The developed LEM-based interpolation framework is intended to be used in conjunction with the existing wealth of available experimental survival fraction data for high Z NP-undoped and NP-doped specific cell line studies (Jain et al. 2012). At a minimum each of these studies possesses a set of in vitro clonogenic assays of a cell line undoped and doped with high Z NPs that have been irradiated by a gamma-/X-ray source with a known energy spectra. The following derivation outlines how these data can be interpolated as a function of NP concentration, up to a maximum concentration corresponding to the NP-doped cell line survival data, within the LEM formalism for a given cell line/incident photon energy spectra combination.

The LEM can be constructed utilising three main assumptions. First, the survival fraction of a cellular colony/system under photon irradiation (SF) can be described via a linear-quadratic response:

$$\begin{aligned} {\text{SF}}[D] = \exp \left( -\alpha D - \beta D^2\right) \end{aligned}$$

(1)

where \(\alpha\) and \(\beta\) are characteristics of the target cell line, and *D* is the mean dose delivered to the entire volume of the cellular colony/system (McMahon et al. 2011; Douglas and Fowler 1976). Second, that cell “inactivation”, e.g. cell death, can be attributed to the creation of a number of lethal lesions within a sensitive small sub-cellular volume such as the cell nucleus (Scholz and Kraft 1996, 2004). Here, a lethal lesion is defined as the local modification of DNA generated from the direct and indirect action of ionisation radiation (i.e. a double-strand break). And finally, any contribution of sub-lethal damage at distances larger than the order of a few microns is ignored as it is assumed that there is no interaction between distant sites (Scholz and Kraft 1996, 2004).

Using these assumptions, it is possible to describe the survival fraction for a cell under photon irradiation in terms of the mean number of lethal lesions (\(\langle N (D) \rangle\)):

$$\begin{aligned}{\text{SF}}[D] = \exp (-\langle N (D) \rangle ) \end{aligned}$$

(2)

and inversely:

$$\begin{aligned} \displaystyle \langle N (D) \rangle = -\log ({\text{SF}}[D]). \end{aligned}$$

(3)

Within each cell under photon irradiation, lethal lesions are generated inhomogeneously and the probability of their creation is a direct function of local dose deposition. These properties mean that total lesion number in a cell’s sensitive region can be given via integration over its whole volume:

$$\begin{aligned} \displaystyle \langle N_{{\text{total}}} (D) \rangle&= \int \frac{-\log ({\text{SF}}[{{d}}(x,y,z)])}{V_{{\text{sens}}}}{\text{d}}V \nonumber \\&= \alpha \int \frac{{ {d}}(x,y,z)}{V_{{\text{sens}}}}{\text{d}}V + \beta \int \frac{{{d}}(x,y,z)^{2}}{V_{{\text{sens}}}}{\text{d}}V \end{aligned}$$

(4)

where \({{d}}(x,y,z)\) is the local dose deposited for a given position within the sensitive region of the cell and \(V_{{\text{sens}}}\) is the total volume of the sensitive region of interest.

For a cellular colony/system doped with a concentration of high Z NPs (*C*), the LEM framework allows for the total local dose deposition within the sensitive region of the cell to be separated into two parts:

$$\begin{aligned} \displaystyle { {d}}(x,y,z) = { {d}}_{\rm U}(x,y,z)+ { {d}}_{{\text{NP}}}(C,x,y,z) \end{aligned}$$

(5)

where \({{d}}_{\rm U}(x,y,z)\) and \({{d}}_{{\text{NP}}}(C,x,y,z)\) are the dose distributions generated within the sensitive region from the direct interaction of radiation with the bulk cell and high Z NPs, respectively. With this separation, Eq. 4 can be expressed as:

$$\begin{aligned} \displaystyle \langle N_{{\text{total}}} (C,D) \rangle&= \alpha \int \frac{{{d}}_{\rm U}(x,y,z)+{{d}}_{{\text{NP}}}(C,x,y,z)}{V_{{\text{sens}}}}{\text{d}}V \nonumber \\&\quad + \beta \int \frac{\left( {{d}}_{\rm U}(x,y,z)+{{d}}_{{\text{NP}}}(C,x,y,z)\right) ^{2}}{V_{{\text{sens}}}}{\text{d}}V \nonumber \\&= \alpha \int \frac{{{d}}_{\rm U}(x,y,z)}{V_{{\text{sens}}}}{\text{d}}V + \beta \int \frac{{{d}}_{\rm U}(x,y,z)^{2}}{V_{{\text{sens}}}}{\text{d}}V \nonumber \\&\quad + \alpha \int \frac{{{d}}_{{\text{NP}}}(C,x,y,z)}{V_{{\text{sens}}}}{\text{d}}V + \beta \int \frac{{{d}}_{{\text{NP}}}(C,x,y,z)^{2}}{V_{{\text{sens}}}}{\text{d}}V \nonumber \\&\quad + 2\beta \int \frac{{{d}}_{\rm U}(x,y,z)\times {{d}}_{{\text{NP}}}(C,x,y,z)}{V_{{\text{sens}}}}{\text{d}}V. \end{aligned}$$

(6)

In addition, over the range of validity of dose in the linear-quadratic model, 1–6 Gy (Joiner and van der Kogel 2009), the probability of two energy deposits within \({{d}}_{\rm U}(x,y,z)\) and \({{d}}_{{\text{NP}}}(C,x,y,z)\) at the same location can be assumed to be negligible. Therefore, their product term in Eq. 6 can be set to zero such that:

$$\begin{aligned} \displaystyle \langle N_{{\text{total}}} (C,D) \rangle&\approx \alpha \int \frac{{{d}}_{\rm U}(x,y,z)}{V_{{\text{sens}}}}{\text{d}}V + \beta \int \frac{{{d}}_{\rm U}(x,y,z)^{2}}{V_{{\text{sens}}}}{\text{d}}V \nonumber \\&\quad + \alpha \int \frac{{{d}}_{{\text{NP}}}(C,x,y,z)}{V_{{\text{sens}}}}{\text{d}}V + \beta \int \frac{{{d}}_{{\text{NP}}}(C,x,y,z)^{2}}{V_{{\text{sens}}}}{\text{d}}V \nonumber \\&= \langle N_{U} (D) \rangle + \langle N_{{\text{NP}}} (C,D) \rangle \end{aligned}$$

(7)

where \(\langle N_{U} (D) \rangle\) is the mean number of lethal lesion generated via photon interaction within an undoped cellular region, and \(\langle N_{{\text{NP}}} (C,D) \rangle\) is the mean number of lethal lesion generated via high Z NP action within the doped cellular region. Here, \(\langle N_{{\text{NP}}} (C,D) \rangle\) encompasses the lethal lesion generated from direct photon interaction with NPs, secondary electron generated from photon–cellular medium interaction collisions with NPs, and secondary electron/photons generated from photon–NP interactions collision with other NPs. If the spatial distribution of NP uptake within the cell line remains approximately constant with concentration, then from a mechanistic perspective the mean number of lethal lesions generated from these effects can be scaled with average NP density up to a critical saturation threshold (McKinnon et al. 2016). Under these assumptions, Eq. 7 can be manipulated to yield:

$$\begin{aligned} \displaystyle \langle N_{{\text{NP}}} (C,D) \rangle&= \langle N_{{\text{total}}} (C,D) \rangle - \langle N_{\rm{U}} (D) \rangle \nonumber \\&\approx \frac{C}{C_{0}}\left( \langle N_{{\text{total}}} (C_{0},D) \rangle - \langle N_{\rm{U}} (D) \rangle \right) \end{aligned}$$

(8)

where \(\langle N_{{\text{total}}} (C_{0},D) \rangle\) is the mean number of lethal lesions for a given dose *D* at a known reference concentration \(C_{0}\). With this, Eq. 7 can be expressed as:

$$\begin{aligned} \displaystyle \langle N_{{\text{total}}} (C,D) \rangle&= \langle N_{\rm{U}} (D) \rangle + \frac{C}{C_{0}}\left( \langle N_{{\text{total}}} (C_{0},D) \rangle - \langle N_{\rm{U}} (D) \rangle \right) \nonumber \\&= -\log ({\text{SF}}_{\rm{U}}[D]) - \frac{C}{C_{0}}\left( \log ({\text{SF}}_{{\text{total}}}[C_{0},D])-\log ({\text{SF}}_{\rm{U}}[D])\right) \nonumber \\&= \left( \alpha _{\rm{U}}+\frac{C}{C_{0}}\Delta \alpha \right) D + \left( \beta _{\rm{U}}+\frac{C}{C_{0}}\Delta \beta \right) D^{2} \end{aligned}$$

(9)

where \(\Delta \alpha = \alpha _{{\text{total}}}(C_{0}) - \alpha _{\rm{U}}\) and \(\Delta \beta = \beta _{{\text{total}}}(C_{0}) - \beta _{\rm{U}}\). The final form of the interpolation framework is then given via the substitution of Eq. 9 into Eq. 2:

$$\begin{aligned} \displaystyle {\text{SF}}[C,D] = \exp \left( -\left( \alpha _{\rm{U}}+\frac{C}{C_{0}}\Delta \alpha \right) D - \left( \beta _{\rm{U}}+\frac{C}{C_{0}}\Delta \beta \right) D^{2} \right) . \end{aligned}$$

(10)