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We propose a methodology for performing the estimation of a key constitutive parameter in a biomechanical heart model - namely, the tissue contractility - using tagged-MRI data. We adopt a sequential data assimilation strategy, and the image data is assumed to be processed in the form of deforming tag planes, which we employ to obtain a discrepancy between the model and the data by computing distances to these surfaces. We assess our procedure using synthetic measurements produced with a model representing an infarcted heart as observed in an animal experiment, and the estimation results are found to be of superior accuracy compared to assimilation based on segmented endo- and epicardium surfaces.

International audience; In this paper we apply specific data assimilation methods in order to estimate regional contractility parameters in a biomechanical heart model, using as measurements real Cine MR images obtained in an animal experiment. We assess the effectiveness of this estimation based on independent knowledge of the controlled infarcted condition, and on late enhancement images. Moreover, we show that the estimated contractility values can improve the model behavior in itself, and that they can serve as an indicator of the local heart function, namely, to assist medical diagnosis for the post-infarct detection of hypokinetic or akinetic regions in the myocardial tissue

A deeper understanding of finite element methods, and the development of improved finite element methods, can only be achieved with an appropriate mathematical and numerical assessment of the proposed techniques. The basis of such an assessment rests on identifying whether certain properties are satisfied by the finite element scheme and these properties depend on the framework within which the finite element method has been formulated.

The influence of the thickness in the finite element analysis of thin structures is a crucial issue, as it is deeply interrelated with the motivation of modeling a 3D continuum as a shell in engineering. Why, indeed, should we use shell models and finite elements – instead of 3D models – to analyze a given structure? The answer to this question seems obvious: firstly, the use of a shell model is to reduce the analysis cost, and secondly, the use of the shell model is to reduce the complexity of the analysis including the interpretation of the results for engineering design. Clearly, the motivation to use shell models rests upon the fact that shell mathematical models and finite elements incorporate kinematical assumptions pertaining to the displacement distribution across the thickness of the structure, see previous chapters.

Implicit in the concept of a “shell” is the idea that the thickness is “small” compared to the other two dimensions. In practice, it is not unusual to deal with structures for which the thickness is smaller by several orders of magnitude, in which case the shell is said to be “thin” (consider, for example, the shell body of a motor car). Considering the role of the thickness parameter t in the shell models that we presented in the previous chapter (see for example Eqs. (4.36) and (4.51)), with different powers of t in the bilinear terms on the left-hand side, it is essential to determine how the properties of the models are affected when this parameter becomes small. Likewise, it is important to know whether the model converges, in some sense to be specified, towards a limit model when the thickness t “tends to zero”, so that this possibly simpler model can be used instead of the original one when t is sufficiently small, i.e. we need to study the asymptotic behavior of the shell models. Of course, our goal is also to investigate the influence of the thickness on the convergence of finite element methods, as we want to be able to identify numerical procedures for which there is no deterioration of convergence when the thickness becomes small. To that purpose, the analysis of the asymptotic behaviors of mathematical shell models clearly also represents a crucial prerequisite on which we concentrate in this chapter, whereas the issues arising in the finite element solutions themselves are addressed in the next chapters.

In Chapter 7 we discussed the difficulties encountered in the formulation of reliable and effective shell elements. These difficulties are summarized in the synopsis of Figure 8.1 in correspondence with the various types of shell asymptotic behaviors that can be encountered, as addressed in Chapter 5. The objective of the present chapter is to propose some strategies to evaluate shell finite element discretizations in the search for improved schemes. With general analytical proofs not available for the convergence behavior, the numerical assessment is a key ingredient in these strategies. As an example we present the formulation of the MITC shell elements and demonstrate how the numerical assessment of these elements can be performed.

We present a novel strategy to perform estimation for a mechanical system defined to feature the same essential characteristics as a heart model, using measurements of a type that is available in medical imaging. We adopt a sequential approach, and the joint state-parameter estimation procedure is constructed based on a robust and effective state estimator inspired from collocated feedback control. The convergence of the resulting joint estimator can be mathematically established, and we demonstrate its effectiveness by identifying localized contractility and stiffness parameters in a test problem representative of cardiac behavior and using synthetic - albeit realistic - measurements.

In this chapter we describe and analyse the linear shell models that we consider in this book. We first describe the fundamental shell kinematics used. Then we discuss the “basic shell model” which is implicitly employed in general finite element solutions and from which other classical shell and plate models can be derived. We summarize the shell models that we call the “shear-membrane-bending model” and the “membrane-bending model”, and introduce the proper mathematical framework in which they define well-posed problems. As special cases of these shell models we obtain well-known plate models.

The nonlinear analysis of shells is today clearly a very large field, in which much research and development has taken place, so that at present many nonlinear analyses can be performed with confidence in engineering practice, see for example (Bathe, 1999, 2001a; Ibrahimbegovic & Kratzig, 2002).

In this chapter, we describe and analyze the main strategies that have been proposed and used to formulate displacement-based finite element procedures for shells. By displacement-based we mean that the finite element solution is obtained by directly applying the variational principle in the finite element space which discretizes the space of admissible displacements for the structure. In particular, this implies that no “numerical trick” – such as reduced integration – is used in the formulation.

We present a 3Dmec hanical model for the behaviour of the cardiac muscle, based on a recently proposed model of myofibre contraction. The construction of the 3D model involves a rheological model similar to that of Hill-Maxwell. We then introduce some discretisation techniques adapted to our mechanical model, and we report on some preliminary numerical results.