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 detachment during an intravitreal injection, the possibility that stress induced by the injection can lead to a tear of the retinal pigment epithelium.
Advanced computational models can provide a critical insight into the underlying physics in many surgical procedures, which may not be intuitive.
The simulations were implemented using COMSOL Multiphysics. We compared the monkey retinal adhesive force of 18 Pa with the results of this study to quantify the maximum retinal stress that occurs during intravitreal injections.
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 pigment epithelial detachment 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. NOTE: Publication of this article is sponsored by the American Ophthalmological Society.
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 before the traditional approach of creating a model. This has reduced both the number of models needed and the cost, but also allowed engineers to evaluate designs for unexpected failure modes and unexpected problems. Although 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 ).
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. Figures 2 to 5 illustrate the steps taken in time-based finite-element calculations.
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 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 the 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 republished, 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 ( 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 published 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 intraoperatively. This theoretical finding was simultaneously confirmed intraoperatively by a different group.
Although our current research concentrates on time-based studies of an intervention (intravitreal injections), critical clinical questions, such as “how does scleral buckling work?” or, similarly, “why does bilateral patching help to reattach the retina in rhegmatogenous retinal detachments?” have been studied using finite-element calculations. Unexpectedly, computational modeling resulted in the finding that the Bernoulli effect (the effect that allows airplanes to fly) plays a key role in the action of a scleral buckle ( Figure 7 ). , 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.
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 perfluron.
Although some problems may be approached in the context of a clinical trial or chart review (eg, comparative effectiveness research), the problems that we address here are more general in nature and provide a global insight into clinical problems that cannot 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 on additional parameters not present in the theoretical model. Thus, these approaches complement rather than compete with clinical trials and biomedical research.
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, allows a 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, corneal disease and surgery, cataract surgery retinal disease, , , glaucoma , (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, and other applications such as COVID-19 spread at the slitlamp and instrument and procedure design. 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 an insight into the 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 pigment 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.
|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 on 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 describe 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 (eg, 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) is carefully determined but is beyond the scope of this article|
|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 Merriam-Webster dictionary, “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 combination 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, etc. 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) vs 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 damping from the surrounding fluid|
|Damped eigenfrequency||It is possible to calculate how rapidly a particular vibration (eigenfrequency) will lose 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 lose energy|
INTRAVITREAL INJECTIONS: THE INFLUENCE OF NEEDLE GAUGE AND INJECTION DEPTH ON RETINAL TRACTION, A FINITE-ELEMENT APPROACH
This section 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 are 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 disease. Although this technique is considered relatively safe, ophthalmologists must constantly be wary of creating a retinal detachment, a rare but vision-threatening complication. The incidence of retinal detachment has been found to occur in between 0.013% and 0.15% of patients receiving intravitreal injections.
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 , 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. Although guidelines for intravitreal injections exist, the needle depth and gauge are left to the discretion of the treating ophthalmologist, resulting in much debate on these parameters to minimize the risk of retinal detachment. Although in vitro experimental injections into animal eyes may provide an insight into drug reflex and vitreous leakage or the ability to form a single intravitreal bubble of gas with an injection, they provide a little insight into the physical mechanisms and processes involved during an injection. Critically, an intravitreal injection, for all physically reasonable situations, should be modeled as a high Reynolds number problem, and this negates the relevance of prior analyses that used techniques appropriate for low Reynolds number conditions to linearize the Navier-Stokes equation. , As such, our computational model provides an insight not available from prior experimental and analytical studies.
Our simulation evaluates the optimal needle gauge and injection depth to reduce the risk of retinal detachment. We believe that this model will help provide ophthalmologists with guidelines for intravitreal injections.
Because this research neither involved human nor animal subjects, no Institutional Review Board or Institutional Animal Care and Use Committee approval was obtained. Likewise, Health Insurance Portability and Accountability Act 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 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 the COMSOL Multiphysics v4.2a finite- element analysis system with the Fluid Structure Interaction module (Stockholm, Sweden), similar to the one previously described. A schematic of this model can be seen in Figure 8 . Because 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 semicircle with a radius of 12 mm, to simulate the dimensions of the eye. Next, because of computational fluid dynamics requirements, we added 30 mm of dead space to the end of the semicircle to create a pressure sink, similar to the remainder of the eye. This assumption can be made because 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.
We used the 3 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/m 3 , the Young modulus was 115.7 × 10 9 Pa, and the Poisson ratio used was 0.321.The needle was also assumed to be placed perpendicular to the pars plana.
For the vitreous fluid, a dynamic viscosity of 1 Pa s was used, which is the average viscosity found in a prior study. An intraocular pressure of 15.5 mm Hg 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 (Reynolds number: 59.8) was found, whereas for a 27-gauge needle, a velocity of 7.218 m/s (Reynolds number: 91.7) was found and for a 30-gauge needle, a velocity of 10.591 m/s (Reynolds 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 (vs 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, 10.7, 8.7, 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, whereas the 6.7 mm represents approximately 50% of the needle being placed in the vitreous cavity.
Previous studies performed on the retinal adhesive forces in living rabbit, cat, and monkey eyes by Kita and Marmor were used in this study to evaluate the relative magnitude of the forces developed during intravitreal injections. Because there have 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 force of 18 Pa with the results of this study to quantify the maximum retinal stress that occurred during intravitreal injections.
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 2 mm from the needle insertion site along the arc length of the retina for all 3 needle gauges ( Figures 9 to 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, whereas the 25-gauge needle produced the least stress on the retina. This held true for all of the injection depths as well.
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 Figures 12 A and B, the intravitreal injection forces can be visualized 3-dimensionally for a 30-gauge needle, with the only difference between the figures being the depth of needle insertion. Figure 12 A shows a higher stress applied to the retina, as opposed to Figure 12 B, which shows a lower amount of stress applied to the retina. The stress is distributed across the retina more evenly in Figure 12 B, when compared with Figure 12 A.