Chemomechanical Mathematical Modeling of Isolated Cardiomyocyte.

Introduction. Cardiovascular functions are usually studied via the contribution of several concurrent scientific fields from a biomechanical point of view [1]. The excitation-contraction mechanisms in the cardiac muscle are coordinated by an autonomous electrical activation spreading through the whole heart [2]. The cardiomyocyte muscular unit is composed of myofibrils bundles containing sarcomeres and consisting of protein myofilaments. At the microscale, the exchange of Calcium between cytosol and sarcoplasmic reticulum stimulates the interaction of these myofilaments inducing the shortening of the sarcomeres and driving the process of excitation-contraction of the whole cardiac cell. Although the complex excitation-contraction mechanism has major evidences both at the theoretical and experimental levels, the full understanding of the exact interplay between the different processes is still lacking [3]. In this work we provide a quantitative simulation study of the behavior of a single cardiomyocyte cell under a precise set of experimental conditions by proposing a novel chemo-active approach. Our mathematical model is based on a continuum active-strain formulation describing the nonlinear mechanical response of biological excitable media [4, 5]. In particular, the active deformation depends directly on the nonlinear dynamics that describe chemical reactions between Calcium species. Methods. Considering the standard continuum mechanics framework, we suppose that the passive material response of the isolated cell can be described through an incompressible transversely isotropic constitutive law. Moreover, in order to account for the activation of the cardiac contraction, we assume a ultiplicative decomposition of the deformation gradient into a passive and an active component, F = Fe*Fa, where this fictitious geometric process is represented by an elastic intermediate configuration. We model the interaction between cytosolic and sarcoplasmic Calcium concentrations following [6, 7]; thus defining the anisotropic mass balance laws via a system of nonlinear reaction-diffusion equations. In our minimal model the activation is assumed to depend solely on the cytosolic Calcium concentration, imposing the active shortening field gf obeying a specific dynamic process. We consider suitable and distinguished boundary conditions for the mechanical problem in order to model different physiological, experimental and pathological behaviors: to this aim the myocyte boundary contains different regions for Dirichlet, Neumann and Robin conditions. Results and Perspectives. The preliminary results we present in this abstract with a prototype model, aimed to study electrophysiology and active cardiac biomechanics with the aim of scaling the overall heart function. The proposed activation mechanism is consistent with a thermodynamic framework entailing a nonlinear nteraction among calcium dynamics and local stretches. Our model reproduces the propagation of Calcium waves and the corresponding spontaneous contraction interacting within the cell. A finite element method is used to discretize the model equations and a set of numerical experiments give evidence of the main features of the model. In particular, the boundary conditions effects on the overall myocyte dynamics have been analyzed simulating self-sustained chemical and mechanical interactions mimicking different scenarios.: fixed regions of the geometrical boundary, representing adhesion regions, or spring-like boundaries, reproducing the presence of contact myocytes. A careful inspection of Calcium waves propagation and stress analysis confirmed several experimental evidences: i) Calcium activation timing and conduction velocities; ii) asymmetric Calcium maxima and stress concentrations; iii) myocyte bending and driving contraction due to internal Calcium sparks. The correct treatment of boundary conditions other than a careful representation of its anisotropic internal structure, its external geometry and its nonlinear chemical dynamics are key aspects to obtain physiological displacements of the cell. In this work an experimental based subcellular calcium dynamics would be implemented analyzing rate-dependent effects, i.e. the positive force-frequency staircase effects. References [1] L.A. Taber, Biomechanics of cardiovascular development, Annu. Rev. Biomed. Eng., 3, 1–25 (2001). [2] J. Keener and J. Sneyd, Mathematical physiology, Springer-Verlag, New York (1998). [3] W.E. Louch, M.K. Stokke, I. Sjaastad, G. Christensen, and O.M. Sejersted, No rest for the weary: diastolic calcium homeostasis in the normal and failing myocardium, Physiology, 27, 308–323 (2012). [4] C. Cherubini, S. Filippi, P. Nardinocchi, and L. Teresi, An electromechanical model of cardiac tissue: Constitutive issues and electrophysiological effects. Prog. Biophys. Mol. Biol., 97, 562–573 (2008). [5] R. Ruiz-Baier, D. Ambrosi, S. Pezzuto, S. Rossi, and A. Quarteroni, Activation models for the numerical simulation of cardiac electromechanical interactions, In: G.A. Holzapfel, E. Kuhl (eds.), Computer Models in Biomechanics: From Nano to Macro, Springer-Verlag, Heidelberg, 189–201 (2013). [6] P. Tracqui, J. Ohayon, and T. Boudou, T. Theoretical analysis of the adaptive contractile behaviour of a single cardiomyocyte cultured on elastic substrates with varying stiffness, J. Theoret. Biol., 255, 92–105 (2008). [7] A. Goldbeter, G. Dupont, and M.J. Berridge, Minimal model for signal-induced Ca2+ oscillations and for their frequency encoding through protein phosphorylation, Proc. Natl. Acad. Sci. USA, 87, 1461–1465 (1990)