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Michael E. DeBakey Institute, Texas A&M University, College Station, Texas
Submitted 8 December 2006 ; accepted in final form 25 April 2007
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ABSTRACT |
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time-varying elastance; propagation; mathematical model
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.
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METHODS |
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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). out was measured with the use of a custom flowmeter used in earlier studies (25).
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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. Pin was maintained at 5.0 mmHg, and Pout 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 (
P) per flow () 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 (d/dt) is a function of r, which is a variable function of time. It also depends on l and lymph density (
), 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, Pt(t), to the chamber volume, V(t), and the dead volume (V0).
 | (3) |
Inlet and outlet valves.
The valves forming the boundaries of lymphangions add another time-varying resistance to lymph flow. However, the resistance (Rvalve) 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 (Pvalve) 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 (Rin and Rout) 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 Pt of each lymphangion was defined as the pressure midway between the lymphangion Pin and Pout. The resulting pressure drop between the Pin and the pressure in the center of the lymphangion is equal to the sum of the pressure drops described by Eqs. 1–5. Thus Pin – Pt 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, in.
 | (6) |
Likewise, the pressure drop from the midline Pt to Pout is the sum of pressure drops described by Eqs. 1–5. In this case, however, all pressure drops are a function of out.
 | (7) |
Assuming a cylindrical shape (i.e., V = 
r2l), Pt 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 out of an upstream lymphangion is set to in of the next lymphangion. Similarly, Pout of an upstream lymphangion is set equal to the Pin 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 (t) can be varied by modifying the empirically derived elastance function in Eq. 3:
 | (10) |
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 Emin and Emax (25). To choose a value of Emin and Emax 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 Emin and Emax (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 Emin and Emax 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 Emin and Emax.
Simultaneous solution of equations.
Equations 6–9 represent four equations with four unknown variables, in, out, r, and Pt, all of which are functions of time. Parameters (Rin, Rout, Rvalve, µ, and ) 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 Pin and Pout with the use of MatLab (The Mathworks) to obtain the values of in, out, r, and Pt 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, Pin was set to 5.0 mmHg and Pout 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 t between the start of contraction of one lymphangion and the start of contraction of the following lymphangion, according to Eq. 10. By setting t either positive or negative, the propagation wave travels in a direction that is either orthograde (positive t) or retrograde (negative t), as described in Fig. 1. To ensure lymphangions pump against a small pressure gradient, Pin was set to 5.0 mmHg and Pout 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 (f1). The second lymphangion was allowed to have a different frequency (f2). The values of f1 and f2 were altered in a way to ensure that the average lymphangion frequency was maintained constant. Pin was set to 5.0 mmHg, and Pout was set to 5.2 mmHg. Mean flow was calculated as a function of relative frequency (i.e., f1 – f2) and plotted with increments of 0.3 min–1.
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RESULTS |
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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 t < 1 s.
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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 f = 0 (see solid square in Fig. 6). Introducing a difference in lymphangion frequencies (i.e., f 
0) yielded a decrease in mean flow of <20%.
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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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 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 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 Pt 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 (Emin and Emax) are chosen. From our isobaric study, lymphangions exhibit a large degree of variability, even when values are normalized to a 1-cm length (Emin ranged from 89 to 125 mmHg/ml, and Emax ranged from 220 to 271 mmHg/ml). Similarly, from biomechanical studies of Ohhashi et al. (24), we estimated Emin to range from 2 to 42 mmHg/ml and Emax to range from 7 to 152 mmHg/ml. From Meisner et al. (20), we estimated Emin to range from 18 to 313 mmHg/ml and Emax to range from 240 to 506 mmHg/ml. The large variations in Emin and Emax 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 (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 (TD) 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 (fmax) 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 Pin (5, 6) and Pout (4, 5, 7, 14). Therefore, attempts to prevent or resolve edema based on changing lymphangion coordination would likely be unrewarding.
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GRANTS |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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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.
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REFERENCES |
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) fall within range of the mean flows (
) measured from 4 different lymphatic vessels. Dashed lines represent the linear regression of the data, as well as the associated 95% confidence intervals. Lymphatic vessel model predictions are similar to experimental data (n = 4).
) when 