HER2+ breast cancer patients, as observed by oncologist Agnes Jager from Erasmus Medical Centre (EMC), often achieve radiologic complete response (rCR) earlier than expected under standard treatments. To address this, Jager has partnered with Delft University of Technology to develop a computational model aimed at personalizing treatment schedules, potentially reducing chemotherapy cycles and minimizing side effects.

Building on previous MSc theses by Nathalie Oudhof, Eva Slingerland, and Rutger Engelberts, this thesis aims to improve the predictive capability of the Drug-Induced Mechanically Coupled Reaction-Diffusion (DI-MRCD) model and test its performance on a larger dataset consisting of 13 patients. The DI-MRCD model combines dynamic contrast-enhanced (DCE) and diffusion-weighted imaging (DWI) magnetic resonance imaging (MRI) data with patient-specific parameters to simulate the reaction of breast cancer tumours on chemotherapy. Key improvements include optimizing and generalizing the pre-processing pipeline for a larger patient cohort and enhancing input reliability by independently computing apparent diffusion coefficients (ADC).

Further refinements to the DI-MRCD model include updates to chemotherapy and shear modulus parameters, switching to a Trust Region Reflective (TRF) optimization method, and introducing a tissue-specific proliferation rate and natural cell death term. Despite these enhancements providing more insight, control, and making the model biologically more realistic, the model struggled to converge, highlighting the challenge of fitting patient-specific parameters with limited data points.

Future improvements could include resolving the convergence issues, incorporating additional calibration parameters, allowing the proliferation rate to travel with tumour cells as they diffuse, incorporating chemotherapy doses into the chemotherapy term, using Bayesian optimization for better parameter estimation, and making the results more explanatory by integrating other patient data.

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.

Every year, thousands of women are diagnosed with breast cancer. To increase their survival chance and improve the treatment plans available, it is important to accurately predict the patient's response to treatment. For HER2+ breast cancer patients, normally a set number of chemotherapy rounds are administered. However, perhaps some patients would benefit from more chemotherapy or conversely, fewer rounds are sufficient. A reaction-diffusion model as well as a mechanically coupled reaction-diffusion model are used to predict the tumour cellularity. Using two magnetic resonance imaging (MRI) scans, a patient-specific model is calibrated to calculate the response for each individual patient. A third MRI scan is used to validate the results. Several preprocessing techniques are used to obtain the necessary information from the MRI scans such as tumour cellularity, segmentation of tissues and breast region.

The model calculations are performed in 2-dimensions and the slice thickness chosen determines how much effect the incorporation of surrounding tissue has. Using thicker slices means that the surrounding tissue has a larger effect on the tumour growth. For patients that respond well to treatment, the models are both able to calculate the downward trend in tumour cellularity. For thin slices (1.6 mm), both models perform similarly. Using thicker slices (6.5 mm), the mechanically coupled model shows better performance. The most important additions to the model are implementation in 3-dimensions so the slice thickness no longer has an effect. Most importantly, the chemotherapy should be incorporated to better obtain a patient-specific prediction of the response to treatment.

HER2+ breast cancer patients suffer from aggressive tumours that often respond well to treatment compared to HER2- patients. Currently, patients are all treated with the same number of anti-tumour treatments, although the required number to eradicate all tumour cells varies between patients. The goal is thus to have an individualised model based on an MRI before the start of therapy and an MRI after several rounds of therapy that predicts how many treatments are needed to eradicate all tumour cells. In the previous MSc thesis by N. Oudhof, a two-dimensional spatiotemporal mechanically coupled reaction-diffusion model was implemented. This was used to model the number of tumour cells in each position of the breast, where the mechanics are included to simulate the behaviour of the surrounding tissue.

The goal of this thesis is to improve that model by improving the calibration in which the patient-specific parameters are optimised, by taking into account the treatment schedule of the patients and by extending the model to three dimensions. For calibration, the first and second MRI scans are used and a third MRI is used for validation. To simulate the chemotherapy, both the Kety-Tofts model and a Normalised Blood Volume Map are applied, of which the latter appeared the best option. In 2D, four models are implemented: the reaction-diffusion models with and without mechanics and with and without chemotherapy term. These were compared by analysing the predicted and measured tumour densities for a cohort of three HER2+ patients. For 3D, only the basic reaction-diffusion and chemotherapy-incorporated reaction-diffusion models are implemented. The models are compared in terms of mean squared error, global relative error and concordance correlation coefficient. For both 2D and 3D, the models without chemotherapy gave slightly better results in terms of these measures, although the differences between the models were rather small. To determine the required number of therapy rounds, it is however better to use the chemotherapy-incorporated models because they offer more possibilities to simulate different treatment strategies. The computation time of the 3D implementation should be reduced before the mechanics can be included, which will allow conclusions to be drawn on which model is most accurate. Other directions for future improvements include using the second MRI to calculate the chemotherapy concentrations after calibration, adding an immunotherapy term and increasing the number of patients.