1180-1181 / 2239
5.2
Making the integrals practical
...d d
E
E
Ω
Ω
′
′
∫ ∫
Setting up equations:
(
)
(
)
(
)
att
prim ph, ,
scat ph, ,
ph
prim ph, ,
scat ph, ,
ph ph,
prim ph, ,
scat ph, ,
,
att
el, ,
el
el,
ph el,
prim ph, ,
scat
,
,
d d
E
E
E
E
E
E
E E
E
E
E
E
E E
E
S
E
Ω
Ω
Ω
Ω
Ω
Ω
Ω Ω
Ω
Ω
Ω
Ω
Ω Ω
Ω Φ
Φ
σ Φ
Φ
σ
Φ
Φ
Ω
Ω Φ σ
Φ
σ
Φ
Φ
′
′
′
′
→
′
′
→ →
′
′
→
′
′
→ →
′
′
⋅∇
+
+
+
=
+
∂
⋅∇ +
−
=
+
∂
∫ ∫
uur
uur
(
)
ph, ,
el el,
el, ,
,
d d
E
E
E E
E
E
Ω
Ω
Ω Ω
Ω
σ
Φ
Ω
′
′
′
′
→
′
′
→ →
′
′
+
⋅
∫ ∫
(
)
(
) (
)
,
2 1
, ,
,
4
a b
a b l
l
l
E E
E E P
σ
σ
π
∞
→
→
+
′
′
′
′
Ω⋅Ω →
Ω⋅Ω
∑
Angular part expanded into Legendre polynomials (cross sections) and spherical harmonics (fluence):
(
)
(
) ( )
1
,
1
,
, ,
d
2
a b l
a b
l
E E
E E P
σ
σ
µ µ µ
→
→
′
′
=
∫
( ) ( )
0
,
0
'
'
l
l
m
m
E
l
l
l
m l
E Y
φ
=
∞
′ ′Ω
= =−
Φ →
Ω
∑ ∑
( )
( )
1
,
'
d
m
m
l
l
E
E Y
φ
−
′ ′Ω
Ω
′
′
= Ω ⋅Φ Ω
∫
where
(
) (
)
( )
(
)
( )
( )
( ) ( )
( )
,
0
0
,
0
2 1
,
,
d d
'
d
4
'
i
i
l
m
m
a b
a b l
i
l
l
i
l
i
E l
l
m l
E
l
m
m
a b l
i
l
i
l
i
E l
m l
l
E
E
E
E P
E Y
E
E E Y
E
σ
σ
φ
π
σ
φ
∞
∞
→
→
′∆
=
= =−
Ω
Ω
∞
→
′∆ = =−
+
′
′
′
′
′ ′
′ ′
′
′ ′
Ω⋅Ω Φ Ω Ω ≈
Ω⋅Ω ⋅
Ω Ω∆
′
′
′
=
Ω ∆
∑ ∑
∑ ∑
∫ ∫
∫
∑∑ ∑
Acuros:
7, # energy bins 25 (photons); 49 (electrons)
∞ ≈
≈
Energy part discretized into energy bins (“groups”).
Combined, the integrals can be rewritten as series with a limited number of terms:
Failla
etal Acuros XB advanced dose calculation for the Eclipse treatment planning system.
Varian white paper
8