In recent years there has been a tremendous amount of research going on in the field of medicine to understand the complex processes behind various mechanisms in the human body, including the development and outcome of disease, multi-scale correlations and dependencies from the genome to the tissue level, and genotype-phenotype couplings of various diseases and syndromes. At the same time the advances in computer vision, data analysis, bioinformatics, and deep learning as well as the computing landscape, have made it possible to process, analyse and interpret complex mechanisms in huge amounts of data. This requires a strong collaboration between all these disciplines ranging from methodological and mathematical studies to clinical applications, so that the complex medical tasks can be solved with approaches from mathematical modelling, numerical analysis, uncertainty quantification, and machine learning. In particular, with the enormous amount of medical data collected today, such as MRI images, blood analyte concentrations, whole genome and exome sequencing, there is a need to apply data science and mathematical methods to handle the data. Since medical applications require high robustness and interpretability of the mathematical models, methods for uncertainty quantification and explainable AI are needed. In this sense, the medical applications give rise to mathematical challenges, that can lead to new methods and algorithms in different areas of mathematics, such as data analysis, mathematical modelling and machine learning. From a medical point of view, there is also a wide range of applications such as the development of patient-specific prevention and treatment strategies, and the prediction of disease development.

In the area of disease diagnosis, such as newborn screening  for rare metabolic diseases, data mining methods can be used to understand the complex relationships within the human metabolism and identify patterns. Due to the low prevalence of these diseases, an accurate diagnosis is a difficult task. Therefore, machine learning and mathematical modelling can be used to assist the diagnostic  process, by handling large amounts of unbalanced data and identifying the important analyte relationships to improve diagnosis.

Mathematical oncology
In cancer research, the complex biological processes of cancer initiation and progression have long been a hidden process. By using mathematical modelling, we are able to gain new, unprecedented medical insights, including the analysis of biological concepts and medical hypotheses about cancer development, and the prediction of clinical outcomes using existing clinical and molecular information. Therefore, the application of mathematics in the field of oncology will facilitate data interpretation and improve our understanding of carcinogenic processes. With this knowledge, we aim to predict the efficacy of current clinical approaches and to improve these prevention and treatment strategies for specific types of cancers.

Personalised medicine
In the field of personalised medicine, radiogenomics combines quantitative data from medical images with molecular genomic markers with the aim to study disease heterogeneity, find non-invasive biomarkers that could replace surgery, and to support biomedical decision making. In our research, we design and construct predictive models using deep learning and statistical methods that can aid in the diagnosis, prognosis, and treatment of patients. We focus our studies on oncology where radiogenomic biomedical data are routinely collected.

Cardiology
In cardiology collaborations between computer scientists and cardiovascular physicians are leading to a joint approach to improving the diagnosis and treatment of cardiovascular disease. The application of computational methods to large data sets enables the simulation of complex disease processes and the prediction of clinical phenotypes and their outcomes. Functional mathematical modelling and simulation allows virtual prediction of the behaviour and functionality of organ tissues, such as the human heart. To obtain reliable simulation results we apply uncertainty quantification techniques to describe noise in the data and assess model simplifications and simulation inaccuracies.

By working closely and sharing knowledge with medical and clinical experts, we aim to solve unanswered challenges in medicine and potentially discover relationships within the human body that have not been understood before. All with the ultimate goal of improving patient care.