In this section, we evaluate the “hybrid” SA-LQR algorithm using a computer simulation environment based on the oral glucose “meal model” of [17
] equipped with a population of 100 in silico
adult patients with T1DM, focusing particularly on the controller’s ability to anticipate a breakfast meal without endangering the patient if the meal is skipped. Because the meal model parameters were developed under the assumption of a fixed “mixed” meal composition, we assume that meals are composed of a fixed percentage of protein, fat, and carbohydrates (45% carbohydrate, 15% protein, and 40% fat). The controller that we test is derived from an instance of the model of Eq. (18)
, representative of an “average” patient with T1DM. In keeping with the discretization used to develop the model, our controller operates by sampling glucose (via CGM) every 15 min and computing insulin infusion rates via Proposition 1, holding the rate constant between samples. In Section 4.3, we show illustrative results for a representative subject with sensor noise. In Section 4.4, we show aggregate results for the population of 100 in silico
subjects. All simulations are conducted with sensor noise, where we use the CGM model proposed in [19
], feeding the corrupted CGM data into the controller. Section 5 compares the SA-LQR strategy to an open loop conventional therapy approach to insulin delivery.
4.1. Experimental scenario
The in silico experiments presented here reveal performance characteristics of the controller in a protocol focused on the anticipation of the breakfast meal. The protocol runs from midnight to 12:30 p.m. and includes one meal at time 420 min (7 a.m.). The breakfast meal, if it arrives at all, will take place at minute 420 of the protocol and will involve 55 g CHO.
4.2. Profiles for the breakfast meal
shows the meal profile information used in the backwards recursion of Proposition 1 in planning insulin delivery prior to breakfast. Note that the profile for breakfast describes an earliest-possible meal arrival time of 240 min and a latest possible meal time of 525 min. The relative frequency distribution fk within this interval is derived from a clipped Gaussian distribution centered at the most likely time for breakfast, 420 min. The standard deviation of the Gaussian distribution before clipping is 30 min. The distribution fk is normalized to achieve different probabilities of skipped breakfast p(skip), as shown in the table, ranging from .0001 (where the meal is almost certain to arrive) to .5 (where the meal arrives or not with equal probability).
Meal profile data for the experiments.
The breakfast meal regime starts at minute 0, and at this time uses the backwards recursion of Proposition 1 to compute a feedback gain matrix K
and set of feedforward gains k
for the 15 min sampling intervals k
through either (1) the last possible time of the meal, if the meal has not arrived, or (2) the time of the breakfast meal arrival, at which time the insulin bolus is delivered. For the purpose of comparison, following (1) the last possible time of the meal or (2) the time of meal arrival and delivery of the open loop bolus, we return the subject to basal rate designed to hold the subject at a blood glucose concentration of 112.5 mg/dl. While this scenario only presents the anticipation and arrival of one meal, another meal regime (lunch, in this case), begins either (1) at the stage following last possible time of the breakfast meal regime in the event that breakfast is skipped, or (2) at the stage just following arrival of the breakfast meal.
4.3. Case study results for a representative subject
Here we illustrate the SA-LQR controller as applied to a representative subject within the in silico population of adult patients with T1DM, using the experimental scenario and meal profile information above. Of course, performance of the controller will depend strongly on the accuracy of the sensor. illustrates the case where the meal arrives at minute 420, with the top plot showing the rate of insulin delivery (U/h) and the bottom plot showing BG (mg/dl). shows the plots of insulin delivery rate and BG for the same subject in the case of a skipped meal, where we focus on the anticipatory insulin effect.
Representative subject: meal taken.
Representative subject: meal skipped.
Notice that in either scenario, whether breakfast is taken or skipped, a profile with p(skip) = .5 injects less insulin in anticipation of the meal. Because there is high level of uncertainty associated with the meal occurrence, there is less anticipatory insulin delivery to guard against a skipped meal scenario. Thus in the case when p(skip) = .5 and the meal is in fact skipped, the subject is able to maintain a safe level of blood glucose, while in the case of p(skip) close to zero, the subject experiences a lower glucose minimum and is at greater risk of hypoglycemia. In any case, there is some level of anticipatory insulin delivery which precedes the meal bolus delivery. gives the meal bolus amounts for the breakfast meal when it does arrive.
Meal bolus insulin amounts (U).
Referring to , in the case where p(skip) = .5 and the meal does occur, the subject experiences the largest meal bolus delivery at meal time, to compensate for insulin that was not delivered in anticipation of the meal; that is, IOB assessment is smallest for the case of p(skip) = .5 at the time of meal arrival. In the case of p(skip) close to zero, a more aggressive anticipatory insulin delivery results in a larger IOB value at the time of meal arrival, thus yielding the smallest meal bolus at mealtime. Note that the insulin trace of (for the case the that breakfast meal is skipped) shows a gradual decline at minute 450 lasting to the end of the breakfast meal regime, which can be explained by the fact that the controller recognizes that the most likely time of the meal has passed, and the chance of the breakfast meal occurring after this most likely time gradually decreases until minute 525, the last possible arrival time for breakfast.
The above representative subject results suggest that, by designing the controller around a meal profile with a significant probability that the meal will be skipped, we can obtain control postprandial performance without the threat of severe hypoglycemia when the skipped-meal event actually occurs. In the case of a skipped meal, we are considering the most extreme scenario. We hypothesize that the case of a delayed meal would lead to results which fall somewhere between the skipped meal and the nominal scenario (when the anticipated meal amount is taken at the most likely time). Alternatively, if a meal is taken which is not accounted for in our behavioral profile, the linear quadratic Gaussian controller will inject insulin in reaction, as opposed to anticipation, to an increase in the blood glucose level.
The results of Section 4.4 confirm this observation for an in silico population of 100 adult patients with T1DM.
4.4. Population study results
Here we illustrate the SA-LQR controller applied to the full in silico population of adults with T1DM, again with sensor noise. summarizes the results of the study reporting max and min glucose concentrations from the onset of the protocol to just following the postprandial peak of the breakfast meal (focusing on controller performance leading up to and just after breakfast).
Results of the population study
Looking at , we see that for larger values for p(skip), the controller delivers less insulin in anticipation of the meal on average, a strategy that protects the subject from hypoglycemia in the event that the meal is skipped. This can be seen in the mean glucose minimum of 90.81 mg/dl in for p(skip) = .5. Because we compensate for a less aggressive anticipatory insulin delivery by delivering a larger meal bolus at meal time (when the meal arrives) in the case of p(skip) = .5, we avoid a higher postprandial peak that could result from this weaker anticipatory insulin delivery. Alternatively, when we consider the results for p(skip) close to zero, results indicate that when the meal is skipped, a greater risk of hypoglycemia results (a mean glucose min of 62.22 mg/dl). A p(skip) close to zero represents an essentially deterministic meal profile, where we control with near certainty the timing and the presence of the meal. Because the controller anticipates that a meal will occur with near certainty, this skipped meal poses an unexpected lack of meal disturbance for the controller. From this population study, we can infer that some optimal value of p(skip) exists so as to control both hypoglycemia (in the event that no meal occurs), as well as hyperglycemia (when a meal does occur). By viewing the probability of a skipped meal in this way, p(skip) is a parameter that can be tuned to optimize control. From the perspective of hypoglycemia avoidance and safety, we choose a larger p(skip) when there is uncertainty as to whether or not a subject will eat in a given regime. Thus choosing a larger p(skip) allows for us to control in a safe way while also taking advantage of our ability to anticipate the arrival of a meal to improve postprandial performance.
presents population outcomes using the Control-Variability Grid Analysis technique of [20
]. The plot shows each patient’s peak post-breakfast glucose concentration (given that breakfast arrives) plotted against the minimum glucose concentration associated with a skipped breakfast, for p
) = .0001 (blue) versus p
) = .5 (pink). (Each patient appears twice, once as a pink dot and once as a blue dot.) Despite the uncertainty about whether breakfast will arrive, a significant fraction of the population lands in the A-zone of the plot where peak glucose is below 180 mg/dl (after breakfast) and above 80 mg/dl (when breakfast is skipped).
Fig. 3 CVGA results for 100 adults comparing p(skip) = .0001 (blue) to p(skip) = .5 (pink); for p(skip) = .0001: (B) 22%, (C) 74% and (D) 4%; for p(skip) = .5: (A) 46% and (B) 54%. (For interpretation of the references to color in this figure legend, the reader (more ...)