Mechanobiology of interfacial growth

A multiscale analysis integrating biomechanics and mechanobiology is today required for deciphering the crosstalk between biochemistry, geometry and elasticity in living materials. In this paper we derive a unified thermomechanical theory coupling growth processes with mass transport phenomena across boundaries and/or material interfaces. Inside a living system made by two contiguous bodies with varying volumes, an interfacial growth mechanism is considered to force fast but continuous variations of the physical fields inside a narrow volume across the material interface. Such a phenomenon is modelled deriving homogenized surface fields on a growing non-material discontinuity, possibly including a singular edge line. A number of balance laws is derived for imposing the conservation of the thermomechanical properties of the biological system. From thermodynamical arguments we find that the normal displacement of the non-material interface is governed by the jump of a new form of material mechanical-energy flux, also involving the kinetic energies and the mass fluxes. Furthermore, the configurational balance indicates that the surface Eshelby tensor is the tangential stress measure driving the material inhomogeneities on the non-material interface. Accordingly, stress-dependent evolution laws for bulk and interfacial growth processes are derived for both volume and surface fields. The proposed thermomechanical theory is finally applied to three biological system models. The first two examples are focused on stress-free growth problems, concerning the morphogenesis of animal horns and of seashells. The third application finally deals with the stress-driven surface evolution of avascular tumours with heterogeneous structures. The results demonstrate that the proposed theory can successfully model those biological systems where growth and mass transport phenomena interact at different length-scales. Coupling biological, mechanical and geometrical factors, the proposed framework represents a powerful multiscale approach for building predictive tools to be used in biological and medical sciences.The research that led to the present paper was partially supported by a grant of the group GNFM of INdAM