Complexity and mathematical tools toward the modelling of multicellular growing systems

A multiscale model for glioma spread including cell-tissue interactions and proliferation

Glioma is a broad class of brain and spinal cord tumors arising from glia cells, which are the main brain cells that can develop into neoplasms. They are highly invasive and lead to irregular tumor margins which are not precisely identifiable by medical imaging, thus rendering a precise enough resection very difficult. The understanding of glioma spread patterns is hence essential for both radiological therapy as well as surgical treatment. In this paper we propose a multiscale model for glioma growth including interactions of the cells with the underlying tissue network, along with proliferative effects. Our current accounting for two subpopulations of cells to accomodate proliferation according to the go-or-grow dichtomoty is an extension of the setting in [16]. As in that paper, we assume that cancer cells use neuronal fiber tracts as invasive pathways. Hence, the individual structure of brain tissue seems to be decisive for the tumor spread. Diffusion tensor imaging (DTI) is able to provide such information, thus opening the way for patient specific modeling of glioma invasion. Starting from a multiscale model involving subcellular (microscopic) and individual (mesoscale) cell dynamics, we perform a parabolic scaling to obtain an approximating reaction-diffusion-transport equation on the macroscale of the tumor cell population. Numerical simulations based on DTI data are carried out in order to assess the performance of our modeling approach.

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

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.

Modelling the lymphatic system: challenges and opportunities

The lymphatic system is a vital part of the circulatory and immune systems, and plays an important role in homeostasis by controlling extracellular fluid volume and in combating infection. Nevertheless, there is a notable disparity in terms of research effort expended in relation to the treatment of lymphatic diseases in contrast to the cardiovascular system. While similarities to the cardiovascular system exist, there are considerable differences in their anatomy and physiology. This review outlines some of the challenges and opportunities for those engaged in modelling biological systems. The study of the lymphatic system is still in its infancy, the vast majority of the models presented in the literature to date having been developed since 2003. The number of distinct models and their variants are few in number, and only one effort has been made thus far to study the entire lymphatic network; elements of the lymphatic system such as the nodes, which act as pumps and reservoirs, have not been addressed by mathematical models. Clearly, more work will be necessary in combination with experimental verification in order to progress and update the knowledge on the function of the lymphatic system. As our knowledge and understanding of its function increase, new and more effective treatments of lymphatic diseases are bound to emerge.

Kinetic theory approach to modeling of cellular repair mechanisms under genome stress

Under acute perturbations from outer environment, a normal cell can trigger cellular self-defense mechanism in response to genome stress. To investigate the kinetics of cellular self-repair process at single cell level further, a model of DNA damage generating and repair is proposed under acute Ion Radiation (IR) by using mathematical framework of kinetic theory of active particles (KTAP). Firstly, we focus on illustrating the profile of Cellular Repair System (CRS) instituted by two sub-populations, each of which is made up of the active particles with different discrete states. Then, we implement the mathematical framework of cellular self-repair mechanism, and illustrate the dynamic processes of Double Strand Breaks (DSBs) and Repair Protein (RP) generating, DSB-protein complexes (DSBCs) synthesizing, and toxins accumulating. Finally, we roughly analyze the capability of cellular self-repair mechanism, cellular activity of transferring DNA damage, and genome stability, especially the different fates of a certain cell before and after the time thresholds of IR perturbations that a cell can tolerate maximally under different IR perturbation circumstances.

Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review

We briefly review hyperbolic and kinetic models for self-organized biological aggregations and traffic-like movement. We begin with the simplest models described by an advection-reaction equation in one spatial dimension. We then increase the complexity of models in steps. To this end, we begin investigating local hyperbolic systems of conservation laws with constant velocity. Next, we proceed to investigate local hyperbolic systems with density-dependent speed, systems that consider population dynamics (i.e., birth and death processes), and nonlocal hyperbolic systems. We conclude by discussing kinetic models in two spatial dimensions and their limiting hyperbolic models. This structural approach allows us to discuss the complexity of the biological problems investigated, and the necessity for deriving complex mathematical models that would explain the observed spatial and spatiotemporal group patterns.

A Mathematical Framework for Cellular Repair Mechanisms under Genomic Stress Based on Kinetic Theory Approach

Generally, a cell can trigger its self-defense mechanism in response to genomic stress under acute perturbations from outer environment. To investigate the dynamic kinetics of cellular repair mechanisms in fighting against genomic stress, a mathematical model of representing and analyzing DNA damage generation and repair process is proposed under acute Ion Radiation (IR) by using the Kinetic Theory of Active Particles (KTAP). In this paper, we focus on describing a mathematical framework of Cellular Repair System (CRS). We also present the dynamic processes of Double Strand Breaks (DSBs) and Repair Protein (RP) generating, DSB-protein complexes (DSBCs) synthesizing, and toxins accumulating under continuous radiation time.