## 2 Models for Photothermic Problem and Agglomerates
### Model for Agglomerates of Individual Nanoparticles  
For a given concentration of metallic nanoparticles, the process of forming agglomerates of nanoparticles is complex [23; 24; 25]. Such metallic nanoparticles can agglomerate by 1-15 nanoparticles/vesicles [26; 27]. In this paper, we consider individual nanoparticles (each one, depending on its size, can produce a sub- or upper-ablation threshold temperature elevation). The relative volumic concentration \(C_{rv}\) and the relative mass concentration \(C_{rm}\) can be defined and related as:  
\[C_{rm} = \rho_{np}V/(\rho_{bm}V_{bm})=\big{(}\rho_{np}/\rho_{bm})C_{rv}, \tag{3}\]  
where \(\rho_{np}\), \(\rho_{bm}\), \(V=NV_{np}\), and \(V_{bm}\) are the mass densities and global volumes of the gold nanoparticles and the biological media (or cell), respectively. Therefore, the number of nanoparticles is \(N\approx C_{rv}V_{bm}/V_{np}\).  
For an agglomerate of \(N\) identical nanoparticles of diameters \(D_{np}\), the inter-distances between the constitutive nanoparticles are additional parameters. Indeed, the electromagnetic coupling between nanoparticles induces an electromagnetic field enhancement and produces a temperature elevation in the biological media/cells. However, such an electromagnetic coupling occurs at a short distance between individual nanoparticles (\(d_{electro}\leq\) 50-70 nm \(\approx D_{np}\)). Moreover, a thermic coupling appears if the inter-distance is  
Figure 1: Slice of the geometry (**a**) for agglomerations of individual nanoparticles (blue) of individual volume \(V_{np}\), embedded in a biological medium/cell of volume \(V_{bm}\). Zoom on an example of the associated mesh domain (**b**) for a particular agglomeration of individual nanoparticles.  
typically lower than \(d_{thermic}\approx 1\) um [21]. Consequently, we can distinguish three spatial distributions of nanoparticles:  
* a dense agglomerate: characterized by strong coupling for \(d\leq d_{electro}\) due to both electromagnetic and thermic couplings;
* a non-dense agglomerate: characterized by soft coupling for \(d_{electro}\leq d\leq d_{thermic}\) only due to thermic coupling;
* a non-agglomerate: characterized by no coupling for \(d\geq d_{thermic}\); each nanoparticle is independant (i.e., isolated individual nanoparticle).  
With this spatial distribution, the number \(N\) of nanoparticles can be written as:  
\[N=\sum_{i=1}^{n_{st}}N_{i}^{st}+\sum_{i=1}^{n_{so}}N_{i}^{so}+\sum_{i=1}^{n_{ mg}}N_{i}^{no}, \tag{4}\]  
with \(N_{i}^{st}\), \(N_{i}^{so}\), and \(N_{i}^{no}\) being the numbers of particles participating in the strong, soft and no coupling (associated with the nanoparticles for the dense agglomerate, the non-dense agglomerate and the non-agglomerate) with occurences \(n_{st}\), \(n_{so}\) and \(n_{no}\), respectively. These numbers are characteristics for each kind of coupling that induces a specific temperature elevation. With such a model for agglomerates and with the photothermic model, the temperature elevation in a biological medium/cell is calculated through FEM. The maximum temperature elevation is directly related to these kinds of agglomerate and is given by:  
\[\Delta T_{N,max}=(T_{N,max}-T_{0}) \geq \Delta T_{N_{max}^{st},max}\geq N_{max}^{st}\Delta T_{1,max}=N_{ max}^{st}(T_{1,max}-T_{0}), \tag{5}\]  
where the \(N_{max}^{st}\) is the higher strong-coupling \(N^{st}\)-agglomerate present in the \(N\)-agglomerate, and \(T_{1,max}\) is the maximum temperature for an isolated nanoparticle with a diameter of \(D_{np}\). First we will consider single dense agglomerates before analyzing statistics on random mixtures of multiple agglomerates.