This talk is about fronts and propagation phenomena for reaction-diffusion equations in non-homogeneous media. I will discuss some specific models arising in population dynamics or in medicine where the medium imposes a direction of propagation.
We analyze the Cahn-Hilliard equation with relaxation boundary condition modeling the evolution of interface in contact with solid boundary. The $L^\infty$ estimate of the solution is established which also enable us to prove the global existen of the soultion. We also study the sharp interface limit of the system. The dynamic of the contact point and the contact angle are derived and the results are compared with the numerical simulations.
Tumor growth involves numerous biochemical and biophysical processes whose interactions can only be understood via a detailed mathematical model. In this talk, I will present a new mathematical model of tumor growth that incorporates both continuum and cell-based descriptions, thereby retains the advantages of each descriptions while circumventing some of their disadvantages. In this model, the cell-based description is used in the region where the majority of growth and cell division occurs, while a continuum description is used for the quiescent and necrotic zones of the tumor and for the extracellular matrix. Reaction-diffusion equations describe the transport and consumption of important nutrients throughout the entire domain. Our novel hybrid model can address single cell-cell adhesion, cell growth, cell division and invasive patterning at the cellular level rather than at the continuum level in the proliferating zone, while simplifying computationally the overall system. Free boundaries arising from this model are different from the standard front propagation characteristic of the usual free boundary. We also show that the model can predict a number of cellular behaviors that have been observed experimentally.
This project is joint work with Hans Othmer, Yangjin Kim and Magda Stolarska
Stochastic generalizations of moving boundary problems appear quite naturally in the continuum description of e.g. solidification problems. Perhaps the simplest example is provided by a so-called one-sided solidification problem in which a condensed (solid) non-diffusing phase grows at the expense of a diluted diffusing phase (vapor or liquid). In this context, noise terms can be introduced to account for fluctuations in the interface kinetics leading to irreversible growth, and in the diffusive currents in the diluted phase. Thus, an effective closed evolution equation for the interface profile can be derived in a systematic way, carrying both deterministic and stochastic contributions, with parameters that can be related to those of the full moving boundary problem. This effective equation provides an interesting instance in which one can study the interplay between noise and non-local effects induced by diffusive interactions. Going beyond the approximations made in this process requires, e.g., formulation of a (stochastic) phase-field description that is equivalent to the original moving boundary problem. In turn, phase-field simulations allow to explore the rich morphological diagram that ensues. Applications will be discussed in the context both of non-living and biological systems subject to diffusion-limited growth, such as surfaces of thin films produced by Chemical Vapor Deposition or by Electrochemical deposition, or bacterial colonies. We will describe work done in collaboration with Mario Castro and Matteo Nicoli.
Blood coagulation is an extremely complex process which is the result of the action of platelets and of a large number of chemicals going through a chemical cascade. Its aim is the formation of a clot, sealing a wound The clot evolution leads to a free boundary problem. It goes in parallel with the process of clot dissolution (fibrinolysis), taking place with a slower time scale. Due to its complexity, the process may fail in various ways because of pathological conditions, leading to thrombosis or bleeding disorders of various types, that have also been the subject of mathematical models. The classical 3-pathway cascade model for blood coagulation, that was formulated in 1964, has been questioned after forty years. Though it is now ascertained to be wrong, its influence has been so strong that many new publications still refer to it. During the last few years a new model has been proposed in the medical literature (the so called cell-based model) and new mathematical papers have been written accordingly. Recently two opposite trends have been observed in mathematical models: on one side a tendency towards "completeness" with an incredible number of pde's describing the biochemistry in great detail (but sometimes ignoring platelets!); on the other side a tendency to focus just on the role of platelets. Those ways of approaching the problem have their own advantages and drawbacks. The "complete models" fail in any case to consider elements of great importance, that, very surprisingly, have been systematically ignored in the huge literature on the subject. The models considering just platelets can be used only for some very early stage of the process. A basic feature of any realistic coagulation model is the coupling between the biochemistry, the evolution of platelets population, and the flow of blood (in turn influenced by the growing clot). Thus blood rheology has a basic role. Blood rheology is known to be a very complicated subject and many different options have been offered. Nevertheless, the main point here is not which rheological model is preferable for blood, but the boundary conditions for blood flow. All models on blood coagulation use a no-slip condition. We prove that even a modest slip can have a dominant influence, depending on the geometry of the growing clot. We will also make a general discussion on the strategy to approach the problem (How many ingredients should be included? How to simplify the description of the chemistry? What targets can be considered realistic? etc.). New perspectives should also account for the most recent discoveries, suggesting that the cell-based model too may need some revision.
We consider a class of excitable system whose dynamics is described by Fitzhugh-Nagumo (FN) equations. We provide a description for rigidly rotating spirals based on the fact that one of the unknowns develops abrupt jumps in some regions of the space. The core of the spiral is delimited by these regions. The description of the spiral is made using a mixture of asymptotic and rigorous arguments. Several open problems whose rigorous solution would provide insight in the problem are formulated. Joint work with M. Aguareles and J. J. L. Velazquez.
Under suitable circumstances, bone tissue is able to self-repair small fractures and to integrate external implants. In doing so, use is made of a tightly regulated sequence of cell processes, which in many aspects resemble embrionary bone development.
In this lecture I will review some steps in such self-repairing mechanisms, and point out a number of modeling problems that arise from their consideration.
We shall discuss the recent progress (joint work with many others) on the PDE tumor models, the linear stability of the tumor, the nonlinear stability of the tumor, the bifurcation diagram near the bifurcation point, the bifurcation diam extensions and the intersection of bifurcation diagram for different bifurcation branch, the numerical solutions along the branches, and possible other types of steady state solutions.
There are a number of interesting and important biological processes that are best modelled as two-phase material mixtures. These include mucin exocytosis and transport, blood clot formation and biofilm formation. These all involve the interplay between flow, physical structure, mechanics and chemistry in a environment with complex dynamic geometry. The mathematical description of these processes requires equations describing multiphase flow, the evolution of composition and structure, and the relationship between stresses and composition/ structure (i.e., constitutive relations). Additionally, these equations of motion must properly account for interactions of the complex materials with dynamic physical boundaries, moving interfaces between materials with markedly different physical properties, and typically include strongly nonlinear constitutive relations or rate expressions.
In this talk, I will describe two features of mucus: the dynamics of mucus vesicle exocytosis and its transport of acid against an acid gradient.
The short story is as follows: Mucin is packaged into vesicles at very high concentration (volume fraction) and when the vesicle is released to the extracellular environment, the mucin expands in volume by two orders of magnitude in a matter of seconds. This rapid expansion is mediated by the rapid exchange of calcium and sodium that changes the crosslinking structure of the mucin polymers, thereby causing it to swell. I will describe a model of gel swelling and deswelling that accounts for these features, and is an interesting free boundary problem.
One of the important functions of the mucus lining of the stomach is to allow digestion of food to take place without the lining of the stomach being digested. An intriguing question is how acid can be released into the lumen of the stomach while maintaining a low concentration of hydrogen ions near the epithelial lining. A possible answer is that the flow of acid against its gradient is mediated by buffering by mucin. When mucin is secreted it rapidly binds hydrogen, but when it reaches the lumen where the pH is low, mucin is degraded by pH-activated pepsin, releasing its acid. The model associated with this process includes a free boundary problem to determine the thickness of the mucus layer and its acid-protective ability.
Mathematical modeling and computational analysis are essential for understanding the dynamics of the complex gene networks that control normal development and homeostasis, and can help to understand how circumvention of that control leads to abnormal outcomes such as cancer. Our objectives here are to discuss the free boundary problems arising from a multi-scale hybrid model. The free boundary evolves not only from the tumor progression but also from mechanical feedbacks from surrounding stromal tissue in the breast duct. The different mechanisms by which the local biochemical and mechanical microenvironment, which is comprised of various signaling molecules, cell types and the extracellular matrix (ECM), affects the progression of potentially-cancerous cells. We present results from a model that deals with the effects of the mechanical properties of the environment on tumor growth, and report results from a model of the signaling pathways and the tumor microenvironment (TME), and how their interactions affect the free boundary of a growing tumor in the duct and development of breast cancer. The results emphasize anew the complexities of the interactions within the TME and their effect on free boundary and growth patterns of a growing tumor, and show that tumor progression is not solely determined by the presence of a clone of mutated immortal cells, but rather that it can be 'community-controlled'.
A variety of free boundary problems pertinent to the growth of tissue on the macroscale will be described and some of their properties highlighted.
In the first part, the extension of the immersed boundary(IB) method is introduced. The fluid-structure interaction is extended to the fluid-solute-structure interaction by the coupling with advection-electrodiffusion. The concentration-dependent cell contraction, osmosis, and ion channel gating mechanisms are introduced as examples. The IB method is also considered in the extension of the swimming environment from Newtonian fluids to non-Newtonian fluids and gels. The swimming of spirochetes is introduced as an example.
In the second part, the collective cellular migration is considered in the wound healing and cancer metastasis problems based on a level-set / finite-volume method. It is based on a crawling cell motility and extended to the tissue domain with the interaction between cell-to-cell and cell-to-substrate.
Cells migrate on surfaces by protruding their front through growth of actin networks, retracting the rear by myosin-driven contraction and adhering to the substrate. Recent experimental and modeling efforts elucidated specific molecular and mechanical processes that allow motile cells to maintain constant distances from front to rear and from side to side while maintaining steady locomotion.
Remarkably, these processes are multiple and redundant, and one of the future modeling challenges is a synthesis of these processes (operating on multiple scales) within a computational framework. Necessarily, such framework have to treat the cell as an object with a free boundary leading to a very nontrivial mathematical problem. I will describe initial successes in modeling the simplest motile cell, fish keratocyte, and discuss future challenges in simulating more complex cells.
We present three models of biomembranes along with their numerical simulation. The first one is purely geometric since the equilibrium shapes are the minimizers of the Willmore (or bending) energy under area and volume constraints. The second model incorporates the effect of the inside (bulk) viscous incompressible fluid and leads to more physical dynamics. The third model describes the interaction of a director field with a membrane, giving rise to an induced spontaneous curvature.
We propose a parametric finite element method for the discretization of these models and examine crucial numerical issues such as dealing with curvature and length constraints within a variational framework. We show several simulations describing the dynamics of purely geometric flows, membrane-fluid interaction, and the dramatic effect of defects of the director field on membrane shape.
This work is joint with S. Bartels, A. Bonito, G. Dolzmann, and M.S. Pauletti.
We study the regularity of almost minimizers for the types of functionals analyzed by Alt, Caffarelli and Freidman. Although almost minimizers do not satisfy an equation using appropriate comparison functions we prove several regularity results. For example in the one phase situation we show that almost minimizers are Lipschitz. Our approach reminiscent of the one used in geometric measure theory to study the regularity of almost minimizers for area. This project is joint work with Guy David.
Chronic wounds represent a major public health problem affecting 6.5 million people in the United States. Wound healing involves complex interactions among different types of cells, different chemical signals, and the extracellular matrix. Ischemia is a major complicating factor in chronic wound healing and primarily caused by peripheral artery diseases. Due to the complexity of the biology, mathematical modeling and computational simulation become essential to understand the dynamics of the whole process. The chronic wound boundary is usually clearly defined and moves as the wound closes or deteriorates, thus mathematical models need to treat the wound boundary as a moving interface, and this approach leads to challenges in analysis and computation of such models.
In this talk, I present a three dimensional mathematical model of chronic wounds. The model consists of a coupled system of partial differential equations that describes the interaction of oxygen, PDGF, VEGF, macrophages, fibroblasts, blood vessels, and the extracellular matrix. The wound boundary is treated as a moving boundary. Simulations of a simplified model demonstrate how ischemic conditions may limit macrophage recruitment to the wound-site and impair wound closure. The results are in general agreement with experimental findings. Open problems include global existence of solution and property of the free boundary. This is joint work with Chandan Sen, Avner Friedman, and Bei Hu.
We derive a novel one-dimensional viscoelastic model of blood vessel capillary growth under nonlinear friction with surroundings, analyze its solution properties, and simulate various growth patterns in angiogenesis. The mathematical model treats the cell density as the growth pressure eliciting viscoelastic response of cells, thus extension or regression of the capillary. Nonlinear analysis provides some conditions to guarantee the global existence of biologically meaningful solutions, while linear analysis and numerical simulations predict the global biological solutions exist as long as the cell density change is sufficiently slow in time. Examples with blow-ups are captured by numerical approximations and the global solutions are recovered by slow growth processes. Numerical simulations demonstrate this model can reproduce angiogenesis experiments under several biological conditions including blood vessel extension without proliferation and blood vessel regression.
We analyze the global existence of classical solutions to the initial boundary- value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting Fokker-Planck equation presents a nonlinearity in the coefficients depending on the probability flux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a source term given by a delta function. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Finally, we will also study the spectrum for the linear problem corresponding to uncoupled networks and its relation to Poincare inequalities for studying their asymptotic behavior.
When a gash or gouge is made in a confluent layer of epithelial cells, the cells move to fill in the "wound". In some cases, such as in wounded embryonic chick wing buds, the movement of the cells is driven by cortical actin contraction (i.e., a purse string mechanism). In adult tissue, though, cells apparently crawl to close wounds. At the single cell level, this crawling is driven by the dynamics of the cell's actin cytoskeleton, which is regulated by a complex biochemical network, and cell signaling has been proposed to play a significant role in directing cells to move into the denuded area. However, wounds made in monolayers of Madin-Darby canine kidney (MDCK) cells still close even when a row of cells is deactivated at the periphery of the wound, and recent experiments show complex, highly-correlated cellular motions that extend tens of cell lengths away from the boundary. These experiments suggest a dominant role for mechanics in wound healing. Here we present a biophysical description of the collective migration of epithelial cells during wound healing based on the basic motility of single cells and cell-cell interactions. This model quantitatively captures the dynamics of wound closure and reproduces the complex cellular flows that are observed. These results provide the mechanism by which wounds can close without purse strings and suggest that wound closure is predominantly a mechanical process that is modified, but not produced, by cell-cell signaling.
During embryonic vasculogenesis, the earliest phase of blood vessel morphogenesis, isolated vascular cell progenitors called angioblasts assemble into a characteristic network pattern. So far, however, the mechanisms underlying the coalescence and patterning of angioblasts remain unclear. In this study we present a hybrid cell-based /continuous model relying on realistic biological assumptions. In particular, we assume that binding of paracrine signals to angioblasts-produced extracellular matrix regulates early vascular patterning, by creating spatially restricted guidance cues required for directed cell migration and coalescence. We use the model to explore the dynamics of network formation as well as the role of cell shape and cell density in this process. This work is part of an ongoing project involving Universities of Dresden (Germany), Málaga and Madrid (Spain). Additional details can be found in: Köhn-Luque et al (2011) PLoS ONE.
We present results on suspensions of self-propelled bacteria where hydrodynamic interactions between particles are taken into account. Bacteria are modeled as point force dipoles subject to two types of interactions: hydrodynamic interactions and excluded volume type interactions introduced through the use of the Lennard-Jones potential. The alignment of asymmetrical particles and the presence of self-propulsion gives rise to a drastic reduction in the effective viscosity of the suspension. An explicit asymptotic formula for the effective viscosity in terms of known physical parameters is derived using a kinetic approach. This model allows for numerical simulations of a large number of particles, which are in agreement with the analytical results and experiment through a full range of concentrations. Current work involves understanding the onset of collective motion in the above model by investigation of the correlation of bacterial velocities and orientations as a function of the interparticle distance. This work is part of a doctoral thesis under the direction of Professor Leonid Berlyand.