**Purpose**:
Laplace's Law, with its compactness and simplicity, has long been employed in ophthalmology for describing the mechanics of the corneoscleral shell. We questioned the appropriateness of Laplace's Law for computing wall stress in the eye considering the advances in knowledge of ocular biomechanics.

**Methods**:
In this manuscript we recapitulate the formulation of Laplace's Law, as well as common interpretations and uses in ophthalmology. Using numerical modeling, we study how Laplace's Law cannot account for important characteristics of the eye, such as variations in globe shape and size or tissue thickness, anisotropy, viscoelasticity, or that the eye is a living, dynamic organ.

**Results**:
We show that accounting for various geometrical and material factors, excluded from Laplace's Law, can alter estimates of corneoscleral wall stress as much as 456% and, therefore, that Laplace's Law is unreliable.

**Conclusions**:
We conclude by illustrating how computational techniques, such as finite element modeling, can account for the factors mentioned above, and are thus more suitable tools to provide quantitative characterization of corneoscleral biomechanics.

^{1}It can also resist physiological alterations due to changes in intraocular pressure (IOP) and ocular pulsations.

^{1}Despite all these mechanical challenges, the corneoscleral shape and the relative positions of the cornea, the lens and the retina are preserved so that the visual ability is unaffected.

^{2–5}

^{6}or low cerebrospinal fluid pressure.

^{7}Keratoconus is characterized by corneal ectasia, which leads to vision distortion, and is associated with altered corneal mechanical properties.

^{8}Axial myopia arises from an elongated corneoscleral shell structure such that the refracted image is not focused on the retina surface.

^{9,10}Overall, quantification of ocular biomechanics is paramount in advancing our understanding of ocular physiology and pathophysiology.

^{11}Searching the online literature repository Scopus

^{12}with the search terms ‘eye and biomechanics' yields more than 10,000 hits from 1964 to 2013 (Fig. 1). The trend is rapidly increasing with approximately half (49.2%) of these documents being published between 2009 and 2013.

**Figure 1**

**Figure 1**

^{13}21 contained one or more manuscripts with the phrase ‘Laplace's Law' or ‘Law of Laplace.' Laplace's Law is an elegant mathematical approach to estimate, for thin vessels, wall stress as a function of vessel pressure, radius of container, and wall thickness. It has been applied in several areas of physiology, often used in explaining the biophysics of hollow organs, such as the alveoli

^{14}and the esophagus.

^{15}It has also been adapted to study the mechanics of the cardiac chambers.

^{16,17}Whereas proponents of Laplace's Law contend that it provides valuable insights to biophysical phenomena,

^{15,18}several authors have also written about misuse or misinterpretation in cardiovascular and respiratory biomechanics.

^{14,19}

**Table 1**

^{20}(1749–1827) was a French scientist and politician (Fig. 2). He made significant contributions in mechanics, calculus, and theory of probability. In Treaty of Celestial Mechanics (Original title in French:

*Traité de Mécanique Céleste*),

^{21}Laplace described a mathematical relationship, Laplace's Law, originally intended to quantify the liquid surface tension in a capillary.

**Figure 2**

**Figure 2**

*σ*), Laplace's Law can be applied to a uniform isotropic hollow sphere with thickness

*t*and radius

*r*, under internal pressure

*p*(Fig. 3).

^{22}It is important to note that the thin shell assumption is applied in Laplace's Law, which postulates that there is no circumferential stress (stress tangential to the surface of corneoscleral shell) variation through the wall thickness. A common way to derive such a formula is to perform a balance of forces between the internal pressure (

*p*) and the resulting wall stress. By examining the cross-section of the sphere, the total force due to the internal pressure is the product of

*p*and the cross-sectional area (

*πr*

^{2}) while the aggregate circumferential tension is the product of

*σ*,

*t*and the cross-sectional circumference (2

*πr*). Equating the two quantities and rearranging them leads to When applied to the corneoscleral shell, Laplace's law is intuitive in many ways: Bigger eyes (larger

*r*) and/or thinner corneoscleral tissue layers (smaller

*t*) will exhibit larger wall stress. In addition, stress increases proportionally to IOP (larger

*p*). Laplace's Law can be derived in other ways, and presented in various formulations such as using the radius of curvature instead of the spherical radius.

^{22}In this work we will discuss Equation 1, as it is the most commonly used formulation in ophthalmology.

**Figure 3**

**Figure 3**

^{23}In humans, both the shape and the size of the corneoscleral shell vary widely among individuals and pathophysiological conditions.

^{23,24}Furthermore, in healthy eyes, the anterior-posterior (AP) and nasal-temporal (NT) diameters of the corneoscleral shell are approximately a millimeter longer than the superior-inferior (SI) diameter,

^{24}indicating that the eye is not a sphere.

^{25–29}a uniform tissue thickness of 0.8 mm,

^{29–31}and exposed to an IOP of 2 kPa (15 mm Hg). Laplace's Law (Equation 1) predicts a circumferential stress of 15.4 kPa that is uniform over the entire shell (Fig. 4a). The situation is quite different once we consider more accurate ocular dimensions. As a case in point, consider an ellipsoidal shell with uniform thickness of 0.8 mm, with AP and NT diameters of 24.6 mm, and an SI diameter of 23.4 mm (as observed physiologically

^{32}). Although different formulations are available for calculating circumferential stress in nonspherical chamber,

^{33}we use an analytical solution developed by Regen

^{34}that enables us to calculate the circumferential wall stress of an ellipsoidal chamber with different diameters. The circumferential stress on the ellipsoidal shell is observed to be heterogeneous across the surface (Fig. 4b). In contrast to the result obtained from Laplace's Law, the circumferential stress peaks at the superior and inferior poles of the ellipsoidal chamber and gradually decreases toward the central transverse (AP-NT) plane. The maximum circumferential stress is 5.1% higher than the value given by Equation 1 while the minimum stress is 5.3% lower. The results indicate that Laplace's Law can both overestimate and underestimate shell stresses at different locations. Note that real corneoscleral shell shapes are more complex than the spheroid considered here, which can cause further heterogeneity in stress fields.

^{35,36}

**Figure 4**

**Figure 4**

^{10,37}Our analysis suggests that if the axial length extends from 24.6 to 26.4 mm (equivalent to a mild myopia case of −5.07 diopters [D]

^{38}), the maximum stress is 12.2% higher than that computed by Laplace's law, while the minimum stress would be 11.8% lower (Fig. 4c). As the various axes of the spheroids are different, the range of differences in stresses will increase. In the case of severe axial myopia, the axial diameter can exceed 33.5 mm (approximately −25.01 D),

^{39}the errors can escalate to 31.3% and 30.7%. The stress distribution also reveals regions of higher stress concentration compared with the healthy eye, particularly at the superior and inferior poles. These localized areas of high stress could potentially be susceptible to material rupture.

^{36}Furthermore, distinct curvatures, thicknesses, porosities, and microstructures between the peripapillary sclera and the neighboring lamina cribrosa (LC) grant the postulation of Laplace inadmissible.

^{40,41}A study has shown that the abrupt change in geometrical and mechanical properties at the scleral canal boundary could lead to an acute stress concentration in the ONH region of up to 25 times the value of IOP.

^{42}The quantity is 300% higher than that estimated through Laplace's Law, yet the latter is still commonly used to predict ONH stresses.

^{29,35,43}The corneal thickness is approximately 520 μm near the center and increases toward the peripheral region (∼660 μm).

^{44}In the sclera, starting from the limbus and traversing posteriorly, the average thickness decreases monotonically until it reaches a minimum near the equator, where the tissue begins to increase in thickness toward the posterior pole. At the ONH, the scleral tissue tapers toward the scleral canal.

^{5,43}The sclera does not have a constant thickness along both NT and SI directions. Peripapillary scleral thickness varies, being thinnest at the inferior and nasal quadrants (∼880 μm), and thickest at the superior and temporal quadrants (∼1050 μm).

^{43}The heterogeneity of corneoscleral tissue thickness can adversely affect the approximation made from Laplace's Law. The thinner parts of the shell can exhibit higher IOP-induced stress than the thicker parts due to a smaller cross-sectional area. Variations in corneoscleral shell thickness, ranging from around 100 μm at the thinnest, to over 1000 μm at the thickest, would imply also an order of magnitude in variations in local stress.

^{45}This stress heterogeneity cannot be predicted by Laplace's Law.

^{44,46}The increased thickness-to-diameter proportion can yield larger stress difference between the inner and outer surfaces; the thin shell assumption from Laplace is no longer valid. In fact, the circumferential stress (across the corneoscleral tissue) is better represented from the thick shell theory.

^{22}The thick shell theory accounts for the pressure difference across the wall and recognizes the variation of circumferential wall stress across the wall thickness. We assume that the IOP is applied on the inner surface with zero external pressure. In this case, the maximum circumferential stress occurs at the inner surface while the minimum at the outer surface. Considering a healthy eye globe with an axial diameter of 24.6 mm and a wall thickness of 0.8 mm (thickness-to-radius ratio of 0.03), the circumferential wall stress obtained from Laplace's Law is 3.3% lower than the maximum stress value and 3.3% higher than the minimum stress value evaluated from the thick shell equation.

^{22}In the case of hyperopia with axial diameter of 17.2 mm

^{44}and wall thickness of 0.8 mm, (thickness-to-radius ratio of 0.047), the differences from the inner wall circumferential stress and the outer wall stress rise to 4.6% and 4.7%, respectively.

^{35}

^{47}Collagen fibrils in the corneal stroma are thin and highly organized.

^{48}The collagen fibers form rings surrounding the limbus that reinforce the region near the cornea.

^{49}The collagenous microstructure of the corneoscleral shell contributes to its unique mechanical characteristics including anisotropy, nonlinearity, viscoelasticity, and inhomogeneity.

^{50}modify Laplace's Law to accommodate varying curvatures,

^{33}or build finite element models incorporating complex geometrical and material features as those described in Table 2.

**Table 2**

^{51}the corneoscleral layer is stiffer along the tangential direction, and is thus well adapted to resist IOP-induced stress. The anterior sclera immediately adjacent to the limbus and the peripapillary sclera that surrounds the optic disc, both exhibit high anisotropy with circumferential organizations.

^{49}Such microstructural organizations are thought to provide reinforcements in order to limit corneal and ONH deformations.

^{52}The varying fiber arrangements are related to change in circumferential stress that cannot be explained by Laplace's Law.

^{45}

^{35}with that from a spherical vessel of similar size computed with Laplace's Law. We focus our comparisons to the peripapillary sclera, a relatively high anisotropic region. Note that the stress estimates from the FE model were obtained based on an anatomically-accurate scleral shell geometry, three-dimensional (3D) experimental deformation measurements, and a realistic constitutive model that took into account collagen fiber anisotropy. Such stresses can be considered more reliable for the monkey eye than those from Laplace's Law.

**Figure 5**

**Figure 5**

^{53}and scleral

^{54}tissues stiffness increases with tissue loading speed. Corneoscleral tissue viscoelasticity is also characterized by the energy dissipated during loading-unloading cycles, defined as hysteresis.

^{53,54}The phenomena of rate-dependent stiffness and hysteresis are postulated to help ocular tissues dissipate the deformation energy and protect them from sudden loading and excessive deformation, such as in blast trauma, in order to prevent injury.

^{36}developed a viscoelastic FE model of the corneoscleral shell to simulate the effect of in vitro PBS injection in expanding ocular volume and elevating IOP. They demonstrated greater and more rapid IOP increases at faster injection rates, as a result of tissue viscoelasticity inducing larger instantaneous tissue stiffness. Volumetric injections of 15 μL at slow (0.1 μLs

^{−1}) and intermediate (1.5 μLs

^{−1}) rates could lead to IOP changes (in their model) of 8 and 11 mm Hg, respectively. In the case of a fast injection (15 μLs

^{−1}), an IOP change of 14 mm Hg was reported, while von Mises stress ranged from 3.74 kPa (5th percentile) to 18.4 kPa (95th percentile). In fact, the distribution of von Mises stress (calculated from the principal stresses) computed from the viscoelastic FE model contrasts with the stress value produced by Laplace's Law, which is evaluated to be 20.8 kPa at IOP of 14 mm Hg (456% and 13.0% higher than the 5th and 95th percentile stress values predicted from viscoelastic FE models, respectively). In addition, Laplace's Law cannot be used to calculate the change of circumferential wall stress due to a change in IOP, as the rate of change of IOP can affect the instantaneous stiffness of the corneoscleral tissue and its deformation. In sum, Laplace's Law cannot be used to make predictions that consider the effect of viscoelasticity.

^{49}and at different tissue depth,

^{51,55}the local mechanical properties are also location-dependent (heterogeneity). Laplace's law is unable to take such complexities into account, and caution should be taken when estimating wall stresses (e.g., in different regions of the scleral shell. In the latter, strong variations in biomechanical properties exist among the anterior, equatorial, and posterior regions).

^{54}

^{45,54,56}exhibiting a nonlinear relationship between circumferential stress and IOP. This is because, as stretch increases with IOP, more and more collagen fibers uncrimp, or straighten, and become recruited to resist the increasing load.

^{35,57–59}Laplace's law does not intrinsically presuppose a linear relationship between stretch and stress. Hence, as long as other assumptions are satisfied, (i.e., a perfect thin spherical shell that is homogeneous and isotropic) Laplace's Law would apply equally well when the material is linear or nonlinear. However, any deviation from these conditions would cause Laplace's Law assumptions to be violated and lead to invalid predictions. Consider, for example, a spherical shell made of a material that is homogeneous, isotropic and nonlinear, with small variations in shell thickness. As IOP increases the stresses on the wall deform the material. In regions where the shell is thinner, the stresses will be slightly higher, which leads to a slightly stiffer material locally.

^{45}In this way, the inhomogeneity in shell thicknesses has been translated into material inhomogeneity.

^{35}with that from a spherical vessel of similar size computed with Laplace's Law. We focus our comparisons to the peripheral sclera (surrounding the scleral canal and further than 1.7 mm from the canal), because it is less anisotropic than other regions of the eye.

^{49,60}The stress data produced from the model deviate from the calculated stress using Laplace's Law and the difference increases with the pressure. The 95th percentile of first principal stress within the peripheral monkey sclera at 10 mm Hg is predicted to be 101% higher than the Laplace's Law wall stress while the 5th percentile of the stress is 71.5% lower. The errors escalate to 250% and −66.5% for 30 mm Hg, and 277% and −63.8% for 45 mm Hg, respectively (Fig. 5b). The comparison shows that the usage of Laplace's Law to estimate the circumferential wall stress can lead to large error as a result of not accounting for material nonlinearity.

^{61,62}In addition, the observed presence of contractile cells (myofibroblasts) within the scleral shell could provide the eye with an additional mechanism to alter its elasticity (and thus its stress levels). The adaptive and active behaviors of the scleral shell (unaccounted for in Laplace's law) are briefly described below.

^{63}The changed scleral material composition and increased laminar architecture

^{64}may lead to a higher probability of scleral interlayer sliding, thus increasing creep

^{65}(faster and larger scleral stretch under load). In glaucoma or in response to chronic IOP elevations, studies have reported scleral thinning,

^{66}structural scleral stiffening,

^{45}and changes in scleral collagen fiber orientation.

^{67}While Laplace's law may be able to consider simplistic remodeling conditions (through changes in shell thickness and/or eye radius), it is unable to account for complex remodeling conditions observed in both myopia and glaucoma.

^{64,68}Such myofibroblast-like cells may be able to alter the biomechanical properties of the sclera through contractile activity.

^{69}The presence of myofibroblasts in tree shrew sclera helped to curb the eye expansion rate after increasing IOP from 15 to 100 mm Hg, and to returned to its pre-expansion shape within an hour. Results from other in vitro studies

^{69,70}suggested that scleral fibroblasts could be differentiated into myofibroblasts through cytokine TGF-β.

^{68,70}Laplace's law, as formulated, can only predict passive stress, and would be unable to take into account active stress as generated by scleral myofibroblasts.

^{23}nonlinear mechanical properties,

^{35,71}collagen microstructural descriptions,

^{35,71,72}and growth and remodeling

^{57}have been considered in these models. However, FE models have their own limitations as well, and validations of these models are still needed. Recent usage of x-ray scattering,

^{60,73}small angle light scattering,

^{49,74–76}and polarized light microscopy

^{59}to decipher the collagen arrangement will enhance the accuracy of the models.

^{49}Implementations of linear,

^{36}and probably nonlinear

^{77,78}viscoelastic models in simulation can also advance sophistication and quality of simulation over time. The use of inverse FE methods

^{35,53,60}or prefitting techniques

^{79}may help bridge the gap between experimental data and simulation results and provide advantages for clinical translations.

^{80}has developed an applet for simple FE modeling of the ONH. The work provides an alternative avenue to understanding the biomechanics of the eye without being limited by technical knowledge.

**C.W. Chung**, None;

**M.J.A. Girard**, None;

**N.-J. Jan**, None;

**I.A. Sigal**, None

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