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5.2.12 Cell Growth
The cell growth material implements a swelling pressure such that the Cauchy stress is given by where Here, represents the referential molar concentration of intracellular solutes (moles of solutes per mixture volume in the reference configuration), is the referential intracellular solid volume fraction, and is the determinant of the deformation gradient, representing the volume ratio in the current configuration. The extracellular osmolarity is . This model assumes that neither intracellular solutes nor extracellular solutes may transport across the cell membrane passively. When a cell divides, it must use active transport mechanisms to bring in membrane-impermeant extracellular solutes inside, some of which are converted into intracellular solid matrix (e.g., cytoskeletal structures). As the intracellular osmolarity increases, water is transported into the cell, thus causing it to swell. The process of cell division is not modeled explicitly in this continuum representation, though the net effect is that cell proliferation leads to an increase in intracellular osmotic pressure, which generally translates into an increase in volume (unless the cell growth is constrained significantly). In a cell growth model, the initial condition (when ) should be selected for and such that , thus
The spatial elasticity tensor associated with this osmotic pressure is where This elasticity tensor has the same form as that of an isotropic elastic material whose effective Young's modulus and Poisson's ratio are given by In the reference configuration, when , it follows that and .