MBI Publications for Edward Green (6)
Squeezing flow of a thin film of incompressible, transversely isotropic fluidPhysics of Fluids (Submitted)
E. Green and A. Friedman
The extensional flow of a thin sheet of incompressible, transversely isotropic fluidEuropean Journal of Applied MathematicsVol. 19 No. 3 (2008) pp. 225-257
AbstractMotivated by the aim of modelling the mechanical behaviour of biological gels (such as collagen gels) which have a fibrous microstructure, we consider the extensional flow of a thin two-dimensional film of incompressible, transversely isotropic viscous fluid. Neglecting inertia, and the effects of gravity and surface tension, leading-order equations are derived from a perturbation expansion of the full flow problem in powers of the (small) inverse aspect ratio. The existence and uniqueness of the solution of the reduced system of equations for small times is then proven. Special cases, in which the solution may be determined explicitly, are considered and we discuss the physical interpretation of the results.
E. Green, S. Waters , K. Shakesheff and H. Byrne
A mathematical model of liver cell aggregation in vitroBulletin of Mathematical BiologyVol. 71 No. 4 (2009) pp. 906-930
AbstractThe behaviour of mammalian cells within three-dimensional structures is an area of intense biological research and underpins the efforts of tissue engineers to regenerate human tissues for clinical applications. In the particular case of hepatocytes (liver cells), the formation of spheroidal multicellular aggregates has been shown to improve cell viability and functionality compared to traditional monolayer culture techniques. We propose a simple mathematical model for the early stages of this aggregation process, when cell clusters form on the surface of the extracellular matrix (ECM) layer on which they are seeded. We focus on interactions between the cells and the viscoelastic ECM substrate. Governing equations for the cells, culture medium and ECM are derived using the principles of mass and momentum balance. The model is then reduced to a system of four partial differential equations, which are investigated analytically and numerically. The model predicts that, provided cells are seeded at a suitable density, aggregates with clearly defined boundaries and a spatially uniform cell density on the interior will form. While the mechanical properties of the ECM do not appear to have a significant effect, strong cell-ECM interactions can inhibit, or possibly prevent, the formation of aggregates. The paper concludes with a discussion of our key findings and suggestions for future work.
E. Green, N. Ovenden and F. Smith
Flow in a multi-branching channel with compliant wallsJournal of Engineering MathematicsVol. 64 No. 4 (2009) pp. 353-365
AbstractThe problem of fluid flow in a compliant-walled channel which branches into two or more daughters is considered with the aim of understanding blood flow through arterio-venous malformations (AVMs) in the brain. The outer walls of the channel are assumed for definiteness to behave as spring-back plates, whilst the divider is taken as rigid. The fluid is assumed to be incompressible and inviscid. When the Strouhal number is small (as occurs in practice in the brain), there are two main axial length scales, one much longer than the vessel width and the other comparable with the vessel width. Also, in the case of small wall displacements, one can analyse the local flow-structure interaction problem using a complex variable method. The flow shows markedly different qualitative features downstream of the branching, depending on the wall stiffness. Keywords Branching channel â?¢ Compliant walls â?¢ Inviscid fluid flow â?¢ Matching A. Matzavinos, C.Y. Kao, J.E.F. Green, A. Sutradhar, M. Miller and A. Friedman, Modelling oxygen transport in surgical tissue transfer, Proceedings of the National Academy of Sciences, 106 (29) p12091-12096, 2009 http://www.pnas.org/content/106/29/12091.abstract Abstract Reconstructive microsurgery is a clinical technique used to transfer large amounts of a patient's tissue from one location used to another in order to restore physical deformities caused by trauma, tumors, or congenital abnormalities. The trend in this field is to transfer tissue using increasingly smaller blood vessels, which decreases problems associated with tissue harvest but increases the possibility that blood supply to the transferred tissue may not be adequate for healing. It would thus be helpful to surgeons to understand the relationship between the tissue volume and blood vessel diameter to ensure success in these operations. As a first step towards addressing this question, we present a simple mathematical model that might be used to predict successful tissue transfer based on blood vessel diameter, tissue volume, and oxygen delivery.
A. Matzavinos, C. Kao, E. Green, A. Sutradhar, M. Miller and A. Friedman
Modelling oxygen transport in surgical tissue transferPNASVol. 106 No. 29 (2009) pp. 12091-12096
E. Green, A. Bassom and A. Friedman
A mathematical model for cell-induced gel compaction in vitroMathematical Models and Methods in Applied Sciences (2012) (Accepted)