Mathematical investigation of diabetically impaired ultradian oscillations in the glucose-insulin regulation

A mechanism for slow rhythms in coordinated pancreatic islet activity

Insulin levels in the blood oscillate with a variety of periods, including rapid (5-10 min), ultradian (50-120 min), and circadian (24 h). Oscillations of insulin are beneficial for lowering blood glucose and disrupted rhythms are found in people with type 2 diabetes and their close relatives. These in vivo secretion dynamics imply that the oscillatory activity of individual islets of Langerhans are synchronized, although the mechanism for this is not known. One mechanism by which islets may synchronize is negative feedback of insulin on whole-body glucose levels. In previous work, we demonstrated that a negative feedback loop with a small time delay, to account for the time required for islets to be exposed to a new glucose concentration in vivo, results in small 3-6 islet populations synchronizing to produce fast closed-loop oscillations. However, these same islet populations could also produce slow closed-loop oscillations with periods longer than the natural islet oscillation periods. Here, we investigate the origin of the slow oscillations and the bistability with the fast oscillations using larger islet populations (20-50 islets). In contrast to what was observed earlier, larger islet populations mainly synchronize to longer-period oscillations that are approximately twice the delay time used in the feedback loop. A mean-field model was also used as a proxy for a large islet population to uncover the underlying mechanism for the slow rhythm. The heterogeneous intrinsic oscillation periods of the islets interferes with this rhythm mechanism when islet populations are small, and is similar to adding noise to the mean-field model. Thus, the effect of a time delay in the glucose feedback mechanism is similar to other examples of time-delayed systems in biology and may be a viable mechanism for ultradian oscillations.

Periodic insulin stimulation of Akt: Dynamic steady states and robustness

The periodic secretion of insulin is a salient feature of the blood glucose control system in vivo. Insulin levels in the blood exhibit oscillations on multiple time scales - rapid, ultradian, and circadian - and the improved metabolic regulation resulting from pulsatile insulin release has been well established. Although numerous mathematical models investigating the causal mechanisms of insulin oscillations have appeared in the literature, to date there has been comparatively little attention given to the influence of periodic insulin stimulation on downstream components of the insulin signalling pathway. In this paper, we explore the effect of high frequency periodic insulin stimulation on Akt (also known as PKB), a crucial crosstalk node in the insulin signalling pathway that coordinates metabolic and mitogenic processes in the cell. We analyse a mathematical model of Akt translocation to the plasma membrane under both single step insulin perturbations and periodic insulin stimulation with an emphasis on - but not limited to - the physiological range of parameter values. It was shown that the system rapidly attains a robust dynamic steady state entrained to the periodic insulin stimulation. Moreover, the translocation of Akt to the plasma membrane in the model permits a sufficient level of phosphorylation to trigger downstream metabolic regulators. However, the modelling also indicated that further investigation of this activation process is required to determine whether the response of Akt is a key determinant of the enhanced metabolic control observed under periodic insulin stimulation.

Dynamics of a glucose-insulin model

Diabetes mellitus is a noncommunicable disease, which is a serious threat to human health around the world. In this paper, we propose a simple glucose-insulin model with Michaelis-Menten function as insulin degradation rate to mimic the pathogenic mechanism of diabetes. By theoretical analysis, a unique positive equilibrium of model exists and it is globally asymptotically stable. The four strategies are designed for diabetes patients based on the sensitivity of parameters, including insulin injection and medicine treatments. Numerical simulations are given to support the theoretical results.

Multi-Timescale Rhythmicity of Blood Glucose and Insulin Delivery Reveals Key Advantages of Hybrid Closed Loop Therapy

BACKGROUND

Blood glucose and insulin exhibit coordinated daily and hourly rhythms in people without diabetes (nonT1D). Although the presence and stability of these rhythms are associated with euglycemia, it is unknown if they (1) are preserved in individuals with type 1 diabetes (T1D) and (2) vary by therapy type. In particular, Hybrid Closed Loop (HCL) systems improve glycemia in T1D compared to Sensor Augmented Pump (SAP) therapies, but the extent to which either recapitulates coupled glucose and insulin rhythmicity is not well described. In HCL systems, more rapid modulation of glucose via automated insulin delivery may result in greater rhythmic coordination and euglycemia. Such precision may not be possible in SAP systems. We hypothesized that HCL users would exhibit fewer hyperglycemic event, superior rhythmicity, and coordination relative to SAP users.

METHODS

Wavelet and coherence analyses were used to compare glucose and insulin delivery rate (IDR) within-day and daily rhythms, and their coordination, in 3 datasets: HCL (n = 150), SAP (n = 89), and nonT1D glucose (n = 16).

RESULTS

Glycemia, correlation between normalized glucose and IDR, daily coherence of glucose and IDR, and amplitude of glucose oscillations differed significantly between SAP and HCL users. Daily glucose rhythms differed significantly between SAP, but not HCL, users and nonT1D individuals.

CONCLUSIONS

SAP use is associated with greater hyperglycemia, higher amplitude glucose fluctuations, and a less stably coordinated rhythmic phenotype compared to HCL use. Improvements in glucose and IDR rhythmicity may contribute to the overall effectiveness of HCL systems.

Examining Type 1 Diabetes Mathematical Models Using Experimental Data

Type 1 diabetes requires treatment with insulin injections and monitoring glucose levels in affected individuals. We explored the utility of two mathematical models in predicting glucose concentration levels in type 1 diabetic mice and determined disease pathways. We adapted two mathematical models, one with β-cells and the other with no β-cell component to determine their capability in predicting glucose concentration and determine type 1 diabetes pathways using published glucose concentration data for four groups of experimental mice. The groups of mice were numbered Mice Group 1–4, depending on the diabetes severity of each group, with severity increasing from group 1–4. A Markov Chain Monte Carlo method based on a Bayesian framework was used to fit the model to determine the best model structure. Akaike information criteria (AIC) and Bayesian information criteria (BIC) approaches were used to assess the best model structure for type 1 diabetes. In fitting the model with no β-cells to glucose level data, we varied insulin absorption rate and insulin clearance rate. However, the model with β-cells required more parameters to match the data and we fitted the β-cell glucose tolerance factor, whole body insulin clearance rate, glucose production rate, and glucose clearance rate. Fitting the models to the blood glucose concentration level gave the least difference in AIC of 1.2, and a difference in BIC of 0.12 for Mice Group 4. The estimated AIC and BIC values were highest for Mice Group 1 than all other mice groups. The models gave substantial differences in AIC and BIC values for Mice Groups 1–3 ranging from 2.10 to 4.05. Our results suggest that the model without β-cells provides a more suitable structure for modelling type 1 diabetes and predicting blood glucose concentration for hypoglycaemic episodes.

Blood glucose concentration control for type 1 diabetic patients: a multiple-model strategy

In this study, a multiple-model strategy is evaluated as an alternative closed-loop method for subcutaneous insulin delivery in type 1 diabetes. Non-linearities of the glucose-insulin regulatory system are considered by modelling the system around five different operating points. After conducting some identification experiments in the UVA/Padova metabolic simulator (accepted simulator by the US Food and Drug Administration (FDA)), five transfer functions are obtained for these operating points. Paying attention to some physiological facts, the control objectives such as the required settling time and permissible bounds of overshoots and undershoots are determined for any transfer functions. Then, five PID controllers are tuned to achieve these objectives and a bank of controllers is constructed. To cope with difficulties of the presence of delays in subcutaneous blood glucose (BG) measuring and in administration of insulin, a glucose-dependent setpoint is considered as the desired trajectory for the BG concentration. The performance of the obtained closed-loop glucose-insulin regulatory system is investigated on the in silico adult cohort of the UVA/Padova metabolic simulator. The obtained results show that the proposed multiple-model strategy leads to a closed-loop mechanism with limited hyperglycemia and no severe hypoglycemia.