Continuous glucose monitoring (CGM), a technique that records blood glucose at a regular intervals. While CGM is more commonly used in type 1 diabetes, it is increasingly becoming attractive for treating type 2 diabetic patients. The time series obtained from a CGM provides a rich picture of the glycemic state of the subjects and may help have tighter control on blood sugar by revealing patterns in their physiological responses to food. However, despite its importance, the biophysical understanding of CGM is far from complete. CGM data series is complex not only because it depends on the composition of the food but also varies with individual physiology. All of these make a full modeling of CGM data a difficult task. Here we propose a simple model to explain CGM data in type 2 diabetes. The model combines a relatively simple glucose-insulin dynamics with a two-compartment food model. Using CGM data of a healthy and a diabetic individual we show that this model can capture liquid meals well. The model also allows us to estimate the parameters in a relatively straightforward manner. This opens up the possibility of personalizing the CGM data. The model also predicts insulin time series from the model, and the rate of appearance of glucose due to food. Our methodology thus paves the way for novel analyses of CGM which have not been possible before.
The evolution of precipitates in stressed solids is modeled by coupling a quasi-steady diffusion equation and a linear elasticity equation with dynamic boundary conditions. The governing equations are solved numerically using a boundary integral method (BIM). A critical step in applying BIM is to develop fast algorithms to reduce the arithmetic operation count of matrix-vector multiplications. In this paper, we develop a fast adaptive treecode algorithm for the diffusion and elasticity problems in two dimensions (2D). We present a novel source dividing strategy to parallelize the treecode. Numerical results show that the speedup factor is nearly perfect up to a moderate number of processors. This approach of parallelization can be readily implemented in other treecodes using either uniform or non-uniform point distribution. We demonstrate the effectiveness of the treecode by computing the long-time evolution of a complicated microstructure in elastic media, which would be extremely difficult with a direct summation method due to CPU time constraint. The treecode speeds up computations dramatically while fulfilling the stringent precision requirement dictated by the spectrally accurate BIM.
In this paper, we propose a space-time rescaling scheme for computing the long time evolution of multiple precipitates in an elastically stressed medium. The algorithm is second order accurate in time, spectrally accurate in space and enables one to simulate the evolution of precipitates in a fraction of the time normally used by fixed-frame algorithms. In particular, we extend the algorithm recently developed for single particle by Li et al. (Li, Lowengrub and Leo, J. Comput. Phys., 335 (2007), 554) to the multiple particle case, which involves key differences in the method. Our results show that without elasticity there are successive tip splitting phenomena accompanied by the formation of narrow channels between the precipitates. In presence of applied elastic field, the precipitates form dendrite-like structures with the primary arms aligned in the principal directions of the elastic field. We demonstrate that when the far-field flux decreases with the effective radius of the system, tip-splitting and dendrite formation can be suppressed, as in the one particle case. Depending on the initial position of the precipitates, we further observe that some precipitates grow while others may shrink, even when a positive far field flux is applied.
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