Simulation of a chain of collapsible contracting lymphangions with progressive valve closure

Abstract

The aim of this investigation was to achieve the first step toward 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 of intersegmental 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 behavior 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 synchronized 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.

Figure 1

Schematic of the lymphangion-chain model. Non-return valves, characterized by through flow-rate Q and flow resistance R_V, alternate with contractile vessel segments characterized by their diameter D. Pressure p is defined at each of the sites joining these elements (effectively, the upstream and downstream end of each segment).

Figure 2

The effect of parameters P_d and D_d on the passive part of the constitutive relation relating diameter D_i to transmural pressure p_tm. Upper panel: P_d = 100 dyn cm⁻², D_d = 0.015 (dashed), 0.020 (solid) and 0.015 (dot-dashed) cm. Lower panel: D_d = 0.02 cm, P_d = 20 (dotted), 50 (dashed), 100 (solid) and 150 (dot-dashed) dyn cm⁻².

Figure 3

The variation of valve resistance R_V with pressure difference across the valve Δp, showing how the constants s_open and P_open control the opening function of the valves. Upper panel: s_open = 0.04 cm² dyn⁻¹, p_open = −20 (dashed), −70 (solid) and −120 (dot-dash) dyn cm⁻². Lower panel: s_open = 0.02 (dashed), 0.04(solid) and 0.08 (dot-dash) cm² dyn⁻¹, p_open = −70 dyn cm⁻².

Figure 4

Cyclic pumping by a chain of four lymphangions, against a pressure difference (600 dyn cm⁻² or 0.612 cm H₂O) sufficient to produce a small negative mean flow-rate (−0.01 ml/hr). Top panel: the waveform of active tension in each lymphangion. Second panel: the four time-varying lymphangion diameters. Third panel: the eight different time-varying pressures—also shown as horizontal black lines are the pressures p_a and p_b. Bottom panel: the flow-rate through each of the five valves. Parameters for this run: M = 3.6 dyn cm⁻¹ (all lymphangions), p_a = 2275, p_b = 2875 dyn cm⁻²; D_d1 = 0.025, D_d2 = 0.022, D_d3 = 0.019, D_d4 = 0.016 cm; P_d1 = 50, P_d2 = 75, P_d3 = 100, P_d4 = 125 dyn cm⁻²; s_open = 0.04 cm² dyn⁻¹, p_open = −70 dyn cm⁻².

Figure 5

Pumping by the same lymphangion chain as in Fig. 4, but now against a pressure difference of 100 dyn cm⁻² (0.102 cm H₂O); the mean flow-rate becomes 0.182 ml/hr. All other parameters as for Fig. 4.

Figure 6

(a) Pump function, i.e. the relation between mean flow-rate Q̄ and overall adverse pressure difference ΔP = p_b − p_a, for the 4-lymphangion model, with p_ae = 175 dyn cm⁻², M = 3.6 dyn cm⁻¹, and two different sets of valve-opening parameters. Apart from variation of p_b, all parameters are otherwise as for Figs. 4 and 5. The 4-letter code attached to individual data points indicates which of lymphangions 1 to 4 were continuously distended [D] or visited both [b] collapsed and distended states during a cycle of pumping. (b) The valve resistance as a function of pressure difference for each valve-parameter set.

Figure 7

(a) For the purposes of investigating regurgitation, a further valve characteristic, V5, was defined. (b) Pump function curves for the single-lymphangion model with these three valve characteristics, at M = 3.6 dyn cm⁻¹. See text for explanation of the dotted lines.

Figure 8

Comparison of the pump function curves for chains of 3 (blue curve, circle symbols), 4 (magenta curve, diamond symbols) and 5 (red curve, star symbols) lymphangions with V3 valves, at p_ae = 175 dyn cm⁻² and M = 3.6 dyn cm⁻¹. The data for 4 lymphangions in the upper panel are as shown in Fig. 6a (diamond symbols). Parameters as for Fig. 4 except as noted in the legend. (a) With D_d reducing downstream. (b) With constant D_d. A single data-point for a corresponding 8-lymphangion model is also shown. Dotted lines indicate where the chain configuration (C, D, b) changes.

Figure 9

Comparison of two values of M, the peak contractile state, for the 3-lymphangion chain with V3 valves when p_a = p_e, i.e. p_a = 2100 dyn cm⁻². Dotted lines indicate where the chain configuration (C, D, b) changes. All other parameters unchanged from those for three lymphangions in Fig. 8a.

Figure 10

Pump function curves for the 3-lymphangion model with V3 valves at p_a − p_e = 0, 87.5 and 175 dyn cm⁻², and M = 3.6 and 5.4 dyn cm⁻¹. Other parameters unchanged. The dotted lines indicate selected contours of constant hydraulic power (product of ΔP and Q̄).

Figure 11

A pump function curve for 3 lymphangions in series, at p_a = 2275 dyn cm⁻² and M = 5.4 dyn cm⁻¹, with V4 valves, D_d1 = 0.029 cm, D_d2 = 0.039 cm, D_d3 = 0.049 cm and P_d1 = P_d2 = P_d3 = 20 dyn cm⁻². Dotted lines indicate where the chain configuration (C, D, b) changes. A single data-point for a corresponding 4-lymphangion model is also shown.

Figure 12

The changes in the pump function curve for the 3-lymphangion model with V3 valves at p_ae = 0 and M = 3.6 dyn cm⁻¹ when the parameters P_d1, P_d2 and P_d3 (dyn cm⁻²) are varied as indicated in the legend. All other parameters as for Fig. 9. Collapsed lymphangions except as noted.

Figure 13

The effect of varying the phase angle by which each lymphangion’s contraction follows that of the one upstream, in the 4-lymphangion model with p_ae = 175 dyn cm⁻², M = 3.6, V5 valves, D_d = 0.021 cm, and P_di = 50, 75, 100 and 125 dyn cm⁻² for i = 1 to 4.