Jérôme Pousin

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.

Diffusion and dissolution in a reactive porous medium: mathematical modelling and numerical simulations

In this work a simple mathematical model for diffusion and dissolution in reactive porous medium is presented. The case of lime and lead in solid phase enclosed in cement matrices is considered more specifically. A numerical method based on finite difference and on a marching technique is proposed and some numerical results are provided. In a simple case, the results obtained are compared with numerical results available in literature.

A nonlinear elastic deformable template for soft structure segmentation: application to the heart segmentation in MRI

This paper proposes a nonlinear 3D deformable model for the image segmentation of soft structures. The template is modelled as an elastic body which is deformed by forces derived from the image. It relies on a template, which is a topological, geometrical and material model of the structure to segment. This model is based on the nonlinear three-dimensional elasticity problem with a boundary condition of pure traction. In addition, the applied forces are successive, as they depend on the displacements. For computations, an incremental algorithm is proposed to minimize the global energy of template deformation. Sufficient conditions of the convergence for the incremental algorithm are given. Finally, a discrete algorithm using the finite element method is presented and evaluated on synthetic images and actual MR images of mouse hearts.

Diffusion and dissolution/precipitation in an open porous reactive medium

In this article a mathematical model is proposed for dissolution/precipitation with diffusion in a reactive open porous medium when many chemical species in solid phase can dissolve. For this purpose, the concept of saturation concentration for a chemical species in liquid phase is introduced and the mathematical properties of the functions representing such saturation concentrations are investigated. Then the equations of dissolution/precipitation are stated and investigated in order to be reformulated as an obstacle problem. Finally, an original mathematical model is derived, and this model can be used for predictive numerical simulations.

F.E.M. implementation for the asymptotic partial decomposition

We consider a Finite Element Method (F.E.M.) implementation for the asymptotic partial decomposition. The advantage of this approach is an important reduction of the number of nodes. The convergence is proved for some model problems. Finally the relation with the “mixed formulation” is discussed.

An efficient numerical scheme for precise time integration of a diffusion-dissolution/precipitation chemical system

A numerical scheme based on an operator splitting method and a dense output event location algorithm is proposed to integrate a diffusion-dissolution/precipitation chemical initial-boundary value problem with jumping nonlinearities. The numerical analysis of the scheme is carried out and it is proved to be of order 2 in time. This global order estimate is illustrated numerically on a test case.

Simultaneous segmentation of the left and right heart ventricles in 3D cine MR images of small animals

New high resolution image techniques allow to capture the anatomy and movement of the heart of small animals. The availability of these in vivo images can be very useful for medical research, however the amount of generated data for large animal studies makes manual analysis a very tedious task. To cope with the problem of automatic analysis of these images, we propose the use of the deformable elastic template method to perform automatic segmentation of the ventricles. To adapt the method to the specificities of high-resolution MRI, several improvements are presented, including an image-context dependent scheme for more robust segmentation. Qualitative results show that our method is able to correctly retrieve the hearts contours in 3D

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 New Dynamic Elastic Model for Cardiac Image Analysis

We believe that 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 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 ID 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.

Incorporating low-level constraints for the retrieval of personalised heart models from dynamic MRI

We have recently presented the dynamic deformable elastic template (DET) model for the retrieval of personalised anatomical and functional models of the heart from dynamic cardiac image sequences. The dynamic DET model is a finite element deformable model, for which the minimum of the energy must satisfy a simplified equation of Dynamics. It yielded fairly accurate results during our valuation process on a 45 patients cine MRI database. However, it experienced difficulties when dealing with very large thickening throughout the cardiac cycle, or on highly pathological cases. In this paper, we introduce prescribed displacements as low level constraints to locally drive the model. Non prescribed contour nodes are displaced according to a combination of forces extracted from prescribed points and image gradient. Prescribing a few points in a whole sequence allows us to retrieve much better segmentations on rather difficult cases.