Evaluation of the Effect of Gain on the Meal Response of an Automated Closed-Loop Insulin Delivery System

Experiments were performed on eight diabetic dogs (26.8 ± 1.7 kg, range 20.3–35.3) housed at Medtronic MiniMed animal facilities. Diabetes was induced by partial pancreatectomy. Dogs were treated with continuous subcutaneous insulin infusion for at least 1 week before experiments to determine the total daily dose (TDD) of insulin required for open-loop glucose control. On the day before experiments began, a new insulin infusion catheter was inserted. Closed-loop experiments were performed using Humalog. Dogs were housed under controlled kennel conditions and used for experiments only if judged to be in good condition (physical appearance, hematocrit, white blood cell count, body temperature, and weight stability). The experimental protocol was approved by the appropriate animal research committee.

Insulin delivery was calculated by an algorithm that responds to glucose with three components: a proportional component (P) that reacts to the difference between plasma glucose and basal glucose (the putative β-cell set point for fasting glucose), an integral component (I) that adapts to persistent hyper- or hypoglycemia, and a derivative component (D) that acts in response to the rate of change of blood glucose. We have previously shown the model to describe insulin secretion during hyperglycemic clamps performed in humans with normal glucose tolerance (16). The model is identical to the well-known proportional-integral-derivative algorithm widely used in engineering (17). It is described by:

where the proportional gain K_P (units/h per mg/dl) determines the rate of insulin delivery in response to glucose above the set point (G_B; mmol/l), T_I (integral time; min) determines the rate at which the underlying basal rate adapts, and T_D (derivative time; min) determines the relative amount of insulin delivered in response to the rate of change of glucose. G_B was fixed at 6.7 mmol/l, and K_P was set in relation to the TDD of insulin under open-loop treatment:

Reference K_P was 0.675 units/h per mg/dl, and reference TDD was 1.0 units · kg⁻¹ · day⁻¹. T_D and T_I were fixed at 66 and 150 min, respectively. These values were based on a computer simulation (computer model in 18) and preliminary experiments performed in dogs during the development of the system. Using these nominal gains, we performed a dose response with K_P increased by 50% (1.5 × TDD) or decreased by 50% (0.5 × TDD).

At the time closed-loop control was initiated (approximately 8:00 a.m.), all preprogrammed basal rates were set to 0, and the initial condition of the integral term (ID_B) (Eq. 1) was set to the 8:00 a.m. basal rate. Setting the initial condition to the 8:00 a.m. basal rate allowed the animals starting closed-loop control at target glucose [P(t) = 0] and stable rate of change of glucose [D(t) = 0] to transfer onto closed-loop control with no change in insulin delivery. For animals starting closed-loop control above target, the initial condition (ID_B) was also set to the 8:00 a.m. value basal rate, and the animals received an insulin bolus calculated by the algorithm: (K_P × T_D) × (glucose − G_B). If PID(t) was calculated to be <0, insulin delivery was suspended, and no changes were made in the integral component. Continuous glucose concentration was calculated by a linear extrapolation of the measured blood glucose samples (referred to in engineering literature as a first-order hold) (19). Insulin delivery was calculated with a discrete form of Eq. 1 and delivered with a Medtronic MiniMed Paradigm 511 pump.

After an overnight period, during which euglycemia was achieved with continuous subcutaneous insulin infusion, a blood-sampling catheter was inserted into a peripheral vein. Blood samples were obtained every 20 min until noon, at which time the dog was fed a standard meal of commercially available dog food (LabDiet). After the meal, blood samples were drawn every 10 min for 3 h and every 20 min thereafter for an additional 3 h. Experiments on the same dog were separated by at least 5 days, and the order was randomized.

The plasma insulin concentration [I_p(t)] in response to a single subcutaneous bolus of insulin was assumed to follow a biexponential curve [h(t)] with the response to multiple boluses linearly summing:

Eq. 3 is the solution to a standard two-compartment (subcutaneous and plasma) insulin model (20). Parameters were identified by nonlinear least squares (Civilized Software, Silver Spring, MD). From the parameters B, τ₁, and τ₂, the time to peak insulin concentration after a subcutaneous bolus and insulin clearance rate were calculated as ln(τ₂/τ₁)/(1/τ₁ − 1/τ₂) and 1/B/(τ₁ − τ₂), respectively.

Data are reported as the means ± SE. Differences in parameters were evaluated with either paired Student’s t tests or one-way ANOVA (GraphPad Prism; GraphPad Software, San Diego, CA).

Blood glucose values were immediately ascertained, using a Chiron 865 automated analyzer (Chiron, Emeryville, CA). Additional blood was drawn into heparinized tubes (∼1 ml) and centrifuged, and the plasma was stored at −20°C. Insulin was measured in duplicate, using an enzyme-linked immunosorbent assay kit (ALPCO Diagnostics, Windham, NH).

Preprandial glucose levels were similar for the different algorithm gains (6.0 ± 0.3, 5.2 ± 0.3, and 4.9 ± 0.5 mmol/l for 0.5 × TDD, TDD, and 1.5 × TDD, respectively; P > 0.05) (Fig. 1A). Peak postprandial glucose significantly decreased as gain increased (12.1 ± 0.6, 9.6 ± 1.0, and 8.5 ± 0.6 mmol/l; P < 0.05). Maximum plasma insulin levels were similar (224.5 ± 50.8, 187.3 ± 27.8, and 194.2 ± 42.6 pmol/l; P > 0.05) (Fig. 1B) as gain increased. Early-phase insulin delivery was substantially higher as gain increased (peak values 2.5 ± 0.9, 4.2 ± 0.8, and 10.1 ± 3.5 units/h, respectively; P < 0.05) (Fig. 1C). Insulin delivery 6 h postmeal was elevated at the 0.5 × TDD level compared with TDD and 1.5 × TDD (2.1 ± 0.3 vs. 0.6 ± 0.2 and 0.7 ± 0.4 units/h; P < 0.05). The lower peak glucose levels achieved with increasing gain resulted in dramatically lower postprandial incremental area under the glucose curve (ΔAUC_GLC) for the 6-h period after the meal (2,009 ± 143, 1,182 ± 158, and 1,000 ± 106 mmol/l × min, for 0.5 × TDD, TDD, and 1.5 × TDD, respectively; P < 0.05) (Fig. 2A), particularly as the gain was increased from 0.5 × TDD to TDD. Insulin delivery over the same period tended to increase with increasing gain, but this did not achieve statistical significance (13.3 ± 1.5, 14.0 ± 0.9, and 15.5 ± 1.1 units, respectively; P > 0.05). All gains resulted in similar incremental insulin AUC (Fig. 2B), but the initial rise in plasma insulin was more rapid with high gain leading to significant differences in the 2-h incremental AUC (P < 0.01) (Fig. 2C). Hypoglycemia (<3.3 mmol) occurred in one experiment for 0.5 × TDD, three experiments for TDD, and two experiments for 1.5 × TDD. For all gains, insulin concentration and insulin delivery remained substantially elevated 6 h after the meal, even though glucose levels had, with the exception of the low gain response, returned to preprandial basal levels by t = 600 min (8.2 ± 0.9, 6.0 ± 0.6, and 6.0 ± 0.5, respectively; P > 0.05) (Fig. 3).

Plasma insulin kinetics were well described by the insulin kinetic model (Eq. 3) for all gains (r² = 0.77, 0.82, and 0.78 for 0.5 × TDD, TDD, and 1.5 × TDD, respectively; correlation calculated between measured and model predicted insulin levels) (Fig. 4). No differences were observed in insulin clearance (44.6 ± 6.5, 48.3 ± 5.0, and 46.9 ± 6.1 ml · min⁻¹ · kg⁻¹) or kinetic time constants (τ₁ = 48.6 ± 7.0, 58.4 ± 5.5, and 53.4 ± 7.5 min; τ₂ = 33.0 ± 6.9, 31.3 ± 6.0, and 36.2 ± 6.1 min) as gain increased (P > 0.05 for all), with predicted time-to-peak insulin concentration after a single bolus to be ∼40–44 min at all gains.

The current study has three main conclusions. First, automated closed-loop glucose control, based on a model of the β-cell, is achievable across a wide range of gains, using subcutaneous insulin delivery. Second, after meals, high plasma insulin levels are required well past the time when plasma glucose is normalized. Third, plasma insulin kinetics during closed-loop insulin delivery are well described by a simple two-compartment insulin model.

The β-cell model/closed-loop algorithm used here (Eq. 1) reproduces the well-established pattern of β-cell insulin secretion in response to a hyperglycemic clamp (16,21). Although the model used here has been shown to describe hyperglycemic clamp data, other models of β-cell secretion have been proposed (22–30). The models are similar insofar as each describes glucose-induced insulin secretion as the sum of components that react immediately to a change in glucose, have a delayed reaction to a change in glucose, and/or react to the rate of change of glucose. Of these models, the one proposed by Cerasi et al. (29) is of particular interest because it also produces a slowly rising second phase without using an integral term per se. Previous closed-loop studies in diabetic dogs have used phases analogous to P(t) and D(t), with intravenous insulin delivery, to show that meal excursions could be obtained that were actually lower than those observed in normal dogs (10,11). More recent studies using insulin analogs have evaluated other administration routes (12–14). The main difference between the algorithms in these earlier studies (10–14) and the algorithm used here is that the proposed β-cell model has an integral component [I(t)] (Eq. 1) that can theoretically adjust insulin delivery (using transient periods of glucose concentration above or below set point) to restore steady-state glucose in the case of changes in insulin sensitivity or endogenous glucose production (result shown using computer simulations based on the Bergman minimal model) (16). Similar adaptation of the β-cell was demonstrated in rats undergoing chronic glucose infusion, whereby an increase in basal insulin secretion was observed without change in glucose (31). The hypothesis is also consistent with earlier observation in humans that adiposity affects fasting insulin concentration independent of fasting glucose (32).

In the PID algorithm, the integral component creates the slow rise in second-phase insulin observed during hyperglycemic clamps (21). In the current study, the integral component maintained an elevated insulin delivery rate after meals, even though near normoglycemia was established (Figs. 1 and 3). The need for elevated insulin delivery was likely caused by a prolonged rate of appearance of glucose after the meal, which has been shown in canines receiving basal insulin alone to lead to elevated plasma glucose levels for 16–20 h (33). Although the maintenance of target glycemia during periods of elevated glucose appearance is desirable, it can be expected that once glucose appearance returns to normal, the elevated value of the integral component will result in a plasma glucose excursion below target (a consideration when deciding to use a high 6.7-mmol/l target rather than a more physiological 5.5 mmol/l).

The reference gain used here was set in direct proportion to open-loop total daily insulin requirement (TDD) in an attempt to keep the product of K_P and insulin sensitivity constant. Underlying this approach is the hypothesis that the β-cell has a similar relationship between insulin secretion and insulin sensitivity—the so-called disposition index (34,35)_. It is not clear, however, whether the “reference K_P” was optimal or whether TDD is a suitable surrogate for insulin sensitivity. The 50% increase in K_P did produce a lower meal excursion without postmeal hypoglycemia (Fig. 1), and it could be argued to be a more optimal gain. However, at this gain, postprandial glucose had a tendency to undershoot/overshoot in some individual experiments, suggesting that a further increase in gain might lead to an oscillatory glucose response.

Also affecting stability was the choice of T_D and T_I. During a hyperglycemic glucose clamp, T_D determines the amount of rapidly released first-phase insulin, and T_I determines the rate at which the second-phase response linearly increases (16). For human subjects, we have estimated β-cell secretion to be well described with values of ∼40 and 100 min (T_D and T_I, respectively) (16). Here, the amount of insulin delivered as first phase was substantially increased (T_D = 66 min), and the second-phase rate of increase was substantially slower (T_I = 150 min). Although these changes were made to compensate for the delays inherent in subcutaneous insulin delivery, they may not be optimal.

Optimization of controller parameters (K_P, T_I, and T_D), can likely only be achieved using a metabolic model (18). We attempted to describe the gain dose response with the minimal model of insulin action (36) and meal models proposed by Hovorka et al. (37) and by Lehmann et al. (38). However, the large change in AUC_GLC observed as the gain was increased from 0.5 × TDD to TDD, versus the much smaller change observed as the gain was further increased to 1.5 × TDD (Fig. 1), could only be fit by introducing nonlinearities, either in insulin-dependent gastric carbohydrate extraction or in minimal model metabolic parameters (data not shown). Support for both mechanisms exists in the literature. Splanchnic glucose uptake (39) and the amount of ingested carbohydrate that is extracted by the gut (37,40) have both been argued to be affected by plasma insulin. Conversely, nonlinearities in insulin sensitivity and glucose effect have also been used in different metabolic models (rev. in 41). Differentiating between these two mechanisms and validating the resulting model describes the reference gain (TDD in this study) response, and perturbations in gain (0.5 × TDD and 1.5 × TDD) will be essential if a model-based optimization of control parameters (K_P, T_I, and T_D) is to be performed. To differentiate between the two mechanisms, it is likely that closed-loop experiments, using glucose tracers, will need to be performed.

Although we were not successful in describing the complete closed-loop dose response, plasma insulin kinetics were well described by the simple two-compartment approximation (r² >0.77) (Fig. 4). Time constants estimated from this approach suggest peak absorption times between 40 and 44 min, similar to the peak time observed in humans (42,43). Insulin clearance was not affected by changes in insulin delivery (P > 0.05) (see results), and no systematic residuals were observed in the fit of the simple two-compartment model, suggesting that a more complicated insulin absorption model (20,44,45) is unnecessary. The ability to describe plasma insulin concentration during closed loop allows for an insulin feedback term to be added to the PID β-cell model. This would allow the putative inhibition of β-cell insulin secretion by plasma insulin (46–48) to be added to the artificial β-cell algorithm used here.

In summary, automated closed-loop insulin delivery, based on a simple PID model of β-cell secretion combined with subcutaneous insulin delivery, results in stable meal response for a wide range of gain. This suggests that the proposed closed-loop system can adjust to day-to-day variability in metabolic processes. Further optimization of algorithm tuning, either by adjusting the relative ratios of the different insulin delivery phases or by inclusion of insulin feedback, may lead to improved meal responses. Optimization is likely to be facilitated by using a more comprehensive metabolic model together with glucose tracer information.

This study was supported by National Institutes of Health Research Grants DK57210 (to K.R.) and DK64567 (to G.M.S.).

The authors thank Natalie Kurtz, Vera Stafekhina, and Gayane Voskanyan for their contribution to the experiments and subsequent laboratory analysis.