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

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.

Multiscale homogenization for dual porosity time-dependent Darcy-Brinkman/Darcy coupling and its application to the lymph node

We study the coupling between time-dependent Darcy-Brinkman and the Darcy equations at the microscale subjected to inhomogeneous body forces and initial conditions to describe a double porosity problem. We derive the homogenized governing equations for this problem using the asymptotic homogenization technique, and as macroscopic results, we obtain a coupling between two Darcy equations, one of which with memory effects, with mass exchange between phases. The memory effects are a consequence of considering the time dependence in the Darcy-Brinkman equation, and they allow us to study in more detail the role of time in the problem under consideration. After the formulation of the model, we solve it in a simplified setting and we use it to describe the movement of fluid within a vascularized lymph node.

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.

A 1D model characterizing the role of spatiotemporal contraction distributions on lymph transport

Lymphedema is a condition in which lymph transport is compromised. The factors that govern the timing of lymphatic contractions are largely unknown; however, these factors likely play a central role in lymphatic health. Computational models have proven useful in quantifying changes in lymph transport; nevertheless, there is still much unknown regarding the regulation of contractions. The purpose of this paper is to utilize computational modeling to examine the role of pacemaking activity in lymph transport. A 1D fluid-solid modeling framework was utilized to describe the interaction between the contracting vessel and the lymph flow. The distribution of contractions along a three-lymphangion chain in time and space was determined by specifying the pacemaking sites and parameters obtained from experimentation. The model effectively replicates the contractility patterns in experiments. Quantitatively, the flow rates were measured at 5.44 and 2.29 [Formula: see text], and the EF values were 78% and less than 33% in the WT and KO models, respectively, which are consistent with the literature. Applying pacemaking parameters in this modeling framework effectively captures lymphatic contractile wave propagations and their relation to lymph transport. It can serve as a motivation for conducting novel studies to evaluate lymphatic pumping function during the development of lymphedema.

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.

Lymphatic Collecting Vessel: New Perspectives on Mechanisms of Contractile Regulation and Potential Lymphatic Contractile Pathways to Target in Obesity and Metabolic Diseases

Obesity and metabolic syndrome pose a significant risk for developing cardiovascular disease and remain a critical healthcare challenge. Given the lymphatic system's role as a nexus for lipid absorption, immune cell trafficking, interstitial fluid and macromolecule homeostasis maintenance, the impact of obesity and metabolic disease on lymphatic function is a burgeoning field in lymphatic research. Work over the past decade has progressed from the association of an obese phenotype with Prox1 haploinsufficiency and the identification of obesity as a risk factor for lymphedema to consistent findings of lymphatic collecting vessel dysfunction across multiple metabolic disease models and organisms and characterization of obesity-induced lymphedema in the morbidly obese. Critically, recent findings have suggested that restoration of lymphatic function can also ameliorate obesity and insulin resistance, positing lymphatic targeted therapies as relevant pharmacological interventions. There remain, however, significant gaps in our understanding of lymphatic collecting vessel function, particularly the mechanisms that regulate the spontaneous contractile activity required for active lymph propulsion and lymph return in humans. In this article, we will review the current findings on lymphatic architecture and collecting vessel function, including recent advances in the ionic basis of lymphatic muscle contractile activity. We will then discuss lymphatic dysfunction observed with metabolic disruption and potential pathways to target with pharmacological approaches to improve lymphatic collecting vessel function.

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.

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.

Fluid pumping of peristaltic vessel fitted with elastic valves

Using numerical simulations, we probe the fluid flow in an axisymmetric peristaltic vessel fitted with elastic bi-leaflet valves. In this biomimetic system that mimics the flow generated in lymphatic vessels, we investigate the effects of the valve and vessel properties on pumping performance of the valved peristaltic vessel. The results indicate that valves significantly increase pumping by reducing backflow. The presence of valves, however, increases the viscous resistance therefore requiring greater work compared to valveless vessels. The benefit of the valves is the most significant when the fluid is pumped against an adverse pressure gradient and for low vessel contraction wave speeds. We identify the optimum vessel and valve parameters leading to the maximum pumping efficiency. We show that the optimum valve elasticity maximizes the pumping flow rate by allowing the valve to block more effectively the backflow while maintaining low resistance during the forward flow. We also examine the pumping in vessels where the vessel contraction amplitude is a function of the adverse pressure gradient as found in lymphatic vessels. We find that in this case the flow is limited by the work generated by the contracting vessel, suggesting that the pumping in lymphatic vessels is constrained by the performance of lymphatic muscle. Given the regional heterogeneity of valve morphology observed throughout the lymphatic vasculature, these results provide insight into how these variations might facilitate efficient lymphatic transport in the vessel's local physiologic context.

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.

A computational model of a network of initial lymphatics and pre-collectors with permeable interstitium

Initial lymphatic vessels are made up of overlapped endothelial cells that act as unidirectional valves enabling one-way drainage of tissue fluid into the lumen of the initial lymphatics when there is a favourable pressure gradient. Initial lymphatics subsequently drain this fluid into the collecting lymphatics. This paper describes a computational model for a network of passive rat mesenteric lymphatic vessels with sparse secondary valves. The network was simulated with the secondary valves both operational and non-operational. The effects on the cycle-mean outflow-rate from the network of both inflammation and the resistance of the surrounding interstitium were considered. The cycle-mean outflow-rate is sensitive to vessel stiffness. If the influence of primary-valve resistance is reduced relative to that of interstitial resistance and intravascular resistance, there is no absolute advantage of extrinsic pumping, since maximum outflow-rate occurs when vessels are rigid. However, there is relative advantage, in that the outflow-rate at intermediate stiffness is higher with the secondary valves functioning than when they are deactivated. If primary-valve resistance dominates, then extrinsic pumping of non-rigid vessels provides absolute advantage. The nonlinear relation between pressure drop and flow-rate of the endothelial primary valves, combined with downstream compliance and pulsatile external pressure, constitutes a separate mechanism of pumping. By enabling the consideration of interactions between multiple phenomena (primary valves, secondary valves, a real network geometry with multiple branches, deformable vessel walls, interstitial resistance and external pressures), the model offers a perspective for delineating physiological phenomena that have not yet been fully linked to the biomechanics of fluid flow through initial lymphatic networks.

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.

Lymphatic Vessel Network Structure and Physiology

The lymphatic system is comprised of a network of vessels interrelated with lymphoid tissue, which has the holistic function to maintain the local physiologic environment for every cell in all tissues of the body. The lymphatic system maintains extracellular fluid homeostasis favorable for optimal tissue function, removing substances that arise due to metabolism or cell death, and optimizing immunity against bacteria, viruses, parasites, and other antigens. This article provides a comprehensive review of important findings over the past century along with recent advances in the understanding of the anatomy and physiology of lymphatic vessels, including tissue/organ specificity, development, mechanisms of lymph formation and transport, lymphangiogenesis, and the roles of lymphatics in disease. © 2019 American Physiological Society. Compr Physiol 9:207-299, 2019.

In-silico dynamic analysis of cytotoxic drug administration to solid tumours: Effect of binding affinity and vessel permeability

The delivery of blood-borne therapeutic agents to solid tumours depends on a broad range of biophysical factors. We present a novel multiscale, multiphysics, in-silico modelling framework that encompasses dynamic tumour growth, angiogenesis and drug delivery, and use this model to simulate the intravenous delivery of cytotoxic drugs. The model accounts for chemo-, hapto- and mechanotactic vessel sprouting, extracellular matrix remodelling, mechano-sensitive vascular remodelling and collapse, intra- and extravascular drug transport, and tumour regression as an effect of a cytotoxic cancer drug. The modelling framework is flexible, allowing the drug properties to be specified, which provides realistic predictions of in-vivo vascular development and structure at different tumour stages. The model also enables the effects of neoadjuvant vascular normalisation to be implicitly tested by decreasing vessel wall pore size. We use the model to test the interplay between time of treatment, drug affinity rate and the size of the vessels’ endothelium pores on the delivery and subsequent tumour regression and vessel remodelling. Model predictions confirm that small-molecule drug delivery is dominated by diffusive transport and further predict that the time of treatment is important for low affinity but not high affinity cytotoxic drugs, the size of the vessel wall pores plays an important role in the effect of low affinity but not high affinity drugs, that high affinity cytotoxic drugs remodel the tumour vasculature providing a large window for the normalisation of the vascular architecture, and that the combination of large pores and high affinity enhances cytotoxic drug delivery efficiency. These results have implications for treatment planning and methods to enhance drug delivery, and highlight the importance of in-silico modelling in investigating the optimisation of cancer therapy on a personalised setting.

One of the main challenges in optimising cancer therapy is understanding the in-vivo cancer environment and how it changes over time. The efficacy of chemotherapeutic drugs is known to be strongly dependent on blood vessel wall properties and the architecture of the developing tumour vasculature, which in turn are dependent on biochemical and mechanical interactions between cancer cells and their microenvironment. Here we present a novel in-silico modelling framework of dynamic tumour growth, angiogenesis and drug delivery, and we use it to explore biophysical factors governing the efficient delivery of cytotoxic drugs to solid tumours. We find that the time of treatment and vessel permeability are important factors for the efficacy of chemical agents with low binding affinity, that high affinity drugs can impact the tumour vasculature remodelling and bring vascular structure back to a more normalised state, and that the combination of large-sized vessel wall pores and high affinity enhances cytotoxic drug delivery and efficacy. These results have implications for treatment planning and optimisation, and show how in-silico models can be used to help understand and optimise cancer therapy.

A one-dimensional mathematical model of collecting lymphatics coupled with an electro-fluid-mechanical contraction model and valve dynamics

We propose a one-dimensional model for collecting lymphatics coupled with a novel Electro-Fluid-Mechanical Contraction (EFMC) model for dynamical contractions, based on a modified FitzHugh-Nagumo model for action potentials. The one-dimensional model for a deformable lymphatic vessel is a nonlinear system of hyperbolic Partial Differential Equations (PDEs). The EFMC model combines the electrical activity of lymphangions (action potentials) with fluid-mechanical feedback (circumferential stretch of the lymphatic wall and wall shear stress) and lymphatic vessel wall contractions. The EFMC model is governed by four Ordinary Differential Equations (ODEs) and phenomenologically relies on: (1) environmental calcium influx, (2) stretch-activated calcium influx, and (3) contraction inhibitions induced by wall shear stresses. We carried out a stability analysis of the stationary state of the EFMC model. Contractions turn out to be triggered by the instability of the stationary state. Overall, the EFMC model allows emulating the influence of pressure and wall shear stress on the frequency of contractions observed experimentally. Lymphatic valves are modelled by extending an existing lumped-parameter model for blood vessels. Modern numerical methods are employed for the one-dimensional model (PDEs), for the EFMC model and valve dynamics (ODEs). Adopting the geometrical structure of collecting lymphatics from rat mesentery, we apply the full mathematical model to a carefully selected suite of test problems inspired by experiments. We analysed several indices of a single lymphangion for a wide range of upstream and downstream pressure combinations which included both favourable and adverse pressure gradients. The most influential model parameters were identified by performing two sensitivity analyses for favourable and adverse pressure gradients.

Contraction of collecting lymphatics: organization of pressure-dependent rate for multiple lymphangions

The paper describes the extension of a previously developed model of pressure-dependent contraction rate to the case of multiple lymphangions. Mechanical factors are key modulators of active lymphatic pumping. As part of the evolution of our lumped-parameter model to match experimental findings, we have designed an algorithm whereby the time until the next contraction depends on lymphangion transmural pressure in the contraction just completed. The functional dependence of frequency on pressure is quantitatively matched to isobaric contraction experiments on isolated lymphatic segments. When each of several lymphangions is given this ability, a scheme for their coordination must be instituted to match the observed synchronization. Accordingly, and in line with an experiment on an isolated lymphatic vessel segment in which we measured contraction sequence and conduction delay, we took the fundamental principle to be that local timing can be overridden by signals to initiate contraction that start in adjacent lymphangions, conducted with a short delay. The scheme leads to retrograde conduction when the lymphangion chain is pumping against an adverse pressure difference, but antegrade conduction when contractions occur with no or a favourable pressure difference. Abolition of these conducted signals leads to chaotic variation of cycle-mean flow-rate from the chain, diastolic duration in each lymphangion, and inter-lymphangion delays. Chaotic rhythm is also seen under other circumstances. Because the model responds to increasing adverse pressure difference by increasing the repetition rate of contractions, it maintains time-average output flow-rate better than one with fixed repetition rate.

Lymphatic System Flows

The supply of oxygen and nutrients to tissues is performed by the blood system, and involves a net leakage of fluid outward at the capillary level. One of the principal functions of the lymphatic system is to gather this fluid and return it to the blood system to maintain overall fluid balance. Fluid in the interstitial spaces is often at subatmospheric pressure, and the return points into the venous system are at pressures of approximately 20 cmH₂O. This adverse pressure difference is overcome by the active pumping of collecting lymphatic vessels, which feature closely spaced one-way valves and contractile muscle cells in their walls. Passive vessel squeezing causes further pumping. The dynamics of lymphatic pumping have been investigated experimentally and mathematically, revealing complex behaviours indicating that the system performance is robust against minor perturbations in pressure and flow. More serious disruptions can lead to incurable swelling of tissues called lymphœdema.

Lymphatic System Flows

The supply of oxygen and nutrients to tissues is performed by the blood system, and involves a net leakage of fluid outward at the capillary level. One of the principal functions of the lymphatic system is to gather this fluid and return it to the blood system to maintain overall fluid balance. Fluid in the interstitial spaces is often at subatmospheric pressure, and the return points into the venous system are at pressures of approximately 20 cmH₂O. This adverse pressure difference is overcome by the active pumping of collecting lymphatic vessels, which feature closely spaced one-way valves and contractile muscle cells in their walls. Passive vessel squeezing causes further pumping. The dynamics of lymphatic pumping have been investigated experimentally and mathematically, revealing complex behaviours indicating that the system performance is robust against minor perturbations in pressure and flow. More serious disruptions can lead to incurable swelling of tissues called lymphœdema.

The relationship between lymphangion chain length and maximum pressure generation established through in vivo imaging and computational modeling

The intrinsic contraction of collecting lymphatic vessels serves as a pumping system to propel lymph against hydrostatic pressure gradients as it returns interstitial fluid to the venous circulation. In the present study, we proposed and validated that the maximum opposing outflow pressure along a chain of lymphangions at which flow can be achieved increases with the length of chain. Using minimally invasive near-infrared imaging to measure the effective pumping pressure at various locations in the rat tail, we demonstrated increases in pumping pressure along the length of the tail. Computational simulations based on a microstructurally motivated model of a chain of lymphangions informed from biaxial testing of isolated vessels was used to provide insights into the pumping mechanisms responsible for the pressure increases observed in vivo. These models suggest that the number of lymphangions in the chain and smooth muscle cell force generation play a significant role in determining the maximum outflow pressure, whereas the frequency of contraction has no effect. In vivo administration of nitric oxide attenuated lymphatic contraction, subsequently lowering the effective pumping pressure. Computational simulations suggest that the reduction in contractile strength of smooth muscle cells in the presence of nitric oxide can account for the reductions in outflow pressure observed along the lymphangion chain in vivo. Thus, combining modeling with multiple measurements of lymphatic pumping pressure provides a method for approximating intrinsic lymphatic muscle activity noninvasively in vivo while also providing insights into factors that determine the extent that a lymphangion chain can transport fluid against an adverse pressure gradient. NEW & NOTEWORTHY Here, we report the first minimally invasive in vivo measurements of the relationship between lymphangion chain length and lymphatic pumping pressure. We also provide the first in vivo validation of lumped parameter models of lymphangion chains previously developed through data obtained from isolated vessel testing.

Opportunities for Studying the Hydrodynamic Context for Breast Cancer Cell Spread Through Lymph Flow

The lymphatic system serves as the primary route for the metastatic spread of breast cancer cells (BCCs). A scarcity of information exists with regard to the advection of BCCs in lymph flow and a fundamental understanding of the response of BCCs to the forces in the lymphatics needs to be established. This review summarizes the flow environment metastatic BCCs are exposed to in the lymphatics. Special attention is paid to the behavior of cells/particles in microflows in an attempt to elucidate the behavior of BCCs under lymph flow conditions (Reynolds number <1).

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.

Synchronization and Random Triggering of Lymphatic Vessel Contractions

The lymphatic system is responsible for transporting interstitial fluid back to the bloodstream, but unlike the cardiovascular system, lacks a centralized pump-the heart–to drive flow. Instead, each collecting lymphatic vessel can individually contract and dilate producing unidirectional flow enforced by intraluminal check valves. Due to the large number and spatial distribution of such pumps, high-level coordination would be unwieldy. This leads to the question of how each segment of lymphatic vessel responds to local signals that can contribute to the coordination of pumping on a network basis. Beginning with elementary fluid mechanics and known cellular behaviors, we show that two complementary oscillators emerge from i) mechanical stretch with calcium ion transport and ii) fluid shear stress induced nitric oxide production (NO). Using numerical simulation and linear stability analysis we show that the newly identified shear-NO oscillator shares similarities with the well-known Van der Pol oscillator, but has unique characteristics. Depending on the operating conditions, the shear-NO process may i) be inherently stable, ii) oscillate spontaneously in response to random disturbances or iii) synchronize with weak periodic stimuli. When the complementary shear-driven and stretch-driven oscillators interact, either may dominate, producing a rich family of behaviors similar to those observed in vivo.

For decades, cardiovascular physiology has been an area of intense research, and we have a fundamental understanding of the mechanisms the heart uses to drive blood flow through the distributed network of vessels in the body. The lymphatic system is now receiving similar attention as more is learned about its functional role in disease processes. The importance of the lymphatic system in collecting excess fluid from tissues and returning it to the blood is well known, but how the lymph flow is regulated without a central pump is poorly understood. Each segment of collecting lymphatic vessel can independently contract yielding a network of distributed pump/conduits. This paper shows how the lymphatic muscle cells that squeeze fluid along the lymphatic vessels can be effectively regulated using only chemical and mechanical signals that they receive from their immediate microenvironment. Using stability theory and the tools of nonlinear dynamics we identify two complementary oscillators that respond to stretch of the vessel wall and shear of fluid flowing over the vessel wall. Numerical simulations of the combined oscillators show that they have characteristics well suited to the regulation of distributed systems in general and may have application in other biological and physical contexts.

A lumped parameter model of mechanically mediated acute and long-term adaptations of contractility and geometry in lymphatics for characterization of lymphedema

A primary purpose of the lymphatic system is to transport fluid from peripheral tissues to the central venous system in order to maintain tissue-fluid balance. Failure to perform this task results in lymphedema marked by swelling of the affected limb as well as geometric remodeling and reduced contractility of the affected lymphatic vessels. The mechanical environment has been implicated in the regulation of lymphatic contractility, but it is unknown how changes in the mechanical environment are related to loss of contractile function and remodeling of the tissue. The purpose of this paper was to introduce a new theoretical framework for acute and long-term adaptations of lymphatic vessels to changes in mechanical loading. This theoretical framework combines a simplified version of a published lumped parameter model for lymphangion function and lymph transport, a published microstructurally motivated constitutive model for the active and passive mechanical behavior of isolated rat thoracic ducts, and novel models for acute mechanically mediated vasoreactive adaptations and long-term volumetric growth to simulate changes in muscle contractility and geometry of a single isolated rat thoracic duct in response to a sustained elevation in afterload. The illustrative examples highlight the potential role of the mechanical environment in the acute maintenance of contractility and long-term geometric remodeling, presumably aimed at meeting fluid flow demands while also maintaining mechanical homeostasis. Results demonstrate that contractility may adapt in response to shear stress to meet fluid flow demands and show that pressure-induced long-term geometric remodeling may attenuate these adaptations and reduce fluid flow. The modeling framework and illustrative simulations help suggest relevant experiments that are necessary to accurately quantify and predict the acute and long-term adaptations of lymphangions to altered mechanical loading.

Pump function curve shape for a model lymphatic vessel

The transport capacity of a contractile segment of lymphatic vessel is defined by its pump function curve relating mean flow-rate and adverse pressure difference. Numerous system characteristics affect curve shape and the magnitude of the generated flow-rates and pressures. Some cannot be varied experimentally, but their separate and interacting effects can be systematically revealed numerically. This paper explores variations in the rate of change of active tension and the form of the relation between active tension and muscle length, factors not known from experiment to functional precision. Whether the pump function curve bends toward or away from the origin depends partly on the curvature of the passive pressure-diameter relation near zero transmural pressure, but rather more on the form of the relation between active tension and muscle length. A pump function curve bending away from the origin defines a well-performing pump by maximum steady output power. This behaviour is favoured by a length/active-tension relationship which sustains tension at smaller lengths. Such a relationship also favours high peak mechanical efficiency, defined as output power divided by the input power obtained from the lymphangion diameter changes and active-tension time-course. The results highlight the need to pin down experimentally the form of the length/active-tension relationship.

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.

Consequences of intravascular lymphatic valve properties: a study of contraction timing in a multi-lymphangion model

The observed properties of valves in collecting lymphatic vessels include transmural pressure-dependent bias to the open state and hysteresis. The bias may reduce resistance to flow when the vessel is functioning as a conduit. However, lymphatic pumping implies a streamwise increase in mean pressure across each valve, suggesting that the bias is then potentially unhelpful. Lymph pumping by a model of several collecting lymphatic vessel segments (lymphangions) in series, which incorporated these properties, was investigated under conditions of adverse pressure difference while varying the refractory period between active muscular contractions and the inter-lymphangion contraction delay. It was found that many combinations of the timing parameters and the adverse pressure difference led to one or more intermediate valves remaining open instead of switching between open and closed states during repetitive contraction cycles. Cyclic valve switching was reliably indicated if the mean pressure in a lymphangion over a cycle was higher than that in the lymphangion upstream, but either lack of or very brief valve closure could cause mean pressure to be lower downstream. Widely separated combinations of refractory period and delay time were found to produce the greatest flow-rate for a given pressure difference. The efficiency of pumping was always maximized by a long refractory period and lymphangion contraction starting when the contraction of the lymphangion immediately upstream was peaking. By means of an ex vivo experiment, it was verified that intermediate valves in a chain of pumping lymphangions can remain open, while the lymphangions on either side of the open valve continue to execute contractions.

Mechanobiological oscillators control lymph flow

The ability of cells to sense and respond to physical forces has been recognized for decades, but researchers are only beginning to appreciate the fundamental importance of mechanical signals in biology. At the larger scale, there has been increased interest in the collective organization of cells and their ability to produce complex, "emergent" behaviors. Often, these complex behaviors result in tissue-level control mechanisms that manifest as biological oscillators, such as observed in fireflies, heartbeats, and circadian rhythms. In many cases, these complex, collective behaviors are controlled--at least in part--by physical forces imposed on the tissue or created by the cells. Here, we use mathematical simulations to show that two complementary mechanobiological oscillators are sufficient to control fluid transport in the lymphatic system: Ca(2+)-mediated contractions can be triggered by vessel stretch, whereas nitric oxide produced in response to the resulting fluid shear stress causes the lymphatic vessel to relax locally. Our model predicts that the Ca(2+) and NO levels alternate spatiotemporally, establishing complementary feedback loops, and that the resulting phasic contractions drive lymph flow. We show that this mechanism is self-regulating and robust over a range of fluid pressure environments, allowing the lymphatic vessels to provide pumping when needed but remain open when flow can be driven by tissue pressure or gravity. Our simulations accurately reproduce the responses to pressure challenges and signaling pathway manipulations observed experimentally, providing an integrated conceptual framework for lymphatic function.

Determining the combined effect of the lymphatic valve leaflets and sinus on resistance to forward flow

The lymphatic system is vital to a proper maintenance of fluid and solute homeostasis. Collecting lymphatics are composed of actively contracting tubular vessels segmented by bulbous sinus regions that encapsulate bi-leaflet check valves. Valve resistance to forward flow strongly influences pumping performance. However, because of the sub-millimeter size of the vessels with flow rates typically <1 ml/h and pressures of a few cmH2O, resistance is difficult to measure experimentally. Using a newly defined idealized geometry, we employed an uncoupled approach where the solid leaflet deflections of the open valve were computed and lymph flow calculations were subsequently performed. We sought to understand: 1) the effect of sinus and leaflet size on the resulting deflections experienced by the valve leaflets and 2) the effects on valve resistance to forward flow of the fully open valve. For geometries with sinus-to-root diameter ratios >1.39, the average resistance to forward flow was 0.95×10(6)[g/(cm4 s)]. Compared to the viscous pressure drop that would occur in a straight tube the same diameter as the upstream lymphangion, valve leaflets alone increase the pressure drop up to 35%. However, the presence of the sinus reduces viscous losses, with the net effect that when combined with leaflets the overall resistance is less than that of the equivalent continuing straight tube. Accurately quantifying resistance to forward flow will add to the knowledge used to develop therapeutics for treating lymphatic disorders and may eventually lead to understanding some forms of primary lymphedema.

Mechanobiology of lymphatic contractions

The lymphatic system is responsible for controlling tissue fluid pressure by facilitating flow of lymph (i.e. the plasma and cells that enter the lymphatic system). Because lymph contains cells of the immune system, its transport is not only important for fluid homeostasis, but also immune function. Lymph drainage can occur via passive flow or active pumping, and much research has identified the key biochemical and mechanical factors that affect output. Although many studies and reviews have addressed how tissue properties and fluid mechanics (i.e. pressure gradients) affect lymph transport [1-3] there is less known about lymphatic mechanobiology. As opposed to passive mechanical properties, mechanobiology describes the active coupling of mechanical signals and biochemical pathways. Lymphatic vasomotion is the result of a fascinating system affected by mechanical forces exerted by the flowing lymph, including pressure-induced vessel stretch and flow-induced shear stresses. These forces can trigger or modulate biochemical pathways important for controlling the lymphatic contractions. Here, I review the current understanding of lymphatic vessel function, focusing on vessel mechanobiology, and summarize the prospects for a comprehensive understanding that integrates the mechanical and biomechanical control mechanisms in the lymphatic system.

Mechanical forces and lymphatic transport

This review examines the current understanding of how the lymphatic vessel network can optimize lymph flow in response to various mechanical forces. Lymphatics are organized as a vascular tree, with blind-ended initial lymphatics, precollectors, prenodal collecting lymphatics, lymph nodes, postnodal collecting lymphatics and the larger trunks (thoracic duct and right lymph duct) that connect to the subclavian veins. The formation of lymph from interstitial fluid depends heavily on oscillating pressure gradients to drive fluid into initial lymphatics. Collecting lymphatics are segmented vessels with unidirectional valves, with each segment, called a lymphangion, possessing an intrinsic pumping mechanism. The lymphangions propel lymph forward against a hydrostatic pressure gradient. Fluid is returned to the central circulation both at lymph nodes and via the larger lymphatic trunks. Several recent developments are discussed, including evidence for the active role of endothelial cells in lymph formation; recent developments on how inflow pressure, outflow pressure, and shear stress affect the pump function of the lymphangion; lymphatic valve gating mechanisms; collecting lymphatic permeability; and current interpretations of the molecular mechanisms within lymphatic endothelial cells and smooth muscle. An improved understanding of the physiological mechanisms by which lymphatic vessels sense mechanical stimuli, integrate the information, and generate the appropriate response is key for determining the pathogenesis of lymphatic insufficiency and developing treatments for lymphedema.

Primary and secondary lymphatic valve development: molecular, functional and mechanical insights

Fluid homeostasis in vertebrates critically relies on the lymphatic system forming a hierarchical network of lymphatic capillaries and collecting lymphatics, for the efficient drainage and transport of extravasated fluid back to the cardiovascular system. Blind-ended lymphatic capillaries employ specialized junctions and anchoring filaments to encourage a unidirectional flow of the interstitial fluid into the initial lymphatic vessels, whereas collecting lymphatics are responsible for the active propulsion of the lymph to the venous circulation via the combined action of lymphatic muscle cells and intraluminal valves. Here we describe recent findings on molecular and physical factors regulating the development and maturation of these two types of valves and examine their role in tissue-fluid homeostasis.

Parameter sensitivity analysis of a lumped-parameter model of a chain of lymphangions in series

Any disruption of the lymphatic system due to trauma or injury can lead to edema. There is no effective cure for lymphedema, partly because predictive knowledge of lymphatic system reactions to interventions is lacking. A well-developed model of the system could greatly improve our understanding of its function. Lymphangions, defined as the vessel segment between two valves, are the individual pumping units. Based on our previous lumped-parameter model of a chain of lymphangions, this study aimed to identify the parameters that affect the system output the most using a sensitivity analysis. The system was highly sensitive to minimum valve resistance, such that variations in this parameter caused an order-of-magnitude change in time-average flow rate for certain values of imposed pressure difference. Average flow rate doubled when contraction frequency was increased within its physiological range. Optimum lymphangion length was found to be some 13-14.5 diameters. A peak of time-average flow rate occurred when transmural pressure was such that the pressure-diameter loop for active contractions was centered near maximum passive vessel compliance. Increasing the number of lymphangions in the chain improved the pumping in the presence of larger adverse pressure differences. For a given pressure difference, the optimal number of lymphangions increased with the total vessel length. These results indicate that further experiments to estimate valve resistance more accurately are necessary. The existence of an optimal value of transmural pressure may provide additional guidelines for increasing pumping in areas affected by edema.

Development of a model of a multi-lymphangion lymphatic vessel incorporating realistic and measured parameter values

Our published model of a lymphatic vessel consisting of multiple actively contracting segments between non-return valves has been further developed by the incorporation of properties derived from observations and measurements of rat mesenteric vessels. These included (1) a refractory period between contractions, (2) a highly nonlinear form for the passive part of the pressure-diameter relationship, (3) hysteretic and transmural-pressure-dependent valve opening and closing pressure thresholds and (4) dependence of active tension on muscle length as reflected in local diameter. Experimentally, lymphatic valves are known to be biased to stay open. In consequence, in the improved model, vessel pumping of fluid suffers losses by regurgitation, and valve closure is dependent on backflow first causing an adverse valve pressure drop sufficient to reach the closure threshold. The assumed resistance of an open valve therefore becomes a critical parameter, and experiments to measure this quantity are reported here. However, incorporating this parameter value, along with other parameter values based on existing measurements, led to ineffective pumping. It is argued that the published measurements of valve-closing pressure threshold overestimate this quantity owing to neglect of micro-pipette resistance. An estimate is made of the extent of the possible resulting error. Correcting by this amount, the pumping performance is improved, but still very inefficient unless the open-valve resistance is also increased beyond the measured level. Arguments are given as to why this is justified, and other areas where experimental data are lacking are identified. The model is capable of future adaptation as new experimental data appear.