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Mathematical modeling is very helpful for noninvasive investigation of glucose-insulin interaction. In this paper, a new time-delay mathematical model is proposed for glucose-insulin endocrine metabolic regulatory feedback system incorporating the $\beta$-cell dynamic and function for regulating and maintaining bloodstream insulin level. The model includes the insulin degradation due to glucose interaction. The dynamical behavior of the model is analyzed and two-dimensional bifurcation diagrams with respect to two essential parameters of the model are obtained. The results show that the time-delay in insulin secretion in response to blood glucose level, and the delay in glucose drop due to increased insulin concentration, can give rise to complex dynamics, such as periodic oscillation. These dynamics are consistent with the biological findings and period doubling cascade and chaotic state which represent metabolic disorder that may lead to diabetes mellitus.

In vivo evaluation of drug properties becomes a crucial in the drug development. Near infrared techniques contributed considerably to the development of protein polypeptide drugs by performing non-invasive in vivo real time measurements on experimental subjects. In this study, a NIR monitoring system comprised of NIR spectroscopy and imaging system for in vivo real time monitoring of the dynamic of protein polypeptide drugs on mouse models was designed. Insulin acted as polypeptide drug model was covalently labeled with NIR fluorescence probe cypate (Ex = 780nm, Em = 830nm). The labeling route was optimized with different factors to obtain the maximum labeling efficiency, which was evaluated by RP-HPLC. The labeling result show that the labeling efficiency was approximated to 95%. Pharmacodynamics study showed that insulin labeled with cypate did not affect its function of hypoglycemic. The half-time of insulin — cypate complex at arteria cruralis blood was 17min, and the dynamic distribution in mouse model indicated that the labeled insulin firstly accumulated in liver and clearance from the enteron system. The NIR monitoring system combined with NIR probe showed a promising way for in vivo monitoring the dynamics of protein polypeptide drugs in small animals.