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Temperature Changes Inside the Human Eye During LTKP

Fig. 12.2. Distribution of the Gaussian laser beam.

12.5.3. Initial-Boundary Conditions

Boundary conditions are specified on both the surfaces of the cornea, C1, and on the sclera, C2. Laser radiation during LTKP is assumed to produce no significant effect on the heat transfer between the environment and the corneal surface. Similarly, heat transfer between the blood flow and the eye is assumed not to be affected by the treatment of LTKP.

On the exterior surface of the cornea, C1, heat is transferred to the environment via convection and radiation. Cooling is aided by the evaporation of tears from the tear film on top of the corneal surface. Mathematically, this condition is given as:

κ1

∂T1

= hamb(T1 Tamb) + εσ(T14

Tamb4

) + Evap,

∂n

 

 

on C1

and

for t > 0, (12.6)

where the first, second, and last term on the right-hand side refer to the heat loss due to convection, radiation, and tear evaporation, respectively, hamb is the ambient convection coefficient, Tamb is the ambient temperature, ε is the corneal emissivity, σ is the Stefan-Boltzmann constant, Evap is the heat loss due to tear evaporation, and ∂T1/∂n is the rate of change of T1 in the outward unit vector normal to the external corneal surface C1.

On the exterior surface of the sclera, C2, heat from blood flow across the sclera enters the eye and diffuses via conduction to the corneal surface.

357

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

Table 12.2. Values of the various control parameters.

Parameter

Value

 

 

Ambient temperature, Tamb (C)

25

Ambient convection coefficient, hamb (Wm2K1)

10

Blood temperature, Tbl (C)

37

Blood convection coefficient, hbl (Wm2K1)

65

Heat loss due to tears evaporation, Evap (Wm2)

40

Emissivity of the cornea surface, ε

0.97

Stefan-Boltzmann constant, σ (Wm2K4)

5.67 × 108

Thus, we may write

 

 

 

∂T6

= hbl(T6

Tbl),

 

 

κ6 ∂n

on C2 and for t > 0,

(12.7)

where hbl is the blood convection coefficient, Tbl is the blood temperature, and ∂T6/∂n denotes the rate of change of T6 in the outward unit vector normal to the external corneal surface, C2. Values of the parameters used in Eqs. (12.6) and (12.7) are the same as those employed by Ooi et al.23 and are summarized in Table 12.2.

The variation of the temperature and heat flux at the interfaces between two contiguous ocular regions, Iij , may be described using the continuity

condition, which is mathematically given as:

 

 

 

 

Ti = Tj and κi

∂Ti

= κj

∂Tj

,

on Iij ,

(12.8)

 

 

∂n

∂n

where Ti and Tj are temperature of regions Ri and Rj , respectively, and ∂Ti/∂n and ∂Tj /∂n are the rate of change of Ti and Tj in the outward unit vector normal to the interface Iij . Since the problem in the present study is a transient one, an initial condition is required to complete the formulation of the problem. For this purpose, the temperature distribution inside the normal unexposed human eye during a steady state is obtained by solving the following:

i Ti(r, z)) = 0, for i = 1, 2, 3, 4, 5 and 6.

(12.9)

The result is subjected to Eqs. (12.6) to (12.8), as an initial, specified condition.

358