Biomedical Engineering Reference
In-Depth Information
T
HE
L
ATERAL
D
OSE
D
ISTRIBUTION OF A
P
ROTON
B
EAM
So far, we have mainly discussed the distribution in depth of the dose
deposited by protons along the central axis of a beam. Now, let us see
what happens at off-axis points.
Pencil beams
In the previous section, in discussing the near-disappearance of
the Bragg peak of an infinitesimal pencil beam, I explained the
phenomenon in terms of broadening of the distal end of the pencil
beam caused by multiple Coulomb scattering of the protons. It is now
time to look more closely at the causes of that broadening. There are
two regions in which broadening occurs - namely, within the patient
and in the material upstream of the patient. The four main effects that
cause the broadening are as follows.
Multiple Coulomb scattering: near-Gaussian core
The details of multiple Coulomb scattering were worked out in
around 1947 by Molière in a pair of comprehensive papers which
have been more often quoted than read - see Gottschalk
et al
. (1993)
for a discussion of Molière's theory. Multiple Coulomb scattering is
the principal cause of the spreading out of an initially infinitesimal
pencil beam. But multiple scattering comes in what can be taken as
two separate parts.
The principal component is a nearly Gaussian distribution, both in the
angle of deviation and in the consequent lateral spread of a pencil
beam. Near the end of range, the standard deviation of the lateral
distribution is approximately 2% of the range.
2
That is, a 150 MeV
-2
proton beam at its end of range (i.e.,
∼
15 g
⋅
cm
) will spread out
sideways to form a near-Gaussian profile whose sigma is about 3mm
and whose full-width at half maximum is about 7mm. As we have
2
The following are some useful relationships. The full-width at half-
maximum of a Gaussian distribution (with a standard deviation of
σ
) is
2.35
; and the
80-20% fall-off of an error function (which is the shape that is generated
when a set of equally weighted Gaussian distributions are summed up) is
1.68
σ
; the 80-20% fall-off down one side of a Gaussian is 1.12
σ
. This last is the number that characterizes the penumbra of a beam
made from a sequence of equally spaced and weighted pencil beams whose
lateral shape is a Gaussian.
σ
