Probabilistic Treatment Planning with Polynomial Chaos Expansion for Proton Therapy

Abstract

Radiotherapy is an important treatment type for patients with cancer. An advantage of state of the art proton therapy with respect to traditional photon therapy is the spatial energy deposition of protons, which is characterized by the Bragg peak. Due to this particular course of energy deposition, the tumor can be irradiated more precisely and as a consequence more healthy tissue can be spared. A drawback is that proton therapy is more sensitive to uncertainties like misalignment of the patient, which is referred to as a set-up error. Robust optimization is the current way to account for such uncertainties during which a few discrete error scenarios are included in the planning. Due to the nature of the uncertainties, probabilistic optimization, on the other hand, is more promising, since it can handle many more scenarios with their occurrence probability taken into account as well. To investigate probabilistic planning, Polynomial Chaos methods were used in this research. Polynomial Chaos can approximate a stochastic response, depending on for example the set-up error, by a series expansion in terms of polynomials. The advantage is that typically the Polynomial Chaos Expansion (PCE) can be evaluated much faster than the stochastic response itself. In this research an investigation is done how PCE can be used with probabilistic treatment planning for proton therapy. First, an overview is given on how PCE can be used to describe probabilistic objective functions. It has been found that in general it is possible to describe random set-up errors with PCE without the need of considerably additional computational power. Moreover, polynomials in dose are especially suitable when fractionation is taken into account. Second, a simple one-dimensional geometry consisting of a tumor with a surrounding organ has been considered. The effects of fractionation on the dose distribution has been calculated with PCE and the results were in accordance with earlier research by Unkelbach et al. (2018). Third, a simplified three-dimensional geometry which mimics a spinal cord with surrounding tumor has been optimized probabilistically with PCE. The expected value of the sum of the quadratic differences between the dose and the prescribed dose has been minimized. The effects of probabilistic optimization became clear from both the nominal and the expected dose distribution. This objective function (i.e. the expected value of a quadratic function of the dose) turned out to be extremely suitable for probabilistic optimization with PCE, however its clinical relevance is less clear.
Last, a real skull base meningioma patient has been considered. As objective function, a percentile of the dose volume parameter has been used with PCE. This function demonstrated a bumpy dependence on the beam intensity and it has been proven that it cannot have a negative gradient. Consequently, a high step size is recommended to approximate the gradient and, if needed, the Hessian. Probabilistic treatment plans were obtained with a single probabilistic objective, with two probabilistic objectives and with nominal constraints and objectives on OARs. The results have been compared with a robust treatment plan according to the recipes of Ter Haar et al. (2018). It has been found that the probabilistic treatment plan shows a more conformal dose distribution, since it does not exhibit any preferred direction, contrary to the robust treatment plan. As a consequence, the automatic extension of the tumor is smaller. A clear benefit has been demonstrated in favor of the probabilistic treatment plan. Besides that, a major advantage of probabilistic optimization with respect to robust optimization is that it makes it possible to directly steer to a probabilistic goal independent of the patient and the treatment site. The computation time of probabilistic planning (in the order of 1-3 weeks) is unfortunately still too high for clinical practice.