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

where nr and nz are the components of the outward unit normal vector on i in the r and z direction, respectively, K and E denote the complete elliptic integral of the first and second kind, respectively, as defined by Abramowitz and Stegun,30 and m, a, and b are, respectively, given as:

2b(r; ξ)

m(r, z; ξ, η) = , a(r, z; ξ, η) + b(r; ξ)

a(r, z; ξ, η) = ξ2 + r2 + (η − z)2, b(r; ξ) = 2rξ.

Note that in the axisymmetric formulation, the z-axis does not form part of the curve boundary i.

The boundary element method is implemented by discretizing the boundary i into small straight-line segments, also known as boundary elements. The domain integral and the time-dependent variable in Eq. (12.10) are treated using the dual-reciprocity boundary element method and the time stepping scheme, respectively. For a detailed derivation, one may refer to the work carried out by Ooi et al.31

12.7. Results

Investigations on the transient temperature distribution inside the cornea during LTKP are carried out for both pulsed and continuous-wave lasers. Only lasers with a Gaussian beam profile are considered. The values of laser parameters used in the present study are selected based on the values used in a typical clinical LTKP treatment. Although a minimum of eight laser spots are usually applied to the corneal surface (see Sec. 12.2), in the present study, only a single laser spot, assumed to be applied to the center of the corneal surface, is considered. This assumption is necessary in order to maintain the axisymmetric feature of the human eye, which is of great computational advantage. With the laser spot placed at the center of the corneal surface, an increase in the temperature near the center of the corneal surface is expected.

To capture the large thermal variation over the small heated area around the center of the corneal surface accurately, the level of boundary discretization used has to be sufficiently fine. Likewise, the number of interior points

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Temperature Changes Inside the Human Eye During LTKP

Fig. 12.3. Interior points selected inside each region of the human eye.

used to implement the dual reciprocity method has to be sufficiently large. In the present study, the human eye model is discretized into 253 boundary elements, with 72 of them located on the boundary of the cornea. A total of 316 interior points are selected inside the human eye model, with 53 and 142 of them placed inside the cornea and anterior chamber, respectively. The interior points selected inside each region of the human eye model are illustrated in Fig. 12.3.

12.7.1. Pulsed Laser

Table 12.3 summarizes the values of laser parameters of a typical LTKP treatment using pulsed laser. These values are obtained from Manns et al.12 Energy per pulse describes the amount of laser energy that is delivered onto the corneal surface during each laser pulse. The laser absorption coefficient of the cornea is obtained by assuming that the cornea has optical properties similar to those in water. The value of peak irradiance, Eo, in Table 12.3 is obtained using the following expression,11

Eo = Epp ,

πw2tp

where Epp is energy per pulse and tp is pulsed duration.

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Table 12.3. Typical laser parameters chosen for the pulsed laser.

Based on the values given in Table 12.3, the function ψ(t) in Eq. (12.3) that describes the period when the laser is on and off may be expressed as:

where t is time taken (in seconds) and J is the time interval defined by:

In carrying out the time-stepping scheme, the time step, t is chosen to be

Figure 12.4 shows the transient temperature changes along the pupillary axis (r = 0) at various depths of the cornea during treatment of pulsed LTKP. The seven temperature peaks seen in Fig. 12.4 corresponds to the seven laser pulses that are applied to the corneal surface. Cooling is observed at intervals between laser pulses where heat that is absorbed inside the cornea is diffused into the environment via convection and radiation and into the other ocular regions inside the eye. Throughout the treatment of LTKP, the corneal surface experiences the largest increase in temperature. At the end of the seventh laser pulse, temperature as high as 111◦C is reached; albeit for only a

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Temperature Changes Inside the Human Eye During LTKP

Fig. 12.4. Transient temperature changes along the pupillary axis at various depth inside the cornea during pulsed LTKP with a Gaussian beam profile.

short duration. At the corneal endothelium, i.e. at z = 588 µm, temperature at the seventh laser pulse is found to be approximately 53◦C. Temperature at the corneal stroma, which is represented by the curves between z = 100 µm and z = 500 µm, increases beyond the threshold for corneal shrinkages after the third laser pulse.

Figure 12.5 illustrates the temperature profile along the pupillary axis at various depths of the cornea during each laser pulse, i.e. for t = 0.0002, 0.2002, 0.4006, 0.6008, 0.801, 1.0012, and 1.2014 s. The vertical lines separate the z-axis into regions occupied by the epithelium, stroma, and endothelium, while the horizontal lines represent the lower threshold for corneal shrinkages (T = 55◦C) and corneal relaxation (T = 90◦C). A large part of the corneal stroma, where the corneal collagens are located, is found to have temperatures that are greater than the threshold for corneal shrinkages. The high temperature indicates corneal shrinkages. At the corneal epithelium, temperature increases beyond the relaxation threshold. However, this may not severely affect the overall shrinkage of the cornea, since the collagens that are responsible for the contraction and relaxation of the cornea are not found in the corneal epithelium. At the end of the seventh laser pulse, the

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