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Based on von Neumann’s model of an economy characterized by processes and goods, we add to that model a component representing capital equipment. We assume that the need for capital equipment by any process is proportional to the rate at which that process is running, and therefore an increase in rate requires that capital equipment be purchased, whereas a decrease in rate allows capital equipment to be sold. We thereby construct a continuous-time dynamical model, which we use to investigate the evolution of economic diversity under two price equilibrium scenarios: the first with non-negative prices and non-positive excess demands; the second with enforced market clearing and with prices allowed to be negative. The second scenario represents an economy in which recycling is required, so that excess supply cannot be discarded. We prove that at any time during the progression of the model economy, the solution to each of the two price equilibrium problems exists, and that non-uniqueness of the solution, if any, does not affect the development of the model economy. We compare matched model economies under the two scenarios by simulating their respective evolutions. In each case, the model economy experiences a process of selection and matures to a state of balanced growth, with a higher growth rate when excess supply is discarded, but with greater economic diversity with enforced recycling. The robustness of these qualitative results is demonstrated by repeated trials of simulations on matched pairs of model economies with different randomly chosen parameters. 

In this paper, we develop a pulsatile compartmental model of the Fontan circulation and use it to explore the effects of a fenestration added to this physiology. A fenestration is a shunt between the systemic and pulmonary veins that is added either at the time of Fontan conversion or at a later time for the treatment of complications. This shunt increases cardiac output and decreases systemic venous pressure. However, these hemodynamic benefits are achieved at the expense of a decrease in the arterial oxygen saturation. The model developed in this paper incorporates fenestration size as a parameter and describes both blood flow and oxygen transport. It is calibrated to clinical data from Fontan patients, and we use it to study the impact of a fenestration on several hemodynamic variables, including systemic oxygen availability, effective oxygen availability, and systemic venous pressure. In certain scenarios corresponding to high-risk Fontan physiology, we demonstrate the existence of a range of fenestration sizes in which the systemic oxygen availability remains relatively constant while the systemic venous pressure decreases. 

Abstract This paper focuses on the derivation and simulation of mathematical models describing new plasma fraction in blood for patients undergoing simultaneous extracorporeal membrane oxygenation and therapeutic plasma exchange. Models for plasma exchange with either veno-arterial or veno-venous extracorporeal membrane oxygenation are considered. Two classes of models are derived for each case, one in the form of an algebraic delay equation and another in the form of a system of delay differential equations. In special cases, our models reduce to single compartment ones for plasma exchange that have been validated with experimental data (Randerson et al., 1982, Artif. Organs, 6, 43–49). We also show that the algebraic differential equations are forward Euler discretizations of the delay differential equations, with timesteps equal to transit times through model compartments. Numerical simulations are performed to compare different model types, to investigate the impact of plasma device port switching on the efficiency of the exchange process, and to study the sensitivity of the models to their parameters. 

For a chemical signal to propagate across a cell, it must navigate a tortuous environment involving a variety of organelle barriers. In this work we study mathematical models for a basic chemical signal, the arrival times at the nuclear membrane of proteins that are activated at the cell membrane and diffuse throughout the cytosol. Organelle surfaces within human B cells are reconstructed from soft X-ray tomographic images, and modeled as reflecting barriers to the molecules’ diffusion. We show that signal inactivation sharpens signals, reducing variability in the arrival time at the nuclear membrane. Inactivation can also compensate for an observed slowdown in signal propagation induced by the presence of organelle barriers, leading to arrival times at the nuclear membrane that are comparable to models in which the cytosol is treated as an open, empty region. In the limit of strong signal inactivation this is achieved by filtering out molecules that traverse non-geodesic paths.