Apparent diffusion coefficient measured by diffusion MRI of moving and deforming domains.

The modeling of the diffusion MRI signal from moving and deforming organs such as the heart is challenging due to significant motion and deformation of the imaged medium during the signal acquisition. Recently, a mathematical formulation of the Bloch-Torrey equation, describing the complex transverse magnetization due to diffusion-encoding magnetic field gradients, was developed to account for the motion and deformation. In that work, the motivation was to cancel the effect of the motion and deformation in the MRI image and the space scale of interest spans multiple voxels. In the present work, we adapt the mathematical equation to study the diffusion MRI signal at the much smaller scale of biological cells. We start with the Bloch-Torrey equation defined on a cell that is moving and deforming and linearize the equation around the magnitude of the diffusion-encoding gradient. The result is a second order signal model in which the linear term gives the imaginary part of the diffusion MRI signal and the quadratic term gives the apparent diffusion coefficient (ADC) attributable to the biological cell. We numerically validate this model for a variety of motions and deformations.

Quantifying the effect of tissue deformation on diffusion-weighted MRI: a mathematical model and an efficient simulation framework applied to cardiac diffusion imaging.

Cardiac motion presents a major challenge in diffusion weighted MRI, often leading to large signal losses that necessitate repeated measurements. The diffusion process in the myocardium is difficult to investigate because of the unqualified sensitivity of diffusion measurements to cardiac motion. A rigorous mathematical formalism is introduced to quantify the effect of tissue motion in diffusion imaging. The presented mathematical model, based on the Bloch-Torrey equations, takes into account deformations according to the laws of continuum mechanics. Approximating this mathematical model by using finite elements method, numerical simulations can predict the sensitivity of the diffusion signal to cardiac motion. Different diffusion encoding schemes are considered and the diffusion weighted MR signals, computed numerically, are compared to available results in literature. Our numerical model can identify the existence of two time points in the cardiac cycle, at which the diffusion is unaffected by myocardial strain and cardiac motion. Of course, these time points depend on the type of diffusion encoding scheme. Our numerical results also show that the motion sensitivity of the diffusion sequence can be reduced by using either spin echo technique with acceleration motion compensation diffusion gradients or stimulated echo acquisition mode with unipolar and bipolar diffusion gradients.

A dynamic elastic model for segmentation and tracking of the heart in MR image sequences.

Strong prior models are a prerequisite for reliable spatio-temporal cardiac image analysis. While several cardiac models have been presented in the past, many of them are either too complex for their parameters to be estimated on the sole basis of MR Images, or overly simplified. In this paper, we present a novel dynamic model, based on the equation of dynamics for elastic materials and on Fourier filtering. The explicit use of dynamics allows us to enforce periodicity and temporal smoothness constraints. We propose an algorithm to solve the continuous dynamical problem associated to numerically adapting the model to the image sequence. Using a simple 1D example, we show how temporal filtering can help removing noise while ensuring the periodicity and smoothness of solutions. The proposed dynamic model is quantitatively evaluated on a database of 15 patients which shows its performance and limitations. Also, the ability of the model to capture cardiac motion is demonstrated on synthetic cardiac sequences. Moreover, existence, uniqueness of the solution and numerical convergence of the algorithm can be demonstrated.

A new dynamic elastic model for cardiac image analysis.

We believe that strong prior models are a pre-requisite for reliable spatio-temporal cardiac image analysis. While several cardiac models have been presented in the past, many of them are either too complex for their parameters to be estimated on the sole basis of MR Images, or overly simplified. In this paper, we present a novel bio-inspired dynamic model, based on the equation of dynamics for elastic materials. The explicit use of dynamics allows us to enforce periodicity and temporal smoothness constraints. We study two different methods for solving the resulting equations, and show them to be equivalent. We show how temporal filtering can help to remove noise and ensure the periodicity and smoothness of solutions. Finally, we show some results in 1D and on a synthetic model to illustrate the benefits of our new dynamic model and to show how it can be used to analyze cardiac MR images.