Phenomenological toy model for flash effect in proton therapy

Abstract

We present a simple model based on general assumptions for the FLASH effect in radiotherapy, leading to a rate equation with only three free parameters. The model can predict the biological effect ratio between healthy and tumoral tissue for arbitrary input treatments, given as a dose rate versus time function. We analyze the behavior of the model and its sensitivity to its free parameters, and decide on suitable parameter values in accordance with available experimental data from the literature. Then we apply our model to study different sets of treatments, modeled as square pulse periodic functions with different pulse peak dose rate, pulse width and repetition period, in order to illustrate how it may be used to guide future experiment design. The model predicts that, for a given average dose rate above the FLASH threshold, a more prominent FLASH effect would be observed for continuous beams than for ultra-pulsated beams with an infinitely short irradiation time. This finding needs to be validated with suitable experiments.

1 Introduction

The FLASH effect, or the sparing of healthy tissues by ultra-high dose rate (UHDR) radiotherapy which, in turn, maintains tumor control, has been at the center of the stage in radiation oncology since its discovery [1]. The field of radiation oncology is collectively pushing for a deeper understanding of the FLASH effect while, at the same time, making a conscious effort to keep up-to-date with the state of the art in Physics [2] and Biology [3, 4]. Proven under different conditions, but lacking a systematic assessment of the effect, the full understanding of its underlying mechanism remains elusive so far. Synergies between FLASH and other novel radiotherapy technologies (such as minibeams) have also been considered [5].

In this context, several models have recently been proposed to predict the extent of the FLASH effect in different conditions. Two nearly-equivalent quantities have been proposed for this purpose: the FLASH sparing effect, or FSE, defined as a ratio of doses D_FLASH/D_CONV for isoeffect; and its nemesis, the Flash Modification Factor, or FMF, equivalent to the concept of RBE or radiobiological effectiveness. Their dependence as a function of dose, LET, average dose rate and instantaneous dose rate, for different biological conditions and oxygenation levels, is indeed subject to discussion.

Modeling, in this context, is necessary for three reasons. First, it constitutes a fair attempt to understand or describe the underlying mechanisms for the FLASH effect [6,7,8,9]. Second, models establish a dialog with experiments as they are not only validated or refuted by them, but they are also used to design future experiments [10,11,12]. Finally, in a field as applied as radiation oncology, some models are designed to guide future clinical implementation, particularly in proton therapy facilities, some have investigated on the minimal FLASH sparing effect needed to compensate the increase in radiobiological damage due to hypofractionation for late-reacting tissues [13], while other have focused on the treatment-planning side of potential patient FLASH facilities based on pencil-beam scanning [14, 15].

Models investigate different physical, chemical and biological mechanisms of radiation damage such as quantification of oxygen depletion [6], inter-track effects [16] or the effect of pulsation schemes at equivalent dose rates [17] during FLASH irradiation. Different in vivo and in-vitro endpoints are modeled; defining for example different RBE quantities for survival and/or mutation induction [10]. In most cases, molecular oxygen is believed to be a strong mediator for the FLASH effect, discussing the role of oxygen tension in tissues [18, 19] and the impact of hypoxia on LET and RBE relationships, namely at the oxygen enhancement ratio [20], 12].

This paper aims at setting forward a very simple phenomenological model which may help to guide experiments and compare to detailed simulations, so as to help identify the biochemical mechanisms which stand as the best candidates to explain the FLASH effect.

Section 2 presents the model, based on a few simple facts and assuming always the simplest possible description which is logically consistent with the boundary conditions of the problem and the known facts (i.e., that there is some sparing effect which activates at high dose rate). In addition we present in this section what arguments make the model plausible as a basis to explore the problem, based on what is known about the biochemical boundary conditions as studied in previous works. Finally this section describes the mathematical calculations necessary to perform predictions in a practical way based on the model. This is done based on an analytical approach (i.e., solving the model differential equation) and on numerical calculations.

Section 3 explores the sensitivity of the model to the values of its free parameters, illustrating how it operates and how its parameters may be determined by comparison to experimental data. At the end of the section this comparison is explicitly made based on recent publications with experimental results.

The purpose of Sect. 4 is to showcase how the model could be used to guide future experiment design. This is done by defining series of experiments in which the time-pattern conditions of the input radiation treatment are varied, while keeping all other elements unchanged. It is shown how a systematic experimental approach guided by the model could allow for further insight into the mechanisms underlying the FLASH effect.

Finally, Sect. 5 presents some conclusions and an outlook to future work developing further these ideas.

2 The model

The main hypothesis is to consider that all the relevant damage to biological tissue, from the point of view of explaining the FLASH effect, proceeds via an intermediate population of a suitable radical(s), activated via radiation and decaying in several ways, some of which produce the biological damage. This may be oversimplifying but is probably a reasonable hypothesis if one wants to focus on the different effects on healthy and tumoral tissue depending on the dose rate (FLASH effect).

The radical concentration is referred to as c(t). In this work we take an average value of c over the whole volume affected by the irradiation, assuming we find ourselves in the chemical (homogeneous) phase of the interaction, as is done in similar studies [7]. To maintain the option of extending the model to a inhomogeneous field c(x, y, z, t) in future work, partial derivative notation will be used systematically.

The rate equation proposed for c(t) in its most general form is:

$$\frac{\partial c(t)}{\partial t}= \frac{{\dot{D}}(t)}{{D}_{0}} \left(1-c(t)\right)- \frac{c(t)}{{\tau }_{r}}- \frac{c(t)}{{\tau }_{d}}- \frac{f\left(c(t)\right)}{{\tau }_{d}}$$

(1)

c(t) is defined in the interval [0,1], that is, it represents a concentration relative to a given saturation level.

${{\dot{D}}}$(t) stands for the instantaneous dose rate on the sample (in Gy/s). Eventually ${\dot{D}}$(t) may be taken as a field Φ(x,y,z; t), as mentioned above, describing the 3D nature of the irradiation field. In such a case one should add to Eq. (1) a suitable diffusive term and include the track structure of the irradiation beam.

${\text{D}}_{0}$ is the first parameter of the model, having the dimensions of dose (Gy).

τ_r and τ_d are the characteristic timescales for decay without damage and decay involving damage, respectively, of the radicals expressed by c. Multiple chemical reactions have been considered in previous works [7, 21], including recombination of radicals (i.e., without damage). In this toy model we consider only these two options so as to obtain the simplest possible description which still retains the most important dynamics of the problem.

f(c) is a function of c to be defined below, linked to a cooperative process generated by the radicals expressed by c and yielding a type of damage specific to tumoral tissue. As this term makes the difference between healthy and tumoral tissue, it will be called in the following differential term. The expression proposed for f(c) is

$$f\left(c(t)\right)={a^{\prime} [c(t)]}^{n}$$

(2)

as it will be explained below. This involves a new model parameter, a´, which is dimensionless, and which encodes the strength of the differential effect between healthy and tumoral tissue. It also involves an exponent n, which relates to the order of the cooperative process behind the differential effect. The simplest choice is n = 2, which would correspond to a two-site cooperative process and which will be used in what follows. It is beyond the scope of this paper which specific biological process may be behind this hypothesis. As an examples any of the order two reactions included in [7, 21] could play this role.

We will further simplify the model by defining τ as 2/τ = 1/τ_d + 1/τ_r and redefining the constant in the quadratic term as a = a´ τ/ τ_d. This leads to a more compact version of (1) as follows:

$$\frac{\partial c(t)}{\partial t}= \frac{{\dot{D}}(t)}{{D}_{0}} \left(1-c(t)\right)- \frac{2c(t)}{\tau }- \frac{{a [c(t)]}^{2}}{\tau }$$

(3)

The rate Eq. (3) for the function c(t) allows to track the evolution of this quantity as a function of time for an arbitrary input function ${\dot{D}}$(t), based on the three free parameters $: {\text{D}}_{0}$ (Gy units), ${\tau }$ (time units) and a (adimensional). Once the irradiation finishes c(t) eventually decays to zero. Then the total damage to healthy or tumoral tissue may be calculated as being proportional to areas under the curve (AUC). Then the ratio of biological effect for the two types of tissue normalized to its value for low dose rate is described by

$$r={\int }_{0}^{\infty }\frac{c(t)}{{\tau }_{d}} \text{d}t/{\int }_{0}^{\infty }\frac{c\left(t\right)+{a c}^{2}\left(t\right)}{{\tau }_{d}} \text{d}t$$

(4)

The numerator and denominator of Eq. (4) are a simplified proxy calculation for the biological effects for healthy and tumoral tissue. The way the ratio is defined guarantees that it tends to unity for very low dose rate, as in that regime quadratic term in the integrand of the denominator is negligible. In practice we will only be interested on how the ratio of biological effect for the two types of tissue evolves with dose rate. Therefore the numerator and denominator alone are not considered as a realistic estimate of the actual biological damage, as in Eq. (3) saturation is introduced as an ad-hoc mathematical artifact to avoid that the sparing of healthy tissue grows in an unbounded way. Necessarily this saturation term leads to a non-realistic behavior of numerator and denominator of Eq. (4) separately, but still their ratio, r, is well behaved and grasps the most important features of the problem under study.

During the rest of the paper we will study r, which is equal to unity when there is no dose rate induced sparing of healthy tissue (i.e., at low dose rate values) and below unity when some dose rate induced sparing is present. At very high dose rate r is closely related to the FMF.

Analyzing the rate equation term by term one sees that the first one, $\frac{{\dot{D}}(t)}{{\text{D}}_{0}} \left(1-c\right)$, is the source for the function c(t), and it has been taken proportional to the instantaneous input of radiation dose, including a multiplicative factor (1-c) which guarantees saturation at c = 1, according to the definition of c. We have included this saturation condition so as to guarantee that the model behaves properly for ultra-high dose rate conditions, as it will be explained later in the text and as supported by experimental evidence (e.g., [3, 22, 23]. The model parameter present in this term, D₀, has the proper dimensionality to make the equation consistent.

The second term is of the type $- \frac{c}{\tau }$, that is, a standard decay (drain) term, just proportional to the concentration c and with a characteristic timescale. Notice at this point that the additional decay term in (1) which does not produce damage, $- \frac{c}{{\tau }_{r}}$, does not play any role in the dynamics of the system, other than renormalizing the total damage produced at the end of a treatment. Indeed, the two linear decay terms in (1) may be combined to a single one, with an effective decay time, and it is easily derived that a fraction of the decays produces damage, whereas the rest does not, resulting that the total damage produced is just rescaled by the ratio between the two time constants, τ_r and τ_d. As the damage calculated is an adimensional quantity, where only comparisons between two different experiments are meaningful, one may safely merge the two decay terms (with and without damage) into a single one, without loss of generality, and redefine 2/τ = 1/τ_d + 1/τ_r, as it has been done in (3).

The third term, $- \frac{{a c}^{2}}{\tau }$, is the most important one in Eq. (3). It is inspired by the fact that several chemical reactions relevant to radiation-induced biological damage, as detailed, for example, in [7] or in [21], are reactions of order higher than one, so that the reaction rate is proportional to a power of the concentration larger than one (the simplest case being two). The rationale for having this term in the equation, representing some process which affects tumoral tissue and not healthy one, is that, being proportional to a power of c larger than unity, it contributes very little to the system dynamics for small values of c, whereas it becomes relevant as c is larger. These are the key ingredients which should generate the right dynamics: no differential effect when c is very small (i.e., for input dose rates small as compared to the ratio D₀/τ); relevant differential effect when c becomes larger, which may only happen at higher dose rates sustained for a long enough period.

The model may be seen as the simplest version of a rate equation based approach to the problem of the FLASH effect, compatible with a wide variety of possible underlying processes at the biochemical level. It is based on applying Eq. (3) to trace the evolution of the function c(t) for an input dose rate function ${{\dot{D}}}$(t), allowing for calculation of the biological damage ratio between healthy and tumoral tissue as a ratio of the integrals corresponding to the proper terms in the right hand side of Eq. (3).

The model may be solved numerically for a completely arbitrary choice of the source function ${\dot{D}}$(t). However, in practice, one is interested on a somewhat simple family of source functions, which may be taken as periodic square waves with a finite total duration. These may be defined with just a few parameters: peak dose rate $({\dot{D}}$_p), pulse width (t_p), pulse repetition period (T) and total number of pulses (N_p). Notice that when this type of input function is chosen, the source term becomes piecewise constant and thus the differential equation is substantially simplified, admitting an analytical solution.

This approach allows us to rewrite the rate equation for each of these intervals and following an iterative process over successive time intervals, we obtain, in general, for an interval $\left({t}_{i},{t}_{i+1}\right)$ the following differential equation to solve:

$$\left\{\begin{array}{c}{c}{\prime}\left(t\right)=A{c}^{2}\left(t\right)+ {B}_{i}c\left(t\right)+ {C}_{i }\; {\textit{for}} \; t\in ({t}_{i},{t}_{i+1})\\ c\left({t}_{i}\right)={c}_{i}\end{array}\right.$$

(5)

which for each part $({t}_{i},{t}_{i+1})$ has a constant value for the source term ${C}_{i}$, and consequently, also for ${B}_{i}$; $A$ is constant and takes the same value for all intervals. The expressions for these coefficients and the analytical solution for the differential equation can be found in detail in Appendix A.

Figure 1a (lower) illustrates how the function c(t) evolves with time as the input radiation comes in (upper) for a given set of standard conditions as given in the figure caption. The points in the lower plot corresponding to the time intervals where the source function is not zero (see the upper plot) are given in red so as to help follow the graph, whereas the intervals with zero source term are shown in black. The function c(t) starts at zero and quickly grows to a value of a few percent with the first pulse. Then it decays until the next pulse arrives. After the second pulse a stationary regime is already achieved and the function becomes practically periodic. We have checked that same results for the time evolution of the function c(t) are obtained by solving Eq. (3) analytically, as explained in this section, and numerically. The calculations have been implemented in Fortran code (numerical method) and Matlab (analytical). The steps for solving the differential equation in the numerical method and for integration in both methods have been chosen small enough so as to get no variation upon further refinement. A systematic comparison of the two independent calculations (analytical and numerical) has been performed to guarantee the robustness of the results presented below.

Fig. 1

a Time evolution of the function c(t) (lower) for a treatment with a periodic beam of squared wave pulses (upper). Pulse width is t_p = 10 ns, peak pulse dose rate is ${\dot{D}}$ _p = 10² Gy s⁻¹ and pulse repetition period is T = 1 µs. Therefore the mean dose rate is 1 Gy/s. The free parameters of the model are tentatively set to: D₀/τ = 40 Gy/s; τ = 1 µs; a = 0.1 (cf. Section 3). Points are shown in red during the time that the beam is on and in black while the beam is off (lower), whereas the time profile of the beam in explicitly given (upper) for reference. b Time evolution of the function c(t) for a treatment with a periodic beam of squared wave pulses (upper). Pulse width is t_p = 10 ns, peak pulse dose rate is ${\dot{D}}$ _p = 10² Gy s⁻¹ and pulse repetition period is T = 1 µs. Therefore the mean dose rate is 1 Gy/s. The free parameters of the model are tentatively set to: D₀/τ = 40 Gy/s; τ = 1 ms; a = 0.1. The inset zooms into the small values of the time variable

As Fig. 1a corresponds to a very specific choice of the parameter τ, we have redone the calculation for τ being three orders of magnitude larger (i.e., 1 ms). In this case the transient behavior of the system to reach a stationary regime takes above one thousand pulses. The scale chosen for the x axis does not allow anymore to distinguish the reaction to single pulses (which may be seen by the zoom at small time values given in the inset), but highlights in this case how the function c(t) takes a time of the order of τ to stabilize. In the subsequent calculations we have arbitrarily chosen to retain τ = 1 µs to illustrate how the model works, but we have included again the option τ = 1 ms in Fig. 7.

The damage ratio between healthy and tumoral can be calculated by Eq. (4). The behavior of r as a function of the mean dose rate has been evaluated for a series of treatments. For each of the treatments we have taken a train of squared pulses with the same total dose and the same values of the pulse width t_p and pulse repetition period T. The value of the peak dose rate ${\dot{D}}$_p is varied from treatment to treatment and r is calculated for each of the cases and represented as a function of the average dose rate of the corresponding treatment. Notice that the average dose rate can be simply calculated as

$$<{\dot{D}}>= {{\dot{D}}}_{p} \frac{{t}_{p}}{T}$$

(6)

The result is shown in Fig. 2, where the model parameters have been fixed to the reference values given in the figure caption. The values chosen may be considered at this stage as a starting point hypothesis, but will be justified as a reasonable choice in the following section based on the experimental evidence available in the literature.

Fig. 2

r as a function of the average dose rate ($<{\dot{D}}>)$ for a set of treatments with a periodic beam of squared wave pulses and the same total dose. Pulse width and pulse repetition period are fixed to 10 ns and 1 µs, respectively. Peak pulse dose rate ${{\dot{D}}}_{p}$ is varied in the range 10²–10⁵ Gy/s to span the range of average dose rates shown in the plot. The free parameters of the model are set to: D₀/τ = 40 Gy/s, τ = 1 µs; a = 0.1

3 Study of the model: parameters and comparison to literature experimental data

In this section we study in further detail the model proposed, starting out from Fig. 2 and making emphasis on its free parameters and the sensitivity of the model predictions to their variation. This study will be performed by concentration on calculations for series of experiments in which the ratio r is calculated as a function of the dose rate for different values of the model parameters, as exemplified by Fig. 2. Finally we will revise the experimental evidence available in the literature to justify a reasonable choice of parameter values for the subsequent calculations presented in the following section.

In Fig. 2 one may observe that for low dose rates, r is very close to unity. When the average dose rate reaches a value comparable to D₀/τ, r starts to decrease until it reaches an asymptotic value close to r = r_∞ = 0.95. In this asymptotic regime saturation comes from the fact that c(t) reaches values close to unity for each of the incoming pulses. As the duration of the pulses is much shorter than the time that the beam is off (i.e., t_p <  < T), and as the quadratic term in the rate equation is small compared to the linear decay term, one may approximate r assuming that c(t) decays exponentially once the pulse is off. Then, if t_p <  < τ, T, the AUCs (numerator and denominator in Eq. (4)) can be calculated by neglecting the contribution of the interval where the beam is on, and therefore r behaves as a ratio of integrals of simple exponentials. For small values of a one may expand to first order in a to obtain:

$${\text{r}}_{\infty } \approx {1}{-}{\text{a}}/{2}$$

(7)

which is very well reproduced by the calculation. This implies that an experimental measurement of a set of treatments with the same squared pulse pattern where only the peak dose rate is varied, and where damage ratio from healthy to tumoral tissue is measured for each of the treatments, with t_p/ τ_d <  < 1 and t_p/T <  < 1, would determine the value of a directly from the asymptotic value of the damage ratio (r_∞).

Figure 3 further illustrates this behavior by reproducing the calculated results of the same set of experiments presented in Fig. 2, now with three different hypotheses for the value of a. The calculated behavior of the system confirms the trend given by Eq. (7). The system is not sensitive to changes in the a parameter for conventional dose rates, but gives rise to important differences as the dose rate approaches the region one would identify as FLASH.

Fig. 3

r as a function of $<{\dot{D}}>$ for three sets of treatments with a periodic beam of squared wave pulses and the same total dose. Pulse width and pulse repetition period are fixed to 10 ns and 1 µs, respectively. Peak pulse dose rate ${{\dot{D}}}_{p}$ ranges between 10²–10⁵ Gy/s to span the range of mean dose rates shown in the plot. The free parameters of the model are set to: D₀/τ = 40 Gy/s, τ = 1 µs; a = 0.1 (black squares)/0.15 (red circles) /0.2 (blue triangles)

Figure 4 explores the role of the parameter D₀. By inspecting the rate equation, it is easy to see that the dynamics of c(t) with respect to the dose rate scale with the ratio D₀/τ, i.e., the differential equation has the same solution if the input dose rate function is multiplied by the same factor as D₀/τ. This is illustrated in Fig. 4, where r is plotted for three sets of treatments. For each set a different value of D₀/τ is fixed and then the only parameter which is varied within the set is the peak dose rate. The three series illustrate the scaling as explained above. This implies that the same set of measurements described above for Fig. 3 could also be used to derive experimentally the value of the ratio D₀/τ. This parameter can be interpreted as the average dose rate for which FLASH effects appear to be relevant.

Fig. 4

r as a function of $<{\dot{D}}>$ for three sets of treatments with a periodic beam of squared wave pulses and the same total dose. Pulse width and pulse repetition period are fixed to 10 ns and 1 µs, respectively. Peak pulse dose rate ${{\dot{D}}}_{p}$ is varied in the range 10²–10⁵ Gy/s to span the range of mean dose rates shown in the plot. The free parameters of the model are set to: D₀/τ = 10 (black squares) /40 (red circles)/150 Gy/s (blue triangles), τ = 1 µs; a = 0.1

The choice of parameters for the model is based on evidence from previous studies, both in vivo and theoretical. For example, at [22], a systematic survey of available data is presented, showing that r at very high dose rates reaches values in the range 0.92–0.95, for total doses below 25 Gy. In [3] many qualitative results are presented and, specifically for protons, results indicating an FMF at the level of 0.95 are also given, although in this case the error bars are limiting the robustness of the conclusion. This leads, according to Eq. (7) to the choice a = 0.1. At [23] G-values (production yields per deposited energy) of suitable chemical species are used as a proxy for dose rate dependent biological damage, yielding G-value vs. dose rate experimental curves with a similar behavior to the one shown in the figures above. We consider all these results a good basis to use the model presented in this paper as guideline to explore and design future experiments in which the input treatment parameters are varied in different ways so as to gain further insight into the FLASH effect.

Since a value of 40 Gy/s is often given as a threshold for FLASH effect reproduced in in vivo experiments, we have chosen this same value for our parameter D₀/τ. To select individual values for D₀ and τ, we consider the characteristic time τ = 1 µs, similar to the kinetic constant of many relevant reactions and often used as a threshold for the homogeneous phase of the reaction process [17, 7]. Therefore, to maintain a dose rate threshold D₀/τ = 40 Gy/s, we select a value of D₀ = 4 10^–5 Gy. As indicated above we have also included some simulations with τ = 1 ms, which is the timescale most likely involved in the relevant biological damage processes. Studying these two extreme cases of the parameter τ is important to characterize fully the model and understand its behavior for further usage.

4 Discussion and predictions of the model

One of the key assumptions leading to Eq. (7) is the fact that the ratio between peak width and characteristic time, t_p/ τ <  < 1 and therefore the integrals in the numerator and denominator of the saturation ratio, r_∞, are fully dominated by the time interval where the beam is off, making the contribution of the beam-on time negligible. This is very easy to visualize in Fig. 1 above. If one assumes that this beam-on period contributes in the saturation regime proportionally to the peak width t_p, that is, that the saturation value c(t) = 1 is reached in a time much shorter than t_p and therefore c(t) may be replaced by unity during the full beam-on interval, then r_∞ may be expanded retaining the first order in both a and the cross product a t_p/τ yielding.

$$r_{\infty } \approx {1}{-}({1}\, + \,{\text{t}}_{{\text{p}}} /\tau ){\text{ a}}/{2}$$

(8)

which reduces to Eq. (7) when t_p/ τ ≈ 0.

Equation (8) suggests that a set of experiments may be designed to separately determine the value of the characteristic time τ and thus allowing for the full determination of the model parameters based on carefully designed experiments. Therefore, a very simplified model, based on general assumptions and leaving aside the specific biochemical channels behind the process under investigation might lead to obtaining this very important chemical parameter, τ, which could allow for comparing to detailed MC simulations and help matching phenomenology with the detailed biochemical underlying processes.

Figure 5 describes three sets of experiments, with the model parameters fixed to specific values as given in the figure caption, and where the three series are identical, except for the fact that t_p has different values for each of them. For the series marked with black full squares and red full circles the t_p/ τ ≈ 0 approximation holds very well, and then r_∞ depends only on a, as described in Eq. (7). However, for the series represented with blue full triangles t_p/ τ is comparable to a and thus Eq. (8) must be used instead. The graph makes evident that the saturation regime now starts to depend on the value of t_p, deviating slightly from the estimate given by Eq. (7). Therefore, a more elaborate set of experiments as the one described in Fig. 5 could eventually allow for determining the third parameter of the model and thus lead to a full picture, which may then be used for predictive modeling. Notice that in practical terms it may be very difficult to obtain conclusive results when performing the type of experiments suggested here, as error bars are likely to be larger than the differences predicted by the model calculation. Comparing model predictions to the actual experimental possibilities may be one of the benefits of this work, so as to choose those experimental options which are more promising to reach robust conclusions.

Fig. 5

r as a function of $<{\dot{D}}>$ for three sets of treatments with a periodic beam of squared wave pulses and the same total dose. Pulse repetition period is fixed to 1 µs. Pulse duration is 1 (black squares) /10 (red circles)/100 ns (blue triangles). Peak pulse dose rate ${{\dot{D}}}_{p}$ is varied to span the range of mean dose rates shown in the plot. The free parameters of the model are set to: D₀/τ = 40 Gy/s, τ = 1 µs; a = 0.1

As a complement to the previous figures, Fig. 6 shows how the response of the system would change if several series of measurements would be done with a different pulse repetition period for each of them. As expected, when the pulse repetition period T becomes much shorter than the characteristic time τ, the function c(t) does not have the time to decay fully after each pulse, and the system remains most of the time at higher values of c(t), yielding a more prominent role of the non-linear term in Eq. (3) and thus producing an increasing sparing effect as T is shorter and shorter. Notice at this point that what is relevant is the ratio between T and τ. Shorter T is equivalent to longer τ from this point of view, as it may be seen below (Fig. 7 and text therein). Eventually one may extrapolate this to the limiting case where the full dose is delivered in a single pulse, and where the sparing would be maximized, according to this model, for a given value of the average dose rate. On the other hand, for dose rates below the saturation regime there is a complex interplay between the parameters T and τ. As the pulse repetition period T becomes larger than τ, the response of the system to the input radiation pulses should start to be independent of how much time it takes for the next pulse to come and therefore the actual response of the system, in terms of r, would depend on the total dose per pulse and not on T. As r is plotted as a function of average dose rate, the r curve drifts linearly toward the left as T becomes higher in this regime. The shape of the r dependence on $<{\dot{D}}>$ is particularly interesting in this case, as the point where two of the series with different T values cross provides a very natural reference to benchmark the behavior of the system. Notice that in this case differences are more pronounced for the high dose rate region that in the case of Fig. 5. Therefore, the options of obtaining conclusive experimental results are more promising.

Fig. 6

r as a function of $<{\dot{D}}>$ for five sets of treatments with a periodic beam of squared wave pulses and the same total dose. Pulse duration is fixed to 10 ns. Pulse repetition period is 0.2 (downwards yellow triangles)/0.5 (green diamonds)/1 (black squares)/1.5 (red circles)/2 µs (upright blue triangles). Peak pulse dose rate ${{\dot{D}}}_{p}$ is varied to span the range of average dose rates shown in the plot. The free parameters of the model are set to: D₀/τ = 40 Gy/s, τ = 1 µs; a = 0.1

Fig. 7

r as a function of $<{\dot{D}}>$ for four sets of treatments with a periodic beam of squared wave pulses and the same total dose. Pulse width and pulse repetition period are fixed to 10 ns and 1 µs, respectively. Peak pulse dose rate ${\dot{D}}_{p}$ is varied to span the range of average dose rates shown in the plot. The free parameters of the model are set to: D₀/τ = 40 Gy/s, τ = 0.5 (black squares) /1 (red circles)/2 µs (blue triangles)/1 ms (green diamonds); a = 0.1

Figure 7 illustrates the interplay of T and τ from a different perspective, now giving four options for the hypothesized value of τ and keeping T fixed. The conclusions are the same as those extracted from Fig. 6. Therefore, in order to draw robust conclusions from experiments along the arguments given above in this work there is a clear indication that T should be chosen to be large enough, so as to guarantee that the interplay with τ is avoided and the approximations given by Eqs. (7) or (8) hold. Alternatively, if τ is much larger (as in one of the cases included in Fig. 7) one may still see a correct behavior of the system and use the comparison between model results and experimental data to determine the other model parameters. Notice that the larger τ is, the more intense and sharp is the change in sparing of the system as a function of the dose rate.

Finally, Fig. 8 shows the same series already given in Fig. 2, where the variation in mean dose rate was obtained by varying the peak dose rate, now complemented with two additional series in which the same dose rate variation is obtained by varying now the peak width and the pulse repetition period. These three series can be seen as three alternative ways of performing the same experiment, when one has full control over the experimental parameters of the input radiation given by the accelerator. The fact that, as explained in the previous paragraphs, the system behaves in a slightly different way depending on which of the input radiation parameters is used to vary the average dose rate is one of the key results of this work. Notice, in particular, that when pulse width and pulse repetition period are varied, the largest value of the average dose rate corresponds to the limit of a continuous beam, i.e., the end of each pulse coincides with the beginning of the next one. Whereas varying of the peak dose rate generates a plateau for the FLASH regime, the model predicts that if one varies either the pulse duration or pulse repetition period, leaving all other treatment parameters fixed, then the response continues varying and generating larger sparing effect up to much larger dose rate values, all the way up to the continuous beam limit. Notice that such limit is in practice a single-shot, ultrafast treatment.

Fig. 8

r as a function of $<{\dot{D}}>$ for three sets of treatments with a periodic beam of squared wave pulses and the same total dose. The range of average dose rates shown in the plot are obtained by varying peak dose rate in the interval 10²–10⁵ Gy/s with t_p = 10 ns and T = 1µs (purple latin crosses, +), pulse duration in the interval 1 ns—1µs with ${{\dot{D}}}_{p}$ = 10³ Gy/s and T = 1µs (green Greek crosses, x) and pulse repetition period in the interval 1 ns—1µs (notice that in this case 1 ns corresponds to the highest dose rate) with ${{\dot{D}}}_{p}$ = 10³ Gy/s and t_p = 1 ns (blue stars, *). The free parameters of the model are set to: D₀/τ = 40 Gy/s, τ = 1 µs; a = 0.1

Therefore, the model predicts that for a fixed mean dose rate it is more efficient (in order to maximize the FLASH effect) to have a smaller ratio between pulse width t_p and pulse repetition period T than to have higher values of the peak pulse dose rate ${{\dot{D}}}_{p}$. This result, which may at first sight look counter-intuitive, emerges naturally from the saturation feature of Eq. (3), implemented in its first term $\frac{{\dot{D}}(t)}{{\text{D}}_{0}} \left(1-c\right)$. When the system is in the saturation regime (i.e., for very high average dose rate) increasing further the value of the peak dose rate does not have any further effect, as the function c(t) can not grow beyond its saturation value. However, increasing the duration of the pulse relative to the pulse repetition period increases the fraction of the total treatment time where the system stays at saturation. This feature of the model gives rise to different predictions from those of other modeling efforts which do not incorporate saturation, such as [17]. This is an important prediction which could eventually be validated or refuted based on future experimental work.

5 Conclusions and outlook

A phenomenological toy model has been presented, based on simple hypotheses and making very general assumptions, which are expected to apply independently of the specific biochemical channels underlying the FLASH effect. Based on this a tool has been developed to calculate isoeffect ratios between healthy and tumoral tissue for a given arbitrary treatment (i.e., time pattern for the dose deposition). Then the tool has been used to study the response of the system to sets of treatments defined as periodic trains of squared pulses. It has been shown that if the basic assumptions of the toy model hold, sets of experiments could be designed to determine separately each of the three free parameters of the model (conditioned to the size of experimental error bars). One of the parameters, τ, has an immediate chemical interpretation, as it is the characteristic time of interaction linked to the biochemical channel underlying the sparing effect of high dose rates. Therefore, determining τ experimentally in this way may help identify which is the most relevant biochemical process behind the FLASH effect. In this work we have chosen two extreme values of the parameter τ: 1 µs, which is the shortest timescale for some of the relevant processes, and 1 ms, which is most likely the relevant value for a realistic modeling of the process under study. The model may be used both to help interpret the results obtained in experiments and to assist on future experiment design.

One of the main predictions of the model is that for high mean dose rate the FLASH effect becomes more efficient if such dose rate is achieved by increasing the pulse width or decreasing the pulse repetition period, as compared to the alternative of increasing the peak pulse dose rate. This prediction may be tested with future experimental work.

Future work points at a few different directions:

• 1. Preparing and performing sets of measurements in a very controlled way so as to test the model. derive estimated values for its free parameters, beyond what is already available in the literature and validate key predictions such of the relative efficacy of increasing pulse width or decreasing pulse repetition period as compared to increasing peak pulse dose rate.

• 2. Exploring further the model opening up the possibility that the exponent n in the non-linear term of (3) adopts values higher than 2, i.e., considering biochemical reactions of order higher than 2 as the candidate for the process underlying the FLASH effect.

• 3. Make a more realistic model where the function c(t) becomes a field and explore the possible interplay between dose fractionation in the time and space domains.

Appendix A

The analytical resolution of the proposed rate equation, considering the source function ${\dot{D}}$(t) as periodic square waves with a finite total duration as explained in Sect. 3, allows us to rewrite the Eq. (3), for one of these intervals, as follows:

$$\frac{dc}{A{c}^{2}+Bc+C}=dt$$

(9)

where:

$$A=\frac{-a}{\tau }<0$$

(10)

$$C=\frac{{{\dot{D}}}_{p}}{ {\text{D}}_{0}}>0$$

(11)

$$B=-(C+\frac{1}{\tau })<0$$

(12)

Noticing that $\Delta ={B}^{2}-4AC>0,$ it becomes a known integral:

$$\int \frac{dc}{A{c}^{2}+Bc+C}=\left\{\begin{array}{c}\frac{-2}{\sqrt{\Delta }}\mathit{atanh}\left(\frac{2Ac+B}{\sqrt{\Delta }}\right)+K if c<\frac{-B-\sqrt{\Delta }}{2A} \\ \frac{-2}{\sqrt{\Delta }}\mathit{acoth}\left(\frac{2Ac+B}{\sqrt{\Delta }}\right)+K if c>\frac{-B+\sqrt{\Delta }}{2A}\end{array}\right.$$

(13)

As the two cases are analogous, we choose the first one to solve. For $t>{t}_{0}$:

$$\frac{2}{\sqrt{\Delta }}atanh\left(\frac{2Ac\left(t\right)+B}{\sqrt{\Delta }}\right)+K=t-{t}_{0}$$

(14)

Using the initial condition c(t = 0) =${c}_{0}$ and taking t₀ = 0:

$$K=\frac{2}{\sqrt{\Delta }}atanh(\frac{2A{c}_{0 }+B}{\sqrt{\Delta }})$$

(15)

Besides, notice that:

$$K\epsilon {\mathbb{R}}\iff \left|\frac{2A{c}_{0}+B}{\sqrt{\Delta }}\right|<1\iff \frac{-B+\sqrt{\Delta }}{2A}<{c}_{0}<\frac{-B-\sqrt{\Delta }}{2A}$$

(16)

As:

$$\frac{B+\sqrt{\Delta }}{2A}<0$$

(17)

and the initial condition must be no negative, we have:

$$0\le {c}_{0}\le \frac{-B-\sqrt{\Delta }}{2A}$$

(18)

which makes sense with the physics of the problem at t = 0, with ${c}_{0}=0$.

Inverting the implicit expression of c(t) from (14):

$$c\left(t\right)=\frac{-1}{2A}\left[B+\sqrt{\Delta }tanh\left(\frac{\sqrt{\Delta }(t-{t}_{0}-K)}{2}\right)\right]$$

(19)

Then, an iterative process follows over the successive time intervals:

$$\left\{\begin{array}{c}{c}{\prime}\left(t\right)=A{c}^{2}\left(t\right)+ {B}_{0}c\left(t\right)+ {C}_{0 } \; \textit{for} \; t\in ({t}_{0},{t}_{1})\\ c\left({t}_{0}\right)={c}_{0}=0\end{array}\right.$$

(20)

$$\left\{\begin{array}{c}{c}{\prime}\left(t\right)=A{c}^{2}\left(t\right)+ {B}_{1}c\left(t\right)+ {C}_{1 }\; \textit{for} \; t\in ({t}_{1},{t}_{2})\\ c\left({t}_{1}\right)={c}_{1}\end{array}\right.$$

(21)

with constant $A$ which is independent of the source function (see 10).

For each interval $({t}_{n},{t}_{n+1})$ one has a constant value for the source term ${C}_{n}$, and consequently, also for ${B}_{n}$.