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Wound healing angiogenesis: An overview on mathematical models
Published in J. Belinha, R.M. Natal Jorge, J.C. Reis Campos, Mário A.P. Vaz, João Manuel, R.S. Tavares, Biodental Engineering V, 2019
A.C. Guerra, J. Belinha, R.M. Natal Jorge
Some authors combined cellular Potts model with PDEs. Merks et al. (2004) implemented a preferential motion for the endothelial cells along gradients of the chemo-attractant. The results obtained with the simulations allowed to study intercellular adhesion and cell morphology and showed that endothelial cell adhesion is essential for stable vessels’ formation. Scianna et al. (2015) established a model that included reaction–diffusion equations to describe VEGF diffusion, decay, uptake and production and oxygen diffusion and consumption. In the model, the endothelial cells, upon VEGF stimulation turn from a quiescent phenotype to a stalk or a tip one. The numerical simulation results showed that the model could reproduce the formation of a functional network by sprouting angiogenesis. van Oers et al. (2014) developed a cellular Potts model to study endothelial cells’ motility, and combined them with the finite element method to study cells’ deformation. This approach allowed to study the interaction between endothelial cells and ECM components and also the cell behaviour that permits the formation of vessels network and sprouting.
Platelet dynamics in blood flow
Published in Annie Viallat, Manouk Abkarian, Dynamics of Blood Cell Suspensions in Microflows, 2019
Jawaad Sheriff, Danny Bluestein
Multiple-platelet continuum models of aggregation consider platelets in terms of their number densities. One approach uses variable flow energies to describe the activation states of platelets, which are allowed to interact with each other, fibrinogen, and the vessel wall through a discrete stochastic cellular Potts model at the microscale, with macroscale blood flow dynamics described by Navier-Stokes equations [235]. Each position of discretized space is occupied by fluid, platelet mass, or other cell types. In another approach, interplatelet bond stress development is tracked in macroscale platelet thrombosis in atherosclerotic arteries using an Oldroyd-B-like evolution equation [57, 58, 60]. Interactions among platelets and coagulation chemistry also play a role in growing thrombi, whose porosity is dependent on the number density of bound platelets [125, 126]. This comprehensive approach utilizes the finite difference approach to discretize equations relating to coagulation biochemistry, chemical activation and deposition of platelets, and two way interaction between plasma dynamics and the platelet plug, whose resistance to flow is determined by a term added to the Navier-Stokes momentum equation [33]. Larger-scale examination of individual platelet behavior in a growing thrombus (i.e. platelet-platelet or platelet-wall interactions) may also be performed using a force coupling method (FCM), where thousands of platelets are treated as rigid spherical Lagrangian particles and two-way coupled to the background flow. The FCM is affected by local hydrodynamics and can be incorporated in the Navier-Stokes equation via a particle body-force term [239]. Attractive or repulsive forces between platelets can also be described by a Morse potential [240].
Coarse-grained modeling of cell division in 3D: influence of density, medium viscosity, and inter-membrane friction on cell growth and nearest neighbor distribution
Published in Soft Materials, 2020
Pranav Madhikar, Jan Åström, Jan Westerholm, Björn Baumeier, Mikko Karttunen
The importance of mechanical properties has been investigated computationally with a variety of models.[8,10,23–33] Current models typically approximate cell membranes by either straight edges, such as in vertex models[23,34–38], flat planes as in Delaunay object dynamics[25,39–41], or as lattice boundaries as in the cellular Potts model.[30,31,42–44] In general, these models do not accurately approximate inter-membrane interactions that are crucial for mechanotransduction and mediated by different proteins. The importance and different approach to model cell–cell contacts is discussed in detail in the review by Van Liedekerke et al.[28]