The aim of this investigation was to achieve the first step towards a comprehensive model of the lymphatic system. A numerical model has been constructed of a lymphatic vessel, consisting of a short series chain of contractile segments (lymphangions) and inter-segmental valves. The changing diameter of a segment governs the difference between the flows through inlet and outlet valves, and is itself governed by a balance between transmural pressure, and passive and active wall properties. The compliance of segments is maximal at intermediate diameters, and decreases when the segments are subject to greatly positive or negative transmural pressure. Fluid flow is the result of time-varying active contraction causing diameter to reduce, and is limited by segmental viscous and valvular resistance. The valves effect a smooth transition from low forward-flow resistance to high backflow resistance. Contraction occurs sequentially in successive lymphangions in the forward-flow direction. The behaviour of chains of one to five lymphangions was investigated by means of pump function curves, with variation of valve opening parameters, maximum contractility, lymphangion size gradation, number of lymphangions, and phase delay between adjacent lymphangion contractions. The model was reasonably robust numerically, with mean flow-rate generally reducing as adverse pressure was increased. Sequential contraction was found to be much more efficient than synchronised contraction. At the highest adverse pressures, pumping failed by one of two mechanisms, depending on parameter settings: either mean leakback flow exceeded forward pumping, or contraction failed to open the lymphangion outlet valve. Maximum pressure and maximum flow-rate were both sensitive to the contractile state; maximum pressure was also determined by the number of lymphangions in series. Maximum flow-rate was highly sensitive to the transmural pressure experienced by the most upstream lymphangions, suggesting that many feeding lymphatics would be needed to supply one downstream lymphangion chain pumping at optimal transmural pressure.

Our published model of a lymphatic vessel consisting of multiple actively contracting segments between non-return valves has been further developed by the incorporation of properties derived from observations and measurements of rat mesenteric vessels. These included (1) a refractory period between contractions, (2) a highly nonlinear form for the passive part of the pressure-diameter relationship, (3) hysteretic and transmural-pressure-dependent valve opening and closing pressure thresholds, (4) dependence of active tension on muscle length as reflected in local diameter. Experimentally, lymphatic valves are known to be biased to stay open. In consequence, in the improved model, vessel pumping of fluid suffers losses by regurgitation, and valve closure is dependent on backflow first causing an adverse valve pressure drop sufficient to reach the closure threshold. The assumed resistance of an open valve therefore becomes a critical parameter, and experiments to measure this quantity are reported here. However, incorporating this parameter value, along with other parameter values based on existing measurements, led to ineffective pumping. It is argued that the published measurements of valve closing pressure threshold overestimate this quantity owing to neglect of micro-pipette resistance. An estimate is made of the extent of the possible resulting error. Correcting by this amount, the pumping performance is improved, but still very inefficient unless the open-valve resistance is also increased beyond the measured level. Arguments are given as to why this is justified, and other areas where experimental data are lacking are identified. The model is capable of future adaptation as new experimental data appear.

We present a scaling theory for unforced inviscid two-dimensional turbulence. Our model unifies existing spatial and temporal scaling theories. The theory is based on a self-similar distribution of vortices of different sizes A. Our model uniquely determines the spatial and temporal scaling of the associated vortex number density which allows the determination of the energy spectra and the vortex distributions. We find that the vortex number density scales as n(A,t)-t(-2/3)/A, which implies an energy spectrum E-k(-5), significantly steeper than the classical Batchelor-Kraichnan scaling. High-resolution numerical simulations corroborate the model.

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An existing lumped-parameter model of multiple lymphangions (lymphatic vascular segments) in series is adapted for the incorporation of recent physiological measurements of lymphatic vascular properties. The new data show very marked nonlinearity of the passive pressure-diameter relation during distension, relative to comparable blood vessels, and complex valve behaviour. Since lymph is transported as a result of either the active contraction or the passive squeezing of vascular segments situated between two one-way valves, the performance of these valves is of primary importance. The valves display hysteresis (the opening and closing pressure-drop thresholds differ), a bias to staying open (both state changes occur when the trans-valve pressure drop is adverse), and pressure-drop threshold dependence on transmural pressure. These properties, in combination with the strong nonlinearity that valve operation represents, have in turn caused intriguing numerical problems in the model, and we describe numerical stratagems by which we have overcome the problems. The principal problem is also generalised into a relatively simple mathematical example, for which solution detail is provided using two different solvers.