Valve-related modes of pump failure in collecting lymphatics: numerical and experimental investigation

Abstract

Lymph is transported along collecting lymphatic vessels by intrinsic and extrinsic pumping. The walls have muscle of a type intermediate between blood-vascular smooth muscle and myocardium; a contracting segment between two valves (a lymphangion) constitutes a pump. This intrinsic mechanism is investigated ex vivo in isolated, spontaneously contracting, perfused segments subjected to controlled external pressures. The reaction to varying afterload is probed by slowly ramping up the outlet pressure until pumping fails. Often the failure occurs when the contraction raises intra-lymphangion pressure insufficiently to overcome the outlet pressure, open the outlet valve and cause ejection, but many segments fail by other means, the mechanisms of which are not clear. We here elucidate those mechanisms by resort to a numerical model. Experimental observations are paired with comparable findings from computer simulations, using a lumped-parameter model that incorporates previously measured valve properties, plus new measurements of active contractile and passive elastic properties, and the dependence of contraction frequency on transmural pressure, all taken from isobaric twitch contraction experiments in the same vessel. Surprisingly, the model predicts seven different possible modes of pump failure, each defined by a different sequence of valve events, with their occurrence depending on the parameter values and boundary conditions. Some, but not all, modes were found experimentally. Further model investigation reveals routes by which a vessel exhibiting one mode of failure might under altered circumstances exhibit another.

Keywords: Fluid–structure interaction; Lymphatic valve; Lymphatic vessel; Pump function.

Figure A1

Opening and closure thresholds for the diameter-operated valves used in model A8g.

Figure 1

A: experimental data. Passive (blue dots) and pharmacologically activated (red dots), pre-twitch and peak-twitch p/D data-points (black dots), at a series of steady pressures. The red/blue lines show the sequence of data-points visited. Twitch amplitude: horizontal black lines. Pre-twitch and peak-twitch p/D data-points are also circled (blue and red respectively); only these data were subsequently used in the model. B: curve fits to the data. Blue circles, vessel diameter immediately prior to each spontaneous twitch contraction; red circles, diameter at the peak of each twitch (data repeated from A). Several contractions were recorded at each of 14 different values of transmural pressure (only those up to 20 cmH₂O are shown here), and the data show good reproducibility from one contraction to the next. Although the data did not extend to Δp_tm < 0, it is necessary to complete the curves to avoid undefined states arising in the model. The curve fitted to the pre-twitch data was extrapolated to negative values of Δp_tm, taking into account the inevitable vessel collapse (Bertram et al. 2011a). The curve fitted to the peak twitch data was folded back to join the pre-twitch curve at a small negative Δp_tm, on the basis that shortening would rapidly become impossible in the collapse regime.

Figure 2

The average rate of spontaneous twitch contractions, as a function of transmural pressure. The fitted curve was used in the model to determine the duration of the relaxation before the next contraction.

Figure 3

Model schematic, and traces from a simulation run under conditions giving rise to behaviour 1. A: pressures p_a and p_b (thin black lines: Pa,b in legend), and intravascular pressures p₀₂ (magenta, upstream of V1), p₁₁ (blue, downstream of V1), p_m (red, mid-lymphangion), p₁₂ (green, upstream of V2), and p_p1 (beige, downstream of V2). Traces are plotted in the order shown in the legend; earlier-plotted traces may be largely over-written by nearly identical ones plotted later. Thus here p₀₂ completely over-writes p_a, and p_p1 completely over-writes p_b. The three intra-lymphangion pressures p₁₁, p_m and p₁₂ are almost identical; only during the first beats when p_b is low are p₁₁ and p_m briefly visible in the rapid filling and rapid ejection phases respectively. Flow-rate traces are here omitted, but the extent of the transient dip of p₁₁, p_m and p₁₂ below p_a in the filling phase, and of their transient peaking above p_b in the ejection phase, is indicative of the high flow-rates achieved initially, before ramping starts. B: the prescribed waveform M(t) controlling the movement between pre- and peak twitch states (cyan, scale 0 to 1 covering the plot height), and the lymphangion diameter (blue). During the time shown, the duration of the relaxation period between contractions shortens in response to increasing mean Δp_tm from 2.22 to 0.50 s. C: valve state for V1 (blue) and V2 (green) − 1 is open, 0 is closed (the two traces are slightly offset vertically for clarity). Closure and opening are both smooth gradual transitions in valve resistance until a threshold which varies according to Δp_tm and the current valve state is passed; see Bertram et al. (2014b) for details. See the Appendix for explanation of the parameters for which values are shown (in c.g.s. units unless otherwise indicated) above the relevant panel.

Figure 4

Pre-twitch (beige) and peak twitch (grey) curves as in Fig. 1B, but here the motion of the (D, Δp_tm)-operating point (red, with the second and last full cycles of contraction and subsequent relaxation over-printed in black) is superimposed as the model executes the cycles shown in Fig. 3. Each cycle of pumping would in the absence of p_b-change be an identical anti-clockwise loop tangent to the pre- and peak twitch curves. The simulation starting point is where V1 opens (blue circle) at Δp_tm < 0; this point is set by the prescribed initial conditions and has no significance; the model rapidly moves to (D, Δp_tm)-coordinates which are in better equilibrium with the external conditions, before the first contraction begins. Contraction causes closure of V1 (blue cross, D = 142 μm) and almost immediately afterwards opening of V2 (green circle, D = 142 μm). That ejection ends with closure of V2 (green cross, D = 85 μm), after brief regurgitation through V2, visible here in the small increase in D just before V2 closes. As can be seen in Fig. 3, that first contraction occurs without complete filling; later cycles start from D ≈ 159 μm. The first and second cycles follow the same end-systolic path. Once the p_b-ramp starts, cycles achieve tangency with the peak-twitch curve further up and to the right, but all pass through the same V1-closure location (superimposed blue crosses, D = 159 μm) until V1 ceases to open. Thereafter, contractions achieve less than nothing; the phases of isometric pressure development and isometric pressure decay (vertical red lines) are almost superimposed, but it can be seen that diameter is slightly greater at V2 closure (green cross) than at V2 opening (green circle). However each successive cycle is then further shifted to the right during diastole, as a result of leakage back through the closed V2, and in consequence the starting point for each contraction climbs up the pre-twitch curve, i.e. both D and Δp_tm increase. Throughout the ramp procedure, the extent of regurgitant flow through V2 before closure also increases, as seen in the growing end-ejection increases in D before V2 closes.

Figure 5

Simulation run under conditions giving rise to behaviour 2 (permanent closure of V1, then later permanent opening of V2). During the 74 s of simulated time shown here, the relaxation period between contractions decreased from 1.26 to 0.36 s. See caption to Fig. 3 for details of panel content and identification of individual traces.

Figure 6

Experiment on an isolated perfused vessel consisting of one complete lymphangion and (beyond each valve) the stump of another lymphangion, used to mount and secure the segment on a cannula. The vessel is here subjected to constant inlet-reservoir pressure p_in (blue) and a ramp increase of outlet-reservoir pressure p_out (red); these correspond to p_a and p_b respectively in the model. The intra-lymphangion pressure p_L (black) is measured by direct puncture of the vessel wall with a micropipette, and servo-null methodology. Diameter at a site downstream of the puncturing micropipette but upstream of the outlet valve was tracked by an optical method applied to the video cine recording of the vessel image as seen by inverted light microscope. The state (open or closed) of each valve (V1 = blue, V2 = red) was likewise assessed by an optical method from the video recording [see Davis et al. (2011) for details]. The result of the application of the p_out ramp is first that V1 ceases opening, then (in this case almost coincident with the end of the ramp) that V2 ceases closing; this is behaviour 2. Modified from fig. 10 of Scallan et al. (2013).

Figure 7

Simulation run under conditions giving rise to behaviour 3 (V1 not opening, then later V2 not closing, then still later V2 operating normally). During the 74 s of simulated time shown here, the relaxation period between contractions decreased from 1.35 to 0.37 s. See caption to Fig. 3 for details of panel content and identification of individual traces.

Figure 8

The path (red, with selected cycles overprinted in black) of the instantaneous (D, Δp_tm)-operating point during a p_out ramp giving rise to behaviour 3 (same run as shown in Fig. 7). See caption to Fig. 4 for further detail of the plotted symbols and curves.

Figure 9

Behaviour 5 (V1 stays closed, V2 immediately follows suit). During the 94 s of simulated time shown here, the relaxation period between contractions decreased from 1.95 to 0.69 s. See caption to Fig. 3 for details of panel content and identification of individual traces.

Figure 10

Behaviour 6 (as 5, but V2 eventually reopens and stays open). For this run, the parameter m which multiplies the rate of contraction activation rise and fall was reduced from 2 to 1. During the 84 s of simulated time shown here, t_r decreased from 2.88 to 0.41 s. See caption to Fig. 3 for details of panel content and identification of individual traces.

Figure 11

A simulation of behaviour 7 (V2 ceases to open). For this run, the two curves governing V1 opening and closure were depressed by 0.3 cmH₂O. During the 62 s of simulated time shown here, the relaxation period between contractions decreased from 3.49 to 1.49 s. See caption to Fig. 3 for details of panel content and identification of individual traces.

Figure 12

Starting at p_out = p_in = 1 cmH₂O, p_out is ramped up. As shown by the intra-lymphangion pressure (black), contractions develop the required systolic pressure up to a certain maximum (at ~203.9 min), beyond which the outlet valve V2 (red) no longer opens; this is behaviour 7. Modified from fig. 1 of Davis et al. (2011).

Figure 13

Two p_out ramps applied to the same vessel, the first starting from p_in = p_out = 2 cmH₂O, the second from p_in = p_out = 1.5 cmH₂O. The result is here termed behaviour 8.