Abstract:Angiogenesis is the process by which new blood vessels develop from existing vasculature. During angiogenesis, endothelial tip cells migrate via diffusion and chemotaxis, loops form via tip-to-tip and tip-to-sprout anastomosis, new tip cells are produced via branching, and a vessel network forms as endothelial cells follow the paths of tip cells. The latter process is known as the snail trail. We use a mean-field approximation to systematically derive a continuum model from a two-dimensional lattice-based cell… Show more
“…Our branching configuration is consistent with that used in Pillay et al. (2017), and we note that other configurations are possible.Fig. 3A tip cell (TC) at site ( i , j ) may branch during a time step by placing daughter TCs at , provided those sites are unoccupied by TCs (Models 1 and 2), or by ECs (Model 2)
…”
Section: Cellular Automatonmentioning
confidence: 99%
“…This differs from the CA framework in Pillay et al. (2017) where anastomosis events occurred every time a TC encountered another cell.Fig. 2 a Tip-to-tip anastomosis (Model 1 and 2): If a tip cell (TC) at site ( i , j ) moves to site , occupied by another TC, then both TCs are removed from the simulation and ECs are placed at sites ( i , j ) and (due to TC movement).…”
Section: Cellular Automatonmentioning
confidence: 99%
“…(2017). Since branching is stimulated by TAFs (Carmeliet and Jain 2011), we assume that the probability with which a TC at site ( i , j ) branches, , increases linearly with TAF concentration as follows: ( is a model parameter, assumed constant, that scales the TAF concentration and ensures that ).…”
Section: Cellular Automatonmentioning
confidence: 99%
“…(2017), we use the column-averaged CA simulation results for the TCs and ECs at some later discrete time step , and [ensemble averages defined in Eqs. (5 and (6)], respectively, as the initial conditions in the continuum model as follows:where .…”
Section: The Continuum Modelmentioning
confidence: 99%
“…(2017), we estimate the parameter controlling tip-to-sprout anastomosis, , now renamed , by fitting the PDE model to averaged CA simulation results. As mentioned in the PDE derivation, this enables us to account for prohibited self-loops, which are not explicitly accounted for in the difference equations.…”
Section: Mean-field Approximation and Behavior Of Pde Modelsmentioning
Angiogenesis is the process by which new blood vessels form from existing vessels. During angiogenesis, tip cells migrate via diffusion and chemotaxis, new tip cells are introduced through branching, loops form via tip-to-tip and tip-to-sprout anastomosis, and a vessel network forms as endothelial cells, known as stalk cells, follow the paths of tip cells (a process known as the snail-trail). Using a mean-field approximation, we systematically derive one-dimensional non-linear continuum models from a lattice-based cellular automaton model of angiogenesis in the corneal assay, explicitly accounting for cell volume. We compare our continuum models and a well-known phenomenological snail-trail model that is linear in the diffusive, chemotactic and branching terms, with averaged cellular automaton simulation results to distinguish macroscale volume exclusion effects and determine whether linear models can capture them. We conclude that, in general, both linear and non-linear models can be used at low cell densities when single or multi-species exclusion effects are negligible at the macroscale. When cell densities increase, our non-linear model should be used to capture non-linear tip cell behavior that occurs when single-species exclusion effects are pronounced, and alternative models should be derived for non-negligible multi-species exclusion effects.
“…Our branching configuration is consistent with that used in Pillay et al. (2017), and we note that other configurations are possible.Fig. 3A tip cell (TC) at site ( i , j ) may branch during a time step by placing daughter TCs at , provided those sites are unoccupied by TCs (Models 1 and 2), or by ECs (Model 2)
…”
Section: Cellular Automatonmentioning
confidence: 99%
“…This differs from the CA framework in Pillay et al. (2017) where anastomosis events occurred every time a TC encountered another cell.Fig. 2 a Tip-to-tip anastomosis (Model 1 and 2): If a tip cell (TC) at site ( i , j ) moves to site , occupied by another TC, then both TCs are removed from the simulation and ECs are placed at sites ( i , j ) and (due to TC movement).…”
Section: Cellular Automatonmentioning
confidence: 99%
“…(2017). Since branching is stimulated by TAFs (Carmeliet and Jain 2011), we assume that the probability with which a TC at site ( i , j ) branches, , increases linearly with TAF concentration as follows: ( is a model parameter, assumed constant, that scales the TAF concentration and ensures that ).…”
Section: Cellular Automatonmentioning
confidence: 99%
“…(2017), we use the column-averaged CA simulation results for the TCs and ECs at some later discrete time step , and [ensemble averages defined in Eqs. (5 and (6)], respectively, as the initial conditions in the continuum model as follows:where .…”
Section: The Continuum Modelmentioning
confidence: 99%
“…(2017), we estimate the parameter controlling tip-to-sprout anastomosis, , now renamed , by fitting the PDE model to averaged CA simulation results. As mentioned in the PDE derivation, this enables us to account for prohibited self-loops, which are not explicitly accounted for in the difference equations.…”
Section: Mean-field Approximation and Behavior Of Pde Modelsmentioning
Angiogenesis is the process by which new blood vessels form from existing vessels. During angiogenesis, tip cells migrate via diffusion and chemotaxis, new tip cells are introduced through branching, loops form via tip-to-tip and tip-to-sprout anastomosis, and a vessel network forms as endothelial cells, known as stalk cells, follow the paths of tip cells (a process known as the snail-trail). Using a mean-field approximation, we systematically derive one-dimensional non-linear continuum models from a lattice-based cellular automaton model of angiogenesis in the corneal assay, explicitly accounting for cell volume. We compare our continuum models and a well-known phenomenological snail-trail model that is linear in the diffusive, chemotactic and branching terms, with averaged cellular automaton simulation results to distinguish macroscale volume exclusion effects and determine whether linear models can capture them. We conclude that, in general, both linear and non-linear models can be used at low cell densities when single or multi-species exclusion effects are negligible at the macroscale. When cell densities increase, our non-linear model should be used to capture non-linear tip cell behavior that occurs when single-species exclusion effects are pronounced, and alternative models should be derived for non-negligible multi-species exclusion effects.
Angiogenesis, arteriogenesis, and pruning are revascularization processes essential to our natural vascular development and adaptation, as well as central players in the onset and development of pathologies such as tumoral growth and stroke recovery. Computational modeling allows for repeatable experimentation and exploration of these complex biological processes. In this review, we provide an introduction to the biological understanding of the vascular adaptation processes of sprouting angiogenesis, intussusceptive angiogenesis, anastomosis, pruning, and arteriogenesis, discussing some of the more significant contributions made to the computational modeling of these processes. Each computational model represents a theoretical framework for how biology functions, and with rises in computing power and study of the problem these frameworks become more accurate and complete. We highlight physiological, pathological, and technological applications that can be benefit from the advances performed by these models, and we also identify which elements of the biology are underexplored in the current state‐of‐the‐art computational models.
This article is categorized under:
Cancer > Computational Models
Cardiovascular Diseases > Computational Models
Angiogenesis is the process wherein endothelial cells (ECs) form sprouts that elongate from the pre‐existing vasculature to create new vascular networks. In addition to its essential role in normal development, angiogenesis plays a vital role in pathologies such as cancer, diabetes and atherosclerosis. Mathematical and computational modeling has contributed to unraveling its complexity. Many existing theoretical models of angiogenic sprouting are based on the “snail‐trail” hypothesis. This framework assumes that leading ECs positioned at sprout tips migrate toward low‐oxygen regions while other ECs in the sprout passively follow the leaders' trails and proliferate to maintain sprout integrity. However, experimental results indicate that, contrary to the snail‐trail assumption, ECs exchange positions within developing vessels, and the elongation of sprouts is primarily driven by directed migration of ECs. The functional role of cell rearrangements remains unclear. This review of the theoretical modeling of angiogenesis is the first to focus on the phenomenon of cell mixing during early sprouting. We start by describing the biological processes that occur during early angiogenesis, such as phenotype specification, cell rearrangements and cell interactions with the microenvironment. Next, we provide an overview of various theoretical approaches that have been employed to model angiogenesis, with particular emphasis on recent in silico models that account for the phenomenon of cell mixing. Finally, we discuss when cell mixing should be incorporated into theoretical models and what essential modeling components such models should include in order to investigate its functional role.This article is categorized under:
Cardiovascular Diseases > Computational Models
Cancer > Computational Models
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