Biomedical Engineering Reference
In-Depth Information
The shape of this distribution arises from a number of cooperating
effects, which I now describe.
Energy loss due to Coulomb interactions with atomic electrons
The principal genesis of the Bragg peak is the slow loss of energy that
protons experience due to Coulomb interactions with atomic electrons
which, you will recall, cause protons to slowly lose energy by
transferring it to atomic electrons. However, they do not give up
energy equally at all depths. At any given point within a stopping
medium, a proton's linear rate of energy loss
−
its “linear energy
transfer” (LET), or “stopping power”
−
which is measured in units of
is given by the Bethe-Block formula.
1
It is
approximately proportional to the inverse of the square of the proton's
mean speed, v:
-2
MeV per g
⋅
cm
−
dE
1
Z
⎛
⎞
(10.1)
2
∝
z
⎝
⎠
2
dx
v
A
where Z and A are, respectively, the atomic and mass numbers of the
target nucleus and z is the charge number of the projectile proton.
The local energy deposition
(i.e., the dose) thus rises
sharply as protons slow down.
This slowing down process,
with its concomitant increase
in dose with depth, is depicted
schematically in Figure 10.6.
Suppose the proton speed at a
A
point such as A is v , then
the dose it would deposit
at A would be given by
Figure 10.6. The contribution to the
Bragg peak from Coulomb scattering
of protons off atomic electrons (see
text).
equation (10.1), substituting v
A
for v. At a deeper point such
1
In describing depths of penetration, the
areal density
is often employed,
with units of g
-2
cm . The areal density of a uniform medium is the product
of path length and density
⋅
or, in inhomogeneous media, the integral of
density over the path length. This integration removes the trivial (from a
physics point of view) influence of density. Trivial because if one could,
say, double the density of some medium and simultaneously halve its
thickness, its effect on a proton beam would be virtually unchanged. For
water, which has almost unit density, the areal density is numerically
equivalent to the depth of penetration measured in centimeters.
−
