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Continuous glucose monitors can measure interstitial glucose concentration in real-time for closed-loop glucose control systems, known as artificial pancreas. These control systems use an insulin feedback to maintain plasma glucose concentration within a narrow and safe range, and thus to avoid health complications. As it is not possible to measure plasma insulin concentration in real-time, insulin models have been used in literature to estimate them. Nevertheless, the significant inter-and intra-patient variability of insulin absorption jeopardizes the accuracy of these estimations. In order to reduce these limitations, our objective is to perform a real-time estimation of plasma insulin concentration from continuous glucose monitoring. Hovorka's glucose-insulin model has been incorporated in an Extended Kalman Filter in which different selected time-variant model parameters have been considered as extended states. The observability of the original Hovorka's model and of several extended models has been evaluated by their Lie derivatives. We have evaluated this methodology with an in-silico study with 100 patients with Type 1 diabetes during 25 hours. Furthermore, it has been also validated using clinical data from 12 insulin pump patients with Type 1 diabetes who underwent four mixed meal studies. Real-time insulin estimations have been compared to plasma insulin measurements to assess performance showing the validity of the methodology here used in comparison with that formerly used for insulin models. Hence, real-time estimations for plasma insulin concentration based on subcutaneous glucose monitoring can be beneficial for increasing the efficiency of control algorithms for the artificial pancreas.

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Insulin therapy in type 1 diabetes aims to mimic the pattern of endogenous insulin secretion found in healthy subjects. Glucose-insulin models are widely used in the development of new predictive control strategies in order to maintain the plasma glucose concentration within a narrow range, avoiding the risks of high or low levels of glucose in the blood. However, due to the high variability of this biological process, the exact values of the model parameters are unknown, but they can be bounded by intervals. In this work, the computation of tight glucose concentration bounds under parametric uncertainty for the development of robust prediction tools is addressed.A monotonicity analysis of the model states and parameters is performed. An analysis of critical points, state transformations and application of differential inequalities are proposed to deal with non-monotone parameters. In contrast to current methods, the guaranteed simulations for the glucose-insulin model are carried out by considering uncertainty in all the parameters and initial conditions. Furthermore, no time-discretisation is required, which helps to reduce the computational time significantly. As a result, we are able to compute a tight glucose envelope that bounds all the possible patient's glycemic responses with low computational effort.

is currently the safest infusion route for a commercial artificial pancreas, as opposed to the endogenous pancreas that secretes insulin directly into portal circulation (the vessels that connect the pancreas and other organs with the liver, which acts as a first blood filter before entering the main circulatory system). Even in approaches with concomitant infusion of glucagon (the socalled dual-hormone artificial pancreas), mechanisms are necessary to avoid an excess of insulin delivery which may lead to late hypoglycemia putting at stake the patient's safety. Independently of how these mechanisms are incorporated into the control schemes, all of them must rely on pharmacokinetic models predicting either circulating plasma insulin or a measure of "insulin-onboard", such as the insulin depot remaining at the subcutaneous tissue before entering circulation. In this article, methods to constrain insulin delivery are reviewed, as well as the subcutaneous insulin pharmacokinetic models and estimators on which they rely upon. A big challenge in the prediction of physiological signals, such as insulin concentration, is the large intra-subject variability that patients suffer. Indeed, in terms of control engineering, a patient is a highly time-varying uncertain plant. The intra-day and day-today patient's behavior change due to circadian rhythms (24-hour rhythmic physiological oscillations driven by the body clock, for instance, daily patterns in insulin sensitivity), and other multiple sources of uncertainty arise in key physiological processes such as meal absorption and subcutaneous insulin absorption. Despite this fact, the use of population models for the prediction of insulin pharmacokinetics, that is, how the infused insulin appears in blood, is still common practice. The impact of variability on the model prediction and its implication in closed-loop performance is analyzed. Nevertheless, large intra-subject variability suggests that real-time state and pharmacokinetic parameters estimation is convenient, even when individualized models are considered. The availability of continuous glucose measurements allows to address this problem should an observable glucose-insulin model be available. Different observer techniques proposed to this purpose are reviewed and discussed. The subcutaneous insulin route The pancreas secretes insulin into the portal vein towards the liver, which acts as a first filter before insulin reaches systemic circulation. In the liver, insulin promotes glucose storage in hepatic cells decreasing glucose production. In the fat and muscle cells, insulin acts as a key that triggers the mobilization of glucose transporters to the cell membrane promoting glucose uptake by the cell. As a result of both actions, plasma glucose concentration decreases. The dynamic lag of insulin action is estimated to be about 30 minutes [3]. An artificial pancreas is a classic closed-loop glucose control system (see Figure 1) that

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The pattern of some real phenomenona can be described by compartmental in-series models. Nevertheless, most of these processes are characterized by their variability, which produces that the exact values of the model parameters are uncertain, although they can be bounded by intervals.The aim of this paper is to compute tight solution envelopes that guarantee the inclusion of all possible behaviours of such processes. Current methods, such as monotonicity analysis, enable us to obtain guaranteed solution envelopes. However, if the model includes non-monotone compartments or parameters, the computation of solution envelopes may produce a significant overestimation.Our proposal consists in performing a change of variables in which the output is unaltered, and the model obtained is monotone with respect to the uncertain parameters. The monotonicity of the new system allows us to compute the output bounds for the original system without overestimation. These model transformations have been developed for linear and non-linear systems. Furthermore, if the conditions are not completely satisfied, a novel method to compute tight solution envelopes is proposed. The methods exposed in this paper have been applied to compute tight solution envelopes for two different models: a linear system for glucose modelling and a non-linear system for an epidemiological model.

BIOMATH h t t p : / / w w w. b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h /Abstract-In this work, the problem of obtaining tight output bounds for compartmental in-series models under parametric uncertainty is addressed. It is well-known that current methods used to compute a solution envelope may produce a significant overestimation. However, monotonicity analysis enables us to estimate a tight solution envelope. Our main aim is to get an equivalent model to the initial one, which is usually non-monotone, by means of a suitable combination of equations. In this new model the system monotonicity with respect to the uncertain parameters depends on the elimination rate values of the original model. If the equivalent model is monotone, no overestimation occurs in the computation of the output bounds.