Biomechanics of joints

G.R. Johnson    Newcastle University, UK

1.1 Introduction

The well-documented rise in the numbers of older people is creating an ever-increasing demand for total joint replacement. At the same time, the increasing health and activeness of these people creates demand for long-lasting reliable joints, minimising the need for costly revision surgery.

The design and development of new joint replacements are highly interdisciplinary activities, calling for the combination of sound biomechanical understanding, detailed knowledge of anatomy and surgical experience and insight. The purpose of this chapter is to provide a solid biomechanical background to the material to be presented in the later chapters. It provides initially an overview of basic mechanics – both kinematic and kinetics followed by basic stress analysis. The second part of the chapter applies the basic mechanical principles to some of the major joints with a particular emphasis on the functional kinematics and of the roles of the major muscles and ligaments.

1.2 Introduction to biomechanics

1.2.1 Defining the biomechanical properties of a joint: degrees of freedom and constraints

Almost without exception, human joints have more than one axis of rotation. The joints of the fingers, while they may superficially be viewed as hinge joints, allow small out-of-plane rotations and translations. Therefore, while the anatomical conventions suffice for clinical discussion, there is a need for a more rigorous set of definitions for biomechanical analysis. In general, the movement of a body is composed of two types: rotation, in which a defined point in the body rotates about a defined axis, and translation, in which motion occurs along a line.

Considering first a simple hinge joint, then, only a single quantity is needed to define the position (e.g., the angle of flexion of a finger joint). However, if translation also takes place (perhaps due to ligamentous laxity) then a second quantity is required to define the relative position of the two bones. These quantities are degrees of freedom which may be defined as the number of independent quantities required to define a position. Thus, a single uncoupled rigid body in three-dimensional space, capable of three translations and three rotations, has six degrees of freedom. Any constraint applied to the rigid body – these constraints may take the form of geometric features (e.g., the approximate ball and socket construction of a hip) or external connecting structures such as a ligament – will reduce the number of degrees of freedom from this maximum of six. It should further be pointed out that the coupling between degrees of freedom (e.g., the translations that accompany flexion/extension of the knee) are kinematic constraints and reduce the number of independent movements. Furthermore, in many cases, the simplified view of a human joint may suggest perhaps a single degree of freedom (e.g., knee) but more detailed studies reveal further movements which are rather smaller but, nevertheless, may be clinically important.

1.2.2 Forces and moments

Basic Newtonian mechanics

According to Newton’s First Law of Motion, a body will continue to move at a constant velocity unless acted upon by a force. Thus, a force may be defined as an action which causes acceleration of a mass. Force is a vector quantity – that is to say that it must be specified in terms of both magnitude and (three-dimensional) direction. According to Newton’s Second Law of Motion, the acceleration of a body is proportional to and occurs in the direction of an applied force:

=ma

where force (F) is measured in newtons, mass (m) is in kilograms (kg) and acceleration (a) is in m/s2.

Vectors and equilibrium

Figure 1.1 shows a system of forces acting on a particle (i.e., a rigid body having no physical size). The resultant force corresponding to a combination of forces can be found as the vector sum – shown graphically in Fig. 1.1; this shows that there is a net force acting on the particle, i.e. it is not in equilibrium. For the particle to be in equilibrium the resultant of the forces must be zero and so the result of the graphical summation of the vectors must be a closed figure (Fig. 1.2).

The solution of the majority of biomechanics problems involves the analysis of equilibrium and a clear understanding is necessary to understand a wide range of problems involving external, joint and muscle forces. It should also be noted that this vector approach can be used ‘in reverse’ so that a vector may be broken down in to components (usually mutually perpendicular). This is particularly useful for solving some equilibrium problems.

Dynamics

In situations where the forces are not in equilibrium, then the particle will experience an acceleration, according to Newton’s Second Law. The acceleration will have a magnitude dependent upon its mass and a direction corresponding to that of the resultant force. Using vector notation:

F→=ma→

This analysis of dynamics is key to the understanding of biomechanical motion. For instance, the detailed calculation of the loading of the lower limb during gait requires this approach.

Rotations and moments

If a system of forces acts on a finite rigid body, then it is important to consider both translation and rotation. In particular, it is possible that, while a set of forces has a zero resultant force, the points of application are such that they cause a rotation. Similarly, where there is a rotational degree of freedom, then the net resultant force may not pass through the centre of rotation and will produce a moment. Moments, which may be thought of as the turning effect of a force, are of particular importance to the mechanics of joints since these are the actions of muscles, e.g. quadriceps at the knee. Mechanically, the moment of a force about a point is defined as the magnitude of the force multiplied by the perpendicular distance between the point and the line of action of the force. Moments have units of newton–metre (Nm). The generation of a moment is shown in Fig. 1.3 illustrating a simplified joint acted upon by a single force which does not pass through the centre of rotation. This leads to a moment about the joint centre equal to F (the magnitude) of the force multiplied by h, the perpendicular distance between the centre of rotation and the line of action of the force.

1.2.3 Equilibrium of a joint: role of joint structures, muscles and ligaments

An arthrodial joint consists of joint surfaces of known (but to some degree variable) geometry, and is crossed by both ligaments and muscles/tendons. For a joint to be in equilibrium after the application of external loads, then the appropriate forces and moments must be produced by these crossing structures. Using the representation of Fig. 1.3 it is now possible to look at the procedure for determining the system of forces acting on a body, e.g. a bone. Equilibrium of forces must be achieved across the joint and the external moment must be balanced by an equal and opposite moment produced by muscle(s). To understand this clearly, it is important to separate the two halves of the joint and to consider free body diagrams of the two bones. It is important to distinguish between the joint contact forces and the external loads. A free body diagram of the ball section of the joint is shown in Fig. 1.4. If we assume that there is no friction at the joint (this is usually realistic for human joints where the coefficients of friction are remarkably small), then the reaction force between the ball and socket must pass though the centre of rotation. In addition, for equilibrium, there must be an external moment M on the joint to balance the moment created by the other forces (which are not collinear).

The major role of muscles is to produce joint moments – the ability to do this is measured by the moment arm which may be defined as the moment produced by a force of 1 N in the muscle. For most joints and muscles, the moment arms are relatively small, so that large muscle forces are commonly required to produce the necessary moments.

1.2.4 Applications to joint mechanics

Elbow flexion

Figure 1.5 is a free body...