Effect of time delay between contractions of adjacent lymphangions in branching networks with <i>n</i>_v = 4.

<p>(A) Pump function curves for synchronized pumping (Δ<i>t</i>_g = Δ<i>t</i>_v = 0 s), time delay only at the junctions (Δ<i>t</i>_v = 0, Δ<i>t</i>_g = 0.5 s), and the baseline case (Δ<i>t</i>_g = Δ<i>t</i>_v = 0.5 s) at <i>p</i>_a − <i>p</i>_e = 4 cmH₂O (solid lines) and <i>p</i>_a − <i>p</i>_e = 3 cmH₂O (dashed). (B) vs. Δ<i>t</i>_v (Δ<i>t</i>_g = Δ<i>t</i>_v): effect of transmural pressure via (variation of <i>p</i>_e) at Δ<i>P</i> = 3 cmH₂O. Solid line: forward propagating contraction wave, dashed line: reverse propagating contraction wave.</p

Effect of number of number of lymphangions (<i>n</i>_v).

<p>(A) Pump function curves for branching networks with <i>n</i>_v = 1–14 (associated <i>L</i>_s = 0.3–4.2 cm). Under favorable pressure differences (Δ<i>P</i> < 0) maximum flow rate was achieved with <i>n</i>_v = 1 because additional valves increased the resistance. Flow rate increased with <i>n</i>_v when Δ<i>P</i> > 16 cmH₂O. (B) Flow waveforms versus time for networks with <i>n</i>_v = 2, 4, and 8 at Δ<i>P</i> = 16 cmH₂O. Networks with larger <i>n</i>_v benefit from additional valves to maintain the pressures required to keep downstream valves open in the presence of an adverse Δ<i>P</i>.</p

Effect of time delay between contractions of adjacent lymphangions in branching networks with <i>n</i>_v = 4 when Δ<i>t</i>_g = Δ<i>t</i>_v.

<p>(A) Pump function curves as Δ<i>t</i>_v increased from 0 to 3.5 by steps of 1.0 s (Increasing <i>p</i>_b). (B) <i></i>vs. Δ<i>t</i>_v: effect of pressure difference across the network. Flow rate response as Δ<i>t</i>_v incrementally increased from 0 to 3.5 s. For reverse travelling waves (dashed line) equivalent time delay in the reverse direction is presented as 3.5 –(Δ<i>t</i>_g = Δ<i>t</i>_v).</p

Simulation conditions for all the cases studied here.

<p>Simulation conditions for all the cases studied here.</p

Timing of contractions.

<p><i>M</i>_<i>t</i> (<i>t</i>) vs. time for branching network with four lymphangions per vessel (<i>n</i>_v = 4). Bottom panel: vessels 0301–0304. Mid panel: vessels 0201 and 0202. Top panel: vessel 0101. Δ<i>t</i>_v is the time delay between contractions of adjacent lymphangions in one vessel; Δ<i>t</i>_g is the time delay between two subsequent generations (at the junction). Specified time course for the contraction of an individual lymphangion (normalized <i>M</i>_<i>t</i> (<i>t</i>) vs. time) is presented.</p

Network structure.

<p>(A) Top: Schematic view of the branching network structure with three generations of bifurcating vessels. The network pumps the fluid from inlets (<i>p</i>_a) to the outlet (<i>p</i>_b). The network is formed by seven vessels arranged in a bifurcating pattern (index 0101–0304). Each vessel is composed of multiple lymphangions and the number of lymphangions per vessel (<i>n</i>_v) is the same for all the vessels in the network. Bottom: Electrical analogy schematic of one of the junctions in the network. (B) Diagram of the algorithm for solving the system of equations and the behavior of the main governing equations. <i>t</i>_ITP is the initial transient period after which a consistent periodic solution is achieved.</p

Effect of transmural pressure (via <i>p</i>_e). vs. external pressure at Δ<i>P</i> = 3, 10, 16 cmH₂O for branching networks with <i>n</i>_v = 2 (dot-dashed), <i>n</i>_v = 4 (solid), and <i>n</i>_v = 8 (dotted).

<p> The top axis indicates transmural pressure. Peak flow rate occurred at higher values of <i>p</i>_eas pressure difference increased. Inlet pressure <i>p</i>_a was constant and equal to 6 cmH₂O.</p

Effect of diastolic period (<i>t</i>_r).

<p>(A) Pump function curves for a branching network with <i>n</i>_v = 4 as <i>t</i>_r increased from 0 to 3.5 s. Diastolic period was advantageous to pumping, particularly at low Δ<i>P</i>, because relaxed vessels present less impedance to incoming flow. (B) Combined effect of diastolic period, contraction period (<i>t</i>_c), and number of lymphangions per vessel (<i>n</i>_v) at Δ<i>P</i> = 3 cmH₂O.</p

Parameters used in the numerical model, including their definition and baseline value.

<p>Parameters used in the numerical model, including their definition and baseline value.</p

Parameters fit to the model in Eq 1, originally presented by Rahbar et al. [3].

<p>The parameters fit to each human vessel specimen in the first six columns with the mean and standard deviation of the human data in the seventh column and the mean and standard deviation of rat mesenteric vessels from Rahbar et al. [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0183222#pone.0183222.ref003" target="_blank">3</a>] in the last column. Rahbar et al. tested vessel segments both upstream and downstream of a secondary lymphatic valve but found no statistical significance between the regions, so we chose to include only upstream data from their work.</p