# A quantitative assessment of Geant4 for predicting the yield and
distribution of positron-emitting fragments in ion beam therapy

Andrew Chacon1, Harley Rutherford1,2, Akram Hamato3, Munetaka Nitta3, Fumihiko
Nishikido3, Yuma Iwao3, Hideaki Tashima3, Eiji Yoshida3, Go Akamatsu3, Sodai
Takyu3, Han Gyu Kang3, Daniel R. Franklin4, Katia Parodi5, Taiga Yamaya3,
Anatoly Rosenfeld2,6, Susanna Guatelli2,6, Mitra Safavi-Naeini1,2,7 1
Australian Nuclear Science and Technology Organisation (ANSTO), NSW, Australia
2 Centre for Medical Radiation Physics, University of Wollongong, Wollongong
NSW 2522 Australia  3 National Institutes for Quantum Science and Technology,
Chiba, Japan  4 School of Electrical and Data Engineering, University of
Technology Sydney, Australia  5 Ludwig-Maximilians-Universität München,
Department of Medical Physics, Faculty of Physics, Garching b. Munich, Germany
6 Illawarra Health and Medical Research Institute, University of Wollongong,
Wollongong NSW 2522 Australia  7 Brain and Mind Centre, University of Sydney,
Sydney, NSW, Australia<EMAIL_ADDRESS>
###### Abstract

Purpose: To compare the accuracy with which different hadronic inelastic
physics models across ten Geant4 Monte Carlo simulation toolkit versions can
predict positron-emitting fragments produced along the beam path during carbon
and oxygen ion therapy.

Materials and Methods: Phantoms of polyethylene, gelatin or poly(methyl
methacrylate) were irradiated with monoenergetic carbon and oxygen ion beams.
Post-irradiation, 4D PET images were acquired and parent 11C, 10C and 15O
radionuclides contributions in each voxel were determined from the extracted
time activity curves. Next, the experimental configurations were simulated in
Geant4 Monte Carlo versions 10.0 to 11.1, with three different fragmentation
models - binary ion cascade (BIC), quantum molecular dynamics (QMD) and the
Liege intranuclear cascade (INCL++) - 30 model-version combinations. Total
positron annihilation and parent isotope production yields predicted by each
simulation were compared between simulations and experiments using normalised
mean squared error and Pearson cross-correlation coefficient. Finally, we
compared the depth of maximum positron annihilation yield and the distal point
at which positron yield decreases to 50% of peak between each model and the
experimental results.

Results: Performance varied considerably across versions and models, with no
one version/model combination providing the best prediction of all positron-
emitting fragments in all evaluated target materials and irradiation
conditions. BIC in Geant4 10.2 provided the best overall agreement with
experimental results in the largest number of test cases. QMD consistently
provided the best estimates of both the depth of peak positron yield (10.4 and
10.6) and the distal 50%-of-peak point (10.2), while BIC also performed well
and INCL generally performed the worst across most Geant4 versions.

Conclusions: Best spatial prediction of annihilation yield and positron-
emitting fragment production during carbon and oxygen ion therapy was found to
be 10.2.p03 with BIC or QMD. These version/model combinations are recommended
for future heavy ion therapy research.

##  1 Introduction

One of the chief advantages of particle therapy as a treatment for cancer is
the high dose gradient between the treatment area and surrounding regions [1].
However, to enable the smallest possible margins, particle range must be
accurately verified relative to the patient anatomy. Many authors have
investigated the post-irradiation imaging of positron-emitting fragments via
positron emission tomography (PET) for range verification of the primary beam
during particle therapy [2, 3, 4, 5, 6, 7, 8, 9], with several prototype in-
beam PET scanners currently under active development [10, 11, 12, 13]. Monte
Carlo simulation methods, implemented using platforms such as Geant4, are now
widely used in particle therapy - applied clinically as a secondary quality
assurance (QA) technique to verify dose during particle therapy [14, 15, 16]
and there is active research in using it for developing and testing range
models [17, 18, 19], PET image-to-dose models [20, 21, 22, 23], and designing
novel PET scanners optimised for range and dose verification [24, 25].

Good models for nuclear fragmentation processes are essential for faithfully
simulating imaging applications in particle therapy, such as PET-based dose
estimation methods for quality assurance, since the production and
distribution of positron-emitting radionuclide fragments directly affects the
quality of the resulting image [20, 21, 22, 26]. Currently, the Geant4 Monte
Carlo simulation framework offers a choice of three hadronic inelastic
fragmentation models that are appropriate for particle therapy - binary ion
cascade (BIC), quantum molecular dynamics (QMD), and Liège intranuclear
cascade (INCL++) [14, 27, 28]111INCL++ is considered the most appropriate
option for neutron spallation simulations, but is included here for
completeness [29].. In our previous study, we evaluated these models by
comparing the spatial distributions of positron-emitting radionuclides
predicted following irradiation of PMMA, gelatin and polyethylene targets by
monoenergetic carbon and oxygen ion beams (simulated using Geant4 10.2.p03) to
equivalent results estimated from experimentally-obtained PET data [30]. The
BIC model provided the best estimates overall; however, none of the models
provided a perfect fit in all evaluated cases, and some significant
discrepancies were observed. Other authors have noted similar variations in
simulation fidelity in different contexts; most recently, Wrońska et al. [31]
evaluated two models, BIC and QBBC, with several versions of Geant4 for
prediction of prompt gamma emission during proton therapy with a PMMA target.
They concluded that Geant4 10.4p02’s BIC implementation with high-precision
neutron data libraries and binary ion fragmentation models provided the best
agreement with the observed experimental data for this use case, with the
simulations performed using other version/model combinations producing
features which were absent in the experimental data. Pei et al. [7] evaluated
the ability of Geant4 to predict the range of particles using Geant4 version
10.6 and the Bertini fragmentation model. They concluded that Geant4 was able
to accurately predict within 1 mm the positron activity peak and distal
falling edge using two carbon monoenergetic beams and a single PMMA phantom.

Since the publication of our previous study, there have been several updates
to Geant4; specifically, six minor releases (versions 10.x) and one major
release (version 11, which has since been updated to version 11.1). Each of
these releases includes modifications to the physics models implemented in
Geant4, which can affect the simulation of positron-emitting fragment
production in particle therapy.

In this work, we have extended the previous study, and present a quantitative
evaluation of Geant4’s ability to predict positron-emitting fragment
production across a total of ten different stable versions (10.0.p04,
10.1.p03, 10.2.p03, 10.3.p03, 10.4.p03, 10.5.p01, 10.6.p03, 10.7.p02, 11.0 and
11.1) which have followed the previous major release (10.0) for each of the
three different fragmentation models [30].

In addition to the normalised mean squared error (NMSE) metric used in the
previous study, three additional metrics - the Pearson cross-correlation
coefficient (CCC), the depth of the positron annihilation peak, and the depth
at which positron annihilation intensity has decreased to 50% of peak are also
used to compare the shape of the predicted positron-emitting fragment
distribution to the experimentally-measured distribution. The experimental
absolute yields of the dominant positron-emitting fragment species (10C, 11C
and 15O) created during irradiation of gelatin, PMMA and polyethylene block
phantoms with beams of 12C ions with energies of 148.5, 290.5 and 350 MeV/u
and 16O ions with energies of 148 and 290 MeV/u were obtained across the full
width at tenth maximum (FWTM) of the beam from a previous study conducted
using the OpenPET scanner at HIMAC, Japan [12, 30]. Akamatsu2019 results in
each of the three target materials and 5 ion/energy combinations were then
compared to those predicted in equivalent simulations performed in Geant4
using each of hadronic fragmentation models (BIC, QMD and INCL++) across the
ten evaluated Geant4 versions for a total of 150 unique
target/ion/energy/version/model test conditions. The total positron yields and
yields of the individual positron-emitting fragment species from each model
and Geant4 version were then compared to experimental measurements using each
of the chosen metrics at the entrance, build-up and Bragg peak, and tail
regions.

##  2 Materials and Methods

This section describes the methods used for obtaining and quantitatively
comparing the experimental and simulated positron annihilation profiles. The
general approach is similar to that used in our previous study (see [30]);
however, it has been extended to include a much wider range of Geant4
versions, and additional comparison metrics are introduced.

The experimental methods used to estimate the total positron annihilation
profile and activity of the dominant positron-emitting fragment isotopes (11C,
10C and 15O) are briefly summarised in Section 2.1. Equivalent simulation
configurations were constructed for each Geant4 version under test, and the
total positron annihilation profile and activity of 11C, 10C and 15O were
predicted for each beam ion/energy, target material, hadronic inelastic
fragmentation model and Geant4 version; the design and parameters of these
simulations are described in detail in Section 2.2.

The simulated annihilation profiles are compared with the experimental
annihilation profiles using the following metrics in each of the entrance,
build-up and Bragg peak and tail regions:

  * •

Normalised mean squared error (NMSE); and

  * •

Pearson cross-correlation coefficient (CCC)

Additionally, the depth of positron annihilation peak and the depth of the
distal point at which the magnitude of the positron annihilation profile
decreases to 50% of the peak value are evaluated. All metrics are described in
detail in Section 2.3.

###  2.1 Experimental configuration

The experimental data obtained in our 2019 paper were used as the ground truth
for this simulation study; a detailed description of the experimental
procedures is presented in that paper [30]. In summary, phantoms constructed
from either pure PMMA, polyethylene or gelatin (encased in a thin-walled PMMA
container), each with dimensions of 100 mm$\times$100 mm$\times$300 mm, were
irradiated with monoenergetic carbon or oxygen ion beams of various energies -
three for carbon ions and two for oxygen (see Table 1). Positron annihilation
profiles (with respect to depth in the target) were obtained using the whole-
body DOI-PET scanner prototype developed at QST [12]. These profiles were
decomposed into the individual population of each of the dominant parent
positron-emitting fragments (11C, 10C and 15O) at $t=0$ (end of irradiation
period) by fitting the observed time-decay curves in each voxel to a
multiexponential decay model.

Table 1: Beam parameters for each ion species and energy. The energy spread is 0.2 % of nominal energy in each case; 95% confidence intervals are given for beam flux. Ion |  Energy (MeV/u) |  $\sigma_{x}$ (mm) |  $\sigma_{y}$ (mm) |  Beam flux (pps)
---|---|---|---|---
12C |  148.5 |  2.77 |  2.67 |  1.8$\times$10${}^{9}\pm 3.8\times$107
12C |  290.5 |  3.08 |  4.70 |  1.8$\times$10${}^{9}\pm 6.4\times$107
12C |  350 |  2.50 |  2.98 |  1.8$\times$10${}^{9}\pm 4.6\times$107
16O |  148 |  2.79 |  2.89 |  1.1$\times$10${}^{9}\pm 2.8\times$107
16O |  290 |  2.60 |  4.90 |  1.1$\times$10${}^{9}\pm 7.0\times$107

###  2.2 Simulation parameters

The same beam parameters, phantom compositions and geometries used in the
experimental measurements were modelled in each version of Geant4. Apart from
minor modifications to the simulation source code required due to version-to-
version changes in certain Geant4 application programming interfaces (APIs),
the code was identical across versions. Simulations were performed using each
of the 10 most recent stable releases of Geant4: 10.0.p04, 10.1.p03, 10.2.p03,
10.3.p03, 10.4.p03, 10.5.p01, 10.6.p03, 10.7.p02, 11.0 and 11.1. For brevity,
the patch number will be dropped when referring to the version of Geant4.

For each version of Geant4, three alternative hadronic ion fragmentation
models were evaluated - Binary Ion Cascade (BIC), Quantum Molecular Dynamics
(QMD) and Liège Intranuclear Cascade (INCL) models222The INCL model was
developed specifically for spallation reactions but is included in this study
as it can also model fragmentation. [27, 28]. All simulations modelled
electromagnetic interactions using the standard option 3 list
(G4EmStandardPhysics_option3). The remaining physics processes, including
hadronic physics models, are listed in Table 10.

The location of each positron annihilation, as well as the identity of the
parent isotope which decayed to emit each positron (principally 11C, 10C and
15O), were scored with a resolution of 1.5 mm3 to match the voxel dimensions
of the experimental OpenPET image reconstruction output. The pristine positron
annihilation profiles were convolved with a 2.3 mm FWHM Gaussian filter to
simulate the point spread function of the PET system.

###  2.3 Evaluation methods

The irradiated target was divided into three separate regions for analysis
since different physics processes dominate in each: the entrance region, the
build-up and Bragg peak region, and the tail region. This segmentation is
defined in the same way as in our previous paper [30]; in summary, the central
build-up and Bragg peak region is defined as follows:

  * •

The proximal edge in the $z$ dimension (along the path of the beam) is defined
as the first point at which the dose deposited along the central axis exceeds
the entrance plateau dose by more than 5% of the difference between peak dose
and the entrance plateau dose; and

  * •

The distal edge in $z$ is defined as the last point at which the deposited
dose is greater than 5% of the absolute peak dose value.

The entrance region is then defined as the region proximal to the build-up and
Bragg peak region, while the tail region is defined as the region distal to
the build-up and Bragg peak region.

The yields of positron-emitting nuclei are defined via (1):

$\mathrm{Yield~{}(Isotope)}=\frac{N~{}(\mathrm{Isotope})}{N~{}\mathrm{(Primary)}}$
(1)

where $N~{}\mathrm{(Isotope)}$ is the yield of the isotope under study in that
region and $N~{}\mathrm{(Primary)}$ is the total number of primary particles.
Yields were calculated in each voxel along the path of the beam.

Three different metrics were chosen to quantify the accuracy of each model in
Geant4: the normalised mean squared error (NMSE), the Pearson cross-
correlation coefficient (CCC), and the range (depth along the path of the
beam) of both the position annihilation peak and the point beyond the peak at
which positron annihilation intensity decreases by 50%. The NMSE was used to
measure the absolute difference between the experimental measurements and
simulation-predicted positron yields in each region:

$NMSE=\frac{\displaystyle\sum_{i=1}^{N_{reg}}|S_{i}-E_{i}|^{2}}{\displaystyle\sum_{i=1}^{N_{reg}}|E_{i}|^{2}}$
(2)

where $S_{i}$ and $E_{i}$ are the simulation and experimental yields in the
$i$th voxel of the $N_{reg}$ voxels in region $reg$ (with a lower value
indicating a better match).

The Pearson CCC was used to compare the shapes of the simulation-predicted
positron-emitting fragment distributions to the experimental measurements:

$CCC=\frac{\sum_{i=1}^{N_{reg}}(S_{norm,i}-\overline{S_{norm}})(E_{i}-\overline{E_{norm}})}{\sqrt{(\sum_{i=1}^{N_{reg}}(S_{norm,i}-\overline{S_{norm}})^{2})(\sum_{i=1}^{N_{reg}}(E_{norm,i}-\overline{E_{norm}})^{2})}}$
(3)

where $S_{norm,i}$ and $E_{norm,i}$ are the normalised simulation and
experimental yields in the $i$th voxel of the $N_{reg}$ voxels in region
$reg$. Normalisation is performed by dividing each $S_{i}$ and $E_{i}$ by the
maximum value in its respective region. $\overline{S_{norm}}$ and
$\overline{E_{norm}}$ are the mean values in each region.

When comparing the models, the closer the CCC between the model prediction and
the experimental estimate of positron-emitting fragment distribution is to +1,
the more accurate the prediction. A CCC greater than 0.95 is considered to
indicate that the simulation has accurately predicted the shape of the yield.

For each version of Geant4, phantom, beam type and energy, the NMSE and CCC
were calculated for both total annihilation photon yield profiles and also for
the profiles of the three main positron-emitting fragment species (10C, 11C
and 15O). The calculation was repeated for each of the $N_{reg}$ regions
(entrance, build-up and Bragg peak, and tail regions). The NMSE and the CCC
were then compared across all evaluated Geant4 versions for each region,
phantom material and beam type.

A total of 5 energy/ion combinations are evaluated (carbon ions at three
energies and oxygen ions at two energies). For oxygen ions, three target
materials (gelatine, PMMA and polyethylene) are evaluated for total positron
annihilation yield and 11C/10C/15O yield. For carbon ions, the same three
target materials are evaluated for total positron annihilation yield and
11C/10C yield and two for 15O yield (polyethylene is omitted since it is not
possible to produce ${}^{15}O$ fragments with a ${}^{12}C$ ion beam and a PE
target which only contains carbon and hydrogen). Thus, there are a total of 15
cases evaluated for total positron annihilation, 11C yield and 10C yield,
while there are 12 cases evaluated for 15O yield.

For range calculations, the difference between the depths at which the
positron annihilation yield reached its maximum value in the experiment and
simulation was calculated (see (4)). Additionally, the point distal to this
maximum at which positron annihilation yield decreases to 50% of the maximum
value was also compared between experiment and simulation. For each version
and model, the mean differences between the experimental and simulation-based
values, as well as the standard deviations and maximum differences were
calculated across all test cases (ion species, energies and target materials).

$\delta_{voxel}=R_{simulation}-R_{experiment}$ (4)

where $R_{x}$ is the range (depth) of the voxel with the maximum value (or,
for distal 50%-of-peak, the first distal voxel to fall below 50% of the
maximum value) in either the simulation or experiment.

##  3 Results and Discussion

The number of cases in which each version/model combination performed the best
or equal-best in terms of each of the evaluated metrics are counted across all
simulations in the entrance, build-up and Bragg peak and tail regions, and
summarised in this section. Detailed results for each experiment are included
in the Supplementary Materials.

For the NMSE metric, equal best is defined by firstly identifying the
version/model with the best mean NMSE, and then accepting any additional
version/model combination whose mean NMSE is within the confidence interval of
the best-performing version/model as being equal. All confidence intervals are
$\pm 2\sigma$ (i.e. 95% assuming a Gaussian distribution).

###  3.1 Entrance region

In the entrance region, positron-emitting fragments are created by target
fragmentation rather than projectile fragmentation. The projectile ions lose
energy via Coulomb interactions, slowing down at an approximately constant
rate as they traverse this region, with only gradual changes to
projectile/target cross sections. As a result, the positron-emitting fragment
distributions are expected to exhibit an approximately flat depthwise profile
in this region.

NMSE and Pearson CCC results between simulation and experimental total
positron annihilation profiles in the entrance region are summarised in Tables
2 and 3, respectively, with corresponding figures shown in Supplementary
Material Section 1.

Table 2: Number of test cases for which each Geant4 version/model combination achieved the lowest or equal-lowest NMSE in the entrance region. Bold text denotes the version/model achieving the highest (or equal-highest) number of best results for each combination of ion/energy/target. Version | Total |  |  11C |  |  10C |  |  15O
---|---|---|---|---|---|---|---
BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL
10 |  6 |  0 |  0 |  |  11 |  3 |  2 |  |  0 |  0 |  2 |  |  6 |  2 |  0
10.1 |  6 |  0 |  0 |  |  11 |  3 |  2 |  |  0 |  0 |  3 |  |  6 |  2 |  0
10.2 |  5 |  0 |  0 |  |  11 |  3 |  2 |  |  0 |  0 |  3 |  |  5 |  2 |  0
10.3 |  6 |  0 |  0 |  |  6 |  0 |  0 |  |  5 |  3 |  6 |  |  4 |  2 |  0
10.4 |  6 |  0 |  0 |  |  1 |  0 |  0 |  |  2 |  3 |  6 |  |  9 |  2 |  0
10.5 |  2 |  0 |  0 |  |  0 |  0 |  0 |  |  2 |  5 |  8 |  |  3 |  1 |  0
10.6 |  4 |  0 |  0 |  |  1 |  0 |  0 |  |  0 |  0 |  5 |  |  5 |  4 |  0
10.7 |  3 |  0 |  0 |  |  1 |  0 |  0 |  |  0 |  0 |  5 |  |  5 |  2 |  0
11 |  4 |  0 |  0 |  |  1 |  0 |  0 |  |  0 |  0 |  5 |  |  5 |  2 |  0
11.1 |  4 |  0 |  0 |  |  0 |  0 |  0 |  |  0 |  0 |  5 |  |  5 |  2 |  0

For the entrance region, the BIC model implemented in Geant4 versions 10,
10.1, 10.3 and 10.4 provided the (equal) lowest NMSE of the yields of total
positron annihilation in 5 out of 15 cases. The BIC model in Geant4 10, 10.1
and 10.2 also provided the (equal) lowest NMSE for 11C fragment production
(11/15 cases), whereas for 10C the best version/model combination was
10.5/INCL (8/15 cases) and for 15O it was 10.6/BIC (9/12 cases).

Table 3: Number of test cases for which each Geant4 version/model combination achieved a CCC greater than 0.95 in the entrance region. Bold text denotes the version/model achieving the highest number of best results for each combination of ion/energy/target. Version | Total |  |  11C |  |  10C |  |  15O
---|---|---|---|---|---|---|---
BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL
10 |  2 |  0 |  1 |  |  4 |  0 |  0 |  |  2 |  2 |  0 |  |  3 |  0 |  3
10.1 |  1 |  0 |  1 |  |  4 |  0 |  0 |  |  2 |  2 |  0 |  |  3 |  0 |  3
10.2 |  2 |  0 |  1 |  |  3 |  0 |  0 |  |  2 |  2 |  0 |  |  3 |  0 |  3
10.3 |  2 |  0 |  3 |  |  3 |  1 |  3 |  |  2 |  2 |  0 |  |  3 |  0 |  2
10.4 |  2 |  0 |  3 |  |  4 |  1 |  3 |  |  1 |  2 |  0 |  |  3 |  0 |  2
10.5 |  3 |  0 |  1 |  |  3 |  2 |  4 |  |  1 |  2 |  0 |  |  3 |  0 |  3
10.6 |  3 |  0 |  1 |  |  2 |  1 |  2 |  |  1 |  2 |  0 |  |  3 |  0 |  2
10.7 |  3 |  0 |  1 |  |  3 |  1 |  3 |  |  1 |  2 |  0 |  |  3 |  0 |  2
11 |  3 |  0 |  1 |  |  2 |  1 |  3 |  |  1 |  2 |  0 |  |  3 |  0 |  2
11.1 |  1 |  0 |  1 |  |  2 |  1 |  2 |  |  1 |  2 |  0 |  |  3 |  0 |  2

Geant4 versions 10.5-11 with BIC and 10.3/10.4 with INCL each achieved a
Pearson CCC greater than 0.95 (3/15 cases) for total positron yield; QMD did
not reach the threshold for any test case in any version of Geant4.

Results for individual radionuclides were also mixed, with 10/BIC, 10.1/BIC,
10.4/BIC and 10.5/INCL achieving the threshold in 4/15 cases for 11C,
10-10.4/BIC and all versions with QMD reaching the threshold in 2/15 cases for
10C, and all versions with BIC and 10/INCL, 10.1/INCL, 10.2/INCL, and
10.5/INCL reaching the threshold for 15O.

###  3.2 Build-up and Bragg peak region

In the build-up and Bragg peak region, positron-emitting fragments are
produced via a combination of target fragmentation and projectile
fragmentation. There is a rapid change in positron-emitting fragment yield
with respect to depth, especially since different positron-emitting fragments
stop at different distances from their point of production.

NMSE and Pearson CCC results between simulation and experimental total
positron annihilation profiles in the build-up and Bragg peak region are
summarised in Tables 4 and 5, respectively, with corresponding figures shown
in Supplementary Material Section 2.

Table 4: Number of test cases for which each Geant4 version/model combination achieved the lowest or equal-lowest NMSE in the build-up and Bragg peak region. Bold text denotes the version/model achieving the highest (or equal-highest) number of best results for each combination of ion/energy/target. Version | Total |  |  11C |  |  10C |  |  15O
---|---|---|---|---|---|---|---
BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL
10 |  4 |  0 |  0 |  |  11 |  6 |  3 |  |  0 |  0 |  2 |  |  3 |  2 |  0
10.1 |  5 |  0 |  0 |  |  11 |  6 |  3 |  |  0 |  0 |  2 |  |  3 |  2 |  0
10.2 |  11 |  1 |  0 |  |  14 |  6 |  3 |  |  0 |  0 |  1 |  |  5 |  1 |  0
10.3 |  1 |  0 |  0 |  |  5 |  0 |  0 |  |  4 |  1 |  2 |  |  3 |  0 |  0
10.4 |  1 |  0 |  0 |  |  1 |  0 |  0 |  |  3 |  2 |  2 |  |  4 |  0 |  0
10.5 |  0 |  0 |  0 |  |  0 |  0 |  0 |  |  3 |  4 |  7 |  |  1 |  2 |  0
10.6 |  1 |  0 |  0 |  |  0 |  0 |  0 |  |  3 |  0 |  9 |  |  2 |  1 |  2
10.7 |  1 |  0 |  0 |  |  0 |  0 |  0 |  |  3 |  2 |  7 |  |  0 |  0 |  1
11 |  2 |  0 |  0 |  |  0 |  0 |  0 |  |  3 |  2 |  9 |  |  2 |  0 |  2
11.1 |  1 |  0 |  0 |  |  0 |  0 |  0 |  |  3 |  2 |  9 |  |  2 |  0 |  2

In the build-up and Bragg peak region, according to the NMSE metric, total
positron yield is most accurately predicted by the BIC model in Geant4 version
10.2, being (equal) best in 11/15 cases. This is much higher than the next-
best combinations (10.1/BIC with 5/11 cases followed by 10/BIC with 4/11).
Similar results are observed for 11C yield, with 10.2/BIC achieving (equal)
best performance in 14/15 cases, and 10/BIC and 10.1/BIC each achieving
(equal) best results in 11/15 cases; QMD also performs reasonably well in this
case with 10, 10.1 and 10.2 achieving wins in 6/15 cases. For 10C, 10.6/INCL,
11/INCL and 11.1/INCL are the best performers (each winning in 9/15 cases).
Finally, for 15O, 10.2/BIC is the best-performing model with 5/12 wins,
followed by 10.4/BIC with 4 wins.

Table 5: Number of test cases for which each Geant4 version/model combination achieved a CCC greater than 0.95 in the build-up and Bragg peak region. Bold text denotes the version/model achieving the highest number of best results for each combination of ion/energy/target. Version | Total |  |  11C |  |  10C |  |  15O
---|---|---|---|---|---|---|---
BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL
10 |  9 |  4 |  3 |  |  3 |  3 |  2 |  |  4 |  3 |  6 |  |  3 |  2 |  3
10.1 |  9 |  4 |  4 |  |  3 |  3 |  3 |  |  4 |  3 |  6 |  |  3 |  2 |  3
10.2 |  10 |  8 |  4 |  |  6 |  6 |  3 |  |  5 |  4 |  5 |  |  3 |  3 |  3
10.3 |  9 |  6 |  6 |  |  6 |  5 |  3 |  |  2 |  2 |  2 |  |  4 |  3 |  3
10.4 |  6 |  8 |  7 |  |  6 |  5 |  3 |  |  1 |  1 |  3 |  |  4 |  3 |  3
10.5 |  8 |  8 |  6 |  |  6 |  5 |  4 |  |  2 |  2 |  3 |  |  3 |  3 |  4
10.6 |  9 |  9 |  6 |  |  6 |  7 |  6 |  |  2 |  1 |  4 |  |  4 |  3 |  4
10.7 |  6 |  6 |  4 |  |  3 |  5 |  2 |  |  2 |  1 |  4 |  |  3 |  2 |  3
11 |  8 |  8 |  6 |  |  4 |  5 |  4 |  |  3 |  2 |  4 |  |  3 |  3 |  3
11.1 |  6 |  5 |  6 |  |  2 |  3 |  4 |  |  3 |  2 |  4 |  |  3 |  3 |  3

Using the Pearson CCC metric, the best-performing version/model combinations
for overall positron yield are 10.2/BIC (10/15 cases), followed by 10/BIC,
10.1/BIC, 10.3/BIC, 10.6/BIC and 10.6/QMD (9/15 cases). Generally, BIC
performed very well, with all Geant4 versions achieving (equal) best
performance in at least 6 cases. 11C yield was best predicted by 10.6/QMD
(7/15 cases) however many version/model combinations did well here also, with
10.2/BIC, 10.2/QMD, 10.3/BIC, 10.4/BIC, 10.5/BIC, 10.6/BIC and 10.6/INCL all
achieving 6/15 wins. 10C yield was best predicted by 10/INCL and 10.1 INCL
(6/15 cases), closely followed by 10.2/BIC and 10.2/INCL which won in 5/15
cases. The best-performing version/model combinations for 15O yield were
10.3/BIC, 10.4/BIC, 10.5/INCL, 10.6/BIC and 10.6/INCL with 4/15 wins each, and
all other version/model combinations achieving 2 or 3 wins.

Table 6 lists difference between the experimental and simulation positron
peak, while Table 7 lists the difference between the 50% fall off point for
the experimental and simulated positron peak.

Table 6: Differences between the depths of the maximum positron annihilation yield in experimental and simulation results. Each voxel has a width of 1.5 mm; the maximum error is always in multiples of 1.5 mm increments. Version | BIC |  | QMD |  | INCL
---|---|---|---|---|---
$\mu$ (mm) |  $\sigma$ (mm) |  max (mm) |  |  $\mu$ (mm) |  $\sigma$ (mm) |  max (mm) |  |  $\mu$ (mm) |  $\sigma$ (mm) |  max (mm)
10 |  1 |  1.85 |  6 |  |  -0.20 |  1.69 |  3 |  |  1.10 |  3.82 |  10.50
10.1 |  1 |  1.85 |  6 |  |  -0.20 |  1.69 |  3 |  |  1 |  3.91 |  10.50
10.2 |  0.60 |  1.37 |  3 |  |  -0.60 |  1.58 |  -3 |  |  0.60 |  3.39 |  9
10.3 |  1.30 |  1.78 |  6 |  |  -0.30 |  1.41 |  -3 |  |  2.30 |  4.12 |  9
10.4 |  1.60 |  1.55 |  6 |  |  -0.10 |  1.44 |  -3 |  |  4 |  4.45 |  10.50
10.5 |  0.69 |  0.99 |  3 |  |  0.60 |  2.03 |  6 |  |  2.20 |  4.24 |  9
10.6 |  0.60 |  1.37 |  3 |  |  -0.10 |  1.55 |  3 |  |  0.10 |  2.50 |  7.50
10.7 |  1.20 |  1.72 |  4.50 |  |  0.60 |  1.95 |  4.50 |  |  0.70 |  3.40 |  10.50
11 |  1.10 |  1.65 |  4.50 |  |  0.60 |  1.95 |  4.50 |  |  0.60 |  3.09 |  9
11.1 |  0.30 |  1.62 |  3 |  |  -0.30 |  1.62 |  3 |  |  0 |  2.78 |  7.50
Table 7: Differences between the distal depths at which the positron annihilation yield has decreased to 50% of the peak value in experimental and simulation results. Each voxel has a width of 1.5 mm; the maximum error is always in multiples of 1.5 mm increments. Version | BIC |  | QMD |  | INCL
---|---|---|---|---|---
$\mu$ (mm) |  $\sigma$ (mm) |  max (mm) |  |  $\mu$ (mm) |  $\sigma$ (mm) |  max (mm) |  |  $\mu$ (mm) |  $\sigma$ (mm) |  max (mm)
10 |  0.70 |  1.49 |  3 |  |  0.30 |  1.41 |  3 |  |  -0.20 |  1.49 |  -3
10.1 |  0.70 |  1.49 |  3 |  |  0.30 |  1.41 |  3 |  |  -0.20 |  1.49 |  -3
10.2 |  0.30 |  1.01 |  1.50 |  |  0 |  1.13 |  1.50 |  |  -0.50 |  1.22 |  -3
10.3 |  0.50 |  1.09 |  3 |  |  0.20 |  1.11 |  1.50 |  |  -0.20 |  1.37 |  -3
10.4 |  0.60 |  1.11 |  3 |  |  0.30 |  1.01 |  1.50 |  |  0.10 |  1.20 |  -3
10.5 |  0.35 |  0.90 |  1.50 |  |  0.20 |  1.11 |  1.50 |  |  -0.50 |  1.22 |  -3
10.6 |  0.40 |  0.89 |  1.50 |  |  0.20 |  1.11 |  1.50 |  |  -0.40 |  1.33 |  -3
10.7 |  1 |  1.46 |  3 |  |  0.70 |  1.59 |  3 |  |  -0.10 |  1.65 |  3
11 |  1 |  1.46 |  3 |  |  0.70 |  1.59 |  3 |  |  0 |  1.60 |  3
11.1 |  0 |  1.60 |  -3 |  |  -0.10 |  1.44 |  -3 |  |  -0.60 |  1.68 |  -3

The smallest differences between experimental and simulation-based depth of
maximum positron annihilation were obtained with Geant4 10.4/QMD ($\mu=-0.1$
mm; max = -3 mm) and 10.6/QMD ($\mu=-0.1$ mm; max = +3 mm). While a smaller
mean value was obtained with 11.1/INCL, the maximum value and standard
deviation were much larger (+7.5 mm and 2.78 mm) compared to 10.4/QMD and
10.6/QMD. Differences in the depth of the distal 50%-of-peak point were much
smaller; the best estimates were obtained with 10.2/QMD ($\mu=0$ mm; max =
+1.5 mm), 11.1/BIC ($\mu=0$ mm; max = -3 mm) and 11/INCL ($\mu=0$ mm; max = +3
mm).

###  3.3 Tail region

In the tail region, positron-emitting radionuclides are primarily produced
through fragmentation of the target material caused by light fragments created
upstream from the primary beam. As such, the production of positron-emitting
fragments in the tail region is highly dependent on fragmentation and
scattering cross sections upstream. Therefore, the yield of positron
annihilation is not expected to rapidly change across this region compared to
the build-up and Bragg peak region.

NMSE and Pearson CCC results between simulation and experimental total
positron annihilation profiles in the tail region are summarised in Tables 8
and 9, respectively, with corresponding figures shown in Supplementary
Material Section 3.

Table 8: Number of test cases for which each Geant4 version/model combination achieved the lowest or equal-lowest NMSE in the tail region. Bold text denotes the version/model achieving the highest (or equal-highest) number of best results for each combination of ion/energy/target. Version | Total |  |  11C |  |  10C |  |  15O
---|---|---|---|---|---|---|---
BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL
10 |  3 |  2 |  2 |  |  5 |  9 |  4 |  |  2 |  2 |  1 |  |  4 |  4 |  4
10.1 |  4 |  2 |  2 |  |  6 |  9 |  4 |  |  1 |  1 |  2 |  |  4 |  4 |  4
10.2 |  12 |  10 |  2 |  |  11 |  12 |  4 |  |  1 |  1 |  3 |  |  7 |  7 |  4
10.3 |  1 |  1 |  1 |  |  1 |  1 |  1 |  |  2 |  4 |  0 |  |  3 |  4 |  4
10.4 |  1 |  1 |  1 |  |  1 |  1 |  1 |  |  2 |  3 |  0 |  |  3 |  4 |  3
10.5 |  1 |  1 |  1 |  |  1 |  1 |  1 |  |  3 |  4 |  2 |  |  2 |  2 |  2
10.6 |  1 |  2 |  1 |  |  1 |  1 |  1 |  |  5 |  3 |  6 |  |  4 |  10 |  4
10.7 |  1 |  1 |  1 |  |  1 |  1 |  1 |  |  3 |  3 |  5 |  |  3 |  4 |  3
11 |  1 |  1 |  1 |  |  1 |  1 |  1 |  |  4 |  3 |  6 |  |  4 |  4 |  3
11.1 |  1 |  1 |  1 |  |  1 |  1 |  1 |  |  4 |  3 |  5 |  |  4 |  4 |  3

Using the NMSE metric, 10.2/BIC was the best-performing version/model
combination for overall positron yield (12/15 cases), with 10.2/QMD being the
second-best (10/15). Results were similar for 11C yield, with the best
version/model combinations being 10.2/QMD (12/15 cases) and 10.2/BIC (11/15).
For 10C, the most wins were obtained by 10.6/INCL and 11/INCL (6/15 cases)
followed by 10.6/BIC, 10.7/INCL and 11.1/INCL (5/15 cases). Finally, for 15O,
the best results were obtained with 10.6/QMD (10/12 cases) followed by
10.2/BIC and 10.2/QMD (7/12 cases).

Table 9: Number of test cases for which each Geant4 version/model combination achieved a CCC greater than 0.95 in the tail region. Bold text denotes the version/model achieving the highest number of best results for each combination of ion/energy/target. Version | Total |  |  11C |  |  10C |  |  15O
---|---|---|---|---|---|---|---
BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL |  |  BIC |  QMD |  INCL
10 |  12 |  11 |  11 |  |  10 |  11 |  11 |  |  4 |  3 |  4 |  |  8 |  7 |  9
10.1 |  12 |  11 |  11 |  |  11 |  11 |  11 |  |  4 |  4 |  4 |  |  8 |  7 |  9
10.2 |  12 |  11 |  11 |  |  10 |  11 |  11 |  |  4 |  4 |  4 |  |  8 |  7 |  7
10.3 |  11 |  11 |  10 |  |  11 |  11 |  11 |  |  2 |  2 |  1 |  |  8 |  7 |  8
10.4 |  10 |  10 |  10 |  |  9 |  11 |  10 |  |  2 |  2 |  0 |  |  7 |  7 |  8
10.5 |  11 |  11 |  11 |  |  11 |  11 |  11 |  |  2 |  2 |  3 |  |  7 |  7 |  8
10.6 |  11 |  11 |  12 |  |  11 |  11 |  11 |  |  2 |  3 |  3 |  |  7 |  7 |  9
10.7 |  12 |  11 |  12 |  |  11 |  11 |  12 |  |  3 |  3 |  3 |  |  7 |  7 |  9
11 |  12 |  11 |  12 |  |  11 |  11 |  13 |  |  3 |  3 |  3 |  |  8 |  8 |  9
11.1 |  12 |  11 |  12 |  |  11 |  11 |  13 |  |  3 |  3 |  3 |  |  8 |  8 |  9

The Pearson CCC results in the tail region were all very similar across Geant4
versions, with only a few wins separating the best and worst-performing
version/model combinations in most instances. All version/models exceeded the
threshold of 0.95 for a clear majority of cases for total positron yield as
well as 11C and 15O production. For total positron annihilation yield, 10/BIC,
10.1/BIC, 10.2/BIC, 10.6/INCL, 10.7/BIC, 10.7/INCL, 11/BIC, 11/INCL, 11.1/BIC
and 11.1/INCL all exceeded the target threshold for 12/15 cases. Even the
worst-performing version/model combinations still exceeded the threshold in
10/15 cases. For 11C yield, 11/INCL and 11.1/INCL reached the threshold in
13/15 cases (with the worst-performing combination scoring 9/15 wins). Fewer
wins were seen with 10C; the best results were obtained with 10/BIC, 10/INCL,
10.1/BIC, 10.1/QMD, 10.1/INCL, 10.2/BIC, 10.2/QMD and 10.2/INCL (4/15 cases).
Finally, 15O yield was best predicted by 10/INCL, 10.1/INCL, 10.6/INCL,
10.7/INCL, 11/INCL and 11.1/INCL (9/12 cases) - again, in this case, even the
worst-performing version/model combinations exceeded the threshold in 7/12
cases.

###  3.4 Overall recommendation

The accuracy of Geant4’s hadronic inelastic physics models (BIC, QMD and INCL)
in predicting both total positron annihilation yield and individual positron-
emitting fragment production is not consistent between different versions of
Geant4; furthermore, later releases do not necessarily provide a more accurate
prediction of experimental observations than preceding versions.

In the entrance region, BIC was clearly the best-performing model, with the
best choice of Geant4 version depending on the particular metric and
fragmentation product of interest. NMSE results generally favoured 10-10.4/BIC
(especially 10.2/BIC), except for 10C yield, which was better predicted by
10.3+/INCL. Pearson CCC performance did not strongly favour any particular
version/model combination, with at most 1/3 of test cases achieving the target
CCC threshold of 0.95 for any version/model.

In the build-up and Bragg peak region and tail region, the results are more
clear-cut. Results for the NMSE metric conclusively show that version 10.2/BIC
is clearly the best choice for total positron yield, 11C production and 15O
yield, while 10.5-11.1/INCL performed the best for 10C. Pearson CCC results
are more mixed, but again, 10.2/BIC gives the best results for total positron
annihilation yield, with most versions of Geant4 with BIC performing well.
10.6/QMD performed the best for 11C, 10/INCL and 10.1/INCL performed the best
for 10C, and there was no clear winner for 15O.

Using the depth-of-maximum-yield metric, the smallest mean differences were
obtained with 10.4/QMD and 10.6/QMD. These versions/models also achieved the
equal-smallest maximum difference (-3 mm and +3 mm, respectively). Across all
versions of Geant4, QMD demonstrated the best overall accuracy (lowest average
mean difference in peak depth) and highest precision (lowest average standard
deviation). INCL was the worst-performing model across all versions, with much
larger maximum differences, and a consistent underestimation of depth of
maximum yield across Geant4 versions, with the exception of version 11.1
(which, despite a mean difference of 0, exhibited a large standard deviation
and maximum value). Standard deviations obtained using INCL were generally
around double those of QMD and BIC. BIC also showed a consistent
underestimation in depth of maximum yield, although the maximum differences
were much smaller than for INCL. For context, the difference between the depth
of the positron annihilation peak and the Bragg peak with monoenergetic ion
beams is of the order of -5.6$\pm$0.8 mm for 12C and -6.6$\pm$0.8 mm for 16O
[32, 33, 6].

Results were generally better for the the distal depth at 50% of peak metric.
In this case, 10.2/QMD, 11.1/BIC and 11/INCL all achieved a mean of zero, with
10.2/QMD also having the equal-lowest maximum value of 1.5 mm (a depth
difference of one voxel). QMD’s maximal values were slightly smaller overall
compared to BIC, and INCL’s were the largest at $\pm$3 mm for all versions.
INCL tended to consistently overestimate the depth of this point, with both
mean and maximum differences being negative in most cases. BIC and QMD both
tended towards underestimating the 50%-of-peak depth, with the exception of
version 11.1 (negative maxima for both, and means of 0 and -0.1 mm,
respectively). Standard deviations were quite small for all versions and
models (with the maximum standard deviation being 1.68 mm, for 11.1/INCL).

Finally, in the tail region, Geant4 10.2 with BIC and QMD again provided the
best prediction of total positron and 11C yield in terms of NMSE, while
10.6/INCL performed the best for 10C and 15O. All version/model combinations
performed well for total positron annihilation, 11C and 15O yield according to
the Pearson CCC metric, while no version/model performed especially well for
10C.

Across all regions, ion species, beam energies, and target materials
evaluated, the combination of Geant4 version 10.2 and BIC is best able to
reproduce experimental results as evaluated using the NMSE and Pearson CCC
metrics - especially in the build-up and Bragg peak region and tail region.
Since the build-up and Bragg peak region is the location where (1) the
majority of the dose resulting from carbon or oxygen ion beam irradiation in
heavy ion therapy is deposited, and (2) where the strongest positron
annihilation signal is observed, the results in this region are the most
relevant to PET image-based QA simulation work. Version 10.2 also provided the
best estimate of the depth of the distal depth at which positron yield
decreased to 50% of peak, although this was obtained with QMD rather than BIC;
the most accurate estimate of the depth of the peak itself was also achieved
with QMD, but with Geant4 versions 10.4 and 10.6. As QMD exhibited the best
accuracy and precision across Geant4 versions, it is the recommended model if
the depth of the yield peak is critical.

The BIC model implemented in Geant4 version 10.5 suffered from a run-time
stability error which resulted in it being unable to simulate all test
scenarios; therefore, we recommend that this version/model combination should
be avoided for future studies.

In the evaluation of individual positron-emitting fragment yield profiles,
predictions of 10C distribution were generally the least accurate in terms of
both the NMSE and Pearson CCC. Interestingly, the INCL model often performed
the best for prediction of 10C fragment yield, although it rarely performed
the best for total positron annihilation and 11C or 15O. Therefore, INCL
should be considered for studies focusing on 10C fragmentation, with the
caveat that range estimation will be less accurate with this model.

Not all models met or exceeded the set threshold of 0.95 for the Pearson CCC
metric. This means that in these cases, the shape of the predicted positron
distribution differs significantly from the experimental measurements. This is
of particular concern if these models are to be used for dose estimation using
a deconvolution approach [20, 21] or for the training of machine learning
models for feature extraction [23].

Finally, it is worth noting that current evaluations of fragmentation cross
sections exhibit uncertainties exceeding 10%, which must be addressed in order
to accurately model positron fragmentation, particularly in the case of
complex fragmentation reactions such as the production of 10C [34, 35]. This
also impacts other Monte Carlo simulation platforms (such as FLUKA, MCNP and
PHITS) which rely on accurate cross section data.

##  4 Conclusion

In this study, the accuracy with which Geant4 is able to predict the
distribution of total positron annihilation yield and the distributions of
individual positron-emitting fragmentation products (11C, 10C and 15O) during
carbon or oxygen ion therapy was compared to experimental data. Three
different hadronic inelastic physics models - Binary Ion Cascade (BIC),
Quantum Molecular Dynamics (QMD) and Liege Intranuclear Cascade model (INCL)
were used with ten different versions of Geant4 - 10.0.p04, 10.1.p03,
10.2.p03, 10.3.p03, 10.4.p03, 10.5.p01, 10.6.p03, 10.7.p02, 11.0 and 11.1, in
three different homogeneous phantoms. The simulated and experimental data were
compared using two different metrics - normalised mean squared error and the
Pearson cross-correlation coefficient. Additionally, the differences between
the simulated and experimental depth of maximum positron annihilation yield,
as well as the distal point at which positron yield declines to 50% of the
peak were evaluated. It was found that the accuracy of the hadronic inelastic
physics models strongly depends on the version of Geant4 in which it was
implemented, and newer versions of Geant4 were not always more accurate at
predicting positron-emitting fragmentation compared to older versions.
Furthermore, it was found that not all version/model combinations were able to
satisfactorily predict the shape of positron annihilation or positron-emitting
fragment distributions, even though they could provide a good estimation of
the total positron annihilation yield and range. For future simulation studies
of therapeutic irradiation using carbon or oxygen ion beams, it is recommended
that Geant4 version 10.2 with the BIC model be used as it is currently the
version/model combination best able to replicate the experimentally-observed
total positron yield and the fragmentation product distributions, while the
depth of the maximum positron yield and distal 50%-of-peak point were best
predicted using the QMD model from Geant4 10.4, 10.6 (peak) and 10.2 (distal
50%-of-peak).

##  5 Acknowledgements

The authors would like to acknowledge the following organisations for
providing access to their high-performance computing resources: the Multi-
modal Australian Sciences Imaging and Visualisation Environment (MASSIVE) “M3”
cluster and Australia’s Nuclear Science and Technology Organisation (ANSTO)
“Tesla” cluster. This research has been conducted with the support of the
Australian government research training program scholarship. The authors
acknowledge the scientific and technical assistance of the National Imaging
Facility, a National Collaborative Research Infrastructure Strategy (NCRIS)
capability at the Australian Nuclear Science and Technology Organisation,
ANSTO.

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##  6 Appendix

Table 10 lists the physics models which were used in the simulations.

Table 10: Hadronic physics processes and models used in all simulations. Interaction |  Energy Range |  Geant4 Model
---|---|---
Radioactive Decay |  All energies |  G4RadioactiveDecayPhysics
Particle Decay |  All energies |  G4Decay
Hadron Elastic |  0–100 TeV |  G4HadronElasticPhysicsHP
Ion Inelastic |  $<$100 MeV |  Binary Light Ion Cascade
|  100 MeV–10 GeV |  BIC or QMD or INCL++
Neutron Capture |  0–20 MeV |  NeutronHPCapture
|  $>$19.9 MeV |  nRadCapture
Neutron Inelastic |  0–20 MeV |  NeutronHPInelastic
|  $>$19.9 MeV |  Binary Cascade
Proton Inelastic |  990 eV–10 TeV |  Binary Cascade


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