Chapter 12

Temperature Changes Inside

the Human Eye During LTKP

Ooi, E.H. and Ng, E.Y.K.

12.1. Introduction to Laser-Thermokeratoplasty (LTKP)

We have developed a complete model of the human eye in the Orz coordinate system to investigate the temperature changes inside the human eye during LTKP. Heat that is absorbed inside the eye during laser radiation is described using the Beer-Lambert law. Two types of lasers that are commonly used in LTKP, which are pulsed and continuous-wave lasers, are investigated. The model is numerically created using the time-stepping dual reciprocity boundary element method. The temperature achieved inside the cornea during pulsed laser radiation is found to be higher than continuous-wave laser radiation, albeit for a short duration. In addition to the analysis of temperature distribution inside the eye, a thermal damage assessment is carried out using the thermal damage model of Henriques and Moritz.

LTKP is a corneal refractive surgery intended to correct vision. LTKP is a subset of the thermokeratoplasty technique that involves heating the cornea via laser treatment to induce corneal shrinkages. When the cornea is subjected to heat, the collagen bonds inside the stroma break, leading to contractions (shrinkages) of the collagen fibrillae.1 The shrinkages of the cornea cause the curvature to change, which ultimately affects the refractive power of the cornea. Based on the aforementioned behavior, ophthalmologists have been able to correlate the amount of corneal shrinkages due to heat, with parameters such as heating duration and laser energy to produce

School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798.

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Ooi, E.H. and Ng, E.Y.K.

the desired level of change in the corneal refractive power. Although LTKP is a less popular method than other laser surgery techniques, such as laser in situ keratomilieusis (LASIK), it is used in clinical practice, especially to treat presbyopia and astigmatism.

One of the main concerns involved in LTKP is the risk of causing irreversible thermal damage that may induce permanent scarring to the cornea. LTKP also suffers from poor predictability and repeatability, which often lead to unreliable clinical outcomes. A proper understanding of the thermal behavior of the cornea during LTKP is, therefore, essential in designing treatment procedures that may help to reduce the risk of thermal injury and to improve the predictability and repeatability of clinical results. Mathematical modeling plays an important role in the studies of corneal temperature changes during LTKP. The role of mathematics in determining the accuracy of the treatment is largely due to the hazardous nature of the treatment procedure, which makes experimental studies on human subjects difficult. Most experimental studies of LTKP have been performed in vitro using samples of animal cornea.1,2

One of the earliest mathematical models that was developed for investigating corneal temperature changes during LTKP was presented by Mainster in 1979.3 Although various studies of the thermal interaction between lasers and cornea have been reported prior to the work of Mainster,3 these investigations mainly focused on the thermal effects of lasers that were not associated with the treatment of LTKP.4,5 In the investigation carried out by Mainster,3 the cornea was modeled as a semi-infinite region where a onedimensional (1D) heat transfer was assumed. By the early 1990s, numerical models that investigated the thermal profile of the cornea during LTKP began to emerge. Most of the models developed during this period considered only the cornea as the solution domain.6,7 The cornea was modeled as a cylinder in the axisymmetrical coordinate system. Numerical solutions were largely obtained using the finite difference method.

Between the early 1990s and the early 2000s, the studies of LTKP were eagerly pursued by Brinkmann et al.8−10 from both the experimental and numerical perspectives. The models, which comprised of the cornea and the anterior chamber, were developed in the axisymmetrical coordinate system and were assumed as a two-layered cylinder. The finite element method was used to obtain a numerical solution. One of the unique features of the

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Temperature Changes Inside the Human Eye During LTKP

models developed by Brinkmann et al.8−10 was the inclusion of intra-corneal focusing of the laser beam, i.e. the distribution of laser energy within the cornea. Various types of lasers used in LTKP, such as Er: Glass, Ho: YAG, and CO2 lasers, have been investigated, and, in most cases, the numerical predictions showed results comparable with those observed during in vitro experiments.

Since the early 2000s, numerical investigations of LTKP have been actively carried out by Manns et al. and Borja et al.11−13 Unlike the models developed by Brinkmann et al.8−10 the Manns et al. and Borja et al. model region consisted only of the cornea and was modeled as a single-layered tissue slab in the two-dimensional (2D) coordinate system. Analytical solutions that were expressed as summations of an infinite series were derived based on the Green’s function approach. The transient temperature profile of the cornea was found to be biologically reasonable.11−13 No comparisons with experimental data have been performed, however. In addition to the investigations on the transient temperature changes of the cornea, Manns et al. and Borja et al.11−13 predicted the degree of thermal damage and the degree of corneal shrinkages inside the cornea during LTKP using the Arrhenius equation, the damage integral, and the second-order kinetic model of corneal shrinkage.

One common feature shared by most previous LTKP studies is the use of simplified models for developing the mathematical models. The choice of using the simplified model approach may be due to the need-to-maintain mathematical simplicity. This approach remains valid since the temperature changes inside the cornea during LTKP are localized to the area where the laser beam is applied. One of the limitations of the simplified model approach, however, is that the assumptions that have to be made when specifying the boundary conditions may not be physiologically correct.

In this study, a model of the human eye is developed to investigate the changes in the temperature distribution inside the human cornea during the treatment of LTKP. In addition to improving our understanding of the thermal behavior of the cornea when subjected to heat, this study is designed to examine the feasibility of using an anatomically more realistic eye model for the studies of ocular heat transfer during LTKP. Problems associated with the increase of computer memory due to the use of a more complex model are avoided by numerically solving the model using the boundary

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Ooi, E.H. and Ng, E.Y.K.

element method. Investigations are carried out for the treatments of LTKP using pulsed and continuous-wave lasers. Additionally, the degree of thermal damage induced onto the cornea during laser radiation is estimated based on the transient temperature profile of the cornea using the thermal damage model of Henriques and Moritz.14

12.2. Characteristics of LTKP

The premise behind the treatment of LTKP is the shrinkage experienced by the corneal collagen when subjecting the cornea to heat. According to Stringer and Parr,15 the corneal collagen shrinks when the cornea is heated to a temperature between 55◦C and 65◦C. However, this observation was made without considering the factors involved in ensuring a successful treatment of LTKP. Subsequent investigations carried out by other researchers have revealed that the cornea has to be heated to a minimum temperature of 64◦C in order to produce permanent corneal shrinkage.16 Likewise, Brinkmann et al.8−10 found that the deep stromal tissues must be heated to temperatures beyond the shrinkage threshold to guarantee long-term effects of corneal shrinkage.

It is necessary to maintain the corneal temperature below 100◦C during LTKP to avoid corneal relaxation.17 The relaxation causes the bonds between the collagen molecules inside the cornea to break, thus countering the intended corneal shrinkage. Another important issue during treatment of LTKP concerns the temperature at the corneal endothelium, which is highly sensitive toward heat.18 According to Wirbelauer,18 endothelial cell deaths occur when the temperature at the endothelium increases to 65◦C. Therefore, an ideal treatment of LTKP is expected to produce the desired amount of permanent shrinkage while minimizing the amount of corneal relaxation and thermal damage inside the cornea.

In a typical clinical treatment of LTKP, eight equally distanced laser spots, arranged concentrically around the center of the corneal surface, are applied to the cornea. Each spot has a radial distance of 3–5 mm from the center of the corneal surface. In some cases, 16 spots are applied, where the additional eight spots are located on an outer ring at a radial distance of approximately 7 mm from the center. Although the threshold for corneal shrinkage has been found to be in the range of 55–65◦C, it is common for the

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