# Optimizing the Traversal Time for Gantry Trajectories for Proton Arc Therapy
Treatment Plans

Viktor Wase, Otte Marthin, Albin Fredriksson, Anton Finnson
RaySearch Laboratories AB
Stockholm
Sweden
<EMAIL_ADDRESS>

###### Abstract

Background: Proton arc therapy is an emerging radiation therapy technique
where either the gantry or the patient continuously rotates during the
irradiation treatment. One of the perceived advantages of proton arc therapy
is the reduced treatment time, but it is still unclear exactly how long these
treatment times will be, given that no machine capable of its delivery is
available at the market at the time of writing.
Purpose: We introduce the algorithm Arc Trajectory Optimization Method (ATOM),
which aims to determine an efficient velocity profile for the gantry for rapid
delivery of a given proton arc treatment plan.
Methods: ATOM computes the trajectory with the shortest delivery time while
ensuring there is enough time to deliver all spots in each energy layer and
switch energy between layers. The feasibility of the dynamic gantry movement
was assured by enforcing maximum and minimum limits for velocity,
acceleration, and jerk. This was achieved by discretizing the gantry velocity
and combining the A* algorithm with the open-source motion generation library
Ruckig. The algorithm was tested on a synthetic data set as well as a liver
case, a prostate case and a head and neck case.
Results: Arc trajectories for plans with 360 energy layers were calculated in
under a second using 256 discrete velocities.
Conclusions: ATOM is an open-source C++ library with a Python interface that
rapidly generates velocity profiles, making it a highly efficient tool for
determining proton arc delivery times, which could be integrated into the
treatment planning process.

_K_ eywords Radiotherapy $\cdot$ Ion Therapy $\cdot$ Proton Arc

##  1 Introduction

Proton arc therapy (PAT) is an emerging radiation therapy treatment technique
delivering the protons in pencil beam scanning mode while the gantry moves, as
opposed to intensity modulated proton therapy (IMPT) where the gantry is
stationary while irradiating. Optimization algorithms for creating PAT
treatment plans often try to reduce the delivery time indirectly by reducing
the number of energy switches to higher energies [1, 2, 3]. The number of
upward energy switches is considered a good predictor of beam delivery time
(BDT), since an upward switch tends to be slower than a downward energy switch
for most delivery systems. Zhao et al. [4] modeled BDT as proportional to the
number of spots that also increases considerably in PAT plans and derived a
method to enforce sparsity in the spot selection. Another group developed a
technique to generate a Pareto front balancing delivery time and plan quality
for proton arcs, using an approximate static BDT model taking into account
energy switches and layer irradiation time [5]. A common dynamic BDT model
assumes a constant gantry rotation speed of
$6\mathrm{\SIUnitSymbolDegree}\mathrm{/}\mathrm{s}$, plus the time added by
irradiation segments and energy switching [2, 6, 7].

In this paper we introduce an algorithm that can be used to design the gantry
velocity curve of a proton arc plan such that the delivery time is minimized.
The algorithm can be used in order to decide the velocity profile of a plan
after the plan has been designed. Alternatively, it can be used repeatedly
during the proton arc plan optimization process to take delivery time into
account when creating the plan. The algorithm is tested on data meant to
simulate a PAT plan, but the algorithm works for other types of ion arcs as
well, since the overall delivery framework is very similar [8]. The algorithm
considers limits on maximum gantry velocity, acceleration as well as jerk (the
derivative of acceleration). Including the jerk is important to create a
smooth velocity profile while limiting the machine wear [9].

At the time of writing there is, to our knowledge, no published algorithm that
designs the velocity curve of the gantry for an ion arc treatment plan. The
inertia could have a significant impact on BDT since the gantries are
incredibly heavy and the delivery time per energy layer varies a lot. Changes
in inertia is important to consider in the cases when the patient is moved and
the gantry is static, since perturbations to the position of the patient
should be minimized. An algorithm that is fast and general enough to be used
by multiple research teams and multiple machines would greatly improve the
effort required to compare the reported delivery times in different research
papers. The present paper seeks to address this need by proposing an algorithm
that meets these criteria.

##  2 Methods and Material

###  2.1 Proton Arc Preliminaries

The main idea of proton arc is to move the gantry while irradiating. It is to
IMPT what VMAT is to IMRT. A proton arc plan is defined in much the same way
as a regular IMPT plan, but with significantly more beams. However, much like
in VMAT planning, these are not refered to as beams, but rather gantry angles
or simply directions. Each such gantry angle corresponds to a gantry angle
window. The gantry angle window is a small angular range of the arc, and each
such window has a set of spots assigned to it, all of which have to be
delivered while the gantry angular position is located inside the gantry angle
window. The idea is that this window should be small enough that it does not
matter at which exact angle a spot is delivered, as long as it is within the
specified window. The dose perturbation should be small enough to be
negligible. See Figure 1 for an illustration of this concept.

Figure 1: Example of a proton arc plan, covering a quarter of a full
revolution and consisting of four energy layers. The gray lines represent the
gantry angle windows.

Throughout this paper we assume that each specified gantry angle contains
exactly one energy layer. However, this assumption is made without loss of
generality to simplify the terminology. If several energy layers are to be
delivered within one gantry window, one can simply consider the combination of
these energy layers and treat them as a single energy layer with irradiation
time equal to the sum of the total irradiation time of all layers and the
internal energy layer switching times.

###  2.2 Model

In this paper we introduce the Arc Trajectory Optimization Method (ATOM),
which is capable of producing optimized velocity profiles for ion arc plans,
given the gantry angles and irradiation times of all energy layers, as well as
the energy switching times between them. The goal of the algorithm is to
create a velocity profile such that the total delivery time is minimized,
while fulfilling all conditions regarding how much time the gantry needs to
spend in and between the gantry angle windows in order to properly deliver the
plan.

In order to keep the deviation between the planned and delivered dose small,
we center the irradiation of each gantry angle window by keeping the gantry
velocity fixed while irradiating and starting/stopping the irradiation equally
long before/after the center of the gantry angle window. This means that the
acceleration is zero during irradiation. Allowing non-zero acceleration during
irradiation could possibly allow for faster delivery, but may increase the
difference between planned and delivered plan. For example, the irradiation
might become systematically skewed towards the latter part of the gantry angle
windows, effectively rotating the dose distribution. Moreover, it increases
the dimensionality of the problem, since both acceleration and velocity must
be determined for each gantry angle window.

The problem to be solved is thus that of choosing a sequence of velocities at
which to start irradiation. The allowed velocities for a given gantry angle
are dependent on the previous and following velocities, and are also
restricted by the limitations on acceleration and jerk. This gives the problem
a combinatorial structure well suited for a dynamic programming formulation.
Such a formulation does, however, require a discretization of the set of
possible velocities for the gantry at start of irradiation. By repeating
experiments with different discretization resolutions we choose a suitable
precision with which to specify these velocities, so that the gain from
further precision is negligible compared to the increase in computation time.

A final assumption in the ATOM solution is to not allow the gantry to switch
direction of travel. This enables quicker solution of the dynamic programming
problem, and while this might increase the optimal delivery time slightly, we
find it reasonable to believe that clinics in practice would want to enforce
this rule, as a heavy proton gantry that switches direction back and forth
throughout the same arc seems problematic with regards to safety and QA. Note
that we investigate the effects of these assumptions on both delivery time and
the algorithm’s computational time in our experiments.

###  2.3 Mathematical Formulation

The problem of optimizing total delivery time can be subdivided into two main
problems. The first problem consists of finding the minimal travel time
between any two arc states, consisting of position, velocity, acceleration,
and jerk. Having solved this first problem, the second problem becomes that of
constructing the fastest path through a sequence of arc states, utilizing that
we know the fastest solution between any two states.

Denote the angle span between the edges of two adjacent gantry angle windows
by $\Delta x$. Given an initial velocity $a$ and a final velocity $b$, let
$t_{i}(a,b)$ denote the traversal time of the fastest velocity profile over
the angle span $\Delta x$ that satisfies the machine constraints and the
required time for energy switching between the $i$th and $(i+1)$th energy
layer, as well as the initial and final velocities. We will also need to
consider the irradiation time of the energy layers. Define
$t^{\textrm{irr}}_{i}$ as the time required to deliver all spots in the $i$th
energy layer.

If there are $N$ gantry angle windows, the velocity profile problem is that of
finding the velocities $v_{1},\ldots,v_{N}$ of all gantry angle windows such
that the total trajectory time

$\sum_{i=2}^{N}t_{i}(v_{i-1},v_{i})+\sum_{i=1}^{N}t^{\textrm{irr}}_{i}$

is minimized. To this end, we construct a directed graph of nodes, each
corresponding to an gantry-angle-window-and-velocity state. Let there be $M$
discrete velocities $s_{1},\ldots,s_{M}$ to choose from, and each node is
denoted by $n_{i,j}$, where the node for $i=2,\ldots,N-1$ and $j=1,\ldots,M$
corresponds to the $i$th gantry angle window and the $j$th discrete velocity
($s_{j}$), and the starting node $n_{1,1}$ and the last node $n_{N,1}$
correspond to a velocity of zero at the first and last gantry angle window,
respectively, i.e., $v_{1}=v_{N}=0$. The nodes have directed edges with
connections corresponding to moving the gantry one angle-step forward, to the
next energy layer. Each edge has an associated cost; the cost between node
$n_{i,j}$ and $n_{i+1,k}$ is given by
$t_{i}(s_{j},s_{k})+t^{\textrm{irr}}_{i}$, the transition time between the
corresponding two states plus the irradiation time for the gantry angle window
we are leaving. Note that the function $t_{i}(v_{i-1},v_{i})$ implicitly
depends on $\Delta x$, $t^{irr}_{i-1}$ and $t^{irr}_{i}$, but they are omitted
for clarity since they are constants and treated as implicit to the functions
$t_{i}$. Further note that the irradiation time of the last layer is ignored
in the problem, and simply added later as a constant. An illustration of this
graph is shown in Figure 3.

Determining the cost of each edge in the graph amounts to solving a sub-
problems to find the optimal trajectory between two velocities, while covering
a given distance. This is a well studied problem [10, 11, 12]. Berscheid et
al. [11] published an open-source C++ library called Ruckig that quickly
solves this problem to optimality under constraints on maximum and minimum
velocity, acceleration and jerk. In fact, this library can solve the more
general problem of having non-zero boundary accelerations, but is still
significantly faster than other algorithms specifically written to find
trajectories with zero boundary acceleration [11]. Therefore, we use Ruckig to
calculate the duration $t_{i}(s_{j},s_{k})$ of the trajectories between the
gantry angle windows. The library also supports enforcing a minimum travel
time, which means that we can make sure that the trajectory is always slower
than the required time for energy layer switching. An example trajectory is
illustrated in Figure 2.

Figure 2: Example of a trajectory generated by Ruckig. The initial velocity is
$0.7\mathrm{\SIUnitSymbolDegree}\mathrm{/}\mathrm{s}$ and the final is
$0.1\mathrm{\SIUnitSymbolDegree}\mathrm{/}\mathrm{s}$, with a delta angle of
$1\mathrm{\SIUnitSymbolDegree}$ and a maximum/minimum jerk of
$0.5\mathrm{\SIUnitSymbolDegree}\mathrm{/}\mathrm{s}^{3}$, maximum/minimum
acceleration of $0.5\mathrm{\SIUnitSymbolDegree}\mathrm{/}\mathrm{s}^{2}$, and
maximum velocity of $5\mathrm{\SIUnitSymbolDegree}\mathrm{/}\mathrm{s}$.

The way Ruckig is adapted to the problem of computing the optimal transition
time between the nodes in consecutive layers of the graph is described in
detail in Algorithm 1. Note that the cost is set to infinity if no feasible
trajectory exists between two nodes. With this construction, the lowest-cost
path through the graph corresponds to the velocity sequence of the time-
optimal arc.

Algorithm 1 Transition Time

Inputs:

$v_{0},v_{1}$ $\triangleright$ The initial and final velocities.

$t_{0},t_{1}$ $\triangleright$ The irradiation times of the previous and next
energy layers.

$t_{s}$ $\triangleright$ The time required to switch from the previous energy
layer to the next one.

$\Delta x$ $\triangleright$ The angular distance between the middle of the
previous and next gantry angle.

Procedure:

window0 := $v_{0}t_{0}$ $\triangleright$ Shrink the gantry angle windows to
the smallest possible size.

window1 := $v_{1}t_{1}$

if window${}_{0}>$ max gantry angle window size OR window${}_{1}>$ max gantry
angle window size then

return $\infty$

end if

angleSpan := $\Delta x-\frac{1}{2}($window${}_{0}+$window${}_{1})$

trajectory := calculateTrajectory($v_{0},v_{1},$ angleSpan, $t_{s}$)
$\triangleright$ Calculated using Ruckig.

if trajectory.isInfeasible then

return $\infty$

end if

return trajectory.transitionTime + $t_{0}$

$n_{1,1}$$n_{2,1}$$n_{2,2}$$n_{2,M}$$n_{i,1}$$n_{i,2}$$n_{i,M}$$n_{N-1,1}$$n_{N-1,2}$$n_{N-1,M}$$n_{N,1}$First
angleSecond angleAngle $i$Angle $N-1$Last angle Figure 3: Illustration of a
network of for determining velocity profiles. Each column represents a gantry
angle that corresponds to the central angle of an energy layer, and each row
$j$ represents a discrete velocity $s_{j}$. The cost between two nodes
$n_{i,j}$ and $n_{i+1,k}$ is equal to $t(s_{j},s_{k})+t^{\textrm{irr}}_{i}$.

Finding the shortest path in a directed graph can be done using a variety of
algorithms. Here, the trajectory problem is solved to optimality using the A*
search algorithm [13].

The A* algorithm requires a heuristic to estimate the remaining path cost from
each node to the end node. If this heuristic estimation is smaller than or
equal to the true value for all nodes, then the A* is guaranteed to find the
optimal solution [13]. We use the sum of the remaining energy layer switching
times plus the sum of the remaining irradiation times as our heuristic. This
means that all nodes that share gantry angle, but not necessarily gantry
velocity, will share heuristic value. This value is always smaller than or
equal to the remaining delivery time, which means that the found path will be
optimal.

In order to reduce the running time of the algorithm we prune the graph. Given
a current velocity $v_{i}$ and delta angle $\Delta x$ we calculate the maximum
and minimum velocity that can possibly be reached for the next gantry angle,
assuming that there is no limit on jerk (i.e. acceleration can change
instantaneously) and that the upcoming delivery window is of width 0. From
these assumptions and using the kinematics equations of motion, we get

$\displaystyle v_{\text{low}}:=\sqrt{v_{i}^{2}-2a_{\max}\Delta x},\quad
v_{\text{high}}:=\sqrt{v_{i}^{2}+2a_{\max}\Delta x}.$

When searching through velocity transitions $v_{i}\to v_{i+1}$ we then only
consider $v_{i+1}$ within $[v_{\text{low}},v_{\text{high}}]$. If
$v_{\text{low}}$ is imaginary, then we simply use 0 as a lower bound.

In summary, ATOM constructs a graph representing all gantry angle windows and
a discretized set of possible velocities during the irraditation; then
computes the traversal time between each pair of gantry angle windows and each
pair of discrete velocities using Algorithm 1; and finally constructs a
sequence of gantry velocities by solving for the shortest path through the
graph.

The code is publicly available for download from the GitHub repository
https://github.com/raysearchlabs/ArcTrajectoryOptimizationMethod under the
liberal MIT license, and can be used by anyone for research purposes. It is
available in both C++ and Python. The run times reported in this paper are
based on the C++ version, running on an Intel Core i9-10940X CPU and using
Windows 10. The code was not parallelized.

###  2.4 Experiments

####  2.4.1 Synthetic Data

The algorithm was run on a synthetic dataset consisting of 1000 plans. Each
plan had 360 energy layers, with one layer every
$1\mathrm{\SIUnitSymbolDegree}$ along the arc, each with a gantry angle window
of width $1\mathrm{\SIUnitSymbolDegree}$. The energy layer switching times
were generated independently with a 90% chance of taking 0.5 s and a 10%
chance of taking 5 s, representing down-switches and up-switches. The
irradiation time per layer is uniformly distributed between 0 and 1.26 s in
order to get the irradiation to represent about 40% of the static time.

The algorithm was tested with two different sets of machine parameters: one
with a small range of allowed accelerations (acceleration-limited machine
parameters) and a second one allowing for more acceleration while limiting the
jerk more (jerk-limited machine parameters). For each set of machine
parameters the algorithms was evaluated both with and without using the
pruning strategy described above. See Table 1.

| $v_{\text{max}}$ | $a_{\text{max}}$ | $j_{\text{max}}$
---|---|---|---
Jerk Limited Machine Parameters | 5.0 | 0.5 | 0.5
Acceleration Limited Machine Parameters | 5.0 | 0.25 | 1.0
Table 1: The algorithm was tested on two datasets, based on these machine
parameters.

The tests were repeated for varying velocity discretization, as well, but with
100 random plans instead of 1000 to make it faster. This was done to get a
sense of how to balance the running time and the accuracy of the algorithm.

####  2.4.2 Patient Data

The algorithm is also tested on three patient cases. The data is a liver case,
a head and neck case, and a prostate case with CTs and ROIs from the TROTS
dataset. The data was imported into a research version of RayStation 2023B and
proton arc plans were using a dynamic full revolution PBSArc with 180 energy
layers were generated. Each case had three objective functions: a robust min
function and robust max function on the PTV, and a non-robust dose fall-off
function on the external. The thresholds for the min and max functions were
set to the target dose $\pm 0.5$ Gy. The dose fall-off function was weighted
1000 times lower than the other functions. The robustness included 21
scenarios, created by 3 mm position uncertainty and 3% density uncertainty.
Each case was allowed to run for 100 iterations until spot filtering, and then
for 50 iterations more. The target dose was 50 Gy for the liver case, 70 Gy
for the prostate case, and 46 Gy for the head and neck case. Each case
consisted of 30 fractions. The dose distribution of the prostate case can be
seen in figure 4.

Figure 4: Proton Arc plan for the prostate case, with 180 energy layers and an
arc spanning the full revolution.

The time required for an upward energy switch was assumed to be 5 s and 0.5
for a down switch. The spot switching time was set to 2 ms and the spot
delivery is 5 ms for one MU. The jerk limited machine parameters were used,
and the the gantry safety window was set to $1^{\circ}$.

Four velocity profiles were designed for each case. The first was produced by
the ATOM algorithm, and the rest were produced by relaxing the first three
assumptions and treating the problem like a continuous optimization problem
with six variables per energy layer: the acceleration, velocity and position
at the start and end of the irradiation of all layers. The velocity and
acceleration before the first layer was set to zero, and the same thing was
done to the velocity and acceleration after the last layer. The start and end
position of the entire arc were set to 0 and 359. All variables in the
optimization that correspond to positions had bounds between zero and one,
where 1 corresponds to the smallest angle where an energy layer can be
delivered and 1 corresponds to the largest angle. The velocity variables were
bounded between 0 and 5, and the acceleration variables where bounded between
0 and 0.5.

The global optimization algorithm Dual Annealing [14] was used to create the
second velocity profile. It used the SciPy [15] implementation and was allowed
to run for 12 h per case. The third and fourth trajectories were created using
a hill climbing optimization algorithm that randomly perturbed $0.5\%$ of the
variables with a Gaussian perturbation with a standard deviation of 0.001. The
difference between the third and fourth velocity profile was only the starting
guesses: one used the solution found by ATOM as a starting guess and the other
used a step and shoot velocity profile which came to a full stop for every
energy layer. These algorithms were allowed to run for 12 h, too. For a fair
comparison between Python scripts and C++ code, we compare the number of calls
to the calculate function in Ruckig rather than running time, since the
calculate function is the computational bottle-neck.

##  3 Results

###  3.1 Synthetic Data

For each plan we calculated the static delivery time as the sum of the energy
layer switches and the irradiation times, representing the beam delivery time
if the gantry were fully static. This was subtracted from the total delivery
time to give the dead time. The static and dynamic delivery times, as
calculated by ATOM, are shown in Figure 5. It should be noted that the dynamic
delivery time is significantly slower than the static delivery one, which
means that using the static delivery time as a model might be ill advised in
some cases. The computational run times are given in Table 2. The algorithm is
a lot faster with pruning than without in both datasets.

| With pruning | Without pruning
---|---|---
Jerk Limited Datase | 0.70 s | 2.04 s
Acceleration Limited Dataset | 0.46 s | 2.11 s
Table 2: The mean computational run times for the plans in the two datasets.
Each plan contains 360 energy layers. Figure 5: The dynamic and static
delivery times of $2\times 1000$ different random plans.

The results from the runs with varying velocity discretization are found in
Figure 6. The time complexity of ATOM with respect to the velocity resolution
is roughly quadratic, which is to be expected given that the number of edges
grows quadratically with the number of velocities (the number of edges is
$O(NM^{2})$ and the number of nodes is $O(NM)$). We think 256 different
velocities is a decent trade-off between running time and accuracy, based on
these results.

Figure 6: Left: The running time as a function of the number of discrete
velocities on a log-log plot. The thin bottom line is the function $x^{2}/c$,
showing that the time complexity of ATOM is roughly quadratic with respect to
the number of velocities. The constant $c$ was set to 200 to roughly mimic the
scaling of the algorithm. Right: The dead time is defined as the total
delivery time minus the time of irradiation, spot switching and energy
switching. This shows the estimated dead time, for all 2$\times$100 plans,
using different velocity grids.

The gantry velocity curve of an example plan can be seen in Figure 7.

Figure 7: The velocity of the gantry for an example plan with 180 energy
layers and 2 degrees between each energy layer. The dark areas symbolize the
time required for upward energy switches. The machine parameters are
$v_{max}=5$, $a_{max}=0.5$ and $j_{max}=0.5$. The zoomed-in window shows that
the curve is smooth, and mostly consists of constant velocity.

###  3.2 Patient Data

The global optimizer struggled to find a decent solution in all three cases,
however it is likely that it would have done so given enough time. It did not
find a feasible solution at all in the liver case. The local optimizer seems
to have gotten stuck in a local minima, which was to be expected. However,
when its initial solution was a start-and-stop arc it failed to find a
solution with a delivery time faster than 900 s, for all three cases. This is
roughly three times slower than the solution found by ATOM. The fact that both
the hill climbing algorithm and the dual annealing algorithm failed to find a
good solution in a reasonable time shows that designing these velocity
profiles is a non-trivial problem and motives the use of the given
assumptions.

Figure 8: The delivery time as a function of number of Ruckig evaluations for
thee patients and four methods. The ATOM case requires a certain number of
Ruckig evaluations and is therefore represented as a single point. A local
optimization algorithm was run with two different starting points: a velocity
profile that starts and stops (S&S) such that every energy layer is delivered
with a static gantry, and the velocity profile generated by ATOM. Note that
the global optimization was unable to find any feasible solutions for the
liver case. All algorithms were given a maximum run time of 12 h.

The local optimization that ignored all our assumptions (except for the one
about reversing gantry direction) and used the solution found by ATOM as its
initial solution managed to improve on that solution in all three cases. In
the head and neck case the delivery time was reduced from 309 s to 279 s. In
the prostate case it went from 288 s to 266 s, and in the liver case it went
from 284 s to 271 s.

##  4 Discussion

We have created a simple, fast and open-source algorithm that can estimate the
fastest delivery time of an ion arc plan. The calculation time for a proton
arc plan is under a second, using a resolution of 256 discrete velocities.

The solutions found by ATOM are optimal under the listed assumptions, but a
local search around the solution found by ATOM showed that it was possible to
reduce the delivery of 5 min plans by roughly 15-30 s.

Without our assumptions the naive solution would entail configuring each state
in the graph to encompass the angular position, velocity, and acceleration of
the gantry during the intervals between irradiation and energy switching. One
could still assume that jerk is zero in all these instantaneous states while
allowing non-zero jerk between them, since there are no bounds on the
derivative of the jerk. This would result in six variables for each energy
layer: position, velocity, and acceleration at the start and end of
irradiation of an energy layer. With a discretization factor of $n=256$, this
yields an approximate count of $256^{6}\approx 2.8\cdot 10^{14}$ possible
states per energy layer.

However, by introducing the assumption that the gantry won’t accelerate during
irradiation, we establish that the acceleration at the start and endpoints are
zero, and the start and end velocities are identical. Furthermore, we can
calculate the end position from the start position, velocity, and irradiation
duration for the layer. Consequently, each layer requires only two variables:
initial position and velocity. Incorporating the assumption that the gantry
angle windows remains centered around the specified gantry angle further
reduces the variables per energy layer to one: velocity. The necessary span of
the layer’s position is determined by multiplying velocity by the irradiation
time, and by assuming symmetry in the position span around the specified
gantry angle, the start and end positions of a layer can be calculated from
its velocity. Thus, we only necessitate $n=256$ states per layer, rendering
the problem computationally feasible.

However, this algorithm is just one part of the jigsaw puzzle. Using ATOM to
compute delivery time for a real plan requires the user to provide a delivery
model of the relevant machine, providing accurate irradiation time per layer
as well as energy layer switching times between layers. However, these kinds
of models already exist for IMPT machines and are starting to be developed for
PAT machines [16, 17].

It is worth noting that a faster algorithm would open up the possibility to
take dynamic beam delivery time into account during the creation of a
treatment plan. For example, algorithms such as ELSA [1] and the MCO-based
algorithm by Wuyckens et al. [5] design plans with short delivery times by
minimizing the number of upward energy switches. Further reduction in the
delivery time may be achievable by tailoring the time estimate to a specific
treatment machine and optimizing it directly accounting for the specific
characteristics and capabilities of the treatment machine in use. However,
this would require thousands, if not tens of thousand, of evaluations of ATOM
which would unfortunately give a run time on the order of an hour. Another
possibility would be to add a penalty/constraint on the BDT during the spot
weight optimization, but this could also significantly increase the running
time of the optimization. We therefore leave the following open research
question: Is it possible to significantly decrease the computational time of
ATOM without sacrificing the fine velocity discretization?

A second open research question concerns the stability of the velocity profile
generated by ATOM. For real plans one never gets exactly the expected delivery
time per layer, or the expected velocity at a point. ATOM does not consider
these uncertainties and might be overly sensitive to them. The second open
question is thus: How sensitive is ATOM generated trajectories to small
perturbations of irradiation time, layer switching time, and velocity? And how
could one adjust ATOM to handle these uncertainties?

##  5 Conclusion

We have developed ATOM, an open-source algorithm for finding velocity profiles
for delivery of ion therapy treatment. ATOM computes the gantry path in under
a second and considers constraints on the magnitude of velocity, acceleration,
and jerk. The gantry velocity profile generated allows for fast delivery of
ion arc plans – in fact, the delivery time is minimized to optimality with
respect to our modelling assumptions. Furthermore, ATOM enables quick
evaluation of delivery times for a variety of candidate plans, giving planners
the option to take delivery time into consideration.

##  6 Acknowledgements

We thank Ivar Bengtsson for discussions on optimization methods, and Sophie
Wuyckens and Erik Engwall for valuable comments on the manuscript.

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Telescopes (MAST). STScI is operated by the Association of Universities for
Research in Astronomy, Inc., under NASA contract NAS5-26555. This work is also
based on observations obtained with the Apache Point Observatory 3.5-meter
telescope, which is owned and operated by the Astrophysical Research
Consortium. Part of our spectroscopic observations were done at the
Observatoire de Haute-Provence. We acknowledge the technical team for their
onsite support as well as the “Programme National de Physique Stellaire”
(PNPS) of CNRS/INSU for their financial support. M.B. acknowledges support
from the “Deutscher Akademischer Austauschdienst” (DAAD) and the Université
Paris Diderot. P.G. was supported by the German space agency (Deutsches
Zentrum für Luft- und Raumfahrt) under PLATO data grant 50OO1501. P.G. and
J.J. acknowledge NASA grant NNX17AF74G for partial support. M.B., R.A.G., and
St.M. have been funded by the PLATO CNES grant. St.M. acknowledges support
from the ERC SPIRE 647383 grant. Sav.M. acknowledges support by the National
Aeronautics and Space Administration under Grant NNX15AF13G, by the National
Science Foundation grant AST-1411685 and the Ramon y Cajal fellowship number
RYC-2015-17697. The research leading to these results has (partially) received
funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (grant agreement N∘670519:
MAMSIE), from the KU Leuven Research Council (grant C16/18/005: PARADISE),
from the Research Foundation Flanders (FWO) under grant agreement G0H5416N
(ERC Runner Up Project), as well as from the BELgian federal Science Policy
Office (BELSPO) through PRODEX grant PLATO. Funding for the Sloan Digital Sky
Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S.
Department of Energy Office of Science, and the Participating Institutions.
SDSS acknowledges support and resources from the Center for High-Performance
Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS
is managed by the Astrophysical Research Consortium for the Participating
Institutions of the SDSS Collaboration including the Brazilian Participation
Group, the Carnegie Institution for Science, Carnegie Mellon University, the
Chilean Participation Group, the French Participation Group, Harvard-
Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The
Johns Hopkins University, Kavli Institute for the Physics and Mathematics of
the Universe (IPMU) / University of Tokyo, the Korean Participation Group,
Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik
Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-
Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für
Extraterrestrische Physik (MPE), National Astronomical Observatories of China,
New Mexico State University, New York University, University of Notre Dame,
Observatório Nacional / MCTI, The Ohio State University, Pennsylvania State
University, Shanghai Astronomical Observatory, United Kingdom Participation
Group, Universidad Nacional Autónoma de México, University of Arizona,
University of Colorado Boulder, University of Oxford, University of
Portsmouth, University of Utah, University of Virginia, University of
Washington, University of Wisconsin, Vanderbilt University, and Yale
University.

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##  Appendix A Power spectra, background fits, and échelle diagrams

|
---|---
|
|
|
Figure 12:  Power spectrum fit of the oscillating stars of the sample. Left: Background fitting. The gray line represents the power spectral density and the black line corresponds to its smoothed version. The sum of the two super-Lorentzian functions and the white noise model is shown in red and the total background, with the Gaussian envelope of the modes, in green. See Sect. 3.1.1 for a description of the method. Right: An échelle diagram. The blue, green, and red lines correspond to the ridges of the radial, dipolar, and quadrupolar modes, respectively. These curves were computed from Eqs. (1), (2), and (3). The dashed horizontal line indicates the value of $\nu_{\mathrm{max}}$. The value of $\Delta\nu$ is reported under the $x$-axis.  |
---|---
|
|
|
Figure 13: Same as Fig. 12. |
---|---
|
|
Figure 14: Same as Fig. 12. |
---|---
|
|
Figure 15:  Power spectrum fit of the non-oscillating stars of the sample. In
each panel, the gray line represents the power spectral density and the black
line corresponds to its smoothed version. A description of the method is given
in Sect. 3.1.1.

##  Appendix B Table with the dynamical masses and radii from the literature

Table 6: Benchmark EBs from the literature. The acronyms F13, G16, B18, and T18 refer to Frandsen et al. (2013), Gaulme et al. (2016a), Brogaard et al. (2018), and Themeßl et al. (2018), respectively. Label | KIC | $M_{\mathrm{dyn}}$ | $R_{\mathrm{dyn}}$ | $\textsl{T}_{\mathrm{eff}}$ | $\log g$ | [Fe/H] | $\nu_{\mathrm{max}}$ | ${\Delta\nu}_{\mathrm{obs}}$ | $S_{\mathrm{ph}}$ | References
---|---|---|---|---|---|---|---|---|---|---
|  | $M_{\odot}$ | $R_{\odot}$ | K | dex | dex |  $\mu$Hz |  $\mu$Hz | % |
$\alpha$ | 5786154 | 1.06(6) | 11.4(2) | 4747 $\pm$ 100 | 2.6(2) | -0.06(6) | 29.75 $\pm$ 0.16 | 3.523 $\pm$ 0.014 | 0.08 | G16
$\beta$ | 7037405 | 1.25(4) | 14.1(2) | 4500 $\pm$ 80 | 2.5(2) | -0.34(1) | 21.75 $\pm$ 0.14 | 2.792 $\pm$ 0.012 | 0.09 | G16, B18
$\gamma$ | 7377422 | 1.05(8) | 9.5(2) | 4938 $\pm$ 110 | 3.1(2) | -0.33(6) | 40.1 $\pm$ 2.1 | 4.643 $\pm$ 0.052 | 1.18 | G16
$\delta$ | 8410637 | 1.56(3) | 10.7(1) | 4699 $\pm$ 91 | 2.7(1) | 0.16(3) | 46.00 $\pm$ 0.19 | 4.641 $\pm$ 0.017 | 0.05 | F13, G16, T18
$\epsilon$ | 8430105 | 1.31(2) | 7.65(5) | 5042 $\pm$ 68 | 3.04(9) | -0.49(4) | 76.70 $\pm$ 0.57 | 7.138 $\pm$ 0.031 | 1.17 | G16
$\zeta$ | 9246715 | 2.149(7) | 8.30(4) | 5030 $\pm$ 45 | 3.0(2) | 0.05(2) | 106.40 $\pm$ 0.80 | 8.310 $\pm$ 0.020 | 0.52 | G16
$\eta$ | 9540226 | 1.33(5) | 12.8(1) | 4680 $\pm$ 80 | 2.2(1) | -0.33(4) | 27.07 $\pm$ 0.15 | 3.216 $\pm$ 0.013 | 0.06 | G16, B18, T18
$\theta$ | 9970396 | 1.14(3) | 8.0(2) | 4860 $\pm$ 80 | 3.1(1) | -0.23(3) | 63.70 $\pm$ 0.16 | 6.320 $\pm$ 0.010 | 0.04 | G16, B18
$\iota$ | 10001167 | 0.81(5) | 12.7(3) | 4700 $\pm$ 66 | 2.6(1) | -0.69(4) | 19.90 $\pm$ 0.09 | 2.762 $\pm$ 0.012 | 0.10 | G16
$\kappa$ | 5640750 | 1.16(1) | 13.12(9) | 4525 $\pm$ 75 | 2.266(6) | -0.29(9) | 24.1 $\pm$ 0.2 | 2.960 $\pm$ 0.006 | 0.07 | T18

##  Appendix C Radial velocities

Table LABEL:tab:radial_velocities contains the radial velocities obtained at
Apache Point and the Haute-Provence Observatories.

##  Appendix D Isochrone fitting of the systems

Figure 16: Isochrone fitting of our SB2 systems (cf. Figure 3) with YaPSI
isochrones. Each panel corresponds to one system. For each panel, the top
subpanel show the giant component (white dot) and the secondary (black dot),
the bottom-right and bottom-left subpanels are zoom-ins of the area around the
primary and secondary components, respectively. The isochrones are computed
for three values of metallicity (see Sect. 3.4).

Figure 17: Same as Fig. 16 for our SB1 systems (cf. Figure 4).

Table 7:  Radial velocity data corresponding to the spectra obtained for the present paper. Dates are mid-exposure times expressed in Kepler Julian dates (KJD), which are barycentric Julian dates BJD with an offset: KJD = BJD $-$ 2,454,833 days. The first spectrum was taken on March 25, 2016 (KJD $=2639.9$), and the last on November 15, 2020 (KJD$=4336.5$). The least significant digit in brackets after the value indicates the statistical uncertainty arising from the position of the peaks in the BF. For the APO data, the dispersion of the data point with respect to the best-fit models indicate that the actual RV uncertainties are actually about 0.5 km s-1 for the giant component and 1 km s-1 for the companion. The statistical error on the BF does not include instrumental effect and atmospheric seeing. The empirical error bars were used to model our data. Dates with a $\star$ symbol correspond with data taken at OHP. The exact timestamps are provided in the electronic table associated to the paper.  Date (KJD) | RV1 ($\mathrm{km.s}^{-1}$) | RV2 ($\mathrm{km.s}^{-1}$)
---|---|---
4054905
2639.9411 | -2.474(28) | 26.00(18)
2670.9044 | -4.058(26) | 26.47(15)
2736.8762 | 4.562(28) | 18.46(14)
2787.6329 | 18.855(29) | 2.62(19)
2817.8751 | 32.108(30) | -12.81(11)
2841.5760 | 38.295(34) | -17.03(12)
2848.6794 | 36.071(27) | -13.26(16)
2850.5840 | 33.567(26) | -13.05(16)
2850.6727 | 35.244(26) | -11.20(17)
3027.9970 | 8.04(52) | 14.68(84)
3178.8424 | -1.292(28) | 23.71(13)
3219.7737 | -4.936(28) | 25.81(11)
3444.9214 | 1.591(27) | 21.73(16)
3446.4615⋆ | 0.8774(34) | 21.2368(34)
3450.4808⋆ | -0.2895(34) | 21.2368(34)
3454.9107 | -0.991(26) | 24.79(16)
3551.6723 | 3.716(27) | 21.50(12)
4360072
2639.9215 | 67.012(23) | …
2670.9419 | 68.593(22) | …
2736.9122 | 68.603(22) | …
2787.6678 | 67.392(25) | …
2841.6178 | 65.674(31) | …
2848.5954 | 62.316(22) | …
2850.6165 | 64.849(23) | …
3028.8859 | 55.935(21) | …
3090.9026 | 53.876(21) | …
3182.6077 | 50.740(22) | …
3203.5560 | 49.593(26) | …
3216.6201 | 49.716(21) | …
3219.7552 | 48.225(22) | …
3398.8290 | 48.800(22) | …
3416.9320 | 50.094(22) | …
3444.8169 | 49.769(21) | …
3568.3692⋆ | 56.7605(34) | …
4473933
2640.8954 | 29.534(56) | …
2670.8507 | 16.835(56) | …
2841.6618 | 25.92(16) | …
2848.7129 | 29.225(57) | …
2868.5606 | 24.067(56) | …
2872.6013 | 21.525(57) | …
3027.8276 | -16.210(54) | …
3060.8789 | 29.585(59) | …
3195.6454 | 8.771(63) | …
3216.6350 | -27.886(55) | …
3217.7197 | -29.736(57) | …
3219.7902 | -34.563(90) | …
3448.6011⋆ | 3.1167(42) | …
3449.4791⋆ | 5.3440(44) | …
3450.3877⋆ | 7.6257(41) | …
3568.4190⋆ | 26.6542(41) | …
4663623
3398.8718 | 14.19(12) | -30.71(47)
3416.9001 | 12.745(58) | -30.86(39)
3444.9399 | 6.997(63) | -26.68(47)
3446.5064⋆ | 7.7468(32) | 1.0176(32)
3454.9435 | 5.804(64) | -24.89(43)
3549.7307 | -12.912(74) | -1.4(2.5)
3557.7885 | -14.781(77) | …
3559.7878 | -14.947(67) | 0.78(46)
3617.5587 | -24.004(56) | 7.21(46)
5193386
2639.9662 | -67.682(70) | -3.67(24)
2850.7793 | -27.684(75) | -50.05(17)
3028.9068 | -85.374(66) | 14.14(17)
3189.7110 | 3.533(66) | -91.43(21)
3203.7668 | -43.01(22) | …
3216.7735 | -65.991(72) | -13.41(19)
3217.5705 | -74.129(71) | -0.52(18)
3219.5748 | -86.034(62) | 16.03(18)
3220.5822 | -86.341(70) | 18.21(19)
3398.9234 | -9.994(63) | -74.73(21)
3416.8781 | -54.906(71) | -19.71(16)
5866138
2639.8914 | -18.388(28) | …
2670.9688 | -17.399(26) | …
2841.7104 | -42.02(12) | …
2848.6626 | -44.002(27) | …
2850.7166 | -42.176(27) | …
3028.9251 | -18.048(26) | …
3189.6639 | -43.789(28) | …
3203.6149 | -30.679(30) | …
3216.5454 | -25.91(24) | …
3217.7478 | -25.621(30) | …
3398.9026 | -18.010(28) | …
3416.9484 | -17.933(27) | …
3447.3921⋆ | -19.0039(43) | …
3448.5436⋆ | -19.0010(36) | …
3454.9293 | -18.582(26) | …
3568.3620⋆ | -23.2137(34) | …
6307537
2639.9993 | -6.294(26) | -27.75(13)
2841.6912 | 25.673(44) | -63.39(24)
2848.7657 | -12.458(47) | -24.09(22)
3028.9415 | -26.591(26) | -5.91(14)
3060.9340 | -43.524(24) | 14.76(13)
3182.6513 | -56.267(26) | 30.67(13)
3189.7300 | -32.230(26) | -0.68(12)
3203.6636 | 4.597(26) | -41.56(11)
3216.6066 | -51.340(25) | 23.69(13)
3217.6647 | -44.899(28) | 16.53(13)
3219.6895 | -29.593(24) | -1.83(12)
3445.4894⋆ | -27.6056(38) | -5.1037(35)
3449.4222⋆ | -53.6294(37) | 26.7354(37)
3568.3835⋆ | -53.5431(34) | 26.5214(34)
6757558
2639.8539 | -16.570(11) | …
2848.6461 | -6.0567(96) | …
2850.6995 | -5.2152(99) | …
3028.9576 | -17.1852(92) | …
3182.6856 | -10.1128(90) | …
3195.6606 | -9.3797(92) | …
3216.6995 | -8.3930(84) | …
3219.7256 | -6.9203(95) | …
3398.8133 | -16.4188(91) | …
3416.8138 | -16.8464(97) | …
3446.5357⋆ | -16.7122(20) | …
7133286
2640.9115 | -62.651(84) | -45.36(49)
2848.7485 | -70.447(72) | -38.01(48)
3027.8437 | -74.521(72) | -35.45(32)
3060.9173 | -40.101(69) | -68.84(33)
3182.7575 | -78.431(69) | -28.90(42)
3189.7462 | -91.840(68) | -15.26(33)
3203.6311 | -22.981(79) | -86.75(37)
3219.5968 | -69.679(69) | -38.18(31)
3220.5595 | -74.694(71) | -32.76(30)
3398.9435 | -16.480(73) | -94.01(34)
3416.8574 | -89.530(72) | -16.39(44)
3445.5692⋆ | -37.7055(36) | -71.1040(36)
7293054
2640.9293 | -21.018(11) | …
2670.9540 | -23.405(11) | …
2736.9583 | -25.230(11) | …
2787.6844 | -27.162(18) | …
2841.6369 | -31.594(15) | …
2848.6102 | -34.627(12) | -13.66(13)
2850.6338 | -32.986(11) | -11.02(14)
3027.8610 | -3.172(11) | -46.650(77)
3090.9186 | -9.315(11) | -39.55(15)
3182.5937 | -14.884(11) | -33.69(12)
3189.6495 | -15.744(11) | -34.209(97)
3195.6016 | -16.260(11) | -33.91(10)
3216.5686 | -17.170(14) | -33.41(17)
3219.7415 | -17.758(12) | -35.07(12)
3444.8321 | -27.831(11) | …
3445.4274⋆ | -27.4945(35) | …
3446.3874⋆ | -27.5233(34) | …
3446.5996⋆ | -27.5093(34) | …
3542.6564 | -34.451(11) | -9.11(12)
3549.7466 | -34.422(12) | -8.49(11)
3557.7730 | -35.512(11) | -7.85(11)
3559.7715 | -35.874(12) | -8.17(11)
3568.3228⋆ | -37.0890(24) | -10.6922(24)
3616.5340 | -44.920(11) | -1.02(14)
3617.5391 | -45.190(11) | 0.79(14)
3843.8042 | -14.308(12) | -34.08(11)
3856.9345 | -15.541(12) | -34.724(98)
4255.6295 | -39.746(12) | -5.69(15)
4264.5895 | -40.650(12) | -6.04(15)
4285.5934 | -43.428(12) | 0.27(15)
4290.5714 | -45.448(12) | -0.04(15)
4291.5834 | -45.622(12) | 1.15(15)
4309.5624 | -50.544(12) | 6.50(15)
4312.5662 | -51.787(12) | 7.54(15)
4319.5615 | -56.589(12) | 11.44(15)
4322.5473 | -56.708(12) | 15.05(15)
4336.5349 | -50.169(12) | 5.53(15)
7768447
2640.9441 | -49.256(17) | …
2817.9294 | -61.837(20) | …
2848.6962 | -76.860(18) | …
2850.7329 | -74.748(17) | …
3027.8761 | -46.108(17) | …
3060.8636 | -58.282(17) | …
3182.7741 | -57.015(16) | …
3203.7440 | -85.316(21) | …
3216.7326 | -76.033(17) | …
3217.7345 | -74.636(19) | …
3220.6397 | -71.364(18) | …
3398.9629 | -45.035(17) | …
3416.9619 | -50.547(17) | …
3448.5590⋆ | -85.0634(30) | …
3450.4275⋆ | -85.0187(34) | …
8435232
2640.9638 | -58.817(50) | 13.53(37)
2848.7298 | -47.126(55) | 2.58(23)
2868.5793 | 9.214(52) | -63.34(34)
2872.6238 | -0.841(56) | -53.87(55)
3027.8946 | -27.263(49) | …
3060.8960 | 6.159(49) | -62.64(65)
3182.6257 | -50.342(49) | 2.79(49)
3189.7674 | -60.643(55) | 15.86(74)
3216.7517 | 5.905(51) | -61.04(25)
3217.6813 | 5.393(51) | -58.07(16)
3219.6181 | -0.033(50) | -53.66(69)
3220.6057 | -3.399(50) | -50.60(33)
3446.5692⋆ | -35.3111(34) | …
9153621
2670.9263 | 47.087(20) | …
2736.8934 | 36.482(21) | …
2787.6519 | 30.816(23) | …
2817.8933 | 26.609(28) | …
2841.5956 | 26.993(31) | …
2848.5805 | 23.190(21) | …
2850.6016 | 25.887(21) | …
3027.9172 | 37.495(20) | …
3060.8244 | 32.605(20) | …
3090.8860 | 30.616(19) | …
3182.7894 | 24.581(20) | …
3189.7854 | 24.062(20) | …
3203.7080 | 26.298(24) | …
3217.5926 | 31.742(23) | …
3219.6417 | 34.107(20) | …
3220.6237 | 34.665(21) | …
3454.8280 | 27.171(20) | 52.1(1.6)
3542.6362 | 74.288(20) | -3.49(91)
3549.7009 | 72.042(21) | -4.92(54)
3450.5511⋆ | 26.9340(34) | 47.5210(33)
3551.6484 | 70.522(20) | -3.52(66)
3557.7332 | 63.727(21) | 4.36(60)
3559.7313 | 61.839(22) | 5.3(1.1)
9904059
2641.0155 | -4.351(89) | …
2817.9114 | -40.027(19) | …
2848.6284 | -6.2598(97) | …
2850.6525 | -3.0982(97) | …
3027.9350 | -38.0051(98) | …
3060.8412 | -5.3891(99) | …
3090.9355 | -20.4216(94) | …
3182.7077 | -14.0052(99) | …
3195.6860 | -21.7066(99) | …
3216.6563 | -33.8285(97) | …
3217.6999 | -34.064(12) | …
3398.9792 | -19.478(11) | …
10015516
2670.8859 | -33.780(28) | …
2736.8589 | -37.545(28) | …
2780.6425 | -24.117(28) | …
2787.6152 | -40.485(28) | …
2817.8599 | -1.359(29) | …
2841.5575 | -3.135(33) | …
2848.5457 | -26.015(29) | …
2850.5685 | -29.245(28) | …
2850.7987 | -29.321(28) | …
2872.7479 | -37.060(30) | …
3027.9546 | 12.687(28) | …
3060.8097 | -45.670(27) | …
3090.8642 | 4.124(28) | …
3445.3837⋆ | 9.4094(41) | …
3445.4556⋆ | 9.3139(37) | …
3446.4008⋆ | 7.5174(32) | …
3449.3888⋆ | 0.4134(32) | …
3568.4058⋆ | 11.6788(31) | …
11235323
2639.8726 | -6.44(12) | -19.22(16)
2872.6573 | 32.503(68) | -59.55(11)
3028.9765 | 37.677(60) | -62.053(90)
3182.6681 | 11.894(58) | -33.987(92)
3189.6800 | 8.907(58) | -32.181(97)
3195.6233 | -60.756(59) | 38.890(84)
3203.6797 | 26.840(58) | -51.318(98)
3216.5852 | -56.590(59) | 36.538(90)
3217.6130 | -49.089(64) | 27.99(11)
3219.7063 | -23.940(62) | -0.573(85)
Table 7: Continued.