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( )
( )
0
s
e
br
p
B B
r
r
r
r
Ο
ρ
−
≈
= +
Simplify the expression known to well model a scatter dose pencil kernel:
Integrate to get scatter dose per incident energy fluence for a circular field of radius
R
The derivation of the equivalent circular field radius using pencil beam convolution
2
circular field radius R
0 0
d d 2
R
B
D
r r
B R
r
π
θ π
≈ ⋅
=
∫ ∫
(
)
4 2
4
Square field side S
0 0
0
1
8
d d 4
d 4 ln 1 2
cos
cos
S
B r
D
r
B S
B S
r
π
π
θ
θ
θ
θ
≈
⋅
=
=
+
∫ ∫
∫
Integrate also to get scatter dose for a square field of side
S
(a square is composed
of 8 triangles 45 degree each)
(
)
4
2
ln 1 2
B S
B R
π
=
+
Set the dose results equal (equivalent field size!) :
and solve for
R
as function of
S
resulting in
(
)
2 ln 1 2
0.5611
R
S
S
π
+
=
=