Development of a model of a multi-lymphangion lymphatic vessel incorporating realistic and measured parameter values

Abstract

Our published model of a lymphatic vessel consisting of multiple actively contracting segments between non-return valves has been further developed by the incorporation of properties derived from observations and measurements of rat mesenteric vessels. These included (1) a refractory period between contractions, (2) a highly nonlinear form for the passive part of the pressure-diameter relationship, (3) hysteretic and transmural-pressure-dependent valve opening and closing pressure thresholds and (4) dependence of active tension on muscle length as reflected in local diameter. Experimentally, lymphatic valves are known to be biased to stay open. In consequence, in the improved model, vessel pumping of fluid suffers losses by regurgitation, and valve closure is dependent on backflow first causing an adverse valve pressure drop sufficient to reach the closure threshold. The assumed resistance of an open valve therefore becomes a critical parameter, and experiments to measure this quantity are reported here. However, incorporating this parameter value, along with other parameter values based on existing measurements, led to ineffective pumping. It is argued that the published measurements of valve-closing pressure threshold overestimate this quantity owing to neglect of micro-pipette resistance. An estimate is made of the extent of the possible resulting error. Correcting by this amount, the pumping performance is improved, but still very inefficient unless the open-valve resistance is also increased beyond the measured level. Arguments are given as to why this is justified, and other areas where experimental data are lacking are identified. The model is capable of future adaptation as new experimental data appear.

Figure 1

A diagram of a version of the multi-lymphangion model with just two lymphangions.

Figure 2

Pressure-diameter relation for a lymphatic vessel in the absence of wall muscle contraction, measured by Davis et al. (2011). Experimentally, diameter depends on transmural pressure, i.e. D = D(Δp_tm), but in the numerical model it is convenient to express the dependence as Δp_tm = Δp_tm(D), and the plot here has the axes arranged to reflect this ordering.

Figure 3

The variation of resistance R_V with the pressure drop Δp_V across a valve in the model. The form of the opening/closing characteristic (low resistance at Δp_V > Δp_o, high at Δp_V < Δp_o) varies with four parameters: Δp_o, s_o, R_Vn and R_Vx. In the valves used here, the threshold pressure drop Δp_o depends on Δp_tm, and the opening threshold differs from the closing threshold.

Figure 4

Experimental data (circles) showing the hysteresis and transmural-pressure dependence of lymphatic valve function, from Davis et al. (2011), and the fitted curves (solid lines) that were used in the lymphangion model. Note the change in definition of the transmural pressure on the abscissa between the two panels. The dotted curve shows the suggested closing threshold explained later in the paper (section 4). Variables here conform to those of Davis et al. (2011); their P_ext is p_e in this paper, while P_in is p_i–1,2 and P_out is p_i1.

Figure 5

A surface-fit to the results of seven experiments to measure the value of open-valve resistance R_Vn, coloured according to the value of the pressure drop Δp_V, which can be read off the vertical axis. Resistance is given by d(Δp_V)/dQ, and varies with both D and Q. The fit is third order, meaning that any cut through the surface at a given D or a given Q will be a curve with at most one reversal of curvature. The statistic r² shows how much of the total data variance about the mean is fitted by the surface. The surfaces representing the 95% confidence limits are also shown in outline.

Figure 6

A cycle of steady-state pumping for the single-lymphangion model which used the closing characteristic to determine the pressure drop threshold for both valve opening and valve closing. In the top panel, the horizontal dotted line shows the value of p_e. The lower solid line shows p_a, the upper one p_b. In the second panel, the waveform of M(t) shows timing only; the magnitude is arbitrary. The binary variables in the bottom panel register whether the inlet or outlet valve (V₁ and V₂ respectively) is open or closed. All parameter values in c.g.s. units except where specified; 6m means 6 × 10⁶ (etc.).

Figure 7

The performance of the corresponding pump with valves using only the opening characteristic for both state-change thresholds. See caption to Fig. 6 for explanation of traces.

Figure 8

The lymphangion’s passive constitutive relation (cyan), showing (blue) the range of diameters visited during the steady-state cycle (not illustrated) when the valves use both opening and closing characteristics. The values of diameter are those which actually occurred, but the values of transmural pressure are not, because the contribution from the active contraction caused the Δp_tm/D-relation to move off the passive curve (the actual Δp_tm-range was from –516.5 to +1918.5 dyn cm⁻² in this case).

Figure 9

The active length-tension relationship. (a) The underlying model is a logistic equation prescribing M_d(D), i.e. how M varies with D; see Appendix 1. (b) Result of dividing M_d(D) by D (green curve), compared with the passive constitutive relation (raw data in black, fitted curve in red). P_d is here the pressure at the normalising diameter indicated by the intersection of dashed lines.

Figure 10

Model run after adding the length-tension relationship and adjusting other parameters. The traces p₁(t) and p_m(t) here follow almost exactly the same path as p₂(t) and are consequently obscured by being overwritten. The curve M_t(t) is normalised by the maximum diameter; the curves M_d(D) and M(D,t) are scaled to the same maximum value, so that M_d(D) and M(D,t) necessarily coincide when M_t(t) reaches its peak. See caption to Fig. 6 for explanation of other traces.

Figure 11

Waveforms for a cycle of steady-state operation of the pump corresponding to the third pair of columns in Table 2. See caption to Fig. 6 for explanation of traces.

Figure 12

The result of increasing valve resistances R_Vx and R_Vn relative to their values in Fig. 11 is restoration of effective pumping. See caption to Fig. 10 for explanation of traces.

Figure 13

The effect of peak active tension M₀ on mean flow-rate Q̄. Left panel: with M_t(t) only. Right panel: with M(D,t). The operating point of Fig. 12 provides the uppermost-but-one point in the right-hand panel.

Figure 14

Pumping by a model of two lymphangions in series incorporating M(D,t) and experimentally justified values of all important parameters except R_Vn, which is artificially raised to circumvent the measured strong bias to remain open of the valves. See caption to Fig. 10 for explanation of traces insofar as one lymphangion is concerned. All three pressures in a lymphangion vary similarly; therefore p₁₂ mostly overwrites p₁₁ and p_1m, and p₂₂ mostly overwrites p₂₁ and p_2m.

Figure 15

Cycle-mean flow-rate Q̄ for a single lymphangion, with and without application of the factor 0.221 to the measured values of valve-closing pressure-drop threshold. Results are shown at various values of R_Vn ranging upward from 5×10⁵ dyn cm⁻²/ml s⁻¹, which approximates the measured value. When the threshold is left at the measured values (see Fig. 4a), Q̄ is negative for all values of R_Vn up to eight times the measured value, owing to leakback through closed valves under the impulsion of Δp_ba (here, 3 cmH₂O). When the closure thresholds are reduced, Q̄ is positive for all values of R_Vn including the measured value; however, a higher value still improves Q̄ . Both Q̄₁ and Q̄₂ are shown; tiny discrepancies represent error due to finite time-step size, not lack of convergence (all runs were continued until Q̄₁ and Q̄₂ stabilised). Other parameters: max. valve resistance R_Vx + R_Vn = 10⁹ dyn cm⁻²/ml s⁻¹, s_o = 0.4 cm²/dyn, t_r = 1.5 s.