Fluid-dynamic modeling of flow in embryonic tissue indicates that lymphatic valve location is not consistently determined by the local fluid shear or its gradient

Abstract

Objective

Intravascular lymphatic valves often occur in proximity to vessel junctions. It is commonly held that disturbed flow at junctions is responsible for accumulation of valve-forming cells (VFCs) at these locations as the initial step in valve creation, and the one which explains the association with these sites. However, evidence in favor is largely limited to cell culture experiments.

Methods

We acquired images of embryonic lymphatic vascular networks from day E16.5, when VFC accumulation has started but the developing valve has not yet altered the local vessel geometry, stained for Prox1, which co-localizes with Foxc2. Using finite-element computational fluid mechanics, we simulated the flow through the networks, under conditions appropriate to this early development stage. Then we correlated the Prox1 distributions with the distributions of simulated fluid shear and shear stress gradient.

Results

Across a total of 16 image sets, no consistent correlation was found between Prox1 distribution and the local magnitude of fluid shear, or its positive or negative gradient.

Conclusions

This, the first direct semi-empirical test of the localization hypothesis to interrogate the tissue from in vivo at the critical moment of development, does not support the idea that a feature of the local flow determines valve localization.

1. INTRODUCTION

Kampmeier noted over 95 years ago that intravascular lymphatic valves tend to occur near lymphatic vessel confluences ⁽¹⁾. More recent observations (e.g. Choi et al. ⁽²⁾; also see review by Bazigou et al. ⁽³⁾) have only confirmed this association.

The commonly accepted hypothesized mechanism behind this is that the lymph flow changes in junction regions to a complex or disturbed pattern. This supposedly causes oscillatory fluid shear at the vessel wall, triggering an accumulation of transcription factors that begin the valve-forming process.

This idea has been particularly advocated by Petrova and co-workers, who have described the cascading involvement of Prox1, Foxc1, Foxc2, connexin37, calcineurin, NFATc1, laminin α5, GATA2, etc. ^(4–7). Bazigou et al. ⁽⁸⁾ found that integrin-α9 is also required, while Kanady et al. ⁽⁹⁾ showed that valve development was disrupted by connexin37 and connexin43 deficiencies. Srinivasan & Oliver ⁽¹⁰⁾ showed that Prox1-expressing endothelial cells are vital to formation of the two lymphovenous valves that guard against blood ingress at the lymphatic vascular system’s exits. Lapinski et al. ⁽¹¹⁾ found that disrupting the Rasa1 gene caused loss of endothelial cells from lymphatic valve leaflets, leading to valve leakage. Choi et al. ⁽²⁾ provided evidence that the oscillatory-shear-stress-sensing ion channel Piezo1 is essential to valve development, although without linking it specifically to valve localization. A number of reviews (e.g. Planas-Paz & Lammert ⁽¹²⁾) have reinforced the near-universal acceptance of the concept of flow-mediated lymphatic valve localization, albeit without adding to the primary evidence. The direct evidence in favor of initial site selection on a flow signal is essentially confined to cell culture, where it is found that oscillatory fluid shear upregulates Foxc2 (but not Prox1) whereas similar levels of steady fluid shear do not.

Valvogenesis is inherently difficult to investigate in vivo, since it occurs in the embryo. In the mouse, with 21-day gestation, the first steps in the process occur at around day E16, with upregulated expression of first Prox1, then laminin α5 and Foxc2, in lymphatic endothelial cells at particular sites along vessels ⁽⁴⁾. Later, with such Prox1/Foxc2-rich cells forming distinct clusters, a localized ring-like constriction of the vessel starts to form, and functional valves are present by E18.5.

The most problematic element in this postulated train of events is the initial flow signal. Experiments in cell culture have tended to involve levels of shear, whether oscillatory or not, that would characterize the mature lymphatic vasculature, rather than the situation existing in the newly developing embryonic lymph vessels, barely yet formed into a stable network. There has also been a tendency to assume that processes determining valve formation and siting are the same as those involved in subsequent valve maintenance. The mature valve, by its very nature, causes a profound disruption in its immediate vicinity, even when open, of what would otherwise be unremarkable laminar flow along the vessel ⁽¹³⁾, and cultured lymphatic endothelial cell responses have been investigated specifically in such situations of highly localized shear gradient ^(14–17). But valvogenesis starts with the selection of a valve site in a vessel or near a vessel junction where such flow as exists is not yet in any way altered by the presence of a valve.

Sabine et al. ⁽⁴⁾ suggested that junction regions were more likely to have recirculating or disturbed flow, supporting this claim with reference to literature dealing with flow in large blood vessels. However, there is a profound difference between the high-speed blood flow which occurs in (say) the carotid bifurcation of a large animal, and the creeping flow of lymph in microscopic developing lymphatic vessels. The defining difference is the Reynolds number of the flow, which involves the product of the fluid flow speed and the diameter of the vessel. In the case of the blood flow, the Reynolds number of the arriving flow along the common carotid artery is of order 500; the flow is accordingly laminar, but recirculates as it passes the carotid sinus. Furthermore, the flow is pulsatile, varying greatly during each heartbeat (this feature is characterized by a second fluid-dynamic dimensionless quantity, the Womersley number). The unsteady flow causes the recirculating eddy to vary cyclically in size and position, such that certain wall locations experience cyclically reversing shear. This latter phenomenon can be taken as a working definition of the critical feature of disturbed flow as it relates to the present discussion; note that it depends on unsteady flow.

In the embryonic lymphatic vascular junction, the situation is completely different. The vessels are tiny and the flow is small, so the Reynolds number is of order 1 or less ⁽¹⁸⁾. Also because of the small vessels, the flow will be quasi-steady (defined by low Womersley number). Quasi-steady flow occurs if the variation over time is slow relative to the rate at which viscosity propagates its control of momentum throughout the flow domain. Consequently the flow pattern does not differ at any instant from that which would pertain if the instantaneous flow were constant; it is identical to that of the corresponding steady flow. Together, these characteristics (low Reynolds and Womersley numbers) produce viscous flow, i.e. a flow which is qualitatively that which would occur if the fluid were highly viscous, irrespective of the fact that lymph has approximately the viscosity of water. Localized flow recirculation cannot occur at vessel junctions under such circumstances. Because there are as yet no valves to prevent reversal, the flow everywhere may well reverse intermittently. The flow through the junction may reverse, consequent upon overall pressure gradient changes resulting from tissue deformation on a larger spatial scale, but it will reverse simultaneously everywhere in the network locality, without special flow features arising at the junction itself. Sabine et al. ⁽⁴⁾ themselves observed such intermittent reversal with the aid of fluorescence lymphangiography.

The self-similar nature of fluid-dynamic viscous flow is the foundation of the present investigation, because it enables simulation of the flow in the absence of direct measurement. To investigate what localizing signals can be derived from the flow by valve-forming cells (VFCs), we do not need to know quantitatively what flow actually occurred in a given lymphatic vascular network; it suffices to simulate qualitatively the viscous flow which we know on universal grounds would have occurred. The simulated flow, and the corresponding spatial distribution of fluid shear stress applied to the walls of the vessels in the network, can then be compared with the actual distribution of VFCs that occurred in the lymphatic vascular network at the start of valvogenesis. Since in blood vessels it has been suggested that endothelial cells sense spatial gradients of wall shear stress ^(19–20), it is also necessary to derive and compare the spatial gradient of wall shear stress along the lymphatic vessels.

In an earlier paper ⁽²¹⁾, we analyzed a small number of images, reaching the conclusion on the basis of these preliminary data that the distribution of VFCs did not appear to co-localize consistently with any signal that could be derived from the viscous flow. Here, we analyze a much larger number of images, and utilize a different and additional statistical technique to make the comparisons between simulated flow and observed VFC distribution. The new statistical approach gives improved estimates of statistical significance.

2. MATERIALS AND METHODS

2.1. Image analysis and flow simulation

As noted above, to investigate the stage of embryonic development when the location of intravascular valves is being determined, it is necessary to consider the lymphatic vessel networks from day E16; any later, and the developing valves themselves start to influence the flow pattern. Accordingly, we obtained from three different laboratories (see Acknowledgments) images of lymphatic networks in small areas of E16 murine dorsal skin. We initially attempted to determine the 3-dimensional geometry of such networks from confocal image stacks, but the information gained normal to the plane of individual images, i.e. normal to the plane of the skin itself, was insufficient. In any case, the process of preparation of such stained-tissue images necessarily flattens the lymphatic vessels (Sabine, private communication). For simulation purposes, we therefore treated the 2-dimensional image information as showing networks of channels extending indefinitely in the direction normal to the image. Although there are important differences between 3D tube flow and 2D channel flow (e.g. the former allows recirculating eddies that do not consist of closed streamlines, and the center-line velocity of the fully developed laminar profile is related by a different constant to the wall shear), in viscous, i.e., fully developed, laminar flow the flow patterns in a network of channels and in the center-line plane of tubes are qualitatively the same. Thus we can compute in 2D the distribution of wall shear that would have existed in a largely planar network of tubes, a description that fits the skin lymphatic vessel networks under consideration here.

To determine the distribution of valve-forming cells in the wall of the lymphatic vessels, we used images of tissue stained for the presence of Prospero-related homeobox protein 1 (Prox1). Although expression of Prox1 in cell culture does not appear to be flow-sensitive ⁽⁴⁾, it co-localizes with forkhead box protein C2 (Foxc2), in which it is, and thus the observed clustering of Prox1 can be used as a surrogate for the supposedly flow-determined clustering of Foxc2. To determine the geometry of the vessel networks, we used images of the same networks, but with the tissue stained instead for vascular endothelial growth factor receptor 3 (Vegfr3).

The schematic of Fig. 1 shows the principal steps in the derivation of information on the spatial distributions of Prox1 density and flow-induced wall shear stress from the input images. The Vegfr3-stained images were manually digitized to obtain the positions in 2D image-space of the vessel walls, a process called segmentation. We tested both existing and own-code image-processing software tools to segment the images in semi-automated ways, but ultimately concluded that the time that had to be spent on each such image cleaning up the effects of automatic segmenting tools outweighed the time taken to digitize by hand. The images varied spatially in intensity and distinctness, and many decisions had to be made about which faint vessels to include and which not. Automated image processing is problematic where such considerations arise. The manual digitization was accomplished using custom code written in Matlab (The Mathworks, Natick MA, USA). Each network image gave rise to some 2−3,000 digitized points, which were imported into the finite-element software ADINA-CFD (formerly from ADINA R&D Inc., Watertown MA, USA, now available from Bentley Systems Inc., Exton PA, USA).

FIGURE 1.

Schematic of main steps in getting from the initial images to arrays of numbers representing Prox1 density and fluid shear stress vs. location along vessels. Where arrow paths cross, there is no connection. (A) Prox1 image. (B) Vegfr3 image. (C) Result of segmentation (digitization of vessel wall points). (D) Binary mask of the flow domain (white). (E) Application of pressure boundary conditions to in/outlets, assuming a linear pressure gradient from left to right (LR). (F) Application of pressure boundary conditions to in/outlets, assuming a linear pressure gradient from top to bottom (UD). (G) Vessel center-lines by skeletonization, with vessels individually distinguished (here by color). (H) Fluid shear stress resulting from the flow through the network with LR boundary conditions. (J) Fluid shear stress resulting from the flow through the network with UD boundary conditions. (K) After use of the mask and the center-lines to define diameters at every pixel along every vessel (for clarity, only a small part of the total network is shown). (L) Diameter-averaged Prox1 image intensity at every axial location along every vessel. (M) Diameter-averaged fluid shear stress image intensity at every axial location resulting from LR boundary conditions. (N) Diameter-averaged fluid shear stress image intensity at every axial location resulting from UD boundary conditions. (O) Axial gradient of image intensity of fluid shear stress at every axial location resulting from LR boundary conditions. (P) Axial gradient of image intensity of fluid shear stress at every axial location resulting from UD boundary conditions. Other examples of some of these sub-image types were previously shown⁽²¹⁾ as separate figures.

Using ADINA pre-processing tools, the points were incorporated into spline-defined curves representing wall outlines (fig. 1 C), while editing point positions to correct any obvious digitization errors. Some of the curves ultimately formed the boundary of a single complex 2D shape representing the outer edge of the flow domain; the rest formed internal islands within this domain. At this time, all vessels entering or leaving an edge of the image window were closed off with lines along the edge, while all internal vessel ends were assumed to be genuine vascular culs-de-sac; these are an anatomical feature of the first generation of lymphatic vessels, called initial lymphatics, responsible for admitting interstitial fluid which thereby becomes termed lymph. On this basis, the flow domain was bounded by one complex outer curve, formed of both sections of vessel wall and lines closing off vessel in/outlets. This was possible only for the great majority of images where all vessels connected; in a few cases where the image consisted of more than one separate network, each part had to be dealt with separately. An example is visible in the schematic, in the top right corner of the network mask (D), where a small sub-network of three vessels is unconnected to the rest, and is absent from the finite-element mesh (E, F). For the analysis with assumed internal culs-de-sac, this sub-domain consisted of two blind ends and one in/outlet at the right edge of the image window. However, an alternative assumption, equally viable, is that vessels coming to an apparent end inside the image actually continue but out of the imaged plane. To accommodate this possibility, all images were re-digitized, converting internal blind ends to in/outlets (the re-digitization involved adjusting point positions so as to define an open conduit rather than a smooth blind stump). In this scenario, all three ends of vessels in the sub-network being used here as example become in/outlets, allowing flow through the three vessels. The sub-network flow was ultimately reincorporated into the computed flow outcomes.

Once a single flow domain had been defined, it was free-form meshed with small triangular elements; each such mesh (only visible in Fig. 1 E&F as a continuous blue domain) consisted of some 10⁵ elements for images of 512×512 pixels, and proportionally more for larger images. The 3 available image sets from Lausanne (see Acknowledgments) were analyzed at this size, since only two were supplied at a higher resolution. The 5 image sets from Michigan, supplied at 900×900 pixels, were reduced to this same size for uniformity with the Lausanne data. In both cases, the side length of the imaged window was approximately 1.5 mm (Lausanne 1.53 mm, Michigan 1.48 mm). The 2 Oklahoma image sets which were used each covered a substantially larger area of murine dorsal skin, and were of higher resolution. To obtain networks consisting of the same order of numbers of vessels as the Lausanne and Michigan sets, four non-overlapping sub-images were carved out from each large image, with the 8 resulting sub-images each consisting of 1080×1080 pixels, covering a square of side length of 1.77 mm.

Along with meshing, the computational fluid-dynamic simulation demanded the application of a flow-driving pressure difference. As explained in our previous paper ⁽²¹⁾, to overcome the lack of information about the actual direction of flow through the networks, we applied a linear pressure gradient across the imaged window of network. Linearity amounts to assuming that the flow through these embryonic vessels is the result of differential tissue compression setting up a pressure gradient on a spatial scale considerably greater than the < 2mm width of the images. We also lack information about the direction of such an applied pressure gradient. To deal with this, we applied the linear pressure gradient in two orthogonal directions: left-to-right (LR) relative to the image, and top-to-bottom (UD). Thus for the LR gradient, each in/outlet was subjected to a pressure proportional to its image x-coordinate, and for the UD gradient, to its image y-coordinate. Because the flow to be simulated is fully viscous, the pattern of (e.g.) left-to-right flow is identical to that for right-to-left, and so RL and DU simulations are not needed. In general, at this stage of development, the individual vessels show essentially random orientation, with no preferred alignment for any given image. The critical driver for the flow simulation is the pressure gradient across each individual vessel. Our imposition of both LR and UD gradients ensures that we sample all possibilities even if the magnitudes of the pressure gradients are not an exact match to those that would have pertained in vivo.

The magnitude of the pressure-gradient boundary conditions applied across the image was organized such that the largest Reynolds number reached in any vessel (channel) was of order unity. In Fig. 1 E, the varying pressure applied LR is shown by the varying length of the red arrows, minimal on the left of the image, maximal on the right (the inward-pointing arrowheads are scarcely visible) at all in/outlets; the width of each arrow equates to the width of the channel at its end. Pressures can be seen applied to ends interior to the image in Fig. 1 E, showing that this is the analysis with all internal in/outlets assumed to continue out-of-plane. Similarly, Fig. 1 F shows the pressures applied to in/outlets under the UD assumption, again with internal in/outlets continuing out-of-plane.

ADINA-CFD then solved the equations of fluid motion (conservation of mass and momentum) for 2D steady flow through the network. Although as described already the flow was set up to be viscous, we did not make any simplifying assumptions such as assuming negligible convective acceleration (acceleration caused by vessel narrowing, and vice versa). The results can be displayed in many different forms, but for the present purpose, what was needed was the distribution in all vessels (channels) of fluid shear stress, which is zero along the center-line of each vessel and maximal at the walls; this is seen in Fig. 1 H and J for the LR and UD assumptions respectively, with shear stress indicated by color intensity on a grey scale, with white being the most intense. These two distributions could then each be compared with the distribution of Prox1 density seen in Fig. 1 A.

For this purpose we utilized information from the network mask (D) and a skeletonization of the mask to obtain the vessel center-lines (G). We first excluded from consideration all particularly short vessels resulting from the skeletonization (length less than 7 pixels in 512×512 images, 12 pixels in 1080×1080 images). Proceeding one pixel at a time along a given vessel center-line, we defined the local diameter of the vessel, normal to the local center-line, as shown in Fig. 1 K. For each such diameter, the corresponding pixels of the Prox1 image and of the shear stress intensity image were identified and an average of the image intensity over those pixels computed, thus yielding data of diameter-averaged Prox1 intensity and shear stress (SS) intensity vs. axial location as shown in Fig. 1 L, M and N. For the particular example shown, there were over 14,000 such locations. To obtain the spatial gradient of fluid shear stress along vessels, the shear stress intensity data were differentiated by taking the difference between successive values of diameter-averaged shear-stress intensity along the center-line of each branch separately. The distance between successive pixels diagonally located was taken as √2 of the distance between adjacent pixels sharing an edge; these differences were divided by √2. This process yielded the shear stress gradient data shown in Fig. 1 O and P for LR and UD boundary conditions respectively.

2.2. Statistical comparison of simulations and image data

Bertram et al. ⁽²¹⁾ devised and applied a scheme to distinguish vessel-junction-region locations from mid-vessel locations. Full details are given in that paper; for the sake of brevity the procedure will only be summarized here. Axial locations along each vessel were expressed as normalized distance from the junction at the start to the junction at the end. A correction was made for all vessels which crossed an image-window border and were accordingly not visible along their whole length. The diameter-averaged data of Prox1 intensity for all vessels were then deposited into 50 bins of normalized distance, and the average of the intensity data in each bin was calculated. Then a second-order polynomial was fitted to these bin-average-intensity vs. normalized distance data, as shown in Fig. 2a. Similarly, the data of shear-stress intensity (or its axial gradient) were deposited into normalized-distance bins for averaging and second-order-polynomial fitting (Fig. 2b). A plot was then formed which compared the averages of intensity in Prox1 bins with their counterpart averages of intensity in SS bins. A line was least-squares fitted to these data, and the strength and significance of the linear correlation were then computed, as shown in the top right panel of Fig. 2c. In our previous paper ⁽²¹⁾ this concluded the analysis; here one further step was taken. Occasionally, a positive correlation of Prox1 and SS is largely due to slope rather than curvature, as in the example seen in Fig. 2c, where both of the second-order fits in Fig. 2a and 2b display overall negative slope. Since this is not for a biological reason (the direction of proceeding along branches is arbitrary here, i.e. normalized locations 0 and 1 could be swapped), such linear trends in the bin-plots of Prox1 and SS against normalized distance were removed (Fig. 2d and 2e). Here the resulting plot of Prox1 against SS (Fig. 2f) then displays a negative correlation, representing the fact that the curvature of the polynomial in Fig. 2d is concave-up (positive coefficient), while that in Fig. 2e is concave-down (negative coefficient). This entire analysis process was repeated for each image set, each direction of applied pressure gradient (LR and UD), with and without moving-average smoothing of the diameter-averaged data, and with and without differentiation of the SS data with respect to axial position along each vessel before binning to obtain the shear stress gradient (SSg).

FIGURE 2.

Analysis of Prox1 and shear stress data in bins of normalized position along each vessel, for one of the Oklahoma images. (a) Bins of diameter-averaged Prox1 intensity vs. normalized position, plus a 2^nd-order polynomial fit as given in the surtitle. (b) As (a), but for diameter-averaged shear stress. (c) Comparison of the Prox1 and shear stress bins, with a linear fit to the data points. (d) As (a), but after removal of the linear trend in the data. Because normalized position is unsymmetrical about zero, the polynomial fit after trend removal still includes a linear term, but describes a curve which begins and ends at the same level. (e) As (b) but with trend removed. (f) A linear fit to the trend-removed data of Prox1 vs. shear stress.

This scheme has now been applied to all sixteen images from all three sources, under all of the above circumstances. During this time it became clear that the computed significance for the Prox1/SS and Prox1/SSg linear fits, as shown by p-value in the surtitles of Fig. 2c and 2f, was spuriously high. The origin of this was the number of bins, which became the number of datapoints for the fits. This had erroneously been taken to be a free parameter. The computed strength of the correlation, as shown by r²-value in the surtitles of Fig. 2c and 2f, was not affected by this flaw; however, inability to assess the significance of each such correlation prevents ranking the correlations and limits the extent to which conclusions can be drawn from the bin analysis about links between flow indices and Prox1 distribution. Accordingly, while we will present results from the bin analysis, we also here present an alternative procedure for assessing the correlations between Prox1 distribution and SS or SSg distribution.

The new procedure uses neither normalized distance nor bins, nor does it utilize the sorting of the Prox1, SS, and SSg data into specific numbered vessels, although the vital correspondence between Prox1-intensity axial locations and SS-intensity axial locations is maintained. Instead we use the statistical method of bootstrapping, following Rowland et al. ⁽²²⁾. For this all the axial locations in an image are randomly sampled with replacement. For each of many such samplings, the Pearson correlation coefficient between Prox1 intensity and SS intensity or its axial spatial derivative SSg is computed. Then a histogram is formed of all the resulting correlations, as shown in the example of Fig. 3. The unbiased strength of the correlation, and whether it is positive or negative, is given by the peak of the histogram. The result is significant if the 95% confidence intervals for the histogram do not straddle zero correlation. More stringent bounds for the probability of the correlation arising by chance can also be calculated, as shown. In the case shown, Prox1 intensity correlates negatively with SSg, i.e. Prox1 intensity is high when SSg is low. SSg can be positive or negative, so the result indicates that Prox1 is denser on average in this image in locations where the axial gradient of fluid shear stress is negative.

FIGURE 3.

A bootstrap histogram, here composed from 50,000 samplings with replacement of unsmoothed Prox1 and SSg data from all axial locations in the Oklahoma image of Fig. 1, with LR applied pressure gradient and internal in/outlets assumed. In this case, Prox1 distribution correlates negatively with SSg; although the correlation is of low strength, it is highly significant if this test is considered in isolation.

We also realized that the analysis of SSg by Bertram et al. ⁽²¹⁾ had failed to account for the direction of flow in individual vessels. This was rectified by deriving from the flow simulation, in addition to SS, the distribution of pressure throughout the network, as determined by the flow caused by the applied pressure gradient (LR or UD). An example of such a pressure distribution, expressed as a grey-scale image, is shown in Fig. 4. After experimentation, it was found that the most accurate way to find the direction of flow via such images was to fit a line to the grey-scale intensities representing pressure along vessel center-line pixels; simulated flow occurred in whichever direction the fitted line descended. The direction of flow was indeterminate in a small proportion of vessels, typically some 5% when internal in/outlets were permitted, and (unsurprisingly) up to 20% when internal vessel culs-de-sac were assumed. In vessels where the flow direction opposed the default computational direction, the SSg values were multiplied by −1. All SSg values in this paper incorporate this improvement.

FIGURE 4.

Pressure distribution throughout a network due to flow under LR boundary conditions. Pressure is indicated by grey shade (out of 256): lowest = black, highest = white. Internal in/outlets are here permitted; the network is that shown in Fig. 1. Although strictly linear variation of pressure at the in/outlets with transverse position is enforced, the pressure within the network varies in a more complex manner according to the local and global geometry and the resulting flow.

Both the bin analysis and the bootstrap procedure can be applied to diameter-averaged Prox1-intensity, SS, and SSg data which are spatially smoothed along individual vessels by applying moving averaging (MA). This step was originally introduced into the analysis protocol as a way to deal with the noise-magnifying property of the spatial differentiation of SS to derive SSg, but it can also be applied to the Prox1 and SS data vs. intra-branch location data, treating each branch separately.

In general, repeated application of the same statistical test will produce some apparently significant findings purely by chance. To estimate and correct for the effect of this, it is necessary to apply a Dunn-Bonferroni correction, by dividing the probability threshold for significance by the number of repeated applications of the test (here, bootstrapping). Excluding the results found after application of moving-average smoothing, since these results will not be presented in detail, the test was applied 128 times (16 images, culs-de-sac vs. internal i/o, UD vs. LR pressure gradient, SS vs. SS gradient). So the probability threshold for significance becomes 3.9×10⁻⁴ (= 0.05/128).

3. RESULTS

When MA-smoothing was applied before the bin analysis, it was found that increasing box-car length (the number of points included in each average) usually produced increasing strength for the correlation of binned Prox1 with either binned SS or binned SSg up to a certain point, then decreasing strength if the box-car length was further increased ⁽²¹⁾. When applied before the bootstrap analysis, the smoothed data usually yielded higher bootstrap correlation strength and correlation significance (lower probability of chance occurrence) than the unsmoothed data.

Over all 16 images, the second-order polynomials fitted to the bins of Prox1 vs. normalized distance along branches showed, by a positive coefficient of the x² term, that Prox1 was more prevalent in junction regions (normalized axial position close to 0 or 1) than in mid-vessel positions (normalized axial position closer to 0.5). Examples of these fits, for Lausanne images, were shown in fig. 12 of Bertram et al. ⁽²¹⁾; the fitted curve was consistently concave upwards.

However, the equivalent bin-plots of SS or SSg vs. normalized distance were much more variable; in many instances the polynomials corroborated the fact evident from inspection of the bin plot that SS or SSg peaked at mid-vessel positions, whilst in other cases it was lowest at mid-vessel positions. This in turn means that, when a line was fitted to a plot of Prox1 bins vs. SS bins, or Prox1 bins vs. SSg bins, the slope was at least as often negative as positive.

The full results are to be found in the spreadsheet which forms the Supporting Information to this paper; a summary is given here. In the absence of a measure of significance, Table 1 shows the strength of each bin-analysis linear correlation between Prox1 and shear stress via the Pearson correlation coefficient, recalling that each datapoint in the correlation has as x-value the average of Prox1 intensity in all vessels at a given normalized along-vessel location, and as y-value the equivalent for shear stress.

TABLE 1.

Correlation strength between Prox1 bins and SS bins, over all 16 image sets: L = Lausanne, M = Michigan, O = Oklahoma. Linear trends removed. Image L3 was segmented for internal i/o in two ways (see Bertram et al. 2021); here, the L3ix segmentation is used. No MA-smoothing.

	SS
	UD		LR
image	culs-de-sac	internal i/o	culs-de-sac	internal i/o
L1	+0.303	+0.430	+0.360	+0.352
L2	−0.170	−0.293	−0.267	+0.153
L3	−0.176	−0.159	−0.323	−0.084
M1	+0.200	+0.669	+0.005	+0.108
M2	+0.061	−0.211	+0.086	−0.346
M3	−0.316	−0.120	+0.271	+0.177
M4	−0.139	+0.203	−0.012	+0.118
M5	+0.200	−0.523	+0.097	−0.283
O1.1	−0.620	−0.560	−0.493	−0.342
O1.2	−0.737	−0.688	−0.671	−0.705
O1.3	−0.275	−0.455	+0.297	−0.247
O1.4	−0.307	−0.371	−0.539	−0.559
O2.1	+0.017	−0.235	+0.012	−0.128
O2.2	−0.397	−0.473	−0.362	−0.531
O2.3	−0.082	−0.160	−0.103	−0.099
O2.4	−0.538	−0.642	−0.574	−0.578

In a majority of cases (22 out of 32), the sign of the correlation with internal in/outlets continuing out-of-plane corresponded to that for the same image but with internal culs-de-sac assumed. The strength of the correlation also tended to be higher when internal in/outlets were assumed (21 out of 32 cases). Treating the cul-de-sac and internal-i/o results as co-dependent, and therefore considering only the internal-i/o results, Prox1 and shear stress were anti-correlated in a majority of cases (24 out of 32). This trend was especially pronounced for the Oklahoma images, where every internal-i/o result displayed a negative Pearson coefficient.

Table 2 displays the corresponding strengths of the correlations between Prox1 and shear stress gradient (computed in the streamwise direction as detailed in the previous section). As for the analyses of Prox1 vs. SS, Table 2 shows that for Prox1 vs. SSg, the sign of the Pearson correlation coefficient for internal in/outlets followed that of the corresponding analysis with internal culs-de-sac in a majority of cases (21 out of 32). And, as for Prox1 vs. SS, the correlation was stronger with assumed internal in/outlets in a majority of cases (19 out of 32). Considering again only the internal-i/o results, the correlation between Prox1 and SSg was negative in 20 cases (out of 32). All eight Oklahoma sub-image sets displayed this property when UD boundary conditions were applied, but positive and negative correlations were found equally often under LR boundary conditions.

TABLE 2.

Correlation strength between Prox1 bins and SSg bins. Linear trends removed. L3ix, not L3im. No MA-smoothing.

	SSg
	UD		LR
image	culs-de-sac	internal i/o	culs-de-sac	internal i/o
L1	+0.039	−0.254	+0.258	−0.088
L2	−0.074	+0.076	+0.008	−0.011
L3	+0.206	+0.316	+0.171	−0.171
M1	−0.249	−0.335	−0.354	−0.314
M2	+0.427	+0.575	−0.187	+0.063
M3	+0.150	+0.197	+0.010	−0.147
M4	−0.066	+0.121	−0.421	−0.295
M5	+0.067	+0.194	−0.064	+0.128
O1.1	−0.148	−0.174	−0.503	−0.492
O1.2	−0.324	−0.203	+0.436	+0.327
O1.3	−0.195	−0.285	−0.339	−0.568
O1.4	+0.022	−0.248	+0.019	+0.021
O2.1	−0.113	−0.077	−0.277	−0.279
O2.2	+0.078	−0.063	+0.439	+0.268
O2.3	−0.051	−0.224	−0.329	−0.224
O2.4	−0.399	−0.562	+0.444	+0.371

With 16 image sets, internal ends assumed either culs-de-sac or continued out-of-plane, LR and UD boundary conditions, SS and SSg, and the option of applying moving-average smoothing to the diameter-averaged data, a very large number of individual bootstrap results have been generated. To reduce the results presentation here to a manageable size, we ranked the significance of those bootstrap results which indicated a significant correlation, by finding via an iterative procedure the percentile at which zero was included in the distribution of sample correlations. This enabled identification of the most significant results for each image, as shown in Table 3.

TABLE 3.

The most significant correlations between unsmoothed Prox1 distribution and unsmoothed shear stress (SS) or shear stress gradient (SSg), over all 16 image sets: L = Lausanne, M = Michigan, O = Oklahoma.

image	culs-de-sac	internal i/o
L1	SS UD: r = +0.158, p = 8.5e-4	SS UD: r = +0.137, p = 2.8e-4
L2	SSg UD: r = +0.036, p = 0.0054	SSg UD: r = +0.042, p = 0.026
L3	SS UD: r = +0.064, p = 0.017 SS LR: r = +0.066, p = 0.012	none significant
M1	none significant	none significant
M2	SSg LR: r = −0.042, p = 0.006	SSg LR: r = −0.041, p = 0.011
M3	SS UD: r = +0.142, p = 0.042	SS UD: r = +0.133, p = 0.046
M4	none significant	SS UD: r = +0.236, p = 0.001
M5	none significant	SS UD: r = −0.157, p = 0.002 SSg UD: r = +0.047, p = 0.004
O1.1	SS LR: r = +0.102, p = 0.001 SSg LR: r = −0.046, p = 3.3e-5	SSg LR: r = −0.057, p = 2.4e-7
O1.2	SS UD: r = −0.179, p = 3.4e-4 SSg LR: r = +0.043, p = 1.6e-5	SSg UD: r = −0.029, p = 3.5e-3 SSg LR: r = +0.028, p = 4.96e-3
O1.3	none significant	SSg LR: r = −0.026, p = 0.036
O1.4	SS LR: r = +0.174, p = 2.4e-6 SSg UD: r = −0.044, p = 8.1e-6 SSg LR: r = +0.045, p = 1.3e-4	SS LR: r = +0.117, p = 3.4e-3 SSg UD: r = −0.048, p = 2.4e-6 SSg LR: r = +0.038, p = 0.003
O2.1	SS LR: r = +0.136, p = 8.7e-4 SSg LR: r = −0.048, p = 1.3e-5	SSg LR: r = −0.040, p = 6.5e-5
O2.2	SS LR: r = +0.135, p = 1.5e-4 SSg UD: r = −0.021, p = 0.020 SSg LR: r = +0.036, p = 1.2e-4	SS LR: r = +0.124, p = 4.0e-4 SSg UD: r = −0.016, p = 0.025 SSg LR: r = +0.046, p = 1.4e-6
O2.3	none significant	SS UD: r = −0.095, p = 0.049 SS LR: r = +0.118, p = 0.011 SSg LR: r = −0.042, p = 2.1e-4
O2.4	SS LR: r = +0.135, p = 1.1e-3 SSg UD: r = −0.027, p = 1.1e-3 SSg LR: r = +0.044, p = 1.5e-6	SS LR: r = +0.118, p = 4.7e-3 SSg UD: r = −0.023, p = 3.7e-3 SSg LR: r = +0.038, p = 1.2e-4

From Table 3 we can discern the following. In those images where a bootstrap correlation of Prox1 and SS reached significance (12/16), the most significant correlation was positive in a majority of cases (9/12). Note that this outcome is contrary to that from the bin analysis, where Prox1 and SS were anti-correlated in a majority of cases (Table 1). In those images where any bootstrap correlation reached significance (15/16), the greatest significance was for a correlation of Prox1 and SSg in a small majority of cases (9/15). Among these nine cases, the greater significance usually occurred (8/9) when the pressure gradient was applied LR. Also among these nine cases, the greater significance was evenly divided between flow simulations with positive (4/9) and with negative (5/9) correlation between Prox1 and SSg. Also among these nine cases, the greater significance was evenly divided between flow simulations with assumed internal culs-de-sac (5/9) vs. those with internal in/outlets (4/9).

The concentration of significance among SSg LR cases suggests some subtle common factor in embryo dorsal skin sample preparation, but does not contribute to answering the main question of whether a flow signal, and if so what, causes valve-forming cell clustering. Nor can a great deal be read into the finding of equipartition of greatest significance of Prox1/SSg correlation between cases analysed with internal culs-de-sac or internal in/outlets, but it does suggest that the prior assumption that both possibilities were equally reasonable is supported by the data. The association of most significant Prox1/SS correlation with positive correlations is reasonably strong (75%), but the finding of significant negative correlation in 25% of these cases cannot be overlooked, and nor can the 25% of all cases in which Prox1 and SS were not significantly correlated. In any case, the results suggest a stronger association between Prox1 distribution and SSg than between Prox1 and SS.

However, all the above degrees of significance neglect the problem of repeated test application. With the much stricter threshold imposed by the Dunn-Bonferroni-correction, a bootstrap-found correlation of Prox1 and SS reached the Dunn-Bonferroni-corrected equivalent of the usually accepted significance threshold (p < 0.05) in only 3 images out of 16 (again amalgamating cul-de-sac and internal-i/o findings); in all three, this correlation was positive. Also under these circumstances, a bootstrap correlation of Prox1 and SSg reached significance in only 7 images out of 16; out of these seven, the correlation was positive in 3 cases and negative in 4. Similar numbers were arrived at after application of MA-smoothing; therefore, those results are not presented here, and will not be discussed further.

4. DISCUSSION

As indicated above, the bin analysis is now deprecated as a means of calculating the significance of Prox1/SS and Prox1/SSg correlations. Nevertheless, it provides the unambiguous finding that, for all sixteen image sets examined here, the polynomial curve fitted to the bins of Prox1 vs. normalized position along vessels had positive curvature (a positive x² coefficient). This means that Prox1 always tended to localize to junction regions. This confirmation of a known fact is a sanity check on the procedure; it also confirms the suitability of the image sets for the investigation of the main question raised in this paper.

However, as shown in Fig. 2, shear stress often displayed the opposite tendency, i.e. a negative x² coefficient for the polynomial fitted to the bins of SS vs. normalized position. Detailed inspection of the individual images of network SS (such as Figs. 1 H and J) revealed the reason: mid-vessel SS peaks occurred when a network consisted largely of vessels which became narrower where further from junctions. Indeed, under the circumstances of viscous flow as here, where complex flow patterns are absent, the axial distribution of shear stress along vessels is necessarily inversely related to the local vessel width.

Table 3 shows that neither possible flow signal was consistently correlated with Prox1 distribution. Why then does every image set but one give rise to any significant correlations at all? This is attributed to the statistical property that a repeated test throws up some apparently significant outcomes by chance. The Dunn-Bonferroni correction for this property is said to be conservative for family-wise Type I error (false positive) rate control, but such criticisms apply broadly to all such methods. However, the correction is a very blunt instrument, and in this case even after its application a minority of significant correlations survive. We argue that the 3 surviving (positive) SS correlations out of 16 do not amount to a convincing case for valve location on the basis of high shear stress. We further contend that the 7 surviving SSg correlations make no case at all for valve location via SSg, since the survivors are equally divided (3 and 4) between positive and negative correlation. The case for SSg being the sought-after flow signal to which VFCs are sensitive is fatally weakened by this finding of equal numbers of positive and negative correlations between Prox1 distribution and SSg. And when we consider the shear stress correlations as well, it is inherently unlikely (Occam’s razor) that the localization signal could be a high shear stress in one network, i.e. piece of dorsal embryonic skin, a positive shear stress gradient in another network, and a negative gradient in a third one.

The present investigation, along with the preliminary work published previously ⁽²¹⁾, is the first, and to our knowledge, only attempt so far to seek to identify a flow variable as the trigger for lymphatic valve localization that uses data from whole embryonic vessel networks. As discussed in the Introduction, there is a well-established narrative that flow is indeed responsible, but the evidence is indirect, coming largely from cell culture. Two problems arise thereby: flow conditions are assumed to exist in vivo that on fundamental fluid-mechanical grounds almost certainly do not, and the magnitude of flow-induced shear is unlikely to be comparable, given that cell culture experiments typically mimic shear stresses characteristic of mature lymphatic networks.

However, the alternative strategy adopted here, of simulating the expected in-vivo network flow, obviously has its own weaknesses. Principal among these is the necessity, given only 2-dimensional images, of treating a 3-dimensional network of tubular lymphatic vessels as a 2-dimensional network of channels. We initially attempted to work with 3D confocal image stacks to reconstruct the original geometry, but concluded that there was essentially no information to be had in the depth direction. The vessel geometry is in any case greatly distorted through flattening in the process of preparation of the biological tissue for imaging. Only the largely two-dimensional nature of the skin lymphatic vessel networks permits the current approach at all. But whereas the real vessels eventually drain to deeper lymphatics, here in the simulations with internal ends assumed to be culs-de-sac, it was assumed that all flow passing through the vessels in the image window occurs entirely parallel to the image plane. However, in our simulations where all such internal ends were assumed to be vessels that continue out-of-plane, we believe that we have largely accounted for depth-directed flow, despite the lack of explicit third-dimension information.

A second necessary assumption in the absence of any experimental information at all about the continuous or intermittent flow through the imaged vessels that would have been occurring at day E16 is that an overall pressure gradient across the network propels flow. While there could be exceptions to this, if for instance a very small region of tissue corresponding to the center of the image window were to be point-compressed, it is a reasonable assumption for non-muscular skin lymphatics lacking valves, in which only extrinsic forces are available to create flow. The likely flow is intermittent, relating to change in the regional dorsal compression of the mouse embryo due to motion of either the embryo itself or its mother. With the imaging window a square of side length less than 2 mm, it is eminently likely that such compression changes would occur on a much larger spatial scale, thereby giving rise to lymphatic-flow-inducing pressure gradients that would approximate the linear ones assumed.

Also necessary to overcome the lack of pressure-field information is an assumption about the direction of any applied gradient. Here we have covered the principal possibilities by computing both the two flows resulting from orthogonal in-plane pressure gradients. In general, at this stage of development, the individual vessels show essentially random orientation, with no preferred alignment for any given image. Even in the occasional instances where the network geometry visible in the image suggests a likely direction of flow, there is no reason to suppose that the pressure fields that would have transiently existed in vivo would have respected that developing network orientation. When such a directed geometry is present, it will be reflected in the magnitude of the simulated flows, but with all such flows limited to those which amount to fully viscous conditions, only the spatial variation of shear engendered by the flow is of importance. Our computation of both orthogonally-originating flows is likely to capture, via one or both simulations, all the significant aspects of spatial shear variation.

As possible flow signal candidates we have examined the spatial variation over the networks of wall shear stress and of its spatial gradient along the vessels (channels). The realization that vessel walls are responsive to fluid shear stress at all started with the observation that atheroma in the aorta occurs in patterns that suggest fluid-dynamic involvement. The intense subsequent investigation of that phenomenon over decades has shown that atheroma is most likely to arise in aortic and other large-arterial locations that experience low or oscillating shear stress ^(23–24), and it was this work that led to the suggested involvement of a flow signal in localizing lymphatic valves. Many other interactions between blood vessel endothelial cells and the fluid shear have subsequently been discovered. The arterial work has focused on locations where the flow pattern is complex, the wall shear varies spatially over short distances, and the flow is disturbed (see, e.g., Sunderland et al. ⁽²⁵⁾). This has allowed the creation of a wide variety of sophisticated candidate flow-signal indices in such vessels (see, e.g., Mazzi et al. ⁽²⁶⁾). However, in minute newly developed lymphatics, there is a dearth of flow signal possibilities given that the flow is viscous. The flow is essentially very simple, with wall shear varying along a given lymphatic vessel exclusively in response to axially varying diameter. The flow division at junctions will reflect the differential resistance to inflow offered by outlets that differ in size and lead to larger and better-connected network regions or smaller, less-well-connected regions. But at the junction itself, the flow remains simple, with a minority of streamlines leading to poorly served vessels and a majority to better-served ones, but all passing the junction without recirculation. This is a consequence of the always locally smooth geometry of lymphatic junctions at this stage of development, examples of which can be seen in Fig. 4. Despite the lack of experimental flow data, we can make this prediction with full confidence, given the overall fluid-dynamic situation as characterized above by the relevant dimensionless numbers.

The number of images examined here is sufficient that it is unlikely that additional image data subjected to the same investigative approach as here would change the outcome. Although the statistics throw up individual instances of apparently significant correlation between the Prox1 distributions and either shear stress magnitude or shear stress gradient, no one such signal is found that consistently so correlates, especially when negative correlation is distinguished from positive correlation. It defies logic to suggest that qualitatively different flow signals are responsible for localizing valves in different instances, so the only reasonable conclusion is that no flow signal that could occur in newly formed cutaneous lymphatic vascular networks can be regarded as responsible.

In turn this demands consideration of what alternative possibilities there could be, if a flow signal is not responsible for the undoubted prevalence of lymphatic valves near vessel junctions. Here we enter upon pure speculation, so these remarks must be treated with caution. It is possible that the cells forming the wall of the lymphatic at junction regions experience a stress field relating to longitudinal tension which differs in kind from that experienced by wall cells located at intermediate points along a single vessel, relatively far from the junctions that occur at each end of all but initial lymphatics. One can imagine that this stress field might be biaxial, with tensile components approaching equality in orthogonal directions in the plane (considered on an extremely local scale) of the wall itself. Elsewhere along the vessels, the equivalent in-plane stress field could be closer to uniaxial. Large arteries are subject to much longitudinal tension, such that they shorten axially by up to 40% when severed and released from surrounding tethering tissue. Mature collecting lymphatics also shorten when so treated, but by a lesser amount. It is not known whether the developing skin lymphatics examined here are under any longitudinal tension at all; their often wandering course (see Fig. 2) suggests that if there is any such tension, it is not at all dominant relative to other factors ^(27–28). Thus, while we offer up this speculative possibility, we do not consider it likely. Resolution of the question addressed here is likely to come only in an era when these most flimsy and so-far most difficult of vessels to observe can be subjected to experimental techniques yet to be developed.

5. PERSPECTIVES

The work tests a popular hypothesis that hemodynamic signals determine where valves will form in the developing lymphatic vasculature, after giving a rationale for expecting the hypothesis to be false. The distribution of Prox1 in embryonic mouse skin as determined by immunofluorescence staining is compared with patterns of hemodynamic force calculated in simulations based on the imaged vessel geometry, and a statistical analysis suggests that the distribution of shear stress and shear stress gradient in the computational model does not correlate well with Prox1 expression. The paper provides evidence for a different view on part of the standard description of valvogenesis that has up to now been accepted on the basis of cell culture findings, where circumstances differ markedly from those which pertain in vivo at the time of embryonic valve localization.

Abbreviations

transcription factor proteins, encoded by similarly named genes:

Foxc1/2

forkhead box C1/C2

Gata2

GATA-binding factor 2 (GATA is a DNA sequence)

Nfatc1

nuclear factor of activated T cells 1

Piezo1

cell stretch sensor, sub-unit of a mechanically activated non-selective ion channel

Prox1

Prospero-related homeobox 1

Rasa1

Ras GTPase-activating protein 1 (GTP, guanosine triphosphate; Ras, rat sarcoma virus)

Vegfr3

vascular endothelial growth factor receptor 3

other abbreviations:

CFD

computational fluid dynamics

LR

left to right (applied pressure gradient)

MA

moving-average (datapoint smoothing)

SS

shear stress

SSg

(axial) shear stress gradient

UD

top to bottom (applied pressure gradient)

VFC

valve-forming cell

Data availability statement

The image sets were kindly donated by other groups, as detailed in the Acknowledgments, and as such are not ours to make available to others. A version of the software used in the analyses will be deposited in a public-access repository.