Cellular signalling effects in high precision radiotherapy

As in previous radiotherapy margin studies (Van Herk et al 2000, van Herk 2004, Selvaraj et al 2013), the irradiation of spherical CTVs was modelled. Dose deliveries are prescribed to a PTV, which is defined as the CTV plus a uniform margin. These volumes are embedded in a larger tissue volume, sized such that the PTV has a margin of at least 30 mm in all directions. For computational purposes, this volume is divided into 1 mm³ voxels.

Spherically symmetric conformal dose distributions were modelled, with radial dose distributions of $D(r)=\frac{{{D}_{0}}}{1+{{\text{e}}^{a\left(r-\left({{r}_{0}}+d\right)\right)}}}$, where D₀ is the prescription dose, r is the distance from the centre of the CTV, and r₀ is the total PTV radius. This gives a sigmoidal dose distribution centred on the target, with 50% dose delivered at a distance of d from the PTV edge. While idealised, these dose distributions are comparable to those which can be delivered clinically through techniques such as VMAT (Selvaraj et al 2013). An illustration of this dose distribution is shown in figure 1.

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The distance d is set to 5 mm, and the steepness of the dose distribution (characterised by a) is set such that the edge of the PTV sees 95% of the prescription dose, which leads to an 80% to 20% penumbral width of 4.8 mm.

Survival incorporating intercellular signalling is investigated using previously published models (McMahon et al 2012, McMahon et al 2013a, 2013b), summarised briefly below.

Within each voxel, cells are exposed to a uniform dose. This causes direct damage (proportional to dose) and induces the secretion of potentially damaging signals. These signals are produced by cells for a time proportional to the physical dose they receive and are assumed to be unstable, being depleted in a simple exponential decay with coefficient λ. In uniformly irradiated populations, this leads to a signal which builds towards an equilibrium concentration before decaying away as cells cease signalling.

Above a certain signal threshold, cells are assumed to suffer the induction of signalling-induced damage at a constant rate. This gives a probability of P_B  =  1  −  e^−κτ for cells experiencing signalling-induced damage, where κ is a response coefficient and τ is the time for which cells are exposed to an above-threshold signal. The level of damage caused is characteristic of the cell-line. Survival is determined from a cumulative damage model. Briefly, both direct radiation-induced and indirect signalling-induced damage translate into a number of microscopic damage events ("hits") within each cell, taken to be Poisson distributed around mean values determined by dose and the time to which cells are exposed to an above-threshold level of signal. This can then potentially lead to cell death either immediately or through mitotic catastrophe at the next cell division, with probabilities that depend on the level of damage within each cell. By calculating average damage levels within each voxel, predictions of cell damage can thus be made. Further detail can be found in previous publications (Partridge 2008, McMahon et al 2012).

In regions which are exposed to modulated fields, signals homogeneously diffuse between voxels. This diffusion leads to shifts in the levels of signalling, and thus cell killing, compared to uniform irradiations. Signal production and diffusion was modelled in a series of timesteps until all cells ceased signalling and the maximum concentration fell below the response threshold, allowing for total exposure times to be calculated for each voxel. These were used to calculate the probability of signalling-induced damage for cells within each voxel, which in turn were used to calculate survival probabilities as outlined above. Input parameters for this study were based on those derived for DU-145 prostate cancer cells (McMahon et al 2013a), giving LQ parameters of α  =  0.15 and $\alpha /\beta =3$ for uniform exposures, as seen experimentally (Butterworth et al 2012).

Signal production rates, ν, were estimated as a value of ν  =  0.05ρ_max min⁻¹ per voxel, where ρ_max is the maximum signal concentration. This is sufficient to drive a response if over 10% of the population has been exposed, as observed experimentally (Butterworth et al 2012). While actual in vitro or in vivo signal concentrations are poorly quantified, in this model survival only depends on the relative values of the maximum signal, production rate and response threshold. Thus, all signal concentrations were modelled in units of the maximum concentration, using a threshold from previous work (McMahon et al 2013a).

The rate of signal diffusion in vivo remains poorly characterised. Here, diffusion is characterised in terms of the equilibrium range, given by $r=\sqrt{D/\lambda}$, where D is the diffusion coefficient. In the examples below, this range is taken as 10 mm, an estimate based on reported measurements of in vivo bystander effects (Mancuso et al 2013). In the cases considered here, predictions would remain unchanged if both this range and physical dimensions are scaled by the same factor.

An analysis of the model's sensitivity to changes in signal range and production parameters is presented in appendix A.

In this analysis, dose delivery is assumed to occur instantaneously, rather than being built up over a period of time through multiple fields as is the case in clinical treatment. While a simplification, experimental evidence suggests that signalling effects are not significantly affected if cells are irradiated using a single acute exposure or a longer exposure delivered using techniques such as IMRT or VMAT (McGarry et al 2012). Additionally, signal kinetics (McMahon et al 2013a) suggest that provided the total treatment duration is smaller than the total signal production time (on the order of 1 h Gy⁻¹ in DU145 cells), the impact of dose rate would be small.

As the model used in this work predicts a survival level for each individual voxel, 'signalling-adjusted' doses were used for comparisons between model predictions and physical plans (McMahon et al 2013b). For a voxel with modelled survival S_i, the signalling-adjusted dose D_S is defined as the dose which would yield the same survival in the limiting case of a large uniform exposure, that is ${{\text{e}}^{-\alpha {{D}_{S}}-\beta {{D}_{S}}2}}={{S}_{i}}$, where α and β are determined from the model's predictions for uniform exposures as noted above.

For comparison between different treatments, CTV Equivalent Uniform Doses (EUDs) were calculated, given as ${{\text{e}}^{-\left(\alpha \text{EUD}+\beta \text{EU}{{\text{D}}^{2}}\right)}}={\sum}^{}{{v}_{i}}{{\text{e}}^{-\left(\alpha {{D}_{i}}+\beta D_{i}^{2}\right)}}$, where v_i is the fraction of the volume exposed to the dose D_i, where D_i can be defined either as the physical dose seen by the voxel or the signalling-adjusted dose predicted by the model, to compare between physical and model predictions.

An illustrative prostate radiotherapy case was analysed to show these effects in a realistic geometry. An initial series of VMAT plans were created with symmetric prostate margins of 0, 3, 5, 10 or 15 mm with a prescription dose of 8 Gy per fraction over 5 fractions. These plans were created using the Eclipse treatment planning system version 11.0.42 (Varian Medical Systems, Palo Alto, CA) for delivery on a Varian Truebeam linear accelerator using a single arc (30 degree collimator rotation) with a 10 MV flattening filter free (FFF) photon beam of dose rate 2400 MU min⁻¹. Plans were normalised so that 98% of the PTV was irradiated to 98% of the prescription dose. An upper limit of 125% was placed on the PTV and was achieved for all margins except for 15 mm. Doses to the urethra, rectum and bladder were minimised as much as possible but the priority was placed on coverage of the PTV. The fraction sizes were scaled to between 2 and 10 Gy per fraction for fractionation analysis.

One of the primary motivations for treatment margins is geometric errors. To investigate their impact on signalling, motion was incorporated in a similar fashion to Selvaraj (Selvaraj et al 2013).

Multi-fraction treatment courses were modelled using a Monte Carlo approach. 100 independent treatment courses were generated for each dose and margin combination to sample possible motion impacts. Within each treatment course, individual fractions were calculated independently, with the position of the CTV shifted to represent both systematic uncertainties (a common offset across all fractions in a given treatment course) and random errors (an additional, independent shift in each fraction). These errors were generated by sampling 3D Gaussian distributions.

The overall impact of motion was quantified by accumulating cell killing across a treatment course, assuming no signal persisted between fractions. This was converted into an EUD per fraction value for ease of comparison. Repopulation was neglected in this analysis, as it would be expected to only provide a constant offset.

Figure 1(b) illustrates the equilibrium signal concentration for targets of various diameters exposed to conformal irradiation with zero margin. Peak concentrations are dependent on target size, with significant fall-off at target edges. For very small volumes (<10 mm) the peak concentration is below the response threshold, preventing any signalling-induced cell killing. Figure 1(c) illustrates this, showing the signal concentration at the edge of the CTV. For irradiations with no margin, signals do not exceed the response threshold until diameters of at least 18 mm. This can lead to significant heterogeneity in cell-killing in these targets, despite uniform physical doses. This effect is reduced by adding margins, leading to higher signal concentrations and better uniformity across the CTV.

Figure 1(d) presents the signalling-adjusted dose volume histograms (DVHs) for such targets exposed to uniform doses of 2 Gy. Despite identical physical DVHs, significant variations occur in modelled cell-killing and thus signalling-adjusted doses. As reported elsewhere (McMahon et al 2012), at these doses intercellular communication drives a large portion of cell killing, so a dramatic reduction in effect is seen in the smallest targets which experience no signalling-driven death. This is mitigated in larger targets due to increasing signalling, but even in larger targets there is a degree of heterogeneity (∼5%) and a loss in overall effect (∼10%) due to reduced signalling compared to uniform doses.

Figure 2 summarises the potential impact of these signalling-driven effects on the CTV EUD following 2 Gy dose delivery to targets of varying sizes and margin prescriptions. For the smallest targets (4 mm) there is a reduction equivalent to nearly a 20% under-dosing of the target volume due to the lack of signalling-driven effects, although even a 1 mm margin significantly abrogated this effect. The scale of this effect reduces gradually with increasing target size, with a 5% reduction in EUD for a 5 cm target irradiated with no additional margins.

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These results suggest that even where therapy could be delivered without geometric errors, there is still a non-trivial benefit of some treatment margin, due to increased cell-killing by signalling from non-tumour cells.

Due to normal tissue toxicities, margins are particularly significant in hypo-fractionated treatments. However, signalling-driven effects are known to saturate at higher doses (Butterworth et al 2011, Butterworth et al 2012), suggesting they may have a reduced importance for larger fractions. Figure 3 presents normalised signalling-adjusted EUDs for varying fraction sizes (i.e. signalling-adjusted EUD divided by the physical plan EUD).

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While similar trends are seen, the scale of the effect is dramatically reduced at higher doses. For zero margin cases, 4 Gy fractions see roughly half the reduction in cell killing at 2 Gy, with the effect reduced to a few percent at the highest doses. This has the additional effect of reducing the sensitivity to margins. While 10 mm margins increased EUDs for 20 mm targets by roughly 5% at 2 Gy, this is on the order of 1% at 4 Gy.

Figure 4 presents the impact of these effects in a clinical prostate radiotherapy plan, planned with zero CTV margin, and a volume equivalent to a target diameter of 48 mm. When compared to physical dose (top), incorporating signalling-driven effects leads to a significant drop in target EUD for 2 Gy fractions (7%, middle), with nearly 20% of the 2 Gy target volume now falling below the prescription dose. However, as fraction size is increased this effect is diminished, and at 8 Gy fractions little impact is seen (bottom, 1%). These differences are highlighted by dose difference plots (right middle & bottom) which show the change in effective dose between signalling-adjusted and physical doses for high-dose voxels (those seeing  >50% of the prescription dose). Outside the target, both fraction sizes see significant increases in cell killing in the lower-dose region immediately surrounding the target. More detailed discussion of signalling effects in low-dose regions can be found in previous work (McMahon et al 2013b).

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Signalling-adjusted EUD analysis was carried out for each of these treatment plans, and is presented in figure 5, normalised to prescription dose as in figure 3. While comparisons between different margin and dosing selections are more difficult due to the increased field complexity needed to deliver treatments with the very smallest margins, the key observations from the simpler model—that the irradiation of tissue outside the target volume or the use of higher doses increases biological effectiveness—remain even in this more complex scenario.

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Figure 6 illustrates the impact of motion in this model. Treatments with an EQD2 of 74 Gy were modelled, either as 37  ×  2 Gy fractions, or fewer fractions (illustrated on the upper x axis) at higher dose per fraction, for a 20 mm diameter CTV with systematic and random uncertainties of 3 mm standard deviation.

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For physical dose distributions alone (figure 6(a)), these results are comparable to previous publications (Selvaraj et al 2013). Geometric uncertainties have a greater impact in hypo-fractionated treatments, due to larger risks of geometric miss and subsequent under-dosing. Margins reduce this effect, but the overall trend remains similar. However, as shown in figure 6(b), signalling effects significantly complicate this analysis.

Because the risk of geometric miss now competes with the loss of signalling-driven killing, normalised EUD no longer strictly decreases with increasing fraction size. Instead, there is a broad maximum at moderate fraction number (5–10, for DU-145 cells), which offers a balance between mitigating signalling effects and offsetting the risk of geometric miss.