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

Lumped parameter simulations of cervical lymphatic vessels: dynamics of murine cerebrospinal fluid efflux from the skull

BACKGROUND

Growing evidence suggests that for rodents, a substantial fraction of cerebrospinal fluid (CSF) drains by crossing the cribriform plate into the nasopharyngeal lymphatics, eventually reaching the cervical lymphatic vessels (CLVs). Disruption of this drainage pathway is associated with various neurological disorders.

METHODS

We employ a lumped parameter method to numerically model CSF drainage across the cribriform plate to CLVs. Our model uses intracranial pressure as an inlet pressure and central venous blood pressure as an outlet pressure. The model incorporates initial lymphatic vessels (modeling those in the nasal region) that absorb the CSF and collecting lymphatic vessels (modeling CLVs) to transport the CSF against an adverse pressure gradient. To determine unknown parameters such as wall stiffness and valve properties, we utilize a Monte Carlo approach and validate our simulation against recent in vivo experimental measurements.

RESULTS

Our parameter analysis reveals the physical characteristics of CLVs. Our results suggest that the stiffness of the vessel wall and the closing state of the valve are crucial for maintaining the vessel size and volume flow rate observed in vivo. We find that a decreased contraction amplitude and frequency leads to a reduction in volume flow rate, and we test the effects of varying the different pressures acting on the CLVs. Finally, we provide evidence that branching of initial lymphatic vessels may deviate from Murray's law to reduce sensitivity to elevated intracranial pressure.

CONCLUSIONS

This is the first numerical study of CSF drainage through CLVs. Our comprehensive parameter analysis offers guidance for future numerical modeling of CLVs. This study also provides a foundation for understanding physiology of CSF drainage, helping guide future experimental studies aimed at identifying causal mechanisms of reduction in CLV transport and potential therapeutic approaches to enhance flow.

The Lymphatic Vascular System: Does Nonuniform Lymphangion Length Limit Flow-Rate?

A previously developed model of a lymphatic vessel as a chain of lymphangions was investigated to determine whether lymphangions of unequal length reduce pumping relative to a similar chain of equal-length ones. The model incorporates passive elastic and active contractile properties taken from ex vivo measurements, and intravascular lymphatic valves as transvalvular pressure-dependent resistances to flow with hysteresis and transmural pressure-dependent bias to the open state as observed experimentally. Coordination of lymphangion contractions is managed by marrying an autonomous transmural pressure-dependent pacemaker for each lymphangion with bidirectional transmission of activation signals between lymphangions, qualitatively matching empirical observations. With eight lymphangions as used here and many nonlinear constraints, the model is capable of complex outcomes. The expected flow-rate advantage conferred by longer lymphangions everywhere was confirmed. However, the anticipated advantage of uniform lymphangions over those of unequal length, compared in chains of equal overall length, was not found. A wide variety of dynamical outcomes was observed, with the most powerful determinant being the adverse pressure difference, rather than the arrangement of long and short lymphangions. This work suggests that the wide variation in lymphangion length which is commonly observed in collecting lymphatic vessels does not confer disadvantage in pumping lymph.

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

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

Transport Barriers Influence the Activation of Anti-Tumor Immunity: A Systems Biology Analysis

Effective anti-cancer immune responses require activation of one or more naïve T cells. If the correct naïve T cell encounters its cognate antigen presented by an antigen presenting cell, then the T cell can activate and proliferate. Here, mathematical modeling is used to explore the possibility that immune activation in lymph nodes is a rate-limiting step in anti-cancer immunity and can affect response rates to immune checkpoint therapy. The model provides a mechanistic framework for optimizing cancer immunotherapy and developing testable solutions to unleash anti-tumor immune responses for more patients with cancer. The results show that antigen production rate and trafficking of naïve T cells into the lymph nodes are key parameters and that treatments designed to enhance tumor antigen production can improve immune checkpoint therapies. The model underscores the potential of radiation therapy in augmenting tumor immunogenicity and neoantigen production for improved ICB therapy, while emphasizing the need for careful consideration in cases where antigen levels are already sufficient to avoid compromising the immune response.

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.

Potentiating glymphatic drainage minimizes post-traumatic cerebral oedema

Cerebral oedema is associated with morbidity and mortality after traumatic brain injury (TBI)1. Noradrenaline levels are increased after TBI2-4, and the amplitude of the increase in noradrenaline predicts both the extent of injury5 and the likelihood of mortality6. Glymphatic impairment is both a feature of and a contributor to brain injury7,8, but its relationship with the injury-associated surge in noradrenaline is unclear. Here we report that acute post-traumatic oedema results from a suppression of glymphatic and lymphatic fluid flow that occurs in response to excessive systemic release of noradrenaline. This post-TBI adrenergic storm was associated with reduced contractility of cervical lymphatic vessels, consistent with diminished return of glymphatic and lymphatic fluid to the systemic circulation. Accordingly, pan-adrenergic receptor inhibition normalized central venous pressure and partly restored glymphatic and cervical lymphatic flow in a mouse model of TBI, and these actions led to substantially reduced brain oedema and improved functional outcomes. Furthermore, post-traumatic inhibition of adrenergic signalling boosted lymphatic export of cellular debris from the traumatic lesion, substantially reducing secondary inflammation and accumulation of phosphorylated tau. These observations suggest that targeting the noradrenergic control of central glymphatic flow may offer a therapeutic approach for treating acute TBI.

Effect of valve spacing on peristaltic pumping

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

A multiscale sliding filament model of lymphatic muscle pumping

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

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.

Characterization of rat tail lymphatic contractility and biomechanics: incorporating nitric oxide-mediated vasoregulation

The lymphatic system transports lymph from the interstitial space back to the great veins via a series of orchestrated contractions of chains of lymphangions. Biomechanical models of lymph transport, validated with ex vivo or in vivo experimental results, have proved useful in revealing novel insight into lymphatic pumping; however, a need remains to characterize the contributions of vasoregulatory compounds in these modelling tools. Nitric oxide (NO) is a key mediator of lymphatic pumping. We quantified the active contractile and passive biaxial biomechanical response of rat tail collecting lymphatics and changes in the contractile response to the exogenous NO administration and integrated these findings into a biomechanical model. The passive mechanical response was characterized with a three-fibre family model. Nonlinear regression and non-parametric bootstrapping were used to identify best-fit material parameters to passive cylindrical biaxial mechanical data, assessing uniqueness and parameter confidence intervals; this model yielded a good fit (R² = 0.90). Exogenous delivery of NO via sodium nitroprusside (SNP) elicited a dose-dependent suppression of contractions; the amplitude of contractions decreased by 30% and the contraction frequency decreased by 70%. Contractile function was characterized with a modified Rachev-Hayashi model, introducing a parameter that is related to SNP concentration; the model provided a good fit (R² = 0.89) to changes in contractile responses to varying concentrations of SNP. These results demonstrated the significant role of NO in lymphatic pumping and provide a predictive biomechanical model to integrate the combined effect of mechanical loading and NO on lymphatic contractility and mechanical response.

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.

A patient-specific modelling method of blood circulatory system for the numerical simulation of enhanced external counterpulsation

Lumped parameter model (LPM) is a common numerical model for hemodynamic simulation of human's blood circulatory system. The numerical simulation of enhanced external counterpulsation (EECP) is a typical biomechanical simulation process based on the LPM of blood circulatory system. In order to simulate patient-specific hemodynamic effects of EECP and develop best treatment strategy for each individual, this study developed an optimization algorithm to individualize LPM elements. Physiological data from 30 volunteers including approximate aortic pressure, cardiac output, ankle pressure and carotid artery flow were clinically collected as optimization objectives. A closed-loop LPM was established for the simulation of blood circulatory system. Aiming at clinical data, a sensitivity analysis for each element was conducted to identify the significant ones. We improved the traditional simulated annealing algorithm to iteratively optimize the sensitive elements. To verify the accuracy of the patient-specific model, 30 samples of simulated data were compared with clinical measurements. In addition, an EECP experiment was conducted on a volunteer to verify the applicability of the optimized model for the simulation of EECP. For these 30 samples, the optimization results show a slight difference between clinical data and simulated data. The average relative root mean square error is lower than 5%. For the subject of EECP experiment, the relative error of hemodynamic responses during EECP is lower than 10%. This slight error demonstrated a good state of optimization. The optimized modeling algorithm can effectively individualize the LPM for blood circulatory system, which is significant to the numerical simulation of patient-specific hemodynamics.

Lymphatic remodelling in response to lymphatic injury in the hind limbs of sheep

Contractile activity in the lymphatic vasculature is essential for maintaining fluid balance within organs and tissues. However, the mechanisms by which collecting lymphatics adapt to changes in fluid load and how these adaptations influence lymphatic contractile activity are unknown. Here we report a model of lymphatic injury based on the ligation of one of two parallel lymphatic vessels in the hind limb of sheep and the evaluation of structural and functional changes in the intact, remodelling lymphatic vessel over a 42-day period. We show that the remodelled lymphatic vessel displayed increasing intrinsic contractile frequency, force generation and vessel compliance, as well as decreasing flow-mediated contractile inhibition via the enzyme endothelial nitric oxide synthase. A computational model of a chain of lymphatic contractile segments incorporating these adaptations predicted increases in the flow-generation capacity of the remodelled vessel at the expense of normal mitochondrial function and elevated oxidative stress within the lymphatic muscle. Our findings may inform interventions for mitigating lymphatic muscle fatigue in patients with dysfunctional lymphatics.

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.

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.

Integrated geometric and mechanical analysis of an image-based lymphatic valve

Lymphatic valves facilitate the lymphatic system's role in maintaining fluid homeostasis. Malformed valves are found in several forms of primary lymphœdema, resulting in incurable swelling of the tissues and immune dysfunction. Their experimental study is complicated by their small size and operation in low pressure and low Reynolds number environments. Mathematical models of these structures can give insight and complement experimentation. In this work, we present the first valve geometry reconstructed from confocal imagery and used in the construction of a subject-specific model in a closing mode. A framework is proposed whereby an image is converted into a valve model. An FEA study was performed to identify the significance of the shear modulus, the consequences of smoothing the leaflet surface and the effect of wall motion on valve behaviour. Smoothing is inherent to any analysis from imagery. The nature of the image, segmentation and meshing all cause attenuation of high-frequency features. Smoothing not only causes loss of surface area but also the loss of high-frequency geometric features which may reduce stiffness. This work aimed to consider these effects and inform studies by taking a manual reconstruction and through manifold harmonic analysis, attenuating higher frequency features to replicate lower resolution images or lower degree-of-freedom reconstructions. In conclusion, two metrics were considered: trans-valvular pressure required to close the valve, ΔP_c, and the retrograde volume displacement after closure. The higher ΔP_c, the greater the volume of lymph that will pass through the valve during closure. Retrograde volume displacement after closure gives a metric of compliance of the valve and for the quality of the valve seal. In the case of the image-specific reconstructed valve, removing features with a wavelength longer than four μm caused changes in ΔP_c. Varying the shear modulus from 10 kPa to 60 kPa caused a 3.85-fold increase in the retrograde volume displaced. The inclusion of a non-rigid wall caused ΔP_c to increase from 1.56 to 2.52 cmH₂O.

Demonstration and Analysis of the Suction Effect for Pumping Lymph from Tissue Beds at Subatmospheric Pressure

Many tissues exhibit subatmospheric interstitial pressures under normal physiologic conditions. The mechanisms by which the lymphatic system extracts fluid from these tissues against the overall pressure gradient are unknown. We address this important physiologic issue by combining experimental measurements of contractile function and pressure generation with a previously validated mathematical model. We provide definitive evidence for the existence of 'suction pressure' in collecting lymphatic vessels, which manifests as a transient drop in pressure downstream of the inlet valve following contraction. This suction opens the inlet valve and is required for filling in the presence of low upstream pressure. Positive transmural pressure is required for this suction, providing the energy required to reopen the vessel. Alternatively, external vessel tethering can serve the same purpose when the transmural pressure is negative. Suction is transmitted upstream, allowing fluid to be drawn in through initial lymphatics. Because suction plays a major role in fluid entry to the lymphatics and is affected by interstitial pressure, our results introduce the phenomenon as another important factor to consider in the study of lymphoedema and its treatment.

The passive biomechanics of human pelvic collecting lymphatic vessels

The lymphatic system has a major significance in the metastatic pathways in women's cancers. Lymphatic pumping depends on both extrinsic and intrinsic mechanisms, and the mechanical behavior of lymphatic vessels regulates the function of the system. However, data on the mechanical properties and function of human lymphatics are lacking. Our aim is to characterize, for the first time, the passive biomechanical behavior of human collecting lymphatic vessels removed at pelvic lymph node dissection during primary debulking surgeries for epithelial ovarian cancer. Isolated vessels were cannulated and then pressurized at varying levels of applied axial stretch in a calcium-free Krebs buffer. Pressurized vessels were then imaged using multi-photon microscopy for collagen-elastin structural composition and fiber orientation. Both pressure-diameter and force-elongation responses were highly nonlinear, and axial stretching of the vessel served to decrease diameter at constant pressure. Pressure-diameter behavior for the human vessels is very similar to data from rat mesenteric vessels, though the human vessels were approximately 10× larger than those from rats. Multiphoton microscopy revealed the vessels to be composed of an inner layer of elastin with an outer layer of aligned collagen fibers. This is the first study that successfully described the passive biomechanical response and composition of human lymphatic vessels in patients with ovarian cancer. Future work should expand on this knowledge base with investigations of vessels from other anatomical locations, contractile behavior, and the implications on metastatic cell transport.

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

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

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

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

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

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

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.

In vivo label-free measurement of lymph flow velocity and volumetric flow rates using Doppler optical coherence tomography

Direct in vivo imaging of lymph flow is key to understanding lymphatic system function in normal and disease states. Optical microscopy techniques provide the resolution required for these measurements, but existing optical techniques for measuring lymph flow require complex protocols and provide limited temporal resolution. Here, we describe a Doppler optical coherence tomography platform that allows direct, label-free quantification of lymph velocity and volumetric flow rates. We overcome the challenge of very low scattering by employing a Doppler algorithm that operates on low signal-to-noise measurements. We show that this technique can measure lymph velocity at sufficiently high temporal resolution to resolve the dynamic pulsatile flow in collecting lymphatic vessels.

Pump function curve shape for a model lymphatic vessel

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

Network Scale Modeling of Lymph Transport and Its Effective Pumping Parameters

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

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

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

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

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

Mechanobiology of lymphatic contractions

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

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

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

Minimally invasive method for determining the effective lymphatic pumping pressure in rats using near-infrared imaging

The ability to quantify collecting vessel function in a minimally invasive fashion is crucial to the study of lymphatic physiology and the role of lymphatic pump function in disease progression. Therefore, we developed a highly sensitive, minimally invasive research platform for quantifying the pumping capacity of collecting lymphatic vessels in the rodent tail and forelimb. To achieve this, we have integrated a near-infrared lymphatic imaging system with a feedback-controlled pressure cuff to modulate lymph flow. After occluding lymphatic flow by inflating a pressure cuff on the limb or tail, we gradually deflate the cuff while imaging flow restoration proximal to the cuff. Using prescribed pressure applications and automated image processing of fluorescence intensity levels in the vessels, we were able to noninvasively quantify the effective pumping pressure (P(eff), pressure at which flow is restored after occlusion) and vessel emptying rate (rate of fluorescence clearance during flow occlusion) of lymphatics in the rat. To demonstrate the sensitivity of this system to changes in lymphatic function, a nitric oxide (NO) donor cream, glyceryl trinitrate ointment (GTNO), was applied to the tails. GTNO decreased P(eff) of the vessels by nearly 50% and the average emptying rate by more than 60%. We also demonstrate the suitability of this approach for acquiring measurements on the rat forelimb. Thus, this novel research platform provides the first minimally invasive measurements of P(eff) and emptying rate in rodents. This experimental platform holds strong potential for future in vivo studies that seek to evaluate changes in lymphatic health and disease.

Minimally invasive method for determining the effective lymphatic pumping pressure in rats using near-infrared imaging

The ability to quantify collecting vessel function in a minimally invasive fashion is crucial to the study of lymphatic physiology and the role of lymphatic pump function in disease progression. Therefore, we developed a highly sensitive, minimally invasive research platform for quantifying the pumping capacity of collecting lymphatic vessels in the rodent tail and forelimb. To achieve this, we have integrated a near-infrared lymphatic imaging system with a feedback-controlled pressure cuff to modulate lymph flow. After occluding lymphatic flow by inflating a pressure cuff on the limb or tail, we gradually deflate the cuff while imaging flow restoration proximal to the cuff. Using prescribed pressure applications and automated image processing of fluorescence intensity levels in the vessels, we were able to noninvasively quantify the effective pumping pressure (P(eff), pressure at which flow is restored after occlusion) and vessel emptying rate (rate of fluorescence clearance during flow occlusion) of lymphatics in the rat. To demonstrate the sensitivity of this system to changes in lymphatic function, a nitric oxide (NO) donor cream, glyceryl trinitrate ointment (GTNO), was applied to the tails. GTNO decreased P(eff) of the vessels by nearly 50% and the average emptying rate by more than 60%. We also demonstrate the suitability of this approach for acquiring measurements on the rat forelimb. Thus, this novel research platform provides the first minimally invasive measurements of P(eff) and emptying rate in rodents. This experimental platform holds strong potential for future in vivo studies that seek to evaluate changes in lymphatic health and disease.