Dose 2016

Static dose surrogate in 3D; convolve over a box function

   

   

   

   

   

   

x f +

x f −

∞ ∞ ∞ ∫ ∫ ∫

1

=

) ( ⋅ ', x y z k x x y y z z x − − − ', )

= ⋅

⋅

−

(

)

(

x

x

" ,0, 0 " D x

" ', ', ' " Ψ

' d dydz

{ 1 1

Erf

Erf

⋅

⋅

2

2

2

σ

σ

if , >> y z f f

σ

p

p

−∞ −∞ −∞

p

where

1  − ≤ ≤   − ≤ ≤   x f x f f y f

x

4

=  

y

y

(

)

" , , " x y z Ψ

f − ≤ ≤  

z f

z

z

 

∈

(

)

0 , ,

x y z

elsewhere

2

and a 3D Gaussian ”kernel”:

2

2

2

(

')

(

')

(

')

x x −

y y −

z z −

0

y

−

−

−

2

2

2

2

2

2

σ

σ

σ

e

e

e

p

p

p

− − − =

(

)

', k x x y y z z ',

'

3/ 2 3 p

2 2

π σ

2

Same result along ”main” axes as for a 1D Gaussian (same σ p )!

4

4

2

0

2

4

x

In margin recipes common to assume dose profile as ”straight out of a large target”. Convolution model assumes infinite number of fractions, where 2 2 p σ σ σ → + p