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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Stability Analysis of Mathematical Modeling of Atherosclerotic Plaque Formation

Journal of Applied Nonlinear Dynamics 9(3) (2020) 361--389 | DOI:10.5890/JAND.2020.09.003

Debasmita Mukherjee, Lakshmi Narayan Guin, Santabrata Chakravarty

Department of Mathematics, Visva-Bharati, Santiniketan, 731235, West Bengal, India

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Abstract

The present theoretical investigation is dealt with stability of the model system characterizing the formation of atherosclerotic plaques emerging from the interactions between several cellular species in the media of blood stream and its surrounding vascular region. The biochemical processes behind the formation of atherosclerotic plaque involve the interactions between several cellular species like low density lipoprotein, free radicals, chemoattractants, monocytes, macrophages, T-cells, smooth muscle cells, foam cells and collagen. Taking all these complex events into account, an appropriate mathematical model depicting the onset of atherosclerotic plaques in the arterial lumen is constructed through the system of ten nonlinear ordinary differential equations (ODEs) for the concentrations of most pertinent components of the atherosclerotic constraints. Besides conducting an in-depth study of the model including several sub-models for their stability criteria, special emphasis is also paid on a reduced model system having adequate relevance to the present system following Quasi Steady State Approximation (QSSA) theory. Both the local and the global stability together with the bifurcation analysis for the reduced model system are carried out analytically. Results of numerical simulation based on the model parameter values reveal the time-series representations for the concentrations of all the interacting species, the local and the global stability and bifurcations with respect to some parameters of significance for the reduced model system under study. The complex features of several subsystems are also examined through several phase portraits in order to explore the clinical implications of atherosclerotic lesion in the arterial lumen. The model validation is also performed through comparison of the present results with those of previous ones [1].

Acknowledgments

The authors gratefully acknowledge the financial support by Special Assistance Programme (SAP-III) sponsored by the University Grants Commission (UGC), New Delhi, India [Grants no. F.510/3/DRSIII/2015(SAP-I)].We would like to express thank the anonymous referee and the editor for supportive remarks and ideas.

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