Introduction

The two- and three-dimensional convection–diffusion equations play an important role in many applications in biomedical engineering. One typical example is the analysis of the effectiveness of different types of bioreactors for tissue engineering. Tissue engineering is a rapidly evolving interdisciplinary research area aiming at the replacement or restoration of diseased or damaged tissue. In many cases, devices made of artificial materials are only capable of partially restoring the original function of native tissues, and may not last for the full lifetime of a patient. In addition, there is no artificial replacement for a large number of tissues and organs. In tissue engineering, new, autologous tissues are grown. The tissue proliferation and differentiation process is strongly affected by mechanical stimuli and transport of oxygen, minerals, nutrients and growth factors. To optimize bioreactor systems, it is necessary to analyse how these systems behave. The convection–diffusion equation plays an important role in this kind of simulating analysis.

Figure 16.1 shows two different bioreactor configurations, both of which have been used in the past to tissue engineer articular cartilage. The work was especially focussed on glucose, oxygen and lactate, because these metabolites play a major role in the biosynthesis and survival of chondrocyte. Questions ranged from: ‘Does significant nutrient depletion occur at the high cells concentrations required for chondrogenesis?’ to ‘Do increasing transport limitations due to matrix accumulation significantly affect metabolite distributions?’ Figure 16.2 shows a typical result for the calculated oxygen distributions in the two bioreactor configurations.

This chapter explains the discretization of the convection-diffusion equation in two or three dimensions. First, the diffusion equation is discussed; thereafter, the convection–diffusion equation is elaborated. The spatial discretization of the weighting function is based on the Galerkin method.

Diffusion Equation

Consider a two- or three-dimensional domain with boundary. As in the onedimensional model problem, the boundary is split into a part along which the essential boundary conditions are specified, and a part along which the natural boundary conditions may be specified. The generic form of the diffusion equation is given by