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, (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.

1 Introduction

We previously reported (Bertram et al. 2011a) on the creation of a numerical model of a lymphatic vessel consisting of several segments or lymphangions, where a lymphangion is that part of a vessel between two valves. Unlike blood vessels, which have smooth muscle in their walls that does not play a motive role in pumping blood around the circulation, lymphatic vessels transport lymph as the result of the volume changes of these segments. Being bounded by one-way valves, the lymphangions constitute pumps. Valves of course also occur in veins, but more sparsely, and most veins do not spontaneously contract. Nevertheless, venous return of blood to the heart is significantly aided by the passive pumping which is the result of valve action in combination with intermittent squeezing of veins by adjacent tissues. This same extrinsic mechanism of pumping is correspondingly more important in lymphatic vessels, which have yet thinner walls than veins, and valves every three to ten diameters of vessel length (Schmid-Schönbein 1990). The muscle in the walls of lymphatic vessels is regarded as a type intermediate between normal vascular smooth muscle and myocardium. In the microlymphatic vessels we consider here (diameter of the order of 200 μm), it is able to mount relatively brisk and powerful twitch contractions which can produce a segment ‘ejection fraction’ (to use cardiac terminology) in excess of 60% (Gashev et al. 2004). Indeed, since this intrinsic method of pumping relies on exactly the same principles as cardiac chamber pumping, the adoption of terms from cardiac physiology is natural and helpful. The relative contributions of extrinsic and intrinsic pumping mechanisms to overall lymph flow probably vary widely around the body, but are currently largely unknown.

For each lymphangion, our model consisted of equations expressing conservation of mass (relating diameter change to the net difference between inflow and outflow), conservation of momentum (relating the pressure drop along a segment to the average of inflow and outflow), and a constitutive relation for the wall (relating diameter to transmural pressure difference Δp_tm). Mass conservation required a first-order differential equation with respect to time. Momentum conservation was satisfied with a Poiseuille equation, since the Reynolds number is of order one or smaller. The wall equation was asymmetrically sigmoidal, combining gradual stiffening with increasing distension when internal pressure was higher than external (Δp_tm > 0) with compliance decrease due to collapse when internal pressure was lower than external (Δp_tm < 0). Volume flow-rate through a valve was related to the pressure drop across the valve via a resistance which varied according to that pressure drop, being low when the pressure drop was positive (promoting flow in the direction of lymph pumping) and high when it was negative. The resistance varied smoothly between these extremes, across the threshold for change-over between open and closed valve states. Intrinsic pumping was induced by imposing a waveform of active wall tension variation over time; this led to an extra term in the constitutive relation, affecting the balance between diameter and Δp_tm. The equations for several lymphangions in series between an inlet reservoir of constant pressure and an outlet reservoir of higher pressure, pumping out of phase with each other, were solved simultaneously and integrated with respect to time. A wide range of parameter values was explored, yielding pump function curves showing how the model vessel reacted to varying pressure loads and other challenges. We refer the reader to our previous paper (Bertram et al. 2011a) for full details.

The model development has continued in many different directions. In conference contributions we have reported on the introduction of two-compartment lymphangion subdivision (Bertram et al. 2011b), to allow realistic simulation of the effects of valve prolapse (breakdown at high adverse pressure drop), and on the added efficiency of lymphangion pumping with a refractory period between contractions (Bertram 2012). But until recently, the values of the several parameters in the model, while chosen to provide reasonable outputs of time-varying diameter, pressure and flow-rate for these tiny vessels, were not specifically linked to measured quantities. We here describe how such parameter values, and improved functional dependencies, have been estimated from our own and published measurements of microlymphatic vessels in vivo, how problems arising from the incorporation of these dependencies and parameter values in the model were overcome, and what have been the model outcomes as a result. In particular, we show that, with all important disposable constants related to experimental measurements, the model points to a possible deficiency in an aspect of the measurements.

Thus the objectives of the paper can be summarised as follows. (1) We incorporate best estimates of experimentally measured lymphangion properties into our existing numerical model of a lymphatic vessel segment consisting of one or lymphangions in series. (2) We determine whether the lymphatic vessel model with these parameters can give rise to effective pumping. (3) We provide focus on the parameters that are not well determined experimentally, and hence provide a possible explanation for the rather poor pumping that we observe when these best estimates are used. The paper innovates in reporting the first lymphatic segment model to incorporate all known relevant experimental measurements, in demonstrating the sensitive dependence of pumping ability on the valve behaviour, and in providing direct indication of where further experimental determination of critical parameters is needed. Overall, we provide a lymphatic-segment model that is capable of being adapted for future comparison with experimental data yet to appear.

The paper proceeds as follows. We will initially present simulations which use parameter values not closely tied to physiological measurements. These simulations also omit the physiological dependence of active tension on muscle length. We will show how these simulations produce effective pumping, in the process identifying several criteria which together define this concept. We also show that these simulations point to certain deficiencies in the predicted pump, representing unlikely behaviour for a physiological system which has been optimised through evolution. To address these we move to parameter values which are as far as possible the result of physiological measurement. We also include a relationship between muscle length (via diameter) and active tension, at each stage of upgrading showing what the consequences are for the pump. It will be shown that the incorporation of parameter values related to physiological measurement leads to a predicted pump which cannot fulfil its function. By means of the dimensionless variables we show how to recover effective pumping, at the cost of at least one parameter departing from the value currently indicated experimentally. Finally we tentatively identify which of the experimental quantities may contain error, and provide an estimate of the putative error.

2 Methods

2.1 Mathematical model and fitting published data

The basic system of two lymphangions in series, with constant-pressure reservoirs at the inlet and outlet (Figure 1), was used to evaluate new adaptations of the model. All the time-variables which the numerical solution must evaluate are labelled in Fig. 1; p denotes pressure, D diameter, Q flow-rate and R_V valve resistance. The equations of the model are as follows.

Figure 1.

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

$$\begin{array}{l}\frac{{dD}_1}{dt}=\frac{2(Q_1-Q_2)}{\pi L{D}_1};\frac{{dD}_2}{dt}=\frac{2(Q_2-Q_3)}{\pi L{D}_2}\\ p_{11}-p_{1m}=\frac{64}{\pi }\frac{\mu {LQ}_1}{{D_1}^4};p_{1m}-p_{12}=\frac{64}{\pi }\frac{\mu {LQ}_2}{{D_1}^4}\\ p_{21}-p_{2m}=\frac{64}{\pi }\frac{\mu {LQ}_2}{{D_2}^4};p_{2m}-p_{22}=\frac{64}{\pi }\frac{\mu {LQ}_3}{{D_2}^4}\\ p_a-p_{11}=R_{V1}{Q}_1,\text{where}{R}_{V1}=R_{Vn}+\frac{R_{Vx}}{1+exp\left(s_o\left(\mathrm{\Delta }{p}_{V1}-\mathrm{\Delta }{p}_{o1}\right)\right)}\text{and}\mathrm{\Delta }{p}_{\mathrm{V}1}=p_{\mathrm{a}}-p_{11}\\ p_{12}-p_{21}=R_{V2}{Q}_2,\text{where}{R}_{V2}=R_{Vn}+\frac{R_{Vx}}{1+exp\left(-s_o\left(\mathrm{\Delta }{p}_{V2}-\mathrm{\Delta }{p}_{o2}\right)\right)}\text{and}\mathrm{\Delta }{p}_{\mathrm{V}2}=p_{12}-p_{21}\\ p_{22}-p_b=R_{V3}{Q}_3,\text{where}{R}_{V3}=R_{Vn}+\frac{R_{Vx}}{1+exp\left(-s_o\left(\mathrm{\Delta }{p}_{V3}-\mathrm{\Delta }{p}_{o3}\right)\right)}\text{and}\mathrm{\Delta }{p}_{\mathrm{V}3}=p_{22}-p_{\mathrm{b}}\\ p_{1m}-p_e=f_p(D_1)+f_a(D_1,t);p_{2m}-p_e=f_p\left(D_2\right)+f_a(D_2,t).\end{array}$$

Each segment of vessel between two valves gives rise to a pressure p_i1 at inlet, p_im at its midpoint and p_i2 at outlet, where i = 1 or 2 is the lymphangion number. The need for three distinct pressures was demonstrated by Bertram et al. (2011b). The pressure external to the vessel, p_e, is here taken as constant, although its variation in vivo is the basis of extrinsic pumping. Additional symbols in the above equations are L (lymphangion length), μ (lymph viscosity), and constants which define the valve properties (explained below). The function f_p(D) describes the passive nonlinear change of vessel diameter D in response to varying Δp_tm. It combines a fit to the published data shown in Figure 2 with a term describing the progressive loss of compliance dD/dΔp_tm as the vessel collapses and further cross-sectional area decrease becomes more difficult. This function, and the values of the constants therein, are detailed in Appendix 1. The function f_a(D,t) describes the contribution of active tension to the relation between D and Δp_tm. Each contraction consists of a waveform of active tension M_t(t), where

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.

$$M_{\mathrm{t}}(t)=M_0[1-cos(2\mathrm{\pi }f(t-t_{\mathrm{c}}))]/2,t_{\mathrm{c}}\le t<t_{\mathrm{c}}+1/f,$$

t = t_c at the start of the contraction and 1/f is its duration. Once t reaches t_c + 1/f , an interval then ensues of duration t_r during which no further contraction starts; thus the total pumping period is 1/f + t_r. In some of the model versions described in this paper, M_t(t) is multiplied by a function M_d(D) describing the length-tension relationship of lymphatic wall smooth muscle. The function M_d(D) is detailed in Appendix 1. The total M(D,t) = M_t(t) M_d(D), or M_t(t) alone, is then inserted into the constitutive relation as the term f_a(D,t) = 2M(D,t)/D which expresses the law of Laplace relating transmural pressure and radius to wall tension in a thin-walled vessel. The units of M(D,t) are dyn cm⁻¹.

The exponential function R_V(Δp_V) describing how valve resistance varies with the pressure drop across the valve under normal opening/closing circumstances¹ is a logistic function, as shown schematically in Figure 3. Its form is determined by four parameters: R_Vn, R_Vx, Δp_o and s_o (maximum valve resistance is actually R_Vx + R_Vn). The valve resistance is halfway between its extremes when Δp_V = Δp_o. In previous versions of the model, the threshold Δp_o was fixed, but it now depends on Δp_tm. The functional dependence for valve opening differs from that for valve closure, following the valve properties experimentally measured by Davis et al. (2011). Figure 4 shows the published data used, along with the curves that were fitted to these data for use in the model. Davis et al. (2011) also re-analyse their data in terms of functional dependence of threshold pressure on diameter, and alternative versions of our model substitute D-dependence for Δp_tm-dependence.

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.

In much of what follows, we also make use of a single-lymphangion version of the model, in which (with reference to Fig. 1) the outlet reservoir at constant pressure p_b immediately follows valve 2. In this case the time-variables are just p₁, p_m, p₂, D, Q₁ and Q₂.

2.2 Open-valve resistance measurements

The measured valve characteristics (Davis et al. 2011) indicate a strong tendency for the valves to stay open. When the transmural pressure is positive (the usual situation, leading to an approximately circular vessel cross-section and the possibility of muscle contraction), the valve opens while the pressure drop across the valve is still slightly adverse. It does not close again until the pressure drop is considerably adverse. This caused the parameter R_Vn, the resistance to flow of an open valve, to become a crucial determinant of the ability of the model to pump effectively. Accordingly experiments were conducted in MJD’s laboratory to measure this quantity. Segments ~600 μm in length of rat mesenteric lymphatic including a valve were cannulated and mounted as described previously (Davis et al. 2011). The maximal passive diameter was estimated with both reservoirs set to 10 cmH₂O; vessel length was adjusted at this pressure to remove axial slack that might interfere with diameter tracking. A constant-flow-rate perfusion pump (Harvard 22) was used to impose known steady flow-rates between 1 and 20 μL/min through the segments in the forward direction, i.e. that causing the valve to open. Measurements were made of pressure difference between the feed reservoir and the sink reservoir, and the segment diameter just upstream of the valve was measured optically as described previously (Davis et al. 2011). The pressure difference was recorded for 30–120 s at a given diameter and flow-rate and an average value noted. The procedure was repeated for flow-rates of 1, 2, 5, 10, 15 and 20 μL/min, and the experiment was conducted at four different diameters (~0.55, 0.65, 0.75 and 0.9 of the maximal passive diameter), manipulating the two reservoir pressures so as to produce the desired value of diameter at all flow-rates. Then a valve-less control segment of rat mesenteric lymphatic vessel of approximately corresponding diameter and length was substituted, using the same pipettes, and the measurements all repeated. In all cases the choice of segments was heavily constrained from above by the size of the rats (which were grown in the laboratory to be as large as possible) and from below by the single pair of cannulae used throughout to mount the vessels. For the control segment, the reservoir pressures were manipulated so as to produce the same four values of segment diameter as were used in the segment with a valve. The importance of the control segment is that a great part of the total measured pressure drop between reservoirs when there is flow consists of pressure drop in the cannulae, which are internally necessarily smaller in diameter at their tips than the segment. The effect of the open valve was deduced by subtracting the measured pressure drop for the valve-less segment from that for the segment with valve at the same diameter and flow-rate. Seven technically successful experiments were conducted.

2.3 Model solution methods

Incorporating a representation of the measured hysteresis and transmural pressure-dependence of the valves into the model was arduous, and is described in detail elsewhere (Bertram et al. 2013). During the process, models were constructed which used only one of the two sides of the hysteretic characteristics. One such model used the curve fitted to the measured valve-closing characteristic Δp_o(Δp_tm) to determine the threshold Δp_V for both opening and closing. The other used the corresponding valve-opening characteristic for both state changes. At the time, these models were thought of purely as simplifications of the total problem, but results obtained with them were sufficiently interesting to warrant reporting herein. For the results presented here, the model equations were solved in Matlab using a fixed-time-step fourth-order-accurate Runge-Kutta integrator for the ordinary differential equations and the Matlab routine FZERO for the algebraic constraint equations.

2.4 Dimensionless variables

In order to compare different pump configurations (combinations of parameter values), we employed dimensionless representations of the important variables and constants. We chose three fundamental quantities: the lymph viscosity μ, and the diameter and pressure scaling parameters for the passive constitutive relation, D_d and P_d. From these we derived dimensional units of time, resistance, active tension and flow-rate

$$t_{\text{unit}}=32\frac{\mu }{P_{\mathrm{d}}};R_{\text{Vunit}}=\frac{64}{\pi }\frac{\mu }{{D_{\mathrm{d}}}^3};M_{\text{unit}}=D_{\mathrm{d}}{P}_{\mathrm{d}};Q_{\text{unit}}=\frac{\pi }{64}\frac{{D_{\mathrm{d}}}^3{P}_{\mathrm{d}}}{\mu }.$$

Note that the magnitude of these normalising quantities varies with the values of P_d and D_d. We then used these quantities, along with P_d and D_d themselves, to non-dimensionalise all the remaining model parameters (for instance dimensionless s_o is just s_oP_d). Excluding a number of minor parameters which affect details such as the shape of functions, there are nine important ones: L, Δp_ae = p_a – p_e, Δp_ba = p_b – p_a, R_Vn, R_Vx, s_o, M₀, f and t_r.

3 Results

3.1 Open-valve resistance

This critical parameter, R_Vn, has not previously been measured, and proved challenging. Data from experiments on seven valves to estimate R_Vn are combined in Figure 5. The data are expressed as a surface of pressure drop Δp_V across the open valve as a function of flow-rate Q and of adjacent vessel diameter D. The diameter is influenced by Δp_tm, and also by the individual vessel. As expected, Δp_V reduces with increasing D. It increases with Q, but less steeply at higher flow-rates; this was anticipated, on the basis that the valve leaflets will increasingly open and lie more parallel to the vessel wall as Q increases, thereby reducing their resistance. The experiments were technically difficult, involving measuring a small difference between two pressure drops, each of which was registered by means of two different transducers. Accordingly the data scatter is quite large, and the fitted curve explains less than half of the variance of the data about their mean value; this is the meaning of the r² value. The figure also shows in outline the surfaces representing the 95% confidence limits. Since the fit was constrained to Δp_V = 0 at Q = 0, these curves coincide with the fitted curve itself at Q = 0. We interpret the results as indicating a rough value for R_Vn of 0.2 cmH₂O per 20 μL/min, or 0.6 × 10⁶ dyn cm⁻²/ml s⁻¹. This value is very approximate; the 95% confidence limits indicate an uncertainty of some ±25%, increasing to ±75% at maximum D, but the experiment shows clearly the correct order of magnitude.

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.

3.2 Modelling results

3.2.1 Effect of valve characteristics

The bias of the valves to stay in the open position caused a considerable drop in pumping efficiency. Figure 6 shows 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. Including a refractory period of 0.5 s, the cycle here lasts 2.5 s. Systole starts at 17.75 s with the beginning of active muscle tension. The pressures p₁, p_m and p₂ all rise in synchrony, and the inflow-rate Q₁ decreases then passes through zero. At 18.05 s the outlet valve opens, and at 18.1 s the inlet valve closes. Systolic ejection (outflow) then occurs. The pressures mount to a peak well in excess of p_b, the outlet reservoir pressure, with the pressure difference p₂ – p_b being dissipated across the open outlet valve, and the pressure difference p_m – p₂ required to overcome the resistance to flow in the outlet half of the lymphangion. In this simulation, the maximum valve resistance was only 17 times the minimum resistance, which was artificially elevated relative to the measured value in order to help achieve valve closure. As a result, the leakage flow through the closed inlet valve is significant relative to the forward flow of systolic ejection. Despite the elevated minimum valve resistance, there is also significant regurgitation through the inlet valve before it closes. Minimum lymphangion diameter and volume is reached at 19 s. Because there is no longer anything to cause elevated pressure in the lymphangion, the inlet valve reopens, and for a while both valves are open. With active tension declining, and lymphangion diameter accordingly starting to increase, fluid is drawn in through both valves, causing a positive flow-rate through the inlet valve and a negative flow-rate through the outlet one. Eventually intra-lymphangion pressure declines far enough that the outlet valve closes. As with the earlier opening of the inlet valve, this causes a jump in pressure, and a sudden change in the corresponding flow-rate. The cycle then completes with gradual lymphangion refilling from the inlet reservoir.

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.).

Modelling valve function using only the opening characteristic for both state-change thresholds reveals additional factors that are detrimental to pumping efficiency. Since the opening characteristic is less biased toward adverse pressure drops, we expected better performance from this pump than the one just presented. However, it transpires that this pump has its own problems; see Figure 7. While many of the same features occur during the cycle of steady-state pumping, this pump is not subject to discontinuities in the pressures or the flow-rates [for reasons analysed in detail elsewhere (Bertram et al. 2013)]. Instead of a period during which both valves are open simultaneously, the cycle here includes two periods when both valves are shut, one on either side of the ejection phase of systole. In cardiac terms, the isovolumic phases of systole have become lengthy. This reduces the efficiency of pumping to the point where the cycle-mean flow-rate is essentially the same (to within 2.5%) as that for the pump with closing-only valves and all other parameters the same.

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.

When the valves are given their full hysteretic characteristics, with use of one characteristic curve to determine opening and another to determine closing, discontinuities in pressure and flow-rate are reintroduced. There are two reasons for this: one is the inherent behaviour of the valves on closing (Bertram et al. 2013), as was manifested already in Fig. 6, and the other is the jump from use of one characteristic to the other at the time of valve state change. Whereas in the former characteristic the prevailing Δp_tm caused a valve resistance near R_Vn + R_Vx/2 at the last time-step before the switch, this same Δp_tm now dictates a valve resistance which is far from that value. In addition, as detailed by Bertram et al. (2013), a change in the definition of the controlling Δp_tm occurs upon switching. The cycle-mean flow-rate for this pump (for brevity these plots are omitted) was essentially unchanged from using closing-only or opening-only valve models; see Table 1.

Table 1.

The critical parameters for each of the model runs described in the text. All quantities in c.g.s. units except for the resulting mean flow-rate Q̄, which is given in ml/hr. The value D_d = 0.008 cm shown here is approximate; the exact value used in the model was 0.0084534 cm. The value P_d = 732 dyn cm⁻² used in the model is an approximation of the exact value (732.2215) for the fit to the data of Fig. 2. The column headed ‘valv’ indicates which of the fitted measured characteristics of Fig. 4 was used to determine the valve switching dependence on transmural pressure.

Fig.	Δpae	Δpba	Pd	Dd	RVn	RVx	so	M0	M(D)	tr	valv	Q̄
6	50	300	35	.025	6×106	96×106	0.2	10	no	0.5	close	.0283
7	50	300	35	.025	6×106	96×106	0.2	10	no	0.5	open	.0280
-	50	300	35	.025	6×106	96×106	0.2	10	no	0.5	both	.0289
run 1	5500	7500	732	.025	106	99×106	0.2	395	no	1.5	both	0.953
10	3924	2943	732	.025	106	99×106	0.2	1100	yes	1.5	both	0.211
run 2	5500	7500	732	.008	2×106	198e5	0.2	120	no	0.5	both	.0557
run 3	6867	2943	732	.025	5×105	9995e5	0.2	1100	yes	1.5	both	–.156
run 4	3924	981	732	.008	5×105	9995e5	0.4	400	yes	1.5	both	–.004
run 5	3924	490.5	732	.008	5×105	9995e5	0.4	400	yes	1.5	both	-1.031

3.2.2 Length/active-tension relation

The parameter values used to illustrate the foregoing behaviours cause the pumping to occur with diameter continuously below the value D_d. D_d controls the scaling of the passive pressure-diameter relationship; it is defined herein as the diameter of the lymphangion when Δp_tm = 0; see Figure 8. If the values of diameter traversed in a cycle are inscribed on the passive pressure-diameter relationship, as shown in Fig. 8, they indicate that the lymphangion was operating continuously in the part of the curve dominated by collapse. What happens if the parameters are adjusted so that the lymphangion is operating in the range D > D_d? This is achieved primarily by increasing Δp_tm, for instance by reducing p_e (offsetting adjustments in the value of M₀ were then necessary). We found that most parameter sets then led to a situation in which the diameter stayed in a small range at the positive-Δp_tm knee of the constitutive relation. Because D was here large, the active tension as defined previously was ineffective in reducing it. This is again the operation of the law of Laplace, as embedded in the definition of f_a(D,t). The change-over from operating points traversing collapsed diameters (left-hand side of the curve) to ones confined to the knee region on the right occurred in a small range of parameter change, such that only very careful selection of parameter values gave pumping with D varying over a significant part of the range between D_d and the knee. Such a high degree of sensitivity is almost certainly unrealistic. This led us to incorporate a length-tension relationship for lymphatic vascular muscle into the model.

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).

Specification of a rational active length-tension relationship was based on well established characteristics of muscle fibre length-tension relationships and the coupling with the highly nonlinear strain-stiffening lymphangion wall. Its essential feature is force generation over a restricted range of diameters, with the low end of that range assumed to be around D_d (Figure 9). The underlying model is a logistic equation prescribing how M varies with D (Fig. 9a), but the shape is modified when M_d(D) is divided by D as in Fig. 9b, for expression as a transmural pressure contribution² for comparison with the passive constitutive relation. The function was chosen to provide three properties for the active component: (1) that it is close to zero at the diameter D = D_d, (2) that it remains quite small except close to the knee of the passive Δp_tm-D curve, and (3) that its contribution to the Δp_tm-D curve reaches a peak just before the maximum diameter imposed by the form of the passive part of the constitutive relation. Fig. 9b shows a value for M₀ of 500 dyn cm⁻¹; later we use values between 600 and 1100 dyn cm cm⁻¹.

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.

3.2.3 Use of experimental parameter values

With the parameters P_d and D_d constrained by the data of Fig. 2, most of the critical model parameters are now based on experimental data. How does the single-lymphangion model behave under these circumstances? We introduced the various changes one by one and observed the results, only some of which can be illustrated with figures for the sake of brevity. Summary results for all the runs mentioned here are brought together in Table 1. Moving to the physiological value of P_d (and adjusting p_a – p_e, p_b – p_a and M₀ to compensate) yielded a reasonably effective pump (run 1 in Table 1), but one which included a spike of transiently very high p₁, p_m and p₂ at the end of ejection, coinciding with very swift reduction of D. The valves exchanged state essentially simultaneously at the end of systole, but both were closed simultaneously for an extended period of time at the beginning of systole.

Adding the length-tension relationship, and again adjusting other parameters to find a near-optimal operating point, removed the undesirable and almost certainly unphysiological spike of high pressure at the end of systole, as shown in Figure 10. The length-tension relationship has a profound effect on the shape and the magnitude of the waveform of active tension, as can be seen by comparing the supplied M_t(t) and the resulting M(D,t); as lymphangion volume falls during ejection, the term M_d(D) is greatly reduced, and so accordingly is the product M(D,t) = M_d(D) M_t(t). The appropriate value of M₀ is therefore much greater than before. In comparison with the runs shown in Figs. 6 and 7, the magnitudes of P_d, p_a – p_e, and p_b – p_a have all also been increased greatly, so the pressure drop along the lymphangion is now relatively insignificant, and p₁, p_m and p₂ follow visibly the same path. The collapse range is now out of reach; intrinsic contraction can no longer reduce D down to D_d. Although the valves swap states at fairly closely aligned times, regurgitation through open valves before they close wastes much of the forward pumping.

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.

When D_d is reduced to the value corresponding to the data of Fig. 2, problems arise. The pump can now barely manage positive Q̄ , even if M_d(D) is removed again (Table 1, run 2). Much of the cycle is spent with both valves open or closed simultaneously, and only the fact that D is small at end-systole prevents massive backflow while both are open.

The pumping is similarly ineffective if D_d reverts to its former value and R_Vn is reduced to 0.5×10⁶ dyn cm⁻²/ml s⁻¹, approximating its measured value. With M_d(D) reincorporated, and p_e and R_Vx adjusted to achieve a good outcome (run 3), the regurgitation before valve closure still exceeds the forward flow. In every other way, this pump operates as it should; the valves swap states almost simultaneously, and systole causes the intra-lymphangion pressure to attain p_b without excessive overshoot.

If all the physiologically based parameters are used, the pump is completely unable to function as such. One of two dysfunctional cycles occurs, depending on the value of p_b. In one of these situations (see Table 1, run 4) the inlet valve stays open perpetually while the outlet valve stays closed, and there is a small negative Q̄ by valve leakage. In the other, occurring at slightly lower p_b, the valves are open throughout, and there is substantial backflow (Table 1, run 5).

3.2.4 Dimensionless variables

More systematic interpretation of the findings demanded the use of dimensionless quantities. Table 2 presents values of the important parameters used in different runs. These particular runs omit the active length-tension relation M_d(D), so active tension is a simple function of time. From left to right, the table shows four different runs, with the parameters for each presented in both dimensional and dimensionless form. The first run has P_d and D_d taking their initial default values. The operating point is one leading to effective pump operation, achieved with a value of R_Vn some eight times higher than the measured value. The high open-valve resistance means that ejection is somewhat impeded, such that relatively high intra-lymphangion pressures are developed in systole. Diastolic filling, which cannot command such large pressure drops in its favour, is also slowed, and accordingly a longer (in fact, more physiologic) refractory period of 2 s (see, e.g., Zawieja et al. (1993)) is specified to allow reasonably complete filling before the next contraction. In the second run, corresponding to run 1 of Table 1, P_d has been increased to the value dictated by Fig. 2. The third pair of columns in Table 2 shows parameters for a run where both P_d and D_d take their values from the curve fitted to the data of Fig. 2 (this run is not in Table 1). The high value of P_d provides substantial passive ‘elastic recoil’, so a long refractory period is no longer needed to achieve complete filling, despite a value of R_Vn which, in order to achieve valve closure, is still four times the measured value.

Table 2.

The conversion of model parameters to dimensionless equivalents. Four different operating points (combinations of parameters) are shown in the four pairs of columns. Four parameters which are used to specify the model but are ultimately of lesser significance here (p_e, p_a, p_b, t₀) are divided off from the nine critical parameters. The main figure of merit for pumping is the resulting mean flow-rate. An auxiliary figure of merit is the ratio of peak to mean Q, where less is better (*excluding spike).

	A6.cy02 (cp1)		A6.cy07 (cp4)		A6.cy09 (cp5)		A6.cy10 (cp6)
	cgs	non-dim	cgs	non-dim	cgs	non-dim	cgs	non-dim
μ	0.01	1	0.01	1	0.01	1	0.01	1
Dd	0.025	1	0.025	1	0.00845	1	0.00845	1
Pd	35	1	732	1	732	1	732	1
t-unit	9.1e-3	1	4.4e-4	1	4.4e-4	1	4.4e-4	1
R-unit	13038	1	13038	1	337237	1	337237	1
M-unit	0.875	1	18.3	1	6.19	1	6.19	1
Q-unit	0.0027	1	0.0561	1	0.0022	1	0.0022	1
pe	2000	57.1	2000	2.7	2000	2.7	2000	2.7
pa	2270	64.9	7500	10.2	5500	7.5	5500	7.5
pb	2570	73.4	15000	20.5	8000	10.9	8000	10.9
t0	0.25	27.3	0.25	571.9	0.25	571.9	0.25	571.9
L	0.3	12	0.3	12	0.3	35.5	0.3	35.5
Δpae	270	7.7	5500	7.5	3500	4.8	3500	4.8
Δpba	300	8.6	7500	10.2	2500	3.4	2500	3.4
RVn	4.0e+6	306.8	1.0e+6	76.7	2.0e+6	5.9	2.0e+7	59.3
RVx	9.6e+7	7363.1	9.9e+7	7593.2	1.98e+8	587.1	1.98e+9	5871.2
so	0.2	7	0.2	146.4	0.2	146.4	0.2	146.4
M0	20	22.9	395	21.6	80	12.9	80	12.9
f	0.5	4.6e-3	0.5	2.2e-4	0.5	2.2e-4	0.5	2.2e-4
tr	2	219	1.5	3431	0.5	1144	1	2288
mean Q	3.46e-5	1.29e-2	2.65e-4	4.72e-3	1.13e-05	5.22e-3	3.00e-05	1.38e-2
peak Q	0.00038	0.14176	0.00361	0.06432	0.00058	0.26619	0.00040	0.18620
peak/mean Q		11.0		13.6*		51.0		13.5

Dimensionless analysis is not needed to realize that if P_d is increased by over 20-fold, the other pressure parameters (Δp_ba and Δp_ae) should be increased accordingly, and similarly the need for increased values of M₀ is readily appreciated. However, less obviously, we have also reduced the effective f, and increased the effective t_r, maintaining a long effective cycle. We have also increased the effective s_o, such that a value which previously gave a gradual change of R_V until switching now gives abrupt change. In earlier versions of the model, such abruptness would have caused numerical problems, but the model is now robust in the face of such perturbations.

In changing D_d to its measured value, we have reduced it only 2.93 times, ostensibly a much less drastic change than that made to P_d. Since L is not altered, the spacing of the valves has now increased; they occur 35 diameters apart, where previously they were 12 diameters apart (taking D_d as representative of the actual lymphangion diameter). However, because the unit of resistance involves D_d³, we have reduced the effective size of the valve resistances R_Vn and R_Vx by a factor of 25. The (dimensional) Poiseuille resistance to flow through the lymphangion has meanwhile increased as some D_d⁻⁴ (again, taking D_d as representative of actual D). Thus, if the dimensional values of R_Vn and R_Vx are left unchanged, they are now very much smaller relative to the Poiseuille resistance within the lymphangion.

The pump of the third pair of columns in Table 2 is illustrated in Figure 11. It is almost completely ineffective, because much of the initial systolic lymphangion volume reduction is used up in useless backflow through the open inlet valve; only later in systole (~18.15 s) does the negative trans-valve pressure drop reach the threshold pressure drop for closure. Because (with reduced D_d) the lymphangion is now much smaller in diameter, a given flow-rate during contraction demands a much greater rate of diameter reduction. The required backflow-rate is not achieved until well into the systolic shortening, despite the development of high p_m to overcome the increased lymphangion flow resistance.

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.

The analysis prescribes that R_Vx can be, and R_Vn should be, increased to restore the previous dimensionless values. This being done (the fourth pair of columns of Table 2), the single-lymphangion pump is restored to effective vigour. If further, the active length-tension relation is also now restored, the result is an effective pump as illustrated in Figure 12, where all important quantities except R_Vn now take a value which is to some extent backed by experimental measurement. The value of M₀ here, 600 dyn cm⁻¹, is supported by using the model to emulate the finding of Davis et al. (2012) that isolated rat mesenteric lymphangions typically cease to pump at a Δp_ba value of 10.9 cmH₂O. The simulations (for brevity not shown here) show that M₀ = 600 dyn cm⁻¹ or more is needed to equal this threshold at Δp_ae = 2.2 cmH₂O, declining to a minimum of 200 dyn cm⁻¹ at higher Δp_ae values.

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.

Keeping the same values of Δp_ba and Δp_ae as in Fig. 12, we can examine how the value of M₀ affects the mean flow-rate, as shown in Figure 13. The physiological result, with M = M(D,t), is shown in the right-hand panel. In the absence of contraction, there is a continuous leakage flow corresponding to the pressure difference Δp_ba applied to the resistance 2(R_Vn + R_Vx) offered by the two closed valves in series. Active contraction has a smoothly progressive effect on Q̄ once sufficient pressure is raised to open the outlet valve. Additional active tension has a gradually diminishing return for values exceeding 200 dyn cm⁻¹. By contrast, when the length-tension relation is omitted and M = M(t) only (left-hand panel), Q̄ rises rather abruptly to a plateau value as soon as M₀ is sufficient to create ejection. In this circumstance, D traverses values ranging all the way from the distension-limiting knee of the passive Δp_tm-D curve down to the low values associated with collapse and concomitant small compliance. The mean flow-rate achieved is accordingly higher than when M = M(D,t), but the result is unphysiological.

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.

Finally, we show in Figure 14 the application of the model as in Fig. 12 to two lymphangions in series. This thus constitutes a physiologically justified model of intrinsic pumping by multiple lymphangions, save for the necessity to raise R_Vn to an elevated value to stave off excessive regurgitation. Relative to Fig. 12, the value of Δp_ba is here doubled, since lymphangions in series can pump against greater Δp_ba, as we have previously demonstrated (Bertram et al. 2011a). Since the contraction of the two lymphangions is staggered, contraction of the first causes an immediate increase in the rate of diastolic filling of the second, as can be seen by comparing D₁(t) and D₂(t).

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.

4 Discussion

Dimensionless variables provide a systematic means of identifying parameter values that restore effective pump function. Having earlier found operating points which yield effective pumping by all our criteria (essentially simultaneous inlet and outlet valve switching at each end of systole, minimal regurgitation before valve closure, avoidance of the development of transient excessively high intra-lymphangion pressures during systole, large systolic ejection flow-rate relative to leakback through the closed inlet valve), we can use the analysis to indicate what needs to be done in an unsatisfactory case.

The methods have here been applied to a model consisting of a single lymphangion. Although we have also developed models of more than two lymphangions, we do not here concentrate on the application of these methods to a multi-lymphangion case, because our central concern here is the necessity to vary R_Vn from its measured value to obtain effective pumping. Our purpose in the present paper was to show how all important model parameters are now related to experimental measurements. As a result of adopting these values, and including an active length-tension relationship, we believe that we have achieved a model of much greater relevance than the earlier one (Bertram et al. 2011a). However, in the process we have shown that the indicated set of values is incompatible with an effective lymphangion pump. We can restore reasonably effective pumping by increasing the minimum resistance to flow though an open valve to some 40 or more times its measured value.

However, this is only a crude fix-up. The set of measurements illustrated in Fig. 5, while subject to rather wide error bounds, fixes the true open-valve resistance for valves in vessels of the diameter in question to a value which is certainly not open to increase by one or two orders of magnitude. Instead, we must look elsewhere for the source of the discrepancy between simulation result and experimental parameter measurement.

We note that the large measured bias to staying open of the valves has not been shown to have any redeeming merit, and is responsible for a large problem of wasted regurgitant flow. It seems that the real lymphatic system has evolved with this property, at least in part; Dixon et al. (2006) have noted backflow in exposed rat mesenteric microlymphatic vessels between contractions. However, in the course of organizing the measurements of the open-valve resistance (Fig. 5), our attention was drawn to the possibility of substantial pressure drops occurring between the reservoir where pressure was regulated and measured, and the vessel where its value needs to be known. The point of minimum diameter of the flow conduit is necessarily the tip of the micropipette which cannulates the vessel, and this therefore constitutes a source of high resistance. The same problem occurs on each side of the valve, there being a similar pipette tip at each end of the mounted vessel segment. This potential source of artefact was taken into account in the design of the protocol for the open-valve-resistance measurements, but was not appreciated at the time of the measurements of valve function (Davis et al. 2011)³. Thus our prediction, on the basis of the simulation outcomes, is that the published measurements of the pressure drop needed to close the valve substantially overestimate the real values. Since the measurements of valve-opening pressure drop threshold were necessarily conducted from the closed-valve no-flow state, this hypothesized source of error does not apply to the measured opening thresholds. We are conducting new measurements of valve function, under a protocol which will allow us to test this idea.

However, in advance of these further experiments, it is possible to estimate what the error might be, on the basis of a reasonable assumption. Recall that the opening and closing threshold data were fitted with the same function (but of course different constants), which gave constant slope against transmural pressure difference Δp_tm once the latter was substantially positive. If we posit that the closing-threshold data were inflated by an artefactual pressure drop from the pipettes, and make use of our existing assumption that all resistive pressure drops increase linearly with flow-rate at the small Reynolds numbers in these tiny vessels, then it is legitimate to reduce all the measured closing pressure drops by the same factor, since the same proportion of artefact would have applied throughout. What is the appropriate factor (between 0 and 1)?

A lower bound is provided by the opening threshold data, since a valve which opens at a larger value of (negative) pressure drop than that at which it closes is logically impossible. This would suggest a factor larger than about 0.1. A somewhat higher lower bound is obtained if it be assumed that the amount of hysteresis in the valve behaviour, i.e. the difference between the opening and the closing thresholds, should be independent of Δp_tm. This assumption dictates that the constant slopes of the curves representing threshold pressure drop vs. Δp_tm far away from Δp_tm = 0 should be equalized, avoiding the possibility of the two curves meeting at some extrapolated Δp_tm. It leads to a factor of 0.22103, i.e. it suggests that 77.9% of the measured closing pressure drop was contributed by sources of hydraulic resistance in series with the valve itself. The resulting suggested closure threshold is illustrated in Fig. 4a by the dotted curve.

When this factor is programmed into the model, it becomes possible for the first time to obtain effective pumping with all parameters at physiological values and without artificially boosting the value of R_Vn. Pumping is still greatly improved if R_Vn is given a value a few times higher than that measured, as shown in Figure 15, but this stratagem is no longer necessary to obtain a positive value of Q̄ .

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.

Thus we have taken the series-lymphangions model initially developed (Bertram et al. 2011a) and related its various parameters to physiological and measured values. We have introduced a refractory period between contractions, changed the lymphangion pressure-diameter relationship to one precisely matching a measured one, incorporated the length-dependence of active wall contraction, and emulated (Bertram et al. 2013) the complex measured properties of microlymphatic valves (Davis et al. 2011). Along the way we have changed the value of open-valve resistance to that which we have ourselves measured, and increased the value of closed-valve resistance to one which prohibits significant leakage. With all model parameters thus linked to experimental measurement, it has been possible to find out from the model itself whether all the experimental measurements were consistent. Finding that they are not, we have through the use of dimensionless analysis found simple ways to restore effective pumping in the model, by increasing open-valve resistance, by reducing the valve-closure bias to staying open, or by a combination of the two. In so doing we have postulated a specific source of experimental error, through a prediction which will in due course be tested.

The incorporation in the model of the measured valves’ bias to staying open (Davis et al. 2011) also leads to instances of the two valves bounding a lymphangion in the model being open at the same time. This is seen in Fig. 6, and again in Fig. 11; there are also periods when the two valves bounding one or other of the two lymphangions modelled in Fig. 14 are open simultaneously. Somewhat similar behaviour has been noted experimentally; Scallan et al. (2012) show in their fig. 4B a case where, with pressure beyond both valves being simultaneously ramped up (equivalent to ramping external pressure down), both valves of an isolated lymphangion remained open for an extended period covering many contractions. Fig. 4 shows explicitly how elevated transmural pressure encourages this behaviour.

We may speculate on the reasons which underlie the dependence of efficient pumping on increasing the value of the open-valve resistance R_Vn above the value estimated from measurement, even when the valve closure threshold is attenuated as in Fig. 15. First, considering Fig. 5, the slope dΔp_V/dQ of the surface diminishes with increasing flow-rate; this we attribute to the valve leaflets increasingly moving back close to the vessel wall. The estimated overall slope of 0.2 cmH₂O per 20 uL/min could perhaps be increased to take that into account, since we are interested in the slope at zero flow-rate, corresponding to the instant of valve closure. Secondly, our measurements were necessarily made with forward flow; it is possible that R_Vn immediately before closure is greater when the flow is retrograde, encouraging the valve leaflets to invade the lumen. Thirdly, it may be that real microlymphatics do not pump efficiently; the system may have evolved to optimise some other criterion involving an open conduit in the absence of significant pressures and flows. It is also possible that the valve bias toward the open position is simply an unavoidable structural limitation arising from the shape needed for low-Reynolds-number operation (see below).

There remains considerable uncertainty in some of the parameter values assumed here. Thus for instance the passive pressure-diameter relation of Fig. 2 is matched to an individual vessel measured by Davis et al. (2011). MJD (Davis, private communication) has encountered other vessels with a smaller range between D_d and the maximum diameter where passive stiffness becomes limiting. A passive pressure-diameter relation based on such vessels would cause even less effective pumping. Similarly, good data on the form of the active contribution to developed pressure, i.e. M_d(D), do not yet exist; M_d(D) here was based on the properties of vascular muscle in general, plus a comparison of the model with the measured (Davis et al. 2012) limiting value of Δp_ba. These tests suggested values of M₀ between 200 and 600 dyn cm⁻¹, depending on Δp_ae. The measurements of Zhang et al. (2007) yielded values of 340 dyn cm⁻¹ (peak tension) and 210 dyn cm⁻¹ (plateau tension) for isolated loops of rat mesenteric microlymphatic, stimulated pharmacologically to maximal contraction. We have used values ranging between 400 and 1100 dyn cm⁻¹ when M_d(D) was in use, although 1100 dyn cm⁻¹ was used only for Fig. 11 where D_d had yet to assume its final value. On the other hand, the shape we have assumed for M_d(D) may underestimate the true extent to which active tension is maintained in lymphatics at low diameters. Because it is problematic to measure circumferential length via ‘diameter’ when strong active contraction may produce circumferential corrugations, there is a dearth of data to continue the experimental curve of active tension vs. length into the region where passive tension is small. Experiments reported so far (Zhang et al. 2007) leave the curve incomplete.

There is also scope for further refinement of the valve description in the model, since the valve dynamics are clearly critical to the pump performance. The valves in these small lymphatics form long funnels which close under purely viscous flow conditions (Mazzoni et al. 1987). When the pressure gradient across the valve is adverse, there will be slightly higher pressure in the sinuses than between the leaflets, giving a small pressure difference across the flexible valve leaflets causing closure of the orifice between them to start. Details of how this transition compares with that shown in Fig. 3 are lacking at present, but the time-scale of closure is likely to be related to the time taken for the fluid between the leaflets at the instant of pressure gradient reversal to move out. As the leaflets come together, the viscous pressure drop through the diminishing orifice may increase, which in turn would be expected to augment the pressure difference causing closure.

There is no evidence that any deficiency of lumped-parameter modelling is responsible for the primary outcome here, i.e. lack of pumping when the parameters adhere to what is known so far about their real values. A higher-order model of a contracting lymphangion (Rahbar and Moore 2011) showed that wall shear stress departed by less than 4% from the Poiseuille values assumed here. Experimental evidence also suggests that unsteady effects are small. The experiments conducted by one of us (MJD) lead to ciné video which shows clearly the events surrounding valve closure (potentially more significant as a source of unsteady transient effects than opening). For such transient effects to arise, there has to be significant inertia. In larger systems, such as cardiac valves, the fluid inertia leads to the valve temporarily moving backwards after closure thanks to the impossibility of halting the regurgitant flow instantaneously, then recoiling elastically back to the valve rest position, with small decaying oscillations of position. But the Reynolds number here is simply too small for inertial effects to be important; the system consists of fluid resistance and tissue elastic compliance only. The videos confirm this impression; the valves close quickly, but entirely without such surge transients.

The model still omits local feedback mechanisms. Microlymphatic vessels are under myogenic control, whereby active contraction is triggered by diastolic lymphangion distension (Gashev et al. 2004). By release of nitric oxide, lymphatic endothelium also mediates the inhibition of intrinsic pumping when fluid shear indicates sufficient flow in its absence (Gashev et al. 2002). In addition, intracellular calcium store release/refilling (Peng et al. 2001; Imtiaz et al. 2010) causes a particularly vigorous contraction after an unusually extended diastole. In the further development of the model, and especially if it be incorporated in a local network of vessels, it will be necessary to include a representation of these items. However, at the level investigated here, i.e. the realistic simulation of individual contractions, these elements are not needed.

Appendix 1

The passive constitutive relation is a nonlinear function Δp_tm = f_p(D), where $f_p(D)=P_d\left[c_1{\left(\frac{D}{c_9}-c_2\right)}^2+c_{3}exp\left(c_4\left(\frac{D}{c_9}-c_5\right)\right)+c_6+c_7\left(\frac{D}{c_9}-c_8\right)+c_{10}{\left(\frac{c_9}{D}\right)}^3\right]$, with $c_9=\frac{D_d}{c_{11}}$. When P_d defines the slope 4P_d/D_d of the function at Δp_tm = 0, the constants have the values c₁ = –2.34457751, c₂ = 1.1262924, c₃ = 3.76013762, c₄ = 79.991135, c₅ = 1.0028029, c₆ = 1.59133174, c₇ = 3.69692633, c₈ = 0.20699868, c₁₀ = –0.0180867408 and c₁₁ = 0.32538081. This system is convenient in the numerical model; however a different scaling system is preferred in experiments. The distension-limiting region of the curve to the right of the knee is reached by Δp_tm = 5 cmH₂O; the corresponding diameter c₉ is used to normalise the measurements of D (D_d then takes the value c₁₁). P_d can be used to refer to this point if so desired (for the defining curve under consideration it then takes the value 4905 = 5 × 981 dyn cm⁻²), in which case the values of c₁, c₃, c₆, c₇ and c₁₀ must be multiplied by 0.7464031.

The active length-tension relation for contraction is a nonlinear function M_d(D), where $M_{\mathrm{d}}(D)=\frac{1}{1+exp(-s_{\mathrm{d}}(D-D_{\mathrm{a}}\left)\right)}+\frac{1}{1+exp\left(s_{\mathrm{d}}\right(D-D_{\mathrm{b}}\left)\right)}-1$, with D_a = 0.85c₉, D_b = 2c₉ and s_d = 3.25/D_d.