Consequences of intravascular lymphatic valve properties: a study of contraction timing in a multi-lymphangion model

Abstract

The observed properties of valves in collecting lymphatic vessels include transmural pressure-dependent bias to the open state and hysteresis. The bias may reduce resistance to flow when the vessel is functioning as a conduit. However, lymphatic pumping implies a streamwise increase in mean pressure across each valve, suggesting that the bias is then potentially unhelpful. Lymph pumping by a model of several collecting lymphatic vessel segments (lymphangions) in series, which incorporated these properties, was investigated under conditions of adverse pressure difference while varying the refractory period between active muscular contractions and the inter-lymphangion contraction delay. It was found that many combinations of the timing parameters and the adverse pressure difference led to one or more intermediate valves remaining open instead of switching between open and closed states during repetitive contraction cycles. Cyclic valve switching was reliably indicated if the mean pressure in a lymphangion over a cycle was higher than that in the lymphangion upstream, but either lack of or very brief valve closure could cause mean pressure to be lower downstream. Widely separated combinations of refractory period and delay time were found to produce the greatest flow-rate for a given pressure difference. The efficiency of pumping was always maximized by a long refractory period and lymphangion contraction starting when the contraction of the lymphangion immediately upstream was peaking. By means of an ex vivo experiment, it was verified that intermediate valves in a chain of pumping lymphangions can remain open, while the lymphangions on either side of the open valve continue to execute contractions.

Fig. 1.

A schematic of a two-lymphangion version of the five-lymphangion model, showing the pressure, diameter, flow-rate, and valve-resistance time variables that the simulation tracked, as well as the constant-pressure reservoirs at each end and their associated fixed resistances. The corresponding lymphangions are shown below.

Fig. 2.

Left: active tension vs. diameter, before (gray) and after (black) modification of the logistic relation for more tension at low diameter. Right: result converted to pressure (black curve) and compared with the curve (gray) fitted to the measured (13) passive pressure/diameter data (black points).

Fig. 3.

The relative timing of active contractions in the five lymphangions, as controlled by t_r and t_d; four examples are shown, arranged in the order of a plot with (t_d, t_r)-axes. Refractory period t_r = 1.5 s (top) and 0 (bottom). Inter-lymphangion delay t_d = 0 (left) and 0.5 s (right). In all cases, the contraction duration is 2 s.

Fig. 4.

Qualitative subdivisions of (t_d, t_r) space.

Fig. 5.

Pump function in terms of Q̄ achieved at a given ΔP, with t_d = 0.5 s. At each operating point, a row of six symbols indicates valve states (left to right = inlet to outlet): ● = valve operates normally, ○ = valve stays open, + = valve stays shut. A: t_r = 1 s. B: t_r = 4 s.

Fig. 6.

Cycle-average pressure in each of the five lymphangions, for two of the operating points in Fig. 5A. ΔP = 6.5 cmH₂O (A), ΔP = 2 cmH₂O (B).

Fig. 7.

Cycle-average flow-rate Q̄ (ml/h) as a function of inter-lymphangion delay t_d and refractory period t_r, at three values of ΔP: from top to bottom, 6, 4, and 2 cmH₂O. Valve function symbols are the same as in Fig. 5, but here, the leftmost of each row of six symbols is placed at the relevant (t_d, t_r) coordinate.

Fig. 8.

Volume pumped per cycle (μl) as a function of inter-lymphangion delay t_d and refractory period t_r, for the same three values of ΔP as in Fig. 7.

Fig. 9.

Reservoir pressure, valve state, and diameter-change traces during the composite ramp procedure built up by superposition of three ramp repetitions. The image at the top shows a photo-montage of the entire segment, with the five valves indicated by numbers.