“…Such methods are typically based upon pencil beam algorithms (Petti 1992, Russell et al 1995, Hong et al 1996, Deasy 1998, Schneider et al 1998, Schaffner et al 1999, Szymanowski and Oelfke 2002, Taylor et al 2017 which perform well in homogeneous media, however they become less accurate when significant inhomogeneities are present, for instance in patients with tumors in the thoracic, head-and-neck, or lung regions (Sawakuchi et al 2008, Paganetti 2012, Yang et al 2012, Bueno et al 2013, Schuemann et al 2014. Monte Carlo (MC) methods, like Geant4 (Agostinelli et al 2003, Allison et al 2006, Fluka (Ferrari et al 2005, Böhlen et al 2014, MCNPX (Waters et al, 2005), etc., which can precisely describe particle transport through matter, are considered the most accurate methods for radiotherapy dose calculations (Schaffner et al 1999, Koch et al 2008, Taddei et al 2009, Harding et al 2014, Giantsoudi et al 2015, Dedes et al 2015. To overcome the limitation of long computing time for traditional MC codes, a variety of fast MC methods have been developed (Jia et al 2012, Giantsoudi et al 2015.…”