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Conceived and designed the experiments: ALB. Performed the experiments: ALB. Analyzed the data: ALB TLJ YJ. Contributed reagents/materials/analysis tools: ALB. Wrote the paper: ALB.

The authors have declared that no competing interests exist.

The extracellular matrix plays a critical role in orchestrating the events necessary for wound healing, muscle repair, morphogenesis, new blood vessel growth, and cancer invasion. In this study, we investigate the influence of extracellular matrix topography on the coordination of multi-cellular interactions in the context of angiogenesis. To do this, we validate our spatio-temporal mathematical model of angiogenesis against empirical data, and within this framework, we vary the density of the matrix fibers to simulate different tissue environments and to explore the possibility of manipulating the extracellular matrix to achieve pro- and anti-angiogenic effects. The model predicts specific ranges of matrix fiber densities that maximize sprout extension speed, induce branching, or interrupt normal angiogenesis, which are independently confirmed by experiment. We then explore matrix fiber alignment as a key factor contributing to peak sprout velocities and in mediating cell shape and orientation. We also quantify the effects of proteolytic matrix degradation by the tip cell on sprout velocity and demonstrate that degradation promotes sprout growth at high matrix densities, but has an inhibitory effect at lower densities. Our results are discussed in the context of ECM targeted pro- and anti-angiogenic therapies that can be tested empirically.

A cell migrating in the extracellular matrix environment has to pull on the matrix fibers to move. When the matrix is too dense, the cell secretes enzymes to degrade the matrix proteins in order to get through. And when the matrix is too sparse, the cell produces matrix proteins to locally increase the “foothold”. How cells interact with the extracellular matrix is important in many processes from wound healing to cancer invasion. We use a computational model to investigate the topography of the matrix on cell migration and coordination in the context of tumor induced new blood vessel growth. The model shows that the density of the matrix fibers can have a strong effect on the extension speed and the morphology of a new blood vessel. Further results show that matrix degradation by the cells can enhance vessel sprout extension at high matrix density, but impede sprout extension at low matrix density. These results can potentially point to new targets for pro- and anti-angiogenesis therapies.

The extracellular matrix (ECM) is a major component of the extravascular tissue region, or stroma, and plays a central role in morphogenesis, including embryogenesis

Cells are equipped with and can upregulate transmembrane receptors that enable them to receive signals from and interact with their environment. Integrins are one such receptor and are stimulated by the various proteins of the ECM

The physical properties of the ECM, such as density, heterogeneity, and stiffness, that affect cell behavior is also an area of current investigation. Matrigel, a popular gelatinous protein substrate for

Understanding how individual cells interpret biochemical and mechanical signals from the ECM is only a part of the whole picture. Morphogenic processes also require multicellular coordination. In addition to the guidance cues cells receive from the ECM, they also receive signals from each other. During new vessel growth, cells adhere to each other through cell-cell junctions, called cadherins, and in order to migrate, cells must coordinate integrin mediated focal adhesions with these cell-cell bonds. This process is referred to as collective or cluster migration

Cells also have the ability to condition the ECM for invasion by producing proteolytic enzymes that degrade specific ECM proteins

In this work, we extend our cellular model of angiogenesis

We previously published a cell-based model of tumor-induced angiogenesis that captures endothelial cell migration, growth, and division at the level of individual cells

The model is two-dimensional. It uses a lattice-based cellular Potts model describing individual cellular interactions coupled with a partial differential equation to describe the spatio-temporal dynamics of vascular endothelial growth factor. At every time step, the discrete and continuous models feedback on each other, and describe the time evolution of the extravascular tissue space and the developing sprout. The cellular Potts model evolves by the Metropolis algorithm: lattice updates are accepted probabilistically to reduce the total energy of the system in time. The probability of accepting a lattice update is given by

The amount of VEGF available at the right hand boundary of the domain is estimated by assuming that in response to a hypoxic environment, quiescent tumor cells secrete a constant amount of VEGF and that VEGF decays at a constant rate. It is reasonable to assume that the concentration of VEGF within the tumor has reached a steady state and therefore that a constant amount of VEGF, denoted

(A) The average extension speeds of our simulated sprouts agree with empirical measurements

Parameter | Symbol | Model Value | Range | Reference |

VEGF Diffusion | 3.6×10^{−4} cm^{2}/h |
|||

VEGF Decay | .6498 h^{−1} |
|||

VEGF Uptake | .06 pg/EC/hr | |||

VEGF Source | S | .035 pg/pixel | ||

Activation Threshold | .0001 pg | fixed | ||

Adhesion | E/L | |||

30 | [10, 50] | |||

76 | I | est | ||

66 | [46, 76] | |||

71 | I | est | ||

85 | I | est | ||

85 | I | est | ||

Membrane Elasticity | E/L^{4} |
|||

0.8 | [0.3, 3] | |||

0.5 | I | |||

0.5 | I | |||

Chemotactic Sensitivity | 1.11·10^{6} E/conc |
[10^{4}, 10^{7}] |
||

est | ||||

Intracellular Continuity | 300 E/L | fixed | est | |

Boltzmann Temperature | kT | 2.5 E | [0.25,11] | est |

The relative value of the cellular Potts model parameters corresponds to referenced physiological measurements and gives rise to cell behavior observed experimentally. Dimensions are given in terms of length, L, and energy, E. EC denotes endothelial cell, ‘est’ indicates an estimated parameter, and I is an insensitive parameter.

To more accurately capture the cell-cell and cell-matrix interactions that occur during morphogenesis, we implement several additional features to this model. One improvement is the implementation of stalk cell chemotaxis. Stalk cells are not inert, but actively respond to chemotactic signals

During the early stages of angiogenesis, cells are recruited from the parent vessel to facilitate sprout extension

As in our previous model, once a cell senses a threshold concentration of VEGF, given by

As described in our previous work

No single model has been proposed that incorporates every aspect of all processes involved in sprouting angiogenesis, nor is this level of complexity necessary for a model to be useful or predictive. It is not our intention to include every bio-chemical or mechanical dynamic at play during angiogenesis. We develop this two-dimensional cell-based model as a step towards elucidating cellular level dynamics fundamental to angiogenesis, including cell growth and migration, and cell-cell and cell-matrix interactions. Consequently, we do not incorporate processes or dynamics at the intracellular level. For example, we describe endothelial cell binding of VEGF to determine cell activation and to capture local variations in VEGF gradients, but neglect intracellular molecular pathways signaled downstream of the receptor-ligand complex. Moreover, our focus is on early angiogenic events and therefore we also do not consider the effects of blood flow on remodeling of mature vascular beds. Numerical studies of flow-induced vascular remodeling have been given attention in McDougall et al.

As is the case in many other simulations of biological systems, when we do not have direct experimental measurements for all of the parameters, choosing these parameter values is not trivial. A list of values and references for our model parameters is provided in ^{5} Pa ^{−6} Pa ^{−6} Pa) and is reflected in the relative values of the corresponding parameters ^{−4}. Therefore, to balance adhesion and growth, ^{6}. We calibrate this parameter to maximize sprout extension speeds. Similarly, the parameter for continuity,

The canonical benchmark for validating models of tumor-induced angiogenesis is the rabbit cornea assay ^{2} area at 1 minute intervals for 10 hours and show sprouting angiogenesis over this period. The average extension speed for newly formed sprouts is 14 µm/hr and ranges from 5 to 27 µm/hr. For cell survival, growth factor is present and is qualitatively characterized as providing a diffuse, or shallow, gradient. No quantitative data pertaining to growth factor gradients or the effect of chemotaxis during vessel growth are reported

We use the above experimental models and reported extension speeds as a close approximation to our model of

Simulated | Observed | Reference | |

Velocity | 10.4±.2 µm/hr | 14 µm/hr (at 10 h) | |

16.0±.6 µm/hr | 10.4–20.8±4.2 µm/hr | ||

Thickness | 16.2±2.4 µm | 15 µm | |

17±4 µm | [pc] | ||

Cell Size | 15–40 µm | 20–40 µm |

Sprout velocity is given at 10 hours for direct comparison to

On average, simulated sprouts migrate 160 µm and reach the domain boundary in approximately 15.6 hours, before any cells in the sprout complete their cell cycle and proliferate. We do not expect to see proliferation in the new sprout because the simulation duration is less than the 18 hour cell cycle and the cell cycle clock is set to zero for newly recruited cells to simulate the very onset of angiogenesis. In our simulations, sprout extension is facilitated by cell recruitment from the parent vessel. Between 15 and 20 cells are typically recruited, which agrees with the number of cells we estimate would be available for recruitment based on parent vessel cell proliferation reported by Kearney et al.

By adjusting key model parameters, we are able to simulate various morphogenic phenomena. For example, by increasing the chemotactic sensitivity of cells in the sprout stalk and decreasing the parameter controlling cellular adhesion to the matrix,

Figure 5 from Oakley

We design a set of numerical experiments allowing us to observe the onset of angiogenesis in extravascular environments of varying matrix fiber density. We consider matrix fiber densities given as a fraction of the total interstitial area,

The average rate at which the sprout grows and migrates, or its average extension speed, is calculated as the total tip cell displacement in time. Average extension speeds in microns per hour (µm/hr) versus matrix fiber density (

(A) Dependence of average sprout extension speed on the density of the extracellular matrix. The model predicts that average extension speeds are maximal in the fiber fraction range

^{2} ^{2}, cells detach from the substrate and lose their viability

From top left to bottom right: (A)

For ^{2}) demonstrating peak migration speeds at intermediate concentrations (5 µg/cm^{2})

As matrix density increases, the network of connected fibers is extensive. Higher fiber density translates into greater matrix homogeneity and a loss of strong guidance cues from fiber heterogeneity. Chemotaxis then plays a stronger role in sprout guidance thereby producing linear sprouts (

Looking at

Based on our earlier observations, the density of the ECM affects capillary sprout migration speeds. As matrix density is increased, a connected fibrous network develops which could be a mechanism for differences in observed average speeds. We hypothesized that peak extension speeds occur when the mechanical properties of the ECM provide contact guidance cues that are aligned with the chemotactic gradients. To examine the effects of matrix fiber alignment on average rates of capillary sprout elongation, we devise another set of numerical experiments. If matrix fiber alignment plays a prominent role in sprout migration, we would expect more rapid rates of sprout elongation when matrix fibers are aligned parallel to VEGF gradients than when fibers are aligned perpendicular to the gradient. We look at three specific cases: matrix fibers aligned perpendicular to VEGF gradients, matrix fibers aligned parallel to the VEGF gradient, and a combination of horizontal and vertical fibers only. We compare these test cases with the baseline simulations of sprout development on matrices of random fiber orientation. We distinguish and refer to these three cases by the angle that is formed between the fiber axis and the axis of the VEGF gradient, that is, 0° denotes a matrix with fibers aligned parallel to the gradient and 90° identifies a matrix of fibers perpendicular to the VEGF gradient. These numerical experiments represent a simplified replica of the matrix fiber restructuring and fiber alignment that occurs as a result of the tractional forces exerted by endothelial cells during migration

As matrix fiber density increases, both the number of focal adhesion binding sites available in the ECM and the connectivity of the fiber network increase. As a measure of connectivity, we consider the network connected if there exists a continuous path along matrix fibers from the parent vessel to the source of chemoattractant. As the density of matrix fibers increases, there will be a density that guarantees network connectedness. This threshold density is known as the percolation threshold. Our model fiber networks are constructed by randomly placing fibers at randomly selected but discrete orientations: 0°, ±30°, ±45°, ±60°, and 90°. Consequently, our fiber network most closely approximates a triangular lattice. We estimate that the percolation threshold in our fiber networks occurs between

(A) At

In light of the above results, we construct patterned matrix topographies to look at the effect of unambiguous contact guidance cues on cell shape, orientation, and sprout morphology. In these numerical experiments, instead of distributing fiber bundles, we engineer matrix cord patterns that vary in width and orientation. As a baseline, we augment a matrix of randomly distributed fibers with horizontal cords 7.2 µm thick (

Sprouts migrate toward higher concentrations of VEGF, however, cells elongate and are clearly oriented in the direction of the matrix cords. (A) Matrix of randomly distributed fibers augmented with horizontal cords 7.2 µm thick, (B) matrix cords 7.2 µm thick aligned horizontally, (C) horizontal cords 2.2 µm thick, (D) vertical cords 2.2 µm thick, and (E) crosshatched cords. Horizontal cords are aligned with to the VEGF gradient (0°); vertical cords are perpendicular to the gradient (90°); crosshatched cords form a ±45° angle with the VEGF gradient. These results demonstrate the important role of contact guidance and tissue structure in determining cell shape and orientation. Snapshots at 12.5 hours.

We find a strong correspondence between fiber alignment and cell shape and orientation. We define cell orientation as the axis of elongation. In

During angiogenesis, endothelial cells not only realign matrix fibers, but they also secrete matrix degrading proteases that break down extracellular matrix proteins and facilitate sprout migration through the stroma ^{2} of matrix each minute. We choose this rate of degradation as a rough approximation based on numerical studies of tip cell collagen proteolysis

Solid lines represent average extension speeds without matrix degradation and the corresponding colored dashed lines show average speeds with tip cell matrix degradation. For

(A) shows that tip cell matrix degradation promotes sprout development at

The effect of degradation is to decrease the density of the ECM and this decrease is entirely localized to the area under and immediately surrounding the sprout body (

In our model, without degradation we observe no branching at matrix fiber densities above

To ascertain the variability and sensitivity of our results to the choice of parameters, we vary one parameter at a time, holding fixed all other

Observation | Parameter Control | |

Unrealistic cell shapes | Strong cell-cell bonds | |

Deform to increase cell-cell contact | ||

Realistic cell shapes & elongation | Balance between cell-cell contact and motility mechanisms | |

Rounder cells | Deform to decrease cell-cell contact | |

Cells migrate away | Little cell-cell adhesion | |

Unrealistic cell shapes | Strong cell-ECM adhesion | |

Deform to increase cell-ECM contact | ||

Thicker, tortuous sprouts | Some focal adhesion release but cells “stick” to ECM | |

Realistic cell shapes | Balance between contact guidance & release of focal adhesion bonds | |

Linear sprouts | Weak contact guidance | |

Chemotaxis dominates | ||

Cells immobile | Inhibition of cell-matrix adhesion | |

Large cells, fewer recruited | Cells easily deviate from target size | |

Realistic cell sizes | Balance between growth & chemotaxis | |

Tip cell detachment | Pressure to keep stalk cell size>chemotactic energy of stalk cells | |

No migration | Chemotactic stimulus too weak | |

Slow migration | Chemotactic stimulus too weak | |

Cells migrate faster with increasing |
Balance between chemotaxis, growth, and contact guidance | |

Cells are pulled apart | Chemotactic stimulus too strong | |

Cells dissociate | Moderated by continuity constraint | |

Faster sprout migration with increasing kT | Lowers probability that large change to total energy is accepted |

In addition, the results do not depend on the compressibility properties of the matrix fibers or interstitial fluid,

Below

The extracellular matrix has attracted a great deal of attention from researchers and experimentalists because of its vital role as a modulator of morphogenic processes. Inspired by our previous finding that the stromal heterogeneity has a strong influence on sprout morphogenesis

During morphogenesis, cells actively restructure and condition the extracellular matrix for migration through proteolytic degradation and fiber reorganization and alignment

Describing matrix fiber cross-linking, viscous interstitial flow, and cell-matrix interactions dynamically within the same modeling framework is currently one of the big challenges in modeling morphological events. A first step is to provide an explicit description of the ECM and cell-matrix interactions, which we have done in this model. Our model is one of few to provide an explicit treatment of the ECM and the only to do so in a cell-based framework. Our model incorporates some key cell-ECM interactions, including adhesion and degradation. We do not, however, consider matrix reorganization or remodeling that can result from endothelial cell matrix secretion, adhesion, and migration. Nor do we consider dynamic matrix fiber cross-linking, which would allow an explicit description of matrix stiffness and the ability to quantify the effects of substrate rigidity on cellular behavior. Instead, we employ a static ECM in this initial investigation so that we can confidently associate vascular morphology with extracellular matrix topology. By doing so, we have shown that matrix topology alone is enough to regulate cell shape and orientation and to initiate sprout branching. Dynamic imaging techniques have recently been developed and are now being used in

It is worth pointing out that at a distance of 100 µm from a tumor 1 mm in diameter, we specify a linear source of VEGF. This choice ensures little or no gradient in the transverse or y–direction and allows us to attribute lateral cell and sprout movement to the mechanical effects of the matrix. Different spatial profiles of VEGF, for example a parabolic source or local sinks and sources of VEGF in the ECM, could also contribute to branching and varied morphological patterns. The effect of different VEGF profiles on angiogenesis has been theoretically modeled by Anderson and Chaplain

Increased understanding leading to the ability to control angiogenesis

The ECM and cell-matrix associations also provide promising possibilities for angiotherapy, but have only more recently received attention as targets and are in less advanced stages of clinical development. Consequently, modeling and simulation have the potential to contribute to and propel further advancement. Current therapeutic interventions aimed at cell-matrix interactions during angiogenesis focus on tissue inhibitors of metalloproteinases (TIMPs) and on integrin-mediated cellular adhesion

Using our model, we regulate cell-matrix binding affinity (

In these collective studies, we use the model to isolate and examine variations in fiber density and structure, and proteolytic matrix degradation as independent mechanisms that control vascular morphogenesis. Nonetheless, the integrin, protease, and growth factors systems are highly connected and provide regulatory feedback for each other

Cell elongation. For a different parameter set, fewer cells are recruited from the parent vessel and cells elongate. Here cells are approximately 40 µm in length and the average extension speed at 14 hours is 6.8 µm/hr. J_{{ee,em,ef}} = {42,76,66}, χ_{tip} = 1.55 χ, χ_{{stalk, prolif}} = 1.45 χ.

(2.30 MB TIF)

Our model accurately captures the cellular response to topographical guidance (no VEGF) on patterned substratum. Compare this image with morphological data of fibroblasts stained for actin and tubulin showing that cells alter their shape, orientation, and polarity to align with the direction of the grooves [see Figure 5e,f from Oakley

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Single cell migration/invasion

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Preferential migration along stretched cells

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Matrix anisotropy induces branching

(5.05 MB MP4)

This work was performed at Los Alamos National Laboratory as part of the PhD dissertation research of ALB.