Optimal postnodal lymphatic network structure that maximizes active propulsion of lymph

Abstract

The lymphatic system acts to return lower-pressured interstitial fluid to the higher-pressured veins by a complex network of vessels spanning more than three orders of magnitude in size. Lymphatic vessels consist of lymphangions, segments of vessels between two unidirectional valves, which contain smooth muscle that cyclically pumps lymph against a pressure gradient. Whereas the principles governing the optimal structure of arterial networks have been identified by variations of Murray's law, the principles governing the optimal structure of the lymphatic system have yet to be elucidated, although lymph flow can be identified as a critical parameter. The reason for this deficiency can be identified. Until recently, there has been no algebraic formula, such as Poiseuille's law, that relates lymphangion structure to its function. We therefore employed a recently developed mathematical model, based on the time-varying elastance model conventionally used to describe ventricular function, that was validated by data collected from postnodal bovine mesenteric lymphangions. From this lymphangion model, we developed a model to determine the structure of a lymphatic network that optimizes lymph flow. The model predicted that there is a lymphangion length that optimizes lymph flow and that symmetrical networks optimize lymph flow when the lymphangions downstream of a bifurcation are 1.26 times the length of the lymphangions immediately upstream. Measured lymphangion lengths (1.14 +/- 0.5 cm, n = 74) were consistent with the range of predicted optimal lengths (0.1-2.1 cm). This modeling approach was possible, because it allowed a structural parameter, such as length, to be treated as a variable.

Pump function curve shape for a model lymphatic vessel

“…(a) Pump function for bovine mesenteric lymphatics in vitro [19]. (b) Pump function, one bovine mesenteric vessel [20].…”

Multiscale Modelling of Lymphatic Drainage

“…In terms of this modelling, lymph vessels differ from arteries in one very important respect; whilst arterial walls are compliant they are also passive, so the response of the system can be modelled by finding suitable fixed values for L, C and R. Lymphangion walls are actively contractile, contributing to the pumping; and in fact the determination and modelling of this is a significant part of modelling the system as a whole. Venugopal, Quick and collaborators [34,35,[55][56][57] developed a more sophisticated lumped parameter representation of the lymph system using the circuit given in Fig. 7 with the parameters being time-varying.…”

“…7 Transmission line description of lymphangeon used by Quick, Venugopal and collaborators [35,56] boundaries are simulated with additional fixed resistances [56]. The model has been validated against experimental results for a single lymphangion [35], and expanded to explore networks of four lymphangions [56] and branching networks [55]. This lumped-parameter approach has proved very successful, allowing investigation of individual pumping behaviour [35], for example demonstrating that whilst lymphatic pumping action is beneficial under normal (positive) pressure gradients, it is counterproductive for reverse (negative) gradients.…”

“…The authors found this coordination of the contraction to have very little impact upon the pumping, and the orthograde and retrograde waves had similar effects, suggesting that individual lymphangions are able to function independently and thus adapt to local conditions as necessary. It has also proved possible to use the model to examine optimal structures of the lymph system [55]. The branching structure of the arterial system is well predicted by Murray's law [33]: in an optimal system, where an artery bifurcates, the cubes of the radii of the daughter vessels must sum to a constant value.…”

Engineering the Lymphatic System

“…Lymphatic network modeling suggests a balance may be achieved between contraction frequency and amplitude to provide an optimal flow rate and that the length of each lymphangion within the network is designed to promote flow [81]. The assumptions that form the basis of this model were that: 1)The maximum shear stress is calculated using the end diastolic radius.…”