Developing transmission line equations of oxygen transport for predicting oxygen distribution in the arterial system

Abstract

The oxygen content in the arterial system plays a significant role in determining the physiological status of a human body. Understanding the oxygen concentration distribution in the arterial system is beneficial for the prevention and intervention of vascular disease. However, the oxygen concentration in the arteries could not be noninvasively monitored in clinical research. Although the oxygen concentration distribution in a vessel could be obtained from a three-dimensional (3D) numerical simulation of blood flow coupled with oxygen transport, a 3D numerical simulation of the systemic arterial tree is complicated and requires considerable computational resources and time. However, the lumped parameter model of oxygen transport derived from transmission line equations of oxygen transport requires fewer computational resources and less time to numerically predict the oxygen concentration distribution in the systemic arterial tree. In this study, transmission line equations of oxygen transport are developed according to the theory of oxygen transport in the vessel, and fluid transmission line equations are used as the theoretical reference for the development. The transmission line equations of oxygen transport could also be regarded as the theoretical basis for developing lumped parameter models of other substances in blood.

Figure 1

Schematic diagrams of the geometrical models of (a) straight vessel, (b) curved vessel and (c) bifurcated vessel. (d) Geometrical parameters of the vessels. D_s and L_s represent the diameter and the length of the straight vessel; d_c, R_c and α_c represent the diameter, the radius of curvature and the angle of curvature of the curved vessel, respectively; D_ccl (l = 1, 2), D_icm (m = 1, 2, 3, 4, 5, 6), D_ecn (n = 1, 2, 3) and β represent the diameters of the common carotid artery, the diameters of the internal carotid artery, the diameters of the external carotid artery and the bifurcation angle of the bifurcated vessel, respectively.

Figure 2

Flowrates at the inlets of three tubes.

Figure 3

Lumped parameter representations of a basic segment. (a) Electrical analogue of blood flow with resistance (R’), inductance (L’), and capacitance (C’) (R’, L’, and C’ represent resistivity, inertance, and compliance, respectively). Current and voltage are related to the flow rate and pressure, respectively. (b) Electrical analogue of oxygen transport with resistance (R), inductance (L), and current sink (R and L represent the properties of oxygen convection). The current sink is related to the diffusive flux of oxygen from the blood to the vascular wall. Current and voltage represent the convective flux and diffusive flux of oxygen, respectively.

Figure 4

Convective flux of oxygen at the end of each tube.