1 Introduction

FLASH radiotherapy is delivered at ultra-high dose rates (e.g., several orders of magnitude higher than what is currently used in clinical settings), and has the potential to revolutionize the future of cancer treatment. Compared to conventional radiotherapy, FLASH radiotherapy may provide equivalent therapeutic effect for tumors while protecting normal tissues. Due to this distinct feature, FLASH has gained prominence in the research community and has the potential to become a paradigm shift for cancer treatment if widely proven [1,2,3,4].

The benefits of FLASH in electron and photon therapy, have already been showcased in organs such as brain, lung, bowel and skin [5,6,7,8,9,10]. The FLASH benefits were achieved at a dose rate higher than 30 Gy/s, with the maximum effect in a range between 100 and 300 Gy/s [3, 6, 10]. By comparison, proton therapy has an intrinsically higher dose rate than conventional electron or photon therapy, through the adjustment of injection current plus its more conformal dose delivery [4, 11,12,13,14,15,16]. One study exposed normal cells in vitro to ultra-high dose rates (1000 Gy/s) and observed less adverse biological effects compared with a dose rate of 0.05 Gy/s [14]. Another study compared two dose rates (100 Gy/s vs. 1 Gy/s), claiming that proton FLASH is able to reduce the levels of both acute and chronic gastrointestinal damages in a mouse model [15]. In contrast to these promising results, contradictory conclusions were also reported. For instance, one study (electron beam) showed an increased toxicity for FLASH (35 Gy/s) compared to a conventional delivery at 0.1 Gy/s [17], by a factor of ~ 1.4. Another study (proton beam) compared the outcomes between two dose rates (0.08 Gy/s and 100 Gy/s) for zebrafish embryos [12], claiming no significant influence of dose rate on toxicity and developmental abnormality. The field while rapidly evolving, is still grappling with new data and has no clear consensus yet.

The underlying mechanism for FLASH benefits remains unclear as well, and are still being heavily debated, which may be linked any or all of 1) the oxygen effect, 2) reactive oxygen species changes and/or 3) immune response [3]. The oxygen effect is mainly linked to the oxygen fixation hypothesis [3, 18,19,20,21,22]. One plausible explanation is that FLASH results in a period of hypoxia, through a series of interactions in between radicals with oxygen occurring at different time scales (~ 10− 5 sec for superoxide anion and ~ 10− 9 sec for hydrogen radical). This consequently results in transient radio-resistance in tissues that were otherwise well oxygenated. For normal tissue (“physoxic”), the physiological level of oxygen decreases rapidly and thus has an important impact on oxygen enhancement ratio (OER) and radio-sensitivity, possibly yielding transient hypoxia for protection (depending on dose and dose rate). By contrast, the oxygen level in tumor tissue is already low (hypoxic) and thus sees no significant change of OER. In a conventional delivery, not only transient oxygen depletion is limited, but also there is sufficient time for oxygen to be resupplied through oxygen resupply. Apart from the oxygen effect, another study proposed that FLASH irradiation may help reduce the level of reactive oxygen species and consequently mitigate radiobiological damage [9, 23]. The immune response has also been linked with FLASH effect [3]. A recent experimental study was carried out through the genome-wide microarray analysis on mice following both FLASH and conventional irradiation, reporting that the immune response was attenuated in FLASH-RT relative to the conventional counterpart [24].

The difference between normal and tumor tissue in terms of the physiological oxygen level is summarized as below. Normal tissue (physoxic) has an oxygen tension typically ranging between 3% and 7.4% (e.g., 21% corresponding to a partial pressure of 160 mmHg for a standard atmospheric pressure of 760 mmHg), physiological hypoxia (~ 2.0%, 15 mmHg), pathological hypoxia (~ 1.0%, 8 mmHg), and radiobiological hypoxia (~ 0.4%, 3 mmHg) [25, 26]. It is well known that tumor tissue is commonly hypoxic and heterogeneous with an oxygen level averaged at ~ 1%. A number of experimental studies tend to support the oxygen depletion hypothesis, demonstrating that the survival curves at ultra-high dose rate mimic those when irradiated in a hypoxic environment [5, 27,28,29,30], including two in vitro studies (0.44–0.7% O2 concentration in [29], 1.6–8.3% O2 concentration in [30], and an in vivo mouse model [5]. However, whether there is a direct causal relationship between oxygen depletion and FLASH benefit still remains controversial. Similar to the two recent studies [31,32,33], we proposed an analytical model to estimate the influence of FLASH-RT on normal tissue sparing, for two cell types (V79 Chinese hamster cells and T1 human kidney cells).

2 OER modeling (oxygen depletion/diffusion) and sparing factor

As illustrated in Fig. 1, normal tissue moves gradually from the aerobic (or physoxic) condition to the hypoxic condition. As the oxygen partial pressure decreases, the survival fraction curve tilt upwards and OER is expected to increase (i.e., becoming more radio-resistant). OER reflects the radioprotective effect of hypoxia under two oxygenation conditions [34,35,36,37]. In this study, the OER was defined as the ratio of the dose at different pressures p1 to that of the hypoxic condition p2, to quantify normal tissue sparing. Based upon the standard linear-quadratic model [35, 38, 39], OER depends on total dose D and tissue-specific radio-sensitivity parameters α and β with the subscripts 1 and 2 representing the low and high oxygen partial pressure conditions, respectively. D2 corresponds to the dose delivered, used as a baseline reference (with fixed α2 and β2). Equation 1 allows for evaluating the differences among different cell types.

$$OER\left({D}_2,{\alpha}_2,{\beta}_2,{\alpha}_1,{\beta}_1\right)\equiv \frac{D_1}{D_2}=\frac{\alpha_1+\sqrt{{\alpha_1}^2-4{\beta}_1\left({\alpha}_2{\textrm{D}}_2+{\beta}_1{{\textrm{D}}_2}^2\right)}}{2{\textrm{D}}_2{\beta}_1}$$
(1)
Fig. 1
figure 1

Illustration of different radiosensitivity under two conditions (aerobic and hypoxic), yielding different OERs. To allow for the transient analysis, the OER associated with each time step (e.g., TA and TB) along the pulse train was analytically calculated

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During treatment delivery, physiological oxygen is gradually depleted in proportion to the radiation dose applied at a rate of L, which is a parameter independent of p0, and typically ranges between 0.21 and 0.42 mmHg/Gy [29, 32, 40, 41] as formulated in Eq. 2. p0 is the oxygen partial pressure at the initial condition (i.e., physoxic for normal tissue) and D is the dose delivered. We selected L to be 0.42 mmHg/Gy in accordance with previous measurements of radiolytic oxygen depletion (ROD) in cells irradiated at ultra-high dose rates [28]. The dependency of oxygen partial pressure on dose rate is not considered here, but is later on.

$$p={p}_0-L\cdot D$$
(2)

The dependence of α1 and β1 in Eq. 1 on oxygen partial pressure was modeled, using the concept of relative radio-sensitivity in the Alper and Howard-Flanders model [35, 36]. For consistent notation, it should be pointed out that α(p) in Eq. 3 corresponds to α1 under a given pressure in Eq. 1. In Eq. 3, αmax and αmin correspond to the maximum value and minimum value (e.g., αmax = αa for aerobic condition, αmin = αh for hypoxic condition), respectively [35]. The parameters for two selected cell types are: V79: αa = 0.135 Gy− 1, βa = 0.032 Gy− 2, αh = 0.06 Gy− 1, βh = 0.003 Gy− 2; T1: αa = 0.10 Gy− 1, βa = 0.047 Gy− 2, αh = 0.02 Gy− 1, βh = 0.009 Gy− 2 [35]. β(p) was modeled in the same manner. p was determined by Eq. 2 for a given condition as dose is gradually deposited. The constant K was selected to be 3 mmHg [35], though strictly speaking, it should be selected according to the oxygen level at which αh and βh were measured.

$$\left\{\begin{array}{c}\ \alpha (p)=\frac{\alpha_{max}\cdot p+{\alpha}_{min}\cdot K}{p+K}\\ {}\ \sqrt{\upbeta (p)}=\frac{\upbeta_{max}\cdot p+{\upbeta}_{min}\cdot K}{p+K}\end{array}\right.$$
(3)

As proposed in the previous studies [31, 32], the effect of transient oxygen depletion can be viewed as a shift along the OER curve, during which oxygen is gradually depleted as radiation dose is deposited. Slightly different from the previous study [31, 32], the sparing factor (always larger than 1) is defined as:

$$Sparing\ Factor=\frac{\frac{1}{N}\sum_{i=1}^N OER\ at\ p=\left({p}_0-\textrm{i}\cdot L\cdot dD\right)}{OER\ at\ p={p}_0}\ \textrm{st}.{p}_0-\textrm{i}\cdot L\cdot dD\ge 0,$$
(4)

where p0 is the initial pressure and i is the index of time step. The calculation step is illustrated in Fig. 1. For each time step, a small amount of dose (dD) is delivered and the oxygen partial pressure is reduced, yielding different α(p) and β(p). The OER within this time step can then be derived. The summation in the numerator in Eq. 4 is linked to the covered area under the OER curve from the initial condition to the end condition (e.g., integration). It should be emphasized that the pulsatile nature of radiation beams was not considered here (e.g., only average dose rate is assumed). N is the number of time steps (i.e., the pulse trains in Fig. 1) of above zero pressure. When the pressure becomes below zero, it is considered as the status of full depletion and the pressure is set to be zero. For the static case (not considering any time dependence such as perfusion), the temporal change of oxygenation is not relevant and so is dose rate. dD is selected to be 0.2 Gy (i.e., further decreasing dD does not improve numerical accuracy).

Using the same diffusion model as proposed in [32], the temporal change of oxygenation due to loco-regional diffusion/metabolism (referred to as the dynamic case) was modeled for a long blood vessel:

$$\frac{\partial p}{\partial t}={D}_o\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial p}{\partial r}\right)-m-\frac{L{D}_p}{T}\ \left(p\ge 0\right)$$
(5)

where T is the total time frame of irradiation and Dp is the total dose delivered during T. The diffusion constant DO is 2 × 10−5cm2s−1 [42, 43] and m represents different levels of oxygen consumption (3, 10, and 20 mmHg/s) [32]. The last term in Eq. 5 is directly linked to Eq. 2. The partial differential equation was solved using the ODESolver™ package in Python, with the following boundary conditions: p(r = 0) = 30 mmHg and p(r = 200μm) = 0mmHg. Δr and Δt were selected to be 5 μm and 3.5 ms, in order to satisfy the stability condition of solving the differential equation. For the dynamic case, dD (dose per time step) depends on both dose rate and time step. For example, a combination of 200 Gy/s and 0.0035 sec (time step) yields a dose deposition of 0.7 Gy. The modeling comprises three steps. The first step is to derive the steady-state of oxygen tension for all distances in absence of irradiation (Dp = 0). The second step is to study the transient oxygen tension as a function of time for multiple dose rates with a fixed total dose of 10 Gy. The third step is to calculate the sparing factor based upon the transient oxygen tension for each distance. Three distances (15, 37.5 and 60 μm) were selected, each distance corresponding to different p0 in Eq. 5.

2.1 Static case

For the static case (a total dose of 10 Gy), the sparing factor demonstrates an initial increase and then reaches a plateau as shown in Fig. 2. The maximum sparing factors (V79/ T1) for different initial oxygen pressures are 1.15/1.13 (1 mmHg), 1.26/1.22 (3 mmHg), 1.25/1.21 (5 mmHg), 1.25/1.21 (8 mmHg) and 1.13/1.11 (10 mmHg), respectively. In case L is reduced by a factor of 2, the dose needs to double to achieve the same sparing factor. Here the concept of dose rate is not relevant.

Fig. 2
figure 2

The sparing factors for multiple combinations of p0 and Dp for a V79 and b T1 for a total dose of 10 Gy, following Eqs. 1-4

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Under different oxygen conditions, the dependence of OER on dose is illustrated in Fig. 3. Different trend between V79 and T1 stems from different radio-sensitivity parameters. Albeit V79 shows no large fluctuation over a dose ranging between 0 and 1.5 Gy, T1 sees a noticeable increase of OER as the dose gets smaller, particularly for hypoxic conditions (e.g., 0.01 mmHg). These results prepare us for the study of the dynamic case, when the concept of time step (e.g., 3.5 ms) is introduced in Section 4, since the change of OER depends on several factors: change of pressure (Eq. 2), change of radio-sensitivity parameters (Eq. 3), and dose per time step during the transition (Eqs. 1 and 2).

Fig. 3
figure 3

The dependency of OER on dose (0-1.5 Gy) are shown for a V79 and b T1, under multiple oxygen partial pressure conditions (with the baseline assumed as 10 mmHg)

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2.2 Dynamic case

The steady state of oxygen pressure for three m values is shown in Fig. 4a. Two exemplary cases (50 Gy/s and 2.5 Gy/s) are shown in Fig. 4b and c, respectively, with the beam delivery starting at t = 1.0 sec. Upon irradiation, the oxygen partial pressure initially drops and then recovers to the steady state within a duration of ~ 1 sec (rising edge). When the dose is delivered over a period of comparable duration to the recovery time (2.5 Gy/s for 4 secs), the resupply of oxygen through diffusion mitigates the sparing effect. In case a larger diffusion constant is applied, the recovery time will be reduced accordingly but should remain in the same order of magnitude (~ 1 sec).

Fig. 4
figure 4

a Illustration of oxygen diffusion (steady-state) as a function of distance and oxygen metabolism rate m. The temporal change of oxygen pressure as a function of time under two dose rates and beam-on time, 50 Gy/s (b) and 2.5 Gy/s (c). For the delivery at 2.5 Gy/s, only the result of distance 3 is shown

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Two cell types exhibit different sparing factors due to different α and β. Different levels of oxygen consumption demonstrate different temporal profiles of oxygen tension, as well as FLASH benefits. The case of 20 mmHg/s yields more noticeable sparing compared to the other two cases (3 mmHg/s and 10 mmHg/s), due to lower oxygen levels at the same depth (Table 1). For the case of 3 mmHg/s, the sparing factor is below 1.01 even for a dose rate up to 500 Gy/s for all three distances, corresponding to an initial pressure of 25.4, 19.1, 13.6 mmHg, respectively. For the case of 10 mmHg/s, the initial pressure at three selected distances is 22.3, 13.0 and 6.1 mmHg, respectively. For the third distance (60 μm), the sparing factor is 1.07/1.10 (V79/T1) for a dose rate between 50 and 100 Gy/s. For m = 20 mmHg/s and a distance of 37.5 μm (initial pressure of 8 mmHg), the sparing factor is less than 1.05 even for a dose rate up to 200 Gy/s. At the distance of 60 μm (initial pressure 1.5 mmHg), the sparing factor demonstrates a continuous increase, reaching 1.03/1.06 (V79/T1) at 2.5 Gy/s and 1.28/1.72 (V79/T1) at 100 Gy/s. T1 shows larger SF than V79, due to its higher OER than that of V79 under the same condition as shown in Fig. 3.

Table 1 The sparing factors (in the form of V79/T1) for m = 3, 10, 20 mmHg/s, and Dp = 10 Gy, calculated from Eq. 5. For the three distances with m = 20 mmHg/s, the initial pressure is 19.5, 8.0 and 1.5 mmHg, respectively
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We reckon that the slight decrease of sparing factor down to 1.62 (for T1 cell) at 200 Gy/s is due to the following two reasons. First, fewer data points along the falling edge are included for the 200 Gy/s case than the 100 Gy/s case, since delivering 10 Gy at a higher dose rate takes less time (i.e., less data points on the falling edge). The full depletion occurs for low initial oxygen pressures (e.g., 1.5 mmHg) and high dose rates (e.g., 200 Gy/s), resulting more steps of zero pressure relative to 100 Gy/s. This results in numerical errors in the computational step, as a form of discretization error. Second, the dependence of OER on dose yields a slightly larger OER for a lower dose rate, a pattern more noticeable for T1 than for V79. Taking the dose rate of 100 Gy/s and 200 Gy/s for examples, the dose deposition during a time step of 3.5 ms is 0.35 Gy and 0.7 Gy (see Fig. 3b), respectively.

3 Experimental study

To help interpret the theoretical model, in vivo oxygen depletion during FLASH irradiation was experimentally measured (see Fig. 5). To achieve this, Oxyphor 2P, an oxygen probe that accounts for oxygen-dependent quenching of phosphorescence was used to quantify oxygen levels in the mouse model [44]. The phosphorescence quantum yield of Oxyphor 2P in deoxygenated aqueous solutions at 22 °C is 0.23, making it an exceptionally phosphorescent sensor. The phosphorescence lifetime of Oxyphor 2P changes with oxygen concentration from 9 μs in air, to 38 μs under fully anaerobic conditions at physiological temperature (36.7 °C). The probe has two absorption bands (near 440 nm and 630 nm), and phosphoresces near 760 nm. More details about the calibration of the phosphorescence lifetime as a function of oxygen levels can be found in [44]. The phosphorescence lifetime was measured by an OxyLED oximeter (Oxygen Enterprises Ltd., Philadelphia PA), designed specifically for oxygen measurements in biological systems. The excitation source is a light emitting diode (peaked at 630 nm), and the detector is an avalanche photodiode (APD). Optical fibers were employed to guide the excitation light to the sample and phosphorescence emission to the detector. When combined with Oxyphor 2P, OxyLED is capable of measuring oxygen in vivo over the entire range of physiological partial pressures (0-160 mmHg).

Fig. 5
figure 5

Experimental setup: a the oximeter (OxyLED), and b the rear view and lateral view (inserted) of the mouse. The black block over the mouse is the circular cut-out used for conforming the electron beam

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All animals were cared for and handled in accordance with National Institutes of Health Guidelines for the care and use of experimental animals and study protocol has been approved by the Dartmouth Institutional Animal Care and Use Committee (IACUC). Nude mice at an age of 6 weeks (Charles River Labs, Wilmington, MA) were injected with 106 MDA-MB 231 cells under the skin on the flank. Oxygen measurements in tumors were performed when the tumors were approximately 8 mm in diameter. Oxyphor 2P (200 μL of 20 μM concentration) was intravenously injected in the tail vein about 24 h prior to the measurements. To fix the mice for accurate delivery of radiation, they were under general anesthesia of inhaled isofluorane at 1.5% (flowing air through a nose cone) during treatment. A heating pad was put underneath the animals to maintain normal body temperature.

The electron FLASH beam was delivered from a Varian Clinac 2100 C/D (Palo Alto, CA) after reversible conversion procedures (see more details in reference [42,43,44]). The mean dose rate of the modified electron FLASH beam was 270 Gy/s (maximum) at isocenter with a pulse repetition rate of 360 Hz. A circular cut-out (radius: ~ 1 cm) was used to irradiate the muscle on the flank of mice. Beam pulses have a width of 3.5 μs (0.75 Gy) and a period of 2.7 ms. A total dose of 10 Gy was delivered. Different mean dose rates (90 Gy/s, 180 Gy/s, 270Gy/s) were obtained by changing the pulse repetition rate (200 MU/min, 400 MU/min, 600 MU/min) in the console.

Figure 6a and b show the comparison of oxygen consumption inside animals between a dose rate of 270 Gy/s (FLASH radiation) and 0.08 Gy/s (conventional radiation). The pO2 decrease is clearly noticed for FLASH radiation but not for conventional radiation, consistent with the trend as exhibited in Fig. 4b and c. FLASH radiation rapidly depletes the oxygen in vivo which then slowly recovers to the initial level, while the oxygen consumption under conventional radiation is too slow to accomplish the same degree of oxygen deletion. Two issues should be emphasized here. First, to complete one measurement (one data point in Fig. 6), it requires 130 ms that is longer than the total delivery time of ~ 37 ms. As a result, the transient oxygen depletion occurs at an extremely short interval tied to the abrupt drop of pO2 (a single data point). Second, the recovery takes much longer (~ 20 secs) compared with that in Fig. 4 (~ 2 secs), which might be due to the measurement comprising a larger spatial scale (a circular area of 1 cm radius) and more complex diffusion than the simplified cylindrical geometry in Section 2.

Fig. 6
figure 6

In vivo oxygen measurement during a FLASH radiation, and b convectional radiation for a total dose deposition of 10 Gy

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The measured oxygen depletions for three different dose rates show no significant difference for a total dose of 10 Gy (Fig. 7), which is consistent with the results given in Table 1 (last two lines). It reveals the independence of oxygen depletion on the dose rate above 90 Gy/s. Unfortunately, no measurement under dose rate below 90 Gy/s was able to be conducted for comparison for the time being, as well as other results related to treatment outcome such as fibrosis under different dose rates.

Fig. 7
figure 7

Quantification of oxygen depletion upon FLASH irradiation at three different dose rates for a total dose rate of 10 Gy. No statistically significant difference was found among three dose rates, analyzed by ANOVA with Tukey’s multiple comparison test (p-value < 0.05, for n = 5)

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4 Discussion

A mechanistic consideration of the FLASH benefits was presented, combined with preliminary experimental results of in vivo animal studies. The analytic model was applied to two cell types (V79 Chinese hamster cells and T1 human kidney cells). There are two reasons for the selection: 1) widely available parameters such as αmax and αmin, and 2) the two cell lines show opposite trend of OER as a function of dose (see Fig. 3).

4.1 Oxygen depletion and dose rate

Our approach is flexible in the derivation of OER incorporating cell-specific radiosensitivity parameter, and yields a direct metric (sparing factor) for evaluating FLASH benefits. In spite of the crude borderlines presented in Table 1, it helps study the influence of dose rate. Sparing factor is tied to the integral area under the OER curve (as a function of oxygen pressure) between the start and end points, largely contributed from points of low oxygen partial pressures. As long as oxygen can be fully and quickly depleted under a fixed dose (e.g., 10 Gy), further boosting the dose rate will have no impact. A back-of-the-envelope calculation based upon Eq. 2 seems to support such a claim as well as the experimental results shown in Fig. 6. Given an initial condition (p0 = 10 mmHg, L = 0.42 mmHg/Gy), Dp is 23.8 Gy for a full depletion and the corresponding dose rate is 23.8 Gy/s for a beam width of 1 sec (i.e., comparable to the recovery time in Fig. 4b). For an initial condition of 30 mmHg, Dp is 71.4 Gy for a full depletion with a dose rate of 71.4 Gy/s. As long as it remains at the status of full depletion, both OER and SF remain constant. Additional evidence is the experimental findings of ultra-short laser accelerated proton pulses (bunch width: nanosecond), which shows no significant difference in radiobiology between two dose rates (total dose: ~ 20 Gy), pulsed mode (1 × 109 Gy/s for ~ 1.3 ns) versus continuous mode (~ 10 Gy/s for 100 ms) [45].

Another issue is with regard to the selection between a linear-quadratic and a linear model for predicting the surviving fraction. In our study, a linear-quadratic response is assumed during each time step (i.e., a small dose step dD). In hindsight, the use of a linear model might be more rigid in this case, since Eq. 4 implicitly assumes a linear model. For a LQ model, the accumulation of sub-lethal damage over the entire dose delivery gives rise to non-linear behavior, that is, the cell survival at a given point depends on both previous and current dose deposition. This is indeed one flaw in our current model, which cannot be easily tackled if one desires to incorporate the time scale. As a remedy, we repeated the study by choosing a linear model (ignoring β in Eq. 1) and a concrete example (distance 3) is presented below. When a linear model is assumed while others are equal, the sparing factor is 1.28/1.84 (vs. 1.28/1.72 in Table 1) for 100 Gy/s, and 1.26/1.80 (vs. 1.28/1.62 in Table 1) for 200 Gy/s. This can be explained as follow: within each time step, dD is small (e.g. 0.35 Gy or 0.7 Gy) and the OER is largely dependent on α in Eq. 1, making only minor difference between a LQ model and a linear model.

The diffusion model suggests that SF relies on the distance away from the blood vessels (e.g., different initial pressures), causing heterogeneous sparing inside normal tissues. A distal location benefits more from FLASH delivery due to lower oxygen tensions (Fig. 4a). Otherwise, if taking into no account of heterogeneity, for an initial condition of 30 mmHg, the change of pO2 is 1.2 mmHg (Fig. 7) is not going to cause any significant change of OER. Second, albeit no clear explanation for the longer recovery in experiments (~ 20 secs in Fig. 6) compared with that in Fig. 4 (~ 2 secs) is known yet, it does not alter the conclusion drawn from Table 1. To elaborate, given a slower recovery, the hypoxic status (e.g., low oxygen partial pressure) will last longer, making high dose rates less necessary for obtaining the same SF.

On the microscopic level, the diffusion/recovery model may be relevant to beam delivery modes in proton FLASH. Taking pencil beam scanning (PBS, regardless of shoot-through or not) as an example, the layer switching time is ~ 0.9 s which is quite comparable to the recovery time of 1 sec in Fig. 4b. This yields a complicated pattern of depletion-recovery-depletion-recovery inside tissues. By contrast, the passive scattering mode irradiates the target in a single shot and causes oxygen depletion simultaneously, despite its relatively lower dose rate relative to the PBS. PBS covers the whole target in multiple spots and the resupply of oxygen from the peripherical region of a spot being irradiated (likely at a much longer time scale as shown in Fig. 6), may reduce the degree of oxygen depletion (e.g., not considered in Eq. 5).

4.2 Response between normal and tumor tissue

In the context of the oxygen depletion hypothesis, the mere difference between normal tissues and tumors is the initial oxygen level and our model is applicable to both of them. Different distances in Fig. 4a can be linked to both normal tissues and tumors of different starting oxygen levels, while tumors are likely to be more hypoxic and experience reduced oxygen diffusion (e.g., smaller diffusion constants in Eq. 5). The trends in Table 1 suggest that given the same dose and dose rate, a lower initial oxygen tension corresponds to a larger sparing factor. Therefore, the depletion of oxygen within tumor tissues is going to result more pronounced radio-resistance relative to normal tissues. However, in several experimental studies, no reduced tumor control under FLASH irradiation is observed [3, 7, 15, 46,47,48,49]. In light of such a paradox, we speculate that the oxygen depletion hypothesis and OER framework may only be applicable in interpreting FLASH benefits, when combined with the postulation of hypoxic stem cell niches in normal tissues [32, 33, 50].

Meanwhile, whether there is a direct causal relationship between oxygen depletion and FLASH benefit is still unclear. As a matter of fact, apart from the oxygen depletion hypothesis and OER framework, other plausible causes have also been proposed. Based on stochiometric calculation, one study hypothesizes that higher level of redox-active iron in tumors compared to normal tissues, as well as more rapid removal and decay of organic hydroperoxides and free radicals in normal tissue, is the cause [48]. Another study proposes that oxygen depletion at ultra-high dose rates promotes the protection of normal tissue by limiting the production of reactive oxygen species (ROS) [51].

Other than oxygen-related processes in a short time scale (less than seconds), the differential response between normal and tumor tissue is likely linked to repair pathways of DNA damage and cell cycles (lasting hours or days). One study finds that FLASH (78 Gy/s) results in more inhibition of proliferating crypt cells compared with standard treatment (0.9 Gy/s) for a total dose of 15 Gy, in 3.5 days after treatment. However, the proliferation at 1-day post-treatment demonstrates no significant difference between two dose rates [15]. Another study reports that for a total dose of 6-10 Gy and treatment time of 50 ms (FLASH: 200 Gy/s and conventional: 3 Gy/s), FLASH induces significant apoptotic effects and G2 arrest (in the order of hours) in cancer cells but not in normal cells, using live-cell fluorescence time lapse imaging [52]. The authors point out that whether such a differential effect is in response to the initial insult or occurs during the recovery from damage, is not clear. When synthesizing the above findings altogether, other causes such as repair-related processes or immune-response may contribute equally, if not more, to FLASH benefits, when compared with oxygen depletion and OER.

4.3 Limitations and future perspective

We are cognizant that the oxygen depletion and diffusion models, as well as a number of selected parameters, may be bit oversimplified. In addition, three other limitations exist in our current study and await further investigation. The first limitation is that although the standard LQ model is reliable in the dose range from 2 to 10 Gy [38], whether it is valid for extreme low dose (e.g., less than 1.0 Gy in Fig. 3), may be questionable. In other words, whether Eq. 1 still provides a valid characterization of OER during each transition (small time step) awaits further validation. Somewhat interleaved with this limitation, the validity of an integration approach (i.e., fixed time scale) to evaluate the change of OER and sparing factor in Eq. 4 is another concern. Nevertheless, in our opinion, to incorporate the time scale as done in the dynamic case is the only avenue to model the role of dose rate and transient behavior. Second, we assume that ROD and fixation on DNA damage occur instantaneously, which may last over a time scale of microseconds and is tied to physicochemical kinetics [40]. Third, L in Eq. 2 (more specifically the rate of ROD) is based upon the result reported in water and cell culture medium. However, L also depends on the specific tissue types [53]. This partially explains the discrepancy between the result in our experiment (0.12 mmHg/Gy) and that in our modeling (0.42 mmHg/Gy).