Glioma follow white matter tracts: a multiscale DTI-based model

Abstract

Gliomas are a class of rarely curable tumors arising from abnormal glia cells in the human brain. The understanding of glioma spreading patterns is essential for both radiological therapy as well as surgical treatment. Diffusion tensor imaging (DTI) allows to infer the white matter fibre structure of the brain in a noninvasive way. Painter and Hillen (J Theor Biol 323:25-39, 2013) used a kinetic partial differential equation to include DTI data into a class of anisotropic diffusion models for glioma spread. Here we extend this model to explicitly include adhesion mechanisms between glioma cells and the extracellular matrix components which are associated to white matter tracts. The mathematical modelling follows the multiscale approach proposed by Kelkel and Surulescu (Math Models Methods Appl Sci 23(3), 2012). We use scaling arguments to deduce a macroscopic advection-diffusion model for this process. The tumor diffusion tensor and the tumor drift velocity depend on both, the directions of the white matter tracts as well as the binding dynamics of the adhesion molecules. The advanced computational platform DUNE enables us to accurately solve our macroscopic model. It turns out that the inclusion of cell binding dynamics on the microlevel is an important factor to explain finger-like spread of glioma.

Lyapunov–Schmidt and Centre Manifold Reduction Methods for Nonlocal PDEs Modelling Animal Aggregations

“…If we substitute this state into (B.1) we recover model (2.22), with the exception of equation (2.22d) which is now a steady-state equation. We note here that while different microscale-macroscale models (Stinner et al, 2014(Stinner et al, , 2015(Stinner et al, , 2016Meral et al, 2015a,b) and mesoscale models (Lorenz & Surulescu, 2014;Engwer et al, 2015;Perthame et al, 2016;Engwer et al, 2017;Hunt & Surulescu, 2017) for tumour-ECM interactions via integrins have been developed over the last two decades, to our knowledge there are not many studies that compare and contrast the outcomes of different ways the integrins are incorporated into these different models.…”

“…Moreover, the coupling of PDEs for cell movement with ODEs for the dynamics of cell receptors has been considered not only for cancer cells but also for other types of cells, such as Dictyostelium cells (Höfer et al, 1995a,b). However, the past 10-15 years have seen the development of structured cell population models (described by mesoscale models) where the macroscopic cell dynamics is dependent on a microscopic variable that characterises the cells at molecular level (Erban & Othmer, 2004Xue et al, 2011;Lorenz & Surulescu, 2014;Engwer et al, 2015;Perthame et al, 2016;Engwer et al, 2017;Hunt & Surulescu, 2017). In Appendix B we present a modified mesoscale version of model (2.13)-(2.16) where we assume that the two cancer cell populations are structured by an internal variable that describes the integrin level.…”

“…To this end, we assume that the two cancer cell populations depend not only on time (t) and space (x), but also on an internal variable c 0 that characterises the level of bound integrins: u i (x,t, c). The new mesoscale model for the dynamics of cancer cell populations structured by cell-bound integrins is given by the following equations: One can eliminate the derivative ∂ /∂ c from equations (B.1a)-(B.1b) by integrating these equations over c, and obtaining new macroscopic equations defined in terms of the averaged tumour cell variables U 1,2 (x,t) = u 1,2 (x,t, c)dc (Erban & Othmer, 2005;Engwer et al, 2015;Domschke et al, 2107). Other studies obtained different macroscopic limits of the mesoscopic structuredpopulation models by considering the assumption that the internal (structure) variable evolves on a much faster time scale that is described by a small parameter ε (Perthame et al, 2016).…”

“…Also, while previous nonlocal models have assumed constant adhesive interactions, here we will assume that these adhesive interactions depend on the level of integrins. Note that instead of assuming a kinetic (mesoscale) approach for cell-dependence on integrins as in Lorenz & Surulescu (2014); Engwer et al (2015); Perthame et al (2016); Engwer et al (2017); Hunt & Surulescu (2017), here we follow a microscale-macroscale approach similar to Stinner et al (2014Stinner et al ( , 2015Stinner et al ( , 2016; Meral et al (2015a,b) and couple the parabolic-hyperbolic model for spatial cell dynamics with an ODE for integrins dynamics. We then investigate this complex multiscale model in terms of pattern formation, with particular attention being payed to the role of mutation rate on the coexistence (or not) of cell sub-populations that form stationary aggregations or travelling wave patterns.…”

“…In particular, Domschke et al (2014) also assumed mutation between different cell populations. A different class of nonlocal models for cancer invasion has been developed in the context of the velocity-jump framework introduced in Othmer et al (1988) and the kinetic models for active particles framework described, for example, in Bellomo et al (2010); see the transport models in Kelkel & Surulescu (2012); Lorenz & Surulescu (2014); Engwer et al (2015Engwer et al ( , 2017; Hunt & Surulescu (2017).…”