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1

Mathematical modeling of short pulsed laser irradiation in the cornea: a dual phase lag model

Autar Ram, Kumar Anuj

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 3

5--16

EN

A simple mathematical model for the temperature evolution in the cornea exposed to short-pulsed Ho: YAG laser under Laser Thermo Keratoplasty (LTK) treatment is developed by incorporating both the heat flux phase-lag and temperature gradient phase-lag in Fourier’s heat transfer model. An analytical solution to the mathematical model is obtained using the Laplace transformation technique. The computational results for the temperature profile and the temperature variation with time are presented through the graphs. The effect of some typical parameters: the heat flux phase-lag and the temperature gradient phase-lag on the temperature distribution and temperature variations are illustrated and discussed.

2

Segmentation of anterior segment boundaries in swept source OCT images

Garcia Marin Yoel, Skrok Marta, Siedlecki Damian, Vincent Stephen J., Collins Michael J., Alonso-Caneiro David

Biocybernetics and Biomedical Engineering

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2021

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Vol. 41, no. 3

903--915

EN

Quantification of the eye’s anterior segment morphology from optical coherence tomography (OCT) images is crucial for research and clinical decision-making, including the diagnosis and monitoring of many ocular diseases. Structural parameters, such as tissue thickness and area are the most common metrics used to quantify these medical images, and tissue segmentation is required before these metrics can be extracted. Currently, swept source OCTallows the capture of cross-sectional images that encompass the entire anterior segment with a high level of detail. However, the manual annotation of tissue boundaries is time-consuming. In this work, an algorithm based on graph-search theory combined with boundary-specific image transformation is applied for the segmentation of anterior segment OCT images. We demonstrate that the method can reliably segment 5 different tissue boundaries in healthy eyes with low boundary error (mean error below 1 pixel across all boundaries). The technique can be used to extract clinically relevant parameters such as central corneal and crystalline lens thickness as well as anterior chamber depth and area, with a high level of agreement with manual segmentation (normalized errors below 1.6%). The proposed method provides a tool that can support clinical and research OCT data analysis.

3

Three-dimensional biomechanical modeling and simulation of trephine cutting cornea for keratoplasty

Su P., Lu D., Deng S., Zhang L., Hao Y., Yang Y.

Acta of Bioengineering and Biomechanics

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2018

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Vol. 20, no. 2

23--33

EN

Trephination is one of the basic operations of keratoplasty, and the biomechanical mechanism of the operation can be revealed based on three-dimensional modeling and simulation of trephine cutting cornea. Methods: Based on the analysis of the physical and biomechanical characteristics of corneal trephination, a three-dimensional numerical model of corneal trephination is built, where the cornea can be simplified to two layers structure including stroma and epithelium, and the trephine cuts the cornea under the vertical motion load and the rotational motion load. A three-dimensional failure criterion of corneal material is proposed based on the yield strength theory. On this basis, trephination simulation is carried out, and the units of corneal material are removed from the model when they meet the defined failure criterion. Results: Under the given parameters including the velocity, the angle and the angular velocity, the trephine force curves, include the linear cutting force and the rotary cutting force are obtained, and show the change of the forces with displacement during the process of trephination simulation. The maps of the equivalent stress show the destruction and deformation of the cornea. Then, the experiment of robotic trephination is carried out under the same parameters and the effectiveness of the simulation is evaluated. Conclusions: Based on mechanics theory and finite element method, the process of trephine cutting cornea has been reproduced, and the interaction mechanism is revealed, which lays the foundation for the development of real-time simulation and virtual system of the corneal surgery.

4

Biomechanical simulation of needle insertion into cornea based on distortion energy failure criterion

Su P., Yang Y., Huang L.

Acta of Bioengineering and Biomechanics

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2016

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Vol. 18, no. 1

65--75

EN

Purpose: This paper is mainly about biomechanical behavior of needle insertion into cornea, and proposes a failure criterion to simulate the insertion process which has attracted considerable attention due to its importance for the minimally invasive treatment. Methods: In the process of needle insertion into cornea, tiny and complex insertion force is generated due to contact between needle and soft tissue. Based on the distortion energy theory, there is proposed a failure criterion of corneal material that can solve contact problem between rigid body and biological tissue in insertion simulation, where Cauchy stress of corneal material is the key to numerical calculation. A finite element model of in vivo cornea is built, and the cornea constrained by sclera is simplified to two layers containing epithelium and stroma. Considering the hyper-viscoelastic property of corneal material, insertion simulation is carried out. Results: By insertion experiment, the insertion force increases with insertion depth accompanying obvious fluctuations. Different insertion forces are generated at different speeds. The punctured locations are obvious in the force-displacement curves. The results of insertion simulation are generally consistent with experimental data. Maps of von Mises stress reflect the tissue injury of the cornea during insertion process, and punctured status corresponds to the point in the curves. Conclusions: The ability of this study to reproduce the behavior of needle insertion into cornea opens a promising perspective for the control of robotic surgery operation as well as the real-time simulation of corneal suture surgery.

5

Corneal hyper-viscoelastic model: derivations, experiments, and simulations

Su P., Yang Y., Song Y

Acta of Bioengineering and Biomechanics

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2015

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Vol. 17, no. 2

73--84

EN

Purpose: The aim of this study is to propose a method to construct corneal biomechanical model which is the foundation for simulation of corneal microsurgery. Methods: Corneal material has two significant characteristics: hyperelastic and viscoelastic. Firstly, Mooney–Rivlin hyperelastic model of cornea obtained based on stored-energy function can be simplified as a linear equation with two unknown parameters. Then, modified Maxwell viscoelastic model of the cornea, whose analytical form is consistent with the generalized Prony-series model, is proposed from the perspective of material mechanics. Results: Parameters of the model are determined by the uniaxial tensile tests and the stress-relaxation tests. Corneal material properties are simulated to verify the hyper-viscoelastic model and measure the effectiveness of the model in the finite element simulation. On this basis, an in vivo model of the corneal is built. And the simulation of extrusion in vivo cornea shows that the force is roughly nonlinearly increasing with displacement, and it is consistent with the results obtained by extrusion experiment of in vivo cornea. Conlusions: This paper derives a corneal hyper-viscoelastic model to describe the material properties more accurately, and explains the mathematical method for determination of the model parameters. The model is an effective biomechanical model, which can be directly used for simulation of trephine and suture in keratoplasty. Although the corneal hyper-viscoelastic model is taken as the object of study, the method has certain adaptability in biomechanical research of ophthalmology.

6

Strain-rate sensitivity of porcine and ovine corneas

Elsheikh A., Kassem W., Jones S.W.

Acta of Bioengineering and Biomechanics

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2011

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Vol. 13, no. 2

25-36

EN

Knowledge of strain-rate sensitivity of corneal tissue is important for improving the understanding of the tissue's response to mechanical actions and the accurate numerical simulation of corneal biomechanical behaviour under the effects of disease and surgery. In the study, fresh and well-preserved porcine and ovine corneal buttons were subjected to uniaxial tension loads with seven different strain rates ranging between 0.8 and 420% per minute. All specimens exhibited increased stiffness (as measured by the tangent modulus) with higher strain rates. However, clear differences in their behaviour were observed. While ovine corneas showed gradual, consistent and mostly statistically significant increases in stiffness with all elevations in strain rate, porcine corneas' response was significant over only a limited range of low strain rates. The effect of strain rate on the material's stress-strain behaviour was considered in the formation of three sets of constitutive models including: (i) a model based on a simple exponential stress-strain relationship, (ii) the Ogden model that considers the tissue's hyperelasticity but not anisotropy, and (iii) a third model by Holzapfel, Gasser and Ogden that considers both hyperelasticity and anisotropy. The three models are introduced to enable consideration of the strain rate effects in simulations employing finite element programs with varying capabilities or in modelling applications in corneal biomechanics which may or may not require consideration of mechanical anisotropy.

7

Goldmann applanation tonometry - not as good as gold

Śródka W.

Acta of Bioengineering and Biomechanics

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2010

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Vol. 12, no. 2

39-47

EN

A thesis that linear mechanics does not apply to the analysis of cornea load during Goldmann applanation tonometry measurement and that the concept of surface tension in the lacrimal fluid is an ineffective attempt at circumventing the associated problems is put forward. The fundamental problem emerging during numerically simulated measurement of pressure on the eyeball, whose dimensions are considered to be calibrated, stems from the fact that the flattening of the cornea at the nominal intraocular pressure leads to a critical state in which the shell loses stability. The consequences are far-reaching. The Goldmann tonometer performs well at low intraocular pressure, but above the nominal pressure its readings are always understated. The cause of the error is not the tonometer itself (its readings can be even very accurate). It is shell “solution” called Imbert–Fick law which is faulty.

8

Model numeryczny gałki ocznej do korekcji pomiarów ciśnienia wewnątrzgałkowego elektronicznymi tonometrami okulistycznymi

Hachoł A., Środka W., Szul E.

Elektronika : konstrukcje, technologie, zastosowania

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2007

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Vol. 48, nr 2

40-44

PL

Zaprezentowano model numeryczny gałki ocznej przeznaczony do korekcji pomiarów ciśnienia wewnątrzgałkowego elektronicznymi tonometrami okulistycznymi. Geometrię oraz parametry materiałowe zostały dobrane tak, aby uczynić model samonastawnym optycznie. Do budowy modelu numerycznego wykorzystano metodę elementów kończonych (pakiet Cosmos/M oraz oprogramowanie własne). Przedstawiono również wyniki badań symulacyjnych, pokazujące wpływ parametrów materiałów struktur oka, grubości rogówki i jej promienia oraz samego ciśnienia wewnątrzgałkowego na wynik pomiaru.

EN

The paper presents numerical model of the human eyeball which can be used for correction of intraocular pressure measurement using electronic ophthalmic tonometers. Investigated model based on the optical self-adjustment efect of the human eyeball for all value of IOP from the physiological range. The FEA method was used for the construction of the model. The exemplary results showing influence of the eye ball materials parameters, central corneal thicknes, radius of cornea curvature and IOP on the tonometric measurements result (obtained with the described model) have been presented.