Uncertainty quantification in a model of tumour invasion

Parameters appearing in mathematical models of tumour invasion are often difficult to assess experimentally and even if experimental values are available their accuracy might not be very good or they might have been obtained in a setting different from that what is modelled. Thus these parameters are uncertain and quantifying the effect of this uncertainty on the model solution or certain derived quantities of interest is beneficial for judging the value of the model and possibly for proposing required dedicated experiments. We present a framework for uncertainty quantification based on fast adaptive stochastic collocation on sparse grids. The advantage of this approach is that it can use an existing simulation environment for the model under investigation in a black-box fashion. We apply the framework to a model of tumour invasion and consider uncertainty in spatially homogeneous as well as heterogeneous parameters. Spatially heterogeneous parameters represent random fields and thus are, in general, infinite-dimensional objects. To make them amenable for a computational analysis, they first must be approximated by finite dimensional objects. In the case of correlated random fields, this can be done, for instance, by a Karhunen-Loeve expansion. Topics covered in this talk also include how uncertainty can be modelled, how the quantification proceeds, and how certain statistics of the quantities of interest are computed efficiently.