Square grid.

We consider a square domain Ω of side L with grid points (x_i, y_j), where x_i = i h, y_j = j h with i, j = 0, …, M − 1, h = L/(M − 1), and M is the number of nodes on a side of the square. The square contains M × M grid points and (M − 1)² elementary squares (pixels), each having an area L²/(M − 1)². To enumerate nodes, we use left-to-right, bottom-to-top order, starting from node 0 on the bottom left corner of the square and ending at node M² − 1 on the rightmost upper corner. In numerical simulations, we use L = 0.495 mm.

Objects, spins and Metropolis algorithm.

Pixels x can belong to different objects Σ_σ, namely ECs, and ECM. The field (called spin in a Potts model) σ(x) denotes the label of the object occupying pixel x [75]. Each given spin configuration for all the pixels in the domain has an associated energy H({σ(x)}) to be specified below. At each Monte Carlo time step (MCTS) t, we select randomly a pixel x, belonging to object Σ_σ, and propose to copy its spin σ(x) to a neighboring (target) pixel x′ that does not belong to Σ_σ(x). The proposed change in the spin configuration (spin flip) changes the configuration energy by an amount △H|_{σ(x)→σ(x′)}, and it is accepted with probability (Metropolis algorithm) [67, 75] (1) The temperature T > 0 is measured in units of energy and it is related to an overall system motility. We have selected T = 4 in our simulations.

Energy functional.

The energy functional H is (2) Here the three first terms are sums over cells and the fourth one sums over all pixels. We have

• a_σ is the area of the cell Σ_σ, A_σ is the target area and ρ_area is the Potts parameter which regulates the fluctuations allowed around the target area. There are two cell types: non-proliferating tip and stalk cells with A_σ = 78.50 μm² and proliferating cells with double target area, A_σ = 157 μm². The target radius of a proliferating cell is times that of a non-proliferating cell.
• p_σ is the perimeter of the cell Σ_σ, P_σ is the target perimeter and ρ_perimeter is the Potts parameter which regulates the fluctuations allowed around the target perimeter. The target perimeters are P_σ = 31.4 μm for non-proliferating cells, and thrice this, P_σ = 94.2 μm, for proliferating cells.
• l_σ is the length of the cell Σ_σ, L_σ = 49.5 μm is the target length of nonproliferating cells, L_σ = 70 μm is the target length of proliferating cells, and ρ_length is the Potts parameter which regulates the fluctuations allowed around the target length. We define the length of the cell from the longest axis of an ellipse that has the same moment of inertia as the cell. The inertia tensor per unit cell area is [83] (3) where N_σ is the number of pixels in cell σ and the distances X in the inertia tensor are measured from the center of mass of the cell. Let us now consider an ellipse whose axes have lengths 2a and 2b (a ≥ b). Its inertia tensor per unit area defined as in Eq (3) has eigenvalues b²/4 and a²/4. Thus, a is twice the square root of the largest eigenvalue of the inertia tensor, (4) We define the length of the cell σ as 2a, where a, given by Eq (4), is calculated from the inertia tensor per unit area of Eq (3). See also [84].
• The Potts parameter is the contact cost between two neighboring pixels. The value of this cost depends on the type of the object to which the pixels belong (cell or medium). Since δ_σ,σ′ is the Kronecker delta, pixels belonging to the same cell do not contribute a term to the adhesion energy.
• The net variation of the durotaxis term H_durot is [67] (5) where ρ_durot is a Potts parameter, g(x, x′) = 1 for extensions and g(x, x′) = −1 for retractions, ϵ₁ and ϵ₂ and v₁ and v₂ (|v₁| = |v₂| = 1) are the eigenvalues and eigenvectors of the strain tensor ϵ_T, respectively. They represent the principal strains and the strain orientation. ϵ_T is the strain in the target pixel for extensions, and the strain in the source pixel for retractions. h(E) = 1/(1 + exp(−ω(E − E_θ))) is a sigmoid function with threshold stiffness E_θ and steepness ω. E(ϵ) = E₀(1 + (ϵ/ϵ_st)1_ϵ≥0) is a function of the principal strains, in which E₀ sets a base stiffness for the substrate, ϵ_st is a stiffening parameter and 1_ϵ≥0 = {1, ϵ ≥ 0; 0, ϵ < 0}: strain stiffening of the substrate only occurs for substrate extension (ϵ ≥ 0), whereas compression (ϵ < 0) does not stiffen or soften the substrate. We have used the parameter values: E_θ = 15 kPa, E₀ = 10 kPa, ω = 0.5 kPa⁻¹, and ϵ_st = 0.1 [67].
• The variation of the chemotaxis term H_chem is (6) where ρ_chem(x, x′) ≥ 0 is a Potts parameter that depends on the type of EC or ECM occupying pixels x and x′ and will be specified later. Its magnitude is measured by a positive constant . We have α_chem = 0.3 and C is the VEGF concentration in the corresponding pixel.

The values of the Potts parameters are listed in Table 1. They are chosen according to those proposed by Bauer et al. [77] and Van Oers et al. [67] and adjusted so as to make that every term of the net variation of the hamiltonian have the same order. The perimeter contribution, absent in Refs. [67, 77], is small compared to the other terms in Eq (2), so that it only affects the computations in extreme cases (e.g., extremely thin cells, thin cells that stick to the blood vessel). We have added a factor in the chemotaxis term to regulate the fluctuation around the resting VEGF concentration. Note that if α_chem is equal to zero, we recuperate the original term of Bauer et al. [77]. The proposed value, α_chem = 0.3, is small.

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Table 1. Dimensionless Potts parameters.

https://doi.org/10.1371/journal.pcbi.1006919.t001

What is the effect of changing the numerical values of the Potts parameters? As previously stated, with the values in Table 1, every term of the net variation of the hamiltonian has the same order. Variations of 10% or smaller in Potts parameters do not change the outcome of the simulations. Variations larger than 10% with respect to those in Table 1 produce unrealistic effects, which are as follows.

• ρ_area. Larger increments force cells to reach their target area faster, thereby increasing cell proliferation. The corresponding term becomes more important than the chemotaxis mechanism, which produces slower evolution of vessels toward the hypoxic zone and large clumps of cells in the vessels. Large reductions of this Potts parameter produce irregular cell proliferation and a much larger variety of cell sizes.
• ρ_perimenter. Large increments produce round cells, whereas large reductions (up to ρ_perimenter = 0) create extremely long and narrow cells stuck to the vessel sprout due to the now dominant effect of the adhesion term.
• ρ_length. Large increments produce elongated cells and force them to reach their target length faster. The corresponding term becomes more important than the adhesion or area term. Then there appear isolated cells that take a long time to reach their target area and proliferate (if they are marked for proliferation). Large reductions (up to ρ_length = 0) produce rounder cells depending on the values of the other Potts parameters.
• ρ_durot. This parameter produces qualitative changes only if it is ten times larger than in Table 1. In such a case, durotaxis overwhelms chemotaxis and the perimeter penalty, leading to cells following the stiffness gradients and sticking to each other, which create very irregular vessels.
• ρ_chem. Larger increments make chemotaxis dominant. Then cells become bigger and elongated and sprouts extend more rapidly. Sometimes tip cells separate from their sprouts as chemotaxis dominates adhesion effects. Larger reductions produce rounder cells that do not polarize along a specific direction, and produce wider and slower sprouts.
• . The adhesion Potts parameter take on different values for cell-cell and cell-ECM boundaries. If these values become equal (e.g., to 16.5), narrower sprouts are produced and there are cells that escape from them. Larger increments of cell-ECM adhesion, makes very costly for ECM to surround cells, which then stick to each other too much. Larger reductions of cell-ECM produces more elongated cells. Reducing cell-cell adhesion favors cells sticking to each other and acquiring irregular shapes since the zero energy for a pixel to be surrounded by other pixels of the same cell would be very similar to the small positive energy for the pixel to be surrounded by pixels of a different cell.

VEGF concentration.

The VEGF concentration obeys the following initial-boundary value problem [77]: (7) (8) (9) In Eq (7), the amount of VEGF bound by an EC per unit time is (10) where υ = 1 h⁻¹ and Γ = 0.02 pg/(μm² h) is the maximum amount of VEGF that it could be consumed by a cell per hour [63, 77]. Other values we use are D_f = 0.036 mm²/h, ν = 0.6498/h [77], and S = 5 × 10⁻¹⁹g/μm² (corresponding to 50 ng/mL [38, 85] for a sample having a 10 μm height [67]).

Strains.

Following Ref. [67], we calculate the ECM strains by using the finite element method to solve the stationary Navier equations of linear elasticity: (11) Here K is the stiffness matrix, u is the array of the x and y displacements of all nodes and f is the array of the traction forces per unit substrate thickness exerted by the cells. For nodes outside ECs, f = 0. For nodes inside ECs, each component represents the traction stress on the kth node, μ_force, times the sum of the distances, d_kj, between the kth node and any node j in the same cell (σ_k is the label of the cell at which node k belongs).

The global stiffness matrix K is assembled from the stiffness matrices K_e of each pixel, (12) in which (13) and B is the strain-displacement matrix for a four-noded quadrilateral pixel (finite element) [67]. B is a 3 × 8 matrix that relates the 8-component node displacement u_e of each pixel to local strains ϵ, (14) where ϵ = (ϵ₁₁, ϵ₂₂, ϵ₁₂) is the 3-component column notation of the strain tensor (15) We have used the numerical values E = 10 kPa, ν = 0.45, and μ_force = 1 N/m². With these definitions and the durotaxis term given by Eq (5), ECs generate mechanical strains in the substrate, perceive a stiffening of the substate along the strain orientation, and extend preferentially on stiffer substrate. The simulated ECs spread out on stiff matrices, contract on soft matrices, and become elongated on matrices of intermediate stiffness [67].

Cell types.

In the model, ECs may be on a tip, hybrid or stalk cell phenotype. In nature, tip cells are characterized by having high levels of Delta-4, VEGFR2, and active VEGF signaling (i.e., high levels of VEGF internalization). They develop filopodia and migrate along the VEGF-A gradient, leading the formation of new branches. Delta-4 proteins at tip cell membranes inhibit the neighboring cells (due to lateral inhibition) to adopt a tip phenotype, thereby forcing them to become stalk cells (with low Delta-4, VEGFR2 and internalized VEGF).

Likewise, in our model, tip cells are distinguished by the number of VEGF molecules they possess. Therefore, a cell that has V larger than all its neighbors and V > 0.5 max_iV_i(t) will acquire the tip cell phenotype and be very motile. To simulate this, tip cells are able to follow the mechanical and chemical cues on the environment, having ρ_chem ≠ 0 and ρ_durot ≠ 0. On the other hand, stalk cells are less motile. We consider two different cases: (A) nonmotile stalk cells with ρ_chem = ρ_durot = 0 in the model (except when they undergo proliferation, as explained below) [77]; and (B) motile stalk cells with the same ρ_durot as for the tip cells, but a smaller ρ_chem than that of the tip cells (see below). Stalk cells, by virtue of the lateral induction, characteristic of Notch-Jagged signaling, are able to induce neighboring cells to adopt a stalk cell phenotype, by promoting a decrease of internal VEGF in them.

In our model we track the cells belonging to each growing vessel. A new sprouting vessel can be formed when a stalk cell acquires the tip phenotype. This cell can then become the leading cell of a new vessel that branches out from out the old one. This is illustrated in Fig 1. If the levels of VEGF inside the tip cells that lead an active growing branch drop to values in the interval 0.2 max_iV_i(t) < V < 0.5 max_iV_i(t), these cells will be in the hybrid phenotype. In spite of the lower amount of Delta-4, VEGFR2 and VEGF, these cells remain with the tip cell characteristics and are able to lead the sprout. Similarly, stalk cells whose internal VEGF increases to the same range acquire the hybrid phenotype and can lead a sprout. The number of cells in the hybrid phenotype is only appreciable for larger Jagged production rates.

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Fig 1. Example of tip cell exchange and branching in the direction of the blue arrow.

Times in MCTS are: (a) 422, (b) 423, (c) 460, (d) 461, (e) 545, (f) 630. The black arrows in this figure represent the directions of largest eigenstrain and, therefore, they point to the likeliest direction of EC motion. The blue arrows indicate the actual direction of motion of a selected tip cell (marked in pink color) for the simulation we have carried out.

https://doi.org/10.1371/journal.pcbi.1006919.g001

Branching.

When a stalk cell acquires the tip cell or the hybrid tip/stalk cell phenotype, this event will lead to the creation of a new active sprouting branch depending on its localization within the existing branch and on its moving direction.

To create a new branch, the boundary of the tip cell must touch the ECM. Moreover, let be the set of ECM pixels that have boundary with the branching tip cell. For each pixel , let the strain vector be v_p = ϵ_jv_j, where ϵ_j is the largest eigenstrain at pixel x_p and v_j is the corresponding unit eigenvector, as defined after Eq (5). The average modulus and argument for the branching cell i are (25)

Let us also assume that the gradient of the chemotactic factor C forms an angle Θ with the x-axis. The new tip cell will branch out, creating a new vessel, if the direction given by θ_i points in the direction of the ECM and if −π/2 < θ_i − Θ < π/2. For other values of θ_i, the tip cell does not leaves the parent vessel, since the chemotactic term of Eq (6) opposes branching. In those cases, the direction given by θ_i points to another cell, not to the ECM. To facilitate branching computationally, we directly exchange the new tip cell with this neighboring cell (see Fig 1(a) and 1(b)). These exchanges may continue in successive MCTS until the new tip cell reaches a boundary of the blood vessel for which the direction given by θ_i points to the ECM, as shown in Fig 1. These cell exchanges in 2D mimic the climbing motion of the new tip cell over the parent vessel in a 3D geometry without merging with it.

We set the branching process to take at least 400 MCTS (incubation time). During this time we implement a persistent motion of the new tip cell in the direction marked by the angle θ_i. During this incubation period, and to provide a good separation from the parent vessel we permit the tip cell to proliferate once (see Fig 1(e) and 1(f), and see below). After this time, the dynamics of the branching vessel follows the same rules as that of any other actively sprouting vessel.

Cell proliferation and duration of one MCTS.

Endothelial cell proliferation in sprouting angiogenesis is regulated by both mechanical tension and VEGF concentration. In sprouting angiogenesis the tip cell creates tension in the cells that follow its lead. On those first stalk cells, this tension produces strain that triggers cell proliferation, if VEGF concentration is high enough [36]. Therefore, in our model, for each active sprouting vessel, one of the stalk cells that is in contact with a tip cell is randomly chosen to undergo proliferation. Only one cell per sprout proliferates. Tip cells in the model cannot proliferate, except only once when they start a new branch. Once a stalk cell attached to a tip cell has been randomly selected as a proliferating cell, its target area in the CPM is set to become twice the size, whereas its target perimeter is set to a value three times that of non-proliferating cells. This cell will then grow in successive MCTS until it reaches this large target area. Then the cell proliferates if the following three conditions hold: (i) C(x_i, t) > ψ_p (external VEGF surpasses a threshold), (ii) the cell belongs to an active blood vessel with cell proliferation, and (iii) the cell is not surrounded completely by other cells. Failure to meet one of these conditions precludes proliferation. If the three conditions are met, we use the unsupervised machine learning algorithm K-means clustering to split the cell. This algorithm calculates the Euclidean distance of each pixel in the cell to the centroid of two groups of pixels and corrects the centroids until the two pixel groups are balanced. These two groups comprise the new cells. Provided the daughter cells share boundary with the tip cell, one of them is randomly chosen to retain the ability to proliferate but the other cell does not proliferate. If the daughter cells do not share boundary with the tip cell, they both become non-proliferating and a different cell that shares boundary with the tip cell is randomly chosen to become a proliferating cell.

The actual proliferation rate depends on the duration of the cell cycle in MCTS and on how many seconds one MCTS lasts. The latter time is fitted so that numerical simulations reproduce the experimentally observed velocity of a sprouting vessel. We consider two cases. In case (A), we drop the elongation constraint ρ_length = 0 and make the stalk cells insensitive to chemo and durotaxis, therefore ρ_chem = ρ_durot = 0 for them. Only tip cells move in this case. In case (B), stalk cells also move, albeit more slowly than tip cells. While ρ_durot = 25 for both types of EC, ρ_chem(x, x′) is given by (26) where i and j are ECs. The level of Delta-4 determines the EC phenotype and, according to Eqs (6) and (26), the strength of their chemotactic drive. Tip cells have a higher level of Delta-4 and, consequently, they are more motile than stalk cells. As the latter also move, it may occur, as shown in Fig 2, that a stalk cell overtakes a tip cell to become the leading cell of its sprout. This has been observed in experiments [37, 38] and in numerical simulations of CPMs different from ours [86]. In our model, the imposed strong gradient of the VEGF concentration C(x, t) precludes ECs to reverse their direction, unlike the CPMs in Ref. [86], which set a constant external VEGF concentration for their simpler Notch-Delta signaling dynamics. In Ref. [86], chemotaxis is guided by the gradients of a chemical signal secreted by the ECs (seemingly different from VEGF). The chemical signal also diffuses and is consumed by the ECM. Local chemical signal gradients may become contrary to the motion of a given EC, thereby facilitating reversal of its motion. In our model on the other hand, numerical simulations show that a growing sprout may separate from the primary blood vessel more than one cell diameter (10 μm). As the primary vessel is a source of ECs, we create a new stalk cell to fill the resulting hole if this happens.

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Fig 2. Examples of stalk cells (light color) overtaking tip cells (dark color) in numerical simulations with case (B) dynamics.

EC centers are marked by dots. Note that ECs are more elongated than those undergoing case (A) dynamics.

https://doi.org/10.1371/journal.pcbi.1006919.g002

To obtain the equivalence between the number of MCTS and the time measured in experiments, we measured the pixel size in Fig. 1H of Ref [85], which is 0.9 μm. According to Fig. 3C of the same reference, the vessel mean elongation is 150 pixels (135 μm) in 36 hours for 50 ng/mL VEGF concentration. In our simulations of case (A) dynamics, the vessel mean elongation is 495 μm in 3001 MCTS. Thus, we set 1 MCTS to be 0.044 hours. A similar calculation for case (B) dynamics yields 1 MCTS = 0.03 hours (vessel mean elongation of 247.5 μm in 2200 MCTS).

We can estimate roughly EC proliferation time by dividing the MCTS one sprout is active by the number of cells created in the sprout during that time. We discard those sprouts whose ECs have never proliferated and average over all numerical simulations with the same dynamics, i.e., cases (A) and (B). The resulting proliferation times are 120 MCTS (5.28 hours) for case (A), and 217 MCTS (6.51 hours) for case (B). The proliferation times thus obtained are lower bounds: although only one EC of a given sprout is selected to proliferate at a given time, we cannot guarantee that the same EC of the same sprout will proliferate again at a later time. It could be a different cell that proliferates next, which would surely occurs when performing different numerical simulations. Then the average time for one EC to complete its cell cycle should be longer than that given by our estimation. With this proviso, our proliferation times are shorter but of the same order of magnitude as previously reported division times for EC in vivo or in vitro (83 minutes for cell mitosis but 17.8 hours to complete the cell cycle [87]). Fine tuning of parameters and better estimations of the cell cycle from numerical simulations might produce better agreement with reported experimental values, which, anyway, present some variability.

Anastomosis.

When an active sprouting blood vessel merges with another active sprouting vessel, i.e. during anastomosis, one of them becomes inactive. If the collision occurs between tip cells of two different vessels, one vessel is randomly chosen to become inactive. If one tip cell merges with a stalk cell of a different active sprouting vessel, the vessel to which the tip cell belongs becomes inactive. The cells of an inactive vessel do not proliferate or branch, although they continue to undergo Notch signaling dynamics.