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The role of material inhomogeneities in biological growth. (English) Zbl 1104.74046
It is shown that, in the presence of anisotropic growth, the source of mass acting within a tissue is ‘modulated’ by the vorticity flux due to the production of material inhomogeneities. Such a modulation occurs through the rearrangement of the source of mass as new material is anisotropically inserted into the tissue. This result may be interpreted as a ‘self-interaction’ between material inhomogeneities and the source that produces them. Enlisting the concentration of nutrients, among the variables which determine the tissue behaviour, it is shown how the availability of these substances is enhanced by the presence of the nonvanishing spin tensor. The global vorticity of the system consists of an elastic contribution, and a contribution due to growth. The presence of the latter term can be explained by the fact that the material inhomogeneities produced by the anisotropic growth act as a source of vorticity on the system.
The study of the interactions between the motion of nutrients (modelled as Brownian particles) and the tissue dynamics may lead to new analogies with solid state physics (e.g., the theory of Berry’s phases). For example, Brownian particles ‘feel’ the vorticity flux as a potential which tends to deflect their trajectories by imprinting a rotational motion. This behavior is remnant of the effect exerted by defects on electronic dynamics in solids. The observed phenomenon of ‘mass renormalization’ is also encountered in solid state physics when the ‘fully-dressed’ Green’s function for a particle propagating in a solid medium can be expressed in terms of a ‘self-energy’ part.
The physical picture presented can also be described in terms of material symmetries. In particular, by interpreting growth as the “breaking” of the body material symmetries, and retrieving the material balance laws with a Noether-like approach, the production of vorticity can be related to the unbalance of material angular momentum due to the generation of anisotropy. If the body is assumed to be isotropic prior to growth, then the anisotropy of the growth tensor breaks the body’s rotational symmetry and brings it into an anisotropic ‘state’. If, however, the body is anisotropic at the start, then its ‘degree’ of anisotropy is changed by growth. In both cases, this process implies a rearrangement of the mass source.

MSC:
74L15 Biomechanical solid mechanics
74E05 Inhomogeneity in solid mechanics
74E10 Anisotropy in solid mechanics
92C10 Biomechanics
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