Newton\u27s Cubic Roots

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A Mathematical Tumor Model with Immune Resistance and Drug Therapy: An Optimal Control Approach

We present a competition model of cancer tumor growth that includes both the immune system response and drug therapy. This is a four-population model that includes tumor cells, host cells, immune cells, and drug interaction. We analyze the stability of the drug-free equilibria with respect to the immune response in order to look for target basins of attraction. One of our goals was to simulate qualitatively the asynchronous tumor-drug interaction known as “Jeffs phenomenon.” The model we develop is successful in generating this asynchronous response behavior. Our other goal was to identify treatment protocols that could improve standard pulsed chemotherapy regimens. Using optimal control theory with constraints and numerical simulations, we obtain new therapy protocols that we then compare with traditional pulsed periodic treatment. The optimal control generated therapies produce larger oscillations in the tumor population over time. However, by the end of the treatment period, total tumor size is smaller than that achieved through traditional pulsed therapy, and the normal cell population suffers nearly no oscillations

Teaching Time Savers: Is Homework Grading on Your Nerves?

You have probably heard it said that we learn mathematics best when we do mathematics, or that mathematics is not a spectator sport. For most of our students, this means that their mathematics courses will involve a fair amount of homework. This homework is often used to evaluate individual student progress, but it can also be used, for example, as a catalyst for discussion, to emphasize a point made in class, and to identify common misunderstandings throughout the class as a whole. There is, however, the matter of grading homework

A Model of Dendritic Cell Therapy for Melanoma

Dendritic cells are a promising immunotherapy tool for boosting an individual’s antigen-specific immune response to cancer. We develop a mathematical model using differential and delay-differential equations to describe the interactions between dendritic cells, effector-immune cells, and tumor cells. We account for the trafficking of immune cells between lymph, blood, and tumor compartments. Our model reflects experimental results both for dendritic cell trafficking and for immune suppression of tumor growth in mice. In addition, in silico experiments suggest more effective immunotherapy treatment protocols can be achieved by modifying dose location and schedule. A sensitivity analysis of the model reveals which patient-specific parameters have the greatest impact on treatment efficacy

B Cell Chronic Lymphocytic Leukemia - A Model with Immune Response

B cell chronic lymphocytic leukemia (B-CLL) is known to have substantial clinical heterogeneity. There is no cure, but treatments allow for disease management. However, the wide range of clinical courses experienced by B-CLL patients makes prognosis and hence treatment a significant challenge. In an attempt to study disease progression across different patients via a unified yet flexible approach, we present a mathematical model of B-CLL with immune response, that can capture both rapid and slow disease progression. This model includes four different cell populations in the peripheral blood of humans: B-CLL cells, NK cells, cytotoxic T cells and helper T cells. We analyze existing data in the medical literature, determine ranges of values for parameters of the model, and compare our model outcomes to clinical patient data. The goal of this work is to provide a tool that may shed light on factors affecting the course of disease progression in patients. This modeling tool can serve as a foundation upon which future treatments can be based

Some Promising Approaches to Tumor-Immune Modeling

Mathematical models of tumor-immune interactions provide an analytical framework in which to address specific questions regarding tumor-immune dynamics. We present a brief summary of several approaches we are currently exploring to model tumor growth, tumor-immune interactions, and treatments. Results to date have shown that simulations of tumor growth using different levels of immune stimulating ligands, effector cells, and tumor challenge, are able to reproduce data from published studies. We additionally present some of our current efforts in the investigation of optimal control to aid in determining improved treatment strategies

Mathematical Modeling of the Regulatory T Cell Effects on Renal Cell Carcinoma Treatment

We present a mathematical model to study the effects of the regulatory T cells (Treg) on Renal Cell Carcinoma (RCC) treatment with sunitinib. The drug sunitinib inhibits the natural self-regulation of the immune system, allowing the effector components of the immune system to function for longer periods of time. This mathematical model builds upon our non-linear ODE model by de Pillis et al. (2009) [13] to incorporate sunitinib treatment, regulatory T cell dynamics, and RCC-specific parameters. The model also elucidates the roles of certain RCC-specific parameters in determining key differences between in silico patients whose immune profiles allowed them to respond well to sunitinib treatment, and those whose profiles did not. Simulations from our model are able to produce results that reflect clinical outcomes to sunitinib treatment such as: (1) sunitinib treatments following standard protocols led to improved tumor control (over no treatment) in about 40% of patients; (2) sunitinib treatments at double the standard dose led to a greater response rate in about 15% the patient population; (3) simulations of patient response indicated improved responses to sunitinib treatment when the patient\u27s immune strength scaling and the immune system strength coefficients parameters were low, allowing for a slightly stronger natural immune response

Model Updating by Adding Known Masses and Stiffnesses

New approaches are developed to update the analytical mass and stiffness matrices of a system. By adding known masses to the structure of interest, measuring the modes of vibration of this mass-modified system, and finally using this set of new data in conjunction with the initial modal survey, the mass matrix of the structure can be corrected. A similar approach can also be used to update the stiffness matrix of the system by attaching known stiffnesses. Manipulating the mass and stiffness correction matrices into vector forms, the connectivity information can be enforced, thereby preserving the physical configuration of the system, which enables successful identification of large structural models with relatively few measured modes

Constructing new optimal entanglement witnesses

We provide a new class of indecomposable entanglement witnesses. In 4 x 4 case it reproduces the well know Breuer-Hall witness. We prove that these new witnesses are optimal and atomic, i.e. they are able to detect the "weakest" quantum entanglement encoded into states with positive partial transposition (PPT). Equivalently, we provide a new construction of indecomposable atomic maps in the algebra of 2k x 2k complex matrices. It is shown that their structural physical approximations give rise to entanglement breaking channels. This result supports recent conjecture by Korbicz et. al.Comment: 9 page

Unital quantum operators on the Bloch ball and Bloch region

For one qubit systems, we present a short, elementary argument characterizing unital quantum operators in terms of their action on Bloch vectors. We then show how our approach generalizes to multi-qubit systems, obtaining inequalities that govern when a ``diagonal'' superoperator on the Bloch region is a quantum operator. These inequalities are the n-qubit analogue of the Algoet-Fujiwara conditions. Our work is facilitated by an analysis of operator-sum decompositions in which negative summands are allowed.Comment: Revised and corrected, to appear in Physical Review