Mathematics Applied to Biology and Medicine
My main area of research at the moment involves developing multiscale mathematical models for the growth and spread of cancer. The research is interdisciplinary and I have collaborations with biologists and clinicians who provide data for my models.
Mathematical modelling of cancer
Cancer is a very complex multi-faceted disease. Despite many efforts, we still cannot explain all the observed
phenomena and, if necessary, make any desirable changes in the dynamics. Experimental methods alone are not sufficient
to build a consistent, systematic theory. However, over the past few years it has become apparent that there is an
opportunity to apply another approach - namely, to complement the traditional, heuristic experimental approach with
mathematical modelling. Mathematical modelling has the potential to overcome the essential difficulty related to
understanding biological processes - namely their spatio-temporal and multi-scale character. It allows one to simulate the
course of the disease and to optimize the therapeutic protocol.
Many cancer therapies are based on the idea of putting some kind of external stress on cancer cells e.g. chemotherapy,
radiotherapy. In order to improve the efficacy of these therapies, recent research has investigated how to combine them
more effectively and optimally - multimodal oncological strategies. A synergistic interaction between heat and radiation
dose as well as various cytotoxic treatments has already been validated in preclinical studies. Nevertheless, the precise
molecular-biological mechanisms of these effects are still under investigation.
Multiscale character of cancer
From a biological and pathological perspective cancer is present over many biological ``scales’’ from genetic,
intra-cellular, cellular, extra-cellular, tissue, organ and finally to whole populations. This requires much deep
understanding of complex processes, feedback mechanisms and ``multi-scale’’ phenomena. In mathematical terms, it
requires the study of highly nonlinear integrated systems.
The system characterization is based on the identification of three natural scales, which are also linked to different
stages of the disease. Therefore, we distinguish between processes at the intracellular, cellular and tissue scale.
The intracellular scale:
In response to changes in the external or internal cellular environment, the expression of particular genes and
consequently protein synthesis may change. Cascades of biochemical reactions that lead to signal transduction from
receptors located on the cell surface to the cell nucleus are called signalling pathways. These signalling pathways
constitute natural regulatory systems that, on the one hand, should ensure cell resistance to random changes in its
condition and on the other hand, should ensure a proper response to external.
The cellular scale:
At the cellular level, the key processes that are modelled are division, differentiation, apoptosis and interactions between
cells. These processes are regulated via signalling pathways. Regulatory proteins, whose production is triggered by
signalling pathways, initiate or modify the processes of cell division and death. In turn, cell differentiation, both normal
and pathological, influences the signalling pathway dynamics, which leads to subsequent changes at the cellular level.
Taking into account links between cellular models and the models described above at the sub-cellular scale is necessary in
order to describe how cells function.
The tissue scale:
Anomalous processes at the cellular level lead to the development of structures such as solid tumours, which, in turn,
affect the proper functioning of tissue and organs. These changes are observed at the level of whole cell populations. This
description is by its very nature phenomenological, but allows for a qualitative understanding of the whole system
depending on key parameters such diffusion coefficients, chemotaxis coefficients or reactions speeds (understanding the
links between these parameters and phenomena occurring at the lower scales is particularly important).
Different mathematical techniques and structures correspond to the above scales. For instance, models at the sub-cellular
scale are most often constructed using ordinary differential equations or Boolean networks. Multicellular systems, i.e.
models at the cellular scale, are usually developed in terms of nonlinear integro-differential equations (of Boltzman type)
or individual-based models, whereas tissue models usually involve free boundary problems and nonlinear partial
differential equations. The choice of the proper mathematical methods is usually linked to the precise biological questions
we want to address and type of available data.
The long term aim of my research is to develop a multiscale "virtual solid tumour", which may have the potential to have a genuine impact in optimizing and adjusting treatments to individual patient needs.