Controlling and stabilizing unpredictable behavior of metabolic reactions and carcinogenesis in biological systems

  • Larysa DzyubakEmail author
  • Oleksandr Dzyubak
  • Jan Awrejcewicz
Original Paper


Developing new designs and optimization of the cancer treatment is extremely important task. In this work, the nonlinear multi-scale diffusion cancer invasion model that describes the interactions of the tumor cells, matrix-metalloproteinases, matrix-degradative enzymes and oxygen is studied. The conditions under which the cancerous biological system exhibits chaotic behavior were obtained by means of the method based on wandering trajectories analysis. Regions of parameters leading to carcinogenesis in the biological system studied were found in control parameter planes ‘number of tumor cells versus diffusion saturation level.’ Significant influence of the biological system initial state to carcinogenesis was ascertained and illustrated by regions in phase planes of initial conditions. Evolution of all regions obtained is presented depending on glucose level.


Tumor Metabolic reactions Carcinogenesis Chaotic attractors Phase spaces Control parameters 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsNational Technical UniversityKharkivUkraine
  2. 2.Ascension All Saint Cancer CenterRacineUSA
  3. 3.Department of Automation, Biomechanics and MechatronicsLodz University of TechnologyLodzPoland

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