Determination of personalized diabetes treatment plans using a two-delay model

Dynamical analysis of a glucose-insulin regulatory system with insulin-degrading enzyme and multiple delays

This paper investigates the dynamics of a glucose-insulin regulatory system model that incorporates: (1) insulin-degrading enzyme in the insulin equation; and (2) discrete time delays respectively in the insulin production term, hepatic glucose production term, and the insulin-degrading enzyme. We provide rigorous results of our model including the asymptotic stability of the equilibrium solution and the existence of Hopf bifurcation. We show that analytically and numerically at a certain value the time delays driven stability or instability occurs when the corresponding model has an interior equilibrium. Moreover, we illustrate the oscillatory regulation and insulin secretion via numerical simulations, which show that the model dynamics exhibit physiological observations and more information by allowing parameters to vary. Our results may provide useful biological insights into diabetes for the glucose-insulin regulatory system model.

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.

Delay-induced uncertainty for a paradigmatic glucose-insulin model

Medical practice in the intensive care unit is based on the assumption that physiological systems such as the human glucose-insulin system are predictable. We demonstrate that delay within the glucose-insulin system can induce sustained temporal chaos, rendering the system unpredictable. Specifically, we exhibit such chaos for the ultradian glucose-insulin model. This well-validated, finite-dimensional model represents feedback delay as a three-stage filter. Using the theory of rank one maps from smooth dynamical systems, we precisely explain the nature of the resulting delay-induced uncertainty (DIU). We develop a framework one may use to diagnose DIU in a general oscillatory dynamical system. For infinite-dimensional delay systems, no analog of the theory of rank one maps exists. Nevertheless, we show that the geometric principles encoded in our DIU framework apply to such systems by exhibiting sustained temporal chaos for a linear shear flow. Our results are potentially broadly applicable because delay is ubiquitous throughout mathematical physiology.

Personalized glucose-insulin model based on signal analysis

Glucose plasma measurements for diabetes patients are generally presented as a glucose concentration-time profile with 15-60min time scale intervals. This limited resolution obscures detailed dynamic events of glucose appearance and metabolism. Measurement intervals of 15min or more could contribute to imperfections in present diabetes treatment. High resolution data from mixed meal tolerance tests (MMTT) for 24 type 1 and type 2 diabetes patients were used in our present modeling. We introduce a model based on the physiological properties of transport, storage and utilization. This logistic approach follows the principles of electrical network analysis and signal processing theory. The method mimics the physiological equivalent of the glucose homeostasis comprising the meal ingestion, absorption via the gastrointestinal tract (GIT) to the endocrine nexus between the liver, pancreatic alpha and beta cells. This model demystifies the metabolic 'black box' by enabling in silico simulations and fitting of individual responses to clinical data. Five-minute intervals MMTT data measured from diabetic subjects result in two independent model parameters that characterize the complete glucose system response at a personalized level. From the individual data measurements, we obtain a model which can be analyzed with a standard electrical network simulator for diagnostics and treatment optimization. The insulin dosing time scale can be accurately adjusted to match the individual requirements of characterized diabetic patients without the physical burden of treatment.

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

We study the effect of diabetic deficiencies on the production of an oscillatory ultradian regime using a deterministic nonlinear model which incorporates two physiological delays. It is shown that insulin resistance impairs the production of oscillations by dampening the ultradian cycles. Four strategies for restoring healthy regulation are explored. Through the introduction of an instantaneous glucose-dependent insulin response, explicit conditions for the existence of periodic solutions in the linearised model are formulated, significantly reducing the complexity of identifying an oscillatory regime. The model is thus shown to be suitable for representing the effect of diabetes on the oscillatory regulation and for investigating pathways to reinstating a physiological healthy regime.

Simulation and qualitative analysis of glucose variability, mean glucose, and hypoglycemia after subcutaneous insulin therapy for stress hyperglycemia

Background

The critically ill can have persistent dysglycemia during the “subacute” recovery phase of their illness because of altered gene expression; it is also not uncommon for these patients to receive continuous enteral nutrition during this time. The optimal short-acting subcutaneous insulin therapy that should be used in this clinical scenario, however, is unknown. Our aim was to conduct a qualitative numerical study of the glucose-insulin dynamics within this patient population to answer the above question. This analysis may help clinicians design a relevant clinical trial.

Methods

Eight virtual patients with stress hyperglycemia were simulated by means of a mathematical model. Each virtual patient had a different combination of insulin resistance and insulin deficiency that defined their unique stress hyperglycemia state; the rate of gluconeogenesis was also doubled. The patients received 25 injections of subcutaneous regular or Lispro insulin (0-6 U) with 3 rates of continuous nutrition. The main outcome measurements were the change in mean glucose concentration, the change in glucose variability, and hypoglycemic episodes. These end points were interpreted by how the ultradian oscillations of glucose concentration were affected by each insulin preparation.

Results

Subcutaneous regular insulin lowered both mean glucose concentrations and glucose variability in a linear fashion. No hypoglycemic episodes were noted. Although subcutaneous Lispro insulin lowered mean glucose concentrations, glucose variability increased in a nonlinear fashion. In patients with high insulin resistance and nutrition at goal, “rebound hyperglycemia” was noted after the insulin analog was rapidly metabolized. When the nutritional source was removed, hypoglycemia tended to occur at higher Lispro insulin doses. Finally, patients with severe insulin resistance seemed the most sensitive to insulin concentration changes.

Conclusions

Subcutaneous regular insulin consistently lowered mean glucose concentrations and glucose variability; its linear dose-response curve rendered the preparation better suited for a sliding-scale protocol. The longer duration of action of subcutaneous regular insulin resulted in better glycemic-control metrics for patients who were continuously postprandial. Clinical trials are needed to examine whether these numerical results represent the glucose-insulin dynamics that occur in intensive care units; if present, their clinical effects should be evaluated.