# CALCULATION OF A SINGLE PENCIL BEAM KERNEL FROM MEASURED PHOTON

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```					CALCULATION OF A SINGLE PENCIL BEAM KERNEL FROM MEASURED PHOTON BEAM DATA

Pascal Storchi Daniel den Hoed Cancer Center University Hospital Rotterdam

Calculation of irregular photon fields by pencil beam convolution
 

D( x, y, z; F) 

 

  F( x', y', z)  K( x  x', y  y', z)dx' dy'

F(x,y,z): fluence of primary photons K(r,z): pencil beam kernel 2 2 1/2 (r=(x + y ) )

Pencil beam model


D( x, y, z; X ) 
2



  F( x', y', z)  K( x  x', y  y', z)dx' dy'

x y

z

Calculation of the pencil beam kernel "theoretical" approach
Pencil beam kernel computed by Monte Carlo techniques: • Energy spectrum must be known: – for the pencil beam kernel – for the primary photon fluence
• Results must be fitted to specific linac.

Calculation of the pencil beam kernel "empirical" approach
Is it possible to extract the pencil beam kernel from measured data ?

Is it possible to extract the pencil beam kernel from measured data ?
Answer: Yes by differentiation of scatter-to-primary ratio, including an electron disequilibrium factor Ceberg, Bjängard and Zhu “Experimental determination of the dose kernel in high-energy x-ray beams” Med. Phys. 1996

Is it possible to extract the pencil beam kernel from measured data ?
Other method, pencil beam kernel computed from:  Phantom Scatter Factor (Sp) of square fields 4x4 up 2 to 40x40 cm  off-axis ratio (penumbra region) of square fields 2 (>5x5 cm )

Present model

depth dose D(x,y,z;X2) = Da(z;X2) 

boundary function Pb(x,y,z;X2) 

envelope profile Pc(r,z)

Pencil beam model


D( x, y, z; X ) 
2



  F( x', y', z)  K( x  x', y  y', z)dx' dy'

x y

z

Phantom Scatter Factor

Phantom Scatter Factor Sp(z,X) is computed from:
 

tabulated Sp(z=10 cm,X) normalized depth dose curve Da(z,X2)

Calculation of the scatter kernel Ks(R,z)
Sp(z,X) given for square fields X4 cm (equivalent circular field radius R2.3 cm). Linear extrapolation is used in the region X<4 cm (R<2.3 cm).
Sp (R, z)   K s (r, z) dr
0 R

K s (R, z) 

dSp (R, z) dR

Fluence of primary photons
first guess: intensity profile Pi = envelope profile Pc
envelope profile only "scatter" measured profile

Fluence of primary photons
deconvolution of scatter kernel Ks from envelope profile Pc => intensity profile Pi

intensity profile

only scatter

measured profile

Boundary kernel Kb
Computed from the boundary profile Pb that has been corrected for the photon scatter:

dPb ( x)  dx



K b (r)

Combination of scatter and boundary kernels in one single pencil beam kernel
K(0,z) = CKb(0,z) K(ri,z) = Ks(ri,z) + CKb(ri,z) ri = (i + ½ ) C = ¼ 2Ks(0,z) ,i=1,...,n

Steps for the calculation of the pencil beam kernel for a given depth
1. Ks: compute the scatter kernel from the phantom scatter factors of square fields 2. Pi: compute the intensity profile such that P c = PiKs 3. Ks: correct the scatter kernel Ks for the influence of Pi 4. Kb: compute the boundary kernel from the profiles of the 10 2, 152, 202, and 252 cm2 fields corrected for the photon scatter 5. K: combine Ks and Kb into one single pencil beam

Results: square field (20x20 cm2)
measurement calculation

Results: asymmetric square field (20x20 cm2)
measurement calculation

Results: rectangular field (30x5 cm2)
measurement calculation

measurement
diode

calculation
pencil beam kernel derived from data measured with ionization chamber

Conclusions
• It is possible to use the PSF and the penumbra region of measured square fields for the derivation of the pencil beam kernel.

• Measurement of the penumbra region must be done with a small detector (diode).

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Lingjuan Ma MS
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