Abstract

Introduction

Intensive insulin therapy in type 1 diabetes mellitus (T1DM) is based on the well-established concept of separate management of basal and bolus insulin. Basal insulin aims at controlling glucose levels between meals and overnight, while bolus insulin aims at controlling postprandial glucose. In the case of multiple daily injections, this is implemented with the use of long-acting insulin analogues for basal insulin, expected to produce a flat profile of plasma insulin concentration throughout the day, and rapid-acting insulin formulations for bolus insulin, expected to have a fast onset of action to overcome postprandial hyperglycemia. In the case of insulin pump therapy, the same rapid-acting insulin is used for both basal and bolus. In this case, a more intelligent use of basal infusion is done with time-varying profiles throughout the day to account for circadian variations of a patient's insulin sensitivity. Also, bolus insulin infusion comes in different forms (e.g., standard, extended/square wave, combination/dual wave) to deal with different types of meals or other circumstances. The reader is referred to the work of Chase et al.¹ for a comparison of different administration modes on postprandial glucose control. Bolus insulin advisors can also be found in current pumps, which calculate bolus insulin dose based on patient's insulin-to-carbohydrate ratio (I:C), correction factor, and insulin on board.²

Insulin pump technology allows richer insulin delivery profiles than those currently used. In this work, the limits of performance of the current strategy used for postprandial glucose control are analyzed in an in silico study with the use of interval analysis (IA) techniques.^3–5 Interval analysis arose in the context of numerical analysis, and the study of computational errors originated by the finite precision of computers. Presently, it is a mature technology to compute the range of a function with uncertain parameters in a mathematically guaranteed way, i.e., the actual solution will be contained in the resulting interval. This property is what makes IA a powerful tool when guaranteed results are sought.

In particular, in this work, the analysis of limits of performance is formulated as a set inversion problem. Given a meal, the feasible set of basal infusion rates and bolus doses that mathematically guarantee postprandial response for a T1DM in silico subject fulfilling the International Diabetes Federation (IDF) guidelines⁶ for postprandial control in T1DM in a 5 h time horizon is computed. It is demonstrated that independent setting of basal and bolus insulin, as currently done, dramatically limits the attainable performance, which is far superior when coordination takes place. This raises the hypothesis that the definition of a “preprandial basal” in combination with bolus insulin may yield a tighter postprandial glucose control in insulin pump therapy. These results support the heuristics of the super bolus concept introduced by Walsh and Roberts^7,8 as a new insulin delivery mode in future smarter insulin pumps, which advocates the administration of future basal insulin at the bolus time to compensate hyperglycemia quicker. This concept is also applied in the acute treatment of episodic hyperglycemia.

Methods

The set inversion via interval analysis (SIVIA) algorithm⁹ has been applied to obtain the feasible set of basal infusion and bolus dose that, for a given meal and initial normoglycemic conditions, mathematically guarantee a postprandial response fulfilling the IDF guidelines (i.e., no hypoglycemia and 2 h postprandial glucose below 140 mg/dl) in a 5 h time horizon for an in silico subject. Hypoglycemia is not explicitly defined in the IDF guidelines. It will be considered here as a glucose value below 70 mg/dl. A lower bound of 0.05 international units of insulin (IU)/h for basal insulin infusion is considered throughout the study based on current insulin pump resolution.¹⁰ However, it must be considered that such a low rate may increase occlusions of infusion sets.

Set inversion via interval analysis is a standard algorithm to solve set inversion problems, i.e., given a set Y in an output space and a map f, a set X in an input space composed of the variables of interest is sought so that f(X)Y. The set Y will be determined, in general, by a list of constraints onto the output space. In the problem under consideration, the output space is postprandial glucose in a 5 h time horizon, and the constraints are given by the IDF guidelines for postprandial glucose control. The input space is the two-dimensional space given by the basal infusion rate (IU/h) and bolus insulin dose (IU). The map f corresponds to a glucoregulatory model that maps basal and bolus insulin onto a glycemic response, given meal size and initial conditions. The Hovorka model,^11,12 with mean parameters and 70 kg body weight, has been used for this purpose. This mathematical model describes the processes of insulin pharmacokinetics and pharmacodynamics, carbohydrate absorption, and glucose transport and elimination. A proportionality factor has been introduced in the insulin-sensitivity parameters to produce in silico subjects with different I:C. Rapid-acting insulin is considered in this model.

Set inversion via interval analysis combines branch-and-bound techniques with IA to build an approximation of the solution set X by means of two subpavings (a set of nonoverlapping boxes or rectangles in the two-dimensional case): the inner subpaving, contained in the actual solution set, and the outer subpaving, which contains the actual solution set (see Figure 1 for an illustration). The range of f for a given box in the input space is computed by means of IA and compared with the constraints. As a result, the input box is classified as true, false, or indeterminate (see Figure 2). Indeterminate boxes are halved along the larger dimension, and the procedure is repeated for each sub-box until some minimum width is reached (tolerance parameter). All the true boxes constitute the inner subpaving. The outer subpaving consists of the true and indeterminate boxes. An initial box must be supplied to the algorithm consisting of the domain of interest.

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Figure 1.

Illustration of inner and outer subpaving for a patient with 1:14 I:C and a 60 g meal. The inner subpaving, corresponding to the collection of rectangles in blue, is guaranteed to contain only solutions of the posed constraint satisfaction problem. Any combination in the blue zone will thus lead to fulfillment of the IDF guidelines in this case. The outer subpaving is given by the blue and red rectangles. Its complementary set (in white) is guaranteed to contain only nonsolutions of the constraint satisfaction problem. Any combination from this set will violate the IDF guidelines in this case. The behavior of combinations in the boundary given by the red rectangles (set difference between outer and inner subpavings) cannot be known a priori. This boundary contains the boundary of the actual solution set X and can be made as thin as wanted with higher computational cost. A lower bound of 0.05 IU/h is considered for basal insulin rate.

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Figure 2.

Basics of the SIVIA algorithm: a box in the input space is classified as true if its image under the map f fulfills the constraints defined on the output space for all its elements (in blue); a box is classified as false if its image violates the constraints for all its elements (in white); and a box is classified as indeterminate otherwise. An indeterminate box will be divided into two boxes, with the process repeated for each sub-box.

To compute the range of the map f for an input box, an interval simulator of the Hovorka model has been developed.¹³ Given uncertain inputs, initial condition, and parameters, the interval simulator is able to produce guaranteed upper and lower envelopes of all possible glucose trajectories. In this case, the input to the simulator will be a box from the input space. If both the upper and lower envelopes fulfill the constraints, the input box will be tagged as true (Figure 3A). If both envelopes violate the constraints, it will be tagged as false (Figure 3B). Otherwise, it will be an indeterminate box.

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Figure 3.

Computing the inner and outer subpaving: (A) a box in the input space is classified as true if all the possible glucose responses for all basal–bolus combinations in the box, computed by an interval simulator of the Hovorka model, fulfill the constraint, and (B) a box is classified as false if all the possible glucose excursions violate the constraint as determined by the interval simulator. In any other case (some trajectories fulfill while some others do not), the box is indeterminate. The results shown correspond to a patient with 1:14 I:C after a meal of 60 g of carbohydrates and initial normoglycemia (90.82 mg/dl). International Diabetes Federation guidelines constraints are superimposed in red.

As a result, any basal–bolus combination contained in the inner subpaving is thus guaranteed to fulfill the IDF guidelines in a 5 h horizon for the considered in silico subject and conditions, while any basal–bolus combination outside the outer subpaving is guaranteed to violate the imposed constraints (see Figure 4) according to the patient's model. It is considered here that bolus insulin is given at mealtime, and basal insulin is kept constant during the 5 h time interval. It can be observed in Figure 4A that combinations close to the boundary yield glucose profiles also close to the constraint and are very fragile. The top boundary of the feasible set is associated with the no hypoglycemia constraint, while the bottom boundary is associated to the 2 h (and onward) glucose concentration below 140 mg/dl. Combinations maximizing the distance to the boundary will thus be better selections, producing responses far from both constraints and therefore more robust. Figure 4B shows how combinations outside the outer subpaving always lead to the violation of the constraints.

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Figure 4.

Postprandial glucose in a 5 h time horizon for a patient with 1:14 I:C after a meal of 60 g of carbohydrates, initial normoglycemia, and different basal–bolus combinations contained (A) in the inner subpaving and (B) outside the outer subpaving. The graph on the left shows the basal–bolus combinations selected. The graph on the right shows the corresponding glucose time profiles coded by color. International Diabetes Federation guidelines constraints are superimposed in red.

A tolerance parameter of 0.05 has been used for both the basal rate and bolus insulin, according to current insulin-pump resolution.¹⁰

Results

Feasible sets of basal and bolus insulin have been computed for meals of 30, 40, 50, 60, 70, and 80 g of carbohydrates and initial normoglycemia. Nominal basal and I:C have been estimated for the in silico subject (Hovorka model with mean parameters, 70 kg body weight, sensitivity proportionality factor 0.5). Nominal basal is taken as the basal infusion normalizing glucose around 100 mg/dl. Insulin-to-carbohydrate ratio is estimated through simulations of the response to a three-meal day.

To ease comparison among strategies, the obtained subpavings have been normalized by the nominal basal (0.8 IU/h) and nominal bolus as computed from the meal size and the estimated I:C (1:14). Therefore, the point (1,1) corresponds to the standard strategy. By drawing a vertical line through the point (1,1), performance for other estimations of I:C can be analyzed while keeping basal infusion constant to the nominal value. This allows us to analyze the best attainable performance of the standard strategy, where basal does not contribute to postprandial glucose control.

To illustrate the total insulin requirements, solutions corresponding to the same total insulin as the standard procedure and an excess/defect of 10% the nominal bolus have also been computed (isolines). Any basal–bolus combination along an isoline will lead to the same total amount of insulin delivered. Figure 5 illustrates the results obtained.

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Figure 5.

Normalized feasible sets for a meal of 30 to 80 g of carbohydrates and initial normoglycemia. It is considered that bolus insulin is given at mealtime, and basal is kept constant for 5 h. All basal–bolus combinations in the blue region will lead to the fulfillment of the IDF guidelines in a 5 h period, according to the model. Combinations along the solid green line will lead to the same total insulin delivered as the standard strategy. Dotted green lines correspond to total insulin with an excess/defect of 10% the nominal bolus.

Discussion

Nominal basal and bolus are not able to get a post-prandial glucose control according to the IDF guidelines since (1,1) does not belong to the feasible set for any computed meal. Although for 30 and 40 g meals, a higher bolus would have yielded a feasible solution for this in silico patient [the vertical line across (1,1) intersects the inner subpaving]. This is not true for meals above 50 g of carbohydrates. In these cases, no intersection exists. This means that, whatever the infused bolus, the constraints will be violated. Computed feasible sets indicate that the limiting factor is the nominal basal insulin infused independently to cover basal glucose. If basal insulin infusion is reduced and bolus insulin is increased accordingly, with respect to their nominal values, IDF guidelines can be achieved. This will be more difficult as the meal size increases, as indicated by the reduction of the volume of the feasible set. Should the computed outer subpaving be the empty set, it is guaranteed that no solution exists and constraints must be relaxed.

Thus, coordinated action of basal and bolus insulin may be an important factor to get tighter postprandial glucose control. This introduces the concept of “preprandial basal.” At mealtime, nominal basal is reduced by a given percentage for a time interval while bolus insulin is increased to compensate. This decrement of basal insulin in favor of bolus insulin allows for more adequate compensation of postprandial hyperglycemia, especially as meal size increases. This is evident from the computed feasible sets, where for 80 g carbohydrates, meal basal should be reduced at least by 50% to obtain a solution.

More robust solutions will be those centered in the feasible set yielding glucose responses far enough from the constraints. This can be achieved, as observed in Figure 5, with an increase of total insulin requirement of 10% the nominal bolus (upper green dotted line) for the considered in silico subject. This is the case for every meal. For 30 g, a delivery of 50% nominal basal is required, 40% for 40 g, 30% for 50 g, 20% for 60 g, and 15% for 70 and 80 g. This must be compensated with an increase of bolus insulin around 50–60% with respect to its nominal value for each meal. These figures will, of course, be specific to an individual patient. Keeping total insulin unchanged may also yield sensible solutions (solid green line in Figure 5) and be easier to implement from the clinical perspective. Although, in this case, constraints will not be strictly fulfilled, solutions close to the bottom boundary of the feasible sets can be obtained, which means that 2 h glucose may be a bit higher than 140 mg/dl, but no hypoglycemia is expected to happen since the distance of the isoline to the upper boundary, which may be considered a measure of the risk, is quite steady.

These results support the heuristics of the super bolus concept, introduced by Walsh and Roberts⁶ as a new insulin delivery mode for future smarter insulin pumps. It consists of the administration of future basal insulin at the bolus time to compensate hyperglycemia faster. According to Figure 5, a super bolus would thus correspond to any point along the isoline depicted in solid green, since total insulin is kept constant. The computed feasible sets provide a theoretical justification of its claimed effectiveness.

It must be kept in mind that IDF guidelines were imposed in a 5 h time horizon. The effect of decreasing postmeal basal must be investigated for a longer duration than 5 h since it might cause rebound hyperglycemia. Hence the time at which to restore basal to its nominal value must be decided to avoid this phenomenon. Figure 6 shows the long-term effect of the duration of the basal insulin decrement on glycemia. It is observed that decreasing basal for the 5 h considered in the feasible sets computation will lead to an undesired rise of glucose after that time. On the other hand, restoring basal to its nominal value 2 h postmeal yields a response close to hypoglycemia. A value of 3 h leads to a good performance. The reasons for this behavior become evident when plasma insulin profiles are investigated (Figure 7). By decreasing basal insulin for 5 h, a plasma insulin concentration of less than half the nominal value is reached. This insufficiency of insulin is reflected, due to the delay of insulin action, approximately 3 h later in a glucose peak. On the other hand, decreasing basal insulin for 2 h produces an excess of insulin due to the superposition of the bolus and restored basal contributions to plasma insulin, reflected in a slower convergence to its nominal value. On the contrary, restoration of nominal basal after 3 h leads to a sharp convergence to nominal plasma insulin with minimal undershoot. At 4 h after the meal, plasma insulin is already at baseline. When compared to the standard therapy (nominal basal and nominal bolus computed from the I:C), this takes places approximately 3 h before, with a higher plasma insulin peak. This allows for compensating post-prandial hyperglycemia without posterior undesirable effects. How long the basal rate can be reduced will depend, of course, on the pharmacodynamic properties of the considered insulin.

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Figure 6.

Effect of the duration of the decrement in basal insulin for postprandial control in a 24 h simulation. The nominal basal is 0.8 IU/h, and the I:C is 1:14. A meal of 60 g of carbohydrates is given at time zero. According to the computed feasible set, a bolus 55% greater than the standard is infused simultaneously, and basal is decreased to 20% the nominal value for 2 to 5 h. Constraints considered in the calculation of the feasible sets are superimposed as a thick red line.

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Figure 7.

Plasma insulin concentration under the conditions described in Figure 6. Plasma insulin concentration for the standard therapy (nominal basal and nominal bolus) is also shown as a dotted line.

Figure 8 shows the performance of the new strategy in a three-meal day, as compared to the standard one. A breakfast of 40 g and a lunch and dinner, both of 60 g of carbohydrates, have been considered in the simulation. Coordination of basal and bolus insulin has been derived according to the computed feasible sets in Figure 5. Note that these feasible sets correspond to initial condition at equilibrium (normoglycemia), which cannot be guaranteed at every meal due to past actions. However, the selection of centered solutions in the feasible sets will allow deviations from the equilibrium, and the method behaves robustly as observed in the results.

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Figure 8.

Comparison of the coordinated basal–bolus and the standard strategy in a three-meal scenario. Breakfast is given at 7:00 (40 g of carbohydrates), lunch at 12:00 (60 g), and dinner at 20:00 (60 g). The top figure shows the obtained glucose profiles for the standard strategy (in red) and the new strategy with basal–bolus coordination (in blue). Meal events are indicated with green triangles. Imposed IDF guidelines are shown as a thick red line. Figures in the middle and at the bottom show the basal infusion profile and bolus insulin dose, respectively. A nominal basal of 0.8 IU/h and a I:C of 1:14 are used for the standard strategy. For basal–bolus coordination, the following selection has been done: nominal basal × 0.4 and nominal bolus × 1.6 for breakfast and nominal basal × 0.2 and nominal bolus × 1.55 for lunch and dinner.

Basal–bolus coordination outperforms standard therapy and allows fulfillment of the IDF guidelines for the three meals, contrary to the standard strategy as summarized in Table 1. A mean peak reduction of 18.1 mg/dl and a mean 2 h postprandial glucose reduction of 36.4 mg/dl are achieved by coordination. The lowest glucose value in the 24 h profile for the standard strategy is 85 mg/dl versus 78.51 mg/dl for the coordinated one. The main impact of basal–bolus coordination is the 2 h glucose value, producing a faster compensation of postprandial hyper-glycemia without producing hypoglycemic episodes. As a result, tighter postprandial glucose control is achieved.

Conclusions

Set inversion methods based on IA have been used to analyze the limits of performance of current insulin pump strategy for postprandial glucose control in an in silico study. International Diabetes Federation guidelines of no hypoglycemia and 2 h postprandial glucose below 140 mg/dl have been set as ideal postprandial performance, and the set of basal insulin rates and bolus insulin doses fulfilling these constraints in a 5 h time horizon have been computed (feasible sets). These sets demonstrate that the current strategy for considering independent roles for basal and bolus insulin is not optimal and dramatically limits the performance. A coordinated use of basal and bolus insulin for postprandial control results in significantly improved performance, allowing fulfillment of the IDF guidelines in a three-meal simulation, contrary to the standard strategy. A decrement of basal insulin rate with respect to nominal value is required (“preprandial basal”) together with an increase in the bolus dose, with respect to the nominal value obtained from the patient's I:C. Nominal basal must be restored to its nominal value after 3 h. This allows a faster delivery of the insulin required to compensate postprandial hyperglycemia with little impact after postprandial period. Clinical trials must be carried out to validate this new approach.

The application of IA for the computation of feasible sets is demonstrated to be a powerful tool for the analysis of attainable performance in glucose control.

Abbreviations

References