Effects of Fractional Dynamics and Time Delay on Nutrient Transport in Blood Flow

In this paper, the fractional-order delay differential equation (FDDE) model is used to explore the dynamics of nutrient transport in a vascular environment to delayed nutrient uptake and pulsatile blood flow. The external nutrient input and nonlinear interaction term and the product of nutrient concentration and blood flow over time are also studied. The delay is introduced to consider the physiological lags which are causes of metabolic processing and absorption of nutrients via the Caputo fractional derivative. The stability of the system is examined at steady and periodically variable blood flow conditions. To obtain constant flow, analytical expressions of the equilibrium points and the critical values of delay are examined. When the blood flow is pulsatile, and considered to be a sinusoidal function, the system has complex oscillatory dynamics, and numerical simulations are used to analyze when the system becomes unstable. A sensitivity analysis is examined to analysis the impact of the main model parameters. Findings show that longer delay times and larger amplitudes of pulsation enhance instability and steady oscillation of nutrient concentration. These results lend some understanding to the physiological connotation of the delayed interactions in nutrient transport and could be used to guide future studies on metabolic control and circulatory mechanisms.