Modeling and Simulation of Viscous Electro-Active Polymers.

Electro-active materials are capable of undergoing large deformation when stimulated by an electric field. They can be divided into electronic and ionic electro-active polymers (EAPs) depending on their actuation mechanism based on their composition. We consider electronic EAPs, for which attractive Coulomb forces or local re-orientation of polar groups cause a bulk deformation. Many of these materials exhibit pronounced visco-elastic behavior. Here we show the development and implementation of a constitutive model, which captures the influence of the electric field on the visco-elastic response within a geometrically non-linear finite element framework. The electric field affects not only the equilibrium part of the strain energy function, but also the viscous part. To adopt the familiar additive split of the strain from the small strain setting, we formulate the governing equations in the logarithmic strain space and additively decompose the logarithmic strain into elastic and viscous parts. We show that the incorporation of the electric field in the viscous response significantly alters the relaxation and hysteresis behavior of the model. Our parametric study demonstrates that the model is sensitive to the choice of the electro-viscous coupling parameters. We simulate several actuator structures to illustrate the performance of the method in typical relaxation and creep scenarios. Our model could serve as a design tool for micro-electro-mechanical systems, microfluidic devices, and stimuli-responsive gels such as artificial skin, tactile displays, or artificial muscle.

The Generalized Hill Model: A Kinematic Approach Towards Active Muscle Contraction.

Excitation-contraction coupling is the physiological process of converting an electrical stimulus into a mechanical response. In muscle, the electrical stimulus is an action potential and the mechanical response is active contraction. The classical Hill model characterizes muscle contraction though one contractile element, activated by electrical excitation, and two non-linear springs, one in series and one in parallel. This rheology translates into an additive decomposition of the total stress into a passive and an active part. Here we supplement this additive decomposition of the stress by a multiplicative decomposition of the deformation gradient into a passive and an active part. We generalize the one-dimensional Hill model to the three-dimensional setting and constitutively define the passive stress as a function of the total deformation gradient and the active stress as a function of both the total deformation gradient and its active part. We show that this novel approach combines the features of both the classical stress-based Hill model and the recent active-strain models. While the notion of active stress is rather phenomenological in nature, active strain is micro-structurally motivated, physically measurable, and straightforward to calibrate. We demonstrate that our model is capable of simulating excitation-contraction coupling in cardiac muscle with its characteristic features of wall thickening, apical lift, and ventricular torsion.

A fully implicit finite element method for bidomain models of cardiac electromechanics.

We propose a novel, monolithic, and unconditionally stable finite element algorithm for the bidomain-based approach to cardiac electromechanics. We introduce the transmembrane potential, the extracellular potential, and the displacement field as independent variables, and extend the common two-field bidomain formulation of electrophysiology to a three-field formulation of electromechanics. The intrinsic coupling arises from both excitation-induced contraction of cardiac cells and the deformation-induced generation of intra-cellular currents. The coupled reaction-diffusion equations of the electrical problem and the momentum balance of the mechanical problem are recast into their weak forms through a conventional isoparametric Galerkin approach. As a novel aspect, we propose a monolithic approach to solve the governing equations of excitation-contraction coupling in a fully coupled, implicit sense. We demonstrate the consistent linearization of the resulting set of non-linear residual equations. To assess the algorithmic performance, we illustrate characteristic features by means of representative three-dimensional initial-boundary value problems. The proposed algorithm may open new avenues to patient specific therapy design by circumventing stability and convergence issues inherent to conventional staggered solution schemes.

A fully implicit finite element method for bidomain models of cardiac electrophysiology.

This work introduces a novel, unconditionally stable and fully coupled finite element method for the bidomain system of equations of cardiac electrophysiology. The transmembrane potential Φ(i)-Φ(e) and the extracellular potential Φ(e) are treated as independent variables. To this end, the respective reaction-diffusion equations are recast into weak forms via a conventional isoparametric Galerkin approach. The resultant nonlinear set of residual equations is consistently linearised. The method results in a symmetric set of equations, which reduces the computational time significantly compared to the conventional solution algorithms. The proposed method is inherently modular and can be combined with phenomenological or ionic models across the cell membrane. The efficiency of the method and the comparison of its computational cost with respect to the simplified monodomain models are demonstrated through representative numerical examples.

Rigid, complete annuloplasty rings increase anterior mitral leaflet strains in the normal beating ovine heart.

BACKGROUND: Annuloplasty ring or band implantation during surgical mitral valve repair perturbs mitral annular dimensions, dynamics, and shape, which have been associated with changes in anterior mitral leaflet (AML) strain patterns and suboptimal long-term repair durability. We hypothesized that rigid rings with nonphysiological three-dimensional shapes, but not saddle-shaped rigid rings or flexible bands, increase AML strains. METHODS AND RESULTS: Sheep had 23 radiopaque markers inserted: 7 along the anterior mitral annulus and 16 equally spaced on the AML. True-sized Cosgrove-Edwards flexible, partial band (n=12), rigid, complete St Jude Medical rigid saddle-shaped (n=12), Carpentier-Edwards Physio (n=12), Edwards IMR ETlogix (n=11), and Edwards GeoForm (n=12) annuloplasty rings were implanted in a releasable fashion. Under acute open-chest conditions, 4-dimensional marker coordinates were obtained using biplane videofluoroscopy along with hemodynamic parameters with the ring inserted and after release. Marker coordinates were triangulated, and the largest maximum principal AML strains were determined during isovolumetric relaxation. No relevant changes in hemodynamics occurred. Compared with the respective control state, strains increased significantly with rigid saddle-shaped annuloplasty ring, Carpentier-Edwards Physio, Edwards IMR ETlogix, and Edwards GeoForm (0.14 ± 0.05 versus 0.16 ± 0.05, P=0.024, 0.15 ± 0.03 versus 0.18 ± 0.04, P=0.020, 0.11 ± 0.05 versus 0.14 ± 0.05, P=0.042, and 0.13 ± 0.05 versus 0.16 ± 0.05, P=0.009), but not with Cosgrove-Edwards band (0.15 ± 0.05 versus 0.15 ± 0.04, P=0.973). CONCLUSIONS: Regardless of three-dimensional shape, rigid, complete annuloplasty rings, but not a flexible, partial band, increased AML strains in the normal beating ovine heart. Clinical studies are needed to determine whether annuloplasty rings affect AML strains in patients, and, if so, whether ring-induced perturbations in leaflet strain states are linked to repair failure.

In vivo dynamic strains of the ovine anterior mitral valve leaflet.

Understanding the mechanics of the mitral valve is crucial in terms of designing and evaluating medical devices and techniques for mitral valve repair. In the current study we characterize the in vivo strains of the anterior mitral valve leaflet. On cardiopulmonary bypass, we sew miniature markers onto the leaflets of 57 sheep. During the cardiac cycle, the coordinates of these markers are recorded via biplane fluoroscopy. From the resulting four-dimensional data sets, we calculate areal, maximum principal, circumferential, and radial leaflet strains and display their profiles on the averaged leaflet geometry. Average peak areal strains are 13.8±6.3%, maximum principal strains are 13.0±4.7%, circumferential strains are 5.0±2.7%, and radial strains are 7.8±4.3%. Maximum principal strains are largest in the belly region, where they are aligned with the circumferential direction during diastole switching into the radial direction during systole. Circumferential strains are concentrated at the distal portion of the belly region close to the free edge of the leaflet, while radial strains are highest in the center of the leaflet, stretching from the posterior to the anterior commissure. In summary, leaflet strains display significant temporal, regional, and directional variations with largest values inside the belly region and toward the free edge. Characterizing strain distribution profiles might be of particular clinical significance when optimizing mitral valve repair techniques in terms of forces on suture lines and on medical devices.

Anterior mitral leaflet curvature during the cardiac cycle in the normal ovine heart.

BACKGROUND: The dynamic changes of anterior mitral leaflet (AML) curvature are of primary importance for optimal left ventricular filling and emptying but are incompletely characterized. METHODS AND RESULTS: Sixteen radiopaque markers were sutured to the AML in 11 sheep, and 4-dimensional marker coordinates were acquired with biplane videofluoroscopy. A surface subdivision algorithm was applied to compute the curvature across the AML at midsystole and at maximal valve opening. Septal-lateral (SL) and commissure-commissure (CC) curvature profiles were calculated along the SL AML meridian (M(SL))and CC AML meridian (M(CC)), respectively, with positive curvature being concave toward the left atrium. At midsystole, the M(SL) was concave near the mitral annulus, turned from concave to convex across the belly, and was convex along the free edge. At maximal valve opening, the M(SL) was flat near the annulus, turned from slightly concave to convex across the belly, and flattened toward the free edge. In contrast, the M(CC) was concave near both commissures and convex at the belly at midsystole but convex near both commissures and concave at the belly at maximal valve opening. CONCLUSIONS: While the SL curvature of the AML along the M(SL) is similar across the belly region at midsystole and early diastole, the CC curvature of the AML along the M(CC) flips, with the belly being convex to the left atrium at midsystole and concave at maximal valve opening. These curvature orientations suggest optimal left ventricular inflow and outflow shapes of the AML and should be preserved during catheter or surgical interventions.

A multiscale model for eccentric and concentric cardiac growth through sarcomerogenesis.

We present a novel computational model for maladaptive cardiac growth in which kinematic changes of the cardiac chambers are attributed to alterations in cytoskeletal architecture and in cellular morphology. We adopt the concept of finite volume growth characterized through the multiplicative decomposition of the deformation gradient into an elastic part and a growth part. The functional form of its growth tensor is correlated to sarcomerogenesis, the creation and deposition of new sarcomere units. In response to chronic volume-overload, an increased diastolic wall strain leads to the addition of sarcomeres in series, resulting in a relative increase in cardiomyocyte length, associated with eccentric hypertrophy and ventricular dilation. In response to chronic pressure-overload, an increased systolic wall stress leads to the addition of sacromeres in parallel, resulting in a relative increase in myocyte cross sectional area, associated with concentric hypertrophy and ventricular wall thickening. The continuum equations for both forms of maladaptive growth are discretized in space using a nonlinear finite element approach, and discretized in time using the implicit Euler backward scheme. We explore a generic bi-ventricular heart model in response to volume- and pressure-overload to demonstrate how local changes in cellular morphology translate into global alterations in cardiac form and function.

How do annuloplasty rings affect mitral leaflet dynamic motion?

OBJECTIVES: To define the effects of annuloplasty rings (ARs) on the dynamic motion of anterior mitral leaflet (AML) and posterior mitral leaflet (PML). METHODS: Fifty-eight adult, Dorsett-hybrid, male sheep (49 + or - 5 kg) had radiopaque markers inserted: eight around the mitral annulus, four along the central meridian (from edge to annulus) of the AML (#A(1)-#A(4)) and one on the PML edge (#P(1)). True-sized Edwards Cosgrove (COS, n=12), St Jude RSAR (St. Jude Medical, St. Paul, MN, USA) (n=12), Carpentier-Edwards Physio (PHYSIO, n=12), Edwards IMR ETlogix (ETL, n=10) or Edwards GeoForm (GEO, n=12) ARs were implanted in a releasable fashion. Under acute open-chest conditions, 4D marker coordinates were obtained using biplane videofluoroscopy with the respective AR inserted (COS, RSAR, PHYSIO, ETL and GEO) and after release (COS-Control, RSAR-Control, PHYSIO-Control, ETL-Control and GEO-Control). AML and PML excursions were calculated as the difference between minimum and maximum angles between the central mitral annular septal-lateral chord and the AML edge markers (alpha(1exc)-alpha(4exc)) and PML edge marker (beta(1exc)) during the cardiac cycle. RESULTS: Relative to Control, (1) RSAR, PHYSIO, ETL and GEO increased excursion of the AML annular (alpha(4exc): 13 + or - 6 degrees vs 16 + or - 7 degrees *, 16 + or - 7 degrees vs 23 + or - 10 degrees *, 12 + or - 4 degrees vs 18 + or - 9 degrees *, 15 + or - 1 degrees vs 20 + or - 9 degrees *, respectively) and belly region (alpha(2exc): 41 + or - 10 degrees vs 45 + or - 10 degrees *, 42 + or - 8 degrees vs 45 + or - 6 degrees , n.s., 33 + or - 13 degrees vs 42 + or - 14 degrees *, 39 + or - 6 degrees vs 44 + or - 6 degrees *, respectively, alpha(3exc): 24 + or - 9 degrees vs 29 + or - 11 degrees *, 28 + or - 10 degrees vs 33 + or - 10 degrees *, 16 + or - 9 degrees vs 21 + or - 12 degrees *, 25 + or - 7 degrees vs 29 + or - 9 degrees *, respectively), but not of the AML edge (alpha(1exc): 42 + or - 8 degrees vs 44 + or - 8 degrees , 43 + or - 8 degrees vs 41 + or - 6 degrees , 42 + or - 11 vs 46 + or - 10 degrees , 39 + or - 9 degrees vs 38 + or - 8 degrees , respectively, all n.s.). COS did not affect AML excursion (alpha(1exc): 40 + or - 8 degrees vs 37 + or - 8 degrees , alpha(2exc): 43 + or - 9 degrees vs 41 + or - 9 degrees , alpha(3exc): 27 + or - 11 degrees vs 27 + or - 10 degrees , alpha(4exc): 18 + or - 8 degrees vs 17 + or - 7 degrees , all n.s.). (2) PML excursion (beta(1exc)) was reduced with GEO (53 + or - 5 degrees vs 43 + or - 6 degrees *), but unchanged with COS, RSAR, PHYSIO or ETL (53 + or - 13 degrees vs 52 + or - 15 degrees , 50 + or - 13 degrees vs 49 + or - 10 degrees , 55 + or - 5 degrees vs 55 + or - 7 degrees , 52 + or - 8 degrees vs 58 + or - 6 degrees , respectively, all n.s); *=p<0.05. CONCLUSIONS: RSAR, PHYSIO, ETL and GEO rings, but not COS, increase AML excursion of the AML annular and belly region, suggesting higher anterior mitral leaflet bending stresses with rigid rings, which potentially could be deleterious with respect to repair durability. The decreased PML excursion observed with GEO could impair left ventricular filling. Clinical studies are needed to validate these findings in patients.