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

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.

Computational fluid dynamic modeling of the lymphatic system: a review of existing models and future directions

Historically, research into the lymphatic system has been overlooked due to both a lack of knowledge and limited recognition of its importance. In the last decade however, lymphatic research has gained substantial momentum and has included the development of a variety of computational models to aid understanding of this complex system. This article reviews existing computational fluid dynamic models of the lymphatics covering each structural component including the initial lymphatics, pre-collecting and collecting vessels, and lymph nodes. This is followed by a summary of limitations and gaps in existing computational models and reasons that development in this field has been hindered to date. Over the next decade, efforts to further characterize lymphatic anatomy and physiology are anticipated to provide key data to further inform and validate lymphatic fluid dynamic models. Development of more comprehensive multiscale- and multi-physics computational models has the potential to significantly enhance the understanding of lymphatic function in both health and disease.

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

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

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.

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

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.

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.

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.

In-vivoviscoelastic properties estimation in subcutaneous adipose tissue by integration of poroviscoelastic-mass transport model (pve-MTM) into wearable electrical impedance tomography (w-EIT)

In-vivoviscoelastic properties have been estimated in human subcutaneous adipose tissue (SAT) by integration of poroviscoelastic-mass transport model (pve-MTM) into wearable electrical impedance tomography (w-EIT) under the influence of external compressive pressure-P.Thepve-MTM predicts the ion concentration distributioncmod(t)by coupling the poroviscoelastic and mass transport model to describe the hydrodynamics, rheology, and transport phenomena inside SAT. Thew-EIT measures the time-difference conductivity distribution∆γ(t)in SAT resulted from the ion transport. Based on the integration, the two viscoelastic properties which are viscoelastic shear modulus of SATGvand relaxation time of SATτvare estimated by applying an iterative curve-fitting between the normalized average ion concentration distributioncˆmod(t)predicted frompve-MTM and the experimental normalized average ion concentration distributioncˆexp(t)derived fromw-EIT. Thein-vivoexperiments were conducted by applying external compressive pressure-Pon human calf boundary to induce interstitial fluid flow and ion movement in SAT. As a result, the value ofGvwas range from 4.9-6.3 kPa and the value ofτvwas range from 27.50-38.5 s with the value of average goodness-of-fit curve fittingR² > 0.76. These values ofGvandτvwere compared to the human and animal tissue from the literature in order to verify this method. The results frompve-MTM provide evidence thatGvandτvplay a role in the predicted value ofcˆmod.

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.

Modelling secondary lymphatic valves with a flexible vessel wall: how geometry and material properties combine to provide function

A three-dimensional finite-element fluid/structure interaction model of an intravascular lymphatic valve was constructed, and its properties were investigated under both favourable and adverse pressure differences, simulating valve opening and valve closure, respectively. The shear modulus of the neo-Hookean material of both vascular wall and valve leaflet was varied, as was the degree of valve opening at rest. Also investigated was how the valve characteristics were affected by prior application of pressure inflating the whole valve. The characteristics were parameterised by the volume flow rate through the valve, the hydraulic resistance to flow, and the maximum sinus radius and inter-leaflet-tip gap on the plane of symmetry bisecting the leaflet, all as functions of the applied pressure difference. Maximum sinus radius on the leaflet-bisection plane increased with increasing pressure applied to either end of the valve segment, but also reflected the non-circular deformation of the sinus cross section caused by the leaflet, such that it passed through a minimum at small favourable pressure differences. When the wall was stiff, the inter-leaflet gap increased sigmoidally during valve opening; when it was as flexible as the leaflet, the gap increased more linearly. Less pressure difference was required both to open and to close the valve when either the wall or the leaflet material was more flexible. The degree of bias of the valve characteristics to the open position increased with the inter-leaflet gap in the resting position and with valve inflation pressure. The characteristics of the simulated valve were compared with those specified in an existing lumped-parameter model of one or more collecting lymphangions and used to estimate a revised value for the constant in that model which controls the rate of valve opening/closure with variation in applied pressure difference. The effects of the revised value on the lymph pumping efficacy predicted by the lumped-parameter model were evaluated.

The effects of valve leaflet mechanics on lymphatic pumping assessed using numerical simulations

The lymphatic system contains intraluminal leaflet valves that function to bias lymph flow back towards the heart. These valves are present in the collecting lymphatic vessels, which generally have lymphatic muscle cells and can spontaneously pump fluid. Recent studies have shown that the valves are open at rest, can allow some backflow, and are a source of nitric oxide (NO). To investigate how these valves function as a mechanical valve and source of vasoactive species to optimize throughput, we developed a mathematical model that explicitly includes Ca²⁺ -modulated contractions, NO production and valve structures. The 2D lattice Boltzmann model includes an initial lymphatic vessel and a collecting lymphangion embedded in a porous tissue. The lymphangion segment has mechanically-active vessel walls and is flanked by deformable valves. Vessel wall motion is passively affected by fluid pressure, while active contractions are driven by intracellular Ca²⁺ fluxes. The model reproduces NO and Ca²⁺ dynamics, valve motion and fluid drainage from tissue. We find that valve structural properties have dramatic effects on performance, and that valves with a stiffer base and flexible tips produce more stable cycling. In agreement with experimental observations, the valves are a major source of NO. Once initiated, the contractions are spontaneous and self-sustained, and the system exhibits interesting non-linear dynamics. For example, increased fluid pressure in the tissue or decreased lymph pressure at the outlet of the system produces high shear stress and high levels of NO, which inhibits contractions. On the other hand, a high outlet pressure opposes the flow, increasing the luminal pressure and the radius of the vessel, which results in strong contractions in response to mechanical stretch of the wall. We also find that the location of contraction initiation is affected by the extent of backflow through the valves.

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.