The effects of valve leaflet mechanics on lymphatic pumping assessed using numerical simulations

Abstract

The lymphatic system contains intraluminal leaflet valves that function to bias lymph flow back towards the heart. These valves are present in the collecting lymphatic vessels, which generally have lymphatic muscle cells and can spontaneously pump fluid. Recent studies have shown that the valves are open at rest, can allow some backflow, and are a source of nitric oxide (NO). To investigate how these valves function as a mechanical valve and source of vasoactive species to optimize throughput, we developed a mathematical model that explicitly includes Ca²⁺ -modulated contractions, NO production and valve structures. The 2D lattice Boltzmann model includes an initial lymphatic vessel and a collecting lymphangion embedded in a porous tissue. The lymphangion segment has mechanically-active vessel walls and is flanked by deformable valves. Vessel wall motion is passively affected by fluid pressure, while active contractions are driven by intracellular Ca²⁺ fluxes. The model reproduces NO and Ca²⁺ dynamics, valve motion and fluid drainage from tissue. We find that valve structural properties have dramatic effects on performance, and that valves with a stiffer base and flexible tips produce more stable cycling. In agreement with experimental observations, the valves are a major source of NO. Once initiated, the contractions are spontaneous and self-sustained, and the system exhibits interesting non-linear dynamics. For example, increased fluid pressure in the tissue or decreased lymph pressure at the outlet of the system produces high shear stress and high levels of NO, which inhibits contractions. On the other hand, a high outlet pressure opposes the flow, increasing the luminal pressure and the radius of the vessel, which results in strong contractions in response to mechanical stretch of the wall. We also find that the location of contraction initiation is affected by the extent of backflow through the valves.

Figure 1

Diagram of the lymphatic vessel. R₀ = 100 μm is the relaxed radius of the vessel.

Figure 2

Hydrodynamic force dF acting on surface element ds.

Figure 3

Simulating intraluminal lymphatic valves. (a) Intravital image of a lymphatic valve in the popliteal collecting lymphatic of a mouse. (b) Parabolic valve shape assumed for the model.

Figure 4

Bending modulus of the valve decreases from the anchor point to the free end.

Figure 5

Schematic of lubrication force between to planes.

Figure 6

Pumping flux is affected by the mechanical properties and rest state of the valves. (a) Output flux is affected by the rigidity parameter A (defined in Fig. 4, increasing A means increased flexibility of the leaflet tip). Flux increases as the tip of the leaflet becomes more flexible, relative to the base. (b) Output flux is increased when the valve rest position is more open (Keeping $A=0.5$, increasing B of Eq. (14)). The inset figure shows how B affects the rest position of the valve.

Figure 7

The ${\mathrm{Ca}}^{2+}$ concentration, diameter of the vessel and flux during periodic contractions, $\mathrm{\Delta }p=0.3015\mathrm{g}\cdot {\mathrm{cm}}^{-1}\cdot {\mathrm{s}}^{-2}$. (a) ${\mathrm{Ca}}^{2+}$ concentration, (b) diameter, and (c) flux. (d and e) are the enlarged curve of ${\mathrm{Ca}}^{2+}$ concentration in (a). The horizontal dotted line indicates the threshold of ${\mathrm{Ca}}^{2+}$ concentration.

Figure 8

(a) NO concentration, (b) pressure and velocity field dynamics during a contraction cycle, $\mathrm{\Delta }p=0.3015\mathrm{g}\cdot {\mathrm{cm}}^{-1}\cdot {\mathrm{s}}^{-2}$. The minimum is blue and maximum is red. The arrows lengths are proportional to the local fluid velocity. See also Supplemental Videos S1 and S2.

Figure 9

Backflow through the gap between the leaflets influences NO distribution.

Figure 10

Pumping state changes with pressure difference $\mathrm{\Delta }{p}_1=0$, $\mathrm{\Delta }{p}_2=6.032\mathrm{g}\cdot {\mathrm{cm}}^{-1}\cdot {\mathrm{s}}^{-2}$, $\mathrm{\Delta }{p}_3=-6.032\mathrm{g}\cdot {\mathrm{cm}}^{-1}\cdot {\mathrm{s}}^{-2}$. (a) Concentration of ${\mathrm{Ca}}^{2+}$, (b) diameter near the outlet of the lymphangion, (c) flux changing with time, and (d) shows the plots in (b) for $t<T_1$ and $T_2\le t<T_3$, shifted to the same baseline time and diameter to show the differences in period and amplitude. (e) Expanded view of the plots in (c), showing details of the waveforms. ${\overline{Q}}_1$, ${\overline{Q}}_2$, ${\overline{Q}}_3$ are the average flux for $\mathrm{\Delta }p=\mathrm{\Delta }{p}_1$, Δp₂, Δp₃, respectively. See also Supplemental Video S3.

Figure 11

NO distribution in the relaxed state. ($\mathrm{\Delta }{p}_2=6.032\mathrm{g}\cdot {\mathrm{cm}}^{-1}\cdot {\mathrm{s}}^{-2}$).

Figure 12

Pumping cycle and average fluxes are affected by Δp.