Atherosclerosis, the leading death in the United State, is a disease in which a plaque builds up inside the arteries. As the plaque continues to grow, the shear force of the blood flow through the decreasing cross section of the lumen increases. This force may eventually cause rupture of the plaque, resulting in the formation of thrombus, and possibly heart attack. It has long been recognized that the formation of a plaque relates to the cholesterol concentration in the blood. For example, individuals with LDL above 190 mg/dL and HDL below 40 mg/dL are at high risk, while individuals with LDL below 100 mg/dL and HDL above 50 mg/dL are at no risk. In this paper, we developed a mathematical model of the formation of a plaque, which includes the following key variables: LDL and HDL, free radicals and oxidized LDL, MMP and TIMP, cytockines: MCP-1, IFN-γ, IL-12 and PDGF, and cells: macrophages, foam cells, T cells and smooth muscle cells. The model is given by a system of partial differential equations with in evolving plaque. Simulations of the model show how the combination of the concentrations of LDL and HDL in the blood determine whether a plaque will grow or disappear. More precisely, we create a map, showing the risk of plaque development for any pair of values (LDL,HDL).

Alzheimer disease (AD) is a progressive neurodegenerative disease that destroys memory and cognitive skills. AD is characterized by the presence of two types of neuropathological hallmarks: extracellular plaques consisting of amyloid β-peptides and intracellular neurofibrillary tangles of hyperphosphorylated tau proteins. The disease affects 5 million people in the United States and 44 million world-wide. Currently there is no drug that can cure, stop or even slow the progression of the disease. If no cure is found, by 2050 the number of alzheimer’s patients in the U.S. will reach 15 million and the cost of caring for them will exceed $ 1 trillion annually. The present paper develops a mathematical model of AD that includes neurons, astrocytes, microglias and peripheral macrophages, as well as amyloid β aggregation and hyperphosphorylated tau proteins. The model is represented by a system of partial differential equations. The model is used to simulate the effect of drugs that either failed in clinical trials, or are currently in clinical trials. Based on these simulations it is suggested that combined therapy with TNF- α inhibitor and anti amyloid β could yield significant efficacy in slowing the progression of AD.

We consider the Bethe equations for the isotropic spin-1/2 Heisenberg quantum spin chain with periodic boundary conditions. We formulate a conjecture for the number of solutions with pairwise distinct roots of these equations, in terms of numbers of so-called singular (or exceptional) solutions. Using homotopy continuation methods, we find all such solutions of the Bethe equations for chains of length up to 14. The numbers of these solutions are in perfect agreement with the conjecture. We also discuss an indirect method of finding solutions of the Bethe equations by solving the Baxter T-Q equation. We briefly comment on implications for thermodynamical computations based on the string hypothesis.

Sarcoidosis is a disease involving abnormal collection of inflammatory cells forming nodules, called granulomas. Such granulomas occur in the lung and the mediastinal lymph nodes, in the heart, and in other vital and nonvital organs. The origin of the disease is unknown, and there are only limited clinical data on lung tissue of patients. No current model of sarcoidosis exists. In this paper we develop a mathematical model on the dynamics of the disease in the lung and use patients’ lung tissue data to validate the model. The model is used to explore potential treatments.