In proton therapy, robust treatment planning is currently used to account for uncertainties in patient alignment and proton beam range. A way to overcome the limitations of robust treatment planning is to use probabilistic treatment planning, which can be computationally expensive due to the calculations of statistical measures of stochastic dose parameters. For this purpose, the method of Polynomial Chaos Expansion (PCE) has been introduced to alleviate computational costs. PCEs are helpful to evaluate the statistical measures analytically, or via sampling, and can usually be evaluated faster than through calculations with the dose engine. In this research, the work of Salverda et al. (2019) is extended upon improving probabilistic treatment planning with Polynomial Chaos Expansion for proton therapy. Probabilistic treatment plan optimizations with patient data in ErasmusMC's iCycle were found to be too costly for clinical use with computation times in the order of 1-3 weeks. The goal of this research is to alleviate the computational cost of probabilistic treatment planning, in which percentiles of stochastic dose volume parameters are used, with the aim of a proof of principle for clinical use.

New methodologies on the calculation of the value, gradient and Hessian of percentiles of dose volume parameters, and treatment plan optimizations with these methods, are tested on a simplified 3D geometry with a tumor and an organ in Matlab. The optimizations are performed with the `fmincon' solver, using the default interior-point algorithm. The construction of PCEs is performed with the OpenGPC package, developed by Z. Perkó et al. (2016).

Three exact improvements on the calculation of the value, gradient and Hessian of percentiles of dose volume parameters have been implemented and show an overall reduction in computation time of 30%. These improvements yield a similar speed-up in the total optimization time of a probabilistic treatment plan, depending on the dose parameters in the objectives and constraints, and their computation times. Furthermore, the accuracy of the gradient and Hessian of percentiles of dose volume parameters has been improved through the introduction of a monotonicity constraint on the PCE coefficients. To reduce the computation time even further, an approximation method is devised for the calculation of the gradient and Hessian of percentiles of dose volume parameters. This method shows a decrease in computation time from 5,600 seconds to only 220 seconds per calculation of the value, gradient and Hessian of a percentile of a dose volume parameter for the 3D geometry and a sample size of 500,000. The problem is that the approximation method is quite heuristic and its accuracy cannot be predicted in general. However, relatively accurate percentile approximations are produced with small absolute errors. Optimizations of probabilistic treatment plans are performed with the exact and approximate method, yielding small differences in terms of dose distributions due to differences in sample size. The cases with the approximation method yield the fastest optimization times and are shown to be 7.6 times faster, from 5.3 days to 16.5 hours, than the case with the exact method and a sample size of 500,000. The cases with the exact method show to be 2.5 and 1.2 times faster for a sample size of 50,000 and 250,000 than for a sample size of 500,000 respectively. The higher number of iterations in cases with the approximation method and cases with the exact method for smaller sample sizes outweigh the rapid increase in computation time for higher sample sizes with the exact method.

A decrease in computation time is shown for probabilistic treatment planning, in which percentiles of dose volume parameters are used, in general, and also in implementations in the 3D geometry with and without the approximation method. In future research, the decrease in computation time with the exact improvements should be investigated in optimizations with patient data in iCycle. Also, the approximation method should be tested on different geometries and for multiple uncertain parameters in the geometry to investigate whether this method is robust enough to be tested in optimizations with patient data in iCycle. As a conclusion, the new methodologies in this research do improve probabilistic treatment planning with Polynomial Chaos Expansion for proton therapy by alleviating computational cost, but a proof of principle for clinical use is not yet achieved.

The wind energy industry is growing more than ever before and wind energy as a renewable energy source has shown quite a potential over the years. Unfortunately, the power yield of a wind farm can fluctuate largely over time, which originates from fluctuating wind speed magnitudes. Convection of air, turbulence, humidity and radiation of heat are processes in the atmospheric boundary layer, the lowest region of the troposphere above Earth’s surface, that are responsible for these fluctuations. This region is characterized by a diurnal cycle in which two examinable cases can be found, namely the convective boundary layer (CBL) case during the daytime period and the stable boundary layer (SBL) case during the nighttime period of the diurnal cycle. In the SBL case, the earlier mentioned physical processes result in a cool surface layer and relatively little turbulence is present, whereas in the CBL case a relatively warm surface layer and high turbulence level are present. The Dutch Atmospheric Large Eddy Simulation (DALES) model is used in which wind turbines are implemented with the help of former TU Delft master student P.A. van Dorp. With DALES, a simulation is performed for each case and the output data of the two simulations are analysed and compared to find the optimal case in terms of total power yield of a configuration of two wind turbines with the second fully in the wake produced by the one in front. An arbitrary distance of 600m between the wind turbines is chosen, which corresponds to 7.5 times the diameter length of the wind turbines, and a hub height of 80m is chosen. First, the simulations are validated by examining the turbulent kinetic energy profiles. Then, the wind speed profiles over the domain are analysed, corresponding to the turbulent kinetic energy profiles and showing that the average wind speed at hub height for the SBL case is larger than for the CBL case. Furthermore, the wake profiles behind the turbines are displayed and compared for the two cases, showing little difference. Finally, the total power yield of this specific wind turbine configuration is calculated. With an average power yield of 1.71MW for the SBL case compared to 0.90MW for the CBL case, the SBL case is shown to be the optimal atmospheric boundary layer case in terms of power production for this specific wind turbine configuration.