An Analog of Einstein’s General Relativity Emerging from Classical Finite Elasticity Theory:Analytical and Computational Issues

The“analogue gravity formalism”,an interdisciplinary theoretical scheme developed in the past for studying several non relativistic classical and quantum systems through effective relativistic curved space-times,is here applied to largely deformable elastic bodies described by the nonlinear theory of solid mechanics.Assuming the simplest nonlinear constitutive relation for the elastic material given by a Kirchhoff-St Venant strain-energy density function,it is possible to write for the perturbations an effective space-time metric if the deformation is purely longitudinal and depends on one spatial coordinate only.Theoretical and numerical studies of the corresponding dynamics are performed in selected cases and physical implications of the results obtained are finally 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。

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.