Inhibition of Contraction Strength and Frequency by Wall Shear Stress in a Single-Lymphangion Model

Abstract

The phasic contractions of collecting lymphatic vessels are reduced in strength and occur at diminished frequency when a favorable pressure difference and the resulting antegrade flow create large fluid shear stresses at the luminal surface. This paper describes a minimal phenomenological model of this mechanism that is applied to a previously validated numerical model of a phasically contracting lymphangion. The parameters of the inhibition model are quantitatively matched to observations in isolated segments of rat lymphatic vessel, first for mesenteric lymphatics then for thoracic duct, and outcomes from the numerical model are then qualitatively compared with recent observations in isolated segments of rat thoracic duct.

1. Introduction

The contractions of collecting lymphatics are subject to a wide range of regulatory mechanisms, both local and centrally mediated. Predominantly sympathetic [1] but also parasympathetic [2] nerve fibers inhabit the vessel walls. Although the fibers are functional and can be stimulated electrically [3], the extent of their normal influence on contractility remains uncertain. As with blood vessels, lymphatics have been demonstrated to be sensitive to a large range of humoral agents; the relevant literature is listed by Zawieja [4]. Turning to local factors, collecting lymphatic vessels develop basal muscle tone, which increases with elevated distending pressure and decreases upon its reduction [5]. In addition, both the frequency and the amplitude of their phasic contractions vary with distending pressure and its rate of change [6]. Both standing tone and phasic contractions are inhibited by nitric oxide (NO) produced by lymphatic endothelial cells in response to fluid shear stress [7,8], and other endothelium-derived factors also play a role in the regulation of contractility by flow [9,10].

Caulk et al. [11] have previously adduced a model of lymphatic contraction inhibition by wall shear. Their model, which uses five adjustable constants, perturbs contraction frequency and active tension to an extent depending on the current departure of wall shear stress from a prescribed set point, in a loop with a 5-min time constant. However, experimental results from the same group [12] indicate that the real shear response can be more rapid. Working with a FitzHugh-Nagumo reduction of the Hodgkin–Huxley ion dynamics model for action potentials, Contarino and Toro [13] added inverse dependence of contraction frequency on wall shear stress. Their model, which does not modulate contraction strength, uses three adjustable constants: a reference wall shear stress, and inhibiting parameters for wall shear stress and for contraction.

Beyond these lumped-parameter models, NO dynamics have been incorporated as the element responsible for ending individual contractions into a two-oscillator model which couples shear-related inhibition by NO and muscle-tension recruitment by calcium ions. The model has been realized in spatially distributed numerical form using the lattice-Boltzmann method [14], and its nonlinear dynamics have been further explored by Baish et al. [15]. However, its concept of the role of time-varying NO concentration has been questioned [16,17].

Our lumped-parameter model of a contracting lymphatic vessel has been progressively developed [18,19], most recently to encompass dependence of contraction frequency on distending pressure for a single lymphangion [20] and in vessel segments consisting of multiple lymphangions [21]. We here sought to incorporate shear-induced contraction inhibition in a lumped-parameter model that minimizes the number of additional parameters and produces results consistent with experimental findings. Much of the biological literature is concerned with molecular signaling and lacks the quantitative data necessary to characterize the system adequately to allow model construction. The exception is the work of Gashev et al. [7], which provides measurements of frequency reduction and contraction strength reduction (expressed in terms of ejection fraction) as a function of the pressure difference causing flow through their isolated segments of rat mesenteric collecting lymphatic and thoracic duct. They maintained mean pressure constant, and diastolic diameter for the mesenteric segments varied insignificantly. Consequently, although the shear stress is unknown, one can be confident at the low Reynolds numbers of lymphatic flows that the flow-rate and the wall shear stress in diastole would have varied in direct proportion to the pressure difference. We accordingly developed the model described in Sec. 2 and matched the model parameters to the data of Gashev et al. [7].

We also make comparison with the data of Kornuta et al. [12], whose apparatus allowed for control of flow-rate. Their findings are for rat thoracic duct and extend the range of contraction-inhibition behaviors documented in lymphatic vessels. Thoracic duct lymphatics differ in many respects (e.g., diameter, segmental contraction nonuniformity) from mesenteric lymphatics, and various differences between the contractile and regulatory behaviors of the two vessels have already been documented [22–23]. Although for this purpose we fitted the shear-inhibition model parameters to the thoracic duct data of Gashev et al. [7], the rest of our lymphatic model uses parameter values that describe passive and active properties of mesenteric vessels, so the resulting comparisons are necessarily indicative only. Our continued extensive use of mesenteric parameter values in the model, even while we compare with rat thoracic data on inhibition by shear, is justified because, despite the mechanical properties provided by Caulk et al. [24], no body of thoracic-duct parameters equivalent to the mesenteric measurements of Davis and coworkers [18,20,25] is available.

2. Methods

The equations and parameters of the one-lymphangion mathematical model to which the contraction–inhibition model is here added have been described previously, and space does not permit detailing here; they are for instance given in the Appendix of Bertram et al. [20].

The algorithm of the contraction–inhibition model is as follows (see Fig. 1). First, the spatially averaged instantaneous wall shear stress S(t) for the lymphangion is computed as

$$S\left(t\right)=\frac{32}{\pi }\frac{\mu Q}{D^3}$$

Fig. 1.

Schematic showing waveforms and variables involved in the calculation of the additional delay t_d of the next contraction caused by wall shear stress. Waveforms of contraction activation M(t) and wall shear stress S(t) are shown; normally, S(t) would reflect at least partly the effect of contractions, but for illustrative purposes here it is shown as independent.

where μ is the lymph viscosity, Q(t) = [Q₁(t) + Q₂(t)]/2, with Q₁ and Q₂ being the flow-rates through the inlet and outlet valves, respectively, and D(t) is the lymphangion diameter. Only the excess wall shear stress S_xs(t) = S(t) − S_thr, where S_thr is a fixed threshold, will be considered; this eliminates the small shear stress transients caused by contractions in the presence of an adverse pressure gradient. Because of the one-way valves in collecting lymphatics, large flows can only occur in the forward direction; therefore, we assume S_xs(t) ≥ 0.

In the absence of inhibitory effects, the next systole would begin after a diastolic relaxation of duration t_r, where the length of t_r is determined at end-systole by consideration of the average transmural pressure over the systole that has just completed. This mechanism has already been described in detail as implemented for both a single lymphangion [20] and multiple lymphangions in series [21].

Consideration of a possible extra delay because of wall shear stress occurs at the time t₀ when the next systole would otherwise begin. If S(t) > S_thr at this time, a first value of added delay is computed as

$$t_{\mathrm{d}0}=k_{\mathrm{tW}}{S}_{\mathrm{xs}}(t_0)$$

where k_tW is a converting constant with units of time per shear stress. At each subsequent time-step, t_d is recalculated using the average of S_xs(t) since t₀, i.e., at some general time t = t₁,

$$t_{\mathrm{d}}=k_{\mathrm{tW}}\frac{1}{t_1-t_0}{\int }_{t_0}^{t_1}{S}_{\mathrm{xs}}\left(t\right)dt$$

The result is compared with the current time. Eventually, after a variable duration which depends on the time-course of S(t), the added delay will expire, i.e., t₁ > t₀ + t_d, and at this time t_c, the next systole is initiated. This formulation means that eventually there will always be a contraction, even if the imposed shear stress stays above the threshold S_thr.

The time-course of systolic contraction activation M(t) has fixed shape, and insofar as described thus far, the duration of systole is fixed; M(t) depends on neither transmural pressure nor wall shear stress, but rises to a peak and decays according to a prescribed waveform such as

$$M\left(t\right)=\left[1-\text{cos}\hspace{0.17em}\left(2\pi f\left(t-t_{\mathrm{c}}\right)\right)\right]/2$$

where t_c ≤ t ≤ t_c + 1/f, and 1/f is the duration of systole; other waveform shapes can be substituted [26,27].

However, in accordance with the findings of Gashev et al. [7], the shear stress also modifies the strength of contraction, i.e., reduces the peak height of M(t), and we choose to keep the waveform shape of M(t) self-similar, by reducing the duration of systole in proportion to the amplitude reduction. This yields reduced systoles that are also abbreviated, which is more faithful to what was seen biologically; see, e.g., Fig. 6 of Kornuta et al. [12].

Specifically, we use the values of t_d calculated as above to influence the amplitude and duration of M(t) by choosing a value for the scaling constant α and calculating from it a dimensionless factor, m_scal, as

$$m_{\text{scal}}={\left(\frac{1/f}{1/f+t_{\mathrm{d}}}\right)}^{\alpha }$$

The exponent α is a scaling parameter, which is applied to the waveform of systolic activation as

$$M\left(t\right)=m_{\text{scal}}\left[1-\text{cos}\left(2\pi f\left(\frac{t-t_{\mathrm{c}}}{m_{\text{scal}}}\right)\right)\right]/2$$

giving a systolic duration m_scal/f. When the scaling constant α = 0, no reduction happens; increasingly positive values of α prescribe increasingly attenuated systoles when the shear stress is sufficient.

We choose the value S_thr = 3 dyn cm⁻² to ensure that shear stresses caused by intrinsic pumping contractions against an adverse pressure difference do not cause shear-induced inhibition of rate or contraction amplitude. The values of the other two constants, k_tW and α, are set to match, respectively, the rate and amplitude reductions observed for rat lymphatics perfused under conditions of favorable pressure difference by Gashev et al. [7].

The results of Gashev et al. [7] on the inhibition by flow of contraction frequency and amplitude in rat mesenteric lymphatics are replotted, with curves fitted, in Fig. 2. They did not measure flow-rate, and the reported circumstances of their experiment do not allow conversion of the applied pressure difference to flow-rate. Although similar values of pressure difference can be applied in a computer model, there is no guarantee that the modeled conduit is similar to the experimental one, so it is not possible to achieve a match in terms of flow-rate or wall shear. However, their results for normalized frequency and for ejection fraction (=1 − D_min²/D_max², where D_max and D_min are the diameters at end-diastole and end-systole, respectively) both indicate with reasonable precision an asymptote representing the maximum inhibition as the pressure difference (and flow-rate) increased. The fitted curves provide approximate values for these asymptotes, which can then be matched in the computer model.

Fig. 2.

Data points: frequency (blue) and ejection fraction (magenta), both normalized by the value at zero pressure difference, versus the pressure difference Δp driving forward flow—data for rat mesenteric lymphatics taken from Fig. 2 of Gashev et al. [7]. Curves of the form y_min + (1 − y_min) exp (−Δp/Δp_c) are fitted to the data, where Δp_c is the pressure constant, and y_min is the asymptote approached as Δp → ∞.

3. Results

For the purpose of matching the model's shear-inhibition parameters to the asymptotes found for the Gashev data, the computer model was run with several steps of inlet/outlet pressure difference while keeping the same mean pressure throughout; see Fig. 3, which uses the eventually found parameter values for mesentery. Each step causes an increase in the diastolic flow-rate and the diastolic shear stress. When the pressure difference is zero, contractions cause transient flow out of each end of the lymphangion (i.e., negative flow-rate at the inlet), immediately followed by transient flow in at each end as the contraction decays. The shear stress used for the inhibition model stays at zero, since it is based on the average of inlet and outlet flow-rates. Once there is a nonzero pressure difference and consequent nonzero diastolic flow-rate, the predominant effect of contraction is to reduce this flow-rate, because the reduction of lymphangion diameter transiently increases the resistance to flow. When examined on an expanded time-axis (not shown), it is found that the outlet flow-rate slightly exceeds the steady diastolic flow-rate right at the beginning of each systole, and the inlet flow-rate slightly exceeds it right at the end. The transiently increased resistance is reflected in the shear stress, which increases during each systole.

Fig. 3.

Inhibition by shear of contraction frequency and strength in the single-lymphangion computer model. Top panel: p_a and p_b (black) are the applied pressures; p_a steps up while p_b steps down, keeping the same average value (p_a + p_b)/2 of 5 cmH₂O while Δp increases in steps of 1 cmH₂O from 0 to 7 cmH₂O. The external pressure p_e (dashed line) is 1 cmH₂O. The pressures within the lymphangion are p₁ (blue), p_m (red), and p₂ (green) at the upstream end, mid-point, and downstream end, respectively. The model includes representations of the resistance of the inlet and outlet micropipettes that would have been used in the experiment, and p_i (magenta) and p_o (beige) are the pressures after the inlet pipette and before the outlet one, respectively. Second panel: lymphangion diameter D (blue) and contraction activation M (cyan). Third panel: flow-rates Q₁ (blue) and Q₂ (green) through the inlet and outlet valves, respectively, and Q_av = (Q₁ + Q₂)/2 (red). Both valves stayed open throughout this procedure. Bottom panel: wall shear stress S computed from Q_av. Dashed line: S_thr.

The results in Fig. 3 were analyzed in terms of normalized frequency and normalized ejection fraction; see Fig. 4. Good fits to the computed values were achieved with curves of the form y = y_min + (1 − y_min)exp[−(Δp − Δp₀)/Δp_c], where y = nfreq, the normalized frequency, in the left panel, and y = nEF, the normalized ejection fraction, in the right panel.

Fig. 4.

Computed reductions of frequency (left) and ejection fraction (right) with increasing pressure difference p_a − p_b, where p_a and p_b are the inlet and outlet pressures, respectively. Both frequency and ejection fraction are normalized by their value at zero pressure difference. The computed values (data-points) are fitted by three-constant curves as indicated in each panel. The values for the asymptotes nfreq_min and nEF_min (given in the figure) closely approximate the experimental values shown in Fig. 2.

The relation between pressure difference and the unreported wall shear stress in the experiments of Gashev et al. [7] would have depended on the empirical setup. There is no likelihood that the same relation applies in our model, and we do not make this assumption. Because the shear stresses implied by the abscissa of Fig. 4 will not correspond to those of Fig. 2, we match only the asymptote values for each ordinate variable. Relative to the equivalent curve fits to the experimental data in Fig. 2, the additional fitted constant Δp₀ here, which is the value of Δp at which y begins to fall, reflects the value of S_thr that we chose. This does not affect the fitted values of nfreq_min and nEF_min as shown in Fig. 4, which approximate the asymptotes of Fig. 2 when the inhibition-model parameters take the values k_tW = 1.2 s/(dyn cm⁻²) and α = 0.65.

Gashev et al. [7] also measured inhibition of contractions as a result of imposing favorable axial pressure differences in segments of rat thoracic duct. In various respects, our numerical model is overall tailored to the properties of rat mesenteric lymphatics, but the model of inhibition by wall shear stress described in this paper is sufficiently self-contained that its properties can be aligned to the thoracic duct findings of Gashev et al. without affecting other aspects of the overall model. Following a similar procedure (not shown) to that described earlier for obtaining inhibition-model parameters appropriate to the rat mesenteric lymphatic results, we obtained values appropriate to thoracic duct of k_tW = 12 s/(dyn cm⁻²) and α = 0.80. Although it may seem that the value of α is relatively little changed, in fact the amplitude attenuation with shear is greatly increased, because k_tW controls t_d, which is used in calculating m_scal.

Using these values, the model was then applied to the observations of Kornuta et al. [12]. In isolated segments of rat thoracic duct, they found that the rate of contractions could be entrained to the frequency of imposed sinusoids of pressure difference versus time, varying between zero and a positive peak (favorable for forward flow). The model as described reproduces this finding, as shown in Fig. 5. For this purpose, the amplitude scaling was turned off by setting α = 0, since their Fig. 4 shows undiminished contraction amplitude during entrainment; however, similar entrainment happens when the inhibition of contraction amplitude is allowed in our model (result not shown). The pressure difference in Fig. 5 is zero until 16.25 s, during which time the frequency of contractions in the model was set by the applied transmural pressure (p_a + p_b)/2 − p_e = 4 cmH₂O as described previously [20]. This pressure (and the diastolic duration so prescribed) did not change subsequently, but the additional diastolic delays computed in response to the time-varying shear stress produced a 1:1 entrainment of contraction rate to pressure-difference frequency. There is a noticeable alternation of contraction phase relative to the shear-forcing cycle, but this gradually dies away when the simulation is continued further than is illustrated in Fig. 5.

Fig. 5.

Response of the computer model to applied pressure-difference sinusoids at a frequency of 0.125 Hz, as used by Kornuta et al. [12] in the experiment shown in their Fig. 4. Traces as described in the caption of Fig. 3.

With the amplitude scaling restored, the model was also applied to the experiment shown in Fig. 6 of Kornuta et al. [12] in which the frequency of applied pressure difference was halved to 0.0625 Hz. Under these conditions, they found that contractions were again prevented during the latter half of each positive shear cycle, while minor contractions of variable amplitude occurred just after each return to zero shear. The equivalent response from the model is shown in Fig. 6. Strong contractions occur only just after the instants of zero pressure difference. Reduced-strength contractions occur at the end of each shear cycle. No contractions occur near the peak itself, i.e., the inhibition is then strong enough to delay contraction to a later time. Overall, the result is as found by Kornuta et al. [12], i.e., the contractile events are constrained to follow the imposed low-frequency shear cycles, and more than one contraction is attempted each time the imposed shear is low.

Fig. 6.

Application of pressure-difference cycles as in Fig. 5, but at half the cycle frequency. Traces as described in the caption of Fig. 3.

Finally, in Fig. 7, the model is used to emulate the experimental procedure shown in Fig. 1 of Kornuta et al. [12], whereby the pressure difference is linearly ramped up from zero, causing contractions to become progressively less frequent. Although again the thoracic duct segment observed by Kornuta et al. [12] did not show any reduction of the strength of contractions during the time-frame of their figure, here we choose to retain the value of α which matches the thoracic duct data of Gashev et al. [7]. Note that we could instead readily match unchanging strength if desired, just by switching off the amplitude attenuation. However, we here increased the (constant) applied average transmural pressure from 4 to 7 cmH₂O, i.e., (p_a + p_b)/2 = 8 cmH₂O. This causes the initial transmural-pressure-dependent contraction rate to be higher, because the length of diastole before the pressure difference starts is thereby reduced from 1.46 to 0.88 s.

Fig. 7.

Application of a linear pressure-difference ramp. Traces as described in the caption of Fig. 3.

The application of low shear (from 10 s onward) does not change the contraction rate. Once the shear threshold is reached, the subsequent reduction of contraction frequency follows a smoothly progressive relation, but the rate of frequency change is sufficiently great that a pause between contractions in response to shear appears to arise quite suddenly. This behavior exactly parallels what Kornuta et al. [12] observed in their Fig. 1. Again, we emphasize that the shear-inhibition model is quantitatively matched to the thoracic duct results of Gashev et al. [7]. The discrepancies in amplitude reduction from the findings of Kornuta et al. [12] in their Figs. 1 and 4 indicate that these thoracic duct preparations behaved qualitatively differently from those of Gashev et al. in respect of contraction strength change. We note that individual specimens can display much variability, viz., the differences in spontaneous frequency of contraction at zero shear seen by Kornuta et al. [12]—their Fig. 5.

4. Discussion

The model of inhibition of contraction strength and frequency implemented here is largely phenomenological; it does not directly represent any individual component of the underlying mechano-sensing and intracellular machinery giving rise to the pharmacological outcomes which ultimately mediate the biological responses. Rather, the strength of the present model is its simplicity; only three constants are needed, values for at least two of which (and arguably all three) are tied down by experiment. Nevertheless, the model successfully emulates qualitatively the currently available data. Our model assumes that there exists a maximal inhibition at large pressure gradient for both frequency and ejection fraction. While there is no experimental proof of this, the data of Gashev et al. [7] strongly suggest it. For thoracic duct frequency, the experimental asymptote is indistinguishable from zero, which the model approximates.

Comparison of Figs. 2 and 4 would suggest that S_thr = 3 dyn cm⁻² may overestimate the value of any equivalent biological threshold which pertained in the circumstances of the mesenteric experiments of Gashev et al. [7]. However, we made no attempt to fit Δp₀ constants to the experimental data, believing that to fit three constants to only four data points per curve (after the use of one for normalization) would verge on overfitting. Inspection of Fig. 2 suggests that the data on reduction of ejection fraction do support the idea of a significant pressure-difference threshold. On the other hand, the data on frequency reduction suggest if anything a negative threshold, which is nonsensical. One can only conclude that the available data for mesentery at this time do not permit accurate specification of a shear-stress threshold above which inhibition of contractions begins; indeed, the possibility that the threshold is zero is not excluded. Kornuta et al. [12] estimated a nonzero threshold for thoracic duct (0.64 dyn/cm² at 3 cmH₂O transmural pressure and 0.97 dyn/cm² at 5 cmH₂O), but since our lymphatic model is dimensioned for mesentery, we did not use these values. The frequency curve in Fig. 2 also matches quite well the data on frequency versus shear stress newly published by Mukherjee et al. [28]. While the shape of the frequency versus pressure difference (or wall shear stress) curve is similar for any given vessel, it seems that there is a shift to the right with decreasing vessel size—see their Fig. 2. It would not have been possible to capture this behavior from the data of Gashev et al. [7] as only the mean of the population is given.

We treated the data of Gashev et al. [7] as a ‘gold standard,’ because (most usefully for the purposes of numerical modeling) they provided values for normalized frequency and ejection fraction at five different levels of imposed pressure difference, a measure which can confidently be expected to be linear with shear stress at low Reynolds number.1 These data clearly show smooth trends of increasing inhibition with increasing pressure difference, allowing the estimation of curve-fit parameters. Furthermore, they investigated both mesenteric and thoracic duct lymphatics (n = 7 and 9, respectively). In contrast, the data of Kornuta et al. [12] are for thoracic duct only and do not provide data on how the level of inhibition varies with the extent of shear stress. Thus, it would not be possible to fit even our simple model of shear inhibition to the Kornuta data.

When the shear-inhibition model is fitted to the thoracic duct data of Gashev et al. [7], and then procedures similar to those used by Kornuta et al. [12] are simulated, the observed entrainment of contraction frequency to the imposed frequency of shear variation is successfully imitated (Fig. 5). Also as in the experiments, when the imposed frequency of shear cycles is low (Fig. 6), contractions are inhibited completely while shear is high, and contractions of varying amplitude occur when it is low. If shear is ramped up (Fig. 7), there is no effect on contraction rate until a threshold is reached, then the interval between contractions rapidly lengthens, just as Kornuta et al. [12] observed. However, the model matched to the thoracic duct data of Gashev et al. [7] predicts simultaneous shear-induced reductions of contraction strength which were absent in some of the observations of Kornuta et al. [12]. This is not a weakness of the model; it is here simply pointing up a qualitative discrepancy between the measurements of Gashev et al. [7] and those of Kornuta et al. [12] for nominally the same lymphatic vessel and species. Further experimental work is needed to explain the genesis of this difference.

The isolated segments of rat thoracic duct used by Kornuta et al. [12] seem to have contracted spontaneously at a rate which could scarcely be increased; no diastolic pause occurs between the contractions illustrated in the absence of shear in their Fig. 4. Likewise, during the post-zero-shear phases where the segment of their Fig. 6 attempted several contractions, no diastole separates the individual contractions. Our model does not reproduce this behavior, because the dependence of rate on transmural pressure [20], which follows the pressure/rate relation observed in an isolated segment of rat mesenteric lymphatic, ordains a relatively low maximum rate at this modest transmural pressure. However, this matter is external to the performance of our contraction–inhibition model.

The model of contraction inhibition by shear implemented here could in principle be applied also to our model of multiple lymphangions in series. This incorporates contraction rate varying inversely with the distending pressure. Since every lymphangion has this ability, lymphangion contractions are coordinated by signals propagated along the chain [21]. A signal is conducted in both directions at the start of systole, and will initiate systole in the adjacent lymphangions unless they are in their refractory state during and shortly after contraction. However, the phenomenon of contraction inhibition by shear is manifested under conditions of favorable pressure difference along the vessel, when there is a large forward flow-rate and all the interlymphangion valves are open. In this situation, there is no material difference between conduits consisting of one or several lymphangions, and both Gashev et al. [7] and Kornuta et al. [12] chose to work with vessel segments that did not include valves. Other possible extensions include the potential for modulation of pumping activity by immune system adaptations, including those within lymph nodes.

Funding Data

• National Institutes of Health (NIH) (Grant No. U01-HL-123420; Funder ID: 10.13039/100000002).