There is an increasing need for replacement tissue and organs as a result of trauma, disease or simply old age. Due to the chronic shortage of available donors, tissue engineering (the cultivation of transplantable human tissue) is becoming a vital avenue of research. However, the complexity of the systems required and high associated cost mean that progress to date has been limited and hence mathematical modelling of the processes involved promises to be of great benefit. There are a number of tissue engineering projects underway in OCIAM.
To successfully grow tissues in the laboratory a better understanding of the complex interaction of processes that underlie tissue growth is required. For mathematical models and computer simulations to provide this insight they must be able to encompass huge changes in the number of cells and considerable heterogeneity in the tissue growth. Hybrid models of tissue growth which combine discrete and continuum models at different times and in different regions of space offer a potential solution to this problem.
A continuum model has been developed to study the effect of fluid and nutrient transport on the growth of a cell layer in a hollow fibre bioreactor (HFB). Asymptotic analysis was used to reduce the governing equations for the fluid and nutrient transport to an analytically tractable system by exploiting the small aspect ratio of the bioreactor. The cell layer was assumed to grow in response to the quasi-steady local nutrient concentration and the growth law solved numerically.
As a first step toward incorporating the effect of the cells on the fluid flow and nutrient distribution in the bioreactor, a one-dimensional model of cell regions growing along a membrane separating an inflow and outflow region was derived, in which the permeability of the membrane to fluid and nutrient is reduced by the presence of the cell regions. Since the cell regions are discrete it is necessary to solve the governing equations numerically.
The continuum model of cell layer growth in a HFB has shown that nutrient delivery to, and waste product removal from, the cells is improved by opening the exit ports on the outside of the bioreactor, which enhances radial flow through the membrane to the cells. The model suggests that this supports greater and more stable growth of the cell layer. The model of cell regions growing on a membrane has been used to investigate the influence of the initial cell density and distribution on the subsequent growth and suggests that higher initial seeding densities reduce the time taken for the cells to reach confluence.
The future aims of the project are to develop more detailed mechanical models for the interaction of cells sparsely seeded on a bioreactor scaffold with the fluid flowing around them; to consider discrete cell-based models of tissue growth and derive continuum approximations of these models that include cell division; and to couple these discrete and continuum models together for tissues in which a continuum approximation is not valid over the whole domain.
For more information please contact Sarah Waters or Lloyd Chapman.
Key references in this area
- R.J. Shipley and S.L. Waters. (2011) Fluid and mass transport modelling to drive the design of cell-packed hollow fibre bioreactors for tissue engineering applications. Mathematical Medicine and Biology.
- J.A. Fozard, H.M. Byrne, O.E. Jensen and J.R. King (2010). Continuum approximations of individual-based models for epithelial monolayers. Mathematical Medicine and Biology 27(1):39-74.
In this project a model has been developed of a hollow-fibre membrane bioreactor (HFMB), which consists of a hollow fibre inserted into a cylindrical glass module (see picture). Cells are seeded in the region outside the fibre and culture medium is driven through the fibre lumen and reaches the cells via the porous fibre walls. Multiphase theory is used to describe the coupled dynamics of the fluid and cells within a simplified bioreactor setup, incorporating nutrients and chemotaxis. The main challenges so far have been the inherent complexity of the biological system and deciding which key features to include in the model, as well as determining the appropriate constitutive laws and parameter values. Existing models and experimental results in the literature have helped to overcome these issues.
Using a combination of asymptotic and numerical techniques the system has been reduced and solved to determine the cell behaviour under different bioreactor conditions, such as varying flow rates and nutrient concentrations. The model may exhibit both cell aggregation and uniform cell growth, depending on the regime considered.
Future directions include the addition of an expanding layer of cells on the fibre surface, and model validation through comparing with experimental results.
Key references in this area
- G. Lemon, J. R. King, H. M. Byrne, O. E. Jensen, and K. M. Shakesheff (2006). Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory. Journal of Mathematical Biology 52: 571-594.
- R. Shipley, S. Waters, and M. Ellis (2010). Definition and validation of operating equations for poly(vinyl alcohol)-poly(lactide-co-glycolide) microfiltration membrane-scaffold bioreactors. Biotechnology and Bioengineering 107(2): 382-392.
One method for growing such tissue is to seed cells in a porous biomaterial scaffold and then to culture this tissue construct within a rotating high aspect ratio vessel (HARV) bioreactor, which in its simplest form is a fluid-filled cylindrical vessel which rotates about its longitudinal axis at a controlled rotation rate. The possible applications of this procedure vary, and include implantation of the engineered tissue into the body to replace damaged tissue and engineering of tumours to test anti-cancer drugs in the laboratory. The main point of interest is modelling the transport of nutrient to the cells within the porous tissue construct.
Novel mathematical models are being developed to determine how the bioreactor geometry and operating conditions (such as rotation rate) affect the resulting tissue growth and the eventual shape of the porous tissue construct. The techniques used include mathematical modelling, asymptotic methods, analytical solutions to partial and ordinary differential equations, and numerical solutions to non-linear ordinary differential equations. A challenge has been adapting the Darcy equations for flow through a stationary porous medium to flow through a moving porous medium in a regime where inertia is non-negligible. This is being combated by considering the problem from a multi-phase point of view.
At present the Navier-Stokes and Darcy Flow equations for the fluid mechanics have been solved analytically in two different flow regimes (under various modelling assumptions). This solution has been coupled with a numerical solution for the tissue construct trajectory, allowing the tissue trajectory and instantaneous streamlines to be determined. Since the fluid velocity inside and outside the tissue construct is then known, the fluid particle paths can be determined, which are a vital element for calculating nutrient transport.
Future directions include the relaxation of some of the modelling assumptions, and the consideration of more biologically relevant cases where the tissue construct porosity varies, both temporally and spatially. Once the fluid flow problems are well understood the nutrient transport and tissue growth problem will be considered, which will enable the prediction of the tissue growth and its resulting shape.
Key references in this area
- Cummings, L.J. & Waters, S.L. (2007). Tissue growth in a rotating bioreactor. Part II: Flow and nutrient transport problems. Math. Med. Biol. 24: 169-208.
- Cummings, L.J., Sawyer, N.B.E., Morgan, S.P., Rose, F.R.A.J. & Waters, S.L. (2009). Tracking large solid constructs suspended in a rotating bioreactor: a combined experimental and theoretical study. Biotech. Bioeng. 104(6): 1224-1234.