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Research Papers

Simulation of a Chain of Collapsible Contracting Lymphangions With Progressive Valve Closure

[+] Author and Article Information
C. D. Bertram, C. Macaskill

School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia

J. E. Moore

Department of Biomedical Engineering, Texas A&M University, College Station, TX 77843-3120

Collapse is here modeled in the simplest possible way, solely as a mechanism of progressive reduction in lymphangion compliance with increasingly negative transmural pressure, ensuring that “diameter” remains positive at all finite transmural pressures. No adjustment is made for the increased resistance to fully developed laminar flow of a noncircular lymphangion cross section.

J Biomech Eng 133(1), 011008 (Dec 23, 2010) (10 pages) doi:10.1115/1.4002799 History: Received May 10, 2010; Revised September 21, 2010; Posted October 15, 2010; Published December 23, 2010; Online December 23, 2010

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.

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References

Figures

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Figure 1

Schematic of the lymphangion-chain model. Nonreturn valves, characterized by through flow-rate Q and flow resistance RV, 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).

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Figure 2

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

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Figure 3

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

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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/h). 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 pa and pb. Bottom panel: the flow-rate through each of the five valves. Parameters for this run: M=3.6 dyn cm−1 (all lymphangions), pa=2275 dyn cm−2, pb=2875 dyn cm−2; Dd1=0.025 cm, Dd2=0.022 cm, Dd3=0.019 cm, Dd4=0.016 cm; Pd1=50 dyn cm−2, Pd2=75 dyn cm−2, Pd3=100 dyn cm−2, Pd4=125 dyn cm−2; sopen=0.04 cm2 dyn−1, popen=−70 dyn cm−2.

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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/h. All other parameters as for Fig. 4.

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Figure 6

(a) Pump function, i.e., the relation between mean flow-rate Q¯ and overall adverse pressure difference ΔP=pb−pa, for the four-lymphangion model, with pae=175 dyn cm−2, M=3.6 dyn cm−1, and two different sets of valve-opening parameters. Apart from variation of pb, all parameters are otherwise as for Figs.  45. The four-letter code attached to individual data points indicates which of lymphangions 1–4 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.

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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 dashed lines.

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Figure 8

Comparison of the pump function curves for chains of three (unfilled circle symbols), four (diamond symbols) and five (filled circle symbols) lymphangions with V3 valves at pae=175 dyn cm−2 and M=3.6 dyn cm−1. The data for four lymphangions in the upper panel are as shown in Fig. 6 (diamond symbols). Parameters as for Fig. 4 except as noted in the legend. (a) With Dd reducing downstream. (b) With constant Dd. A single data-point for a corresponding eight-lymphangion model is also shown. Dotted lines indicate where the chain configuration (C, D, and b) changes.

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Figure 9

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

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Figure 10

Pump function curves for the three-lymphangion model with V3 valves at pa−pe=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 Q¯).

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Figure 11

A pump function curve for three lymphangions in series at pa=2275 dyn cm−2 and M=5.4 dyn cm−1, with V4 valves, Dd1=0.029 cm, Dd2=0.039 cm, Dd3=0.049 cm, and Pd1=Pd2=Pd3=20 dyn cm−2. Dotted lines indicate where the chain configuration (C, D, and b) changes. A single data-point for a corresponding four-lymphangion model is also shown.

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Figure 12

The changes in the pump function curve for the three-lymphangion model with V3 valves at pae=0 and M=3.6 dyn cm−1 when the parameters Pd1, Pd2, and Pd3(dyn cm−2) are varied as indicated in the legend. All other parameters as for Fig. 9. Collapsed lymphangions except as noted.

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Figure 13

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

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