168 unifying physics of accelerators, lasers and plasma
the destined target volume (see Fig. 9.2), this minimizes the
impact on healthy tissues — especially in cases where the target volume is located close to critical organs.
Quantitatively, the Bragg peak can be explained by the
following formula for the mean energy loss of moderately relativistic heavy particles (Bethe equation):
(
)
dE
1 1 2m e c 2 β 2 γ 2 W max
(− ) ≈ Kz
2 Z
ln
− β
2
(9.1)
dx
A β 2 2
I 2
Here, z is the charge number of an incident particle, Z
and A are the charge number and atomic mass of the absorber, respectively, and β and γ are the relativistic factors
of the incident particle. The parameters under the logarithm
are: I — the mean excitation energy of the atom’s electron,
and W max — the maximum energy transfer in a single colPenetration ranges for pro- lision. For a particle with mass M, the latter is defined
tons in water:
as W max = 2m e c 2 β 2 γ 2 /(1 + 2γm e /M + (m e /M) 2 ). The coeffi2
250 MeV — 38 cm;
cient K is defined as follows: K = 4πN A r m e c 2 where N A is
e
200 MeV — 26 cm;
Avogadro’s number. The coefficient K approximately equals
150 MeV — 15.6 cm;
0.3 MeV · cm 2 /mol.
100 MeV — 7.6 cm;
The usefulness of the Bragg peak for treating tumors was
50 MeV — 2.2 cm.
first realized by Robert R. Wilson in 1946. Overlaying several
Bragg’s peaks described by Eq. 9.1 creates a uniform dose distribution in a given volume (as illustrated in Fig. 9.2) — this is
often called a spreadout Bragg peak. Such overlaying requires
an adjustment to the energy and intensity of each individual
proton beam.
The ideally sharp Bragg peak is, in practice, somewhat
spread — firstly due to the statistical character of interaction,
and secondly due to nuclear interactions between the protons
and absorber, which happen with some probability.
The sharpness of the Bragg peak is an enabling feature of
proton therapy, but simultaneously it is a factor that increases
the sensitivity of the method to errors, especially to the errors
in the predicted depth range. In a particular case when the
target volume is located near a critical organ, ideally, one can
completely eliminate irradiation of the critical organ while
filling the entire target volume uniformly. In practice, however, one cannot obtain a sharp irradiation boundary of the
irradiated volume, due to the necessity to allow for some uncertainties of the depth range. Possible motion of the critical
organs during irradiation — as well as shrinkage of the tumor
(and possibly corresponding shift of the critical organs) as
the treatment progresses — are also important factors, which
need to be taken into account in proton therapy planning.
The above-mentioned sensitivity to errors places a particularly strong requirement on the energy of protons in the
cases when plasma acceleration is used. The beam needs to
have a well-defined energy. This can be ensured either via
predictable plasma acceleration or by an appropriate energyselection system.
the destined target volume (see Fig. 9.2), this minimizes the
impact on healthy tissues — especially in cases where the target volume is located close to critical organs.
Quantitatively, the Bragg peak can be explained by the
following formula for the mean energy loss of moderately relativistic heavy particles (Bethe equation):
(
)
dE
1 1 2m e c 2 β 2 γ 2 W max
(− ) ≈ Kz
2 Z
ln
− β
2
(9.1)
dx
A β 2 2
I 2
Here, z is the charge number of an incident particle, Z
and A are the charge number and atomic mass of the absorber, respectively, and β and γ are the relativistic factors
of the incident particle. The parameters under the logarithm
are: I — the mean excitation energy of the atom’s electron,
and W max — the maximum energy transfer in a single colPenetration ranges for pro- lision. For a particle with mass M, the latter is defined
tons in water:
as W max = 2m e c 2 β 2 γ 2 /(1 + 2γm e /M + (m e /M) 2 ). The coeffi2
250 MeV — 38 cm;
cient K is defined as follows: K = 4πN A r m e c 2 where N A is
e
200 MeV — 26 cm;
Avogadro’s number. The coefficient K approximately equals
150 MeV — 15.6 cm;
0.3 MeV · cm 2 /mol.
100 MeV — 7.6 cm;
The usefulness of the Bragg peak for treating tumors was
50 MeV — 2.2 cm.
first realized by Robert R. Wilson in 1946. Overlaying several
Bragg’s peaks described by Eq. 9.1 creates a uniform dose distribution in a given volume (as illustrated in Fig. 9.2) — this is
often called a spreadout Bragg peak. Such overlaying requires
an adjustment to the energy and intensity of each individual
proton beam.
The ideally sharp Bragg peak is, in practice, somewhat
spread — firstly due to the statistical character of interaction,
and secondly due to nuclear interactions between the protons
and absorber, which happen with some probability.
The sharpness of the Bragg peak is an enabling feature of
proton therapy, but simultaneously it is a factor that increases
the sensitivity of the method to errors, especially to the errors
in the predicted depth range. In a particular case when the
target volume is located near a critical organ, ideally, one can
completely eliminate irradiation of the critical organ while
filling the entire target volume uniformly. In practice, however, one cannot obtain a sharp irradiation boundary of the
irradiated volume, due to the necessity to allow for some uncertainties of the depth range. Possible motion of the critical
organs during irradiation — as well as shrinkage of the tumor
(and possibly corresponding shift of the critical organs) as
the treatment progresses — are also important factors, which
need to be taken into account in proton therapy planning.
The above-mentioned sensitivity to errors places a particularly strong requirement on the energy of protons in the
cases when plasma acceleration is used. The beam needs to
have a well-defined energy. This can be ensured either via
predictable plasma acceleration or by an appropriate energyselection system.