Biomedical Engineering Reference
In-Depth Information
Z
1
D
Q
D
yf .y/Q.y/dy;
(18.4)
where
y
is the lineal energy, the energy deposited in a volume of mean cord-
length, l . At high doses Q should be generalised to utilise the specific energy
quantity, i.e.,
Z
1
D
Q
D
z
f.
z
/Q.
z
/d
z
:
(18.5)
Many in vitro studies have shown that Q.
z
/ is essentially linear in
z
, until
z
ap
pro
aches 100 keV/micron where saturati
on
effects become important [
38
]. That
is Q
y
D
. These studies lend support
to the theory of dual radiation action and the value of y
D
in predicting relative
biological effect (RBE) at low dose limits. The theory states that there is a quadratic
dependence on
z
on the formation of critical lesions, where
z
is considered to the
concentration of molecular sublesions in the relevant volume, i.e. a chemical process
in the nucleus. The spectrum of sublesions is E.
z
/
z
D
, and in the limit of zero dose Q
ˇ
z
2
. It can be shown that the
D
average yield of lesions as a function of dose is,
D
2
/
ˇD
2
:
D
C
D
C
E.
z
/
ˇ.
z
D
D
˛D
(18.6)
where ˛ and ˇ are the coefficients usually associated with the linear-
q
uadratic
theory. Accordingly ˛=ˇ
D
D
y
D
l=m.The
theory says nothing about the value of the ˇ coefficient and is therefore independent
of radiation type within the microdosimetric framework. y
D
is often said to be
proportional to ˛, however this may not be appropriate given the larger uncertainties
asssociated with determining ˇ, and its influence on the calculated value of ˛ for
the high-doses typically needed for in vitro studies.
z
D
, and in the limit of zero dose ˛=ˇ
18.3
Calculating high-dose response: Specific energy
Lineal energy spectra and moments calculated in water from in-air spectra for a
range of X-ray energies are published in the PhD thesis of Verhaegen (Ghent U.)
who utilised the TRION track-structure code [
39
]. This data has been used as the
basic input data for single event, lineal energy spectra. TRION utilises gaseous water
cross-sections. More recently TRION has been been updated to include liquid water
cross-sections [
40
] however these cross-sections are expected to be less reliable.
Specific energy has been calculated from lineal energy distributions with a Monte
Carlo sampling method that is easily implemented in Matlab code. The Poisson
distribution of events traversing the volume of interest is determined by calculating
the mean number of traversing events, n, as a function of dose. The average number
of independent tracks per Gy is shown in Table
18.1
for 10 and 1000 nm spherical
volumes and for X-ray energies of 10,100 and 1250 keV.
