Philip Batchelor

A Rigorous Framework for Diffusion Tensor Calculus

In biological tissue, all eigenvalues of the diffusion tensor are assumed to be positive. Calculations in diffusion tensor MRI generally do not take into account this positive definiteness property of the tensor. Here, the space of positive definite tensors is used to construct a framework for diffusion tensor analysis. The method defines a distance function between a pair of tensors and the associated shortest path (geodesic) joining them. From this distance a method for computing tensor means, a new measure of anisotropy, and a method for tensor interpolation are derived. The method is illustrated using simulated and in vivo data. Magn Reson Med 53:221–225, 2005.

Anisotropic noise propagation in diffusion tensor MRI sampling schemes

The subject of this study is the controversial choice of directions in diffusion tensor MRI (DT‐MRI); specifically, the numerical algebra related to this choice. In DT‐MRI, apparent diffusivities are sampled in six or more directions and a least‐squares equation is solved to reconstruct the diffusion tensor. Numerical characteristics of the system are considered, in particular the condition number and normal matrix, and are shown to be dependent on the relative orientation of the tensor with respect to the laboratory frame. As a consequence, noise propagation can be anisotropic. However, the class of icosahedral direction schemes is an exception, and icosahedral directions have the same condition number and normal matrix for direction encoding as the ideal scheme with an infinite number of directions. This normal matrix and its condition number are rotationally invariant. Numerical simulations show that for icosahedral schemes with 30 directions the standard deviation of the fractional anisotropy is both low and nearly independent of fiber orientation. The recommended choice of directions for a DT‐MRI experiment is therefore the icosahedral set of directions with the highest number of directions achievable in the available time. Magn Reson Med 49:1143–1151, 2003.

Matrix description of general motion correction applied to multishot images

Motion of an object degrades MR images, as the acquisition is time‐dependent, and thus k‐space is inconsistently sampled. This causes ghosts. Current motion correction methods make restrictive assumptions on the type of motions, for example, that it is a translation or rotation, and use special properties of k‐space for these transformations. Such methods, however, cannot be generalized easily to nonrigid types of motions, and even rotations in multiple shots can be a problem. Here, a method is presented that can handle general nonrigid motion models. A general matrix equation gives the corrupted image from the ideal object. Thus, inversion of this system allows us to get the ideal image from the corrupted one. This inversion is possible by efficient methods mixing Fourier transforms with the conjugate gradient method. A faster but empirical inversion is discussed as well as methods to determine the motion. Simulated three‐dimensional affine data and two‐dimensional pulsation data and in vivo nonrigid data are used for demonstration. All examples are multishot images where the object moves between shots. The results indicate that it is now possible to correct for nonrigid types of motion that are representative of many types of patient motion, although computation times remain an issue. Magn Reson Med, 2005.

Motion corrected compressed sensing for free-breathing dynamic cardiac MRI

Compressed sensing (CS) has been demonstrated to accelerate MRI acquisitions by reconstructing sparse images of good quality from highly undersampled data. Motion during MR scans can cause inconsistencies in k‐space data, resulting in strong motion artifacts in the reconstructed images. For CS to be useful in these applications, motion correction techniques need to be combined with the undersampled reconstruction. Recently, joint motion correction and CS approaches have been proposed to partially correct for effects of motion. However, the main limitation of these approaches is that they can only correct for affine deformations. In this work, we propose a novel motion corrected CS framework for free‐breathing dynamic cardiac MRI that incorporates a general motion correction formulation directly into the CS reconstruction. This framework can correct for arbitrary affine or nonrigid motion in the CS reconstructed cardiac images, while simultaneously benefiting from highly accelerated MR acquisition. The application of this approach is demonstrated both in simulations and in vivo data for 2D respiratory self‐gated free‐breathing cardiac CINE MRI, using a golden angle radial acquisition. Results show that this approach allows for the reconstruction of respiratory motion corrected cardiac CINE images with similar quality to breath‐held acquisitions. Magn Reson Med 70:504–516, 2013.

Quantification of the shape of fiber tracts

The fiber tracts generated using diffusion MRI are usually simply displayed and assessed visually for a specific clinical or medical research purpose. This paper proposes computational techniques that can be used to study the shape of the tracts and make interindividual comparisons. These methods make use of fundamental geometric invariants, such as curvatures and torsions, or Fourier descriptors, together with the link of a pair of curves. Intersubject comparisons only require that the starting and ending points of the tracts can be defined and do not require point‐by‐point correspondences such as obtained using image registration. Principal component analysis‐based shape analysis is also investigated. The invariants are tested on simulations and in vivo datasets, and the scale dependence and noise sensitivity of the measures are assessed. The potential for these techniques to be used in neuroscience research and clinical applications is demonstrated. Magn Reson Med, 2006.

In vivo human cardiac fibre architecture estimation using shape-based diffusion tensor processing

In vivo imaging of cardiac 3D fibre architecture is still a practical and methodological challenge. However it potentially provides important clinical insights, for example leading to a better understanding of the pathophysiology and the follow up of ventricular remodelling after therapy. Recently, the acquisition of 2D multi-slice Diffusion Tensor Images (DTI) of the in vivo human heart has become feasible, yielding a limited number of slices with relatively poor signal-to-noise ratios. In this article, we present a method to analyse the fibre architecture of the left ventricle (LV) using shape-based transformation into a normalised Prolate Spheroidal coordinate frame. Secondly, a dense approximation scheme of the complete 3D cardiac fibre architecture of the LV from a limited number of DTI slices is proposed and validated using ex vivo data. Those two methods are applied in vivo to a group of healthy volunteers, on which 2D DTI slices of the LV were acquired using a free-breathing motion compensated protocol. Results demonstrate the advantages of using curvilinear coordinates both for the anaylsis and the interpolation of cardiac DTI information. Resulting in vivo fibre architecture was found to agree with data from previous studies on ex vivo hearts.

k-t group sparse: A method for accelerating dynamic MRI

Compressed sensing (CS) is a data‐reduction technique that has been applied to speed up the acquisition in MRI. However, the use of this technique in dynamic MR applications has been limited in terms of the maximum achievable reduction factor. In general, noise‐like artefacts and bad temporal fidelity are visible in standard CS MRI reconstructions when high reduction factors are used. To increase the maximum achievable reduction factor, additional or prior information can be incorporated in the CS reconstruction. Here, a novel CS reconstruction method is proposed that exploits the structure within the sparse representation of a signal by enforcing the support components to be in the form of groups. These groups act like a constraint in the reconstruction. The information about the support region can be easily obtained from training data in dynamic MRI acquisitions. The proposed approach was tested in two‐dimensional cardiac cine MRI with both downsampled and undersampled data. Results show that higher acceleration factors (up to 9‐fold), with improved spatial and temporal quality, can be obtained with the proposed approach in comparison to the standard CS reconstructions. Magn Reson Med, 2011.

In vivo human 3D cardiac fibre architecture: reconstruction using curvilinear interpolation of diffusion tensor images

In vivo imaging of the cardiac 3D fibre architecture is still a challenge, but it would have many clinical applications, for instance to better understand pathologies and to follow up remodelling after therapy. Recently, cardiac MRI enabled the acquisition of Diffusion Tensor images (DTI) of 2D slices. We propose a method for the complete 3D reconstruction of cardiac fibre architecture in the left ventricular myocardium from sparse in vivo DTI slices. This is achieved in two steps. First we map non-linearly the left ventricular geometry to a truncated ellipsoid. Second, we express coordinates and tensor components in Prolate Spheroidal System, where an anisotropic Gaussian kernel regression interpolation is performed. The framework is initially applied to a statistical cardiac DTI atlas in order to estimate the optimal anisotropic bandwidths. Then, it is applied to in vivo beating heart DTI data sparsely acquired on a healthy subject. Resulting in vivo tensor field shows good correlation with literature, especially the elevation (helix) angle transmural variation. To our knowledge, this is the first reconstruction of in vivo human 3D cardiac fibre structure. Such approach opens up possibilities in terms of analysis of the fibre architecture in patients.

Nonlinear phase correction of navigated multi‐coil diffusion images

Cardiac pulsatility causes a nonrigid motion of the brain. In multi‐shot diffusion imaging this leads to spatially varying phase changes that must be corrected. A conjugate gradient based reconstruction is presented that includes phase changes measured using two‐dimensional navigator echoes, coil sensitivity information, navigator‐determined weightings, and data from multiple coils and averages.

Perfusion phantom : an efficient and reproducible method to simulate myocardial first-pass perfusion measurements with cardiovascular magnetic resonance

The aim of this article is to describe a novel hardware perfusion phantom that simulates myocardial first‐pass perfusion allowing comparisons between different MR techniques and validation of the results against a true gold standard. MR perfusion images were acquired at different myocardial perfusion rates and variable doses of gadolinium and cardiac output. The system proved to be sensitive to controlled variations of myocardial perfusion rate, contrast agent dose, and cardiac output. It produced distinct signal intensity curves for perfusion rates ranging from 1 to 10 mL/mL/min. Quantification of myocardial blood flow by signal deconvolution techniques provided accurate measurements of perfusion. The phantom also proved to be very reproducible between different sessions and different operators. This novel hardware perfusion phantom system allows reliable, reproducible, and efficient simulation of myocardial first‐pass MR perfusion. Direct comparison between the results of image‐based quantification and reference values of flow and myocardial perfusion will allow development and validation of accurate quantification methods. Magn Reson Med, 2013.