# Introduction to biomechanics

The study of biomechanics ranges from the inner workings of a cell to the movement and development of limbs, the vasculature, and bones. As we develop a greater understanding of the physiological behavior of living tissues, researchers are able to advance the field of tissue engineering, as well as develop improved treatments for a wide array of pathologies.

Biomechanics as a sports science, kinesiology, applies the laws of mechanics and physics to human performance in order to gain a greater understanding of performance in athletic events through modeling, simulation, and measurement.

## Continuum mechanics

It is often appropriate to model living tissues as continuous media. For example, at the tissue level, the arterial wall can be modeled as a continuum. This assumption breaks down when the length scales of interest approach the order of the microstructural details of the material. The basic postulates of continuum mechanics are conservation of linear and angular momentum, conservation of mass, conservation of energy, and the entropy inequality. Solids are usually modeled using "reference" or "Lagrangian" coordinates, whereas fluids are often modeled using "spatial" or "Eulerian" coordinates. Using these postulates and some assumptions regarding the particular problem at hand, a set of equilibrium equations can be established. The kinematics and constitutive relations are also needed to model a continuum.

Second and fourth order tensors are crucial in representing many quantities in biomechanics. In practice, however, the full tensor form of a fourth-order constitutive matrix is rarely used. Instead, simplifications such as isotropy, transverse isotropy, and incompressibility reduce the number of independent components. Commonly-used second-order tensors include the Cauchy stress tensor, the second Piola-Kirchhoff stress tensor, the deformation gradient tensor, and the Green strain tensor. A reader of the biomechanics literature would be well-advised to note precisely the definitions of the various tensors which are being used in a particular work.

## Biomechanics of circulation

Under most circumstances, blood flow can be modeled by the Navier-Stokes equations. Whole blood can often be assumed to be an incompressible Newtonian fluid. However, this assumption fails when considering flows within arterioles. At this scale, the effects of individual red blood cells becomes significant, and whole blood can no longer be modeled as a continuum.

## Biomechanics of the bones

Bones are anisotropic but are approximately transversely isotropic. In other words, bones are stronger along one axis than across that axis, and are approximately the same strength no matter how they are rotated around that axis.

The stress-strain relations of bones can be modeled using Hooke's Law, in which they are related by linear constants known as the Young's modulus or the elastic modulus, and the shear modulus and Poisson's ratio, collectively known as the Lamé constants. The constitutive matrix, a fourth order tensor, depends on the isotropy of the bone.

$\sigma_{ij}=C_{ijkl}\epsilon_{kl}$

## Biomechanics of the Muscle

• Skeletal muscle (striated): Unlike cardiac muscle, skeletal muscle can develop a sustained condition known as tetany through high frequency stimulation, resulting in overlapping twitches and a phenomenon known as wave summation. At a sufficiently high frequency, tetany occurs, and the contracticle force appears constant through time. This allows skeletal muscle to develop a wide variety of forces. This muscle type can be voluntarily controlled. Hill's Model is the most popular model used to study muscle.
• Cardiac muscle (striated): Cardiomyocytes are a highly specialized cell type. These involuntarily contracted cells are located in the heart wall and operate in concert to develop synchronized beats. This is attributable to a refractory period between twitches.
• Smooth muscle (smooth - lacking striations): The stomach, vasculature, and most of the digestive tract are largely composed of smooth muscle. This muscle type is involuntary and is controlled by the enteric nervous system.

## Soft tissue biomechanics

Soft tissues such as tendon, ligament and cartilage are combinations of matrix proteins and fluid. In each of these tissues the main strength bearing element is collagen, although the amount and type of collagen varies according to the function each tissue must perform. Elastin is also a major load-bearing constituent within skin, the vasculature, and connective tissues. The function of tendons is to connect muscle with bone and is subjected to tensile loads. Tendons must be strong to facilitate movement of the body while at the same time remaining compliant to prevent damage to the muscle tissues. Ligaments connect bone to bone and therefore are stiffer than tendons but are relatively close in their tensile strength. Cartilage, on the other hand, is primarily loaded in compression and acts as a cushion in the joints to distribute loads between bones. The compressive strength of collagen is derived mainly from collagen as in tendons and ligaments, however because collagen is comparable to a "wet noodle" it must be supported by cross-links of glycosaminoglycans that also attract water and create a nearly incompressible tissue capable of supporting compressive loads.

Recently, research is growing on the biomechanics of other types of soft tissues such as skin and internal organs. This interest is spurred by the need for realism in the development of medical simulation.