## 3 Numerical Results and Discussion
### Influence of the Concentration on the Mixture of Multiple Agglomerates
to the fact that the strong couplings (electro-thermic couplings) are less present for smaller relative concentrations. By controlling the relative mass concentration \(C_{rm}\) (or relative volume concentration \(C_{r\gamma}\)) of the agglomerate, it becomes possible to limit the range of maximum temperature \(T_{N^{st},max}\) produced by such agglomerates. For both minimizing such a risk of overpassing the upper temperature limit \(T_{lim}\) (inducing damage to the neighboring biological media/cells) and ensuring an efficiency to the PA above the ablation threshold, the relevant criteria are on the minimum and maximum numbers of dense agglomerates (i.e., only considering the strong coupling), relative to the mass concentration \(C_{rm}\). Therefore, \(N^{st}_{min}\) and \(N^{st}_{max}\) must satisfy \(T_{N^{st}_{min}}\geq T_{abl}\) and \(T_{N^{st}_{max}}\leq T_{lim}\), and their values can be obtained by:  
\[N^{st}_{min}=Int\bigg{(}\frac{T_{abl}-T_{0}}{\Delta T_{1,max}}\bigg{)},\ \ \ \text{and}\ \ \ N^{st}_{max}=N^{st}_{min}+1, \tag{7}\]  
where \(Int(. )\) is the integer part. With such \(N^{st}_{min}\) and \(N^{st}_{max}\), a range of relative mass concentrations can be managed with a probability of succes. The probabilities of producing agglomerates are deduced from statistical analyses obtained from multiple relative mass concentrations and random generations of the spatial distributions of the agglomerates. We use 100 relative mass concentration values, varying in (0.02; 35.0)%, and for each value, 1000 random spatial distributions are produced for the agglomerates before solving the photo-thermic problem. With these random generations, statistical analysis of the \(N^{st}\)-agglomerates permit assigning probabilities of producing \(N^{st}\)-agglomerates and their uncertainties. Figure 6a shows the probability of producing an \(N\)-agglomerate as a function of the relative mass concentration \(C_{rm}\) for various strong couplings. As expected, for a given concentration, the larger the \(N^{st}\), the smaller the probability of producing such an \(N^{st}\)-agglomerate. Such a probability increases with the relative mass concentration. Figure 6b gives the ranges of temperatures (minimum and maximum) as a function of the relative mass concentration \(C_{rm}\) for mixtures of agglomerates and for three diameters \(D_{np}\) of nanoparticles. The larger the diameter \(D_{np}\) of nanoparticles, the larger the range of temperatures and the higher the maximum temperature (that could overpass the maximum temperature limit). The obtained results are consistent with the expected behavior. Indeed, the trends of Figure 6a,b are similar in \(C_{rm}\) and to the sizes of the nanoparticles forming the mixture of agglomerates: an increasing concentration induces an increase of the probability of producing at least one agglomerate with \(N^{st}\geq N^{st}_{min}\). The larger the diameter \(D_{np}\) of the nanoparticles forming the mixture of agglomerates, the higher the maximum temperature that rapidly overpasses the temperature limit. The main parameter governing the efficiency of the PA and limiting the risk of temperature limit \(T_{lim}\) is directly associated with the production of one \(N^{st}_{min}\)-agglomerate for a given concentration. A compromise between the sizes of the nanoparticles and the relative mass concentration must be found. Here, \(D_{np}=(85.0\pm 5.0)\) nm and \(C_{rm}=(2.0\pm 1.0)\)% ensure an active PA with a realistic concentration by limiting the exceeding of the temperature limit.