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We study dynamics emergent from a two-dimensional reaction--diffusion process modelled via a finite lattice dynamical system, as well as an analogous PDE system, involving spatially nonlocal interactions. These models govern the evolution of cells in a bioactive porous medium, with evolution of the local cell density depending on a coupled quasi--static fluid flow problem. We demonstrate differences emergent from the choice of a discrete lattice or a continuum for the spatial domain of such a process. We find long--time oscillations and steady states in cell density in both lattice and continuum models, but that the continuum model only exhibits solutions with vertical symmetry, independent of initial data, whereas the finite lattice admits asymmetric oscillations and steady states arising from symmetry-breaking bifurcations. We conjecture that it is the structure of the finite lattice which allows for more complicated asymmetric dynamics. Our analysis suggests that the origin of both types of oscillations is a nonlocal reaction-diffusion mechanism mediated by quasi-static fluid flow.
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A contemporary procedure to grow artificial tissue is to seed cells onto a porous biomaterial scaffold and culture it within a perfusion bioreactor to facilitate the transport of nutrients to growing cells. Typical models of cell growth for tissue engineering applications make use of spatially homogeneous or spatially continuous equations to model cell growth, flow of culture medium, nutrient transport, and their interactions. The network structure of the physical porous scaffold is often incorporated through parameters in these models, either phenomenologically or through techniques like mathematical homogenization. We derive a model on a square grid lattice to demonstrate the importance of explicitly modelling the network structure of the porous scaffold, and compare results from this model with those from a modified continuum model from the literature. We capture two-way coupling between cell growth and fluid flow by allowing cells to block pores, and by allowing the shear stress of the fluid to affect cell growth and death. We explore a range of parameters for both models, and demonstrate quantitative and qualitative differences between predictions from each of these approaches, including spatial pattern formation and local oscillations in cell density present only in the lattice model. These differences suggest that for some parameter regimes, corresponding to specific cell types and scaffold geometries, the lattice model gives qualitatively different model predictions than typical continuum models. Our results inform model selection for bioactive porous tissue scaffolds, aiding in the development of successful tissue engineering experiments and eventually clinically successful technologies.
Mesoporous bioactive glasses (MBGs) in the system SiO2-CaO-P2O5-Ga2O3 have been synthesized by the evaporation induced self-assembly method and subsequent impregnation with Ga cations. Two different compositions have been prepared and the local environment of Ga(III) has been characterized using 29Si, 71Ga and 31P NMR analysis, demonstrating that Ga(III) is efficiently incorporated as both, network former (GaO4 units) and network modifier (GaO6 units). In vitro bioactivity tests evidenced that Ga-containing MBGs retain their capability for nucleation and growth of an apatite-like layer in contact with a simulated body fluid with ion concentrations nearly equal to those of human blood plasma. Finally, in vitro cell culture tests evidenced that Ga incorporation results in a selective effect on osteoblasts and osteoclasts. Indeed, the presence of this element enhances the early differentiation towards osteoblast phenotype while disturbing osteoclastogenesis. Considering these results, Ga-doped MBGs might be proposed as bone substitutes, especially in osteoporosis scenarios.
Macroporous scaffolds made of a SiO2-CaO-P2O5 mesoporous bioactive glass (MBG) and epolycaprolactone (PCL) have been prepared by robocasting. These scaffolds showed an excellent in vitro biocompatibility in contact with osteoblast like cells (Saos 2) and osteoclasts derived from RAW 264.7 macrophages. In vivo studies were carried out by implantation into cavitary defects drilled in osteoporotic sheep. The scaffolds evidenced excellent bone regeneration properties, promoting new bone formation at both the peripheral and the inner parts of the scaffolds, thick trabeculae, high vascularization and high presence of osteoblasts and osteoclasts. In order to evaluate the effects of the local release of an antiosteoporotic drug, 1% (%wt) of zoledronic acid was incorporated to the scaffolds. The scaffolds loaded with zoledronic acid induced apoptosis in Saos 2 cells, impeded osteoclast differentiation in a time dependent manner and inhibited bone healing, promoting an intense inflammatory response in osteoporotic sheep.
We investigate the chemical dissolution of porous media using a network model in which the system is represented as a series of interconnected pipes with the diameter of each segment increasing in proportion to the local reactant consumption. Moreover, the topology of the network is allowed to change dynamically during the simulation: as the diameters of the eroding pores become comparable with the interpore distances, the pores are joined together thus changing the interconnections within the network. With this model, we investigate different growth regimes in an evolving porous medium, identifying the mechanisms responsible for the emergence of specific patterns. We consider both the random and regular network and study the effect of the network geometry on the patterns. Finally, we consider practically important problem of finding an optimum flow rate that gives a maximum increase in permeability for a given amount of reactant.
The motion of active polymers in a porous medium is shown to depend critically on flexibilty, activity and degree of polymerization. For given Peclet number, we observe a transition from localisation to diffusion as the stiffness of the chains is increased. Whereas stiff chains move almost unhindered through the porous medium, flexible ones spiral and get stuck. Their motion can be accounted for by the model of a continuous time random walk with a renewal process corresponding to unspiraling. The waiting time distribution is shown to develop heavy tails for decreasing stiffness, resulting in subdiffusive and ultimately caged behaviour.
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