Fixed Points and Stability Analysis in the Motion of Human Heart Valve Leaflet Emagbetere, Eyere 1,* , Salau, A.O. Tajudeen 2 and Oluwole, Oluleke 2 Mechanical Engineering Department, Federal University of Petroleum Resources, Effurun, Delta State, Nigeria Mechanical Engineering Department, University of Ibadan, Ibadan, Nigeria emagbetere.eyere@fupre.edu Keywords: Heart valve model, phase portraits, numerical simulation, transcritical bifurcation Abstract. This work was set out to gain further insight into the kinetics of the human heart valve leaflet. The Korakianitis and Shi lumped parameter model was adopted for this study. The fixed points were determined, and then, their stability properties were assessed by evaluating eigenvalues of the Jacobian matrices. Normal physiological parameters for the valve model were simulated; based on which, a local bifurcation diagram was generated. Phase portraits were plotted from simulated responses, and were used to observe the qualitative properties of the valve leaflet motion. The evaluated fixed points were found to be dependent on pressure and flow effects, and independent on friction or damping effect. Observed switching of stability between the two fixed points indicated that the leaflet motion undergoes transcritical bifurcation. Of the two fixed points, one is always either a stable spiral or generative node while the other is a saddle. Numerical simulations were carried out to verify the analytical solutions. Subject Classification Numbers: (MSC2010) 74L15, 70E17, 37N15 1. Introduction The human heart valves have leaflets which open and close passively to control unidirectional blood flow. Although normal leaflets motion is expected for both native and prosthetic valves, loss in mobility has been observed in certain valve types [1-4]. Whether severe or mild, any form of abnormal leaflet motion is said to render the valve structurally dysfunctional [3]. While it is important to characterise the clinical incidence and risk of abnormal leaflets motion, there is also the need for further investigation into the dynamics of valve leaflets motion. Experimental assessment of valve leaflets’ motion started out with the attachment of film probes to animal models and obtaining velocity profile using anemometer [5, 6]. This was followed by the attachment of markers to animal models and tracking with x-rays to obtain velocity profiles of the valve leaflets [7; 8]. These techniques were applied mainly on animal models and cannot sufficiently study human heart valve dynamics. Additionally, they are not feasible for clinical studies. The introduction of laser scanners for quantifying valve motion [9] was a good replacement for attached markers. Via laser profiling, the deformations of the valve leaflets under different flow conditions were successfully studied [10, 11]. Over the years, clinical scanners had gained improvements and were used to assess valve motion both for clinical studies and diagnostic purposes. There are current sophisticated imaging techniques such as the Computed Tomography (CT). Very recently, leaflet motion of certain prosthetics heart valves was widely studied using CT scanners [2, 3, 12, 13]. It is noteworthy that imaging techniques cannot be used to assess the kinetics (forces and effect on motion) of the valve leaflets. Furthermore, assessment of long term variation in dynamical variables of the heart valve motion cannot be achieved using current imaging techniques. Combination of imaging techniques and mathematical modelling has been extensively used to achieve impressive success in the study of heart valve dynamics. There are a number of review articles that reported some of these computational models; their development, applications and limitations [14-16]. There are recent highly sophisticated 3D fluid structure interaction models for heart valves International Frontier Science Letters Submitted: 2018-01-19 ISSN: 2349-4484, Vol. 14, pp 1-18 Revised: 2018-06-12 doi:10.18052/www.scipress.com/IFSL.14.1 Accepted: 2018-07-19 2019 SciPress Ltd., Switzerland Online: 2019-03-06 SciPress applies the CC-BY 4.0 license to works we publish: https://creativecommons.org/licenses/by/4.0/