Estimation of the oxygen enhancement ratio for charged particle radiation

The substantial benefits of molecular oxygen on treatment response has long been known and clinically exploited. For conventional x-ray therapy, the presence of oxygen can boost cell kill by up to a factor of approximately three (Evans and Koch 2003, Vaupel and Mayer 2007, Wilson and Hay 2011, Grimes et al 2017). This improved kill with oxygen is known as the oxygen enhancement ratio (OER), typically defined as the ratio of cell kill under highly oxygenated conditions relative to the cell kill under anoxia (Hall and Giaccia 2006). The non-linear quasi-hyperbolic variation of OER with oxygen partial pressure is a consequence of the Oxygen Fixation Hypothesis (OFH, (Bertout et al 2008)). Under the OFH model, direct damage by hydroxyl radicals and related products is considered relatively easy to chemically reverse. However, when these radical produces encounter oxygen molecules, they have the capacity to 'fix' damage, preventing easy chemical repair and resulting in improved kill. This has been described mechanistically in recent years (Grimes and Partridge 2015), and as a strictly chemical process manifests in all cell types.

While this is well-understood for x-ray radiation, OER changes markedly as a function of Linear Energy Transfer (LET) in charged particle therapy, decreasing to unity at very high LETs (Hall and Giaccia 2006). The likely reason for this is that under increasing LET, direct damage to DNA due to interactions with charged particles becomes the dominant process, and the contribution of oxygen fixation towards cell-kill diminishes as direct kill increases (Grimes et al 2017). A major appeal of charged particle therapy is that high doses can be very effectively delivered to chosen regions, whilst largely sparing healthy tissue. At the target site, direct kill typically suffices so that oxygen enhancement is not required. However, LET is non-zero along the particle range, and can have interactions with oxygen which would increase the cell-kill along the particle range, potentially resulting in increased damage to healthy physoxic cells in many instances. To the author's knowledge, this is a facet of particle therapy that has not yet been extensively studied, and may have ramifications for treatment. In this work, we establish an empirical model for this process, and explore the implications of this effect on therapy.

For the interaction of x-ray photons, a model has been previously derived (Grimes and Partridge 2015) which quantifies the relationship between OER and oxygen tension, given by

$$\begin{eqnarray} &&OER(p) = 1 + \frac{\phi_O}{\phi_D}\left(1 - e^{- \varphi p} \right) \end{eqnarray} \tag{ 1 }$$

where p is the oxygen partial pressure. The rate constant ϕ can be theoretically derived from the thermal velocity of oxygen molecules in water, the mean-free path of oxygen molecules and the probability of an interaction between oxygen molecule and ionized DNA, and has previously been measured to be 0.26 ± 0.02 mmHg⁻¹ (Grimes and Partridge 2015). Oxygen partial pressure here is measured in units of mmHg, in line with historical medical convention, but this can readily be converted to other units if required (Grimes et al 2014). In this model, φ_O denotes the fraction of cells killed under oxic conditions and φ_D those killed by direct interaction, which have been experimentally determined previously for x-ray photons, with ϕ having no known dependence on LET. Here we derive a method to empirically estimate how this will vary under changing LET. We initially assume that cell kill events are independent, and direct cell kill events (when a charged particle impinges directly on DNA) occur at a rate χ_D per particle LET. Indirect cell kill occurs where a charged particle means an oxygen molecule under OFH, at a rate χ_I per particle LET. We initially assume these two processes are independent for simplicity. In this case, the probability of these processes occuring and resulting in a cell-kill event are given by Poisson statistics as

$$\begin{eqnarray} &&p_{D}(L) = \chi_{D}L\exp\left(-\chi_{D}L \right) \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} &&p_{I}(L) = \chi_{I}L\exp\left(-\chi_{I}L \right) \end{eqnarray} \tag{ 3 }$$

where L is LET in keV/µm. The number of cells killed is a function of of particle fluence along the track length, and the LET specific cell kill probability. This complex expression however does not have to be explicitly calculated here. Instead, we note that the fraction of cells killed under oxic conditions over direct kill is congruent to the fraction of indirect kill probability over direct, so that

$$\begin{eqnarray} &&\frac{\phi_{O}}{\phi_{D}} = \frac{p_{I}}{p_{D}} = \frac{\chi_{I}}{\chi_{D}}\exp \left(-L(\chi_{I} - \chi_{D})\right). \end{eqnarray}$$

In this case, we may re-write equation (1) empirically in terms of LET and oxygen partial pressure as

$$\begin{eqnarray} &&OER(L,p) = 1 + \left(\frac{\chi_{I}}{\chi_{D}}\exp \left(-L(\chi_{I} - \chi_{D})\right)\right)\left(1 - e^{- \varphi p} \right). \end{eqnarray} \tag{ 5 }$$

2.1. Parameter fitting

2.2. OER variation with range for proton and carbon-ion therapy

Model fitting results are given in table 2, and shown over the various data sets in figure 1. The weighted fit achieved a higher co-efficient of determination than the unweighted approach. Weighted fit values were used to estimate the maximum OER achievable with protons and carbon ions of varying energy along their track, as depicted in figure 2.

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The model derived herein suggests that for protons even up to relatively high energy, the maximum OER is still close to the theoretical maximum, and typically above 2.74 until right before the terminus of the track. Carbon ions had lower maximum OERs than protons, due to their significantly higher LET along the path length, which also fell rapidly on approach the terminal end of the particle's path. However, along most of the path, OER remained close to maximum value, typically over 2.5 for the energies considered in this work.

This has potential clinical implications - as charged particles traverse physoxic tissue, their effective cell kill is amplified by the oxygen enhancement ratio possible. As healthy tissue is typically well-oxygenated, then the potential for increased cell-kill of healthy tissue along the path should be considered. OER for different LET can be explicitly taken from the model outlined here, which is itself a multiplier for projected cell-kill. This suggests that without correction, it is likely that dose to healthy tissue would be under-estimated.

The model outlined here is empirical, and not explicitly mechanistic. One might reasonably enquire as to whether the derived parameters are physically realistic, or simply phenomenological. It's worth examining the parameters χ_D and χ_I to ascertain whether they have any physical significance. The probability functions outlined in equations (2) and 3 are plotted in figure 3. Function maximums occur where $\frac{\partial p_{D}}{dL} = 0$ and $\frac{\partial p_{I}}{dL} = 0$, respectively, so that the values of L which maximises p_D and p_I are $\chi_{D}^{-1}$ and $\chi_{I}^{-1}$. This yields a maximum probability for direct kill at 99.4 keV / µ m (95% confidence interval: 91.2 -109.2 keV / µ m). This agrees well with literature values for optimal cell-kill of 100 keV / µ m ((Joiner and Van der Kogel 2009), (Hasan 2019)).

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It is important to note that there are limitations to such a model - because it is not mechanistic, there are other physical processes that might result in more nuanced effects at certain energies. Nor does this approach explicitly address the impacts of varying oxygen on predicted α / β ratios, which can be empirically estimated by other means (Wenzl and Wilkens 2011). The chief benefit of this simple model is that it provides a basis for rapid estimation of likely OER for different LET and partial pressure, generalising previous work (Grimes and Partridge 2015) on photons to account for charged particles. This makes it useful as a basis for simulations of dose to healthy and tumour tissue, or a starting point for more mechanistic investigation of oxygen and charged particle interactions on therapy outcome.

The author would like to profoundly thank Drs Tatiana Wenzl and Jan J Wilkens for kindly sharing the raw OER data referred to in this work.