## 3 Numerical Results and Discussion
### Influence of the Concentration on the Mixture of Multiple Agglomerates  
By increasing the concentration of the nanoparticles, the probability of producing a mixture of various agglomerates increases. Such a mixture of agglomerates will induce a temperature elevation that is expected to be larger than the temperature elevation of a single strong coupling \(N^{st}\)-agglomerate. In order to protect the neighboring biological media/cells, the control of the maximum temperature, relative to the mixture of agglomerates, must be achieved. Therefore, numerical simulations for various mass concentrations (or volumic concentrations) of nanoparticles are performed in order to analyze the concentration effects on the maximum temperature elevation and on its spatial extension. The range of the relative mass concentrations are given in Table 1. For fixed relative mass concentrations \(C_{rm}\), multiple configurations of agglomerates can occur, mixing strong, soft, and no thermic coupling (i.e., corresponding to various occurences \(n_{st}\geq 1\), \(n_{so}\geq 0\), and \(n_{no}\geq 0\)). Figure 3 shows two examples of agglomerate configurations and temperature maps for two relative mass concentrations, \(C_{rm}=11.8\%\) (a), (c), and \(C_{rm}=1.5\%\) (b), (d), respectively. These two concentrations are relative to the local volume of the biological media/cells \(V_{bm}\) (illustrated in Figure 3a,b). In these examples, the agglomerate is constituted by mixtures of \(N^{st}\) varying in (1; 6) for the relative mass concentration \(C_{rm}=11.8\%\), and with \(N^{st}\) in (1; 4) for the relative mass concentration \(C_{rm}=1.5\%\), respectively. As expected (Equation (5)), for a mixture of agglomerates, the maximum temperature \(T_{N,max}\) is larger than the maximum temperature obtained with a single strong-coupling \(N^{st}\)-agglomerate (i.e., \(T_{N,max}>T_{N^{st},max}\)). Here, the maximum is \(T_{N,max}=53.2\ ^{\circ}\mathrm{C}\geq T_{N^{st},max}=49.9\ ^{\circ}\mathrm{C}\) (with \(N^{st}=6\)) for \(C_{rm}=11.8\%\) and \(T_{N,max}=45.8\ ^{\circ}\mathrm{C}\geq T_{N^{st},max}=44.2\ ^{\circ}\mathrm{C}\) (with \(N^{st}=4\)) for \(C_{rm}=1.5\%\). Figure 3: Agglomerates of nanoparticles (blue disks) for two relative mass concentrations, (**a**) \(C_{rm}=11.8\%\) (i.e., relative volumic concentration \(C_{rm}=0.66\%\)) and (**b**) \(C_{rm}=1.5\%\) (i.e., relative volumic concentration \(C_{rm}=0.08\%\)), and their respective temperature maps (**c**,**d**). Figures 4a-c and 5a-c show the temperature variations above the ablation threshold as a function of the extensions along the x-, y-, and z-axes for the two mixtures of agglomerates with \(C_{rm}=11.8\%\) and \(C_{rm}=1.5\%\), respectively. Figure 4: Temperature (in \({}^{\circ}\)C) as a function of the spatial distributions of the agglomerated electromagnetically coupled nanoparticles along the (**a**) x-, (**b**) y-, and (**c**) z-axes for the relative mass concentrations \(C_{rm}=11.8\%\) (i.e., relative volumic concentration \(C_{rm}=0.66\%\)). These temperature distributions correspond to the agglomeration of nanoparticles given in Figure 3a. The zones of higher temperatures reveal the strong coupling for dense agglomerates and the zones of soft coupling revealed in the vicinity of multiple \(N^{st}\)-agglomerates (shown in the magenta circles). The larger the \(N^{st}\) value, the higher the maximum temperature. The effects of soft couplings (thermic coupling) are more easily demonstrated due to lower maximum temperatures (i.e., with small \(N^{st}\) values). These lower temperatures are related  
Figure 5: Temperature (in \({}^{\circ}\)C) as a function of the spatial distributions of the agglomerated electromagnetically coupled nanoparticles along the (**a**) \(\times\)-, (**b**) \(y\)-, and (**c**) \(z\)-axes for the relative mass concentrations \(C_{\mathit{m}}=1.5\%\) (i.e., relative volumic concentration \(C_{\mathit{m}}=0.08\%\)). These temperature distributions correspond to the agglomeration of nanoparticles given in Figure 3b.