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).

THE SCOPE OF MACROMOLECULAR INTERACTIONS

Interactions between proteins constitute the best-characterized and perhaps largest family of interactions between biological macromolecules. This is partly because of the long-recognized diversity of the protein constituents of cells. The significance of diversity in nucleic acids, lipids, and carbohydrates is becoming more apparent, but proteins have held pride of place as the carriers of biological specificity for many decades. It is also partly because the diversity of protein structure and function is so easily observed in contrast to the more subtle variations that determine the specificity of other macromolecules. It is not surprising, therefore, that the bulk of biomolecular interaction studies performed with optical biosensors involves protein–protein binding. The technology can in principle be used to monitor any molecular interactions, and studies involving nucleic acids answer for a significant fraction of the published literature.

Proteins lend themselves well to biomolecular interaction work with optical biosensors. The wealth of functional groups on a “standard” protein molecule offers a selection of approaches for attaching interactants to the sensor surface, ranging from amine- or thiol-based chemical coupling to specific high-affinity capturing methods. Membrane-associated proteins can be embedded in lipid layers on the sensor surface to preserve a hydrophobic environment. Commonly, proteins are relatively large molecules with molecular weights from tens up to hundreds of kilodaltons, so the response obtained in mass-concentration detection methods like surface plasmon resonance (SPR) is usually well above the detection limit of the instrumentation. In addition, most protein–protein interactions can be broken by exposure to moderately mild conditions, offering good opportunities for efficient surface regeneration.

INTRODUCTION

Label-free detection is a key factor contributing to the increasing popularity of biosensors, in particular those based on surface plasmon resonance (SPR). By measuring changes in refractive index close to a sensor surface, SPR biosensors allow the user to study the interaction between immobilized molecules (often referred to as ligands) and analytes in solution, in real time and without analyte labeling. Observed binding rates and levels can be interpreted in different ways to provide information on the specificity, kinetics, and affinity of the interaction or for determination of the concentration of the analyte. The ease by which this information is obtained has changed customer workflows in antibody and small-molecule interaction analysis, and in screening. There is now a clear shift from label-based and affinity/IC50-based workflows to a label-free and kinetic-based workflow.

Over the last 20 years I have obtained my knowledge and experience in the biosensor field in development and use of Biacore systems (GE Healthcare Companies), but the information provided in this chapter is general and therefore relevant to many label-free platforms.

Biacore systems commonly integrate detection, sensor surface, and liquid handling technologies and have multispot capability, that is, they allow independent and simultaneous measurements on several discrete spots on a single surface. Ligands can be attached to the sensor surface through a variety of chemistries, and the liquid handling system makes it possible to inject reagents and samples into different flow cells. Samples can be injected in sequence to build multimolecular complexes, but in most cases formation of the complex between the immobilized ligand and one analyte is investigated.

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.

INTRODUCTION

Cell-based assays are now well established in nearly all stages of drug discovery and development. Indeed, they are commonly applied to all the critical steps of the process from receptor target identification and validation, compound screening and structure–activity relationship (SAR) through to toxicology. This is due in a large part to their functional nature and their ability to introduce biological complexity to the drug discovery process at a much lower cost than in vivo testing. Additionally, these assays have been successfully scaled to meet the throughput needs of screening labs. Cellular assays complement and extend the knowledge of the more mechanistic interactions of receptors and ligands as understood from biochemical assays. Indeed, cell-based assays are commonly used iteratively with biochemical assays to inform the process and direct screening efforts.

Label-free cell-based assays are gaining broader acceptance in screening labs, as they provide novel read-outs of cellular signaling and carry with them many practical advantages to the drug discovery workflow. For example, the lack of labels greatly simplifies assay development, removing many steps, each of which requires optimization. Effort is also saved in the engineering of chimeric or tagged molecules, which are often altered to the point that they no longer maintain natural function, leaving results somewhat in question relative to the native biology of the system being tested.

Electrical impedance assays are novel label-free assays that fit well into the screening workflow. Originally used to study the basic physical properties of materials, impedance technology has evolved over time to yield instrumentation for the biophysical evaluation of living cells.

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.

Introduction

We experience the effects of force in everyday life and have an intuitive notion of force. For example, we exert a force on our body when we lift or push an object while we continuously (fortunately) feel the effect of gravitational forces, for instance while sitting, walking, etc. All parts of the human body in one way or the other are loaded by forces. Our bones provide rigidity to the body and can sustain high loads. The skin is resistant to force, simply pull on the skin to witness this. The cardiovascular system is continuously loaded dynamically due to the pulsating blood pressure. The bladder is loaded and stretched when it fills up. The intervertebral discs serve as flexible force transmitting media that give the spine its flexibility. Beside force we are using levers all the time in our daily life to increase the ‘ force ’ that we want to apply to some object, for example by opening doors with the latch, opening a bottle with a bottle-opener. We feel the effect of a lever arm when holding a weight close to our body instead of using a stretched arm. These experiences are the result of the moment that can be exerted by a force. Understanding the impact of force and moment on the human body requires us to formalize the intuitive notion of force and moment. That is the objective of this chapter.

INTRODUCTION

The ability to study interactions among biomolecules, to observe the activity of cells, and to detect analytes specifically from bodily fluids, manufacturing processes, or environmental samples are cornerstones of life science research, pharmaceutical discovery, medical diagnosis, and food/water safety assurance. These capabilities and many others are enabled by the ability to perform biochemical and cell-based assays that allow scientists to ask basic questions about whether one analyte interacts with another, how strong the binding affinity is between two proteins, whether a chemical compound will affect the proliferation rate of cancer cells, and the concentration of a biomarker for cancer within a patient's blood sample. The development of technology to meet these requirements is challenging because biochemical analytes, which can include drug compounds with molecular weights below 500 Da, DNA oligomers, peptides, enzymes, anti-bodies, and viral particles, are exceedingly small and sometimes present within a test sample at concentrations in the fg/ml to pg/ml concentration range that simultaneously contains thousands of other molecules at concentrations orders of magnitude greater. Larger biochemical analytes – such as bacteria, spores, cells, and cell clusters – are less difficult to observe if they can be stained with a colored or fluorescent dye. However, such treatments generally result in the death of the specimen, thus preventing the ability to study a single population repeatedly over a long time period.

INTRODUCTION

Selection of promising, well-characterized hits and leads is essential for success in the drug discovery process. To this end, new technologies that enable screening for new hits and profiling them with respect to their interaction with the targeted biological system are highly demanded in the pharmaceutical industry. The biological system can be a single biomolecule or a cascade of biomolecules including the target or a whole cellular system. Independent from the targeted biological system, detailed information on the interaction of potential drug candidates with the targeted biomolecule form the basis for understanding more complex schemes. Binding assays provide such information on affinity, kinetics, and thermodynamics.

Until recently, obtaining reliable data was often difficult and time-consuming because most of the methods available used labeled compounds that had to be specially synthesized. Biophysical binding assays generate label-free, high-quality data on the interaction between a target and a potential drug candidate. Label-free screening methods include isothermal titration calorimetry (ITC), analytical ultracentrifugation (AUC), nuclear magnetic resonance spectroscopy, mass spectroscopy, and biosensor.

Biosensors, as described in this book, are the tools most often used. They offer rapid access to these relevant binding data without the need for labeling interacting molecules. These biosensors measure in real time the quantity of the complex formed between a molecule immobilized on the sensor surface and a molecule in solution.