Diabetes: Models, Signals, and Control

Abstract

The control of diabetes is an interdisciplinary endeavor, which includes a significant biomedical engineering component, with traditions of success beginning in the early 1960s. It began with modeling of the insulin-glucose system, and progressed to large-scale in silico experiments, and automated closed-loop control (artificial pancreas). Here, we follow these engineering efforts through the last, almost 50 years. We begin with the now classic minimal modeling approach and discuss a number of subsequent models, which have recently resulted in the first in silico simulation model accepted as substitute to animal trials in the quest for optimal diabetes control. We then review metabolic monitoring, with a particular emphasis on the new continuous glucose sensors, on the analyses of their time-series signals, and on the opportunities that they present for automation of diabetes control. Finally, we review control strategies that have been successfully employed in vivo or in silico, presenting a promise for the development of a future artificial pancreas and, in particular, discuss a modular architecture for building closed-loop control systems, including insulin delivery and patient safety supervision layers. We conclude with a brief discussion of the unique interactions between human physiology, behavioral events, engineering modeling and control relevant to diabetes.

Figures

Fig. 1

Disciplines which contribute to diabetes control.

Fig. 2

Scheme of the glucose-insulin system.

Fig. 3

Compartmental model of glucose kinetics in steady state. Upper panel: three compartment model with glucose utilization taking place in plasma and rapidly equilibrating tissues; Middle panel: three compartment model with glucose utilization taking place in slowly and rapidly equilibrating tissues; Lower panel: compartment model with glucose utilization taking place in plasma + rapidly equilibrating tissue and slowly equilibrating tissues.

Fig. 4

Model of glucose kinetics in steady state: model-derived parametric representation at basal (upper panel) and elevated insulin (lower panel). EGP denotes endogenous glucose production.

Fig. 5

Comparison among insulin concentration in plasma (compartment 1), rapidly (compartment 2) and slowly (compartment 3) equilibrating tissues with glucose utilization measured with glucose clamp technique. Compartment 3 mimics the time course of glucose infusion (=utilization).

Fig. 6

Compartmental models of insulin (upper panel), glucose (middle panel), and tracer glucose (lower panel) kinetics.

Fig. 7

Plasma glucose (upper) and insulin (lower) concentrations measured during IVGTT (right), OGTT (middle), and MTT (left, panel).

Fig. 8

Decomposition of glucose-insulin system into glucose and insulin subsystems.

Fig. 9

Left panel: IVGTT glucose minimal model. Right panel: OGTT/MTT glucose minimal model.

Fig. 10

Two compartment model of tracer glucose kinetics. The insulin-independent glucose utilization takes place in the accessible compartment ( Q1∗) while insulin-dependent glucose utilization consists of two components, one constant, Rd0, and the other proportional to glycemia. Insulin-dependent glucose utilization is parametrically controlled by insulin in a compartment remote from plasma (I′).

Fig. 11

Endogenous glucose production estimated with the dual tracer technique (open circles, vertical bars represents standard deviation) and with deconvolution (continuous line).

Fig. 12

Comparison between endogenous glucose production (upper panel) and rate of appearance of glucose (lower panel) during a meal reconstructed with models and triple tracer model-independent method.

Fig. 13

Major glucose processing: diffusion to/from the intersitium, active transport in and out of the cell, and phosphorylation/metabolism.

Fig. 14

The 5 k model of [18F]FDG in skeletal muscle: Cp is [18F]FDG plasma arterial concentration, Cc extracellular concentration of [18F]FDG normalized to tissue volume, Ce [18F]FDG tissue concentration, Cm [18F]FDG – 6 – P tissue concentration, total 18F activity concentration in the ROI, K1 [ml/ml/min] and k2 [min−1] the exchange between plasma and extracellular space, k3 [min−1] and k4 [min−1] transport in and out of cell, k5 [min−1] phosphorylation.

Fig. 15

Employment of PET tracers to study glucose diffusion through capillary membrane, active transport into the cells and metabolism.

Fig. 16

Compartmental models of insulin kinetics. Upper panel: two compartment model with insulin degradation in the accessible compartment; Upper panel: two compartment model with insulin degradation in the remote compartment; Upper panel: one compartment model.

Fig. 17

Schematic representation of insulin (upper) and C-peptide (lower) pancreatic secretion and kinetics. Insulin is secreted by the beta-cells in the portal vein and extracted by the liver before it appears in plasma; C-peptide is secreted by the beta-cells, equimolarly to insulin, passes through the liver, before it appears in plasma, but is not extracted.

Fig. 18

Left panel: IVGTT C-peptide minimal model. Right panel: OGTT/MTT C-peptide minimal model.

Fig. 19

Disposition index paradigm. Left panel: a normal individual could be represent by state I; if beta-cells respond to a decrease in insulin sensitivity by adequately increasing insulin secretion (state II) the product of beta-cell function and insulin sensitivity (the disposition index) is unchanged, and normal glucose tolerance is retained. In contrast, if there is not an adequate compensatory increase in beta-cell function to the decreased insulin sensitivity (state 2) the individual develops glucose intolerance. Right panel: importance of segregating glucose tolerance into its individual components of beta-cell responsivity and insulin sensitivity. Subject x is intolerant due to its poor beta-cell function while subject y has poor insulin sensitivity; these two individuals need opposite therapy vectors.

Fig. 20

Scheme of the glucose-insulin control system which relates measured plasma concentrations, i.e., glucose and insulin, to glucose fluxes, i.e., rate of appearance, production, utilization, renal extraction, and insulin fluxes, i.e., secretion and degradation.

Fig. 21

Mixed meal data base (average of 204 nondiabetic subjects, grey area represents mean± 1SD range). Top panel: glucose (left) and insulin (right) concentrations. Middle panel: endogenous glucose production (left) and glucose rate of appearance (right). Bottom panel: glucose utilization (left) and insulin secretion (right).

Fig. 22

Unit process models and forcing function strategy: endogenous glucose production (top left panel); glucose rate of appearance (top right panel); glucose utilization (bottom left panel); insulin secretion (bottom right panel). Entering arrows represent forcing function variables, outgoing arrows are model output.

Fig. 23

Example of daily glucose concentration in some generated in silico subjects.

Fig. 24

Employment of the type 1 diabetes simulator for testing closed-loop control algorithm for insulin infusion.

Fig. 25

Upper panel: overview of the in silico model of insulin secretion which includes mobilization of secretory granules from a very large reserve pool to the cell periphery, where they attach to the plasma membrane (docking). The granules can mature further (priming) and attach to calcium channels, thus entering the “readily releasable pool” (RRP). Calcium influx provides the signal triggering membrane fusion. Lower panel: mathematical formulation of the model.

Fig. 26

Simulation results: insulin secretion (SR) in response to the staircase glucose stimulation (G).

Fig. 27

Transforming BG data into risk space equalizing the hypoglycemic and hyperglycemic blood glucose ranges.

Fig. 28

Two simulated CGM profiles obtained with different noise variance. Noisy (gray line) versus Kalman filtered (black) signals. Shaded areas correspond to the burn-in intervals.

Fig. 29

Modular layered architecture of the artificial pancreas. The layers work on different time scales: the fastest one deal with safety maintenance, the middle one with real-time closed-loop control, and the top one with tuning and supervision on a daily or longer time scale. Three main functionalities are included in each layer: control, estimation, and data management. Decision flow is from top to bottom and information flow is from bottom to top. A layer can override decisions suggested by its upper layer, e.g., for safety reasons.

Fig. 30

Block diagram of a closed-loop glucose control systems including a feedforward action. Information on meal time and amount is used to generate a feedforward action, typically under the form of a premeal bolus. The insulin control signal is obtained as the sum of the feedforward action and the feedback computed by the controller on the basis of glucose sensing.

Fig. 31

MPC prediction scheme: given the model, past inputs and outputs, the future outputs are predicted as a function of future inputs.

Fig. 32

Results of a recent clinical trial that compared conventional open-loop therapy to closed-loop glucose control using Linear Model Predictive Control. Closed-loop control achieved an increase of overnight percent time within the target range and an almost five-fold reduction of the number of nocturnal hypoglycemic episodes.

Fig. 33

An example of CVGA plot. Each patient is represented by a point in the CVGA plane whose coordinates correspond to the minimal and maximal glycemia reached during the monitored time period (note that the axis of the minimal glycemia is reverted). Regulation improves as the points get closer to the lower left corner, corresponding to ideal euglycemia. The glycemic regulation of two populations (white and black circles) is compared. The greater percentage of patients within the A region indicates that the white-dot population achieves better glycemic control compared to the black-dot population.

Fig. 34

Dual-layer bio-behavioral structure of diabetes modeling and control: Layer 1 includes the puzzle of physiologic and behavioral characteristics that determine the specifics of each individual. Layer 2 includes the engineering approaches available to support the optimization of diabetes control.