If Only I Had a Few Negative X-rays, I’d be All Set

In Chapter 16 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss intensity-modulated radiation therapy. The goal is to fire x-rays from many different directions, each direction having a different distribution of intensity, to provide a large dose to the tumor while sparing the surrounding normal tissue.

In classical radiotherapy, the beam was either of uniform fluence across the field, or it was shaped by an attenuating wedge placed in the field. Intensity-modulated radiation therapy (IMRT) is achieved by stepping the collimator leaves during exposure so that the fluence varies from square to square in Fig. 16.45 (Goitein 2008; Khan 2010, Ch. 20)

It was originally hoped that CT [computed tomography] reconstruction techniques could be used to determine the collimator settings at different angles. This does not work because it is impossible to make the filtered radiation field negative, as the CT reconstruction would demand. IMRT with conventional treatment planning improves the dose pattern (Goitein 2008; Yu et al. 2008), providing better sparing of adjacent normal tissue and allowing a boost in dose to the tumor.

What’s up with this talk in IPMB about “negative” radiation? To gain insight, read what Steve Webb wrote in his review article “The Physical Basis of IMRT and Inverse Planning” (British Journal of Radiology, Volume 76, Pages 678-689, 2003).

With the glorious wisdom of hindsight some very early developments—we may call them pre-history—might be considered part of the development of IMRT. The mathematician George Birkhoff showed in 1940 that any drawing could be made up of lines of varying pencil thickness so long as negative pencils were allowed [20]. If we read ‘‘X-rays’’ for ‘‘pencils’’ and ‘‘dose distribution’’ for ‘‘picture’’ the analogy with IMRT is clear. Sadly there are no negative X-rays or uncomplicated tumour control would be 100% guaranteed...

[20]. Birkhoff GD (1940) “On Drawings Composed of Uniform Straight Lines,” Journal de Mathématiques Pures et Appliquées, Volume 19, Pages 221–236.

Oh well,

The best laid schemes o' mice an' men 
Gang aft a-gley

If only I had a few of those negative x-rays, I’d be all set.

An example of how a drawing can be made up of lines (in this case, of uniform thickness), based on a figure from “On Drawings Composed of Uniform Straight Lines” by George Birkhoff. An analogous problem is faced when designing a treatment plan using intensity-modulated radiation therapy.