Estimating and visualizing myocardial active stress wave patterns is crucial to understanding the mechanical activity of the heart and provides a potential non-invasive method to assess myocardial function. These patterns can be reconstructed by analyzing 2D and/or 3D tissue displacement data acquired using medical imaging. Here we describe an application that utilizes a 3D finite element formulation to reconstruct active stress from displacement data. As a proof of concept, a simple cubic mesh was used to represent a myocardial tissue "sample" consisting of a 10 x 10 x 10 lattice of nodes featuring different fiber directions that rotate with depth, mimicking cardiac transverse isotropy. In the forward model, tissue deformation was generated using a test wave with active stresses that mimic the myocardial contractile forces. The generated deformation field was used as input to an inverse model designed to reconstruct the original active stress distribution. We numerically simulated malfunctioning tissue regions (experiencing limited contractility and hence active stress) within the healthy tissue. We also assessed model sensitivity by adding noise to the deformation field generated using the forward model. The difference image between the original and reconstructed active stress distribution suggests that the model accurately estimates active stress from tissue deformation data with a high signal-to-noise ratio.

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Various technologies, such as electrocardiography, optical mapping, and patch clamping, have been developed to monitor cardiac electrophysiological behavior in live tissue. One limitation is that none of the available measurement methods is capable of monitoring simultaneously all quantities, such as intracellular ionic concentrations and ion-channel gating states, that may be important contributors to arrhythmia formation. Data assimilation strategies such as Kalman filtering can be used to fill in missing measurements, but to our knowledge, there have been few comparisons of different state estimation algorithms applied to the same cardiac action potential model. To help develop a framework for comparing performances of estimators, we applied two estimation algorithms, an unscented Kalman filter (UKF) and a gainscheduled Kalman filter (GSKF), to a two-variable Karma model of a cardiac cell. We generated simulated data from the model and compared the abilities of the algorithms to infer the slow variable of the model from measurements of the fast variable and vice versa. The UKF performed well when the process noise variance was low relative to measurement noise, while the opposite was often true for the GSKF, and estimation errors tended to be smaller when the fast variable was chosen as the measurement.

Various models exist to predict the active stresses and membrane potentials within cardiac muscle tissue. However, there exist no methods to reliably measure active stresses, nor do there exist ways to measure transmural membrane potentials that are suitable for in vivo usage. Prior work has devised a linear model to map from the active stresses within the tissue to displacements [1]. In situations where measurements of tissue displacements are entirely precise, we are able to naively solve for the active stresses from the measurements with ease. However, real measurement processes always carry some associated random error and, in the presence of this error, our naive solution to this inverse problem fails. In this work we propose the use of the Ensemble Transform Kalman Filter to more reliably solve this inverse problem. This technique is faster than other related Kalman Filter techniques while still generating high quality estimates which improve on our naive solution. We demonstrate, using in silico simulations, that the Ensemble Transform Kalman Filter produces errors whose standard deviation is an order of magnitude smaller than the least-squares solution.