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I would like to make a comment on the “4π radiotherapy”; mentioned by Tarn et al. regarding the 4π radiotherapy for prostate cases. The concept of 4π radiotherapy was originally floated by Dong et al. in 2013; and subsequently used by several authors; calming to have delivered a radiotherapy technique which look into a tumour from 4π solid angle [1‐8].
The geometrical constriction of a teletherapy machine/linear accelerator mechanically represent a Cantilever, where head anchored at only one end with a vertical support from which it is protruding. A teletherapy machine having two additional degree of freedoms; a full arc gantry rotation of (0-2πc) and a half arc couch rotation (0-πc).
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Geometry of three dimensional Euclidian space, solid geometry, defines the angle obtained by a surface in terms of solid angle presented as following.where ds is the surface area and r is the radius vector can obtained a solid angle of 4πc at its centre as described below.
$$ \mathrm{d}\Omega =\frac{ds}{r^2} $$
Solid angle at the centre of a spherewhere, r, θ and φ are radius vector polar and azimuthal angle.
$$ \Omega =\frac{\mathrm{Area}}{r^2}=\frac{1}{r^2}\left[{\int}_{\theta =0}^{\pi }{\int}_{\upvarphi =0}^{2\pi}\left( rSin\theta d\varphi \right).\left( rd\theta \right)\right] $$
$$ =\frac{4{\pi r}^2}{r^2}=4{\uppi}^{\mathrm{c}}\ \left[={12.56}^{\mathrm{c}};{}^{\mathrm{c}}\ \mathrm{is}\ \mathrm{steradian}\right]. $$
Under the geometrical boundary condition of the linear accelerator rotational degree of freedom (gantry: 00–3600-00 and couch 900–00-2700; however 900–1800–2700 is inaccessible to couch) azimuthal angle integration reduces to 0-πc. Therefore maximum accessible solid angle for a linear accelerator machine is
$$ =\frac{1}{r^2}{\int}_{\theta =0}^{\pi }{\int}_{\upvarphi =0}^{\pi}\left( rSin\theta d\varphi \right).\left( rd\theta \right)=2{\uppi}^{\mathrm{c}};\mathrm{solid}\ \mathrm{angle}\ \mathrm{obtained}\ \mathrm{by}\ \mathrm{a}\ \mathrm{hemisphere}. $$
This type of hemispherical therapy delivery is only possible for two ends of the human that is either brain or foot. Rest of the length (head neck-thorax-abdomen-pelvis) of the human body is not accessible even for a 2πc radiotherapy. Therefore claimed to have “4π radiotherapy” for prostate does not hold geometrically.
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I would like to mention that, as an example, the solid angle created by a full arc (0-2πc) gantry rotation with a 40 × 40 cm2 field opening and couch angle at zero degree iswhich is (1/5)th of the 4πc. Solid angle further reduces with the multileaf collimator shaped or blocked fields.
$$ \Omega\ \mathrm{Full}\ \mathrm{ARC}=\frac{Area}{r^2}=\frac{1}{100\ {cm}^2}\ \left[2\uppi\ 100\ \mathrm{cm}\times 40\ \mathrm{cm}\right]={2.51}^{\mathrm{c}}, $$
To perform a “4π radiotherapy” a patient need to be point and radiotherapy machines should be able to move to any point on the surface of a spare; under the present design of any teletherapy machines like linear accelerator, tele-cobalt, tomotherapy (Accuray Inc., Madison, WI) or Cyber knife (Accuray Inc., Madison, WI) cannot perform a “4π radiotherapy”. Probably only Brachytherapy can be near to a “4π radiotherapy” approximating (highly) the source as a point source.
A generalised geometrical misconception of “4π radiotherapy” was floated in 2013 by Dong et al. and propagating up to date (Victoria et al.) [1‐8].
The technique described by the listed authors in this article could have been identified (or nomenclated) by something else but definitely not by “4π radiotherapy”. “4π Radiotherapy” is a geometrically non-viable and scientifically wrong concept; tagged with a fancy name to establish its superiority over generalized non-coplanar technique. Therefore the misconception about “4π radiotherapy” need to be corrected should not be used in future.
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