11.2. Quantities Related to Dosimetry 623
11.2.I BeamSize.............................
An important parameter, that must be known while making dose calculations for
radiation beams, is the size of the beam. The reason is that the absolute exposure or
dose can not be calculated without knowing the area of the beam. To clarify further,
let us take the example of a radiotherapy setup. The radiation beam, produced by
a source, such as an accelerator, is made to pass through a set of collimators before
the subject is exposed to it. The collimators are the ones that define the shape
of the beam. If one knows the beam flux (number of particles per unit time per
unit area), one can multiply it with the area of the beam to find the total number
of particles per unit time. The situation for dose calculations is not that simple,
though. In most of the formulae derived for dosimetry, one generally assumes a
circular or a square beam. However, in practice, a beam can have different shapes:
square, rectangular, circular, or even irregular. The problem is that one can not
simply calculate the area of a beam having arbitrary shape and substitute the result
in these equations. One needs to know theeffectivearea that would have the same
impact as a beam having circular or square area.
The case of an irregularly shaped beam is fairly complex but, to a very good
approximation, any irregular shape can be assumed to be circular, square, or rect-
angular. A square beam can be equated to a circular beam having the same area,
that is
Asq = Acir
⇒x^2 =πr^2
⇒x=√
πr, (11.2.43)wherexrepresents a side of the square andris the radius of the equivalent circle.
Now, the area of a rectangular beam can not be simply equated to the area of
a circular or a square beam. However, one cantranslatea rectangular beam into a
square beam such that the ratio of their areas to their perimeters remain the same,
that is
Asq
Psq
=
Arect
Prect, (11.2.44)
whereAandP represent area and perimeter, and the subscriptsrectandsqrefer
to rectangular and square beams respectively. In terms of sides, we can then write
x^2
4 x=
yz
2(y+z)⇒x =2 yz
y+z, (11.2.45)
wherexrepresents a side of the square, andyandzrepresent the adjacent sides of
the rectangle.
What these equations imply is that one can translate essentially any beam shape
into either a square or a circular shape (see example below). This greatly simpli-
fies computations of the parameters related to dosimetry, specially in the field of
radiation therapy.