Mass transport in morphogenetic processes: a second gradient theory for volumetric growth and material remodeling

64/2013

Mass transport in morphogenetic processes: a second gradient theory for volumetric growth and material remodeling

Monday 16th December 2013

Ciarletta, P; Ambrosi, D.; Maugin, G.A.

In this work, we derive a novel thermomechanical theory for growth and remodeling of biological materials in morphogenetic processes. This second gradient hyperelastic theory is the first attempt to describe both volumetric growth and mass transport phenomena in a single-phase continuum model, where both stress- and shape-dependent growth regulations can be investigated. The diffusion of biochemical species (e.g. morphogens, growth factors, migration signals) inside the material is driven by configurational forces, enforced in the balance equations and in the set of constitutive relations. Mass transport is found to depend both on first- and on second-order material connections, possibly withstanding a chemotactic behavior with respect to diffusing molecules. We find that the driving forces of mass diffusion can be written in terms of covariant material derivatives reflecting, in a purely geometrical manner, the presence of a (first-order) torsion and a (second-order) curvature. Thermodynamical arguments show that the Eshelby stress and hyperstress tensors drive the rearrangement of the first- and second-order material inhomogeneities, respectively. In particular, an evolution law is proposed for the first-order transplant, extending a well-known result for inelastic materials. Moreover, we define the first stress-driven evolution law of the second-order transplant in function of the completely material Eshelby hyperstress. The theory is applied to two biomechanical examples, showing how an Eshelbian coupling can coordinate volumetric growth, mass transport and internal stress state, both in physio- logical and pathological conditions. Finally, possible applications of the proposed model are discussed for studying the unknown regulation mechanisms in morphogenetic processes, as well as for an optimizing scaffold architecture in regenerative medicine and tissue engineering.

This report, or a modified version of it, has been also submitted to, or published on
J. Mech. Phys. Solids