With breast cancer being the leading cause of death in the Netherlands, while also being expected to have double the amount of cases in the next ten years, it is vital that treatment is optimised. Over the last decade, research has been done to incorporate mathematical modelling in this process, especially in the case of HER2+ breast cancer patients. This aggressive form has poor chances of survival, but responds well to chemotherapy and is expected to be quite predictable. In previous works of N. Oudhof and E. Slingerland, a mechanically coupled reaction-diffusion model with an extension of chemotherapy was implemented in 2D, using three MRI scans to predict tumour response. The first two scans, taken right before and during treatment, are used to find patient-specific parameters corresponding to proliferation, tumour movement and chemotherapy efficacy. This calibrated model is then used to predict the third scan, taken at the end of treatment. Calibrating this model to individual patients takes up to a day, however. \\

This thesis aims to extend this model to a higher resolution 3D setting, which should also hand doctors predictions within a working day. To increase speed, the linear-elastic sub-problem of finding shear stress due to tissue types and tumour growth was first optimised. With a novel Laplacian preconditioner in the Conjugate Gradient method, using FFT's and a tridiagonal solver, the time needed was vastly improved. Second, the maximum order of error needed for accurate temporal integration was confirmed to be quite high. Hence a state-of-the-art Parareal implementation was made, using Runge-Kutta 4 and Crank Nicholson as the respective fine and course solver, which succeeded in being both faster and more accurate than simple first-order methods. Then it was found that the underdeterminedness of the problem is best tackled using Total Variation Regularisation on the proliferation parameter and Tikhonov Regularisation on the other two parameters. This ensures that unique solutions can be found in reasonable time, that properly reflect expectations of these parameters in practical settings. The best set of parameters were found fastest with Powell's Dog-leg method, for which a novel way of finding the Jacobian analytically was used. \\

On simulated data, the error in the amount of tumour cells in the third image went down to single-digit percentage rate, with a maximal shape correlation coefficient. This prediction can be made within a few hours, which means that the feasibility of solving this problem in practical settings has been established successfully. On the real data, the calibration succeeded in calibrating the model on the first two scans, but the predictions still have room of improvement. The most important suggestion of this work is that more research must be done in the verification of the suitability of this model to the available real data, using the techniques presented here. Further improvements can be made by exploring more possibilities of using parallelisation, and by obtaining more data. Before obtaining more data, one should investigate the impact of the timing of the scans on the calibration results. ... Read more

As long as a wave has a large enough wavelength, it should reflect off of smooth surfaces specularly: that is what the law of reflection states. This phenomenon is widely known, and used in for instance sonars. Electrons that reflect off of boundaries within conductors should abide this law as well, due to the particle-wave duality. However, it was recently discovered that for disordered graphene boundaries electrons scatter diffusively even when their wavelengths should be long enough not to. Graphene is a 2-dimensional semiconductor with linear dispersion at low energies, making it a widely researched topic as its applications in future electronic parts are promising. Its bilayer counterpart is made by connecting two layers on top of each other, which in contrast has a quadratic dispersion at low energies. In this work, the specularity of electron reflections within bilayer graphene were studied. This was done by constructing a tight-binding model, where random potentials on the outermost atoms represented imperfections of the lattice. Subsequently the variance of the scattering angle and the scattering phase was analysed, which can be derived using the scattering matrix of themodel. This was done numerically for both, and analytically for the phase as well. It was discovered that the variance of the scattering angle increases cubically with the Fermi wavelength of the electrons. This happens no matter what the distance between the Fermi energy and the disorder mean is, as long as the disorder strength is nonzero. The behaviour of these reflections is the exact opposite of what the law of reflection states. Furthermore, the variance of the scattering phase remains constant when the Fermi energy is close to the disorder mean, and changes inversely proportional to the boundary periodicity if the Fermi energy is further away. Consequently, the variance of the scattering phase versus the Fermi wavelength and the variance of the scattering phase versus the boundary periodicity do not share the same trend in bilayer graphene. More research is needed to verify this phenomenon, which could be done using the continuum description or a magnetic focusing device. ... Read more