Control Limitations in Models of T1DM and the Robustness of Optimal Insulin Delivery

Abstract

Background:

In insulin therapy, the blood glucose level is constrained from below by the hypoglycemic threshold, that is, the blood glucose level must remain above this threshold. It has been shown that this constraint fundamentally limits the ability to lower the maxima of the blood glucose level predicted by many mathematical models of glucose metabolism. However, it is desirable to minimize hyperglycemia as well. Hence, a desirable insulin input is one that minimizes the maximum glucose concentration while causing it to remain above the hypoglycemic, or higher, threshold. It has been shown that this input, which we call optimal, is characterized by glucose profiles for which either each maximum of the glucose concentration is followed by a minimum or each minimum is followed by a maximum.

Methods:

We discuss the implication of this inherent control limitation for clinical practice and test, through simulation, the robustness of the optimal input to a number of different model and parameter uncertainties. We further develop guidelines on how to design an optimal insulin input that is robust to such uncertainties.

Results:

The optimal input is in general not robust to uncertainties. However, a number of strategies may be used to ensure the blood glucose level remains above the hypoglycemic threshold and the maximum blood glucose level achieved is less than that achieved by standard therapy.

Conclusions:

An understanding of the limitations on the controllability of the blood glucose level is important for future treatment improvements and the development of artificial pancreas systems.

Recent research has focused on the development of artificial pancreas systems¹ for the management of type 1 diabetes. Due to the narrow and variable therapeutic range and slow response time of insulin, reliable and accurate models of glucose dynamics are essential to the development and operation of such systems. In addition, understanding and modelling of the dynamics of glucose regulation, will lead to further treatment improvements, such as timing of insulin delivery and the avoidance of hypoglycemia.

A number of models of glucose regulation have been proposed.^2-4 Each is typically comprised of subsystems describing different physiological processes such as insulin kinetics and glucose absorption. However, despite the recent focus on the modelling of glucose metabolism not much work has been done exploring the limitations on glucose regulation that arise from these models.

The available research has focused on comprehensive models of glucose dynamics which are generally preferred to test treatment policies and control algorithms, for example the UVA/PADOVA type 1 diabetes simulator.⁵ Typically, these models are high order dynamic systems with many parameters to ensure robustness to interindividual variability. However, simpler models are useful to establish general theoretical properties that would otherwise be difficult to investigate analytically. Indeed, most models of glucose dynamics share certain mathematical properties—such as meals having a positive effect on the plasma glucose. Thus analytic results obtained for simpler models can give insights into the behavior of more comprehensive models.

The Bergman minimal model⁶ is a simplified model of glucose metabolism frequently used for virtual patient simulations and as the basis of more comprehensive models such as the Kanderian model⁷—considered by Goodwin et al,⁸ the Fabietti model,⁹ and the extensions of Roy and Parker.^10,11

The control limitations explored by Townsend et al^12,13 contribute to the theoretical understanding of the Bergman minimal model by characterising the magnitude, delivery time, and duration of insulin bolus inputs that give the lowest maximum glucose concentration while avoiding hypoglycemia. A detailed explanation of these results is given in this paper after presenting some mathematical preliminaries. These results stem from the fact that the desired blood glucose level is effectively constrained from below by the hypoglycemic threshold. However, due to the complexity of glucose and metabolic regulation most models do not always accurately predict blood glucose responses but remain within a range of actual blood glucose responses. This can be accounted for by assuming the existence of some model uncertainty the implications of which are investigated in this article.

The effect of a fixed constraint on the control of plasma glucose concentrations was also investigated by Goodwin et al⁸ who use a model derived from the Bergman model, to propose a nonlinear insulin bolus dosing algorithm. However, the bolus is constrained to be an impulse applied contemporaneously with an impulsive food input. This is a specific example of the cases considered by Townsend et al.^12,13

Here, we provide a nonmathematical explanation of the technical results of Townsend et al^12,13 together with a number of examples illustrating the glucose response to some typical disturbances. In addition, we investigate, through simulation, the robustness of the glucose response to uncertainty in the ingestion time, uncertainty in the quantity of a meal, and uncertainty in the response time of insulin. We compare the glucose response of Townsend et al^12,13 with that produced by current treatment policies. Finally, we provide recommendations based on the results of this work and Townsend et al^12,13 and reflect on implications of these results on clinical practice.

Mathematical Preliminaries

The Bergman minimal model,^2,10 is a nonlinear continuous-time model comprising a set of first order linear ordinary differential equations which govern the concentration and effectiveness of insulin:

$$\frac{d}{dt}{I}_{sc}\left(t\right)=-\frac{1}{{\tau }_1}{I}_{SC}\left(t\right)+\frac{1}{{\tau }_1}\frac{ID\left(t\right)}{C_l}$$

$$\frac{d}{dt}{I}_P\left(t\right)=-\frac{1}{{\tau }_2}{I}_P\left(t\right)+\frac{1}{{\tau }_2}{I}_{SC}$$

$$\frac{d}{dt}{I}_{eff}\left(t\right)=-p_2{I}_{eff}\left(t\right)+p_2{S}_I{I}_p\left(t\right)$$

and a nonlinear ordinary differential equation which governs the plasma glucose concentration $g\left(t\right):$

$$\frac{d}{dt}g\left(t\right)=-g\left(t\right).\left(I_{eff}+G\right)+r\left(t\right)+E$$

where $\frac{d}{dt}$ is the derivative with respect to time and:

• $ID\left(t\right),I_{SC}\left(t\right),I_p\left(t\right)$ and $I_{eff}\left(t\right)$ are the delivery, subcutaneous concentration, plasma concentration, and insulin effectiveness, respectively.

• $\frac{1}{{\tau }_1}$and $\frac{1}{{\tau }_2}$are time constants.

• $C_l,S_I$and $p_2$are the clearance rate, insulin sensitivity and the insulin motility.¹¹

• $g\left(t\right)$ is the plasma glucose concentration.

• E and G are the endogenous glucose production and the effect of glucose on the uptake of plasma glucose and the suppression of endogenous glucose production, respectively.

• $r\left(t\right)$ is the glucose absorption from meals.

A variety of physiological values for the above may be found in Table 1 of Goodwin et al.⁸ For notational convenience, we denote the input function by $u\left(t\right)=ID\left(t\right).$ We concentrate on square-wave inputs with a constant basal input as this is typical of basal-bolus management and as insulin pumps are only capable of delivering square-wave inputs. Thus, we assume, throughout, that the insulin input to the model is of the form:

Table 1.

Parameters for Bergman Minimal Model Used in Examples.

Parameter	Value	Unit
$p_2$	$1.6\times {10}^{-2}$	$mi{n}^{-1}$
$S_I$	$9\times {10}^{-4}$	$mL\mu U^{-1}$
${\tau }_2^{-1}$	$2.44\times {10}^{-2}$	$mi{n}^{-1}$
${\tau }_1^{-1}$	$2.5\times {10}^{-2}$	$mi{n}^{-1}$
$C_l^{-1}$	$8\times {10}^{-4}$	$mLmi{n}^{-1}$
E	2	$mgd{L}^{-1}mi{n}^{-1}$
G	$3.19\times {10}^{-3}$	$mi{n}^{-1}$
V	$1\times {10}^{-2}$	dL

$$u\left(t\right)=\overline{u}+\widehat{u}{\chi }_A\left(t\right)$$ (1)

where $\overline{u}$ is a constant basal flow and $\widehat{u}{\chi }_A\left(t\right)$ is a square-wave input of arbitrary duration which we refer to as a bolus. This may represent either a standard bolus, that is, injection or impulse of insulin, or a square wave or extended bolus depending on the duration of the square wave. We take the function r to be a given bounded function, typically, a positive meal response. However, the function r may be negative so long as the assumptions listed below are met.

Despite our focus on the Bergman minimal model the theoretical results discussed here apply to other models of type 1 diabetes, for example, the Hovorka model.¹⁴ Our assumptions are that for the considered model:

1. Glucose decreases with insulin, for example, if two possible boluses are delivered then the blood glucose level that would result from the larger bolus is lower than the blood glucose level resulting from the smaller.

2. If zero insulin is delivered then the blood glucose level does not go below the selected lower bound.

3. All inputs are bounded and the response to external disturbances and the insulin effectiveness eventually decay to their basal states.

More formally for a model of the form:

$$g=f\left(g,u,r,t\right)$$

where g is the plasma glucose concentration, u is the insulin input and r is some bounded function, we assume:

1. g is monotonically decreasing in u.

2. $r\to 0$ as $t\to \infty$.

3. $f\left(g,0,r,t\right)\ge \lambda$ for all t, where $\lambda$ is the desired lower bound.

Throughout, the initial conditions are taken such that the steady-state glucose value is $108mgd{L}^{-1}.$ In all numerical examples the parameters of the Bergman minimal model are taken, unless otherwise stated, to be as in Table 1.

Finally, unless otherwise stated, the meal disturbance function $r\left(t\right)$ is taken to be:

$$r\left(t\right):=V{f}_1\left(t\right)$$

where V is a positive constant representing the distribution volume for glucose equilibria⁸—which can be interpreted as an individual’s “gain” to carbohydrate or food, that is, the number of $mgd{L}^{-1}$ that 1g of carbohydrate raises an individual’s blood glucose level—and is either the response to the system:

$$\left(\begin{array}{c}\frac{d}{dt}{f}_1(t)\\ \frac{d}{dt}{f}_2(t)\end{array}\right)=\left(\frac{1}{40}\right)\left(\begin{array}{cc}-1& 1\\ 0& -1\end{array}\right)\left(\begin{array}{c}{f}_1\left(t\right)\\ f_2\left(t\right)\end{array}\right)+\left(\begin{array}{c}0\\ \delta \left(t\right)\end{array}\right)$$ (2)

Or:

$$\left(\begin{array}{c}\frac{d}{dt}{f}_1(t)\\ \frac{d}{dt}{f}_2(t)\end{array}\right)=\left(\frac{1}{100}\right)\left(\begin{array}{cc}-1& 1\\ 0& -1\end{array}\right)\left(\begin{array}{c}{f}_1\left(t\right)\\ f_2\left(t\right)\end{array}\right)+\left(\begin{array}{c}0\\ \delta \left(t\right)\end{array}\right)$$ (3)

In (2) and (3) the ingestion of the meal is represented by $\delta \left(t\right)=25{\chi }_{\left[100,103\right]},$ that is, is a square wave with a duration of $3\left(min\right)$ and magnitude of $25\left(g\right)$ applied at time 100. Note that all times and durations are given in minutes. This is as ${\chi }_{\left[100,103\right]}$ is a square wave with duration 3 of magnitude 1. The magnitude of $\delta \left(t\right)$ corresponds to the glycemic load of the meal which we take to be the quantity of carbohydrate (CHO), in grams. We refer to (2) as a typical meal—since the glycemic effect of the meal occurs rapidly—and (3) as a slow meal—since the glycemic effect of the meal occurs over a long period. In practice, the typical meal would usually be high in carbohydrate whereas the slow meal may be higher in fat or protein.¹⁵ Plots of the meal disturbance function for both the typical and slow meal are shown in the third plot of Figure 1 by the solid and dashed line respectively. In each simulation we specify which meal disturbance function, typical or slow, is used. As noted above, the presented theoretical results are not limited to the Bergman minimal model nor the specific food absorption dynamics used here, for the purpose of simulation. Indeed, the function, $r\left(t\right),$ may be replaced by any other function, provided it meets the assumptions mentioned above. For example, the food absorption dynamics could be simulated using the food model found in the UVA/PADOVA model.⁵

Figure 1.

The optimal glucose response to the typical, shown by the solid line, and slow meals, shown by the dashed line. The functions of each insulin and food response are shown in the second and third plots respectively. A bolus of magnitude 10.86 IU is delivered at time 67 for the typical response, shown by the solid line, and a bolus of magnitude 10.07 IU is delivered at time 114 for the slow response, shown by the dashed line.

We stress, that the specific magnitude, constants and values chosen for both the insulin and food absorption dynamics are not essential to the results presented here and are selected to best represent the theoretical results of Townsend et al.^12,13 Indeed, the parameters, such as insulin sensitivity, of the model can vary with time. Provided these variations are known a priori the theoretical results of Townsend et al^12,13 still apply. How a posteriori variation in the parameters, that is, model uncertainty, influences these results is investigated in the current article.

Despite this, we note that, with the constants specified in Table 1 the insulin dynamics used here are similar to those of a rapid acting insulin such as Insulin Aspart.¹⁶

Lastly, for completeness, we quote a result, Theorem 20, of Townsend et al¹² which is invoked later in the paper.

Theorem (Theorem 20). Consider the following two cases for the glucose response $g\left(t\right)$:

1. no global maximum of $g\left(t\right)$ occurs after a global minimum.

2. no global minimum of $g\left(t\right)$ occurs after a global maximum.

Fix the bolus delivery time $t^{\prime }.$ Let $u\left(t^{\prime }\tau \right)$ and $v\left(t^{\prime },\sigma \right)$ be two distinct boluses delivered at $t^{\prime }$, with durations $\tau$ and $\sigma$ respectively. Suppose either: u and $v$ satisfy A, u and $v$ satisfy B or u satisfies A and $v$ satisfies B. Then for each respective case:

1. $\gamma \left(u\right)<\gamma \left(v\right)$ if and only if$\tau <\sigma .$

2. $\gamma \left(u\right)<\gamma \left(v\right)$ if and only if$\tau >\sigma .$

3. there exists $\alpha \in \left(\sigma ,\tau \right)$ such that $\gamma \left(m\left(t^{\prime },\alpha \right)\right)<\min \left\{\gamma \left(u\right),\gamma \left(v\right)\right\}.$ Furthermore, for all $\alpha \notin \left(\sigma ,\tau \right)$ the maximum $\gamma \left(m\left(t^{\prime },\alpha \right)\right)\ge \min \left\{\gamma \left(v\right),\gamma \left(u\right)\right\}.$

where $\gamma \left(u\right):=\max \left\{g\left(t,u\right)\right\}$, i.e., the global maximum of the glucose response when the insulin input is u.

Control Limitations and Example

The controllability of the glucose response predicted by models of glucose metabolism meeting the above assumptions was investigated by Townsend et al.^12,13 Specifically, the control objective was to lower the maximum glucose concentration subject to a fixed minimum glucose concentration. In this context, an optimal insulin input gives the lowest possible maximum glucose concentration while avoiding hypoglycemia. The constraint on the minimum glucose concentration is required because the risks associated with hypoglycemia are, generally, far greater than those associated with hyperglycemia. The fact that the minimum glucose concentration is constrained induces a fundamental limitation on the controllability of the maximum glucose concentration when the control input is a square wave. This fundamental limitation also allows the optimality, of the input, to be determined from the shape of the glucose response and specifies the effect of changes to the parameters, that is, the magnitude, input time and duration, of a bolus input on the maximum plasma glucose concentration. This limit can act as metric for the optimality of control algorithms designed for artificial pancreas systems and assist in the determination of bolus guidelines.

The following sections discuss and illustrate the implications of the fundamental limitation on the optimal glucose concentration for different types of bolusing strategies.

Bolus With Fixed Duration and Optimal Application Time

For a bolus of insulin having a fixed duration but for which the application time could be optimized, Townsend et al¹³ showed that two types of optimal glucose concentration responses were possible. That is, the lowest possible maximum glucose concentration is produced, either, by the input for which the glucose response achieves the lower bound before and after the maximum glucose concentration, or by the input for which there are two equal maxima in the glucose concentration surrounding a minimum. Any attempt to further lower the maximum glucose concentration will result in hypoglycemia or require additional control action. The first case of optimal glucose concentration shape, or “min-max-min” shape, is illustrated by the blue solid curve in the top plot of Figure 1. The second case, or “max-min-max” shape, is illustrated by the orange dotted curve in the top plot of Figure 1.

In the examples presented in Figure 1, the solid lines represent the response to the typical meal and the dashed lines the response to the slow meal. For both examples the insulin input is a bolus of the form (1) with duration 1, that is, an impulse. The minimum glucose concentration is chosen to be $81mgd{L}^{-1},$ with the magnitude and timing of the bolus designed for the response to reach the minimum value but not go below it.

The first plot in Figure 1 shows the plasma glucose responses to the function $r\left(t\right),$ that is, the meal responses, and an optimal bolus input. The latter two responses, shown in the second and third plots of the same figure, correspond to the typical and slow meals ingested at time 100—see the description of $\delta \left(t\right)$ after (3)—and insulin boluses delivered at the computed optimal application times for each meal. For the solid line, that is, the typical meal response, the magnitude of the bolus is $10.86IU$ delivered at time 67. Two minima occur at times 106 and 319 bounding the unique maximum which occurs at time 153. For the dashed line, that is, the slow response, the magnitude of the bolus is $10.07IU$ delivered at time 114. Two maxima occur at times 139 and 508 bounding the unique maximum which occurs at time 231.

Figure 2 shows the maximum glucose concentration, and the magnitude of a bolus input, which attains but does not go below the minimum glucose concentration, as a function of the input time for both the typical, solid line, and slow, dashed line, meals. We see that the maximum glucose concentration is minimized at the optimal input times specified above and given by the vertical black lines. Interestingly, the magnitude of the optimal bolus does not necessarily coincide with the maximum possible bolus. This can be particularly seen in the case of the slow meal.

Figure 2.

The maximum of the glucose response to the typical, shown by the solid blue line, and slow meals, shown by the dashed blue line, as a function of bolus delivery time. The magnitude of the bolus, shown by the orange solid, and dashed lines for the typical and slow responses respectively, varies to ensure the glucose response attains but does not go below the minimum glucose concentration chosen to be $81mgd{L}^{-1}.$ The bolus magnitude (IU) is indicated by the axis on the right.

Clinical Implications

This example demonstrates that, for an impulse of insulin, it is better to prebolus to minimize postprandial hyperglycemia from typical meal disturbances. But for slower responses, for example, the slow meal, it is better to postbolus. However, the exact timing, magnitudes, and glucose values reached are heavily dependent on the model and parameters used. As will be explored in the sequel, when the duration of the bolus can be controlled then prebolusing becomes the optimal strategy in all situations.

Bolus With Optimal Duration and Timing

When both the duration and time of administration of a bolus of insulin are controllable, Townsend et al¹² showed that an insulin input is optimal if and only if the resulting glucose concentration response has two equal maxima interlaced between two equal minima. An example of this optimal response is illustrated by the purple dashed curve in Figure 3. Again, any attempt to further lower the glucose concentration will result in hypoglycemia. However, as the duration of a bolus can never be shorter than an impulse, for certain disturbances it is not possible to achieve the optimal shape of two maxima interlaced between two minima. For example, it is not possible to further lower the maximum glucose concentration of the response to the typical meal. This is as the response to the typical meal, when the bolus input is an impulse, has a minimum followed by a maximum followed by a minimum. Thus extending the duration of the insulin input would move the minima further from the maximum resulting in a higher maximum as the peak insulin effectiveness from a longer duration bolus would be lower.

Figure 3.

Selected plasma glucose concentration g(t) profiles for various input durations. The blue, orange, dashed purple, and green lines correspond to the durations 1, 120, 291, and 400, respectively, and each square wave is nested inside the one with the closest higher duration.

An example of this is presented in Figures 3 and 4. In the first plot of Figure 3 the blue, orange, dashed purple and green lines correspond to the glucose response to the slow meal and boluses of durations 1, 120, 291 and 400 respectively. These boluses are shown, in corresponding colors, in the second plot of Figure 3. We note that each square-wave bolus is nested inside the one with the closest higher duration. The magnitudes of each bolus are $10.05,0.09,0.0422,$ and $0.03$, respectively, implying that the total insulin delivered is $10.05IU,10.80IU,10.13IU$and $12.00IU$ for each duration. This shows that the maximum of the blood glucose response does not monotonically decrease with total insulin delivered even when the treatment policy is designed to be optimal for a given duration.

Figure 4.

Maximum plasma glucose concentration as a function of the duration of the insulin input.

As the duration approaches 291, which corresponds to the dashed purple profile of Figure 3, the maximum glucose concentration decreases. This is shown in Figure 4 in which the duration of the insulin input is varied, from a bolus to a square-wave input with a duration of 400, and the maximum of each corresponding blood glucose response is found. As the duration increases from a bolus the maximum blood glucose level decreases monotonically until a duration of 291 at which the maximum blood glucose concentration of the response attains its minimum. The maximum then increases monotonically for durations greater than 291.

Clinical Implications and Recommendations

It should be noted that for typical meal disturbances and current available insulin responses the meal disturbance acts on the plasma glucose much more rapidly than the insulin. Thus, in almost all situations, the optimal input is an impulse of insulin delivered before the ingestion of a meal. This results in a minimum in the plasma glucose followed by the maximum followed by a minimum.¹² However, longer duration square waves may be appropriate in some circumstances. For example, for safety or robustness reasons, as explored in the sequel. Also, meals with longer responses such as those seen from high fat and protein meals, may require the bolus to be supplemented with a square wave of longer duration.^15,17 Although, as can be seen in Figure 3 once the duration of the square wave exceeds the optimal duration of 291 a minimum of the glucose concentration precedes the maximum. This may be used to help identify if the duration of an extended bolus is too long.

Bolus With Fixed Input Time but Optimal Duration

Finally, Theorem 20—quoted above, showed that should the input time of a bolus be fixed but the duration is controllable then a longer duration square wave results in a lower maximum blood glucose level if the blood glucose level attains its lower bound strictly before attaining its global maximum. Analogously, a shorter duration results in a lower maximum blood glucose level if the blood glucose level attains its maximum before its minimum. This accords with the result derived by Goodwin et al,¹⁸ where it was shown for a bolus delivered contemporaneously with a meal disturbance that an impulse produces the lowest maximum blood glucose concentration.

Examples of this are shown for the typical and slow meals in Figures 5 and 6, respectively. In both cases, the input time of the bolus is set to 85 and the upper plots show example glucose responses for the durations 1,68 and 120. For the typical meal, in Figure 5, an impulse gives the lowest maximum glucose concentration as all responses attain their maximum before their minimum. In Figure 6, responses to insulin boluses with durations less than 68 attain their minimum before their maximum, as shown by the dashed blue response. Thus longer durations give lower maximum glucose levels. However, the input with duration 68, producing the dashed orange response, is optimal in that it has two equal maxima about the minimum glucose concentration. Hence this duration minimizes the maximum glucose concentration for the input time 85. All inputs with durations greater than 68 attain their maximum before their minimum, as shown by the dashed green response. Thus increasing the duration of the input increases the maximum glucose concentration.

Figure 5.

Example glucose responses to inputs of different durations and maximum plasma glucose concentration as a function of the input duration for a fixed input time of 85 and the typical meal. The magnitude of the bolus is varied so that the minimum of the blood glucose response attains but otherwise remains above the selected minimum concentration of $81mgd{L}^{-1}.$

Figure 6.

Example glucose responses to inputs of different durations and the maximum plasma glucose concentration as a function of the input duration for a fixed input time of 85 and the slow meal. The magnitude of the bolus is varied so that the minimum of the blood glucose response attains but otherwise remains above the selected minimum concentration of $81mgd{L}^{-1}.$

Robustness and Simulations

The optimal bolus is, in general, not robust to uncertainties in the model, model parameters nor disturbances. This is as it attains the lower bound at least once and for typical meal disturbances, twice. Thus uncertainty or variation in the parameters will cause the blood glucose level to fall below the specified minimum concentration or have unequal global maxima. In addition, as is shown by the orange lines in Figure 2 the magnitude of the bolus can vary significantly around the optimal input time, especially for the typical meal. Hence uncertainty in this time may have a large influence on the total delivered insulin. However, given a range of parameters or disturbances, strategies, such as the following may be employed to ensure the blood glucose level stays above the desired minimum in all cases and is nearly optimal in the most likely case:

1. raising the lower bound or

2. lowering the flowrate of insulin, such as setting $u=0$

3. lowering the preprandial blood glucose level by, for example, increasing the basal flow.

When the input is restricted to be of the form (1) the only alternative is to design the magnitude and timing of the bolus to avoid going below the lower bound for the potential parameter values which would produce the lowest blood glucose level.

However, Theorem 20 may be used to inform the choice of duration of the bolus that will minimize the expected value of the maximum glucose value should all responses, other than the one optimized for, satisfy either of the conditions of the theorem, that is, all maxima occur before all minima or all minima occur before all maxima, as outlined above.

Uncertainty in Meal Timing

A situation in which Theorem 20 would apply would be if there were uncertainty in the timing of the typical meal.

Suppose the timing of the bolus input is chosen such that the latest possible ingestion time, of the meal, produces an optimal response, for example, if the meal may be ingested at any time in the interval from 100 to 250 then the bolus is designed such that the glucose response is optimal if the meal is consumed at time 250. Then the maximum glucose level will precede the minimum glucose level of all other responses, that is, responses for which the meal is consumed earlier. Thus, in this case, an impulse would be optimal.

Analogously, should the timing of the bolus be such that the earliest ingestion time is optimal, for example, if the bolus is designed to be optimal for a meal ingestion time of 100, then the minimum glucose concentration will precede the global maximum for all other responses. Thus a longer duration square wave would produce the lowest expected maximum blood glucose response. An example of this is shown in Figure 7, in which the typical meal has a range of possible ingestion times from 100 to 250. The blue and orange lines show the expected maximum glucose concentration as a function of the duration of the bolus when the timing of the bolus is chosen so that, respectively, an ingestion time of 200 and 100—that is, a later possible and the earliest possible—ingestion times produce an optimal glucose response. The expected maximum in Figure 7 is computed by simulating glucose responses to all integer meal ingestion times in the interval $\left[100,250\right]$ and the bolus of specified duration, that is, for each duration the responses to all meal ingestion times from 100 to 250 are simulated and their maximum blood glucose concentrations averaged to give the blue and orange lines in Figure 7. As expected the longer duration square wave produces a lower expected maximum glucose value for the orange case. In the blue case a square wave with duration 25 gives the lowest expected value. This is as the timing of the bolus is designed to be optimal for a meal ingestion time of 200 and not 250. Thus for durations less than 25 the minimum glucose concentration precedes the maximum glucose concentration.

Figure 7.

The expected maximum glucose value for the typical meal where the ingestion time is within the range 100-250. The orange line is when the timing of the bolus is such that the input is optimal for an ingestion time of 100 and the blue is such that the bolus is optimal when the ingestion time is 200. Note that the delivery time of the insulin bolus varies with the duration of the bolus.

Clinical Implications

It is often uncertain exactly when a meal will be consumed. In such cases, a bolus of insulin designed to optimize the glucose response to the latest possible ingestion time of the meal is the most robust. An additional benefit of this strategy is that, should the meal occur before the injection time of such a bolus, then a bolus can be given contemporaneously with the meal. An alternative option is to lower the preprandial glucose level to the selected lower bound—for example, by increasing the basal flow—and then bolusing with the meal. This input would produce a “min-max-min” type glucose response and is unaffected by uncertainty in meal timing. However, such an insulin input is not of the form (1) and thus the optimality of such an input is not covered by the mathematical results of Townsend et al.^12,13

Uncertain Meal Amount

In Figure 8 we show the possible responses for a $25\%$ uncertainty in ingested carbohydrate, where the timing and amount of the bolus is chosen based on the expected amount of carbohydrate, that is, the mean. The blood glucose level will drop below the lower bound whenever the amount of carbohydrate consumed is less than the expected quantity. It is possible to mitigate this by either: raising the lower bound so that in all cases the blood glucose level remains above the true lower bound or equivalently deciding the magnitude of the bolus based on the minimum amount of consumed carbohydrate, or setting the insulin flow to zero when the maximum blood glucose level occurs. This latter policy minimizes the expected maximum blood glucose level.

Figure 8.

Range of possible BGL responses for a ±25% variation in ingested carbohydrate. A bolus of insulin with magnitude 13.86 IU is delivered at time 64 for all responses. For the dashed response the basal insulin flow, $\overline{u},$ is set to 0 over the interval $\left[184,251\right].$ The second plot shows the plasma insulin concentration for each treatment policy.

It is possible to extend the range of uncertainty over which hypoglycemia will not occur by extending the duration of the bolus input. However, as seen in Figure 9, the increase in the allowable uncertainty is minor in comparison to the increase in duration.

Figure 9.

Minimum amount of carbohydrate which can be ingested and hypoglycemia avoided given a bolus of varying duration, which is designed to be optimal for an input of 25 g of carbohydrate and zero insulin occurring when the maximum glucose level is attained.

Clinical Implications

Accurate estimation of the glycemic load of a meal, for example, through carbohydrate counting, is, typically, subject to large uncertainties. Thus it is important to develop robust strategies to deal with such situations.

Prebolusing to lower the preprandial glucose level and setting the insulin flow to 0 for a short period of time when the glucose concentration achieves its maximum—identifiable by the point at which the rate of increase of the glucose profile decreases, or is 0—would be the most effective strategy. However this is not possible in the absence of continuous glucose monitoring or pump therapy. In this case, it is best to underestimate the glycemic load of the meal and bolus accordingly. As shown in the example presented in Figure 8, this will produce a higher maximum blood glucose level should the glycemic load be greater than estimated. However, with quite large uncertainties in glycemic load, it is not necessary to take postprandial corrective action as the bolus designed to achieve the optimal glucose profile has sufficient insulin to restore euglycemia.

Uncertainty in the Response Time of Insulin

In the Bergman minimal model it may be assumed that the parameters $S_I$ and $C_l$ are known from the steady-state glucose value. Thus uncertainty in the action of insulin would be contained in the parameters $p_2,{\tau }_1^{-1}$ and ${\tau }_2^{-1}.$ As the model for insulin is linear, any uncertainty in either of these parameters has the same effect. Specifically, an increase shifts the peak of the insulin activity to an earlier time and any decrease shifts it to a later time, that is, to the left and right, respectively.

As shown in Figure 10 this uncertainty can result in either pre- or postprandial hypoglycemia. However, for a glucose response with a single maximum it is possible to mitigate postprandial hypoglycemia arising from slower than expected insulin action and retain the same maximum glucose concentration. This is achieved by stopping the insulin flow once the blood glucose level has reached its maximum value and recommencing it after it has passed the second minima. An example of this is shown in Figure 11.

Figure 10.

Range of possible blood glucose responses, to the typical meal and a bolus of magnitude $13.81IU$ delivered at time 64, for a 25% uncertainty in $p_2$.

Figure 11.

Blood glucose responses, to the typical meal and a bolus as in Figure 10, where zero insulin is delivered after maximum for a value of $p_2$ that is 75% of the expected value.

To avoid the preprandial hypoglycemia which results from an increase in insulin response time the magnitude and timing of the delivered bolus need to be determined according to the shortest possible response time. This is shown in Figure 12. This bolus also needs to be tested against the longest response time to ensure the postprandial hypoglycemia is avoidable. If it is not then the lower bound for the blood glucose concentration needs to be raised to accommodate this.

Figure 12.

Blood glucose responses, to the typical meal, where the magnitude, $11.46IU$, and timing, 70, of the insulin bolus is based on the shortest possible insulin response time with a lower bound of $81mgd{L}^{-1}$ which avoids postprandial hypoglycemia in the case of a longer response time.

Safety Optimality Trade-Off

As shown in Figure 13 for the typical meal, the maximum blood glucose level attained increases super-linearly with the chosen lower bound. Thus there is a large benefit in selecting the lower bound on the blood glucose level to be as low as possible.

Figure 13.

Maximum blood glucose level attained for the typical meal and optimal insulin bolus, where the magnitude and timing of the bolus is designed so that the minimum of the glucose response attains but does not go below the specified minimum target.

Clinical Implications

This reinforces that lowering the preprandial glucose level as much as possible significantly improves glycemic control. However, as demonstrated in other examples, the closer the lower bound is set to the hypoglycemic threshold the more likely it is that hypoglycemia will occur. Thus this lower bound should be selected based on individual and situational circumstances. For example, this bound would be set higher during or before exercise.

Comparison With Current Algorithms for Typical Meal Disturbances

In Figure 14 we compare the responses to the optimal treatment policy with a contemporaneous bolus for the typical and slow meals. Curves corresponding to the typical and slow meals are shown by the solid and dashed lines respectively. Also curves corresponding to the optimal and standard policies are shown by the blue and purple lines, respectively. For the slow meal, the optimal policy, the response to which is shown by the dashed blue line, gives a maximum glucose concentration of $124mgd{L}^{-1}$ whereas the contemporaneous bolus, shown by the dashed purple line, attains a maximum of $126mgd{L}^{-1}.$ These values are close since the optimal injection time is 111. However, it is possible to further decrease the maximum blood glucose level of the optimal response in this case by increasing the duration of the insulin input.

Figure 14.

Comparison of glucose responses to the optimal treatment policy, shown in blue, and a contemporaneous bolus, shown in purple, for the typical and slow meals, shown by the solid and dashed lines, respectively. The dashed black line indicates the maximum blood glucose concentration of the glucose response to the optimal treatment policy and the slow meal. The magnitudes of the contemporaneous bolus are $4.74IU$ and $5.91IU$ for the typical and slow meals, respectively, both are delivered at time 100. The magnitudes and timings of the optimal boluses for the typical and slow meals are $13.86IU$ at time 64 and $11.82IU$ at time 111, respectively.

For the typical meal, the optimal policy, shown by the solid blue line, achieves a maximum blood glucose concentration of $118mgd{L}^{-1},$ whereas the contemporaneous bolus, shown by the solid purple line, attains a maximum blood glucose level of $205mgd{L}^{-1}.$ This demonstrates there is a significant benefit to prebolusing insulin, that is, delivering the insulin well before the ingestion of a meal.

In Figure 15 we compare the optimal treatment policy with uncertainty in ingestion time of the typical meal to a contemporaneous bolus which is calculated to attain a minimum of $72mgd{L}^{-1}.$

Figure 15.

Glucose profiles for a bolus delivered such that the response to the typical meal, ingested at time 100 is optimal but hypoglycemia does not occur with a ±30% uncertainty in ingestion time. For the first plot the bolus, with a duration of 90, is delivered at time 33 with magnitude 0.075, resulting in $6.75IU$ of insulin being delivered in total. For the second plot a bolus, impulse, of magnitude $6.01IU$ is delivered at time 72.

The optimal bolus is designed so that the response, shown by the solid green line in Figure 15, to the typical meal, ingested at time 100, is optimal but for which no hypoglycemia occurs should the meal, instead, be ingested any time in the interval from 70 to 130. The glucose responses to an ingestion time of 70 and a bolus designed to optimize an ingestion time of 100 is shown in blue. Similarly, the glucose response to an ingestion time of 130 and a bolus designed to optimize an ingestion time of 100 is shown in purple. As can be seen in Figure 15, the optimal bolus is designed so that the blood glucose level remains above $72mgd{L}^{-1}$ for all possible meal ingestion times, that is, the optimal bolus is robust to a $\pm 30\%$ uncertainty in the meal timing. To ensure robustness, the lower bound of the optimal bolus is raised, to $92mgd{L}^{-1}$ in the first plot and $98mgd{L}^{-1}$ in the second plot of Figure 15. In both plots the glucose response when the meal is ingested at time 130 has a global minimum glucose concentration of $72mgd{L}^{-1}.$

In both plots the response to the optimal bolus is shown by the solid line and the response to the contemporaneous bolus by the dashed line. In the first plot the duration of the optimal input is 90 while in the second the input is an impulse of insulin.

The expected maximum blood glucose level for the optimal bolus with a duration of 90 is $203mgd{L}^{-1}$ while for the optimal impulse it is $202mgd{L}^{-1}.$ In all cases the contemporaneous bolus attains a maximum of $205mgd{L}^{-1}.$ That is, even while remaining robust to a $\pm 30\%$ variation in ingestion time the optimal policy has, on average, a lower maximum blood glucose level than current treatment.

When the contemporaneous bolus is instead delivered 15 minutes prior to the ingestion time, simulating a typical prebolus, the maximum blood glucose level is $161mgd{L}^{-1}.$ With a $\pm 10\%$ variation in ingestion time the expected maximum blood glucose level for an optimal bolus of insulin which avoids hypoglycemia is $153mgd{L}^{-1}.$

Conclusion

The fundamental control limitation inherent in many models of glucose metabolism has important consequences for insulin administration. Even though, in general, the optimal glucose response is not robust to model and parameter uncertainties using knowledge of the optimal glucose response to design treatment produces a lower maximum glucose concentration than standard therapy.