Local weighting of nanometric track structure properties in macroscopic voxel geometries for particle beam treatment planning

Data for the analysis have been produced using the Geant4 Monte Carlo toolkit (Agostinelli et al 2003) version 9.6 and, in particular, the Geant4-DNA processes (Incerti et al 2010a, 2010b) that are included in the low energy package. The simulation with the Geant4-DNA processes is performed on step-by-step basis (i.e. avoiding the condensed history approximation). Due to the low energy cut for the electron transport (7.4 eV in this work) it is possible to simulate the energy depositions with nanometric resolution, i.e. to perform a detailed track structure simulation. For more details, the reader is referred to the Geant4-DNA documentation⁵. We simulated tracks of protons within a homogeneous liquid water cube of 20 μm side length. The primary particles were started in the centre of the water cube at point (0, 0, 0). Traveling in the z-direction this resulted in tracks with a length of 10 μm. In a second step we cut a track of 1 μm length out of the full track starting 1 μm behind the point where the simulation was started. We found that this procedure ensures that the considered track segment also contains energy depositions by secondary electrons created further upstream. The position and the amount of deposited energy was recorded for each individual ionization of a water molecule. The simulations were performed for protons in a clinical relevant energy range between 1 MeV and 100 MeV (1 MeV, 3 MeV, 5 MeV, 10 MeV, 25 MeV, 50 MeV, 100 MeV). As the energy loss of the primary particles along the first 2 μm of the track is lower than 6% for the lowest proton energy simulated, the projectile energy can be considered as constant. The number of simulated tracks was between 100 and 500, leading to at least 13 000 ionization events for high proton energies such as 50 MeV and 100 MeV. The number of ionization events for lower proton energies was even higher.

It is generally accepted that damage to the DNA in the cell nucleus is the primary reason for the occurrence of biological effects after irradiation (Goodhead 1994, Olive 1998). The field of nanodosimetry aims at describing the stochastic nature of particle track structure on the nanometer scale in order to explain radiobiological phenomena with the help of the concept of local radiation quality (Grosswendt 2004, 2005, Grosswendt 2006, Garty et al 2006, 2010, Rabus and Nettelbeck 2011, Bug et al 2014). Therefore we calculated ionization cluster size distributions (ICSDs) which describe the probabilities for ionization clusters of a particular size to occur in nanometric volumes. The ionization cluster size is defined as the number of ionizations produced by a particular passing particle track in a considered target volume. The size of these volumes was chosen to be cylindrical with dimensions comparable to a DNA segment of ten base pairs (one convolution of the DNA double helix). The higher the probability for large cluster sizes—which may produce irreparable double strand breaks—the higher is the probability for the occurrence of radiobiological effects. Statistical parameters of ionization cluster size distributions (de Nardo et al 2002) are promising regarding their potential to link the ionization pattern of the particle tracks to the radiobiological outcome.

In order to obtain data for parameterizing the track structure properties, we calculated ICSDs for all simulated proton tracks for all considered energies. The cluster size was determined by scoring the number of ionizations in cylindrical target volumes. As usual in nanodosimetry we chose cylinders which resemble the geometry of a DNA segment of 10 base pairs with a height of 3.4 nm and a diameter of 2.3 nm. For each track, a cuboid was placed around the primary particle trajectory, the size of which was determined by the outermost ionizations and a margin of 10 nm that was added to avoid effects at the edge of this volume. Then 10 million cylinders were placed in the cuboid at randomly distributed locations. The orientation of the cylinders was chosen randomly as well, and cylinders could overlap. Conditional ICSDs were then determined by counting the ionizations in each cylinder where at least one ionization occurred. The distribution was then normalized so that the probability to obtain at least one ionization became a hundred percent. With the resulting relative frequency distribution $P\left(\nu |Q\right)$, where Q is the radiation quality and ν the ionization cluster size, the conditional probability to obtain a particular cluster size can be derived. Figure 1 shows the resulting relative frequency of cluster sizes between 1 and 10. For 1 MeV protons the relative frequencies for large cluster sizes are significantly higher and decrease with increasing energy of the incident proton.

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As mentioned above statistical moments of the ICSDs may have the potential to explain radiobiological effects. In this analysis we make use of the conditional mean ionization cluster size M₁(Q) which is calculated by summing up the products of the cluster size ν and the corresponding conditional probability P:

$$\begin{eqnarray}&&{{M}_{1}}(Q)=\underset{\nu =1}{\overset{\infty}{\mathop \sum}}\,\nu \centerdot P\left(\nu |Q\right)\end{eqnarray} \tag{ 1 }$$

Other moments of the ICSD used in this analysis are the complementary cumulative probabilities to obtain a cluster size larger than two or three, respectively:

$$\begin{eqnarray}&&{{F}_{2}}(Q)=\underset{\nu =2}{\overset{\infty}{\mathop \sum}}\,P\left(\nu |Q\right)\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{{F}_{3}}(Q)=\underset{\nu =3}{\overset{\infty}{\mathop \sum}}\,P\left(\nu |Q\right)\end{eqnarray} \tag{ 3 }$$

These moments of the distribution are potentially correlated to the probability to obtain simple or complex double strand breaks of the DNA molecule (Grosswendt 2005, Garty et al 2006, 2010) and hence candidates as input quantities for a treatment planning system based on measurable track structure properties.

First, we will analyze the overall track structure properties in a nucleus which is traversed by two different particles. We assume that the nucleus is the primary target and the DNA molecule is homogeneously distributed in the nucleus. The sensitive volume (e.g. the nucleus or a part of it) is modeled by a spherical volume with a diameter of 1 μm (green dots in figure 5). The direction of the tracks traversing this nucleus was chosen randomly. Hence the length of the tracks in the nuclei is different for each track. Blue and red dots in figure 5 show the ionizing interactions produced within the nucleus by the tracks of two protons with energies of 1 MeV and 10 MeV respectively.

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Figure 6 shows the individual ICSDs of two single proton tracks with energies of 1 MeV (blue dashed line) and 10 MeV (red dashed line). The relative frequency to obtain a large cluster of ionizations is larger for the 1 MeV proton compared to the 10 MeV proton. In the next step we analyzed the overall track structure in the nucleus, with both tracks present. By scoring the number of ionizations in 100 million randomly distributed cylinders, the situation of a mixed field was simulated. The result shows that the overall ICSD is dominated by the ICSD of the 1 MeV proton. In order to calculate the overall ICSD from the ICSDs of the single tracks, the trivial approach would be a simple track averaging (i.e. fluence weighting) of the ICSDs of the single tracks. The overall ICSD for a set of tracks ${{t}_{1}},{{t}_{2}},...{{t}_{N}}$ is then given by:

$$\begin{eqnarray}&&\text{ICS}{{\text{D}}_{t}}\left({{t}_{1}},{{t}_{2}},...{{t}_{N}}\right)=\frac{1}{N}\underset{i=1}{\overset{N}{\mathop \sum}}\,\text{ICSD}\left({{t}_{i}}\right)\end{eqnarray} \tag{ 7 }$$

This trivial approach (orange circles in figure 6) underestimates the influence of the 1 MeV proton. A more sophisticated approach is a dose averaging (i.e. specific energy weighting) according to

$$\begin{eqnarray}&&\text{ICS}{{\text{D}}_{d}}\left({{t}_{1}},{{t}_{2}},...{{t}_{N}}\right)=\frac{1}{D}\underset{i=1}{\overset{N}{\mathop \sum}}\,{{d}_{i}}\text{ICSD}\left({{t}_{i}}\right)\end{eqnarray} \tag{ 8 }$$

where d_i is the dose (or specific energy) contribution of track i in the investigated volume and $D=\sum_{i}{{d}_{i}}$. Figure 6 shows that the dose averaged ICSD (turquoise triangles in figure 6) agrees very well with the ICSD resulting from the overall track structure (green triangles). Hence the overall ICSD in a single nucleus can be calculated with the knowledge of the energy of the tracks traversing this nucleus and the dose deposited by these tracks.

As shown in section 4.2 the ICSD for a nucleus hit by two particles can be calculated by performing a specific energy weighting of the ICSDs of the individual particles. In the clinical situation our knowledge is restricted to the type of particles hitting the voxel and their energy spectra. Therefore we have to show that a stopping power weighting reproduces averaged values which are in agreement with the results of the overall track structure in the nuclei which were hit. The difference to the situation in section 4.2 is that—in the clinical situation—we do not know the exact, individual ICSDs of the single particles hitting the nuclei (nor their length respectively dose contribution). Hence we will use a modified version of the specific energy weighting, where the contributions of the individual tracks are weighted by the respective stopping power (since the average specific energy will be the stopping power times the average track length in the nucleus).

In a future planning system, the calculations can be performed with the statistical moments of the ionization cluster size distribution as the parameterizations presented in section 3 enable a fast calculation. Thus the following analysis will be based on the statistical moments already parameterized in section 3.

Figure 7 shows the result for different combinations of proton energies. The data points connected with solid lines show the results of the analysis of the Monte Carlo generated data set. Each data set corresponding to a data point in the plot consists of one hundred nuclei containing two randomly orientated tracks each. The data points marked with asterisks in figure 7 correspond to the 1 MeV series, i.e. the average of hundred nuclei that are hit by a 1 MeV proton and by a proton of the energy indicated on the x-axis. Data points marked with squares corresponds to the 5 MeV series and circles to 25 MeV. The overall track structure was determined in each nucleus, a mean ICSD was determined for all nuclei and the statistical moments presented in section 2.2 were calculated. The results of these simulations were then compared to the track structure properties calculated with the modified stopping power weighting method:

$$\begin{eqnarray}&&{{P}_{d}}\left({{t}_{1}},{{t}_{2}},...{{t}_{N}}\right)=\frac{1}{S}\underset{i=1}{\overset{N}{\mathop \sum}}\,{{S}_{i}}P\left({{t}_{i}}\right)\end{eqnarray} \tag{ 9 }$$

where S is the sum of stopping powers of all tracks, S_i is the stopping power of track i and P(t_i) is the track structure property of track i which was obtained from the parameterizations of the original simulation data presented in section 3.

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