## Materials and Methods
### Models
#### 4.3.2 In-Vivo Predictions  
The computational model is a multiphysics FEM model, thus, it took into account optical and thermal effects. Light distribution was based on the diffusion approximation of the transport theory [29] and temperature distribution by Pennes bio-heat transfer equation [30], which takes into account the effect of cell death on blood perfusion and the dependence of cell death and properties of the tissue. Cell death was determined by an Arrhenius based integral injury model [53]. _Light Distribution._ The optical diffusion approximation of the transport theory [29] was used to describe light distribution due to the dominance of scattering over absorption in biological tissues. It is defined by:  
\[\frac{1}{c_{n}}\frac{\partial}{\partial t}\varphi=D\
abla^{2}\varphi-\mu_{a} \varphi+S \tag{6}\]  
\(c_{n}\) (m s\({}^{-1}\)) is the speed of light in a medium, \(\varphi\) (W m\({}^{-2}\)) is the fluence rate, \(\mu_{a}\) (m\({}^{-1}\)) is the absorption coefficient, \(S\) (W m\({}^{-3}\)) is the light source term and \(D=\mu_{a}/\mu_{\text{eff}}^{2}\) (m) is the diffusion coefficient. \(\mu_{\text{eff}}=\sqrt{3\mu_{a}(\mu_{a}+\mu_{s}^{\prime})}\) (m\({}^{-1}\)) is the effective attenuation coefficient and \(\mu_{s}^{\prime}\) (m\({}^{-1}\)) is the reduced scattering coefficient. Assuming that the light source was a continuous wave Gaussian NIR laser beam that was incident onto the breast model, the \(\varphi\) can be defined by  
\[\phi(\overline{r})=\frac{P_{0}\exp(-\mu_{\text{eff}}\overline{r}\cdot\hat{n})} {4\pi Dr} \tag{7}\]  
where \(P_{0}\) is the laser power and \(\hat{n}\) is the direction of the beam. A summary of the values of the optical properties of the tissue used in the simulation is presented in Table 2.  
_Temperature Distribution._ The Pennes bio-heat transfer equation [30] was used to estimate the temperature distribution. An additional term was added to account for the external heat source. The resulting equation is given by:  
\[\rho c_{\text{P}}\frac{\partial T}{\partial t}=\lambda(T)\
abla^{2}T+\rho_{\text {b}}c_{\text{b}}\omega_{\text{b}}(\Omega)(T_{\text{b}}-T)+Q_{\text{met}}+Q \tag{8}\]  
where \(\rho\) (kg m\({}^{-3}\)) is the density, \(c_{\text{P}}\) (J kg\({}^{-1}\) K\({}^{-1}\)) is the specific heat capacity at constant pressure. \(\lambda(T)\) (W m\({}^{-1}\) K\({}^{-1}\)) is the temperature dependent thermal conductivity, which is assumed to be a linear function defined by [54]:  
\[\lambda(T)=\lambda_{(37\ ^{\circ}C)}[1+0.0028(T-293.15K)] \tag{9}\]  
where \(T\) (K) and \(T_{\text{b}}\) (K) are the normal body and arbitrary temperatures, respectively. \(\rho_{\text{b}}\) is the density of blood, \(c_{\text{b}}\), the specific heat capacity of blood and \(\omega_{\text{b}}(\Omega)\) is the coefficient of blood perfusion assumed to be dependent on the cell damage, \(\Omega\), and defined by [33; 38]:  
\[\omega_{b}(\Omega)=\left\{ \begin{array}{ll}\omega_{b}^{0}&\text{if }\Omega=0\\ (1+25\Omega-260\Omega^{2})\omega_{b}^{0},&\text{if }0<\Omega\leq 0.1\\ (1-\Omega)\omega_{b}^{0},&\text{if }0.1<\Omega\leq 1\\ 0,&\text{if }\Omega>1\end{array} \right. \tag{10}\]  
\(\omega_{b}^{0}\) (s\({}^{-1}\)) is the baseline coefficient of blood perfusion. \(Q_{\text{met}}\) (W m\({}^{-3}\)) is the metabolic heat. \(Q\) accounts for external heat sources, which varies for the different domains of geometry. The heat generated after the absorption of NIR light is defined as \(\mu_{a}\varphi(r)\) (W m\({}^{-3}\)) and \(N\sigma_{a}\varphi(r)\) (W m\({}^{-3}\)) for the tissue and tumor domains respectively. \(N\) is the number volume of Fe\({}_{3}\)O\({}_{4}\) NPs and \(\sigma_{a}\) (m\({}^{2}\)) is the absorption cross-section of nanoparticles. Table 3 presents a summary of the values of the thermo-physical properties that were used in the simulation.