  Fast and accurate predictions of uncertainties in the computed dose are
crucial for the determination of robust treatment plans in radiation therapy.
This requires the solution of particle transport problems with uncertain
parameters or initial conditions. Monte Carlo methods are often used to solve
transport problems especially for applications which require high accuracy. In
these cases, common non-intrusive solution strategies that involve repeated
simulations of the problem at different points in the parameter space quickly
become infeasible due to their long run-times. Intrusive methods however limit
the usability in combination with proprietary simulation engines. In our
previous paper [51], we demonstrated the application of a new non-intrusive
uncertainty quantification approach for Monte Carlo simulations in proton dose
calculations with normally distributed errors on realistic patient data. In
this paper, we introduce a generalized formulation and focus on a more in-depth
theoretical analysis of this method concerning bias, error and convergence of
the estimates. The multivariate input model of the proposed approach further
supports almost arbitrary error correlation models. We demonstrate how this
framework can be used to model and efficiently quantify complex auto-correlated
and time-dependent errors.
