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 within the human body, including development and outcome of diseases, 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, analyze and interpret complex mechanisms in huge amount of data. Hence, a strong collaboration between all these disciplines ranging from methodological and mathematical studies to clinical applications is needed such that the complex medical tasks can be solved with approaches from mathematical modeling, numerical analysis, uncertainty quantification, and machine learning. In particular, with the enormous amount of medical data collected nowadays, such as MRI images, blood analyte concentration, 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 of Uncertainty Quantification and Explainable AI are in need. In this sense, the medical applications give rise to mathematical challenges, which can lead to new methods and algorithms in various fields of mathematics, like data analysis, mathematical modeling and machine learning. From a medical point of view, there is also a wide range of application such as the development of patient-specific prevention and treatment strategies, and disease course predictions.

In the area of disease diagnosis, such as newborn screening  for rare metabolic diseases, data mining methods can be applied to understand the complex relationships within the human metabolism and detect patterns. Due to the low prevalence of these disease, an accurate diagnosis is a difficult task. Therefore, methods of machine learning and mathematical modeling can be applied to support the diagnosis procedure, by handling huge amount of imbalanced data and detecting the important analyte relationships to enhance the diagnosis.

In cancer research, the complex biological processes of cancer development and progression are for a long time a hidden process and can thus, not be observed in the human body. Using mathematical modeling, we are able to gain novel, unprecedented medical insights, including the analysis of biological concepts and medical hypotheses about cancer evolution, and the prediction of clinical outcomes using existing clinical and molecular information. Therefore, applying 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 those prevention and treatment strategies for specific types of cancer.

In the field of personalized medicine, radiogenomics combines quantitative data from medical images with molecular genomic markers with the aim to study heterogeneities in diseases, to find non-invasive biomarkers that might replace surgical interventions, and to support biomedical decisions. In our research, we design and construct predictive models through deep learning and statistical methods that can aid in patient diagnosis, prognosis, and treatment. We focus our studies on oncology where radiogenomic biomedical data are routinely collected.

In cardiology collaborations between computer scientists and cardiovascular physician scientists lead to a joint approach to enhance the diagnosis and treatment of cardiovascular diseases. The application of computational methods for large data sets makes it possible to simulate complex disease processes as well as predict clinical phenotypes and their outcomes. Functional mathematical modeling and simulations allow to virtually reproduce or predict patient-individually the behavior and functionality of organ tissues, e.g. within the human heart. To obtain reliable simulation results we apply techniques of uncertainty quantification to describe noise in the data and assess model simplifications and simulation inaccuracies.

Through a strong collaboration and exchange with medical and clinical experts, utilizing their comprehensive knowledge, we follow the aim of solving unanswered challenges in medicine, and possibly to discover links within the human body, which could not be understood before. This is all done with keeping the overall goal of improving patients' care in mind.