Section: New Results

Model-Data Interaction

Displacement Reconstructions in Ultrasound Elastography

Participant : Sébastien Imperiale.

In collaboration with Guillaume Bal (Columbia University, New York, USA), we have considered the reconstruction of internal elastic displacements from ultrasound measurements, which finds applications in the medical imaging modality called elastography. By appropriate interferometry and windowed Fourier transforms of the ultrasound measurements, we have proposed a reconstruction procedure of the vectorial structure of spatially varying elastic displacements in biological tissues. This provides a modeling and generalization of scalar reconstruction procedures routinely used in elastography. The proposed algorithm has been justified using a single scattering approximation and local asymptotic analysis. Its validity has been assessed by numerical simulations.

Recursive joint state and parameter estimation

Participants : Atte Aalto, Philippe Moireau [correspondant] .

We propose a method for estimating the parameters of a linear dynamical system from noisy measurements over a given, finite time, interval. For this purpose we develop a recursive modification of the joint state and parameter estimation method proposed in [7] . As the time interval is fixed, any errors in the initial state of the system may cause a significant error in the parameter estimate. Therefore, the parameter estimator is complemented by the so called back and forth nudging (BFN) method for estimating the system's initial state. The proposed strategy can also be regarded as a hybrid least squares optimization method for minimizing the quadratic discrepancy between the measured and simulated outputs over the set of all possible initial states and system parameters.

The optimality of the BFN method with colocated feedback has been considered as well. We have shown that in the case when the system's dynamics are governed by a skew-adjoint generator, the initial state estimate given by the BFN method converges to the minimizer of the quadratic output discrepancy – provided that the observer gains are chosen suitably. If the system’s generator is essentially skew-adjoint and dissipative, a certain modification of the feedback operator is required in order to obtain such convergence.

Convergence of discrete-time Kalman filter estimate to continuous-time estimate

Participant : Atte Aalto [correspondant] .

The Kalman(-Bucy) filter gives the optimal (minimum variance) solution to the state estimation problem for linear systems with Gaussian initial state, and white input and output noise processes. The implementation of the discrete-time Kalman filter is straightforward as it is readily formulated in an algorithmic manner. Thus, it may be tempting to use the discrete-time filter on the time-sampled continuous-time system. We study the convergence of the state estimate obtained from the discrete-time Kalman filter to the continuous-time estimate as the temporal discretization is refined. The convergence follows from the martingale convergence theorem, but surprisingly, no results exist on the rate of convergence. We derive convergence rate estimates for the discrete-time estimate under a number of different sets of assumptions starting from finite-dimensional systems and infinite-dimensional systems with bounded output operators and then proceeding to systems with unbounded output operators and systems with analytic semigroups. The proofs are based on applying the discrete-time Kalman filter on a dense, numerable subset of the time interval of interest, and bounding the change in the state estimate as the new data points are being added. These bounds, in turn, are based on smoothness estimates of the noise-free output.

Observers for the wave equation in unbounded domains

Participants : Sébastien Imperiale, Philippe Moireau [correspondant] , Antoine Tonnoir, Sonia Fliss [Poems team] , Karim Ramdani [Sphinx team] .

We are interested in the reconstruction of initial data for the wave equation problem in unbounded domains using an observer strategy. A major advantage of this method for problems set in bounded domains is the exponential convergence of the algorithm of reconstruction. In our case, the specificity is the unboundedness of the domain which requires to bound it with artificial boundaries for numerical computations. To avoid spurious reflections due to these artificial boundaries, we consider transparent boundary conditions. The difficulty then is to adapt the classical observers technique to this case. Indeed, after enough time, the outgoing waves have left the computational domain and the related information is in some sense “lost”.

First results have been obtained for the 1D case: the theoretical proof of the (exponential) convergence of the algorithm has been done, and the method has been numerically validated. We are currently working on the extension to the 2D case, which raises new difficulties. In particular, the construction of the transparent boundary condition is not obvious and implies a non-local operator in both time and space. Due to this non-local operator, the theoretical analysis of the convergence of the method is then much more difficult.

A Luenberger observer for reaction-diffusion models with front position data

Participants : Dominique Chapelle, Annabelle Collin, Philippe Moireau [correspondant] .

We propose a Luenberger observer for reaction-diffusion models with propagating front features, and for data associated with the location of the front over time. Such models are considered in various application fields, such as electrophysiology, wild-land fire propagation and tumor growth modeling. Drawing our inspiration from image processing methods by considering a data similarity measure of Mumford-Shah type, we start by proposing an observer for the eikonal-curvature equation that can be derived from the reaction-diffusion model by an asymptotic expansion. We then carry over this observer to the underlying reaction-diffusion equation by an “inverse asymptotic analysis”, and we show that the associated correction in the dynamics has a stabilizing effect for the linearized estimation error. We also discuss the extension to joint state-parameter estimation by using the earlier-proposed ROUKF strategy. We published a first work [17] where the observer feedback is derived from the shape-derivative of the data similarity measure. Then, in [21] , in order to improve the observer formulation, we followed a strategy of increasing importance in shape optimization or “level-set”-based image segmentation by complementing the required shape derivatives, used to modify the shape contours, by a topological derivative that represents the sensitivity of the similarity measure when removing a small part of the domain. Both results are illustrated with test problems pertaining to electrophysiology modeling, including with a realistic model of cardiac atria. Our numerical trials show that state estimation is directly very effective with the proposed Luenberger observer.

Identification of weakly coupled multiphysics problems. Application to the inverse problem of electrocardiography

Participants : Cesare Corrado [Reo team] , Jean-Frédéric Gerbeau [Reo team] , Philippe Moireau [correspondant] .

This work addresses the inverse problem of electrocardiography from a new perspective, by combining electrical and mechanical measurements. Our strategy relies on the definition of a model of the electromechanical contraction which is registered on ECG data, but also on measured mechanical displacements of the heart tissue typically extracted from medical images. In this respect, we establish in this work the convergence of a sequential estimator which combines for such coupled problems various state-of-the-art sequential data assimilation methods in a unified consistent and efficient framework. Indeed, we aggregate a Luenberger observer for the mechanical state and a Reduced-Order Unscented Kalman Filter applied on the parameters to be identified and a POD projection of the electrical state. Then, using synthetic data we show the benefits of our approach for the estimation of the electrical state of the ventricles along the heart beat, compared with more classical strategies that only consider an electrophysiological model with ECG measurements. Our numerical results actually show that the mechanical measurements improve the identifiability of the electrical problem, allowing to reconstruct the electrical state of the coupled system more precisely. Therefore, this work is intended to be a first proof of concept, with theoretical justifications and numerical investigations, of the advantage of using available multi-modal observations for the estimation and identification of an electromechanical model of the heart.

Data assimilation of cine-MR images by a biophysical model

Participants : Radomir Chabiniok, Dominique Chapelle [correspondant] , Alexandra Groth, Philippe Moireau, Juergen Weese.

Within the European project VP2HF, we participated in extending the image segmentation tool developed by Philips Hamburg (Alexandra Groth, Jürgen Weese) to process clinically routine cine-MR images for creating anatomical models of heart. Secondly, together with A. Groth and J. Weese we defined a discrepancy operator – between a biomechanical heart model and cine-MR images – that does not require segmenting MR images prior to data assimilation. Initial results of the state estimation using this discrepancy operator were presented at the 2nd VP2HF evaluation meeting (December 2015), and extending these results into a journal paper is a joint objective of the M3DISIM team and of Philips Hamburg.