  The high precision and conformity of intensity-modulated particle therapy
(IMPT) comes at the cost of susceptibility to treatment uncertainties in
particle range and patient set-up. Dose uncertainty quantification and
mitigation, which is usually based on sampled error scenarios, however becomes
challenging when computing the dose with computationally expensive but accurate
Monte Carlo (MC) simulations. This paper introduces an importance
(re-)weighting method in MC history scoring to concurrently construct estimates
for error scenarios, the expected dose and its variance from a single set of MC
simulated particle histories. The approach relies on a multivariate Gaussian
input and uncertainty model, which assigns probabilities to the initial phase
space sample, enabling the use of different correlation models. Exploring and
adapting bivariate emittance parametrizations for the beam shape, accuracy can
be traded between that of the uncertainty or the nominal dose estimate. The
method was implemented using the MC code TOPAS and tested for proton IMPT plan
delivery in comparison to a reference scenario estimate. We achieve accurate
results for set-up uncertainties ($\gamma_{3mm/3\%} \geq 99.99\%$) and
expectedly lower but still sufficient agreement for range uncertainties, which
are approximated with uncertainty over the energy distribution ($\gamma_{3
mm/3\%} \geq 99.50\%$ ($E[\boldsymbol{d}]$), $\gamma_{3mm/3\%} \geq 91.69\%$
($\sigma(\boldsymbol{d})$) ). Initial experiments on a water phantom, a
prostate and a liver case show that the re-weighting approach lowers the CPU
time by more than an order of magnitude. Further, we show that uncertainty
induced by interplay and other dynamic influences may be approximated using
suitable error correlation models.
