# Design of HIFU Treatment Plans Using Thermodynamic Equilibrium Algorithm
## 3 Results & Discussions  
### Optimization Performance  
The proposed TEA was tested with 25 optimization test functions. All of them were minimization problems. Details of the functions are listed in Table 1. The first problem \(f_{1}\) is Ackley's function, which was frequently used as an optimization benchmark function. Figure 6: Heat distribution along the main focus axis during a sonication with \(t_{on}\) = 3 s and \(t_{off}\) = 10 s at time \(t\) = 3 s, 4 s, 6 s, 8 s, 10 s.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Function Name**} & \multirow{2}{*}{**3D Pilot**} & \multirow{2}{*}{**Function Formula**} & \multirow{2}{*}{**Minimum Value**} & \multirow{2}{*}{**Search Domain**} & \multicolumn{2}{c}{**Optimum Value of**} \\  & & & & & **Max Cart Function** & \\ \hline \multirow{4}{*}{ \begin{tabular}{c} [PERSON]’s function \\ \end{tabular}  } & \multirow{4}{*}{ \begin{tabular}{c} \(f_{1}(x,y)=-2\) \\ \(-2\)(\(\Delta(\Delta x^{2}+y^{2})\)) \\ \(-2\)(\(\Delta y(\Delta x^{2}+y^{2})\)) \\ \(-2\)(\(\Delta y(\Delta x^{2}+y^{2})\)) \\ \(-2\)(\(\Delta y(\Delta x^{2}+y^{2})\)) \\ \end{tabular}  } & \multirow{4}{*}{ \begin{tabular}{c} \(f_{1}(0,0)=0\) \\ \(-5<xy<5\) \\ \(-5\)(\(\Delta y(\Delta y(\Delta x^{2}+y^{2})\)) \\ \(-2\)(\Delta y(\Delta x^{2}+y^{2})\)) \\ \end{tabular}  } & \multirow{4}{*}{ \begin{tabular}{c} \(f_{1}(0,0)=0\) \\ \(-5<xy<5\) \\ \(-5\)(\Delta y(\Delta y(\Delta x^{2}+y^{2})\)) \\ \(-2\)(\Delta y(\Delta x^{2}+y^{2})\) \\ \end{tabular}  } & \multirow{4}{*}{ \begin{tabular}{c} \(-5\) \\ \(-5\) \\ \(-5\)(\Delta y(\Delta y(\Delta x^{2}+y^{2})\)) \\ \(-2\)(\Delta y(\Delta x^{2}+y^{2})\) \\ \end{tabular}  } & \multirow{4}{*}{ \begin{tabular}{c} \(-5\) \\ \(-5\) \\ \(-5\) \\ \(-5\)(\Delta y(\Delta y(\Delta x^{2}+y^{2})\)) \\ \end{tabular}  } & \multirow{4}{*}{\begin{tabular}{c} \(-5\) \\ \(-5\) \\ \(-5\) \\ \(-5\) \\ \(-5\) \\ \(-5\) \\ \(-5\) \\ \(-5\) \\ \(-5\) \\ \hline \multirow{4}{*}{ \begin{tabular}{c} Achary’s function \\ \end{tabular}  } & \multirow{4}{*}{ \begin{tabular}{c} \(f_{2}(x,y)=-2\) \\ \(-2\)(\(\Delta y(\Delta x^{2}+y^{2})\)) \\ \(-2\)(\(\Delta y(\Delta x^{2}+y^{2})\)) \\ \(-2\)(\(\Delta y(\Delta x^{2}+y^{2})\)) \\ \end{tabular}  } & \multirow{4}{*}{ \begin{tabular}{c} \(f_{2}(0,0)=0\) \\ \(-5<xy<5\) \\ \(-5\)(\Delta y(\Delta x^{2}+y^{2})\) \\ \(-2\)(\Delta y(\Delta x^{2}+y^{2})\) \\ \end{tabular}  } & \multirow{4}{*}{\begin{tabular}{c} \(-5\) \\ \(-5\) \\ \(-5\) \\ \(-5\) \\ \(-5\) \\ \(-5\) \\ \(-5\) \\ \hline \multirow{4}{*}{ \begin{tabular}{c} [PERSON]’s function \\ \end{tabular}  } & \multirow{4}{*}{\ \begin{tabular}{c} \(f_{2}(x,y)=-(1+x-y^{2})\) \\ \(-1\)(\(\Delta y(\Delta x^{2}+y^{2})\)) \\ \(-1\)(\(\Delta y(\Delta x^{2}+y^{2})\)) \\ \(-2\)(\(\Delta y(\Delta x^{2}+y^{2})\)) \\ \end{tabular}  } & \multirow{4}{*}{
\begin{tabular}{c} \(f_{2}(0,0)=0\) \\ \(-10<xy<For \(f_{1}\), a 3D plot of the function is shown in Figure 7. The global minimum of the function in interval \(-5<x,y<5\) is located at \(f_{1}(0,0)\) and the cost of 0. In the optimization process using TEA, the initial population of 100 systems was chosen in Figure 8a. Every coupled system individually explored the minimum point. The systems are indicated by \(\times\) marks. Figure 8b-d show the movements of systems at iterations 17, 34, and 50. It can be seen that at iteration 17, while most of the systems were located at local minima of the function, a couple of them had reached the global minimum point, at which the best cost is \(3.591\times 10^{-6}\). At iteration 34, although a couple of systems were trapped, the majority of them reached the equilibrium and the best cost was \(8.141\times 10^{-11}\). At iteration 50, almost all systems had reached the optimal location. The final answer was \(9.251\times 10^{-15}\), which is very close to zero. As examples of other optimization algorithms, continuous GA and PSO were also applied to this problem. To make a fair comparison of the performance among these methods, the initial population of 100 was chosen for all of them.