Parameter sensitivity analysis of a lumped-parameter model of a chain of lymphangions in series

Abstract

Any disruption of the lymphatic system due to trauma or injury can lead to edema. There is no effective cure for lymphedema, partly because predictive knowledge of lymphatic system reactions to interventions is lacking. A well-developed model of the system could greatly improve our understanding of its function. Lymphangions, defined as the vessel segment between two valves, are the individual pumping units. Based on our previous lumped-parameter model of a chain of lymphangions, this study aimed to identify the parameters that affect the system output the most using a sensitivity analysis. The system was highly sensitive to minimum valve resistance, such that variations in this parameter caused an order-of-magnitude change in time-average flow rate for certain values of imposed pressure difference. Average flow rate doubled when contraction frequency was increased within its physiological range. Optimum lymphangion length was found to be some 13–14.5 diameters. A peak of time-average flow rate occurred when transmural pressure was such that the pressure-diameter loop for active contractions was centered near maximum passive vessel compliance. Increasing the number of lymphangions in the chain improved the pumping in the presence of larger adverse pressure differences. For a given pressure difference, the optimal number of lymphangions increased with the total vessel length. These results indicate that further experiments to estimate valve resistance more accurately are necessary. The existence of an optimal value of transmural pressure may provide additional guidelines for increasing pumping in areas affected by edema.

the lymphatic system works in parallel to veins and, besides fluid return, organizes immune response and lipid absorption. It is also implicated in the spread of cancer cells. The lymphatic system collects some 4–8 liters of fluid per day from the interstitial space and pumps it back to the venous circulation (13). The system is required to pump viscous fluid against gravity and pressure. In contrast to the blood circulatory system, there is no primary pump in the lymphatic network.

Lymphatic pumping occurs through the combination of valves that prevent backflow and the change in volume of the vessel segment in between (referred to as a lymphangion). There are two pumping mechanisms at work in the lymphatic system. Extrinsic pumping is the result of pressure changes or movements outside the lymphatic vessel, due to (e.g.) respiration or skeletal muscle contractions, compressing lymphangions and expelling the lymph, whereas intrinsic pumping is caused by the active contractions of muscle in the lymphangion wall. The extrinsic and intrinsic pumping mechanisms together allow the system to generate forward flow. Their relative contribution, which varies in different regions of the body, is only partly understood.

Disruption of the lymph due to infection, trauma, or injury results in fluid buildup in the tissues (lymphedema), which affects more than 90 million people worldwide (22). The lack of an effective cure for this disease can be attributed in part to our insufficient knowledge of the system and its transport mechanisms. Despite its importance, the system has so far received little attention from biomechanicians. A more extensive and physiologically based numerical model is necessary to expand our knowledge of the system's performance (15).

In the first lymphatic modeling effort, Reddy and colleagues (20, 21) developed a one-dimensional model for seven generations of large lymphatic vessels. The model did not include smaller vessels that are responsible for much of the pumping. Quick and colleagues (18, 27) developed a lumped model of a single lymphangion and a chain of lymphangions in series, using the approach that was previously developed for cardiac ventricular contractions. Macdonald et al. (14) simulated the lymphangion divided between valves into four segments. Recently, Bertram et al. (2) created a lumped-parameter model for a chain of lymphangions in series. Equations of conservation of mass, conservation of momentum, and vessel wall force balance are solved for each lymphangion. The model accounts for both passive behavior of the vessel and active vessel contractions. Lymphatic valves are defined by flow resistance that varies with pressure difference across the valve, a concept closer to their real behavior than in any previous model.

The model developed by Bertram et al. (2) improved on previous models in terms of modeling the valve behavior, the active contraction, and the passive behavior of the vessel, and recent refinements (1, 4) have taken these improvements further. However, more accurate estimates of the parameters in the model are crucial if it is to be applied to understanding normal and pathological function. Due to the difficulty of isolated-vessel and in situ experiments, it is worthwhile first to determine the parameters that have most effect on system response and then to focus the experimental studies on those parameters. Our goal here was to conduct a parameter sensitivity analysis to determine the parameters with the greatest effect on the system outcomes.

METHODS

Modifications of the Model

We performed a parameter sensitivity analysis for a lumped-parameter model of a chain of lymphangions in series, as previously developed (2). Extensive details of the equations and method of solution used are available in that paper; the equations are also summarized in Appendix. The model analyzed in this study differs from the original model in the definition of pressure variables. Previously, intra-lymphangion pressure was only defined at inlet (p₁) and at outlet (p₂). Here, as in Bertram et al. (4), we define an additional pressure p_m at the center of each lymphangion, which is related to the external pressure p_e for the calculation of the vessel wall force balance. Two Poiseuille relations then relate p₁ − p_m and p_m − p₂ to upstream and downstream flow rates, respectively. The modification in the definition of pressure variables increases the number of equations in the model but permits more realistic simulation of backflow upon valve failure at extreme adverse pressure differences (3).

The solution was computed by solving equations of conservation of mass, conservation of momentum, and vessel wall force balance, resulting for each lymphangion in a nonlinear ordinary differential equation for diameter and two algebraic equations, together comprising a differential-algebraic equation system. Whereas we originally (2) developed our own computational scheme in MatLab to solve these equations, in this study the differential-algebraic equation system was solved with a standard MatLab (R2010b, MathWorks) function. The revised method of solution replicated the former results in a fast and robust manner. The parameter values and baseline conditions (see Table 1) were as used before (2), and the values assigned to each parameter were the same for all the lymphangions in the chain unless it is stated otherwise. The outcome chosen as the primary criterion for judging contribution to pumping was the average flow rate of the last lymphangion. The simulations ran until the average flow rate was stable and independent of the initial conditions; this was achieved after different numbers of active contraction cycles depending on the values of the parameters in the model.

Table 1.

Parameters used in the numerical model, including their definition and baseline value (i= 1:n)

Description	Parameter	Value	Units
Valve parameters
Valve failure pressure	Δpf	−18.4	cmH2O
Valve failure slope	sf	0.049	cm2/dyn
Minimum valve resistance	RVn	600	g/(cm4·s)
Manimum valve resistance	RVx	1.2 × 107	g/(cm4·s)
Valve opening pressure	Δpo	−0.07	cmH2O
Valve opening slope	so	0.04	cm2/dyn
Nonvalve parameters
Fluid viscosity	μ	0.01	g/(cm·s)
Lymphangion length	L	0.3	cm
Vessel length	Ls	1.2	cm
Number of lymphangions	n	4	—
Pressure constant in vessel wall force balance relation	Pdi (i = 1 to 4)	50, 75, 100, 125	dyn/cm2
	(i = 1 to 8)	50, 75,..., 225
Diameter constant in vessel wall force balance relation	Ddi (i = 1 to 4)	0.025	cm
	(i = 1 to 8)	0.021 (for Fig. 6)
Active tension	M	3.6	dyn/cm
Contraction frequency	f	0.5	Hz
Refractory period	tr	0	S
Inter-lymphangion phase	(via t0i)	90	Degree
External pressure	pe	2.14	cmH2O
Inlet pressure	pa	2.32	cmH2O
Outlet pressure	pb	2.42 (pb,1)	cmH2O
		2.67 (pb,2)
		2.92 (pb,3)
Pressure difference between the two ends of the chain	ΔP	pb − pa	cmH2O

Parameter Sensitivity

Most of the sensitivity analysis study was performed for a chain of four lymphangions in series. We also investigated chains with 2 to 15 lymphangions to determine the optimum number of lymphangions for a certain vessel length. The effect of refractory period was investigated in a chain of eight lymphangions with a modified pressure-diameter relation (see Appendix). The pressure difference ΔP = p_b − p_a across the chain of lymphangions was initially determined by varying outlet pressure p_b over a range of 2.0–3.6 cmH₂O, while inlet pressure p_a remained constant at 2.32 cmH₂O and external pressure p_e at 2.14 cmH₂O, i.e., p_a − p_e = 0.18 cmH₂O; for the eight-lymphangion study, p_a − p_e = 0.1784 cmH₂O. Then a pump function curve was created to illustrate the capability of the system to generate flow under different pressure differences (Fig. 1). Based on that curve, we conducted the parameter sensitivity study for ΔP values of 0.1, 0.35, and 0.6 cmH₂O, corresponding to high, medium, and low or negative flow rates, respectively (still at the same p_a and p_e).

Fig. 1.

See Glossary for the definition of each algebraic variable. Pump function curve for a chain of 4 lymphangions in series, at p_e = 2.14 cmH₂O. The horizontal axis shows the average flow rate of the last lymphangion in the chain; the vertical axis shows the pressure difference between the 2 ends of the chain. Three values of pressure difference (ΔP) were chosen at different regions of the pump function curve, and the parameter sensitivity study was performed for these values (ΔP = 0.1, 0.35, and 0.6 cmH₂O).

The parameters were analyzed in two groups: those related to the valves and those related to the intervalve vessel segments. One-at-a-time parameter sensitivity analysis (i.e., variation of one parameter, whereas the others remained constant) was performed for minimum and maximum valve resistance (R_Vn and R_Vx) and for the lymphangion-chain parameters of external pressure (p_e) and contraction frequency (f). (Maximum valve resistance is actually R_Vx + R_Vn. Since valve resistance varies gradually across the transition, R_Vx was varied in concert with Δp_o so that resistance to forward flow was the same for all cases.) The values of the parameters were varied within what was estimated as their physiological range. When such information was not available parameters were varied until the system outcome reached a plateau. For example, p_e was varied between 1.7 and 3.0 cmH₂O, whereas all the other parameters remained constant and equal to the values shown in Table 1.

Due to the difficulties associated with experiments on lymphatic vessels, experimental data are only available for a limited group of parameters. It is known that the contraction frequency of lymphatic vessels varies in different tissues. For example, collecting lymphatic vessels contract at a frequency of 0.0016–0.16 Hz (15). Other investigators reported contraction frequencies as high as 0.5 Hz (24). In the rat mesentery, Zhang et al. (30) measured a peak active tension of 340 dyn/cm in vessels with nominal diameters of ∼150 μm. Telinius et al. measured a peak active tension of 6,240 dyn/cm in the human thoracic duct, with average vessel diameter of 2.21 mm, and 1.4 contractions per min in vitro (26); others have observed 4–6 contractions per min in situ (8).

Dimensions of the lymphangions in this study (determined by L and D_di) are in the range of those observed in the rat mesentery. The diameter of lymphangions in human lymphatic vessels varies from 10–60 μm in initial lymphatics, to 2 to 3 mm in major lymphatic ducts (25). The diameters used here (250 μm) correspond roughly to small collecting lymphatic vessels in humans.

Lymphangion length in the rat mesentery is around 0.3 cm, the baseline value used in this model. In humans, the precollecting lymph vessels in the head and neck region have a diameter of 0.1–0.3 mm with valves every 1 to 2 mm, suggesting that lymphangions are some 6–10 diameters long. The diameter of the larger lymphatic ducts in this region is about 2 mm, with valves every 1.5 diameters or so (17). Valve spacing in the human thoracic duct is reported to be about 1 cm (12).

Two-parameter sensitivity study (varying 2 parameters simultaneously) was conducted for active tension (M) and lymphangion length (L). The values assigned to M in our model at this stage were chosen to generate reasonable outputs for diameter, pressure, and flow rate for these small lymphangions. We repeated this two-parameter study at several different values of the diameter constant D_di, to investigate how the optimal values of M and L varied with average lymphangion diameter. In almost all simulations, active contraction started immediately after the end of the previous contraction, such that the waveform of active tension was a continuous sine wave, but the effect of adding a refractory period between contractions was investigated as well. The number of lymphangions in the chain was also varied, in combination with lymphangion length (L) and overall vessel length (L_s). In these cases, the pressure constant in the vessel wall force balance relationship (P_di) increased by 25 dyn/cm² in each lymphangion compared with the one upstream. The sensitivity to each parameter was evaluated by the degree of change in average flow rate Q̄ from the system for a fixed adverse pressure difference (ΔP).

RESULTS

Sensitivity to Valve-Related Parameters

Sensitivity analysis of a chain of four lymphangions in series showed that among the parameters related to the valves, the system was more sensitive to R_Vn than to R_Vx (Fig. 2). Variation of R_Vn affected forward flow dramatically, particularly near the lower end of the investigated range. For example, decreasing R_Vn from 8×10⁶ to 1×10⁶ g/(cm⁴·s) under the lowest imposed pressure difference resulted in an order-of-magnitude increase in mean flow rate. At ΔP = 0.10 and 0.35 cmH₂O, increasing R_Vn reduced Q̄ to below zero and near zero, respectively. At ΔP = 0.6 cmH₂O, backflow dropped by 60% of its initial value as R_Vn increased from zero to 8×10⁶ g/(cm⁴·s) (Fig. 2A).

Fig. 2.

Mean flow rate Q̄ vs. the minimum (A) and maximum (B) possible resistance of each valve, for ΔP = 0.10 (solid), 0.35 (dot-dashed), and 0.60 (dashed) cmH₂O. For the simulations in B, Δp_o was slightly adjusted along with R_Vx so as to maintain the same resistance to forward flow at low flow rates.

Increasing R_Vx from 6×10⁶ to 3×10⁷ g/(cm⁴·s) increased Q̄ by 15 and 52% of its initial value for ΔP = 0.10 and 0.35 cmH₂O, respectively. At ΔP = 0.6 cmH₂O, the chain of lymphangions failed to generate forward flow, but backflow decreased as R_Vx increased (Fig. 2B). Comparison of Fig. 2, A and B, shows that in the cases with forward flow where the valves open, increasing R_Vn from its baseline value of 600 g/(cm⁴·s) caused at least seven times more change in Q̄ than increasing R_Vx from its baseline value of 1.2×10⁷ g/(cm⁴·s).

Sensitivity to Nonvalve-Related Parameters

Sensitivity to contraction frequency.

At low frequency Q̄ increased almost in proportion with f (Fig. 3); for instance, at ΔP = 0.10 cmH₂O, increasing f from 0.2 to 0.4 Hz resulted in a doubling of Q̄. The effect tapered off at frequencies higher than physiological. Behavior was similar at ΔP = 0.35 cmH₂O, but with lower Q̄. At ΔP = 0.6 cmH₂O, the system was unable to generate forward flow at any frequency.

Fig. 3.

Variation of mean flow rate Q̄ with contraction frequency f, at 3 values of ΔP.

Sensitivity to external pressure.

Q̄ went through a peak as transmural pressure was varied (by means of p_e) while leaving ΔP fixed (Fig. 4A). Increasing p_e is equivalent to reducing both p_a and p_b for the lymphangion chain, while leaving their difference unaltered. The optimum transmural pressure (transmural pressure that maximized Q̄) was close to zero, depended on ΔP, and was higher (i.e., lower p_e) at lower values of ΔP. The peak Q̄ was 0.28 ml/h at ΔP = 0.10 cmH₂O, less than half that at ΔP = 0.35 cmH₂O, and only just above zero at ΔP = 0.6 cmH₂O.

Fig. 4.

Sensitivity to transmural pressure. A: Q̄ vs. external pressure p_e at 3 values of ΔP. Peak Q̄ occurred at a value of p_e that gave inlet transmural pressure p_a − p_e close to zero. Loops of midlymphangion transmural pressure (Δp_tm = p_m − p_e) vs. diameter D for the fourth (last) lymphangion were computed for the 4 locations indicated on the curve for ΔP = 0.1 cmH₂O. B: Δp_tm-D loops for the 4 points chosen in A, superimposed on the passive pressure-diameter curve. C: magnification of the 4 Δp_tm-D loops.

To understand this behavior better, we chose four points on the curve of Q̄ vs. p_a − p_e at ΔP = 0.10 cmH₂O, and superimposed the pumping loops (similar to cardiac pressure-volume loops) on a curve of transmural pressure Δp_tm = p_m − p_e versus diameter. These points were chosen in the ascending, maximum, descending, and far right-end regions of the curve, corresponding to p_e = 1.8, 2.0, 2.3, and 2.9 cmH₂O, respectively. The slope of the Δp_tm-D curve is inversely related to the compliance of the tube (Fig. 4, B and C). A comparison of Fig. 4, A and B, shows that peak Q̄ is generated when the vessel is most compliant, that is, at the flattest point on the Δp_tm-D curve. Moving away from that point in either direction, the vessel becomes less compliant and Q̄ decreases. The rapid decrease in Q̄ at high p_e is due to the tube collapsing, which reduces compliance and makes further cross-sectional area reduction by muscular contraction more difficult (D in this region should be thought of as hydraulic diameter). Collapse also increases the input impedance of the downstream lymphangion (as affected by the instantaneous extent of active contraction) to pumping from upstream lymphangions. These results are for the fourth lymphangion in the chain; in other lymphangions, peak Q̄ occurred near the most compliant state of the vessel as well, only shifting slightly to the right on the Δp_tm-D curve.

Sensitivity to lymphangion length and active tension.

Pumping showed complex sensitivity to combined variations in lymphangion length (L) and active tension (M) (Fig. 5). At ΔP = 0.10 cmH₂O, peak Q̄ occurred at L ≈ 0.27 cm. At low values of L, increasing M caused stronger contractions, and up to M = 9 dyn/cm, this behavior was monotonic up to L = 0.14 cm. Beyond this length, Q̄ progressively folded over as M continued to increase, such that the highest Q̄ of 0.25 ml/h was registered at M = 5 dyn/cm. Higher values of M resulted in peak Q̄ as low as 0.2 ml/h, because the vessel remained at small diameters for a longer time, increasing input impedance to upstream pumping. At all values of M, increasing L was initially beneficial to Q̄, but then detrimental. Peak Q̄ occurred at smaller L as M increased, ranging from 0.4 cm at low M to 0.25 cm at M = 9 dyn/cm. At higher values of ΔP, the lymphangion chain exhibited qualitatively similar behavior but gave lower values of Q̄. Peak Q̄ still occurred at L = 0.3 cm, but the highest values of Q̄ (0.24 and 0.22 ml/h) were achieved at M = 6 and 8 dyn/cm for ΔP = 0.35 and 0.60 cmH₂O, respectively.

Fig. 5.

A: Q̄ vs. lymphangion length L at various values of active tension M, for ΔP = 0.10 cmH₂O. B: optimum (max. Q̄) value of L vs. average diameter over a cycle D_av. C: active contribution to transmural pressure (2M_opt/D_av) vs. average diameter.

The values of M and L which cause maximum Q̄ increase with vessel diameter. This was proved by changing nominal vessel diameter (by varying the diameter constant D_di for all lymphangions, while keeping the same overall adverse pressure difference ΔP across the lymphangion chain). As the caliber of the vessel increased, the system reached optimum Q̄ with longer lymphangions (Fig. 5B) and larger M. Thus, as average D over a cycle increased from 0.02 to 0.045 and then 0.07 cm (with D_di = 0.025, 0.06, and 0.1 cm, respectively), the peak Q̄ increased from 0.24 to 2.54 to 9.64 ml/h. These optima were achieved with M = 4.8, 14.8, and 25.4 dyn/cm, and L = 0.27, 0.61, and 1.02 cm, respectively. Increases in optimal L were roughly proportional to average D, but optimal M increased more. This is illustrated in Fig. 5C, where the optimal values of M are shown in terms of 2M_optimal/D_average, which is a time-averaged measure of the instantaneous active contribution to balancing the transmural pressure in the constitutive relation (see Appendix).

Effect of Refractory Period

A refractory period imposes a period of smooth muscle cell relaxation before the next contraction is initiated. Addition of a refractory period to a model of a chain of eight lymphangions in series caused maximum pump power to fall as the refractory period increased. Figure 6A shows the pump function curves for four refractory periods (t_r = 0, 1, 2, and 3 s), while the duration of contraction (smoothly from zero to peak and back to zero) remained fixed at 2 s. The modified pressure-diameter relation used here causes the curves to be convex toward the origin, unlike that shown in Fig. 1, and the maximum ΔP for which Q̄ > 0 is higher because of increased n. The thin gray lines are curves of constant hydraulic power drawn tangent to the point of maximum power in each case. Maximum output power defined thus is a number that characterizes the pump performance for a given set of parameter values, irrespective of the pressure difference ΔP which the pump faces. Figure 6B shows the variation in maximum power as refractory period increases. Maximum power dropped by 44% when refractory period increased from zero to 3 s.

Fig. 6.

Effect of addition of a refractory period. A: pump function (ΔP vs. Q̄) at 4 values of the refractory period. Thin gray curves: constant hydraulic power, tangent to each curve at maximum power. B: maximum pumping power from the 8 lymphangion chain vs. refractory period. C: maximum power vs. time-average active tension, which reduces as the refractory period increases.

Although maximum power dropped with t_r, there was a gain in pumping efficiency. Figure 6C shows how the maximum pumping power varied with the time-average active tension [2M/(2 + t_r), where M = 3.6 dyn/cm] in the presence of refractory period t_r, relative to what was produced with no refractory period. Time-average active tension relates to the metabolic cost of pumping in terms of time-average effort by muscle. With a refractory period of 1 s, the average value of M(t) dropped from 3.6 to 2.4 dyn/cm, suggesting that maximum output power might reasonably be expected to fall to 66.7%. But in fact maximum power was still more than 87.5% of that generated with no refractory period. As the refractory period increased further, maximum power continued to fall, but the pump was able to maintain a clear efficiency advantage, in terms of output hydraulic power for a given time-average-tension input cost, over a pump with no refractory period.

Effect of the Number of Lymphangions in the Chain

Simulations with varying numbers n of lymphangions in the chain showed that increasing n does not necessarily increase Q̄. To study this effect we considered two cases. In the first (Fig. 7), the number of lymphangions and the lymphangion length were varied simultaneously; this determined the number of lymphangions that generated the highest Q̄ at each lymphangion length. In the second case (Fig. 8), the results of varying n and overall chain length (L_s) were used to find the optimum number of lymphangions for a specific L_s.

Fig. 7.

Q̄ vs. lymphangion length L, as the number of lymphangions in the chain n is varied. A: ΔP = 0.10 cmH₂O. B: ΔP = 0.35 cmH₂O. C: ΔP = 0.60 cmH₂O.

Fig. 8.

Q̄ vs. overall chain length L_s, as the number of lymphangions in the chain n is varied. A: ΔP = 0.10 cmH₂O. B: ΔP = 0.35 cmH₂O. C: ΔP = 0.60 cmH₂O.

For the first case (Fig. 7A), we observed that at ΔP = 0.10 cmH₂O, the system reached the highest Q̄ (0.22 ml/h) with four lymphangions, at L ≈ 0.27 cm; more lymphangions gave a lower peak Q̄. However, beyond L ≈ 0.7 cm, Q̄ increased monotonically with n. At ΔP = 0.35, the system reached highest Q̄ (0.17 ml/h) with eight lymphangions, at L ≈ 0.38 cm; beyond L ≈ 0.55 cm, Q̄ increased monotonically with n (Fig. 7B). At 0.60 cmH₂O, the system reached the highest Q̄ of 0.12 ml/h at L ≈ 0.50 cm, with more lymphangions (Fig. 7C).

An optimum number of lymphangions was found for each L_s at ΔP = 0.1 cmH₂O (e.g., n = 6 at L_s = 2 cm); see Fig. 9. As expected, the optimal number was small at low L_s (e.g., 3 at L_s = 1 cm) and large at high L_s. Beyond L_s = 7 cm, the optimal number was 15, and monotonic advantage over the chains with fewer lymphangions was gained (Fig. 8A). The plot shows that 15 lymphangions yield the greatest Q̄ if distributed over L_s = 5.5 cm. At ΔP = 0.35 and 0.60 cmH₂O, the system reached the highest Q̄ of 0.17 and 0.13 ml/h, at L_s = 3 cm and L_s = 7 cm, with n = 8 and n = 14, respectively (Fig. 8, B and C).

Fig. 9.

The optimum (in terms of maximizing Q̄) number n of lymphangions at each vessel length L_s, for ΔP = 0.10 cmH₂O.

DISCUSSION

We performed parameter sensitivity analysis for the model developed by Bertram et al. (2) for a chain of lymphangions in series. This model differed from previous modeling efforts (14, 18, 20) in considering the pressure-dependent behavior of the valves and also in its treatment of the passive and active behavior of the vessel. More accurate measurements of some of the large number of parameters are required to improve the model. Parameter sensitivity analysis can help prioritize the difficult experiments involved in estimating these parameters.

Our results demonstrated that the system was more sensitive to minimum valve resistance (R_Vn) than to maximum valve resistance (R_Vx). R_Vn, the resistance to flow of an open valve, is a necessary parameter in the model, because it is only through backflow in the face of R_Vn that the pressure drop required to close the valve (6, 16) is generated. However, raising R_Vn also increases the impedance to forward flow. R_Vx is nominally only related to pumping efficiency through backflow prevention, but in this model its influence may still be felt at small favorable pressure drops. High sensitivity to R_Vn implies that while preventing backflow is crucial for the overall function of the lymphatic system, the system is also highly sensitive to any source of impedance to forward flow. Despite the importance of R_Vn, measurement of this parameter remains a challenge, and only recently have the first measurements become available. Davis et al. (6) characterized valve gating and behavior in collecting lymphatic vessels from rat mesentery. Their results showed that valves are biased to the open position, with the axial pressure gradients necessary to open and close the valves dependent on transmural pressure [behaviors not represented in this version of the model, but see Bertram et al. (1, 4)]. However, measurement of R_Vn requires knowledge of the flow rate, which was not measured by Davis et al. Bertram et al. (1) imposed flow rates of 0–20 μl/min through vessels of 100–225 μm diameter and concluded that R_Vn was around 0.6 × 10⁶ g/(cm⁴·s).

Among the lymphangion parameters, pumping output was most sensitive to contraction frequency (f), external pressure (p_e), active tension (M), and lymphangion length (L). Q̄ increased with f, with sensitivity to f being highest at the relatively low frequencies typical of in vivo performance. Experiments by Davis et al. (7) showed that lymphatic vessels use variations in contraction frequency as an adaptation mechanism to different levels of adverse pressure difference, increasing f (or rather, reducing t_r) as adverse pressure difference increases. This behavior is similar to that observed in the cardiovascular system (e.g., heart rate increases with exercise).

In this study we deliberately chose to examine three operating points (combinations of ΔP and Q̄), one of which (ΔP = 0.6 cmH₂O) was located at a point on the pump function curve corresponding to a small retrograde Q̄ by leakage through closed valves, a physiologically relevant case. No amount of variation in the frequency of active contraction changed the retrograde flow to forward flow; only an optimal value of p_e prevented the backflow when active contraction parameters took their baseline values in the four-lymphangion chain. The implication is that optimal p_e combined with increased M can create forward flow in the face of this high, unfavorable value of ΔP. The situation can be regarded as an approximate model of combined intrinsic lymphatic pumping and carefully gauged external compression. In the ranges of p_e above and below that corresponding to optimal compression, the intrinsic mechanism was overloaded at ΔP = 0.6 cmH₂O and in need of more lymphangions (see below).

The identification of an optimal value of p_e was unexpected. Superposition of pressure-diameter loops on the pressure-diameter curves suggested that peak time-average flow rate occurs when the vessel is in its most compliant state, with a slight shift depending on the adjacent lymphangion impedances. The existence of an optimum value of external pressure, at values slightly greater than inlet pressure p_a, may explain why external compression as therapy for lymphedema works for only a small percentage of patients (28). However, recent experiments by Rahbar et al. (19) showed that the vessel remains in its most compliant state over a wider range of diameters than is the case for the pressure-diameter relationship used in this study. Consequently, the range of optimal external pressures in vivo may be wider than that which we have shown here. More accurate modeling of the passive behavior of the vessel would be required to investigate further the concept of an optimal degree of external compression. Any such effort should of course include the effects of external load on lymph production at the capillary level, which is not represented here.

Increasing active tension initially increased the pumping. Experiments by Davis et al. (7) also showed augmented pumping activity with increasing active contraction. However, in our study, further increases (Fig. 5) caused the vessel to stay longer in the constricted state, thus increasing the input impedance of downstream lymphangions to incoming forward flow. The optimum value of M increased as the pressure difference across the chain increased. Further increases in M were definitely counterproductive. The optimum value of M (M_opt) also increased as the caliber of the vessel increased. Since active tension contributes to the vessel-wall force balance as 2M(t)/D by the law of Laplace, M_opt would be expected to increase proportionately with D, just to maintain the same ability to cope with a given distending pressure in larger lymphangions. Since presumably wall thickness (and in particular that part occupied by muscle cells) also increases with vessel diameter, extra active stress from individual muscle cells would not be required to bring this about. However, the data in Fig. 5C show that maximum Q̄ was obtained with an active contribution to transmural pressure that increased from ∼0.5 to ∼0.75 cmH₂O with diameter. This must reflect the fact that larger-diameter lymphangion chains were doing more pumping work, as a result of achieving much higher maximum Q̄ against the same ΔP (and unchanged R_Vx and R_Vn). In turn, this suggests if this behavior of the model is reflected in vivo, that larger lymphatic vessels might be expected to have rather more abundant muscle cells than in mere proportion to their size, all other things being equal.

Increasing contraction frequency increased flow rates monotonically (but with diminishing returns) over the range of frequencies tested here, which exceeded the physiological range. However, these simulations involved contractions succeeding each other without an intervening refractory period and a somewhat unphysiological ability to vary the rapidity of both contraction and relaxation. We have also examined conditions of fixed contraction duration and variable refractory period. In some situations involving high contraction frequency, pumping efficiency suffers due to incomplete filling (unpublished results).

For all three tested pressure differences (ΔP = 0.10, 0.35, 0.60 cmH₂O), time-average flow rate peaked at L ≈ 0.3 cm, regardless of the value of M. This length corresponds closely to the physiological range for the lymphatic vessels on which our baseline models were based. These results, however, were for a chain of four lymphangions in a certain diameter range. Vessels with larger diameter reached the highest mean flow rate at higher values of L. Over the range of diameters tested, the system thus displayed a preference for lymphangions of length between 13 and 14.5 diameters.

Addition of a refractory period to the model showed that pumping held up well despite the reduction in the duty cycle of lymphatic muscle cells. Time-average active tension is likely to be a meaningful measure of lymphatic energy consumption, by analogy with the behavior of myocardium, where Sarnoff et al. (23) found such a correlation. Specifically, they found that myocardial oxygen consumption was better predicted by the area under the systolic portion of the ventricular (or aortic) pressure waveform, which they dubbed the “tension time index,” than by external stroke work. Assuming a similar fundamental concept for lymphatic muscle, the result shown in Fig. 6C implies that the lymphatic system uses metabolic energy efficiently by contracting less often than the maximum rate (no refractory period). The addition of a refractory period also serves to reduce the input impedance of downstream lymphangions to flow coming from upstream ones.

We also investigated chains with varying numbers of lymphangions. The physiological question addressed was, For a given vessel length, into how many lymphangions should the vessel be subdivided to generate the maximum average flow rate? We first looked at the effect of simply adding/removing lymphangions to/from the chain of four lymphangions. This approach, of course, resulted in different overall chain lengths for each case. Our results demonstrated that adding lymphangions to the chain does not necessarily increase Q̄, because the additional valves necessarily impose a cost in added input impedance. But this disadvantage is outweighed when the valves are sufficiently spaced out in longer lymphangions. The simulations also showed that at higher pressure differences, optimum pumping is achieved by using more and longer lymphangions. We then considered the more physiological case in which the overall chain length remained constant and the number of lymphangions varied, resulting in different lymphangion lengths for each case. These simulations produced the optimum number of lymphangions for a given vessel length. Increasing the number of lymphangions can improve the pumping if the effect of the pumping/contractile power added to the system by that lymphangion is greater than that of the added impediment to flow. In varying the number of lymphangions, we have gone beyond the physiological range and studied vessels with lengths up to 12 cm to test our model. We ignored the effect of gravity because the parameter values are based on the measurements in rat mesentery and a long chain of lymphangions with vertical orientation is unlikely in rats. The effect of gravity will be included in future models encompassing more extensive series and parallel vessel networks. In Poiseuille flow, the gravitational term adds linearly to the pressure gradient.

Most of our current knowledge about lymph regulation comes from animal experiments. Comparison of the available data suggests that there are differences in flow and pressure sensitivity in different tissues. Furthermore, the experimental results from different species are sometimes inconsistent with each other (9). Thus it would not be surprising if the lymph transport in humans differs from animal studies in some aspects.

Understanding the lymph regulation process is a critical step toward discovering an effective treatment for lymphatic diseases such as lymphedema. Unfortunately, there is no comprehensive source of data for lymph transport in humans. Due to the upright posture of the body, the regional variation in contractility is expected to be greater in humans. Different activities during the day will also greatly affect lymph regulation. Exposure to different movements and tissue pressures in different parts of the body will lead to regional variability in lymphatic transport (9, 10).

It should be noted that the version of the model examined in this paper has limitations in addition to those mentioned above. The model does not take into account some of the physiological behaviors of the system observed experimentally. These include length-dependent active tension (1, 4, 30), irregular contractions (29), transmural-pressure-dependent activation, and shear-stress-dependent active tension (5, 11). We have also assumed that the fluid is homogeneous, with no cellular content, and Newtonian.

In summary, our results suggest that further experimental measurements are required to refine our knowledge of open valve resistance and of vessel geometry (especially valve location), as well as of contraction frequency and magnitude. In particular, work involving other versions of the model (1, 4) strongly indicates a need for further investigation of the relation between active tension and circumferential length in lymphatics, especially at low length where the vessel may depart from a circular cross-section but active tension can apparently still be exerted (30). It is also crucial to know how these parameters vary in lymphatic vessels from different anatomic locations.

GRANTS

We gratefully acknowledge the support of National Heart, Lung, and Blood Institute Grant R01-HL-094269. J. E. Moore, Jr., also acknowledges support from the Royal Academy of Engineering and a Royal Society-Wolfson Research Merit Award.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

AUTHOR CONTRIBUTIONS

S.J., C.D.B., and W.J.R. performed experiments; S.J., C.D.B., and W.J.R. analyzed data; S.J. and C.D.B. interpreted results of experiments; S.J. and C.D.B. prepared figures; S.J. drafted manuscript; S.J., C.D.B., and J.E.M. edited and revised manuscript; C.D.B. and J.E.M. approved final version of manuscript; J.E.M. conception and design of research.

Glossary

D_di

Diameter constant in vessel wall force balance relation

f

Contraction frequency

i

Suffix identifying a lymphangion in the chain

L

Lymphangion length

L_s

Overall chain length

M

Active tension

n

Number of lymphangions in the chain

p_a

Pressure at the start of the chain

p_b

Pressure at the end of the chain

p_e

External pressure

p₁

Pressure at the lymphangion inlet

p_m

Mid-lymphangion pressure

p₂

Pressure at the lymphangion outlet

P_di

Pressure constant in vessel wall force balance relation

Q̄

Average flow rate of the last lymphangion in the chain

R_Vn

Minimum valve resistance

R_Vx

Maximum valve resistance

s_f

Valve failure slope

s_o

Valve opening slope

t_0i

Delay in starting contractions of lymphangion i

t_r

Refractory period

Δp

Pressure difference across valve i

Δp_f

Valve failure pressure difference

Δp_o

Valve opening pressure difference

ΔP

Pressure difference between the two ends of the chain

μ

Fluid viscosity

Appendix:

EQUATIONS OF THE MODEL

See Glossary for the definition of each algebraic variable. Conservation of mass:

$${\mathit{\text{Q}}}_{\mathit{\text{i}}+1}={\mathit{\text{Q}}}_i-\frac{\pi {\mathit{\text{D}}}_{\mathit{\text{i}}}{\text{L}}_{\mathit{\text{i}}}}{\text{2}}\frac{\mathrm{d}{\mathit{\text{D}}}_{\mathit{\text{i}}}}{\mathrm{d}\mathit{t}}.$$

Conservation of momentum:

$${\text{p}}_{\mathit{\text{i}}-1,2}-{\text{p}}_{\mathit{\text{i}}1}={\mathit{R}}_{\mathrm{V}\mathit{i}}{\mathit{\text{Q}}}_{\mathit{\text{i}}}\text{;}\hspace{0.17em}\frac{{\text{p}}_{\mathit{\text{i}}1}-{\text{p}}_{\mathit{\text{im}}}}{{\mathit{\text{L}}}_{\mathit{\text{i}}}}=\frac{64\mu {\mathit{\text{Q}}}_{\mathit{\text{i}}}}{\pi {\mathit{\text{D}}}_{\mathit{\text{i}}}^4};$$

$$\frac{{\text{p}}_{\mathit{\text{im}}}-{\text{p}}_{\mathit{\text{i}}\text{2}}}{{\mathit{\text{L}}}_{\mathit{\text{i}}}}=\frac{64\mu {\mathit{\text{Q}}}_{\text{i}+1}}{\pi {\mathit{\text{D}}}_{\mathit{\text{i}}}^{\text{4}}}\text{;}\hspace{0.17em}{\text{p}}_{\mathit{\text{i}}2}-{\text{p}}_{\mathit{\text{i}}\text{+1,1}}={\mathit{\text{R}}}_{\text{V}\mathit{\text{i}}\text{+1}}{\mathit{\text{Q}}}_{\mathit{\text{i}}\text{+1,}}$$

$$\text{where}\hspace{0.17em}{\mathit{R}}_{\text{V}\mathit{i}}={\mathit{\text{R}}}_{\text{V}\mathit{i}}\left({\text{p}}_{\mathit{\text{i}}-1,2}-{\text{p}}_{\mathit{\text{i}}1}\right)\hspace{0.17em}\text{and}\hspace{0.17em}{\mathit{\text{R}}}_{\text{V}\mathit{\text{i}}+1}={\mathit{\text{R}}}_{\text{V}\mathit{\text{i}}+1}\left({\text{p}}_{\mathit{\text{i}}2}-{\text{p}}_{\mathit{\text{i}}+1,1}\right).$$

Vessel wall force balance:

$$\Delta {\text{p}}_{\text{t}\text{m}}={\text{p}}_{\mathit{\text{i}}\text{m}}-{\text{p}}_{\mathrm{\text{e}}}=\frac{\mathit{\text{M}}\left(1-\cos \left(2\pi \mathit{\text{f}}\left(\mathit{\text{t}}-{\mathit{\text{t}}}_{0\mathit{\text{i}}}\right)\right)\right)}{{\mathit{\text{D}}}_{\mathit{\text{i}}}}+{\text{P}}_{\text{d}\mathit{\text{i}}}\left({\mathit{\text{e}}}^{\frac{{\mathit{\text{D}}}_{\mathit{\text{i}}}}{{\mathit{\text{D}}}_{\text{d}\mathit{\text{i}}}}}-{\left(\frac{{\mathit{\text{D}}}_{\text{d}\mathit{\text{i}}}}{{\mathit{\text{D}}}_{\mathit{\text{i}}}}\right)}^3\right)\hspace{0.17em}\text{when}\hspace{0.17em}n=4;$$

$$\Delta {\text{p}}_{\text{tm}}={\text{p}}_{\mathit{\text{i}}\text{m}}-{\text{p}}_{\mathrm{\text{e}}}=\frac{\mathit{\text{M}}\left(1-\cos \left(2\pi \mathit{\text{f}}\left(\mathit{\text{t}}-{\mathit{\text{t}}}_{\mathit{\text{ci}}}\right)\right)\right)}{{\mathit{\text{D}}}_{\mathit{\text{i}}}}+\frac{4}{15}{\text{P}}_{\text{d}\mathit{\text{i}}}\left({\text{12}\mathit{\text{e}}}^{\frac{{\mathit{\text{D}}}_{\mathit{\text{i}}}}{{\mathit{\text{D}}}_{\text{d}\mathit{\text{i}}}}-1}-11-{\left(\frac{{\mathit{\text{D}}}_{\text{d}\mathit{\text{i}}}}{{\mathit{\text{D}}}_{\mathit{\text{i}}}}\right)}^3\right)\hspace{0.17em}\text{when}\hspace{0.17em}n=8;$$

where t_c ≤ t ≤ t_c + 1/f and t_c defines the start of a contraction; the following contraction begins at t_c + 1/f + t_r.

MatLab's ode15s was used to solve the system of equations.