Basic concepts in mechanics

The following are fundamental terms and concepts from mechanics that may not be familiar to clinicians ( Box 20.2 ). The interested reader may pursue these ideas in greater depth by referring to appropriate textbooks.

Stress is a measure of the forces transmitted through, or carried by, a material or tissue. Specifically, stress is the force divided by the cross-sectional area over which it acts. For example, pressure is a stress and can be expressed in pounds per square inch (psi).

Strain is a measure of the local deformation induced by an applied stress. It is computed as the change in length of a material divided by its resting length, and is often expressed as a percentage. For example, a wire that was originally 10 mm long that is stretched an additional 1 mm exhibits 10% tensile strain.

In addition to tension and compression, a material can undergo shear. Tension, compression, and shear are often referred to as the three modes of stress and strain. However, these three modes are not independent, as shown in Figure 20.2 .

Schematic illustration of tissue straining in two dimensions. (A) A square tissue region abcd is deformed by forces that act normal to the faces of the square, represented by the black arrows. The tissue experiences tensile stretching in one direction and compression in another. Superimposed on the square tissue region is a circle that deforms with the tissue. (B) The same tissue region is deformed by shearing forces that act tangential to the faces of the square. However, as this region deforms it not only experiences shear, but also extension and compression, as can be verified by noting how the distances between ac and bd change.

(Adapted from Sigal IA, Flanagan JG, Tertinegg I, et al. Predicted extension, compression and shearing of optic nerve head tissues. Exp Eye Res 2007;85:312–322.)

It has been established that the biologic response of tissues and cells depends strongly on the mode of the strain stimulus (tension, compression, or shear), as well as on their magnitudes and temporal profiles. It is therefore of interest to determine which modes of strain and stress the tissues of the ONH are exposed to as IOP is elevated. Note that strains are generally not homogeneous. When the LC deforms, some regions could be highly strained in different modes, whereas others remain largely unaffected. This is important because the biological effects on cells are likely to be strongly dependent on the local levels of strain or stress rather than on global levels.

We would also like to emphasize that mechanical stress, which represents forces, is not synonymous with notions of stress typically used in physiologic or metabolic contexts (e.g., ischemic or oxidative stress). Mechanical stress cannot be measured directly, and we believe it is strain that damages tissues. However, stress is often used to predict the sites of and failure in engineering structures and has been correlated to damage in tissues, so it may be that, while strain is causing the damage, stress is a better predictor of the sites of that damage.

Stress and strain (i.e., forces and deformation) in a material are related to each other through material properties, and this constitutive relationship is intrinsic to each material. For a given load, a material that exhibits large strains is thought of as compliant. Conversely, a material that exhibits small strains for the same load is termed stiff. A stiff tissue such as sclera can have high stress but low strain, while an equal volume of compliant tissue, like retina, might have high strain even at low levels of stress. The simple description above does not account for many of the complexities in material properties that occur in soft biologic tissues, such as anisotropy, nonlinearity, and viscoelasticity ( Box 20.3 ). These complexities are likely to be fundamental to understanding ocular mechanics and will be discussed in the context of scleral biomechanics below.

Scleral biomechanics

From a mechanical perspective, the eye is a pressure vessel, on which IOP produces deformation, strain, and stress. Through computational modeling, the sclera has been shown to have a strong influence on how the LC deforms when IOP changes ( Box 20.4 ). Therefore, understanding the mechanical behavior of the sclera is essential to understanding IOP-induced LC deformations. One of the mechanisms by which scleral biomechanics can influence the response of the LC to IOP is illustrated in Figure 20.3 .

Influence of scleral mechanics on optic nerve head mechanics. Intraocular pressure (IOP) induces large deformations on a compliant sclera (top); these deformations are transmitted to the scleral canal, resulting in a large scleral canal expansion that pulls the lamina cribrosa (LC) taut despite the direct posterior force of IOP on the LC. Conversely, a stiff sclera deforms little under IOP (bottom), with small scleral canal expansions and little lateral stretching of the LC, thus allowing the LC to be displaced posteriorly by the direct action of IOP on its anterior surface.

Alas, describing the mechanical behavior of soft tissue such as the sclera is a formidable undertaking that demands extensive experimental and mathematical efforts. The first step in characterizing scleral material properties is the development of an experimental mechanical test to measure the deformation of the tissue subjected to loads (e.g., uniaxial, biaxial, or pressurization tests). The second step is the development of a constitutive model (i.e., relationship between stresses and strains) which describes the tissue mechanical behavior as observed in the experiment and provides a mathematical representation of the tissue’s material properties.

Historically, the sclera has been described as a thin-walled, spherical pressure vessel obeying the analytical equation known as Laplace’s law. Laplace’s law is useful to estimate the state of stress in nonbiological pressure vessels, but it is inadequate for describing many aspects of the eye’s mechanical response to variations in IOP. The sclera has been shown to exhibit several properties that violate the assumptions of Laplace’s law. First, the eye is a pressure vessel of nonuniform thickness. Second, in terms of its material properties, the sclera is nonlinear, anisotropic, and viscoelastic. These concepts are fundamental to understanding scleral and ONH biomechanics, and below we present them in greater detail.

Nonlinearity is a property exhibited by most soft tissues, often as a consequence of collagen fibers within the tissue ( Figure 20.4 ). In a pressure vessel with linear material properties, the stiffness would remain constant during pressure-induced deformation, whereas a nonlinear pressure vessel would experience either softening or stiffening as it deforms. Recent experiments have shown that the sclera exhibits a considerable increase in stiffness, at least fivefold, when exposed to an acute elevation of IOP from 5 to 45 mmHg. This dramatic change shows the substantial impact that collagen fibers can have on scleral stiffness and deformation.

(A) Nonlinearity is a property that sclera exhibits, in which the relationship between loading and deformation is not linear. At low intraocular pressure (IOP), collagen fibers are initially crimped, which makes the sclera more compliant. As IOP increases, the scleral collagen fibers uncrimp and eventually become straight, resulting in a dramatic increase in scleral stiffness (an increase in the amount of IOP elevation necessary to produce the same deformation). (B) Schematic illustration of the various degrees of planar anisotropy present in thin soft tissues. Skin has a highly disorganized arrangement of collagen fibers and therefore resists loads similarly in many directions, a property known as isotropy. In contrast, tendons have well-organized collagen fibers running principally in the longitudinal direction, and therefore these tissues sustain loads differently along and across their length, a property known as anisotropy. The sclera is thought to have a collagen fiber alignment that is between those of skin and ligaments.

Anisotropy, as opposed to isotropy, is the property by which materials exhibit different stiffness in different directions. For thin biological tissues, anisotropy is primarily dictated by the organization of their fibrous structure, which is confined within the plane of the tissue, as illustrated in Figure 20.4 . Unlike the cornea, scleral collagen fiber orientation has not been fully characterized. However, it has been shown through computational modeling that collagen fiber orientation and distribution are major determinants of scleral deformation. Further experimental work is needed to characterize better scleral anisotropy and its effects on LC biomechanics.

Viscoelastic materials, such as the sclera, exhibit higher resistance to deformation when loaded quickly rather than slowly. Downs and coworkers characterized the viscoelastic material properties of normal rabbit and monkey peripapillary sclera and found that the material properties of peripapillary sclera are highly time-dependent (viscoelastic). This behavior protects ocular tissues from large deformations during short-term spikes in IOP, which occur during blinks, eye rubbing, or high-speed impacts.

These three aspects of the sclera’s material properties – nonlinearity, anisotropy, and viscoelasticity – affect IOP-induced scleral deformations, but they are not the only important determinants of the eye’s mechanical response. Material properties may be combined with thickness and shape to define another useful concept, that of structural, or effective, stiffness.

Studies of the scleral thickness of human and monkey eyes show that, on average, the human sclera is about twice as thick as the monkey sclera. The sclera is thinnest near the equator (as thin as 100 µm in both species) and thickest in the peripapillary region (average of 1000 µm in the human and 450 µm in the monkey). Large variations in peripapillary scleral thickness occur naturally and in pathologic conditions (e.g., myopia ), and have been hypothesized to be an important determinant of individual susceptibility to IOP. Figure 20.5 illustrates how IOP-related stress is distributed in the peripapillary sclera in two situations: homogeneous thickness with a circular scleral canal, and inhomogeneous thickness with an elliptical scleral canal.

The thickness of the peripapillary sclera and the size and shape of the scleral canal influence the magnitude and distribution of intraocular pressure-related stress within the peripapillary sclera, and within the optic nerve head (ONH). Plots of stress within three-dimensional biomechanical models of the posterior sclera and ONH illustrate how stress concentrates around the scleral canal. (A) An idealized model with a circular canal and a perfectly spherical scleral shell wall of uniform thickness. (B) An elliptical scleral canal with anatomic variations in scleral shell thickness. In both cases, the stresses concentrate around the canal, but the more realistic model shows stress that varies substantially around the canal and can extend further out into the sclera. For clarity, only the scleral tissues are shown.

Optic nerve head and lamina cribosa biomechanics

Models of the optic nerve head

Initial experimental studies of ONH biomechanics were often designed to examine and quantify a posterior deformation of the LC in response to an acute increase in IOP. Unfortunately, it is difficult to take measurements of the LC directly because it is fragile and relatively inaccessible, and therefore experimental efforts used one of two approaches: histological examination of ONH tissues fixed at different IOPs or measurement of acute deformations of the ONH surface through imaging, and using these deformations as a surrogate for the deformations of the underlying LC.

Both approaches produced interesting results. For example, histology-based studies found that acutely elevating IOP produced a posterior deformation of the LC (12–79 µm in humans, and 10–23 µm in monkeys ). However, these studies also highlighted the large variability in ONH geometry between individuals and the difficulties inherent in histomorphometry, which complicates distinguishing the effects of IOP from the natural differences between eyes. Imaging-based studies were subject to the assumption that IOP-induced deformations of the ONH surface are a good surrogate for deformations of the LC. Some experiments, as well as the modeling studies we describe below, have suggested that IOP-induced deformations of the ONH surface are not good predictors of LC deformations ( Box 20.5 ).

Box 20.5

The role of imaging

• •
  

  Models have suggested that the intraocular pressure-induced displacements of the optic nerve head surface might not be a good surrogate for those of the underlying lamina

  
  
• •
  

  Advances in imaging such as deep-scanning ocular coherence tomography show promise for imaging the acute deformations of the lamina

Recently, there have been advances in direct imaging of the acute deformations of the LC itself using second harmonic imaging or deep-scanning ocular coherence tomography ( Figure 20.6 ). Although promising, these technologies are still in development and have been unable to characterize the response of the ONH tissues to IOP.

High-resolution Spectralis (Heidelberg Engineering, Heidelberg, Germany) ocular coherence tomographic (OCT) B-scans of a normal monkey eye obtained at 10 mmHg (top) and 45 mmHg (bottom) intraocular pressure (IOP). Bruch’s membrane and the anterior lamina cribrosa (LC) surface have been delineated (red and green dots, respectively). The area enclosed by the anterior surface of the LC and a plane defined at Bruch’s membrane (BM) opening (area shaded green) is larger in the scan at 45 mmHg. There was no detectable lateral expansion of the scleral canal (vertical green lines), suggesting that the increase in IOP produced posterior laminar deformation. Using a second reference plane parallel to BM opening (dashed red lines), it is also visible that the BM is outwardly bowed at high IOP, which suggests that there is IOP-induced posterior deformation of the peripapillary sclera. While choroidal compression may contribute to this finding, we believe that the behavior of BM is principally related to the behavior of the sclera. The parameters computed from the images are volumetric, but are shown here in cross-section for clarity.

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