A Review on the Use of Grid-Based Boltzmann Equation Solvers for Dose Calculation in External Photon Beam Treatment Planning

Table 1

A summary describing information of some previous investigations for the accuracy of D-LBTE solvers in predicting the doses in heterogeneous simple geometric phantoms using single open fields.

Published investigations	Gifford et al. 2006 [22]	Vassiliev et al. 2010 [25]	Bush et al. 2011 [27]	Han et al. 2011 [28]	Kan et al. 2012 [30]	Lloyd and Ansbacher 2013 [31]
Beam energy	18 MV	6 and 18 MV	6 and 18 MV	6 and 18 MV	6 MV	6 and 18 MV
Field sizes	1.5×1.5 cm^2	2.5×2.5 cm^2; 5.0×5.0 cm^2; 10.0×10.0 cm^2	4.0×4.0 cm^2; 10.0×10.0 cm^2; 15.0×10.0 cm^2	2.5×2.5 cm^2; 5.0×5.0 cm^2; 10.0×10.0 cm^2	2.0×2.0 cm^2; 3.0×3.0 cm^2; 5.0×5.0 cm^2	10.0×10.0 cm^2
Phantom(s); geometry	One multilayer phantom: water (0–3 cm),; aluminium, Al (3–5 cm),; lung (5–12 cm),; water (12–30 cm)	One multilayer phantom: water (0–3 cm),; bone (3–5 cm),; lung (5–12 cm),; water (12–30 cm)	Two phantoms: (i) one with a single insert of normal lung, light lung, or air in water,; (ii) a bone/lung phantom with several disk-shaped bony structures	One multilayer phantom: water (0–3 cm),; bone (3–5 cm),; lung (5–12 cm),; water (12–30 cm)	30.0×30.0×30.0 cm^3 of water containing 5.0×5.0×30.0 cm^3 of air	20.0×20.0×20.0 cm^3 of muscle cube containing 2.0×2.0×18.0 cm^3 of stainless steel or titanium alloy
Monte carlo simulation	EGS4/Presta,; 0.3% statistical uncertainty,; resolution: 0.5×0.5×0.2 cm^3 voxels	DOSXYZnrc,; <0.1% statistical uncertainty, resolution: 0.2×0.2×0.2 cm^3 voxels, 0.1 cm laterally in penumbra region	DOSXYZnrc; ~1% statistical uncertainty in media except up to 4.5% in air, resolution: 0.25×0.25×0.25 cm^3 voxels	DOSXYZnrc,; <1% statistical uncertainty,; resolution: 0.2×0.2×0.2 cm^3 voxels for most volume, 0.1×0.1×0.2 cm^3 near water/bone and bone/lung interfaces	EGS4/Presta,; 2.0% statistical uncertainty,; resolution: 1/10 of field dimensions with 0.2 mm bin thickness	DOSXYZnrc,; ~1% statistical uncertainty,; resolution: 0.2×0.2×0.2 cm^3 voxels
D-LBTE solver	Attila code	Acuros; (Transpire, Inc.)	AXB of version 10	AXB of version 10	AXB of version 10	AXB of version 11
Dose distribution examined	PDD	PDD and lateral profiles	PDD and lateral profiles	PDD, lateral profiles, and 3D gamma evaluation	PDD	PDD and lateral profiles
Difference between D-LBTE solver and Monte Carlo simulation	Average discrepancy is 1.4%, with 2.2% maximum discrepancy observed at water/Al interface	For 6 MV, max. discrepancy < 1.5%, with DTA < 0.7 mm in the build-up region. For 18 MV, max. discrepancy < 2.3% with DTA < 0.3 mm in the build-up region	Discrepancies were within 2% in lung, 3% in light lung, up to 4.5% in air, 1.8% in bone, with slightly larger discrepancy (up to 5%) at interfaces	For 6 MV, average discrepancy of 1.1% in PDD and 1.6% in dose profiles. For 18 MV, average discrepancy of 1.6% in PDD and 3.0% and dose profiles	Discrepancies are mostly within 2%, with slightly higher discrepancy (up to 6%) at the air/tissue interface in the secondary build-up region	In general good agreement between AXB and MC, with an average gamma agreement with a 2%/1mm criteria of 91.3% to 96.8%