Simulation of a chain of collapsible contracting lymphangions with progressive valve closure

The aim of this investigation was to achieve the first step toward a comprehensive model of the lymphatic system. A numerical model has been constructed of a lymphatic vessel, consisting of a short series chain of contractile segments (lymphangions) and of intersegmental valves. The changing diameter of a segment governs the difference between the flows through inlet and outlet valves and is itself governed by a balance between transmural pressure and passive and active wall properties. The compliance of segments is maximal at intermediate diameters and decreases when the segments are subject to greatly positive or negative transmural pressure. Fluid flow is the result of time-varying active contraction causing diameter to reduce and is limited by segmental viscous and valvular resistance. The valves effect a smooth transition from low forward-flow resistance to high backflow resistance. Contraction occurs sequentially in successive lymphangions in the forward-flow direction. The behavior of chains of one to five lymphangions was investigated by means of pump function curves, with variation of valve opening parameters, maximum contractility, lymphangion size gradation, number of lymphangions, and phase delay between adjacent lymphangion contractions. The model was reasonably robust numerically, with mean flow-rate generally reducing as adverse pressure was increased. Sequential contraction was found to be much more efficient than synchronized contraction. At the highest adverse pressures, pumping failed by one of two mechanisms, depending on parameter settings: either mean leakback flow exceeded forward pumping or contraction failed to open the lymphangion outlet valve. Maximum pressure and maximum flow-rate were both sensitive to the contractile state; maximum pressure was also determined by the number of lymphangions in series. Maximum flow-rate was highly sensitive to the transmural pressure experienced by the most upstream lymphangions, suggesting that many feeding lymphatics would be needed to supply one downstream lymphangion chain pumping at optimal transmural pressure.     

Determinants of valve gating in collecting lymphatic vessels from rat mesentery

Secondary lymphatic valves are essential for minimizing backflow of lymph and are presumed to gate passively according to the instantaneous trans-valve pressure gradient. We hypothesized that valve gating is also modulated by vessel distention, which could alter leaflet stiffness and coaptation. To test this hypothesis, we devised protocols to measure the small pressure gradients required to open or close lymphatic valves and determine if the gradients varied as a function of vessel diameter. Lymphatic vessels were isolated from rat mesentery, cannulated, and pressurized using a servo-control system. Detection of valve leaflet position simultaneously with diameter and intraluminal pressure changes in two-valve segments revealed the detailed temporal relationships between these parameters during the lymphatic contraction cycle. The timing of valve movements was similar to that of cardiac valves, but only when lymphatic vessel afterload was elevated. The pressure gradients required to open or close a valve were determined in one-valve segments during slow, ramp-wise pressure elevation, either from the input or output side of the valve. Tests were conducted over a wide range of baseline pressures (and thus diameters) in passive vessels as well as in vessels with two levels of imposed tone. Surprisingly, the pressure gradient required for valve closure varied >20-fold (0.1-2.2 cmH(2)O) as a passive vessel progressively distended. Similarly, the pressure gradient required for valve opening varied sixfold with vessel distention. Finally, our functional evidence supports the concept that lymphatic muscle tone exerts an indirect effect on valve gating.     

Incorporating measured valve properties into a numerical model of a lymphatic vessel

An existing lumped-parameter model of multiple lymphangions (lymphatic vascular segments) in series is adapted for the incorporation of recent physiological measurements of lymphatic vascular properties. The new data show very marked nonlinearity of the passive pressure–diameter relation during distension, relative to comparable blood vessels, and complex valve behaviour. Since lymph is transported as a result of either the active contraction or the passive squeezing of vascular segments situated between two one-way valves, the performance of these valves is of primary importance. The valves display hysteresis (the opening and closing pressure drop thresholds differ), a bias to staying open (both state changes occur when the trans-valve pressure drop is adverse) and pressure-drop threshold dependence on transmural pressure. These properties, in combination with the strong nonlinearity that valve operation represents, have in turn caused intriguing numerical problems in the model, and we describe numerical stratagems by which we have overcome the problems. The principal problem is also generalised into a relatively simple mathematical example, for which solution detail is provided using two different solvers.     

Valve-related modes of pump failure in collecting lymphatics: numerical and experimental investigation

Lymph is transported along collecting lymphatic vessels by intrinsic and extrinsic pumping. The walls have muscle of a type intermediate between blood-vascular smooth muscle and myocardium; a contracting segment between two valves (a lymphangion) constitutes a pump. This intrinsic mechanism is investigated ex vivo in isolated, spontaneously contracting, perfused segments subjected to controlled external pressures. The reaction to varying afterload is probed by slowly ramping up the outlet pressure until pumping fails. Often the failure occurs when the contraction raises intra-lymphangion pressure insufficiently to overcome the outlet pressure, open the outlet valve and cause ejection, but many segments fail by other means, the mechanisms of which are not clear. We here elucidate those mechanisms by resort to a numerical model. Experimental observations are paired with comparable findings from computer simulations, using a lumped-parameter model that incorporates previously measured valve properties, plus new measurements of active contractile and passive elastic properties, and the dependence of contraction frequency on transmural pressure, all taken from isobaric twitch contraction experiments in the same vessel. Surprisingly, the model predicts seven different possible modes of pump failure, each defined by a different sequence of valve events, with their occurrence depending on the parameter values and boundary conditions. Some, but not all, modes were found experimentally. Further model investigation reveals routes by which a vessel exhibiting one mode of failure might under altered circumstances exhibit another.     

Probing the effect of morphology on lymphatic valve dynamic function

The lymphatic system is vital to the circulatory and immune systems, performing a range of important functions such as transport of interstitial fluid, fatty acid, and immune cells. Lymphatic vessels are composed of contractile walls and lymphatic valves, allowing them to pump lymph against adverse pressure gradients and to prevent backflow. Despite the importance of the lymphatic system, the contribution of mechanical and geometric changes of lymphatic valves and vessels in pathologies of lymphatic dysfunction, such as lymphedema, is not well understood. We develop a fully coupled fluid–solid, three-dimensional computational model to interrogate the various parameters thought to influence valve behavior and the consequences of these changes to overall lymphatic function. A lattice Boltzmann model is used to simulate the lymph, while a lattice spring model is used to model the mechanics of lymphatic valves. Lymphatic valve functions such as enabling lymph flow and preventing backflow under varied lymphatic valve geometries and mechanical properties are investigated to provide an understanding of the function of lymphatic vessels and valves. The simulations indicate that lymphatic valve function is optimized when valves are of low aspect ratio and bending stiffness, so long as these parameters are maintained at high enough values to allow for proper valve closing. This suggests that valve stiffening could have a profound effect on overall lymphatic pumping performance. Furthermore, dynamic valve simulations showed that this model captures the delayed response of lymphatic valves to dynamic flow conditions, which is an essential feature of valve operation. Thus, our model enhances our understanding of how lymphatic pathologies, specifically those exhibiting abnormal valve morphologies such as has been suggested to occur in cases of primary lymphedema, can lead to lymphatic dysfunctions.     

A fully coupled fluid-structure interaction model of the secondary lymphatic valve

Abstract

The secondary lymphatic valve is a bi-leaflet structure frequent throughout collecting vessels that serves to prevent retrograde flow of lymph. Despite its vital function in lymph flow and apparent importance in disease development, the lymphatic valve and its associated fluid dynamics have been largely understudied. The goal of this work was to construct a physiologically relevant computational model of an idealized rat mesenteric lymphatic valve using fully coupled fluid-structure interactions to investigate the relationship between three-dimensional flow patterns and stress/deformation within the valve leaflets. The minimum valve resistance to flow, which has been shown to be an important parameter in effective lymphatic pumping, was computed as 268?g/mm⁴?s. Hysteretic behavior of the lymphatic valve was confirmed by comparing resistance values for a given transvalvular pressure drop during opening and closing. Furthermore, eddy structures were present within the sinus adjacent to the valve leaflets in what appear to be areas of vortical flow; the eddy structures were characterized by non-zero velocity values (up to ～4?mm/s) in response to an applied unsteady transvalvular pressure. These modeling capabilities present a useful platform for investigating the complex interplay between soft tissue motion and fluid dynamics of lymphatic valves and contribute to the breadth of knowledge regarding the importance of biomechanics in lymphatic system function.     

Network Scale Modeling of Lymph Transport and Its Effective Pumping Parameters

The lymphatic system is an open-ended network of vessels that run in parallel to the blood circulation system. These vessels are present in almost all of the tissues of the body to remove excess fluid. Similar to blood vessels, lymphatic vessels are found in branched arrangements. Due to the complexity of experiments on lymphatic networks and the difficulty to control the important functional parameters in these setups, computational modeling becomes an effective and essential means of understanding lymphatic network pumping dynamics. Here we aimed to determine the effect of pumping coordination in branched network structures on the regulation of lymph flow. Lymphatic vessel networks were created by building upon our previous lumped-parameter model of lymphangions in series. In our network model, each vessel is itself divided into multiple lymphangions by lymphatic valves that help maintain forward flow. Vessel junctions are modeled by equating the pressures and balancing mass flows. Our results demonstrated that a 1.5 s rest-period between contractions optimizes the flow rate. A time delay between contractions of lymphangions at the junction of branches provided an advantage over synchronous pumping, but additional time delays within individual vessels only increased the flow rate for adverse pressure differences greater than 10.5 cmH₂O. Additionally, we quantified the pumping capability of the system under increasing levels of steady transmural pressure and outflow pressure for different network sizes. We observed that peak flow rates normally occurred under transmural pressures between 2 to 4 cmH₂O (for multiple pressure differences and network sizes). Networks with 10 lymphangions per vessel had the highest pumping capability under a wide range of adverse pressure differences. For favorable pressure differences, pumping was more efficient with fewer lymphangions. These findings are valuable for translating experimental measurements from the single lymphangion level to tissue and organ scales.     

A multiscale sliding filament model of lymphatic muscle pumping

The lymphatics maintain fluid balance by returning interstitial fluid to veins via contraction/compression of vessel segments with check valves. Disruption of lymphatic pumping can result in a condition called lymphedema with interstitial fluid accumulation. Lymphedema treatments are often ineffective, which is partially attributable to insufficient understanding of specialized lymphatic muscle lining the vessels. This muscle exhibits cardiac-like phasic contractions and smooth muscle-like tonic contractions to generate and regulate flow. To understand the relationship between this sub-cellular contractile machinery and organ-level pumping, we have developed a multiscale computational model of phasic and tonic contractions in lymphatic muscle and coupled it to a lymphangion pumping model. Our model uses the sliding filament model (Huxley in Prog Biophys Biophys Chem 7:255–318, 1957) and its adaptation for smooth muscle (Mijailovich in Biophys J 79(5):2667–2681, 2000). Multiple structural arrangements of contractile components and viscoelastic elements were trialed but only one provided physiologic results. We then coupled this model with our previous lumped parameter model of the lymphangion to relate results to experiments. We show that the model produces similar pressure, diameter, and flow tracings to experiments on rat mesenteric lymphatics. This model provides the first estimates of lymphatic muscle contraction energetics and the ability to assess the potential effects of sub-cellular level phenomena such as calcium oscillations on lymphangion outflow. The maximum efficiency value predicted (40%) is at the upper end of estimates for other muscle types. Spontaneous calcium oscillations during diastole were found to increase outflow up to approximately 50% in the range of frequencies and amplitudes tested.

     

Active lymphatic pumping and sheep lung lymph flow

Active (intrinsic) lymphatic pumping may be an important factor determining lymph flow from the lungs. Unfortunately, in most experiments, it is very difficult to determine the influence of active pumping vs. passive factors on lymph flow. However, 1) the pumping activity (stroke volume and frequency) of isolated lymphatic segments varies nonlinearly with transmural pressure, and 2) the lung lymph flow from awake sheep varies nonlinearly with lymphatic outflow pressure. Accordingly, if lymphatic pumping significantly influences lung lymph flow, then it should be possible to describe the sheep lung lymph flow vs. outflow pressure data with the pumping activity data. To test this, we used published lymphatic pumping activity data to develop a mathematical model of the lymphatic pump for a segment of lymphatic vessel. Flow vs. outflow pressure relationships obtained from simulations with this model were very similar to the data from sheep. Our results indicate that both passive factors and active lymphatic pumping contribute to lymph flow, and our model may allow investigators to distinguish the effects of active pumping vs. passive factors in the regulation of lymph flow.     

Lumped parameter model for computing the minimum pressure during mechanical heart valve closure

The cavitation inception threshold of mechanical heart valves has been shown to be highly variable. This is in part due to the random distribution of the initial and final conditions that characterize leaflet closure. While numerous hypotheses exist explaining the mechanisms of inception, no consistent scaling laws have been developed to describe this phenomenon due to the complex nature of these dynamic conditions. Thus in order to isolate and assess the impact of these varied conditions and mechanisms on inception, a system of ordinary differential equations is developed to describe each system component and solved numerically to predict the minimum pressure generated during valve closure. In addition, an experiment was conducted in a mock circulatory loop using an optically transparent size 29 bileaflet valve over a range of conditions to calibrate and validate this model under physiological conditions. High-speed video and high-response pressure measurements were obtained simultaneously to characterize the relationship between the valve motion, fluid motion, and negative pressure transients during closure. The simulation model was calibrated using data from a single closure cycle and then compared to other experimental flow conditions and to results found in the literature. The simulation showed good agreement with the closing dynamics and with the minimum pressure trends in the current experiment. Additionally, the simulation suggests that the variability observed experimentally (when using dP/dt alone as the primary measure of cavitation inception) is predictable. Overall, results from the current form of this lumped parameter model indicate that it is a good engineering assessment tool.     

Evidence for a second valve system in lymphatics: endothelial microvalves.

The mechanism for interstitial fluid uptake into the lymphatics remains speculative and unresolved. A system of intralymphatic valves exists that prevents reflow along the length of the lymphatic channels. However, these valves are not sufficient to provide unidirectional flow at the level of the initial lymphatics. We investigate here the hypothesis that initial lymphatics have a second, separate valve system that permits fluid to enter from the interstitium into the initial lymph channels but prevents escape back out into the tissue. The transport of fluorescent microspheres (0.31 microm) across endothelium of initial lymphatics in rat cremaster muscle was investigated with micropipette manipulation techniques. The results indicate that microspheres can readily pass from the interstitium across the endothelium into the lumen of the initial lymphatics. Once inside the lymphatic lumen, the microspheres cannot be forced out of the lumen even after elevation of the lymphatic pressure by outflow obstruction. Reaspiration of the microspheres inside the lymphatic lumen with a micropipette is blocked by the lymphatic endothelium. This blockade exists whether the aspiration is carried out at the microsphere entry site or anywhere along the initial lymphatics. Nevertheless, puncture of the initial lymphatic endothelium with the micropipette leads to rapid aspiration of intralymphatic microspheres. Investigation of lymphatic endothelial sections fixed during lymph pumping shows open interendothelial junctions not found in resting initial lymphatics. These results suggest that initial lymphatics have a (primary) valve system at the level of the endothelium. In conjunction with the classical (secondary) intralymphatic valves, the primary valves provide the mechanism that facilitates the unidirectional flow during periodic compression and expansion of initial lymphatics.     

A model of a radially expanding and contracting lymphangion

The lymphatic system is an extensive vascular network featuring valves and contractile walls that pump interstitial fluid and plasma proteins back to the main circulation. Immune function also relies on the lymphatic system's ability to transport white blood cells. Failure to drain and pump this excess fluid results in edema characterized by fluid retention and swelling of limbs. It is, therefore, important to understand the mechanisms of fluid transport and pumping of lymphatic vessels. Unfortunately, there are very few studies in this area, most of which assume Poiseuille flow conditions. In vivo observations reveal that these vessels contract strongly, with diameter changes of the order of magnitude of the diameter itself over a cycle that lasts typically 2-3s. The radial velocity of the contracting vessel is on the order of the axial fluid velocity, suggesting that modeling flow in these vessels with a Poiseuille model is inappropriate. In this paper, we describe a model of a radially expanding and contracting lymphatic vessel and investigate the validity of assuming Poiseuille flow to estimate wall shear stress, which is presumably important for lymphatic endothelial cell mechanotransduction. Three different wall motions, periodic sinusoidal, skewed sinusoidal and physiologic wall motions, were investigated with steady and unsteady parabolic inlet velocities. Despite high radial velocities resulting from the wall motion, wall shear stress values were within 4% of quasi-static Poiseuille values. Therefore, Poiseuille flow is valid for the estimation of wall shear stress for the majority of the lymphangion contractile cycle.     

A Model for Fluid Drainage by the Lymphatic System

This study investigates the fluid flow through tissues where lymphatic drainage occurs. Lymphatic drainage requires the use of two valve systems, primary and secondary. Primary valves are located in the initial lymphatics. Overlapping endothelial cells around the circumferential lining of lymphatic capillaries are presumed to act as a unidirectional valve system. Secondary valves are located in the lumen of the collecting lymphatics and act as another unidirectional valve system; these are well studied in contrast to primary valves. We propose a model for the drainage of fluid by the lymphatic system that includes the primary valve system. The analysis in this work incorporates the mechanics of the primary lymphatic valves as well as the fluid flow through the interstitium and that through the walls of the blood capillaries. The model predicts a piecewise linear relation between the drainage flux and the pressure difference between the blood and lymphatic capillaries. The model describes a permeable membrane around a blood capillary, an elastic primary lymphatic valve and the interstitium lying between the two.     

A model for mechanics of primary lymphatic valves

Recent experimental evidence indicates that lymphatics have two valve systems, a set of primary valves in the wall of the endothelial cells of initial lymphatics and a secondary valve system in the lumen of the lymphatics. While the intralymphatic secondary valves are well described, no analysis of the primary valves is available. We propose a model for primary lymphatics valves at the junctions between lymphatic endothelial cells. The model consists of two overlapping endothelial extensions at a cell junction in the initial lymphatics. One cell extension is firmly attached to the adjacent connective tissue while the other cell extension is not attached to the interstitial collagen. It is free to bend into the lumen of the lymphatic when the lymphatic pressure falls below the adjacent interstitial fluid pressure. Thereby the cell junction opens a gap permitting entry of interstitial fluid into the lymphatic lumen. When the lymphatic fluid pressure rises above the adjacent interstitial fluid pressure, the endothelial extensions contact each other and the junction is closed preventing fluid reflow into the interstitial space. The model illustrates the mechanics of valve action and provides the first time a rational analysis of the mechanisms underlying fluid collection in the initial lymphatics and lymph transport in the microcirculation.     

Modulation of fluid pumping in isolated bovine mesenteric lymphatics by a thromboxane/endoperoxide analogue

A thromboxane/endoperoxide analogue (compound U46619) is known to stimulate phasic and tonic contractions in quiescent bovine lymphatic vessels and enhance contractile activity in spontaneously active vessels. In order to determine how these effects relate to changes in fluid propulsion by the lymphatics, we have assessed the effects of U46619 on the ability of isolated bovine mesenteric lymphatics to pump fluid in vitro. Bovine lymphatic segments (up to 8 cm in length with a minimum of 4 valves) were cannulated at both ends and fluid input provided from a reservoir. Flow through the vessels was regulated by intraluminal pressures. On average, changes in transmural pressures up to 8 cm H2O resulted in enhanced pumping; pressures above this level depressed flow. The dominant effect of U46619 (added to the reservoir) was to depress pumping; 10(-7) and 10(-9)M decreased flow at all transmural pressures tested; 10(-8)M had a dual effect, slightly inhibiting flow at low transmural pressures and enhancing flow at higher pressures. These results suggest that thromboxane may stimulate or inhibit lymphatic pumping depending on the concentration of the agent and the transmural pressure applied to the vessel. These effects may relate to its ability to induce variable changes in luminal diameter and frequency and force of contractions.     

Nitric oxide formation by lymphatic bulb and valves is a major regulatory component of lymphatic pumping

Microscopic lymphatics produce nitric oxide (NO) during contraction as flow shear activates the endothelial cells. The valve leaflets and bulbous valve housing contain a large amount of endothelial nitric oxide synthase (eNOS) due both to many endothelial cells and increased expression of eNOS. Direct NO measurements indicate the valve area has a 30-50% higher NO concentration ([NO]) than tubular regions although both regions generate equivalent relative increases in [NO] with each contraction. We hypothesize that 1) the greater eNOS and [NO] of the bulb region would have greater effects to lower pumping activity of the overall lymphatic than occurs in tubular regions and 2), the elevated [NO] in the bulb region may be because of high NO production in the valve leaflets that diffuses to the wall of the bulb. Measurement of [NO] with a micropipette inside the lymphatic bulb revealed the valve leaflets generate ~50% larger [NO] than the bulb wall in the in vivo rat mesenteric lymphatics. The valves add NO to the lymph that quickly diffuses to the bulb wall. Bradykinin locally released iontophoretically from a micropipette on both bulbs and tubes increased the [NO] in a dose-dependent manner up to ~50%, demonstrating agonist activation of the NO pathway. However, pumping output determined by contraction frequency and stroke volume decreased much more for the bulb than tubular areas in response to the bradykinin. In effect, NO generation by the bulb area and its valves limits the pumped flow of the total lymphatic by lowering frequency and stroke volume of individual contractions.     

Intrinsic pump-conduit behavior of lymphangions

Lymphangions, segments of lymphatic vessels bounded by valves, have characteristics of both ventricles and arteries. They can act primarily like pumps when actively transporting lymph against a pressure gradient. They also can act as conduit vessels when passively transporting lymph down a pressure gradient. This duality has implications for clinical treatment of several types of edema, since the strategy to optimize lymph flow may depend on whether it is most beneficial for lymphangions to act as pumps or conduits. To address this duality, we employed a simple computational model of a contracting lymphangion, predicted the flows at both positive and negative axial pressure gradients, and validated the results with in vitro experiments on bovine mesenteric vessels. This model illustrates that contraction increases flow for normal axial pressure gradients. With edema, limb elevation, or external compression, however, the pressure gradient might reverse, and lymph may flow passively down a pressure gradient. In such cases, the valves may be forced open during the entire contraction cycle. The vessel thus acts as a conduit, and contraction has the effect of increasing resistance to passive flow, thus inhibiting flow rather than promoting it. This analysis may explain a possible physiological benefit of the observed flow-mediated inhibition of the lymphatic pump at high flow rates.     

Computational model for the transition from peristaltic to pulsatile flow in the embryonic heart tube

Early in development, the heart is a single muscle-wrapped tube without formed valves. Yet survival of the embryo depends on the ability of this tube to pump blood at steadily increasing rates and pressures. Developmental biologists historically have speculated that the heart tube pumps via a peristaltic mechanism, with a wave of contraction propagating from the inflow to the outflow end. Physiological measurements, however, have shown that the flow becomes pulsatile in character quite early in development, before the valves form. Here, we use a computational model for flow though the embryonic heart to explore the pumping mechanism. Results from the model show that endocardial cushions, which are valve primordia arising near the ends of the tube, induce a transition from peristaltic to pulsatile flow. Comparison of numerical results with published experimental data shows reasonably good agreement for various pressure and flow parameters. This study illustrates the interrelationship between form and function in the early embryonic heart.     

Novel characterization of lymphatic valve formation during corneal inflammation

Lymphatic research has progressed rapidly in recent years. Though lymphatic dysfunction has been found in a wide array of disorders from transplant rejection to cancer metastasis, to date, there is still little effective treatment for lymphatic diseases. The cornea offers an optimal site for lymphatic research due to its accessible location, transparent nature, and lymphatic-free but inducible features. However, it still remains unknown whether lymphatic valves exist in newly formed lymphatic vessels in the cornea, and how this relates to an inflammatory response. In this study, we provide the first evidence showing that lymphatic valves were formed in mouse cornea during suture-induced inflammation with the up-regulation of integrin alpha 9. The number of corneal valves increased with the progression of inflammatory lymphangiogenesis. Moreover, we have detected lymphatic valves at various developmental stages, from incomplete to more developed ones. In addition to defining the average diameter of lymphatic vessels equipped with lymphatic valves, we also report that lymphatic valves were more often located near the branching points. Taken together, these novel findings not only provide new insights into corneal lymphatic formation and maturation, but also identify a new model for future investigation on lymphatic valve formation and possibly therapeutic intervention.     

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

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.