ODE models for the management of diabetes: A review

Multivariable identification based MPC for closed-loop glucose regulation subject to individual variability

The controller is important for the artificial pancreas to guide insulin infusion in diabetic therapy. However, the inter- and intra-individual variability and time delay of glucose metabolism bring challenges to control glucose within a normal range. In this study, a multivariable identification based model predictive control (mi-MPC) is developed to overcome the above challenges. Firstly, an integrated glucose-insulin model is established to describe insulin absorption, glucose-insulin interaction under meal disturbance, and glucose transport. On this basis, an observable glucose-insulin dynamic model is formed, in which the individual parameters and disturbances can be identified by designing a particle filtering estimator. Next, embedded with the identified glucose-insulin dynamic model, a mi-MPC method is proposed. In this controller, plasma glucose concentration (PGC), an important variable and indicator of glucose regulation, is estimated and controlled directly. Finally, the method was tested on 30 in-silico subjects produced by the UVa/Padova simulator. The results show that the mi-MPC method including the model, individual identification, and the controller can regulate glucose with the mean value of 7.45 mmol/L without meal announcement.

Mathematical models of cystic fibrosis as a systemic disease

Cystic fibrosis (CF) is widely known as a disease of the lung, even though it is in truth a systemic disease, whose symptoms typically manifest in gastrointestinal dysfunction first. CF ultimately impairs not only the pancreas and intestine but also the lungs, gonads, liver, kidneys, bones, and the cardiovascular system. It is caused by one of several mutations in the gene of the epithelial ion channel protein CFTR. Intense research and improved antimicrobial treatments during the past eight decades have steadily increased the predicted life expectancy of a person with CF (pwCF) from a few weeks to over 50 years. Moreover, several drugs ameliorating the sequelae of the disease have become available in recent years, and notable treatments of the root cause of the disease have recently generated substantial improvements in health for some but not all pwCF. Yet, numerous fundamental questions remain unanswered. Complicating CF, for instance in the lung, is the fact that the associated insufficient chloride secretion typically perturbs the electrochemical balance across epithelia and, in the airways, leads to the accumulation of thick, viscous mucus and mucus plaques that cannot be cleared effectively and provide a rich breeding ground for a spectrum of bacterial and fungal communities. The subsequent infections often become chronic and respond poorly to antibiotic treatments, with outcomes sometimes only weakly correlated with the drug susceptibility of the target pathogen. Furthermore, in contrast to rapidly resolved acute infections with a single target pathogen, chronic infections commonly involve multi-species bacterial communities, called "infection microbiomes," that develop their own ecological and evolutionary dynamics. It is presently impossible to devise mathematical models of CF in its entirety, but it is feasible to design models for many of the distinct drivers of the disease. Building upon these growing yet isolated modeling efforts, we discuss in the following the feasibility of a multi-scale modeling framework, known as template-and-anchor modeling, that allows the gradual integration of refined sub-models with different granularity. The article first reviews the most important biomedical aspects of CF and subsequently describes mathematical modeling approaches that already exist or have the potential to deepen our understanding of the multitude aspects of the disease and their interrelationships. The conceptual ideas behind the approaches proposed here do not only pertain to CF but are translatable to other systemic diseases. This article is categorized under: Congenital Diseases > Computational Models.

A Glucose-Only Model to Extract Physiological Information from Postprandial Glucose Profiles in Subjects with Normal Glucose Tolerance

BACKGROUND

Current mathematical models of postprandial glucose metabolism in people with normal and impaired glucose tolerance rely on insulin measurements and are therefore not applicable in clinical practice. This research aims to develop a model that only requires glucose data for parameter estimation while also providing useful information on insulin sensitivity, insulin dynamics and the meal-related glucose appearance (GA).

METHODS

The proposed glucose-only model (GOM) is based on the oral minimal model (OMM) of glucose dynamics and substitutes the insulin dynamics with a novel function dependant on glucose levels and GA. A Bayesian method and glucose data from 22 subjects with normal glucose tolerance are utilised for parameter estimation. To validate the results of the GOM, a comparison to the results of the OMM, obtained by using glucose and insulin data from the same subjects is carried out.

RESULTS

The proposed GOM describes the glucose dynamics with comparable precision to the OMM with an RMSE of 5.1 ± 2.3 mg/dL and 5.3 ± 2.4 mg/dL, respectively and contains a parameter that is significantly correlated to the insulin sensitivity estimated by the OMM (r = 0.7) Furthermore, the dynamic properties of the time profiles of GA and insulin dynamics inferred by the GOM show high similarity to the corresponding results of the OMM.

CONCLUSIONS

The proposed GOM can be used to extract useful physiological information on glucose metabolism in subjects with normal glucose tolerance. The model can be further developed for clinical applications to patients with impaired glucose tolerance under the use of continuous glucose monitoring data.

Ultimate Bounds for a Diabetes Mathematical Model Considering Glucose Homeostasis

This paper deals with a recently reported mathematical model formulated by five first-order ordinary differential equations that describe glucoregulatory dynamics. As main contributions, we found a localization domain with all compact invariant sets; we settled on sufficient conditions for the existence of a bounded positively-invariant domain. We applied the localization of compact invariant sets and Lyapunov’s direct methods to obtain these results. The localization results establish the maximum cell concentration for each variable. On the other hand, Lyapunov’s direct method provides sufficient conditions for the bounded positively-invariant domain to attract all trajectories with non-negative initial conditions. Further, we illustrate our analytical results with numerical simulations. Overall, our results are valuable information for a better understanding of this disease. Bounds and attractive domains are crucial tools to design practical applications such as insulin controllers or in silico experiments. In addition, the model can be used to understand the long-term dynamics of the system.

A methodical survey of mathematical model-based control techniques based on open and closed loop control approach for diabetes management

Disturbance of blood sugar level is controlled through well-known biomechanical feedback loops: high levels of glucose in blood facilitate to release insulin from the pancreas which accelerates the absorption rate of cellular glucose. Low glucose levels encourage to release pancreatic glucagon which induces glycogen breakdown to glucose in the liver. These bio-control systems do not function properly in diabetic patients. Though the control of disease seems intuitively easy, in real life, due to many differences in structure by diet and fasting, exercise, medications, patient’s profile and other stressors, it is not that easy. The mathematical models of the glucose-insulin regulatory system follow the patient’s physiological conditions which make it difficult to identify and estimate all the model parameters. In this paper, we have given a systematic literature review on mathematical models of the diabetic patients, and various kinds of disease control techniques through the development of open and closed loop insulin deliver command system and optimization of exogenous insulin rate. It demonstrates the open and closed loop type model-based control strategies underlying the assumptions of the concerned models. The combination of mathematical model with control strategies such as genetic algorithm (GA), neural network (NN), sliding mode controller (SMC), model predictive controller (MPC), and fuzzy logic control (FLC) has been considered, which provides an overview of this area, highlighting the control profile over the diabetic model with promising clinical results, outlining key challenges, and identifying needs for the future research. Also, the significance of these control algorithms has been discussed in the presence of the noises, the controller’s robustness and various other disturbances. It provides substantial information on diabetes management through various control techniques.

The Effect of a Non-Local Fractional Operator in an Asymmetrical Glucose-Insulin Regulatory System: Analysis, Synchronization and Electronic Implementation

For studying biological conditions with higher precision, the memory characteristics defined by the fractional-order versions of living dynamical systems have been pointed out as a meaningful approach. Therefore, we analyze the dynamics of a glucose-insulin regulatory system by applying a non-local fractional operator in order to represent the memory of the underlying system, and whose state-variables define the population densities of insulin, glucose, and β-cells, respectively. We focus mainly on four parameters that are associated with different disorders (type 1 and type 2 diabetes mellitus, hypoglycemia, and hyperinsulinemia) to determine their observation ranges as a relation to the fractional-order. Like many preceding works in biosystems, the resulting analysis showed chaotic behaviors related to the fractional-order and system parameters. Subsequently, we propose an active control scheme for forcing the chaotic regime (an illness) to follow a periodic oscillatory state, i.e., a disorder-free equilibrium. Finally, we also present the electronic realization of the fractional glucose-insulin regulatory model to prove the conceptual findings.

A Glucose-Insulin Mixture Model and Application to Short-Term Hypoglycemia Prediction in the Night Time

OBJECTIVE

Insulin-induced hypoglycemia is recognized as a critical problem for diabetic patients, especially at night. To give glucose prediction and advance warning of hypoglycemia of at least 30 minutes, various glucose-insulin models have been proposed. Recognizing the complementary nature of the models, this research proposes a Glucose-Insulin Mixture (GIM) model to predict the glucose values for hypoglycemia detection, by optimally fusing different models with its adjusted parameters to address the inter- and intra-individual variability.

METHODS

Two types of classic glucose-insulin models, the Ruan model, with single-compartment glucose kinetics, and the Hovorka model, with two-compartment glucose kinetics, are selected as two candidate models. Based on Bayesian inference, GIM is introduced with quantified contributions from the models with the associated parameters. GIM is then applied to predict the glucose values and hypoglycemia events.

RESULTS

The proposed model is validated by the nocturnal glucose data collected from 12 participants with type 1 diabetes. The GIM model has promising fitting of RMSE within 0.3465 mmol/L and predicting of RMSE within 0.5571 mmol/L. According to the literature, the hypoglycemia is defined as 3.9 mmol/L, and the GIM model shows good short-term hypoglycemia prediction performance with the data collected within the last hour (accuracy: 95.97%, precision: 91.77%, recall: 95.60%). In addition, the probability of hypoglycemia event in 30 minutes is inferred.

CONCLUSION

GIM, by fusing various glucose-insulin models via Bayesian inference, has the promise to capture glucose dynamics and predict hypoglycemia.

SIGNIFICANCE

GIM based short-term hypoglycemia prediction has potential clinical utility for timely intervention.