2200-2201 / 2239
Static dose surrogate in 3D; convolve over a box function
(
)
(
)
1
" , , "
0 , ,
elsewhere
x
x
y
y
z
z
f
x f
f
y f
x y z
f
z f
x y z
Ψ
− ≤ ≤
− ≤ ≤
=
− ≤ ≤
∈
(
)
(
) (
)
{
p
if , >>
p
p
1
" ,0, 0 "
" ', ', ' "
',
',
' d dydz
1 1
Erf
Erf
2
2
2
y z
x
x
f f
x f
x f
D x
x y z k x x y y z z x
σ
Ψ
σ
σ
∞ ∞ ∞
−∞ −∞ −∞
+
−
=
⋅
− − −
= ⋅
⋅
−
⋅
⋅
∫ ∫ ∫
where
2
4
and a 3D Gaussian ”kernel”:
(
)
2
2
2
2
2
2
p
p
p
(
')
(
')
(
')
2
2
2
3/ 2 3
p
e
e
e
',
',
'
2 2
x x
y y
z z
k x x y y z z
σ
σ
σ
π σ
−
−
−
−
−
−
− − − =
Same result along ”main” axes as
for a 1D Gaussian (same
σ
p
)!
In margin recipes common to
assume dose profile as ”straight out of a large target”. Convolution
model assumes infinite number of fractions, where
4
2
0
2
4
4
2
0
x
y
2
2
p
p
σ σ σ
→ +