A Mathematical Model of Valveless Pumping: A Lumped Model with Time-Dependent Compliance, Resistance, and Inertia

Abstract

A new lumped model of flow driven by pumping without valves is presented, motivated by biomedical applications: the circulation of the human fetus before the development of the heart valves and mechanism of blood flow during the external cardiopulmonary resuscitation (CPR). The phenomenon of existence of a unidirectional net flow around a loop of tubing that consists of two different compliances is called valveless pumping. The lumped parameter model of valveless pumping in this paper is governed by the ordinary differential equations for pressure and flow, with time-dependent compliance, resistance, and inertia. This simple model can represent the essential features of valveless pumping we observed in earlier mathematical models and physical experiments of valveless pumping. We demonstrate that not only parameters of the driving function, such as frequency or amplitude, but also physical parameters, such as wall thickness and tube stiffness, are important in determining the direction and magnitude of a net flow. In this system, we report two new and interesting phenomena of valveless pumping: One is that the shifted peak frequency can be predicted by the pulsewave speed and the other is that time-dependent resistance is a crucial factor in generating valveless pumping. We also demonstrate that this lumped model can be extended to a one-dimensional flow model of valveless pumping and explain why a linear case, the case of the constant compliance, resistance, and inertia, generates almost zero net flow. This emphasizes that the nonlinearity of valveless pumping is also an important factor to generate a net flow in a closed loop model of valveless pumping.

Multidimensional Open System for Valveless Pumping

“…From Fig. 4, it is obvious that VP strongly depends on the pumping frequency, as observed in previous studies [4,11,12,14,15,17,22,24,26,35,37]. Two positive peaks at around between two tanks as functions of the frequency and the compression duration.…”

“…The IB method has been applied to diverse fluid-structure interaction problems in the areas of biophysics and biomedicine, such as two-and three-dimensional simulations of blood flow in the heart [8,27,28,29,30], design of prosthetic cardiac valves [25], wave propagation in the cochlea [2], VP in a closed loop [11,12,14,15,17,22], simulations of whirling instability [23], higher-order IB method [9], single-cell modeling of tumor dynamics [33], flow of suspensions [7], peristaltic pumping of solid particles [5], and aquatic animal locomotion [6]. Although the IB method can be used to deal with sophisticated and time-dependent materials that interact with the fluid, it has been observed that the volume is not conserved strictly.…”

“…For instance, a VP system can generate an efficient net flow by controlling the parameters related to the driving force or tube materials. Therefore, VP has been studied extensively through experiments [10,19,20,21,34], analytical theories [1,26,32,36], and mathematical models [3,4,11,12,14,15,17,22,24,26,35,37].…”

Valveless Pumping in Open Tank System Using Energy Conserving Compartment Model

“…To understand the mechanism of valveless pumping phenomena discussed by Liebau [14], many attempts have been made for recent decades in theoretical [1,17,21], experimental [4,7,8,20,22], and numerical [3,6,9,12,13] point of view. Kenner [10] points out that the energy-driven source of fluid movement, asymmetry of conduits, and an asymmetry of the course of movement are conditions for valveless pumping.…”

“…The equations derived from the energy principle on each compartment of the system thus satisfy the energy conserving property in the entire system, so that it is called the energy conserving compartment model or briefly the ECCM. It is motivated by Jung's lumped model [9] which is a lumped model for a closed loop system of valveless pumping. Based on the Ohm's law and the definition of compliance, Jung's model is designed for the flow system in a closed loop with soft and stiff parts on an elastic tube.…”