scans were ECG triggered, with the proposed reconstruction compensating for respiratory motion. A majority of black blood RARE pulse sequences were used in 2D experiments (black blood FSE, TE = 36 ms, ETL = 16, TI = 650 ms). Through-plane motion effects were minimized by programming relatively thick slices (10 mm), mainly in sagittal orientation. Short axis slices from the heart were also tested. 3D data were acquired with a balanced SSFP sequence (3D FIESTA, TE = 2 ms, TR = 3 ms), with Cartesian sequential sampling of the k-space.

Table 1. Description of Motion Corrupted Scans Used for Validation, Including 2D and 3D Scans Targeted on Various Organs
Subject numberScan numberNEXOrientationOrganPulse sequencek-space matrix# Coil elements
Phantom-4Sagittal-Black blood FSEFSE 256 × 2568
Subject 1A3SagittalHeartBlack blood FSE256 × 2568
 B3SagittalHeartBlack blood FSE512 × 5128
 C3Short axisHeartBlack blood FSE512 × 5128
Subject 2A2SagittalHeartBlack blood FSE256 × 2568
 B2SagittalHeartBlack blood FSE512 × 5128
 C2SagittalLiverBlack blood FSE256 × 2568
 D2SagittalLiverBlack blood FSE512 × 5128
 E5Short axisHeart3D FIESTA128 × 128 × 648
Subject 3A3SagittalLiverBlack blood FSE256 × 2568
 B3SagittalHeartBlack blood FSE256 × 2568
Subject 4A3SagittalLiver/kidneyGRASS256 × 25612
Subject 5A3SagittalLiver/kidney3D FIESTA256 × 256 × 328
Subject 6A3Short axisHeartBlack blood FSE256 × 2568
 B3SagittalHeartBlack blood FSE256 × 2568
 C3SagittalLiver/kidney3D FIESTA256 × 256 × 328
 D5Short axisHeart3D FIESTA128 × 128 × 648

Coil arrays of 8 elements or more were employed. Mainly fully sampled scans were carried out. Indeed we focussed on putting forward motion artifact reduction, without being hindered by possible SNR loss caused by high g-factors in parallel imaging. Moreover, the proposed method aims at recovering higher SNR images, since it is not limited by acquisition time—on the contrary, the reconstruction improves with the acquisition time, as a multiple NEX overdetermines system [11].

A moving phantom setup was also used for our validations, comprised of both static and moving phantoms. The moving part achieved 1D periodic translation, with a pneumatic command generated by a ventilator simulating respiratory motion. The cardiac synchronization signal was generated by an ECG simulator device, in order that the configuration was the same as in our experiments with volunteer subjects. The imaging plane was chosen so that the moving phantom remains in plane. The setup produced displacements of 30 mm in amplitude in the up/down direction, and 5 mm in the right/left direction, and the period for translational motion was set to 5 s. These settings are typical of adult respiratory motion.

External Sensors Used as Model Inputs

Physiological signals were collected using a modified version of the Maglife (Schiller Médical, Wissembourg, France) patient monitoring system. These signals were converted into optical signals, transmitted outside the scan room through optical fibers, and then recorded on the signal analyzer and event controller (SAEC) computer and electronics system (18). The SAEC system allows synchronous acquisition and recording of physiological signals (respiration and ECG) and MR signals (image acquisition windows and MR gradients).

In 2D reconstructions, four input signals were used, including two respiratory bellows (one each on the thorax and abdomen), and their derivatives. Using the derivatives can help to compensate for dephasing between true motion of an organ and external sensor measurements. In 3D reconstructions, only one respiratory bellow was used, in order to limit the reconstruction time, and was chosen according to the organ of interest.

Numerical Implementation Details

Practical implementation required certain simplifications for relatively fast resolution. Spatial transformation operators were applied through linear interpolation, offering a good compromise between regularity and the computation time needed. The number of terms in sum [4] was reduced by grouping, as much as possible, k-space data being acquired in similar spatial configurations. This was done by clustering K-tuples of sensor measurements, after quantization of all model input signals onto 8 quantization levels. The number of quantization levels was chosen to remain of the same order as maximal motion amplitude (in voxel units).

The following resolution levels (expressed in terms of image size) were used in 2D examples: 32 × 32, 64 × 64, 128 × 128, 256 × 256. The last resolution level is, by far, the more demanding in terms of computational power. It is proposed to optimize the model until the penultimate level. Thus, in the ultimate level, only the reconstruction step is performed.

The regularization parameters involved in the two inverse problems were empirically set to λ = 0.1 and μ = 0.1 ∥ε(a more systematic method could be employed, e.g. based on the study of L-curves (19)). The GMRES was chosen as the iterative system solver for both inversions. Not all inner iterations were performed inside the model optimization step (GMRES iterations) in order to speed up resolution further. System coupling makes this possible: indeed, it is most advantageous to update the solution image and the model often, rather than waiting for complete convergence of every single optimization step.

GRICS reconstructions were executed on a recent workstation (AMD Opteron 265, 8 GB RAM), using Matlab (Mathworks, Natick, MA) code. Reconstruction times, for 2D 256 × 256 images, ranged from 3 min (phantom data with 1 model input) to 10 min (with four model inputs). In 3D, with only one model input used, reconstruction times ranged from 40 min (128 × 128 × 32 volume) to 210 min (256 × 256 × 32 volume). Our Matlab implementation was single threaded; however, the reconstruction is highly parallelizable. Indeed the main computations in both optimization problems ( equation image and equation image) can be parallelized as they both involve a sum over independent terms.

RESULTS

Moving Phantom Experiment

Reconstructions from the moving phantom setup are shown in Fig. 3. Images reconstructed by standard Fourier transform show typical ghost artifacts produced by the moving part of the setup in the phase encoding direction (1). The GRICS reconstruction was able to remove most artifacts efficiently. A residual artifact can be seen in the center of the resolution phantom, in which only the finest details appear with residual blurring.

Figure 3.

Reconstruction from the moving phantom, achieving in-plane periodic piecewise translation: standard reconstruction (a), GRICS (b); static acquisition for reference (c). The moving part of the setup achieved periodic translation with a maximum amplitude of 29.5 mm in the SI direction (up/down) and 5.1 mm in the AP direction (right/left). Motion artifacts are removed efficiently. Only the finest details in the resolution phantom appear with slight residual blurring in the GRICS reconstruction compared to the static scan.

Free-breathing subject data

Example reconstructions from healthy volunteers are shown in Fig. 4 (2D, short axis slice from the heart) and Fig. 5 (3D volume in the liver and left kidney). Visual comparison indicates a significant improvement with the GRICS reconstruction compared to the standard Fourier reconstruction. A reference scan acquired in breath-hold is also shown for comparison.

Figure 4.

Reconstruction from an ECG triggered RARE sequence (short axis slice, Subject 6A), acquired in free breathing: standard reconstruction (a), GRICS with full sampling (b), GRICS with undersampling (R = 1.8) (c); breath-hold for reference (d).

Figure 5.

Reconstruction from a 3D balanced SSFP sequence (liver and right kidney, Subject 5A), acquired in free breathing. One sagittal slice is represented on top: standard reconstruction (a), GRICS (b), breath-hold for reference (c). One axial slice is represented on the bottom: standard reconstruction (d), GRICS (e), breath-hold for reference (f).

Table 2. Assessment of Reconstructed Image Quality with Standard Fourier Based Reconstruction and with GRICS (Full Sampling)
 Entropy (less is better)Gradient entropy (more is better)
FourierGRICSFourierGRICS
Phantom3.443.050.750.7
Subject 1A4.194.140.940.97
Subject 1B4.534.511.051.19
Subject 1C43.980.760.78
Subject 2A3.923.821.11.13
Subject 2B3.813.780.790.83
Subject 2C3.753.581.130.99
Subject 2D3.733.570.590.6
Subject 2E4.13.931.591.7
Subject 3A3.343.350.80.83
Subject 3B3.533.490.720.73
Subject 4A4.134.211.321.5
Subject 5A4.224.031.631.81
Subject 6A4.184.141.021.1
Subject 6B3.813.771.041.09
Subject 6C4.053.881.541.68
Subject 6D4.194.051.61.75

GRICS results using undersampled data are also displayed in Fig. 4. Undersampling was performed retrospectively, by removing one k-space line out of two, except the 32 central phase encodes that were necessary for coil sensitivity autocalibration. This resulted in a reduction factor of R = 1.8. In that example, undersampling did not cause reducing the total number of shots (only the amount of data within each shot), so the temporal overdetermination needed by the motion model was preserved. The undersampled reconstruction exhibited more residual artifacts than the fully sampled GRICS reconstruction, although being significantly improved compared to the Fourier reconstruction.

Quantitative assessment of reconstructed image quality is given in Table 2. Two criteria commonly used in image processing are indicated: entropy quantifies signal dispersion in the image (motion artifacts generally cause signal dispersion and thus increase entropy), and gradient entropy quantifies singularity of the image, that is, the amount of fine details (blurring artifacts affect mainly sharp edges and thus decrease gradient entropy). GRICS reconstructions show improvements in the sense of both image quality criteria in most cases. All images were improved in the sense of at least one of the two criteria used. T2

Precalibrated VS Autocalibrated Models

In one example, we compared results obtained by GRICS reconstruction with those obtained with a fixed motion model, providing prior calibration (free-breathing calibration scan), using the method proposed in (8). Displacement fields predicted by both models (prior calibration and autocalibration), are shown in Fig. 6, in one typical respiratory cycle. Slight differences can be seen, however, similar values are found in both prediction fields in corresponding areas/organs (in the present problem, no ground truth is available). The example in Fig. 6 is a typical situation in which rigid or affine models would be suboptimal, whether considering prior calibration or autocalibration results. Corresponding reconstructions are shown in Fig. 7.

Figure 6.

Comparison between displacement fields obtained by prior motion model calibration (method in (8)) and by autocalibration (GRICS method), in data from Subject 6B. Reconstructed images are shown in Fig. 7.

Figure 7.

Reconstruction from an ECG triggered RARE sequence (sagittal slice from the heart, Subject 6B): standard reconstruction (a), motion-compensated reconstruction with fixed model obtained by prior calibration (b), GRICS (c); breath-hold for reference (d). The displacement fields used in (b) and (c) are shown in Fig. 6.

Combining the GRICS method with prior calibration as initialization was also tested. These attempts led to intermediate results in terms of predicted displacement fields (between those found by prior calibration and autocalibration), although this had minor influence on reconstructed image. Therefore, optimal solution models may rely on good initialization in certain situations.

Convergence

Practical convergence of the GRICS reconstruction can be assessed though evolution curves, as shown in Fig. 8. These plots show that the residue is decreased significantly at the different resolution levels. The method could still be optimized by stopping iterations earlier, i.e. as soon as the decrease is inferior to a certain fixed tolerance.

Figure 8.

Monitoring of GRICS convergence with the proposed multiresolution fixed-point algorithm (Appendix A), in 256 × 256 reconstructions (8 pulse sequences, see details in Table 2). Plots show the decrease of the reconstruction residue ε over iterations, observed at each resolution level. The model was optimized until the penultimate level (128 × 128) before final reconstruction at full resolution (256 × 256).

DISCUSSION

Signal Conservation Assumption

Optimizing reconstruction with respect to the motion model, in our formulation, relies on the MR signal conservation assumption. Although this assumption is generally not stated clearly, it is implicitly done in any reconstruction method, even in a standard Fourier reconstruction. Indeed, if the signal is not conserved, inconsitencies will lie in the k-space data, thus modifying the reconstructed image (i.e. creating artifacts).

It has been proposed to use the optical flow equation, in order to impose this constraint locally. An analogy can be made with the law of mass conservation used in continuum mechanics—in the present problem, only the mass “visible” in MRI appears in the balance. The optical flow equation is equivalent to the law of mass conservation for velocity fields having constant zero divergence. In continuum mechanics, the latter hypothesis applies to incompressible media such as deformable solids or incompressible fluids. Such a description of motion in the human body should be relevant in most cases.

Differences between Precalibrated and Autocalibrated Motion Correction

Differences have been observed between displacement fields found in prior calibration and in autocalibration. The main reason for that comes from the model optimization problem being underdetermined by nature, like optical flow motion detection techniques. As can be seen from Eq. [10], motion from homogeneous areas has few consequences on the reconstruction residue, yielding an indetermination. Similarly to optical flow methods, an additional constraint has to be imposed to raise this indetermination. We chose a simple Tikhonov constraint for the resolution, imposing regularity on the displacement fields (smoothness of their spatial gradients). But other techniques inspired from optical flow processing or image inpainting problems might improve the resolution further. It is possible either to use a calibration step providing an initial estimate of the motion model, or implement the method in full autocalibration. In the latter case, increasing NEX reduces the underdetermination.

Optimal Acquisition Strategy

The proposed approach is fundamentally different from standard reconstruction techniques, in particular with regard to multiple excitations. With Fourier reconstructions, repeated excitations are averaged, resulting in blurring. This effect can be described by a convolution kernel whose width is approximately the range of motion (10). On the contrary, increasing NEX in the proposed approach improves both the motion model and the solution image, since the conditioning of both inverse problems is increased (operator E in [1], and operator R in [12]), by adding new linearly independent data to both linear systems of equations. As a result, high-resolution and high-SNR scans, requiring a high-NEX, and that cannot be accomplished within a breath-hold, will benefit most advantageously from the proposed method.

The main limitation of the method is related to the classes of displacements that are allowed by the model. As the time dimension is constrained by input signals, only motion that is correlated with these input signals will be compensated for. This assumption is the cost to pay for allowing spatially free deformations. Describing time evolution under less restrictive assumptions would also be possible, by adding linearly independent signals to the model inputs. The reason for keeping the number of model inputs very low is essentially due to computational issues. If enough inputs can be employed, it should also be possible to consider the resulting motion model for diagnostic purposes, as it is done with k-t sampling methods used in dynamic MRI reconstructions (20, 21). The particularity of our approach is that MR signals from every time sample are registered and thus combined to form the reconstructed image. On the contrary, k-t methods do not combine these data, as they aim at reconstructing missing, unacquired data, using prior knowledge about expected temporal and/or spatial frequencies in the k-t space.

Another issue is associated with the number of quantization levels applied to the model inputs in order to reduce the computational complexity. If L is the number of quantization levels, and K the number of model inputs, a maximum number of LK motion states can be represented by the model (actually there are less because of the partial correlation between these inputs). To correct for motion on the order of magnitude of a voxel (or subvoxel motion), the model should describe at least as many motion states as the number of voxels between the reference position and the position of maximal motion amplitude. The optimal value of L depends on the number of inputs, the partial correlation between these inputs, and the maximal amplitude of motion. Roughly, in the worst case (with only one input, or if all inputs are collinear), L should be at least the maximal motion amplitude, in voxel units.

Other Sources of Bad Conditioning

Sources of bad conditioning, other than motion, may be present in the reconstruction problem. Hence the reconstruction residue may contain additional terms. More generally, it is possible to optimize other sources of perturbation in the same way as it has been done for motion. If a model for that perturbation can be added to the acquisition pipeline, it will be possible to optimize that particular model as well. The GRICS reconstruction will then be comprised of more than two coupled systems of equations (one system for image reconstruction + one system for optimization of each perturbation model), which may be solved in a similar manner (fixed-point multiresolution scheme, with one model being optimized at the time, while other models are fixed). Further work should be done to investigate the theoretical convergence of such a method. One possible limitation is related to practical implementation, as not all sources of perturbations may be formulated as a linear optimization problem like in the present application.

A favorable situation occurs when a perturbation model, described by certain parameters α, can be reduced to a diagonal operator, placed at the beginning of the acquisition pipeline (see Fig. 1). Then, taking advantage of the commutation property equation image, it is possible to optimize (linearly) the reconstruction process with respect to both ρ0 and α. Thus, other possible applications of the GRICS framework include parallel imaging (coil sensitivity determination), correction of artifacts induced by B0 or B1 inhomogeneities (by introducing B0 or B1 maps in the acquisition pipeline), correction of artifacts specific to certain pulse sequences (dephasing in EPI or multi-echo sequences, trajectory deformation in spiral scans…), and modeling of signal intensity variations during the scan (e.g. due to a contrast agent). Our method has similarities with joint optimization problems proposed in (22, 23). The main difference is that GRICS uses different cost functions for each optimization problem, arising from the reconstruction residue expression.

CONCLUSION

In conclusion, a framework has been proposed for correcting, in reconstruction, motion artifacts induced by spatial encoding errors (interview motion). In this formulation, image reconstruction (in a general way) and motion-model optimization are treated as two inverse problems which are coupled. The framework overcomes many limitations of existing techniques, as its scope is very large and it makes fewer asumptions about motion. Indeed, the theory applies to arbitrary k-space sampling, it is compatible with multiple coil acquisition as it includes SENSE reconstruction, and it allows arbitrary motion to be corrected (without restriction to rigid or affine motion). The latter property is of the highest relevance for cardiac or abdominal imaging. However, a model for patient motion is required to reduce the number of parameters describing motion, in order to make the optimization process practically feasible. A few monitoring signals, provided by external sensors (like respiratory bellows or ECG (24)) or navigator echoes, reasonably correlated with motion, are already sufficient to drive the model and improve image quality significantly. As in the results presented here, the method may be fully autocalibrated, in the sense of both multiple coil acquisition (determination of coil sensitivity maps) and motion correction (determination of model coefficient maps). Therefore the framework might allow providing accurate MR images from thoracic or abdominal organs, even in patients for whom breath holding is impossible or inaccurate (young children, patients with dyspnea or cerebral disease…). Moreover, the GRICS method may be applied to scans recorded during an unlimited time period and thus, with a potential for higher SNR than conventional images acquired during breath holding. This property is highly favorable for investigations requiring high spatial resolution, such as coronary angiography.

Acknowledgements

The authors would like to thank Schiller Médical for providing an important part of the instrumentation required.

Appendix A

Multiresolution Fixed-Point Algorithm

Initialize motion model: e.g. α(0) ≡ 0

For each resolution level r (low to high resolution)

 Do

  Optimize residue /ρ0 (motion-compensated recon-struction):

   

equation image

  Compute residue:

   

equation image

  Optimize residue /δα (motion model optimization):

   

equation image

  Update motion model:

   

equation image

 until stopping condition is reached (e.g. residue stops decreasing)

end for

Solution image and model (ρ0, α) to the coupled system, at final resolution level

Appendix B

Implementation of Motion Model Optimization

The model optimization step aims at solving the problem defined by Eq. [12]. In the following we use simplified notations: R stands for equation image and α stands for α(k), k being the iteration index. A solution is found by writing the Euler-Lagrange equation:

equation image(B1)

An explicit expression of the R operator is given below:

equation image(B2)

and for each dimension equation image:

equation image(B3)

Therefore δα is solution of:

equation image(B4)

The problems amounts to a linear system of the form A δα = b. As for the Eq. [1], inversion is performed using an iterative system solver.

The system in Eq. B4 is complex, however the model coefficients should be real, since they account for displacement values. In our implementation, we did not impose any additional constraint in order to obtain pure real solutions, and imaginary part was simply truncated. A more rigorous implementation may be proposed, as similar problems can be found in partial Fourier techniques (25, 26) and in parallel imaging (27). This simplification did not prevent convergence in our experimental results, as this optimization was part of an iterative process.

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