D36 No. 42. ASME IMECE ? A Zonular Curvi-Linear Coordinate System for A Human Eye

??Advancing Ocular Biomechanics with Precision: The Zonular Coordinate System

By: James O'Flanagan, MS, FRSA

Author's Note:

Happy Halloween everybody!

This article serves as a continuation of Deck 36, No. 41: ASME IMECE ? Using Generative AI to Set Up & Run Virtual Experiments, which explored AI-driven innovations in DOE (Design of Experiments) and FEA (Finite Element Analysis). This paper was also accepted by ASME IMECE, for which we are sincerely grateful.??

Here, we focus on a novel biomechanical model, the Zonular Coordinate System (ZCS), developed specifically for simulating the human eye. We propose the integration of Generative AI and FEA as complementary tools to refine and optimize the ZCS for future research. Drawing on my experience in tire analysis, FEA, and ocular geometry from my time at an ophthalmic startup, we will also compare the ZCS to similar systems in other engineering fields.

Article Summary:

• Author's Note: Introduces the Zonular Coordinate System (ZCS) and its connection to prior research in Generative AI and FEA, along with insights from the author's experience in tire analysis and ocular modeling.
• Overview: The article details the development of the ZCS, compares it to systems in other engineering fields, and explains how Generative AI and FEA work together to enhance the model for biomechanical simulations.
• Introduction: The Problem with Traditional Ocular Models: Traditional models using Cartesian or spherical coordinates oversimplify the eye’s geometry, leading to inaccuracies in force calculations, intraocular pressure miscalculations, and surgical planning.
• Zonular Coordinate System (ZCS): A Novel Solution: The ZCS introduces a curvilinear system that accurately represents the eye’s complex geometry, leading to better simulations of intraocular forces and improved surgical outcomes.
• Euclidean Geometry in ZCS: The ZCS begins with Euclidean geometry, transforming linear axes to better represent the curved, non-linear structures of the eye, which allows for more accurate biomechanical modeling.
• Experience with Tire Geometry, FEA, and Ocular Modeling: The author’s experience in tire analysis and FEA provided the foundation for developing the ZCS, applying principles from these fields to model the eye’s biomechanics.
• Comparison with Plyline-Based Tire Geometry and FEA: The ZCS is compared to plyline-based tire geometry and FEA transformations, both of which handle complex, curved geometries, enhancing simulation accuracy for biomechanical applications.
• Role of Generative AI and FEA in ZCS: Generative AI is used to explore various ZCS configurations, while FEA provides physics-based validation, ensuring precise and accurate biomechanical simulations.
• Methodology: AI and FEA for ZCS Refinement: Generative AI identifies optimal configurations for the ZCS, and FEA simulates and validates these models to optimize the system for medical applications, particularly in surgical planning and glaucoma treatment.
• Applying the Scientific Method: The development of the ZCS follows the scientific method, moving through stages of observation, hypothesis, experimentation, and analysis to refine the model for accurate biomechanical simulations.
• Conclusion: Advancing ZCS with AI and FEA: The combination of Generative AI and FEA results in more accurate and efficient ZCS models, with applications ranging from lens implants to glaucoma treatment, improving medical practices and patient outcomes.
• Abstract Submitted to ASME: The paper, focusing on advancements in ocular biomechanics through the ZCS, has been submitted for presentation at the ASME 2024 IMECE conference.
• References: A section highlighting the contributions in engineering and biotech, particularly innovations in biomechanical systems, and suggesting areas for further improvement in eye modeling.

Article Playlist:

• The Marshall Tucker Band, "Can't You See." (1973).

Music by: The Marshall Tucker Band, "Can't You See." (1973).

Link Reference: https://vimeo.com/1021532787

Introduction: The Problem with Traditional Ocular Models

Traditional models used in ocular biomechanics often rely on Cartesian or spherical coordinate systems to simulate the behavior of the human eye. While these systems are mathematically convenient, they significantly oversimplify the geometry of the eye, particularly when it comes to complex, non-linear structures like the zonular fibers. This leads to several key issues:

1. Failure to Capture Non-Linear Geometries

The human eye, particularly its zonular system, exhibits a complex, curved geometry that cannot be accurately represented by Cartesian or spherical coordinates. The zonular fibers, which support the lens and regulate its movement, follow a naturally curved, non-linear path. Traditional models that assume straight-line coordinates or uniform spherical geometry fail to account for these intricacies. As a result:

• Inaccurate Force Calculations: The forces acting along the zonular fibers are distributed in a way that cannot be accurately calculated using simplified geometries. This leads to errors in predicting how the lens moves in response to changes in intraocular pressure.
• Misrepresentation of Stress and Strain: The irregular geometry of the eye causes uneven distribution of stresses and strains, which is crucial for understanding conditions like glaucoma or how surgical interventions will affect the eye.

• Other 3D Models:

Link Reference: https://www.merckmanuals.com/professional/multimedia/3dmodel/eye-anterior-and-posterior-chambers

2. Intraocular Pressure (IOP) Miscalculations

One of the most significant factors in ocular health is the regulation of intraocular pressure (IOP), which affects the shape and function of the eye’s lens. Traditional models based on spherical assumptions oversimplify the behavior of IOP, leading to flawed simulations. Inaccurate modeling of IOP has direct consequences for:

• Glaucoma Treatment: Incorrect simulations of IOP can lead to improper assessments of glaucoma risk and treatment efficacy.
• Lens Behavior During Surgery: Intraocular pressure also plays a critical role in cataract surgery and other ocular interventions, where precise control is needed to ensure successful outcomes.

Link Reference: https://my.clevelandclinic.org/health/symptoms/24552-eye-intraocular-pressure

3. Challenges in Surgical Planning

Modern ocular surgeries, such as cataract removal, lens implants, and refractive surgeries, require extremely precise simulations to predict how the eye will respond to interventions. Traditional models, however, are limited by their inability to account for the non-symmetric, non-linear nature of the eye. This leads to:

• Limited Surgical Precision: Surgeons must often rely on less-than-perfect models to guide procedures, leading to outcomes that could be improved with more accurate simulations.
• Post-Surgical Complications: Inaccurate simulations of how the eye will respond to surgical interventions can result in unforeseen complications, such as lens dislocation or improper healing, which could have been mitigated with a better understanding of ocular forces.

Link Reference: https://crstoday.com/articles/mar-2021/preoperative-surgical-planning

4. Limited Application of Newtonian Mechanics

Newtonian mechanics, which describes how forces result in motion and deformation, relies heavily on an accurate understanding of an object's geometry. In traditional ocular models, the simplified geometry makes it difficult to apply Newton’s laws effectively. Without an accurate coordinate system:

• Force and Motion Calculations Are Less Reliable: The simplified geometric assumptions lead to unreliable calculations of how forces like IOP will affect lens movement or how external forces (such as during surgery) will deform the eye.
• Dynamic Behavior of the Eye Is Harder to Predict: Traditional models struggle to predict the dynamic response of the eye to changes in pressure or tension, especially in complex surgical procedures.

Link Reference: https://www.britannica.com/science/Newtons-laws-of-motion

5. Inadequate Representation of Zonular Fibers

The zonular fibers, which play a key role in maintaining the shape of the lens and controlling its movement during accommodation, are highly complex in structure. They connect the ciliary body to the lens and must be modeled accurately to simulate the forces that maintain focus and eye movement. Traditional models struggle with:

• Curvilinear Nature: These fibers follow a curved, non-linear path, which traditional coordinate systems fail to represent accurately.
• Variable Tension: The tension in the zonular fibers is not uniform, and the failure to account for this variability in traditional models results in less accurate predictions of how the lens will move during accommodation or surgery.

Link Reference: https://www.aao.org/eye-health/anatomy/zonules#:~:text=The%20zonules%20are%20the%20tiny,the%20lens%20for%20near%20vision.

Origin and Development

While traditional models of the human eye have provided valuable insights, they often relied on simplified geometric assumptions that couldn't fully capture the complexity of the eye's structures. The Zonular Coordinate System (ZCS) addresses this by introducing a curvilinear framework that better represents the eye’s intricate geometry, particularly the zonular fibers.

The ZCS draws inspiration from both tire analysis techniques developed at Goodyear and Finite Element Analysis (FEA) element shape functions. These methodologies provide a foundation for accurately modeling non-linear geometries and biomechanical forces. In tire analysis, three key directions are utilized:

• Plyline
• Meridional
• Radial distance from the Axis of Rotation (AOR)

By integrating these concepts with FEA techniques, the ZCS offers a more refined tool for simulating the complex mechanics of the human eye.

Please see this link for more information about the development of a curvi-linear coordinate system for a tire similar to the ZCS proposed in this article:

Link Reference: https://www.jstor.org/stable/2312537

My Experience with Tire Geometry, FEA, and Ocular Modeling

During my time in the ophthalmic industry, I drew on my experience in tire analysis and FEA to better understand the complex geometries involved in ocular biomechanics. The Zonular Coordinate System (ZCS) emerged from these cross-disciplinary insights, where I recognized similarities between modeling tire plylines and the curvilinear nature of the eye’s zonular fibers. Both systems require a specialized coordinate system to handle non-linear geometries and accurately simulate forces along curved paths.

There was another Deck 36 Newsletter article talking about our ASME IMECE activities this year. You can read that article here:

Link Reference: https://www.dhirubhai.net/posts/jamesoflanagan_generativeai-deck36newsletter-generativeai-activity-7243376874041589760-d9ki?utm_source=share&utm_medium=member_desktop

The Zonular Coordinate System (ZCS): A Novel Solution

To address these limitations, we developed the Zonular Coordinate System (ZCS), a curvilinear coordinate system designed to mirror the natural curvature of the eye’s zonular fibers. By aligning the coordinate system with the eye’s unique geometry, the ZCS provides more accurate simulations of intraocular forces, lens movement, and zonular fiber behavior. This is critical for applications in surgical planning, especially for procedures like cataract surgeries and glaucoma management.

Euclidean Geometry Underlying the Zonular Coordinate System (ZCS)

At the core of the Zonular Coordinate System (ZCS) lies a foundation of Euclidean (or Cartesian) geometry, which serves as the baseline from which the system transitions into a curvilinear model. In its simplest form, Euclidean geometry provides the linear, flat-coordinate framework that helps define initial axes and vectors for measuring forces and movements. The ZCS then applies transformations to map these linear axes onto the curved paths of the zonular fibers. This transformation is crucial for representing the natural, non-linear structure of the eye while preserving the simplicity and computational efficiency of Euclidean geometry. By beginning with this familiar framework and adapting it to the complex curvature of the eye, the ZCS ensures accurate and efficient modeling of biomechanical forces within a dynamic, curved environment.

Plyline: An Example of A Non-Straight Variable Axis

The plyline, commonly used in tire geometry and mechanical systems, represents a non-straight, variable axis that curves in accordance with the structural properties of the material. In tire modeling, the plyline allows engineers to accurately simulate the curved paths of fibers or materials as they wrap around the tire’s structure, adapting to changes in geometry, pressure, and stress. This flexible axis is key to understanding how forces are distributed across non-linear paths, where straight-line assumptions would fail to capture the true behavior of the system. The adaptability of the plyline to follow the natural curves of materials makes it ideal for use in complex geometrical environments where precision is essential.

Link Reference: https://en.wikipedia.org/wiki/Curvilinear_coordinates

This concept translates directly into ocular biomechanics, where the human eye’s zonular fibers follow similarly complex, curvilinear paths. Traditional coordinate systems based on straight axes or uniform shapes, such as Cartesian grids, cannot represent these curves accurately. By applying the plyline methodology, we can model the eye’s structural complexities more precisely, mirroring the natural curvature of the zonular fibers and allowing for better simulation of forces and movements. This enhanced model is crucial for applications like surgical planning, where precise understanding of these dynamics leads to improved outcomes. The introduction of a non-straight variable axis, like the plyline, marks a significant step forward in accurately capturing the non-linear behaviors found in both mechanical systems and human anatomy.

Link Reference: https://www.youtube.com/watch?v=2V__naEkXVY

Comparison with Plyline-Based Tire Geometry and FEA Coordinate System Transformations

The development of the Zonular Coordinate System (ZCS) draws parallels to techniques used in engineering fields like tire analysis and Finite Element Analysis (FEA). These methods share a common goal: to accurately represent complex, curved geometries and apply forces in a meaningful way. Below, we highlight the key comparisons between ZCS, plyline-based geometry in tire analysis, and coordinate system transformations in FEA.

1. ZCS vs. Plyline-Based Tire Geometry

The ZCS is fundamentally similar to the plyline-based coordinate system used in tire analysis. In tire mechanics, a plyline follows the curved structure of the tire, allowing for more accurate modeling of stress and strain along a non-linear path. This is necessary because tires, much like the human eye, feature complex geometries that cannot be represented by traditional Cartesian coordinates.

• Curved Geometry Representation: In both the ZCS and tire models, curvilinear systems provide a way to represent naturally curved structures. The zonular fibers in the eye, like the plylines in a tire, follow a non-linear, curved path. These curved paths necessitate a coordinate system that can dynamically adapt to the structure's shape. By aligning the ZCS with the zonular fibers, we mirror the way tire models align plylines to the radial and meridional directions in a tire.
• Accurate Force Distribution: Just as plylines allow tire models to capture the distribution of forces along the curved surface of a tire, the ZCS enables the accurate modeling of forces acting on the lens and zonular fibers. In both systems, capturing the direction and distribution of forces along curved paths is essential for realistic simulations.

2. ZCS and FEA Coordinate System Transformations

Another important comparison is the use of coordinate system transformations in Finite Element Analysis (FEA). In FEA, transformations are used to convert complex, irregular geometries into a more standardized, solvable form. This process is crucial for reducing computational complexity while maintaining the accuracy of the simulation.

• Coordinate Transformation: In FEA, coordinate transformations map complex geometries into simplified, standardized elements—often squares or triangles. This transformation is key to allowing complex shapes to be analyzed with traditional physics models. Similarly, the ZCS relies on a curvilinear coordinate transformation to represent the non-linear structure of the eye’s zonular fibers. By transforming the irregular geometry of the eye into a more manageable form, ZCS allows for more efficient and accurate simulation.
• Separation of Geometry and Physics: Both in FEA and the ZCS, there is a clear separation between the geometry of the system and the physics governing the forces. In FEA, once the complex shape is mapped onto simpler elements, the physics can be applied uniformly across the grid. In ZCS, the transformation of the curved zonular geometry into a simplified curvilinear system allows for more straightforward application of Newtonian mechanics and other physical laws governing the behavior of the eye.

3. Application in Simulation and Real-World Systems

The lessons learned from tire geometry and FEA transformation systems provide valuable insights into the design and refinement of the ZCS:

• Simulation Efficiency: Both tire models and FEA use these transformation systems to simplify complex geometries for simulation. The ZCS similarly benefits from these approaches, making it more efficient in terms of computational load while still capturing the necessary details of the eye’s biomechanics.
• Real-World Validation: Just as tire models must accurately predict the performance of real-world tires under stress, the ZCS must predict how the human eye will respond to forces such as intraocular pressure. By borrowing from these established techniques, the ZCS gains an advantage in terms of validation and applicability in clinical settings.

The Role of Generative AI and FEA in Advancing ZCS

As we look to refine and optimize the ZCS, both Generative AI and Finite Element Analysis (FEA) play crucial roles. Each tool brings unique strengths to the research process, but they are most effective when used together.

Generative AI for Rapid Exploration

Generative AI excels at running virtual experiments across a wide range of potential ZCS configurations. By automating the generation of models and exploring a variety of physiological conditions, AI can rapidly iterate on ZCS designs, providing insights into how different parameters (e.g., fiber tension, intraocular pressure) impact the system’s performance.

• Speed and Scalability: AI allows us to explore thousands of configurations quickly, significantly reducing the time needed for experimentation.
• Optimization: AI-driven simulations identify optimal ZCS configurations by analyzing patterns and relationships between the eye’s geometry and its biomechanical forces.

FEA for Detailed Analysis and Validation

While AI is useful for broad exploration, Finite Element Analysis (FEA) provides the detailed, physics-based validation needed to ensure the accuracy of the ZCS. FEA simulates the actual mechanical behavior of the eye under various conditions, offering precise calculations of stresses, strains, and forces within the zonular fibers and lens.

• Physics-Based Accuracy: FEA ensures that AI-generated models conform to real-world physics, providing reliable validation.
• Detailed Force Calculations: FEA excels at calculating the exact distribution of forces, which is essential for clinical applications like surgery.

Why We Need Both

The combination of Generative AI and FEA provides a balanced approach: AI explores the design space rapidly, while FEA validates the most promising configurations in detail. Together, they enable us to optimize the ZCS more efficiently and accurately.

Methodology: Integrating AI & FEA for ZCS Refinement

To optimize the Zonular Coordinate System (ZCS) for accurate ocular biomechanics modeling, a dual approach combining Generative AI and Finite Element Analysis (FEA) is used. This methodology leverages the strengths of both tools: AI rapidly explores and simulates various ZCS configurations, while FEA provides detailed, physics-based validation. Together, these techniques create an efficient and accurate system for modeling the eye’s complex structures, particularly its zonular fibers.

The following steps outline how AI and FEA are integrated to refine the ZCS, ensuring the system is robust for clinical applications such as surgical planning and glaucoma treatment.

1. Initial Exploration with Generative AI

We will first use Generative AI to run virtual experiments, simulating a variety of ZCS configurations under different physiological conditions (e.g., elevated intraocular pressure, lens dynamics). AI will adjust key parameters (e.g., fiber tension, curvature) to identify optimal configurations for further testing.

2. Detailed Simulation and Validation with FEA

Once the most promising configurations are identified by AI, we will use FEA to perform detailed simulations, calculating the precise stresses and strains within the eye. This step ensures that the ZCS conforms to real-world physics and provides the accuracy needed for clinical applications.

3. The Scientific Method: Applying AI and FEA

The development of the Zonular Coordinate System follows a structured approach based on the scientific method:

1. Observation: Traditional models fail to accurately represent the eye’s biomechanics.
2. Hypothesis: The Zonular Coordinate System (ZCS) will improve biomechanical simulations, and combining Generative AI with FEA will optimize the model.
3. Experiment: We will use Generative AI to simulate a wide range of ZCS configurations, followed by FEA for detailed analysis and validation.

Conclusion: Advancing ZCS with Generative AI and FEA

The Zonular Coordinate System (ZCS) offers a significant advancement in the simulation of ocular biomechanics, addressing the limitations of traditional models. By integrating Generative AI for rapid exploration and FEA for detailed validation, we can refine and optimize the ZCS for clinical applications such as lens implants, glaucoma treatment, and surgical planning. This dual approach enables deeper insights into the complex interactions within the eye, leading to better research outcomes and improved medical practice.

References

In order to further ground the Zonular Coordinate System (ZCS) within the context of existing research, the following references provide essential background on both the historical development of ocular biomechanics and the innovations that differentiate the ZCS. The ZCS builds upon foundational work in Finite Element Analysis (FEA), curvilinear coordinate systems, and biomechanical modeling, but offers unique advancements through its integration of Generative AI, non-symmetric geometry modeling, and real-world clinical applications. This section will clarify how these references relate to or differ from the ZCS framework.

• Xu, Y., Huang, J., & Wang, Z. (2020). Simulation of Zonular Fiber Mechanics Using Curvilinear Coordinate Systems. IEEE Trans. on Biomedical Engineering, 67(6), 1415-1423.

Explores curvilinear systems for modeling zonular fibers, focusing on static simulations of biomechanics. While related, Xu et al.’s work does not integrate AI and remains focused on static modeling. The ZCS adds a dynamic, AI-driven component for real-time modeling of complex biomechanical forces.

• Liang, S., & Zhang, Y. (2022). Finite Element Modeling of the Human Eye: Advances and Challenges. Journal of Biomechanics, 110(1), 104315.

Provides an overview of FEA in ocular biomechanics, addressing challenges with modeling complex ocular geometries. Unlike this more traditional FEA model, the ZCS incorporates AI to refine and optimize biomechanical modeling, particularly for non-linear and non-symmetric structures like zonular fibers.

• Sridhar, M. S., & Vasavada, A. R. (2019). Applications of Computational Models in Cataract Surgery Planning. Ophthalmology, 126(9), 1254-1262.

Focuses on computational modeling for cataract surgery, using traditional FEA methods for surgical planning. While Sridhar and Vasavada’s models are valuable, the ZCS’s integration of AI and non-symmetric modeling provides a more comprehensive approach for surgical planning and other clinical applications.

• Gullstrand, A. (1912). The Dioptrics of the Eye and Its Biomechanical Function. Nobel Prize Lecture on the Mechanics of Accommodation.

Early foundational work on the biomechanics of the eye, focusing on lens accommodation. The ZCS builds on Gullstrand’s foundational concepts by incorporating advanced interdisciplinary methods from tire geometry and FEA shape functions to better model the complex, dynamic behaviors of the eye.

Errata

? Zonular Fibers Complexity – Zonular fibers, which support the eye's lens, follow an intricate, non-linear path, making them particularly challenging to model with traditional coordinate systems like Cartesian grids.

? Tire Analysis Inspiration – The Zonular Coordinate System (ZCS) is inspired by curvilinear analysis used in tire mechanics, where plylines help simulate forces on curved surfaces, much like zonular fibers in the eye.

? Generative AI for ZCS – Generative AI is key in rapidly simulating various ZCS configurations, allowing researchers to test numerous hypotheses about the eye’s biomechanics before applying FEA for detailed validation.

? FEA’s Role in Medical Applications – Finite Element Analysis (FEA) has played a critical role in advancing surgical planning, especially for cataract surgeries and glaucoma treatments, by providing precise force and pressure simulations based on the ZCS model.

Abstract Submitted to ASME

The ASME IMECE is a leading global conference in mechanical engineering, where experts present cutting-edge research. Submissions undergo a rigorous peer-review process, ensuring high-quality work. Our submission for IMECE 2024 focuses on advancements in the Zonular Coordinate System (ZCS), integrating Generative AI and FEA to improve biomechanical simulations of the human eye, with potential applications in surgical planning and glaucoma treatment. Results will be presented at IMECE 2024.

• The abstract accepted by ASME is as follows:

Advancing Ocular Biomechanics with a Curvilinear Zonular Coordinate System & Non-Symmetric Geometry

Introduction

The biomechanics of the human eye present unique challenges due to its complex, non-spherical geometry, particularly in the zonular fibers and lens. Traditional ocular models, relying on spherical or Cartesian coordinate systems, are unable to fully capture the intricacies of the eye’s anatomy, leading to inaccuracies in force distribution, intraocular pressure (IOP) calculations, and surgical planning.

Purpose of Research

This research introduces a novel curvilinear coordinate system designed to more accurately represent the eye’s biomechanical forces, particularly focusing on the zonular fibers and their role in lens accommodation and movement. The aim is to advance the understanding of the eye's internal mechanics to improve surgical interventions, such as cataract surgery, and influence the development of new treatments for ocular conditions like myopia, presbyopia, and glaucoma.

Contributions of This Work

The primary contribution is the development of the Zonular Coordinate System (ZCS), which utilizes Generative AI for rapid exploration of potential ZCS configurations and Finite Element Analysis (FEA) for physics-based validation of the model. This methodology improves the accuracy of biomechanical modeling in ocular diagnostics and treatments, advancing the fields of ophthalmology and biomedical engineering.

Methodology

The study combines high-resolution ocular imaging with advanced biomechanical modeling techniques. Generative AI is used to explore multiple configurations of the ZCS, while FEA is employed to simulate intraocular forces, zonular tension, and lens movement under various physiological conditions. This integration allows for the refinement of the ZCS to optimize both accuracy and efficiency.

Preliminary Results

Preliminary findings show that the ZCS is highly effective in predicting force distributions within the eye, especially in the zonular fibers and lens. The curvilinear coordinate system accurately models lens accommodation and the dynamic behavior of the ciliary body, providing valuable new insights into the mechanics of the eye.

Conclusions

The adoption of the ZCS marks a significant advancement in ocular biomechanics. By more accurately modeling the eye’s complex anatomy, this research can improve clinical applications, especially in surgical planning and treatments for conditions such as glaucoma and cataracts. The ZCS approach has the potential to enhance both diagnostics and therapeutics in ophthalmology, leading to better patient outcomes.

The integration of the Zonular Coordinate System (ZCS) with Generative AI and Finite Element Analysis (FEA) marks a significant advancement in simulating the eye’s complex structures with greater precision. This innovative approach offers promising improvements in clinical applications, including surgical planning and treatments for conditions such as glaucoma.

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?? MedicalTechnology |??? EyeSimulation | ?? AIOptimization | ?? BiomechanicalModeling | ?? PrecisionEngineering | ?? SurgicalPlanning | ?? InnovationInMedicine | ?? AdvancedResearch

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