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Retina  |   December 2012
Primary Blast Injury to the Eye and Orbit: Finite Element Modeling
Author Affiliations & Notes
  • Tommaso Rossi
    From the Ospedale Oftalmico di Roma, Rome, Italy;
  • Barbara Boccassini
    Fondazione Bietti IRCCS, Rome, Italy; and
  • Luca Esposito
    University of Cassino, School of Engineering, Department of Mechanics, Cassino, Italy.
  • Chiara Clemente
    University of Cassino, School of Engineering, Department of Mechanics, Cassino, Italy.
  • Mario Iossa
    From the Ospedale Oftalmico di Roma, Rome, Italy;
  • Luca Placentino
    From the Ospedale Oftalmico di Roma, Rome, Italy;
  • Nicola Bonora
    University of Cassino, School of Engineering, Department of Mechanics, Cassino, Italy.
Investigative Ophthalmology & Visual Science December 2012, Vol.53, 8057-8066. doi:10.1167/iovs.12-10591
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      Tommaso Rossi, Barbara Boccassini, Luca Esposito, Chiara Clemente, Mario Iossa, Luca Placentino, Nicola Bonora; Primary Blast Injury to the Eye and Orbit: Finite Element Modeling. Invest. Ophthalmol. Vis. Sci. 2012;53(13):8057-8066. doi: 10.1167/iovs.12-10591.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Purpose.: Primary blast injury (PBI) mostly affects air-filled organs, although it is sporadically reported in fluid-filled organs, including the eye. The purpose of the present paper is to explain orbit blast injury mechanisms through finite element modeling (FEM).

Methods.: FEM meshes of the eye, orbit, and skull were generated. Pressure, strain, and strain rates were calculated at the cornea, vitreous base, equator, macula, and orbit apex for pressures known to cause tympanic rupture, lung damage, and 50% chance of mortality.

Results.: Pressures within the orbit ranged between +0.25 and −1.4 MegaPascal (MPa) for tympanic rupture, +3 and −1 MPa for lung damage, and +20 and −6 MPa for 50% mortality. Higher trinitrotoluene (TNT) quantity and closer explosion caused significantly higher pressures, and the impact angle significantly influenced pressure at all locations. Pressure waves reflected and amplified to create steady waves resonating within the orbit. Strain reached 20% along multiple axes, and strain rates exceeded 30,000 s−1 at all locations even for the smallest amount of TNT.

Conclusions.: The orbit's pyramidlike shape with bony walls and the mechanical impedance mismatch between fluidlike content and anterior air-tissue interface determine pressure wave reflection and amplification. The resulting steady wave resonates within the orbit and can explain both macular holes and optic nerve damage after ocular PBI.

Introduction
Primary blast injury (PBI) refers to biological damage caused by the peak incident overpressure (PIO) wave generated by an explosion. Initially described exclusively in hollow, air-filled organs, 1,2 the spectrum of PBI progressively enlarged to encompass fluid-filled organs, including the central nervous system 3,4 and the eye. 36  
Macular holes, choroidal rupture, and optic nerve damage occur relatively frequently after blunt trauma, 7 and although the pathogenic mechanism remains unclear, vitreous traction, 8 globe deformation, differential eye layer stiffness, 9 shockwave propagation, and multiaxial strain 10 likely participate. 
Finite element modeling (FEM) is a numerical analysis method largely used to simulate multiphysics problems. Previous applications to ophthalmology 1113 and, specifically, to blunt eye trauma 10 have been reported. 
The purpose of the present paper is to create a computational model of ocular PBI, simulate the propagation of blast waves through the orbit, and explain its pathogenesis. The study was prompted by the observation of macular holes, multiple choroidal ruptures, subretinal bleeding, and optic atrophy after blast exposure, in the absence of any contact with splinters and/or flying objects (Fig. 1). 
Figure 1. 
 
Pure PBI in a patient exposed solely to blast overpressure wave, in the absence of any contact with splinters and/or flying objects. Two months after PIO exposure, subretinal fibrosis is well evident temporal to the fovea, and some subretinal heme is still present. Note that the temporal aspect of the optic disc is pale.
Figure 1. 
 
Pure PBI in a patient exposed solely to blast overpressure wave, in the absence of any contact with splinters and/or flying objects. Two months after PIO exposure, subretinal fibrosis is well evident temporal to the fovea, and some subretinal heme is still present. Note that the temporal aspect of the optic disc is pale.
Materials and Methods
Finite Element Modeling
The eye model has been previously described, 10,14 while orbit and retrobulbar fat tissue were reconstructed based on patient computed tomography scan and magnetic resonance imaging (Fig. 2; see the Appendix for further technical details on FEM). 
Figure 2. 
 
(A) Exploded view of the complete FEM: skull + orbit + eyeball. (B) Transverse (horizontal; a) and sagittal (vertical; b) sections of the orbit FEM model. Selected locations are reported: 1, cornea (corneal apex); 2, vitreous base; 3, equator; 4, macula; 5, orbit; 6, orbit apex. Visual axis (continuous line) and orbit axis (dashed line) are reported as well.
Figure 2. 
 
(A) Exploded view of the complete FEM: skull + orbit + eyeball. (B) Transverse (horizontal; a) and sagittal (vertical; b) sections of the orbit FEM model. Selected locations are reported: 1, cornea (corneal apex); 2, vitreous base; 3, equator; 4, macula; 5, orbit; 6, orbit apex. Visual axis (continuous line) and orbit axis (dashed line) are reported as well.
Constitutive Models and Mechanical Properties
Constitutive parameters were determined through a reverse engineering calibration procedure that was previously reported, 10 and MSC Dytran 2010 (MSC Software, Santa Ana, CA) was used to run simulations. The orbit volume was assumed to be filled with a viscoelastic solid representing the fat tissue. 15 (See the Appendix for details on the constitutive model.) 
Detonation Model
Trinitrotoluene (TNT) was used as a reference explosive. A detonation wave is assumed to be a discontinuity that propagates through the unreacted material, instantaneously releasing energy. A spherical detonation wave propagating in air has been modeled; detonation parameters used in the present work were published by Lee et al. 35 (See the Appendix for details on the detonation model.) 
Unless otherwise specified, the progression of blast waves was assumed to be perpendicular to the corneal surface and the eye in primary position, (i.e., staring at the explosion). For the purpose of studying the effects of different angles of propagation, blast waves striking the cornea at ±30° and ±45° angle with the visual axis (being 0°) and along the anatomic axis of the bony orbit (i.e., along the line traced between the geometric center of the anterior aperture of the orbit and the orbit apex) (Fig. 2B) were also modeled. In our model, the orbit axis line was angled −20° on the transverse (horizontal) plane and 10° on the sagittal (vertical) plane. 
Angles were arbitrarily defined as negative if the blast progressed toward the nasal side in a right orbit and positive if the blast wave progressed toward the temporal side. 
Reference Pressures
The human tolerance of blast has been studied extensively, 16 and pressure levels and duration associated with benchmark biological effects: tympanic rupture, lung damage, and 50% mortality (Fig. 3). 
Figure 3. 
 
Benchmark biological effects associated with PBI, plotted as a function of PIO (y-axis) and overpressure duration (x-axis). Data for TNT quantities used in our simulation are reported on the graph for both 1 m and 0.5 m distance from the explosion. It is assumed that the patient is staring at the explosion (i.e., propagation of pressure waves is perpendicular to visual axis).
Figure 3. 
 
Benchmark biological effects associated with PBI, plotted as a function of PIO (y-axis) and overpressure duration (x-axis). Data for TNT quantities used in our simulation are reported on the graph for both 1 m and 0.5 m distance from the explosion. It is assumed that the patient is staring at the explosion (i.e., propagation of pressure waves is perpendicular to visual axis).
We conducted a parametric analysis of TNT quantities necessary to produce the above-mentioned reference biological damage (Fig. 3) when detonating at 1 and 0.5 meters from the eye, in order to simulate inadvertent explosion during manipulation. 
Pressure, strain, strain rates, and volumetric changes as defined 10 elsewhere were calculated at discrete locations known to be clinically relevant: the corneal apex, the vitreous base, the retinal equator, the macula, the posterior projection of the macula on the bony orbit wall (referred to hereinafter as “orbit”), and the orbit apex (Fig. 2B). 
Statistical Analysis
Statistical analysis used paired-sample t-test for pressure values. Significance was set at the 0.05 level. 
Results
Orbit pressures secondary to explosions generating reference biological damage are reported in Figures 4A and 4B for explosions occurring, respectively, at 0.5 m and 1 m from the cornea. Pressures generated at 1 m were significantly higher throughout the simulation at each location considered (paired t-test, P < 0.001). Regardless of the distance from ignition, all locations posterior to the vitreous base (equator, macula, orbit, and orbit apex) suffered pressures significantly higher than those reached at the cornea. All locations showed sinusoidal pressure variation after the initial peak, although only those posterior to the equator shifted toward negative values (traction), as shown in Figures 4A and 4B (movies simulating the explosion of 32 g of TNT at 0.5 m are also available for both transverse and vertical sections; see Supplementary Movies S1 and S2 [movie1.jpg and movie2.jpg], respectively). 
Figure 4. 
 
(A) Pressure graph for explosions occurring 0.5 m from the eye, as a function of time (x-axis), recorded at selected ocular and orbit locations. Overall time span is 0.5 ms. Note that multiple y-axis data sets were plotted, each on a different scale of magnitude referring to, respectively, 2.5 g TNT (tympanic rupture threshold; leftmost y-axis), 32 g TNT (lung damage threshold; middle y-axis), and 177 g TNT (50% mortality threshold; rightmost y-axis). Grey-shaded areas (for each y-axis magnitude level, as described above) represent pressure levels known in the literature to cause retinal and/or choroidal damage (Table 2). Dashed horizontal line at the 0.15-MPa level of the leftmost y-axis refers to the PIO generated by the explosion in air. (B) Pressure graph for explosions occurring 1 m from the eye, as a function of time, recorded at selected ocular and orbit locations. Overall time span is 0.5 ms. Note that multiple y-axis data sets were plotted, each on a different scale of magnitude referring to, respectively, 20 g TNT (tympanic rupture threshold; leftmost y-axis), 250 g TNT (lung damage threshold; middle y-axis), and 1500 g TNT (50% mortality rate; rightmost y-axis). Grey-shaded areas (for each y-axis magnitude level, as described above) represent pressure levels known in the literature to cause retinal and/or choroidal damage (Table 2). Dashed horizontal line at the 0.15-MPa level of the leftmost y-axis refers to the PIO generated by the explosion in air.
Figure 4. 
 
(A) Pressure graph for explosions occurring 0.5 m from the eye, as a function of time (x-axis), recorded at selected ocular and orbit locations. Overall time span is 0.5 ms. Note that multiple y-axis data sets were plotted, each on a different scale of magnitude referring to, respectively, 2.5 g TNT (tympanic rupture threshold; leftmost y-axis), 32 g TNT (lung damage threshold; middle y-axis), and 177 g TNT (50% mortality threshold; rightmost y-axis). Grey-shaded areas (for each y-axis magnitude level, as described above) represent pressure levels known in the literature to cause retinal and/or choroidal damage (Table 2). Dashed horizontal line at the 0.15-MPa level of the leftmost y-axis refers to the PIO generated by the explosion in air. (B) Pressure graph for explosions occurring 1 m from the eye, as a function of time, recorded at selected ocular and orbit locations. Overall time span is 0.5 ms. Note that multiple y-axis data sets were plotted, each on a different scale of magnitude referring to, respectively, 20 g TNT (tympanic rupture threshold; leftmost y-axis), 250 g TNT (lung damage threshold; middle y-axis), and 1500 g TNT (50% mortality rate; rightmost y-axis). Grey-shaded areas (for each y-axis magnitude level, as described above) represent pressure levels known in the literature to cause retinal and/or choroidal damage (Table 2). Dashed horizontal line at the 0.15-MPa level of the leftmost y-axis refers to the PIO generated by the explosion in air.
Table 1. 
 
Review of Retinal and Choroidal Pressure and Strain Values Compatible with Tears and Ruptures
Table 1. 
 
Review of Retinal and Choroidal Pressure and Strain Values Compatible with Tears and Ruptures
Author, y Tissue Young's Modulus Young's Modulus, MPa Stress at RD
Zauberman, 40 1969 Retina 80 mg (retinal strip to detachment)
Zauberman, 41 1972 Retina 150 mg/mm elevation at 10 mm/min speed
Graebel, 42 1977 Choroid 9.98 × 104 Pa tangent modulus (stress 5 × 103 Pa) 0.0998
Wu, 43 1987 Choroid 6.52 × 104 Pa tangent modulus (stress 4.8 × 103 Pa) 0.0652
Wu, 43 1987 Retina 0.46 × 104–0.58 × 104 Pa tangent (stress 1.2 × 103 Pa) 0.0046 to 0.0058
Jones, 44 1992 Retina 2 × 104 Pa 0.02
Wollensak, 45 2004 Retina 1.1 × 105 Pa 0.11 11.3 × 103 Pa at 99 mm/min
Friberg, 46 1988 Choroid 6 × 105 Pa 600
Wollensak, 45 2004 Choroid 106 Pa 100 1730 × 103 Pa at 99 mm/min
Franze, 11 2011 Retina 900 to 1800 Pa 0.0009 to 0.0018
Table 2. 
 
TNT Quantity and PIO Associated with Benchmark Biological Effects
Table 2. 
 
TNT Quantity and PIO Associated with Benchmark Biological Effects
Distance from Explosion 50% Mortality 50% Lung Damage Tympanic Rupture
TNT, g PIO, MPa TNT, g PIO, MPa TNT, g PIO, MPa
0.5 m 177 1.49 32 0.43 2.5 0.15
1 m 1500 1.49 250 0.43 20 0.15
The angle between blast wave progression and orbit orientation significantly influenced the pressure (Fig. 5) at each location tested (paired t-test, P < 0.001). Zero degrees yielded the highest and lowest pressures at all locations except for the orbit apex, where pressure peaked when blast wave progressed exactly along the orbit axis. 
Figure 5. 
 
Peak pressures as a function of incident angle (relative angle between blast wave expansion and orbit orientation). Note that 0° (blast wave expansion along the visual axis; i.e., the patient is staring at the explosion with the eye in primary position) resulted in the highest pressures at all locations except for the orbit apex, where the highest positive pressure was reached only when the blast wave propagated exactly along the bony orbit axis.
Figure 5. 
 
Peak pressures as a function of incident angle (relative angle between blast wave expansion and orbit orientation). Note that 0° (blast wave expansion along the visual axis; i.e., the patient is staring at the explosion with the eye in primary position) resulted in the highest pressures at all locations except for the orbit apex, where the highest positive pressure was reached only when the blast wave propagated exactly along the bony orbit axis.
Multiaxial strain is reported in Figure 6 as the trace of the elastic tensor and reflects the volumetric change imposed by pressure to the orbit's contents. Since stress and strain are proportional, obviously, strain also proved significantly higher for explosions occurring at 1 m than at 0.5 m at each location considered (paired t-test, P < 0.001). Strain proved relevant for each TNT amount considered: even 2.5 g of TNT at 0.5 m resulted in average deformation per each axis of up to −8.4% at the cornea and ranging between −5.8% and +4.6% elsewhere, while 32 g yielded, respectively, −14.4% at the cornea and a range between −10% and +7.9% elsewhere, and 177 grams produced −32% at the cornea and between −17% and +29% elsewhere. 
Figure 6. 
 
(A) Trace of the elastic tensor expressed as the algebraic sum of maximum, medium, and minimum strain versus time, calculated at each location for explosions occurring at 0.5 m from the eye. Overall time span is 0.5 ms. Multiple y-axis data sets refer to scales of magnitude of deformation occurring for 2.5 g TNT (leftmost y-axis), 32 g TNT (middle y-axis), and 177 g TNT (rightmost y-axis). Note that the cornea is subject to the greatest deformation, followed by the macula and orbit apex. Note also that posterior structures only expand to reach positive strain levels, as a result of resonance phenomena. (B) Trace of the elastic tensor expressed as the algebraic sum of maximum, medium, and minimum strain versus time, calculated at each location for explosions occurring at 1 m from the eye. Overall time span is 0.5 ms. Multiple y-axis data sets refer to scales of magnitude of deformation occurring for 2.5 g TNT (leftmost y-axis), 32 g TNT (middle y-axis), and 177 g TNT (rightmost y-axis). Note that the cornea is subject to the greatest deformation, followed by the macula and orbit apex. Note also that posterior structures only expand to reach positive strain levels, as a result of resonance phenomena. An amount of 2.5 g TNT resulted in axial deformation of up to −8.4% at the cornea, ranging between +4.6% and −5.8% elsewhere.
Figure 6. 
 
(A) Trace of the elastic tensor expressed as the algebraic sum of maximum, medium, and minimum strain versus time, calculated at each location for explosions occurring at 0.5 m from the eye. Overall time span is 0.5 ms. Multiple y-axis data sets refer to scales of magnitude of deformation occurring for 2.5 g TNT (leftmost y-axis), 32 g TNT (middle y-axis), and 177 g TNT (rightmost y-axis). Note that the cornea is subject to the greatest deformation, followed by the macula and orbit apex. Note also that posterior structures only expand to reach positive strain levels, as a result of resonance phenomena. (B) Trace of the elastic tensor expressed as the algebraic sum of maximum, medium, and minimum strain versus time, calculated at each location for explosions occurring at 1 m from the eye. Overall time span is 0.5 ms. Multiple y-axis data sets refer to scales of magnitude of deformation occurring for 2.5 g TNT (leftmost y-axis), 32 g TNT (middle y-axis), and 177 g TNT (rightmost y-axis). Note that the cornea is subject to the greatest deformation, followed by the macula and orbit apex. Note also that posterior structures only expand to reach positive strain levels, as a result of resonance phenomena. An amount of 2.5 g TNT resulted in axial deformation of up to −8.4% at the cornea, ranging between +4.6% and −5.8% elsewhere.
Strain rates are reported in Figure 7. The difference between explosions occurring at 0.5 m and those at 1 m from the eye appeared significant (P < 0.01) for the ranges of pressures causing lung damage and 50% mortality, while the difference in pressures causing tympanic rupture did not reach a statistically significant level. 
Figure 7. 
 
Maximum strain rate (s−1) calculated for each location and reference PIO.
Figure 7. 
 
Maximum strain rate (s−1) calculated for each location and reference PIO.
Discussion
Explosion-related eye trauma is predominantly due to flying debris, and “pure” PBI is such an unusual mechanism 17,18 that it has been questioned as a whole. 19,20 Although blast effects on hollow, gas-filled organs have been comprehensively investigated, the consequences for fluid-filled organs remain poorly understood. 
Our model suggests that even the least amount of TNT we considered (2.5 g at 0.5 m) can cause profound harm to the retina, choroid, and optic nerve, since the calculated pressures (Fig. 4) are orders of magnitude higher than published data on the tensile strengths (Tables 1, 2) of those tissues. 
Regardless of magnitude and distance from ignition, pressure showed two distinct patterns: anterior structures such as the cornea and vitreous base (which are noticeably outside of the bony orbit, in our model) quickly reached their peak pressure and quickly decayed, as expected. Locations posterior to the vitreous base (and inside the bony shell of the orbit) behaved differently, with much higher positive pressures alternating with significant negative waves, in a peculiar sinusoidal fashion with a much lower tendency to damping (Fig. 4). 
We believe these results can be explained according to the theory of wave reflection at impedance interfaces 21 : when a denser surface is encountered (e.g., tissue-bone interface), pressure waves reflect, maintaining the same polarity (i.e., a positive pulse reflects as positive), while a “free surface,” where mechanical impedance drops to almost zero (e.g., cornea-air interface), reflects pressure with an inverted sign (i.e., a positive pressure reflects as negative). 
Blast-generated positive pressure waves, therefore, travel the orbit at the speed of sound and rebound on the orbit wall to generate multiple reflected waves of the same (positive) sign. Such waves turn negative when, after travelling the orbit backwards, they reflect onto the free cornea-air interface. This would explain why the incoming positive pressure builds up 5-fold within the orbit and why more-posterior locations experience higher positive and negative pressures (Fig. 4). 
The pyramid-like orbit geometry with homogeneous fluidlike content (eye bulb and retrobulbar fat tissue retain negligible mechanical impedance difference), much denser bony walls, and an air-tissue anterior interface (the cornea-air boundary), represent the ideal environment for amplification, allowing multiple reflections and channeling pressure toward the geometric apex. We hypothesize that pressure wave interference generates two stress waves travelling the orbit in opposite directions simultaneously and resulting in a resonating “steady wave” (see the Appendix and Supplementary Movies S1 and S2, for more details on steady waves). 
This had been already suggested when ocular PBI was initially reported in 1945 22,23 but never tested any further, although even Duke-Elder 24 postulated that both positive and negative pressures contributed to eye damage. 
Interestingly, varying the angle of blast propagation yielded significantly different pressures (Fig. 6), as predicted by Rones and Wilder in 1947, 25 who stated that blast effects depended upon wave direction. When blast propagation occurred along the visual axis (0° angle), pressure peaked at all locations except for the orbit apex, which reached its maximum pressure only when the blast propagated exactly along the orbit axis. This is consistent with our hypothesis of pressure amplification due to orbit shape and impedance mismatch. When the patient is staring at the explosion, in fact (0° angle), all locations tested are directly exposed to the shock front through the anterior orbit aperture, except the apex, because the visual and orbit axis are misaligned. When blast waves travel exactly along the orbit axis, on the contrary, pressures converge toward the geometric apex, thanks to a more favorable reflection path. 
Optic nerve damage in the absence of ocular signs can therefore also be explained by the higher pressures reached at the orbit apex that selectively impair the vascular and nervous structures located in this area. Zuckerman 26 reported retrobulbar hemorrhage after PBI, and Shelah et al. 27 described a dog who suffered blindness after blast exposure. Chalioulias et al. 6 also reported mydriasis in a soldier who was leaning his head on the metallic door pillar of an armored vehicle hit by an explosion. 
All orbit structures suffered intense strain (Fig. 6) and extremely high strain rates (Fig. 7). Strain-related mechanisms of cellular damage have been investigated and include direct neuronal injury, axonal transport interference, 28 hemoglobin-mediated pressure-dependent oxidative stress, 29 and visual pathway degeneration 30 and ischemia. 31  
In summary, we believe PBI can result in different clinical pictures ranging from mild concussion to optic atrophy. It should be emphasized that even a limited amount of explosive can generate high pressures if the ignition point is close enough and the eye points at it, as is often the case when manipulating explosives. 
Pitfalls of the present study reside in the schematization typical of numerical simulation, a particularly delicate process when dealing with biological models. All tissues, in fact, show anisotropic behavior (i.e., respond differently under the same loading conditions applied in different directions 32,33 ), and the constitutive parameters remain matter for debate. Such controversies, although capable of altering numerical results, cannot affect the basic mechanism of wave reflection and the concept of steady wave amplification and resonance. The overall ocular response, moreover, should be considered rather solid since it has been deducted through reverse engineering techniques 10 from an ex vivo ocular model. 
The pressure and strain measures, although subject to refinement, seem reasonably accurate on a larger scale and for the general purpose of introducing a new theory on the pathogenic mechanism of PBI. 
We therefore believe the concept of a resonating steady wave due to the peculiar orbit geometry and impedance mismatch can be validly proposed and is strongly supported by our numerical simulation and by clinical evidence. 
Supplementary Materials
Appendix: Technical Specifications
Finite Element Model
The eye and orbit were meshed with 3-dimensional (3-D) 8-node brick elements. The complete FEM model was achieved by connecting the ocular globe to the orbit and inserting both into a schematic skull modeled as a rigid body weighing 4 kg and meshed with shell elements (Fig. 2). 
Constitutive Models and Mechanical Properties
Constitutive models and mechanical properties are reported in Tables A1 and A2
Figure A1. 
 
Comparison between the FEM result and the proposed formulation at the macula for the case of 32 g TNT at 0.5 m. Continuous line represents pressure at the macula when 32 g TNT explodes at 0.5 m from the eye, and dashed line represents the Equation 4 prediction. Note that the two lines almost overlap, signifying a good accuracy of the proposed equation.
Figure A1. 
 
Comparison between the FEM result and the proposed formulation at the macula for the case of 32 g TNT at 0.5 m. Continuous line represents pressure at the macula when 32 g TNT explodes at 0.5 m from the eye, and dashed line represents the Equation 4 prediction. Note that the two lines almost overlap, signifying a good accuracy of the proposed equation.
Table A1. 
 
Constitutive Model and Mechanical Parameters of Solid Materials
Table A1. 
 
Constitutive Model and Mechanical Parameters of Solid Materials
Material Constitutive Model Young Modulus, MPa Shear Modulus, MPa Poisson Modulus Bulk Modulus, MPa Density, kg/m3
Sclera Linear elastic 29.5 0.454 1200
Cornea Linear EOS + shear modulus 1.66 145 1149
Lens Linear EOS 1550 1100
Retina Linear EOS + shear modulus 0.035 1000 1100
Aqueous Linear EOS 2200 1000
Table A2. 
 
Constitutive Model and Mechanical Parameters of Viscous Materials
Table A2. 
 
Constitutive Model and Mechanical Parameters of Viscous Materials
Material Constitutive Model G 0, MPa G , MPa Viscosity η 0, MPa β Bulk Modulus, MPa Density, kg/m3
Vitreous Linear EOS + viscoelastic linear shear 7.6E-6 4.16E-6 5E-6 0.0538 2000 950
Orbital fat1 Linear EOS + viscoelastic linear shear 9E-4 5E-4 50 2100 918
Detonation Model
Modeling the response to blast relies on accurate descriptions of the blast loading pressure profiles. The explosive-produced blast profile is calculated using detonation modeling of the high-explosive event. The equation of state for the detonation products is the primary modeling description of the work output from the explosive that causes the subsequent air blast. The Jones-Wilkins-Lee (JWL) model was used for the explosive gaseous products. 34,35 The equation of state based on this empirical model is probably the one currently most used for detonation and blast modeling and allows the calculation of high-energy-explosive detonation product pressure as follows:  where e is the specific internal energy per unit mass, ρ 0 is the reference density, ρ is the overall material density, η is the ρ/ρ 0 ratio, p 0 is the initial pressure, and A, B, ω, R 1 and R 2 are constants whose values have been determined from dynamic experiments for many common explosives and are available in the literature. 3639  
The environment surrounding the explosive was assumed to behave as an ideal gas using the gamma law equation of state: where γ is the ratio of specific heats (cp/cv ). 
The effect of a detonation on the environment can be simulated by assuming the detonated material as a sphere of hot gas with a homogeneous density and specific internal energy. This approach is suited for problems in which the processes inside the explosive material are not to be investigated. 
The gas generated by the explosion propagates radially from the ignition point. Assuming spherical symmetry for the charge and the resulting gas expansion, it is possible to reduce the simulation from the 3-D spatial domain to a 1-dimensional (1-D) Eulerian wedge-shaped domain. In this way, the computational time can be also reduced, significantly speeding up the solution. 
The transition from the 1-D domain to the 3-D space of the eye model was obtained by a mapping process. The space surrounding the eye model was meshed in such a way that the fully developed pressure profile can be used as the initial condition for the 3-D domain. 
In the 3-D model, the Lagrangian mesh of the eye and the orbit are comprised into an Eulerian mesh through which the blast wave expands. The general coupling algorithm was used to enable the interaction between the Eulerian and Lagrangian meshes. This is based on the creation of a coupling surface on the Lagrangian structure. Here, forces are calculated and transferred between two solver domains. At the same time, stress in the Eulerian elements generates force, causing deformation of Lagrangian elements. 
Discussion of Steady Wave Equation
Given the generic equation of a steady wave where A is the amplitude, λ is the wavelength, and f is the frequency of the interfering sinusoidal waves, based on the above-described assumptions and on our FEM data, we propose the following steady wave equation for our specific case: where Pi(t) is the incident pressure. 
A comparison between pressure values at the macula (x = 24 mm) (Fig. 5; 32 g/0.5 m) and the equation prediction is shown in Figure A1, where λ was assumed to be 140 mm (orbit length), f approximately 15,120 Hz, and Pi (t) to have the exponential behavior where P d max is the maximum value of the dynamic pressure, related to the motion of the air particles behind the blast front. 
The overlap of FEM data and the Equation 4 prediction in Figure A1 is impressive and suggests that a steady wave formulation can indeed explain pressure behavior within the orbit. 
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Footnotes
 The authors alone are responsible for the content and writing of the paper.
Footnotes
 Disclosure: T. Rossi, None; B. Boccassini, None; L. Esposito, None; C. Clemente, None; M. Iossa, None; L. Placentino, None; N. Bonora, None
Figure 1. 
 
Pure PBI in a patient exposed solely to blast overpressure wave, in the absence of any contact with splinters and/or flying objects. Two months after PIO exposure, subretinal fibrosis is well evident temporal to the fovea, and some subretinal heme is still present. Note that the temporal aspect of the optic disc is pale.
Figure 1. 
 
Pure PBI in a patient exposed solely to blast overpressure wave, in the absence of any contact with splinters and/or flying objects. Two months after PIO exposure, subretinal fibrosis is well evident temporal to the fovea, and some subretinal heme is still present. Note that the temporal aspect of the optic disc is pale.
Figure 2. 
 
(A) Exploded view of the complete FEM: skull + orbit + eyeball. (B) Transverse (horizontal; a) and sagittal (vertical; b) sections of the orbit FEM model. Selected locations are reported: 1, cornea (corneal apex); 2, vitreous base; 3, equator; 4, macula; 5, orbit; 6, orbit apex. Visual axis (continuous line) and orbit axis (dashed line) are reported as well.
Figure 2. 
 
(A) Exploded view of the complete FEM: skull + orbit + eyeball. (B) Transverse (horizontal; a) and sagittal (vertical; b) sections of the orbit FEM model. Selected locations are reported: 1, cornea (corneal apex); 2, vitreous base; 3, equator; 4, macula; 5, orbit; 6, orbit apex. Visual axis (continuous line) and orbit axis (dashed line) are reported as well.
Figure 3. 
 
Benchmark biological effects associated with PBI, plotted as a function of PIO (y-axis) and overpressure duration (x-axis). Data for TNT quantities used in our simulation are reported on the graph for both 1 m and 0.5 m distance from the explosion. It is assumed that the patient is staring at the explosion (i.e., propagation of pressure waves is perpendicular to visual axis).
Figure 3. 
 
Benchmark biological effects associated with PBI, plotted as a function of PIO (y-axis) and overpressure duration (x-axis). Data for TNT quantities used in our simulation are reported on the graph for both 1 m and 0.5 m distance from the explosion. It is assumed that the patient is staring at the explosion (i.e., propagation of pressure waves is perpendicular to visual axis).
Figure 4. 
 
(A) Pressure graph for explosions occurring 0.5 m from the eye, as a function of time (x-axis), recorded at selected ocular and orbit locations. Overall time span is 0.5 ms. Note that multiple y-axis data sets were plotted, each on a different scale of magnitude referring to, respectively, 2.5 g TNT (tympanic rupture threshold; leftmost y-axis), 32 g TNT (lung damage threshold; middle y-axis), and 177 g TNT (50% mortality threshold; rightmost y-axis). Grey-shaded areas (for each y-axis magnitude level, as described above) represent pressure levels known in the literature to cause retinal and/or choroidal damage (Table 2). Dashed horizontal line at the 0.15-MPa level of the leftmost y-axis refers to the PIO generated by the explosion in air. (B) Pressure graph for explosions occurring 1 m from the eye, as a function of time, recorded at selected ocular and orbit locations. Overall time span is 0.5 ms. Note that multiple y-axis data sets were plotted, each on a different scale of magnitude referring to, respectively, 20 g TNT (tympanic rupture threshold; leftmost y-axis), 250 g TNT (lung damage threshold; middle y-axis), and 1500 g TNT (50% mortality rate; rightmost y-axis). Grey-shaded areas (for each y-axis magnitude level, as described above) represent pressure levels known in the literature to cause retinal and/or choroidal damage (Table 2). Dashed horizontal line at the 0.15-MPa level of the leftmost y-axis refers to the PIO generated by the explosion in air.
Figure 4. 
 
(A) Pressure graph for explosions occurring 0.5 m from the eye, as a function of time (x-axis), recorded at selected ocular and orbit locations. Overall time span is 0.5 ms. Note that multiple y-axis data sets were plotted, each on a different scale of magnitude referring to, respectively, 2.5 g TNT (tympanic rupture threshold; leftmost y-axis), 32 g TNT (lung damage threshold; middle y-axis), and 177 g TNT (50% mortality threshold; rightmost y-axis). Grey-shaded areas (for each y-axis magnitude level, as described above) represent pressure levels known in the literature to cause retinal and/or choroidal damage (Table 2). Dashed horizontal line at the 0.15-MPa level of the leftmost y-axis refers to the PIO generated by the explosion in air. (B) Pressure graph for explosions occurring 1 m from the eye, as a function of time, recorded at selected ocular and orbit locations. Overall time span is 0.5 ms. Note that multiple y-axis data sets were plotted, each on a different scale of magnitude referring to, respectively, 20 g TNT (tympanic rupture threshold; leftmost y-axis), 250 g TNT (lung damage threshold; middle y-axis), and 1500 g TNT (50% mortality rate; rightmost y-axis). Grey-shaded areas (for each y-axis magnitude level, as described above) represent pressure levels known in the literature to cause retinal and/or choroidal damage (Table 2). Dashed horizontal line at the 0.15-MPa level of the leftmost y-axis refers to the PIO generated by the explosion in air.
Figure 5. 
 
Peak pressures as a function of incident angle (relative angle between blast wave expansion and orbit orientation). Note that 0° (blast wave expansion along the visual axis; i.e., the patient is staring at the explosion with the eye in primary position) resulted in the highest pressures at all locations except for the orbit apex, where the highest positive pressure was reached only when the blast wave propagated exactly along the bony orbit axis.
Figure 5. 
 
Peak pressures as a function of incident angle (relative angle between blast wave expansion and orbit orientation). Note that 0° (blast wave expansion along the visual axis; i.e., the patient is staring at the explosion with the eye in primary position) resulted in the highest pressures at all locations except for the orbit apex, where the highest positive pressure was reached only when the blast wave propagated exactly along the bony orbit axis.
Figure 6. 
 
(A) Trace of the elastic tensor expressed as the algebraic sum of maximum, medium, and minimum strain versus time, calculated at each location for explosions occurring at 0.5 m from the eye. Overall time span is 0.5 ms. Multiple y-axis data sets refer to scales of magnitude of deformation occurring for 2.5 g TNT (leftmost y-axis), 32 g TNT (middle y-axis), and 177 g TNT (rightmost y-axis). Note that the cornea is subject to the greatest deformation, followed by the macula and orbit apex. Note also that posterior structures only expand to reach positive strain levels, as a result of resonance phenomena. (B) Trace of the elastic tensor expressed as the algebraic sum of maximum, medium, and minimum strain versus time, calculated at each location for explosions occurring at 1 m from the eye. Overall time span is 0.5 ms. Multiple y-axis data sets refer to scales of magnitude of deformation occurring for 2.5 g TNT (leftmost y-axis), 32 g TNT (middle y-axis), and 177 g TNT (rightmost y-axis). Note that the cornea is subject to the greatest deformation, followed by the macula and orbit apex. Note also that posterior structures only expand to reach positive strain levels, as a result of resonance phenomena. An amount of 2.5 g TNT resulted in axial deformation of up to −8.4% at the cornea, ranging between +4.6% and −5.8% elsewhere.
Figure 6. 
 
(A) Trace of the elastic tensor expressed as the algebraic sum of maximum, medium, and minimum strain versus time, calculated at each location for explosions occurring at 0.5 m from the eye. Overall time span is 0.5 ms. Multiple y-axis data sets refer to scales of magnitude of deformation occurring for 2.5 g TNT (leftmost y-axis), 32 g TNT (middle y-axis), and 177 g TNT (rightmost y-axis). Note that the cornea is subject to the greatest deformation, followed by the macula and orbit apex. Note also that posterior structures only expand to reach positive strain levels, as a result of resonance phenomena. (B) Trace of the elastic tensor expressed as the algebraic sum of maximum, medium, and minimum strain versus time, calculated at each location for explosions occurring at 1 m from the eye. Overall time span is 0.5 ms. Multiple y-axis data sets refer to scales of magnitude of deformation occurring for 2.5 g TNT (leftmost y-axis), 32 g TNT (middle y-axis), and 177 g TNT (rightmost y-axis). Note that the cornea is subject to the greatest deformation, followed by the macula and orbit apex. Note also that posterior structures only expand to reach positive strain levels, as a result of resonance phenomena. An amount of 2.5 g TNT resulted in axial deformation of up to −8.4% at the cornea, ranging between +4.6% and −5.8% elsewhere.
Figure 7. 
 
Maximum strain rate (s−1) calculated for each location and reference PIO.
Figure 7. 
 
Maximum strain rate (s−1) calculated for each location and reference PIO.
Figure A1. 
 
Comparison between the FEM result and the proposed formulation at the macula for the case of 32 g TNT at 0.5 m. Continuous line represents pressure at the macula when 32 g TNT explodes at 0.5 m from the eye, and dashed line represents the Equation 4 prediction. Note that the two lines almost overlap, signifying a good accuracy of the proposed equation.
Figure A1. 
 
Comparison between the FEM result and the proposed formulation at the macula for the case of 32 g TNT at 0.5 m. Continuous line represents pressure at the macula when 32 g TNT explodes at 0.5 m from the eye, and dashed line represents the Equation 4 prediction. Note that the two lines almost overlap, signifying a good accuracy of the proposed equation.
Table 1. 
 
Review of Retinal and Choroidal Pressure and Strain Values Compatible with Tears and Ruptures
Table 1. 
 
Review of Retinal and Choroidal Pressure and Strain Values Compatible with Tears and Ruptures
Author, y Tissue Young's Modulus Young's Modulus, MPa Stress at RD
Zauberman, 40 1969 Retina 80 mg (retinal strip to detachment)
Zauberman, 41 1972 Retina 150 mg/mm elevation at 10 mm/min speed
Graebel, 42 1977 Choroid 9.98 × 104 Pa tangent modulus (stress 5 × 103 Pa) 0.0998
Wu, 43 1987 Choroid 6.52 × 104 Pa tangent modulus (stress 4.8 × 103 Pa) 0.0652
Wu, 43 1987 Retina 0.46 × 104–0.58 × 104 Pa tangent (stress 1.2 × 103 Pa) 0.0046 to 0.0058
Jones, 44 1992 Retina 2 × 104 Pa 0.02
Wollensak, 45 2004 Retina 1.1 × 105 Pa 0.11 11.3 × 103 Pa at 99 mm/min
Friberg, 46 1988 Choroid 6 × 105 Pa 600
Wollensak, 45 2004 Choroid 106 Pa 100 1730 × 103 Pa at 99 mm/min
Franze, 11 2011 Retina 900 to 1800 Pa 0.0009 to 0.0018
Table 2. 
 
TNT Quantity and PIO Associated with Benchmark Biological Effects
Table 2. 
 
TNT Quantity and PIO Associated with Benchmark Biological Effects
Distance from Explosion 50% Mortality 50% Lung Damage Tympanic Rupture
TNT, g PIO, MPa TNT, g PIO, MPa TNT, g PIO, MPa
0.5 m 177 1.49 32 0.43 2.5 0.15
1 m 1500 1.49 250 0.43 20 0.15
Table A1. 
 
Constitutive Model and Mechanical Parameters of Solid Materials
Table A1. 
 
Constitutive Model and Mechanical Parameters of Solid Materials
Material Constitutive Model Young Modulus, MPa Shear Modulus, MPa Poisson Modulus Bulk Modulus, MPa Density, kg/m3
Sclera Linear elastic 29.5 0.454 1200
Cornea Linear EOS + shear modulus 1.66 145 1149
Lens Linear EOS 1550 1100
Retina Linear EOS + shear modulus 0.035 1000 1100
Aqueous Linear EOS 2200 1000
Table A2. 
 
Constitutive Model and Mechanical Parameters of Viscous Materials
Table A2. 
 
Constitutive Model and Mechanical Parameters of Viscous Materials
Material Constitutive Model G 0, MPa G , MPa Viscosity η 0, MPa β Bulk Modulus, MPa Density, kg/m3
Vitreous Linear EOS + viscoelastic linear shear 7.6E-6 4.16E-6 5E-6 0.0538 2000 950
Orbital fat1 Linear EOS + viscoelastic linear shear 9E-4 5E-4 50 2100 918
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