The disposition index: from individual to population approach

To correctly evaluate the glucose control system, it is crucial to account for both insulin sensitivity and secretion. The disposition index (DI) is the most widely accepted method to do so. The original paradigm (hyperbolic law) consists of the multiplicative product of indices related to insulin sensitivity and secretion, but more recently, an alternative formula has been proposed with the exponent α (power function law). Traditionally, curve-fitting approaches have been used to evaluate the DI in a population: the algorithmic implementations often introduce some critical issues, such as the assumption that one of the two indices is error free or the effects of the log transformation on the measurement errors. In this work, we review the commonly used approaches and show that they provide biased estimates. Then we propose a novel nonlinear total least square (NLTLS) approach, which does not need to use the approximations built in the previously proposed alternatives, and show its superiority. All of the traditional fit procedures, including NLTLS, account only for uncertainty affecting insulin sensitivity and secretion indices when they are estimated from noisy data. Thus, they fail when part of the observed variability is due to inherent differences in DI values between individuals. To handle this inevitable source of variability, we propose a nonlinear mixed-effects approach that describes the DI using population hyperparameters such as the population typical values and covariance matrix. On simulated data, this novel technique is much more reliable than the curve-fitting approaches, and it proves robust even when no or small population variability is present in the DI values. Applying this new approach to the analysis of real IVGTT data suggests a value of α significantly smaller than 1, supporting the importance of testing the power function law as an alternative to the simpler hyperbolic law.

The 23 November 1980 southern Italy earthquake

abstract The seismic activity associated with the catastrophic southern Italy earthquake was monitored by 11 seismic stations operating before this event, within an epicentral distance of 200 km, and by 32 additional short-period seismometers installed soon after the main shock. The hypocenter of this event was located at 40°46′N and 15°18′E, at 16 km depth. The fault-plane solution reveals normal faulting, with tensile axis dipping 18° and oriented orthogonal to the axis of the Apennines chain. This mechanism is in good agreement with the stress pattern inferred from some previous earthquakes and the local seismotectonics. The hypocenter locations of more than 600 aftershocks, with local magnitudes greater than 2.4, show a pronounced alignment extending for about 70 km, oriented north 120° and scattered laterally less than 15 km. These events are mostly concentrated between 8 and 16 km depth. A cluster of aftershocks occurred close to the hypocenter of the main shock covering a region elongated 25 km which corresponds also to the highly damaged area. No significant spreading of the aftershock area with time is observed, but one of the events with higher magnitude (ML = 4.8, 14 February 1981) is displaced 20 km NW from the tip of the aftershock region. The time evolution of the number of aftershocks fits well Omori's hyperbolic law with a decay coeffcient of 1.07 ± 0.06. The possibility of a future delayed multiple sequence of large events, as already observed in the past along the central and southern Apennines, is discussed. In particular, a relatively high seismic potential seems to exist along the northern boundary of the 1980 rupture segment.

Velocity Field Induced by Singularities in a Reaction Turbine Fluid Filament of Varying Thickness Approximated by Hyperbolic Law

The flow in a hydraulic reaction turbine can be analyzed by considering the blade surfaces to be surfaces of discontinuity. For such an analysis the velocity field induced by a system of distributed singularities (sources and vortices) is to be determined. In this paper the runner space is divided into elementary runners, assuming the absolute flow in the runner to be irrotational and the stream surfaces to be surfaces of revolution. The axisymmetric fluid filament thus obtained is then conformally mapped on to a plane. The variable thickness of the resulting plane filament is approximated by hyperbolic law. The velocity field due to a singularity is obtained by solving the governing differential equation in a closed form. The solution is extended for a chain of singularities.

THE ELASTIC EXTENSIBILITY OF MUSCLE

Measurements of the extensibility of muscles and other tissues were first made, as far as the authors are aware, by Wertheim in 1846. He proposed for them a hyperbolic law of elasticity of the form: y2 = ax2 + bx. Further measurements were made at much later dates by Marey and by Howell, who did not attach to them any mathematical law of elasticity.The present writers have investigated the elasticity of muscles again, and have found that their own data, as well as those of previous investigators, conform to the logarithmic law,[Formula: see text]where E is the extension, W the stretching force, and k and c are constants.This law is found to hold for muscles both striped and plain, and for nerve tissue. It seems to be true for all tissues except bone which, according to Wertheim, follows Hooke's law of elasticity for inorganic elastic matter.