Mathematical Modelling of the Structure and Function of the Lymphatic System

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.

Multiscale homogenization for dual porosity time-dependent Darcy-Brinkman/Darcy coupling and its application to the lymph node

We study the coupling between time-dependent Darcy-Brinkman and the Darcy equations at the microscale subjected to inhomogeneous body forces and initial conditions to describe a double porosity problem. We derive the homogenized governing equations for this problem using the asymptotic homogenization technique, and as macroscopic results, we obtain a coupling between two Darcy equations, one of which with memory effects, with mass exchange between phases. The memory effects are a consequence of considering the time dependence in the Darcy-Brinkman equation, and they allow us to study in more detail the role of time in the problem under consideration. After the formulation of the model, we solve it in a simplified setting and we use it to describe the movement of fluid within a vascularized lymph node.

Progenitor Cell Sources for 3D Bioprinting of Lymphatic Vessels and Potential Clinical Application

The lymphatic system maintains tissue fluid homeostasis and it is involved in the transport of nutrients and immunosurveillance. It also plays a pivotal role in both pathological and regenerative processes. Lymphatic development in the embryo occurs by polarization and proliferation of lymphatic endothelial cells from the lymph sacs, that is, lymphangiogenesis. Alternatively, lymphvasculogenesis further contributes to the formation of lymphatic vessels. In adult tissues, lymphatic formation rarely occurs under physiological conditions, being restricted to pathological processes. In lymphvasculogenesis, progenitor cells seem to be a source of lymphatic vessels. Indeed, mesenchymal stem cells, adipose stem cells, endothelial progenitor cells, and colony-forming endothelial cells are able to promote lymphatic regeneration by different mechanisms, such as direct differentiation and paracrine effects. In this review, we summarize what is known on the diverse stem/progenitor cell niches available for the lymphatic system, emphasizing the potential that these cells hold for lymphatic tissue engineering through 3D bioprinting and their translation to clinical application.

Pathomechanisms of Disturbances Underlying Chronic Disorders

Chronic disorders' complexity poses enormous challenges to our understanding of such disorders [...].

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.

Graph theory applications in congenital heart disease

Graph theory can be used to address problems with complex network structures. Congenital heart diseases (CHDs) involve complex abnormal connections between chambers, vessels, and organs. We proposed a new method to represent CHDs based on graph theory, wherein vertices were defined as the spaces through which blood flows and edges were defined by the blood flow between the spaces and direction of the blood flow. The CHDs of tetralogy of Fallot (TOF) and transposition of the great arteries (TGA) were selected as examples for constructing directed graphs and binary adjacency matrices. Patients with totally repaired TOF, surgically corrected d-TGA, and Fontan circulation undergoing four-dimensional (4D) flow magnetic resonance imaging (MRI) were included as examples for constructing the weighted adjacency matrices. The directed graphs and binary adjacency matrices of the normal heart, extreme TOF undergoing a right modified Blalock-Taussig shunt, and d-TGA with a ventricular septal defect were constructed. The weighted adjacency matrix of totally repaired TOF was constructed using the peak velocities obtained from 4D flow MRI. The developed method is promising for representing CHDs and may be helpful in developing artificial intelligence and conducting future research on CHD.

Holistic View on the Structure of Immune Response: Petri Net Model

The simulation of immune response is a challenging task because quantitative data are scarce. Quantitative theoretical models either focus on specific cell-cell interactions or have to make assumptions about parameters. The broad variation of, e.g., the dimensions and abundance between lymph nodes as well as between individual patients hampers conclusive quantitative modeling. No theoretical model has been established representing a consensus on the set of major cellular processes involved in the immune response. In this paper, we apply the Petri net formalism to construct a semi-quantitative mathematical model of the lymph nodes. The model covers the major cellular processes of immune response and fulfills the formal requirements of Petri net models. The intention is to develop a model taking into account the viewpoints of experienced pathologists and computer scientists in the field of systems biology. In order to verify formal requirements, we discuss invariant properties and apply the asynchronous firing rule of a place/transition net. Twenty-five transition invariants cover the model, and each is assigned to a functional mode of the immune response. In simulations, the Petri net model describes the dynamic modes of the immune response, its adaption to antigens, and its loss of memory.

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.

A Mathematical Description of the Flow in a Spherical Lymph Node

The motion of the lymph has a very important role in the immune system, and it is influenced by the porosity of the lymph nodes: more than 90% takes the peripheral path without entering the lymphoid compartment. In this paper, we construct a mathematical model of a lymph node assumed to have a spherical geometry, where the subcapsular sinus is a thin spherical shell near the external wall of the lymph node and the core is a porous material describing the lymphoid compartment. For the mathematical formulation, we assume incompressibility and we use Stokes together with Darcy-Brinkman equation for the flow of the lymph. Thanks to the hypothesis of axisymmetric flow with respect to the azimuthal angle and the use of the stream function approach, we find an explicit solution for the fully developed pulsatile flow in terms of Gegenbauer polynomials. A selected set of plots is provided to show the trend of motion in the case of physiological parameters. Then, a finite element simulation is performed and it is compared with the explicit solution.

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.