Malignant Invasion Model with Small Amount of Diffusion in the Framework of Scale Relativity Theory
We study a particular model of tumor progression, Perumpanani's malignant invasion model with a small amount of diffusion, in the framework of Scale Relativity theory, namely, we assume the invasive cells, the connective tissue and the proteases are moving through a non-differential medium governed by the Scale Relativity theory. Furthermore, we consider an action-reaction type law acting on the system formed by the extracellular matrix and the non-differential medium. As a result, our modeled/artificial cancer cell proliferation satisfies a logistic law accounting for the competition for space with the non-differential medium and the time evolution of the concentration of the connective tissue increases proportional to the real fractal velocity, squared. Also, we find that over small distances, even in avascular stages, malignant tumors might propagate and invade healthy tissues. Finally, we get exact solutions for Perumpanani's malignant invasion model with a small amount of diffusion in terms of Scale Relativity theory, using the factorization and the tanh methods. For small diffusion coefficients, we may conclude that there is a gap between the invasive cells front and the degraded connective tissue, the extracellular matrix is not continuously degraded by the concentration of proteases and the later one shows amplified followed by amortized oscillations and jumps between two distinct levels.
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Document Type: Review Article
Publication date: December 1, 2015
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