## 3 Numerical Results and Discussion
### Single Dense Agglomerate of Identical Nanoparticles  
The case of a single dense agglomerate corresponds to \(N=N^{st}\) (i.e., \(n_{st}=1\), \(n_{so}=0\) and \(n_{no}=0\) in Equation (4)). From Equation (5), the maximum temperature elevation produced by such an agglomerate is given by [21]:  
\[\Delta T_{N^{st},max}=(T_{N^{st},max}-T_{0})\geq N^{st}\Delta T_{1,max}=N^{st}( T_{1,max}-T_{0}). \tag{6}\]  
Therefore, the maximum temperature of a \(N^{st}\)-agglomerate is directly related to the dense agglomerate constituted by \(N^{st}\) nanoparticles (i.e., related to the number of nanoparticles participating only in a strong coupling: electromagnetic and thermic coupling). We achieve FEM simulations of the elevation of temperature produced by \(N=N^{st}\) identical nanoparticles for \(1\leq N^{st}\leq 10\) identical nanoparticles and diameters of particles \(D_{np}\) varying in (15; 180) nm by a step of 5 nm. We solve, numerically, the photothermic problem of dense agglomerates and analyze the maximum temperature as function of the number of nanoparticles (see Figure 2).  
The temperature increases and is larger than \(N^{st}\) times the temperature elevation produced by one uncoupled nanoparticle (Equation (6)). We can note that the larger the diameter \(D_{np}\), the smaller the \(N^{st}\)-agglomerate satisfying a maximum temperature overpassing the thermal ablation threshold, but the higher the maximum temperature that could overpass a reasonable maximum temperature limit \(T_{lim}\) (e.g., 55-60 \({}^{\circ}\)C) [31]. On the other hand, overpassing the thermal ablation threshold with small \(D_{np}\) requires a larger \(N^{st}\)-agglomerate, but reduces the risk of overpassing a reasonable maximum temperature limit. See, for example, the temperature elevations for agglomerates of identical nanoparticles with diameters of \(D_{np}=100\) nm and \(D_{np}=80\) nm (the two magenta vertical dashed lines in Figure 2). The \(N^{st}\)-agglomerate of \(D_{np}=100\) nm overpasses the ablation threshold with only \(N^{st}>1\), but overpasses the temperature limit with only \(N^{st}>3\). This contrasts with a \(N^{st}\)-agglomerate of \(D_{np}=80\) nm, which is overpassing the ablation threshold with \(N^{st}>2\) and which is still under the temperature limit with \(N^{st}<10\). Therefore, the calculation of the temperature as a function of \(N^{st}\) is useful.  
For the dense agglomerate, the electromagnetic and thermic coupling has a tremendous influence on temperature elevation, on the risk of inactive PA (for small concentrations or small agglomerates/particles), or of PA causing damage to the surrounding biological media/cells (for large concentrations or large agglomerates/particles). This is depends mainly on the incident laser power, the size of the nanoparticles, and the concentration  
Figure 2: Temperature evolution as a function of the diameter \(D_{np}\) of individual nanoparticles and for dense-agglomerates of \(N^{st}\) identical nanoparticles (\(N^{no}=1\), \(N^{st}=2,4\), and \(8\)), producing coupling (i.e., due to electromagnetic and thermic coupling).  
of such nanoparticles. For a fixed incident laser power, a balance between the size of the nanoparticles and the concentration of nanoparticles (i.e., associated with the production of the \(N^{st}\)-agglomerate that induces a maximum temperature) should be found. This balance can ensure an active PA (overpassing the thermal ablation threshold), limit the risk of maximum temperature, and protect the neighboring biological media/cells. However, in real life, for a given concentration of nanoparticles, the control of agglomerates is impossible. Therefore, it becomes necessary to consider the mixture of multiple agglomerates of various kinds of coupling.