Introduction

The two- and three-dimensional convection-diffusion equation plays an important role in many applications in biomedical engineering. One typical example from recent research 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 outside the human body by seeding cultured cells on scaffolds and further developed in a bioreactor for later implantation. 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.

Fig. 16.1 shows two different bioreactor configurations, which both 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 chondrocyte biosynthesis and survival. Questions ranged from: ‘Does significant nutrient depletion occur at the high cell concentrations required for chondrogenesis?’ to ‘Do increasing transport limitations due to matrix accumulation significantly affect metabolite distributions?

Introduction

This chapter extends the formulation of the previous chapter for the one-dimensional diffusion equation to the time-dependent convection-diffusion equation. Although a good functioning of the human body relies on maintaining a homeostasis or equilibrium in the physiological state of the tissues and organs, it is a dynamic equilibrium. This means that all processes have to respond to changing inputs, which are caused by changes of the environment. The diffusion processes taking place in the body are not constant, but instationary, so time has to be included as an independent variable in the diffusion equation. Thus, the instationary diffusion equation becomes a partial differential equation.

Convection is the process whereby heat or particles are transported by air or fluid moving from one point to another point. Diffusion could be seen as a process of transport through immobilized fluid or air. When the fluid itself moves, particles in that fluid are dragged along. This is called convection and also plays a major role in biomechanics. An example is the loss of heat because moving air is passing the body. The air next to the body is heated by conduction, moves away and carries off the heat just taken from the body. Another example is a drug that is released at some spot in the circulation and is transported away from that spot by means of the blood flow. In larger blood vessels the prime mechanism of transportation is convection.

Introduction

Up to this point all treated problems were in a certain way one-dimensional. Indeed, in Chapter 3 we have discussed equilibrium of two- and three-dimensional bodies and in Chapters 4 the fibres were allowed to have some arbitrary orientation in three-dimensional space. But, when deformations were involved, the focus was on fibres and bars, dealing with one-dimensional force/strain relationships. Only one-dimensional equations have been solved. In the following chapters, the theory will be extended to the description of three-dimensional bodies and it is opportune to spend some time looking at the concept of a continuum.

Consider a certain amount of solid and/or fluid material in a three-dimensional space. Although in reality for neighbouring points in space the (physical) character and behaviour of the residing material may be completely different (because of discontinuities at the microscopic level, becoming clearer by reducing the scale of observation) it is common practice that a less detailed description (at a macroscopic level) with a more gradual change of physical properties is used. The discontinuous heterogeneous reality is homogenized and modelled as a continuum. To make this clearer, consider the bone in Fig. 7.1. Although one might conceive the bone at a macroscopic level, as depicted in Fig. 7.1(a), as a massive structure filling all the volume that it occupies in space, it is clear from that at a smaller scale the bone is a discrete structure with open spaces in between (although the spaces can be filled with a softer material or a liquid).

Over the past two decades the benefits of biosensor analysis have begun to be recognized in many areas of analytical science, research, and development, with analytical systems now used routinely as mainstream research tools in many laboratories in many fields. Simplistically, biosensors can be defined as devices that use biological or chemical receptors to detect analytes (molecules) in a sample. They give detailed information on the binding affinity and in many cases also the binding stoichiometry, thermodynamics, and kinetics of an interaction. Label-free biosensors, by definition, do not require the use of reporter elements (fluorescent, luminescent, radiometric, or colorimetric) to facilitate measurements. Instead, a receptor molecule is normally connected in some way to a transducer that produces an electrical signal in real time. Other techniques such as isothermal titration calorimetry (ITC), nuclear magnetic resonance (NMR), and mass spectrometry require neither reporter labels nor surface-bound receptors. In all cases detailed information on an interaction can be obtained during analysis while minimizing sample processing requirements. Unlike label- and reporter-based technologies that simply confirm the presence or absence of a detector molecule, label-free techniques can provide direct information on analyte binding to target molecules typically in the form of mass addition or depletion from the surface of a sensor substrate or via changes in a physical bulk property (such as the heat capacity) of a sample. Until recently, label-free technologies have failed to gain widespread acceptance due to technical constraints, low throughput, high user expertise requirements, and cost. Whereas they have proved to be powerful tools in the hands of a skilled user, they have not always been readily adapted to everyday lab use in which simple-to-understand results are a prerequisite.

INTRODUCTION

Many interactions studied in the biological and biomedical sciences occur with receptors at membrane surfaces. Prominent examples are neuroreceptors, cytokine receptors, ligand-gated ion channels, G protein-coupled receptors (GPCRs), and antibody and cytokine receptors. Interactions with these receptors are especially important to academics and the pharmaceutical industry as almost half of the 100 best-selling drugs on the market are targeted to a membrane receptor. To better understand the binding mechanisms of ligands with these receptors, the ligand-receptor interactions must be probed directly in vivo or in reconstituted membrane systems. Most techniques for detailed kinetic analysis of molecular recognition events are applied in solution phase using a truncated, soluble form of the receptor. Membrane receptors, however, possess significant hydrophobic domains and are likely to have different tertiary structures and binding affinities in solution relative to those occurring in a membrane environment. This approach is limited to receptors containing a single transmembrane domain and does not allow the study of signaling cascades triggered by ligand binding to a receptor or the investigation of complex membrane proteins that often homo- or heterodimerize. However, in the last 10 years there has been significant progress in the development of techniques that allow the analysis of membrane-associated ligand–receptor interactions in a model resembling their native membrane environment.

Biophysical techniques such as patch clamping, magic angle spinning nuclear magnetic resonance (MAS-NMR), fluorescence correlation spectroscopy, fluorescence resonance energy transfer, and analytical ultracentrifugation have been applied to the analysis of binding to whole cells, membrane protoplasts, and proteoliposomes.

The cover contains images reflecting biomechanics research topics at the Eindhoven University of Technology. An important aspect of mechanics is experimental work to determine material properties and to validate models. The application field ranges from microscopic structures at the level of cells to larger organs like the heart. The core of biomechanics is constituted by models formulated in terms of partial differential equations and computer models to derive approximate solutions.

Introduction

In the previous chapter the shape functions Ni have hardly been discussed in any detail. The key purpose of this chapter is first to introduce isoparametric shape functions, and second to outline numerical integration of the integrals appearing in the element coefficient matrices and element column. Before this can be done it is useful to understand the minimum requirements to be imposed on the shape functions. The key question involved is, what conditions should at least be satisfied such that the approximate solution of the boundary value problems, dealt with in the previous chapter, generated by a finite element analysis, converges to the exact solution at mesh refinement. The answer is:

1. (i) The shape functions should be smooth within each element Ωe, i.e. shape functions are not allowed to be discontinuous within an element.

2. (ii) The shape functions should be continuous across each element boundary. This condition does not always have to be satisfied, but this is beyond the scope of the present book.

3. (iii) The shape functions should be complete, i.e. at element level the shape functions should enable the representation of uniform gradients of the field variable(s) to be approximated.

Conditions (i) and (ii) allow that the gradients of the shape functions show finite jumps across the element interface. However, smoothness in the element interior assures that all integrals in which gradients of the unknown function, say u, occur can be evaluated.

INTRODUCTION

The interface between a sensor surface and the chemical or biological systems to be studied is a vital component of all sensor systems, including label-free biosensors. With the exception of solution-phase systems such as calorimetry or analytical ultracentrifugation, receptors must be attached to some form of solid support to transduce a binding event to the sensor. During this process, receptors must retain their native conformation and binding activity, attachment to the sensor must be stable over the course of the assay, and binding sites must be presented to the solution phase to interact with the analyte to generate a detectable signal. Most importantly, the support must be resistant to nonspecific binding of analyte and other sample components that could mask a specific binding signal.

Many coupling strategies utilize a bespoke chemical linker layer between the sensor and the biological component to achieve these ends. Functionalized alkane thiols and alkoxy silanes, which form stable monolayers on planar surfaces act as ideal linkers. The alkyl termini of these molecules can be derivatized with ethyleneglycol subunits to produce a protein-resistant planar surface or can be mixed with molecules that possess suitable reactivity for receptor capture, for example, –epoxy, –carboxyl, –amino, –biotinyl, –nitrilotriacetic acid. The larger binding partner (e.g., a protein target) is normally immobilized on the surface, and the smaller binding partner (e.g., a drug candidate) is allowed to bind to this surface from free solution.