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.
Much of the skeleton of sharks, skate and rays (Elasmobranchii) is characterized by a tessellated structure, composed of a shell of small, mineralized plates (tesserae) joined by intertesseral ligaments overlaying a soft cartilage core. Although tessellated cartilage is a defining feature of this group of fishes, the significance of this skeletal tissue type - particularly from a mechanical perspective - is unknown. The aim of the present work was to perform stress relaxation experiments with tessellated cartilage samples from the jaws of blue sharks to better understand the time dependent behavior of this skeletal type. In order to facilitate this aim, the resulting relaxation behavior for different loading directions were simulated using the transversely isotropic biphasic model and this model combined with generalized Maxwell elements to represent the tessellated layer. Analysis of the ability of the models to simulate the observed experimental behavior indicates that the transversely isotropic biphasic model can provide good predictions of the relaxation behavior of the hyaline cartilage. However, the incorporation of Maxwell elements into this model can achieve a more accurate simulation of the dynamic behavior of calcified cartilage when the loading is parallel to the tessellated layer. Correlation of experimental data with present combined composite models showed that the equilibrium modulus of the tessellated layer for this loading direction is about 45 times greater than that for uncalcified cartilage. Moreover, tessellation has relatively little effect on the viscoelasticity of shark cartilage under loading that is normal to the tessellated layer.
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