The Probabilistic Description of Ionization–Cluster Size Formation
The behavior of the cumulative distribution functions
at different site sizes can be interpreted on the basis of a probabilistic theory of ionization–cluster size formation [
9]. For ionizing particles of radiation quality
Q crossing a nanometer-sized spherical target volume
V along its diameter
D, it is assumed that the ionization cluster-size caused within
V is exclusively determined by the average number
of ionizing interactions of a primary particle along
D and by the behavior of
δ-electrons within the target volume.
Based on these basic assumptions, the probability
of ionization–cluster size
ν is given by a compound Poisson process [
9] described by Equation (6).
Here, is the probability that in the event of exactly κ primary ionizations due to primary particles of radiation quality Q, a cluster size ν is formed within the target volume (the ionizations due to δ-electrons are included). The average number of ionizing interactions of a primary particle along D is given as the quotient , where is the mean free ionization path length of the primary particles in matter.
In view of the fact that in a short track segment, an ionization process due to a primary particle is independent of the number of previously formed ions, the
distribution is given by the
κ-fold convolution of the probability distribution
,
ν = 1, 2, 3, …, in the case of a single primary ionization (
κ = 1), which is referred to in the following single-ionization distribution, i.e.,
where the convolution operation, indicated by the asterisk, is performed
κ times (
κ-fold convolution) and is defined for two discrete functions
and
,
ν = 0, 1, 2, …, as
.
The single-ionization distributions , ν = 0, 1, 2, … represent the cluster-size formation due to a single primary ionization event; therefore, they are independent of , but at least in principle, they depend on the spectral distribution of secondary electrons set in motion by impact ionization and, thus, on the particle’s velocity. In contrast, is determined by the mean free ionization path length of the primary particles in matter and, thus, on the particles’ charge state and velocity.
To relate the ionization–cluster size probabilities defined by Equation (6) to the members of the single-ionization distribution, the formalism of folding discrete distributions can be applied:
Here, Equation (8) represents the folding of the single-ionization distribution
with the distribution
in the case of exactly
κ-1 primary ionizations. For
κ = 0, the expression
reflects the fact that, if no primary ionizations take place, no
δ-electrons are produced in the target volume, and the only possible cluster size is
ν = 0. This assumption neglects the contribution to the total ionization by
δ-electrons that are produced outside of the target volume
V and enter it. For
κ ≥ 1, by successive application of the convolution operation, all members of the
-distribution can be written as a sum of products, which exclusively consists of members of the
-distribution in the case of a single primary ionization. For particles directly crossing the target volume
V:
because in volume
V, there are at least
primary ionizations. As a consequence, in case of a particle traversing the target volume, the superior limit of the summation in Equation (8) can be substituted by
:
The first values of
, for
, can be easily obtained:
The cumulative distributions
and
and
can be simply derived and written in terms of the members
and
of the single-ionization distribution:
can also be written in terms of
and the mean value
of the
distribution, as derived in detail in [
9]:
It can be observed that, under the hypothesis
,
depends only on the quotient
, while
and
depend on the single ionization distribution values
and
and therefore on the spectral distribution of secondary
δ-electrons. However,
Figure 1 shows only negligible dependency of
,
and
on particle type and thus also on the primary particle velocity if
is the same.
Comparing Equations (14)–(16) with the fitting Equations (3)–(5), the following relations exist for the fitting parameters
:
As defined in Equation (18), the reciprocal of
represents the mean ionization yield per single primary ionization; thus, it is a measure of the additional contribution by the
δ-electrons to the primary ionization. According to
Table 1, the contribution of
δ-electrons to the total average ionization yield amounts to about 10%, 17%, 27% and 33% of the total for site sizes of 1, 2, 5 and 10 nm, respectively.
Figure 1 shows, in particular, that all
values for protons, helium and carbon ions lie on a unique curve. The parameter
is independent of particle type and velocity, which can be interpreted in the sense that, on average, each
δ-electron contributes to the ionization cluster with an additional mean number of ionizations that is largely independent of particle type and velocity. To confirm this finding,
Figure 7 shows the cluster size distributions in 1 nm site size, due to the total contribution by primary particles and
δ-electrons, and that due to ionizations of the primary particle only, for different radiation qualities. The results for the mean cluster sizes and their ratios are given in
Table 2. The values of the quotient
found by direct simulation of ICSD for several radiation qualities are almost invariant with particle velocity and are in very good agreement with the value of the parameter
, given in
Table 1 for 1 nm site size. This implies that
also depends negligibly on radiation quality.
Equation (19) implies that the probability that the single primary ionization results in a cluster of size ν = 1, , is also largely independent of particle type and velocity, but depends on site diameter D. Recursively, the same conclusion can be drawn for the probability .
In the works by Conte et al. [
16,
17], a good correlation was found between radiobiological cross sections at 5% survival and
measured in a volume of 1 nm diameter, and between the cross sections at low doses,
, and
, measured in a volume of diameter 1.5 nm. The proportionality factors were slightly different in the two cases. These results were based on the assumption of a unique dependence of
and
on the mean cluster size
, also independent of target size. Consistently, only the values
were simulated for radiation qualities at which biological data were available, and then, values of
and
were assigned, based on experimental results obtained at larger target volumes, neglecting the dependence of the functions
on target size. In this work, more accurate simulations of ionization cluster size distributions and derived cumulative distributions were performed at the different radiation qualities investigated. It was found out that the biological cross sections,
and
at least for V79 and HSG cells, depend on linear combinations of
and
calculated in a simulated spherical volume of 1 nm in diameter:
Note that the proportionality factor is the same for both cross sections and corresponds to their unique saturation value. It is a parameter that depends on the specific cell line and corresponds approximately to the nucleus size; it was found to be 50 μm2 for V79 cells and 75 μm2 for HSG cells.
In terms of
,
and
, Equations (20) and (21) can be rewritten as:
Equations (23) and (24) express that the biological cross sections are strongly correlated with the probability of cluster sizes ν = 0, 1 and 2.