Analytical probabilistic modeling for radiation therapy treatment planning.

This paper introduces the concept of analytical probabilistic modeling (APM) to quantify uncertainties in quality indicators of radiation therapy treatment plans. Assuming Gaussian probability densities over the input parameters of the treatment plan quality indicators, APM enables the calculation of the moments of the induced probability density over the treatment plan quality indicators by analytical integration. This paper focuses on analytical probabilistic dose calculation algorithms and the implications of APM regarding treatment planning. We derive closed-form expressions for the expectation value and the (co)variance of (1) intensity-modulated photon and proton dose distributions based on a pencil beam algorithm and (2) the standard quadratic objective function used in inverse planning. Complex correlation models of high dimensional uncertain input parameters and the different nature of random and systematic uncertainties in fractionated radiation therapy are explicitly incorporated into APM. APM variance calculations on phantom data sets show that the correlation assumptions and the difference of random and systematic uncertainties of the input parameters have a crucial impact on the uncertainty of the resulting dose. The derivations regarding the quadratic objective function show that APM has the potential to enable robust planning at almost the same computational cost like conventional inverse planning after a single probabilistic dose calculation. Beneficial applications of APM in the context of radiation therapy treatment planning are feasible.