Computational Modeling of Ophthalmic Procedures

Abstract

PURPOSE

In order to explore how finite-element calculations can continue to contribute to diverse problems in ophthalmology and vision science, we describe our recent work on modeling the force on the peripheral retina in intravitreal injections and how that force increases with shorter, smaller gauge needles.

We also present a calculation that determines the location and stress on a retinal pigment epithelial (RPE) detachment during an intravitreal injection the possibility that stress induced by the injection can lead to a tear of the RPE.

BACKGROUND

Advanced computational models can provide critical insight into the underlying physics in many surgical procedures, which may not be intuitive.

METHODS

The simulations were implemented using COMSOL Multiphysics. We compared the monkey retinal adhesive force¹ of 18 Pa to the results of this study to quantify the maximum retinal stress that occurs during intravitreal injections.

CONCLUSION

Currently used 30-gauge needles produce stress on the retina during intravitreal injections that is only slightly below the limit that can create retinal tears. As retina specialists attempt to use smaller needles, the risk of complications may increase.

In addition, we find that during an intravitreal injection, the stress on the retina in a PED occurs at the edge of the detachment (found clinically) and the stress is sufficient to tear the retina. These findings may guide physicians in future clinical research.

INTRODUCTION

FINITE ELEMENT CALCULATIONS FOR OPHTHALMOLOGISTS AND VISION SCIENTISTS

It has become important for the well-informed ophthalmologist to understand the fundamental principles of a number of fields, such as genetics, developmental biology, or angiogenesis, and these fields have continued to expand at breathtaking speed. Similarly, engineering, particularly computational modeling, has seen a revolution in recent years. Many devices and products are initially developed and tested in silico prior to the traditional approach of creating a model. This has both reduced the number of models needed and the cost, but also allowed engineers to evaluate designs for unexpected failure modes and unexpected problems. While computational methods such as artificial intelligence and machine learning have received a great deal of attention recently, finite element methods promise to revolutionize the development of new devices and interventions and to improve current treatments by allowing unique insights into physics at different length scales, particularly at the micron and nanoscale. The number of publications found in PubMed under the search query (Eye OR Retina OR Ophthalmology) AND Finite-Element have increased nearly exponentially in recent years (Figure 1).

Figure 1.

The number of publications found in PubMed under the search query (Eye OR Retina OR Ophthalmology) AND Finite-Element.

Figure 1 The exponential growth in the number of publications found in PubMed under the search query (Eye OR Retina OR Ophthalmology) AND Finite-Element.

To help the reader to better understand the work described in this thesis, a brief overview of finite element methods, directed at the practicing ophthalmologist or vision scientist, is included here. The following figures (Figure 2–Figure 5) illustrate the steps taken in time-based finite element calculations.

Figure 2.

Create a drawing, a to-scale model of the system that you would like to model. Consider symmetry, as a problem that can be reduced to a 2-dimensional problem requires much less computing power and is more likely to find a stable solution.

Figure 5.

Starting at time zero, take small, incremental steps in time and, at each time point compute the new position of each point on the grid.

Figure 2. Create a drawing, a to-scale model of the system that you would like to model. It is important to consider symmetry as a problem that, for example, can be reduced to a 2-dimensional problem requires much less computing power and is more likely to find a stable solution. Figure from https://www.comsol.com/model/fluid-structure-interaction-361 downloaded 2021/07/03.

Figure 3. Determine the physical properties of each element in the system (water, neural tissue, sclera, titanium,…). Many finite-element systems contain a database of the physical properties of common materials. Determine the boundary conditions for all boundaries in the system (water flowing over a surface has a no-slip boundary condition while an entry-site for fluid might have a fixed fluid velocity). The relevant ‘physics’ or equations will depend upon the problem being addressed. For example, for an intravitreal injection, the equations of fluid mechanics and structural mechanics must be simultaneously solved in the relevant materials (fluid mechanics in the fluids and structural mechanics in solid materials). Figure from https://www.comsol.com/model/fluid-structure-interaction-361 downloaded 2021/07/03.

Figure 3.

Determine the physical properties of each element in the system (water, neural tissue, sclera, titanium,…). Many finite-element systems contain a database of the physical properties of common materials. Determine the boundary conditions for all boundaries in the system (water flowing over a surface has a no-slip boundary condition while an entry-site for fluid might have a fixed fluid velocity). The relevant ‘physics or equations will depend upon the problem being addressed. For example, for an intravitreal injection, the equations of fluid mechanics and structural mechanics must be simultaneously solved in the relevant materials (fluid mechanics in the fluids and structural mechanics in solid materials).

Figure 4. Divide the model into a grid of points where there is likely to be only a small change between neighboring points. Starting at time zero, take small, incremental steps in time and, at each time point compute the new position of each point on the grid. Figure from https://www.comsol.com/model/fluid-structure-interaction-361 downloaded 2021/07/03.

Figure 4.

Divide the model into a grid of points where there is likely to be only a small change between neighboring points.

Figure 5. Repeat this process to calculate the behavior of the system. Figure from https://www.comsol.com/model/fluid-structure-interaction-361 downloaded 2021/07/03.

Note that this is one form of finite element calculation. We have discussed an example of failure of a fragmatome in this thesis to highlight that, for example, one could measure the response of a system as a function of vibrational frequencies driving a device such as a fragmatome, rather than studying the behavior of the system as a function of time. Clearly, ‘the devil is in the details’ and a strong background in solving partial differential equations is important for obtaining consistent and reliable results.

This thesis has been submitted to fulfill requirements of the American Ophthalmological Society and is a compilation of research conducted over several years. It illustrates the ability to apply computational methods to a diverse array of clinically important problems in ophthalmology. It should be noted that some of the figures included in this thesis are re-published, with permission of the publishers, and are also available at no cost through PubMed because of the National Institutes of Health’s Public Access Policy (https://publicaccess.nih.gov).

COMPUTATIONAL APPROACHES TO PROBLEMS IN OPHTHALMOLOGY

The approach taken in this thesis to the study of intravitreal injections uses a time-based calculation. An alternative approach is to study a device or system as a function of frequency. By characterizing the frequencies (modes or eigenfrequencies) that a device can vibrate at and how different driving forces can drive the device to move in a particular manner, it is possible to study critical topics such as device failure. For example, we have previously published² a study of small-gauge (ie, small diameter) fragmatomes to predict device behavior.

Unexpectedly, there are stresses generated within the titanium fragmatome (Figure 6) sufficient to cause fracture of the fragmatome intra-operatively. This theoretical finding was simultaneously confirmed intraoperatively by a different group³.

Figure 6.

Von Mises stress (left) and volumetric strain (right) for a 30 mm long, 23-gauge hollow fragmatome at a frequency of 45 kHz.

While our current research concentrates upon time-based studies of an intervention (intravitreal injections), critical clinical questions, such as “how does scleral buckling work?”⁴ or, similarly, “why does bilateral help to re-attach the retina in rhegatogenous retinal detachments?”⁵ have been studied using finite element calculations. Unexpectedly, computational modeling resulted in the finding that the Bernoulli effect (the effect allows airplanes to fly) plays a key role in the action of a scleral buckle (Figure 7)^6,4. The critical decision in formulating such an approach is to simplify the computation sufficiently to make it tractable while still not losing the fundamental physics present in the clinical condition.

Figure 7.

Simulated fluid flow from left to right. Note the elevated pressure to the left of the buckle indentation and the decreased pressure (signified by the blue color) to the right of the indentation.

Finally, some problems may be approached more analytically, relying less on computational models. One example of this approach is our study of the use of pneumatic retinopexy and perfluron⁷.

While some problems may be approached in the context of a clinical trial or chart review (for example, comparative effectiveness research) the problems that we address here are more general in nature and provide global insight into clinical problems that can not be obtained through a clinical trial. The disadvantage of such computational approaches is that the threshold where a given finding becomes clinically relevant may depend upon additional parameters not present in the theoretical model. Thus, these approaches complement, rather than compete with clinical trials and biomedical research.

LITERATURE REVIEW

Typically, computational modeling takes a different approach than many other problem-solving techniques used in Ophthalmology and Visual Sciences. A critical aspect of such quantitative research that many reviewers find challenging is that, rather than develop a comprehensive model that explains all possible variations or causes of a problem, such physics-based approaches to understanding attempt to simplify and extract the underlying physical principles. Such simplification, with additional complexities, perhaps included in a later model as a perturbation, allow great insight but is clearly different than the approach normally taken in biomedical research. A review of published work in this area helps to illustrate this issue more completely.

Finite element calculations have been applied to problems in ophthalmology and vision science increasingly, as the computational power necessary to perform these calculations has become more widely available.

A review of the literature on finite element analysis in ophthalmology reveals numerous studies in ocular trauma^{8,9,10,11,12,13,14,15,16,17,18,19}, corneal disease and surgery^{20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41}, cataract surgery^{42,43,44,45,46,47,48,49,50,51,52} retinal disease^{2,53,54,55,56,57,58,59,60,61,6,62,63,} glaucoma^{64,65,66,42,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92} (both the biomechanics of the iris in pigment dispersion and angle closure of the trabecular meshwork in open angle glaucoma, corneal biomechanics and measurement of intraocular pressure), pediatric ophthalmology and eye movements^93,94,95,96, and other applications such as COVID-19 spread at the slit lamp⁹⁷ and instrument and procedure design^{98,99,100,101}. This literature is growing exponentially, illustrating the importance of these methods.

There are a number of poorly understood questions in vitreoretinal surgery that are not amenable to study using animal models or standard biological techniques that we sought to model both here and previously. This thesis provides insight on injection technique in performing intravitreal injections and in addition, the influence of an intravitreal injection on stress in the retina with the goal of evaluating if such stress is sufficient to tear the retina in a pigmental epithelial detachment (PED).

This thesis has also included some results that the authors have been actively involved in developing. A glossary of technical terms (Table 1) is included to assist the non-engineer.

Table 1.

Glossary

Term	Definition
Strain	Change in length divided by length
Stress	Force per unit Area
Compressive Stress	Stress when a material is compressed
Tensile Stress	Stress when a material is placed under tension (pulled or stretched)
Shear Stress	Stress when a material is sheared (force applied parallel to the surface)
Von Mises Stress	A number corresponding to the overall stress
Von Mises yield criterion	A material starts yielding when the von Mises stress reaches the yield stress. There are more mathematical but consistent definitions.
Eigenvalue	A characteristic value, for example, a frequency that a guitar string can vibrate at would be an eigenvalue
Pressure	Force per unit area (note the similarity to Stress)
Differential equation	An equation that relates changes in one quantity to the amount of other physical quantities. For example, changes in the velocity of an object over time (the acceleration) might depend upon the position of that object and this relationship can be written down as an equation that can be solved if additional information about the system (like the initial position and velocity of the object) are known.
Navier-Stokes Equations	A set of commonly used partial differential equations that describes the motion of viscous fluids. It is possible to think of the equations as the fluid-mechanics equivalent of Force = Mass * Acceleration (Newton’s Second Law) and conservation of mass (you do not make or remove fluid).
Finite Element calculation	A form of performing calculations using a computer in which the system being modeled (for example, a fragmatome needle in water) is broken up into a fine grid where the motion of each point in the grid is computed using the equations that describe structural mechanics, fluid mechanics, and also acoustical physics at the same time (ie coupled). The motion of the entire system is calculated over a short period of time and, in this way, the motion of the entire system is predicted. Discussion of technical details (such as the size of the grid or the time step) are carefully determined but are beyond the scope of this manuscript.
Eigenfrequency	The characteristic or resonant frequency at which an item will ‘ring’ or vibrate. Note that the name is from German, where eigen means inherent or characteristic. The term eigenfrequency is, according to the Miriam-Wester dictionary1, “One of the frequencies with which a given oscillatory system is capable of vibrating”. The motion of a given system, such as a fragmatome, can be described by a combinaton of its eigenfrequencies in the same way as a musical sound can be described by the frequencies that make up that sound. Consider that a stick that can oscillate at a given frequency, such that the entire stick moves except the ends; or it can oscillate at a frequency such that there is a point in the center of the stick that also does not move,... Each such frequency at which the stick can oscillate is an eigenfrequency and the stick, when struck, will oscillate at a combination of these eigenfrequencies. Another way to think about this is the equalizer on a stereo system. There is often a screen that displays frequency (along the x-axis) versus power (along the y-axis). This shows that sound (or any function) can be broken down into a number of frequencies. Critically, many frequencies will be quickly damped out and only certain frequencies, the eigenfrequencies, will persist and it is these frequencies that determine the behavior of the system.
Undamped eigenfrequencies	The eigenfrequencies that are present and would be present if there were no dampling from the surrounding fluid.
Damped eigenfrequency	It is possible to calculate how rapidly a particular vibration (eigenfrequency) will loose energy to the surrounding fluid, and thus will have lower amplitude vibration because of how the fragmatome moves. We use this term to denote eigenfrequencies that loose energy.

https://www.merriam-webster.com. Accessed September 6, 2021.

INTRAVITREAL INJECTIONS: THE INFLUENCE OF NEEDLE GAUGE AND INJECTION DEPTH ON RETINAL TRACTION, A FINITE-ELEMENT APPROACH

This chapter illustrates how computational modeling allows the investigator to explore a difficult to measure stress that occurs in a surgical procedure, in particular intravitreal injections. Such studies are important as they may guide future modifications of the procedure while minimizing risk to the patient. Brian William Berg, a medical student at Temple University, collaborated with William J Foster on this research project.

Intravitreal injections of medication directly into the vitreous cavity is a commonly performed procedure to deliver a targeted dose of a drug to the eye. This technique avoids the potential systemic adverse effects of some medications, such as thromboembolic events, myocardial infarction, stroke, hypertension, gastrointestinal perforations, and kidney disease¹⁰². While this technique is considered relatively safe, ophthalmologists must constantly be wary of creating a retinal detachment, a rare but vision-threatening complication¹⁰³. The incidence of retinal detachment has been found to occur in between 0.013% to 0.15% of patients receiving intravitreal injections^104,105,106.

With new monoclonal antibody-based therapeutics being developed each year to treat retinal diseases, the number of intravitreal injections continues to rapidly increase. In 2002, a total of 15,000 intravitreal injections were performed in the United States. In 2012, a total of 2,354,753 injections were performed. It is estimated that, in 2016, the number of injections per year was 5.9 million^107,108 and this growth in injections has continued. This rapid increase in the number of injections raises the absolute risk for retinal detachment, further justifying the need to investigate the optimal technique to minimize retinal traction. While guidelines for intravitreal injections exist, the needle depth and gauge are left to the discretion of the treating ophthalmologist¹⁰⁹, resulting in much debate on these parameters to minimize the risk of retinal detachment. While in vitro experimental injections into animal eyes may provide insight into drug reflex and vitreous leakage¹¹⁰ or the ability to form a single intravitreal bubble of gas with an injection¹¹¹, they provide little insight into the physical mechanisms and processes involved during an injection. Critically, an intra-vitreal injection, for all physically reasonable situations, should be modeled as a high Reynold’s number problem and this negates the relevance of prior analyses that utilized techniques appropriate for low Reynolds number conditions to linearize the Navier-Stokes equation^111,112. As such, our computational model provides insight not available from prior experimental and analytical studies.

Our simulation evaluates the optimal needle gauge and injection depth in order to reduce the risk of retinal detachment. We believe that this model will help provide ophthalmologists with guidelines for intravitreal injections.

METHODS

Because this research neither involved human nor animal subjects, no Institutional Review Board (IRB) or Institutional Animal Care and Use Committee (IACUC) approval was obtained. Likewise, HIPAA compliance and Clinical Trial registration are not relevant to this research. The research was conducted in a manner that adhered to the Declaration of Helsinki and all applicable laws in both the United States of America and Canada.

Our model aims to simulate the in vivo conditions of the vitreous cavity during intravitreal injection of medication. We developed a simplified model of the eye, modeled using COMSOL Multiphysics v4.2a finite element analysis system with the Fluid Structure Interaction module (Stockholm, Sweden), similar to the one previously described¹¹³. A schematic of this model can be seen in Figure 8. Since we are interested in forces in the retina near the needle, a 2D axisymmetric model of the eye was used. The model consists of a semi-circle with a radius of 12mm, to simulate the dimensions of the eye. Next, due to computational fluid dynamics requirements, we added 30 mm of dead space to the end of the semi-circle to create a pressure sink, similar to the remainder of the eye. This assumption can be made since the risk of a retinal tear occurs transiently during injection, and we are only interested in injection-induced acute forces in the anterior retina that can result in retinal tears during injection.

Figure 8.

2D Schematic of the model system used to simulate intravitreal injections. The X and Y axis are in units of mm.

We utilized the three most common needle gauges used in intravitreal injections, with parameters for 25 gauge (0.26 mm inner diameter and 0.1270 mm wall thickness), 27 gauge (0.21 mm inner diameter and 0.1016 mm wall thickness), and 30 gauge (0.159 mm inner diameter and 0.0762 mm wall thickness) needles. Simulating a stainless steel needle, the needle’s density was 4506 kg/m³, the Young’s modulus was 115.7*10⁹ Pa, and the Poisson ratio used was 0.321. The needle was also assumed to be placed perpendicular to the pars plana¹¹⁴.

For the vitreous fluid, a dynamic viscosity of 1 Pa*s was used, which is the average viscosity found in a prior study¹¹⁵. An intraocular pressure of 15.5 mmHg¹¹⁶ was used. An injected volume of 0.05 cc of fluid was used because this is the standard amount of drug given during intravitreal injection. It was assumed that the fluid properties of the vitreous and the injectable fluid were the same. The injection time for this simulation was 0.2 seconds. The velocity of the fluid leaving the needle during injection can be calculated by the volumetric flow rate divided by the cross sectional area. For a 25 gauge needle, a velocity of 4.709 m/s (Reynold’s Number: 59.8) was found while, for a 27 gauge needle, a velocity of 7.218 m/s (Reynold’s Number: 91.7) was found and, for a 30 gauge needle, a velocity of 10.591 m/s (Reynold’s Number: 134.5) was found. The Reynolds Number, the ratio of inertial to viscous forces, is mentioned because it characterizes the system in that it demonstrates that the flow in the system is ‘smooth’/laminar (versus turbulent).

The simulations were implemented using COMSOL Multiphysics v4.2 with the Fluid Structure Interaction module to facilitate modeling fluid and structural equations. Finite element calculations were performed by simultaneously solving the equations for fluid mechanics and fluid-structure interaction with the retina and the needle. At the fluid-solid boundaries of the retina and the needle, a standard no-slip assumption was made, meaning fluid will have zero velocity relative to the boundary. By using COMSOL Multiphysics v4.2 to solve these equations simultaneously, a point of maximum stress on the retina could be found. A maximum element mesh size of 0.2 mm and a minimum element mesh size of 0.00135 mm were used. This allows us to see the physical properties of the vitreous cavity before, during, and after injection of a fluid.

Needle injection depths were chosen assuming a standard one-half inch long needle. For each needle gauge used, injection depths of 12.7 mm, 10.7 mm, 8.7 mm, and 6.7 mm were used. An injection depth of 12.7 mm correlates with the full length of the one-half inch needle being fully placed into the vitreous cavity, while the 6.7mm represents approximately 50% of the needle being placed in the vitreous cavity.

Previous studies done on the retinal adhesive forces in living rabbit, cat, and monkey eyes by Kita and Marmor¹ were used in this study to evaluate the relative magnitude of the forces developed during intravitreal injections. Because there has not been any human studies on the retinal adhesive force, using monkey eyes was chosen as the best surrogate marker. We compared the monkey retinal adhesive force¹ of 18 Pa to the results of this study to quantify the maximum retinal stress that occurred during intravitreal injections.

RESULTS

Finite-element calculations of the vitreous and structural components of the eye were performed during the entire duration of the intravitreal injection. During intravitreal injections, the maximal force on the retina for each of the needle gauges was found.

The highest retinal traction forces occur approximately 2mm from the needle insertion site along the arc length of the retina for all three needle gauges (Figure 9, Figure 10, Figure 11). This was true regardless of the injection depth of the needle, making the location of maximal retinal stress independent of injection of needle gauge and injection depth. In addition, the greatest retinal stress was found when the needle was only shallowly inserted into the eye, so that the end of the needle was closest to the retina. When comparing the different needle gauges for maximal retinal stress, it was found that the 30 gauge needle produced the highest stress on the retina, while the 25 gauge needle produced the least stress on the retina. This held true for all of the injection depths as well.

Figure 9.

Retinal shear stress along the arc length of the retina adjacent to the 25-gauge needle at four different injection depths at injection time of 0.2 seconds.

Figure 10.

Retinal shear stress along the arc length of the retina adjacent to the 27-gauge needle at four different injection depths at injection time of 0.2 seconds.

Figure 11.

Retinal shear stress along the arc length of the retina adjacent to the 30-gauge needle at four different injection depths at injection time of 0.2 seconds.

With a 25 gauge needle, the maximal retinal stress of 9 Pa occurs approximately 2 mm from the injection site at an injection depth of 6.7 mm (approximately 50% of a half-inch needle inserted into the vitreous cavity). The least amount of retinal stress of 4.6 Pa occurs approximately 2 mm from the injection site at an injection depth of 12.7 mm (fully inserted half-inch needle).

With a 27 gauge needle, the maximal retinal stress of 13 Pa occurs approximately 2 mm from the injection site at an injection depth of 6.7 mm (approximately 50% of a half-inch needle inserted into the vitreous cavity). The least amount of retinal stress of 6.8 Pa occurs approximately 2 mm from the needle at an injection depth of 12.7 mm (fully inserted half-inch needle).

With a 30 gauge needle, the maximal retinal stress of 15 Pa occurs approximately 2 mm adjacent to the injection site at an injection depth of 6.7 mm (approximately 50% of a half-inch needle inserted into the vitreous cavity). The least amount of retinal stress of 8 Pa occurs approximately 2 mm from the needle occurs at an injection depth of 12.7 mm (fully inserted half-inch needle).

Comparing Figure 12 A and B, the intravitreal injection forces can be visualized three-dimensionally for a 30 gauge needle, with the only difference between the figures being the depth of needle insertion. Figure 12A shows a higher stress applied to the retina, as opposed to Figure 12B, which shows a lower amount of stress applied to the retina. The stress is distributed across the retina more evenly in Figure 12B, when compared to Figure 12A.

Figure 12.

Calculated pressure gradient at an infusion length of (A) 6.7 mm and (B) 12.7mm with a 30-gauge needle. This figure is a three-dimensional representation of the data shown in figure 4.

Even with the maximal retinal shear stress of 15 Pa occurring with a 30 gauge needle at an injection depth of 6.7 mm, the force induced by the injection was just below the retinal adhesive force¹ of 18 Pa in monkey eyes. For all of these injections, the injection-induced retinal stress is within an order of magnitude of the retinal adhesive forces in monkey eyes.

Critically, the maximal retinal stress increases linearly with needle gauge (Figure 13).

Figure 13.

Calculated maximal retinal stress as a function of needle gauge.

DISCUSSION

In this study, we have created and applied a computational model for intravitreal injections that allows us to have a better understanding of the forces created during injection. Our results provide insight about the stress placed on the retina during injections with varying parameters of needle gauge and depth. Based upon our calculations, the use of larger needles (smaller gauge) and placing the end of the needle deeper into the eye were both found to decrease the stress on the adjacent retina.

We believe that the results of this study make physical sense. With the vitreous cavity containing a very viscous fluid of 1 Pa*s (similar to that of motor oil), it can be thought that any force that is applied to this cavity will cause tugging on the adjacent retina. During an intravitreal injection, the velocity of the fluid is anywhere from 4.709 m/s to 10.591 m/s, depending on the gauge of the needle. The injected fluid will impart its momentum to the viscous vitreous which will, in turn, apply a strain perpendicular to the retina.

We can also appreciate that the further the needle is inserted into the vitreous cavity, the lower the amount of stress that will be applied to the retina. This makes physical sense. It can be seen that when the needle is only 6.7 mm into the vitreous cavity, the needle bevel is closer to the retina. Physically, this means that in any disturbance applied to the vitreous (an injection, for example), the tugging of the vitreous and subsequent strain on the retina is more pronounced than if the needle was inserted deeper into the eye and thus further from the retina.

Based on our calculations, larger-bore (smaller gauge) needles, requiring slower fluid velocities, and placing the needle deeper into the globe each reduce retinal traction. For example, in the range of parameters that we have studied, by using a (larger diameter) 25 gauge needle and inserting the needle 12.7mm in (one-half inch, the length of the needle) into the eye, the least amount of force is applied to the retina.

These results become critical as devices utilizing smaller and shorter needles are developed to attempt to automate the task of performing intravitreal injections. These seemingly logical changes to miniaturize the needle, may result in greater traction on the retina and an increased risk of retinal detachment. For example, if an injection were performed on a patient with a 32 gauge needle, as is currently done in some practices using widely available needles, one can extrapolate the current data to predict a stress of 17.8 Pa, similar to the stress needed to tear the retina and the stress would clearly exceed this limit for smaller diameter needles. Even though the percentile risk of retinal detachment is relatively low, as devices for semi-automated delivery of drugs to the vitreous cavity are developed¹¹⁷, non-retina specialists perform intravitreal injections¹¹⁸, and the number of intravitreal injections continues to increase, the risk for an increased absolute number of cases of retinal detachment will continue to increase. It is thus critical to provide engineers and surgeons with insight into how to most safely and atraumatically deliver intravitreal medications.

While our computational modeling allowed us to account for the retinal shear forces, a limitation of the model is that we did not account for mechanical force during insertion of the needle or for reflux that occurs during injection¹¹⁰. In addition, no needle bevel was used. While these parameters may seem to be irrelevant, they may also provide a mechanism to apply extra stress on the retina, increasing the risk of detachment.

The vitreous is known to change in viscosity as a function of both patient age and location within the vitreous cavity. Changes in the vitreous can be corrected for in these calculations by considering the Reynolds number of the system (the ratio of inertial forces to viscous forces or density*velocity*diameter/viscosity). If the viscosity of the vitreous is decreased, as often occurs in an older patient, the resulting physical problem is equivalent to increasing the diameter of the needle. This allows our calculations to be relevant to changes in both needle dimeter and fluid viscosity.

Finally, many biological fluids, including human vitreous humour are non-Newtonian fluids. The viscosity varies with fluid shear (think silly putty which is very elastic and bounces at high shear rates while it can be molded with your fingers at low shear rate). There exists a number of mathematical models that attempt to take into account such changes in viscosity with shear, for example the Carreau viscosity model¹¹⁹. We evaluated a number of such models but found that, even in a fairly symmetric system such as we are studying, each model led to inconsistencies and singularities in the calculations that were not physically reasonable. We have discussed this issue with a number of engineering colleagues and have found that this area of fluid mechanics research is currently best used for simple flow geometries. We don’t believe that this is a problem in the current calculations as the shear stress is greater than 1 Pa and it is known¹²⁰ that the viscosity of vitreous drops precipitously at a shear of 0.1 Pa and then remains stable at higher shear.

In summary, we believe that the findings of this study may aid physicians and engineers in determining the gauge and the depth of the needle in patients undergoing conventional or semi-automated intravitreal delivery of medications. Critically, while we are not yet at the point where needle gauge influences the risk of retinal detachment, as injection technologies progress, these considerations may become increasingly important.

SIMULATED STRESS ON A PIGMENT EPITHELIAL DETACHMENT DURING AN INTRAVITREAL INJECTION

This chapter illustrates further how computational modeling allows the investigator to explore a difficult to measure stress, stress in a retinal pigment epithelial detachment (PED) that occurs in a surgical procedure, intravitreal injections. Such studies are important as they may guide future modifications of the procedure while minimizing risk to the patient. This study arose out of a conversation between William J. Foster and Drs Amir Hadayer and Shlomit Schaal. William J. Foster worked with Steven N. Luminais, a medical student at Temple University, to conduct the calculations for this original research project.

A Retinal pigment epithelium (RPE) tear is a well-recognized complication of age-related macular degeneration (AMD), with a prevalence of 10–12% in natural-history reports.^121,122,123 RPE tears can also complicate other chorioretinal disorders including traumatic chorioretinopathy, pathologic myopia, angioid streaks, choroidal tumors, central serous chorioretinopathy and polypoidal choroidal vasculopathy.¹²⁴ The tear usually starts at the edges of a PED in the temporal aspect of the macula, where the RPE retracts, either in a folded fashion or rolls back, taking a semi-circular form.^122,125,126 This condition can be easily diagnosed clinically, with fluorescein angiography (FA) or optical coherence tomography OCT. While this condition can occur spontaneously, it is often triggered by a procedure, usually an intravitreal injection. Soon after the introduction of anti-vascular endothelial growth factor (anti-VEGF) biologics, reports appeared to suggest an increased risk of PED tears related to intravitreal injections.^{124,126,127,128,129,130,131,132} The reported prevalence of PED tear after intravitreal injection is 14–20%.^{129,133,134,135}

The mechanisms of this pathology and how intraocular procedures may trigger an RPE tear are not yet fully understood and are likely multifactorial. Previous publications suggested a few theoretical explanations including increased choroidal hydrostatic pressure, contraction of neovascular tissue after treatment and globe deformation during insertion of the needle into the eye. A few risk factors have been identified for RPE tears associated with intravitreal injections: PED height, surface area, diameter, presence of subretinal fluid and RPE folds.^{124,125,132,136}

Interestingly, very little is found in the literature regarding the temporal evolution of the RPE tear, specifically, the time from the triggering event (or injection) to the beginning of the RPE tear, and whether it is an acute event or a slowly progressing process.

The purpose of this study is to investigate the biomechanical forces which occur in a PED in the settings of an intravitreal injection, with the goal of obtaining a better understanding of this blinding pathology and suggesting strategies to reduce its risk. As there is no realistic animal model of a PED, animal and human studies of the acute changes in a PED after an intraviteal injection are not feasible.

METHODS

A serous PED between Bruch’s membrane and the retina was modeled using COMSOL Multiphysics v5.2a finite element analysis system with the Fluid Structure Interaction module (Stockholm, Sweden) as shown in Figure 14. A 3-dimensional, axisymmetric, spherical model of the posterior eye was developed. The model included the vitreous humor modeled as a viscoelastic fluid, a partially spherical PED with a 500 µm radius with serous fluid within the PED, and 3 layers for the sclera, Bruch’s membrane, and the retina pigmented epithelium (RPE). Because the choriocapillaris is much ‘softer’ than the sclera, the mechanical properties of the combined choriocapillaris and sclera is modeled by a scleral layer. Physical constants and dimensions can be found in Table 2. The model was centered on the PED and limited to 30 degrees radially for efficient computation.

Figure 14.

A schematic showing the sclera, Bruchs membrane, RPE (2), serous PED (3), and vitreous humor. The model is axisymmetric about r = 0. Distance is in micrometers.

Table 2.

Dimensions and physical constants used in the simulation.

Tissue	Dimension or Physical Constant	Value
Eye	Outer Radius Intraocular pressure	12 mm 17 mmHg
Sclera	Thickness Young’s Modulus Poisson Ratio	750 µm (Ref 132) 2 MPa (Ref 12122) 0.45 (Ref 12122)
Bruchs Membrane	Thickness Young’s Modulus Poisson Ratio	20 µm 10 kPa 0.45 (Ref 12122)
RPE	Thickness Young’s Modulus Density	20 µm (Ref 133) 1 kPa 1.05 g/cm3 (Ref 12122)
PED	Radius Material	500 µm Water (included in COMSOL)

A 0.05 mL intravitreal injection was simulated by calculating the increase in the circumference of the eye, given the volume of injection, then moving the simulated sclera, Bruch’s membrane, and RPE radially outward at a constant velocity for the duration of the simulated injection. Injection durations of 0.25 s, 0.5 s, 0.75 s, 1.0 s, and 2.5 s were simulated and the injection started 0.01 seconds after the simulation started, to avoid artifacts. The simulation was continued until the simulated von Meiss stress in the RPE reached a steady state. The simulation used an extremely fine, physics controlled mesh generated by COMSOL. The additional von Meiss stress on the RPE was calculated at every time point and multiple virtual stress probes were placed along the RPE spanning the junction of the PED and non-detached RPE.

Because this research neither involved human nor animal subjects, no Institutional Review Board (IRB) or Institutional Animal Care and Use Committee (IACUC) approval was obtained. Likewise, HIPAA compliance and Clinical Trial registration are not relevant to this research. The research was conducted in a manner that adhered to the Declaration of Helsinki and all applicable laws in both the United States of America and Canada.

RESULTS

The maximal computed stress present within any part of the entire RPE is shown in Figure 15 and Table 3 while the maximum stress present within any part of the entire RPE at each time point of the simulation is shown in Figure 16. As may be expected for a membrane, there are transient periods of oscillating stresses on the RPE, most prominent at the start and end of the simulated injection (Figure 17 and Figure 18, respectively) with an approximately linear increase in stress between the oscillations. The maximum stress was found near the junction between the PED and the normal retina tissue for all injection durations. The maximal RPE stress was found near the start of the injection for injection durations 0.25, 0.5 and 1.0 s (80.7, 40.6 and 21.5 N/m² respectively) while the post injection oscillation produced the largest stress for injection durations 0.75 and 2.5 s (28.2 and 6.7 N/m² respectively). Injection durations longer than 2.5 s did not differ significantly from the results of 2.5 s.

Figure 15.

Maximal RPE stress as a function of injection duration.

Table 3.

Maximal stress within any part of the entire RPE during the simulated injection.

Injection Duration (sec)	Maximal RPE Stress (N/m2)
0.25	80.7
0.5	40.6
0.75	28.2
1.0	21.5
2.5	6.7

Figure 16.

Computed maximal von Mises stress within any part of the RPE.

Figure 17.

Maximal computed von Mises stress within any part of the RPE. The start of the injection is at t = 0.

Figure 18.

Maximal computed von Mises stress present within any part of the RPE. The end of the injection is at t = 0.

DISCUSSION

A PED tear is a serious condition with possibly devastating visual consequences. Clinical information is available about the natural history of this condition, but little is known about its etiology or pathogenesis. Unfortunately, to-date, no specific treatment can be offered to repair or improve vision in PED tears after they have occurred. Anti-VEGF agents have been used to prevent further progression of the underlying disease, with varying results.^124,125,127 RPE tears have been clinically associated with prior anti-VEGF injections, however the time sequence and the mechanism for such tears has not been elucidated.

We hypothesized that a PED tear can be, in some cases, induced by a purely mechanical phenomenon. To further investigate this hypothesis, a computer model of a PED has been designed to quantify the effect of an intravitreal injection on the stresses that are imposed on the RPE. We report that the point of maximum stress during a standard 0.05 ml intravitreal injection is located at the junction of the PED with the attached RPE. The point of maximum stress is independent of the speed of injection. The stress imposed on the RPE is 10 times greater when the injection duration is 0.25 seconds compared with injection duration of 1 second or longer.

Our results suggest that the following sequence of events that may result in an RPE tear: 1) a PED exists 2) following an intravitreal injection, the vitreous volume acutely expands, the sclera is stretched, and the stress in the RPE reaches a critical level 3) the RPE breaks abruptly at the junction of the PED and the normal retina, and 4) the RPE contracts or rolls up.

It is our hope that this thesis encourages other colleagues to publish their experience in managing PEDs and, in particular, their observations regarding the timing of a RPE tear with respect to the injection of an anti-VEGF compound. Such clinical observations would be helpful in evaluating this mechanical mechanism as a cause of RPE tears.

Given the results of our mathematical model of the stresses imposed on a RPE during an intravitreal injection, we recommend that intravitreal injections should be administered slowly, especially where a large PED exists. Note that, as injection time increases, the maximal stress on the PED reaches a lower limit. Smaller injection volumes may also be beneficial in avoiding PED tears.

In addition, an anterior chamber paracentesis prior to the injection can be considered.

Critically, as investigators seek to make smaller and “less invasive” devices to perform intravitreal injections, they may increase the risk of retinal detachment. The use of a device that provides a shorter injection time for the same volume of fluid is predicted to produce greater stress on the retina.

This thesis, to the best of our knowledge, is the first manuscript to utilize computational methods to investigate PED tears related to intravitreal injections and to suggest a probable time sequence for the evolution of the RPE tear. Our results support a mechanical hypothesis as one of many possible etiologies for the formation of a PED tear, and we propose practical ways that may reduce the risk of PED tears related to intravitreal injections.