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

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.

The Value of Percutaneous Contrast-Enhanced Ultrasound in Sentinel Lymph Node Identification, Metastatic Status and Burden Diagnosis in Early Breast Cancer

OBJECTIVE

The aim of this study was to evaluate the value of percutaneous contrast-enhanced ultrasound (PCEUS) in the identification and characterization of sentinel lymph node (SLN).

METHODS

A total of 102 breast cancer patients were collected and underwent preoperative PCEUS, which was used to identify SLN and lymphatic drainage. SLNs were classified into 4 enhancement patterns, including 6 subtypes: homogeneous (I), featured inhomogeneous (II) including inhomogeneous hypoenhancement (IIa) and annular or semi-annular enhancement (IIb), focal filling defect (III) including filling defect area < 50% (IIIa) and filling defect area ≥ 50% (IIIb), and no enhancement (IV). The enhancement patterns of SLNs were compared with the final pathological diagnosis.

RESULTS

The identification rate of SLNs using PCEUS was 100% (102/102); the rate of identification of LCs was 100% (102/102), and the coincidence rate was 98.0% (100/102). Four lymphatic drainage patterns (LDPs) including 5 subtypes were found: single LC/single SLN(74.5%), multiple LCs/ single SLN (13.7%) including 2 subtypes:2 LCs/1 SLN and 3 LCs/1 SLN, single LC/multiple SLNs (7.8%), and multiple LCs/multiple SLNs (3.9%). A total of 86.3% (44/51) of patients without axillary metastasis could be safely selected for types I, IIa, and IIb, while the axillary metastasis rates of types III and IV were 74.4% and 87.5%, respectively (P < .001). Compared with grayscale US, the PCEUS significant improvement in diagnosing metastatic SLNs (.794 versus .579, P < .001). For the SLN metastatic burden, Types I, IIa, IIb, and IIIa had ≤2 SLNs metastases, with a pathological coincidence rate of (64/67, 95.5%), and types IIIb and IV had >2 SLNs metastases, with a pathological coincidence rate of (25/35, 71.4%) (P < .001). The AUC of PCEUS for the diagnosis of SLN metastatic status and burden was .794 and .879, respectively (P < .001).

CONCLUSION

PCEUS has a high identification rate for SLN and has good potential for diagnosing SLN metastatic status and burden by enhancement patterns.

Predictive value for axillary lymph node metastases in early breast cancer: Based on contrast-enhanced ultrasound characteristics of the primary lesion and sentinel lymph node

OBJECTIVE

To evaluate the value of contrast-enhanced ultrasound (CEUS) characteristics based on primary lesion combined with lymphatic contrast-enhanced ultrasound (LCEUS) patterns of SLN in predicting axillary lymph node metastasis (ALNM) with T1-2N0 breast cancer.

METHODS

A retrospective study was conducted in 118 patients with clinically confirmed T1-2N0 breast cancer. Conventional ultrasound (CUS) and CEUS characteristics of the primary lesion and enhancement patterns of SLN were recorded. The risk factors associated with ALNM were selected by univariate and binary logistic regression analysis, and the receiver operating characteristic (ROC) curve was drawn for the evaluation of predictive ALNM metastasis performance.

RESULTS

Univariate analysis showed that age, HER-2 status, tumor size, nutrient vessels, extended range of enhancement lesion, and the enhancement patterns of SLN were significant predictive features of ALNM. Further binary logistic regression analysis indicated that the extended range of enhancement lesion (p <  0.001) and the enhancement patterns of SLN (p <  0.001) were independent risk factors for ALNM. ROC analysis showed that the AUC of the combination of these two indicators for predicting ALNM was 0.931 (95% CI: 0.887-0.976, sensitivity: 75.0%, specificity: 99.8%).

CONCLUSION

The CEUS characteristics of primary lesion combined with enhancement patterns of SLN are highly valuable in predicting ALNM and can guide clinical axillary surgery decision-making in early breast cancer.

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.

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.

Contrast-enhanced lymphatic US can improve the preoperative diagnostic performance for sentinel lymph nodes in early breast cancer

OBJECTIVES

To evaluate the preoperative diagnostic value of contrast-enhanced lymphatic ultrasound (CEUS) for the sentinel lymph node (SLN) status in early breast cancer.

MATERIALS AND METHODS

We prospectively recruited 102 consecutive patients with clinically node-negative early breast cancer from July 2021 to October 2021. All patients underwent conventional US and percutaneous CEUS examinations. The CEUS of SLNs were classified into four enhancement patterns: homogeneous (I), featured inhomogeneous (II), focal defect (III), and no enhancement (IV). The diagnostic performance of conventional US and CEUS for SLN metastasis was assessed by receiver operating characteristic (ROC) curves and decision curves.

RESULTS

A total of 78 women were enrolled in this study, including 55, 18, and 5 patients with negative axilla, 1-2, and ≥ 3 metastastic SLNs pathologically, respectively. The identification rate of SLNs by CEUS was 100%. Patterns I and II can select 91.7% (44/48) of patients with disease-free axilla, while patterns III and IV had higher percentages of metastasis (65.2%, p < 0.001 and 57.1%, p < 0.002, respectively). For the SLN metastatic burden, 100% (48/48) of patients with pattern I/II had ≤ 2 metastatic SLNs. Compared with conventional US, the CEUS enhancement patterns showed significant improvement in diagnosing metastatic SLNs (0.813 vs 0.601, p < 0.001). CEUS had greater clinical benefits and correctly reclassified 48% of metastatic SLNs (p < 0.001) without sacrificing the classification accuracy of negative SLNs (p = 0.25), and could improve prediction accuracy by 0.42 (p < 0.001).

CONCLUSIONS

CEUS demonstrated better diagnostic performance and greater clinical benefits than conventional US for the preoperative diagnosis of SLNs, showing its potential to select candidates for precluding axillary surgery in early breast cancer.

KEY POINTS

• The homogeneous and featured inhomogeneous enhancement of SLNs are highly suggestive of negative LNs, while focal defect (p < 0.001) and no enhancement (p < 0.002) patterns had higher percentages of metastasis. • The proportion of SLNs with highly suspicious signs on conventional US increases as the type of enhancement pattern increases (no suspicious signs in pattern I/II, 34.8% in pattern III, and 85.7% in pattern IV). • Compared with conventional US, CEUS improved the area under the receiver operating characteristic curve (0.813 vs. 0.601, p < 0.001) and had greater clinical benefits (IDI = 0.42, p < 0.001) for the diagnosis of axillary metastasis.

Integrative lymph node-mimicking models created with biomaterials and computational tools to study the immune system

The lymph node (LN) is a vital organ of the lymphatic and immune system that enables timely detection, response, and clearance of harmful substances from the body. Each LN comprises of distinct substructures, which host a plethora of immune cell types working in tandem to coordinate complex innate and adaptive immune responses. An improved understanding of LN biology could facilitate treatment in LN-associated pathologies and immunotherapeutic interventions, yet at present, animal models, which often have poor physiological relevance, are the most popular experimental platforms. Emerging biomaterial engineering offers powerful alternatives, with the potential to circumvent limitations of animal models, for in-depth characterization and engineering of the lymphatic and adaptive immune system. In addition, mathematical and computational approaches, particularly in the current age of big data research, are reliable tools to verify and complement biomaterial works. In this review, we first discuss the importance of lymph node in immunity protection followed by recent advances using biomaterials to create in vitro/vivo LN-mimicking models to recreate the lymphoid tissue microstructure and microenvironment, as well as to describe the related immuno-functionality for biological investigation. We also explore the great potential of mathematical and computational models to serve as in silico supports. Furthermore, we suggest how both in vitro/vivo and in silico approaches can be integrated to strengthen basic patho-biological research, translational drug screening and clinical personalized therapies. We hope that this review will promote synergistic collaborations to accelerate progress of LN-mimicking systems to enhance understanding of immuno-complexity.

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.

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.

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.

Ultra High-frequency Ultrasonographic Imaging with 70 MHz Scanner for Visualization of the Lymphatic Vessels

Supplemental Digital Content is available in the text.

Background:

Identification and localization of functional lymphatic vessels are important for lymphaticovenular anastomosis. Conventional high-frequency ultrasound (CHFUS) has been reported to be useful for them, but it has some disadvantages. In this article, we present new capabilities of ultra high-frequency ultrasound (UHFUS) for imaging of the lymphatic vessels, which may overcome the weakness of CHFUS.

Methods:

Thirty unaffected extremities in 30 unilateral secondary lymphedema patients (13 upper limbs and 17 lower limbs) were examined. Identification of the lymphatic vessels using UHFUS and CHFUS were performed at 3 sites in each unaffected extremity. Number and diameter of the detected lymphatic vessels were compared between UHFUS and CHFUS groups. At the same time, new characteristics of the lymphatic vessels seen with UHFUS were investigated.

Results:

One hundred sixty-nine lymphatic vessels were detected with UHFUS, and 118 lymphatic vessels with CHFUS. The number of lymphatic vessels found in upper and lower extremities was significantly larger with UHFUS than with CHFUS. The diameter of lymphatic vessels found in upper and lower extremities was significantly smaller with UHFUS than with CHFUS. All lymphatic vessels that were detected in UFHUS were less likely to collapse when the transducer was against the skin of the examined sites.

Conclusions:

Detection rate of the lymphatic vessels in nonlymphedematous extremities with UHFUS was higher than that with CHFUS. UHFUS provides images with extremely high resolution, demonstrating new characteristics of the lymphatic vessels.

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.

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).

The advection of microparticles, MCF-7 and MDA-MB-231 breast cancer cells in response to very low Reynolds numbers

The lymphatic system is an extensive vascular network that serves as the primary route for the metastatic spread of breast cancer cells (BCCs). The dynamics by which BCCs travel in the lymphatics to distant sites, and eventually establish metastatic tumors, remain poorly understood. Particle tracking techniques were employed to analyze the behavior of MCF-7 and MDA-MB-231 BCCs which were exposed to lymphatic flow conditions in a 100 μm square microchannel. The behavior of the BCCs was compared to rigid particles of various diameters (η = d_p/H= 0.05-0.32) that have been used to simulate cell flow in lymph. Parabolic velocity profiles were recorded for all particle sizes. All particles were found to lag the fluid velocity, the larger the particle the slower its velocity relative to the local flow (5%-15% velocity lag recorded). A distinct difference between the behavior of BCCs and particles was recorded. The BCCs travelled approximately 40% slower than the undisturbed flow, indicating that morphology and size affects their response to lymphatic flow conditions (Re < 1). BCCs adhered together, forming aggregates whose behavior was irregular. At lymphatic flow rates, MCF-7s were distributed uniformly across the channel in comparison to the MDA-MB-231 cells which travelled in the central region (88% of cells found within 0.35 ≤ W ≤ 0.64), indicating that metastatic MDA-MB-231 cells are subjected to a lower range of shear stresses in vivo. This suggests that both size and deformability need to be considered when modelling BCC behavior in the lymphatics. This finding will inform the development of in vitro lymphatic flow and metastasis models.

Spatio-temporal Models of Lymphangiogenesis in Wound Healing

Several studies suggest that one possible cause of impaired wound healing is failed or insufficient lymphangiogenesis, that is the formation of new lymphatic capillaries. Although many mathematical models have been developed to describe the formation of blood capillaries (angiogenesis), very few have been proposed for the regeneration of the lymphatic network. Lymphangiogenesis is a markedly different process from angiogenesis, occurring at different times and in response to different chemical stimuli. Two main hypotheses have been proposed: (1) lymphatic capillaries sprout from existing interrupted ones at the edge of the wound in analogy to the blood angiogenesis case and (2) lymphatic endothelial cells first pool in the wound region following the lymph flow and then, once sufficiently populated, start to form a network. Here, we present two PDE models describing lymphangiogenesis according to these two different hypotheses. Further, we include the effect of advection due to interstitial flow and lymph flow coming from open capillaries. The variables represent different cell densities and growth factor concentrations, and where possible the parameters are estimated from biological data. The models are then solved numerically and the results are compared with the available biological literature.

Estimation of the Pressure Drop Required for Lymph Flow through Initial Lymphatic Networks

BACKGROUND

Lymphatic function is critical for maintaining interstitial fluid balance and is linked to multiple pathological conditions. While smooth muscle contractile mechanisms responsible for fluid flow through collecting lymphatic vessels are well studied, how fluid flows into and through initial lymphatic networks remains poorly understood. The objective of this study was to estimate the pressure difference needed for flow through an intact initial lymphatic network.

METHODS AND RESULTS

Pressure drops were computed for real and theoretical networks with varying branch orders using a segmental Poiseuille flow model. Vessel geometries per branch order were based on measurements from adult Wistar rat mesenteric initial lymphatic networks. For computational predications based on real network geometries and combinations of low or high output velocities (2 mm/s, 4 mm/s) and viscosities (1 cp, 1.5 cp), pressure drops were estimated to range 0.31-2.57 mmHg. The anatomical data for the real networks were also used to create a set of theoretical networks in order to identify possible minimum and maximum pressure drops. The pressure difference range for the theoretical networks was 0.16-3.16 mmHg.

CONCLUSIONS

The results support the possibility for suction pressures generated from cyclic smooth muscle contractions of upstream collecting lymphatics being sufficient for fluid flow through an initial lymphatic network.

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.

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.

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.

A model for fluid drainage by the lymphatic system

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

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.

Lymphatic pump treatment enhances the lymphatic and immune systems

The osteopathic medical profession has long advocated the use of osteopathic lymphatic pump treatments (LPT) to improve lymphatic circulation, reduce edema and combat infectious disease. However, until recently, there was no scientific evidence that LPT enhances function of the lymphatic and immune systems. This review discusses the physiological functions of the lymphatic system, the ability of LPT to increase lymph flow under normal and experimental conditions, the clinical benefits of LPT, current research models for the study of LPT and the potential mechanisms by which LPT enhances lymphatic and immune function.

Engineering the Lymphatic System

The recent advances in our understanding of lymphatic physiology and the role of the lymphatics in actively regulating fluid balance, lipid transport, and immune cell trafficking has been furthered in part through innovations in imaging, tissue engineering, quantitative biology, biomechanics, and computational modeling. Interdisciplinary and bioengineering approaches will continue to be crucial to the progression of the field, given that lymphatic biology and function are intimately woven with the local microenvironment and mechanical loads experienced by the vessel. This is particularly the case in lymphatic diseases such as lymphedema where the microenvironment can be drastically altered by tissue fibrosis and adipocyte accumulation. In this review we will highlight contributions engineering and mechanics have made to lymphatic physiology and will discuss areas that will be important for future research.

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.

The structure and mechanical properties of collecting lymphatic vessels: an investigation using multimodal nonlinear microscopy

This study employed nonlinear microscopy on fresh, unstained and unfixed collecting lymphatic vessels to determine the wall structure and its relationships to the mechanical properties of the tissue. Fresh bovine mesenteric collecting lymphatic vessels were mounted in a vessel bath and imaged under different luminal pressures (0-30 cmH(2)O pressure head), and longitudinal tensions. The entire wall thickness was imaged, using two-photon fluorescence to visualize elastin, second harmonic generation to image the collagen, and coherent anti-Stokes Raman scattering to image the cell membrane. The adventitial fat cells were coupled to the wall within the elastin-rich network of fibres. The medial smooth muscle cells were too densely packed to resolve the boundaries of individual cells in en face images, but in tissue sections their appearance was consistent with electron microscopic data. Two distinct populations of collagen fibre were revealed. Large fibre (15-25 microm diameter) bundles were present in the inner media and small fibres (2-5 microm diameter) were distributed throughout the wall. The responses to longitudinal tension and luminal pressure indicated that the larger fibres resist the longitudinal strain and the smaller oppose pressure forces. Individual elastin fibres were of uniform thickness (1-3 microm) and interwove amongst themselves and between the collagen fibres. The network was probably too sparse directly to support mechanical loads and we speculate that its main function is to maintain the organization of collagen bundles during recovery from contraction.