Nonlinear lymphangion pressure-volume relationship minimizes edema

Abstract

Lymphangions, the segments of lymphatic vessel between two valves, contract cyclically and actively pump, analogous to cardiac ventricles. Besides having a discernable systole and diastole, lymphangions have a relatively linear end-systolic pressure-volume relationship (with slope E_max) and a nonlinear end-diastolic pressure-volume relationship (with slope E_min). To counter increased microvascular filtration (causing increased lymphatic inlet pressure), lymphangions must respond to modest increases in transmural pressure by increasing pumping. To counter venous hypertension (causing increased lymphatic inlet and outlet pressures), lymphangions must respond to potentially large increases in transmural pressure by maintaining lymph flow. We therefore hypothesized that the nonlinear lymphangion pressure-volume relationship allows transition from a transmural pressure-dependent stroke volume to a transmural pressure-independent stroke volume as transmural pressure increases. To test this hypothesis, we applied a mathematical model based on the time-varying elastance concept typically applied to ventricles (the ratio of pressure to volume cycles periodically from a minimum, E_min, to a maximum, E_max). This model predicted that lymphangions increase stroke volume and stroke work with transmural pressure if E_min < E_max at low transmural pressures, but maintain stroke volume and stroke work if E_min= E_max at higher transmural pressures. Furthermore, at higher transmural pressures, stroke work is evenly distributed among a chain of lymphangions. Model predictions were tested by comparison to previously reported data. Model predictions were consistent with reported lymphangion properties and pressure-flow relationships of entire lymphatic systems. The nonlinear lymphangion pressure-volume relationship therefore minimizes edema resulting from both increased microvascular filtration and venous hypertension.

lymphatic vessels actively pump lymph to prevent edema. One of the primary purposes of the lymphatic system is to return fluid from the lower-pressured interstitial space to the higher-pressured veins of the neck (33). Lymphangions, the fundamental functional unit of the lymphatic system, consist of a section of lymphatic vessel bounded by two unidirectional valves (20). Lymphangions contract cyclically and actively pump lymph, analogous to contracting cardiac ventricles, and can be characterized by systolic and diastolic periods, stroke volume (11, 16), and even stroke work (16). The lymphangion pump has to adapt to keep interstitial pressure (and thus interstitial volume) low on the one hand and maintain lymph flow when faced with high transmural pressures caused by venous hypertension on the other. Although these functional challenges are met in part by responding to changes in lymph flow (13), they are primarily overcome by responding to changes in pressure (19). Failure of lymphangions to functionally adapt can exacerbate interstitial edema formation, the accumulation of excess fluid in the interstitial space.

Lymphangion mathematical models based on analogies to cardiac ventricles and blood vessels have played critical roles when experimental approaches were intractable.

Reddy et al. (25) were the first to develop a model of an entire lymphatic system based on the Navier-Stokes equations, classically used to predict the flow through blood vessels from fundamental physical principles. The resulting model determined how lymph viscosity, inertia, and cyclical contraction affect pressures and flows throughout the entire lymphatic system (26), which are not feasible to measure experimentally. In a less ambitious approach, Quick et al. (23) and Venugopal et al. (32) developed a lymphangion model based on the time-varying elastance concept (i.e., the ratio of transmural pressure and volume), classically used to characterize cardiac ventricles (27, 28). The slope of the end-diastolic pressure-volume relationship (E_min) acts as an index of diastolic tone, and the slope of the end-systolic pressure-volume relationship (E_max) acts as an index of contractility. This model was originally derived based on simplifying the Navier-Stokes equations as in Reddy et al. (25) and retained critical terms quantifying the contribution of lymph inertia and lymph viscosity. This model reproduced measured pressure-flow relationships (23) and correctly predicted that lymphangions transition from active pumps to passive conduits when inlet pressure rises above outlet pressure (22). This model was further extended by Venugopal et al. (32) to address controversies that arose from the inability to completely control variables experimentally. For instance, Venugopal et al. demonstrate that coordination of contraction has minimal effect on lymph flow, a result that has since been corroborated by other mathematical models (17). They also illustrated that the “effective resistance” of a lymphatic system is actually a relatively simple function of lymphangion contractility and contraction frequency (24). A relatively simple principle was also revealed by comparing a simple algebraic version of the model (24) that neglected lymph inertia and viscosity to a more detailed numerical version (32) that included these complexities. The results were nearly identical, indicating that inertial and viscous effects on lymphangion pressure-flow relationships are negligible, a result later corroborated by a particle imaging velocimetry (4, 5). Taken together, these approaches illustrate the advantages of mathematical models of lymphangions when they 1) are based on measured lymphangion properties, 2) incorporate fundamental physical properties, and 3) selectively simplify complex phenomena.

Lymphangion diastolic function is notably different than that of ventricles.

Although critical for developing mathematical models of lymphatic vessels, analogies to blood vessels and cardiac ventricles do not completely capture lymphatic vessel behavior. For instance, because Reynolds numbers are exceptionally low (<5) (5), models incorporating lymph inertia to predict lymph flow are therefore needlessly complex to describe larger conducting vessels. Furthermore, heart-arterial system coupling results from an active pump emptying into a passive arterial network. This is qualitatively different from lymphangion-lymphangion interaction arising from two active pumps (18, 32, 34). This interaction becomes all the more complex when one considers that the afterload of one lymphangion forms the preload of the next. Although the lymphatic mathematical models developed by Drake et al. (11) and Quick et al. (24) assumed pressure-volume relationships similar to those of ventricles, with relatively linear end-diastolic pressure-volume relationships, the lymphangion end-diastolic pressure-volume relationship is highly nonlinear (2, 16, 21) and results from an active tone (15). At low transmural pressures, small changes in pressure result in large increases in the end-diastolic volume. As transmural pressure rises, the end-diastolic volume becomes much less sensitive to pressure (21). The functional implications for this high degree of nonlinearity of the lymphangion end-diastolic pressure-volume relationship have yet to be elucidated.

Lymphatic vessels must perform two different functions to accommodate two edemagenic conditions.

On the one hand, increases in microvascular filtration can act as an edemagenic stress by raising the interstitial pressure (30, 31). The result is to raise the lymphangion transmural pressure while lowering the axial pressure gradient (31). For instance, a classic experiment illustrated that increasing microvascular filtration via venous pressure elevation causes interstitial pressure in the dog paw to transiently rise at least to 2 mmHg (30). To limit edema formation in such cases, lymphatic vessels must respond to relatively low increases in pressures by increasing lymph flow. On the other hand, increases in central venous pressure, which increase microvascular filtration, can act as an edemagenic stress by raising both lymphatic system inlet and outlet pressures (6). The result is to markedly raise lymphangion transmural pressure, while having minimal impact on lymphangion axial pressure gradients. In fact, venous hypertension can raise lymphatic transmural pressure as much as 25 cmH₂O (6). To limit edema formation, lymphatic vessels must respond to increases in these relatively high transmural pressures by maintaining lymph flow at a constant level. Indeed, Drake et al. (10) reported that increasing capillary pressure (and thus decreasing lymphangion axial pressure gradient and increasing transmural pressure) led to a rapid increase in lymph flow up to a “transition pressure”. Further increases in pressure above this transition pressure resulted in very small changes in lymph flow. These two edemagenic conditions imply a need for lymphangions to have different responses in two transmural pressure ranges. At low pressures, lymphangions need to pump greater volumes of lymph to accommodate higher microvascular filtration. At high pressures, lymphangions do not have to pump more, but instead must maintain a constant stroke volume against an elevated outlet pressure (8). We therefore hypothesized that the nonlinear lymphangion pressure-volume relationship allows transition from a predominant transmural pressure-dependent stroke volume to a predominant transmural pressure-independent stroke volume as pressure increases.

THEORY

Algebraic solution to predict lymphangion flow.

Following the approach used to describe ventricular function (29), the stroke volume (SV) of a lymphangion was approximated, assuming that the pressure-volume loops were rectangular (24)

$$\text{SV}=({\text{V}}_{\text{ed}}-{\text{V}}_{\text{es}})=\left(\frac{{\text{P}}_{\text{in}}}{{\text{E}}_{\min }}-\frac{{\text{P}}_{\text{out}}}{{\text{E}}_{\max }}+{\text{V}}_{o,\text{ed}}-{\text{V}}_{o,\text{es}}\right),$$ (1)

where V_ed and V_es are the end-diastolic volume and end-systolic volume, P_in and P_out are the lymphangion inlet and outlet pressures, E_min and V_{o, ed} are the slope and intercept of the end-diastolic pressure-volume relationship, and E_max and V_{o, es} are the slope and intercept of the end-systolic pressure-volume relationship. E_max is a commonly used index for characterizing contractility (27, 29), and E_min is used for characterizing diastolic tone (27, 29). In the case that the pressure-volume loops are not rectangular, the calculation of stroke volume would not be affected (29). By multiplying Eq. 1 by the contraction frequency, lymphangion flow can be approximated. The validity and limitations of the model characterized by Eq. 1 were discussed in detail previously, having been tested by comparing its predictions to a more complex mathematical model that incorporates complexities such as inertial and resistive effects (23) and to experimentally measured lymph flow (32). Model results were consistent with measured lymphangion function, despite the fairly large variation in lymphatic function from lymphangion to lymphangion. The simple analytical solution (i.e., algebraic formula) expressed by Eq. 1 provides insight into the relationship of lymphangion structure and function.

Stroke work.

The lymphangion stroke work (i.e., external work done to propel lymph) can be calculated as the product of change in pressure (P_out − P_in) and stroke volume

$$\text{SW}=({\text{P}}_{\text{out}}-{\text{P}}_{\text{in}})\cdot \text{SV.}$$ (2)

This description has been commonly used by many investigators to describe the external stroke work by a cardiac ventricle (3, 14, 29). Equation 2 includes the inherent assumption that the lymphangion pressure-volume relationship is rectangular. This assumption is valid in vitro when inlet and outlet pressures are set constant values. In vivo, however, this assumption would tend to underestimate stroke work, considering that the boundary conditions are never truly isobaric. For example, Li et al. (16) reported that sheep mesenteric lymphangions exhibit hexagonal pressure-volume loops in vivo. Assuming a rectangular stroke work in these lymphangions would underestimate stroke work by as much as 25%. Although this approximation systematically underestimates stroke work in vivo, Eq. 2 can be easily adjusted to accommodate different shapes of the pressure-volume loop.

Linear representation of nonlinear end-diastolic pressure-volume relationship.

The time-varying elastance model used in earlier lymphangion models (23, 32) assumed a single, linear end-diastolic pressure-volume relationship. The lymphangion end-diastolic pressure-volume relationship, however, is curvilinear, and has been described as exponential (16, 21). To describe the transition from transmural pressure-dependent stroke volume to transmural pressure-independent stroke volume, the end-diastolic pressure-volume relationship was characterized with two segments having different slopes above and below a pressure (P*).

$$V_{ed}=\{\begin{array}{c}{V}_{o,es}+\frac{P_{in}}{E_{\min 1}},P_{in}<P^{*}(zone1)\\ V_{o,ed2}+\frac{P_{in}}{E_{\max }},P_{in}>P^{*}(zone2)\end{array}$$ (3)

In the low-pressure zone (P_in < P*), E_min1 and V_{o, es} are the slope and intercept of the end-diastolic pressure-volume relationship, and in the high-pressure zone (P_in > P*), E_max and V_{o, ed2} are the slope and intercept of the end-diastolic pressure-volume relationship, respectively.

Behavior of lymphangion pressure-volume relationship in zone 1.

Quick et al. (24) originally used a version of Eq. 1 in which V_{o, ed} and V_{o, es} in Eq. 1 were set equal. Stroke volume can be expressed in terms of an axial pressure gradient (ΔP), which is equal to P_out − P_in, as well as a transmural pressure (P_trans), approximated by (P_in+P_out)/2.

$$\text{SV}=({\text{V}}_{\text{ed}}-{\text{V}}_{\text{es}})=\left(\frac{{\text{P}}_{\text{in}}}{{\text{E}}_{\min }}-\frac{{\text{P}}_{\text{out}}}{{\text{E}}_{\max }}\right)={\text{P}}_{\text{trans}}\left(\frac{{\text{E}}_{\max }-{\text{E}}_{\min }}{{\text{E}}_{\max }{\text{E}}_{\min }}\right)-\Delta \text{P}\left(\frac{{\text{E}}_{\max }+{\text{E}}_{\min }}{2{\text{E}}_{\max }{\text{E}}_{\min }}\right),$$ (4)

Stroke volume thus increases with an increase in transmural pressure and decreases with an increase in the axial pressure gradient. Figure 1 illustrates that as inlet and outlet pressure increases an equivalent amount (thus raising the average transmural pressure), the resulting stroke volume increases.

Fig. 1.

Illustration of pressure-volume relationship in zones 1 and 2. Axial pressure gradients (ΔP = P_out − P_in) are assumed constant. Pressure-volume relationship of a lymphangion is sensitive to transmural pressure (Eq. 4) below P* (zone 1). Slope of the end-systolic pressure-volume relationship (E_max) is greater than the slope of the end-diastolic pressure-volume relationship (E_min), but their intercepts (V_{o, es} and V_{o, ed}) are equal. Increases in inlet pressure result in increased stroke volume (SV₁ to SV₂). Pressure-volume relationship of a lymphangion is insensitive to transmural pressure (Eq. 5) above P* (zone 2). In this case, E_max = E_min, and V_{o, es} < V_{o, ed}. Stroke volume (SV₃ to SV₄) does not change with increases in transmural pressure.

Behavior of lymphangion pressure-volume relationship in zone 2.

Unlike in Quick et al. (24), if V_{o, ed} and V_{o, es} are not set equal, but E_max and E_min are instead set equal, the resulting end-systolic and end-diastolic pressure-volume relationships would be parallel. Figure 1 illustrates that as inlet and outlet pressure increases an equivalent amount (thus raising transmural pressure while maintaining a constant axial pressure gradient), the resulting stroke volume remains constant. In this case, Eq. 1 degenerates, and the stoke volume becomes sensitive only to axial pressure gradient, ΔP,

$$\text{SV}=\frac{{\text{P}}_{\text{in}}-{\text{P}}_{\text{out}}}{{\text{E}}_{\max }}+{\text{V}}_{o,\text{ed}2}-{\text{V}}_{o,es}=-\Delta \text{P}\left(\frac{1}{{\text{E}}_{\max }}\right)+{\text{V}}_{\text{o, ed}2}-{\text{V}}_{\text{o, es}}.$$ (5)

Determination of pressure gradient and stroke work along a series of lymphangions.

Each lymphangion in a lymphatic vessel can add a pressure increment (ΔP) along its length. The amount of ΔP added by a particular lymphangion depends on whether it is subjected to low (zone 1) or high (zone 2) pressures. Equation 1 was therefore used to calculate the pressure increments added by each lymphangion in a vessel. To yield a general result, it was assumed that a lymphatic vessel with inlet pressure P_in consisted of a number (n) of lymphangions. For simplicity, it was assumed that the contraction wave traveled in an antegrade direction and that when an upstream lymphangion contracted, the downstream lymphangion was in diastole. Using a similar model. Venugopal et al. (32) reported that changing the timing of contractions in a chain of lymphangions had little effect on lymph flow, a result that was independently reported by Macdonald et al. (17). Assuming that the inlet pressure of an upstream lymphangion is equal to the outlet pressure of a downstream lymphangion, the resulting pressure step of the nth lymphangion (ΔP_n) becomes a function of P_in, E_min, E_max, V_{o, es}, and V_{o, ed2}, and the ejection fraction (EF) of the first lymphangion calculated from SV/V_ed. The resulting ΔP_n is a geometric series, which can be expressed in a simple analytical form.

$$\Delta {\text{P}}_{\text{n}}=\{\begin{array}{l}{\left(\frac{{\text{E}}_{\max }}{{\text{E}}_{\min }}\right)}^n+\left[1-\text{EF}-\frac{{\text{E}}_{\min }}{{\text{E}}_{\max }}\right]\cdot {\text{P}}_{\text{in}},{\text{P}}_{\text{in}}<\text{P}*\text{ (}zone1\text{)}\\ \left[\frac{{\text{V}}_{\text{o},\text{ed}2}-{\text{V}}_{\text{o},\text{es}}}{{\text{V}}_{\text{ed}}}-\text{EF}\right]\cdot {\text{P}}_{\text{in}},{\text{P}}_{\text{in}}>\text{P}*(zone2)\end{array}$$ (6)

In Zone 1, the term (E_max/E_min)ⁿ causes the pressure step added by each subsequent to increase dramatically. This pressure step increases very quickly if the lymphangions have a high level of systolic contractility (E_max) or a low level of diastolic tone (E_min). The last lymphangion not only raises lymph pressure the most, but also does the most stroke work (Eq. 2). In zone 2, in contrast, the pressure step is constant, i.e., each lymphangion raises the lymph pressure an equal amount. This is evident in Eq. 6, because n does not appear in the equation in zone 2. Consequently, in zone 2, all lymphangions contribute an equal amount of stroke work.

METHODS

Analyzing previously reported data.

Bovine mesenteric lymphangion pressure-volume relationships previously reported by Ohhashi et al. (21), which share critical similarities with other reported relationships (2, 16), were chosen for analysis. Data were digitized and imported into Mathematica 6 (Wolfram Research). Similarly, a dog lymphatic system interstitial pressure-lymph flow relationship (30), also sharing critical similarities with other reported relationships (9, 31), was chosen for analysis. Data were digitized and plotted.

Stroke volume and stroke work for two pressure zones.

Stroke volume and stroke work were calculated for a single lymphangion subjected to both low and high pressures (Eq. 3). In pressure zone 1, E_min1 was assumed to equal the slope of the end-diastolic pressure-volume relationship from Ohhashi et al. (21). In pressure zone 2, the slope of the end-diastolic pressure-volume relationship was assumed to be equal to the slope of the end-systolic pressure-volume relationship. To illustrate how this pressure-volume relationship affects stroke volume and stroke work, inlet pressure and outlet pressure were set assuming a differential pressure of 0.2 cmH₂O. The resulting stroke volume and stroke work were calculated from Eq. 3 and plotted as a function of inlet pressure.

Determining whether lymphangions exhibit parallel end-systolic and end-diastolic pressure-volume relationships at high pressures.

To ensure that lymphangions respond differently to transmural pressures above and below 2 mmHg (30), Eq. 3 was fit to the data from Ohhashi et al. (21). The slope (i.e., E_max) and intercept (i.e., V_{o, es}) of the end-systolic pressure-volume relationship were found by simple linear regression. The slope of the end-diastolic pressure-volume relationship in zone 1 (i.e., E_min1) was found by simple linear regression of the data below the assumed pressure (P*). The value of V_{o, ed2} was calculated to ensure that the pressure-volume relationships in both pressure zones were contiguous. The model results were plotted with the data from Ohhashi et al.

Identifying the best-fit end-diastolic pressure-volume relationship.

Instead of assuming a priori which pressure zones may be appropriate to characterize lymphatic function as transmural pressure dependent or transmural pressure independent, the question can be reversed. To determine the value of P* in Eq. 3 that best fits the data from Ohhashi et al. (21), a simple linear regression and nonlinear parameter estimation was employed. To test whether the end-diastolic pressure-volume relationship could adequately be described by a line with the same slope as the end-systolic pressure-volume relationship (E_max) over a range of pressures, E_max was obtained from the slope of the linear regression of the end-systolic pressure-volume relationship. Nonlinear parameter estimation methods were applied to yield best-fit values for E_min1 and P*. Briefly, an error function was defined as the sum of the square of the residuals (difference between model and data values), according to convention (1). To ensure that we found a global minimum, the error function was tabulated as a function of all three parameters for the entire range of possible values, and the minimum was located.

Determination of pressure gradient and stroke work along a series of lymphangions.

To illustrate the distribution of pressures and stroke work along a lymphatic vessel characterized by Eqs. 3 and 6, the pressure steps along a lymphatic vessel consisting of three lymphangions was calculated. First, contraction frequencies and lengths were assumed to be equal. Assuming conservation of mass, lymphangion stroke volumes were also necessarily equal. Furthermore, the outlet pressure of an upstream lymphangion was set equal to the inlet pressure for the subsequent lymphangion. For simplicity, it was assumed that the contraction wave traveled in an antegrade direction and that when an upstream lymphangion contracted, the downstream lymphangion was in diastole. Furthermore, although Eqs. 1–5 do not include the effects of either lymph viscosity or mass, Quick et al. (24) reported that resistance to lymph flow or lymph inertia has negligible effects on flow. Thus the change in pressure along the chain of lymphangions is due to pumping only and can be characterized by Eqs. 1–5 alone. The values for E_max, E_min1, P*, V_{o, es}, and V_{o, ed2} were set to values obtained from the nonlinear parameter estimation described above. For illustrative purposes, in the low-pressure zone 1, the inlet pressure was set at 0.25 cmH₂O and stroke volume was set to 0.02 ml. For the high-pressure zone 2, the inlet pressure was set at 4 cmH₂O and stroke volume was set to 0.175 ml. The stroke volumes and the inlet pressures were chosen such that all three lymphangions function in either zone 1 or zone 2. The pressure steps and stroke work were calculated using Eqs. 2–5 for each lymphangion, and the outlet pressures and stroke works were plotted.

RESULTS

Stroke volume and stroke work in two pressure zones.

The pressure-volume relationship characterized by Eq. 3 results in significantly different function at low and high pressures. For zone 1, the stroke volume and stroke work increase with transmural pressure (Fig. 1). However, in zone 2, there is no change in stroke volume or stroke work with transmural pressure (Fig. 1).

Lymphatic system flow as a function of interstitial fluid pressure.

Figure 2 illustrates a previously reported increase in lymph flow resulting from increases in interstitial fluid pressure by Taylor (30). As interstitial fluid pressure rises, lymph flow initially increases significantly. As interstitial pressure rises above 2 mmHg, lymph flow becomes very insensitive to interstitial pressure.

Fig. 2.

Relationship of interstitial tissue pressure and relative lymph flow in a dog paw digitized from Taylor (30), representing the effects of increasing microvascular filtration caused by increased venous pressure elevation. Two functional zones are apparent. In the low-pressure zone, as interstitial pressure increases, lymph flow increases dramatically with pressure. In the high-pressure zone, as interstitial pressure increases, lymph flow plateaus and maintains a constant value.

Determining whether lymphangions have parallel end-diastolic and end-systolic pressure-volume relationships above an assumed pressure (P*).

Figure 3 illustrates the fit of Eq. 3 to the data from Ohhashi et al. (21) assuming a P* of 2 mmHg a priori. The average deviation of the end-diastolic pressure-volume relationship from the data was 8.6%. More importantly, the model falls within the reported SD, indicating the model is consistent with measured data.

Fig. 3.

Pressure-volume relationship of bovine postnodal mesenteric lymphangions digitized from Ohhashi et al. (21) (circles) representing the end-systolic pressure-volume relationship (top curve) and the end-diastolic pressure-volume relationship (bottom curve). Solid lines represent the model (Eq. 3) when slope of end-diastolic pressure-volume relationship in the higher pressure range (i.e., zone 2) was set equal to the slope of end-systolic pressure-volume relationship (E_max). Fit of model to data using a value of the transmural pressure, P* (2 mmHg), demarcating low- and high-pressure zones was assumed a priori. Fit falls within SD, indicating the assumption that E_min= E_max is consistent with the data.

Parameter estimation to delineate low- and high-pressure zones.

Assuming that the slopes of the end-systolic and end-diastolic pressure-volume relationships are equal above P*, the resulting values obtained from nonlinear parameter estimations for E_max, V_{o, es}, E_min1, P*, and V_{o, ed2} were 101.4 cmH₂O/ml, 0.05 ml, 11.5 cmH₂O/ml, 2.4 cmH₂O, and 0.24 ml, respectively. The average deviation of the end-diastolic pressure-volume relationship from the data was 5.8%.

Equal pressure steps along a lymphatic vessel.

In zone 1, where E_min < E_max, the amount of pressure increment contributed by each lymphangion increases as transmural pressure increases along the length of the lymphatic vessel (Fig. 4A). As a result, each lymphangion contributes an increasing amount of stroke work (Fig. 4B). However, there is an equal contribution of pressure from each lymphangion for zone 2 (Fig. 4C), where E_min = E_max. Since adjacent lymphangions must have equal SV, from Eq. 2 they will also contribute equal stroke work (Fig. 4D).

Fig. 4.

Illustration of lymphangion transmural pressure and stroke work for a vessel consisting of 3 lymphangions. A: for lymphangions in the low-pressure zone 1, the incremental pressures contributed by lymphangions are not equal. B: unequal pressure increments in zone 1 lead to unequal lymphangion stroke work, with stroke work increasing down the length of the vessel. C: pressure steps are equal for all 3 lymphangions in higher-pressure zone 2. D: equal pressure increments lead to equal stroke work in each lymphangion in zone 2.

DISCUSSION

The present work illustrates that the nonlinear pressure-volume relationship allows lymphangion stroke volume to be sensitive to transmural pressure and the axial pressure gradient at low pressures and to be insensitive to transmural pressures at higher pressures. The sensitivity at lower pressures occurs, because the slope of the end-systolic pressure-volume relationship (E_max) is greater than the slope of the end-diastolic pressure-volume relationship (E_min) (Fig. 1). As a result, lymphangion stroke volume and stroke work increase with transmural pressure (Fig. 1). This fundamental behavior of lymphangions helps limit edema formation from increased microvascular filtration, because a relatively small increase in interstitial fluid pressure (7, 30, 31) decreases the axial pressure gradient (ΔP) while increasing the transmural pressure (P_trans) (Eq. 4). In contrast, because E_min is nearly equal to E_max at higher pressures (Fig. 1), lymphangion stroke volume and stroke work are insensitive to transmural pressure (Fig. 1). This fundamental behavior of lymphangions helps limit edema formation from venous hypertension, because venous hypertension, which raises both lymphatic system inlet and outlet pressures (11), causes little effect on the lymphangion axial pressure gradient (Eq. 5). This insensitivity to transmural pressure has the additional benefit of ensuring that the stroke work is consistent along a string of lymphangions (Fig. 4). To arrive at these conclusions, we first identified a maximal transmural pressure P* that can likely occur from increased microvascular filtration alone. Then, we hypothesized that lymphangions were transmural pressure dependent below P* and transmural pressure independent above P*. We found that this could be accomplished by having a pressure-dependent end-diastolic pressure-volume relationship with different slopes in low- and high-pressure zones (Fig. 1). Our predicted function of a lymphangion is consistent with reported flow from a lymphatic system (Fig. 2) as well as the reported pressure-volume relationships of lymphangions (Fig. 3). This deductive approach to revealing the effect of highly nonlinear lymphangion properties was based on fundamental principles and simple mathematical modeling.

Inductively addressing the functional consequences of nonlinear lymphangion properties.

The difficulty in understanding the particular impact of nonlinear lymphangion properties from an inductive method is illustrated by analyzing lymphangion pressure-volume relationships without a priori assumption of a pressure that demarcates the two zones of lymphangion function. Through nonlinear parameter estimation, we found that the best-fit value for P* was 2.4 cmH₂O, which is close to our assumed value. The fit of the model to the data, however, is not particularly sensitive to the particular value of the assumed P*. For instance, in the extreme case of setting the pressure to 7 cmH₂O, the average error in the fit would only increase to 16.4%. Thus there is a relatively smooth transition from zone 1 to zone 2, where lymphangion transmural pressures may result from either increased marked microvascular filtration alone or mild venous hypertension. However, with moderate venous hypertension, lymphangions are expected to be exposed to transmural pressures well within zone 2. Although the value of P* was used to demarcate the two pressure zones, its absolute value is not important physiologically. Instead, the present work elucidates the principle that the nonlinear lymphangion pressure-volume relationship minimizes edema resulting from both increased microvascular filtration and venous hypertension.

Pump capacity depends on the slopes and the intercepts of lymphangion pressure-volume relationships.

The simple analytical formula for lymphangion stroke volume (Eq. 1) provides unique insights with which to explore the relative importance of lymphangion structural properties. Because lymphangion properties are different at high and low transmural pressures, the sensitivity to changes in the parameters is strikingly different. To illustrate, a simple “parameter sensitivity analysis” was performed to determine how much stroke volume would change for a 10% change in model parameters. At very low transmural pressures (i.e., zone 1), the dead volume (i.e., the intercept of the end-diastolic pressure-diameter relationship) does not have any effect on stroke volume. At really high transmural pressures (i.e., zone 2), however, a 10% increase in either the end-diastolic (V_{o, ed2}) or end-systolic (V_{o, es}) dead volumes results in a change of >100% in stroke volume. Combining these parameters, the differential dead volume (ΔV_o = V_{o, ed2} − V_{o, es}) arises as an independent variable in Eq. 1. The sensitivity of stroke work to ΔV_o (45%) is similar to the sensitivities to E_min (53%) or E_max (45%). The sensitivity of the system to ΔV_o is further illustrated in Eq. 6, in which ΔV_o appears as an explicit parameter in zone 2, but is absent in zone 1. Although E_min has been used as an index of lymphangion diastolic tone (11), and E_max has been used as an index of systolic contractility (16, 23), ΔV_o emerges as an independent, novel index of pump function.

Simple approximations allow prediction of global behavior.

The present work extended a previously reported lymphangion model (24) that was tested by comparing its predictions to experimental data on the one hand, and a more complex numerical model on the other (32). This model neglects lymph inertia and viscosity, since they have negligible effects on developed pressure gradients when the lymphangion acts as a pump. However, this assumption makes the model inappropriate to characterize conduit behavior that arises when the normal axial pressure gradient reverses (22, 23). More importantly, the present model does not include the effects of endothelial shear stress, which is known to cause dilation and even pump inhibition at high flows (13). The assumption of a constant time-varying elastance implicitly assumes that baseline lymphangion contractility and tone do not change with flow. To use the present model to predict lymph flow over very wide ranges, it would be necessary to adjust the tone and contractility (E_min, E_max) and the dead volumes (V_{o, es}, V_{o, ed2}) appropriately. The most important simplification of this model is the effect of contraction frequency, which increases with transmural pressure (19). To derive stroke work, it was assumed that contraction frequency does not change. When there is no change in contraction frequency with pressure, insensitivity of stroke volume to transmural pressure would result from E_min= E_max. When there is an increase in contraction frequency with pressure, this requirement is relaxed, and E_min can be higher than E_max. In this case, the end-systolic and end-diastolic pressure-volume relationships would no longer need to be parallel but could begin to approach each other at higher pressures. Indeed, this trend can be found in some other pressure-volume relationships such as those reported by Ohhashi et al. (21) and Li et al. (16).

Theoretical analysis is consistent with observed behavior.

Using cardiac analogies and assuming that E_min and E_max were equal at higher transmural pressures have provided key insights. In particular, the present work has elucidated the functional consequences of a curvilinear end-diastolic pressure-volume relationship (16), how lymphangions are able to maintain flows with changes in central venous pressure (8), and why lymphangions in series maintain equal pressure increments despite the higher transmural pressures in downstream lymphangions (12). Furthermore, this insight shows how lymphangions ensure a minimum necessary flow for protein turnover, immune function, and lipid absorption. None of these insights can arise from experimental approaches alone but instead from mathematical modeling based on measured lymphangions properties and fundamental physical principles.

GRANTS

This work was supported by National Heart, Lung, and Blood Institute Grants K25 HL-070608 (C. M. Quick) and R01 HL-092916 (R. H. Stewart), American Heart Association Grants 0565116Y (C. M. Quick) and 0365127Y (R. H. Stewart), and Centers for Disease Control and Prevention Grant 623086 (G. A. Laine).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.