## 2 Models for Photothermic Problem and Agglomerates
### Photothermic Model  
When illuminated by electromagnetic waves, absorbing materials are sources of temperature elevation. The gold nanoparticles absorb the light energy and play the role of a heat source. The heat density of such heat sources depends on the local electromagnetic field \(\mathbf{E}(\mathbf{x})\) as follows:  
\[Q(\mathbf{x})=\frac{\omega}{2}\varepsilon_{0}Im(\varepsilon_{r}(\mathbf{x}))| \mathbf{E}(\mathbf{x})|^{2},\quad\forall\mathbf{x}\in\Omega, \tag{1}\]  
where \(\omega\) is the incident angular frequency, \(\varepsilon_{0}\) is the permittivity of a vacuum, and \(\varepsilon_{r}\) is the complex relative permittivity of materials. The electromagnetic field \(\mathbf{E}(\mathbf{x})\) is the solution of the Helmholtz equation with a radiation boundary condition [20; 22] (see Appendix A.1 for electromagnetic and thermic equations). With such a heat source, the temperature elevation is governed through the heat equation (Equation (2)):  
\[\frac{\partial T(\mathbf{x},t)}{\partial t}=\frac{1}{\rho(\mathbf{x})C_{p}( \mathbf{x})}[\
abla\cdot(\kappa(\mathbf{x})\
abla T(\mathbf{x},t))+Q(\mathbf{ x})]\quad\quad\forall\mathbf{x}\in\Omega, \tag{2}\]  
with the time detoned by \(t\), the temperature by \(T\), the volumic mass density by \(\rho\), the specific heat capacity by \(C_{p}\), the thermal conductivity by \(\kappa\), and \((\
abla\cdot)\) and \(\
abla\) denoting the divergence and gradient operators, respectively. The solution of the full problem requires solving the heat equation (Equation (2)) by including the electromagnetic source \(Q\) (Equation (1)).  
The distribution of the temperature is obtained by solving the weak-coupled photothermic problem with an illumination duration \(t\) longer than the thermic characteristic time of materials \(\tau_{thermic}\) (i.e., \(\tau_{thermic}\approx 1\) ms). The solution of Equation (2) converges to the stationary solution (i.e., \(t\geq\tau_{thermic}\), then \(\partial T(\mathbf{x},t)/\partial t\longrightarrow 0\), corresponding to a continuous illumination). The characteristic electromagnetic time \(\tau_{electro}\) (e.g., \(1.9\leq\tau_{electro}\leq 2.9\) fs in the NIR domain of 650-950 nm) is much shorter than the characteristic thermic time \(\tau_{thermic}\)Therefore, the electromagnetic and thermic problems induce a weak coupling and could be solved sequentially.  
Equations (1) and (2) are solved through a 3D FEM with an adaptive remeshing process, ensuring the convergence of the numerical solution to a stable solution [18; 19; 20]. The remeshing process (see Appendix A.2) ensures the automatic and systematic control of the accuracy of computation in the multiscale domain (e.g., the typical size of the nanoparticles is small, between 10-100 nanometers, and that of the cell is 1-10 microns). Figure 1a shows a schematic of the referenced problem: agglomerates of individual gold nanoparticles (blue circles) on individual volume \(V_{np}\) and permittivity \(\epsilon_{np}\), embedded in a volume of biological media \(V_{bm}\) with relative permittivity \(\epsilon_{bm}\). The computations are achieved in a domain \(\Omega\) with boundary \(\Gamma\). Figure 1b shows an example of the mesh domain associated with such an agglomerate of nanoparticles.  
With such a numerical model, the temperature variations that are induced by the electromagnetic and thermic coupling in the agglomerate of nanoparticles in the biological media/cells are studied.