Zholtkevych G.N.¹, Bespalov G. Y. ¹, Nosov K. V. ¹, Zhukov V.I.², Visotskaya E.V.³, Pecherska A.I. ³

¹ – V.N. Karazin Kharkiv National University, ² – Kharkov National Medical University, ³ – Kharkov National University of Radioelectrinics

A NEW CLASS OF MATHEMATICAL MODELS IN CANCER RESEARCH — OPPORTUNITIES AND PROSPECTS

Problems of diagnosis, treatment and prevention of cancer in large measure can only be addressed with use of systematic approach. Nowadays a necessary element of this approach is mathematical modeling of characteristic aspects of structure and dynamics of relationships of biochemical parameters, distinguishing the clinical norm from pathology. To simulate the systemic aspects we need to use all available actual data, both experimental and clinical, including ones, which are small in size and with gaps. In the latter case, the application of recently developed with authors participation mathematical models is promising. These models are based on the theory of dynamical systems and are called the discrete modeling of dynamical systems (DMDS) and dynamic models for dichotomous attributes (DMDA).

The DMDS model allows on the bases of correlation matrix between variables to reveal the structure of between- and inter-component relationships, specified by pairwise positive and negative mutual influences (or lack of influences) in the system [1-4]. Here, we are dealing with the following six relationships — "+, +", "-,-", "+,-", "+,0", "-,0", and "0,0" for relations between component and three symmetric relationships "+,+", "-,-", and "0,0" for inter-component relations. Having the relationships structure, for certain initial conditions an idealized trajectory can be constructed [4]. It reflects the cycle of components' changes expressed in conditional units and shows a time sequence of componens' values.

In the framework of the DMDA model it is assumed, that all components are dichotomous, that is they take the values 0 and 1, where 0 and 1 are just different levels of an attribute. This model allows to represent the dependence of the value of each variable at the current step of the trajectory from a set of variables (which may includes target variable itself) on the previous step.

This work aims the study of possibilities of the DMDS and DMDP models for investigating systemic aspects of cancer development. Specifically, we compare the structures of relationships between biochemical parameters associated with L-tryptophan for two groups: 1) relatively healthy patients, 2) patients at the second stage of adenocarcinoma of the stomach.

With use of DMDS the relationships between tryptophan, serotonin, 5-oxyindoleacetic acid and ammonia in blood serum of the two groups were revealed. They allow us to demonstrate a topical difference between the groups, related with regulation mechanism of content of 5-oxyindoleacetic acid playing an important role in the mechanism of cancer occurrence. More precisely, the relationships of the type "+,-" between tryptophan and 5-oxyindoleacetic acid were found in the healthy group. On the second stage of adenocarcinoma of the stomach these relationships disappear, being replaced by relationships of the type "+,0", where tryptophan positively influences onto 5-oxyindoleacetic acid. There does not exist that effect in the healthy group. It should be noted, that the relationships " +, -" in systems of different nature are an important factor in homeostasis support, which specific change is present in development of various diseases, including oncological ones.

It is well known, that content of tryptophan, serotonin, 5-oxyindoleacetic acid, melatonin, indican, and ammonia in organism presents important aspects of homeostasis mechanisms. Using the DMDA model, in two above mentioned groups the patterns of oscillation of six listed above substances around their sample means were studied.

For the group on second stage of adenocarcinoma of the stomach the oriented graph, built according to results of modeling, has only one edge corresponding to transition from high content of 5-oxyindoleacetic acid to low content and only one edge corresponding to transition from low content to high one.

The graph of the healthy group has four edges corresponding to transition from high content of 5-oxyindoleacetic acid to low content and three edges for transition from low content to high one.

Thus, the results of simulation with use of DMDA suggest that the development of adenocarcinoma of the stomach is accompanied by reducing variability of mechanisms for maintaining such important oncological aspect of homeostasis, as oscillation of 5-oxyindoleacetic acid content around its average values.

In the hole it can be assumed that our examples of applying the DMDS and DMDA models for description of homeostasis aspects related with development of adenocarcinoma of the stomach attests the abilities of these new classes of mathematical models in cancer research.

References:

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