# Thermal Analysis of Cornea Heated with Terahertz Radiation
## 2 Materials and Methods  
In order to investigate the time-dependent heat transfer and temperature distributions in the cornea illuminated by terahertz radiation, we developed a two-dimensional (2D) axisymmetric model, as illustrated in Figure 1. This model includes the surrounding fluids, like tears and aqueous humor, similar to the models for thermal effects in the cornea under microwaves [2] or RF heating [5]. A continuous-wave terahertz beam of 3 mm radius and 1 THz frequency impinges onto the center of the cornea. Based on the structure of the human eye, the cornea is treated as a homogeneous and curving layer with 0.4\(\sim\)0.7 mm thickness [21] and 7.5 mm radius [2,5]; the tear film with a thickness of 5\(\sim\)15 \(\upmu\)m, and the aqueous humor with a thickness of 2 mm [2,5] were modeled as the major water content. We made use of the bioheat equation to govern the terahertz bioheat effects in the cornea, which can be written as [22]:  
\[\rho C\frac{\partial T}{\partial t}=\
abla(k\
abla T)+q\,. \tag{1}\]  
Here, \(\rho\) is the density of the corneal tissue (kg/m\({}^{3}\)), \(C\) is the specific heat of the corneal tissue(J/(kg\(\cdot\)K)), \(T\) is the corneal tissue's local temporal temperature (K), and \(k\) is the thermal  
Figure 1: Schematic diagram of THz thermal treatment of the cornea (out of scale). conductivity of the corneal tissue (W/(m\(\cdot\)K)). Heat source \(q\) is related to absorbed THz radiation (W/m\({}^{3}\)), which can be given by the Beer-Lambert law [23]:  
\[q(z,r)=\alpha(1-R_{f})I_{0}\exp{(-\alpha|z|)}\,. \tag{2}\]  
Here, \(R_{f}\) is the reflectivity of THz radiation by the corneal tissue, which is taken to be \(R_{f}=0.02\) @1 THz [11], \(I_{0}\) is the induced localized THz power intensity (\(I_{0}(z,r)=\frac{2P}{\pi w_{0}^{2}}\exp(\frac{-2r^{2}}{w_{0}^{2}})\), \(P\) is THz laser power in Watts, \(w_{0}\) is the radius of the THz beam with Gaussian distribution), \(\alpha\) is the absorption coefficient of the biotissue, and \(\{z,r\}\) denotes the coordinates. For convenience in the following discussion (i.e., influence of the THz beam radius and power density), \(P_{\mathrm{d}}\) represents the power density of the THz beam in Watts per square centimeter (\(P_{\mathrm{d}}=\frac{P}{\pi w_{0}^{2}}\)). Heated by THz radiation, the cornea experiences a rise in temperature, and could suffer irreversible thermal damage if temperature rises above a threshold. According to the Arrhenius formulation, the damage degree can be quantified by a parameter, \(\Omega(t)\), which is expressed as [24]  
\[\Omega(t)=\ln{\left(\frac{c(t_{0})}{c(t)}\right)}=\frac{k_{\mathrm{B}}}{\hbar} \exp{\Delta S/R}\int{T(t)\exp{(-\frac{\Delta E}{R\cdot T(t)})}}\mathrm{d}t\,. \tag{3}\]  
where \(c(t0)\) is the initial concentration of undamaged corneal-tissue molecules, \(c(t)\) is the concentration of remaining corneal tissue at time \(t\), \(k_{\mathrm{B}}\) is [PERSON]'s constant, \(\hbar\) is [PERSON]'s constant, \(R=8.313\,\mathrm{J/(mol\cdot K)}\) is the universal gas constant, \(\Delta S=106\,\mathrm{\ kJ/mol}\) and \(\Delta E=39\,\mathrm{J/(mol\cdot K)}\) are the entropy and the enthalpy of collagen denaturation [22, 24], respectively. With Equation (3), the spatial distribution of thermal damage can be obtained. The bioheat equation, i.e., Equation (1), is solved using the finite-element method (COMSOL). The initial temperature and the temperature at the bottom surface of the aqueous humor were set to be 35 \({}^{\circ}\)C. We recorded the temperature as the damage degree at each time step. Physical and thermal parameters of the water (tear and aqueous humor) and the cornea [5, 22, 24] that were used in the model are summarized in Table 1.  
Additionally, we used an axisymmetry boundary condition at \(r=0\) in our model. The effects of heat convection in all interfaces (tear-air, cornea-tear, cornea-aqueous humor) of the model were taken into account using thermal-transfer coefficients.