In this paper, we revisit our previous work in which we derive an effective macroscale description suitable to describe the growth of biological tissue within a porous tissue-engineering scaffold. The underlying tissue dynamics is described as a multiphase mixture, thereby naturally accommodating features such as interstitial growth and active cell motion. Via a linearization of the underlying multiphase model (whose nonlinearity poses a significant challenge for such analyses), we obtain, by means of multiple-scale homogenization, a simplified macroscale model that nevertheless retains explicit dependence on both the microscale scaffold structure and the tissue dynamics, via so-called unit-cell problems that provide permeability tensors to parameterize the macroscale description. In our previous work, the cell problems retain macroscale dependence, posing significant challenges for computational implementation of the eventual macroscopic model; here, we obtain a decoupled system whereby the quasi-steady cell problems may be solved separately from the macroscale description. Moreover, we indicate how the formulation is influenced by a set of alternative microscale boundary conditions.

We derive an effective macroscale description for the growth of tissue on a porous scaffold. A multiphase model is employed to describe the tissue dynamics; linearisation to facilitate a multiple-scale homogenisation provides an effective macroscale description, which incorporates dependence on the microscale structure and dynamics. In particular, the resulting description admits both interstitial growth and active cell motion. This model comprises Darcy flow, and differential equations for the volume fraction of cells within the scaffold and the concentration of nutrient, required for growth. These are coupled with Stokes-type cell problems on the microscale, incorporating dependence on active cell motion and pore scale structure. The cell problems provide the permeability tensors with which the macroscale flow is parameterised. A subset of solutions is illustrated by numerical simulations.

The swelling and shrinkage of biological tissues are modelled by a four-component mixture theory in which a deformable and charged porous medium is saturated with a fluid with dissolved ions. Four components are defined: solid, liquid, cations and anions. The aim of this paper is the construction of the Lagrangian model of the four-component system. It is shown that, with the choice of Lagrangian description of the solid skeleton, the motion of the other components can be described in terms of Lagrangian initial system of the solid skeleton as well. Such an approach has a particularly important bearing on computer-aided calculations. Balance laws are derived for each component and for the whole mixture. In cooperation of the second law of thermodynamics, the constitutive equations are given. This theory results in a coupled system of nonlinear parabolic differential equations together with an algebraic constraint for electroneutrality. In this model, it is desirable to obtain an accurate approximation of the fluid flow and ions flow. Such an accurate approximation can be determined by the mixed finite element method. Part II is devoted to this task.

A mathematical model of the tumour growth along a blood vessel is proposed. Themodel employs the mixture theory approach to describe a tissue which consists of cells, extracellularmatrix and liquid. The growing tumour tissue is supposed to be surrounded by the hosttissue. Tumours where complete oxydation of glucose prevails are considered. Special attention ispaid to consistent description of oxygen consumption and growth processes based on the energybalance. A finite difference numerical method is proposed. The level set method is used to trackan interface between the tissues. The simulations show localization of the tumour within a limiteddistance from the vessels and constant expansion velocity along the vessels.

Dehydration melting of crustal rocks may commonly occur in response to the intrusion of mafic magma in the mid- or lower crust. However, the relative importance of melt buoyancy, shear or dyking in melt generation and extraction under geologically relevant conditions is not well understood. A numerical model of the partial melting of a metapelite is presented and the model results are compared with the Ivrea-Verbano Zone in northern Italy. The numerical model uses the mixture theory approach to modelling simultaneous convection and phase change and includes special ramping and switching functions to accommodate the rheology of crystal-melt mixtures in accordance with the results of deformation experiments. The model explicitly includes both porous media flow and thermally and compositionally driven bulk convection of a restitecharged melt mass. A range of melt viscosity and critical melt fraction models is considered. General agreement was found between predicted positions of isopleths and those from the Ivrea-Verbano Zone. Maximum melt velocities in the region of porous flow are found to be 1 × 10−7 and 1 × 10−1m per year in the region of viscous flow. The results indicate that melt buoyancy alone may not be a sufficient agent for melt extraction and that extensive, vigorous convection of partially molten rocks above mafic bodies is unlikely, in accord with direct geological examples.