An analytical model for gold nanoparticle radiosensitisation

In this paper, we derive a variance-driven Local-Effect-Model ($\sigma$-LEM) to predict radiosensitization due to gold nanoparticles (AuNP). Assuming that the number of Au photo-ionisations scales strictly with particle volume $V_{\mathrm{NP}}\propto R^{3}$, a linear relation between dose-enhancement ratio and concentration is achieved ($DER = 1 + K_c,c$), in which $K_c$ is a beam-quality and nucleus-size-specific term, and $c$ is the concentration in mM. Furthermore, assuming that the cascade energy deposition is log-normally distributed, the enhanced dose in each target voxel can be written as $D_{\text{enh}} = D_{0}\exp(\sigma Z)$ with $Z \sim \mathcal{N}(0,1)$ and width $\sigma = \sqrt{2\ln(1+Kc)}$. Assuming a linear-quadratic (LQ) dose response, a relation between cell survival and dose can be derived. Despite no closed form for the log-normal distribution, averaging over the entire domain using first- and second-order moments leads to three possible closed forms: variance-only, mixed-term, and second-order. These three variants adapt well to low-concentration, mid-concentration, and high-concentration regimes. The model was tested for Bovine aortic endothelial cells (BAEC) results taken from a Local Effect Model (LEM) and experimental values. The model agrees within $\le 2.5$ with the experimental and LEM data, but presents significant changes to the conceptual results obtained with the LEM, in particular indicating that AuNP dose enhancement is mostly $\alpha$-driven, as posited previously by other authors. These findings are further developed in the manuscript. The theoretical framework presented here collapses radiobiological outcomes to three experimentally controllable variables -- beam quality, nucleus size, and intracellular concentration $c$ -- while retaining mechanistic fidelity. Additional tests should be made to further confirm the validity of the model.