Avner Friedman

Higher order nonlinear degenerate parabolic equations

This paper is concerned with nonlinear degenerate parabolic equations of the form ut + (−1)m − 1 D(f(u) D2m + 1u) = 0 with f(u) ~ ¦u¦n (n ⩾ 1) near u = 0 and D = ∂∂x. Under appropriate boundary conditions it is shown that there exists a weak solution u. Some of the main results of the paper are that u ⩾ 0 if u0 ⩾ 0, and that the support of u(·, t) (when u0 ⩾ 0) increases with t (for the last property we require that n ⩾ 2 and m = 1).

MicroRNA regulation of a cancer network: Consequences of the feedback loops involving miR-17-92, E2F, and Myc

The transcription factors E2F and Myc participate in the control of cell proliferation and apoptosis, and can act as oncogenes or tumor suppressors depending on their levels of expression. Positive feedback loops in the regulation of these factors are predicted—and recently shown experimentally—to lead to bistability, which is a phenomenon characterized by the existence of low and high protein levels (“off” and “on” levels, respectively), with sharp transitions between levels being inducible by, for example, changes in growth factor concentrations. E2F and Myc are inhibited at the posttranscriptional step by members of a cluster of microRNAs (miRs) called miR-17-92. In return, E2F and Myc induce the transcription of miR-17-92, thus forming a negative feedback loop in the interaction network. The consequences of the coupling between the E2F/Myc positive feedback loops and the E2F/Myc/miR-17-92 negative feedback loop are analyzed using a mathematical model. The model predicts that miR-17-92 plays a critical role in regulating the position of the off–on switch in E2F/Myc protein levels, and in determining the on levels of these proteins. The model also predicts large-amplitude protein oscillations that coexist with the off steady state levels. Using the concept and model prediction of a “cancer zone,” the oncogenic and tumor suppressor properties of miR-17-92 is demonstrated to parallel the same properties of E2F and Myc.

The Stefan problem in several space variables

Introduction. The Stefan problem is a free boundary problem for parabolic equations. The solution is required to satisfy the usual initial-boundary conditions, but a part of the boundary is free. Naturally, an additional condition is imposed at the free boundary. A two-phase problem is such that on both sides of the free boundary there are given parabolic equations and initial-boundary conditions, and neither of the solutions is identically constant. In case the space-dimension is one, there are numerous results concerning existence, uniqueness, stability, and asymptotic behavior of the solution; we refer to [1] and the literature quoted there (see also [8]). In the case of several space variables the problem is much harder. The difficulty is not merely due to mathematical shortcomings but also to complications in the physical situation. Thus, even if the data are very smooth the solution need not be smooth, in general. For example, when a body of ice having the shape

Hypoxia inducible microRNA 210 attenuates keratinocyte proliferation and impairs closure in a murine model of ischemic wounds

Ischemia complicates wound closure. Here, we are unique in presenting a murine ischemic wound model that is based on bipedicle flap approach. Using this model of ischemic wounds we have sought to elucidate how microRNAs may be implicated in limiting wound re-epithelialization under hypoxia, a major component of ischemia. Ischemia, evaluated by laser Doppler as well as hyperspectral imaging, limited blood flow and lowered tissue oxygen saturation. EPR oximetry demonstrated that the ischemic wound tissue had pO2 <10 mm Hg. Ischemic wounds suffered from compromised macrophage recruitment and delayed wound epithelialization. Specifically, epithelial proliferation, as determined by Ki67 staining, was compromised. In vivo imaging showed massive hypoxia inducible factor-1α (HIF-1α) stabilization in ischemic wounds, where HIF-1α induced miR-210 expression that, in turn, silenced its target E2F3, which was markedly down-regulated in the wound-edge tissue of ischemic wounds. E2F3 was recognized as a key facilitator of cell proliferation. In keratinocytes, knock-down of E2F3 limited cell proliferation. Forced stabilization of HIF-1α using Ad-VP16- HIF-1α under normoxic conditions up-regulated miR-210 expression, down-regulated E2F3, and limited cell proliferation. Studies using cellular delivery of miR-210 antagomir and mimic demonstrated a key role of miR-210 in limiting keratinocyte proliferation. In summary, these results are unique in presenting evidence demonstrating that the hypoxia component of ischemia may limit wound re-epithelialization by stabilizing HIF-1α, which induces miR-210 expression, resulting in the down-regulation of the cell-cycle regulatory protein E2F3.

Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence

We consider an electrostatic problem for a conductor consisting of finitely many small inhomogeneities of extreme conductivity, embedded in a spatially varying reference medium. Firstly we establish an asymptotic formula for the voltage potential in terms of the reference voltage potential, the location of the inhomogeneities and their geometry. Secondly we use this representation formula to prove a Lipschitz-continuous dependence estimate for the corresponding inverse problem. This estimate bounds the difference in the location and in certain geometric properties of two sets of inhomogeneities by the difference in the boundary voltage potentials corresponding to a fixed current distribution.

Convexity of solutions of semilinear elliptic equations

On considere le probleme elliptique suivant Δu=f(u) dans Ω, u=M sur ∂Ω ou Ω est un domaine convexe de R 2 et M est soit une constante reelle soit +∞. On etablit que, pour une bonne fonction monotone g(t), pout toute solution u on a g(u) est strictement convexe dans Ω

Glioma Virotherapy: Effects of Innate Immune Suppression and Increased Viral Replication Capacity

Oncolytic viruses are genetically altered replication-competent viruses that infect, and reproduce in, cancer cells but do not harm normal cells. On lysis of the infected cells, the newly formed viruses burst out and infect other tumor cells. Experiments with injecting mutant herpes simplex virus 1 (hrR3) into glioma implanted in brains of rats show lack of efficacy in eradicating the cancer. This failure is attributed to interference by the immune system. Initial pretreatment with immunosuppressive agent cyclophosphamide reduces the percentage of immune cells. We introduce a mathematical model and use it to determine how different protocols of cyclophosphamide treatment and how increased burst size of the mutated virus will affect the growth of the cancer. One of our conclusions is that the diameter of the cancer will decrease from 4 mm to eventually 1 mm if the burst size of the virus is triple that which is currently available. The effect of repeated cyclophosphamide treatment is to maintain a low density of uninfected cells in the tumor, thus reducing the probability of migration of tumor cells to other locations in the brain.

Wound angiogenesis as a function of tissue oxygen tension: a mathematical model.

Wound healing represents a well orchestrated reparative response that is induced by injuries. Angiogenesis plays a central role in wound healing. In this work, we sought to develop the first mathematical model directed at addressing the role of tissue oxygen tension on cutaneous wound healing. Key components of the developed model include capillary tips, capillary sprouts, fibroblasts, inflammatory cells, chemoattractants, oxygen, and the extracellular matrix. The model consists of a system of nonlinear partial differential equations describing the interactions in space and time of these variables. The simulated results agree with the reported literature on the biology of wound healing. The proposed model represents a useful tool to analyze strategies for improved healing and generate a hypothesis for experimental testing.

Mathematics in industrial problems

Mathematics in industrial problems , Mathematics in industrial problems , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی

Modeling the immune rheostat of macrophages in the lung in response to infection

In the lung, alternatively activated macrophages (AAM) form the first line of defense against microbial infection. Due to the highly regulated nature of AAM, the lung can be considered as an immunosuppressive organ for respiratory pathogens. However, as infection progresses in the lung, another population of macrophages, known as classically activated macrophages (CAM) enters; these cells are typically activated by IFN-γ. CAM are far more effective than AAM in clearing the microbial load, producing proinflammatory cytokines and antimicrobial defense mechanisms necessary to mount an adequate immune response. Here, we are concerned with determining the first time when the population of CAM becomes more dominant than the population of AAM. This proposed “switching time” is explored in the context of Mycobacterium tuberculosis (MTb) infection. We have developed a mathematical model that describes the interactions among cells, bacteria, and cytokines involved in the activation of both AAM and CAM. The model, based on a system of differential equations, represents a useful tool to analyze strategies for reducing the switching time, and to generate hypotheses for experimental testing.