  Purpose: Variations in proton Relative Biological Effectiveness (RBE) with
Linear Energy Transfer (LET) remain one of the largest sources of uncertainty
in proton radiotherapy. This work seeks to identify physics-based metrics which
can be applied to reduce this biological uncertainty.
  Materials and Methods: Three different physical metrics - dose, dose $\times$
LET and a complexity-weighted dose (CWD, Dose $\times$ (1+$\kappa LET_D$) )
were compared with in vitro experimental studies of proton RBE and clinical
treatment plans analysed using RBE models. The biological effects of protons in
each system were plotted against these metrics to quantify the degree of
biological uncertainty introduced by RBE variations in each case.
  Results: When the biological effects of protons were plotted against dose
alone, significant biological uncertainty was introduced as the LET-dependence
of RBE was neglected. Plotting biological effects against dose $\times$ LET
significantly over-estimated the impact of LET on cell survival, leading to
similar or greater levels of biological uncertainty. CWD, by contrast,
significantly reduced biological uncertainties in both experiments and clinical
plans. For prostate and medulloblastoma treatment plans, biological
uncertainties were reduced from $\pm$ 5% to less than 1%.
  Conclusions: While not a replacement for full RBE models, physics-based
metrics such as CWD have the potential to significantly reduce the
uncertainties in proton planning which result from variations in RBE. These
metrics may be used to identify regions in normal tissues which may see
unexpectedly high effects due to end-of-range elevations of RBE, or as a tool
in optimisation to deliver uniform biological effects.
