Introduction

The balance of fluid and proteins in the body is maintained by transport of lymph from low or negative pressure tissue spaces through the lymph nodes and on to the subclavian vein near the heart (central venous pressure), against gravity (when standing upright) and without a central pump. This feat is accomplished mainly by contraction of lymphatic muscle cells in the walls of collecting lymphatic vessels. The forward propulsion of lymph is achieved via numerous unidirectional intraluminal valves throughout the system that segment the vessels into smaller contracting units (lymphangions) [1].

Lymph formation in the tissues and consistent flow throughout the lymphatic network is required to maintain tissue homeostasis and prevent edema [2]. Thus, it is important to identify the parameters that play major roles in efficient lymphatic pumping. Experimental studies on isolated lymphatic vessels have long aimed to determine the functional importance of such parameters at the lymphangion and tissue level. Since Reddy’s studies in 1970s, computational models have complemented the experimental evidence and provided additional insights where experimental measurements could not be performed [3, 4].

For lymphangions in a sequential series, the overall function and efficiency of lymph pumping is affected by coordination of the spontaneous, active contraction waves [5, 6]. Experimental observations of rat mesenteric lymphatics in situ indicate that contractions of adjacent lymphangions are coordinated along the vessels >80% of the time (i.e., same frequency with small time delay between contractions) with a slight preference for an orthograde propagation direction [5]. Mechanical blockade of the lymph bolus into the subsequent downstream lymphangion did not significantly alter the propagation characteristics, whereas a gap junction blocker significantly decreased coordination and propagation [5]. In vivo, both orthograde (in the flow direction of the valves) and retrograde (in the reverse direction) propagation along the length of the vessel have been observed in collecting mesenteric lymphatic vessels from bovine [7], rat [5] and guinea-pig [8]. Zawieja et al. reported similarly propagating contraction phenomena for branching vessels in situ [5]. Crow et al. found a greater likelihood of the upstream lymphangion constricting before the downstream lymphangion in guinea pigs mesentery (with a time delay about 0.5 s) [8]. They suggested that transmission of both mechanical and electrical signals between adjacent lymphangions could contribute to the coordination of contractile activity. Davis et al. observed both synchronous and conducted contractions with about 0.7 s time delay in isolated vessels from rat thoracic duct. In those preparations, the axial pressure difference modulated the time delay between contractions [9]. All of these studies emphasized the importance of quantifying the effect of propagation and coordination of spontaneous lymphatic contractions along vessel networks on lymphatic pumping. McHale also hypothesized that this coordination should be much more complex in a highly branched lymphatic network structure [7]. Previous computational models of multiple lymphangions in series have reported minimal change in average flow rate between forward and reverse propagating waves of contraction [10]. One study reported a small advantage in peak flow rate via the reverse propagation of the contraction wave [11]. In a lumped parameter model of lymphangions in series, Bertram et al. found that the time delay between contractions was more advantageous than the synchronization of contractions for pumping [6]. More recently, they reported that under adverse pressure differences lymphatic valve function could also be affected by the timing of contractions [12].

In addition to the effects of lymphangion contraction, external loads such as pressure and tissue tethering can modulate lymphatic pumping by influencing the effective transmural pressure. Mislin first described the stimulation of the lymph pump by pressure [1]. McHale et al. observed an increase in flow output of cannulated bovine mesenteric lymphatic vessels following a controlled rise in transmural pressure [13]. Gashev et al. reported that lymphatic vessels are very sensitive to transmural pressure and that most effective pumping is achieved under different levels of transmural pressures in lymphatics from different tissues; for example, optimum transmural pressure (resulting in maximum flow) for rat thoracic duct is ~1–3 cmH₂O while that in the mesenteric lymphatics is ~5 cmH₂O [14]. Eisenhoffer et al. also observed that cannulated bovine mesenteric lymphatic vessels exhibited optimal pumping over a certain range of transmural pressures. Additionally they observed that upon increase of outflow pressure, flow output decreases nonlinearly with pressure. They reported a lower variation in lymph flow in response to elevation of outflow pressure at lower transmural pressures [15]. We previously performed a parameter sensitivity study for a lumped parameter model of lymphangions in series that similarly demonstrated the existence of an optimum transmural pressure that maximized the flow rate as well as a non-linear pump function curve describing the flow rate drop as outflow pressure increased [16].

Due to the complexity of experimental measurements with the isolation and cannulation of lymphatic vessels, experimental data are limited in several functional aspects of the lymphatic system. Rigorous control over many of the parameters such as the timing of contractions, diastolic period, external pressure, inlet/outlet pressures (in situ), and network size is not possible experimentally. Additionally most of the experiments are performed for chains of lymphangions in series, while from anatomic observations we know that lymphatic networks are highly branched [17, 18]. Reddy 1974 was the first to attempt and develop a network model of post-nodal lymphatic vessels. The model however, did not include several key features of collecting lymphatic vessels such as highly non-linear pressure diameter relationship, realistic valve behavior, and phasic contractions. Here we have expanded our lumped parameter model of lymphangions in series [16] to a bifurcating network structure of collecting lymphatic vessels where each vessel contains multiple lymphangions. We have studied the collective network behavior (in terms of mean flow rate ) in response to variations in pressure difference (via outlet pressure p_b), the size of the network (via number of lymphangions per vessel n_v), transmural pressure (via p_e), timing of contractions via variation of diastolic period for an individual lymphangion (t_r), time delay between contraction of adjacent lymphangions (Δt_v), and time delay at the junctions (Δt_g).

Our results offer several novel predictions as to the function and pumping regulation of branched lymphatic networks and provide further insight into translation of experimental data from single lymphangions to the network scale.

Materials and Methods

Network structure

We have developed a computational model for fluid transport in branching networks of lymphatic vessels in which each vessel is itself composed of a number of lymphangions n_v (Fig 1A). Individual lymphangions (along with their adjacent valves) are modelled using a lumped parameter approach, following the algorithm presented in Fig 1B and with more detail than in our previous studies [6, 16]. Briefly, equations of conservation of mass, conservation of momentum, and a force balance on the vessel wall are solved for each lymphangion, resulting in a nonlinear ordinary differential equation for diameter (D). The system of equations is then solved computationally, using the Matlab subroutine ode15s. The force balance on the vessel wall accounts for the force generated by active contraction of lymphatic muscle cells embedded in the wall of the lymphatic vessel as well as the passive behavior of the vessel. The tension generated by lymphatic muscle cells (M) is dependent on muscle length; therefore M varies with diameter of the vessel via a sigmoidal, diameter-dependent function (M_d(D)). Muscle contraction and muscle relaxation are modelled using a time dependent sinusoidal function (M_t(t)). Subsequent contractions of each lymphangion are separated by a period of relaxation (referred to as the diastolic period, duration t_r). Time delays between contractions of adjacent lymphangions in one vessel (Δt_v) and in two subsequent generations (Δt_g) are implemented as well (Fig 2). The passive behavior of the vessel is dictated by the pressure diameter relationship obtained experimentally from isolated rat mesenteric lymphatic vessels [19]. Valves are modeled using a sigmoidal function with minimum resistance to forward flow which transitions to maximum resistance at a slightly negative pressure difference (Δp_o) to prevent backflow. This valve behavior was prescribed based on the experimental observations indicating that lymphatic valves are biased in the open position and there is a small amount of backflow before completely closing [20]. This is implemented via the small negative pressure difference Δp_o required to close the valve (Δp_o remains constant in all situations, whereas other studies have explored manipulations of this parameter and its effect on pumping of small chains of lymphangions [12, 19, 21]). The baseline values of the parameters of the model are presented in Table 1. The equations of the model are as follows:

1. Conservation of mass:
2. Conservation of momentum: where and . With , and .
3. Pressure at the junctions (e.g., junction in Fig 1A): Thus .
4. Vessel wall force balance: With and . 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. with .

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Fig 1. Network structure.

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

https://doi.org/10.1371/journal.pone.0148384.g001

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Fig 2. Timing of contractions.

M_t (t) vs. time for branching network with four lymphangions per vessel (n_v = 4). Bottom panel: vessels 0301–0304. Mid panel: vessels 0201 and 0202. Top panel: vessel 0101. Δt_v is the time delay between contractions of adjacent lymphangions in one vessel; Δt_g is the time delay between two subsequent generations (at the junction). Specified time course for the contraction of an individual lymphangion (normalized M_t (t) vs. time) is presented.

https://doi.org/10.1371/journal.pone.0148384.g002

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Table 1. Parameters used in the numerical model, including their definition and baseline value.

https://doi.org/10.1371/journal.pone.0148384.t001

To construct the computational model of a tree-like network, individual lymphangions were added first in series then arranged to create branching network structures. Vessel junctions in the network were modeled by equating the pressures and satisfying conservation of mass. The model predictions for a network with three generations of bifurcated vessels (Fig 1) are presented here. To facilitate a systematic analysis, model parameters describing individual lymphangions (valve and non-valve parameters, Table 1) and the number of lymphangions per vessel (n_v) were equal throughout the network (unless stated otherwise) and remained constant during each simulation except for the parameters studied (further details in the next section). The time averaged flow rate of the outlet lymphangion () in the network was chosen to represent the system output.

Parameter variations

The impact of network size on its pumping capability was investigated by generating pump function curves for bifurcated networks with 1 to 14 lymphangions per vessel (n_v = 1–14). In these studies lymphangion length (L) remained constant; as a result variation of n_v contributes to longer vessels. All the other network geometry parameters remained constant at their baseline values. Based on these pump function curves we chose four lymphangions per vessel (n_v = 4) as our baseline network size. We investigated the flow rate response to manipulations in p_out and transmural pressure (Δp_tm), via variation of external pressure (p_e), for three network sizes of 2, 4, and 8 lymphangions per vessel.

Pumping coordination within the network is determined by 1) the diastolic period and 2) time delay between contractions of adjacent lymphangions. In the branching network structure we recognize two categories of time delays: the time delay between contraction of adjacent lymphangions in one vessel (Δt_v) and between two subsequent generations (Δt_g: the time delay at the junction) (Fig 2).

Diastolic periods on the order of 9 s have been observed in collecting rat mesenteric lymphatic vessels [22, 23]. Here we obtained pump function curves for bifurcating networks with n_v = 4 when t_r increased by steps of 0.5 s from 0 to 3.5 s. Additionally, we determined the relative importance of time delay within vessels and time delay at the junction for effective pumping under varying levels of outlet pressure. In studying lymphatic branch points, Zawieja et al. observed little difference in the propagation of contraction waves between lymphangions in a parent vessel (primary vessel segment) vs. at the junctions, suggesting that under physiologic conditions Δt_g = Δt_v is a valid assumption [5]. Based on this knowledge, we investigated the impact of the time delay between contractions when Δt_v = Δt_g under different external pressures and pressure differences. Δt_v = Δt_g was incrementally increased from 0 to t_c + t_r. Time delays smaller than half of total cycle time (Δt_g = Δt_v < ½(t_c + t_r) = 1.75 s) represent the forward propagating contraction wave and larger time delays are equivalent to the reverse propagating waves (Fig 2). For reverse travelling waves the equivalent time delay in the reverse direction can be calculated as t_c + t_r–(Δt_g = Δt_v). For example, Δt_g = Δt_v = 3.25 s is equivalent of 0.25 s time delay in the reverse direction, when total cycle time is 3.5 s. The simulation conditions for all the cases studied here are summarized in Table 2; the parameters not listed in this table take their baseline values (see Table 1).

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Table 2. Simulation conditions for all the cases studied here.

https://doi.org/10.1371/journal.pone.0148384.t002

Results

Number of lymphangions per vessel

Pump function curves of networks with one to fourteen lymphangions per vessel showed that networks with smaller n_v were able to generate greater flow under favorable pressure differences. Specifically, at ΔP = –3 cmH₂O (ΔP < 0 represents a favorable axial pressure difference), the network with a single lymphangion in each vessel had the highest flow rate of 3.29 ml/hr (this curve is cut off in the figure at ΔP = –1.5 cmH₂O and flow rate of 2.3 ml/hr to allow for better representation of the other curves.), whereas the lowest flow rate of 1.40 ml/hr occurred for the network with n_v = 14. Under adverse pressure differences up to 10 cmH₂O, n_v = 10 was optimal and resulted in flow rates of 1.37 ml/hr at 0 pressure difference and 0.82 ml/hr at ΔP = 10 cmH₂O. At ΔP = 12 and 14 cmH₂O peak flow rates of 0.58 and 0.33 ml/hr were achieved with 6 and 8 lymphangions per vessel, respectively, but the advantage over n_v = 10 was less than 2%. Using n_v = 10 produced the highest flow rate (0.21 ml/hr) at ΔP = 16 cmH₂O (Fig 3A).

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Fig 3. Effect of number of number of lymphangions (n_v).

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

https://doi.org/10.1371/journal.pone.0148384.g003

Higher flow rates were achieved with more lymphangions under larger adverse pressure differences (ΔP >16 cmH₂O). For example at ΔP = 18 cmH₂O, the network with a single lymphangion in each vessel produced only = 0.012 ml/hr, yet the network with n_v = 14 was able to produce 12× more flow. Flow waveforms for networks of 2, 4, and 8 lymphangions per vessel revealed that the more efficient pumping under high adverse pressure differences in networks with larger n_v results from the large number of pumping elements that build up the pressures required for the valves to remain open for a longer period in the face of an adverse pressure difference (Fig 3B).

Effect of external pressure

The flow rate was maximized at certain values of transmural pressure, ranging from 2 to 4 cmH₂O, depending on ΔP (Fig 4). At lower values of transmural pressure (corresponding to high external pressure in this figure), productive pumping was essentially eliminated. Transmural pressures higher than this range (low external pressure) minimally affected the flow output. These findings correspond well with previous experimental observations on the heterogeneity of lymphatic pumps from different regions of the body [14]. Peak flow rate occurred at higher transmural pressures as pressure load increased. At ΔP = 3 cmH₂O, flow rate was maximized at p_e ≈ 2 cmH₂O. External pressures resulting in peak flow rate ranged between 2.2 and 2.6 cmH₂O at ΔP = 10 cmH₂O for all network sizes tested. At ΔP = 16 cmH₂O, maximum occurred when p_e ≈ 2.8 cmH₂O for n_v = 2, and p_e ≈ 3.6 cmH₂O for networks with four and eight lymphangions per vessel. For n_v = 2, peak flow rates of 0.90, 0.51, and 0.11 ml/hr were achieved at ΔP = 3, 10, and 16 cmH₂O, respectively. Peak increased to 0.14, 0.74, 0.21 ml/hr for n_v = 4, and 1.25, 0.84, and 0.26 ml/hr for n_v = 8. In these studies the inlet pressures were always set at 6 cmH₂O, thus optimum pumping was achieved under external pressures lower than the inlet pressure (p_e<p_a).

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Fig 4. Effect of transmural pressure (via p_e). vs. external pressure at ΔP = 3, 10, 16 cmH₂O for branching networks with n_v = 2 (dot-dashed), n_v = 4 (solid), and n_v = 8 (dotted).

The top axis indicates transmural pressure. Peak flow rate occurred at higher values of p_eas pressure difference increased. Inlet pressure p_a was constant and equal to 6 cmH₂O.

https://doi.org/10.1371/journal.pone.0148384.g004

Pumping coordination in branching networks

Effect of diastolic period.

Pump function curves of networks with four lymphangions per vessel demonstrated that the addition of a diastolic period (t_r) between contractions is advantageous for pumping. Under favorable pressure differences, we observed the highest flow rate of 1.49 ml/hr with the longest t_r (= 3.5 s here), whereas maximum pumping output was achieved with t_r = 1.5 s under adverse pressure differences. When ΔP > 10 cmH₂O, little advantage was gained with variation of t_r (Fig 5A). Simultaneous variation of n_v and t_r revealed that longer diastolic periods were required to maximize in larger networks. At ΔP = 3 cmH₂O, maximal flow rate occurred with longer diastolic periods as the number of lymphangions increased (with higher flow rates at larger n_v). The changing pattern observed here is the result of interplay between variation in diastolic period and time delay between contractions of adjacent lymphangions (Δt_v = Δt_g = 0.5 s). When Δt_v = Δt_g = 0 s (n_v = 4), flow rate peaked at t_r = 0.5 s and continuously dropped for longer diastolic periods. At this pressure, peak flow rate achieved with synchronous pumping yielded values close to the maximum outflow of the network with n_v = 4 and t_c = 1.25 s. In the branching network structures with four lymphangions per vessel, simultaneous variation of t_r and period of contraction (t_c = 1/f) at ΔP = 3cmH₂O showed that faster contractions (shorter period of contraction, higher frequency) reduces the diastolic period required to achieve peak . Peak flow rates were observed when diastolic period was 1.8, 1.5, and 1.2 s, for contraction periods of 2.5, 2.0, and 1.25 s, respectively. Peak flow rate was higher when vessel contracted at a higher frequency (Fig 5B).

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Fig 5. Effect of diastolic period (t_r).

(A) Pump function curves for a branching network with n_v = 4 as t_r increased from 0 to 3.5 s. Diastolic period was advantageous to pumping, particularly at low ΔP, because relaxed vessels present less impedance to incoming flow. (B) Combined effect of diastolic period, contraction period (t_c), and number of lymphangions per vessel (n_v) at ΔP = 3 cmH₂O.

https://doi.org/10.1371/journal.pone.0148384.g005

Effect of time delay between contractions of adjacent lymphangions.

Pump function curves of branching networks with four lymphangions per vessel showed that the presence of a time delay at the junctions (Δt_v = 0, Δt_g = 0.5 s) is advantageous over synchronous pumping of all lymphangions (Δt_v = 0, Δt_g = 0). For adverse pressure differences above 10.5 cmH₂O, additional time delay between lymphangions in one vessel (Δt_v = 0.5, Δt_g = 0.5 s) improved the pumping. For ΔP < 5.5 cmH₂O, synchronous contractions were advantageous over the baseline case (Δt_v = 0.5, Δt_g = 0.5 s) (Fig 6A, solid lines). Under lower transmural pressure (achieved by elevating p_e from 2 to 3 cmH₂O), however, synchronous contractions resulted in a less efficient pump (lower , except for ΔP = –3 cmH₂O), but addition of a time delay at the junctions (Δt_v = 0, Δt_g = 0.5 s) restored pumping. For ΔP > 6.6 cmH₂O the highest flow rates were achieved by additional time delay between contraction of lymphangions in one vessel (Δt_v = 0.5, Δt_g = 0.5 s) (Fig 6A, dashed lines).

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Fig 6. Effect of time delay between contractions of adjacent lymphangions in branching networks with n_v = 4.

(A) Pump function curves for synchronized pumping (Δt_g = Δt_v = 0 s), time delay only at the junctions (Δt_v = 0, Δt_g = 0.5 s), and the baseline case (Δt_g = Δt_v = 0.5 s) at p_a − p_e = 4 cmH₂O (solid lines) and p_a − p_e = 3 cmH₂O (dashed). (B) vs. Δt_v (Δt_g = Δt_v): effect of transmural pressure via (variation of p_e) at ΔP = 3 cmH₂O. Solid line: forward propagating contraction wave, dashed line: reverse propagating contraction wave.

https://doi.org/10.1371/journal.pone.0148384.g006

Lowering of transmural pressure (in the form of p_a–p_e) resulted in less pronounced peaks and valleys and a more uniform response to an incremental increase of Δt_g = Δt_v (Fig 6B). For a bifurcating network with n_v = 4 pumping against ΔP = 3 cmH₂O, maximum occurred when contractions were 0.15 s apart for p_a−p_e = 4 and 5 cmH₂O ( = 1.47 and 1.58 ml/hr, respectively); but the flow rate response was not monotonic and peaked again at Δt_g = Δt_v ≈ 1.6 s ( = 1.08 and 1.30 ml/hr). Further compressing the vessel with p_a−p_e = 3 cmH₂O produced a peak flow rate of 0.94 ml/hr at Δt_g = Δt_v = 0.3 s, with additional peaks at Δt_g = Δt_v = 1.4 and 3.4 s (= 0.1 s retrograde). Longer time delays (Δt_g = Δt_v = 0.45 s) were required to achieve peak (0.35 ml/hr) for the highest external pressure tested (p_a−p_e = 2 cmH₂O, vessels are not yet collapsed), a second lower peak was also present at Δt_g = Δt_v = 3.2 s (= 0.3 s retrograde) (Fig 6B). Forward propagation of contraction waves consistently produced higher flow rates. The reverse propagating wave became advantageous only in a narrow dynamic range of Δt_g = Δt_v < 0.16 s when p_a−p_e = 3 cmH₂O and Δt_g = Δt_v < 0.21 s for p_a−p_e = 2 cmH₂O. The rise in flow rate was small with a maximum increase of 0.085 ml/hr (at Δt_g = Δt_v = 0.05 s) and 0.04 ml/hr (at Δt_g = Δt_v = 0.1 s) under p_a–p_e = 3 and 2 cmH₂O, respectively.

For branching networks with n_v = 4, pumping capability (in terms of ) increased as Δt_g = Δt_v increased from 0. Δt_g = Δt_v = 0.15 s was optimal for pumping at ΔP < 8 cmH₂O (curve not shown); for higher pressure differences Δt_g = Δt_v = 0.25 s maximized (curve not shown). Flow rate dropped with further time delay between contractions (larger Δt_g = Δt_v) and the curvature of the pump function curves was altered. Δt_g = Δt_v = 2 s resulted in lowest flow rates under favorable pressure differences, but produced higher for ΔP > 12 cmH₂O compared to all larger values of Δt_g = Δt_v. Finally, at Δt_g = Δt_v = 3.5 s the pump function curve reverted to that with Δt_g = Δt_v = 0 s (total cycle time: t_c + t_r = 3.5 s) (Fig 7A).

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Fig 7. Effect of time delay between contractions of adjacent lymphangions in branching networks with n_v = 4 when Δt_g = Δt_v.

(A) Pump function curves as Δt_v increased from 0 to 3.5 by steps of 1.0 s (Increasing p_b). (B) vs. Δt_v: effect of pressure difference across the network. Flow rate response as Δt_v incrementally increased from 0 to 3.5 s. For reverse travelling waves (dashed line) equivalent time delay in the reverse direction is presented as 3.5 –(Δt_g = Δt_v).

https://doi.org/10.1371/journal.pone.0148384.g007

Incremental increases of Δt_g = Δt_v from 0 to 3.5 s showed that under pressure differences of −3, 0, and 6 cmH₂O, Δt_g = Δt_v = 0.15 s was most advantageous for pumping, resulting in peak flow rates of 2.01, 1.77, and 1.20 ml/hr, respectively. Additional peaks were also observed at Δt_g = Δt_v = 0.8, 1.6, 2.5 s (= 1 s retrograde) (ΔP = −3 and 0 cmH₂O), and 3.4 s (= 0.1 s retrograde) (ΔP = −3 cmH₂O). These regional maxima never exceeded the initial peak observed at a small non-zero Δt_g = Δt_v. For ΔP = 12 and 18 cmH₂O, Δt_g = Δt_v = 0.3 s resulted in peak flow rates of 0.62 and 0.11 ml/hr, respectively. The response was more uniform for larger adverse pressure differences of 18 and 24 cmH₂O. The system was unable to pump against pressure differences of 18 and 24 cmH₂O (Fig 7B). When the number of lymphangions n_v doubled from 4 to 8, still peaked for Δt_g = Δt_v = 0.15 s, but for n_v = 2 peak flow rate occurred with longer time delays of Δt_g = Δt_v = 0.25 s (figure not shown).

Discussion

We have constructed a network scale model of contracting collecting lymphatic vessels in a symmetric branching pattern in order to understand how the characteristics of these networks determine their pumping abilities. Each vessel is itself divided into multiple lymphangions separated by lymphatic valves. This study is the first to analyze a branching network of lymphatic vessels since Reddy in the 1970s [24], and benefits from a greater understanding of lymphatic pumping, mainly from experimental work, in the intervening years. Given the highly variable patterns of coordinated lymphangion contractions in vivo, and our inability to modulate their coordination experimentally, a modeling approach is currently the best means to assess the effects of contraction coordination on pumping dynamics. Our computational approach incorporates some of the most recent measurements of parameter values and essential functional mechanisms (e.g., passive pressure diameter relationship) of these vessels, and enables us to control many of the parameters of the system that cannot be controlled experimentally. One of the main areas of focus in this study was the effect of pumping coordination in tree-like network structures on the regulation of lymph flow. This was investigated by means of a diastolic period (i.e., time delay between subsequent contractions of one lymphangion) and a time delay between contractions of adjacent lymphangions (or the conduction delay; in one vessel Δt_v and at the junctions Δt_g). Additionally, we quantified the pumping capability of the system under increasing levels of steady transmural pressure and outflow pressure for different network sizes.

Spontaneous contractions are highly coordinated in isolated lymphatic vessels with small time delays of approximately 0–0.5 s between contractions of two adjacent lymphangions [8]. Although lymphatic vessels form highly branched networks, the complexities of ex vivo and in situ experiments pose a limit on functional studies of such structures. Our computational model of tree-like networks builds on existing data from isolated vessel experiments to explore the effects of contraction coordination on network scale behavior. We demonstrated that system output was more sensitive to the time delay at the junction Δt_g compared to time delay between lymphangions in one vessel Δt_v. When time delay at the junction was in place (Δt_g ≠ 0), non-zero values of Δt_v only provided a small additional advantage under high adverse pressure differences. Thus, a small time delay only at the junctions (Δt_g ≠ 0, Δt_v = 0) was sufficient to maximize flow rate under a wide range of pressure differences. When transmural pressure was lowered (via higher p_e) flow rate dropped but non-zero values of Δt_v became advantageous at a lower pressure difference. At this lower transmural pressure, the presence of any form of time delay resulted in higher flow rates compared to synchronous pumping (Δt_g = 0, Δt_v = 0), except at the highest favorable pressure difference tested. This result could imply that under favorable pressure differences more synchrony between contractions is beneficial for lymph transport. In agreement with this finding, Davis et al. reported a 24% drop in time delay between contractions of lymphangions in rat thoracic duct when the favorable pressure difference increased from 0 to 3 cmH₂O [9]. Thus, the effect of time delay between contractions of adjacent lymphangions on pumping efficiency is highly influenced by transmural pressure and the axial pressure difference. The advantage gained by imposing a time delay between contractions of adjacent lymphangions agrees generally with our previous studies of contracting lymphangions in series [6]. Our results further demonstrated that the change in flow rate in response to variations in outlet pressure decreased at lower transmural pressures; similar to the observations of Eisenhoffer et al. in isolated bovine mesenteric lymphatic vessels [15].

Experimental measurements of lymphangions in series ex vivo and in situ as well as branching segments (in situ) have confirmed the presence of both forward and reverse propagating waves of contraction from one lymphangion to another, with forward propagation appearing to be slightly more common. In the results presented here, time delays smaller than half of total cycle time (Δt_g = Δt_v < ½(t_c + t_r)) represent forward propagating waves and larger time delays are equivalent to the reverse propagating contraction waves. For reverse travelling waves the equivalent time delay in the reverse direction can be calculated as t_c + t_r–(Δt_g = Δt_v). Overall, higher flow rates were observed with a forward propagating contraction wave relative to its equivalent reverse propagating wave. The unidirectional nature of the valves favors fluid movement in the direction of forward flow but it is not known if this directional advantage is necessarily reflected in the conduction of the electrical signal that coordinates the contractions. This is in agreement with previous modeling efforts that found forward propagating contraction waves advantageous for efficient pumping in series of lymphangions [10–12]. When all the other parameters took the baseline values, the combination of Δt_g = Δt_v = 0.15 s maximized pumping. Under more challenging pumping conditions (high adverse pressure difference, high external pressure) the system was less sensitive to variations in time delay. While pumping can be maximized by imposing an optimum timing between contractions of adjacent lymphangions under moderate pressure loads, these mechanisms lose their effectiveness in the presence of challenging pressure conditions.

Another determinant of pumping coordination is the diastolic period, which is the time delay between subsequent contractions of an individual lymphangion. In general, longer diastolic periods yielded higher flow rates under favorable and small adverse pressure differences. This is in agreement with recent observations in a model of lymphangions in series with transmural pressure-dependent valve behavior [12]. Additionally, we found that the lymphatic network performed less efficiently when the system was forced to pump against high adverse ΔP. This can be attributed to the fact that lymphatic contractions are less crucial under favorable pressure differences when the pressure head drives the fluid, and actually serve to increase impedance to incoming flow. The baseline value of t_r = 1.5 s resulted in the highest flow rate over a wide range of pressure differences. At a moderate pressure difference ΔP of 3 cmH₂O, peak flow rate was achieved with longer diastolic periods as the number of lymphangions increased. Peak flow rate occurred with shorter diastolic periods as contraction frequency increased, suggesting that the ratio of diastolic period and contraction period is an influential factor. Importantly, the advantage gained (in terms of ) by doubling the contraction frequency was 4.7 times greater than that achieved by doubling the number of lymphangions, which shows the significance of timing of contractions for efficient pumping.

Transmural pressure is an important classical modulator of lymph transport [13, 25]. In these studies we identified a certain range of transmural pressures that maximized flow under different pressure differences and network sizes. The possibility of an optimum transmural pressure (associated with peak flow) has been suggested in previous functional studies of lymphangions in series in both experiments and modeling [14, 16, 25]. A bound on the functional range of transmural pressure is set by the fact that vessels collapse when the external pressure is higher, i.e., effective pumping requires positive transmural pressure. Then, as ΔP increases, flow rate decreases, but can recover to some degree with an increase in transmural pressure. In this study, flow rate maximized at lower transmural pressures when the number of lymphangions was increased. We previously showed that most effective pumping (maximum ) is achieved under transmural pressures associated with the most compliant state of the vessel on the pressure diameter curve, as it is in this range that contractions are most effective in generating stroke volume [16]. In this study, the tension generated by lymphatic muscle cells varies with diameter of the vessel and is a contributing factor as well. Thus, the most efficient pumping is achieved when there is an optimum balance of vessel compliance and contractile tension throughout the entire network. During lymphedema, accumulation of fluid in the tissue may result in higher external pressures that can collapse collecting lymphatics and inhibit effective pumping. Application of compression garments as a treatment method for this condition in some cases mitigates further limb swelling, but our results consistently indicate that application of additional external pressure actually impedes pumping in collecting vessels.

The pressure difference that the lymph pump has to overcome is another important hydrodynamic factor that regulates lymph flow. We observed that larger lymphatic networks demonstrate less variation in under increasing adverse pressure difference. On the other hand, larger networks perform less efficiently under favorable pressure differences. The disadvantage of larger networks stems from the additional resistances presented by longer vessels and more valves. Thus, at each level of pressure difference, it is the balance between the additional pumping power vs. increased resistance that determines whether higher flow rates can be achieved. Under adverse pressure differences, increasing the number of lymphangions generally improved pumping but a smaller advantage was gained for n_v > 4. The large number of pumping elements helped maintain the pressures that enabled the valves to remain open longer in the face of an adverse pressure difference, thus producing greater forward flow. For the pressure differences 10–15 cmH₂O the advantage from additional pumping power is eliminated by higher resistances, which is why the curves are very close for n_v = 4–14.

There are only very few reports of experimental assessment of lymphatic pump curves, in part because the experiments are extremely difficult and subject to considerable uncertainties. Detailed comparison of modeling predications and experimental measurements is not feasible because it is impossible to measure all of the necessary pressure and flow data simultaneously to any functional degree of precision. In the few reports that do exist, similar nonlinear pump function curves have been observed experimentally [15]. The nonlinear shapes of the curves predicted by our model are the result of the complex, non-linear equations that govern the behavior of the system (e.g., pressure-diameter relationship, diameter dependent tension, valve behavior). Because many of the system parameters vary with changes in pressure and diameter, it is expected that the curves considerably change and even cross over one another.

The active and passive components of lymph transport as well as multiple regulatory mechanisms cooperate to sustain tissue fluid homeostasis. In the current computational model we described the passive behavior of the vessel using a pressure-diameter relationship obtained from isolated lymphatic vessels from rat mesentery. The tension generated during active contractions was dependent on the diameter of the vessel, to represent the physiological behavior of muscle cells. In the lymphatic system, several autoregulatory mechanisms such as stretch-dependent or shear-dependent responses modulate lymph flow by altering lymphatic muscle cell behavior (frequency and magnitude of contraction). Such regulatory mechanisms are lacking from the current model but will be included in the future refinements. In the current model we did not consider inhomogeneities in function (e.g. in contractions) or structure of the network (e.g., subsequent vessel generation via anastomoses). Experimental observations have reported regional differences in function of isolated lymphatic vessels from rat [14]. For example, mesenteric lymphatic vessels are stronger pumps whereas the thoracic duct largely relies on passive mechanisms for lymph transport [25]. Substantial differences are expected between functional aspects of lymphatic vessels in different species. Thus, obtaining more data on the biomechanical properties of human lymphatic vessels is crucial for improving the translational interpretations of this work.

In summary, we have constructed computational models of branching networks composed of contracting lymphangions. Lymph flow conditions vary in different parts of the body and under different hydrodynamic conditions. Among the many factors that influence lymph flow, we studied the effect of pumping coordination, external pressure, and axial pressure difference on lymphatic pumping. We found that the effects of these functional parameters are intertwined. For example, the dependence of the optimum timing of contractions on axial pressure difference and transmural pressure can suggest that under physiologic conditions the system uses pumping coordination as an adaptation mechanism to different levels of pressure. We confirmed that flow rate can be maximized under a certain range of transmural pressures for the network of lymphangions, in agreement with previous observations in series of lymphangions. The network scale behavior resulted in a more complex response to variations in timing of contractions between adjacent lymphangions. Inclusion of a diastolic period between the subsequent contractions of each lymphangion was advantageous for pumping, suggesting that the lymphatic system favors efficient use of metabolic energy by imposing resting periods between contractions. Additionally, the presence of this diastolic period allowed higher flow rates under a favorable pressure difference compared to the case with t_r = 0. Finally, this study provides further insight into the important factors that regulate lymph flow on a network scale. Our results demonstrate that parameters that control pumping coordination (diastolic period and time delay between contraction of adjacent lymphangions) are major determinants of effective pumping and are highly influenced by both transaxial and transmural pressure differences. Thus, efficiency of the lymphatic pump is highly dependent on the coordination of the contraction wave. Further investigation of parameters that control the direction and conduction of lymphatic pacemaking wave as well as understanding mechanisms leading to muscle contraction dysfunction are of high importance. Our results further suggest that lymphatic pumping parameters are normally interrelated and that pumping efficiency under a certain set of parameter values can vary to a large extent depending on the hydrostatic pressure conditions (transaxial and transmural pressure differences). Understanding the important mechanisms involved in muscle contraction and coordination paves the way for pharmacological treatment of underlying causes of lymphatic pumping failure in diseases like lymphedema. These findings could be potentially useful for better understanding and treatment of human lymphatic diseases but it is crucial to have better estimates on how these important pumping parameters vary in different regions of the human body.