# Modeling interspur interactions as a potential mechanism of the FLASH
effect.

Alexander Baikalov  Technical University of Munich, School of Medicine and
Klinikum rechts der Isar, Department of Radiation Oncology, Munich, Germany
Helmholtz Zentrum München GmbH, German Research Center for Environmental
Health, Institute of Radiation Medicine, Neuherberg, Germany  University of
Texas MD Anderson Cancer Center, Department of Radiation Physics, Houston,
Texas, United States of America  Ramin Abolfath  University of Texas MD
Anderson Cancer Center, Department of Radiation Physics, Houston, Texas,
United States of America  Radhe Mohan  University of Texas MD Anderson Cancer
Center, Department of Radiation Physics, Houston, Texas, United States of
America  Emil Schüler  University of Texas MD Anderson Cancer Center,
Department of Radiation Physics, Houston, Texas, United States of America
Graduate School of Biomedical Sciences, University of Texas, Houston, Texas,
United States of America  Jan J. Wilkens  Technical University of Munich,
School of Medicine and Klinikum rechts der Isar, Department of Radiation
Oncology, Munich, Germany  Stefan Bartzsch  Technical University of Munich,
School of Medicine and Klinikum rechts der Isar, Department of Radiation
Oncology, Munich, Germany  Helmholtz Zentrum München GmbH, German Research
Center for Environmental Health, Institute of Radiation Medicine, Neuherberg,
Germany

(August 28, 2024)

## Abstract

The mechanism responsible for the FLASH effect, normal tissue sparing by
ultra-high dose rate (UHDR) irradiation with isoeffective tumor control
compared to conventional dose rate (CDR) irradiation, remains undetermined.
Here we investigate the contribution of interspur interactions (interactions
between radiolytic species of individual particle tracks) to overall
radiochemical interactions as a function of irradiation parameters, and
suggest an increase in interspur interaction as a potential mechanism for
tissue sparing in FLASH radiation therapy. We construct a model that
analytically represents the spatiotemporal distribution of spurs in a target
volume as a function of irradiation parameters (e.g. dose, dose rate, linear
energy transfer), and quantifies the effect of interspur interactions on the
ongoing radiochemistry. Spurs evolve under a simplified reaction-diffusion
equation with parameters based on Monte Carlo simulations, and interspur
interaction is quantified by calculating the expected values of interspur
overlap in the target.

The model demonstrates that for any set of irradiation parameters, a minimum
critical dose and dose rate are necessary to induce significant interspur
interaction, and that interspur interactions correlate negatively with beam
linear energy transfer at a fixed dose. The model suggests optimal beam
parameters, including dose, dose rate, linear energy transfer, and pulse
structure, to maximize interspur interactions. Depending on the rate of
radical scavenging in the target, which limits interspur interaction, this
model predicts that the irradiation parameters necessary to elicit the FLASH
effect may coincide with an onset of significant interspur interactions,
suggesting that interspur interaction may be the underlying mechanism of the
FLASH effect.

##  1 Introduction

The past decade has witnessed a surge in investigation into the effects of
high dose rate irradiation on tissue toxicity. Many studies, both in vitro and
in vivo, and recently in a human patient, have reported normal tissue sparing
by irradiation at ultra-high dose rates (UHDRs,
$\gtrapprox$40\text{\,}\mathrm{G}\mathrm{y}\mathrm{/}\mathrm{s}$$) relative to
irradiation at conventional dose rates (CDRs,
$\approx$0.01\text{\,}\mathrm{G}\mathrm{y}\mathrm{/}\mathrm{s}$$) while
maintaining equivalent tumor control probability [1, 2, 3, 4, 5, 6]. This
biological effect, known as the FLASH effect, presents significant benefits to
clinical radiotherapy applications. However, clinical application of the FLASH
effect is currently hindered by an incomplete understanding of both the
precise irradiation parameters necessary to elicit a reproducible FLASH effect
as well as the underlying mechanisms [7, 8, 2].

Analyses of studies reporting a FLASH effect or lack thereof have provided
some approximate constraints on the irradiation parameters required to induce
an observable FLASH effect: an average dose rate
$\gtrapprox$40\text{\,}\mathrm{G}\mathrm{y}\mathrm{/}\mathrm{s}$$, a total
dose $\gtrapprox$10\text{\,}\mathrm{G}\mathrm{y}$$, and thus an exposure time
of $\lessapprox$0.2\text{\,}\mathrm{s}$$ [9, 10]. These constraints are by no
means absolute nor do they guarantee an observable FLASH effect, as evidenced
by multiple studies unable to observe a FLASH effect [9]. In fact, many other
parameters, such as pulse width, pulse frequency, tissue type and oxygenation,
radiation modality, field size, etc., may play a role. Notwithstanding, the
FLASH effect has been reported in many different tissue types including mouse
brain [11, 5, 12, 13], zebrafish embryos [11], mini-pig skin [3], cat skin
[3], and human skin [14], and for different radiation modalities such as
electrons [12, 1, 3], x-rays [5], and protons [15, 16, 17].

Unraveling the mechanism responsible for the FLASH effect (FLASH mechanism)
would greatly accelerate the clinical translation of FLASH radiotherapy (RT),
e.g. by defining optimum irradiation parameters. Comparing the time scale of
radiation effects in tissue (see Figure 1) to irradiation parameters that
elicited a FLASH effect in experiments, radiochemical interactions seem to be
the most likely source of the FLASH effect. For instance, Montay-Gruel et al.
showed an induction of the FLASH effect in vivo when a
$10\text{\,}\mathrm{G}\mathrm{y}$ exposure was shortened from
$0.5\text{\,}\mathrm{s}$ to approximately $0.2\text{\,}\mathrm{s}$ [12].
Biological responses, although certainly subsequently affected by FLASH
irradiation, are unlikely to be the source of the FLASH effect as they occur
at time scales too large (e.g. $\approx${10}^{3}\text{\,}\mathrm{s}$$ for DNA
repair [18]) to account for temporal irradiation changes on the order of
$0.50.2\text{\,}\mathrm{s}$. On the other hand, considering that the physical
and physico-chemical stages of a single track are finished within a picosecond
and take place on length scales of a few nanometers, interactions between
separate tracks at these extremely early stages are negligible. For
comparison, only approximately one particle track arrived per
$3.2\text{\,}\mu\mathrm{m}^{2}$ and per $26\text{\,}\mathrm{n}\mathrm{s}$ on
the target on average in the aforementioned FLASH study for a dose of
$10\text{\,}\mathrm{G}\mathrm{y}$ and an exposure time of
$0.2\text{\,}\mathrm{s}$. These quantities were derived assuming a cubical
water target of $5\text{\,}\mathrm{m}\mathrm{m}$ side length to emulate a
mouse brain, the target used in the study, and an average energy deposition in
that target per primary electron of
$1\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$, which was calculated via electron
stopping power data [19].

The most prominent hypotheses of FLASH mechanisms in the literature do indeed
point to radiochemical effects caused by the shortened time scale of
production of water radiolysis products. An overview of the relevant processes
of the radiochemical phase is given in Figure 1. For example, the oxygen
depletion hypothesis suggests that radiolytic oxygen depletion at UHDR occurs
at a rate significantly quicker than tissue oxygen replenishment via diffusion
can take place [11, 20, 21, 22, 23, 24, 25, 26, 27]. Therefore, in contrast to
CDR, a transient hypoxia of the irradiated tissue may be achieved, rendering
it less radiosensitive due to the well-documented hypoxia-induced
radioresistance of cellular tissue [28]. Although oxygen concentrations have
been shown to significantly affect the onset of the FLASH effect [20, 11],
many studies indicate that oxygen depletion alone cannot explain the FLASH
effect as proposed [29, 30, 31, 32].

A variety of different FLASH mechanisms involving altered interaction
phenomena among water radiolysis products at UHDR have been proposed. For
example, increased rates of radical recombination interactions (e.g.
$\ce{{}^{.}OH+^{.}OH->H2O2}$) could decrease the yields of radicals available
for biological damage [24, 32, 33]. Additionally, higher radical
concentrations directly after UHDR irradiation may maximize the differences
between normal and tumor tissue metabolism and radical removal efficiency
[27].

Figure 1: Relevant processes of the different approximate time scales of
radiation effects in tissue compared with the standard exposure times for
FLASH and conventional (CONV) irradiation. The chemical stage can be divided
into the heterogeneous and homogeneous chemical stages, as is shown in this
figure; however, the listed processes that occur during these phases are not
strictly divided and can occur throughout the entire chemical phase. Figure
adapted from Le Caër [34]. Figure 2: Spatial distribution of a
$1\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ proton track in pure water (after
${10}^{-15}\text{\,}\mathrm{s}$) and its resulting spur at different time
points (${10}^{-12}\text{\,}\mathrm{s}$ \- ${10}^{-6}\text{\,}\mathrm{s}$).
The spur is shifted spatially along the x-axis for each time point for better
visualization. In this figure, the proton track (blue) is hidden by the large
number of $\delta$ electrons (orange) it produces, which are primarily
responsible for the production of radiochemical species (red). The beam axis
is along the positive z axis. Simulations were performed with TOPAS n-Bio
[35].

##  2 Methods

###  2.1 Motivation

In this work, we examine radiochemical changes at UHDR that could cause the
FLASH effect at the transition from the heterogeneous to the homogeneous
chemical stage, as described in Figure 1. While radiochemical species are
initially highly localized around the particle track (heterogeneous chemical
stage), thermal diffusion dissolves this structure leading to a more
homogeneous spread of these radiolysis products. We refer to the collection of
radiochemical species produced by a single particle track as that particle’s
spur. The interactions that these species undergo can be classified into
intraspur, spur-environment, and interspur interactions. Every spur
experiences intraspur interactions (i.e. interactions among the radiochemical
species of a single spur) while undergoing an outward diffusion due to the
thermal motion of its species, as depicted in Figure 2. Intraspur interactions
result in a net concentration decay of the species the spur comprises, most
notably of the hydroxyl radical ($\ce{{}^{.}OH}$), which is responsible for
the majority of radiation-induced DNA damage [36]. Each spur also experiences
interactions with its local aqueous environment, e.g. with dissolved oxygen or
cellular scavengers. Lastly, if two or more spurs are located close enough in
space and time, they can undergo interspur interactions.

When considering the aforementioned UHDR to FLASH irradiation transition
($0.50.2\text{\,}\mathrm{s}$) from the perspective of the chemical stage of
spurs, the average spatial distribution of spurs does not change. Instead,
only their temporal distribution is changed, as spurs arrive in quicker
succession. While intraspur interactions remain unaffected by a change in dose
rate, spur-environment and interspur interactions may change with dose rate.
The oxygen depletion hypothesis is an example of a spur-environment
interaction affected by dose rate. Increasing the temporal density of spurs
increases the probability of interspur interactions. Hypotheses for FLASH
mechanisms involving increased radical interactions due to shortened exposure
times (increased dose rates at a fixed dose) must therefore attribute the
increase in radical interactions to either an indirect effect of altered spur-
environment interactions, or to increased interspur interactions. In this
context, interspur interactions also refer to interspur interactions that
occur during the homogeneous chemistry phase, long after the original track
structures have dissolved.

Here, we present an analytical model to quantify intraspur and interspur
interactions as a function of irradiation parameters. A discussion of the
validities and consequences of the assumptions made in this model is given in
Section 4.

###  2.2 Spur Definition

In this model, spurs are assumed to be perfectly straight and homogeneous
along the beam axis, and can thus be described in the 2-dimensional plane
orthogonal to the beam axis with the position vector $\vec{r}=(x,y)$. An
analogous 3-dimensional model, more representative of sparsely ionizing
radiation, is discussed in Section 4.4.

We consider a system of $N_{s}$ spurs distributed across a target. An
arbitrary spur at position $\vec{r_{i}}$ and creation time $t_{i}$ is defined
by a probability density function (PDF) $c_{i}(\vec{r},t)$ describing the
spatio-temporal distribution of the radiochemical species that make up the
spur. The spatio-temporal evolution of the spur is governed by a reaction-
diffusion equation comprising a cylindrically-symmetric diffusion term with
diffusion coefficient $\alpha$, a decay term with decay constant $k_{s}$, and
an interspur interaction term with reaction rate $k_{r}$:

$\frac{\partial}{\partial
t}c_{i}(\vec{r},t)=\alpha\nabla^{2}c_{i}(\vec{r},t)-k_{s}c_{i}(\vec{r},t)-k_{r}\sum_{i\neq
j}^{N_{s}}c_{i}(\vec{r},t)c_{j}(\vec{r},t)\ .$ (1)

The decay term represents the effective result of all intraspur and spur-
environment interactions; thus, the spur decay constant $k_{s}$ is a model
parameter that fixes the lifetime of a spur depending on the spur’s own
characteristics and those of its environment. The interspur interaction term
considers the interaction with every other spur on the target. In this work we
only consider the limit of weakly-interacting spurs and thus neglect this
interspur interaction term.

If we set $c_{0}$ to be the initial total number of species in any spur, the
solution to this partial differential equation (see Section A.1), neglecting
interspur interactions, is a normal distribution about the arrival point of
the spur that broadens and decays with time:

$c_{i}(\vec{r},t)=c_{0}\cdot\frac{e^{-\frac{\absolutevalue{\vec{r}-\vec{r_{i}}}^{2}}{4\alpha(t-t_{i})}-k_{s}(t-t_{i})}}{4\pi\alpha(t-t_{i})}\
,$ (2)

One disadvantage of describing the spur by this PDF is that at very short time
scales, the normal distribution becomes infinitely dense, which is an
unrealistic representation of the spur, and leads to problems when calculating
intraspur interactions. In reality, the species of a spur already have a
finitely spread distribution at the moment of production, caused by spatial
variance in the primary particle’s path as well as a spatial spread in the
production of chemical species in the physico-chemical phase. In order to
address this problem, we apply a temporal shift to give each spur a minimum
effective age $\tau_{0}$, which represents its non-zero spatial distribution
upon creation. Thus, the PDF of an arbitrary spur centered at $\vec{r_{i}}$
for $t>t_{i}$ is

$c_{i}(\vec{r},t)=c_{0}\cdot\frac{e^{-\frac{\absolutevalue{\vec{r}-\vec{r_{i}}}^{2}}{4\alpha(t-t_{i}+\tau_{0})}-k_{s}(t-t_{i})}}{4\pi\alpha(t-t_{i}+\tau_{0})}\
.$ (3)

Quantitative values for the diffusion coefficient $\alpha$, minimum spur age
$\tau_{0}$, and decay constant $k_{s}$ can be approximated by fitting
characteristics of the analytical spur to the MC simulated spur, as shown in
Figures 3a and 3b.

###  2.3 Spur Distribution

For simplicity, the beam is assumed to have a constant dose rate over the
exposure time $T$, and thus the PDF of the creation time $t_{i}$ of the $i$th
spur at time $t$ is

$P_{t}(t_{i})=\begin{cases}\frac{1}{\min(t,T)}\quad&\text{for}\quad 0\leq
t_{i}\leq\min(t,T)\\\ 0\quad&\text{elsewhere}\end{cases}\ .$ (4)

The expected total number of spurs on the target over time $N_{s}(t)$ can be
expressed with the total energy deposited in the target $E$ and the average
energy deposited per spur $E_{s}$ for a target of density $\rho$, thickness
$z$ (measured along the beam axis), and cross-sectional area $A$:

$\displaystyle N_{s}(t)$
$\displaystyle=\frac{E}{E_{s}}\cdot\frac{\min(t,T)}{T}$ (5)
$\displaystyle=\frac{D\cdot A\cdot\rho\cdot z}{E_{s}}\cdot\frac{\min(t,T)}{T}$
(6) $\displaystyle=\frac{D\cdot A\cdot\rho}{L}\cdot\frac{\min(t,T)}{T}\ .$ (7)

In the last step we introduce the linear energy transfer (LET) $L=E_{s}/z$,
which will depend on the radiation quality.

Assuming a spatially homogeneous beam across the target area $A$, the expected
number of spurs within any circular area of radius $R$ can be expressed as a
fraction of the total number of spurs on the target:

$N_{R}(t)=N_{s}(t)\cdot\frac{\pi R^{2}}{A}\\\ $ (8)

The PDF of the displacement $s$ between one arbitrary spur and any other spur
within a radius $R$ is given by the ratio between the area element $2\pi s$
and the total area $\pi R^{2}$:

$P_{s}(s)=\frac{2s}{R^{2}}\ ,\quad s\leq R\ .$ (9)

###  2.4 Interaction Quantification

The quantification of interspur interaction is a critical component of this
model. A measure is needed that takes into account the spatial variations of
different spurs, and reflects a physical quantity relevant to the
radiochemistry at hand. To this end, the interaction rate $\omega$ is first
defined, which yields the rate of change of the quantity of species at time
$t$ due to the interaction of two spurs assuming second order reactions with
reaction rate $k_{r}$:

$\displaystyle\omega_{1,2}(t)$ $\displaystyle=\int k_{r}\cdot
c_{1}(\vec{r},t)\cdot c_{2}(\vec{r},t)\,dV$ (10) $\displaystyle=k_{r}\cdot
c_{0}^{2}\cdot\frac{e^{-\frac{\absolutevalue{\vec{r_{1}}-\vec{r_{2}}}^{2}}{4\alpha(2t-t_{1}-t_{2}+2\tau_{0})}-k_{s}(2t-t_{1}-t_{2})}}{4\pi\alpha(2t-t_{1}-t_{2}+2\tau_{0})}\
.$ (11)

Integrating the interaction rate in Equation 11 over time yields the total
interspur conversion $I$, which represents the net change in the quantity of
species due to the interspur interaction over the relevant time period:

$I_{1,2}(t)=\int_{0}^{t}\omega_{1,2}(t^{\prime})\,dt^{\prime}\ .$ (12)

$I$ will be the measure of interest in this model, as it represents an
empirically measurable quantity.

###  2.5 Interspur Interactions

The expected interaction rate of an arbitrary spur $a$ with all neighboring
spurs within an interaction volume of radius $R$ can be found by multiplying
the number of spurs inside that interaction volume $N_{R}$ by the expected
interaction rate between one arbitrary spur and another within that volume,

$\sum_{i}\,\omega_{a,i}(t)=N_{R}(t)\cdot\langle\omega_{1,2}(t)\rangle\ ,$ (13)

The total interspur conversion of all spurs in the target volume is then

$I_{\text{inter}}(t)=\frac{N_{s}(t)}{2}\cdot\int_{0}^{t}\lim_{R\rightarrow\infty}N_{R}(t^{\prime})\cdot\langle\omega_{1,2}(t^{\prime})\rangle\,dt^{\prime}\
,$ (14)

where $R$ approaches infinity under the assumption that each spur is
effectively within an infinite isotropic volume. The factor $\frac{1}{2}$
ensures that each interspur interaction is not double-counted.

Evaluating this expression (see Section A.2) yields

$I_{\text{inter}}(t)=\begin{cases}\lim_{\epsilon\rightarrow 0}\frac{B}{2}\cdot
t\cdot\left[\ln(\frac{t}{\epsilon})-\Gamma(0,2k_{s}t)+2\Gamma(0,k_{s}t)+\Gamma(0,2k_{s}\epsilon)-2\Gamma(0,k_{s}\epsilon)\right]&\text{for}\quad
t\leq T\\\\[10.0pt]
I_{\text{inter}}(T)+\frac{B}{2}\cdot\frac{(e^{k_{s}T}-1)^{2}}{2k_{s}}\cdot(e^{-2k_{s}T}-e^{-2k_{s}t})&\text{for}\quad
t>T\end{cases}\ ,$ (15)

where

$B=\frac{N_{s}(t)^{2}}{A}\cdot\frac{k_{r}}{k_{s}^{2}}\cdot
c_{0}^{2}\cdot\left(\frac{1}{\min(t,T)}\right)^{2}\ ,$ (16)

and $\Gamma(s,x)$ is the incomplete upper gamma function given by

$\Gamma(s,x)=\int_{x}^{\infty}t^{s-1}e^{-t}\,dt\ .$ (17)

###  2.6 Intraspur Interactions

The total intraspur conversion across the entire target volume is simply a sum
of each spur’s self-interaction

$I_{\text{intra}}(t)=\sum_{i}^{N_{s}(t)}\int_{0}^{t}\omega_{i,i}(t^{\prime})\,dt^{\prime}\
.$ (18)

This expression can be simplified with a suitable approximation, removing the
need for tedious summation over all spurs. Due to a spur’s outward diffusion,
the amount of intraspur conversion converges over time to a maximum value. If
this convergence time is negligible compared to the total exposure time, the
total intraspur conversion can be approximated as a linear increase over the
exposure time towards the total maximum amount of intraspur conversion of all
spurs on the target:

$\displaystyle I_{\text{intra}}(t)$ $\displaystyle\approx
N_{s}(t)\cdot\int_{0}^{\infty}\omega_{i,i}(t)\,dt$ (19)
$\displaystyle=N_{s}(t)\cdot\int_{0}^{\infty}k_{r}\cdot
c_{0}^{2}\cdot\frac{e^{-2k_{s}t}}{8\pi\alpha(t+\tau_{0})}\,dt$ (20)
$\displaystyle=N_{s}(t)\cdot k_{r}\cdot
c_{0}^{2}\cdot\frac{e^{2k_{s}\tau_{0}}}{8\pi\alpha}\cdot\Gamma\left(0,\,2k_{s}\tau_{0}\right)\
.$ (21)

###  2.7 Ratio of Interspur to Intraspur Interactions

In order to quantify the extent to which interspur interactions affect the
overall radiochemistry, the ratio of interspur to intraspur interaction is
used in this model as the main measure of interest:

$\Phi(t)=\frac{I_{\text{inter}}(t)}{I_{\text{intra}}(t)}\ ,$ (22)

whereby we define significant interspur interaction as $\Phi(t)\geq 1$.

Evaluating this ratio with Equations 14, 18, and 7 yields

$\Phi(t)=2\pi\alpha\cdot\frac{D\cdot\rho}{L}\cdot\frac{e^{-2k_{s}\tau_{0}}}{k_{s}^{2}\cdot\Gamma(0,2k_{s}\tau_{0})}\cdot\frac{1}{T}\cdot\begin{cases}f_{1}(t,k_{s})&\text{for}\quad
t\leq T\\\\[10.0pt]
f_{1}(T,k_{s})+\frac{(e^{k_{s}T}-1)^{2}}{k_{s}T}\left(e^{-2k_{s}T}-e^{-2k_{s}t}\right)&\text{for}\quad
t>T\end{cases}$ (23)

where

$f_{1}(t^{\prime},k_{s})=2\cdot\lim_{\epsilon\rightarrow
0}\left[\ln(\frac{t^{\prime}}{\epsilon})-\Gamma(0,2k_{s}t^{\prime})+2\Gamma(0,k_{s}t^{\prime})+\Gamma(0,2k_{s}\epsilon)-2\Gamma(0,k_{s}\epsilon)\right]\
,$ (24)

and $\Gamma(s,x)$ is defined as in Equation 17. Table 1 summarizes all
variables and their definitions.

The primary particle fluence can be expressed using Equation 7 as

$\displaystyle F$ $\displaystyle=\frac{N_{s}}{A}$ (25)
$\displaystyle=\frac{D\cdot\rho}{L}\ .$ (26)

The value of $\Phi(t)$ for $t\gg T$ is of interest in order to analyze the
final ratio of interspur to intraspur interactions after all spurs have
decayed and no more interactions can occur. This value, using Equations 23 and
26, is given by

$\displaystyle\Phi_{t\gg T}$ $\displaystyle=\lim_{t\rightarrow\infty}\Phi(t)$
(27) $\displaystyle=\alpha\cdot F\cdot f_{2}(T,k_{s},\tau_{0})\ ,$ (28)

where

$f_{2}(T,k_{s},\tau_{0})=\frac{2\pi}{T}\cdot\frac{e^{-2k_{s}\tau_{0}}}{k_{s}^{2}\cdot\Gamma(0,2k_{s}\tau_{0})}\cdot\left[f_{1}(T,k_{s})+\frac{(e^{-k_{s}T}-1)^{2}}{k_{s}T}\right]\
.$ (29)

Variable | Description
---|---
$\alpha$ | Spur diffusion coefficient
$\tau_{0}$ | Spur minimum age
$k_{s}$ | Spur decay constant
$\rho$ | Target density
$L$ | LET
$D$ | Dose
$T$ | Exposure time
$F$ | Primary particle fluence
Table 1: Variables described in this model.

###  2.8 Monte Carlo Simulation of a Particle Spur

Monte Carlo (MC) simulations, with the MC radiolysis toolkit TOPAS n-Bio [35],
were performed in order to characterize the evolution dynamics of a spur. A
single $1\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ proton spur in a pure water
target, as depicted in Figure 2, was analyzed from the physical stage through
the end of the chemical stage ($1\text{\,}\mu\mathrm{s}$). The cubical scoring
volume, with a $2\text{\,}\mu\mathrm{m}$ side-length, was embedded in the
target volume to ensure diffusive equilibrium within the scoring volume at all
time points. The resulting diffusive broadening of the spur and concentration
decay of the spur due to intraspur interactions are depicted in Figures 3a and
3b, respectively. Fitting the analytical descriptions (Equation 3) of a spur
to this MC-simulated spur allows for approximation of the spur parameters
$\alpha$ and $\tau_{0}$.

(a)

(b)

Figure 3: Evolution characteristics of the species $\ce{{}^{.}OH}$,
$\ce{e_{aq}^{-}}$, $\ce{H^{.}}$, $\ce{H2O2}$, and $\ce{OH^{-}}$ from a
$1\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ proton spur simulated with TOPAS
n-Bio [35]. (a) Standard deviations of the radial displacements of the spur
(blue points) alongside the theoretical standard deviations (dashed lines)
assuming a normal distribution ($\sigma=\sqrt{2\alpha(t+\tau_{0})\,}$).
Including the minimum spur age correction $\tau_{0}$ results in a better fit
of the simulated spur, due to its initial non-zero spatial variance. Fitted
values for these parameters to the simulated data are $\alpha=$(4.35\pm
0.04)\text{\times}{10}^{-9}\text{\,}\mathrm{m}^{2}\mathrm{/}\mathrm{s}$$ and
$\tau_{0}=$(1.99\pm 0.08)\text{\times}{10}^{-8}\text{\,}\mathrm{s}$$. (b)
Concentration decay of the same simulated
$1\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ proton spur relative to its initial
concentration due to intraspur interactions (blue line) alongside an
exponential decay of the form $e^{-kt}$ for reference.

##  3 Results

For the following analysis of Equations 23 and 28, results are shown
graphically for different values of the spur decay constant $k_{s}$ and LET
$L$. A large range of potential values of $k_{s}$ is used, as its realistic
value in water or in tissue is uncertain. For the LET,
$L\approx$0.2\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}\mathrm{/}\mu\mathrm{m}$$
corresponds to a $6\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ electron beam in a
$5\text{\,}\mathrm{m}\mathrm{m}$ thick water target, which replicates the beam
used in an empirical FLASH study [12], while
$L\approx$26\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}\mathrm{/}\mu\mathrm{m}$$
corresponds to a $1\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ proton. Values for
the diffusion constant $\alpha$ and the minimum spur age $\tau_{0}$ are taken
from the fits described in Figure 3a. The values used for these parameters are
discussed further in Section 4.3.

Figure 4a depicts the increase in $\Phi(t)$, i.e. the ratio of interspur to
intraspur interactions, during and after a single $1.8\text{\,}\mu\mathrm{s}$
pulse exposure of $10\text{\,}\mathrm{G}\mathrm{y}$ for different values of
the spur decay constant $k_{s}$. Initially, $\Phi(t)\approx 0$ as the first
few spurs undergo intraspur interactions. As the exposure continues and more
spurs are created, begin to broaden, and overlap, interspur interactions
become more and more prominent. Once the exposure has ended (indicated by the
vertical dashed line), interspur interactions continue to increase until the
final remaining spurs have decayed, at which point $\Phi(t)$ approaches its
final value, $\Phi_{t\gg T}$. Increasing the spur decay constant $k_{s}$
reduces interspur interactions more than it reduces intraspur interactions,
which can result in interspur interactions never becoming significant, i.e.
$\Phi_{t\gg T}<1$, as seen in Figure 4a. Similarly, increasing the beam LET
reduces the significance of interspur interactions.

Figure 4b depicts the final ratio of interspur to intraspur interactions,
$\Phi_{t\gg T}$, across a large range of dose rates at a fixed dose of
$10\text{\,}\mathrm{G}\mathrm{y}$. At very low dose rates, where the creation
time between spurs is large, interspur interactions are negligible as
individual spurs decay before subsequent spurs are created. As the dose rate
increases, $\Phi_{t\gg T}$ grows as spurs are more temporally concentrated and
interspur interactions become more prominent. If the spur decay constant is
sufficiently low, interspur interactions may become significant ($\Phi_{t\gg
T}\geq 1$) at some critical dose rate. At very high dose rates, all spurs
arrive almost instantaneously relative to their decay lifetimes, and the
amount of interspur interaction is determined only by their spatial
separation, i.e. the fluence, and the spur decay constant $k_{s}$. A higher
LET results in a decreased fluence, thereby decreasing interspur interaction
and increasing the critical dose rate needed to achieve significant interspur
interaction. A similar effect occurs when increasing the spur decay constant.
In fact, if the spur decay constant or LET are too high, significant interspur
interaction may never be possible at any dose rate.

To illustrate the dose and dose rate dependencies of significant interspur
interaction simultaneously, Figure 5a depicts the value of $\Phi_{t\gg T}$ for
a large range of dose and dose rate combinations. This heatmap demonstrates
the transition from intraspur-dominated regions (red) to interspur-dominated
regions (blue) of the dose/dose rate parameter space. The isovalue curve at
$\Phi_{t\gg T}=1$ (white line) indicates the minimum doses and dose rates
necessary to achieve significant interspur interaction. Figure 5b shows this
isovalue curve for different values of the spur decay constant $k_{s}$ and LET
$L$. Apparent in this figure is that increasing the spur decay constant or the
LET raises the critical doses and dose rates for significant interspur
interaction.

(a)

(b)

Figure 4: (a) The ratio of interspur to intraspur interactions, $\Phi(t)$,
during and after a $1.8\text{\,}\mu\mathrm{s}$,
$10\text{\,}\mathrm{G}\mathrm{y}$ exposure. (b) The final total ratio of
interspur to intraspur interactions, $\Phi_{t\gg T}$, after a
$10\text{\,}\mathrm{G}\mathrm{y}$ exposure as a function of dose rate at a
fixed dose. Results are shown for different spur decay constants $k_{s}$
(indicated by color, given in $s^{-1}$) and beam LET (indicated by line style,
given in $\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}\mathrm{/}\mu\mathrm{m}$).

(a)

(b)

Figure 5: (a) Heatmap showing the value of $\Phi_{t\gg T}$ for different dose
and dose rate combinations. Interspur interactions are considered significant
when $\Phi_{t\gg T}\geq 1$; therefore, the white strip indicates the
transition from intraspur-dominated interactions (red) and interspur-dominated
interactions (blue). (b) Isovalue curves showing where $\Phi_{t\gg T}=1$ for
different spur decay constants and LETs. Results are shown for different spur
decay constants $k_{s}$ (indicated by color, given in $s^{-1}$) and beam LET
(indicated by line style, given in
$\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}\mathrm{/}\mu\mathrm{m}$). Increasing
the spur decay constant or the LET results in increased doses and dose rates
necessary to achieve significant interspur interaction.

##  4 Discussion

###  4.1 Results

Equation 28 expresses the significance of interspur interactions as a function
of irradiation parameters. It demonstrates that the final ratio of interspur
to intraspur interactions, $\Phi_{t\gg T}$, is only dependent on two beam
parameters: the fluence $F$ and the exposure time $T$, whereby the fluence is
determined by the dose $D$ and LET $L$. All other dependencies pertain to
parameters specific to the spur: the spur diffusion constant $\alpha$, the
minimum spur age $\tau_{0}$, and the spur decay constant $k_{s}$.

At fixed values for these spur parameters and a fixed LET, Figure 5a
illustrates how interspur interactions depend on the dose and dose rate. For
any set of irradiation parameters, there is a minimum dose and a minimum dose
rate necessary for significant interspur interactions. These correspond to a
minimum fluence and a maximum exposure time, respectively. Thus, interspur
interactions are limited in both the spatial (fluence) and the temporal
(exposure time) domain. At low dose rates, extremely high doses are necessary
to elicit significant interspur interactions. As the dose rate is increased,
this critical dose is strongly reduced; a one order of magnitude increase in
dose rate may result in a three or more order of magnitude decrease in
critical dose. However, once a certain dose rate is reached, interspur
interactions becomes dose-rate-independent. Here, the temporal separation
between spur arrivals is so short relative to their evolution and interaction,
that all spurs arrive effectively instantaneously; thus, interspur
interactions are purely dependent on the spurs’ spatial separation (determined
by the fluence).

Figure 5b depicts the effects of the beam LET and the spur decay constant on
$\Phi_{t\gg T}$. A higher-LET beam has a reduced fluence (each particle
deposits more energy in the target), and thereby requires a higher minimum
dose and dose rate necessary for significant interspur interactions. Applied
to proton beams, this model thus predicts that, for the same total dose, the
plateau region of a proton beam would induce much stronger interspur
interaction than the Bragg peak region. Similarly to an increased LET, an
increased spur decay constant $k_{s}$ results in less interspur interactions
due to a quicker decay rate of each spur.

#### Quantitative Considerations

The descriptions of the results of this model have thus far been purposefully
qualitative. Precise quantitative predictions with this model in its current
form are not possible due to uncertainties in the values of multiple critical
parameters as well as assumptions made in the construction of the model. For
example, interspur interactions were arbitrarily set to become significant
when $\Phi_{t\gg T}=1$; however, interspur interactions may have significant
impacts on the radiochemistry at much lower values of $\Phi_{t\gg T}$. Figure
5a illustrates how such a change would affect the critical doses and dose
rates to achieve interspur interaction. Another crucial variable which
strongly influences the qualitative results of the model, but whose value
remains very uncertain, is the spur decay constant $k_{s}$ (see Figure 5b).
This is discussed in detail in Section 4.3 alongside other limitations of the
model.

It remains clear that experimental verification of the radiochemical phenomena
predicted by this model is necessary to fit the parameters of the model before
quantitative predictions can be made. Notwithstanding, the model makes
interesting qualitative predictions, provides insight into the irradiation
parameters vital to interspur interaction, and produces easily testable
hypotheses.

###  4.2 Application to FLASH-RT

#### Parameter Dependencies

If interspur interactions are responsible for the normal tissue sparing
observed in the FLASH effect, one would expect this sparing to have similar
parameter dependencies as the onset of significant interspur interactions.
Indeed, parameter dependencies predicted by the model, such as a minimum dose
and minimum dose rate, are reflected in empirical FLASH studies [9, 10, 12].
Assuming values of $\Phi=1$ and
$k_{s}=$8\text{\times}{10}^{3}\text{\,}\mathrm{s}^{-1}$$, and keeping in mind
the aforementioned uncertainties in quantitative prediction by this model, the
onset of significant interspur interactions predicted by this model reflects
the minimum dose and dose-rate dependencies shown in some empirical FLASH
studies of $\approx$10\text{\,}\mathrm{G}\mathrm{y}$$ and
$\approx$40\text{\,}\mathrm{G}\mathrm{y}\mathrm{/}\mathrm{s}$$, respectively
(see Figure 5a) [9, 10, 12].

Most FLASH experiments in the literature, including the one referenced here,
use pulsed electron beams, whereby dose rates within each pulse may reach
values of
$\approx${10}^{6}\text{\,}\mathrm{G}\mathrm{y}\mathrm{/}\mathrm{s}$$. As is
discussed in detail in Section 4.5, a pulsed beam greatly increases interspur
interaction, further strengthening the probability that interspur interactions
play a role in the FLASH effect. In a similar fashion, the very high local
dose rates of scanned proton pencil beams should be taken into consideration
[37].

Assuming again that interspur interactions are responsible for the FLASH
effect, this model makes a few crucial predictions for eliciting a FLASH
effect at a fixed total treatment dose.

  * •

The total treatment dose must be above a threshold value, regardless of how
high the dose rate is.

  * •

The dose rate must be increased (i.e. exposure time decreased) beyond a
critical value, which depends on the dose (lower doses require higher dose
rates), to begin eliciting the FLASH effect.

  * •

Increasing the dose rate far beyond that critical value will have diminishing
benefits.

  * •

The beam LET should be reduced as much as possible; i.e., higher energies at
the target should be used for particle beams.

  * •

Beams should be pulsed with the lowest number of pulses and lowest pulse
widths possible (maximize the dose per pulse).

Even if the parameter dependencies of the FLASH effect are indeed well-
described by this model, such a correlation cannot rule out the effects of
other phenomena besides interspur interaction, such as oxygen depletion or
other spur-environment interactions which are also expected to have a dose and
dose-rate dependency. In order to distinguish such effects from interspur
interactions, careful analyses of the relevant types of radiochemical
interactions must be performed. The exact mechanism by which interspur
interactions could induce normal tissue sparing is beyond the scope of this
work, but may be due to increased radical-radical interactions [24, 32, 33].

#### Differential Sparing of Normal and Tumor Tissue

Critical to the FLASH effect is the sparing of normal tissue without a
coincidental sparing of tumor tissue; without this differential sparing, the
therapeutic benefit of FLASH-RT would be lost. The mechanism by which normal
tissue is spared by UHDR irradiation must thus be negated within tumor tissue.
With respect to interspur interactions, this may be explained by different
effective spur decay constants ($k_{s}$) in normal and tumor tissue caused by
differing oxygen, labile iron, or radical scavenger concentrations, leading to
reduced interspur interaction. Alternatively, differing biological responses
of normal and tumor tissue to the radiochemical effects of interspur
interactions may explain this differential sparing. Determining the exact
consequences of these differences is beyond the scope of this work but remains
vital to be elucidated by future investigation.

###  4.3 Model

#### Spur Characteristics

The model presented in this work makes use of multiple assumptions based on
the ensemble average of a very large number of spurs, such as the straightness
and homogeneity of individual spurs along the beam axis. This assumption is
valid in the very densely ionizing radiation limit, but may no longer hold
when considering sparsely ionizing radiation; the relevant corrections to the
model are described in Section 4.4. Another assumption of this model is that
the border effects of the radiation field can be neglected, i.e. each spur is
assumed to be situated within an infinite, isotropic volume of other spurs. A
simple back-of-the-envelope calculation (see Section A.3) indicates that for a
square radiation field size of side length $5\text{\,}\mathrm{m}\mathrm{m}$,
less than $0.03\%$ of spurs would experience such border effects, and can
therefore be ignored.

Spurs in this model evolve under a reaction-diffusion equation (Equation 1),
which comprises a diffusion term, an intraspur and spur-environment
interaction term, and an interspur interaction term. An estimation of
quantitative values for the diffusion coefficient $\alpha$ and temporal shift
parameter $\tau_{0}$ is performed in Section 2.8 by analyzing a
$1\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ proton spur simulated with the
Monte Carlo code TOPAS n-Bio. The simulation yields very good agreement with
the spatial variance over time of the model’s spur. This particle and energy
were chosen due to the straightness of its track, important for the assumption
of straight spurs. Although the calculated value of $\alpha$ should be
independent of the particle type and energy, since it depends on the simulated
diffusion constants of the spur’s constituents, the value of $\tau_{0}$ may be
dependent on the radiation quality. For example, spurs of higher-energy
charged particles may have a larger initial spatial variance due to physical
effects such as the larger ranges of secondary particles, Coulomb explosion,
and thermal spike [38]. Recently, the effects of thermal spike on the
shockwave-like outward propagation of chemical species have been described
[39], which could induce more interspur interaction and less intraspur
interaction compared to when only thermal diffusion is considered. as
different particles will exhibit different spatial ionization patterns.
However, the effect of $\tau_{0}$ on $\Phi_{t\gg T}$ in this model, compared
to parameters like $k_{s}$, is minimal; a change over many orders of magnitude
(${10}^{-15}{10}^{-6}\text{\,}\mathrm{s}$) of $\tau_{0}$ changes $\Phi_{t\gg
T}$ by less than one order of magnitude. Therefore, these values for $\alpha$
and $\tau_{0}$ will be assumed to be valid.

#### Spur Interaction

In this model, the effects of interspur interactions on the evolution of spurs
are ignored under the assumption of weakly-interacting spurs; i.e. assuming
interspur interactions do not significantly affect the distributions of
species within spurs (see Equation 1). However, once interspur interactions
become significant, this assumption is no longer valid. Thus, this simplified
model is best used to determine when interspur interactions become
significant, but is less accurate at describing the dynamics of these
interactions past that point. Quantification of both interspur and intraspur
interaction is performed assuming second order reaction kinetics with only one
reaction rate constant, $k_{r}$. Although many relevant reactions do indeed
follow second order reaction kinetics, this simplification neglects the
different types of interactions that may occur due to the varying spatial
distributions of different species within a spur. For instance, the
consumption of hydroxyl radicals in intraspur interactions is likely dominated
by the self-recombination of hydroxyl radicals due to their high local
concentrations within the spur; however, when considering interspur
interactions, the reaction of hydroxyl with faster-diffusing species such as
$\ce{e_{aq}-}$ or $\ce{H^{.}}$ is more probable.

#### Intraspur and Spur-environment Interactions

The intraspur and spur-environment interaction term (see Equation 1) is
critical to the outcomes of this model. In this model, this term comprises an
exponential decay with decay constant $k_{s}$. In reality, this decay is
likely not exponential as it is a combination of many different processes,
e.g. multiple different intraspur reactions, scavenging by different
environmental reagents, and production (as opposed to depletion) of different
radiochemical species via these reactions. In addition, many of these
reactions may be in competition with each other [32]. The exponential
simplification used in this model poses to approximately fit these complex
interactions with one useful parameter, the decay constant $k_{s}$, to
qualitatively analyze the effects of varying spur decay due to intraspur or
spur-environment interactions. For the same reasons the decay processes are
complex and varied, determining an appropriate value for $k_{s}$ in tissue is
non-trivial. In order to get an idea of its approximate value, we examine the
decay of a spur simulated in pure water and discuss the effects of scavengers
typically found within a cellular environment.

Due to the highly localized concentrations of reactive species within the
spur, purely intraspur interaction is initially the most prevalent type of
interaction after the creation of a spur. Figure 3b depicts a simulated
$1\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ proton spur in
$2\text{\,}\mu\mathrm{m}$ of water undergoing solely intraspur interactions
(no other spurs and no environmental reagents such as oxygen were included).
The spur’s original concentration is initially rapidly halved within
$\approx$100\text{\,}\mathrm{n}\mathrm{s}$$, after which intraspur
interactions barely affect the spur. An exponential decay clearly
underestimates the simulated decay rate at very short times while
overestimating the decay rate at longer times. Likely, accurately reflecting
the spur’s decay dynamics in this model at longer time scales is more
important than at shorter time scales when interspur interactions are minimal
(see Figure 4a). To this end, it could be advantageous to represent the
effects of intraspur interaction by halving the initial concentration $c_{0}$
instead of including them in the decay constant $k_{s}$.

After the initial rapid self-consumption of the spur due to intraspur
interactions, spur-environment interactions likely begin to play a larger
role. Glutathione is an important hydroxyl radical scavenger, with a
scavenging rate of $\approx${10}^{10}\text{\,}\mathrm{s}^{-1}\mathrm{M}^{-1}$$
[40, 41], present in the cell cytosol at concentrations of
$\approx$1\text{\,}\mathrm{m}\mathrm{M}$$ [42]. Assuming an excess of
glutathione, the resulting hydroxyl decay constant would be
$\approx${10}^{7}\text{\,}\mathrm{s}^{-1}$$. If we instead consider the
scavenging of hydrated electrons by an excess of oxygen (scavenging rate
$\approx${10}^{10}\text{\,}\mathrm{s}^{-1}\mathrm{M}^{-1}$$ [43]) in a 5%
($\approx$50\text{\,}\mu\mathrm{M}$$) oxygen solution to form the superoxide
anion $\ce{O_{2}^{.}-}$, the decay constant would be
$\approx$5\text{\times}{10}^{5}\text{\,}\mathrm{s}^{-1}$$. It should also be
considered that spur-environment interactions not only consume radicals, but
can produce more radicals. For example, stable hydrogen peroxide initially
formed by recombination of hydroxyl radicals may later form hydroxyl radicals
again via the Fenton reaction, effectively decreasing the decay rate of the
spur. Within tissue, many factors such as scavenger concentrations, labile
iron concentrations, oxygenation, and pH may affect the spur decay rate. These
characteristics may change significantly between tissue types, and thus may
differ in different organs and in tumor tissue. Thus, $k_{s}$ remains a model
parameter that may assume a large range of values and can vary to represent
different tissue environments. However, as the results of this model are
critically dependent on the value of $k_{s}$ (see Section 3), experimental
verification of its value is necessary in determining the validity of this
model’s assumptions.

###  4.4 Sparsely Ionizing Radiation

The assumptions of perfectly straight and homogeneous spurs do not hold for
sparsely ionizing (low-LET) radiation. For example, the distribution of
ionization events in photon tracks is much more scattered and sparse, and the
spatial distinction between individual photon tracks is minimal. In order to
address these issues, the model can be adjusted to describe a 3-D distribution
of “clusters” of reactive species formed from these ionization events. Their
evolution and interaction is handled in a similar way as the original model
for densely ionizing radiation. The sparsely ionizing radiation model is
detailed in Section A.4.

The results of the sparsely ionizing radiation model have very similar
characteristics to that of the densely ionizing radiation model. One important
difference is the replacement of the LET ($E_{s}/z$) with the energy deposited
per cluster, $E_{c}$. Otherwise, there is only a slightly modified dependency
on the diffusion constant $\alpha$, the spur decay constant $k_{s}$, and the
minimum spur age $\tau_{0}$, all of which arise from the change in intraspur
interactions from 2-D to 3-D diffusion. The dependencies on time $t$, the
exposure time $T$, and the dose $D$ are identical between the two models.
Since results are comparable between strictly 2-D, straight spurs of the
densely ionizing radiation model and the 3-D, uniformly distributed clusters
of the sparsely ionizing radiation model, it can be assumed that realistic
spurs of particles, which lay in between the characterizations of these two
extreme assumptions, share similar interspur and intraspur interaction
behavior.

###  4.5 Pulsed Beam

Thus far, the construction of this interspur interaction model has only
considered a continuous (constant dose rate) beam. However, many FLASH studies
take advantage of pulsed beams in order to achieve the ultra-high mean dose
rates characteristic of FLASH irradiation. The pulsatile nature of a beam
likely critically affects the dynamics of interspur interaction. Assuming a
beam of total dose $D$ and total exposure time $T$ pulsed at a frequency $f$
comprising identical pulses of width $w$, the dose per pulse $D_{p}$ and dose
rate within each pulse $\accentset{\mbox{\Large.}}{D}_{p}$ can be expressed as

$D_{p}=\frac{D}{f\cdot
T}\qquad\text{and}\qquad\accentset{\mbox{\Large.}}{D}_{p}=\frac{D}{f\cdot
T\cdot w}\ .$ (30)

Each pulse can be considered to be a continuous beam irradiation with total
dose $D_{p}$ and mean dose rate $\accentset{\mbox{\Large.}}{D}_{p}$. Thus, by
substituting these values for the doses and dose rates depicted in Figure 5,
the significance of interspur interaction within each pulse can be analyzed.
For a fixed total dose and mean dose rate, decreasing the pulse frequency and
decreasing the pulse width results in a higher dose rate per pulse yielding
more significant interspur interaction. This approach assumes that
interactions between spurs of separate pulses (interpulse) are negligible, and
thus represents the lower limit of pulsed-beam interspur interaction. The
authors intend to extend this model to include interpulse interaction in order
to determine its significance for the irradiation parameters typically used in
FLASH studies.

Some of the results shown in Section 3 reflect the irradiation parameters used
in the aforementioned study by Montay Gruel et al. [12], assuming a continuous
exposure. However, this beam was actually pulsed, and the onset of the
observed FLASH effect occurred when the exposure time was decreased from
$0.50.2\text{\,}\mathrm{s}$ at a constant pulse frequency of
$100\text{\,}\mathrm{H}\mathrm{z}$ and pulse width of
$1.8\text{\,}\mu\mathrm{s}$. Thus, the dose per pulse and dose rate per pulse
were changed from $0.20.5\text{\,}\mathrm{G}\mathrm{y}$ and from
$1.1\text{\times}{10}^{6}2.8\text{\times}{10}^{6}\text{\,}\mathrm{G}\mathrm{y}\mathrm{/}\mathrm{s}$,
respectively, to elicit a FLASH effect. Although not shown in Figure 5, a spur
decay constant of $k_{s}=${10}^{5}\text{\,}\mathrm{s}^{-1}$$ would place the
isovalue curve of $\Phi_{t\gg T}=1$ at this parameter transition. Considering
the pulsatile nature of this beam results in a large increase in the
significance of interspur interaction.

##  5 Conclusion

The objective of this work was to analyze the effects of interspur
interactions on the overall radiochemistry at ultra-high dose rates, and
examine whether interspur interactions may play a role in the FLASH effect. To
this end, an analytical model was developed to describe the spatiotemporal
distribution, evolution, and interaction of an ensemble of spurs in a target
during and after irradiation. Individual spurs were defined by a 2-D reaction-
diffusion equation comprising the radial broadening of the spur due to thermal
diffusion and an exponential decay of the spur due to intraspur and spur-
environment interactions. The parameters modulating the spur’s diffusive
broadening were approximated by fitting the theoretical description of the
spur to Monte Carlo simulations. The exponential decay was modulated by a
model parameter, the spur decay constant $k_{s}$. The effects of interspur
interactions on the evolution of individual spurs were ignored under the
assumption of weakly-interacting spurs. Spurs were distributed throughout the
target assuming a spatially homogeneous beam with a constant dose rate.
Intraspur and interspur interactions were quantified across the ensemble of
spurs by their spatio-temporal overlap assuming second-order reaction
kinetics.

The ratio of interspur to intraspur interactions, $\Phi$, was then derived as
a function of irradiation parameters (e.g. dose, dose rate, LET). Interspur
interactions were considered significant to the overall radiochemistry when
this ratio reached unity, i.e. when interspur interactions led to as much
species conversion as did intraspur interactions ($\Phi=1$). This model
demonstrated that $\Phi$ is predominantly dependent on three parameters: the
beam fluence, the exposure time, and the spur decay constant. For any set of
parameters, a minimum critical dose and dose rate were shown to be necessary
to induce significant interspur interactions. This critical dose and dose rate
could be reduced by decreasing the LET or spur decay constant. A similar
effect could be achieved by lowering the value of $\Phi$ defining significant
interspur interactions.

This model demonstrates that significant interspur interaction may occur at
the same threshold doses and dose rates characteristic of the observed FLASH
effect [9, 10], depending on the values of the model parameters $\Phi$ and
$k_{s}$. This model additionally suggests optimal irradiation parameters
necessary to induce interspur interactions, and, by extension, potentially the
FLASH effect. The differential sparing effect between normal and tumor tissue
could be explained by a different effective spur decay constant within the
different tissue types. Significant interspur interaction could lead to normal
tissue sparing by, for example, increased radical-radical interaction [24, 32,
33]. Further analysis of this effect will need to consider interactions
between spurs of separate pulses from pulsed beams, as well as the physical
spur-broadening effects of high-LET radiation. In addition, the decay rate and
dynamics of spurs in this model should be further developed and analyzed for
more accurate description. Investigations of this effect will need to
discriminate between interspur interactions and other interaction phenomena
induced by high dose rates, such as oxygen depletion. Radiochemical analyses
of the yields of various radiolytic species in different media as a function
of beam parameters are currently underway and will help determine values for
the critical parameters of this model.

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##  Appendix A Appendix

###  A.1 Spur Reaction Diffusion Equation

The general partial differential equation governing the spatio-temporal
evolution of spurs undergoing diffusion and linear consumption, ignoring
interspur interactions as described in Section 2.2, is

$\frac{\partial}{\partial
t}c(\vec{r},t)=\alpha\nabla^{2}c(\vec{r},t)-k_{s}c(\vec{r},t)\ .$ (31)

Using the Fourier transform

$\hat{P}(k_{1},...,k_{n})=\int_{\textbf{R}^{n}}\,e^{-2\pi i(k_{1}\cdot
x_{1}+...+k_{n}\cdot x_{n})}\cdot P(x_{1},...,x_{n})\ dx...dx_{n}\ $ (32)

and the identity

$\frac{\partial^{n}}{\partial x^{n}}P=(2\pi ik_{x})^{n}\hat{P}\ ,$ (33)

Equation 31 can be rewritten as

$\frac{\partial}{\partial
t}\hat{c}=[-4\alpha\pi^{2}(k_{x}^{2}+k_{y}^{2})-k_{s}]\hat{c}\ ,$ (34)

thereby losing its spatial dependency. Solving for $\hat{c}$ thus becomes
trivial,

$\hat{c}=A\cdot e^{[-4\alpha\pi^{2}(k_{x}^{2}+k_{y}^{2})-k_{s}]\cdot t}\ ,$
(35)

and its real-space equivalent can be constructed using the inverse Fourier
transform

$P(x_{1},...,x_{n})=\int_{\textbf{R}^{n}}\,e^{2\pi i(k_{1}\cdot
x_{1}+...+k_{n}\cdot x_{n})}\cdot\hat{P}(k_{1},...,k_{n})\ dk...dk_{n}$ (36)

and the initial condition

$\int c(\vec{r},t=0)\,dV=c_{0}\ ,$ (37)

yielding

$c(\vec{r},t)=c_{0}\cdot\frac{e^{-\frac{\absolutevalue{\vec{r}}^{2}}{4\alpha
t}-k_{s}t}}{4\pi\alpha t}\ .$ (38)

###  A.2 Interspur Interaction Measure

Evaluating Equation 13 considering the limit $R\rightarrow\infty$ with the
formulations derived in Sections 2.2, 2.3, and 2.4 yields

$\displaystyle\sum_{i}\,\omega_{a,i}(t)$
$\displaystyle=\lim_{R\rightarrow\infty}N_{R}(t)\cdot\langle\omega_{1,2}(t)\rangle$
(39) $\displaystyle=\lim_{R\rightarrow\infty}N_{s}(t)\cdot\frac{\pi
R^{2}}{A}\cdot\int\\!\\!\\!\int\\!\\!\\!\int P_{t}(t_{1})\cdot
P_{t}(t_{2})\cdot P_{s}(s)\cdot\omega_{1,2}(t)\ ds\,dt_{2}\,dt_{1}$ (40)
$\displaystyle=\lim_{R\rightarrow\infty}\frac{N_{s}(t)}{A}\cdot\int\\!\\!\\!\int\\!\\!\\!\int_{0}^{R}P_{t}(t_{1})\cdot
P_{t}(t_{2})\cdot 2\pi s\cdot\frac{k_{r}\cdot c_{0}^{2}\cdot
e^{-\frac{s^{2}}{4\alpha(2t-t_{1}-t_{2}+2\tau_{0})}-k_{s}(2t-t_{1}-t_{2})}}{4\pi\alpha(2t-t_{1}-t_{2}+2\tau_{0})}\
ds\,dt_{2}\,dt_{1}$ (41) $\displaystyle=\frac{N_{s}(t)}{A}\cdot k_{r}\cdot
c_{0}^{2}\cdot\int\\!\\!\\!\int P_{t}(t_{1})\cdot P_{t}(t_{2})\cdot
e^{-k_{s}(2t-t_{1}-t_{2})}\,dt_{2}\,dt_{1}$ (42)
$\displaystyle=\frac{N_{s}(t)}{A}\cdot k_{r}\cdot
c_{0}^{2}\cdot\int\\!\\!\\!\int\left(\frac{1}{\min(t,T)}\right)^{2}\cdot
e^{-k_{s}(2t-t_{1}-t_{2})}\,dt_{2}\,dt_{1}$ (43)
$\displaystyle=\frac{N_{s}(t)}{A}\cdot k_{r}\cdot
c_{0}^{2}\cdot\left(\frac{1}{\min(t,T)}\right)^{2}\cdot\begin{cases}\int_{0}^{t}\int_{0}^{t}e^{-k_{s}(2t-t_{1}-t_{2})}\
dt_{2}\,dt_{1}&\text{for}\quad t\leq T\\\\[10.0pt]
\int_{0}^{T}\int_{0}^{T}e^{-k_{s}(2t-t_{1}-t_{2})}\
dt_{2}\,dt_{1}&\text{for}\quad t>T\end{cases}$ (44)
$\displaystyle=\frac{N_{s}(t)}{A}\cdot\frac{k_{r}}{k_{s}^{2}}\cdot
c_{0}^{2}\cdot\left(\frac{1}{\min(t,T)}\right)^{2}\cdot\begin{cases}(e^{-k_{s}t}-1)^{2}&\text{for}\quad
t\leq T\\\\[10.0pt] (e^{k_{s}T}-1)^{2}\cdot e^{-2k_{s}t}&\text{for}\quad
t>T\end{cases}$ (45)

Thus, the total interspur conversion of all spurs in the target volume,
following Equation 14, is

$I_{\text{inter}}(t)=\begin{cases}\lim_{\epsilon\rightarrow 0}\frac{B}{2}\cdot
t\cdot\left(\ln(\frac{t}{\epsilon})-\Gamma(0,2k_{s}t)+2\Gamma(0,k_{s}t)+\Gamma(0,2k_{s}\epsilon)-2\Gamma(0,k_{s}\epsilon)\right)&\text{for}\quad
t\leq T\\\\[10.0pt]
I_{\text{inter}}(T)+\frac{B}{2}\cdot\frac{(e^{k_{s}T}-1)^{2}}{2k_{s}}\cdot(e^{-2k_{s}T}-e^{-2k_{s}t})&\text{for}\quad
t>T\end{cases}\ ,$ (46)

where

$B=\frac{N_{s}(t)^{2}}{A}\cdot\frac{k_{r}}{k_{s}^{2}}\cdot
c_{0}^{2}\cdot\left(\frac{1}{\min(t,T)}\right)^{2}\ ,$ (47)

and $\Gamma(s,x)$ is the incomplete upper gamma function given by

$\Gamma(s,x)=\int_{x}^{\infty}t^{s-1}e^{-t}\,dt\ .$ (48)

###  A.3 Consideration of Border Effects

Assuming the spatial concentration distribution of a spur is described by a
bivariate normal distribution (Equation 3), approximately $99.7\%$ of a spur
is contained within a radius of $r=3\sigma$ where $\sigma=\sqrt{2\alpha t}$ is
the standard deviation of the distribution, i.e. the mean diffusion distance
of any one of the spur’s species. Assuming
$\alpha=$4.3\text{\times}{10}^{-9}\text{\,}\mathrm{m}^{2}\mathrm{/}\mathrm{s}$$
and $t=$1\text{\,}\mu\mathrm{s}$$, this radius is
$r\approx$0.3\text{\,}\mu\mathrm{m}$$. Assuming a spur cannot interact with
any other spur that is centered farther than $2r$ away, then, for a
homogeneous square radiation field size of side length
$5\text{\,}\mathrm{m}\mathrm{m}$, at least $99.97\%$ of spurs in the field do
not experience any border effects.

###  A.4 Model Application to Sparsely Ionizing Radiation

Here, the calculations performed in Section 2, which applied to densely
ionizing radiation (assuming spurs are straight and homogeneous along the beam
axis), are adjusted to apply to sparsely ionizing radiation. Under this
assumption, each primary particle may create multiple sparsely distributed
ionization events, and each ionization event creates a cluster of species. The
ionization events are assumed to be uniformly randomly distributed throughout
the target volume. This calculation thereby follows the distribution,
evolution, and interaction of 3-dimensional clusters as opposed to
2-dimensional spurs.

####  A.4.1 Cluster Definition

The spatio-temporal evolution of a cluster is given by the same reaction
diffusion equation as that of a spur (Equation 1) with diffusion coefficient
$\alpha$, decay rate $k_{s}$, and intercluster interaction rate $k_{r}$:

$\frac{\partial}{\partial
t}c_{i}(\vec{r},t)=\alpha\nabla^{2}c_{i}(\vec{r},t)-k_{s}c_{i}(\vec{r},t)-k_{r}\sum_{i\neq
j}^{N_{s}}c_{i}(\vec{r},t)c_{j}(\vec{r},t)\ .$ (49)

which, solved in 3 spatial dimensions, assuming only weakly-interacting
clusters, applying the temporal shift $\tau_{0}$ to define the cluster’s
minimum age, and setting $c_{0}$ to be the initial quantity of species in a
cluster, yields the PDF of an arbitrary cluster,

$c_{i}(\vec{r},t)=\frac{e^{-\frac{\absolutevalue{\vec{r}-\vec{r_{i}}}^{2}}{4\alpha(t-t_{i}+\tau_{0})}-k_{s}(t-t_{i})}}{8(\pi\alpha(t-t_{i}+\tau_{0}))^{3/2}}\
.$ (50)

####  A.4.2 Cluster Distribution

Assuming a beam of constant dose rate over an exposure time $T$, the PDF of
the creation time $t_{i}$ of the $i$th cluster at time $t$ is

$P_{t}(t_{i})=\begin{cases}\frac{1}{\min(t,T)}\quad&\text{for}\quad 0\leq
t_{i}\leq\min(t,T)\\\ 0\quad&\text{elsewhere}\end{cases}\ .$ (51)

The expected total number of clusters in the target over time $N_{c}(t)$ can
be expressed with the total energy deposited in the target $E$ and the average
energy deposited per cluster $E_{c}$ for a target of density $\rho$ and volume
$V$:

$\displaystyle N_{c}(t)$
$\displaystyle=\frac{E}{E_{c}}\cdot\frac{\min(t,T)}{T}$ (52)
$\displaystyle=\frac{D\cdot V\cdot\rho}{E_{c}}\cdot\frac{\min(t,T)}{T}\ .$
(53)

$E_{c}$ will depend on the radiation quality.

Assuming a spatially homogeneous beam, clusters arrive uniformly at random
throughout the target volume, so the expected number of clusters within any
spherical volume of radius $R$ can be expressed as a fraction of the total
number of clusters on the target:

$N_{R}(t)=\frac{4\pi R^{3}}{3V}\cdot N_{c}(t)\\\ $ (54)

Similarly, the PDF of the displacement $s$ between one arbitrary cluster and
another cluster within a radius $R$ is given by the ratio between the volume
element $4\pi r^{2}$ and the total volume $\frac{4}{3}\pi R^{3}$:

$P_{s}(s)=3\frac{r^{2}}{R^{3}}\ ,\quad s\leq R\ .$ (55)

####  A.4.3 Cluster Interaction Measure

The interaction rate $\omega$ between two individual clusters is:

$\displaystyle\omega_{1,2}(t)$ $\displaystyle=\int k_{r}\cdot
c_{1}(\vec{r},t)\cdot c_{2}(\vec{r},t)\,dV$ (56) $\displaystyle=k_{r}\cdot
c_{0}^{2}\cdot\frac{e^{-\frac{\absolutevalue{\vec{r_{1}}-\vec{r_{2}}}^{2}}{4\alpha(2t-t_{1}-t_{2}+2\tau_{0})}-k_{s}(2t-t_{1}-t_{2})}}{8(\pi\alpha(2t-t_{1}-t_{2}+2\tau_{0}))^{3/2}}\
.$ (57)

####  A.4.4 Intercluster Interactions

The total intercluster interaction rate of an arbitrary cluster $a$ with all
neighboring clusters is therefore

$\displaystyle\sum_{i}\omega_{a,i}(t)$
$\displaystyle=\lim_{R\rightarrow\infty}N_{R}(t)\cdot\langle\omega_{1,2}(t)\rangle$
(58) $\displaystyle=\lim_{R\rightarrow\infty}N_{c}(t)\cdot\frac{4\pi
R^{3}}{3V}\cdot\int\\!\\!\\!\int\\!\\!\\!\int P_{t}(t_{1})\cdot
P_{t}(t_{2})\cdot P_{s}(s)\cdot\omega_{1,2}(t)\ ds\,dt_{2}\,dt_{1}$ (59)
$\displaystyle=\frac{N_{c}(t)}{V}\cdot k_{r}\cdot
c_{0}^{2}\cdot\left(\frac{1}{\min(t,T)}\right)^{2}\cdot\begin{cases}\int_{0}^{t}\int_{0}^{t}e^{-k_{s}(2t-t_{1}-t_{2})}\
dt_{2}\,dt_{1}&\text{for}\quad t\leq T\\\\[10.0pt]
\int_{t-T}^{t}\int_{t-T}^{t}e^{-k_{s}(2t-t_{1}-t_{2})}\
dt_{2}\,dt_{1}&\text{for}\quad t>T\end{cases}$ (60)
$\displaystyle=\frac{N_{c}(t)}{V}\cdot\frac{k_{r}}{k_{s}^{2}}\cdot
c_{0}^{2}\cdot\left(\frac{1}{\min(t,T)}\right)^{2}\cdot\begin{cases}(e^{-k_{s}t}-1)^{2}&\text{for}\quad
t\leq T\\\\[10.0pt] (e^{k_{s}T}-1)^{2}\cdot e^{-2k_{s}t}&\text{for}\quad
t>T\end{cases}$ (61)

Thus, the total intercluster conversion is

$\displaystyle I_{\text{inter}}(t)$
$\displaystyle=\frac{N_{c}(t)}{2}\cdot\int_{0}^{t}\lim_{R\rightarrow\infty}N_{R}(t^{\prime})\cdot\langle\omega_{1,2}(t^{\prime})\rangle\,dt^{\prime}$
(62) $\displaystyle=\begin{cases}\lim_{\epsilon\rightarrow 0}\frac{B}{2}\cdot
t\cdot\left(\ln(\frac{t}{\epsilon})-\Gamma(0,2k_{s}t)+2\Gamma(0,k_{s}t)+\Gamma(0,2k_{s}\epsilon)-2\Gamma(0,k_{s}\epsilon)\right)&\text{for}\quad
t\leq T\\\\[10.0pt]
I_{\text{inter}}(T)+\frac{B}{2}\cdot\frac{(e^{k_{s}T}-1)^{2}}{2k_{s}}\cdot(e^{-2k_{s}T}-e^{-2k_{s}t})&\text{for}\quad
t>T\end{cases}\ ,$ (63)

where

$B=\frac{N_{c}(t)^{2}}{A}\cdot\frac{k_{r}}{k_{s}^{2}}\cdot
c_{0}^{2}\cdot\left(\frac{1}{\min(t,T)}\right)^{2}\ .$ (64)

####  A.4.5 Intracluster Interactions

The total intracluster conversion in a continuous beam is approximated using
the same method as described in Section 2.6,

$\displaystyle I_{\text{intra}}(t)$
$\displaystyle=\sum_{i}^{N_{c}(t)}\int_{0}^{t}\omega_{i,i}(t)\,dt^{\prime}$
(65) $\displaystyle\approx N_{c}(t)\cdot\int_{0}^{\infty}\omega_{i,i}(t)dt$
(66) $\displaystyle=N_{c}(t)\cdot\int_{0}^{\infty}k_{r}\cdot
c_{0}^{2}\cdot\frac{e^{-2k_{s}t}}{8(2\pi\alpha(t+\tau_{0}))^{3/2}}\,dt$ (67)
$\displaystyle=N_{c}(t)\cdot k_{r}\cdot
c_{0}^{2}\cdot\frac{\sqrt{k_{s}}}{16(\pi\alpha)^{3/2}}\cdot
e^{2k_{s}\tau_{0}}\cdot\Gamma\left(-\frac{1}{2},\,2k_{s}\tau_{0}\right)$ (68)

####  A.4.6 Comparison to Densely Ionizing Radiation

In order to compare the sparsely ionizing radiation application of this model
with clusters to the densely ionizing radiation application with spurs, the
subscripts “clusters” and “spurs” will be used to denote the relevant
quantities, which differ as follows:

$\displaystyle I_{\text{inter, clusters}}(t)$ $\displaystyle=I_{\text{inter,
spurs}}(t)\cdot\frac{L^{2}}{E_{c}^{2}}\cdot\frac{V}{A}$ (69) $\displaystyle
I_{\text{intra, clusters}}(t)$ $\displaystyle=I_{\text{intra,
spurs}}(t)\cdot\frac{L}{E_{c}}\cdot\frac{V}{A}\cdot\sqrt{\frac{k_{s}}{4\pi\alpha}}\cdot\frac{\Gamma(-\frac{1}{2},\,2k_{s}\tau_{0})}{\Gamma(0,\,2k_{s}\tau_{0})}$
(70) $\displaystyle\Rightarrow\Phi_{\text{clusters}}(t)$
$\displaystyle=\Phi_{\text{spurs}}(t)\cdot\frac{L}{E_{c}}\cdot\sqrt{\frac{4\pi\alpha}{k_{s}}}\cdot\frac{\Gamma(0,\,2k_{s}\tau_{0})}{\Gamma(-\frac{1}{2},\,2k_{s}\tau_{0})}\
.$ (71)