How quickly does FLASH need to be delivered? A theoretical study of radiolytic oxygen depletion kinetics in tissues

Purpose. Radiation delivered over ultra-short timescales (‘FLASH’ radiotherapy) leads to a reduction in normal tissue toxicities for a range of tissues in the preclinical setting. Experiments have shown this reduction occurs for total delivery times less than a ‘critical’ time that varies by two orders of magnitude between brain (∼0.3 s) and skin (⪆10 s), and three orders of magnitude across different bowel experiments, from ∼0.01 to ⪆(1–10) s. Understanding the factors responsible for this broad variation may be important for translation of FLASH into the clinic and understanding the mechanisms behind FLASH. Methods. Assuming radiolytic oxygen depletion (ROD) to be the primary driver of FLASH effects, oxygen diffusion, consumption, and ROD were evaluated numerically for simulated tissues with pseudorandom vasculatures for a range of radiation delivery times, capillary densities, and oxygen consumption rates (OCR’s). The resulting time-dependent oxygen partial pressure distribution histograms were used to estimate cell survival in these tissues using the linear quadratic model, modified to incorporate oxygen-enhancement ratio effects. Results. Independent of the capillary density, there was a substantial increase in predicted cell survival when the total delivery time was less than the capillary oxygen tension (mmHg) divided by the OCR (expressed in units of mmHg/s), setting the critical delivery time for FLASH in simulated tissues. Using literature OCR values for different normal tissues, the predicted range of critical delivery times agreed well with experimental values for skin and brain and, modifying our model to allow for fluctuating perfusion, bowel. Conclusions. The broad three-orders-of-magnitude variation in critical irradiation delivery times observed in in vivo preclinical experiments can be accounted for by the ROD hypothesis and differences in the OCR amongst simulated normal tissues. Characterization of these may help guide future experiments and open the door to optimized tissue-specific clinical protocols.


Introduction
A substantial body of work over the past century has established that radiation response depends on the magnitude of dose as well as the radiation delivery time τ IR .Radiation delivered over long time periods (minutes to hours) loses efficacy with increasing τ IR due to DNA repair mechanisms that act over this timescale.At the same time, beginning with high dose-rate experiments on bacteria (Dewey and Boag 1959), pollen grains (Kirby-Smith and Dolphin 1958), and mice (Lindop and Rotblat 1960, Lindop and Rotblat 1963, Hornsey and Alper 1966), it was discovered that radiation efficacy reaches a maximum when τ IR is smaller than the characteristic DNA repair time and eventually decreases as τ IR is reduced further (Kirby-Smith and Dolphin 1958, Lindop andRotblat 1963, Hornsey andAlper 1966).Experiments that have delivered radiation over a range of delivery times have identified a 'critical' radiation delivery time below which radiation efficacy is substantially reduced.Remarkably, this critical time varies by three orders-of-magnitude in vivo and five ordersof-magnitude in vitro; examples of such experiments where at least five delivery times were used (offering sufficient temporal resolution to discern this time) are shown in see table 1.This paper seeks to provide a potential explanation for this range of critical times in experiments where high-dose rate sparing is observed.
With the rediscovery of the normal tissue sparing effects of ultra-high dose rate radiation in 2014 by Favaudon and collaborators and the surprising finding that such radiation may not lead to an appreciable reduction in tumour response (Favaudon et al 2014) (the 'FLASH' effect), there is now a large effort to move FLASH therapy into the clinic (Bourhis et al 2019, Wilson et al 2019, Esplen et al 2020, Hendry 2020, Taylor et al 2022a, Vozenin et al 2022).To this end, understanding the delivery time-dependence of radiation efficacy is potentially important for two reasons: first, to determine the delivery parameters needed to observe FLASH effects in the clinical setting (Rothwell et al 2021) and second, to give insight into the mechanisms underpinning them.As an example of the first, skin sparing has been observed for relatively long delivery times, τ IR  10 s (Field and Bewley 1974, Inada et al 1980, Hendry et al 1982), while brain sparing has only been observed for much faster irradiations, with τ IR  0.3 s (Montay-Gruel et al 2017).Vexingly, the critical delivery time for bowel toxicity to be diminished varies between ∼ 0.01 s (Ruan et al 2021) and ∼(1-10) s (Lindop andRotblat 1963, Hornsey andAlper 1966).Fully characterizing this large variation amongst different tissues (and within bowel) will be important for designing preclinical experiments and optimizing clinical delivery protocols in a tissuespecific way.
The second reason for understanding the variation in critical delivery times is that it may give insight into the mechanisms responsible for FLASH.Observing that the dose-modifying effects of ultra-high dose radiation could be modulated by varying the oxygen levels in these systems (Kirby-Smith and Dolphin 1958, Dewey and Boag 1959, Hornsey 1970, Hornsey and Bewley 1971), it was initially hypothesized that this reduction in efficacy was due to radiolytic oxygen depletion (ROD), since oxygen is radiosensitizing.Recently, the ROD hypothesis has come under scrutiny, in large part due to the small reduction in oxygen tensions for typical dose regimens (Favaudon et al 2014, Boscolo et al 2021, Cao et al 2021, Jansen et al 2021).Others have argued, however, thatin addition to other factors such as intrinsic radiosensitivity-, radiation response is largely determined by a proportionately small population of cells at radiobiologically low oxygen tensions that exist even in otherwise well-perfused and oxygenated tissue (Hornsey 1970, Field and Bewley 1974, Hendry 1979, Pratx and Kapp 2019a), and for which their radiosensitivity varies rapidly with even small changes in mean tension (Pratx Table 1.Summary of preclinical experiments where radiation was delivered over a range (5) of delivery times, allowing for a crossover behaviour of radiotherapy efficacy to be discerned between FLASH and conventional dose-rates.The critical delivery time τ IR below which FLASH effects are evident was estimated from the data reported in these papers; the range of possible was determined by the spacing between the two delivery time values surrounding this value.* Double-pulse study where t IR is the time between pulses.and Kapp 2019b, Taylor et al 2022b).An important test of any proposed FLASH mechanism-ROD or otherwise-is its ability to describe the variation shown in table 1.
Here, assuming that ROD is the mechanism primarily responsible for the FLASH effect, we studied the impact of the delivery time on radiotherapy efficacy using the theoretical formalism developed in (Taylor et al 2022b).Within the ROD hypothesis, it is commonly claimed that tissue sparing effects occur when radiation is delivered faster than can be replenished via diffusion (Dewey and Boag 1959, Lindop and Rotblat 1960, Epp et al 1973, Ling et al 1978).Comparatively little work has been done to quantify this, however, by e.g.solving the diffusion equation for oxygen transport.Ling and colleagues did this for a simulated in vitro monolayer cellular system, finding excellent agreement with empirical oxygen depletion kinetics and cell-survivial versus deliverytime dependencies (Ling et al 1978).Pratx and Kapp studied the kinetics of oxygen depletion by solving a reaction-diffusion equation for oxygen that incorporated oxygen diffusion with consumption and ROD for a single-capillary geometry (Pratx and Kapp 2019b).Rothwell and colleagues simulated a similar geometry to assess the impact of parameter values-oxygen diffusivity, consumption rate, and those associated with RODon the magnitude of the FLASH effect (Rothwell et al 2021), Zou and collaborators studied oxygen transport for simulated proton FLASH deliveries (Zou et al 2022).In (Taylor et al 2022b), we applied the formalism of Pratx and Kapp to a simulated tissue model with pseudo-randomly allocated capillaries to determine the impact of ROD on differential tumour and normal tissue cell survival.
The present study extends this work by simulating oxygen kinetics in a range of in silico normal tissues, characterized by different capillary densities and oxygen consumption rates (OCRs) corresponding to mean oxygen tensions found in typical normal tissues, for a range of radiation delivery times.Simulated radiotherapy efficacy was quantified by the cell survival fraction using the linear quadratic model modified to account for a time-dependent oxygen tension during irradiation.Literature values for the OCR in skin, bowel, and brain were used in these simulations to compare with experimental results for the critical radiation delivery times observed for these tissues.We also make predictions for the critical delivery times in tissues such as breast and liver for which we are unaware of any experiments to date.
We emphasise that the FLASH effect formally refers to both normal-tissue sparing as well as apparent isoefficacy with conventional radiation for tumours.Because most experiments have not observed a difference between conventional and ultra-high dose rate irradiation for tumours in vivo, we can only account for the observed time-dependencies of normal tissue sparing in this work; see the Discussion for further comment on this.It is not our aim here to provide a comprehensive assessment of all FLASH experiments in terms of the ROD hypothesis, including those that apparently conflict with it (reviewed in Lin et al 2022, Vozenin et al 2022, Limoli andVozenin 2023).Rather, we only focus here on explaining experiments where the irradiation-time dependence of sparing was studied.

Methods
Radiotherapy efficacy was quantified by the ratio

/
of cell survival fraction (SF) for FLASH radiotherapy delivered over a time t IR and the survival fraction for conventional radiotherapy using a representative single-fraction D = 20 Gy dose regimen.SF for conventional radiotherapy was calculated by neglecting ROD; see below.We also computed the dose-modifying factor That is, the ratio of a representative FLASH dose, = D 20 Gy, delivered over an interval t , IR and the dose delivered conventionally that gives rise to the same survival fraction.To the extent that FLASH radiotherapy is expected to increase cell survival, both the relative survival fraction in equation (1) the DMF are expected to be 1, monotonically increasing with decreasing t .

IR
Although it is common in the literature to refer to the (mean) 'dose rate', equal to t D , IR / this quantity conflates two quantities that separately impact FLASH response, dose and total delivery time.Previous work, including our own (Taylor et al 2022b), has examined the dose-dependency.Our analysis here will be focused instead on the dependence of radiation response on t IR for a fixed dose, which is less studied The methodology for calculating the survival fraction is the same as that used in Taylor et al (2022b), which we briefly review below.Survival fractions were calculated using the linear-quadratic (LQ) model for a single fraction of radiotherapy, modified to account for oxygen enhancement ratio (OER) effects by convolving the LQ expression for survival fraction with the time-dependent distribution ( ) f p t , of oxygen partial pressure values within the tissue:

/
is the biologically effective dose corresponding to a dose D, modified by the oxygen partial pressure-dependent OER (Howard-Flanders and Alper 1957, Wouters and Brown 1997, Carlson et al 2006) OER max = 3 is the maximum OER value, achieved when  p K , m the partial pressure K m at which OER achieves half its maximum value.α and β in equations (3) and (4) quantify the linear and quadratic dependencies of radiosensitivity with respect to dose; values for these parameters and all others used in the calculation of SF are shown in table 2.
During irradiation, the oxygen partial pressure (tension) distribution ( ) f p t , evolves in time due to ROD and oxygen replenishment from capillaries.It was calculated by discretely sampling the solution of the reactiondiffusion equation (Pratx and Kapp 2019b, Taylor et al 2022b) for the oxygen partial pressure ( )  p r t , on simulated tissue substrates.Here, D O 2 is the diffusivity of oxygen, OCR is the maximum OCR,  D is the instantaneous dose rate, and G 0 is the radiolytic oxygen yield.Michaelis- Menten forms were assumed for oxygen consumption and ROD.The remaining parameters are defined in table 2. Simulated tissues comprised two-dimensional domains of size 3 × 3 mm −2 with capillaries of diameter 10 μm placed randomly with areal density n c , ensuring that no capillaries were ever closer than 20 μm to eachother ('pseudorandom').( )  p r t , was taken to be equal to the capillary oxygen tension p c (40 mmHg) at the surface of the capillaries; zero-flux Neumann boundary conditions were imposed along the edges of the domain boundaries.Equation (6) was solved over the radiation delivery time t   t 0 IR using a finite-element technique for three values of OCR, 5, 15, and 40 mmHg s −1 , t IR between 2 ms and 200 s, encompassing the range of delivery times of the in vivo experiments shown in  Ling (1975), the simplified first-order kinetics of the ROD term in equation ( 6) only provides an accurate description of ROD for pulses longer than ∼ 1 ms, while most experiments using pulsed radiation employ pulses with durations on the order of several ms's.Rothwell and colleagues combined the second-order model of Ling's with oxygen diffusion calculations from a simulated capillary to investigate the sensitivity of the magnitude of ROD to model parameter values (Rothwell et al 2021).Importantly for our purposes, the nano-to-millisecond temporal structure of radiation-including wether it is pulsed or continuous-affects the predicted magnitude of ROD at a quantitative, but not qualitative, level.Since the oxygen replenishment times scales for in vivo systems shown in table 1 are all much greater than this 1 ms timescale and also typical ms pulse durations, our conclusions regarding the t IR -dependence of irradiation when  t IR 1 ms are not sensitive to the temporal structure of radiation.
SF conv was computed using the same formalism described above, but with the oxygen partial pressure distribution computed from the static solution of equation ( 6), with = G 0, 0 i.e. without ROD.Justification for all parameter values shown in table 2 is given in Taylor et al (2022b).However, we note that our three chosen OCR values are considered to be representative values of slow-consuming tissues such as skin and breast (OCR ∼ 5 mmHg s −1 ), intermediate consuming tissues including intestine (basal OCR ∼ 10-20 mmHg s −1 ), tumours (OCR ∼ 15 mmHg s −1 ), and fast-consuming tissues such as liver, brain, and kidney (OCR  40 mmHg s −1 ); see table 3. The values of n c were chosen to give mean tissue oxygen tensions of (20-30) mmHg, typical of normal tissues, and represents a reduced set of values as compared to Taylor et al (2022b), where a broader range of mean tensions were simulated, albeit at a single dose rate.
In using a pseudorandom model of capillary architecture-discussed at length in Taylor et al (2022b)-the model described above is restricted to tissues for which the functional subunits are larger than the mean distance, µ - n , c 1 2 / between capillaries ('homogeneously vascularised').This includes many radiosensitive tissues such as liver, brain, skin (dermal layer, at least), kidney, breast, but excludes lung which, has a specialized vascular architecture that varies spatially over short lengthscales, on the order of a cell diameter.At first glance, bowel is also not described by this model insofar as the radiosensitive subunits are the crypts of Lieberkuhn, each surrounded by its own capillary plexus and hence, comparable in size to the mean distance between capillaries.As reviewed in the Discussion and Supplementary Materials, however, in the basal state, many of the capillaries are transiently occluded and the mean distance between patent capillaries can be substantially larger than the crypts.Hence, we hypothesize that, with modifications to allow for transient fluctuations in perfusion, the above model also describes bowel toxicity effects.

Oxygen depletion timescales
Before presenting numerical solutions of the above model, we introduce two timescales that will be used to interpret our results.As noted in the Introduction, within the ROD hypothesis, FLASH effects arise when the delivery time is shorter than the time it takes oxygen to diffuse from an external 'source', replenishing the oxygen depleted by ROD (Dewey and Boag 1959, Lindop and Rotblat 1960, Epp et al 1973, Ling et al 1978).This is the time ( ) / it takes oxygen to diffuse a distance l (The constant of proportionality in this expression is geometry-dependent and of order unity; in equations (7)-(10) below we will set it equal to one.) and hence, FLASH effects require

/
In vivo, capillaries provide the oxygen source and thus, cells a distance l from the nearest patent capillary will experience radiation sparing as long as equation ( 7) is satisfied.This expression also implies that the degree to which sparing effects arise depends on the distance of a given cell to the nearest capillary and thus, there will be a spectrum of FLASH sparing in tissue.As argued by us Taylor et al (2022b) and others (Hornsey 1970, Field and Bewley 1974, Hendry 1979, Pratx and Kapp 2019a), however, FLASH effects in vivo occur for cells that are hypoxic even before ROD occurs.Such hypoxic 'niches' exist even in well-perfused normal tissues such as liver Table 3.Rates of oxygen metabolism for various tumours and normal tissues.References refer to metabolic rates in units of μl O2 /g tissue /min; conversions to mmHg/s utilize the conversion factor employed by Thomlinson and Gray (1955) and brain (Vaupel et al 1989), bowel (Hornsey 1970), bone marrow (Spencer et al 2014), skin (Hendry 1979, Hendry et al 1982), and lung (Down et al 1984, Travis andLuca 1985), and exist at a distance (Thomlinson and Gray 1955) from the nearest capillary.A key finding of Taylor et al (2022b) was that even if these cells account for a relatively small fraction of all cells in a tissue, they can dominate the radiotherapy response.Assuming as much and consequently setting = l l h in equation (7), we thus hypothesize that FLASH effects arise when ( ) t  p OCR. 9 between capillaries, providing another potential timescale below which FLASH effects occur: In numerical simulations of our model (equations ( 3)-( 6)), we vary both the oxygen consumption rate OCR as well as the mean capillary density n c in order to validate the hypothesis shown in equation (9), namely FLASH kinetics in vivo are determined by the OCR.

Results
In figures 1(a) and (c), the ratio, equation (1), of FLASH and conventional radiotherapy survival fractions and the DMF are plotted as functions of the total delivery time t IR for the three values of OCR.In these plots, we chose representative capillary densities of n c = 76 mm −2 , 191 mm −2 , and 392 mm −2 for the low, intermediate, and high OCR rates, respectively.These were chosen to give approximately the same mean tissue oxygenation in the absence of ROD, around 26 mmHg, a typical value for normal tissues such as liver and brain (Vaupel et al 1989).Using other values of the capillary densities yielded the same behaviour shown in figure 1 in the time domain, but with a different magnitude for the relative survival fraction ratio; see below.
Both the survival fraction and dose modifying factors exhibit quasi-sigmoidal behaviour as a function of the total delivery time (figure 1), varying slowly with respect to t IR for ultra-short and ultra-long times, and rapidly in the vicinity of the characteristic critical time, on the order of 0.1-10 s in our simulations, depending on the OCR.In figures 1(b) and (d), we plot the relative survival fraction and DMF as functions of the delivery time scaled by the oxygen consumption time, Even though the magnitudes differ, all curves converge in the temporal domain, confirming our hypothesis shown in equation (9) that ROD kinetics are consumption limited. In

/
The change ∆ ̅ p in mean tension is scaled by • G D, 0 which is the expected oxygen depletion for a homogeneous system (e.g.liquid bath) in the limit where there is no replenishment or consumption (Pratx and Kapp 2019b).As with the survival function and DMF curves in figure 1, these plots converge in the temporal domain when plotted as a function of 0 Conversely, in the opposite limit where the delivery time is very long, t , cons ROD is completely replenished by oxygen from capillaries during irradiation and ∆ ̅ p approaches zero.
As a further test of our hypothesis that it is the oxygen consumption time (equation ( 9)) and not the mean inter-capillary diffusion time (equation ( 10)) that determines the critical FLASH delivery time, figure 3 shows an example plot of the relative survival fraction and ∆ ̅ p for two capillary density values, n c = 127 mm −2 and 191 mm −2 using a common OCR value, 15 mmHg s −1 .At these capillary densities, the mean tissue oxygen tensions (without ROD) are 20 mmHg and 26 mmHg, respectively.Although the magnitudes of ∆ ̅ p and the relative survival fraction differ, the curves exhibit essentially the same time dependence.Hence, ROD kinetics in our simulated tissues are not sensitive to the time it takes oxygen to diffuse to the mean distance to nearest capillary, equation (10), since that time differs by a factor of 191/127 ≈ 1.5 between the two curves shown in figure 3.

Discussion
Although most modern FLASH experiments have studied radiation response at two dose rates ((Montay-Gruel et al 2017, Ruan et al 2021, Cooper et al 2022) and are the only exceptions of which we are aware), representing conventional and FLASH regimens, many early experiments measured response for a range of delivery times in order to elucidate the mechanisms of radiation therapy efficacy.In table 1, we show the results of experiments where a sufficient number (5) of delivery times were used to be able to discern the critical value below which a reduction in radiotherapy efficacy is evident.In addition to a two order-of-magnitude difference in the critical τ IR between mouse brain s) and skin (∼10 s) and the three order-of-magitude difference (∼0.01 s to 1 s) between bowel experiments, we also include in this table the five order-of-magnitude variation observed in in vitro experiments, from 10 −5 s for bacterial monolayers to ∼20 s for pollen grains.We argue below that these differences can be accounted for within the ROD hypothesis from differences in tissue oxygen consumption rates and, for in vitro systems, diffusion geometries.
The numerical results shown in figures 1-3 for simulated in vivo normal tissues implicate the oxygen consumption time, equation (9), as setting the critical total delivery time below which cell survival and ROD begin to differ substantially from radiation delivered over long times (conventional radiotherapy).p c is fairly constant from person-to-person, under normal physiological conditions (Ortiz-Prado et al 2019), whereas the OCR varies by nearly two orders of magnitude across tissues, see Vaupel et al (1989) and table 3. From the data shown in table 1, radiation sparing of the skin becomes evident at relatively long delivery times (10 s), whereas FLASH must be delivered in a fraction of a second to observe sparing of brain.Skin is a slow-consuming tissue, with OCR values 5 mmHg/s.Conversely, brain is a fast-consuming tissue, with a much larger OCR (40 mmHg s −1 ).Using 40 mmHg as a representative capillary oxygen tension (corresponding to the venous end; in Taylor et al (2022b), we argued that the cells proximal to the venous end should be impacted most strongly by FLASH), the predicted oxygen consumption times for brain and skin are thus, ∼1 s and ∼10 s, respectively, consistent with these results.Other fast-consuming tissues where we predict that a sub-second τ IR is needed to observe FLASH effects include liver and kidney.Similar to skin, normal breast tissue has a low OCR and we predict that sparing effects should be evident over longer delivery times, 10 s.
During preparation of this manuscript for submission, we became aware of earlier work by Zhou and colleagues who applied dimensional analysis to the question of the minimum dose rate needed to achieve the FLASH effect (Zhou et al 2020).As part of their analysis, they concluded that equation (7) sets the critical 0 This product is the expected change in oxygen partial pressure in a homogenous liquid without consumption or replenishment.As with the survival fraction and dose modifying factor shown in figure 1, the critical delivery time for there to be a substantial reduction in mean tissue partial pressure is set by the oxygen consumption time.Note that ∆ ̅ p does not quite achieve the 'homogenenous bath' depletion value • G D 0 in the limit of very small delivery times since tissue in the immediate vicinity of capillaries will always be completely replenished during irradiation.for a single oxygen consumption rate, 15 mmHg s −1 , but different capillary densities.While the magnitude of the FLASH effect is impacted by the capillary density, the time-dependence is not.Owing to its exponential dependence on small hypoxic regions, the magnitude of the survival fraction (a) is more sensitive to differences in capillary density than the change in mean oxygen partial pressure (b).The independence of depletion kinetics on capillary density is thus clearer for the mean partial pressure.
irradiation time, albeit choosing a fixed literature value for the characteristic distance to the nearest capillary.There is no conflict with our results since, in physiological systems, this distance is not constant across tissues but likely scales to match the OCR.(In this regard, the results shown in figure 3-while valid-likely do not correspond to a physiologically accurate scenario since two tissues with different capillary densities would not have the same OCR value.)We prefer to use OCR as our key differentiating parameter since it seems to be better characterized across tissues than capillary density (see table 3 and references therein).
Although bowel, like lung, has a highly-structured vasculature optimized for nutrient exchange, with some modifications, we believe that our results can explain the nearly three order-of-magnitude difference in the critical radiation delivery time that has been observed for bowel, spanning ∼ 0.01 s (Ruan et al 2021) to ∼(1-10) s (Lindop andRotblat 1963, Hornsey andAlper 1966).Bowel radiation response is largely determined by a single layer of epithelial cells lining the intestinal crypts which are closely enveloped by the pericryptal capillary network.This network is not uniformly perfused, however, and, in the basal resting state, up to 80% of the arterioles feeding them are occluded (Matheson et al 2000).The basal OCR is likewise relatively low, 7-14 mmHg s −1 (Lutz et al 1975, Desai et al 1996) (see table 3), consistent with equation (9) and the (1-10) s critical delivery times observed in (Lindop andRotblat 1963, Hornsey andAlper 1966).In response to stimuli including food consumption, however, the arterioles feeding the crypts become patent, enhancing perfusion dramatically (Matheson et al 2000, Zheng et al 2015).It is well known that this increase in perfusion is accompanied by an increase in the mean OCR, maintaining oxygen levels (Desai et al 1996, Zheng et al 2015).In the Supplementary Materials, we present a simple geometric argument that the local OCR can increase by as much as three orders of magnitude in this well-perfused state, providing a potential explanation for the ultrashort ∼10 ms critical delivery time observed by Ruan and colleagues (Ruan et al 2021).
In vitro systems present an interesting contrast to our simulated in vivo results.In most cases, the critical delivery time is set by the oxygen diffusion time, equation (7), with l the distance that oxygen must diffuse from the atmosphere/oxygen source to DNA molecules within cells.In vitro kinetics are thus 'geometry limited' rather than consumption limited.As argued by Ling in their study of monolayer Chinese hamster cell systems (Ling 1975) (2022), for which we could not determine details of the system in order to compute oxygen diffusion times) can be accounted for within the ROD hypothesis by differences in oxygen replenishment kinetics.
To our mind, this fact, in conjunction with experiments that found FLASH effects are modulated by varying the oxygen level in tissues (Kirby-Smith and Dolphin 1958, Dewey and Boag 1959, Hornsey 1970, Hornsey and Bewley 1971) suggests an important role for ROD in the observed normal-tissue sparing effects of FLASH radiation.That being said, there are significant challenges for the ROD hypothesis to explain FLASH effects (Limoli and Vozenin 2023).First amongst these in our opinion is the apparent iso-efficacy of FLASH and conventional irradiation in controlling preclinical tumours (Favaudon et al 2014, Montay-Gruel et al 2021).Although ROD predicts a smaller sparing effect for hypoxic tumours (Taylor et al 2022b) (and we remind the reader that in the present work, only tissues with mean oxygen tensions between 20 and 30 mmHg were simulated, corresponding to expected normal tissues values), not all tumours are hypoxic.Within the ROD hypothesis, the robust iso-efficacy across many experiments may additionally require there to be differences in radiochemistry between cancer and normal tissue cells (Spitz et al 2019) to account for this.A less appreciated challenge to the ROD hypothesis is the fact that strategies deployed clinically to increase tissue oxygen to combat tumour hypoxia-notably carbogen (Janssens et al 2012) and metabolism modification therapies (Koritzinsky 2015)-have not brought with them substantial increases in toxicities, which our simple oxygendependent radiosensitivity based model of cell survivial would naïvely predict.Nonetheless, the primary conclusion of this manuscript is that the observed timescales of radiation efficacy-in all the experimental works we could find where this is quantified-are consistent with expected differences in oxygen diffusion and consumption times.We further note that although we have adopted a specific model of toxicity here using cell survival as a surrogate with oxygen-enhancement ratio (OER) effects responsible for sparing (Pratx andKapp 2019b, Taylor et al 2022b), our conclusions regarding oxygen kinetics would hold for any proposed FLASH mechanism reliant on ROD.
A greater appreciation for the differences in the diffusion and consumption kinetics of oxygen in tissues may lead to an increase in the 'FLASH signal' in preclinical experiments and help identify optimal clinical use scenarios.Several recent works (Smyth et al 2018, Buonanno et al 2019, Venkatesulu et al 2019, Zhang et al 2023) have not found sparing effects at ultra-high dose rates, possibly due to overly long delivery times.Buonanno and colleagues, for instance, found no discernible difference in clonogenic survival curves between FLASH and conventional radiotherapy in a normal lung fibroblast cell line, irradiated in vitro (Buonanno et al 2019).Their experiments used delivery times of  10 ms which, as noted above, may not be fast enough to observe an effect in mammalian cell lines for which the diffusion time is expected to be a few ms.Similarly, and taking into account perfusion variability in bowel, robust bowel sparing effects may only reproducibly (i.e.independent of perfusion at the time of irradiation) arise for ultra-fast deliveries, 0.01 s (Levy et al 2020, Ruan et al 2021), potentially explaining studies that have found no FLASH sparing effect for bowel (Smyth et al 2018, Venkatesulu et al 2019, Zhang et al 2023).For irradiation times of 3.5 μs, Adrian et al found variable and sometimes not-significant survival curve differences for a range of in vitro cell lines, including lung fibroblasts (Adrian et al 2021).In contrast to Buonanno et al and the in vitro experiments shown in table 1 (save for Cooper et al (2022)), they irradiated adherent cells in medium, which introduces a heterogeneous, time-dependent oxygen distribution (Place et al 2017), complicating the interplay between oxygen diffusion, consumption, and ROD (manuscript in preparation).
Ultimately, the analysis presented here may help guide efforts to translate FLASH into the clinic and lead to ways to optimize the FLASH therapeutic window.We predict a substantial FLASH effect for normal breast tissue, for instance, achievable for delivery times on the order of (1-10) s owing to its slow rate of oxygen consumption.Liver-another site which has so far not been explored in preclinical FLASH experiments as far as we know-may also benefit from FLASH, but only at ultra high dose rates, t  IR 0.1 s, as it is a fast metabolizer of oxygen.We hypothesize that bowel sparing is sensitive to transient perfusion, opening the door to testing pharmacological modifications of bowel perfusion (Zheng et al 2015).

Conclusions
The normal tissue sparing effects of radiation delivered at ultra-high dose rates have been observed over a wide range of total delivery times.Here, we have argued that the five order-up-magnitude variation in the maximal radiation delivery times for these effects to occur can be explained by differences in oxygen consumption and diffusion between different tissues as well as between in vivo and in vitro experiments.Efforts should be made to model these effects quantitatively in preclinical experiments in order to maximize the FLASH signal and optimize delivery protocols.
. the time it takes oxygen leaving a capillary to be completely consumed by the tissue.On dimensional grounds, the only other length scale in our problem is the mean distance ̅ µ -

Figure 1 .
Figure 1.Impact of radiation delivery time on survival fraction (SF; top row) and the dose modifying factor (bottom row).Curves are shown for different oxygen consumption rates, 5 mmGy s −1 ('Low'), 15 mmHg Gy −1 ('Intermediate'), and 40 mmHg Gy −1 ('Fast').Tissue sparing effects, manifested as an increased cell survival fraction and dose modifying factor, increase with decreasing delivery time.The 'critical' delivery time, below which normal tissue sparing effects become pronounced (e.g. ( )  t SF SF 1 IR conv / ) range from ∼0.1 to 10 s in (a) and (c).Normalizing the delivery time by the oxygen consumption time p OCR c / (in (b) and (d)) the curves approximately collapse onto single curves, demonstrating that the critical delivery time below which FLASH effects become evident is set by this time.

Figure 2 .
Figure 2. Change ∆ ̅ p in mean tissue oxygen tension due to 20 Gy of FLASH scaled by the dose D multiplied by the ROD yield G .0 This product is the expected change in oxygen partial pressure in a homogenous liquid without consumption or replenishment.As with the survival fraction and dose modifying factor shown in figure 1, the critical delivery time for there to be a substantial reduction in mean tissue partial pressure is set by the oxygen consumption time.Note that ∆ ̅ p does not quite achieve the 'homogenenous bath' depletion value • G D

Figure 3 .
Figure 3. Relative survival fraction (a) and change in mean tissue oxygen partial pressure (b)for a single oxygen consumption rate, 15 mmHg s −1 , but different capillary densities.While the magnitude of the FLASH effect is impacted by the capillary density, the time-dependence is not.Owing to its exponential dependence on small hypoxic regions, the magnitude of the survival fraction (a) is more sensitive to differences in capillary density than the change in mean oxygen partial pressure (b).The independence of depletion kinetics on capillary density is thus clearer for the mean partial pressure.
Insufficient resolution to discern a critical duration, only a range. ** table 1, and a range n c = (64 → 392) mm −2 of capillary densities, chosen to reproduce physiological mean tissue oxygen tension values.
IR As shown in detailed calculations by

Table 2 .
Parameter definitions and values.* Representative values from table 3. ** Representative value of quoted references.
; see supplementary materials for details.
, the relevant geometric distance is the cellular radius.Using = D O 2 2000 μm 2 s −1 and l = 4 μm gives t  IR 2 ms, consistent with the experimental values shown in table 1.Note that this is much smaller than typical in vivo oxygen consumption times.In contrast, Kirby-Smith and Dolphin found an increase in chromosomal breakages in a (presumably) single layer of Tradescantia microspores for radiation delivered in 20 s (Kirby-Smith and Dolphin 1958).Typical microspores are several hundred microns in diameter, potentially accounting for this much longer diffusion time, which scales as the square of the diffusion distance.Similarly, the experiments by Weiss and collaborators (Weiss et al 1975) used Serratia Marcescens bacteria, which are much smaller than mammalian cells.Assuming a cellular radius of 0.4 μm (Weiss et al 1975) gives a critical diffusion time of ∼4 × 10 −5 s, again consistent with the value shown in table 1.Thus, in contrast to homogeneously vascularised tissues, in vitro monolayer systems exhibit a much larger variation in critical radiation delivery times, a result of the three order-of-magnitude variation in system size, between bacterial cells and pollen grains.While one may question the relevance of bacterial and plant radiobiological data to in vivo FLASH results, we include them here because irradiation time data is available (table 1 is believed to be comprehensive, including all such data) and-if one believes in vitro experiments have anything to say about in vivo FLASH effects, they provide data that should be explained by any putative FLASH mechanism.Combined with our simulated in vivo tissues, we thus argue that all experiments shown in table 1 (with the possible exception of the whole blood experiment of Cooper et al