"Advanced Modeling and Simulation in Engineering Sciences"
, Volume 8 (1) – Jan 9, 2021

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philippe.moireau@inria.fr Project-Team M3DISIM, Inria Tagged Magnetic Resonance images (tagged-MRI) are generally considered to be the Saclay-Ile-de-France, Inria, 91128 gold standard of medical imaging in cardiology. By imaging spatially-modulated Palaiseau, France Full list of author information is magnetizations of the deforming tissue, indeed, this modality enables an assessment of available at the end of the article intra-myocardial deformations over the heart cycle. The objective of the present work is to incorporate the most valuable information contained in tagged-MRI in a data assimilation framework, in order to perform joint state-parameter estimation for a complete biomechanical model of the heart. This type of estimation is the second major step, after initial anatomical personalization, for obtaining a genuinely patient-speciﬁc model that integrates the individual characteristics of the patient, an essential prerequisite for beneﬁtting from the model predictive capabilities. Here, we focus our attention on proposing adequate means of quantitatively comparing the cardiac model with various types of data that can be extracted from tagged-MRI after an initial image processing step, namely, 3D displacements ﬁelds, deforming tag planes or grids, or apparent 2D displacements. This quantitative comparison—called discrepancy measure—is then used to feed a sequential data assimilation procedure. In the state estimation stage of this procedure, we also propose a new algorithm based on the prediction–correction paradigm, which provides increased ﬂexibility and eﬀectiveness in the solution process. The complete estimation chain is eventually assessed with synthetic data, produced by running a realistic model simulation representing an infarcted heart characterized by increased stiﬀness and reduced contractility in a given region of the myocardium. From this simulation we extract the 3D displacements, tag planes and grids, and apparent 2D displacements, and we assess the estimation with each corresponding discrepancy measure. We demonstrate that—via regional estimation of the above parameters—the data assimilation procedure allows to quantitatively estimate the biophysical parameters with good accuracy, thus simultaneously providing the location of the infarct and characterizing its seriousness. This shows great potential for combining a biomechanical heart model with tagged-MRI in order to extract valuable new indices in clinical diagnosis. Keywords: Data assimilation, Estimation, Cardiac modeling, Tagged-MRI, Patient-speciﬁc model © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 0123456789().,–: volV Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 2 of 47 Introduction Cardiac biomechanical modeling has made tremendous progress over the past decades, and some accurate models are now available to represent the complex deformations of the organ—among other quantities of interest—over full heartbeats, frequently based on multi-physics and multi-scale formulations, see e.g. [1,2] and references therein. As for all natural systems—as e.g. in geophysics—a great challenge consists in dealing with the many unknown or uncertain quantities—initial conditions, boundary conditions, and various physical parameters—that need to be prescribed for running model simula- tions [3]. In this work, we decide to rely on a data assimilation strategy [4]toestimatethe uncertain quantities while allowing predictive simulations. Concerning the speciﬁc problem of estimation in cardiac biomechanical modeling, dif- ﬁculties arise from both (1) the complexity of the models considered, and (2) the nature of the available measurements, often relying on medical imaging [5]. An eﬀective estima- tion methodology has been proposed by [6] for this type of model, based on a so-called sequential approach—also known as observer method. In this approach, the dynamical model is corrected at each time using the computed discrepancy between the current simulation and the actual measurements, see also [7]. This strategy was designed to be applicable to measurements concerning displacements, whether they be given internally— in a sub-region of the system—or on a boundary or a part thereof. It was also shown to be extendable to data consisting of segmented surfaces as obtained by processing various types of medical imaging dynamical sequences. In this paper, we focus on estimation based on data provided by tagged-MR imaging sequences [8,9]. Tagged-MR is generally considered to be the “gold standard” in cardiac imaging, in particular as regards the assessment of so-called “cardiac mechanical indi- cators”, namely, indicators pertaining to displacements, strains, and volumes [10]. As a matter of fact, tagged-MR images visualize the deformations of grids associated with the actual tissues, which is of course most valuable for clinical purposes, both from a qual- itative standpoint as assessed by the physician’s eye, and with a view to obtaining such quantitative indicators. However, the problem of extracting actual 3D material displace- ments from a tagged-MR sequence gives rise to serious diﬃculties [11,12]. In fact, in many cases only 2D “apparent” displacements are obtained, which introduces speciﬁc errors in the displacement-based quantitative indicators, in addition to usual inaccura- cies pertaining to image processing. Of course, these diﬃculties are also of concern when extracted displacements are to be used in an estimation setup, hence this justiﬁes looking more closely into tagged-MR modalities to devise and analyze strategies to adequately employ them for estimation purposes. In this regard, the contributions presented in this communication are twofold. First, we propose a systematic approach to incorporate within an estimation frame- work a wide range of data, potentially obtained from prior processing steps applied on tagged-MR images. These data vary in their nature, covering the cases of: full mechanical displacements in a subdomain; sequences of deforming tag planes and tag grids; and 2D apparent displacements. Extracting state—and parameter—corrections from these data is an intricate task. To address this challenge, we devise for every case relevant discrepancy measures. The soundness of our approach is corroborated by a complete mathematical Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 3 of 47 analysis of the state observer in an idealized fully linear case, provided as complementary material in Appendix A. Secondly, we propose a relevant time-discretization scheme for the state observer, which is a particularly crucial point in the context of sequential data estimation. This scheme is built upon a “prediction–correction” strategy, where the former step corresponds to genuine model time marching, and the latter to discrepancy-based adjustments. This clean decomposition of these steps oﬀers numerous practical beneﬁts. Additionally, we are able to provide evidence that the obtained time-discrete observer retains the convergence properties of the time-continuous observer, with rigorous proofs detailed in Appendix B. The outline of the paper is as follows. In the forthcoming section we recall the main principles underlying the design of observers, and we provide a quick overview of the mechanical model of a beating heart, as an example of a model formulation. The next section is dedicated to describing the potential information extracted from tagged-MR images and to proposing—for each type of data—the discrepancy measures. We then address the issue of space and time discretization of the observer in order to perform joint state and parameter estimation. Finally, in the last section we present several numerical experiments in which we performed parameter estimation based on synthetic measure- ments. Position of the problem Principles of sequential estimator design The aim of a sequential estimator—also called observer—is to approximate a real tra- jectory, in spite of various uncertainties, using the knowledge provided by the mea- surements obtained on this speciﬁc real trajectory. Let us consider a real trajectory ref y (t),t ∈ [0, +∞), belonging to the so-called state space Y and solution, in our case, of a—possibly inﬁnite-dimensional—dynamical system summarized in the state space form ref ref y˙ = F(y ,t), with an uncertain initial condition ref y (0) = y + ζ , 0 y where y is a known a priori and ζ is the uncertain part in the initial condition. Therefore, 0 y any simulation of y—based on the discretization of the dynamical system—starting only from y will be aﬀected by the propagation of this error made in the initial condition. To circumvent this diﬃculty, we can beneﬁt from the measurements at our disposal on the trajectory. We denote by z these measurements—also called observations and belonging to the observation space Z—which are assumed to be generated by a mapping H on the real trajectory, up to additional measurement errors ref z = H(y ,t) + χ. The observer denoted by y is a system that starts from the only part known in the initial condition—namely y —and uses in time the available measurements z to generate a tra- ref jectory y(t),t ∈ [0, +∞) that converges to y as fast as possible. Therefore, simulating y instead of y from y gives a better approximation of the targeted system. 0 Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 4 of 47 The main categories of observers addressed here are computed by a feedback law based on the measurements in the form y = F( y,t) + G(z − H( y,t)) y(0) = y , where G is called the gain operator, also referred to as ﬁlter. The dynamics of y is corrected when a discrepancy is observed between the actual measurements z and the measurement H( y) that would have been produced by y. This discrepancy J( y,t) = z − H( y) is also called innovation since it not only expresses an observation error, but also a source of improvement for the observer. We point out that with certain types of measurements— as is typically the case with image-based observations—it is sometimes diﬃcult to deﬁne an adequate observation operator but easier to directly compute a discrepancy [6]. This is not a problem for the observer deﬁnition since only the discrepancy appears. In a fully linear situation, namely, when the dynamics is linear with F(y,t) = A(t)y + R and H(y,t) = H(t)y,the mostwell-knowngainoperatorisgiven by theKalmangain, see e.g. [13,14] and references therein. This operator is expressed as G(t) = P(t)H(t) where P is an operator following the Riccati evolution equation ∗ ∗ P = AP + PA − PH HP, P(0) = P , and H is the adjoint of H. Although P is computable for any dynamics operator A, it leads after spatial discretization to a discrete operator which is intractable in practice. This phenomenon has been known for decades and called “curse of dimensionality” [14,15]. Therefore, for speciﬁc dynamics other types of gains have been investigated as initiated by ref [16]. They are based on the fact that, when computing the estimation error y = y − y, we get in a fully linear setting the following dynamics y = (A − GH) y − Gχ y(0) = ζ . Hence, G should be designed to stabilize the estimation error dynamics operator A − GH, so that the homogeneous system tends to 0, namely, t→+∞ y − −−− → 0. χ =0 In the presence of noise in the measurements, this would also control the error dynamics. This strategy is referred to as the Luenberger observer or nudging—see [17] for a survey. For the elastodynamics system—a particular case of second-order hyperbolic systems— [6] has shown that a very simple choice of G = γ H , (1) with γ a scalar coeﬃcient can be suﬃcient. In a nonlinear conﬁguration, fewer theoretical results are available. However, an accepted strategy is to replace in the gain the use of the adjoint of the observation operator, ∗ ∗ namely H , by the adjoint of the tangent operator DH( y) around the estimated trajectory. Therefore for small errors we can expect that the linearized error around the trajectory is stable. Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 5 of 47 One last fundamental aspect that we need to describe in this introduction to observer design is how parameter estimation—also called identiﬁcation—can be carried out. Let us denote by θ the uncertain parameter to be identiﬁed. Note that θ may be a vector of components or even a distributed ﬁeld. The main idea is to introduce an augmented state vector and dynamics operator y F(y, θ,t) y = , F( y,t) = , θ 0 such that we still have y˙ = F( y,t). Then, a Kalman observer can be directly deﬁned on this augmented model leading to a covariance operator and gain P P G yy yθ y P = , G = . P P G θy θθ θ However, it is more intricate to deﬁne a relevant Luenberger observer for the augmented system as the observations are frequently linked to the parameters through the state only. Therefore, there is little hope that γ H will lead to an eﬃcient gain. An alternative strategy was proposed by [18] as a generalization of the adaptive ﬁltering strategy of [19,20]. The idea is to retain the Luenberger observer on the state while using a Kalman-like gain on the parameters. This strategy can be very eﬀective in practice, since it is common to consider a parameter described much more coarsely than the state discretization, thus alleviating the curse of dimensionality associated with optimal ﬁltering. The complete observer reads ˙ ˙ y = F( y, θ,t) + γ DH (z − H( y,t)) + Lθ, y(0) = y ⎪ 0 −1 ∗ ∗ θ = U L DH (z − H( y,t)), θ(0) = θ (2) ⎪ L = (D F − γ DH DH)L + D F, L(0) = 0 y θ ∗ ∗ −1 U = L DH DHL, U(0) = P (0), θθ −1 where U is a reduced covariance operator on the parameter space and L is a “sensitivity” operator from the parameter space to the state space. We see in the dynamics (2) that the state gain is the combination of the Luenberger gain and a gain directly inferred from the parameter ﬁlter so that −1 ∗ ∗ G = (γ 1 + LU L )DH . In [18] the convergence of the complete observer is also established—at least in a linear conﬁguration—based on the idea that the Luenberger state observer reduces the uncer- tainty to the parameter space where the optimal ﬁlter operates. Moreover, the eﬀective- ness of this approach has already been applied to biomechanical identiﬁcation problems by [7,21]. Example of model formulation We consider here a model of beating heart involving a large displacement solid formulation with active stresses driven by an electrical input. Let us now introduce some notations in order to summarize the model equations. We denote the heart domain by (t) at any time t. This domain is the image of a reference conﬁguration through the solid deformation mapping × [0,T] −→ (t) ϕ : (ξ,t) −→ x = ϕ(ξ,t) = ξ + u(ξ,t), Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 6 of 47 where u denotes the solid displacement, so that the solid velocity is given by v = u˙.The deformation gradient tensor F is given by F(ξ,t) = ∇ ϕ = 1 + ∇ u. ξ ξ Furthermore, we introduce the right Cauchy-Green deformation tensor C = F · F.We ﬁnally recall that the Green-Lagrange strain tensor denoted by e is deﬁned by 1 1 e = (C − 1) = ∇ u + (∇ u) + (∇ u) · ∇ u . ξ ξ ξ ξ 2 2 Regarding the mechanical quantities notation, we denote by ρ the tissue mass per unit volume, and by σ the Cauchy stress tensor associated with the deformed conﬁguration. In the reference conﬁguration, we respectively deﬁne the associated ﬁrst and second Piola- − −1 −1 − Kirchhoﬀ stress tensors as T = J σ · F and = F · T = JF · σ · F , where J = det F. The constitutive law can be considered as a nonlinear rheological combination of a passive part and an active part T = + . The passive part is described by a p a p hyperelastic law of potential W and a viscous component chosen proportional to the strain rate e˙ ∂W (e, e˙) = (e) + η e˙. p s ∂e Concerning the hyperelastic law, there exists some experimental evidence—based on detailed ex-vivo tri-axial shear testing—in favor of a complete orthotropic passive behavior [22], with a so-called sheet structure providing a second privileged direction, namely, after the muscle ﬁber direction. However, the sheet direction cannot be characterized in- vivo for patient-speciﬁc modeling purposes. Moreover, various studies have shown good agreements of transversely isotropic models with experimental data obtained at the organ level, see e.g. [23,24]. We thus consider a transversely isotropic law of exponential type earlier proposed by [25], and inspired from the fully orthotropic model of [26], viz. 2 2 W = C exp C (J − 3) + C exp C (J − 1) + κ(J − 1) − κ ln(J), 0 1 1 2 3 4 where J is the standard ﬁrst reduced invariant, J is the reduced invariant accounting 1 4 for the anisotropy of the material in the ﬁber direction τ,namely J = J tr(C),J = 1 4 J τ · C · τ, and κ is the bulk coeﬃcient. For the active part , we rely on the model proposed in [27], with internal variables deﬁning the active strain e , the active stiﬀness k and the associated active stress τ , c c c along the ﬁber direction τ in a chemically-controlled constitutive law describing myoﬁbre mechanics [27,28]. Therefore, we have = σ (e ,k , τ )τ ⊗ τ. We ﬁnally end up with a 1D c c c the following second Piola-Kirchhoﬀ stress tensor ∂W (e,e ,k , τ ) = (e) + η e˙ + σ (e ,k , τ )τ ⊗ τ. c c c s 1D c c c ∂e Concerning the boundary conditions, following [7] we model the interactions with the surrounding organs by visco-elastic boundary conditions on a subpart of the boundary, which gives in the reference conﬁguration T · n = k u + c v on , where n denotes the s s n surface normal in the reference conﬁguration. Regarding the pressure load, we consider a uniform following pressure on the left and right endocardium easily written in the deformed conﬁguration σ · n =−p n on (t),i ={1, 2}, where, here, n denotes the t v,i t n,i t normal of the deformed conﬁguration boundary. Finally, the complete mechanical model Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 7 of 47 reads u˙ = v, in ⎪ 0 ⎪ ρv˙ − div(T) = 0, in T · n = k u + c v, on (3) s s n T · n =−Jp F · n, on v,i c,i T · n = 0, on ∂ \((∪ ) ∪ ). 0 i c,i n Together with the internal variable dynamics given in [27], it constitutes a general deﬁni- tion of the dynamical operator denoted by F in our above summarized description for a state y corresponding to (u, v,e ,k , τ ). c c c Filtering data available from imaging modalities For assessing the physiological condition of a patient’s heart, physicians usually seek stan- dard indicators such as mass, volume or ejection fraction. Additionally intra-myocardial deformations are of great importance to assess the cardiac function in a localized man- ner. Even though the former three global indicators can be obtained using various types of medical image modalities—e.g. cine-MRI—the latter are more intricate to capture by standard procedures. Magnetic resonance imaging with tissue marking—referred to as tagged-MRI—was introduced in the late 80’s [8]. By non-invasively imprinting a pattern in the acquired images—through speciﬁc magnetization of the tissue—it reveals myocardial deformations. Various types of tagged image modalities have arisen since its inception. They diﬀer by the orientation, the temporal persistence or even the shape of the pattern. For instance, the SPAcial Modulation of Magnetization (SPAMM) modality—introduced by [9]—generates a grid-like pattern, whereas the ﬁrst tagging images included a radial pattern—see e.g. [29]. The temporal persistence of the pattern in SPAMM images covers the complete heart systole. Figure 1a shows an example of a SPAMM image (in short axis view) at marking time. We observe the regular pattern within the image domain. Figure 1b shows the same image slice obtained during contraction. In this second image we observe the deformation of the originally regular pattern subject to the material displacements. Even though SPAMM images are the most popular type of tagged-MRI, other modalities exist. For example, we can cite the DANTE sequence—initially introduced by [30]—which a 2D slice at marking time b 2D slice during systole Fig. 1 Example of SPAMM images in short axis view Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 8 of 47 provides a thinner tag grid pattern. Another example is the CSPAMM modality that aims at decreasing the tag pattern fading using two sequences of SPAMM images—see [31]. Combinations of 2D images, e.g. by superposition in a orthogonal direction, are histori- cally the ﬁrst type of tagged image modalities that was proposed. However, they are natu- rally aﬀected by through-the-plane motion. This particular aspect will also be referred to as “2D apparent displacement” in the following. It corresponds to the shift in the position of the intersection of the deforming material with the image plane. Note that this displace- ment is neither a full displacement measurement nor a projection of the displacement onto the image plane. To circumvent this limitation, later works have led to the production of complete three-dimensional tagged-MRI—see [11]. 3D tagging (3D SPAMM) is an imag- ing modality of major interest since it can provide truly three-dimensional information on the heart strain. From the most direct type of data to more complex observations, our work is dedi- cated to the design of observers based on 2D or 3D SPAMM images. Such an endeavor requires—see “Principles of sequential estimator design” section—to be able to compute the discrepancies between the various types of pre-processed data and the model. First, we assume that this image processing step leads to the reconstruction of the fully three dimensional deformation of the heart—from 3D SPAMM for instance [11,32], or from the collection of various 2D SPAMM [33,34]. In this context, following [6]wepropose an eﬃcient way to assimilate this direct displacement measurement. However, this may introduce a new source of error pertaining to displacement tracking, in addition to those inherent to the imaging modality per se. Hence, we further consider three distinct situa- tions aiming at gradually decreasing our demands on this prior processing step. To start with, we propose a discrepancy measure based on the assumption that we are able to reconstruct the tag planes ﬁtting the tag pattern [35–37]. In the case of two-dimensional images, obtaining these surfaces may require a complex interpolation scheme in the image transverse direction. Therefore, in a second step, we consider the case of 2D tag grids lying within the image planes [38]. In a ﬁnal step, we propose means of comparison between the model and 2D apparent displacements extracted from 2D MR images, see [39–42]. Filtering 3D displacements from 3D grids By constructing the tagging pattern in three directions, 3D SPAMM is a powerful image modality that potentially leads to a reconstruction of the complete three dimensional heart motion [11]. For instance, [12] proposed to adapt the HARmonic Phase (HARP) method—which performs tag patterns tracking by analyzing the frequency contents of the image—to 3D images in order to extract these data. However, this modality suﬀers from long acquisition times, requiring multiple breath-holds from the patient. Note that recent works [43] have shown that this diﬃculty can be partially overcome, typically using signal under-sampling. Another technique to extract three dimensional displacements of the heart from SPAMM images is to acquire two orthogonal sets of 2D images in short and long axis—see for instance [33,44]or[45]. It should be noted that this process is likely to suﬀer from slice misregistration and through-the-plane motion. However, as a ﬁrst step, it seems reasonable to assume that the observations take the form of the 3D tissue displacements with a resolution corresponding to the tag pattern spacing. In a (forthcoming) discrete setting, this entails the use of a space interpolation Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 9 of 47 operator between the mesh model and the observation region. Nevertheless, in a contin- uous formalism, we can assume that we have at our disposal some measurements of the displacements in a subdomain ω of the geometry. We introduce the observation operator H = (H 0) ∈ L(Y,Z) such that |Y Y → Z H : |Y u → 1 u, u 1 3 where Y = H ( ) is the displacement space. Note that H apply to the state space Y gathering the displacement space and the velocity space (and eventually additional vari- ables in more general mechanical conﬁguration). By contrast, H u is the same operator of |Y input space restricted to the displacement space Y . We then need to specify the obser- 2 3 vation space Z. One possible choice is to consider L (ω ) , the space of square-integrable vector ﬁelds. However, this space does not characterize the maximum amount of informa- tion we have on the system, since the observations typically comes from a displacement in 1 3 H ( ) , the space of square-integrable vector ﬁelds with square-integrable ﬁrst deriva- 1 3 tives. We should rather consider Z = H (ω ) , and we propose in Appendix A a more complete mathematical justiﬁcation of this choice. Hence, following [6], we introduce a convenient way to deﬁne a norm in this space. Let us consider the following extension lin − div(σ (ψ)) = 0, in \ω ⎪ 0 0 ψ = ϕ, in ω (4) lin ⎪ σ (ψ) · n = k ψ, on s n lin σ (ψ) · n = 0, on ∂ \ , 0 n lin where σ denotes the stress tensor given by linearized isotropic elasticity. In particular, we denote the corresponding linear constitutive law by lin σ (ψ) = A : ε(ψ), where ε denotes the usual linearized strain tensor. The boundary conditions on ∂ are adequately obtained from (3). In the following, we will denote the extension operator by ψ = Ext (ϕ). Note that an equivalent variational characterization of the extension is given by ∀v ∈ Y s.t. v = 0, (Ext (ϕ), v ) = 0, (5) ω E 0 0 |ω where the energy dot-product is here deﬁned by lin (u , u ) = σ (u ): ε(u )d + k u · u d. 1 2 E 1 2 s 1 2 0 n We can prove—see Appendix A for a similar dot product (·, ·) —that (Ext (ϕ), Ext (ϕ)) ω ω 0 0 1 3 isanormin Z = H (ω ) . It is now possible to deﬁne the adjoint of the observation operator that is needed in (1). We ﬁnd in [6] and Appendix A that H is given by Z → Y ∗ ∗ |Y H = , with H : |Y z → Ext (z). 0 Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 10 of 47 Note that in the rest of the document, by a slight abuse of notation, we will denote by H the operator applying either on the state space or on the displacement ﬁeld extracted from the complete state y. We can now deﬁne, in a continuous formalism, the state observer. As recalled in “Position of the problem” section, this state observer is the ﬁrst ingredient of our state-parameter estimation procedure as the state is corrected by physically-based feedback and the parameters by a Kalman feedback. Indeed, following [6], the state observer corresponding to (1)with G = γ H is given in strong formulation by: u = v + γ Ext (z − 1 u), in ω ω 0 ⎪ 0 0 ⎪ ˙ ρ v − div(T) = 0, in ⎨ 0 (6) T · n = k u + c v, on s s n ⎪ T · n =−Jp F · n, on v,i c,i T · n = 0, on , where = ∂ \((∪ ) ∪ ). In order to justify why such a simple feedback is in fact 0 0 i c,i n very eﬀective for controlling state errors in our formulation we propose: (1) a complete mathematical analysis in a simpliﬁed elastodynamics conﬁguration in Appendix A;(2) an energy estimate of the estimation error proving at least the decrease of the estimation error in a general framework. Indeed, let us compute ref ref y = u, v = u − u, v − v , in the simpliﬁed case of linearized visco-elasticity without activation internal variables, namely lin σ = σ (u) + η ε(v). The estimation error satisﬁes the following weak formulation, for any v ∈ Y , ρ v · v d + ( u, v ) + η ε( v): ε(v )d + c v · v d = 0, E s s 0 0 n with the additional observer-based relation u = v − γ Ext (1 u), ω ω 0 0 assuming zero measurement error to ﬁx the ideas. Weighing the latter relation by u and using the energy dot-product yields 1 d 1 d 2 2 ( v, u) = + γ (Ext (1 u), u) = + γ Ext (1 u) , E ω ω E ω ω 0 E 0 0 0 E 0 0 0 0 E 2 dt 2 dt where we have used the orthogonality property (5). We can now substitute this expression in the above variational formulation applied with the test function v = v, which gives d 1 2 2 K E dt 2 =− η ε( v): ε( v)d − c v d − γ Ext (1 u) , (7) s s ω ω 0 0 0 n where = (ρ v, v) denotes the kinetic energy of the error. We can see that the 2 3 K L ( ) total energy of the error—namely, elastic energy of the deformation plus kinetic energy of the velocity—decreases. Due to the observer correction term, we can expect a faster stabilization than with the sole natural dissipation. Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 11 of 47 Filtering tagged-MR planes and grid The limitations of 3D SPAMM that we have previously mentioned make this imaging modality diﬃcult to use in clinical routine. In most clinical cases only 2D tagged-MRI is available. These datasets can be treated plane by plane to extract apparent displacements. Prior to proposing a corresponding observer for this type of data, we assume that a ﬁrst step of image processing leads to the construction of geometrical objects taking the form of tag planes or tag grids and following in time the deformations of the tag patterns—see [36–38] for examples of tag planes constructions and [38,46,47] for tag grids. Extension of surface data For the deﬁnition of the spaces and norms associated with the discrepancy measures needed in this section, following [6] and the approach detailed in the previous section, we will use an extension operator mapping data provided on a surface to the whole solid domain. More precisely, we denote by S a surface embedded in the reference domain 1 2 1 2 and e a vector ﬁeld deﬁned on S ,with(e , e )deﬁnedsothat(e , e , e) gives an 0 0 ⊥ ⊥ ⊥ ⊥ orthonormal basis at any point in S . We deﬁne the extension ψ = Ext (e ; ϕ) from S 0 S 0 of the scalar ﬁeld ϕ in the direction e as lin − div(σ (ψ)) = 0, in ⎪ 0 ψ · e = ϕ, on S lin 1 lin 2 (8) (σ (ψ) · n) · e = (σ · n) · e = 0, on S ⊥ ⊥ lin ⎪ σ (ψ) · n = k ψ, on s n lin σ (ψ) · n = 0, on ∂ \ . 0 n An equivalent variational characterization of this extension operator is ∀v ∈ Y s.t. v · e = 0on S , (Ext (e ; ϕ), v ) = 0. (9) 0 S E 0 0 We can deﬁne a norm on the surface-based data using this extension, namely, ψ . Typically, considering the following linear observation operator Y → Z H: u → u| · e, we can use this norm in the observation space. Accordingly, observer terms in the form H (z − H u) will give in a variational setting (Ext (e ;z − H u), Ext (e ;H v )) = (Ext (e ;z − H u), v ) , S S E S E 0 0 0 0 0 where the second expression is obtained by using the characterization (9) when observing that on S Ext (e ;H v ) · e = v · e. Therefore, we have in this case H (z − H u) = Ext (e ;z − H u). (10) Now, when dealing more generally with a discrepancy operator J( u, z) pertaining to the displacements on a surface S , we will generalize this strategy by using the observer correction given by −Ext (D J( u, z) ; J( u, z)), S u 0 Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 12 of 47 obtained by directly substituting in (10)J( u, z) for (z − H u), and −D J( u, z)—associated with a vector ﬁeld on S —for e since in the linear case we have H u = u · e on S .Of 0 0 course, this also easily generalizes to a measurement made of a collection of such surface- i S i based data associated with N surfaces (S ) and associated vector ﬁelds e , for which 0 i=1 the correction will be given (in the linear case) by ∗ i H (z − H u) = Ext i (e ;z − H u). (11) i=1 Tag planes P i We consider data consisting of a set of N tag planes T = P deforming over time. Following the original ideas of [6] the discrepancy between the model and the data will be measured using the signed distances between the tag planes and the corresponding model data. These model data are deforming surfaces obtained by applying the model displacements to the initial conﬁguration of the tag planes. Let us then denote by T = P i P the set of tag planes in the reference conﬁguration, mapped by the estimated P i i trajectory u to T = P . For any point in a model tag plane x = ξ + u(ξ) ∈ P for some ξ ∈ P , we can compute the signed distance to the corresponding target tag plane P by dist( x,P ) = ( x − x) · n , (12) i i P P where x is a point on the ith model tag plane P , x is the projection of this point on the corresponding observed tag plane, and n is the normal of the observed tag plane at the projection point—see Fig. 2. The discrepancy operator is then the application mapping the displacement ﬁeld to this collection of (scalar) distance ﬁelds deﬁned over the planes of T . When diﬀerentiating with respect to the displacement ﬁeld we have u i ∀v ∈ Y ,D dist( x,P ) · v = n · v . ab Fig. 2 Illustration of tag planes discrepancy measure Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 13 of 47 a Deforming tag grid b 2D apparent displacements Fig. 3 Example of a tag grid and apparent displacements extracted from real tagged-MR images. Apparent displacements obtained from inTag plugin of OsiriX software [48] Hence, the application of the above-described strategy gives an observer that follows the mechanical system of equations (3), except for the ﬁrst equation modiﬁed into u = v − γ Ext n ( x); dist( x,P ) . (13) i i P P i=1 Tag grids We now consider the data in the form of a collection of tag lines deforming within the set of (2D) image slices—see Fig. 3a for an example. We thus assume that we have N ij j such lines L in each 2D image I ,with1 ≤ j ≤ N . We cannot directly design the i=1 discrepancy operator based on the corresponding model lines, since displacement ﬁelds are not well-deﬁned along lines in the variational space. To circumvent this diﬃculty, we again consider the tag planes in the model and project each point of the planes onto the neighboring image slices. Denoting by j the Euclidean orthogonal projection onto the image I , we compute the signed distance of the projected point to the corresponding tag line within each image—see Fig. 4—i.e. ij dist( j x,L ) = j x − ij j x · n ij. I I L I L Then, we can interpolate the signed distances thus-obtained in the various images concerned—which provides interpolated distance ﬁelds over the model tag planes as a discrepancy operator, namely, ij J ( u, z) = J dist( x,L ) , (14) i (j) for each plane P , where J denotes the interpolation operator. When diﬀerentiating (j) this expression, we have u ij ∀v ∈ Y ,D dist( x,L ) · v = n · v , j ij I L but we also have a contribution coming from the interpolation operator derivative. Since this interpolation only depends on the coordinate of the point considered along the axis orthogonal to all image slices, denoting by J the derivative with respect to this coordi- (j) nate, a straightforward computation ﬁnally yields u ij ∀v ∈ Y ,D J ( u, z) · v = J n + J dist( x,L ) n · v , ij j u i (j) I L (j) I Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 14 of 47 Fig. 4 Illustration of tag grids discrepancy measure where n denotes the direction vector of the orthogonal axis. Note that when consid- ering e.g. linear interpolation, the derivative J is directly given by the ﬁnite diﬀerence (j) expression computed between the two adjacent planes. This ﬁnally gives for the observer correction equation i ij u = v − γ Ext i e ; J dist( j x,L ) , (15) (j) i=1 i ij with e = J n ij + J dist( j x,L ) n . (j) L I (j) Filtering 2D apparent displacements Finally, we consider the measurements corresponding to apparent displacements obtained from the processing of 2D tagged images—see Fig. 3b for an example. In the literature we can distinguish two main corresponding extraction methods. A ﬁrst manner consists in tracking the tag pattern directly in the image plane. For instance [49,50]consideran optical ﬂow methodology that takes into account the fading of the tag pattern during the acquisition process. Another solution proposed by [41] is to perform non-rigid image registration. A second family of methods consists in working in the frequency domain. The most popular method is the HARP technique—see [51,52]—which tracks the phase of the tag pattern. Following this trend, recent works proposed by [42,53]use theGabor ﬁlter to obtain a better estimation of local deformations in late systole—which appears to be a slight limitation of the HARP methodology. In any case, we can assume that this pre-processing step enables us to track—throughout the dynamic sequence—the intersections of material ﬁbers originally orthogonal to the image planes. This holds e.g. for such ﬁbers corresponding to the intersection of tag planes, but 2D-tag processing techniques generally provide a (2D) displacement ﬁeld all over the image slices—as depicted in Fig. 3b. Note that the material point located at the intersection between the ﬁber and the image changes over time due to through-the- plane motion. Hence, the measurement is not a material displacement, see Fig. 5. This induces serious complications in the exact form of the tangent observation operator DH, therefore we will use an approximate form based on a small displacements assumption. With this assumption, the observation operator reduces to the components of the material displacements tangential to the image plane. In this case, the measurement is a two- Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 15 of 47 app Fig. 5 Illustration of the diﬀerence between material displacement u and apparent displacement u component ﬁeld over a plane—instead of a scalar ﬁeld for the above distances. Hence, we resort to a slightly diﬀerent extension operator, namely, ψ = Ext (e ; ϕ)deﬁnedby lin − div(σ (ψ)) = 0, in ψ = ϕ, on S e 0 lin (16) (σ · n) · e = 0, on S lin ⎪ σ · n = k ψ, on s n lin σ · n = 0, on ∂ \ , 0 n where denotes the projection onto the plane orthogonal to e, plane in which ϕ is assumed to lie. Finally, the correction equation for the observer reads u = v + γ Ext n ;z − H( u) , (17) j=1 where n denotes the normal to the image planes, and the innovation term z − H( u)will be computed—see “Generating apparent displacements” section—based on the actual tracking of material ﬁbers, i.e. without small displacements assumption. Time discretization using a prediction–correction scheme Time discretization of the state observer In this section we address the issue of the time discretization of the observer. Here, we focus our eﬀort on ensuring that the dissipative behavior of the (time discrete) estimation error dynamics is preserved, up to some consistency terms inherent to any discretiza- tion. A speciﬁc numerical time scheme based on a mid-point scheme was proposed by [6] for similar observers. Our present approach, however, diﬀers in the sense that the time-discrete observer is built on a prediction–correction paradigm. Consequently, the prediction part—in practice, iterations of the direct model—and the correction part— i.e. the action of exploiting the discrepancies between the data and the model—can be managed in separate ways. Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 16 of 47 In order to incorporate the observation operators described in the previous section, we consider the case of nonlinear operators. More precisely, neglecting the observation noise, we assume that the observations are obtained by z = H(y(nt)), (18) where H(·) is a nonlinear and suﬃciently smooth observation operator. Following [6]we propose to deﬁne the time-discrete observer as n+1 n n+1 n y − y y + y − − + + ⎪ = F (19a) t 2 n+1 n+1 y − y + − e ∗ n+1 e e n+1 e = γ DH z − H( y ) − DH ( y − y ) (19b) + + + + + y = y n+1 n+1 where we denote by y and y the prediction and correction steps respectively. In − + e e the correction relation (19b), we use y an extrapolated trajectory, and DH the tangent + + e e e operator of the observation operator evaluated at y ,i.e. DH = DH( y ). While other + + + n+1 choices are possible—see [6]—the most simple approach is to set y = y . + − Even though fewer theoretical results can be obtained in the case of nonlinear obser- vation operators, this numerical procedure is based on a linearization argument. This leads, after a local analysis around the trajectory used in the linearization, to a dissipative behavior similar to that of a linear problem. To fully understand this crucial aspect, we derive in Proposition B.2 of Appendix B the energy estimate satisﬁed by the estimation error, in the fully linear case. Building on this ﬁrst result, we deduce in Proposition B.4 of Appendix B a similar relation satisﬁed by the linearized estimation error. Due to the lin- earization procedure proposed in (19b)—compared to an explicit scheme—this estimate shows desirable dissipation properties, up to two (natural) source terms: a ﬁrst consis- tency term emanating from the time discretization process, and a second term due to the linearization process. Hence, assuming that the initial condition is reasonably close to the target initial condition, the latter will have little inﬂuence on the overall stability of the numerical scheme. In the case of discrepancy measures—see “Filtering tagged-MR planes and grid” section for practical examples—the observations and the real trajectory are linked through the implicit relation J y(nt), z = 0. In this case, the time-discrete observer can be directly inferred from system (19a)–(19b) n+1 n+1 n n y − y y + y − + − + ⎪ = F (20a) t 2 n+1 n+1 y − y + − ∗ n+1 e e n+1 e e =−γ DJ J( y , z ) + DJ ( y − y ) (20b) + + + + + ⎪ t y = y The corresponding linearized estimation error satisﬁes the estimate in Proposition B.4 e e n+1 of Appendix B, but the operator DJ = DJ( y , z ) appears instead of the tangent of the + + observation operator. Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 17 of 47 Characterization and iterative resolution of the correction step We now introduce a fully discrete model—solved in practice—by considering the vectors of degrees of freedom associated with a standard ﬁnite element spatial discretization. We denote by capital letters the vectors of degrees of freedom, and by italic operators the matrices associated with the functional operators used until now. For example, we denote dof by U ∈ R the vector of degrees of freedom—of dimension N —associated with the dof u u function u deﬁned in the ﬁnite-element space Y ⊂ Y . We also deﬁne a state vector as the concatenation of the displacement degrees of freedom and the velocity degrees of freedom Y = . This ﬁnal space discretization procedure enables us to deﬁne a fully- discrete transition operator F , such that the prediction step (19a)—or (20a)—can be n+1|n rewritten as n+1 Y = F (Y ), (21) n+1|n − + where F represents the time-discrete ﬂow from time t to time t . Concerning the n+1|n n n+1 correction step, some speciﬁc elements need to be addressed due to the presence of the adjoint of the tangent observation operator in (19b)—or of the discrepancy operator in (20b). This aspect is particularly important—as extensively detailed in [6]—when dealing u u with displacement-based observations. To that end, we deﬁne for all (u , u ) ∈ Y ×Y h,1 h,2 h h U MU = (ρu , u ) 2 3 = ρu · u d, 2 h,1 h,2 h,1 h,2 L ( ) 1 0 and U KU = (u , u ) = ε(u ): A : ε(u )d + k u · u d, 2 h,1 h,2 E h,1 h,2 s h,1 h,2 1 0 0 n the mass and stiﬀness matrix respectively, and we consider the associated norm K 0 N = . 0 M In the same manner, we denote the linear operator after space discretization by H,and by DH the tangent of a nonlinear observation operator. By a slight abuse of notation we use the same notation H—or DH—when applied to U or to Y , despite the fact that it corresponds in this latter case to H 0 —or DH 0 . Concerning the observation norm, we consider the matrix S computed through the extension operators as detailed in the previous section. For example, when considering 3D displacements, let us consider Z ⊂ Z a suitable discretization of the observation space. The (linear) observation operator boils down to an interpolator between the mesh and the observation subdomain ω , whereas S is deﬁned for all (ψ , ψ ) ∈ Z × Z 0 h ,1 h ,2 h h by S = (ψ , ψ ) = (Ext (ψ ), Ext (ψ )) . 2 h ,1 h ,1 Z ω h ,1 ω h ,2 E 1 0 0 0 We recall that, in this particular case, the extension operator is deﬁned by (4). This extends directly to tag planes (or tag grids) and to apparent displacements using the deﬁnition of the extension operators (8)and (16) respectively. Taking into account this spatial discretization, we can write the correction step (19b) after space discretization as n+1 n+1 Y − Y + − n+1 e n+1 e e e N = γ DH S Z − H(Y ) − DH (Y − Y ) , + + + + + t Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 18 of 47 where DH is the tangent of the observation operator around the extrapolated trajectory n+1 e e Y . To simplify the presentation, we set Y = Y , so that the correction step can be + + − expressed in a more practical form as n+1 n+1 e e −1 e n+1 n+1 Y = Y + γ (N + γ DH S DH ) DH S (Z − H(Y )), (22) t t + − − + + + where S = tS. We remark that, in a fully linear setting, (22) corresponds exactly n+1 to the Best Linear Unbiased Estimator (BLUE) associated with the observation Z ,the −1 n+1 −1 observation covariance S ,the a priori state Y and the a priori state covariance γ N t − [15]. Alternatively, it can be interpreted as solving the following minimization problem n+1 2 n+1 n+1 e n+1 2 min Y − Y + γ Z − H(Y ) − DH (Y − Y ) − N − + − S where and are the norms induced by the matrices N and S respectively. From N S t this alternate characterization, we see that the correction step corresponds essentially to a n+1 linearized version of a least-squares minimization around the a priori state Y .Notethat similar characterizations can be deduced from (20b) in the case of discrepancy operators. The complete system (21)–(22) can be seen as a prediction–correction discrete-time sequential estimator as is the case for the discrete-time Kalman ﬁlter [15,54], but here −1 the a priori state covariance remains constant equal to γ N . Therefore, whereas the discrete-time Kalman ﬁlter is not computable for systems arising from PDEs, our ﬁlter is, since N is sparse. Let us now give some additional methodological key points to solve the correction step (22) in a very eﬀective way. We remark that e e N ≤ N + γ DH S DH ≤ N 1 + O(t) , + + −1 which proves that N is a good preconditioner to solve Eq. (22) with an iterative solver. This reveals to be very helpful as typically we do not want to store the opera- e e tor DH S DH . Indeed, with an iterative solver, we only need to be able to compute + + −1 e e for any vector Y quantities like N Y and (N + γ DH S DH )Y . Using an iterative + + solver is particularly eﬀective in the case of apparent displacements where the observation space is the concatenation of the set of image planes having a potentially high resolution, yielding a very dense operator S. Finally, a practical diﬃculty may arise in our speciﬁc case where quantities of the form e e DH S DH Y may be cumbersome to compute because of the choice of the obser- + + vation norm S, obtained from various extension operators. Authors in [6] demonstrated that a judicious approximation of the extension can be found by considering a penalized minimization problem. For instance, in the case of 3D displacement measurements, the computation of the adjoint ψ = H ϕ = Ext (ϕ) for ϕ ∈ Z can be approximated, after space discretization, by solving min ε K + ( − H ) M ( − H ) , (23) obs dof 2 ∈R where ε> 0 is the (small) penalization parameter, and are vectors of degrees of freedom, H is an interpolation operator between the mesh and the subdomain ω ,and M is the matrix associated with the L -norm on the observation space obs M = φ · φ d, ∀(φ , φ ) ∈ Z × Z . obs 2 h h h ,1 h ,2 h ,1 h ,2 0 Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 19 of 47 From the penalized form (23) of the extension operator, we derive a more convenient expression of the observation norm S. To do so, we start by remarking that, for all u u (u , u ) ∈ Y × Y , h,1 h,2 h h U H SHU = (1 u , 1 u ) 2 ω h,1 ω h,2 Z 1 0 0 = (Ext (1 u ), Ext (1 u )) ω ω h,1 ω ω h,2 E 0 0 0 0 0 = (u , Ext (1 u )) , h,1 ω ω h,2 E 0 0 0 where we have used the orthogonality property (5) to obtain the last relation. Then, replacing the extension operator by its penalized approximation (23)yields −1 U H SHU = U K (εK + H M H) H M HU , 2 2 obs obs 1 1 from which we retrieve the relation given in [6], −1 H S = K (εK + H M H) H M . (24) obs obs Similarly, when dealing with tag planes, tag grid or apparent displacement, one can approx- imate the extension operators (8)and (16) using a penalization strategy. Time discretization of the joint state and parameter observer Having deﬁned the discrepancy operator and designed the state observer, we can now consider the additional stage of parameter identiﬁcation through the state and parame- ter observer introduced at the end of “Principles of sequential estimator design” section. We should now deﬁne the adequate discretization of System (2) compatible with the discretization of the state observer already deﬁned in (20a)–(20b). To that purpose, two discretizations are available in the literature. The ﬁrst one is based on the fact that (2)cor- responds to applying a continuous-time Reduced-Order Extended Kalman Filter (RoEKF) to the parametric space, hence a proper discretization is clearly the prediction–correction scheme deﬁned by the discrete-time RoEKF [18]. The second one proposed by [55]is not directly an exact discretization but rather an extension at the discrete-time level. In fact, the parameter dependency makes the joint state and parameter system nonlinear even if the state dynamics is linear. Therefore, the RoEKF ﬁlter on the parameters is only an approximate optimal ﬁlter. Other choices of approximate reduced-order optimal ﬁlter can therefore be used when available. It is typically the case of the Reduced-Order Unscented Kalman Filter (RoUKF) derived by [55]. This ﬁlter replaces at the discrete-time level the tangent computations in (2) by ﬁnite diﬀerence computations which appear to be better adapted to large nonlinearities. In addition, there is no need to implement the tan- gent operators, as instead the original dynamics and observation operators are applied on so-called “sampling points”. This algorithm thus combines accuracy and computational eﬃciency, and it has already been successfully applied in real biomechanical applications, indeed, see [7,21,56,57]. Moreover, it reduces to the Reduced-Order Kalman Filter upon linearization, which allows to validate its stability with an error linearization study as achieved by [55]. Indeed, both algorithms reduce to the reduced-order Kalman ﬁlter after linearization. For completeness we here recall the complete algorithm in our case, before proceeding to the results section. Following [55,58], for θ discretized in R , we introduce r + 1 so-called unitary simplex [i] r sampling points I in the space R and the associated weights α with the following rules r+1 r+1 α I = 0, α I I = 1, (25) i [i] i [i] [i] i=1 i=1 Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 20 of 47 i.e. with zero mean and unit covariance so that, at each time step, the sampling points can be generated around the estimated values based on the actual covariance estimation. Given an adequate sampling rule, we store the corresponding weights in the diagonal matrix M and precompute these unitary sigma-points. We also denote by M = tM ,and by t obs [I ] the matrix concatenating the (I ) vectors side by side, and similarly for other [∗] [i] 1≤i≤r+1 matrices aggregating some particle vectors. We then perform at each time step 1. Sampling (1 ≤ i ≤ r + 1): n −1 Q = (U ) n + n (26a) Y = Y + L Q I n y [i] [i]+ n + n θ = θ + L Q I n [i] [i]+ θ 2. State prediction (1 ≤ i ≤ r + 1): n+1 n n Y = F (Y , θ ) ⎪ n+1|n [i]− [i]+ [i]+ n+1 (26b) θ = θ [i]− [i]+ n+1 r+1 n+1 θ = α θ − i=1 [i]− 3. State correction (1 ≤ i ≤ r + 1): n+1 n+1 n+1 Z = H (Y ) [i]− [i]− ⎨ n+1 n+1 Y = Y [i]−+ [i]− (26c) e e −1 e n+1 n+1 ⎪ +γ (N + γ DH S DH ) DH S (Z − Z ) t t ⎪ + + + [i]− ⎩ r+1 n+1 n+1 Y = α Y −+ i=1 [i]−+ 4. Parametric correction: n+1 n+1 ⎪ L = [Y ]M [I ] α [∗] ⎪ Y [∗]−+ n+1 n+1 L = [θ ]M [I ] α [∗] ⎪ [∗]−+ n+1 r+1 n+1 Z = α Z ⎪ i − i=1 [i]− ⎨ n+1 n+1 = [Z ]M [I ] [∗] [∗]− (26d) n+1 n+1 n+1 ⎪ U = 1 + ( ) M n+1 n+1 n+1 n+1 n+1 ⎪ ϒ = U ( ) M (Z − Z ) ⎪ n+1 n+1 n+1 n+1 Y = Y − L ϒ + −+ n+1 n+1 n+1 ⎩ n+1 θ = θ − L ϒ + − Results In order to illustrate and assess the above data assimilation method, we propose to per- form parameter estimation in a synthetic data context. More precisely, in this applicative example we will extract from the direct simulation of an infarcted heart the tag planes, tag grids and apparent displacements. The infarct is represented in the model by increasing the stiﬀness and decreasing the contractility in the septum, see Fig. 6.Tothatpurpose we deﬁne two parameters θ and θ such that the constant values (C ,C ), appearing in 0 2 Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 21 of 47 Fig. 6 Model geometry (top row), location of the infarct used in direct simulations and 2-region estimation (bottom-left), and geometry with AHA-regions in the left ventricle (bottom-right) Table 1 Set parameters for the biomechanical heart model (SI units) Valve Base Left Vent. Right Vent 3 3 3 3 C 85.5 · 10 28.5 · 10 5.7 · 10 5.7 · 10 −1 −1 −1 −1 C 1.1 · 10 1.1 · 10 1.1 · 10 1.1 · 10 3 3 3 3 C 57 · 10 28.5 · 10 5.7 · 10 5.7 · 10 −1 −1 −1 −1 C 1.1 · 10 1.1 · 10 1.1 · 10 1.1 · 10 1 1 1 1 η 7 · 10 7 · 10 7 · 10 7 · 10 5 5 5 5 κ 2 · 10 2 · 10 2 · 10 2 · 10 α 1.5 1.5 1.5 1.5 5 5 5 5 σ 6.2 · 10 6.2 · 10 6.2 · 10 7.44 · 10 5 5 5 5 k 1.0 · 10 1.0 · 10 1.0 · 10 1.0 · 10 1 1 1 1 μ 7 · 10 7 · 10 7 · 10 7 · 10 7 7 7 7 E 3.0 · 10 3.0 · 10 3.0 · 10 3.0 · 10 The last ﬁve parameters concern the active part of the constitutive law, and are deﬁned in details in [27] the hyperelastic potential, and the contractility of the tissue σ , appearing in the model proposed in [27], are transformed into θ θ (C ,C ) → 2 (C ,C ) , σ → 2 σ . 0 2 0 2 0 0 This exponential dependence allows to keep these parameters positive, which is crucial from a physical standpoint, and to ensure stability of the numerical scheme during the estimation procedure. Using the calibration strategy of [25]—based on a reduced modeling approach—we propose a complete set of model parameters, see Table 1, for which direct simulations are Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 22 of 47 in good agreement with standard values of physiological and mechanical indicators—see Fig. 7. For the parameters describing the infarct we choose (θ , θ) = (1, −1) in the septum, and (θ , θ) = (0, 0) otherwise. Synthetic data generation Generating the synthetic measurements required to assess our proposed tagged-MRI data assimilation strategy is an intricate procedure in itself. In this section we discuss the various methodological steps used to build, from the biomechanical heart model, the tag planes, the tag grids and the 2D apparent displacements. Generating tag planes and tag grids A natural idea to build the set of tag planes from a direct simulation is to produce, in the deformed conﬁguration at marking time, a set of two-dimensional triangular meshes associated with the planes and to consider the nodal interpolation operator between the model tetrahedral mesh and the set of tag meshes. Denoting by J i the interpolation operator of a single tag plane, its displacement u i is given in this context by J i (u(ξ,t) − u ), if ξ ∈ ∩ P m m P m m u i (ξ ,t) = P m 0, otherwise, where u is the model displacement at marking time t , ξ denotes the associated m m deformed coordinate, namely, ξ = ϕ(ξ,t ), and = ϕ( ,t ). This leads to signif- m m 0 m icantly irregular displacements near the intersections between the model boundary and the tag planes. One way to circumvent this limitation is to consider the tag planes as made of an elastic material and to regularize the interpolated displacement using an appropriate elastic model. However, as the geometry at hand is two-dimensional, a shell model would be required. To simplify this task—which is, in essence, an issue of data regularization— P i we consider a set of elastic (3D) tag layers V . In practice, each tag layer is built so i i i that P ⊂ V . Hence, the displacement of a tag plane P is derived from m m m u i (ξ ,t) = u i (ξ ,t)| i , (27) m m P V P m m m where u is the displacement of the tag layer V , verifying V m V i i ⎪ − div(σ (u )) = 0, in V \ ∩ P ⎨ V m m (28) u i = J i (u − u ), in ∩ P m m V P m m m V i σ (u i ) · n = 0, on ∂V V m The procedure described in (28) is in fact the extension—in the sense of “Filtering tagged- MR planes and grid” section—of the interpolated displacement, namely, u = Ext J (u − u ) . i i V P m m ∩P In (28) we denoted by σ the Cauchy stress tensor describing the tag layer material. In practice, a relevant choice is a linearization around a given trajectory of the heart material since, within the image, the tag pattern follows the heart material points. We show in Fig. 10a an example of synthetic tag planes extracted from a direct simulation. As far as the tag grids are concerned, they are obtained by clipping the tag plane meshes with the image planes. Figure 8 illustrates the complete procedure of construction of a tag plane and of several tag lines, while Fig. 10b gives an example of extracted synthetic tag grids. Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 23 of 47 ·10 Atrial Cavity 1.5 Proximal Distal 0.5 00.20.40.60.8 time (s) a Pressures −4 ·10 Healthy Infarcted 0.9 0.8 0.7 0.6 00.20.40.60.8 time (s) b Cavity volumes ·10 Healthy Infarcted 1.5 0.5 0.60.70.80.91 −4 Volume (m ) ·10 c P-V loops Fig. 7 Some indicators obtained from direct simulations for a healthy and infarcted heart Volume (m ) Pressure (Pa) Pressure (Pa) Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 24 of 47 Fig. 8 Illustration of synthetic tag plane and tag lines construction ab c Fig. 9 Illustration of synthetic apparent displacement computation Generating apparent displacements Once the tag grids are created, the apparent displacement ﬁeld can be approximated by tracking the displacements of the tag lines intersection points. More precisely, at marking time, we compute the intersection point of every tag lines ((red) crosses in Fig. 9a). During the simulation, as the tag lines deform, we track the displacements of the intersection points, leading to the (green) vectors in Fig. 9b. Once the displacements of the intersection points are computed, a global apparent displacement ﬁeld of the image plane is produced ((blue) vectors in Fig. 9c) by standard interpolation. We show in Fig. 10c an example of synthetic apparent displacements extracted from a direct simulation. Discrepancy measure in practice The tagging process is not performed in the reference conﬁguration—never observed in reality—but at the previously mentioned marking time. For this reason, the tagging pattern is necessarily built over an already deformed conﬁguration. Hence, the information obtained from a set of tagged-MR images should be considered of Eulerian nature. Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 25 of 47 (a) Tag planes (b) Tag grids (c) Apparent displacements Fig. 10 Example of computations of synthetic data extracted from a direct simulation of an infarcted heart Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 26 of 47 However, assuming the displacement at marking time u(ξ,t ) is given, we can circum- vent this diﬃculty by introducing this additional information in the ﬁltering procedure. From an algorithmic standpoint, the discrepancy measure is computed as follows: P i 1. [oﬄine] Build at marking time the set of tag planes P , 2. [oﬄine] Build at marking time the interpolation operator {J i } from the i=1 P i deformed conﬁguration (by u(ξ,t )) to the tag planes P , P i 3. [online] From the estimated displacement, deform the tag planes P using (28), P i 4. [online] From the estimated (deformed) tag planes P compute the innova- tion terms appearing in (12) (planes data), in (14)(taggrids)orin (17)(apparent displacement). In our context of synthetic data assimilation, we directly provided the displacement u(ξ,t ). In real cases, the task of estimating the displacement at marking time could be carried out using, for instance, the segmentation of the endo- and epicardium of the left ventricle—obtained typically from cine-MR images. In any case it requires another source of information on the system and this points out a certain limitation of the tagged-MRI data for estimation purposes. Spectral analysis of the observer associated with (3D) tag planes In this section we discuss the observer built using a set of tag planes that can be decom- posed into three distinct families. Each tag plane in a given family shares—at marking time—the same orthogonal direction and, only for the sake of simplicity, we assume that the three directions are orthogonal. This type of data set may be referred to, in what fol- lows, as (3D) tag planes. As already mentioned and emphasized in Appendix B, the quality of the state ﬁltering procedure can be assessed by the amount of damping we introduce in the otherwise conservative or weakly damped system. For this reason we propose to conduct a numerical study of the spectra of the operators driving the target dynamical system and the estimation error dynamical systems. For simplicity, we consider a linear elastic model, typically obtained from the linearization around the null trajectory of the calibrated passive cardiac model. To facilitate this analysis we also decrease the natural viscosity of the target system. Hence, we consider the solutions of the following spectral problem 0 K K 0 Y = λ Y, −K −C 0 M which corresponds to the operator without ﬁlter and where, additionally to the stiﬀness K and mass M matrices, we denote by C the damping matrix obtained after spatial discretization. For the case of complete displacement observer we consider the spectral problem −1 γ K (εK + H M H) H M HK K 0 obs obs Y = λ Y, −K −C 0 M where H corresponds to the identity matrix or an interpolation operator. Note that the form of the stabilization operator is the one derived in (24) where the extension operator has been approximated by its penalized counterpart. For the (3D) tag plane observer, the Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 27 of 47 Fig. 11 Spectra of the time continuous operators. (North-West) : In (cyan) operators without ﬁlter, in (red) complete displacement observer. (North-East): In (green) (3D) tag planes with spacing 8 mm. (South-West): In (green) (3D) tag planes with spacing 3.5 mm. (South-East): In (green) (3D) tag planes with spacing 0.25 mm spectral problem reads −1 γ K (εK + D J M D J) D J M D JK K 0 u obs u u obs u Y = λ Y, −K −C 0 M where the discrepancy operator corresponds to the one detailed in (13). Using the optimal criterion on the gain γ provided by [6], in Fig. 11 we show the spectra obtained for the operators without ﬁlter, with complete displacement feedback and with (3D) tag planes. In these plots we also vary the spacing between two consecutive tag planes from 8 mm, to 3.5mm and 0.25 mm. In the three situations we observe that for low frequencies the observer using tag planes acts as the direct displacement observer. For coarse tag patterns we naturally observe that the higher frequencies are less stabilized than with the direct displacement observer. This phenomenon disappears as the tag pattern becomes denser, and the overall eﬃciency of the tag planes observer tends towards the full displacement observer. This result is of major importance since it comforts the intuition—and hopes—that tagged-MRI could indeed provide information on intra-myocardial deformations. Even though a more thorough mathematical analysis should be carried out to corroborate these results, a ﬁrst explanation for these outstanding performances is that the tangent of the discrepancy operator is the concatenation of the normal vector ﬁelds of the tag planes, which in the case of 3D tag planes span all directions in space. Estimation results Since we have evaluated—through spectral analysis—the state estimation capabilities of our proposed observer, we can now assess the joint state and parameter estimator by Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 28 of 47 identiﬁcation of the infarct location and intensity presented in the beginning of the results section. We distinguish two situations. First we provide the exact location of the infarcted region but not the values of the parameters characterizing the pathology. Hence, we have two separate regions, for each of which we identify a contractility parameter and the main passive stiﬀness parameter. This case may be referred to as the 2-Regions case. Sec- ondly, we use the AHA-Regions—roughly corresponding to the territories of the coronary arteries, see [59]—to partition the left ventricle and retrieve both active contractility and passive stiﬀness parameters in each AHA-Region. In this case, the infarct location will be inferred from the parameters spatial variation over the AHA subdivision. In all cases, for the sake of simplicity, the reference conﬁguration and the intra-cavity pressures are assumed to be known and the initial condition is deﬁned by solving an equilibrium state with the lowest pressure sustained before the atrial contraction. We point out that, since the stiﬀness is globally modiﬁed between the target system and the observer, an error in the initial condition will be introduced during the estimation. The 2-Regions case We start with the simpler 2-Regions case and we only seek to assess the observability provided by the data. We choose a ﬁne spatial distribution of the tag planes by setting the space between two consecutive tag planes to 3.5 mm. Denoting by M the surface mass matrix computed on the set of tags, we deﬁne the measurement observation norm S by setting, in (23) M = M . obs The parameter m represents the square of the standard deviation of the discrepancy measure. In the perspective of only assessing the method capabilities, the observations are extracted from the direct simulations—as explained in “Synthetic data generation” section—with a high temporal resolution of 1 output every 25 simulation time steps— −4 set in our simulations to 2.5 · 10 s—and no noise is added. Therefore, following [7]we rescale obs m = m = 25 m , obs obs and set a high conﬁdence in the observations with m = (0.65 mm) . The results are obs presented in Figs. 12, 13, 14 and 15, where the dashed curves visualize the estimated standard deviation. In Fig. 12, we consider the most optimistic conﬁguration where three- dimensional tag planes are available, whereas in Fig. 13 we consider a more realistic conﬁg- uration where only two directions of tag planes—here a short axis grid only—are available. Then, in Fig. 14 we proceed with the corresponding bi-dimensional grid as described in “Synthetic data generation” section. We thus rely on the grid-based observer discrepancy. Finally,we present in Fig. 15 the results where extracted 2D apparent displacements are deﬁned as the available measurements. In this particular case, since the innovation corre- sponds to the comparison of two vector ﬁelds deﬁned within the image planes, we take the observation norm as a piecewise constant mass matrix in the image domain. The behav- ior of the estimation procedure is very similar for all types of processed data considered, and the parameter values are accurately estimated in the two regions, both for the active (contractility) and passive (stiﬀness) parameters. It should be noted, in particular, that the estimations produced based on 2D tagged data—namely, with tag planes and grids, and apparent displacements—are as eﬀective and accurate as those obtained with 3D tags. Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 29 of 47 Fig. 12 Estimation of active (top) and passive (bottom) parameters using (3D) tag planes The AHA-Regions case We now consider a conﬁguration that can be encountered in a more realistic context. First, we only generate from the direct simulation approximately 20 “synthetic” images in the cardiac cycle from which we extract 2D tag planes. Then, for the estimation, we set up an AHA-based partition, which deﬁnes 17 regions, namely, the 16 AHA regions and the remaining part of the heart. The evolution of the joint state and parameter estimation is presented in Figs. 16, 17 and 18. The state convergence is demonstrated through the evolution of the volume curves and P–V loop plots, whereas the evolution of each parameter is presented in Figs. 17 and 18. The results are divided into three groups of parameters associated with the three diﬀerent long axis elevations of the AHA partitions, namely basal (regions 1–6), intermediate (regions 7–12) and apical (regions 14–16). We visualize in Fig. 19 the ﬁnal parameter identiﬁcation diagram in a bull’s eye representation as used in a clinical context. In addition, we recall the identiﬁcation that would be obtained when only exploiting cine- MRI segmentations of the endocardium and epicardium as presented in the previous work of [7]. This directly illustrates the identiﬁcation beneﬁts obtained with the tagged-MRI measurements, in particular concerning the passive stiﬀness parameters. Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 30 of 47 Fig. 13 Estimation of active (top) and passive (bottom) parameters using tag planes Discussion First, concerning state estimation, our results conﬁrm the remarkable eﬀectiveness of the strategy previously introduced by [6], and the excellent adequacy of both tagged-MR as an imaging modality of choice for estimation purposes, and of our herein-proposed strategies for incorporating such data via speciﬁcally designed discrepancy operators. Indeed, the above spectral analysis gives a very clear indication as to how fast state estimation errors are being damped when using this estimation chain. In particular, we have observed the convergence of the spectrum—with respect to tag spacing—towards that of the observer with full 3D observation, while the spectrum obtained with coarse tags shows that standard tagspacingisamplysuﬃcienttoobtainuniformdampingrates.Thisisalsoconﬁrmedwith the results obtained in the joint state-parameter estimation trials, in which mechanical indicators are eﬀectively and accurately retrieved, recall Fig. 16. As regards parameter estimation, the additional estimation stage provided by RoUKF ﬁltering—combined with the state observer—also shows very good performance. In the 2- Regions estimation setup, in particular, both active and passive parameters are very accu- rately estimated, and in a very short time as soon as the parameters concerned become observable in the type of behavior that is encountered along the cardiac cycle. Namely, passive parameters are mostly observable during the—rather short—initial diastolic phase Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 31 of 47 Fig. 14 Estimation of active (top) and passive (bottom) parameters using tag grids associated with atrial contraction, while of course contractility parameters can only be revealed once the electrical activation actually starts. Note that this 2-Regions setup gives a realistic strategy in clinical perspectives, as cardiac MR performed for infarct diagnosis frequently includes late-enhancement sequences, which can be segmented to provide the desired subdivision into healthy and diseased regions. In case late-enhancement images (or the associated segmentations) are not available, or when additional concurrent local- ization information is desired, estimation can be performed based on the AHA subdi- vision. The corresponding estimation results exhibit the same general features as with two regions—namely, rather fast convergence during diastole and systole for passive and active parameters, respectively —albeit as expected the estimation is less accurate for each individual parameter. Nevertheless, active parameters are still quantitatively retrieved, and passive parameters are quite discriminately detected within the infarcted region, and much more so than with estimation based on Cine-MR. Of course, fundamental identiﬁability issues are of concern in this multiple parameter estimation context, and we can expect that identiﬁability would be improved—hence estimation would be more accurate—when using segmented Cine sequences in addition to tagged images in the estimation procedure. Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 32 of 47 Fig. 15 Estimation of active (top) and passive (bottom) parameters using apparent displacements Fig. 16 Time evolution of the volume and P-V loop for the observer (in (green)) using tag planes (with coarse time sampling of the observations) compared with the healthy direct model (in (blue)) and the reference infarcted model (in (red)) Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 33 of 47 Fig. 17 Time evolution of active estimated parameters using tag planes with coarse time sampling of the observations (dashed lines give reference values) Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 34 of 47 Fig. 18 Time evolution of and passive estimated parameters using tag planes with coarse time sampling of the observations (dashed lines give reference values) Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 35 of 47 Fig. 19 Estimation of active (top row) and passive (bottom row) parameters using tag planes (middle column) and cine surfaces (right column) with coarse time sampling of the observations Towards clinical applications Despite its eﬃciency with synthetic data, our data assimilation strategy still needs to be validated using real clinical data. To address this challenge, it remains to complete the proposed approach with the following requirements: 1. Obtaining patient speciﬁc geometries and ﬁber orientations. Even if segmenting the ventricles is a well studied problem, it remains an issue [3,60]. It is even more the case for obtaining ﬁber orientation maps, directly from images [61], or by combining geometrical models [62] whose parameters could also be assimilated [63]. Moreover, with large deformation systems such as the heart complete bio-mechanical model, we ultimately need a reference conﬁguration which needs to be inferred from the images at hand, typically using inverse methods [64]. 2. Registering the tagged-MR sequence with respect to the initial geometry, hence to the reference conﬁguration by transitivity. 3. Including blood pressure measurements. Note that we should either rely on direct but invasive measurements in the cavities or on estimating these blood pressures from non-invasive distal pressure using typically signal processing [65,66]ordata assimilation. Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 36 of 47 4. Specifying the boundary conditions, or using additional measurements to estimate them. In our modeling choice, viscoelastic boundary conditions are deﬁned on which needs to be geometrically designated. Moreover, this simpliﬁed model should be ultimately jointly estimated with similar data assimilation methods [21]. One tremendous diﬃculty is that each error in the steps listed above can introduce irre- ducible errors in the estimation procedure. This was typically our experience in [7,21,67] where we discussed how boundary conditions modeling errors can pollute the result- ing estimation in a “clinical” context. Therefore, even if our method paves the way for carefully integrating tagged-MRI measurements in a data assimilation strategy, the com- plete model-data fusion pipeline in clinical cardiac applications necessitates to combine multiple technologies and expertise from modeling to data processing and associated registrations [3,5]. Conclusions We have proposed speciﬁc methods for integrating tagged-MR sequences in a data assim- ilation framework with a beating heart model. Tagged-MRI represents the “gold standard” in cardiac imaging, and great beneﬁts are expected from using the corresponding rich kine- matical information for performing the joint state-parameter estimation of the system, and of various modeling parameters of high potential value in terms of clinical diagnosis assistance. In this data assimilation framework, a crucial ingredient lies in the adequate formula- tion of a discrepancy operator to compare the model and the data. We have considered several options, based on: (1) extracted 3D displacements; (2) tag planes in the 3D vol- ume; (3) tag grids in 2D slices; (4) apparent displacements in 2D slices. In practice, the speciﬁc choice of discrepancy operator could be based on the type of tagged sequence and on the available corresponding post-processing tools, albeit our uniﬁed framework also allows detailed comparative assessments. For the purpose of state estimation, each deﬁnition of discrepancy operator was accompanied by the formulation of an adapted ﬁltering operator. We have also proposed well-adapted discretization strategies. As regards time dis- cretization, in particular, a two-step “prediction–correction” type algorithm was designed for the proposed estimation systems, allowing to completely dissociate the operations related to the model and those performed for estimation purposes, e.g. with two diﬀerent— coupled—software codes. This is very valuable from a software architecture perspective, and in particular makes the estimation strategy compatible with modular concepts such as those underlying the Verdandi data assimilation library [68]. Mathematical analyses have been provided at the various stages of construction, to substantiate the convergence of the overall observer strategy based on a simpliﬁed—albeit illuminating—linear example, and also to assess the eﬀects of discretization procedures. Finally, some detailed numerical assessments of the overall estimation framework have been performed, based on synthetic data produced by a reference cardiac simulation rep- resenting the behavior of an infarcted heart in a realistic manner. The assessment results show that state estimation is extremely eﬀective, while the performance of parameter esti- mation depends on the speciﬁc estimation objectives, as can be expected from the point of view of observability. In particular, when the diseased region is pre-determined prior Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 37 of 47 to estimation, active and passive parameters are very accurately and quickly retrieved in the infarcted and healthy regions. When the more challenging objective of estimation in an AHA subdivision is considered without any prior on the diseased region, the conver- gence of each individual parameter value is less accurate, but the overall distribution of parameters is very adequately retrieved, allowing for eﬀective localization and quantita- tive assessment of the disease. This provides a great improvement over similar estimation based on using Cine-MR alone, which only gives adequate results for active parameters. All major ingredients are thus in place for using this methodological framework in a patient-speciﬁc context with actual data, which is of course a most natural perspective of this work. Addressing this challenge will imply combining diﬀerent measurements and image modalities in order to estimate patient-speciﬁc modeling components. Other perspectives concern the consideration of alternative discrepancy operators, such as with the formalism of currents [69–71] which would allow to dispense with using sophisticated image post-processing tools on tagged-MR as a prerequisite for data assimilation. Authors’ contributions AI, DC and PM contributed to the conception, design, analysis, interpretations’, and the drafting and revising of the manuscript. Moreover, AI and PM contributed to the software engineering of the method. All authors read and approved the ﬁnal manuscript. Funding The authors gratefully acknowledge support from the European Union’s Seventh Framework Programme for research, technological development and demonstration, under Grant agreement #611823 (VP2HF Project). Competing interests The authors declare that they have no competing interests. Author details 1 2 CEA List, CEA, 91190 Saclay, France, Project-Team M3DISIM, Inria Saclay-Ile-de-France, Inria, 91128 Palaiseau, France, LMS, Ecole Polytechnique, CNRS, Institut Polytechnique de Paris, 91128 Palaiseau, France. Appendix A: The illuminating example of eﬃcient state feedback in elastodynamics In the following very simpliﬁed example, we aim at mathematically proving why a simple state feedback can have a comparable eﬃciency with respect to a Kalman-based feedback in the context of elastodynamics problems. Let us then consider a simpliﬁed conﬁguration where the model and the observation operator are linear. More precisely, we assume a linear elastic system in which the Cauchy stress tensor is given by σ = A : ε(u), where the elasticity tensor A is assumed to be constant and isotropic. Deﬁning (·) = div(A : ε(·)), our model simply reads ⎪ u˙ = v, in × (0,T) ρv˙ − u = f , in × (0,T) (29) e 0 u = 0, on ∂ × (0,T). 1 2 The external load is a time-dependent regular function f ∈ C ([0,T],L ( )). We intro- v 2 3 u 1 3 u v duce Y = L ( ) , the displacement space Y = H ( ) ,and Y = Y ×Y . Using the 0 0 Korn and Poincaré inequalities, Y is an Hilbert space with the following scalar product ∀(u , u ) ∈ Y , (u , u ) = ε(u ): A : ε(u )d. 1 2 1 2 E 1 2 0 Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 38 of 47 In a semi-group theory context we introduce the semi-group generator A ∈ L(D(A),Y) with 0 1 A = , 0 u 1 v and we can prove that (29) admits a classical solution C ([0,T],Y ) ∩ C ([0,T],Y ) for every initial condition in the domain [72] u v v D(A) = (u, v) ∈ Y × Y , div(A : ε(u)) ∈ Y . Moreover, the operator A is skew-adjoint, implying that ∀t > 0, y(t) = y(0) , Y Y corresponding to the energy balance on the system (29)with E = y . Using the 2 Y semi-group theory, we rewrite the dynamics of the model in the abstract state-space form y˙ = Ay + R, y(0) = y + ζ . 0 y Concerning this model, we assume that we have at our disposal some measurements of the displacements. We introduce the observation operator H = (H 0) such that |Y Y → Z H : |Y u → 1 u, 1 3 where ω is an internal open subdomain of and Z = H (ω ) .HereH andH as 0 0 0 |Y H = (H u 0) correspond to the same operators with diﬀerent input spaces. |Y Using the extension ψ = Ext (ϕ)deﬁnedby − (ψ) = 0, in e 0 (30) ψ = ϕ, in ω ψ = 0, on ∂ , we can prove the following property. Proposition A.1 For any (ϕ , ϕ ) ∈ Z , the bilinear form (Ext (ϕ ), Ext (ϕ )) ω ω E 1 2 0 1 0 2 0 1 3 deﬁnes a scalar product on Z = H (ω ) . Proof The proof is a simple extension of the property proven by [73] for scalar equations. Let ϕ be an element of Z. The only diﬃculty lies in proving the norm equivalence with .First,wehave 1 3 H (ω ) 2 2 2 ∇ ϕ 1 3 2 3 2 3 H (ω ) L (ω ) L (ω ) 0 0 0 2 2 ≤ ∇ Ext (ϕ) + Ext (ϕ) ω 2 3 2 3 0 0 L ( ) L ( ) 0 0 ≤ (1 + C ) ∇ Ext (ϕ) p ω 2 3 L ( ) ≤ C (1 + C ) Ext (ϕ) , k p ω with C given by the Poincaré inequality and C given by Korn inequality and a bound C p k a on the elasticity tensor. Conversely, by continuity of the extension on \ω with respect 0 0 to the data, there exists a constant C > 0 such that for any ψ = Ext (ϕ)wehave d ω ε(ψ): A : ε(ψ)d ≤ C d 1 |∂ω H (∂ω ) \ω 0 0 0 Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 39 of 47 Hence, denoting by C the constant arising from the continuity of the trace operator, we have Ext (ϕ) ≤ ε(ψ): A : ε(ψ)d + C ϕ ω d 1 H (∂ω ) ω 0 2 2 ≤ C ∇ ϕ + C a 2 3 d 1 |∂ω L (ω ) 0 H (∂ω ) ≤ (C + C C ) a d t 1 3 H (ω ) which completes the proof. It is now possible to deﬁne the adjoint of the observation operator. Proposition A.2 The operator H is bounded from Y to Z and H is given by Z → Y ∗ |Y ∗ H = with H u : |Y ϕ → Ext (ϕ) Proof Let us ﬁrst prove that H is bounded. We consider ψ ∈ Y and ϕ such that ϕ = H ψ. We have directly, from norm equivalences, |Y 2 2 2 = Ext (ϕ) ≤ C ≤ C ω 1 1 3 1 1 Z 0 3 E H (ω ) H ( ) 0 0 0 Then,wehavethatfor all ϕ ∈ Z and v ∈ Y (ϕ, H u v ) = ε(Ext (ϕ)) : A : ε(Ext (v )) d. |Y Z ω ω 0 0 |ω By the variational characterization of the extension (5)wehave ε(Ext (ϕ)) : A : ε(Ext (v )) d = ε(Ext (ϕ)) : A : ε(v )d, ω ω ω 0 0 0 |ω 0 0 ∗ ∗ since v − Ext (v ) = 0on ω . Therefore (ϕ, H v ) = (H ϕ, v ) , and H is ω 0 |Y Z u E 0 0 |ω |ω |Y 0 0 given by Z → Y H : Ext (ϕ) ϕ → . We can now deﬁne the observer by the dynamics ⎨ ∗ y = A y + R + γ H (z − H y) y = y , which in strong form reads ⎪ u = v + γ Ext (z − 1 u), in ω ω 0 ⎨ 0 0 ρ v − ( u) = f , in (31) e 0 u = 0, in ∂ , which converges to the solution of (29) under the observability condition given by the next theorem, see e.g. [74] for a proof. Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 40 of 47 Theorem A.3 Let A be a time-independent skew-adjoint operator generating a group and H ∈ L(Y,Z). The error system y of dynamics y = (A − γ H H) y is exponentially stable if the following observability condition is satisﬁed: there exists two constants (C ,T) such that for any solution y of y˙ = Ay,wehave st 2 2 Hy(s) ds ≥ C y(0) . (32) st Z Y We can now make explicit the speciﬁc observability condition in our conﬁguration that will allow us to invoke Theorem A.3 of Appendix A. Theorem A.4 If there exists a constant C and a time T such that every solution of st u˙ = v,in ρv˙ − u = 0,in e 0 u = 0,on ∂ satisﬁes the observability condition 2 2 2 st Ext (1 u) dt ≥ c u(0) + v(0) , (33) ω ω 2 0 0 E E L 0 0 then, in the absence of observation error, the observer given by the dynamics (31) converges ref to the solution y of (29) such that ref z = Hy . Proof We have deﬁned the reference trajectory as the solution of ref ref y˙ = Ay + R, and the observer as the solution of y = A y + R + γ H (z − H y). ref The error y = y − y isthensolutionof y = (A − γ H H) y. which, from Theorem A.3 of Appendix A, converges exponentially to 0 for every initial condition when the observability condition (33) is veriﬁed. Following [75], we deﬁne the elastic geometric control condition: Deﬁnition A.1 (Elastic Geometric Control Condition) The elastic geometric control con- dition is satisﬁed if every combination of pressure (P) and shear (S) waves ray encounters— in the sense of [75]—the subdomain of observation. Readers may refer to [75] for a complete description of such rays. This condition gener- alizes to the vectorial case the so-called geometric control condition (GCC) introduced by [76], allowing to control any solution of the acoustic wave equation from the observations of the time derivative of the wave in a subdomain. Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 41 of 47 Theorem A.5 The observability condition of Theorem A.4 holds on a subdomain ω as soon as the elastic geometric control condition is satisﬁed with ω ˇ and dist( \ω , ω ˇ ) > 0. 0 0 0 0 Proof For technical reasons, we assume that the elastic geometric control condition is satisﬁed for an observation domain ω ˇ slightly smaller than ω , namely, with ω ˇ ⊂ ω and 0 0 0 0 dist( \ω , ω ˇ ) > 0. We ﬁrst recall the classical observability result when the velocity is 0 0 0 observed. In fact there exists a constant C and a time T such that every solution of (29) st satisﬁes the observability condition 2 2 dt ≥ C u(0) + v(0) , (34) 2 3 st 2 3 L (ˇ ω ) E L ( ) 0 0 0 with T = T − δ for δ> 0 suﬃciently small, as soon as the elasticity geometric control condition is veriﬁed in the time interval [0,T[[75]. Following what was already done for acoustic waves by [73] we will use a property of equirepartition (over time) of the total energy localized within the observation subdomain between the kinetic and elastic contributions to infer (33)from (34). Let ψ ∈ C ( ) be a cutoﬀ function satisfying 0, if ξ ∈ \ω 0 0 ψ(ξ) = 1, if ξ ∈ ω ˘ 2 2 and 0 ≤ ψ(ξ) ≤ 1 for every ξ ∈ . Denote also φ(t) = t (T − t) . Then, by repeated integrations by parts we obtain 0 = φψ(¨u − u) · u d dt 0 ω ˘ ˘ T 2 T |u| = φψ d dt − φψ |u˙ | d dt 0 ω 0 ω 0 0 − φ ε(u): A :(∇ψ ⊗ u)d dt 0 ω + φψ ε(u): A : ε(u)d dt. 0 ω Moreover, ε(u): A :(∇ψ ⊗ u)d ≤ C ε(u) 1,∞ 2 3 st 2 3 W L (ω ) L (ω ) 0 ≤ C 1,∞ st 1 3 H (ω ) where C represents a diﬀerent constant in each line. This identity combined with the st properties of the cutoﬀ functions φ and ψ provides, for any strictly positive ε, the existence of a constant C > 0 such that st ˘ ˘ T −ε T 2 2 |u˙ | d dt ≤ C u(·,t) dt. st 1 3 H (ω ) ε ω ˇ 0 ω 0 0 Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 42 of 47 ˘ ˘ Substituting T + 2ε for T in all the above computations gives ˘ ˘ T +ε T +2ε 2 2 |u˙ | d dt ≤ C u(·,t) dt. st 1 3 H (ω ) ε ω ˘ 0 We proceed by making the change of variable τ = t − ε in the left-hand side integral, yielding ˘ ˘ T T +2ε 2 2 |u˙(ξ, τ + ε)| d dτ ≤ C dt. (35) st 1 3 H (ω ) 0 ω ˘ 0 Noting that u(ξ,t + ε) satisﬁes the elastodynamics system with initial data (u(ξ, ε), u˙(ξ, ε)) and applying (34) with this shifted solution, we obtain that there exists also C such that st 2 2 2 |u˙(ξ, τ + ε)| d dτ ≥ C u(ε) u˙(ε) . (36) st 2 3 0 L ( ) 0 ω ˘ Combining (35), (36) and the fact that the energy of the solution of the elastodynamics equation is exactly conserved over time, we have our observability inequality (33)upon choosing ε = . Appendix B: Analysis of the prediction–correction scheme This appendix is dedicated to the analysis of the prediction–correction scheme proposed in “Time discretization of the state observer” section. More speciﬁcally, we are interested in showing that the linearization procedure applied in the context of nonlinear observation operators induces a dissipative behavior for the linearized time-discrete estimation error. This property is a crucial aspect when building relevant state observers. Linear model and linear observation operator To start with, we assume that the dynamical system satisﬁed by the target trajectory is given by y( ˙ t) = (A + ηV)y(t) (37) y(0) = y + ζ , 0 y where A is a skew-adjoint operator, V is a self-adjoint and semi-negative operator and η ≥ 0 is a viscosity coeﬃcient. Note that (37) can be interpreted as a linearization of (3) around the stress-free conﬁguration. Additionally, we consider a linear observation operator and we neglect, for simplicity, the observation noise. Denoting by t the (constant) time step of the numerical procedure, the time-discrete observations read z = Hy(nt). The prediction–correction scheme for the observer reads n+1 n+1 n n y − y y + y − + − + ⎪ = (A + ηV) (38a) t 2 n+1 n+1 y − y + − n+1 ∗ n+1 = γ H z − H y (38b) ⎪ t y = y . Relation (38a) corresponds to the prediction step, with the operators driving the target system, while (38b) is the correction step. Deﬁning the discrete estimation error from the correction step n n y = y(nt) − y , (39) + + Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 43 of 47 and associating a prediction error with n+1 n+1 y = y((n + 1)t) − y , (40) − − we can determine the time-discrete dynamics satisﬁed by the estimation error. Proposition B.1 Assuming that y ∈ C ([0,T],Y), then the estimation error satisﬁes the following discrete dynamical system n+1 n+1 n n y − y y + y − + − + ⎪ n = (A + ηV) + ε (41a) t 2 n+1 n+1 y − y + − ∗ n+1 =−γ H H y (41b) ⎪ t y = y − y 0 0 ... n 2 with ε = O t . C ([0,T],Y) Proof (41b) is directly inferred from the deﬁnition of the prediction estimation error and using (38b), namely, n+1 n+1 y = y((n + 1)t) − y − − n+1 ∗ n+1 = y((n + 1)t) − y + γtH H y + + ∗ n+1 = (1 + γtH H) y . We now have to work our way to (41a). First, we remark that n+1 n+1 n n y − y y − y y((n + 1)t) − y(nt) − + − + = − . (42) t t t Using centered Taylor expansions we can easily see that y((n + 1)t) − y(nt) y((n + 1)t) + y(nt) = (A + ηV) + ε , (43) t 2 ... n 2 with ε = O t . Therefore, feeding equation (42)with (43)and C ([nt,(n+1)t],Y) (38a), we obtain n+1 n n+1 n y − y y((n + 1)t) − y y(nt) − y − + − + = (A + ηV) + (A + ηV) + ε , (44) t 2 2 hence, (41a)holds. We can now establish the energy estimate associated with (41a)–(41b) Proposition B.2 The norm of the estimation error, namely n n E = y , + + satisﬁes the following estimate n+1 n+1 n n E − E y + y + + − + n+1 =−η −V − γ H y t 2 Z n+1 y + y n+1 − + 2 ∗ n −γ H H y + (ε , ) . (45) 2 2 Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 44 of 47 n+1 n+1 Proof Denoting E = y , we have, from system (41a)-(41b), − − 2 Y n+1 n+1 n+1 n n n E − E √ y + y y + y − + − + − + ⎨ =−η −V + ε , t 2 2 (46) n+1 n+1 n+1 n+1 E − E y + y + − n+1 + − ⎩ ∗ =−γ H H y , t 2 Y Equation (41b)leads to n+1 n+1 E − E γ + − ∗ n+1 n+1 ∗ n+1 =−γ (H H y , y + tH H y ) + + + t 2 2 2 n+1 2 ∗ n+1 =−γ H y − γ H H y , + + Z 2 Y which, by regrouping both equations in (46), entails the desired estimate. In (45) we see the eﬀect of the correction step (38b) leading to some dissipation terms brought by the observation operator. The expression is the abstract and discrete version of expression (7), perturbed with natural consistency terms. Linear model and nonlinear observation operator As a second step, we consider the case of a nonlinear observation operator, so that the observations are obtained through (18), while the model operator remains linear. In this case, the related prediction–correction time-scheme is n+1 n+1 n n ⎪ y − y y + y − + − + = (A + ηV) (47a) t 2 n+1 n+1 y − y + − e ∗ n+1 e e n+1 e = γ DH z − H( y ) − DH ( y − y ) (47b) + + + + + ⎪ t y = y . As for the linear case, one can derive the time-discrete dynamics satisﬁed by the estimation error. Proposition B.3 Assuming that y ∈ C ([0,T],Y), then the estimation error associated with (47a)–(47b) satisﬁes the discrete dynamical system n+1 n n+1 n y − y y + y − + − + ⎪ n = (A + ηV) + ε (48a) t 2 n+1 n+1 y − y + − e ∗ e n+1 n =−γ DH DH y + λ (48b) + + + y = ζ , with the source term in (48a) is identical to the one in (41a), and the source term in (48b) is given by n e λ = O( y((n + 1)t) − y ). Proof Similarly to the linear case, the prediction estimation error is n+1 n+1 y = y((n + 1)t) − y . − − Imperiale et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:2 Page 45 of 47 This entails (48b) since from equation (47b) and the linearization of H(y((n + 1)t)) around the extrapolated trajectory y we have n+1 n+1 e ∗ n+1 e e n+1 e y = y((n + 1)t) − y + γtDH z − H( y ) − DH ( y − y ) − + + + + + + e ∗ e n+1 n = (1 + γtDH DH ) y + tλ . + + + All the other computations previously presented to prove Proposition B.1 still hold, so that we obtain the dynamical system (48a)–(48b) satisﬁed by the estimation error. We can now establish the energy estimate associated with (48a)–(48b). Proposition B.4 The norm of the estimation error in the case of a nonlinear observation operator satisﬁes the following estimate n+1 n n+1 n E − E y + y + + − + =−η −V t 2 2 2 e n+1 2 e e n+1 −γ DH y − γ DH DH y + + + + + Z 2 Z n+1 n n+1 n+1 y + y y + y − + + − n n + ε , + λ , . 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