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On the basis of two universal power-law scaling laws, i.e. the scaling dynamic hysteresis in physics and the allometric scaling metabolism in biosystem, we studied the dynamic response and the evolution of an immunosurveillant anti-tumor system subjected to a periodic external intervention, which is equivalent to the scheme of a radiotherapy or chemotherapy, within the framework of the growth dynamics of tumor. Under the modulation of either an abrupt or a gradual change external intervention, the population density of tumors exhibits a dynamic hysteresis to the intervention. The area of dynamic hysteresis loop characterizes a sort of dissipative-therapeutic relationship of the dynamic responding of treated tumors with the dose consumption of accumulated external intervention per cycle of therapy. Scaling the area of dynamic hysteresis loops against the intensity of an external intervention, we deduced a characteristic quantity which was defined as the theoretical therapeutic effectiveness of treated tumor and related with the destructive metabolism of tumor under treatment. The calculated dose-effectiveness profiles, namely the dose cumulant per cycle of intervention versus the therapeutic effectiveness, could be well scaled into a universal quadratic formula regardless of either an abrupt or a gradual change intervention involved. We present a new concept, i.e., the therapy-effect matrix and the dose cumulant matrix, to expound the new finding observed in the growth and regression dynamics of a modulated anti-tumor system. [ABSTRACT FROM AUTHOR]

To investigate the proliferation and invasion of a tumor within an inhomogeneous matrix, we studied the spatiotemporal dynamics of two types of growth-diffusion systems (GDSs) with logistic or Allee growth occurring on a two-dimensional square site percolation lattice via numerical computation and finite-size scaling approaches. A critical percolation threshold exists in the two systems, but becomes obscure with an increasing Allee effect in Allee growth. The two systems evidently differ in their short-time spatiotemporal patterns: The tumor number density in the logistic model grows and spreads continuously and subdiffusively or weakly superdiffusively while that in the Allee model does so discretely and strongly superdiffusively. This difference is attributed to a lack of cooperation between sites for growth and diffusion in the logistic model as compared to its Allee counterpart. The Allee growth pattern is characterized by a rougher border and more inhomogeneous interior than its logistic counterpart. Judging from their growth-diffusion feature in combination with a clinical image analysis, we conclude that Allee growth is more suitable for modeling the proliferation and invasion of an early-stage malignant tumor than is logistic growth. A phase diagram that correlates a tumor's growth and diffusion on a percolation lattice with a site occupation fraction and Allee effect was established to reveal the sensitivity on proliferation and spreading of a tumor towards the above parameters. The Allee effect was also found to induce diverse dynamic features on its short-time growth and diffusion in the GDS, which brings in an opposite trend toward a tumor's growth and diffusion. [ABSTRACT FROM AUTHOR]