Research
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 antiangiogenic 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 Kortewegde 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
Quantumwell 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 integrodifferential 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.
last modified:
Apr 2017
