Passive and Active Deformation Processes of 3D Fibre-Reinforced Caricatures of Cardiovascular Tissues Antonio DiCarlo 1 , Paola Nardinocchi 2 , Tom´as Svatoˇ n 3 , and Luciano Teresi *,1 1 Modeling & Simulation Lab, University “Roma Tre”, Italy; 2 Dept. Structural Engineering & Geotechnics, University of Rome “La Sapienza”, Italy; 3 Dept. Mathematics, University of West Bohemia, Czech Republic. * Corresponding author: Via C. Segre 2, 0146 Roma, Italy; teresi@uniroma3.it Abstract: In this paper, we present a math- ematical model of contractile elastic solids meant to simulate various districts of the car- diovascular system, and based on the concepts of active deformation and embedded muscle fi- bres. Specifically, here we deal with the mod- elling of the gross mechanics of the left ventri- cle (LV) which is strictly related to its pump function. The muscle fibres embedded in the LV walls govern, through their contraction and relaxation, the characteristic phases of the car- diac cycle. Moreover, muscle fibres define the anisotropy directions of the LV wall, and the collagene fibres determine the material proper- ties along these directions; thus, to model the mechanical behaviour of the LV, both the pas- sive and the active material properties of the must be accurately accounted for. As is well known, the effectiveness of the pumping ac- tion is well represented by the pressure-volume (PV) diagrams that relate the blood pressure to the volume of the LV during the cardiac cy- cle. Here, we aim at reproducing realistic PV by specifying appropriate sequence of muscle contraction Keywords: Biomechanics, muscle modelling, fibre-reinforcedd materials, active contrac- tions. 1 Introduction A key issue in the modelling of biological tis- sues is the active nature of muscle fibres, in other words, their ability to contract and re- lax in response to biochemical signals. Such a behaviour is commonly accounted for through an additive decomposition of the stress into a standard component, representing the passive response of the tissue, typically (visco-)elastic, and a nonstandard active component, meant to represent the dynamical effects of muscle contraction [1]. We favour a different modelling perspec- tive, in which muscle contraction is accounted for by introducing the notion of active defor- mation : we assume that the contraction ex- perienced by a muscle fibre under stimulus is described at the macroscopic scale by a (stress- free) change in the length of the fibre; the visi- ble length of the fibre, in turn, depends on the amount of stress it sustains. To avoid a key misleading, it is worth saying that the notion of “active state”–a physiologic notion for mus- cles, coincides with that of “ground state”–a mechanical notion for elastic bodies. Thus, the active deformation describes how a muscu- lar tissue shortens once activated and left free to contract, while the visible deformation de- scribes the state that a muscular tissue attains once contracted, loaded and/or kinematically constrained (as in isometric activation). The corresponding material model is based on a two-layer kinematics, comprising the clas- sical vector-valued displacement field, plus a tensor-valued field parameterizing the evolv- ing stress-free state of muscular tissue. This viewpoint, anticipated by the linearized 1D model introduced in [2], was developed in [3], [4], and [5] into a full-fledged nonlinear 3D the- ory, along the lines set forth by the theory of material remodelling [6], [7]. 2 Muscle Modeling Let the body B be a smooth region (with boundary ∂ B) of the three dimensional Eu- clidean space E , and V the linear space of translations associated to E ; a displacement of B is described by a smooth vector field u : B→V ; (2.1) thus, x = X + u(X) denotes the position of a material point X ∈B. The visible deforma- Excerpt from the Proceedings of the COMSOL Conference 2009 Milan