| Abstract | Multiphase modelling is a natural framework for studying many biological systems, such as tissue growth. Biological tissue can be grown external to the body in a perfusion bioreactor, a device to replicate the in vivo environment. The bioreactor system comprises a cell-seeded porous scaffold, which is placed within a culture medium filled cylinder and a flow is driven across the scaffold allowing the mechanical stimulation of cells via pressure and/or fluid shear. The stimuli affects whether the cells proliferate or deposit Extra Cellular Matrix (ECM). Using the multiphase framework, different phases represent the constituents of the system, e.g. the scaffold, ECM, cells, interstitial fluid and culture medium. The resulting model comprises non-standard mixed systems of non-linear Partial Differential Equations (PDEs). For example, multiphase models of the perfusion bioreactor consist of: (i) viscous fluid flow equations for each phase; (ii) hyperbolic PDEs for mass conservation; and (iii) elliptic or parabolic PDEs for nutrient concentrations. Analytical progress with such systems is usually only possible if additional model assumptions, such as radial symmetry or small aspect ratio of the bioreactor, are made. The numerical solution of these equations presents numerous challenges: the numerical methods for solving fluid flow equations and hyperbolic PDEs are notoriously prone to complications such as instability and large computational time. Advanced numerical algorithms are therefore required in order to guarantee an accurate and efficient solution. In this talk we utilise a numerical and computational framework based upon the Galerkin finite element method to present numerical solution of the coupled systems of parabolic, elliptic and hyperbolic PDEs described above in two or more dimensions. This enables us to investigate the effect of interactions between constitutive phases in the tissue model. In particular we investigate the two-dimensional structure of tissue constructs grown in a perfusion bioreactor, external to the body, under varying growth stimuli, showing: (i) when and where dimensional simplifications, such as the long wavelength limit, are appropriate; and (ii) how different seeding strategies affect the final composition of the resulting tissue constructs. This work is of both theoretical and practical interest utilising advanced numerical methods to create biologically relevant conclusions. |
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