The computation times for each run were recorded. The average running time was then taken between the normoxic and hypoxic oxygen level runs for each pulse dose rate and frequency.
These values are shown in Table 8 below.
Average simulation computation time
D˙p[Gy/s] Total dose [Gy] Pulse frequency [Hz] Average running time [s]
25 0.02 400 2681
Table 8: The average computation times for all simulations.
5 Discussion
This project sought to examine to which degree different dose delivery patterns of ionizing radi-ation cause indirect DNA damage from a radio-chemical perspective. Furthermore, the purpose of the project was to investigate if certain dose delivery patterns exist where there is an increase or reduction of toxic radicals, in particular the concentration of OH ([OH ]), for unchanged total delivered physical dose. The investigation has found that simulating the radiolysis of water with RadChemModel produced results which could suggest a difference in toxicity be-tween CONV-RT and FLASH-RT. The main result from the simulations with FLASH beam parameters is that when the pulse frequency is large enough, [OH ] decreases with every pulse.
However, there are many areas of the program and parameters used which could be improved to achieve more reliable results.
Overall, the obtained results from RadChemModel show that the concentration of the toxic OH 31
radical at ultra-high dose rates evolves very differently with time compared to conventional dose rates. [OH ] behaved as expected when the simulation was provided with conventional beam parameters. There was a steady but slow increase in [OH ] for every pulse, and after an initial steep growth the rate of change began to pan out. Although Table 6 shows that the total increase of [OH ] by the last pulse is of the order 10−11 mol/l, which is very small compared to values in Table 7, it is possible to assume that if the total dose continued to increase, the total accumulated number of radicals produced would be more significant. At ultra-high FLASH dose rates however, [OH ](t) changes appearance. As expected, [OH ] increases rapidly as each pulse is delivered. When the pulse dose rate is larger, more water molecules will be ionized which in turn produces many more radicals. Immediately after the pulse is turned off, [OH ] drops and reaches equilibrium. In the FLASH case, the rate of decline in [OH ] seemed to increase as the dose rate increases. Equally, it can be seen in Figure 7b that at conventional dose rates, reaching equilibrium takes an estimated 2.5 ms. In comparison, it took much less than 0.5 ms for [OH ] to reach equilibrium state with FLASH dose rates in Figure 11. This suggests that the chemical reactions in the radiolysis of water has some kind of dependence on the dose rate of the beam, and that FLASH dose rates encourages certain reactions to take place which causes [OH ] to reach equilibrium at a faster rate.
The most significant observation from this project was that [OH ](t) showed a surprising de-pendence on pulse frequency. At lower pulse frequencies (100-1000 Hz) with FLASH dose rates, [OH ] has time to reach equilibrium before the next pulse arrives. It was also detected that by the last pulse, the concentration of OH approached a slightly larger value of equilibrium than the first pulse. By contrast, when the frequency of the pulse is high enough (10000 Hz) so that the next pulse happened before the concentration from the previous pulse had reached equilibrium, [OH ] drops relatively rapidly with each pulse. In general, it was observed that a larger increase of [OH ] lead to [OH ] reaching equilibrium quicker after the pulse is turned off. Therefore, when the peaks of [OH ] begin overlapping and OH accumulates, it seeks equi-librium even more rapidly and the net effect is a decrease in [OH ]. This behaviour suggests that some reactions in radiolysis of water are prioritized when there is an overlapping of each radical population from each pulse causing OH to be used up in some reactions quicker than it is being produced by other reactions. The phenomenon did not seem to change much between the different pulse dose rates ranging bewteen 106 - 5 × 106 Gy/s, but there is clearly no decline in [OH ] in the results for ˙Dp =25 Gy/s suggesting that the dose rate as well as the frequency are important factors.
Differential sparing between normal tissue and the tumour using FLASH-RT has been reported in several previous in vivo studies. The sparing (reduced damage) effect on normal tissue with FLASH-RT has been named the FLASH-effect [1]. This project has looked at different initial oxygen conditions for each of the beam parameters to investigate if any distinctions could be made between normoxia (5% dissolved O2) and hypoxia (≤ 1% dissolved O2) which correspond to normal tissue and cancerous tissue, respectively. Although both the pulse dose rate and the frequency of the beam seem to be important parameters when determining whether there was a decrease in toxic radicals produced, there was very little difference in toxicity between
different initial oxygen levels corresponding to normoxia and hypoxia. In fact, it was found that conventional beam parameters induced more toxicity in hypoxic conditions which contradicts the established theory that hypoxic cells are more radioresistant than normoxic cells. On the other hand, more reasonable differences in the relative change in [OH ] were found for FLASH beam parameters. Normoxic conditions received a higher relative increase in [OH ] than hypoxic conditions. Furthermore, in Table 7 it can be seen that ∆[OH ] decreases with frequencies up to 1000 Hz, whereas ∆[OH ] increases with pulse frequency in hypoxic conditions. In addition, the results suggest that ∆[OH ] is also dependent on the pulse dose rate. A higher pulse dose rate seems to correspond to a lower relative change in [OH ]. In summary, very little can be said about the difference toxicity between normoxic and hypoxic conditions based on the results from this simulation.
During the process of analysing the results, several questions regarding the validity of the data emerged. For instance, comparing the toxicity between different initial O2 levels would have been possible if the results displayed the total number of [OH ] after each run and not just how [OH ] varied with time. Before the project was initiated, the aim was to integrate [C](t) in the time domain to obtain the cumulative radical concentration for each simulation run to identify the total concentration of OH . Due to time constraints, this step was not included in the code.
Instead, the only conclusion which can be made is whether the function [OH ](t) showed any differential behaviour between normoxia and hypoxia, although, as previously discussed, it was not possible to determine any definite distinctions. In the absence of cumulative radical con-centration, the behaviour of [OH ] was too similar to make any substantial conclusion whether normoxic or hypoxic conditions showed any differential toxicity.
Additionally, not having the means of comparing FLASH-RT and CONV-RT in isodose con-ditions was major drawback in the analysis of the results, since this was one of the important aspects of the simulation. The conventional dose rate was only tested up to a total dose of 0.02 Gy, instead of 10 Gy as for the other simulations. This was because the program took too long (hours) to compute any duration of dose longer than 1 second. It is difficult to approxi-mate the shape of [OH ](t) in Figure 7a during the additional time it would take to deliver 10 Gy. The rate of [OH ] is likely to remain constant up until a certain total dose. After that, assuming that the rate behaves the same must be done with caution. Therefore, in addition to not having time to include a function in the code which would calculate the cumulative con-centration of OH , not acquiring a complete profile of how [OH ](t) behaves with conventional beam parameters for unchanged physical dose has had a great impact on any comparisons made.
6 Outlook
Throughout this project, how to improve the simulation program has been an ongoing discus-sion. The main drawback has been the running time of the code which has put a limit on the beam parameters which could be tested. Table 8 shows the average running time for each
33
simulation with different beam parameters. To deliver a total dose of 0.02 Gy with ˙Dp = 25 Gy/s (see Table 5 for the conventional beam parameters used), the simulation took over 40 minutes to compute. This means that if the total dose to be delivered were to be 10 Gy using the same average dose rate, the simulation would take days. The problem with running the program for longer periods of time is related to the error of the ODE-solver. If there is an error of 107 for each time step and the simulation run 106 steps in total, the error may build up to 10
% in the worst case. Therefore, if the program is left running over a time interval of days there was a higher risk of accumulating errors. Even letting the program run for a couple of hours was not recommended for this reason. This is why the simulations could not be performed with isodose conditions. To improve the computation time, an alternative choice for the ODE-solver would have been preferable.
The version of the code used in this project utilised a built-in Python 3 ODE-solver. The ODE-solver uses a standard method to solve the system of ODEs numerically. Since the solu-tion is in the temporal domain and the time frames of the actual pulses of the order µs, the ODE-solver has to take very small steps to avoid missing out any critical data points. The step size has a major effect on the computation time and has to remain small enough to solve the system of ODEs accurately, yet the majority of the irradiation time is the time between pulses where nothing is happening. The solver will automatically increase the step size when the rate of change of concentration is very small, but if the step size is larger than a pulse then that event may be completely missed out by the solver. To avoid this problem, the maximum step size had been set to be a fraction of the pulse width, which will of course compromise the running time. To speed up the computation, it is therefore desired that the ODE-solver should be able to leave out periods of time of no changes in radical concentrations. This so-lution to the problem is still being investigated and did not fit into the time frame of this project.
Another challenge was to determine any differential toxicity between normoxic and hypoxic conditions, which has been observed in previous studies. A valuable addition to the code would therefore be a function which computes the cumulative concentration of OH as a function of time. This way of presenting the results would be very useful for determining exactly how many toxic radicals are produced and if there is a significant difference between normoxia and hypoxia. Calculating the cumulative concentration of [OH ](t) for FLASH-RT and CONV-RT would have been a useful contribution to the results to confirm that a there is a difference in toxicity. Figure 7a showing [OH ](t) for conventional beam parameters indicates that the cumu-lative concentration indeed may be different from the cumucumu-lative concentration with FLASH, but this was not quantified during this project.
The values for initial O2concentrations in the unit mol/l corresponding to normoxia and hypoxia in this project were sufficient to make simulations with, but there are no rigorous calculations behind them. The oxygen levels in normal tissue are described to range between 3 - 7 % in several studies. However, how this percentage would then be converted to a concentration of the unit mol/l is not stated. In a paper by Wenger et al., this type of problem is brought up to
attention as a frequently asked question in hypoxia research. Biologists prefer to describe oxy-gen levels in tissue using a unit of pressure called millimeters of mercury [mmHg], also known as O2 partial pressure [20]. For example, normoxic conditions is equivalent to a partial pressure of around 38 mmHg [20]. Converting oxygen partial pressure to the unit mol/l is dependent on which solution the oxygen is dissolved into, the temperature of the solution, the phase of the dissolved oxygen as well as the outside pressure. The oxygen levels also fluctuate within the tumour, due to a poorly grown network of blood vessels. In light of this, it is possible that the values used for simulating normoxic and hypoxic conditions in this project may therefore not be exact, as they were found without taking any of the parameters from Wenger et al.’s paper into account. Although the project sought to look for only qualitative dependencies, doing suf-ficient research into how to find accurate values for different oxygen concentrations in different types of tissue would have perhaps increased the reliability of the results obtained in this project.
Regarding the future of sharing progress in FLASH radiotherapy research, it is mentioned in the paper by Wilson et al. that there is generally a lack of consistency when describing which parameters and equipment are being used to detect the FLASH-effect in previous studies [1].
For example, the distinction between pulse dose rate and average dose rate is not made in some of the studies, which causes confusion about what the conditions were when the FLASH-effect was being tested. If the experiment cannot be repeated because the exact dose rates which were used are not described with clarity, future research will be at a disadvantage. It was difficult to decide on which parameters should be used in this project because of this. Another example is that normoxia and hypoxia do not seem to be comprehensively defined, other than as a percentage, in the papers which have been published about the FLASH-effect. Since the majority of the previous studies on FLASH-RT have been in vivo, the exact oxygen concentra-tion in the tissue is not necessarily stated because it is not a controlled variable. In the case of a computer generated simulation, where all conditions need to be defined beforehand, this is very important information to be able to compare results with other findings.
In summary, a simple model like RadChemModel could be developed further to achieve more comprehensive and reliable results. First of all, a function which calculates the cumulative con-centration of each radical is needed to confirm a difference in toxicity between runs. Secondly, the ODE-solver needs to be modified to speed up computation time so that beam parameters can be compared in isodose conditions. If conventional beam parameters were able to run for longer, the significance of any difference in toxicity between CONV-RT and FLASH-RT would have been easier to judge. On the other hand, with this simple model, it can be shown that [OH ] not only changes with time but there is also a noticeable difference in [OH ](t) between FLASH-RT and CONV-RT. Therefore, this model could be a good start for investigating what the underlying mechanism of the FLASH-effect could be.
35
7 Conclusions
• The simulations with RadChemModel show that [OH ] increases steadily with conven-tional beam parameters whilst [OH ] does not behave the same way with FLASH beam parameters, suggesting that FLASH-RT induces less toxicity than CONV-RT in this model.
• The simulations with FLASH beam parameters show a frequency dependence on [OH ].
At a high pulse frequency (10000 Hz) the equilibrium state of [OH ] decreases with time whereas at lower frequencies (100 - 1000 Hz) it increases with time.
• There was no clear difference in toxicity between simulations with normoxic and hypoxic O2 conditions for both FLASH and conventional beam parameters. Without calculat-ing the cumulative concentration of OH in this project, it was difficult to confirm any differential toxicity.
• RadChemModel has been proven to be a simple but sufficient model to show how [OH ](t) changes with time. However, the main limitation with using the simulation has been the computation time being too long. Changing to a faster ODE-solver would be favourable for future investigations.
8 References
[1] Wilson JD, Hammond EM, Higgins GS, Petersson K. Ultra-High Dose Rate (FLASH) Radiotherapy: Silver Bullet or Fool’s Gold? Frontiers in Oncology. 2020;9.
[2] Bassler N, Grzanka L. RadChemModel; 2021. Available from: https://github.com/
APTG/radchem.
[3] Berglund E, J¨onsson BA. Medicinsk fysik. 1st ed. Lund: Studentlitteratur; 2007.
[4] Hall EJ, Giaccia AJ. Radiobiology for the Radiologist. Philadelphia, United States:
Wolters Kluwer Health; 2011.
[5] Attix FH. Introduction to Radiological Physics and Radiation Dosimetry. 1st ed. John Wiley & Sons, Ltd; 1986.
[6] Thomas DJ. ICRU report 85: fundamental quantities and units for ionizing radiation.
Radiation Protection Dosimetry. 2012 Jul;150(4):550–552.
[7] Lilley JS. Nuclear physics : principles and applications. The Manchester physics series.
Chichester: Wiley; 2001.
[8] Spitz DR, Buettner GR, Petronek MS, St-Aubin JJ, Flynn RT, Waldron TJ, et al. An in-tegrated physico-chemical approach for explaining the differential impact of FLASH versus conventional dose rate irradiation on cancer and normal tissue responses. Radiotherapy and Oncology. 2019 Oct;139:23–27.
[9] Pedro Andreo, David T Burns, Alan E Nahum, Jan Seuntjens, Frank Herbert Attix. Fun-damentals of Ionizing Radiation Dosimetry. vol. 2nd edition. Weinheim, Germany: Wiley-VCH; 2017.
[10] Klassen NV, Ross CK. Water Calorimetry: The Heat Defect. Journal of Research of the National Institute of Standards and Technology. 1997 Jan;102 No. 1. Last Modified:
2017-02-17T13:12-05:00.
[11] Labarbe R, Hotoiu L, Barbier J, Favaudon V. A physicochemical model of reaction kinetics supports peroxyl radical recombination as the main determinant of the FLASH effect.
Radiotherapy and Oncology. 2020 Dec;153:303–310.
[12] Cardelli L. From Processes to ODEs by Chemistry. In: IFIP International Federation for Information Processing. vol. 273. Boston: Springer; 2008. p. 261–281.
[13] Lai Y, Jia X, Chi Y. Modeling the effect of oxygen on the chemical stage of water radiolysis using GPU-based microscopic Monte Carlo simulations, with an application in FLASH radiotherapy. Physics in Medicine & Biology. 2021 Jan;66(2):025004.
[14] Montay-Gruel P, Acharya MM, Petersson K, Alikhani L, Yakkala C, Allen BD, et al. Long-term neurocognitive benefits of FLASH radiotherapy driven by reduced reactive oxygen species. Proceedings of the National Academy of Sciences. 2019 May;116(22):10943–10951.
[15] PTB. Linear accelerators; 2016. Available from: https://www.ptb.de/cms/en/ptb/
37
fachabteilungen/abt6/fb-62/621-high-energy-photon-and-electron-radiation/
electron-accelerator-facility-for-radiotherapy-dosimetry/
linear-accelerators.html.
[16] Petersson K, Adrian G, Butterworth K, McMahon SJ. A Quantitative Analysis of the Role of Oxygen Tension in FLASH Radiation Therapy. International Journal of Radiation Oncology, Biology, Physics. 2020 Jul;107(3):539–547.
[17] Favaudon V, Lentz JM, Heinrich S, Patriarca A, de Marzi L, Fouillade C, et al. Time-resolved dosimetry of pulsed electron beams in very high dose-rate, FLASH irradiation for radiotherapy preclinical studies. Elsevier. 2019 Nov;944:162537.
[18] Vozenin MC, Hendry JH, Limoli CL. Biological Benefits of Ultra-high Dose Rate FLASH Radiotherapy: Sleeping Beauty Awoken. Clinical Oncology. 2019 Jul;31(7):407–415.
[19] SciPy. SciPy v1.6.3 Reference Guide; 2021. Available from: https://docs.scipy.org/
doc/scipy/reference/generated/scipy.integrate.solve_ivp.html.
[20] Wenger RH, Kurtcuoglu V, Scholz CC, Marti HH, Hoogewijs D. Frequently asked questions in hypoxia research. Hypoxia. 2015 Sep;3:35–43.
A Appendix A: Additional graphs of results
(a)Graph showing [OH ](t) at ˙Dp= 106 in physxic conditions. (b)Graph showing [OH ](t) at ˙Dp= 106 in hypoxic conditions.
(c)Graph showing [OH ](t) at ˙Dp= 2 × 106in normoxic condi-tions.
(d)Graph showing [OH ](t) at ˙Dp= 2 × 106 in hypoxic condi-tions.
(e)Graph showing [OH ](t) at ˙Dp= 5 × 106in normoxic condi-tions.
(f) Graph showing [OH ](t) at ˙Dp= 5 × 106 in hypoxic condi-tions.
Figure 16: Comparing [OH ](t) for a total dose of 10 Gy delivered as a single pulse at different FLASH dose rates for normoxic and hypoxic initial oxygen conditions.
39
B Appendix B: Concentration of O
2data and discus-sion
∆[O2] [µmol/l] Relative ∆[O2] (%) D˙p[Gy/s] Pulse frequency [Hz] Normoxic Hypoxic Normoxic Hypoxic
25 400 −1.17 × 10−2 −6.04 × 10−3 −4.21 × 10−7 −2.17 × 10−6 Table 9: The change in [O2] in water for conventional beam parameters in normoxic and hypoxic initial oxygen levels.
∆[O2] [µmol/l] Relative ∆[O2] (%) D˙p[Gy/s] Pulse frequency [Hz] Normoxic Hypoxic Normoxic Hypoxic
106 DC -3.91 -3.72 −1.41 × 10−4 −1.34 × 10−3 Table 10: The change in [O2] between the first and last pulse for FLASH beam parameters in
106 DC -3.91 -3.72 −1.41 × 10−4 −1.34 × 10−3 Table 10: The change in [O2] between the first and last pulse for FLASH beam parameters in