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   Luis Bonilla
   Gregorio Millan Institute for Fluid Dynamics, Nanoscience &
   Industrial Mathematics
   Universidad Carlos III de Madrid

Modeling angiogenesis

Angiogenesis is a multiscale process by which blood vessels grow from existing ones and carry oxygen to distant organs. Angiogenesis is essential for normal organ growth and wounded tissue repair but it may also be induced by tumors to amplify their own growth. Mathematical and computational models contribute to understanding angiogenesis and developing anti-angiogenic drugs, but most work only involves numerical simulations and analysis has lagged. A recent stochastic model of the early stages of tumor induced angiogenesis including branching, elongation, and anastomosis (fusion) of blood vessels captures some of its intrinsic multiscale structures. Vessel tips proliferate due to branching, elongate following Langevin dynamics and, when they meet other vessels, join them by anastomosis and stop moving. Stalk endothelial cells follow the tip cells, so that the trajectories thereof constitute the advancing blood vessel. We have been able to extract a deterministic integropartial differential description of the vessel tip density that includes vessel proliferation and anastomosis for the first time. Anastomosis keeps the number of vessel tips relatively small, so that we cannot use the law of large numbers to derive equations for their density. Nevertheless, we show that ensemble averages over many replicas of the stochastic process correspond to the solution of the deterministic equations with appropriate boundary conditions. Most of the time, the density of tips sprouting from a primary blood vessel advances chemotactically towards the tumor driven by a soliton similar to the famous Korteweg-de Vries soliton. There are two collective coordinates whose slow dynamics changes the shape and velocity of the soliton. Analyzing the equations for the collective coordinates paves the way for controlling angiogenesis through the soliton, the engine that drives this process. We are working to extend our results to more realistic models using our results for the simpler model as a template for future research.

Defects and ripples in graphene

We have studied the stability and evolution of various elastic defects in a flat graphene sheet and the electronic properties of the most stable configurations. Our stability studies use elementary periodized discrete elasticity models of graphene sheets. Two types of dislocations are found to be stable: 'glide' dislocations consisting of heptagon–pentagon pairs, and 'shuffle' dislocations, an octagon with a dangling bond. Unlike the most studied case of carbon nanotubes, Stone Wales defects seem to be dynamically unstable in the planar graphene sheet. This fact has recently been corroborated by experimental observations. Similar defects in which one of the pentagon–heptagon pairs is displaced vertically with respect to the other one are found to be dynamically stable. Shuffle dislocations will give rise to local magnetic moments that can provide an alternative route to magnetism in graphene. In a suspended graphene sheet, we have included bending effects and the possibility that the sheet be locally curved upwards or downwards. We find a critical temperature above which a flat sheet is stable and below which stable ripples appear. Rippling in the presence of defects is also being studied.

Nonlinear transport in nanostructures

Nonlinear charge and spin transport in nanostructures is essential to propose and model electronic and spintronic devices and yet there are fundamental issues that are poorly understood. We are working to derive quantum and semiclassical kinetic equations describing charge and spin transport in simple nanostructures (semiconductor superlattices and lateral superlattices) in order to obtain and solve reduced equations for electron densities and electric field and compare and validate the results obtained using these different descriptions.

Multiscale methods

Studying the impact of defects in the macroscopic properties of a solid material is a multiscale problem that involves processes taking place at scales ranging from the atomic scale to the macroscale. Finding a way to transfer the relevant information from the lower scales to the upper scales is a largely unsolved problem. A basic problem in this context consists in developing hybrid schemes that couple an atomistic description in a localized region with a continuum description around it. Discretizing the surrounding continuum by means of finite element schemes may give rise to singularities and reflections by an abrupt change in the mesh. We are exploring different ways to couple the atomistic and the continuum region: nonreflecting coupling conditions based on discrete Green functions, perfectly matched layers, meshless methods...

Homogeneous and heterogeneous vapor condensation in cold walls

We are studying by a combination of singular perturbation and numerical methods the problem of vapor condensation and deposition in a cold wall within simple laminar boundary layer flows. This problem has interest for vapor deposition in combustion chambers, fouling and corrosion in biofuel plants, chemical vapor deposition, outside vapor deposition and aerosol capture by cold plates or rejection by hot ones.

Quantum well semiconductor detectors

Quantum-well based semiconductor detectors may be used to improve astrophysical instrumentation such as used by the ESA Herschel space laboratory that may work in the 3 to 5 terahertz frequency range (beyond the range of current detectors). The appearance of time and space charge inhomogeneities limit the applicability of the quantum well concept and need modelling and understanding. Models may consist of nonlocal and nonlinear integro-differential equations for the Wigner function or for the nonequilibrium Green function. We are developing appropriate numerical methods and analysis for infrared sensors such as the UC Santa Barbara TACIT sensor. This project is a collaboration with University of Iceland funded by a NILS Mobility Program.

Spontaneous chaos at room temperature and true random number generators

Recently, spontaneous chaos at room temperature has been discovered in semiconductor superlattices, which opens the possibility to build fast true random number generators. In the lab, these devices have been shown to produce bit rates up to 80Gbits per second, way above the usual generators that produce 1Gb/s. We have proposed and we are analyzing models for charge transport in weakly coupled semiconductor superlattices that explain different possible types of spontaneous chaos at room temperature depending on the device configuration, randomness in doping density and configuration, and thermal and shot noise. This work is being carried out in collaboration with the groups led by Bjorn Birnir (UCSB and University of Iceland), Holger Grahn (Paul Drude Institute, Berlin) and Yaohui Zhang (SINANO, Suzhou, China). The collaboration with University of Iceland was funded by a NILS Mobility Program.

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last modified: Apr 2017