## Abstract

The lymphatic system returns interstitial fluid to the central venous circulation, in part, by the cyclical contraction of a series of “lymphangion pumps” in a lymphatic vessel. The dynamics of individual lymphangions have been well characterized in vitro; their frequencies and strengths of contraction are sensitive to both preload and afterload. However, lymphangion interaction within a lymphatic vessel has been poorly characterized because it is difficult to experimentally alter properties of individual lymphangions and because the afterload of one lymphangion is coupled to the preload of another. To determine the effects of lymphangion interaction on lymph flow, we adapted an existing mathematical model of a lymphangion (characterizing lymphangion contractility, lymph viscosity, and inertia) to create a new lymphatic vessel model consisting of several lymphangions in series. The lymphatic vessel model was validated with focused experiments on bovine mesenteric lymphatic vessels in vitro. The model was then used to predict changes in lymph flow with different time delays between onset of contraction of adjacent lymphangions (coordinated case) and with different relative lymphangion contraction frequencies (noncoordinated case). Coordination of contraction had little impact on mean flow. Furthermore, orthograde and retrograde propagations of contractile waves had similar effects on flow. Model results explain why neither retrograde propagation of contractile waves nor the lack of electrical continuity between lymphangions adversely impacts flow. Because lymphangion coordination minimally affects mean flow in lymphatic vessels, lymphangions have flexibility to independently adapt to local conditions.

- time-varying elastance
- propagation
- mathematical model

a major function of the lymphatic system is to return interstitial fluid to the central venous circulation. Interstitial fluid, called lymph once it has entered an initial lymphatic vessel, is transported via prenodal lymphatic vessels to the lymph nodes. Larger, postnodal lymphatic vessels transport lymph from lymph nodes to the upper thorax and great veins in the neck. Transport of lymph has been viewed as a passive process, governed by lymphatic vessel radius, length, and tone (11). However, it is now recognized that an active process is required to transport lymph from the lower pressure initial lymphatics to the higher pressure central veins (2, 11, 21, 22, 24). Many investigators thus have focused on the function of the fundamental unit of the lymphatic system, the lymphangion, which is the vessel segment between two valves (9, 15, 19). Like cardiac ventricles, lymphangions cyclically contract, and the amount of fluid pumped is governed by preload, afterload, contractility, and contraction frequency. Increasing lymph flow by manipulating lymphangion properties is a potential avenue for development of treatments to resolve edema, the accumulation of excess interstitial fluid (2). The fundamental challenge, therefore, has been to relate lymphangion properties (characterized by radius, length, tone, contractility, and contraction frequency) to lymphatic system function (characterized by lymph flow) (10, 16, 17, 19).

Although it is recognized that lymphatic vessel function is determined by the properties of its constituent lymphangions (11), some behaviors only become evident when their interactions are considered (27). As long as there is electrical continuity along the lymphatic smooth muscle layer, contractions of adjacent lymphangions tend to be coordinated (3, 32). Even if lymphangions contract at the same frequency (synchronously), they rarely contract simultaneously because the depolarization wave travels at 4–8 mm/s (24, 32). The effect of the wave propagation time (i.e., time delay between contractions of adjacent lymphangions) has been extraordinarily difficult to characterize because the afterload of an upstream lymphangion is coupled to the preload of a downstream lymphangion. Increases in preload increase lymph flow (5), but increases in afterload decrease lymph flow (4, 7). Although lymphangion radius, length, tone, contractility, and contraction frequency affect lymph flow, only two parameters characterize lymphangion coordination and the resulting contraction waves: *1*) the time delay between the onset of contractions of adjacent lymphangions (for synchronous contractions) and *2*) the relative frequency between adjacent lymphangions (for asynchronous contractions).

Of the many properties governing lymphatic function, lymphangion time delays and relative frequencies are particularly difficult to control experimentally (3, 18, 32). Zawieja et al. (32) characterized changes in a “pump index” resulting from altering lymphangion coordination with gap junction blockers. Focusing on the mechanisms of coordination, they did not attempt to directly control lymphangion time delays and relative frequencies. McHale and Meharg (18) differentially cooled interconnected lymphangions to affect coordination of lymphangion contractions. By changing the temperature of the lymphatic segments, they were able to cause contraction waves to propagate in both an “orthograde” direction (with flow) and a “retrograde” direction (opposing flow) (Fig. 1). Changing the direction of wave propagation had little effect on lymph flow. Like most conceivable experimental approaches, these pioneering studies were limited in their ability to determine how coordination affects lymphatic function because lymphangion time delays and relative frequencies could not be rigorously controlled. Furthermore, interventions affect lymphangion radius, tone, and contractility, each of which can affect lymph flow. Without intervention, investigators are constrained to observing naturally occurring discoordination.

A recently developed mathematical model of a lymphangion (25) provides a novel basis to determine how lymphangion coordination impacts lymphatic function. By adaptation of accepted mathematical models of ventricles (28–30) and blood vessels (23, 31), the lymphangion model was based on both experimental data and fundamental physical principles. Measured pressures and volumes obtained from a postnodal bovine mesenteric lymphangion were used to create a time-varying elastance function, classically used to characterize contracting ventricles (29, 30). The effects of lymph viscosity and density on hydraulic resistance and inertia were then characterized by the principles governing flow through a blood vessel (23). The model was experimentally validated and used to elucidate the dual nature of lymphangions as both pumps and conduits (25). Because it is based on fundamental principles, this model can predict lymph flow resulting from changes in any particular lymphangion parameter, including radius, length, tone, contractility, and contraction frequency. The purpose of the present work is to adapt this lymphangion model to determine the effect of synchronous and asynchronous lymphangion contraction on lymphatic vessel function.

## METHODS

#### Experimental preparation to construct and validate the mathematical model.

In vitro experiments were performed on isolated postnodal bovine mesenteric lymphatic vessels to validate the lymphatic vessel mathematical model. Vessels were obtained from a cattle abattoir within minutes after slaughter and were placed in a tubular organ bath (Fig. 2). Vessels were perfused with an oxygenated polyionic solution (20) maintained at pH 7.4. The length of the organ bath was set at the approximate resting length of the lymphatic vessels. Vessels were submerged to a depth of 1 cm and thus subjected to an external hydrostatic pressure of 1 cmH_{2}O. The measured parameters were the inlet pressure (P_{in}), outlet pressure (P_{out}), lymphangion radius (*r*), and outlet flow (Q̇_{out}). A charge-coupled device camera (Sony XC-ST50) was used to acquire images of the vessel, which were captured by an image acquisition board (National Instruments). The *r* values were measured in the middle of each lymphangion using a custom video dimension analyzer developed in LabVIEW (National Instruments). Q̇_{out} was measured with the use of a custom flowmeter used in earlier studies (25).

#### Experimental protocol to construct the mathematical model.

The first experimental protocol utilized an isobaric preparation to characterize lymphangion tone and contractility. Lymphangion segments that had no valves were chosen, thus allowing control of transmural pressure (P_{t}). That is, P_{in} and P_{out} were set to the same values and were assumed to be equal to P_{t}. In accordance with previous experiments to characterize lymphangion contractility (20, 25), pressures were then varied from 3 to 10 mmHg. The diameters of each lymphangion (*n* = 4) were measured for 1 min after every change in pressure. Average end-systolic and end-diastolic volumes were determined for each pressure step, assuming a cylindrical lymphangion.

#### Experimental protocol to validate the mathematical model.

The second experimental protocol was designed to characterize lymphatic vessel function by measuring the relationship of mean flow and the axial pressure gradient. Vessel segments were chosen that contained two lymphangions (and thus 3 valves) contracting synchronously. P_{in} was maintained at 5.0 mmHg, and P_{out} was varied from 5.0 to 7.5 mmHg. Mean flow from each vessel was measured for 1 min after every change in pressure gradient. Data from all vessels (*n* = 4) were pooled, and mean flow was plotted as a function of axial pressure gradient. The results of a linear regression and associated 95% confidence intervals were also plotted.

#### Basis of lymphangion mathematical model.

The mathematical model of the lymphatic vessel was based on a lymphangion model previously developed by Quick et al. (25). Briefly, the classical transmission line characterization of blood vessels (23) was modified so that three important hydrodynamic properties (vessel resistance, inertance, and compliance) were expressed as functions of time. The same formulation for a lymphangion was developed by Reddy et al. (26), who derived an axial pressure-flow relationship starting from the Navier-Stokes equation. In their approach, the second-order and nonlinear terms of the Navier-Stokes equation were assumed to be negligible, and an equation of continuity was assumed based on conservation of mass. In the present formulation, each lymphangion was bounded by valves, and inlet and outlet resistances were assumed.

#### Time-varying resistance.

With the assumption of a cylindrical shape, the time-varying resistance, R(*t*), was calculated using Poiseuille's law (23). The pressure drop (ΔP) per flow (Q̇) through the vessel depends on *r*, which is a variable function of time. It also depends on the length of the vessel (*l*) and blood viscosity (μ), which are both predefined constant parameters. (1)

#### Time-varying inertance.

The effect of lymph inertia (23) was characterized by a time-varying inertance, L(*t*). The pressure drop due to fluid acceleration (dQ̇/d*t*) is a function of *r,* which is a variable function of time. It also depends on *l* and lymph density (ρ), which are both predefined constant parameters. (2)

#### Time-varying elastance.

To describe the contractile behavior of the lymphangion, the time-varying elastance description originally developed by Suga et al. (29) to characterize ventricular function was employed. Time-varying elastance, E(*t*), is the inverse of a time-varying compliance, and relates the transmural pressure, P_{t}(*t*), to the chamber volume, V(*t*), and the dead volume (V_{0}). (3) V_{0} is the volume at zero pressure. Although E(*t*) is a function of time, it was defined before simulation. The minimum (E_{min}) and maximum values (E_{max}) of E(*t*) characterize the tone and contractility of the lymphangion (24).

#### Inlet and outlet valves.

The valves forming the boundaries of lymphangions add another time-varying resistance to lymph flow. However, the resistance (R_{valve}) varies from a small value when the valves are open to infinity as the valves close, as in Reddy et al. (26). The valves are assumed to open or close as the pressure gradient across the valve (ΔP_{valve}) changes. Any changes in upstream or downstream fluid velocities indirectly affect the valve by altering the axial pressure gradient across it. (4)

#### Inlet and outlet resistances.

Lymphangion flow depends not only on the fluid dynamics within a lymphatic vessel but also on its environment (i.e., the vessel boundary conditions). To characterize the upstream and downstream boundary conditions, a pressure in series with a resistance was assumed, consistent with the characterization of a lymphatic network originally described by Drake et al. (5). (5)

To mimic the in vitro setup, the inlet and outlet resistances (R_{in} and R_{out}) were set to constant values equal to the resistance of the inlet and outlet tubing (25).

#### Formulating model equations.

The vessel model was developed so that the P_{t} of each lymphangion was defined as the pressure midway between the lymphangion P_{in} and P_{out}. The resulting pressure drop between the P_{in} and the pressure in the center of the lymphangion is equal to the sum of the pressure drops described by *Eqs. 1*–*5*. Thus P_{in} − P_{t} is equal to the pressure drop due to the inlet resistance, the valve resistance, the viscous pressure drop, and the pressure drop due to inertia. In this case, all pressure drops are a function of the inlet flow, Q̇_{in}. (6)

Likewise, the pressure drop from the midline P_{t} to P_{out} is the sum of pressure drops described by *Eqs. 1*–*5*. In this case, however, all pressure drops are a function of Q̇_{out}. (7)

Assuming a cylindrical shape (i.e., V = πr^{2}*l*), P_{t} can be formulated as a function of E(*t*) and *r* (*Eq. 3*). (8)

Assuming conservation of mass, the volume of a lymphangion is equal to the integral of inflow minus outflow. (9)

*Equations 6*–*9* completely characterize the physics of lymph flow in individual lymphangions and are consistent with the model originally presented by Quick et al. (25). To expand this model to characterize multiple lymphangions, conservation of mass is assumed and Q̇_{out} of an upstream lymphangion is set to Q̇_{in} of the next lymphangion. Similarly, P_{out} of an upstream lymphangion is set equal to the P_{in} of the next.

#### Modulating frequency and time delay.

The time-varying elastance function defines the strength and timing of lymphangion contraction in the mathematical model. Hence, by alteration of the elastance function, the chronotropic characteristics of lymphangions can be varied. The frequency of contraction (*f*) and time delay between the start of the contraction of adjacent lymphangions (Δ*t*) can be varied by modifying the empirically derived elastance function in *Eq. 3*: (10) where E′(*t*) is the modified elastance function and *T* is the period of the experimentally derived elastance function.

#### Lymphatic vessel model.

To determine the effect of time delay and relative frequency on lymph flow, lymphangions described by *Eqs. 6*–*9* were theoretically connected in series. Simulations were performed for either two or three lymphangions in series (depending on the particular protocol); the behavior of the vessel did not change appreciably by increasing the number of lymphangions to four or more. Most model parameters in the present work are identical to those reported by Quick et al. (25). However, the mean flow predicted from the mathematical model proved to be sensitive to the assumed values of E_{min} and E_{max} (25). To choose a value of E_{min} and E_{max} most appropriate for the present study, the data resulting from the experimental protocol to construct the mathematical model (i.e., isobaric protocol) were analyzed. In accordance with procedures commonly used to characterize ventricular E_{min} and E_{max} (29), the slopes of the average end-diastolic pressure-volume relationship and the end-systolic pressure-volume relationship for each lymphangion were determined. The resulting values of E_{min} and E_{max} were normalized for a 1-cm vessel segment. To create a standard function for E(*t*), complete cycles of pressure and volume from a representative lymphangion were substituted into *Eq. 3*. The resulting function was then scaled so that its minimum and maximum values corresponded to the average values of E_{min} and E_{max}.

#### Simultaneous solution of equations.

*Equations 6*–*9* represent four equations with four unknown variables, Q̇_{in}, Q̇_{out}, *r*, and P_{t}, all of which are functions of time. Parameters (R_{in,} R_{out}, R_{valve}, μ, and ρ) have known, constant values that have been reported earlier (25). E(*t*) is a periodic function that was defined before simulation (25) as described above. *Equations 6*–*9* were solved simultaneously for given values of P_{in} and P_{out} with the use of MatLab (The Mathworks) to obtain the values of Q̇_{in}, Q̇_{out}, *r*, and P_{t} as functions of time. The program used a multistep solver with an implementation of the trapezoidal rule.

#### Validation of mathematical model.

A lymphatic vessel model with two lymphangions (and thus 3 valves) was constructed. To match the experimental protocol used to validate the mathematical model, P_{in} was set to 5.0 mmHg and P_{out} was varied from 5.0 to 7.5 mmHg. Mean flow from the model was calculated with *Eqs. 6*–*9* and plotted as a function of axial pressure gradient.

#### Synchronous contractions: changing time delay and direction of wave propagation.

To model the effects of propagation time delay, three contiguous lymphangions were made to pump with a set Δ*t* between the start of contraction of one lymphangion and the start of contraction of the following lymphangion, according to *Eq. 10*. By setting Δ*t* either positive or negative, the propagation wave travels in a direction that is either orthograde (positive Δ*t*) or retrograde (negative Δ*t*), as described in Fig. 1. To ensure lymphangions pump against a small pressure gradient, P_{in} was set to 5.0 mmHg and P_{out} to 5.2 mmHg. Mean flow was calculated as a function of time delay and plotted in increments of 0.2 s.

#### Asynchronous contractions: changing relative lymphangion contraction frequency.

The effect of eliminating coordination in the lymphatic vessel model (consisting of 3 lymphangions) was determined by introducing a relative frequency between adjacent lymphangions. To simplify, the third lymphangion contraction, frequency was set to the same frequency as the first lymphangion (*f*_{1}). The second lymphangion was allowed to have a different frequency (*f*_{2}). The values of *f*_{1} and *f*_{2} were altered in a way to ensure that the average lymphangion frequency was maintained constant. P_{in} was set to 5.0 mmHg, and P_{out} was set to 5.2 mmHg. Mean flow was calculated as a function of relative frequency (i.e., *f*_{1} − *f*_{2}) and plotted with increments of 0.3 min^{−1}.

## RESULTS

#### Standard time-varying elastance function.

Figure 3 illustrates the time-varying elastance function used for the model simulations. Values of E_{min} and E_{max} were estimated to be 246 ± 21 and 106 ± 15 mmHg/ml, respectively (*n* = 4).

#### Lymphatic vessel model validation.

As P_{out} was decreased from 7.5 to 5.0 mmHg (*n* = 4), experimental values of mean flow increased, with a linear regression of the pooled data having a slope of 0.040 ml·min^{−1}·mmHg^{−1} (Fig. 4). The results of linear regression with 95% confidence intervals are also indicated. Model results fall well within the 95% confidence interval.

#### Orthograde and retrograde directional effects on flow.

For synchronized lymphangions, switching the direction of contraction propagation has little effect on the predicted mean flow (arrows, Fig. 5). That is, the left half of the plot in Fig. 5 (retrograde propagation) is similar to the right half (orthograde propagation) when Δ*t* < 1 s. Mean flow was greater when the propagated contraction wave travels in the orthograde direction than when it travels in the retrograde direction, although the percent difference between maximum and minimum flows was <5.1% when Δ*t* < 1 s.

#### Effect of noncoordinated lymphangion contraction on lymph flow.

Mean flow was optimal when lymphangions were coordinated (i.e., Δ*f* = 0; Fig. 6). Allowing lymphangions to contract with even a small difference in frequency causes asynchronous contraction. The result is that lymphangions periodically contracted both in phase and out of phase. This abrupt change in behavior with a small difference in frequency manifests as a discontinuity in Fig. 6 at Δ*f* = 0 (see solid square in Fig. 6). Introducing a difference in lymphangion frequencies (i.e., Δ*f* ≠ 0) yielded a decrease in mean flow of <20%.

## DISCUSSION

#### Coordination has minimal impact on flow.

The present work uses a predictive model to illustrate that lymphangion coordination minimally affects flow in lymphatic vessels. By manipulation of two parameters describing synchronous and asynchronous contractions (i.e., time delay and relative frequency), two particular behaviors were identified. First, mean flow (Fig. 5) is similar with both positive and negative time delays (i.e., orthograde and retrograde wave propagation). Second, a zero frequency difference (coordinated case) results in a similar mean flow (Fig. 6) as a nonzero frequency difference (noncoordinated case). These two results elucidate two striking experimental observations: *1*) retrograde propagation of contraction waves and *2*) the existence of electrical discontinuities.

#### Explaining orthograde and retrograde contractile propagation.

Contractile waves in lymphatic vessels have been observed to propagate in both orthograde and retrograde directions (3, 18, 32). Model results indicate that there is little effect of changing the relative time delay between lymphangion contractions with orthograde propagation (positive Δ*t*, right side of Fig. 5). In this case, increasing the time delay increases both the preload of downstream lymphangions (tending to increase flow) and afterload of upstream lymphangions (tending to decrease flow). These effects tend to balance. There is a greater effect of changing the relative time delay between lymphangion contractions with retrograde propagation (negative Δ*t*; Fig. 5, *left*). In this case, retrograde propagation increases lymphangion afterload (since the downstream lymphangion is still in systole) without a concomitant increase in lymphangion preload. Although orthograde propagation could result in higher flow than retrograde propagation if the time delay is large, model results indicate that the difference in orthograde and retrograde propagation is negligible (∼5%) if time delays are <1 s.

#### Explaining electrical discontinuity.

Although electrical discontinuity has not been previously documented for conducting bovine mesenteric lymphatic vessels, investigators have observed discontinuities in the smooth muscle layer lining the lymphatic vessel across the valves separating lymphangions of rat and guinea-pig lymphangions (3, 32). This anatomic discontinuity results in an electrical discontinuity, disrupting the ability of adjacent lymphangions to electrically synchronize contractions. It would be tempting to view discontinuities as aberrations, not only because they are rarely reported and may only exist in a few species but also because the structure of the lymphatic vessel does not appear to support integration of lymphangion function. This view, however, presupposes a positive benefit of lymphangion synchronization. The present work illustrates that the lack of synchronous contraction has little effect on lymph flow (Fig. 6).

#### Modeling tools predict behavior.

The lymphatic vessel model, based on fundamental physics and measured lymphangion properties, not only can mimic behavior observed in vitro but also can allow theoretical manipulation of parameters that are difficult to alter experimentally. Two different approaches were used to validate the mathematical model. The first approach tested predictions of a model of a single lymphangion and was previously reported (25). Briefly, predicted instantaneous diameters, instantaneous flows, and average flows resulting from different axial pressure gradients were found to be similar to those observed from in vitro experiments (25). The second approach taken here tested predictions of a model of a lymphatic vessel consisting of multiple lymphangions (*Eqs. 6*–*9*). The model's predicted mean flow-axial pressure gradient was compared with measured values resulting from coordinated contraction (Fig. 4). Although the measured data exhibited significant variation, the simulation results were consistent with the data. To ensure that we fully exercised the model, the mathematical model was validated with measured data (Fig. 4) that were different from data used to create the model (i.e., isobaric experiments to characterize contractility; Fig. 3). These complementary approaches to validate the model increase confidence in other predictions (i.e., Figs. 5 and 6) that are not possible to validate directly.

#### Model caveats.

There are a number of issues that can limit the generalizability of the results. The time-varying elastance model (*Eq. 3*) does not include the effects of pump failure (19) or shear stress-mediated inhibition (10). Hence, care has to be taken to simulate lymphatic vessel behavior in the appropriate pressure and flow ranges where assumed lymphangion properties are valid. Similarly, this theoretical model excluded the confounding effects of P_{t} on contraction frequency (19). Furthermore, the volume calculations assumed cylindrical vessels. Although there are regional variations in diameter along the length of postnodal bovine mesenteric lymphangions, they tend to be small. For this reason, a cylindrical approximation has been made in several previous studies (2, 15, 24, 25). Care has to be taken if the present model is used to characterize smaller prenodal collecting vessels, which tend to have a bulblike structure (13). Furthermore, the calculation of resistance and inertance implicitly assumed laminar flow, potentially causing pressure gradients to be underestimated. These approximations, however, may not play as important a role as how the parameter values describing lymphangion tone and contractility (E_{min} and E_{max}) are chosen. From our isobaric study, lymphangions exhibit a large degree of variability, even when values are normalized to a 1-cm length (E_{min} ranged from 89 to 125 mmHg/ml, and E_{max} ranged from 220 to 271 mmHg/ml). Similarly, from biomechanical studies of Ohhashi et al. (24), we estimated E_{min} to range from 2 to 42 mmHg/ml and E_{max} to range from 7 to 152 mmHg/ml. From Meisner et al. (20), we estimated E_{min} to range from 18 to 313 mmHg/ml and E_{max} to range from 240 to 506 mmHg/ml. The large variations in E_{min} and E_{max} values suggest not only a high degree of variation among lymphangions but also a sensitivity of parameter values to the particular experimental protocol (i.e., isobaric, isometric, and isovolumetric).

#### Lymphangion length limits maximum contraction frequency.

The time delay between adjacent lymphangions (Δ*t*) is not generally reported but can be calculated as the ratio of lymphangion length (*l*) to contractile wave propagation velocity (*v*). For instance, using particular reported values for length of 25 mm (8) and propagation velocity of 5 mm/s (24) results in time delay (*l*/*v*) of 5 s. This is unlikely to be a physiological value of time delay with common lymphangion contraction frequencies because it would result in ineffective pumping. That is, if the propagation velocity is too slow or the diastolic period (*T*_{D}) is too long, the proximal end of the lymphangion could start to relax before the contraction wave reaches the distal end. If the net volume of a chamber does not change, a pressure pump will not be able expel any lymph. To avoid a behavior where the proximal end of the lymphangion starts to relax before the contraction wave has propagated to the distal end, a fundamental maximum contraction frequency (*f*_{max}) can be postulated. (11)

To ensure that our model does not include phenomena that may not be physiological, we first assumed uniform contraction and then limited the range of time delays simulated to 2 s (3) (Fig. 5). Perhaps more importantly, *Eq. 11* suggests a fundamental limitation for all lymphangions. Because there is little variation in the propagation velocity from microlymphatics to the collecting lymphatics (24, 32), we can predict that maximum contraction frequency is inversely proportional to lymphangion length (*Eq. 11*). This may explain the observation that (shorter) microlymphatic vessels contract with a higher frequency than (longer) postnodal lymphatic vessels (9).

#### Implications to network adaptability.

The observed insensitivity of lymph flow to asynchrony of adjacent lymphangions is not surprising considering the structure of a lymphatic network (27). Because lymphangions respond to different local conditions by changing frequency, they necessarily must be asynchronous at junctions. The insensitivity to the lack of coordination is thus a manifestation of a larger principle: lymphangions are highly adaptable to local conditions, yet they can maintain global lymph flow through a lymphatic network. Therefore, lymphangion coordination is not nearly as important as lymphangion contractility (16) and P_{in} (5, 6) and P_{out} (4, 5, 7, 14). Therefore, attempts to prevent or resolve edema based on changing lymphangion coordination would likely be unrewarding.

## GRANTS

This work was supported by National Heart, Lung, and Blood Institute Grant K25 HL-070608 (C. M. Quick), American Heart Association Grants 0565116Y (C. M. Quick) and 0365127Y (R. H. Stewart), and Centers for Disease Control and Prevention Grant 620069 (G. A. Laine).

## Acknowledgments

We thank our Research Assistant, Melinda Webster, for procuring viable lymphatic vessels and ensuring proper setup of the vessel bath.

## Footnotes

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

*advertisement*” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

- Copyright © 2007 by the American Physiological Society