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

Numerical investigation on flow-induced wall shear stress variation of metastatic cancer cells in lymphatics with elastic valves

Hematogenous metastasis occurs when cancer cells detach from the extracellular matrix in the primary tumor into the bloodstream or lymphatic system. Elucidating the response of metastatic tumor cells in suspension to the flow conditions in lymphatics with valves from a mechanical/fluidic perspective is necessary. A physiologically relevant computational model of a lymphatic vessel with valves was constructed using fully coupled fluid-cell-vessel interactions to investigate the effects of lymphatic vessel contractility, valve properties, and cell size and stiffness on the variations in magnitude and gradient of the flow-induced wall shear stress (WSS) experienced by suspended tumor cells. Results indicated that the maximum WSSmax increased with the increments in cell diameter, vessel contraction amplitude, and valve stiffness. The decrease in vessel contraction period and valve aspect ratio also increased the maximum WSSmax. The influence of the properties of the valve on the WSS was more significant among the factors mentioned above. The maximum WSSmax acting on the cancer cell when the cell reversed the direction of its motion in the valve region increased by 0.5-1.4 times that before the cell entered the valve region. The maximum change in WSS was in the range of 0.004-0.028 Pa/µm depending on the factors studied. They slightly exceeded the values associated with breast cancer cell apoptosis. The results of this study provide biofluid mechanics-based support for mechanobiological research on the metastasis of metastatic cancer cells in suspension within the lymphatics.

Computational fluid dynamic modeling of the lymphatic system: a review of existing models and future directions

Historically, research into the lymphatic system has been overlooked due to both a lack of knowledge and limited recognition of its importance. In the last decade however, lymphatic research has gained substantial momentum and has included the development of a variety of computational models to aid understanding of this complex system. This article reviews existing computational fluid dynamic models of the lymphatics covering each structural component including the initial lymphatics, pre-collecting and collecting vessels, and lymph nodes. This is followed by a summary of limitations and gaps in existing computational models and reasons that development in this field has been hindered to date. Over the next decade, efforts to further characterize lymphatic anatomy and physiology are anticipated to provide key data to further inform and validate lymphatic fluid dynamic models. Development of more comprehensive multiscale- and multi-physics computational models has the potential to significantly enhance the understanding of lymphatic function in both health and disease.

Effect of valve spacing on peristaltic pumping

Peristaltic fluid pumping due to a periodically propagating contraction wave in a vessel fitted with one-way elastic valves is investigated numerically. It is concluded that the valve spacing within the vessel relative to the contraction wavelength plays a critical role in providing efficient pumping. When the valve spacing does not match the wavelength, the valves open asynchronously and the volume of the vessel segments bounded by two consecutive valves changes periodically, thereby inducing volumetric fluid pumping. The volumetric pumping leads to higher pumping flowrate and efficiency against an adverse pressure gradient. The optimum pumping occurs when the ratio of valve spacing to contraction wavelength is about2/3. This pumping regime is characterized by a longer period during which the valves are open. The results are useful for further understanding the pumping features of lymphatic system and provide insight into the design of biomimetic pumping devices.

An Enhanced 3D Model of Intravascular Lymphatic Valves to Assess Leaflet Apposition and Transvalvular Differences in Wall Distensibility

Lymphatic valves operate in a fluid-dynamically viscous environment that has little in common with that of cardiac valves, and accordingly have a different, axially lengthened, shape. A previously developed 3D fluid/structure interaction model of a lymphatic valve was extended to allow the simulation of stages of valve closure after the leaflets come together. This required that the numerical leaflet be prevented from passing into space occupied by the similar other leaflet. The resulting large deflections of the leaflet and lesser deflections of the rest of the valve were mapped as functions of the transvalvular pressure. In a second new development, the model was reconstructed to allow the vessel wall to have different material properties on either side of where the valve leaflet inserts into the wall. As part of this, a new pre-processing scheme was devised which allows easier construction of models with modified valve dimensions, and techniques for successfully interfacing the CAD software to the FE software are described. A two-fold change in wall properties either side of the leaflet made relatively little difference to valve operation apart from affecting the degree of sinus distension during valve closure. However, the numerically permitted strains were modest (<14%), and did not allow examination of the large-scale highly nonlinear elastic properties exhibited by real lymphatic vessels. A small series of murine popliteal, mesenteric, and inguinal-axillary lymphatic vessel segments containing a valve were experimentally investigated ex vivo. The pressure–diameter curves measured just upstream and just downstream of the valve were parameterised by computing the difference in tubular distensibility at three values of transmural pressure. In the popliteal and mesenteric segments, it was found that the distensibility was usually greater just downstream, i.e., in the sinus region, than upstream, at low and intermediate transmural pressure. However, there was wide variation in the extent of difference, and possible reasons for this are discussed.

Pump efficacy in a two-dimensional, fluid-structure interaction model of a chain of contracting lymphangions

The transport of lymph through the lymphatic vasculature is the mechanism for returning excess interstitial fluid to the circulatory system, and it is essential for fluid homeostasis. Collecting lymphatic vessels comprise a significant portion of the lymphatic vasculature and are divided by valves into contractile segments known as lymphangions. Despite its importance, lymphatic transport in collecting vessels is not well understood. We present a computational model to study lymph flow through chains of valved, contracting lymphangions. We used the Navier-Stokes equations to model the fluid flow and the immersed boundary method to handle the two-way, fluid-structure interaction in 2D, non-axisymmetric simulations. We used our model to evaluate the effects of chain length, contraction style, and adverse axial pressure difference (AAPD) on cycle-mean flow rates (CMFRs). In the model, longer lymphangion chains generally yield larger CMFRs, and they fail to generate positive CMFRs at higher AAPDs than shorter chains. Simultaneously contracting pumps generate the largest CMFRs at nearly every AAPD and for every chain length. Due to the contraction timing and valve dynamics, non-simultaneous pumps generate lower CMFRs than the simultaneous pumps; the discrepancy diminishes as the AAPD increases. Valve dynamics vary with the contraction style and exhibit hysteretic opening and closing behaviors. Our model provides insight into how contraction propagation affects flow rates and transport through a lymphangion chain.

Microfluidic valvular chips and a numerical lymphatic vessel model for the study of lymph transport characteristics

Lymph transport inside lymphatic vessels is highly complex and not yet fully understood. So far, a consensus has not been reached among existing analytical models on how spatiotemporal coordination of contracting adjacent lymphangions affects lymph transport. To understand complex lymph transport, we created a novel microfluidic valvular chip with flexible bicuspid valves and segmental pneumatic pumps based on a microfluidic device with an inside 3D structure made of hydrogels. Inside the chip, water moved unidirectionally when the microfluidic channel was locally compressed, with its velocity profile closely resembling the waveform of lymph observed in vivo. Furthermore, for a systematic and mechanistic study, we constructed a numerical model based on fluid-structure interaction and validated the model via demonstration of similarities in water transport characteristics between the model and the chip. Using this model, we examined various mechanical and time-dependent parameters, such as period, phase delay, sequence, and strength of contractions, valve compliance, fluid viscosity, and pressure differences, for their effects on water transport. Although our model is simplified, it enabled a parametric study that helped clarify the mechano-temporal correlations between compressions of adjacent chambers via transmissions of hydrodynamic forces, which regulate complex lymph transport. Moreover, our chip demonstrated technical advances that enable unidirectional discrete movement of fluid in the picoliter range by phenumatic pumping. The velocity profile is also similar to the pulse waveform of arteries under pathological conditions such as increased aortic stiffness, allowing our chip to be used for in vitro mechanobiology studies of endothelial cells.

EphrinB2-EphB4 signalling provides Rho-mediated homeostatic control of lymphatic endothelial cell junction integrity

Endothelial integrity is vital for homeostasis and adjusted to tissue demands. Although fluid uptake by lymphatic capillaries is a critical attribute of the lymphatic vasculature, the barrier function of collecting lymphatic vessels is also important by ensuring efficient fluid drainage as well as lymph node delivery of antigens and immune cells. Here, we identified the transmembrane ligand EphrinB2 and its receptor EphB4 as critical homeostatic regulators of collecting lymphatic vessel integrity. Conditional gene deletion in mice revealed that EphrinB2/EphB4 signalling is dispensable for blood endothelial barrier function, but required for stabilization of lymphatic endothelial cell (LEC) junctions in different organs of juvenile and adult mice. Studies in primary human LECs further showed that basal EphrinB2/EphB4 signalling controls junctional localisation of the tight junction protein CLDN5 and junction stability via Rac1/Rho-mediated regulation of cytoskeletal contractility. EphrinB2/EphB4 signalling therefore provides a potential therapeutic target to selectively modulate lymphatic vessel permeability and function.

Lymph vessels are thin walled tubes that, similar to blood vessels, carry white blood cells, fluids and waste. Unlike veins and arteries, however, lymph vessels do not carry red blood cells and their main function is to remove excess fluid from tissues. The cells that line vessels in the body are called endothelial cells, and they are tightly linked together by proteins to control what goes into and comes out of the vessels. The chemical, physical and mechanical signals that control the junctions between endothelial cells are often the same in different vessel types, but their effects can vary.

The endothelial cells of both blood and lymph vessels have two interacting proteins on their membrane known as EphrinB2 and its receptor, EphB4. When these two proteins interact, the EphB4 receptor becomes activated, which leads to changes in the junctions that link endothelial cells together. Frye et al. examined the role of EphrinB2 and EphB4 in the lymphatic system of mice. When either EphrinB2 or EphB4 are genetically removed in newborn or adult mice, lymph vessels become disrupted, but no significant effect is observed on blood vessels. The reason for the different responses in blood and lymph vessels is unknown.

The results further showed that lymphatic endothelial cells need EphB4 and EphrinB2 to be constantly interacting to maintain the integrity of the lymph vessels. Further examination of human endothelial cells grown in the laboratory revealed that this constant signalling controls the internal protein scaffold that determines a cell’s shape and integrity. Changes in the internal scaffold affect the organization of the junctions that link neighboring lymphatic endothelial cells together.

The loss of signalling between EphrinB2 and EphB4 in lymph vessels reflects the increase in vessel leakage seen in response to bacterial infections and in some genetic conditions such as lymphoedema. Finding ways to control the signalling between these two proteins could help treat these conditions by developing drugs that improve endothelial cell integrity in lymph vessels.

Modelling secondary lymphatic valves with a flexible vessel wall: how geometry and material properties combine to provide function

A three-dimensional finite-element fluid/structure interaction model of an intravascular lymphatic valve was constructed, and its properties were investigated under both favourable and adverse pressure differences, simulating valve opening and valve closure, respectively. The shear modulus of the neo-Hookean material of both vascular wall and valve leaflet was varied, as was the degree of valve opening at rest. Also investigated was how the valve characteristics were affected by prior application of pressure inflating the whole valve. The characteristics were parameterised by the volume flow rate through the valve, the hydraulic resistance to flow, and the maximum sinus radius and inter-leaflet-tip gap on the plane of symmetry bisecting the leaflet, all as functions of the applied pressure difference. Maximum sinus radius on the leaflet-bisection plane increased with increasing pressure applied to either end of the valve segment, but also reflected the non-circular deformation of the sinus cross section caused by the leaflet, such that it passed through a minimum at small favourable pressure differences. When the wall was stiff, the inter-leaflet gap increased sigmoidally during valve opening; when it was as flexible as the leaflet, the gap increased more linearly. Less pressure difference was required both to open and to close the valve when either the wall or the leaflet material was more flexible. The degree of bias of the valve characteristics to the open position increased with the inter-leaflet gap in the resting position and with valve inflation pressure. The characteristics of the simulated valve were compared with those specified in an existing lumped-parameter model of one or more collecting lymphangions and used to estimate a revised value for the constant in that model which controls the rate of valve opening/closure with variation in applied pressure difference. The effects of the revised value on the lymph pumping efficacy predicted by the lumped-parameter model were evaluated.

The effects of valve leaflet mechanics on lymphatic pumping assessed using numerical simulations

The lymphatic system contains intraluminal leaflet valves that function to bias lymph flow back towards the heart. These valves are present in the collecting lymphatic vessels, which generally have lymphatic muscle cells and can spontaneously pump fluid. Recent studies have shown that the valves are open at rest, can allow some backflow, and are a source of nitric oxide (NO). To investigate how these valves function as a mechanical valve and source of vasoactive species to optimize throughput, we developed a mathematical model that explicitly includes Ca²⁺ -modulated contractions, NO production and valve structures. The 2D lattice Boltzmann model includes an initial lymphatic vessel and a collecting lymphangion embedded in a porous tissue. The lymphangion segment has mechanically-active vessel walls and is flanked by deformable valves. Vessel wall motion is passively affected by fluid pressure, while active contractions are driven by intracellular Ca²⁺ fluxes. The model reproduces NO and Ca²⁺ dynamics, valve motion and fluid drainage from tissue. We find that valve structural properties have dramatic effects on performance, and that valves with a stiffer base and flexible tips produce more stable cycling. In agreement with experimental observations, the valves are a major source of NO. Once initiated, the contractions are spontaneous and self-sustained, and the system exhibits interesting non-linear dynamics. For example, increased fluid pressure in the tissue or decreased lymph pressure at the outlet of the system produces high shear stress and high levels of NO, which inhibits contractions. On the other hand, a high outlet pressure opposes the flow, increasing the luminal pressure and the radius of the vessel, which results in strong contractions in response to mechanical stretch of the wall. We also find that the location of contraction initiation is affected by the extent of backflow through the valves.