An orthotropic active–strain model for the myocardium mechanics and its numerical approximation

In the wide literature devoted to the cardiac structural mechanics, the strain energy proposed by Holzapfel and Ogden exhibits a number of interesting features: it has suitable mathematical properties and it is based on few material parameters that can, in principle, be identified by standard laboratory tests. In this work we illustrate the implementation of a numerical solver based on such a model for both the passive and active mechanics of the heart. Moreover we discuss its performance on a few tests that can be regarded as preliminary to the adoption of the Holzapfel-Ogden model for a real cardiac simulation. While the passive behavior of the cardiac muscle is modeled as an orthotropic hyperelastic material, the active contraction is here accounted for a multiplicative decomposition of the deformation gradient, yielding the so-called "active strain" approach, a formulation that automatically preserves the ellipticity of the stress tensor and introduces just one extra parameter in the model. We adopt the usual volumetric-isochoric decomposition of the stress tensor to obtain a mathematically consistent quasi-incompressible version of the material, then the numerical approximation applies to a classical Hu-Washizu three fields formulation. After introduction of the tangent problem, we select suitable finite element spaces for the representation of the physical fields. Boundary conditions are prescribed by introduction of a Lagrange multiplier. The robustness and performance of the numerical solver are tested versus a novel benchmark test, for which an exact solution is provided. The curvature data obtained from the free contraction of muscular thin films are used to fit the active contraction parameter. (C) 2014 Elsevier Masson SAS. All rights reserved

Finite element simulations of the active stress in the imaginal disc of the Drosophila Melanogaster

International audienceDuring the larval stages of development, the imaginal disc of Drosphila Melanogaster is composed by a monolayer of epithelial cells, which undergo a strain actively produced by the cells themselves. The well-organized collective contraction produces a stress field that seemingly has a double morphogenetic role: it orchestrates the cellular organization towards the macroscopic shape emergence while simultaneously providing a local information on the organ size. Here we perform numerical simulations of such a mechanical control on morphogenesis at a continuum level, using a three-dimensional finite model that accounts for the active cell contraction. The numerical model is able to reproduce the (few) known qualitative characteristics of the tensional patterns within the imaginal disc of the fruit fly. The computed stress components slightly deviate from planarity, thus confirming the previous theoretical assumptions of a nonlinear elastic analytical model, and enforcing the hypothesis that the spatial variation of the mechanical stress may act as a size regulating signal that locally scales with the global dimension of the domain

Solid Tumors Are Poroelastic Solids with a Chemo-mechanical Feedback on Growth

The experimental evidence that a feedback exists between growth and stress in tumors poses challenging questions. First, the rheological properties (the \u201cconstitutive equations\u201d) of aggregates of malignant cells are still a matter of debate. Secondly, the feedback law (the \u201cgrowth law\u201d) that relates stress and mitotic\u2013apoptotic rate is far to be identified. We address these questions on the basis of a theoretical analysis of in vitro and in vivo experiments that involve the growth of tumor spheroids. We show that solid tumors exhibit several mechanical features of a poroelastic material, where the cellular component behaves like an elastic solid. When the solid component of the spheroid is loaded at the boundary, the cellular aggregate grows up to an asymptotic volume that depends on the exerted compression. Residual stress shows up when solid tumors are radially cut, highlighting a peculiar tensional pattern. By a novel numerical approach we correlate the measured opening angle and the underlying residual stress in a sphere. The features of the mechanobiological system can be explained in terms of a feedback of mechanics on the cell proliferation rate as modulated by the availability of nutrient, that is radially damped by the balance between diffusion and consumption. The volumetric growth profiles and the pattern of residual stress can be theoretically reproduced assuming a dependence of the target stress on the concentration of nutrient which is specific of the malignant tissue

A Comparison Between Active Strain and Active Stress in Transversely Isotropic Hyperelastic Materials

Active materials are media for which deformations can occur in absence of loads, given an external stimulus. Two approaches to the modeling of such materials are mainly used in literature, both based on the introduction of a new tensor: an additive stress Pact in the active stress case and a multiplicative strain Fa in the active strain one. Aim of this paper is the comparison between the two approaches on simple shears. Considering an incompressible and transversely isotropic material, we design constitutive relations for Pact and Fa so that they produce the same results for a uniaxial deformation along the symmetry axis. We then study the two approaches in the case of a simple shear deformation. In a hyperelastic setting, we show that the two approaches produce different stress components along a simple shear, unless some necessary conditions on the strain energy density are fulfilled. However, such conditions are very restrictive and rule out the usual elastic strain energy functionals. Active stress and active strain therefore produce different results in shear, even if they both fit uniaxial data. Our results show that experimental data on the stress-stretch response on uniaxial deformations are not enough to establish which activation approach can capture better the mechanics of active materials. We conclude that other types of deformations, beyond the uniaxial one, should be taken into consideration in the modeling of such materials

Segregated Algorithms for the Numerical Simulation of Cardiac Electromechanics in the Left Human Ventricle

We propose and numerically assess three segregated ( partitioned) algorithms for the numerical solution of the coupled electromechanics problem for the left human ventricle. We split the coupled problem into its core mathematical models and we proceed to their numerical approximation. Space and time discretizations of the core problems are carried out by means of the Finite Element Method and Backward Differentiation Formulas, respectively. In our mathematical model, electrophysiology is represented by the monodomain equation while the Holzapfel-Ogden strain energy function is used for the passive characterization of tissue mechanics. A transmurally variable active strain model is used for the active deformation of the fibers of the myocardium to couple the electrophysiology and the mechanics in the framework of the active strain model. In this work, we focus on the numerical strategy to deal with the solution of the coupled model, which is based on novel segregated algorithms that we propose. These also allow using different time discretization schemes for the core submodels, thus leading to the formulation of staggered algorithms, a feature that we systematically exploit to increase the efficiency of the overall computational procedure. By means of numerical tests we show that these staggered algorithms feature (at least) first order of accuracy. We take advantage of the efficiency of the segregated schemes to solve, in a High Performance Computing framework, the cardiac electromechanics problem for the human left ventricle, for both idealized and subject-specific configurations

Modeling Pathologies of Diastolic and Systolic Heart Failure

International audienceChronic heart failure is a medical condition that involves structural and functional changes of the heart and a progressive reduction in cardiac output. Heart failure is classified into two categories: diastolic heart failure, a thickening of the ventricular wall associated with impaired filling; and systolic heart failure, a dilation of the ventricles associated with reduced pump function. In theory, the pathophysiology of heart failure is well understood. In practice, however, heart failure is highly sensitive to cardiac microstructure, geometry, and loading. This makes it virtually impossible to predict the time line of heart failure for a diseased individual. Here we show that computational modeling allows us to integrate knowledge from different scales to create an individualized model for cardiac growth and remodeling during chronic heart failure. Our model naturally connects molecular events of parallel and serial sarcomere deposition with cellular phenomena of myofibrillogenesis and sarcomerogenesis to whole organ function. Our simulations predict chronic alterations in wall thickness, chamber size, and cardiac geometry, which agree favorably with the clinical observations in patients with diastolic and systolic heart failure. In contrast to existing single- or bi-ventricular models, our new four-chamber model can also predict characteristic secondary effects including papillary muscle dislocation, annular dilation, regurgitant flow, and outflow obstruction. Our prototype study suggests that computational modeling provides a patient-specific window into the progression of heart failure with a view towards personalized treatment planning

Electro-mechanical modeling and simulation of reentry phenomena in the presence of myocardial infarction

In this work we present a parallel solver for the numerical simulation of the cardiac electro-mechanical activity. We first review the most complete mathematical model of cardiac electro-mechanics, the so-called electro-mechanical coupling (EMC) model, which consists of the following four sub-models, strongly coupled together: the Bidomain model for the electrical activity at tissue scale, constituted by a parabolic system of two reaction-diffusion partial differential equations (PDEs); the finite elasticity system for the mechanical behavior at tissue scale; the membrane model for the bioelectrical activity at cellular scale, consisting of a stiff system of ordinary differential equations (ODEs); the active tension model for the mechanical activity at cellular scale, consisting of a system of ODEs. The discretization of the EMC model is performed by finite elements in space and an operator splitting strategy in time, based on semi-implicit finite differences. As a result of the discretization techniques adopted, the most computational demanding part at each time step is the solution of the non-linear algebraic system, deriving from the discretization of the finite elasticity equations, and of the linear system deriving from the discretization of the Bidomain equations. The former is solved by a Newton-GMRES-BDDC solver, i.e. the Jacobian system at each Newton iteration is solved by GMRES accelerated by the Balancing Domain Decomposition by Constraints (BDDC) preconditioner. The latter is solved by the Conjugate Gradient method, preconditioned by the Multilevel Additive Schwarz preconditioner. The performance of the resulting parallel solver is studied on the simulation of the induction of ventricular tachycardia in an idealized left ventricle affected by an infarct scar. The simulations are run on the Marconi-KNL cluster of the Cineca laboratory