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02S-58 Stapp Car Crash Journal, Vol. 46 (November 2002), pp. Copyright © 2002 The Stapp Association

A Nonlinear Finite Element Model of the Eye with Experimental Validation for the Prediction of Globe Rupture Joel D. Stitzel, Stefan M. Duma, Joseph M. Cormier, Ian P. Herring Virginia Tech, Impact Biomechanics Laboratory

__________________________________ ABSTRACT – Over 2.4 million eye injuries occur each year in the US, with over 30,000 patients left blind as a result of the trauma. The majority of these injuries occur in automobile crashes, military operations and sporting activities. This paper presents a nonlinear finite element model of the eye and the results of 22 experiments using human eyes to validate for globe rupture injury prediction. The model of the human eye consists of the cornea, sclera, lens, ciliary body, zonules, aqueous humor and vitreous body. Lagrangian membrane elements are used for the cornea and sclera, Lagrangian bricks for the lens, ciliary, and zonules, and Eulerian brick elements comprise the aqueous and vitreous. Nonlinear, isotropic material properties of the sclera and cornea were gathered from uniaxial tensile strip tests performed up to rupture. Dynamic modeling was performed using LS-Dyna. Experimental validation tests consisted of 22 tests using three scenarios: impacts from foam particles, BB's, and baseballs onto fresh eyes used within 24 hours postmortem. The energies of the projectiles were chosen so as to provide both globe rupture and no rupture tests. Displacements of the eye were recorded using high speed color video at 7100 frames per second. The matched simulations predicted rupture of the eye when rupture was seen in the BB and baseball tests, and closely predicted displacements of the eye for the foam tests. Globe rupture has previously been shown to occur at peak stresses of 9.4 MPa using the material properties included in the model. Because of dynamic effects and improvements in boundary conditions resulting from a more realistic modeling of the fluid in the anterior and posterior chambers, the stresses can be much higher than those previously predicted, with the globe remaining intact. The model is empirically verified to predict globe rupture for stresses in the corneoscleral shell exceeding 23 MPa, and local dynamic pressures exceeding 2.1 MPa. The model can be used as a predictive aid to reduce the burden of eye injury, and can serve as a validated model to predict globe rupture. KEYWORDS – Eye, Modeling, Rupture, Injury, Cornea, Sclera, Limbus __________________________________ INTRODUCTION Eye injuries are expensive to treat, affect a large proportion of the population, and often result in long term disability (Schein, 1988). Over 2.4 million eye injuries occur yearly in the United States alone, with over 30,000 of these patients left blind in at least one eye as a result of the trauma (Lueder, 2000; Parver, 1986). The resulting potential for impaired sight or even blindness carries an extreme societal cost. Severe eye injuries, such as ruptured globes and retinal detachments, are typically the result of high speed blunt trauma such as impact with sports equipment, projectile and goggle loading in military operations, and contact with an airbag or steering wheel in automobile crashes. The most frequent

sporting objects involved in blunt ocular trauma include baseballs, squash balls, the ends of hockey sticks, and elbows (Chisholm, 1969; Vinger, 1999). Military databases report that the percentage of eye injuries sustained by soldiers in war and peacetime has increased significantly in this century relative to the total number of injuries (Biehl, 1999;DoswaldBeck, 1993). In Operation Desert Storm, the incidence of ocular injury was 13% of the total combat injuries (Heier, 1993). Furthermore, the increasing use of night vision goggles has the potential to increase this percentage (Power, 2002b). In automobile crashes, windshield glass has been identified as a serious eye injury mechanism with 40% of these cases resulting in blindness in at least

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one eye (Muller-Jensen, 1970). With the recent widespread implementation of airbag, fatal and severe injuries have been reduced, while the risk of less severe injuries such as eye injuries has increased (Jernigan, 2001). Moreover, the medical literature is replete with case studies on airbag induced eye injuries (Duma, 1996; Ghafouri, 1997; Vichnin, 1995: Stein, 1999). These case studies include a wide range of ocular injuries from corneal abrasions to ruptured globes. Eye Anatomy The exterior of the eye, referred to as the corneoscleral shell, consists of the transparent cornea on the anterior surface and the white sclera that comprises the rest of the globe surface (Figure 1) (Westmorland, 1997). The anterior surface of the cornea is covered with epithelial cells while the posterior, or interior, surface of the cornea is covered by a layer of endothelial cells. The ciliary body supports the lens through the zonules and provides the mechanism for accommodation. The anterior chamber contains the volume of fluid between the cornea and the lens, while the vitreous is the volume of fluid posterior to the lens. The retina is the light-transducing inner layer of the sclera.

FIGURE 1. Primary ocular structures: A) aqueous humor, B) ciliary body, C) cornea, D) choroid, I) iris, L) lens, N) optic nerve, OO conjunctiva, P) pupil, R) retina, S) sclera, V) vitreous humor, Z) zonules

Previous Eye Research An extensive amount of both experimental and computational research studies have been published on eye injuries from blunt trauma. Blunt objects such as BB’s, metal cylinders, foam particles, paintballs,

golf balls, squash balls, and baseballs have all been used to investigate ocular trauma (Delori, 1969; Duma, 2000; Galler, 1995; Green, 1990; Preston, 1980; Scott, 2000; Umlas, 1995; Vinger, 1994, Vinger, 1997; Vinger, 1999). In each study, the kinetic energy associated with the chosen blunt objects was calculated and used to predict injury. Many studies note that less energy is required of smaller objects than of larger objects to produce similar injuries. The data suggest a local stress level in the eye would be the best predictor of injury, although none of the studies specifically state this idea. There is a large amount of data, but no uniform predictor of eye injury in all these cases. Many authors focus on kinetic energy as predictive of eye injury, but its use varies greatly depending on the size, mass, and stiffness of the projectile. In regards to airbag induced eye injuries, there exists a paucity of experimental data compared to the number of individual case study publications. Fukagawa (1993) found that increased inflator aggressivity contributed to increased endothelial cell damage. The most recent airbag related study examined the injury potential of high-speed foam particles released during airbag deployment (Duma, 2000). This study illustrates the compounding risk of eye injuries from not only airbag contact, but also from particles released from the module during deployment. However, the previous experimental work on blunt trauma and airbag related eye injuries lacks quantifiable measurements needed for the validation of a computational eye model. Although computational modeling of the human eye has been studied, the majority of the research to date has centered on the investigation of corneal incisions used for corrective eye surgery (Bryant, 1996; Hanna, 1989; Sawusch, 1992; Wray, 1994). These models are designed for static solutions and not useful for impact trauma studies. More recently, two finite element models of the human eye have been presented for use in dynamic events. First, Kisielewicz (1998) and Uchio (1999) presented an eye model for the investigation of ocular impacts from grinder debris and airbag injuries after radial keratotomy procedures. Nonlinear, isotropic material properties of the sclera and cornea were gathered from uniaxial tensile strip tests performed up to rupture. While this model provided useful insights as the first dynamic model, it is limited by nonbiofidelic boundary conditions and lack of thorough validation experiments. The second finite element model of the eye was presented by Power et al (2002a and 2002b). This model was developed to examine eye injuries in military pilots from the night

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vision goggles and to investigate eye injuries from high speed foam particles. Although this model included biofidelic boundary conditions, it is limited by the relatively large mesh size and lack of fluid elements in the anterior chamber and vitreous. In summary, an extensive amount of experimental and computation research has been performed on the human eye; however, useful validation tests and an accurate finite element model are not available. The purpose of this study is to present a new finite element model of the eye that will be useful in predicting the most severe eye injuries, globe ruptures. A corresponding goal of this study is to present a new series of eye experiments with foam particles, BBs, and baseball that provide the quantifiable validation data needed for the model development. METHODS Twenty-two (22) experimental tests with fresh human eyes and six matched simulations were performed in which the eye was struck at the apex of the cornea. A range of energies were employed with three different types of projectiles: BBs, baseballs, and foam. The type of impacting objects were chosen so as to provide only blunt loading for the prediction of globe rupture, with speeds resulting in both no rupture and rupture tests. Rupture of the eye due to sharp edge and puncture type loading was not tested. Experimental Nine foam tests were performed from 10 m/s to 32 m/s, named EF1-EF9 (Table 1). BB tests were performed within a range from 53.0 m/s to 123 m/s, with eight tests, E1-E8, within this range. Five baseball tests, EB1-EB5, were performed within a velocity range from 30 m/s to 43 m/s. TABLE 1. Experimental Test Matrix

Test

Object

E1-E8 EF1-EF9 EB1-EB5

BB Foam Baseball

Diameter (mm) 4.50 6.35 76.10

Mass (g) 0.375 0.077 146.5

Experimental Specimens Eyes were obtained from unembalmed human cadavers and tested within 24 hours of death. Eyes were enucleated within 8 hours postmortem. Institutional review board approval was obtained for all test procedures. Screening for Hepatitis A, B, C, and HIV was conducted on the tissue prior to acceptance into the research program.

Test Configuration and Instrumentation The test configuration was designed to strike the cornea centrally using each of the three projectile types. This was viewed as a starting point to predict globe rupture, and enabled the measurement of displacement with high speed video for lower speed tests. Eyes were mounted in gelatin in a 12.7 mm thick polycarbonate and polypropylene eye mount, designed to approximate the fatty tissue and dimensions of a human orbit. The eye mount was attached to a steel and aluminum backing plate mounted on casters to allow anterior-posterior translation of the whole unit when necessary. This was only an issue, however, for the baseball tests. The use of gelatin has been proposed by Vinger (1999) to simulate the fatty tissue surrounding the eye. For the testing performed herein, the gelatin also serves to hold the eye in place for the tests. The gelatin filled the entire orbital space, and enough was used so as to cover up just posterior to the limbus of the eye. The anterior-posterior dimension of the 'orbit' is 53 mm, the superior-inferior 41 mm, and the medial-lateral 48 mm. The orbit size was consistent for all tests. Since the size of eyes among all persons is fairly consistent, the assumption of a constantsized orbit resulted in the same ratio between relative size of the eyes vs. the eye mounts (Vinger, 1999). Polycarbonate was used to facilitate viewing the impact event from the side. Prior to each test, the eyes were repressurized by inserting a 27 gauge needle into the anterior chamber at the limbus. Repressurization was performed only to bring the interior pressure of the eyes in accordance with previous work performed by Vinger (1999). Foam particles were launched using a pneumatic gun with an intermediate air chamber to limit the airflow out of the gun (Figure 2). This was done so as to avoid pushing the eye out of the gelatin with too much air. The BB tests used a 0.177 caliber American Classic™ air powered BB gun, Model 1377. The gun could deliver a BB repeatably at 56 m/s, 92 m/s, and 122 m/s. Baseballs were fired using the same pneumatic gun used for the foam tests, but with a solenoid valve attached to a 110 liter tank. Initial pressure of the tank was measured prior to each test. This setup allowed baseballs to be delivered at a range of speeds greater than the fastest major league pitch (44.7 m/s) to slower pitch speeds. Events were filmed using high speed video for qualitative data such as rupture/no rupture and quantitative data such as corneal apex displacement. All video was taken at 7100 frames per second (Phantom 4, Vision Research, Inc., Wayne, NJ). Eye injury evaluation was performed via visual and

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Check Valve Foam Particle

A Small Volume Tank

Air Tank

Eye in Gelatin Solenoid Valve

Polycarbonate Artificial Orbit

B Air Tank

Baseball

BB

C Air Gun

Trigger

FIGURE 2. Diagram of experimental setup for eye tests: (A) Foam particle tests, (B) Baseball tests, (C) BB tests

photographic techniques in order to describe the presence and type of globe rupture.

assumptions used in creating the eye model, mesh creation, and the matrix of simulations used in the computational work.

Computational A detailed eye model was developed to predict globe rupture based on the experimental data, which will be called the Virginia Tech Eye Model (VTEM). The model was designed to support both large and high rate deformation, and includes the corneoscleral shell, including the cornea, sclera, and limbus, represented by the region that the cornea and sclera join in the anterior portion of the eye. These ocular structures are represented by a mesh with Lagrangian formulations for element properties, and this portion of the eye sits within a mesh with an Eulerian description of fluid flow. Also, the three projectile types simulated in the current work, a BB, baseball, and foam particle, were modeled using Lagrangian elements. The model was developed using FEMB (Finite Element Model Builder, Engineering Technology Associates, Troy, MI) and Gridgen (Version 13, Pointwise, Fort Worth, TX). Simulations were run on a 2.2 GHz Pentium IV with 1.5 GB of memory (SCSI Workstation) using LSDyna version 2.7 (Livermore Software Technology Corporation, Livermore, CA). Simulations took an average of 6 hours to run, depending on the length of time required to reach the peak deflection and/or peak stress for a given test. The geometry of the eye model is presented first, followed by the material

Geometry The eye was first drawn in cross section, then revolved to form a quarter cylinder geometry, using two planes of symmetry. Dimensions of the corneoscleral shell were similar to the Woo (1972) 2D model (Figure 3). One geometry was used for all simulations, as the geometry of the normal human eye has been shown to be about the same irrespective of age once adulthood has been achieved, and irrespective of size and sex of the person (Vinger, 2000). The model cross section is based on two spherical geometries, with the anterior chamber formed by a 7.8 mm radius circle whose center is located 5 mm anterior to the posterior chamber, with radius 12 mm. Thicknesses were then assigned at the corneal apex of 0.52 mm, the posterior pole of 1.0 mm, the equator of 0.55 mm, and two thicknesses at the limbus, 0.66 mm anterior to the limbus and 0.8 mm posterior to the limbus. Fillets were used to smooth out the geometry at the limbus, making it more realistic. Dimensions and location of the lens were taken from LeGrand's Full Theoretical Eye (LeGrand, 1946). The Full Theoretical Eye gives the distance along the anterior-posterior axis of the eye from the corneal pole to the anterior surface of the lens as 3.6 mm, and the distance to the posterior

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surface of the lens as 7.6 mm. The radius of curvature of the anterior surface of the lens is described as 10.2 mm, and the posterior radius of curvature is 6 mm. The superior portion of the lens (in cross section) was then filleted to form a continuous curve. The ciliary body and zonules were included by beginning the ciliary body just posterior the limbus, and continuing it posteriorly. 0.55 mm

0.8 mm 0.66 mm R = 12 mm

determined that the cornea could be considered to behave very nearly as a membrane, with little ability to resist bending stresses. It has also been shown that the compressive and bending rigidity of the cornea is two magnitudes smaller than the in-plane, or membrane tensile rigidity (Pinsky and Datye, 1991). This finding is also supported for scleral tissue by Battaglioli and Kamm (1984), who measured the compressive modulus of the sclera and compared this to the in-plane tensile modulus, finding it to be about 1/100th of the tensile modulus value. More recent approaches to modeling biological membranes and soft ocular tissues have taken into account this lack of resistance to bending and compressive loads (Heys and Barocas, 1999).

R = 10.2 mm R = 6 mm 0.52 mm

R = 7.8 mm 3 4 mm mm

1 mm 5 mm

12.4 mm 24.8 mm

FIGURE 3. Dimensions of the corneoscleral shell and lens

Material Definitions The corneoscleral shell was modeled as a membrane material using an orthotropic membrane formulation. This formulation allows for the input of a nonlinear stress strain curve in the local element 1,2 axes (orthogonal directions in the plane) and also the specification of a nonlinear shear modulus. The membrane formulation uses the nonlinear elastic material properties that are input into the model for tensile loads, and uses a separate constant material modulus for compressive loads. This allowed for the input of a different modulus in compression than in tension. This approach to modeling the eye as a membrane is supported by Hoeltzel (1992), who

The cornea and sclera material properties were taken from Uchio (1999), which were nonlinear, elastic material properties from uniaxial strip tests taken to failure (Table 2). These stress strain curves begin with stiffnesses of 124 MPa and 358 MPa for the cornea and sclera, respectively, then the stiffness decreases with increasing strain. For both components, the stress-strain curve is defined only up to a stress of 9.4 MPa which occurs at a strain of 15.2% for the cornea and 6.2% for the sclera. Therefore, in previous work this stress and strain criteria has been used as the criteria for failure. In simulations, the stress-strain curves beyond this level are extrapolated using the last two points in the stress-strain curve, giving the cornea and sclera stiffnesses of 14 MPa and 35 MPa beyond the peak defined stress of 9.4 MPa. In compression, Uchio (1999) mentioned the fact that the cornea and sclera have compressive strengths that are two orders of magnitude below their tensile strengths. However,

TABLE 2. Material properties and number of elements used in the simulation (for one quarter revolution)

Ocular or Test Component

Element Type

Formulation

Number of Elements

Elastic Modulus (MPa) Nonlinear Nonlinear 6.888 357.78 11 N/A N/A Rigid 2.208 12.0 N/A

Density (kg/m3)

Cornea Shell Lagrangian 315 Sclera Shell Lagrangian 1540 Lens Solid Lagrangian 343 Zonules Solid Lagrangian 490 Ciliary Solid Lagrangian 2632 Aqueous Solid Eulerian 800* Vitreous Solid Eulerian 3900* BB Solid Lagrangian 126 Foam Solid Lagrangian 256 Baseball Solid Lagrangian 350 Eulerian Mesh Solid Eulerian 7420* Total 18172 * For the Aqueous and Vitreous, this number represents the initially filled The remaining elements form the initially unfilled Eulerian mesh.

1400 1400 1078 1000 1600 1006 1006 7860 192 636 1006 volume.

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they did not show whether this was incorporated into the eye model they developed. For the VTEM, the compressive stiffness of the cornea and sclera are 1/100 of their stiffnesses in tension, or 1.24 MPa and 3.58 MPa. Since the shear properties were not reported by Uchio, the shear modulus was calculated piecewise based on the incremental elastic modulus of the cornea and sclera, with an assumption of isotropy and the corresponding relationship between elastic modulus, Poisson's ratio (assumed incompressible), and shear modulus. The lens was modeled as linearly elastic, isotropic, and incompressible using force-displacement data from Czygan and Hartung (1995) who performed tests of human eye lens nuclei. The remainder of the lens was assumed to have the same properties as the nucleus (Fisher, 1971). The ciliary body was modeled as isotropic and linearly elastic, with the properties of muscle the same as those used by Power (2001). The ciliary body in the model was inserted into the inner surface of the scleral shell just posterior to the limbus. The ciliary body extended 2/3 of the perpendicular distance between its midway point of insertion on the scleral shell, and the closest point on the lens. The zonules connecting the ciliary body to the lens were modeled as isotropic and linearly elastic. In reality, the zonules are a very fine network of strands connecting the ciliary body to the lens, transferring force between the lens and the ciliary body. Because of the compliance of the ciliary muscle and the intended purpose of the zonules, the zonules were modeled with a stiffness equal to that of the stiffest modulus of the corneoscleral shell. Large deformation of Lagrangian elements, even those intended for modeling of fluids such as in the anterior chamber and the vitreous of an eye model, presents a problem. Given the extremely large deformation of the eye, an Eulerian approach to fluid modeling was used for the fluids in the anterior chamber and the vitreous. The Eulerian approach is more specifically an arbitrary lagrangian-eulerian (ALE) approach, wherein the fluid portions of the eye are deformed during the course of a timestep in a lagrangian manner (involving mesh deformation), followed by an advection step during which a remapping of the eulerian mesh occurs. During this step, the eulerian mesh is deformed back to its original configuration, with a concurrent calculation of the fluid flow from element to element. A constraint was defined between the Lagrangian and Eulerian elements defined at the corneoscleral shell and also along the outer surfaces of the intraocular structures modeled. The LS-Dyna constraint *constrained_lagrange_in_solid was used to

kinematically constrain the Lagrangian solid within the Eulerian mesh. This approach was taken over an approach of using an ALE formulation for both the solid and fluid portions because the nonlinear membrane material model used was available for lagrangian shell elements only. The Gruneisen equation of state (Equation 1) was chosen to describe the behavior of the Eulerian fluid (Hallquist, 1998). This equation of state defines pressure for a material in terms of relative volume and other parameters. The Gruneisen equation of state has the following form:



p=

ρ oC 2 µ ⎢1 + ⎛⎜1 −

γo ⎞

a 2⎤ ⎟µ − µ ⎥ 2⎠ 2 ⎦

⎣ ⎝ ⎡ µ2 µ3 ⎤ ( ) − − − − µ S S S 1 1 1 2 ⎢ µ + 1 3 (µ + 1)2 ⎥⎦ ⎣

(1)

where C is the intercept of the Us-Up curve, S1, S2, and S3 are the coefficients of the slope of the Us-Up curve, γo is the Gruneisen gamma, and a is the first order volume correction to γo. The compression, µ, is defined in terms of the relative volume, V, by µ=1/V1. Us is the speed of a shockwave through the material, and Up is the speed of the shocked material, both with respect to the rest frame of the unshocked material. So, C is the speed of sound through the material, or the velocity at which a stress wave will propagate through the material at rest. As a comparison, standard values for water are C=1482.9 m/s, S1=2.1057, S2 = -0.1744, S3=0.010085, γo = 1.2 (Dobratz, 1981). The densities of aqueous and vitreous, nominally 1003 kg/m3 and 1009 kg/m3, were averaged to a density of 1006 kg/m3 (Duck, 1990). The speed of sound for the aqueous, reported to be 1481-1525 m/s at 25.5 °C, is very close to the speed of sound through the vitreous, 1523-1532 m/s at 37 °C (Duck, 1990). For the VTEM, the speed of sound through the aqueous and vitreous was approximated as the average of the aqueous sound speed, 1503 m/s. The remaining variables were taken to be those used in standard simulations involving water. The Gruneisen equation of state does not support shear in the same way that is thought of with a shear viscosity in a fluid. The Jaumann stress rate is used with equations of state in LS-DYNA. The approach computes the deviatoric stress components based on the deviatoric strain rate tensor, and is outlined in the LS-DYNA theory Manual (Hallquist, 1998). The bulk viscosity is used in the deviatoric strain rate tensor, and is computed, as in nearly all wave propagation codes, by adding a viscous term to the pressure to smear shock discontinuities into rapidly

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varying but continuous transition regions (Hallquist, 1998). The foam was modeled using properties given by Duma (2000) as foam that might be covering a dashboard in an automobile. The BB was modeled as rigid because of its much greater modulus in comparison to the eye. For the baseball hardness, an inverse dynamic finite element approach was used to obtain the elasticity of a softer variety of baseball, the Compression-Displacement (CD) 25. A target displacement of 6.25 mm at 25 pounds of force (CD25), similar to the testing performed by the National Operating Committee on Standards for Athletic Equipment (NOCSAE) was modeled in a separate finite element study, using rigid plates properly constrained. The elastic modulus was altered until the peak displacement matched that found in NOCSAE tests. Six iterations were required to determine this isotropic stiffness that would best represent the ball, which was 12 MPa. The ball was given a Poisson's ratio of 0.28. Mesh Creation The computer model of the eye was modeled using quarter symmetry, to simulate impacts into the center of the eye only. The corneoscleral shell (cornea and sclera), lens, zonules, and ciliary body were modeled, using shell elements for the corneoscleral shell, and solid elements for the lens, zonules, and ciliary body. The symmetry is modeled using global symmetry planes, which automatically control the symmetry of all points falling on the chosen plane. The Lagrangian mesh was then embedded in an Eulerian fluid mesh, a technique which will be justified and explained further. The corneoscleral shell and lens were created in Gridgen. The lens grid was initially smoothed using an elliptic mesh solver routine within Gridgen to achieve the maximum amount of orthogonality for the sides at corners of all solid elements. This increases the stability of the solution and lessens the chance that simulations will terminate due to 'negative volume elements', or elements that invert due to non-physical deformation taking place during a time step. The mesh density, excluding exterior eulerian parts, was anywhere from 4 to 11 times greater than the density used in previous three-dimensional models, depending on the part (Uchio, 1999). The meshing procedure used to define the corneoscleral shell consisted of meshing the eye in circumferential rings, using an incremental angle along the meridian of the eye (Figure 4). The meshing procedure for the shell resulted in a thickness for the corneoscleral shell that was continuous from the corneal apex to the posterior

pole. The thickness was made to vary like previous models (Uchio, 1999), but in a continuous manner. This was accomplished by using only one line of shell elements per ring. The thickness was then specified for that segment on both the anteriormost side of the ring segment and on the posteriormost side. The first 13 rings starting from the corneal apex form the cornea, the remainder of the rings in the model (59 additional) form the sclera. The nonlinear Meridional Direction

Sclera

Cornea

Thickness continuous at ring intersections, varies on each side of a ring

Circumferential Direction

FIGURE 4. Rings forming the corneoscleral shell, definition of directions, and breakup of corneoscleral shell into cornea and sclera

material properties were applied in the shell local 1 and 2 directions, corresponding to the meridional and circumferential directions due to the orientation of the mesh. Because the model was intended to accurately represent the different failure modes of the globe that can result in globe rupture, it had to be able to accommodate realistic physical contacts that might occur during impact with the three types of objects tested. For example, in the case of a BB contact, the possibility of contact between the posterior surface of the cornea and the anterior surface of the lens had to be included. Fukagawa (1993) has demonstrated that the epithelial cells on the anterior surface of the cornea can regenerate to cover the cornea after abrasions that remove them, but endothelial cells on the posterior surface of the cornea cannot. Ophthalmologists can assess eye trauma partly by looking at the anterior surface of the lens to see if cells have been transferred to this surface. For the baseball contact, the possibility of large deformation and displacement had to be taken into account. For foam contact, where small displacements of the cornea were expected, accurate modeling of the lens, ciliary, zonules, and anterior chamber contents should result in precise reproduction of the

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displacements. For these reasons, isotropic material approximations for the fluid in the interior of the eye, and also Lagrangian approaches to modeling fluid had to be abandoned. For large deformations resulting in the pinching off of a portion of a fluidfilled volume, dimensional limitations imposed by the Courant timestep criterion require the use of a different approach to fluid modeling than a Lagrangian one. In order to accommodate large deformation, the fluid filled anterior and posterior chambers were modeled using Eulerian elements. All initially filled spaces and any location into which the Eulerian fluid may ultimately be transported during a simulation have to be meshed. For this reason, it was anticipated that the fluid from the anterior chamber could be transported anterior to the cornea due to bulging of areas close to the point of impact, as well as backward into the areas that are occupied by the lens, zonules, and ciliary body at t=0 ms. The worst case scenario for posterior translation of the fluid within the anterior and posterior chambers was estimated to take place during the baseball simulations. To mesh the region around the eye in anticipation of equatorial expansion and posterior translation of the eye, an external grid is generated posterior to the limbus, that fans out as it moves posteriorly, and is defined 15 mm behind the posterior pole. The Lagrangian mesh (Figure 5) sits within the Eulerian mesh (Figure 6), with the anterior and posterior chambers centered over the filled regions.

Coincident nodes were used to tie the ciliary body to the sclera, and to tie the lens to the zonules. However, the number of axisymmetric divisions or layers forming the zonules exceeded the number of divisions in the ciliary body. Since these surfaces did not match up exactly, a tied contact interaction was created between the ciliary body and zonules. This type of contact matches the velocity of nodes and surfaces defined in the contact. It does not allow for the separation of the two bodies either by moving together or apart. For BB and foam tests, contact was defined between the cornea with the impacting object and lens using the *contact_automatic_surface_to_surface algorithm, one of the contact algorithms defined in LS-Dyna. For the baseball tests, contact was included with the sclera since the impacting area of a baseball is much larger than that occurring with a BB or foam particle. The default treatment within LS-Dyna of interface friction for this type of contact was used. LS-Dyna uses a Coulomb type friction, when contact friction is invoked. For the simulations reported herein, the static and dynamic coefficients of friction were 0.0, signifying 'frictionless' contacts. Since the impacts occurred at the corneal apex, and were not expected to result in 'glancing' blows, the effect of contact friction was assumed to be negligible. However, simulations were not performed with contact friction to investigate the changes that would occur with contact friction. This was done to minimize the number of varied parameters for the study to those certain of having a direct influence, including object incoming velocity. The eye was mounted in a virtual Initially unfilled volume

Corneoscleral Shell Anterior Chamber

Globe

Lens Zonules

Posterior Chamber Ciliary Body

FIGURE 5. Lagrangian mesh of eye showing corneoscleral shell, lens, zonules, and ciliary body

Initially filled volume

FIGURE 6. Eulerian mesh showing initially filled volume (dark gray) and initially unfilled volume (light gray)

Stitzel et al. / Stapp Car Crash Journal 46 (November 2002)

"gel" to simulate the experimental boundary conditions, the properties of which are given by Power (2001). The dimensions of this gel matched the size of the gelatin-filled orbit used in the experimental tests. The anterior and posterior chambers are separated given that the filled Eulerian fluid is constrained to the movement of the Lagrangian solid portions of the eye. Fluid flow through the region of the zonules was not allowed. Clinically, fluid flow can occur through this region, as the iris allows flow between the iris and the lens. However, the amount is very small, and dynamically it was assumed that no fluid can flow past the iris. Because the iris was not modeled in the simulation, the effects of its presence are modeled by not allowing fluid flow through the zonules. There is support for this assumption in the reverse pupillary block theory originally proposed by Karickhoff (1992). Reverse pupillary block implies that aqueous humor is able to flow forward through the pupil under normal conditions, but when the pressure in the anterior chamber exceeds that in the posterior chamber, the iris is forced against the lens, preventing flow from the anterior chamber into the posterior chamber (Heys and Barocas, 2002). The total number of elements used in the simulation is 18,172. This includes the initially unfilled Eulerian mesh, and the three objects modeled. The total number of elements devoted to modeling the Lagrangian geometry of the quarter cylinder eye and the initially filled space, which represent the eye at time t=0, is 10,020. Were the eye modified to fill all four quadrants, the number of elements would therefore total 40,080. Simulation Test Matrix For each type of projectile, two simulations were run in which the projectile struck the corneal apex centrally (Table 3). The magnitude of the velocity used for the incoming projectile was based on the lower and upper speeds used in the experiments. In the case of the BB and baseball simulations, the two speeds were intended to provide stress-strain,

pressure, and strain rate data for experimental tests that resulted in no rupture and rupture. In the case of the foam simulations, the speeds chosen represent the lower and upper range of speeds used in the experimental tests. Data Analysis All computational stress and strain data was filtered to SAE 3000 Hz using a built-in filter in LS-DYNA's post processor, LS-Post. Strain rate data was filtered to SAE 10000 Hz, and displacement data was left unfiltered. This had little effect on the foam and baseball tests, and was done to circumvent noise in the BB tests. Previous studies have used CFC 180 (Shah, 2001) for 30 ms simulations, but that appeared to overfilter the data in this application. Strain rate data is included only for the purpose of inferring the usefulness of the static material test assumptions. Peak principal stress was recorded for all simulations, as well as strain in the local element x and y directions. Peak principal stress was used for comparison between simulations due to the lack of stress development in the compressive, or local z, direction. The membrane element formulation in LSDYNA does not compute the compressive stresses in the shell thickness direction. However, strains are computed. Therefore, it was possible to report peak principal stress as the peak stress in the plane of the membrane, but peak principal strain could not be assumed to be in-plane. For this reason, the strain is shown in the local element x and y directions for comparison. Because the element orientation was controlled, the resulting x and y element directions represent the meridional and circumferential directions, respectively. RESULTS Experimental results will be presented first, followed by results from the simulations, and then comparisons of experimental and computational results will be made.

TABLE 3. Simulation Test Matrix

Simulation #

Object

S1 S2 SF1 SF2 SB1 SB2

BB BB Foam Foam Baseball Baseball

Length of Simulation (ms) 0.3 0.3 0.4 0.4 0.6 0.6

Mass (g)

Velocity (m/s)

0.375 0.375 0.077 0.077 146.5 146.5

56.0 92.0 10.0 30.0 34.4 41.2

Kinetic Energy (Joules) 0.588 1.587 0.004 0.035 86.7 124.3

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Experimental For the BB tests, kinetic energies in the range from 0.527 Joules to 2.807 Joules were achieved, which represent velocities from 53.0 m/s to 122.4 m/s (Table 4). Four of the eight tests resulted in globe rupture. The results from the BB tests demonstrate that with BB's, at lower kinetic energies close to 0.55 Joules, globe rupture never occurred. At higher kinetic energies in the range of 1.5 Joules, globe rupture always occurs. Tests E1-E4 did not result in rupture, while tests E5 to E7 did. For BB tests with rupture, the exact time can be determined within a window of several frames of video. Peak deflection is not reported. The maximum frame rate at which the high speed video could be run and still give adequate resolution was 7100 frames per second. A typical impact, judging from computational results and inferring from the experimental videos, lasted less than one millisecond. From the time of initial contact of the BB with the eye to maximum deformation of the eye, only about 0.3 ms elapsed. Within this time frame, only three frames can be taken at 7100 frames per second. It is unlikely that the frame closest to peak deformation of the eye actually was taken at the time of peak deformation of the eye. Therefore, high speed video at this rate was

used only to quantify exactly the speed of the incoming projectile, and film the rupture event. For the foam tests, globe rupture was not observed (Table 4), The primary purpose of the foam tests was to provide low-energy, small-displacement data by which the computational model could be validated. The kinetic energies of the foam pellets all fall far below the energy range of the BB's, in a range from 0.004 Joules to 0.037 Joules. Because of the lower speed of the foam pellets, a greater number of frames could be taken during a foam test. This allowed the peak corneal apex displacement to be more accurately determined for the foam tests. The data indicate that the progression of peak corneal apex displacement with incoming velocity of the foam particle is linear (Figure 7). Therefore, a straight line approximation was made to the data for all the foam tests, EF1-EF9. The goodness of fit of this displacement vs. incoming speed line is better than expected given experimental error, R2 = 0.9185. For the baseball tests, significant differences in the kinetic energies achieved versus the BB tests are apparent (Table 4). Test EB1, with a kinetic energy of 66.2 Joules did not result in globe rupture, and tests EB2 - EB 5, with kinetic energies from 86.7

TABLE 4. Eye injury data for experimental tests

Test

Object

Mass (g)

E1 E2 E3 E4 E5 E6 E7 E8 EF1 EF2 EF3 EF4 EF5 EF6 EF7 EF8 EF9 EB1 EB2 EB3 EB4 EB5

BB BB BB BB BB BB BB BB Foam Foam Foam Foam Foam Foam Foam Foam Foam Baseball Baseball Baseball Baseball Baseball

0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.077 0.077 0.077 0.077 0.077 0.077 0.077 0.077 0.077 146.5 146.5 146.5 146.5 146.5

Velocity, (m/s) 53.0 55.8 53.8 59.7 85.2 91.7 90.4 122.4 10.6 14.2 14.3 18.9 23.0 26.8 28.6 26.7 31.0 30.1 34.4 35.5 41.2 42.8

Kinetic Energy (Joules) 0.527 0.583 0.542 0.669 1.361 1.578 1.532 2.807 0.004 0.008 0.008 0.014 0.020 0.028 0.028 0.032 0.037 66.2 86.7 92.1 124.3 134.5

Globe Rupture No No No No Yes Yes Yes Yes No No No No No No No No No No Yes Yes Yes Yes

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splits are apparent; however, in test EB4, a split appears at the equator visible from the side, but the split continues across the center of the front of the cornea (Figure 8). In test EB5, rupture has occurred at the limbus and the eye has also split across the superior portion of the cornea (Figure 9). Tests EB4 and EB5 both resulted in gaping holes in the eye, and resulted in almost total extrusion of the intraocular contents, while in tests EB2 and EB3, most of the intraocular contents remained intact though the eye split.

3.5

Eye Displacement (mm)

3

2.5

2

Velocity Disp. (m/s) (mm) 10.6 1.2 14.2 1.7 14.3 1.4 18.9 2.1 23.0 2.2 26.8 2.4 28.6 2.3 26.7 2.8 31 3.2

2

R = 0.9185

1.5

1 10.0

Computational 15.0

20.0 25.0 Foam Speed (m/s)

30.0

Iterations of the model were checked using the energy measures available with LS-DYNA for conservation of energy, low element hourglassing, and no excessive distortion of elements. The results of computational simulations, including peak stress in the corneoscleral shell, time of peak stress, peak strain, peak strain rate, peak deflection when applicable, and peak pressure were recorded (Table ). For each of the three impact types, the peak principal stress data was viewed graphically and the location of

FIGURE 7. Peak corneal apex displacement vs. foam speed at impact

Joules to 134.5 Joules did. There appears to be a progression of severity of injury between tests 2-3 and 4-5. In EB2 and EB3, similar injuries occurred, with short ruptures beginning at the limbus and heading towards the equator. In test EB2, one such split in the sclera is apparent, and in test EB3, two

Splits at limbus, heading posteriorly FIGURE 8. Test EB3 - Two ruptures beginning at limbus, extending to equator

FIGURE 9. Test EB5 - Rupture through cornea across equator to posterior of eye

TABLE 5. Eye stress and injury data for computational runs

Test

Object

Peak Stress (MPa)

S1 S2 SF1 SF2 SB1 SB2

BB BB Foam Foam Baseball Baseball

22.812 32.757 3.180 7.830 22.153 24.617

Peak Stress Time (ms) 0.125 0.090 0.245 0.160 0.490 0.440

Peak Strain

Peak Strain Rate (strains / ms)

Peak Deflection (mm)

Peak Pressure (MPa)

0.437 0.732 0.009 0.025 0.309 0.360

27.8 26.8 0.0445 0.198 2.62 4.14

6.42 7.79 1.30 2.67 N/A N/A

1.832 2.478 0.124 0.386 2.099 2.548

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peak stress was determined. The peak principal stresses were all below 35 MPa for the BB tests, below 10 MPa for the foam tests, and below 25 MPa for the baseball tests. Therefore, the range from 0-35 MPa is used for shading BB results, 0-10 is used for shading foam results, and 0-25 is used for shading baseball results. It was assumed that some of the highest peak strain rates in the corneoscleral shell would be attained for the same regions that resulted in the peak stresses. This is supported by the ramping of the peak stresses: the higher peak stresses generally had higher onset rates. Also, the peak strain rates are also reported only to provide insight for the exploration of future material models. For this reason, peak strain rates are reported for the same region and element that the peak stresses were acquired. Peak pressure is reported for the region immediately adjacent to the location of peak stress. So, in the case of the BB tests, the peak pressure is reported for the anterior chamber just proximal to the corneal apex. For the foam tests, the pressure is reported in the aqueous humor just posterior to the limbus, behind the region of peak stress. For the baseball tests, the pressure is reported just inside the corneoscleral shell in the posterior chamber at the equator. The quarter cylinder geometry results have been reflected about two global symmetry planes in order to show the full frontal picture of the impact. For BB impacts, the first 13 rings of the eye model starting at the corneal apex, which comprise the cornea are shown. For the BB tests, the peak stresses were always in the anteriormost portion of the cornea. (Figures 10 and 11) This is an indication that the peak stresses are at the location of impact and are a direct result of contact with the incoming projectile. The simulations of BB impacts onto the corneal apex

Direction of Impact

FIGURE 10. Peak principal stress, MPa, for low speed BB simulation (cornea shown), t=0.125 ms

resulted in a peak stress of 22.812 MPa for the lowspeed BB simulation, and 32.757 MPa for the high speed BB simulation. The peak strains were roughly equal in both the meridional and circumferential directions, and the peak strain rates were noticeably higher than in the other test types, with strain rates of 27.8 strains/ms for the low speed BB and 26.8 strains/ms for the high speed BB. For foam impacts, the first 26 rings of the cornea are shown, along with stress shading (Figures 12 and 13). The stresses are shown at the time of peak stress. The first 13 rings represent the cornea, while the next 13 rings represent the sclera, with the ring sets adjoining in the region of the limbus. The region of peak stress begins in the cornea and terminates prior to the insertion of the ciliary body. The results demonstrate that the stresses are due to pressure effects in the anterior chamber. The stresses cannot be a result of bending in the region, since the membrane formulation does not compute compressive or bending stresses. But, the region of stress begins outside of the location of foam impact, and so is not a local effect due directly to impact with the foam. The peak stresses approach 3.2 MPa for the low speed foam impact, and 7.830 MPa for the high speed foam impact. Peak strain rates are lower than shown for the BB tests, 0.0445 strains/ms and 0.198 strains/ms for the low and high speed foam tests, respectively. The peak deflection of the corneal apex for the low speed foam simulation was 1.3 mm, while the peak deflection for the high speed foam impact was 2.67 mm. For baseball impacts, the entire corneoscleral shell is shown, along with stress shading (Figures 14 and 15). The shading results are qualitatively the same for the low speed and high speed baseball simulations, the

Direction of Impact

FIGURE 11. Peak principal stress, MPa, for high speed BB simulation (cornea shown), t=0.09 ms

Stitzel et al. / Stapp Car Crash Journal 46 (November 2002)

FIGURE 12. Peak principal stress, MPa, for low speed foam simulation, t=0.245 ms

FIGURE 13. Peak principal stress, MPa, for high speed foam simulation, t=0.160 ms

FIGURE 14. Peak principal stress, MPa, for high speed baseball simulation, t=0.10 ms

FIGURE 15. Peak principal stress, MPa, for high speed baseball simulation, t=0.44 ms

only difference being that the circumferential stresses are higher at the time of peak stress due to equatorial expansion (Figure 15). The shading results are presented for two different times during the tests, once at an early stage of the impact and again at a later stage of the impact. At 0.11 ms, the peak principal stress in the region of the limbus (just as in the foam tests previously shown) is 8.03 MPa (Figure 14). At a later time, 0.440 ms for the baseball (Figure 15), the peak principal stress is 24.6 MPa, and is primarily due to equatorial expansion. The peak principal stress for the low speed baseball test during the early part of the event is 8.349 MPa, occurring at t=0.12 ms. Again, the peak principal

stress occurring during the later part of the event is 22.153 MPa, and due to equatorial expansion. Experimental and Computational Results: Comparison For the BB tests, a time sequence is shown of one of the BB tests with failure, E7 (Figure 16). The sequence shows the BB as it first makes contact with the eye, t=0 ms, then presumably shortly after failure (t=0.14 ms), then at t=0.56 ms. This third frame is shown only to demonstrate that rupture has occurred and the BB has begun exiting the posterior of the eye once more. At t=0.56 ms, fluid is beginning to exit the corneal apex where the BB has punctured the eye, and the BB has begun to exit the eye at the posterior

Stitzel et al. / Stapp Car Crash Journal 46 (November 2002)

BB

BB inside

t = 0 ms

t = 0.14 ms

Fluid out

BB exiting

t = 0.56 ms

FIGURE 16. Time lapse sequence of BB test E7 with failure (V=90.4 m/s)

Stress, MPa

t = 0 ms

t = 0.14 ms

FIGURE 17. Time lapse sequence of BB simulation S2 (V=92 m/s)

pole. Simulation of the high speed BB (S2) is also shown, with frames at t=0 ms and t=0.14 ms (Figure 17). T=0.14 ms is the closest camera frame that existed at the time of peak stress predicted by the simulation, t=0.09 ms. Since the model does not fail, and was run only long enough to determine peak stress in the corneoscleral shell, a matched simulation picture is not shown for t=0.56 ms.

corneoscleral shell during this time. The maximum stress shading is in the region from 10 MPa to 12.5 MPa at t=0.14 ms, and from 17.5 MPa to 20 MPa at t=0.28 ms. Additionally, this peak stress is located in the anterior portion of the eye, near the site of impact with the baseball. This is also the location that rupture occurred in the experiments.

A sequence of frames taken during the foam tests is shown (Figure 18). Given the lack of globe rupture, it was possible to show frames matching in time, with the highest stresses in this sequence occurring at the limbus, at 0.14 ms. The frame at 0.14 ms for the computational run shown here, SF2, is very close to the time of overall peak stress found for the eye in simulations (0.16 ms, Table ). By t=0.28 ms, in test SF2, the foam has begun to withdraw from the eye, and the peak stresses in the limbus have propagated rearward towards the equator of the eye. (Figure 19)

The straight-line fit from Figure 7 was used to determine the experimental results at a level matching that of the simulations run. The results demonstrate that the model of the eye predicts a peak corneal apex deflection of 1.30 mm for an incoming foam particle at 10 m/s, when the experimental results show this level to be 1.19 mm. The experimental results show a peak corneal apex displacement of 2.85 mm at 30 m/s, while the simulation with a 30 m/s foam particle resulted in a peak corneal apex displacement of 2.67 mm (Figure 22).

In the baseball tests shown, EB4 (Figure 20) and SB2 (Figure 21), the time of initial contact, t=0 ms, is shown, as is the displacement and stress shading at t=0.14 ms, and globe rupture has occurred by 0.28 ms. The experimental video frames are three consecutive frames, so the exact moment that rupture occurred is uncertain, but it is clear that rupture of the eye along a 5-10 mm length occurred in less than 1/7100th of a second. The simulation model shows a dramatic increase in peak principal stress in the

Finally, looking at all three projectile types, at both high and low incoming velocities, the stress history for the region chosen in Figures10 through 15, or the region where the peak principal stress in the corneoscleral shell was reached for that particular type of test, is shown (Figure 23). For the BB, the region of peak principal stress during an impact event was the apex of the cornea in the region of impact. For the foam, it was just anterior to the limbus,

Stitzel et al. / Stapp Car Crash Journal 46 (November 2002)

t = 0 ms

t = 0.14 ms

t = 0.28 ms

FIGURE 18. Time lapse sequence of foam test EF7, no failure (V=26.8 m/s)

Stress, MPa

t = 0 ms

t = 0.14 ms

t = 0.28 ms

FIGURE 19. Time lapse sequence of foam simulation SF2 (V=30 m/s)

Initial contact

t = 0 ms

Globe rupture

t = 0.14 ms

t = 0.28 ms

FIGURE 20. Time lapse sequence of baseball test EB4, with failure (V=41.2 m/s)

Stress, MPa

t = 0 ms

t = 0.14 ms

t = 0.28 ms

FIGURE 21. Time lapse sequence of baseball simulation EB2 (V=41.2 m/s)

outside of the actual location of impact of the foam. For the baseball simulations, the region of peak principal stress was the equator of the eye. The peak principal stress occurs sooner for the higher speed tests, and the peak stress is higher for higher projectile incoming velocities. The BB tests, however, show a greater difference in the magnitude

of peak principal stress than do the foam or baseball tests (Table 5). The strain information is included here because the inclusion of peak principal strain may provide insight into the injury mechanisms involved in rupture of the globe (Figure 24). The peak strains for the foam tests

Stitzel et al. / Stapp Car Crash Journal 46 (November 2002)

are not shown, as experimental foam tests did not result in rupture, and rupture was not predicted. The peak strains were far lower than occurred for the BB 3.0

X-Displacement (mm)

2.5 2.0

DISCUSSION

1.5

The discussion will present a summary of the accomplishments of the current study, followed by more specific discussion presented in three sections: (1) Improvements versus existing models of the eye, (2) Comparison of results with currently existing data, and (3) Suggestions for future work.

1.0 SF1 - Foam 10 m/s SF2 - Foam 30 m/s Experimental Prediction, 10 m/s Experimental Prediction, 30 m/s

0.5 0.0 0.00

0.10

0.20 Time (ms)

0.30

0.40

FIGURE 22. Peak corneal apex displacement vs. time for simulation foam tests, with experimental prediction 35 Maximum Principal Stress (MPa)

- Peak Stresses 30 25 20 15

Foam 10 m/s Foam 30 m/s BB 56 m/s BB 92 m/s Ball 34.4 m/s Ball 41.2 m/s

10 5 0 0

0.1

0.2

0.3 Time (ms)

0.4

0.5

0.6

FIGURE 23. Maximum principal stress vs. time for all three projectile types, for low and high velocities 0.80 BB 56 m/s X BB 56 m/s Y BB 92 m/s X BB 92 m/s Y Ball 34.4 m/s X equator Ball 34.4 m/s Y equator Ball 41.2 m/s X equator Ball 41.2 m/s Y equator

Strain

0.60

0.40

0.20

0.00 0.00

and circumferential (Y) strain directions show greater variance in strain for the baseball tests than they do for the BB tests. Peak strains of greater than 40% are shown for the low speed BB simulation, while peak strains of greater than 70% are shown for the high speed BB simulation.

In the current work, a model of the eye including the corneoscleral shell and interior structures is presented. This model is the first 1/4 symmetry model of the eye to include many of the interior structures of the eye, and is an attempt to predict rupture of the eye due to dynamic loading. The changes in the model versus previous models include the incorporation of eulerian fluid elements for the anterior and posterior chambers. This change has allowed the ability to model large deformation, since the fluid-filled space in the anterior chamber is capable of becoming pinched off, allowing contact between the cornea and lens. It incorporates a continuously variable corneal and scleral thickness, where some previous approaches have used discontinuous thicknesses. Additionally, the mapped mesh of the corneoscleral shell, where local element circumference and meridional directions are a property of the mesh, will allow for the future incorporation of orthotropic and viscoelastic material properties when these become available. The mesh also allows for a continuously variable thickness duplicating anatomical thicknesses, without discontinuity. The model has been empirically validated using several different types of impacting projectiles, to determine its suitability for many different types of loading. Improvements versus existing models of the eye

0.10

0.20

0.30 Time, ms

0.40

0.50

0.60

FIGURE 24. Strain vs. time, meridional (X) and circumferential (Y) directions, for BB and baseball tests

and baseball tests. The peak strains for both the high speed and low speed foam tests were higher in the meridional direction than in the circumferential one. For the low speed foam test, SF1, the peak strain was 0.881% (0.00881) at 0.245 ms, and for the high speed foam test, the peak strain was 2.46% (0.0246) at 0.16 ms. For the BB and baseball tests, the meridional (X)

The use of a continuously variable corneal thickness removes the concern about discontinuities in corneal thickness resulting in non-physical stresses. This is made possible by segmenting the eye into circumferential rings, and using quadrilateral elements with thickness varying on the anterior and posterior ends to represent the thickness of the corneoscleral shell. Another benefit is that this approach to meshing will also allow for the future incorporation of orthotropic material properties,

Stitzel et al. / Stapp Car Crash Journal 46 (November 2002)

differing in the meridional and circumferential directions. The modeling of the eye using Lagrangian elements that are then inserted into an Eulerian mesh, with the fluid modeled using an equation of state to describe its behavior, is the first known by the authors for modeling high rate loading of the eye. Most previous eye models have used a Lagrangian mesh for the fluid or applied a static pressure to the inner surfaces of the eye to determine the stresses and strains in the corneoscleral shell (Uchio 1999, 2001; Woo, 1977). Some of the most recent work modeling fluid filled volumes for high rate loading has included the modeling of the aorta for traumatic aortic rupture (Shah, 2001). This work was performed in LS-Dyna as well, and used the *linear_fluid airbag control volume approach to model the fluid-filled aorta and adjacent vessels. It is an important distinction to make that many of the airbag modeling approaches include only a pressure-volume relationship to calculate the resultant loads on the surfaces defining the control volume. Airbag control volume approaches for modeling have been developed on the assumption that the fluid filling them is gas, which has a significantly lower density than water or blood. Therefore, the density of the 'fluid' used in an airbag control volume may be inconsequential to the kinematics and kinetics of the solution. It does not contribute to inertial loading in the model, which is a drawback when the volume of the fluid is a large proportion of the region of interest. In the case where the fluid-filled volume is the inside of the aorta, and the aorta is part of a larger model of the thorax, the need to include inertial effects of the fluid as well as pressure effects is unclear. However, the need to include both inertial and pressure effects is very clear when the fluid-filled structure is the eye. The fluid filling the eye is a large proportion of the total mass of the eye, and can be expected to contribute significantly to its inertial response. For the large deformations seen in the experiments and simulations, it became apparent that a purely Lagrangian approach would not suffice. There is an inherent limitation to using solid elements approximated as a fluid in a fluid-structure interface. The use of Lagrangian solid elements in the anterior chamber to interact with the cornea is an assumption that bears little resemblance to reality. A fluid provides lower resistance to shear and deformation, particularly when it is encased within a deformable chamber such as the eye. When the fluid in the anterior chamber is modeled using isotropic solid elements, this type of formulation can allow for the development of stresses in the local Lagrangian

mesh, and therefore detract from those developed in the membrane of the eye. However, there are Lagrangian approaches to fluid body modeling that are acceptable for small deformations. An Eulerian representation, and not a Lagrangian one, is still required in the present application by the fact that there is large deformation in the anterior chamber. The VTEM allows contact of the posterior surface of the cornea with the anterior surface of the lens, a finding which is observed in clinical practice. It is only allowed because fluid can be transported out of a space that becomes closed off due to large deformation. Comparison of results with existing data A great deal of work has been performed comparing the energies of different types of objects required to rupture the globe. Bullock (1996) presented a summary of much of the relative literature in an investigation of ocular and orbital trauma from water balloon slingshots. Our findings with respect to BB impacts are consistent with those presented by previous investigators (Weidenthal, 1966; Tillett, 1962). These previous studies show a threshold of 0.7 J for globe rupture in the anterior of the cornea due to impact with a BB, and experimental results presented here show agreement with this, an intact globe for energies up to 0.669 Joules, and globe rupture at all energies above that. Based on the computational results, and their comparison with matched experimental testing, the VTEM predicts globe rupture for principal stresses exceeding 23 MPa in the corneoscleral shell. However, this differs greatly from that reached by Uchio (1999, 2001). Uchio predicts globe rupture for strains of 18.0% in the cornea and 6.8% in the sclera, corresponding to a stress of 9.4 MPa for both tissues. This criteria is based on uniaxial strip tests of cornea and sclera at quasistatic strain rates, and so may not be applicable to high rate loading of the corneoscleral shell. In addition, the work presented by Uchio did not model the fluid in the anterior and posterior chambers using fluid elements, instead using a solid element approach. The ability of a solid element to absorb local stresses to a greater extent than a fluid could reduce the peak tensile stress reached in the corneoscleral shell. Also, without the incorporation of viscoelastic material properties, a material model may overpredict the probability of globe rupture, because higher stresses may be achieved than those found in quasistatic material testing. Additionally, Uchio (1999) used a representation of impacting foreign bodies and performed simulations using 30 m/s and 60 m/s impacts. The geometry of these foreign bodies was similar to a pyramid with the top

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removed, and a large cubic base for mass. This resulted in a blunt impacting foreign body. The nature of the impacting bodies in the work described herein involved slightly curved surfaces in the BB and baseball tests, which could result in somewhat different stress readings. Also, the mass range for the projectiles used by Uchio (1999) was from 0.05 g, just less than the foam particle mass used for the VTEM, to 0.30 g, just less than the mass of the BB. This difference and the fact that the low and high speed BB tests ranged were 56 m/s and 92 m/s allows for some justification of why the peak stresses differed from the results of previous authors. Another potential reason for the difference in peak stress and strain, and the inability of the 9.4 MPa or 18.0%/6.8% rupture criteria to predict rupture involves the transverse isotropy of the stress-strain curves used. Since the stress-strain characteristics are not direction-dependent, they cannot be expected to predict rupture in a tissue that has properties that are direction dependent. The experimental results, and the results of previous studies, confirm this. In the higher speed baseball tests, tears across the front of the cornea in test EB5 and along the limbus, then tearing back across the eye at the equator in test EB6, demonstrate this. Many previous authors have pointed out the directional dependence of the properties of the eye, demonstrating that the eye ruptures in the region of the limbus are often circumferentially oriented, and ruptures at the equator are often meridionally oriented, revealing the predominant direction of collagen fibers in those two regions, which is circumferential at the limbus and meridional at the equator (Meek, 1982, 1987, 1999). As a first attempt to investigate the anisotropic behavior of the cornea, Shin (1997) performed membrane inflation tests to examine the distribution of strain in the human cornea. In summary, most of the previous work investigating anisotropy of the eye has focused on the cornea only, due to concerns about radial keratotomy (RK) and other vision correction procedures. The results of baseball simulations, when compared to the experimental work, point toward a mechanism of injury that begins at the posterior aspect of the limbus and then moves posteriorly. This is due to the timing of stresses in the simulations compared to the timing of rupture in the simulations. The time of rupture for most of the baseball tests was 0.14 ms to 0.28 ms, at which time during the simulation SB2, the predominant stress was occurring in the cornea was just anterior to the insertion of the ciliary body. It is not unlikely that the onset of globe rupture is a result of stress in this region, and not the stresses appearing later on during the simulation, at the equator. However, the stresses

never exceeded even that predicted by Uchio (1999, 2001) for globe rupture in this region, during this time frame. This highlights the need for more focused experimental work. The strain rates resulting from the BB simulations greatly exceeded that found in the foam and baseball simulations, by over an order of magnitude versus the low speed baseball tests, and just less than a magnitude for the high speed baseball tests. There was agreement between the Uchio injury criteria for the foam tests, which did not predict failure, and failure did not occur. However, the peak stress criterion of 9.4 MPa is again not applicable for the BB simulations S1 and S2. Four separate tests (E1E4) with low speed BB's at an average of 55.6 m/s failed to rupture the eye. The simulation with a 56 m/s BB resulted in a peak stress of 22.8 MPa, well above the level predicted to rupture the globe. With the peak strain rates an average of 27.3 strains/ms for the BB tests, the use of a viscoelastic material characterization is called into question. In terms of an intraocular pressure rupture criterion, previous work has shown human cadaver eyes to rupture at intraocular pressures between 0.441 and 0.931 MPa (Dyster-Aas, 1962). It has previously been reported that the eye requires pressures up to 0.586 MPa for rupture, with 24 and 44 diopters of laser correction, involving the reduction of corneal thickness (Burnstein, 1995). Since much of the recent literature on rupture pressure of eyes involves operated eyes with photorefractive keratectomy (PRK), or laser-assisted in situ keratomileusis (LASIK), the rupture pressure of an intact, unoperated eye with no surgical correction during a dynamic event could be much higher. The peak pressure encountered within the simulations for which there is a matching test without failure is 2.099 MPa in simulation SB2. It is not unreasonable to assume that regions within the eye could encounter overpressures of 2.25 times that required to rupture an eye in quasistatic tests. However, pressure was not measured experimentally, and the reader is cautioned against comparing computational pressure results with results from other authors. Further material testing is warranted to explore this effect. Overpressure in all simulations remained within 90% of its peak value for only about 0.04 ms, confirming that this is a transient phenomenon and not quasistatic. While previous authors such as Bullock (1996) have pointed out differences in kinetic energy of different types of impacting projectiles with the globe, and alluded to stress and strain as factors in predicting

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orbital trauma, the most recent attempt to try to use stress and strain as a predictor for the threshold of globe rupture (Uchio 1999, 2001) carries with it many assumptions that could be non-biofidelic. The use of energy as a predictor for globe rupture, as well as pressure, is only part of the solution. This fact has been noted by previous authors, but not explored using the same approach as in the current work.

accuracy. However, the authors make no claim that the pressures match those that would be found experimentally in a human eye. In particular, the pressure would be expected to drop after rupture occurs, and since rupture is not modeled, pressures that are reached may be higher than could ever be achieved experimentally. Suggestions for future work

Limitations of the current study The fact that failure is not incorporated into the current model is a limitation of the work. The lack of failure calls into question the predictive capabilities of the model once globe rupture has occurred. However, we have attempted to clarify that the model is only intended to predict globe rupture, not to model rupture. The model will require further adjustments before it is capable of actually rupturing. Two main assumptions of the current model are the assumption of transverse isotropy and the assumption of quasistatic material properties in the model. These could also be viewed as limitations of the biofidelity of the model. Therefore, the model could be interpreted as predictive only up to the time of failure in a matched test. However, limitations on the type and direction of material properties mean that the model would not necessarily 'fail', were failure incorporated, at the right time. Therefore, the model has been empirically validated by comparing peak stresses throughout a computational run with the results from matched tests. Closer attention to the 23 MPa failure criteria presented in the current study is certainly warranted. Changes in local material properties from the assumption of material orthotropy or transverse isotropy, and the incorporation of highrate material properties, could alter the criteria for failure. It has been noted that pressure readings were not taken from the interior of the eye during the testing performed herein. Therefore, the results of pressures from the analytical model have not been checked against experimentally measured results. The authors caution against using the analytical pressure data for comparison with other published results until matched tests have been performed to actually measure the pressure dynamically in the anterior chamber during an impact. The act of actually placing a transducer into the anterior chamber of the eye for the current work would have confounded the experimental setup beyond acceptable levels. The pressures achieved with the model do differentiate between the rupture and no rupture cases in the experimental work, and so they could be used within the model to predict rupture with some degree of

Suggestions for future study all include items that can be viewed as original assumptions, that also are limitations of the modeling work performed herein. Future work should involve performing high-rate testing of corneal and scleral strips, testing corneal and scleral strips using biaxial testing or uniaxial testing in the circumferential and meridional directions, and testing the eye's response to dynamic overpressure. It is believed that the incorporation of high-rate material testing, particularly that takes into account the orthotropy of the corneoscleral shell, could significantly alter the outputs from the model. While the model is currently validated, the outputs from the model once it includes high-rate and directional material properties could substantially differ, but also reflect further improved biofidelity. It may turn out that the 'ultimate' criteria for failure is one that is local, and takes into account directional differences in strength and sensitivity to rate of loading. Future modification of the VTEM could also incorporate failure into the current model. With failure modeling, elements would be deleted when they exceeded a certain threshold stress, and with the current modeling approach, this would allow for the flow of fluid out of the model and release of stress due to pressure. This could result in the accurate prediction of splits in the corneoscleral shell such as that seen in experiments such as those shown in Figure 20. One of the primary reasons for originally using a 1/4 geometry, and not a 2-D model, was for the eventual incorporation of anisotropy into layers of the model. Future models are intended to include anisotropic material properties, with properties in the meridional and circumferential directions differing. The current membrane element formulation actually uses an orthotropic assumption, but for the work presented herein the material properties were taken to be the same in both directions in the plane. The results from experimental high-rate testing could also be incorporated into the VTEM. CONCLUSIONS The VTEM is the current state of the art in eye models for the prediction of globe rupture due to high

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speed blunt impact. The model predicts very closely the level of deformation due to low speed impacts with foam particles, and shows a marked difference in the stresses achieved by different velocities of blunt objects such as BBs and baseballs. The model is empirically verified to predict globe rupture for stresses in the corneoscleral shell exceeding 23 MPa, and local dynamic pressures exceeding 2.1 MPa. It allows contact between the posterior cornea and the anterior lens, a finding described in clinical practice. It allows very large deformations due to contact with large blunt objects, and shows an increase in circumferential stress due to equatorial expansion. A greater level of detail has been achieved through the incorporation of various improvements over existing models, detailed previously. The VTEM is ideally suited to the incorporation of properties from future material testing. A truly biofidelic model should use an injury criteria that will be applicable for different projectile types, different loading rates, and different boundary conditions. Only by looking at local stress in the material can this be achieved. It is believed that the VTEM represents the first step in this direction. It will reduce the need for experimentation every time a question is asked as to what types of injury can be expected due to a given type of loading, such as that predicted by an airbag. ACKNOWLEDGMENTS This research was sponsored by the Eye Injury Biomechanics Program, Virginia Tech, the United States Army Aeromedical Research Laboratory (USAARL), the Jeffress Foundation, and the ASPIRES program at Virginia Tech. Thanks to Livermore Software Technology Corporation (LSTC), for support with LS-Dyna. REFERENCES Battaglioli, J.L., Kamm, R.D. (1984). Measurements of the compressive properties of scleral tissue. Investigative Ophthalmology and Visual Sciences, 25: 59-65. Bryant, M.R., McDonnell, P.J. (1996) Constitutive laws for biomechanical modeling of refractive surgery. Journal of Biomechanical Engineering, 118: 473-481. Bullock, J.D., Johnson, D.A., Ballal, D.P., Bullock, R.J. (1996) Ocular and orbital trauma from water balloon slingshots: a clinical, epidemiological, experimental, and theoretical study. Transactions of the American Ophthalmological Society, XCIV, 105-134.

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