Research interests:

mathematical and computational modeling of

systems_biology Mathematical models of tumor dynamics have become more accurate and accepted in recent years and enable a better prediction of biological pathways that may be involved in the initiation and development of a tumor. The big aim for theoretical and practical oncologists is to find ways to treat the disease or improve the life of patients. Mathematical models help to identify crucial mechanisms to compare different treatments or design new treatment strategies. With the growing acceptance of models of tumor development the subsequent application of treatment planning will play an increasingly important role in the clinic. Using models one can compare different approaches or design new treatment strategies, which then can be tailored to individual patient data. With more information on cancer relevant to modelling becoming available the new well parameterised models have the power to predict responses to various treatment techniques such as drug scheduling in chemotherapy, immunotherapy, and radiotherapy as well as combinations of these.

self_metastases Cancer stem cells and self-metastases.
Tumors are intrinsically heterogeneous. The majority of tumor cells have limited life span and replicative potential, and only a small minority — so-called cancer stem cells — live forever, divide infinitely and potentially produce more such stem cells. It is these stem cells that determine tumor formation, and their dynamics are counterintutively inhibited by their non-stem progeny. Only a high migration rate can liberate stem cells and enable their migration to seed new clones in the vicinity of the original cluster. In this manner, the tumour continually ‘self-metastasizes’.

We use computer models to define the behavior of single cells, and then let single cells populate a computational domain. As the number of cancer cells increases over time, competition for environmental resources (such as space) defines population dynamics. A result is a cancer cell population — a tumor — growing sub-exponentially. Tumor progession is dictated by the ability of stem cells to form self-metastases that together form a malignant invasive morphology.

Numerical analysis and computational simulation of partial differential equation models in mathematical biology is now an integral part of the research in this field. Increasingly, we are seeing the development of partial differential equation models in more than one space dimension, and it is therefore necessary to generate a clear and effective visualization platform between mathematicians and biologists to communicate the results. The mathematical extension of models to three spatial dimensions from one or two is often a trivial task, whereas the visualization of the results is more complicated. We apply the established Marching Cubes volume rendering technique to the study of a mathematical model of malignant solid tumor growth and invasion in an irregular heterogeneous three-dimensional domain, i.e. the female breast. Due to the different variables that interact with each other, more than one data set may have to be displayed simultaneously which can be realized through transparency blending.