Page 71 - ARNM-2-4

Advances in Radiotherapy
& Nuclear Medicine Different approaches for the computation of BED

Appendix

BED mean vs. BED nud
As mentioned previously, BED mean defined in Eq. (16) always exceeds or equals the true BED (i.e., BED ) defined in Eqs.
nud
(3) and (4). To prove this statement for an arbitrary dose distribution in the target, one can express BED mean as the natural
logarithm of a geometric mean, that is,

  N voxel  1 N voxels 
−1
()
BED mean =−α ln   ∏ S j     (A1)
 
   j=1   
where S(i) = exp(−αBED voxel (i)). Conversely, the BED for the same dose distribution is given by the natural logarithm
nud
of the arithmetic mean, that is,
 1 N voxel 
−1
BED nud =−α ln   ∑ Sj   (A2)
()
 N voxel j=1 
Because the arithmetic mean is always equal to or greater than the corresponding geometric mean, we have

 1 N voxel    N voxel  1 1 N voxel 
ln
−  ∑ Sj ()  ≤−ln   ∏ Sj ()    (A3)
 
  N voxel j =1      j =1    
which implies that BED nud ≤ BED mean . Note that BED nud = BED mean if and only if the probability of survival is the same for
different voxels. That is, S(j) = const, j = 1, 2,…, N voxel .

Volume 2 Issue 4 (2024) 11 doi: 10.36922/arnm.4826