An integrated Monte Carlo track-structure simulation framework for modeling inter and intra-track effects on homogenous chemistry

Abstract

Objective:

The TOPAS-nBio Monte Carlo track structure simulation code, a wrapper of Geant4-DNA, was extended for its use in pulsed and longtime homogeneous chemistry simulations using the Gillespie algorithm.

Approach:

Three different tests were used to assess the reliability of the implementation and its ability to accurately reproduce published experimental results: (1) A simple model with a known analytical solution, (2) the temporal evolution of chemical yields during the homogeneous chemistry stage, and (3) radiolysis simulations conducted in pure water with dissolved oxygen at concentrations ranging from $10\mathrm{\mu }\mathrm{M}$ to 1mM with [H₂O₂] yields calculated for 100 MeV protons at conventional and FLASH dose rates of 0.286 Gy/s and 500 Gy/s, respectively. Simulated chemical yield results were compared closely with data calculated using the Kinetiscope software which also employs the Gillespie algorithm.

Main results:

Validation results in the third test agreed with experimental data of similar dose rates and oxygen concentrations within one standard deviation, with a maximum of 1% difference for both conventional and FLASH dose rates. In conclusion, the new implementation of TOPAS-nBio for the homogeneous long time chemistry simulation was capable of recreating the chemical evolution of the reactive intermediates that follow water radiolysis.

Significance:

Thus, TOPAS-nBio provides a reliable all-in-one chemistry simulation of the physical, physico-chemical, non-homogeneous, and homogeneous chemistry and could be of use for the study of FLASH dose rate effects on radiation chemistry.

1. Introduction

The development of innovative radiotherapy (RT) modalities to improve the therapeutic ratio has been the focus of modern medical radiation research. In recent years, FLASH RT research has gained popularity due to the benefits on sparing healthy tissue while maintaining similar tumor control response to conventional RT (Favaudon et al., 2014). This phenomenon has been called the “FLASH-effect” (Dewey & Boag, 1959; Lin et al., 2021). FLASH RT encompasses doses of several Gray delivered in short pulses of 1–2 μs leading to total dose rates higher than ~40 Gy/s (Wilson et al., 2020). In contrast, conventional RT uses much lower doses per pulse at total dose rates of less than ~0.2 Gy/s (Karzmark C. J. et al., 1993; Sharma, 2011). It has been observed that FLASH RT produces fewer negative effects on healthy tissue, which could be used to treat tumors located in sensible or vital organ regions like the lungs or brain (Favaudon et al., 2014, Montay-Gruel et al., 2019); however, the exact mechanism behind the FLASH effect remains under investigation. A potential hypothesis relies on the depletion of oxygen caused by its reaction with chemical species produced in the radiolysis process (Spitz et al., 2019).

A powerful tool to study such fundamental processes is the Monte Carlo (MC) method. MC simulations are one of the more accurate tools to conduct detailed studies of the stochastic effects of radiation on biological tissue. MC track-structure (MCTS) codes allow the modelling of the non-homogeneous chemistry stage following radiolysis that includes the diffusion of chemical species and their reaction kinetics (Karamitros et al., 2014; Ramos-Méndez et al., 2018; Ramos-Méndez, Shin, et al., 2020). In addition, MCTS codes have been used to study the intertrack effects existing in pulses of radiation delivered in liquid water (Alanazi et al., 2020; Kreipl et al., 2009; Ramos-Méndez, et al., 2020). The downside of the MCTS approach is the time it takes to simulate both the non-homogeneous and homogeneous chemistry. Current codes use the Step-By-Step (SBS) or the Independent Reaction Times (IRT) methods, both based on the Smoluchowsky theory of diffusion kinetics (Clifford et al., 1982; Pimblott & Green, 1992; Plante, 2011; Plante & Devroye, 2015) to simulate both the non-homogeneous and the homogeneous chemical stages. For the latter, both SBS and IRT methods model the reaction of chemical species with the bulk by random sampling using an exponential decay function that depends on the concentration of chemical species in the bulk. It is assumed that the concentration of solutes in the bulk does not change and that their spatial distribution is continuous. This approach has the downside of greatly increasing the computation time when increasing the sampling time, number of chemical species, and/or reactions. These downsides have limited the simulation to only a few pulses of radiation (Ramos-Méndez, et al., 2020). At the same time, these approaches might incorrectly estimate experimental chemical yields for cases where changes in the concentrations of solutes in the medium occur.

Homogeneous chemical systems are defined by a finite number of reactions that are modeled using a set of coupled differential equations. This set of differential equations is known as the “chemical master equation” (Gillespie et al., 2013; McQuarrie, 1967) and its solution describes the time evolution of the concentration of all the molecule types present in the system. To solve the master chemical equation, integrator codes have been used effectively and MC algorithms like the Stochastic Simulation Algorithm (SSA) also known as the Gillespie algorithm (Gillespie, 1977) were developed. Those methods rely on modeling all the chemical species as part of the medium by considering only their individual “concentrations” or “number of species”. Examples of software that allow for the simulation of the homogeneous chemistry trough integration algorithms are FACSIMILE (Curtis A. R. & Sweetenham W. P., 1987) and using the SSA algorithm like Kinetiscope (Wiegel et al., 2015). Indeed, these codes cannot simulate the physical stage as detailed and accurate as MCTS does. Moreover, Kinetiscope is unable to simulate the physical and physico-chemical processes while FACSIMILE has inadequate physical models that give results too far from experimental measurements. For these reasons, input to the homogeneous stage is usually obtained from MCTS simulations or experimental data.

TOPAS-nBio (Schuemann et al., 2019) is an MCTS extension of TOPAS (Faddegon et al., 2020; Perl et al., 2012), which are built on top of Geant4-DNA (Incerti et al., 2010) and Geant4 toolkits (Agostinelli et al., 2003), respectively. The aim of TOPAS-nBio is to provide its users with the tools necessary to study the effects of radiation at the cellular and sub-cellular scale. One of the features of TOPAS-nBio is its own implementation of the IRT method for the simulation of chemical diffusion and reaction kinetics for the non-homogeneous chemistry stage (Ramos-Méndez, Shin, et al., 2020). This implementation has been validated for radiation chemistry studies of both liquid water and biologically relevant media at different temperatures (Ramos-Méndez et al., 2022; Ramos-Mendez et al., 2021).

In this work we integrated a variant of the SSA into TOPAS-nBio known as the direct Gillespie algorithm(Gillespie, 1976). This implementation allows for a computationally efficient and continuous simulation of the physical and non-homogeneous chemistry stages followed by the homogeneous chemistry stage. In this way, effects produced by the accumulation of particle tracks within short pulses of radiation can be accounted for (Ramos-Mendez 2020). The implementation was validated for conventional and ultra-high dose rate (FLASH) scenarios using experimental data from the literature for H₂O₂ yields. This extension provides a modeling framework for assisting research in FLASH radiotherapy with TOPAS-nBio.

2. Materials and methods.

2.1. Homogeneous Chemistry modeling

2.1.1. The Gillespie algorithm

The homogeneous chemistry algorithm implemented into TOPAS-nBio in this study is the direct Gillespie method (Gillespie, 1977). We chose the direct method due to its higher accuracy when compared against other variants of the SSA that make use of a leaping approximation algorithm (Cao et al., 2006) to decrease calculation time. The algorithm used in the direct method involves the calculation of the propensity of the system, defined as:

$$A=\sum _i{a}_i;a_i=k_i\cdot S_{i1}\cdot S_{i2}$$ (1)

where $k_i$ is the reaction rate of the $i$-th reaction in the model, and $S_{i1}$ and $S_{i2}$ are the concentrations of the reactive intermediates involved in the $i$-th reaction. If the reaction is a first order (background) reaction or dissociation reaction, $S_{i2}=1$. The time increase is calculated using:

$$\mathrm{\Delta }t=\frac{1}{A}\mathrm{l}\mathrm{n}\left(\frac{1}{1-U_1}\right)$$ (2)

where ${\mathrm{U}}_1$ is a uniformly distributed random number between 0 and 1. Lastly, the reaction selected to occur is the first $i$-th reaction that satisfies:

$$\sum _{i=0}^j{a}_i\ge U_{2}A$$ (3)

where ${\mathrm{U}}_2$ is a uniformly distributed random number between 0 and 1. Equations 1–3 comprise a step in the Gillespie algorithm. Between each time step, the time is increased by $\mathrm{\Delta }t$ and the $j$-th reaction is conducted by reducing the reactive and increasing the products by one. This process is repeated until the simulation time reaches a user-specified end time or the propensity of the system is zero.

2.1.2. IRT – Gillespie Transition Scheme.

Different schemes to transition from the IRT algorithm to the Gillespie algorithm were used in this work for independent histories and accumulated histories, the latter to account for the interaction of histories within a single pulse. In this context, we define a history as the interaction of the primary particle and all its secondaries with the media and the subsequent production of all the chemical species present at the end of the non-homogeneous chemistry stage. For the independent histories scheme, all chemical species remaining from the simulation at the end of each history were fed into the Gillespie algorithm, which continued the simulation until the user defined end time was reached. For the accumulated histories scheme, histories were stored in bunches that comprised a single pulse of around $1-2\mathrm{\mu }\mathrm{s}$. To simulate the temporal pulse shape, we followed the methodology described in (Ramos-Méndez, et al., 2020). A brief description of the method is as follows; the total prescribed dose was divided into the number of simulated pulses and histories were transported until enough dose for each pulse was accumulated. To simulate the shape of the pulse, a time t_H was assigned to each history by sampling a random time from a normal distribution of full width at half maximum and mean time determined by the pulse number and dose rate. This procedure was reasonable as the physical stage lasts only a few femto-seconds, a much shorter time than a typical radiation pulse of a few micro-seconds (Ramos-Méndez, et al., 2020). Each pulse was simulated using the IRT method assuming no inter-pulse contributions. We found that for the pulse frequency and dose rates used in this work, the inter-pulse effects were insignificant in the calculation of chemical yields and thus were not considered in order to lower the simulation time. However, the inter-pulse functionality was left in the code as a user parameter for its use in future works with validation still pending. The chemical species that survived at the end of the non-homogeneous chemistry, also known as escape yields, from individual pulses were passed to the Gillespie algorithm to indicate an increment in the concentration of chemical species at the corresponding time at which the non-homogeneous kinetics finished.

2.2. Verification of the TOPAS-nBio Gillespie Implementation

Three different verifications tests were conducted to ensure that the Gillespie algorithm implementation in TOPAS-nBio was working properly:

1. Solving a simple chemistry model with a known analytical solution.

2. Comparing the temporal evolution of chemical yields during the homogeneous chemistry stage using independent histories with results calculated with the direct Gillespie method of TOPAS-nBio against the Kinetiscope software.

3. Comparing the final chemical yields of the homogeneous chemistry stage using accumulated histories to form radiation pulses obtained from TOPAS-nBio against the Kinetiscope software. This test was conducted for conventional and FLASH dose rates.

2.2.1. Chemical Setup

For Test 1, the simplified model used for the verification of the TOPAS-nBio Gillespie’s algorithm implementation was the following:

$$A\to B;k_{obs}^1=1.00\times {10}^6{M}^{-1}{s}^{-1},$$ (4)

$$B\to C;k_{obs}^2=1.05\times {10}^6{M}^{-1}{s}^{-1},$$ (5)

where $\mathrm{A},\mathrm{B}$, and $\mathrm{C}$ are dummy chemical species. The initial concentrations were $A_0=2\times {10}^{-6}M,B_0=0$ and $C_0=0$. This verification test was run until the system ran to completion at about 0.1 ms. Results were compared with the analytical solution given by equations 6, 7 and 8.

$$A(t)=A_0\mathrm{e}\mathrm{x}\mathrm{p}\left(-k_{obs}^{1}t\right)$$ (6)

$$B(t)=\frac{A_0{k}_{obs}^1}{k_{obs}^2-k_{obs}^1}[\mathrm{e}\mathrm{x}\mathrm{p}\left(-k_{obs}^{1}t\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(-k_{obs}^{2}t\right)]$$ (7)

$$C(t)=A_0-A(t)-B(t)$$ (8)

For Test 2, chemical yields produced using independent histories were calculated. Because Kinetiscope is neither capable of simulating the physical nor the non-homogeneous chemistry stages, we ran the TOPAS-nBio IRT simulations until the end of the non-homogeneous chemical stage and fed the output yields into Kinetiscope. The same chemistry model, described below, was used in TOPAS-nBio and Kinetiscope. The transition time point between the non-homogeneous and homogenous chemistry stage which triggered the Gillespie algorithm was determined as follows. TOPAS-nBio output yields were obtained at three different times $1\mathrm{\mu }\mathrm{s},10\mathrm{\mu }\mathrm{s}$ and $100\mathrm{\mu }\mathrm{s}$. Subsequently, the time evolution of chemical yields using these input values was compared using Kinetiscope and full TOPAS-nBio simulations including the physical stage until 1000 s. For Test 3, chemical yields produced by radiation pulses were calculated. The Kinetiscope software allows input of initial chemical yields in a discrete way using a time step function. In this way, the TOPAS-nBio results were compared against Kinetiscope to verify the proper implementation that mimics multiple radiation pulses and longer times. The chemical model used in this work for both Kinetoscope and the verification Tests 2 and 3 is shown in Table 1. The table includes validated reaction rates using TOPAS-nBio for the non-homogeneous chemistry using low LET radiation (Ramos-Mendez et al., 2021; Ramos-Méndez et al., 2018). In addition, the reactions and reaction rates for the longer times were obtained from (Pastina & LaVerne, 2001; Plante, 2021).

Table 1:

Reaction list used for the simulation of G values and concentrations obtained from (Pastina & LaVerne, 2001; Plante, 2021; Ramos-Mendez et al., 2021). R1B and R2B use concentrations corresponding to pH 7 while REB2 and REB3 used specific concentrations of dissolved O₂. 1 M = 1 mol/dm³

	TOPAS Default reactions for pure liquid water		Extra Reactions for H2O2 Production
Reaction Index	Reaction	Reaction Rate (M−1s−1)	Reaction Index	Reaction	Reaction Rate (M−1s−1)
R1	$H_3{O}^{+}+{OH}^{-}\to H_{2}O$	14.3 × 1010	RE1	$H_3{O}^{+}+O_2^{-}\to {HO}_2$	5 × 1010
R2	$H^{•}+H^{•}\to H_2$	7.8 × 109	RE2	$e_{\mathit{aq}}^{-}+O_2\to O_2^{-}$	1.9 × 1010
R3	$H^{•}+{}^{•}OH\to H_{2}O$	2 × 1010	RE3	$H+O_2\to {HO}_2$	2.1 × 1010
R4	$H+H_2{O}_2\to {}^{•}OH$	9 × 107	RE4	${HO}_2+{HO}_2\to H_2{O}_2+O_2$	8.3 × 105
R5	$H^{•}+e_{aq}^{-}\to H_2+{OH}^{-}$	2.5 × 1010	RE5	${HO}_2+O_2^{-}\to H_2{O}_2+O_2+{OH}^{-}$	9.7 × 107
R6	${}^{•}OH+{}^{•}OH\to H_2{O}_2$	5.5 × 109	RE6	$O_2^{-}+O_2^{-}\to H_2{O}_2+O_2+{OH}^{-}$	4.9 × 103
R7	${}^{•}OH+e_{aq}^{-}\to {OH}^{-}$	3 × 1010	RE7	${HO}_2+OH\to O_2+H_{2}O$	6 × 109
R8	$H_2{O}_2+e_{aq}^{-}\to {OH}^{-}+{}^{•}OH$	1.1 × 1010	RE8	$O_2^{-}+OH\to O_2+{OH}^{-}$	8.2 × 109
R9	$e_{aq}^{-}+e_{aq}^{-}\to 2{\mathit{O}\mathit{H}}^{-}+{\mathit{H}}_2$	5.5 × 109	RE9	$e_{\mathit{aq}}^{-}+{\mathit{H}\mathit{O}}_2\to {\mathit{H}}_2{\mathit{O}}_2+{\mathit{O}\mathit{H}}^{-}$	2 × 1010
R10	$e_{aq}^{-}+H_3{O}^{+}\to H$	2.3 × 1010	RE10	$e_{\mathit{aq}}^{-}+{\mathit{O}}_2^{-}\to {\mathit{H}}_2{\mathit{O}}_2+2{\mathit{O}\mathit{H}}^{-}$	1.3 × 1010
			RE11	$\mathit{H}+{\mathit{H}\mathit{O}}_2\to {\mathit{H}}_2{\mathit{O}}_2$	1.5 × 1010
			RE12	$\mathit{H}+{\mathit{O}}_2^{-}\to {\mathit{H}}_2{\mathit{O}}_2+{\mathit{O}\mathit{H}}^{-}$	1.8 × 1010
Background reactions for pure liquid water			RE13	${\mathit{H}\mathit{O}}_2+{\mathit{H}}_2{\mathit{O}}_2\to \mathit{O}\mathit{H}+{\mathit{O}}_2$	5 × 10−1
Reaction Index	Reaction	Reaction Rate (s−1)	RE14	${\mathit{O}}_2^{-}+{\mathit{H}}_2{\mathit{O}}_2\to \mathit{O}\mathit{H}+{\mathit{O}}_2+{\mathit{O}\mathit{H}}^{-}$	1.3 × 10−1
R1B	$e_{aq}^{-}+H_3{\mathit{O}}^{+}\to H$	2.3 × 1010	RE15	$\mathit{O}\mathit{H}+{\mathit{H}}_2\to \mathit{H}+{\mathit{H}}_2\mathit{O}$	4.3 × 107
R2B	${\mathit{H}}_3\mathit{O}+{\mathit{O}\mathit{H}}^{-}\to {\mathit{H}}_2\mathit{O}$	14.3 × 1010	RE16	$\mathit{O}\mathit{H}+{\mathit{H}}_2{\mathit{O}}_2\to {\mathit{O}}_2^{-}+{\mathit{H}}_3{\mathit{O}}^{+}$	2.7 × 107
Extra background and dissociation reactions for H2O2 production
Reaction Index	Reaction			Rate Constant (s −1)
REB1	${\mathit{H}\mathit{O}}_2\to {\mathit{H}}^{+}+{\mathit{O}}_2^{-}$			8.05 × 105
REB2	$e_{\mathit{aq}}^{-}+{\mathit{O}}_2\to {\mathit{O}}_2^{-}$			1.9 × 1010
REB3	$\mathit{H}+{\mathit{O}}_2\to {\mathit{H}\mathit{O}}_2$			2.1 × 1010

Conventional and FLASH simulations (Test 3) were conducted using four different oxygen concentration levels: $10\mathrm{\mu }\mathrm{M},50\mathrm{\mu }\mathrm{M}$, 0.25 mM, and 1mM that corresponds to different saturation levels with respect of 1 atm and 20°C liquid water (~0.25 mM) (Chapman et al., 1970). The different oxygen concentration levels corresponded to the experimental conditions from the literature.(Anderson & Hart, 1961; Montay-Gruel et al., 2019; Roth & Laverne, 2011; Sehested et al., 1968).

2.2.2. Physical setup

Test 2 and 3 used a 100 MeV proton beam normally incident on one of the surfaces of a $27\mathrm{\mu }{\mathrm{m}}^3$ cubic box of water, we chose this volume as a compromise between computational efficiency and data accuracy for FLASH dose rate simulations, allowing us to compare Conventional and FLASH dose rates in a fair manner. The radiation field size was chosen as $2.8\times 2.8\mathrm{\mu }{\mathrm{m}}^2$ to avoid particles being transported outside of the cube while delivering radiation to the majority of the cube’s face. For test 3 the radiation pulses for conventional and high dose rates were generated with the configuration shown in Table 2, recreated from (Montay-Gruel et al., 2019). Conventional simulations consisted of up to 700 pulses of 0.02857 Gy each while FLASH simulations consisted of up to four pulses of 5 Gy each, for a maximum dose of 20 Gy. The particle transport part of our simulations was carried out considering a pure liquid water environment without any considerations of oxygen ionizations at the different saturation levels. We believe this assumption is reasonable when considering that the proportion of oxygen to water molecules, with concentrations of 0.005 – 1mM and 55.5M respectively, gives an oxygen ionization probability in the range of 1.8 × 10⁻⁵ – 1.8 × 10⁻³ %, which is low enough to be negligible for the interests of this work.

Table 2:

Pulse Information for the Conventional and FLASH RT dose rates comparison. Pulses were simulated using a Gaussian distribution with the FWHM equal to the pulse width.

Modality	Pulse Width (μs)	Pulse Frequency (Hz)	In Pulse Dose Rate (Gy s−1)	Total Dose Rate (Gy s−1)
Conventional	1	10	2.857×104	0.2857
FLASH	1.8	100	2.78×106	500

3. Results

3.1. Verification

3.1.1. Analytical Gillespie algorithm verification

The temporal evolution of generic chemical species A, B, and C used for the verification of the Gillespie algorithm implementation, Test 1, is shown in figure 1. As depicted, the results from TOPAS-nBio agreed with the analytical results within one standard deviation of 5.2%, 1.99% and 0.0006% for A, B and C respectively, measured at the end of the simulation. This was determined from one thousand repetitions of the simulation. The equilibrium for both was reached at about $8\mathrm{\mu }\mathrm{s}$ at which point only C chemical species remained.

Figure 1:

TOPAS-nBio direct Gillespie algorithm validation against the analytical solution. Results from TOPAS-nBio are shown for the concentration of A (continuous line), B (long dashed line) and C (dotted line). Analytical results are also shown for A (open circles), B (open squares) and C (open triangles). TOPAS-nBio standard deviation was kept under 0.9% of the calculated value.

3.1.2. Homogeneous Chemistry Algorithm Verification for Independent Tracks

Independent track scheme simulation results are shown in figure 2 for six different chemical species: •OH, e_aq⁻, H₂, H₂O₂, H• and HO₂. We tested three different time values (1 us, 10 us and 100 us) for the non-homogeneous to the homogeneous chemistry transition point. The results for the three transition times are shown against its Kinetiscope equivalent. Differences between TOPAS-nBio and Kinetiscope were within one standard deviation with a value of 0.36% for $1\mathrm{\mu }\mathrm{s},10\mathrm{\mu }\mathrm{s}$ and $100\mathrm{\mu }\mathrm{s}$. The previously reported standard deviation is for H₂O₂ at 1000 s, this is because H₂O₂ was the molecule of interest for this work. No significant differences were found for the final yields at 1000 s between the three transition points with differences also within one standard deviation. Note that these transition points are only valid for pure water and will be different for more concentrated systems as encountered in a cell.

Figure 2:

Independent ionizing particle track simulations. Results for TOPAS-nBio are shown for a $1\mathrm{\mu }\mathrm{s}$ (black band), $10\mathrm{\mu }\mathrm{s}$ (blue band) and $100\mathrm{\mu }\mathrm{s}$ (red band) non-homogeneous to homogeneous transition time respectively using $10\mathrm{\mu }\mathrm{M}$ of oxygen concentration. Kinetiscope results are shown for $1\mathrm{\mu }\mathrm{s}$ (black open squares), $10\mathrm{\mu }\mathrm{s}$ (blue open triangles) and $100\mathrm{\mu }\mathrm{s}$ (red open stars) respectively. TOPAS-nBio results are shown with its standard deviations represented by the width of the bands. *H₂ yields are shown with a 10x factor on the Y axis.

3.1.3. Homogeneous Chemistry Algorithm Verification for Radiation Pulses

Simulations using TOPAS-nBio were done for up to 700 pulses of conventional and up to 4 pulses of FLASH dose rates. For the pulse radiolysis verification, we compared up to10 conventional dose rate pulses and the 4 pulses of FLASH dose rates. Results showing the difference in H₂O₂ yield behaviors between TOPAS-nBio and Kinetiscope are shown in figure 3. Differences for H₂O₂ yields between both approaches are within one standard deviation with a value of 1.01% and 0.06% measured at the end of our simulations for ten pulses of conventional and four pulses of FLASH dose rates respectively, with the standard deviation for the FLASH results being noticeable lower compared with the conventional ones due to the lower number of pulses.

Figure 3:

Pulse radiolysis verification. TOPAS-nBio results are shown as dashed lines and Kinetiscope results are shown as gray transparent bands. Results were calculated using the chemical model from table 1 with $50\mathrm{\mu }\mathrm{M}$ oxygen concentration.

3.2. Conventional and FLASH dose rates Validation

Simulation results for the [H₂O₂] yield with respect to the dose are presented in figure 4. Results include four oxygen concentrations for both conventional and FLASH dose rates. The concentrations of H₂O₂ for conventional and FLASH RT pulses are shown in figure 5 for different oxygen concentrations. Experimental results from the literature are also shown for conventional (Montay-Gruel et al., 2019; Roth & Laverne, 2011) and FLASH (Anderson & Hart, 1961; Montay-Gruel et al., 2019; Sehested et al., 1968) dose rates. Standard deviations were within 1.97% and 0.083% for conventional and FLASH dose rates respectively, these results exhibit the same behavior from section 3.1.3, with the FLASH results having a lower standard deviation than the conventional ones due to the lower number of pulses needed to get a dose of 20 Gy. The results shown in figure 5 were obtained by taking the results from figure 4a and conducting a linear fit of the form $\left[H_2{O}_2\right](D)=mD$ where $D$ is the dose and $m$ is the [H₂O_2] yield per gray for each oxygen concentration, giving four points for conventional and FLASH. Figure 5 results are shown alongside the standard deviation from the fitting procedure with a maximum value of 0.4% for conventional and 0.88% for FLASH.

Figure 4:

[H₂O₂] production calculated at 1000 s as a function of dose. a) Results for total H₂O₂ yields and b) H₂O₂ production. Results for conventional (gray) and FLASH (black) dose rates for different oxygen concentrations of $10\mathrm{\mu }\mathrm{M}$ (circles), $50\mathrm{\mu }\mathrm{M}$ (squares), $250\mathrm{\mu }\mathrm{M}$ (diamonds) and 1mM (triangles) are shown.

Figure 5:

[H₂O₂] yield per gray. Results are shown for O₂ concentrations ranging from $10\mathrm{\mu }\mathrm{M}$ to 1 mM for conventional (open pentagons with gray solid line) and FLASH (open plus sign with black dashed line) dose rates. Experimental data for conventional dose rates is shown with gray circles (Montay Gruel, 2019) and gray squares (Roth & LaVerne, 2011). Experimental data for FLASH dose rates is shown with black diamons (Montay Gruel, 2019), black triangles (Sehested, 1968) and black stars (Anderson, 1961).

TOPAS-nBio results agreed within 0.93% for the conventional dose rate results when compared against (Roth & Laverne, 2011) and 1% for the FLASH dose rate when compared against (Sehested et al., 1968).

Figure 6 shows the main reactions (reactions that contribute with more than 1% to the final yields) that contributes to H₂O₂ yields for the different number of pulses in the conventional dose rates.

Figure 6:

Main reactions contributing to the H₂O₂ yield. Only reactions with more than 1% of contribution at any point are shown. These curves are calculated with $50\mathrm{\mu }\mathrm{M}$ of [O₂] and conventional dose rates.

4. Discussion

In this work the Gillespie algorithm was implemented in TOPAS-nBio. The verification of the algorithm followed a three-stage method (using Tests 1–3) to verify that the implementation was working correctly and establish its accuracy. For the first verification test, we found that the Gillespie algorithm implementation in TOPAS-nBio recreated the analytical solution of a simple three molecule model within one standard deviation of MC statistical uncertainties. We conclude that there are no significant differences between the result predicted by the implementation of the Gillespie algorithm in TOPAS-nBio and the analytical solution.

4.1. Homogeneous Chemistry Algorithm Verification for Independent Tracks and Pulse Radiolysis

Verification results for independent histories and pulses of radiation were in reasonable agreement, considering statistical uncertainty, of the results from Kinetiscope in both, behavior, and final yields. With TOPAS-nBio, the simulation of the physical and non-homogeneous chemical stages for each individual particle track included some variability in the escape yields caused by the random nature of the ionization events of each track. For Kinetiscope, this variability was not present since we only passed the mean escape yields of multiple particle tracks from TOPAS to Kinetiscope and the program was only run once. For all the chemical species, a smooth time evolution was observed by both TOPAS-nBio and Kinetiscope, except for the HO₂ as shown on the last panel of figure 2. The point where the step decrease of HO₂ yields begins corresponds with the transition between the non-homogeneous and homogeneous chemistry. It is caused by the nature of the IRT method which does not allow for changes in scavenger concentrations. The IRT implementation of TOPAS-nBio treats dissociation reactions as scavenger reactions. Since these reactions depend on a variable concentration, they are ignored during the non-homogeneous stage and only come into play at the homogeneous stage. The implementation of dissociation reactions is left to the future.

4.2. Conventional and FLASH dose rate Validation

The model used in this work shows that the yield of H₂O₂ depends on dose-rate as shown in figure 5. This dependence, discussed by others (Wardman, 2020), is caused by the accumulation of the remaining chemical species generated by the radiation pulses, which affects the competition kinetics for the chemical species generated by the following pulses. This effect increases as more pulses start to accumulate more H₂O₂, O₂, O₂⁻ and H₂ molecules. These have all been observed to survive for hours after irradiation (Pastina & LaVerne, 2001). This buildup of species causes a decrease in H₂O₂ production between pulses due to the competition kinetics of chemical species for •OH consumption. This effect is shown in Figure 6, where the contribution of the main reactions includes reaction RE6. The contribution of this reaction is not significant until enough O₂⁻ buildup has been achieved, in this study at around 200 pulses. Figure 6 also shows that reactions R8 and RE16 are mostly unaffected by the pulse dynamic, which can be explained by their fast reaction times, meaning that in our model, these reactions occur mostly at short time scales and are thus unaffected by the changes in the medium composition that is being considering in the homogeneous stage.

The main H₂O₂ generating channel is the reaction R6 from Table 1. During the first pulse, •OH will react with itself to create H₂O₂ with very little competition with O₂⁻ and H₂O₂ (reactions R8, RE6, and R16). Once the buildup of O₂⁻ and H₂O₂ is high enough, H₂O₂ generation will occur mostly during the homogeneous chemistry with little contribution from the non-homogeneous chemistry yields. The experimental value of 1.0 molecules per 100 eV at 0.250 mM of oxygen falls in a middle point between the escape yield of 0.7 molecules per 100 eV and the single pulse conventional dose rate yield of 1.30 ± 0.01 molecules per 100 eV as predicted by both TOPAS and Kinetiscope. These results suggest that consecutive pulses do lower the production of H₂O₂ little by little. However, proof that this actually occurs requires experimental measurements showing the yields of H₂O₂ in pure liquid water for only a few pulses, if possible.

On the other hand, FLASH dose rates do not show this behavior at the dose rates used in this work due to the short time duration of the number of pulses involved. In addition, the short pulse separation is not enough to complete O₂⁻ and H₂O₂ generation and start the competition kinetics of these chemical species with OH to generate H₂O₂. Although O₂⁻ and H₂O₂ generation is not high enough at one or two pulses to reduce the total H₂O₂ yield per gray in a significant manner, this reduction is still present, as can be seen from our simulation results in figure 4b as the H2O2 yield decreases with increasing dose/number of pulses. The same trends are shown in the experimental results from Montay-Gruel (Montay-Gruel et al., 2019). More complete evidence of the prediction by our simulations would benefit from measurement of pulse-by-pulse conventional and FLASH dose rate H₂O₂ yields in pure liquid water, measured at different oxygen concentrations. Although H₂O₂ contribution towards FLASH therapeutic effects can be debated (Wardman, 2020), having a tool capable of recreating its chemical yields at long times is nonetheless an important step towards creating a tool capable of simulating FLASH dose rates effects on cellular environments which will be done in future works by proposing chemical models that take into account the cellular environment composition following a similar approach to (Labarbe et al., 2020).

5. Conclusion

In this work, we verified the integration of the Gillespie algorithm into TOPAS-nBio for the efficient simulation of pulses of radiation at conventional and ultra-high dose rate regimes. The integrated implementation was validated by successfully recreating the experimental results for H₂O₂ yield within 1% for both conventional and FLASH dose rates, with no significant difference found between the implementation in TOPAS-nBio with both analytical, and Kinetiscope results. Thus, we have developed and verified a tool for studying long time chemistry effects that, as shown in this work, are correlated with FLASH dose rates, and can be used to study FLASH dose rate effects in liquid water.