Steady state analysis for effects of tumor microenvironmental factors on tumor growth dynamics

Many biological systems are often subjected to random environmental influences that cannot be understood from the deterministic theoretical approach. Theoretical description of these systems can only be correctly understood from the probabilistic (stochastic) view point, even though the source of randomness may vary depending on the nature of the process and its physical origin. For instance, random processes that evolve with a system intrinsically are best modeled by master equation which is in the form of nonlinear integro partial differential equation with discrete jump moments at short times. However, for systems subject to external random effects, and for which the jump moments in the transition probability approaches zero, the master equation description approaches the so-called Fokker Planck equation with continuous state space. Tumor growth system subject to random micro environmental factors effect within the tumor site is the main focus of this thesis. We have considered one-dimensional tumor model in the form of Langevin equation subject to influence from the surrounding tumor micro environmental factors effect. The tumor micro environmental factors are the random biological processes existing within the immediate neighbourhood of the tumor cells, and whose effects influence tumor growth greatly by either promoting growth, inhibiting growth or sometimes neutral to malignancy. Moreover, the tumor model consist of the logistic model as the deterministic evolution equation for tumor growth, and the stochastic component consisting of additive and multiplicative noise terms respectively. The additive noise term represent the surrounding tumor microenvironmental factors effect which are external to the tumor, while the multiplicative noise term rep-resent tumor response to the surrounding micro environmental factors effect, and which effects are proportional to the state of tumor growth. In addition, the two noise terms are correlated having originated from the same source. The tumor model is firstly considered to be driven by correlated additive and multiplicative white noises respectively, where the additive noise term represent the non-immunogenic micro environmental factors effects within the tumor site. The under-lying transition probability for the tumor model satisfies the Fokker Planck equation, and of which the steady state distribution corresponding to the long-term limit solution for the tumor growth system is obtained. The study revealed that the surrounding non-immunogenic tumor micro environmental factors have a diffusive effect on tumor growth as indicated by the tumor response parameter. The tumor model is further considered to be driven by correlated noises with non-zero correlation time (colored noise case), of which consequence yield a non-Markovian tumor model. Consequently, the underlying transition probability for the tumor model does not obey the Markovian Fokker Planck equation, and using the Novikov theorem, Fox approach and the Ansatzof Hanggi, an approximate Fokker Planck equation in the steady state regime is obtained. Further, the steady state properties for the tumor growth system is explored using numerical simulations, where it is observed that the strength of the correlation time has a strong influence on the growth effects exerted by the non-immunogenic component of tumor microenvironment on tumor growth. Finally, the deterministic component of the tumor model is extended to include the tumor-immune interaction potential. This allows us to study the tumor response to the dual effects of immuno-genic and non-immunogenic tumor micro environmental factors within the tumor site.It is observed that in the presence of adequate immune response, the growth effects exerted by the non-immunogenic tumor micro environmental factors are opposed, and instead the tumor growth is reduced towards extinction. The research in this thesis is not directly focused on the biological aspect of tumor growth, but rather on the theoretical study of complex properties and behaviors likely exhibited by tumors in response to the surrounding tumor micro environmental factors effects, which has great influence on tumor evolution and progression. This type of research is particularly important towards understanding the tumor growth process at micro level for the design of an effective treatment strategy for tumor diagnosis, and for necessary medical precautions

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