iLogDemons: A Demons-Based Registration Algorithm for Tracking Incompressible Elastic Biological Tissues

Tracking soft tissues in medical images using non-linear image registration algorithms requires methods that are fast and provide spatial transformations consistent with the biological characteristics of the tissues. LogDemons algorithm is a fast non-linear registration method that computes diffeomorphic transformations parameterised by stationary velocity fields. Although computationally efficient, its use for tissue tracking has been limited because of its ad-hoc Gaussian regularisation, which hampers the implementation of more biologically motivated regularisations. In this work, we improve the logDemons by integrating elasticity and incompressibility for soft-tissue tracking. To that end, a mathematical justification of demons Gaussian regularisation is proposed. Building on this result, we replace the Gaussian smoothing by an efficient elastic-like regulariser based on isotropic differential quadratic forms of vector fields. The registration energy functional is finally minimised under the divergence-free constraint to get incompressible deformations. As the elastic regulariser and the constraint are linear, the method remains computationally tractable and easy to implement. Tests on synthetic incompressible deformations showed that our approach outperforms the original logDemons in terms of elastic incompressible deformation recovery without reducing the image matching accuracy. As an application, we applied the proposed algorithm to estimate 3D myocardium strain on clinical cine MRI of two adult patients. Results showed that incompressibility constraint improves the cardiac motion recovery when compared to the ground truth provided by 3D tagged MRI.     

Triangular springs for modeling nonlinear membranes

This paper provides a formal connection between springs and continuum mechanics in the context of one-dimensional and two-dimensional elasticity. In the first stage, the equivalence between tensile springs and the finite element discretization of stretching energy of planar curves is established. Furthermore, when the strain is a quadratic function of stretch, this energy can be described with a new type of springs called tensile biquadratic springs. In the second stage, we extend this equivalence to nonlinear membranes (St Venant-Kirchhoff materials) on triangular meshes leading to triangular biquadratic and quadratic springs. Those tensile and angular springs produce isotropic deformations parameterized by Young modulus and Poisson ratios on unstructured meshes in an efficient and simple way. For a specific choice of the Poisson ratio, 1/3, we show that regular spring-mass models may be used realistically to simulate a membrane behavior. Finally, the different spring formulations are tested in pure traction and cloth simulation experiments.     

Real-time elastic deformations of soft tissues for surgerysimulation

We describe a novel method for surgery simulation including a volumetric model built from medical images and an elastic modeling of the deformations. The physical model is based on elasticity theory which suitably links the shape of deformable bodies and the forces associated with the deformation. A real time computation of the deformation is possible thanks to a preprocessing of elementary deformations derived from a finite element method. This method has been implemented in a system including a force feedback device and a collision detection algorithm. The simulator works in real time with a high resolution liver model     

General Object Reconstruction Based on Simplex Meshes

In this paper, we propose a general tridimensional reconstruction algorithm of range and volumetric images, based on deformable simplex meshes. Simplex meshes are topologically dual of triangulations and have the advantage of permitting smooth deformations in a simple and efficient manner. Our reconstruction algorithm can handle surfaces without any restriction on their shape or topology. The different tasks performed during the reconstruction include the segmentation of given objects in the scene, the extrapolation of missing data, and the control of smoothness, density, and geometric quality of the reconstructed meshes. The reconstruction takes place in two stages. First, the initialization stage creates a simplex mesh in the vicinity of the data model either manually or using an automatic procedure. Then, after a few iterations, the mesh topology can be modified by creating holes or by increasing its genus. Finally, an iterative refinement algorithm decreases the distance of the mesh from the data while preserving high geometric and topological quality. Several reconstruction examples are provided with quantitative and qualitative results.     

Building maps of local apparent conductivity of the epicardium with a 2-D electrophysiological model of the heart

In this paper, we address the problem of estimating the parameters of an electrophysiological model of the heart from a set of electrical recordings. The chosen model is the reaction-diffusion model on the transmembrane potential proposed by Aliev and Panfilov. For this model of the transmembrane, we estimate a local apparent two-dimensional conductivity from a measured depolarization time distribution. First, we perform an initial adjustment including the choice of initial conditions and of a set of global parameters. We then propose a local estimation by minimizing the quadratic error between the depolarization time computed by the model and the measures. As a first step we address the problem on the epicardial surface in the case of an isotropic version of the Aliev and Panfilov model. The minimization is performed using Brent method without computing the derivative of the error. The feasibility of the approach is demonstrated on synthetic electrophysiological measurements. A proof of concept is obtained on real electrophysiological measures of normal and infarcted canine hearts.     

Personalization of cardiac motion and contractility from images using variational data assimilation

Personalization is a key aspect of biophysical models in order to impact clinical practice. In this paper, we propose a personalization method of electromechanical models of the heart from cine-MR images based on the adjoint method. After estimation of electrophysiological parameters, the cardiac motion is estimated based on a proactive electromechanical model. Then cardiac contractilities on two or three regions are estimated by minimizing the discrepancy between measured and simulation motion. Evaluation of the method on three patients with infarcted or dilated myocardium is provided.     

Tumor growth parameters estimation and source localization from a unique time point: Application to low-grade gliomas

Coupling time series of MR Images with reaction–diffusion-based models has provided interesting ways to better understand the proliferative-invasive aspect of glial cells in tumors. In this paper, we address a different formulation of the inverse problem: from a single time point image of a non-swollen brain tumor, estimate the tumor source location and the diffusivity ratio between white and gray matter, while exploring the possibility to predict the further extent of the observed tumor at later time points in low-grade gliomas. The synthetic and clinical results show the stability of the located source and its varying distance from the tumor barycenter and how the estimated ratio controls the spikiness of the tumor.     

Realistic simulation of the 3-D growth of brain tumors in MR images coupling diffusion with biomechanical deformation

We propose a new model to simulate the three-dimensional (3-D) growth of glioblastomas multiforma (GBMs), the most aggressive glial tumors. The GBM speed of growth depends on the invaded tissue: faster in white than in gray matter, it is stopped by the dura or the ventricles. These different structures are introduced into the model using an atlas matching technique. The atlas includes both the segmentations of anatomical structures and diffusion information in white matter fibers. We use the finite element method (FEM) to simulate the invasion of the GBM in the brain parenchyma and its mechanical interaction with the invaded structures (mass effect). Depending on the considered tissue, the former effect is modeled with a reaction-diffusion or a Gompertz equation, while the latter is based on a linear elastic brain constitutive equation. In addition, we propose a new coupling equation taking into account the mechanical influence of the tumor cells on the invaded tissues. The tumor growth simulation is assessed by comparing the in-silico GBM growth with the real growth observed on two magnetic resonance images (MRIs) of a patient acquired with 6 mo difference. Results show the feasibility of this new conceptual approach and justifies its further evaluation.