Theoretical and Numerical Modeling of Nonlinear Electromechanics with applications to Biological Active Media

We present a general theoretical framework for the formulation of the nonlinear electromechanics of polymeric and biological active media.The approach developed here is based on the additive decomposition of the Helmholtz free energy in elastic and inelastic parts and on the multiplicative decomposition of the deformation gradient in passive and active parts.We describe a thermodynamically sound scenario that accounts for geometric and material nonlinearities.In view of numerical applications,we specialize the general approach to a particular material model accounting for the behavior of fiber reinforced tissues.Specifically,we use the model to solve via finite elements a uniaxial electromechanical problem dynamically activated by an electrophysiological stimulus.Implications for nonlinear solid mechanics and computational electrophysiology are finally discussed.

Modelling Thermo-Electro-Mechanical Effects in Orthotropic Cardiac Tissue

In this paper we introduce a new mathematical model for the active contraction of cardiac muscle,featuring different thermo-electric and nonlinear conductivity properties.The passive hyperelastic response of the tissue is described by an orthotropic exponential model,whereas the ionic activity dictates active contraction in-corporated through the concept of orthotropic active strain.We use a fully incompressible formulation,and the generated strain modifies directly the conductivity mechanisms in the medium through the pull-back transformation.We also investigate the influence of thermo-electric effects in the onset of multiphysics emergent spatiotem-poral dynamics,using nonlinear diffusion.It turns out that these ingredients have a key role in reproducing pathological chaotic dynamics such as ventricular fibrillation during inflammatory events,for instance.The specific structure of the governing equations suggests to cast the problem in mixed-primal form and we write it in terms of Kirchhoff stress,displacements,solid pressure,dimensionless electric potential,activation generation,and ionic variables.We also advance a new mixed-primal finite element method for its numerical approximation,and we use it to explore the properties of the model and to assess the importance of coupling terms,by means of a few computational experiments in 3D.

On Spiral Waves Arising in Natural Systems

Spiral waves appear in many different natural contexts:excitable biological tissues,fungi and amoebae colonies,chemical reactions,growing crystals,fluids and gas eddies as well as in galaxies.While the existing theories explain the presence of spirals in terms of nonlinear parabolic equations,it is explored here the fact that selfsustained spiral wave regime is already present in the linear heat operator,in terms of integer Bessel functions of complex argument.Such solutions,even if commonly not discussed in the literature because diverging at spatial infinity,play a central role in the understanding of the universality of spiral process.In particular,we have studied how in nonlinear reaction-diffusion models the linear part of the equations determines the wave front appearance while nonlinearities are mandatory to cancel out the blowup of solutions.The spiral wave pattern still requires however at least two cross-reacting species to be physically realized.Biological implications of such a results are discussed.

A Bistable FieldModel of Cancer Dynamics

Cancer spread is a dynamical process occurring not only in time but also in space which,for solid tumors at least,can be modeled quantitatively by reaction and diffusion equations with a bistable behavior:tumor cell colonization happens in a portion of tissue and propagates,but in some cases the process is stopped.Such a cancer proliferation/extintion dynamics is obtained in many mathematical models as a limit of complicated interacting biological fields.In this article we present a very basic model of cancer proliferation adopting the bistable equation for a single tumor cell dynamics.The reaction-diffusion theory is numerically and analytically studied and then extended in order to take into account dispersal effects in cancer progression in analogy with ecological models based on the porous medium equation.Possible implications of this approach for explanation and prediction of tumor development on the lines of existing studies on brain cancer progression are discussed.The potential role of continuum models in connecting the two predominant interpretative theories about cancer,once formalized in appropriatemathematical terms,is discussed。