Temperature Changes Inside the Human Eye During LTKP

cornea to be heated beyond this threshold. The temperature that the cornea is eventually heated to depends on the type of lasers used. Two types of lasers, pulsed and continuous-wave, can be used for LTKP.

12.3. Pulsed Laser

Each laser pulse delivers a large amount of energy at a duration that is usually less than 0.25 s. This output is irregular.19 One of the most commonly used pulsed lasers in the treatment of LTKP is the Ho: YAG laser, which emits a laser beam at a wavelength of 2.1 µm.7,16 Other types of pulsed lasers that have been used in LTKP include the Er: Glass and the Tm: YAG laser. In a typical LTKP procedure using a pulsed laser, seven laser pulses, each with the duration of 200 µs, are applied to the corneal surface at a repetition rate of 5 Hz.12

12.4. Continuous-Wave Laser

Unlike pulse lasers, continuous-wave lasers produce steady and continuous output.19 Radiation is usually carried out at a low-energy rate. Among the type of continuous-wave lasers that have been used in LTKP are laser diodes, CO2 lasers, and CoMgF2 lasers. Laser diodes are usually preferred over CO2 and CoMgF2 lasers due to their tunability. This tunability allows the laser beam to be emitted at different wavelengths, hence, making it a more versatile choice for LTKP. Commonly used wavelengths range from 1.85 to 1.87 µm.9 LTKP treatment using continuous-wave lasers may be carried out using two approaches, the 10-s coagulation, where the cornea is heated continuously for 10 s at a laser power of 0.125 W, and the minute coagulation, where the cornea is heated continuously for 60 s are both carried out using a 0.10-W laser.10

12.5. Mathematical Model

12.5.1. Model Description

Our model of the human eye is developed based on dimensions found in literature.20,21 Figure 12.1 displays the model developed with reference

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Fig. 12.1. Model of the human eye used in the study of LTKP.

to the axisymmetrical coordinate system, Orz. The model comprises six regions, which are the cornea, the anterior chamber, the lens, the posterior chamber, the vitreous, and the sclera. These regions are denoted by R1, R2, R3, R4, R5, and R6, respectively.

The retina and choroid, which are relatively thin when compared to the sclera, have been modeled with the sclera as one homogeneous region.22,23 Similarly, the iris and sclera are treated as one homogeneous region, since they have been found to exhibit the same thermal properties.24 The exterior surfaces of the cornea and sclera are denoted by boundaries C1 and C2, respectively.

12.5.2. Governing Equations

Assuming that blood perfusion and metabolic heat generation inside the eye are negligible, and treating each ocular region as a solid, the transient temperature distribution inside the human eye during LTKP, with reference to the model illustrated in Fig. 10.1, may be described using

where ρi, ci, and κi are the density, specific heat, and thermal conductivity of region Ri, respectively, Ti is the temperature distribution in Ri, t is time, and Si is the heat generated inside region Ri due to the absorption of laser

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Temperature Changes Inside the Human Eye During LTKP

Table 12.1. Thermal properties of the human eye.

energy. Each region inside the human eye is assumed to be homogeneous and thermally isotropic. Note that the temperature distribution is now a function of space and time.

Values of the thermal conductivity, density, and specific heat of each ocular region may be found in literature.24−28 These values are tabulated in Table 12.1.

Heat that is generated inside the eye, Si due to the absorption of laser energy may be described using the Beer-Lambert law.19 For the typical range of laser wavelengths used in LTKP (10.85–2.1 µm), Si may be mathematically expressed as:

=

(12.2)

where µ is the corneal absorption coefficient, which is dependent on the wavelength of the laser (unit of m−1), F is the Fresnel reflectance of the corneal surface, E(r, z) is the incident irradiance at the center of the corneal surface, and ψ(t) is given as:

which dictates the time that the laser is being applied onto the cornea. When the laser is off, ψ(t) has a value of zero. Consequently, the value of

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Si becomes zero; implying that no heat generation takes place inside the human eye.

Equation (12.2) implies that the laser beam that is incident on the corneal surface generates heat only inside the cornea. This implication is largely attributed to the penetration depth of laser, cornea dp, and thickness. Penetration depth generally describes the depth inside the ocular media where radiation from the laser approaches zero, as it decreases exponentially from the irradiated surface.19 Mathematically, the penetration depth is given by:

The typical laser wavelengths used in the treatment of LTKP (as described in Sec. 12.2) ranges from 1.87 to 2.1 µm. These values correspond to those of the corneal absorption coefficient of 1900–2000 m−1. Using Eq. (12.4), the range of dp corresponding to the range of the corneal absorption coefficient is calculated as 0.48–0.50 mm. This calculation suggests that if a laser with a wavelength of 1.87 µm is applied to the corneal surface, the laser radiation approaches zero at a depth of 0.50 mm from the corneal surface. Similarly, if a laser with a wavelength of 2.1 µm is used, radiation at regions beyond the depth of 0.48 mm from the corneal surface becomes negligible. Since the thickness of the cornea in the present model is approximately 0.588 mm, no radiation is expected in the ocular regions located beyond the cornea.

The mathematical expression of the incident irradiance, E(r, z), in Eq. (12.2) depends on the profile of the laser beam that arrives at the corneal surface. Ideally, the laser beam may be assumed to have a flat profile so that the distribution of irradiance is uniform. In most practical cases, the laser beam has a distribution that follows the profile of a Gaussian-type Equation,19 such as shown in Fig. 12.2. For laser beams that are centered near the center of the corneal surface, (r, z) = (0, 0), the function of E(r, z) for a Gaussian beam profile is mathematically given as:

where Eo denotes the peak irradiance and w denotes the radius of the laser beam where the laser irradiance decreases to exp(−1) times its maximum value.19

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