Some issues related to growth and goal functions for continuum biological systems

Abstract

The growth of biological tissues is described as the volumic production of mass of so-called germs, considered as potential sites for tissue renewal/resorption. The growth is characterized by a growth transformation gradient. The continuous change of the domain occupied by each germ during growth is analysed, hence, the mechanical balance laws are written considering the additional terms due to the domain variation. The principle of virtual power is then expressed, leading to the balance of momentum, assuming that the power of internal forces originates from both volumic and surfacic potentials. The equilibrium equations of the growing germ split into volumic, surfacic and lineic contributions. Viewing growth as an irreversible phenomenon, the writing of the second principle of thermodynamics for a germ continuously receiving matter due to transport phenomena allows to identify the driving forces linked to growth. In the second part, the growth and evolution are viewed as finding the best configuration of the growing solid in terms of its metric. Thereby, the four-dimensional space–time is structured in the form of isogrowth surfaces, the evolution of growth being described as a motion from one slice to the next. A Lagrangian formalism is established in the space of metric, whereby both the geometrical and physical contents of the growing solid are considered. Adopting this higher level of description, the stationnarity of the total action integral renders the dynamics of the metric field, while the kinematic compatibility condition prescribes its spatial form.