Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

High‐resolution, three‐dimensional simulations from the Massachusetts Institute of Technology general circulation model ocean model are used to calculate the subaqueous melt rate of the calving faces of Umiamako, Rinks, Kangerdlugssup, Store, and Kangilerngata glaciers, west Greenland, from 1992 to 2015. Model forcing is from monthly reconstructions of ocean state and ice sheet runoff. Results are analyzed in combination with observations of bathymetry, bed elevation, ice front retreat, and glacier speed. We calculate that subaqueous melt rates are 2–3 times larger in summer compared to winter and doubled in magnitude since the 1990s due to enhanced subglacial runoff and 1.6 ± 0.3°C warmer ocean temperature. Umiamako and Kangilerngata retreated rapidly in the 2000s when subaqueous melt rates exceeded the calving rates and ice front retreated to deeper bed elevation. In contrast, Store, Kangerdlugssup, and Rinks have remained stable because their subaqueous melt rates are 3–4 times lower than their calving rates, i.e., the glaciers are dominated by calving processes.

We develop a multispecies continuum model to simulate the spatiotemporal dynamics of cell lineages in solid tumors. The model accounts for protein signaling factors produced by cells in lineages, and nutrients supplied by the microenvironment. Together, these regulate the rates of proliferation, self-renewal and differentiation of cells within the lineages, and control cell population sizes and distributions. Terminally differentiated cells release proteins (e.g., from the TGFβ superfamily) that feedback upon less differentiated cells in the lineage both to promote differentiation and decrease rates of proliferation (and self-renewal). Stem cells release a short-range factor that promotes self-renewal (e.g., representative of Wnt signaling factors), as well as a long-range inhibitor of this factor (e.g., representative of Wnt inhibitors such as Dkk and SFRPs). We find that the progression of the tumors and their response to treatment is controlled by the spatiotemporal dynamics of the signaling processes. The model predicts the development of spatiotemporal heterogeneous distributions of the feedback factors (Wnt, Dkk and TGFβ) and tumor cell populations with clusters of stem cells appearing at the tumor boundary, consistent with recent experiments. The nonlinear coupling between the heterogeneous expressions of growth factors and the heterogeneous distributions of cell populations at different lineage stages tends to create asymmetry in tumor shape that may sufficiently alter otherwise homeostatic feedback so as to favor escape from growth control. This occurs in a setting of invasive fingering, and enhanced aggressiveness after standard therapeutic interventions. We find, however, that combination therapy involving differentiation promoters and radiotherapy is very effective in eradicating such a tumor.

No abstract

In this paper, we consider Wigner-Yanase-Dyson information as a measure of quantum uncertainty of a mixed state. We study some of the interesting properties of this generalized measure.The construction is reminiscent of the generalized entropies that have shown to be useful in many applications.