Time dynamics of the dose deposited by relativistic ultra-short electron beams

Ultra-short electron beams are used as ultra-fast radiation source for radiobiology experiments aiming at very high energy electron beams (VHEE) radiotherapy with very high dose rates. Laser plasma accelerators are capable of producing electron beams as short as 1 fs and with tunable energy from few MeV up to multi-GeV with compact footprint. This makes them an attractive source for applications in different fields, where the ultra-short (fs) duration plays an important role. The time dynamics of the dose deposited by electron beams with energies in the range 50–250 MeV have been studied and the results are presented here. The results set a quantitative limit to the maximum dose rate at which the electron beams can impart dose.


Introduction
Radiotherapy using very high energy electron (VHEE), 50-250 MeV electron beams, is studied as a possible cancer treatment (DesRosiers et al 2000), exploiting the fact that electron beam focusing could lead to favorable dose profiles (Kokurewicz et al 2019).The development of a VHEE radiotherapy device with conventional RF technology has been carried out in recent years (Maxim et al 2019).
Recently, also electron beams accelerated using the laser wakefield acceleration, LWFA, technique have been studied for possible application in medicine (Malka et al 2008) and radiobiology (Yang et al 2011a(Yang et al , 2011b)).For these experiments, where it is often assumed that the time resolution is limited by the electron beam time duration, LWFA represent an attractive source due to their record-low few fs time duration (Gauduel et al 2010, Gauduel 2011).
Both, conventionally generated and laser plasma generated beams, can produce extremely short (fs scale) bunch length (Lundh et al 2011, Islam et al 2015, Stephan et al 2022).
The dosimetric properties of such beams have been characterized in previous works (Subiel et al 2014(Subiel et al , 2017(Subiel et al , 2020)), where the measurement of very high dose rates represented a challenge (McManus et al 2020).
In this work, by means of Monte Carlo simulations with the FLUKA code (Battistoni et al 2015, Ahdida et al 2022), two features that affect the timescale over which the dose is imparted on the target, have been investigated: first the bunch elongation of the primary beam as it propagates within the target, and second the time dynamics of the dose deposition.

Methods
The bunch elongation and time dynamics of the dose deposition by electron beams inside the target have been investigated by means of FLUKA Monte Carlo simulations.FLUKA is one of the most widespread generalpurpose Monte Carlo transport code and it is regularly used in many laboratories worldwide (NEA 2021).The details of the simulations are discussed below and reported in table 1.
The geometry of the used phantom target is a cylinder (R = 20.0 cm, Z = 30.0cm) of water (without walls).Water was chosen as the material since it is widely used in dosimetry and radiobiology experiments and could approximate the human body.The beam is directed along the axis of the cylinder, as can be seen in figure 1.The simulated electron beam energies are 50 MeV, 150 MeV and 250 MeV (only 150 MeV beam energy has been considered for the bunch elongation study).The transverse distribution of the cylindrical electron beams is flat with a 1 cm diameter.No beam divergence has been considered.All primary particles were started at t = 0s.The electromagnetic transport thresholds are set to 100 keV.This value is a compromise between the CPU time and the bin size, the latter being very sensitive to the MC limitations described in 2.2.The selection of this value is also supported by figure 11 in Plante and Cucinotta (2009).

Bunch elongation
The simulations were done similarly to the one described in section 2.1.1 of Subiel et al (2017) and Subiel et al (2020), to estimate the distribution of the time needed by the electrons above 5 MeV energy to reach predefined scoring planes.The bunch length was defined as the FWHM of the peak of each distribution.The scoring planes are perpendicular to the beam direction, located every 5 cm along the beam axis.

Time dynamics of the dose deposition.
FLUKA only provide limited built-in time scoring capabilities to investigate the time of the energy deposition.To access the actual time of the energy deposition, it is necessary to use FLUKA's user routines, where a user can access all parameters of the simulations.In most cases, it is enough to record the required parameters in an external file and process them after the simulation is finished.However, in this case, the recorded information would have been unmanageable, because energy is deposited at each interaction and particle step.That is why a custom 3D histogram scoring was programmed.This custom scoring relies on the definition of a cylindrical mesh made defined over small finite volumes in radius (R), depth (Z), and time (t).The accuracy of the custom scoring was tested against the built-in time cut-off scorings of FLUKA.
The bins have a longitudinal length of 3 mm and a radial length of 2 mm, corresponding to a mesh of 100 x 100 bins.The time step is set to 400 fs (12 500 time bins).The size of the volume imposes a limitation on the accessible resolutions.
The number of finite volumes in the mesh is limited due to the memory allocation needed to run the simulation.The chosen mesh has been considered as a reasonable compromise between precision and computing requirements.Furthermore, due to the continuous energy deposition, an intrinsic limitation also arises.A high-energy primary would require significant time to cross the length of the bin.For example, it would take about 10 ps for a relativistic particle to cross a 3 mm bin.Therefore the choice of the bin dimensions implicitly constrains the time estimates.

Bunch elongation
Table 2 shows the calculated bunch lengths at different depths in the water phantom.
The important result is that even though the simulations assumed all the primaries started instantaneously, after a few centimetres the beam has a duration of the picosecond scale.This represents a lower limit on the time needed to impart the dose, which is intrinsic in the physics of the propagation of the beam within the phantom.This lower limit is particularly relevant for FLASH beams regardless of their acceleration process.Bunches having a femtosecond scale length would elongate during the propagation to get similar to those accelerated in conventional accelerators.

Time dynamics of the dose deposition
The maps, as a function of the position inside the phantom, of the dose deposited and of the time necessary to deposit 99% of it are shown in figure 2. Figures 3 and 4 are the one-dimensional projection of figure 2 for chosen radii.
These three figures highlight the different effects due to the primary and secondary particles.Within the beam channel, the primary electrons are the driving contributors to the dose deposition.The secondary particles generated give a smaller contribution which results in the slight increase visible in figure 4 panel (a).As the electromagnetic shower cease to be sustained by the primary electrons because of the energy loss, then the deposited dose decreases.The 'flatness' of the first part of the dose deposition curves, is reflected in the relatively constant value of the time necessary to impart 99% of the dose along the z axis, see figure 4 panel (a).
By definition, outside the beam channel, the main contributors to the dose deposition are the secondary particles.The dose deposition does not reach a plateau but peaks at a specific depth.The dose imparted is one order of magnitude lower, see figure 3 panel (b), and it is imparted on a longer timescale, see figure 4

panel (b).
The very high value of the time to impart the dose at small depth is due to the fact that only back-scattered secondaries contribute to the dose deposition in this portion of the R-Z plane.
In case of 50 MeV beam, the primaries are not energetic enough to cross the entire phantom.They are absorbed within about 20 cm, while basically only secondaries deposit dose beyond that depth.This leads to the two lobed structure, see figure 2 panel (b).The decrease of the time necessary to deposit 99% of the dose at the end of the phantom see figure 2 panel (b) and figure 4 panel (a), is due to the fact that no secondaries are backscattered from beyond the end of the phantom.This effect can be observed on a smaller scale for the 150 and 250 MeV beams, see figure 4 panel (a). Figure 5 allows another insight on the relation between the imparted dose and the time necessary for it to be imparted.In panel (a), for Z = 0 cm, the whole dose is deposited by the primary beams, therefore it is the same all over the beam channel R < 1 cm and of the order of about 20 ps.As the beam penetrates the target phantom, an electromagnetic shower is generated, secondaries also contribute to the dose deposition therefore increasing the time necessary to deposit the dose.Additionally, because of Multiple Coulomb Scattering, the primary beam gets larger; this can be seen in the progression from panel (a) to (f) for each beam energy.The electromagnetic shower that is generated has an important effect on the minimum time necessary to deposit 99% of the dose.First, the time increases because of the large number of secondaries produced, that can propagate backward and deposit energy after the primary beam has passed.Second, the time decreases at the end of the target because the number of secondary decreases as no more are produced from beyond the end of the target.Figure 6 shows the evolution of the deposited dose at different depths within the phantom and for the innermost radial layer.In each panel, t = 0 corresponds to the moment when the first energy deposition occurs in the considered spatial bin.These distributions are affected by the limitation of the Monte Carlo simulations discussed in section 2.2.Two interesting considerations can be made about these pictures.First, the time scale of the dose deposition is on the ps scale, longer than the time needed by the primary beam to cross the scoring bin.This is due to the contribution of secondaries created in other scoring bins which have to propagate within the phantom and deposit dose at a later time.The pictures suffer from the MC limitations discussed above, therefore the linear increase in the first half of the plot is just a linear artefact due to this limitation.Nevertheless, departures from the linear increase are real and are due to the contribution of secondary particles.In panel (a) the transitions at the start and end of the primary contribution are sharp.Starting from panel (b) these transitions soften up due to the contribution from secondary particles.The softening at the beginning of the energy deposition is due to the secondary bremsstrahlung photons created at shallower depths, which propagate at a higher speed than the primary electrons.

Conclusions
In this work it is shown that the dose deposition by a relativistic electron beam inside a water phantom, is not instantaneous because of two different reasons.First, the electron bunch elongation when traversing a medium and second, the contribution to the dose by secondary particles.It is also important to remember that the simulations are stopped when the radiation falls below the chosen Monte Carlo transport threshold, therefore implying that the real timescale is even longer.It can therefore be concluded that to calculate accurately the dose  The scoring bin size is 2 mm (radial) × 3 mm (longitudinal).Time scoring is done with 400 fs time bins.On-axis dose deposition is faster for more energetic electrons, while the one off-axis has a more complex behaviour.rate it cannot be simply assumed that the dose is deposited in a time interval corresponding to the electron bunch time duration at the source.
It is also shown and quantified how the time necessary to deposit the dose in the water phantom is dependent on the location considered: its depth within the phantom (in correlation with the beam energy), whether it is within the beam channel, and whether it is close to the edge of the phantom.
Furthermore, it is important to remember that chemical and biological effects may further lead to an increase of the time over which physiological effects happen.Last, it is important to stress that the results obtained in this work applied to whichever electron beam, regardless of the acceleration mechanism.

Figure 1 .
Figure 1.3D cutaway geometry of the water phantom.The red arrow indicates the direction and axis of the beam.

Figure 2 .
Figure 2. Panels (a), (c) and (e): dose deposition as function of the spatial position in the water phantom.Panels (b), (d) and (f): time required to deposit 99% of the total dose.The scoring bin size is 2 mm (radial) × 3 mm (longitudinal).Time scoring is done with 400 fs time bins.As in the cuvette case, the time required for the dose deposition is strongly dependent on the electron beam energy.

Figure 3 .
Figure 3. Dose deposited on-axis (a) and off-axis (b) by the electron beam in the water phantom as a function of the depth for different beam energy.The scoring bin size is 2 mm (radial)×3 mm (longitudinal).The on-axis dose profile consists of a flat region, whose size is increasing with the beam energy, followed by a decrease.The off-axis profile has about one order of magnitude less dose and exhibits a local maximum at a depth depending on the beam energy.

Figure 4 .
Figure 4. Time needed to deposit 99% of the dose at different radii of the phantom as function of the depth for different beam energy.The scoring bin size is 2 mm (radial) × 3 mm (longitudinal).Time scoring is done with 400 fs time bins.On-axis dose deposition is faster for more energetic electrons, while the one off-axis has a more complex behaviour.

Figure 5 .
Figure 5.Time needed to deposit 99% of the dose at different depths of the water phantom as function of the radius for different beam energy.The scoring bin size is 2 mm (radial) × 3 mm (longitudinal).Time scoring is done with 400 fs time bins.

Figure 6 .
Figure 6.Dose deposition time profiles at the center of the phantom and at different depths along the beam axis.The scoring bin size is 2 mm (radial) × 3 mm (longitudinal).Time scoring is done with 400 fs time bins.In each panel, t = 0 corresponds to the moment when the first energy deposition occurs in the considered spatial bin.The linear increase in the first part of the curves is the plotting artefact already discussed.Only secondary particles contribute to the second part of the curves.The non-linear behaviour, close to t = 0 ps, visible in panels from (b) onward, is due to bremsstrahlung photons produced at smaller depths, that reach the scoring bin before the primary beam.

Table 2 .
Bunch lengths at different depths as calculated with FLUKA.