Multiscale Modelling of Lymphatic Drainage

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.

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.

Correlative 3D Imaging and Microfluidic Modelling of Human Pulmonary Lymphatics using Immunohistochemistry and High-resolution μCT

Lung lymphatics maintain fluid homoeostasis by providing a drainage system that returns fluid, cells and metabolites to the circulatory system. The 3D structure of the human pulmonary lymphatic network is essential to lung function, but it is poorly characterised. Image-based 3D mathematical modelling of pulmonary lymphatic microfluidics has been limited by the lack of accurate and representative image geometries. This is due to the microstructural similarity of the lymphatics to the blood vessel network, the lack of lymphatic-specific biomarkers, the technical limitations associated with image resolution in 3D, and sectioning artefacts present in 2D techniques. We present a method that combines lymphatic specific (D240 antibody) immunohistochemistry (IHC), optimised high-resolution X-ray microfocus computed tomography (μCT) and finite-element mathematical modelling to assess the function of human peripheral lung tissue. The initial results identify lymphatic heterogeneity within and between lung tissue. Lymphatic vessel volume fraction and fractal dimension significantly decreases away from the lung pleural surface (p < 0.001, n = 25 and p < 0.01, n = 20, respectively). Microfluidic modelling successfully shows that in lung tissue the fluid derived from the blood vessels drains through the interstitium into the lymphatic vessel network and this drainage is different in the subpleural space compared to the intralobular space. When comparing lung tissue from health and disease, human pulmonary lymphatics were significantly different across five morphometric measures used in this study (p ≤ 0.0001). This proof of principle study establishes a new engineering technology and workflow for further studies of pulmonary lymphatics and demonstrates for the first time the combination of correlative μCT and IHC to enable 3D mathematical modelling of human lung microfluidics at micrometre resolution.

A Mathematical Model of Lymphangiogenesis in a Zebrafish Embryo

The lymphatic system of a vertebrate is important in health and diseases. We propose a novel mathematical model to elucidate the lymphangiogenic processes in zebrafish embryos. Specifically, we are interested in the period when lymphatic endothelial cells (LECs) exit the posterior cardinal vein and migrate to the horizontal myoseptum of a zebrafish embryo. We wonder whether vascular endothelial growth factor C (VEGFC) is a morphogen and a chemotactic factor for these LECs. The model considers the interstitial flow driving convection, the reactive transport of VEGFC, and the changing dynamics of the extracellular matrix in the embryo. Simulations of the model illustrate that VEGFC behaves very differently in diffusion and convection-dominant scenarios. In the former case, it must bind to the matrix to establish a functional morphogen gradient. In the latter case, the opposite is true and the pressure field is the key determinant of what VEGFC may do to the LECs. Degradation of collagen I, a matrix component, by matrix metallopeptidase 2 controls the spatiotemporal dynamics of VEGFC. It controls whether diffusion or convection is dominant in the embryo; it can create channels of abundant VEGFC and scarce collagen I to facilitate lymphangiogenesis; when collagen I is insufficient, VEGFC cannot influence the LECs at all. We predict that VEGFC is a morphogen for the migrating LECs, but it is not a chemotactic factor for them.

A Model for Interstitial Drainage Through a Sliding Lymphatic Valve

This study investigates fluid flow and elastic deformation in tissues that are drained by the primary lymphatic system. A model is formulated based on the Rossi hypothesis that states that the primary lymphatic valves, which are formed by overlapping endothelial cells around the circumferential lining of lymphatic capillaries, open in response to swelling of the surrounding tissue. Tissue deformation and interstitial fluid flow through the tissue are treated using the Biot equations of poroelasticity and, the fluid flux (into the interstitium) across the walls of the blood capillaries, is assumed to be linearly related to the pressure difference across the walls via a constant of proportionality (the vascular permeability). The resulting model is solved in a periodic domain containing one blood capillary and one lymphatic capillary starting from a configuration in which the tissue is undeformed. On imposition of a constant pressure difference between blood and lymphatic capillaries, the solutions are found to settle to a steady state. Given that the magnitude of pressure fluctuations in the lymphatic system is much smaller than this pressure difference between blood and lymph, it is postulated that the resulting steady-state solution gives a good representation of the state of the tissue under physiological conditions. The effects of changes to the Young's modulus of the tissue, the blood-lymphatic pressure difference, vascular permeability and valve dimensions on the steady state are investigated and discussed in terms of their effects on oedema in the context of age- and pregnancy-related changes to the body.

Multiscale modelling in immunology: a review

One of the greatest challenges in biomedicine is to get a unified view of observations made from the molecular up to the organism scale. Towards this goal, multiscale models have been highly instrumental in contexts such as the cardiovascular field, angiogenesis, neurosciences and tumour biology. More recently, such models are becoming an increasingly important resource to address immunological questions as well. Systematic mining of the literature in multiscale modelling led us to identify three main fields of immunological applications: host-virus interactions, inflammatory diseases and their treatment and development of multiscale simulation platforms for immunological research and for educational purposes. Here, we review the current developments in these directions, which illustrate that multiscale models can consistently integrate immunological data generated at several scales, and can be used to describe and optimize therapeutic treatments of complex immune diseases.