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

A quasilinear hyperbolic one-dimensional model of the lymph flow through a lymphangion with valve dynamics and a contractile wall

This paper presents a one-dimensional model that describes fluid flow in lymphangions, the segments of lymphatic vessels between valves, using quasilinear hyperbolic systems. The model incorporates a phenomenological pressure-cross-sectional area relationship based on existing literature. Numerical solutions of the differential equations align with known results, offering insights into lymphatic flow dynamics. This model enhances the understanding of lymph movement through the lymphatic system, driven by lymphangion contractions.

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

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

Effect of valve spacing on peristaltic pumping

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

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

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

A multiscale sliding filament model of lymphatic muscle pumping

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

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

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

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.

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.

A novel mouse tail lymphedema model for observing lymphatic pump failure during lymphedema development

It has been suggested that many forms of secondary lymphedema in humans are driven by a progressive loss of lymphatic pump function after an initial risk-inducing event. However, the link between pump failure and disease progression has remained elusive due to experimental challenges in the clinical setting and a lack of adequate animal models. Using a novel surgical model of lymphatic injury, we track the adaptation and functional decline of the lymphatic network in response to surgery. This model mimics the histological hallmarks of the typical mouse tail lymphedema model while leaving an intact collecting vessel for analysis of functional changes during disease progression. Lymphatic function in the intact collecting vessel negatively correlated with swelling, while a loss of pumping pressure generation remained even after resolution of swelling. By using this model to study the role of obesity in lymphedema development, we show that obesity exacerbates acquired lymphatic pump failure following lymphatic injury, suggesting one mechanism through which obesity may worsen lymphedema. This lymphatic injury model will allow for future studies investigating the molecular mechanisms leading to lymphedema development.

Inhibition of contraction strength and frequency by wall shear stress in a single-lymphangion model

The phasic contractions of collecting lymphatic vessels are reduced in strength and occur at diminished frequency when the favourable pressure difference and the resulting antegrade flow create large fluid shear stresses at the luminal surface. This paper describes a minimal phenomenological model of this mechanism, that is applied to a previously validated numerical model of a phasically contracting lymphangion. The parameters of the inhibition model are quantitatively matched to observations in isolated segments of rat lymphatic vessel, first for mesenteric lymphatics then for thoracic duct, and outcomes from the numerical model are then qualitatively compared with recent observations in isolated segments of rat thoracic duct.

Entrainment of Lymphatic Contraction to Oscillatory Flow

Lymphedema, a disfiguring condition characterized by an asymmetrical swelling of the limbs, is suspected to be caused by dysfunctions in the lymphatic system. A possible source of lymphatic dysfunction is the reduced mechanosensitivity of lymphangions, the spontaneously contracting units of the lymphatic system. In this study, the entrainment of lymphangions to an oscillatory wall shear stress (OWSS) is characterized in rat thoracic ducts in relation to their shear sensitivity. The critical shear stress above which the thoracic ducts show a substantial inhibition of contraction was found to be significantly negatively correlated to the diameter of the lymphangion. The entrainment of the lymphangion to an applied OWSS was found to be significantly dependent on the difference between the applied frequency and the intrinsic frequency of contraction of the lymphangion. The strength of the entrainment was also positively correlated to the applied shear stress when the applied shear was less than the critical shear stress of the vessel. The ejection fraction and fractional pump flow were also affected by the difference between the frequency of the applied OWSS and the vessel's intrinsic contraction frequency. The results suggest an adaptation of the lymphangion contractility to the existing oscillatory shear stress as a function of its intrinsic contractility and shear sensitivity. These adaptations might be crucial to ensure synchronized contraction of lymphangions through mechanosensitive means and might help explain the lymphatic dysfunctions that result from impaired mechanosensitivity.

In vitro model reveals a role for mechanical stretch in the remodeling response of lymphatic muscle cells

OBJECTIVE

Using primary LMCs in vitro, we sought to characterize the impact of LMC remodeling on their functional and molecular response to mechanical loading and culture conditions.

METHODS

Primary "wounded leg" LMCs were derived from the hindlimb of three sheep who underwent lymphatic injury 6 weeks prior, while "control leg" LMCs were derived from the contralateral, unwounded, limb. Function of the LMCs was characterized in response to media of variable levels of serum (10% vs 0.2%) and glucose (4.5 vs 1 g/L). Functional and proteomic data were evaluated in LMCs exposed to cyclic stretch (0.1 Hz, 7.5% elongation) for 1 week.

RESULTS

LMCs were sensitive to changes in serum levels, significantly reducing overall activity and collagen synthesis under low serum conditions. LMCs from the remodeled vessel had higher baseline levels of metabolic activity but not collagen synthesis. Cyclic loading induced cellular alignment perpendicular to the axis of stretch and alterations in signaling pathways associated with metabolism. Remodeled LMCs had consistently higher levels of metabolic activity and were more resistant to strain-induced apoptosis.

CONCLUSIONS

LMCs exist on a functional spectrum, becoming more active in response to stretching and maintaining phenotypic remodeling in response to local lymphatic/tissue damage.

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

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

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

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

Lymphatic System Flows

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

Lymphatic System Flows

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

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

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

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.

An augmented iterative method for identifying a stress-free reference configuration in image-based biomechanical modeling

Continued advances in computational power and methods have enabled image-based biomechanical modeling to become an important tool in basic science, diagnostic and therapeutic medicine, and medical device design. One of the many challenges of this approach, however, is identification of a stress-free reference configuration based on in vivo images of loaded and often prestrained or residually stressed soft tissues and organs. Fortunately, iterative methods have been proposed to solve this inverse problem, among them Sellier's method. This method is particularly appealing because it is easy to implement, convergences reasonably fast, and can be coupled to nearly any finite element package. By means of several practical examples, however, we demonstrate that in its original formulation Sellier's method is not optimally fast and may not converge for problems with large deformations. Fortunately, we can also show that a simple, inexpensive augmentation of Sellier's method based on Aitken's delta-squared process can not only ensure convergence but also significantly accelerate the method.