## 2 Materials and Methods
### Mathematical Models  
The PID control algorithm contains proportional control, integral control, and differential control of deviation. It works as shown in Figure 2, which can achieve intelligent control of laser power automatic adjustment by adjusting parameters. To control the temperature in the center of the fiber targeting area, which relies solely on the working experience of the medical personnel and lacks a precise temperature control mechanism, the PID control algorithm is introduced to control the maximum fiber temperature at 90 \({}^{\circ}\)C. The specific control equation is shown in Equations (1) and (2) [14]:  
\[P_{in}=K_{p}\cdot e(t)+K_{i}\cdot\int_{\theta}^{t}e(t)dt+K_{d}\cdot\frac{d[e(t )]}{dt} \tag{1}\]  
Figure 1: Model geometry. \[e(t)=T_{\text{set}}-T_{\text{tip}}=90\ ^{\circ}\text{C}-T_{\text{tip}} \tag{2}\]  
where \(P_{in}\) is the feedback power (W) under PID control, \(K_{p}\) is the proportional control coefficient, \(K_{i}\) is the integral control coefficient, \(K_{d}\) is the differential control coefficient, \(e(t)\) is the temperature deviation (\({}^{\circ}\)C) at the time \(t\)(s), \(T_{\text{set}}\) is the set target temperature (90 \({}^{\circ}\)C), and \(T_{\text{tip}}\) is the actual measured temperature (\({}^{\circ}\)C). After adjustment, the temperature diagram of the domain point probe and the temperature distribution diagram of LITT is shown in Figures 3 and 4. Figure 3: Temperature diagram of domain point probe after adjustment. Figure 2: PID control principle. In the LITT simulation process, light enters the tissue due to scattering to form the angle of diffuse light approximation into an isotropic state, and the laser irradiance is calculated by applying the diffuse approximation equation as shown in Equations (3) and (4) [15,16]. The light propagates in the fiber and diffuses outward after reaching the target head, mainly divided into radial and axial diffusion in spatial distribution. For simplicity, the current model assumes that the laser intensity consists of two directional laser powers: the radial direction of the diffusion part, P1 (assuming 95% of the incident laser power, with the incident power being the feedback power Pin under PID control), in a fiber super-Gaussian distribution [15]; and the axial direction, P2 (assuming 5% of the incident laser power), in a Gaussian distribution [17], as shown in Equations (5) and (6). **Laser irradiance**  
\[\mu_{a}\Phi(x,y,z)-D\
abla^{2}\Phi(x,y,z)=s(x,y,z) \tag{3}\]  
\[D=\frac{1}{3(\mu_{a}+\mu_{s}(1-g))},\;s(x,y,z)=\mu_{s}(1-g)(QJ+QZ) \tag{4}\]  
**Radial super-Gaussian distribution equation**  
\[QJ=k\cdot\frac{P1}{2\pi wl}\cdot e^{-2(\frac{z}{2})^{p}}\cdot e^{-(\mu_{a}+\mu _{s}(1-g))\sqrt{x^{2}+y^{2}}} \tag{5}\]  
**Axial Gaussian distribution equation**  
\[QZ=\frac{P2}{2\pi\sigma^{2}}\cdot e^{-\frac{(x^{2}+y^{2})}{2\sigma^{2}}}\cdot e ^{-(\mu_{a}+\mu_{s}(1-g))z} \tag{6}\]  
where \(\Phi(x,\,y,\,z)\) is the laser irradiance (W/m\({}^{2}\)), \(QJ\) is the laser irradiance in the radial direction (W/m\({}^{2}\)), and \(QZ\) is the laser irradiance in the axial direction (W/m\({}^{2}\)); \(\mu_{a}\) is the absorption coefficient (1/m), \(\mu_{s}\) is the scattering coefficient (1/m), \(g\) is the anisotropic factor, \(D\) is the diffusion constant in the tissue (m), \(w\) is the fiber radius (mm), 1 is the fiber length  
Figure 4: PID control LITT temperature distribution chart. (mm), \(\sigma\) is the Gaussian distribution deviation [18], \(p\) is the super-Gaussian distribution order [18], and \(k\) is the loss factor [19].