The derivation of so-called `effective descriptions' that explicitly incorporate microscale physics into a macroscopic model has garnered much attention, with popular applications in poroelasticity, and models of the subsurface in particular. More recently, such approaches have been applied to describe the physics of biological tissue. In such applications, a key feature is that the material is active, undergoing both elastic deformation and growth in response to local biophysical/chemical cues.
Here, two new macroscale descriptions of drug/nutrient-limited tissue growth are introduced, obtained by means of two-scale asymptotics. First, a multiphase viscous fluid model is employed to describe the dynamics of a growing tissue within a porous scaffold (of the kind employed in tissue engineering applications) at the microscale. Secondly, the coupling between growth and elastic deformation is considered, employing a morpho-elastic description of a growing poroelastic medium. Importantly, in this work, the restrictive assumptions typically made on the underlying model to permit a more straightforward multiscale analysis are relaxed, by considering finite growth and deformation at the pore scale.
In each case, a multiple scales analysis provides an effective macroscale description, which incorporates dependence on the microscale structure and dynamics provided by prototypical `unit cell-problems'. Importantly, due to the complexity that we accommodate, and in contrast to many other similar studies, these microscale unit cell problems are themselves parameterised by the macroscale dynamics.
In the first case, the resulting model comprises a Darcy flow, and differential equations for the volume fraction of cells within the scaffold and the concentration of nutrient, required for growth. Stokes-type cell problems retain multiscale dependence, incorporating active cell motion [1]. Example numerical simulations indicate the influence of microstructure and cell dynamics on predicted macroscale tissue evolution. In the morpho-elastic model, the effective macroscale dynamics are described by a Biot-type system, augmented with additional terms pertaining to growth, coupled to an advection--reaction--diffusion equation [2].
[1] HOLDEN, COLLIS, BROOK and O'DEA. (2018). A multiphase multiscale model for nutrient limited tissue growth, ANZIAM (In press)
[2] COLLIS, BROWN, HUBBARD and O'DEA. (2017). Effective Equations Governing an Active Poroelastic Medium, Proceedings of the Royal Society A. 473, 20160755