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).

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 goal of the present chapter is to describe a procedure to formally determine solutions for solid mechanics problems, fluid mechanics problems and problems with filtration and diffusion. Mechanical problems in biomechanics can be very diverse and most problems are so complex, that it is impossible to derive analytical solutions and often very complicated to determine numerical solutions. Fortunately, in most cases it is not necessary to describe all phenomena related to the problem in full detail and simplifying assumptions can be made, thus reducing the complexity of the set of equations that have to be solved. The present chapter deals with formulating problems and solution strategies, starting from the most general set of equations and gradually reducing the generality by imposing simplifying assumptions. In Section 13.2 this will be done for solids. Section 13.3 is devoted to solving fluid mechanics problems. The last section of this chapter discusses diffusion and filtration.

Solution strategies for deforming solids

In this section it is assumed that the material (or material fraction) to be considered can be modelled as a deforming solid continuum. This implies that it is possible and significant to define a reference configuration or reference state. With respect to the reference state, the displacement field as a function of time supplies a full description of the deformation process to which the continuum is subjected (at least under the restrictions given in previous chapters, such as for example a constant temperature).

Introduction

In the previous chapter on fibres the material behaviour was constantly considered to be elastic, meaning that a unique relation exists between the extensional force and the deformation of the fibre. This implies that the force versus stretch curves for the loading and unloading path are indentical. There is no history dependency and all energy that is stored into the fibre during deformation is regained during the unloading phase. This also implies that the rate of loading or unloading does not affect the force versus stretch curves. However, most biological materials do not behave elastically!

An example of a loading history and a typical response of a biological material is shown in Figures 5.1(a) and (b). In Fig. 5.1(a) a deformation history is given that might be used in an experiment to mechanically characterize some material specimen. The specimen is stretched fast to a certain value, then the deformation is fixed and after a certain time restored to zero. After a short resting period, the stretch is applied again but to a higher value of the stretch. This deformation cycle is repeated several times. In this case the length change is prescribed and the associated force is measured. Fig. 5.1(b) shows the result of such a measurement. When the length of the fibre is kept constant, the force decreases in time. This phenomenon is called relaxation. Reversely, if a constant load is applied, the length of the fibre will increase. This is called creep.

In September 1997 an educational programme in Biomedical Engineering, unique in the Netherlands, started at the Eindhoven University of Technology, together with the University of Maastricht, as a logical step after almost two decades of research collaboration between both universities. This development culminated in the foundation of the Department of Biomedical Engineering in April 1999 and the creation of a graduate programme (MSc) in Biomedical Engineering in 2000 and Medical Engineering in 2002.

Already at the start of this educational programme, it was decided that a comprehensive course in biomechanics had to be part of the curriculum and that this course had to start right at the beginning of the Bachelor phase. A search for suitable material for this purpose showed that excellent biomechanics textbooks exist. But many of these books are very specialized to certain aspects of biomechanics. The more general textbooks are addressing mechanical or civil engineers or physicists who wish to specialize in biomechanics, so these books include chapters or sections on biology and physiology. Almost all books that were found are at Masters or post-graduate level, requiring basic to sophisticated knowledge of mechanics and mathematics. At a more fundamental level only books could be found that were written for mechanical and civil engineers.

We decided to write our own course material for the basic training in mechanics appropriate for our candidate biomedical engineers at Bachelor level, starting with the basic concepts of mechanics and ending with numerical solution procedures, based on the Finite Element Method.

In the present and following chapters extensive use will be made of a simple finite element code mlfem_nac. This code, including a manual, can be freely downloaded from the website: www.mate.tue.nl/biomechanicsbook.

The code is written in the program environment MATLAB. To be able to use this environment a licence for MATLAB has to be obtained. For information about MATLAB see: www.mathworks.com.

Introduction

It will be clear from the previous chapters that many problems in biomechanics are described by (sets of) partial differential equations and for most problems it is difficult or impossible to derive closed form (analytical) solutions. However, by means of computers, approximate solutions can be determined for a very large range of complex problems, which is one of the reasons why biomechanics as a discipline has grown so fast in the last three decades. These computer-aided solutions are called numerical solutions, as opposed to analytical or closed form solutions of equations. The present and following chapters are devoted to the numerical solution of partial differential equations, for which several methods exist. The most important ones are the Finite Difference Method and the Finite Element Method. The latter is especially suitable for partial differential equations on domains with complicated geometries, material properties and boundary conditions (which is nearly always the case in biomechanics). That is why the next chapters focus on the Finite Element Method. The basic concepts of the method are explained in the present chapter.

Introduction

Fibres and fibre-like structures play an important role in the mechanical properties of biological tissues. Fibre-like structures may be found in almost all human tissues. A typical example is the fibre reinforcement in a heart valve, Fig. 4.1(a). Another illustration is found in the intervertebral disc as shown in Fig. 4.1(b).

Fibre reinforcement, largely inspired by nature, is frequently used in prosthesis design to optimize mechanical performance. An example is found in the aortic valve prosthesis, see Fig. 4.2.

Fibres are long slender bodies and, essentially, have a tensile load bearing capacity along the fibre direction only. The most simple approximation of the, often complicated, mechanical behaviour of fibres is to assume that they behave elastically. In that case fibres have much in common with springs. The objective of this chapter is to formulate a relation between the force in the fibre and the change in length of a fibre. Such a relation is called a constitutive model.

Elastic fibres in one dimension

Assume, for the time being, that the fibre is represented by a simple spring as sketched in Fig. 4.3. At the left end the spring is attached to the wall while the right end is loaded with a certain force F. If no load is applied to the spring (fibre) the length of the spring equals l0, called the reference or initial length.