Introduction

During embryogenesis, endothelial precursor cells, angioblasts, migrate and differentiate to form the first blood vessels. This process of blood vessel formation by de novo endothelial cell (EC) production, is called vasculogenesis [1]. After vasculogenesis, the primary vascular network is expanded by angiogenesis. Angiogenesis, the formation of new blood vessels from existing ones, is crucial for physiological and pathological mechanisms such as embryonic development, wound healing and tumour growth. Pro-angiogenic factors, including vascular endothelial growth factor (VEGF)-A, stimulate sprouting in ECs, which make up the inner lining of blood vessels [2–4]. The ECs proliferate and migrate into the extracellular environment and form a vascular network that supplies the surrounding tissue with nutrients and oxygen and removes waste products. A key regulatory factor determining the final structure of the new blood vessel network is the composition of the extracellular matrix (ECM), the scaffold of proteins that surrounds the blood vessels and provides structural support to the newly formed multicellular network [5]. Specific integrin binding sites on ECM proteins like fibronectin and laminin allow cells to exert stresses on the ECM and sense the mechanical properties of their environment [6]. Cells also secrete enzymes which digest the ECM, releasing ECM-retained growth factors, including VEGF-A, that stimulate EC migration [7]. Additionally, cell-cell interactions, like Notch signalling and vascular endothelial cadherin (VE-cadherin) mediated adhesion, regulate individual cell behaviour within the network and separate ECs into leading tip- and proliferating stalk cells [4]. In response to VEGF-A, cells upregulate Delta-like 4 (DLL4), a transmembrane Notch ligand that activates NOTCH1 in neighbouring cells [8]. NOTCH1 activation subsequently downregulates VEGF receptor (VEGFR)-2 expression, reducing the neighbouring cell’s susceptibility to VEGF-A, resulting in less angiogenic potential of the cells [8]. Similarly, VE-cadherin binding inhibits VEGFR-2 signalling in neighbouring cells [9]. The combination of extracellular, intercellular and intracellular regulation makes angiogenesis a complex and multiscale process. An understanding of how each of these regulatory mechanisms contribute to the final network formation is necessary for biomedical applications where angiogenesis needs to be stimulated, steered or reduced, like tissue engineering [10] or vascular renormalization [11].

One of the more commonly used in vitro models to test the effect of environmental changes on EC network formation is the tube formation assay (TFA). In this assay ECs are placed on top of a basement membrane-like substrate, where the ECs, under the right conditions, spontaneously organise into networks of interconnected tubes [12]. With this assay the influence of various chemical and mechanical stimuli on EC networks has been investigated by comparing the branch density of the final EC network between conditions [13]. In the analysis of TFA outcomes it is often assumed that a higher density of branches indicates a higher angiogenic potential of the cells (i.e., more sprouting). However, the temporal development of the network over time may contain key information on the dynamic processes regulating angiogenesis. Merks et al. (2006), therefore, measured the density of branch points in developing vascular networks in vitro at regular intervals and observed that the network density dropped slowly over time [14]. Parsa et al. (2011), analysed time-lapse videos of in vitro EC network formation in more detail, and described the development of EC networks as a carefully orchestrated sequence of five subevents: (1) rearrangement and aggregation, (2) spreading, (3) elongation and formation of cell–cell contacts, (4) plexus stabilization, and (5) plexus reorganization [15]. The first two stages describe the attachment of the cells to the substrate directly after cell seeding. The rate and the extent of cell spreading on the substrate have been linked to substrate stiffness and composition [6], and are assumed to be influenced by the traction forces cells are able to exert on the substrate [16]. Later stages in the tube formation assay rely on a combination of cell-cell and cell-ECM interactions: Rüdiger et al. (2020) observed a collapse of EC networks on soft, laminin matrices, confirming the importance of the ECM as a scaffold to provide support for the network to stabilise [5].

Computational modelling is a helpful tool for proposing hypotheses and testing whether (a combination of) proposed developmental mechanisms are sufficient to explain biological observations [17]. Different computational models of angiogenesis all consider angiogenesis as the assembly of ECs into a vascular-like network structure [18–20]. For example, in Manoussaki et al. (1996), the in vitro matrix remodelling behaviour of ECs on a compliant substrate is described with a continuum model, in which ECs are described as local densities using a system of differential equations [21]. Similarly, in Serini et al. (2003), ECs are modelled to migrate upwards a chemoattractant gradient with a continuum model [22]. In Palachanis et al. (2015), ECs are considered as identical 2D ellipses, using a single-particle model [23]. Köhn-Luque et al., (2013) used a hybrid continuum and cell-based model to model the effect of VEGF retention in the ECM on EC network formation [7]. In cell-based models of angiogenesis, ECs are described as discrete areas of variable shapes, where multicellular behaviour arises from the response of single cells to inputs from their microenvironment [24,25]. Similarly, Vega et al. (2020) used a hybrid continuum and cell-based model to model the interaction between Notch- and VEGF signalling and cell-ECM adhesion [26].

Interestingly, both continuum models, particle-based models, as well as cell-based models are successful in mimicking EC network formation, and different hypotheses, or model inputs, can in some cases lead to a qualitatively similar output. In this work we focus on three cell-based models as previously developed in our group [14,27,28]. The “cell elongation model” assumes that ECs are attracted towards one another through autocrine/paracrine signalling by secreting a chemoattractant [14]. In this model it was observed that cells self-organise into vascular-like structures if the cells have an elongated shape. Observations by Parsa et al. (2011) support the necessity for ECs to elongate to form networks, as cells elongate before they start forming a network and continue to elongate as the network develops [15]. The “contact inhibition model” also assumes that ECs are attracted to one another through a secreted chemoattractant, but here it was assumed that VE-cadherin binding mediates contact inhibition of chemotaxis, following observations by Dejana et al. (2004) that VE-cadherin-VEGFR2 complex forming limits cell proliferation and promotes quiescence [29] and that in the yolk sac of VE-cadherin double-knockout mice and in in vitro mouse allantois explants ECs failed to form vascular networks and aggregated into isolated islands [28,30]. In disagreement with VEGF guided cell-cell attraction-based models, Rüdiger et al. (2020) observed persistent network formation in the absence of a VEGF-A gradient and they argue that the in vitro network formation is driven by mechanical communication [5]. The importance of cell-ECM interactions is further supported by observations by Stéphanou et al. (2007), where they observed faster and more lacunae in network formation on fibrinogen gels of intermediate rigidity [31]. In light of these observations, the “mechanical cell-cell communication model” assumes that ECs are attracted to one another solely through mechanical interaction with the ECM [27]. In this cell-ECM interaction model ECs induce strain in the ECM through contractile forces and migrate up the strain gradient. Thus, a variety of models can successfully reproduce multicellular network formation starting from different assumptions, all inspired by observed EC behaviour. Therefore, we need to critically examine these models to see which of these assumptions, if any, are correct.

Thus, for a more complete understanding of the integration of different mechanisms driving EC network formation, we must quantitatively compare these computational models to in vitro experiments by systematically changing model parameters and in vitro experimental setup, and by iteratively refining the models based on the outcomes of the comparison. As an initial step, we will critically examine three models originating from our own research group: (1) The cell elongation model [14]; (2) the contact inhibition model [28]; and (3) the mechanical cell-cell communication model [27]. We select model parameters from the original work [14,27,28] and from published global sensitivity analyses [32]. Then we fit the models to our own in vitro observations, and finally we present additional parameter examinations. For the comparison, dynamical analyses of in vitro experiments are required. Here we present new dynamic time-lapse videos of tube formation assays and a new analysis pipeline based on ImageJ, and demonstrate its use for model selection for network formation. Commercial and open-source image analysis tools are available for quantitative analyses of EC networks [33–36]. However, these tools are designed for single frame end-point analyses and therefore less suitable for high-throughput dynamical analyses of large quantities of data. Our novel image analysis pipeline is suitable to dynamically analyse and compare in vitro and simulated networks. The pipeline allows us to reliably extract network features over time, such as the number of branches and their length and the average size and number of lacunae. In this study we use this pipeline to compare the output of three cellular Potts models of 2D EC network formation [14,27,28] to in vitro experiments over time at different cell densities and find that the distance at which cells are able to communicate with each other determines the features of the final, stabilised network, but the speed and manner at which the networks stabilise depends on other characteristics of the cell communication, like the speed at which the cell attracting signal spreads.

Results

Quantification of in vitro endothelial cell network dynamics

To understand and compare the dynamic behaviour during in vitro and in silico EC network formation, we developed an image analysis pipeline to extract network characteristics from 2D EC network formation time-lapses and applied it to in vitro and in silico videos. To capture in vitro EC network formation, ECs were seeded on a Matrigel matrix and imaged in phase contrast for 24 hours with an interval of 15 minutes (Fig 1A). The ECs in the time-lapse images were segmented from the background using a combination of Gaussian denoising and local variance extraction, followed by Huang thresholding (S1A Fig). Network features were extracted from the segmented time-lapses for each time step using FIJI plugins Analyze Skeleton [37] and Analyze Particles (Fig 1B and S1 Video). For the in silico networks, we selected three cell-based computational models of 2D EC network formation [14,27,28] and extracted the network characteristics directly from the output images using the same image analysis pipeline as the segmented in vitro time-lapses.

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Fig 1. Image analysis pipeline for in vitro phase contrast time-lapses. A) Graphic summary of in vitro experiment. B) Networks were segmented from the background and labelled in FIJI. Insert is 350 x 350 µm². C) ECs were seeded on Matrigel, and network formation is captured in phase contrast for 24 hours with intervals of 15 minutes. Insert is 500 x 500 µm². D-G) Quantitative analysis of (D-F) lacunae and (G-H) branches under standard conditions. Shaded area represents a standard deviation of eight wells on one slide.

https://doi.org/10.1371/journal.pcbi.1012965.g001

In the in vitro culture, under standard conditions, the ECs started forming cell-cell connections within the first hour after seeding on a Matrigel matrix and thereafter self-organised into fully connected networks within six hours (Fig 1C). We were able to divide the development of the network characteristics over time in three distinct parts, which we describe following Parsa et al. (2011) [15], as: (1) The first hour resembled stages 1 and 2, “rearrangement and aggregation”, followed by “spreading” (Fig 1C). In these stages the ECs were randomly dispersed on top of the Matrigel and started to form attachments to the ECM. (2) The next 6 hours matched Parsa’s stage 3, “elongation and formation of cell-cell contact”. During this stage new connections formed between ECs, which is reflected in an increase in the number of lacunae from to per mm² (Fig 1D). (3) Hereafter, the number of lacunae decreased linearly in time (, p-value < 0.001), resembling Parsa’s stage 4 and 5: “plexus stabilization” and “reorganization”. The average lacuna area displayed a linear increase due to a higher occurrence of lacuna merging and smaller lacunae closing compared to the formation of new lacunae (Fig 1E) (, p-value < 0.001). The number of branches within the network continuously decreased and their average length increased (Fig 1G and 1H). During the first three stages branches predominantly increased in length due to merging of branch sprouts, but during stages 4 and 5 we observed an increase in merging of existing branches.

In silico models can be adjusted to resemble in vitro networks

To compare the dynamics of in silico networks to the in vitro networks we selected model parameters that mimic the in vitro experiment. To represent the inner well of the in vitro experiments (Fig 1A), ECs were simulated within a circular field with a diameter of 1900 lattice sites for the two chemotaxis models (1 lattice site = 2 x 2 µm) and 760 lattice sites for the mechanical model (1 lattice site = 5 x 5 µm) (Fig 2A and S2-4 Videos). To determine the target area and length of the simulated cells, we measured cell areas and diameters in fluorescently labelled ECs within network time-lapses (S2A Fig). We observed an average cell area of µm² after three hours (S2B Fig), with a cell length of µm for elongated cells (S2C Fig). Based on these measurements we set the target area for the simulated cells to 250 lattice sites (chemotaxis models) and 40 lattice sites (mechanical model), which corresponds to a physical area of 1000 µm². In the cell elongation model, we set the target length to 45 lattice sites, which corresponds to 90 µm.

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Fig 2. Quantitative comparisons between two chemotaxis-based models and a mechanical model of 2D EC network formation to in vitro time-lapse results. A) The model parameters are adjusted to resemble the in vitro situation and network characteristics are extracted using the same image analysis pipeline as the in vitro networks. B) Simulated ECs were placed on a circular grid and run for 2880 Monte Carlo steps (MCS). In the chemotaxis models ECs (red) secrete a chemoattractant (grey) that diffuses through the medium (white). In the mechanical model ECs (white) exert strain on their environment (green/blue heatmap) (Field area is 500 x 500 µm²). C) A comparison of the number of lacunae per mm² for chemoattractant diffusion lengths ranging from 10 µm to 100 µm for the cell elongation model and the contact inhibition model after 2880 MCS. In magenta the in vitro number of lacunae per mm² after 24 hours. Inserts show the cell elongation model for a diffusion length of 10, 50 and 100 µm. D-G) Quantitative analysis of (D, E) lacunae and (F, G) branches in vitro (magenta) and in the cell elongation model (blue), the contact inhibition model (golden) and the mechanical model (green). H) The number of sprouts, branches connected to a single node in vitro, in the cell elongation model, the contact inhibition model and the mechanical model. . Shaded area represents a standard deviation of eight simulations or eight in vitro time-lapses.

https://doi.org/10.1371/journal.pcbi.1012965.g002

In the cell elongation model, ECs elongated towards a target length, secreted a chemoattractant and migrated upwards the chemoattractant gradient [14]. In the contact inhibition model ECs secreted a chemoattractant and migrated upwards the chemoattractant gradient, but locally inhibited chemotaxis in their neighbours (Fig 2A) [28]. To find the parameters that describe the chemoattractant in the two chemotaxis models for which the models most resemble the in vitro networks (i.e., the diffusion coefficient, the decay rate and the secretion rate) we investigated the effect of the diffusion length of the chemoattractant on the network formation. Diffusion length () depends on the diffusion coefficient (D) and the decay rate (∊) of the chemoattractant according to:

(1)

Therefore, we altered the diffusion length from to with steps of by varying the decay rate of the chemoattractant, while keeping the diffusion coefficient constant at . (Fig 2C). For a full list of the selected parameters, see S1 Table. In vitro, the average number of lacunae per mm² after 24 hours was 3.30 ± 0.83, with an average lacuna area of 0.18 ± 0.04 mm² (Fig 1D and 1E). When we increased the diffusion length in the two chemotaxis model, we observed that the average number of lacunae decreased (Fig 2C), in agreement with previous work [14,28,32].

Based on the outcome of the comparison of the number of lacunae (Fig 2C) and branches (S2D Fig) per mm², we selected a decay rate of , which corresponds to a diffusion length of µm. This resulted in an average number of lacunae per mm² after 2880 MCS of , with an average area of mm² for the cell elongation model and lacunae per mm², with an average area of mm² for the contact inhibition model (Fig 2D and 2E). With the selected diffusion length of µm, we observed some similarities between the dynamics of the model outcomes and the in vitro networks: During the initial relaxation time (100 MCS) the ECs in the models reached their target size and shape, which mimics stages 2 and 3 of the 5 stages described in [15]: cell spreading and elongation. Next, we looked at the formation of cell-cell connections as ECs started excreting a chemoattractant. The increase in cell-cell connections is reflected by a decrease in number of branches (Fig 2F). The transition towards stage 4, stabilization of the network, was more gradual in the contact inhibition model than in the in vitro networks and in the cell elongation model (Fig 2E). The number of lacunae decreased faster in the cell elongation model than in the contact inhibition model (Fig 2E) and for the first 1680 MCS we observed fewer and larger lacunae in the contact inhibition model than in the cell elongation model. The number of branches was initially larger in the cell elongation model than in the contact inhibition model. In agreement with the in vitro networks, the branch number decreased quasi-exponentially in the cell elongation model, whereas in the contact inhibition model the number of branches decreased more linearly (Fig 2F). Both chemotaxis-based models have longer branches than the in vitro networks (Fig 2G). This difference in number of branches and their length could partially be explained by a higher density of sprouts detected in vitro than in the models (Fig 2H). Sprouts are branches that are connected to a single node and the Analyze Skeleton plugin splits an existing branch into two where a sprout appears, resulting in two shorter branches.

In the mechanical model, the ECs are assumed to stiffen the ECM through the exertion of traction forces on their environment and move upwards the stiffness gradient (Fig 2B) [27]. We selected a Young’s modulus of 12 kPa, with a Poisson’s ratio of 0.45, based on the original work. For a full list of the selected parameters, see S2 Table. For the selected parameters, cells formed a dense network of branches with small lacunae. The composition of this network was similar to an early-stage EC network. This suggests that in the present form of the mechanical model, the mechanical model successfully reproduces the network formation phases (phases 1-3), but it did not describe the remodelling phase well.

Another noticeable difference between the three cellular Potts models and the in vitro EC networks was the composition and thickness of the branches (Figs 1B and 2B, and S3). In vitro cells are able to move on top of each other to form a semi-3D structure, consisting of thin branches and thick nodes. This is reflected in a decrease in the cell-covered area over time (S2E Fig). In the strictly 2D in silico models, however, cells are unable to move on top of each other and the cell-covered area remains constant (S2E Fig). Therefore, the models form increasingly thick branches and larger nodes as the cell density increases (S3 Fig).

Increasing cell density results in an initial increase in number of branches and lacunae in vitro

To investigate whether the models respond similarly to a change in the initial conditions as in vitro networks, we next analysed network formation at different cell densities. Previous in vitro studies on primary ECs observed an inability for the cells to form tubes under or over a threshold density, similar to a percolation threshold [5,22]. Consistent with these previous results, we observed that HMEC-1 cells retained the ability to form cell-cell connections and short branches at 1000 ECs per well (80 cells/mm²) (Fig 3A). However, they were not able to form a network spanning the full well. At a cell density of more than 2000 ECs per well (160 cells mm²), we observed the development of a fully connected network spanning the entire well (Fig 3A). With an increase in cells density a higher number of lacunae and branches formed within the first hours (Fig 3B and 3D). However, for over 4000 cells per well (320 cells mm²), the networks remodelled to the same number of lacunae and branches and average lacuna and branch sizes (Fig 3B-3D).

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Fig 3. A comparison of the response of in vitro networks and model outcomes to higher EC seeding density. A) Segmented in vitro networks with an inner well cell count of 1000 (top) and 4000 (bottom). B-E) Quantitative analysis of (B, C) branches and (D, E) lacunae for different cell densities. Scale bar = 500 µm. Shaded area represents standard deviation of 5 or 6 wells from two separate experiments.

https://doi.org/10.1371/journal.pcbi.1012965.g003

The cell elongation model resembles the in vitro networks in its response to changes in cell density

Whereas a higher cell density caused an initial increase in the number of branches per mm² for all three in silico models, this difference diminished as the network remodelled in the two chemotaxis-based models at a diffusion length of 70 µm (Fig 4B and 4E), similar to what we observed in vitro (Fig 3B). The mechanical model had a much higher number of branches per mm², and the difference between branches per mm² for different cell densities converged much slower than in the chemotaxis-based models and the in vitro time-lapses (Figs 3B and 4A–4I). Out of the three models, the later dynamics of the cell elongation model most resembled the in vitro network remodelling, which appears to be independent to a change in cell density, even when the model is subjected to a range of diffusion lengths (Figs 4A–4C, and S4).

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Fig 4. A comparison of the response of model outcomes to a higher EC density. Quantitative analysis of the number of branches for different cell densities with different diffusion lengths ) in the (A-C) cell elongation model and (D-F) the contact inhibition model, and (G-I) different stiffness sensitivity (β) in the mechanical model. . Shaded area represents the standard deviation of six simulations.

https://doi.org/10.1371/journal.pcbi.1012965.g004

Discussion

In biomedical applications, like tissue engineering [10] or tumour vascular normalization [11], angiogenesis is stimulated, redirected or reduced to properly vascularise the tissue of interest. Computational models can help understand how different environmental factors and cellular cues affect various aspects of EC migration and network formation, and help identify effective regulatory targets. Many computational models of EC network formation, based on assumptions like cell-cell attraction [28], cell shape effects [14,23], cell-ECM interaction [21,27], or a combination of the former [38], have been successful at simulating the formation of EC networks that qualitatively resemble those observed in vitro. To obtain more insight into the extent to which the mechanisms described by these computational models are a sufficient explanation for EC network formation in vitro we selected three previously published, cell-based models of 2D EC network formation and compared these with time-lapse data of EC network formation in vitro: (1) the cell elongation model [14]; (2) the contact-inhibited chemotaxis model [28]; and (3) the mechanical cell-cell communication model [27]. We examined their dynamical behaviour using a range of quantifications and compared it to time-lapses data of in vitro EC network formation.

To quantify dynamical features of both in vitro EC network formation and outcomes of cellular Potts models of 2D EC network formation, we developed a custom image analysis pipeline (Figs 1B and 2A, and S1A). To compare the models to our in vitro time-lapses we selected parameters from the original work [14,27,28] and from published sensitivity analyses [39], and fitted parameters to our own in vitro observations (S2A-S2C Fig). We used the outcome of our initial dynamical analysis of the in vitro and simulated EC networks to find a parameter space in which the network features of the chemotaxis-based model outcomes resembled those of the in vitro EC networks (Figs 2C and S2D). For the mechanical cell-cell communication model we were not able to find such a parameter regime. The mechanical model was very sensitive and often lost its ability to form networks after small changes in the explored parameters. Its high computational cost also made it difficult to perform large scale parameter sweeps.

We then asked how the in vitro model responded to systematic changes in the culture conditions, with the aim of mimicking the same conditions in silico. Among the conditions tested (S5 Fig) only cell seeding density affected network formation in silico and in vitro to a sufficient degree to allow for assessment of the correctness of the models (Figs 3 and 4). Interestingly, both chemotaxis-based models showed similar dynamic behaviour as the in vitro networks for the selected parameter regime (Figs 3 and 4, and S4). They showed an initial cell-cell contact forming phase, followed by a remodelling phase, similar to Parsa’s stages 3 to 5 of in vitro network formation: “elongation and formation of cell-cell contacts”, “plexus stabilization”, and “plexus reorganization”. In the mechanical model, however, the ECs quickly form connections, mimicking early in vitro network formation, but there is little network remodelling in the model outcomes.

We found that, among the models tested, the dynamic behaviour of the lacunae in the cell elongation model matched the dynamics of the in vitro networks most closely (Fig 2D and 2E). In the cell elongation model [14] the ECs actively elongate before they migrate upwards a chemoattractant gradient. Because of their shape, the elongated ECs can form long-range connections to other ECs and can thus form branches and close lacunae more rapidly than rounded ECs (Fig 2E and 2F). Due to the slower movement along the short axis for elongated cells, branches are not very mobile and merge slowly as the network stabilises [40]. This results in a gradual decrease in lacunae per mm², which mimics the remodelling phase of in vitro network formation quite well (Fig 2E). In the contact inhibition model [28] however, network remodelling proceeds more slowly than in the elongated EC model (Fig 2D and 2E). Also, in the contact inhibition model, for a short diffusion length ( µm), the number of branches per mm² remains dependent on the cell density (Fig 4D), which is not observed in vitro or in the cell elongation model. The cell elongation model also predicts that networks can form at lower cell densities than the contact inhibition model (Figs 4 and S4). Thus, the cell elongation seems more robust in reproducing the remodelling stage of in vitro EC network formation than the contact inhibition model.

There were also clear discrepancies between the cell elongation model and in vitro network formation. For example, Parsa et al. (2011) observed that in vitro ECs aggregate and spread first, before elongating [15]. Thus, in vitro, the earliest stages of network formation seem not driven by elongation, as assumed by the cell elongation model. Also, the cell elongation model was unable to explain the observation that network formation depends on substrate stiffness [5,41,42], as well as substrate density (S5F Fig) [6] and substrate thickness [43]. Such observations may be better explained by models based on mechanical cell-cell communication [21,27]. Rüdiger et al. (2020) argue that mechanical forces are the main driver behind in vitro EC network formation [5]. In this study they continuously perfused the ECs during in vitro network formation to wash away any possible chemoattractant gradient, and network formation still occurs. To further argue for mechanical regulation, they show that ECs compress the substrate up to a distance of 30 µm, with measurable ECM displacement up to 100 µm. These distances fall within the chemical signalling range of what we observed in the two chemotaxis models similar to the in vitro networks (Figs 2C and S2D). In the mechanical model we observed strains up to 40 µm from the cell boundary in the investigated parameter regime (S2F Fig). However, this was not enough to form networks similar to those observed in vitro (Fig 2). Finding parameter regimes for the mechanical model in which it quantitatively reproduces the in vitro dynamics might make it possible to explore the effect of ECM straining on EC network formation in more detail. To find new parameter regimes in the mechanical model we could combine our presented image-analysis pipeline with topological data analysis [44] and novel high-throughput parameter inference methods such as FitMultiCell [45], which combines the iteratively changing of parameters, the running of simulations and the assessment of the model outcomes in one tool, to streamline the model fitting process. Also, a number of improvements are possible in the mechanical model, which so far focused on synthetic, fully elastic substrates such as polyacrylamide gels, which cannot be deformed or remodelled. In cell cultures using natural ECM materials such as Matrigel, the ECs tend to deform and remodel the substrate over time [5,46], altering the substrate’s material properties and its response to stress. The absence of ECM remodelling in the mechanical model could explain why the EC networks in the mechanical model stabilises relative quickly, whereas in vitro and in the two chemotaxis-based models the EC networks continuously evolve as the simulated chemoattractant gradient gradually stabilises over time. The effect of matrix remodelling, in the form of strain-stiffening, on EC network formation dynamics could be investigated in vitro with synthetic gels with tuneable stiffness [47,48]. With these gels the travel distance of mechanical perturbations on the system can be altered systematically, to measure its effect on EC network formation [16]. With the use of synthetic gels, it is also possible to wash away any chemoattractant from the system, without the possible interference of ECM retention, which might have influenced the network formation in Rüdiger et al. (2020). In silico models have considered matrix remodelling by using continuum equations of matrix advection due to pulling by the ECs [21,49]. A similar approach was integrated in a recent hybrid CPM-PDE model of network formation [38]. In these models, the number of junctions was reported to increase over time, in contrast to our in vitro observations (Fig 1C-1H) and those of others [15,50], where the network remodels into a less dense network within 24 hours. In our ongoing work, we are integrating detailed models of the structural remodelling of fibrous ECMs into models of angiogenesis [51,52], which in the future may give more insight into the potential role of matrix remodelling.

In the two chemotaxis-based models we assumed that each MCS represents 30 seconds in real time and each pixel is 2 x 2 µm. This was chosen such that the mean speed of the cells became , which, it was previously argued, complies with in vivo observations [14,53]. This allows us to translate the physical processes like chemoattractant diffusion and decay rate to physical units (). We fixed the diffusion coefficient at . The main driver behind angiogenesis is generally assumed to be VEGF-A [54]. The diffusion coefficient of VEGF-A is in the order of in collagen-I, in Matrigel and in water [55–57], which is 100 to 10000-fold higher than what we implemented in the model. However, ECM and EC membrane receptor retention of VEGF-A could affect VEGF-A diffusion, resulting in a smaller effective diffusion coefficient and decay rate [7]. Köhn-Luque et al., (2013) show an aggregation of fluorescently labelled VEGF in the substrate surrounding HUVECs on Matrigel within 5-10 minutes after VEGF addition [7]. Additionally, it is yet unclear whether VEGF-A diffusion is the main driver behind angiogenic network formation in vitro. Especially since Rüdiger et al. (2020) observed EC network formation under constant perfusion to avoid the emergence of a chemical gradient [5]. With a diffusion coefficient of we observe simulated networks most like in vitro at a diffusion length of (Fig 2C). With a diffusion coefficient of , the decay rate would need to be for a diffusion length of , since , which is much faster than the 90 minute half-life of VEGF-A measured in vitro [58]. In vivo, VEGF-A has been shown to have a half-life of 33.7 minutes [59].

There are some steps in this image analysis pipeline that require optimization. For example, we implemented a watershed step in the pipeline to correct for breaks in EC branches where thin stretched-out ECs are misinterpreted as background during segmentation. This watershed step can result in an overrepresentation of connections between branches, but overall, we observe an accurate and consistent result (S1B Fig). To circumvent the need for segmentation corrections and other suboptimal parameter settings in the segmentation a neural network could be trained to perform the segmentation steps necessary for the network quantification [60]. This network can be trained to recognise thin stretches of ECs where conventional image analysis methods, which rely mostly on contrast, might mistake them for background. However, the training of a neural network requires a lot of correctly annotated training data and undertrained networks will give suboptimal results. More interestingly, a neural network could be trained to recognise network features that are more difficult to extract from segmented images automatically, like cell division events. Another interesting addition to the image analysis pipeline would be the inclusion of topological data analysis (TDA) [44]. The time-lapses of the segmented branches and lacunae could be regarded as multidimensional graphs suitable for further analysis using TDA. TDA could also be expanded to work on 3D time-lapses to analyse 3D in vitro EC sprouting assays [61,62], 3D vessels-on-a-chip sprouting assays [63] or even in vivo systems, like the developing zebrafish embryo [64]. 3D in vitro systems have the benefit of both having a high degree of control over the experimental environment, as allowing for 3D vascular growth and EC migration. ECs in vivo migrate both in 2D over the basal membrane, as in 3D during sprouting into the ECM [65]. 2D models of EC network formation can be regarded as a projection of a 3D system and adapted to study different migration types, but recent improvements in computational power have allowed the increased usage of more complex 3D models of angiogenesis [66,67]. 3D models, both in vitro, in vivo and in silico allow for a greater understanding of the complex interactions ECs have with the fibrous ECM during sprouting [68].

Conventional tube formation assays to assess the sprouting ability of ECs are usually only examined for a single characteristic (e.g. the number of branches) at a single time point [33]. However, we observed that network features converged over time (Fig 3B–3E). This indicates that an endpoint observation might not suffice to examine the influence of a compound or knockout on collected cell behaviour before a stable network has been formed. We also observed variation between different samples of the same experimental conditions. This can be explained by variations in fitness of different passage numbers of ECs [13], handling of the ECs prior to the experiment, and small fluctuations in cell seeding density. Cell density naturally affects the distance between cells right after seeding, and can therefore affect the ability of cells to sense their neighbours at the start of the experiment [5,13,15]. Cells also influence the experimental environment by digesting nutrients and excreting signalling molecules [69], but also by actively remodelling the ECM [5,70]. Therefore, it is important that the exact number of seeded ECs is also considered when comparing samples within 6 hours after seeding or at low cell densities.

In this work we have shown the importance of dynamic analyses of EC network formation by showing how EC network features converge over time. We hope to have paved the way for a more systematic comparison of computational models to biological experiments, and we have indicated areas of improvement for existing cellular Potts models of 2D EC network formation. In the future we plan to expand our analysis of EC networks through more sophisticated data-analysis techniques such as topological data analysis [44] and Sobol’ analysis [32] for falsifying and refining our understanding of EC network formation using in vitro and in silico modelling in hopes that such techniques can be generalised to the dynamics of other models of biological morphogenesis.

Methods

Computational model

The three computational models of EC network formation [14,27,28] are all based on hybrid cellular Potts models (CPMs). Hybrid CPMs couple a dynamical description of cell behaviour based on the CPM with a detailed model of the cellular micro-environment, specifically the concentration of extracellular signalling molecules or strains and stresses in the ECM. The cellular Potts model is a cell-based model that focuses on modelling cell-cell interactions and predicting collective cell behaviour [73]. Cells are represented as collections of lattice sites with an identical index on a grid, where each index labels a single cell. Each lattice site represents an area of 2 x 2 µm² in the chemotaxis models and 5 x 5 µm² in the mechanical model. To mimic the circular in vitro dishes, we labelled all lattice sites outside of a circle with a diameter of 3800 µm as boundary sites in the cell elongation and contact inhibition models and as a fixed node in the mechanical model.

Apart from the shapes of the simulation domains, models were used as described previously [14,27,28]. Briefly, during a Monte Carlo step (MCS), a time-step in the simulation representing 30 seconds, cells change their shape and move through the lattice by attempting to copy its index to a neighbouring lattice site. There are N copy attempts per MCS, with N the number of lattice sites. If the copy attempt results in a lower effective energy of the system, the copy attempt is accepted. With an effective energy H:

(2)

where represents the bond energy between a lattice site and one of its eight, second-order neighbours , and and represent energy penalties for deviation from a designated target area () or target length (), which we set at 50 and 5.0 respectively.

If the copy attempt does not result in a lower effective energy, the copy is accepted with Boltzmann probability:

(3)

where T is the “cellular temperature” of the system, a measure of cell motility, which we set at 50, and the change in the effective energy function.

In the cell elongation [14] and contact inhibition [28] models cells secrete a chemoattractant , which diffuses and degrades in the ECM:

(4)

where outside the cells, is the secretion rate and the degradation rate of the chemoattractant and the diffusion constant of the chemoattractant.

An additional term is added to the energy function, which promotes copies upwards the chemoattractant gradient:

(5)

Where is the strength of the chemotactic response, which we set at 1000. For details see [14,28].

In the mechanical cell-cell communication model [27], cells were able to deform the ECM through cell-shape dependent traction forces. The deformations of the ECM are calculated using the finite element method. An additional term is added to the energy function, such that cells respond to the ECM stiffness:

(6)

where for cell extensions and for cell retractions, is a parameter which we set at 24, mimics the influence of the ECM stiffness as a function of and , the principal strains, on the tension in the cell-ECM connection, gives the copy direction, and and are the strain orientations.

The preference for higher stiffness is implemented as a sigmoid function:

(7)

where sets the strength of the durotactic response, determines the stiffness where half this strength is reached and represents the stiffness sensitivity which determines the steepness of the curve. For details see [27].

Supporting information

Acknowledgments

This work was performed using the compute resources from the Academic Leiden Interdisciplinary Cluster Environment (ALICE) provided by Leiden University.