A computer model of the lymphatic system

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.

Numerical studies of the lymphatic uptake rate

Lymphatic uptake is essential for transporting nutrients, wastes, immune cells, and therapeutic proteins. Despite its importance, the literature lacks a quantitative analysis of the factors that affect lymphatic uptake, including interstitial pressure, downstream pressure, and tissue deformation. In this paper, we present a coupled model of a poroelastic tissue with initial lymphatics and quantify the impact of these factors on the rate of lymphatic uptake. Our results indicate that the lymphatic uptake increases with the amplitude of the oscillating downstream pressure when the amplitude exceeds a threshold. Additionally, the cross-sectional area of initial lymphatics increases with the volumetric strain of the tissue, while the interstitial pressure increases when the strain rate becomes negative. Therefore, the lymphatic uptake reaches its maximum when the tissue has positive volumetric strain while being compressed. We have also investigated the effect of intersection angles and positions of two initial lymphatics and concluded that they have minor impacts on lymphatic uptake. However, the lymphatic uptake per unit length of initial lymphatics decreases with their total length. These findings advance our understanding of lymphatic uptake and can guide the development of strategies to accelerate the transport of therapeutics.

In Vitro, In Vivo, and In Silico Models of Lymphangiogenesis in Solid Malignancies

Simple Summary

Lymphangiogenesis is the formation of new lymphatic vessels in physiological conditions but has also been found to be associated with pathologies. For example, it has been proven to be involved in cancer progression and metastatic dissemination through the body. Thus, it became a key element to study in the management of this widespread disease. To date, the study of lymphangiogenesis takes place at the biological (in vitro and in vivo) and computational (in silico) levels. The association of these complementary fields combined with imaging techniques constitutes a real toolbox in pathological lymphangiogenesis understanding.

Abstract

Lymphangiogenesis (LA) is the formation of new lymphatic vessels by lymphatic endothelial cells (LECs) sprouting from pre-existing lymphatic vessels. It is increasingly recognized as being involved in many diseases, such as in cancer and secondary lymphedema, which most often results from cancer treatments. For some cancers, excessive LA is associated with cancer progression and metastatic dissemination to the lymph nodes (LNs) through lymphatic vessels. The study of LA through in vitro, in vivo, and, more recently, in silico models is of paramount importance in providing novel insights and identifying the key molecular actors in the biological dysregulation of this process under pathological conditions. In this review, the different biological (in vitro and in vivo) models of LA, especially in a cancer context, are explained and discussed, highlighting their principal modeled features as well as their advantages and drawbacks. Imaging techniques of the lymphatics, complementary or even essential to in vivo models, are also clarified and allow the establishment of the link with computational approaches. In silico models are introduced, theoretically described, and illustrated with examples specific to the lymphatic system and the LA. Together, these models constitute a toolbox allowing the LA research to be brought to the next level.

Cerebrospinal fluid dynamics coupled to the global circulation in holistic setting: Mathematical models, numerical methods and applications

This paper presents a mathematical model of the global, arterio-venous circulation in the entire human body, coupled to a refined description of the cerebrospinal fluid (CSF) dynamics in the craniospinal cavity. The present model represents a substantially revised version of the original Müller-Toro mathematical model. It includes one-dimensional (1D), non-linear systems of partial differential equations for 323 major blood vessels and 85 zero-dimensional, differential-algebraic systems for the remaining components. Highlights include the myogenic mechanism of cerebral blood regulation; refined vasculature for the inner ear, the brainstem and the cerebellum; and viscoelastic, rather than purely elastic, models for all blood vessels, arterial and venous. The derived 1D parabolic systems of partial differential equations for all major vessels are approximated by hyperbolic systems with stiff source terms following a relaxation approach. A major novelty of this paper is the coupling of the circulation, as described, to a refined description of the CSF dynamics in the craniospinal cavity, following Linninger et al. The numerical solution methodology employed to approximate the hyperbolic non-linear systems of partial differential equations with stiff source terms is based on the Arbitrary DERivative Riemann problem finite volume framework, supplemented with a well-balanced formulation, and a local time stepping procedure. The full model is validated through comparison of computational results against published data and bespoke MRI measurements. Then we present two medical applications: (i) transverse sinus stenoses and their relation to Idiopathic Intracranial Hypertension; and (ii) extra-cranial venous strictures and their impact in the inner ear circulation, and its implications for Ménière's disease.

Mathematical Modeling of Lymph Node Drainage Function by Neural Network

The lymph node (LN) represents a key structural component of the lymphatic system network responsible for the fluid balance in tissues and the immune system functioning. Playing an important role in providing the immune defense of the host organism, LNs can also contribute to the progression of pathological processes, e.g., the spreading of cancer cells. To gain a deeper understanding of the transport function of LNs, experimental approaches are used. Mathematical modeling of the fluid transport through the LN represents a complementary tool for studying the LN functioning under broadly varying physiological conditions. We developed an artificial neural network (NN) model to describe the lymph node drainage function. The NN model predicts the flow characteristics through the LN, including the exchange with the blood vascular systems in relation to the boundary and lymphodynamic conditions, such as the afferent lymph flow, Darcy’s law constants and Starling’s equation parameters. The model is formulated as a feedforward NN with one hidden layer. The NN complements the computational physics-based model of a stationary fluid flow through the LN and the fluid transport across the blood vessel system of the LN. The physical model is specified as a system of boundary integral equations (IEs) equivalent to the original partial differential equations (PDEs; Darcy’s Law and Starling’s equation) formulations. The IE model has been used to generate the training dataset for identifying the NN model architecture and parameters. The computation of the output LN drainage function characteristics (the fluid flow parameters and the exchange with blood) with the trained NN model required about 1000-fold less central processing unit (CPU) time than computationally tracing the flow characteristics of interest with the physics-based IE model. The use of the presented computational models will allow for a more realistic description and prediction of the immune cell circulation, cytokine distribution and drug pharmacokinetics in humans under various health and disease states as well as assisting in the development of artificial LN-on-a-chip technologies.

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.

Graph Theory for Modeling and Analysis of the Human Lymphatic System

The human lymphatic system (HLS) is a complex network of lymphatic organs linked through the lymphatic vessels. We present a graph theory-based approach to model and analyze the human lymphatic network. Two different methods of building a graph are considered: the method using anatomical data directly and the method based on a system of rules derived from structural analysis of HLS. A simple anatomical data-based graph is converted to an oriented graph by quantifying the steady-state fluid balance in the lymphatic network with the use of the Poiseuille equation in vessels and the mass conservation at vessel junctions. A computational algorithm for the generation of the rule-based random graph is developed and implemented. Some fundamental characteristics of the two types of HLS graph models are analyzed using different metrics such as graph energy, clustering, robustness, etc.

Mathematical Modelling of the Structure and Function of the Lymphatic System

This paper presents current knowledge about the structure and function of the lymphatic system. Mathematical models of lymph flow in the single lymphangion, the series of lymphangions, the lymph nodes, and the whole lymphatic system are considered. The main results and further perspectives are discussed.

Multiscale Agent-Based and Hybrid Modeling of the Tumor Immune Microenvironment

Multiscale systems biology and systems pharmacology are powerful methodologies that are playing increasingly important roles in understanding the fundamental mechanisms of biological phenomena and in clinical applications. In this review, we summarize the state of the art in the applications of agent-based models (ABM) and hybrid modeling to the tumor immune microenvironment and cancer immune response, including immunotherapy. Heterogeneity is a hallmark of cancer; tumor heterogeneity at the molecular, cellular, and tissue scales is a major determinant of metastasis, drug resistance, and low response rate to molecular targeted therapies and immunotherapies. Agent-based modeling is an effective methodology to obtain and understand quantitative characteristics of these processes and to propose clinical solutions aimed at overcoming the current obstacles in cancer treatment. We review models focusing on intra-tumor heterogeneity, particularly on interactions between cancer cells and stromal cells, including immune cells, the role of tumor-associated vasculature in the immune response, immune-related tumor mechanobiology, and cancer immunotherapy. We discuss the role of digital pathology in parameterizing and validating spatial computational models and potential applications to therapeutics.

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.

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).

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.

Quantification of the passive and active biaxial mechanical behaviour and microstructural organization of rat thoracic ducts

Mechanical loading conditions are likely to play a key role in passive and active (contractile) behaviour of lymphatic vessels. The development of a microstructurally motivated model of lymphatic tissue is necessary for quantification of mechanically mediated maladaptive remodelling in the lymphatic vasculature. Towards this end, we performed cylindrical biaxial testing of Sprague-Dawley rat thoracic ducts (n = 6) and constitutive modelling to characterize their mechanical behaviour. Spontaneous contraction was quantified at transmural pressures of 3, 6 and 9 cmH2O. Cyclic inflation in calcium-free saline was performed at fixed axial stretches between 1.30 and 1.60, while recording pressure, outer diameter and axial force. A microstructurally motivated four-fibre family constitutive model originally proposed by Holzapfel et al. (Holzapfel et al. 2000 J. Elast. 61, 1-48. (doi:10.1023/A:1010835316564)) was used to quantify the passive mechanical response, and the model of Rachev and Hayashi was used to quantify the active (contractile) mechanical response. The average error between data and theory was 8.9 ± 0.8% for passive data and 6.6 ± 2.6% and 6.8 ± 3.4% for the systolic and basal conditions, respectively, for active data. Multi-photon microscopy was performed to quantify vessel wall thickness (32.2 ± 1.60 µm) and elastin and collagen organization for three loading conditions. Elastin exhibited structural 'fibre families' oriented nearly circumferentially and axially. Sample-to-sample variation was observed in collagen fibre distributions, which were often non-axisymmetric, suggesting material asymmetry. In closure, this paper presents a microstructurally motivated model that accurately captures the biaxial active and passive mechanical behaviour in lymphatics and offers potential for future research to identify parameters contributing to mechanically mediated disease development.

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.

Confocal image-based computational modeling of nitric oxide transport in a rat mesenteric lymphatic vessel

The lymphatic system plays important roles in protein and solute transport as well as in the immune system. Its functionality is vital to proper homeostasis and fluid balance. Lymph may be propelled by intrinsic (active) vessel pumping or passive compression from external tissue movement. With regard to the former, nitric oxide (NO) is known to play an important role modulating lymphatic vessel contraction and vasodilation. Lymphatic endothelial cells (LECs) are sensitive to shear, and increases in flow have been shown to cause enhanced production of NO by LECs. Additionally, high concentrations of NO have been experimentally observed in the sinus region of mesenteric lymphatic vessels. A computational flow and mass transfer model using physiologic geometries obtained from confocal images of a rat mesenteric lymphatic vessel was developed to determine the characteristics of NO transport in the lymphatic flow regime. Both steady and unsteady analyses were performed. Production of NO was shear-dependent; basal cases using constant production were also generated. Simulations revealed areas of flow stagnation adjacent to the valve leaflets, suggesting the high concentrations observed here experimentally are due to minimal convection in this region. LEC sensitivity to shear was found to alter the concentration of NO in the vessel, and the convective forces were found to profoundly affect the concentration of NO at a Péclet value greater than approximately 61. The quasisteady analysis was able to resolve wall shear stress within 0.15% of the unsteady case. However, the percent difference between unsteady and quasisteady conditions was higher for NO concentration (6.7%). We have shown high NO concentrations adjacent to the valve leaflets are most likely due to flow-mediated processes rather than differential production by shear-sensitive LECs. Additionally, this model supports experimental findings of shear-dependent production, since removing shear dependence resulted in concentrations that are physiologically counterintuitive. Understanding the transport mechanisms and flow regimes in the lymphatic vasculature could help in the development of therapeutics to treat lymphatic disorders.

In vivo quantification of lymph viscosity and pressure in lymphatic vessels and draining lymph nodes of arthritic joints in mice

Rheumatoid arthritis (RA) is a chronic inflammatory joint disease with episodic flares. In TNF-Tg mice, a model of inflammatory-erosive arthritis, the popliteal lymph node (PLN) enlarges during the pre-arthritic 'expanding' phase, and then 'collapses' with adjacent knee flare associated with the loss of the intrinsic lymphatic pulse. As the mechanisms responsible are unknown, we developed in vivo methods to quantify lymph viscosity and pressure in mice with wild-type (WT), expanding and collapsed PLN. While no differences in viscosity were detected via multiphoton fluorescence recovery after photobleaching (MP-FRAP) of injected FITC-BSA, a 32.6% decrease in lymph speed was observed in vessels afferent to collapsed PLN (P < 0.05). Direct measurement of intra-lymph node pressure (LNP) demonstrated a decrease in expanding PLN versus WT pressure (3.41 ± 0.43 vs. 6.86 ± 0.56 cmH2O; P < 0.01), which dramatically increased to 9.92 ± 1.79 cmH2O in collapsed PLN. Lymphatic pumping pressure (LPP), measured indirectly by slowly releasing a pressurized cuff occluding indocyanine green (ICG), demonstrated an increase in vessels afferent to expanding PLN versus WT (18.76 ± 2.34 vs. 11.04 ± 1.47 cmH2O; P < 0.01), which dropped to 2.61 ± 0.72 cmH2O (P < 0.001) after PLN collapse. Herein, we document the first in vivo measurements of murine lymph viscosity and lymphatic pressure, and provide evidence to support the hypothesis that lymphangiogenesis and lymphatic transport are compensatory mechanisms to prevent synovitis via increased drainage of inflamed joints. Furthermore, the decrease in lymphatic flow and loss of LPP during PLN collapse are consistent with decreased drainage from the joint during arthritic flare, and validate these biomarkers of RA progression and possibly other chronic inflammatory conditions.

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.

Modelling the lymphatic system: challenges and opportunities

The lymphatic system is a vital part of the circulatory and immune systems, and plays an important role in homeostasis by controlling extracellular fluid volume and in combating infection. Nevertheless, there is a notable disparity in terms of research effort expended in relation to the treatment of lymphatic diseases in contrast to the cardiovascular system. While similarities to the cardiovascular system exist, there are considerable differences in their anatomy and physiology. This review outlines some of the challenges and opportunities for those engaged in modelling biological systems. The study of the lymphatic system is still in its infancy, the vast majority of the models presented in the literature to date having been developed since 2003. The number of distinct models and their variants are few in number, and only one effort has been made thus far to study the entire lymphatic network; elements of the lymphatic system such as the nodes, which act as pumps and reservoirs, have not been addressed by mathematical models. Clearly, more work will be necessary in combination with experimental verification in order to progress and update the knowledge on the function of the lymphatic system. As our knowledge and understanding of its function increase, new and more effective treatments of lymphatic diseases are bound to emerge.

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.

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.

Nonlinear lymphangion pressure-volume relationship minimizes edema

Lymphangions, the segments of lymphatic vessel between two valves, contract cyclically and actively pump, analogous to cardiac ventricles. Besides having a discernable systole and diastole, lymphangions have a relatively linear end-systolic pressure-volume relationship (with slope E(max)) and a nonlinear end-diastolic pressure-volume relationship (with slope E(min)). To counter increased microvascular filtration (causing increased lymphatic inlet pressure), lymphangions must respond to modest increases in transmural pressure by increasing pumping. To counter venous hypertension (causing increased lymphatic inlet and outlet pressures), lymphangions must respond to potentially large increases in transmural pressure by maintaining lymph flow. We therefore hypothesized that the nonlinear lymphangion pressure-volume relationship allows transition from a transmural pressure-dependent stroke volume to a transmural pressure-independent stroke volume as transmural pressure increases. To test this hypothesis, we applied a mathematical model based on the time-varying elastance concept typically applied to ventricles (the ratio of pressure to volume cycles periodically from a minimum, E(min), to a maximum, E(max)). This model predicted that lymphangions increase stroke volume and stroke work with transmural pressure if E(min) < E(max) at low transmural pressures, but maintain stroke volume and stroke work if E(min)= E(max) at higher transmural pressures. Furthermore, at higher transmural pressures, stroke work is evenly distributed among a chain of lymphangions. Model predictions were tested by comparison to previously reported data. Model predictions were consistent with reported lymphangion properties and pressure-flow relationships of entire lymphatic systems. The nonlinear lymphangion pressure-volume relationship therefore minimizes edema resulting from both increased microvascular filtration and venous hypertension.

Biofluid mechanics of special organs and the issue of system control. Sixth International Bio-Fluid Mechanics Symposium and Workshop, March 28-30, 2008 Pasadena, California

In the field of fluid flow within the human body, focus has been placed on the transportation of blood in the systemic circulation since the discovery of that system; but, other fluids and fluid flow phenomena pervade the body. Some of the most fascinating fluid flow phenomena within the human body involve fluids other than blood and a service other than transport--the lymphatic and pulmonary systems are two striking examples. While transport is still involved in both cases, this is not the only service which they provide and blood is not the only fluid involved. In both systems, filtration, extraction, enrichment, and in general some "treatment" of the fluid itself is the primary function. The study of the systemic circulation has also been conventionally limited to treating the system as if it were an open-loop system governed by the laws of fluid mechanics alone, independent of physiological controls and regulations. This implies that system failures can be explained fully in terms of the laws of fluid mechanics, which of course is not the case. In this paper we examine the clinical implications of these issues and of the special biofluid mechanics issues involved in the lymphatic and pulmonary systems.

Optimal postnodal lymphatic network structure that maximizes active propulsion of lymph

The lymphatic system acts to return lower-pressured interstitial fluid to the higher-pressured veins by a complex network of vessels spanning more than three orders of magnitude in size. Lymphatic vessels consist of lymphangions, segments of vessels between two unidirectional valves, which contain smooth muscle that cyclically pumps lymph against a pressure gradient. Whereas the principles governing the optimal structure of arterial networks have been identified by variations of Murray's law, the principles governing the optimal structure of the lymphatic system have yet to be elucidated, although lymph flow can be identified as a critical parameter. The reason for this deficiency can be identified. Until recently, there has been no algebraic formula, such as Poiseuille's law, that relates lymphangion structure to its function. We therefore employed a recently developed mathematical model, based on the time-varying elastance model conventionally used to describe ventricular function, that was validated by data collected from postnodal bovine mesenteric lymphangions. From this lymphangion model, we developed a model to determine the structure of a lymphatic network that optimizes lymph flow. The model predicted that there is a lymphangion length that optimizes lymph flow and that symmetrical networks optimize lymph flow when the lymphangions downstream of a bifurcation are 1.26 times the length of the lymphangions immediately upstream. Measured lymphangion lengths (1.14 +/- 0.5 cm, n = 74) were consistent with the range of predicted optimal lengths (0.1-2.1 cm). This modeling approach was possible, because it allowed a structural parameter, such as length, to be treated as a variable.

Modeling flow in collecting lymphatic vessels: one-dimensional flow through a series of contractile elements

The lymphatic system comprises a series of elements, lymphangions, separated by valves and possessed of active, contractile walls to pump interstitial fluid from its collection in the terminal lymphatics back to the main circulation. Despite its importance, there is a dearth of information on the fluid dynamics of the lymphatic system. In this article, we describe linked experimental and computational work aimed at elucidating the biomechanical properties of the individual lymphangions. We measure the static and dynamic mechanical properties of excised bovine collecting lymphatics and develop a one-dimensional computational model of the coupled fluid flow/wall motion. The computational model is able to reproduce the pumping behavior of the real vessel using a simple contraction function producing fast contraction pulses traveling in the retrograde direction to the flow.

First-order approximation for the pressure-flow relationship of spontaneously contracting lymphangions

To return lymph to the great veins of the neck, it must be actively pumped against a pressure gradient. Mean lymph flow in a portion of a lymphatic network has been characterized by an empirical relationship (Pin − Pout = −Pp + RLQL), where Pin − Pout is the axial pressure gradient and QL is mean lymph flow. RL and Pp are empirical parameters characterizing the effective lymphatic resistance and pump pressure, respectively. The relation of these global empirical parameters to the properties of lymphangions, the segments of a lymphatic vessel bounded by valves, has been problematic. Lymphangions have a structure like blood vessels but cyclically contract like cardiac ventricles; they are characterized by a contraction frequency ( f) and the slopes of the end-diastolic pressure-volume relationship [minimum value of resulting elastance ( Emin)] and end-systolic pressure-volume relationship [maximum value of resulting elastance ( Emax)]. Poiseuille's law provides a first-order approximation relating the pressure-flow relationship to the fundamental properties of a blood vessel. No analogous formula exists for a pumping lymphangion. We therefore derived an algebraic formula predicting lymphangion flow from fundamental physical principles and known lymphangion properties. Quantitative analysis revealed that lymph inertia and resistance to lymph flow are negligible and that lymphangions act like a series of interconnected ventricles. For a single lymphangion, Pp = Pin ( Emax − Emin)/ Emin and RL = Emax/ f. The formula was tested against a validated, realistic mathematical model of a lymphangion and found to be accurate. Predicted flows were within the range of flows measured in vitro. The present work therefore provides a general solution that makes it possible to relate fundamental lymphangion properties to lymphatic system function.

Lymphangion coordination minimally affects mean flow in lymphatic vessels

The lymphatic system returns interstitial fluid to the central venous circulation, in part, by the cyclical contraction of a series of “lymphangion pumps” in a lymphatic vessel. The dynamics of individual lymphangions have been well characterized in vitro; their frequencies and strengths of contraction are sensitive to both preload and afterload. However, lymphangion interaction within a lymphatic vessel has been poorly characterized because it is difficult to experimentally alter properties of individual lymphangions and because the afterload of one lymphangion is coupled to the preload of another. To determine the effects of lymphangion interaction on lymph flow, we adapted an existing mathematical model of a lymphangion (characterizing lymphangion contractility, lymph viscosity, and inertia) to create a new lymphatic vessel model consisting of several lymphangions in series. The lymphatic vessel model was validated with focused experiments on bovine mesenteric lymphatic vessels in vitro. The model was then used to predict changes in lymph flow with different time delays between onset of contraction of adjacent lymphangions (coordinated case) and with different relative lymphangion contraction frequencies (noncoordinated case). Coordination of contraction had little impact on mean flow. Furthermore, orthograde and retrograde propagations of contractile waves had similar effects on flow. Model results explain why neither retrograde propagation of contractile waves nor the lack of electrical continuity between lymphangions adversely impacts flow. Because lymphangion coordination minimally affects mean flow in lymphatic vessels, lymphangions have flexibility to independently adapt to local conditions.

Anatomy of the lymphatics

The lymphatic system is perhaps the most complicated system of Homo sapiens. An introduction to the anatomy, embryology, and anomalies of the lymphatics is presented. The overall anatomy and drainage of the lymphatic vessels in outlined. The topographic anatomy, relations, and variations of the principle vessels of the lymphatic system (the right lymphatic duct, the thoracic duct, and the cisterna chyli) are presented in detail.

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.

Optimal lymphatic vessel structure

Lymphatic vessels transport excess interstitial fluid from the low-pressure tissues to the higher pressure veins. The basic structural unit of lymphatic vessels is the lymphangion, a segment of the vessel separated by two unidirectional valves. Lymphangions cyclically contract like ventricles and can actively pump lymph. Lymphangions, as conduit vessels, also can act as arteries, and resist lymph flow. Functional parameters such as pressures, flow, and efficiency are determined by structural parameters like length, radius, and wall thickness. Since these structural parameters are unalterable experimentally, we developed a computational model to study the effect of a particular structural parameter, lymphangion length, to a particular functional variable, lymph flow. The model predicts that flow is a bimodal function of length, exhibiting an optimal length in the same order of magnitude as that observed experimentally. In essence, when the length to radius ratio is small, lymphangions act more like ventricles, where longer lengths yield greater chamber volume and thus lymph pumped. When the length to radius ratio is large, lymphangions act more like arteries, where longer lengths yield greater resistances to flow. This approach provides the means to explore how lymphatic vessel structure is optimized in a variety of conditions.

Current topics of physiology and pharmacology in the lymphatic system

We have reviewed physiological significance of rhythmical spontaneous contractions of collecting lymph vessels, which play a pivotal role in lymph transport and seem to control lymph formation through changing the pacemaker sites of the rhythmic contractions and contractile patterns of the lymphangions. A characteristic feature that the rhythmic pump activity works in vivo physiologically under the specific environment of lower oxygen tension in lymph (25-40 mm Hg) has been evaluated. With the characteristic feature, generation of endogenous nitric oxide (NO) from lymphatic endothelial cells and/or activation of ATP-sensitive potassium channels (K(ATP)) are reviewed to play crucial roles in the regulation of lymph transport at physiological or pathophysiological conditions. Chemical substances released from malignant tumor cells and tumor-derived parathyroid hormone-related peptide (PTHr-P) are also shown to cause a significant reduction of lymphatic pump activity through generation of endogenous NO and activation of K(ATP) channels. Finally, we have discussed physiological significance and roles of the lower oxygen tension in lymph, generation of endogenous NO, and activation of K(ATP) in lymph formation, lymph transport, and the functions of lymph nodes.

A mathematical model of flow through the terminal lymphatics

Proper understanding of the mechanisms of fluid absorption and flow through the terminal lymphatics is essential for the control of several pathological conditions such as edema, bedsores and cancer. A mathematical model of the terminal lymphatics was developed using the principles of mechanics. Computer simulation results substantiate the hypothesis that fluid absorption and flow through the terminal lymphatics occur due to suction mechanisms of the adjacent contractile lymphatic segments and due to periodic fluctuations in the interstitial fluid pressure. In addition, the results suggested that increasing the length of a terminal lymphatic vessel beyond a certain limit does not cause further increase in fluid flow into the terminal lymphatic.

A model of the lung interstitial-lymphatic system

Our model of the pulmonary interstitial-lymphatic system is based on the assumption that the lung interstitial space can be divided into two compartments. The first compartment (C1) contains the terminal lymph vessels. Increases in the fluid pressure within this compartment, along with increased pressure generated by lymph vessel pumping, cause the lymph flow rate to increase. The lymph vessels run through the second compartment (C2) which we believe represents the perivascular spaces. Increases in the fluid volume of C2 cause the lymph vessels to dilate and this causes lymph vessel resistance to decrease. Normally the lymph flow rate equals the microvascular filtration rate so that lung fluid volume is constant. According to our model, increases in filtration rate cause fluid to collect in C1 and C2. The resulting increase in fluid pressure in C1, increased lymph vessel pumping, and the decrease in lymph vessel resistance in C2 cause lymph flow to increase. Eventually, the lymph flow rises to equal the filtration rate and lung fluid volume becomes constant again. The results of simulations with our model indicate that decreases in lymph vessel resistance are essential for lymph flow to increase substantially as edema develops.

Perspectives in non-traditional biofluid mechanics

Although biofluid mechanics has been studied extensively, most of the studies have concentrated on cardiovascular biofluid mechanics. Very little attention has been paid to the other important problems in biomedicine. Several non-traditional areas which offer interesting and challenging problems remain unexplored, and fluid mechanics can have fruitful interaction with these disciplines. This paper brings into focus some of the important areas of biomedicine which offer fertile grounds for biofluid mechanics studies.