Basic biomechanics terminology

Biomechanics deals with the study of forces and displacements acting on biological materials and their resulting effects. In this section, basic biomechanical terminology and principles are defined and applied to the disc. A more detailed explanation of these biomechanical concepts can be found in two excellent texts written for nonengineers by Low and Read⁵ and Panjabi and White.⁶

FORCE: Force is an action that moves or deforms a body – a push or a pull. It is a vector quantity often depicted using an arrow which indicates a magnitude, direction, and a point of application. Figure 78.3 demonstrates these actions applied to a spinal motion segment. The unit of force is a Newton. Complex forces are usually broken down into their ‘components.’ For example, the generic applied force in Figure 78.3A is broken down into its components of normal force and shear force as shown in Figure 78.3B.

Fig. 78.3 Force vector acting on the motion segment. (A) Generic direction. (B) Components for force in normal (compression) and shear loading.

BENDING MOMENT OR TORQUE: When a force is applied at some distance from the center of an object it may cause a bending or turning effect, which is known as a moment or a torque (Fig. 78.4). Moment and torque are both defined as the magnitude of the force times the lever arm (the perpendicular distance between the force and the chosen center of rotation). The units for bending and torque are N-mm or N-m, depending on the scale of the objects considered. Bending moment and torque are the same except that generally in bending the force and lever arm are in the plane parallel to the axis of the structure (Fig. 78.4A), while for a torque they are perpendicular to the axis of the structure (Fig. 78.4B). An example of a bending moment is forward flexion, where the weight of the upper body times its distance from the spine is the applied bending moment. An example of torque is the torso twisting along the long axis of the body. Another example is when a screwdriver is turned; in this case, the force applied to the handle times the diameter of the handle is the torque. The clinical application of moments and torques in the spine explains how seemingly small forces may generate significant moments as a result of large lever arms.

Fig. 78.4 Moments and torques acting on the motion segment. (A) Bending. (B) Torsion.

STRESS: Force is generally not applied at a single point, but is distributed over a contact area. This distribution of force, or load intensity, is defined as stress. Stress is the ratio of force divided by the area of application and usually has the units of N/mm² (MPa or Megapascals) or N/m² (Pascals), depending on the size scale. Stress is sometimes defined in Imperial units as pounds per square inch (psi). Stress can be broken down into its components, as is done for force. The force components shown in Figure 78.3B, when divided by the disc surface area, can be used to calculate the applied compressive and shear stress. Externally applied stresses on an object give rise to internal stresses.

PRESSURE: There are two types of pressures: contact pressure and fluid pressure (Fig. 78.5). Contact pressure occurs in solids and is an applied compressive stress. For example, the contact of the two facet joint surfaces imposes a compressive stress that is sometimes called a contact pressure. Importantly, fluid pressure, also known as hydrostatic pressure, occurs in contained fluids and should not be confused with solid stress or contact stress, even though it has the same units. Fluid pressure is the intensity of loading within a contained fluid. The fluid pressure is generated by the interactions of the fluid molecules with the surface of its container. Hydrostatic pressure is isotropic, which means that it is the same in all directions. The nucleus pulposus of the disc is considered to be a fluid (or fluid-like) substance and therefore generates a hydrostatic pressure when the disc is loaded. This pressure is the same in all directions: vertically against the endplates and radially against the containing walls of the anulus fibrosus, as shown in Figure 78.5.

Fig. 78.5 Nucleus pulposus hydrostatic pressure and facet joint contact.

DISPLACEMENT AND STRAIN: Displacement refers to the motion of particles within an object. Similar to force, displacement is a vector quantity and therefore has both a magnitude and a direction. The units of displacement are millimeters (mm) or meters (m), depending on the size of the object, and are sometimes further decreased to micrometers. Like force, displacement is often broken down into its components. The application of force to a body usually causes a displacement of the body. Strain is defined as the ratio of displacement of the object to its original length and, therefore, is unitless. Strain is sometimes multiplied by 100 and given as a percent strain. Externally applied displacements lead to internal displacements and strains.

Soft tissue mechanical behaviors

LOADING CONFIGURATIONS: There are three standard mechanical testing configurations for soft tissues: tension, compression, and shear. For tension and compression experiments the specimen ends are pulled away from each other (tension) or pushed towards each other (compression). In a tension test, the specimen is generally long and thin (Fig. 78.6A) and is held in the testing machine at both ends. The specimen ends are pulled apart leading to an increase in the length and decrease in width. In a compression test, the sample is generally a short cylinder, and the two ends of the specimen are pushed towards each other resulting in a decrease in the height of the specimen (Fig. 78.6B). For shear loading tests, a sample may be either rectangular or cylindrical and the load is applied parallel to the surface (Fig. 78.6C). The sample then changes its shape without undergoing either compression or tension.

Fig. 78.6 Standard loading configurations in tension, compression, and shear. (A) Tension. (B) Compression. (C) Shear.

Compression tests can have three distinct loading configurations because more complex experiments are used to quantify the loading mechanisms that resist the applied load: hydrostatic pressure in the interstitial fluid and the solid matrix stress. These mechanisms are quantified by considering the tissue as a multiphase material consisting of a solid, a fluid, and sometimes an ionic charge phase. Compression tests are performed in either an unconfined manner, where the boundaries of the cylinder expand upon loading, or in a confining chamber. In the confined compression experiment, a cylindrical sample bathed in a fluid is constrained within a chamber of the same diameter, above which a porous platen is placed. While the sample is held in this position, a swelling pressure is generated within the tissue. Next, stepwise increases in compressive displacements are applied to confined samples and the resultant force is recorded. Each displacement is held for several minutes or hours until equilibrium and a steady state is reached. The force, displacement, and time data are analyzed. Confined compression mechanical tests are typically analyzed using a biphasic model and the material properties of solid matrix modulus and permeability are determined from the experiment. The solid matrix modulus is the normalized stiffness of the material and has units of MPa. The permeability is defined as the ease by which the fluid phase flows through the solid phase and has units of m⁴/Ns. Another type of compression test is ‘indentation’ where a probe is compressed against any flat surface and the amount of indentation is recorded. One advantage of this technique is that it can be applied to the tissue sample in situ (without excising and precisely preparing a test sample). A disadvantage, however, is that sophisticated numerical analyses are needed to calculate material properties from the measured data.

STRUCTURAL AND MATERIAL PROPERTIES: A biomechanical study typically involves experimentation and curve-fitting to a mathematical equation to determine the relationship between the force and displacement. However, for specimens of the same material with different dimensions, the force–displacement curves will be different, as shown in Figure 78.7A. This is because both force and displacement are ‘as measured’ quantities and not normalized with respect to the specimen geometry (e.g. cross-sectional area and height). Thus, it is important that the data for force and displacement be normalized to enable comparisons among different-sized samples. These principles have led to the development of normalized quantities such as stress and strain. The stress–strain response of the same material prepared into samples of different sizes is the same (Fig. 78.7B). The slope of this curve is defined as the modulus of the material and has the units of MPa or N/mm². Similarly, the slope of the force versus displacement curve is defined as the stiffness and has the units of N/mm. For tension and compression testing, Poisson’s ratio is defined as the negative of the lateral contraction strain divided by the applied longitudinal strain. In shear testing, the shear modulus is calculated as the slope of the shear stress (applied parallel to the sample surface) versus the angle of displacement.

Fig. 78.7 Representative response for structural and material properties. (A) Force–displacement. (B) Stress–strain response.

NONLINEARITY: A material is nonlinear if the force–displacement or stress–strain response does not have constant slope. While many materials are linear, most biological materials are nonlinear. Consequently, a single stiffness or modulus cannot describe the tissue’s behavior over all loading levels. In motion segment mechanics, this nonlinearity is exemplified by the dramatically different behavior in the low-stiffness neutral zone compared to the high-stiffness loading zones (Fig.78.8A). In soft tissue mechanics, nonlinearity is observed by the low ‘toe region’ modulus and the larger ‘linear region’ modulus (Fig. 78.8B). The large differences in compressive and tensile properties are another form of material nonlinearity.

Fig. 78.8 Nonlinearity (A) Nonlinear motion segment response showing neutral zone. (B) Nonlinear stress–strain response.

Nonlinearity in a spine motion segment is observed as the very large changes in stiffness as the motion segment is moved through the range of motion in any single loading direction. This is an important mechanical behavior because it contributes to the overall range of motion and because an increase in the neutral zone length is consistently observed in degenerated discs.⁷ A typical cyclic load response of an intervertebral disc is shown in Figure 78.8A. The curve is characterized by high stiffness regions at the extreme ends of loading and by a low stiffness region at low loads. The region of the response in the low load (or moment) and low deformation (or rotation) region is called the ‘neutral zone.’ The neutral zone is defined as the region on either side of the neutral position where there is little or no resistance to motion.^7,8 The neutral zone mechanical properties are used to understand the stability of the disc motion segment. Although early studies reported the neutral zone as simply the displacement (rotation) over which loads (moments) are negligible, recent studies have focused on quantifying this important region of disc mechanical function.^9,10

VISCOELASTICITY: All biological tissues, including disc, bone, cartilage, muscle, tendon, and ligaments, exhibit viscoelastic behaviors. A material is viscoelastic if it has time-dependent mechanical behaviors. For example, if a constant compression load (load-control) is applied to a spine segment it will immediately deform (the elastic response) and it will slowly continue to deform over the next several minutes and hours (the viscous response). The most commonly used test to evaluate tissue viscoelasticity in the spine is called a creep test (Fig. 78.9A). The analogous test applied under a constant displacement (displacement-control) is called a stress-relaxation test (Fig. 78.9B). In this case, a constant displacement is applied to the spine segment which immediately experiences a stress (the elastic response). Over the next few minutes or hours, the segment relaxes to a lower stress value (the viscous response). In a viscoelastic experiment, the stress (relaxation) or strain (creep) data are applied to a best-fit curve, the constants from which permit comparisons among different specimens. Another viscoelastic response is hysteresis, which describes a difference of cyclic loading in the load-displacement curve in the loading and unloading phases. The intervening area in this curve is described as the hysteresis area, and represents energy lost as heat due to viscous effects (see Fig. 78.8A).

Fig. 78.9 Viscoelastic response of soft tissues. (A) Creep. (B) Stress–relaxation.

ANISOTROPY: A material is anisotropic if its mechanical behaviors vary under different directions of loading. Most soft tissues, including the anulus fibrosus, are highly anisotropic due to their collagen fiber structure. The stiffness and modulus when the material is stretched along the fiber direction is much larger than when stretched in any other direction. The nucleus pulposus is generally considered isotropic because its mechanical behaviors are the same in all directions. This is likely a result of the random orientation of the collagen fibers and because hydrostatic pressure dominates its mechanical response.

INHOMOGENEITY: A material is inhomogeneous if its mechanical behaviors vary with location within the tissue. The anulus fibrosus and most other soft tissues are inhomogeneous. Variations in the underlying structure and composition of soft tissues generate inhomogeneous mechanical properties. Consequently, when measurements of tissue properties are performed, it is necessary to specify the location from which the samples were obtained.

Nucleus pulposus mechanics

The nucleus pulposus is a highly hydrated gel with important material properties that include swelling pressure, compression modulus, permeability, and shear modulus. The nucleus has a large water and proteoglycan content. Proteoglycans have negatively charged glycosaminoglycans attached, which attract positive ions and generate high osmotic pressures. If the nucleus pulposus is removed and placed in a hydrating solution it swells to at least two times its volume. The nucleus swelling pressure has been measured using a pressure transducer in vivo and in excised cadaveric spines under mechanical loading. The swelling pressure of a healthy nucleus is 0.1–0.2 MPa in a recumbent position and may reach as high as 1–3 MPa when standing or lifting.^11,12 Similarly high pressures have been measured in cadaveric motion segments under externally applied loads.^13,14 The swelling pressure has also been measured using osmometry methods¹⁵ and in confined compression,¹⁶ producing values consistent with in vivo measurements. The nucleus pulposus swelling pressure has been shown to be correlated with the glycosaminoglycans content.^15,16

The large propensity of the nucleus pulposus to swell makes it difficult to prepare test samples and to measure its mechanical properties. As a result, there is a paucity of studies in the literature which have examined these properties. This dearth of mechanical data has made it difficult to properly model the nucleus pulposus in finite element studies of the spine. Lack of mechanical data has also hampered the development of tissue-engineered and artificial disc replacements because clear targets for mechanical function are needed. Fortunately, recent studies have provided measures of the tissue properties and their changes with degeneration. The equilibrium shear modulus of nondegenerate human nucleus pulposus is 0.2 kPa and increases to 0.6–0.8 kPa with degeneration.¹⁷ The high water content of the nucleus pulposus contributes to the large viscoelastic effects observed. Further, the dynamic shear modulus is almost two orders of magnitude larger than the equilibrium modulus, ranging 5–60 kPa¹⁷ and increases with degeneration. When dynamic loading is applied in shear, the nucleus behaves like a viscoelastic solid, suggesting that finite element models that describe the nucleus as a fluid may miss important behaviors.¹⁸ The compression modulus measured at equilibrium in a confined compression test is three orders of magnitude higher than the shear modulus at 0.5 MPa for nondegenerate human tissue.¹⁶ The viscoelastic behavior, as measured by the hydraulic permeability, was calculated to be 3 × 10⁻¹⁶ m⁴/Ns.¹⁶

The nucleus pulposus is the first disc substructure to exhibit degenerative changes: the shear stiffness increases,¹⁸ the pressure decreases,¹⁵ the compression stiffness increases,¹⁶ the permeability decreases,¹⁶ and its compositional changes lead to a shift from a fluid-like behavior to a solid-like behavior.¹⁷ When the nucleus is degenerated, the swelling pressure decreases dramatically to 0.03 MPa or less.^11,12,16,19 While stiffness increases with degeneration and the behaviors become more solid-like, osmometry and mechanical compression tests show that even with degeneration the overall behavior of the nucleus pulposus is still dominated by the swelling component.^16,20 These biochemical changes observed in the degenerating nucleus pulposus and their biomechanical consequences may be key factors in further progression of degeneration.

Anulus fibrosus mechanics

The anulus fibrosus surrounds the nucleus pulposus and has a highly organized fibrocartilagenous structure. Although compression is considered the primary loading configuration of the anulus, it is also loaded under tension, compression, and shear, and its structural organization and biochemical composition reflect these complex loading configurations.

COMPRESSION: The anulus fibrosus undergoes large compression loads during the activities of daily living due to both axial loading and as a result of bending and torsional loading. The primary directions for compression loading in the anulus are in the axial direction, due to body weight, and in the radial direction, due to nucleus pulposus swelling. The swelling pressure represents the pressure generated when the anulus is held at the in situ displacement. Swelling pressure correlates with the biochemical content of glycosaminoglycan due to the osmotic pressures generated by balancing their negative charges. Since the glycosaminoglycan content is high in the anulus fibrosus (and nucleus pulposus), the swelling pressure is an important material parameter. Confined compression experiments and biphasic analyses have revealed the material properties of solid matrix modulus and permeability. The anulus compressive modulus is 0.6 MPa and the permeability is 2 × 10⁻¹⁶ m⁴/Ns.^21,22

The anulus behavior in compression is nonlinear and inhomo-genous. Nonlinearity is observed by the increase in the compressive modulus and the decrease in permeability with increasing amounts of applied compressive strain.²¹ The nonlinear properties of the anulus are more dramatic for the compressive stiffness than for the permeability.²³ Inhomogeneity is demonstrated by regional and radial variations in both material properties and biochemical composition.^21,23 Unlike the tensile material properties, the compression properties are not strongly anisotropic.²¹ Very little compression anisotropy is observed for samples oriented in the axial and radial direction. This suggests that the collagen alignment does not strongly influence the anulus behavior in compression. That is, this loading configuration is primarily supported by the fluid pressurization mechanism. In contrast, the anulus fibrosus permeability is direction dependent, with the highest permeability in the radial direction.²⁴ Anisotropic swelling of the anulus has been observed, where the tissue swells more in the radial direction than in the axial direction.²⁵ Similarly, anisotropy in the swelling behavior under confined compression for radial and axial tissue samples has been observed.^21,26

With disc degeneration, the swelling pressure of the anulus fibrosus decreases dramatically from 0.13 MPa to 0.05 MPa.^21,23 The permeability also changes with degeneration: the radial permeability decreases and the axial and circumferential permeabilities increase.²⁴ The combination of an inability to generate hydrostatic pressure when loaded in compression and altered permeability transfers more of the compressive load onto the anulus solid matrix. When this occurs, the anulus is likely to accelerate degeneration due to both mechanical fatigue and altered cell response to this altered loading. Interestingly, the loss of swelling pressure is partially compensated for by an increase in stiffness of the solid matrix from 0.5 MPa in nondegenerated anulus to 1.1 MPa in degenerated anulus.^21,23 This increased modulus is likely due to a combination of increased matrix density following loss of water, to increased collagen cross-linking, and to cellular remodeling in response to altered load.