SPEAKERS:
[Go To Abstract] Mark Alber (U. of California Riverside)
[Go To Abstract] Vincenzo Capasso (U. of Milan)
[Go To Abstract] Jaume Casademunt (U. of Barcelona)
[Go To Abstract] Avner Friedman (Ohio State U.)
[Go To Abstract] Benjamin Gess (Max Planck Institute for Mathematics)
[Go To Abstract] Anna Gilbert (U. of Michigan)
[Go To Abstract] Alexander Grosberg (New York U.)
[Go To Abstract] Thomas Lepoutre (Université Claude Bernard, Lyon 1)
[Go To Abstract] Anna Marciniak-Czochra (U. of Heidelberg)
[Go To Abstract] Cyrill Muratov (New Jersey Institute of Technology)
[Go To Abstract] Fernando Peruani (Université Nice Sophia Antipolis)
[Go To Abstract] Benedetto Piccoli (Rutgers U.)
[Go To Abstract] Nastassia Pouradier Duteil (Jacques-Louis Lions Lab)
[Go To Abstract] John Schotland (U. of Michigan)
[Go To Abstract] Angela Stevens (U. of Muenster)
[Go To Abstract] Lei Zhang (Shanghai Jiao Tong U.)
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Multi-scale models of deformation of blood clots
Mark Alber, University of California Riverside
Abstract: Thromboembolism, one of the leading causes of morbidity and mortality worldwide, is characterized by formation of obstructive intravascular clots (thrombi) and their mechanical breakage (embolization). A novel two-dimensional multi-phase computational model will be described that simulates active interactions between the main components of the clot, including platelets and fibrin. It can be used for studying the impact of various physiologically relevant blood shear flow conditions on deformation and embolization of a partially obstructive clot with variable permeability. Simulations provide new insights into mechanisms underlying clot stability and embolization that cannot be studied experimentally at this time. In particular, model simulations, calibrated using experimental intravital imaging of an established arteriolar clot, show that flow-induced changes in size, shape and internal structure of the clot are largely determined by two shear-dependent mechanisms: reversible attachment of platelets to the exterior of the clot and removal of large clot pieces [1]. Model simulations also predict that blood clots with higher permeability are more prone to embolization with enhanced disintegration under increasing shear rate. In contrast, less permeable clots are more resistant to rupture due to shear rate dependent clot stiffening originating from enhanced platelet adhesion and aggregation. Role of platelets-fibrin network mechanical interactions in determining shape of a clot will be also discussed and quantified using analysis of experimental data [2,3]. These results can be used in future to predict risk of thromboembolism based on the data about composition, permeability and deformability of a clot under specific local haemodynamic conditions.
[1] S. Xu, Z. Xu, O.V. Kim, R.I. Litvinov, J.W. Weisel, M. Alber (2017) Model predictions of deformation, embolization and permeability of partially obstructive blood clots under variable shear flow. J. R. Soc. Interface, 14: 20170441.
[2] O.V. Kim, R.I. Litvinov, M. Alber, J.W. Weisel (2017) Quantitative structural mechanobiology of platelet driven blood clot contraction, Nature Communications, 8: 1274.
[3] S. Britton, O.V. Kim, F. Pancaldi, Z. Xu, R.I. Litvinov, J.W. Weisel, M. Alber (2019) Contribution of nascent cohesive fiber-fiber interactions to the non-linear elasticity of fibrin networks under tensile load, Acta Biomaterialia.
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On the Regional Control Of Spatially Structured Epidemic Systems. Application to a Malaria Model.
Vincenzo Capasso, University of Milan, Italy
Abstract: The talk will concern a unified review of a set of previous papers by the same author and collaborators concerning the mathematical modelling and control of malaria epidemics. The presentation moves from a conceptual mathematical model of malaria as a spatially structured system. Significant model reductions have been introduced, in order to make the resulting reaction-diffusion system mathematically affordable. Only the dynamics of the infected mosquitoes and of the infected humans has been included, so that a two-component reaction-diffusion system is finally taken.
The spread of the disease is controlled by three actions (controls): killing mosquitoes, treating the infected humans and reducing the contact rate mosquitoes-humans. The public health concern consists of providing methods for the eradication of the disease in the relevant population, as fast as possible. On the other hand, very often the entire domain of interest for the epidemic, is unknown, or difficult to reach for the implementation of suitable environmental sanitation programs. This has led the authors to suggest that implementation of such programs might be done only in a given subregion, conveniently chosen so to lead to an effective eradication of the epidemic in the whole habitat (“Think globally, act locally”).
To start with, the problem of the eradicability of the disease is considered. We have proven that it is possible to decrease exponentially both the human and the vector infective population everywhere in the relevant habitat by acting only in a suitable subdomain. Later the regional control problem of reducing the total cost of the damages produced by the disease, of the controls and of the intervention in a certain subdomain is treated for the finite time horizon case.
In order to take the logistic structure of the habitat into account the level set method is used as a key ingredient for describing the subregion of intervention.
[1] S. Aniţa, V. Capasso (2002) A stabilizability problem for a reaction-diffusion system modelling a class of spatially structured epidemic model, Nonlinear Anal. Real World Appl., 3: 453–464.
[2] S. Aniţa, V. Capasso (2012) Stabilization of a reaction-diffusion system modelling malaria transmission. Discrete and Continuous Dynamical Systems, Series B, 17: 1673–1684.
[3] S. Aniţa, V. Capasso, G. Demetriu (2019) Regional Control for a spatially structured malaria model. Mathematical Methods in the Applied Sciences, 42: 2909–2933.
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Hydrodynamics of epithelia: waves, wetting and fingering.
Jaume Casademunt, University of Barcelona, Spain
Abstract: Collective migration of cohesive groups of cells is a hallmark of tissue remodeling events underlying embryonic morphogenesis, wound repair and cancer invasion. In this collective migration, supra-cellular properties such as collective polarization or force generation emerge and eventually control large-scale tissue organization. This suggests that a coarse-grained approach based on a hydrodynamic description of tissues as continuous active materials may shed some light into our understanding of tissue dynamics. Specifically, an appealing open question is to what extent the complex biological regulation at play can be encoded in a series of material parameters within a purely mechanical description. Here we present an overview of hydrodynamic modeling of cell tissues as active polar fluids, and discuss some examples where this approach has been instrumental to elucidate physical mechanisms behind collective cell behavior in epithelia: the occurrence of elastic-like waves, the wetting-dewetting transition in spreading monolayers, and the understanding of morphological instabilities at tissue edges.
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Using mathematical models in the design of cancer clinical trials
Avner Friedman, Ohio State University
Abstract: Before designing a cancer clinical trial with combination of two drugs, we need to consider the following questions:
- How to avoid antagonism between the two drugs?
- How to schedule the order of injections of the drugs?
- How to reduce negative side-effects without decreasing efficacy? ‘ How to reduce drug resistance?
We shall use mathematical models to address these questions with some specific drugs. The models are represented by systems of PDEs in the tumor, while the boundary of the tumor is a free boundary. We shall consider combinations of immune therapy (e.g. PD-I inhibitor) with a chemotherapy or oncolytic therapy. This work is joint work with Xilan Lai from Renmin University.
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Nonlinear stochastic partial differential equations in non-equilibrium statistical mechanics
Benjamin Gess, Max Planck Institute for Mathematics in the Sciences, Leipzig Universität Bielefeld, Germany
Abstract: In this talk we will revisit the occurrence of nonlinear stochastic partial differential equations in the description of fluctuations in non-equilibrium statistical mechanics. More precisely, the hydrodynamic limit of the zero range process and its fluctuations will be recalled, drawing the link to stochastic quasilinear diffusion equations with conservative, nonlinear noise. We will then present a well-posedness theory for these equations.
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Metric repair
Anna Gilbert, University of Michigan
Abstract: For many machine learning tasks, the input data lie on a low dimensional manifold embedded in a high-dimensional space and, because of this high-dimensional structure, most algorithms inefficient. The typical solution is to reduce the dimension of the input data using a standard dimension reduction algorithms such as ISOMAP, LAPLACIAN EIGENMAPS or LLES. This approach, however, does not always work in practice as these algorithms require that we have somewhat ideal data. Unfortunately, most data sets either have missing entries or unacceptably noisy values. That is, real data are far from ideal and we cannot use these algorithms directly. In this talk, we focus on the case when we have missing data. Some techniques, such as matrix completion, can be used to fill in missing data but these methods do not capture the non-linear structure of the manifold. Here, we present a new algorithm MR-MISSING that extends these previous algorithms and can be used to compute low dimensional representation on data sets with missing entries. We demonstrate the effectiveness of our algorithm by running three different experiments. We visually verify the effectiveness of our algorithm on synthetic manifolds, we numerically compare our projections against those computed by first filling in data using nlPCA and mDRUR on the MNIST data set, and we also show that we can do classification on MNIST with missing data. We also provide a theoretical guarantee for MR-MISSING under some simplifying assumptions.
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To Knot or Not to Knot
Alexander Grosberg, New York University
Abstract: While topological ideas are widely popular in physics, topology of classical linear threads of polymers presents steep mathematical and conceptual challenges, with applications in both biopolymers and materials. I will start with the simplest phenomenon in the field: if you try to walk a dog on (unreasonably) long leash, it is likely that leash will soon be heavily wound around your legs. Viewing topological constraints as a form of quenched disorder, I will formulate a few known classical results about “knot entropy” and minimal thermodynamic work needed to untie all knots. Continuing with increasingly sophisticated models and phenomena, I will review several more recent theoretical and experimental achievements, and conclude with the discussion of a controversial concept of “topological glass”.
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Mathematical models of polarizing cells
Thomas Lepoutre, Université Claude Bernard, Lyon 1, France
Abstract: We consider drift diffusion models arising from mathematical modelling of cell polarization. They take the form of drift diffusion equations with nonlinear boundary conditions on the half line. In such models, we can study how different regimes might appear that will represent sharp polarization (represented in some models by blow up), absence of polarization (diffusion dominated regime). In the case of two dialoguing cells, a bistable structure can be observed.
This is joint work with Nicolas Meunier, Nicolas Muller and Vincent Calvez.
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Mathematical models of mechano-chemical pattern formation
Anna Marciniak-Czochra, University of Heidelgerg
Abstract: Cells and tissues are objects of the physical world, and therefore they obey the laws of physics and chemistry, notwithstanding the molecular complexity of biological systems. A natural question arises about the mathematical principles at play in generating such complex entities from simple laws. In this talk, I show how different pattern formation concepts may stand challenges arising from the current experimental research. Specifically, Turing-style morphogen-based models are compared to mechano-chemical models exhibiting de novo pattern formation. Patterning potential of mechano-chemical interactions is investigated using two classes of mathematical models coupling dynamics of diffusing molecular signals with a model of tissue deformation. The first class of models is based on energy minimization that leads to 4-th order partial differential equations of evolution of infinitely thin deforming tissue (pseudo-3D model), coupled with a surface reaction-diffusion equation. The second approach (full-3D model) consists of a continuous model of large tissue deformation coupled with a discrete description of spatial distribution of cells to account for active deformation of single cells. Model simulations showing patterns of tissue invagination or evagination are compared to experimental results.
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The mathematics of charged liquid drops
Cyrill Muratov, The New Jersey Institute of Technology
Abstract: In this talk, I will present an overview of recent analytical developments in the studies of equilibrium configurations of liquid drops in the presence of repulsive Coulombic forces. Due to the fundamental nature of Coulombic interaction, these problems arise in systems of very different physical nature and on vastly different scales: from femtometer scale of a single atomic nucleus to micrometer scale of droplets in electrosprays to kilometer scale of neutron stars. Mathematically, these problems all share a common feature that the equilibrium shape of a charged drop is determined by an interplay of the cohesive action of surface tension and the repulsive effect of long-range forces that favor drop fragmentation. More generally, these problems present a prime example of problems of energy driven pattern formation via a competition of long-range attraction and long range repulsion. In the talk, I will focus on two classical models – Gamow’s liquid drop model of an atomic nucleus and Rayleigh’s model of perfectly conducting liquid drops. Surprisingly, despite a very similar physical background these two models exhibit drastically different mathematical properties. I will discuss the basic questions of existence vs. non-existence, as well as some qualitative properties of global energy minimizers in these models, and present the current state of the art for this class of geometric problems of calculus of variations.
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Intermittent collective motion and information propagation fronts in biological systems
Fernando Peruani, Université Nice Sophia Antipolis, France
Abstract: Intermittent motion is observed in biological systems at all scales, from bacterial systems to sheep herds. First, I will discuss how Escherichia coli and Merino sheep explores surfaces by alternating stop and moving phases. Specifically, I will show that the mathematical description of the observed intermittent motion requires the use of three behavioral states in both biological systems. Then I will focus on large groups of individuals (e.g. sheep) to show that the emergence of intermittent collective motion involves an activation wave that spreads over a collection of initially static agents. We will see that the velocity of active agents, where both, the magnitude and direction of the agent’s velocity play a crucial role. Furthermore, we will learn that when the individual that initiates the collective motion phase is located at the group boundary and moves away from the group, the information front speed is lower than when the agents move towards the center of the group. Finally, we will see that as agent’s moving speed is increased above a threshold the physics of the problem changes, and a shock wave emerges.
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Nonlinear and measure-theoretic tools for bionetworks
Benedetto Piccoli, Rutgers University
Abstract: In this talk we will present two new techniques developed to study complex biological networks. First we will analyze new methods for the simulation of large biochemical network with focus on QSP (Quantitative Systems Pharmacology) and virtual patient populations. The main idea is to combine knowledge from the fields of: Markov Chains, Compartmental Systems, Control Theory and others to give general conditions for existence and uniqueness of equilibria. Main applications are to cholesterol metabolism and tuberculosis.
Secondly, we will describe a general framework to study reaction-diffusion equations on time-evolving manifold. This approach is useful to study problems in Developmental Biology when various ligands diffuse on growing embryos (or egg chambers) to activate morphogenic pathways.
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Sparse control of Hegselmann-Krause models: Black hole and declustering
Nastassia Pouradier Duteil, Jacques-Louis Lions Lab
Abstract: We elaborate control strategies to prevent clustering effects in opinion formation models. This is the exact opposite of numerous situations encountered in the literature where, on the contrary, one seeks controls promoting consensus. In order to promote declustering, instead of using the classical variance that does not capture well the phenomenon of dispersion, we introduce an entropy-type functional that is adapted to measuring pairwise distances between agents. We then focus on a Hegselmann-Krause-type system and design declustering sparse controls both in finite-dimensional and kinetic models. We provide general conditions that characterize whether clustering can be avoided as function of the initial data. Such results include the description of black holes (where complete collapse to consensus is not avoidable), safety zones (where the control can keep the system far from clustering), basins of attraction (attractive zones around the clustering set) and collapse prevention (when convergence to the clustering set can be avoided).
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Superresolution and Inverse Problems with Internal Sources
John C. Schotland, University of Michigan, Ann Arbor
Abstract: I will discuss a method to reconstruct the optical properties of a scattering medium with subwavelength resolution. The method is based on the solution to an inverse scattering problem with internal sources. Applications to photoactivated localization microscopy are described. This is joint work with Anna Gilbert, Jeremy Hoskins and Howard Levinson.
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Signaling gradients in surface/boundary dynamics as basis for regeneration in flatworms
Angela Stevens University of Münster, Germany
Abstract: We introduce and analyze a mathematical model for the regeneration of planarian flatworms. This system of differential equations incorporates dynamics of head and tail cells which express positional control genes that in turn translate into localized signals that guide stem cell differentiation. Orientation and positional information is encoded in the dynamics of a long range wnt related signaling gradient. We motivate our model in relation to experimental data and demonstrate how it correctly reproduces cut and graft experiments. In particular, our system improves on previous models by preserving polarity in regeneration, over orders of magnitude in body size during cutting experiments and growth phases. Our model relies on tristability in cell density dynamics, between head, trunk, and tail. In addition, key to polarity preservation in regeneration, our system includes sensitivity of cell differentiation to gradients of wnt-related signals measured relative to the tissue surface. This process is particularly relevant in a small tissue layer close to wounds during their healing, and modeled here in a robust fashion through dynamic boundary conditions. This is joint work with A. Scheel and C. Tenbrock.
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Modeling and simulation of an active swimmer in nematic liquid crystal
Lei Zhang, Shanghai Jiao Tong University, China
Abstract: Living liquid crystal (LLC) is a class of active matter that combines active particles such as swimming bacteria with a lyotropic liquid crystal. The interaction of active motion with orientation order of liquid crystal (LC) leads to striking optical, hydrodynamical, and electrical properties of LLC, as well as collective behavior and emergence of intriguing patterns. In this work, we aim to understand how the orientation order of liquid crystal affects the motion of a single swimmer. We study a nonlinearly coupled PDE model which combines the well-known Edwards-Beris model for liquid crystal hydrodynamics with a squirmer model describing active swimmer. Analytical and numerical study reveals the competition between the dissipation dynamics of liquid crystal flow and the active propulsion. Numerical results show how the shape and anchoring strength may affect the stable squirming direction. This is a joint work with Hai Chi, Leonid Berlyand, and Igor Aronson.
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