Some implications of Scale Relativity theory in avascular stages of growth of solid tumors in the presence of an immune system response

Abstract

We present a traveling-wave analysis of a reduced mathematical model describing the growth of a solid tumor in the presence of an immune system response in the framework of Scale Relativity theory. Attention is focused upon the attack of tumor cells by tumor-infiltrating cytotoxic lymphocytes (TICLs), in a small multicellular tumor, without necrosis and at some stage prior to (tumor-induced) angiogenesis. For a particular choice of parameters, the underlying system of partial differential equations is able to simulate the well-documented phenomenon of cancer dormancy and propagation of a perturbation in the tumor cell concentration by cnoidal modes, by depicting spatially heterogeneous tumor cell distributions that are characterized by a relatively small total number of tumor cells. This behavior is consistent with several immunomorphological investigations. Moreover, the alteration of certain parameters of the model is enough to induce soliton like modes and soliton packets into the system, which in turn result in tumor invasion in the form of a standard traveling wave. In the same framework of Scale Relativity theory, a very important feature of malignant tumors also results, that even in avascular stages they might propagate and invade healthy tissues, by means of a diffusion on a Newtonian fluid.

Solitary solutions to an androgen‐deprivation prostate cancer treatment model

“…where 1 , … , m are constants and 1 , … , m are characteristic roots from a polynomial that is analogous to Equation (21). Such computations lead tô(…”

“…15 More recently, solitary solutions have been found to be significant in biomedical research. A cold laser therapy method based on soliton tweezers is developed in Alavi et al 16 Models demonstrating that angiogenesis is driven by and can be controlled using solitons are considered in Bonilla et al 17 Applications of solitary solutions in multiphoton microscropy are discussed in Wang et al 18 A multiple scale soliton perturbation analysis of DNA systems with inhomogeneity effects is performed in Okaly et al 19 The role of solitary solutions in intracellular signaling is considered in Barvitenko et al 20 Soliton-like traveling wave solutions to a model describing the growth of a solid tumor in the presence of an immune system response are constructed in Buzea et al 21 Algorithms for the construction of traveling-wave (soliton-like) solutions have been extensively discussed in literature. Techniques for constructing traveling-wave solutions to various partial differential equations (PDEs) used to model heat transfer are developed in Gao et al 22 Traveling wave solutions falling within the scope of local fractional derivative formulation for the Korteweg-de-Vries equation and the class of two-dimensional Burgers-type equations are considered in Yang and Machado et al 23 25 A new technique for constructing traveling-wave solutions based on factorization is proposed in Yang, Gao, and Srivastava.…”

“…More recently, solitary solutions have been found to be significant in biomedical research. A cold laser therapy method based on soliton tweezers is developed in Alavi et al Models demonstrating that angiogenesis is driven by and can be controlled using solitons are considered in Bonilla et al Applications of solitary solutions in multiphoton microscropy are discussed in Wang et al A multiple scale soliton perturbation analysis of DNA systems with inhomogeneity effects is performed in Okaly et al The role of solitary solutions in intracellular signaling is considered in Barvitenko et al Soliton‐like traveling wave solutions to a model describing the growth of a solid tumor in the presence of an immune system response are constructed in Buzea et al…”

A dynamic model of the immune response to the onset of a tumor

“…They are divided into point models [4,7,8,13,14], which describe the tumor growth mechanism and the immune response to the tumor, and three dimensional models, which describe not only the growth mechanism but also the spatial growth of the tumor. In the latest models, the description of the growth mechanism is significantly simplified com pared to the first models; the spatial aspect is described by cellular automates [9,10], a simple diffusion term [3,5,11], or by the Navier-Stokes equations [12]. The growth mechanism is usually described by two to six (up to 12) equations that are similar to the equa tions of homogeneous chemical kinetics, whereas any adaptive responses of the immune are ignored.…”