Glucose Rate-of-Change and Insulin-on-Board Jointly Weighted Zone Model Predictive Control

Abstract

We present design and evaluation of closed-loop insulin delivery using zone model predictive control (MPC) featuring an adaptive weighting scheme to address prolonged hyperglycemia due to changes in insulin sensitivity, underdelivery from profile mismatch, and meal composition. In the MPC cost function, the penalty on predicted glucose deviation from the upper zone boundary is weighted by a joint function of predicted glucose rate-of-change (ROC) and insulin-on-board (IOB). The asymmetric weighting gradually increases when glucose ROC and IOB were jointly low, independent of glucose magnitude, to limit hyperglycemia while aggressively reduces for negative glucose ROC to avoid hypoglycemia. The proposed controller was evaluated using two simulation scenarios: an induced resistance scenario and a nominal scenario to highlight the performance over a reference zone MPC with glucose ROC weighting only. The continuous adaption scheme resulted in consistent improvement for the entire glucose range without incurring additional risk of hypoglycemia. For the induced resistance and no feedforward bolus scenario, the percent time in 70-180 mg/dL was higher (53.5% versus 48.9%, p<0.001) with larger improvement in the overnight percent time in tighter glucose range 70-140 mg/dL (70.9% versus 52.9%, p<0.001). The results from extensive simulations, as well as clinical validation in three different outpatient studies demonstrate the utility and safety of the proposed zone MPC.

I. Introduction

EXOGENOUS insulin delivery is required for the management of type 1 diabetes (T1D) to mitigate hyperglycemia and glucose variability from meal disturbances and basal requirements. With advances in device technology and closed-loop algorithms, safe and effective glucose regulation is now increasingly possible [1]–[5]. Among the important considerations in closing the loop [2] is prolonged hyperglycemia that is defined in this work as glucose elevated from the desired range longer than its natural dynamics irrespective of the magnitude of difference from that desired range. This could occur due to systematic changes in insulin sensitivity from circadian dynamics [6], [7] or physiological stressors [8], underdelivery of insulin from mismatches in baseline profiles such as basal rates and insulin-to-carbohydrate ratios, limitations of the model for control, varying meal macronutrient composition [9]–[11], or a combination of these factors.

For improved glucose regulation, while considering the fundamental trade-off between hyperglycemia and hypoglycemia and inter-person variability, the changing requirements necessitate that the controller adapts on shorter time scales (hours) in the case of suboptimal control leading to prolonged hyperglycemia, as well as on longer time scales (weeks or months) for metabolic changes, such as, during pregnancy and adolescence. Model predictive control (MPC) [12] is suited for closed-loop insulin delivery through a customized design of tractable receding-horizon minimization of a cost function addressing the aforementioned trade-off subject to model predictions and critical safety constraints including nonnegativity of insulin values. For the shorter time scales, various adaptation and personalization schemes for MPC have been proposed, including internal model update [13]–[15], time varying setpoints [13], [15]–[17], augmented states and cost [17]–[19], control penalty weight [17], [20]–[22] and output penalty weight [17], [22], [23].

In practice, a controller for closed-loop insulin delivery operates in conditions ranging from no knowledge of meal disturbance to complete knowledge of meal carbohydrate content with an (optimal) feedforward bolus. Since insulin can only be added and cannot be removed, and given the lagged effect of insulin, it is useful to consider insulin-on-board (IOB) as the amount of fast-acting insulin that has yet to have an effect. A zero IOB would imply only basal insulin in the recent history. On the other hand, the direction of effect of meal disturbance and IOB on glucose can be ascertained by glucose rate-of-change (ROC) or glucose velocity, which is a measure of the net movement of glucose.

The main technical contribution of this work is a zone MPC [24] formulation with an adaptive weighting scheme using physiological states indicative of prolonged hyperglycemia. The design builds on and improves the clinically-validated constrained zone MPC with weighting based on glucose ROC [23] by extending the weighting on predicted glucose deviation from the upper zone boundary to be a joint function of glucose ROC and IOB. The asymmetric weighting is designed to increase the weight > 1 only under conditions of low glucose ROC and IOB when additional insulin may be required during any cause of deviations from equilibrium conditions such as for later part of extended disturbance response [25]. The same design also retains the feature of aggressive reduction in the weight ≪ 1 for negative glucose ROC values to avoid hypoglycemia. The results from numerical simulations and clinical validation from three different outpatient clinical studies in adults (n = 10, n = 35) and pregnant women (n = 11) indicate that the proposed controller can safely increase insulin delivery when required while backing off when necessary, with larger effects during the overnight period.

The paper is structured in the following sections: Section II describes glucose regulation and physiological states under prolonged hyperglycemia. Zone MPC using the joint asymmetric weighting is described in Section III. Numerical simulations on the US Food and Drug Administration (FDA) accepted UVA/Padova metabolic meal simulator for T1D with 100 virtual subjects [26] are presented in Section IV and selected details of controller performance during clinical validation are presented in Section V. The paper ends with conclusion in Section VI.

II. Glucose Regulation and Prolonged Hyperglycemia

In the body, glucose values are tightly regulated in the presence of disturbances such as meal and exercise where insulin is the primary anabolic hormone to reject the meal disturbance by enabling glucose utilization and suppressing endogenous glucose production. The effect of meal and insulin on glucose can be described using interconnected compartment models [27]. In the following, we describe selected compartments from the UVA/Padova T1D simulator whose parameters were modified to simulate prolonged hyperglycemia, and subsequently we compare resulting change in trajectories in the glucose ROC–IOB plane. The complete simulator equations are described in [26].

The glucose compartments can be described as:

$${\dot{G}}_p(t)=EGP(t)+Ra(t)-U_{ii}(t)-E(t)-k_1{G}_p(t)+k_2{G}_t(t),$$ (1)

$${\dot{G}}_t(t)=-U_{id}(t)+k_1{G}_p(t)-k_2{G}_t(t),$$ (2)

where $G_p$ and $G_t$ are glucose in the plasma and tissue compartments, EGP is endogenous glucose production as described in (9), $Ra$ is the glucose rate of appearance from a meal as described in (10), $U_{ii}$ is insulin-independent glucose utilization, E is renal excretion and $U_{id}$ is insulin-dependent glucose utilization as described in (8), and $k_{\mathrm{*}}$ are the virtual subject-specific parameters. The continuous glucose monitor (CGM) sensor noise can be described as:

$${\dot{G}}_s(t)=-k_{sc}\left(G_s(t)-G_p(t)\right),$$ (3)

$$G_{\text{CGM}}(t)=G_s(t)+{ϵ}_s(t),$$ (4)

$$e_s(t)=0.7\left(e_s(t-1)+v_s(t)\right),v_s(t)~\mathcal{N}(0,1),$$ (5)

$${ϵ}_s(t)=-5.47+15.96\text{sinh}\left(\frac{e_s(t)+0.54}{1.68}\right),$$ (6)

where $G_s$ is glucose in the interstitial compartment and $k_{sc}$ is virtual subject-specific parameter. The non-Gaussian noise ${ϵ}_s$ is generated through a nonlinear transformation of a Gaussian distributed autoregressive time series $e_s$ of order one with zero mean and unit variance as described in [28].

The plasma insulin compartment can be described as:

$${\dot{I}}_p(t)=-\left(m_2+m_4\right)I_p(t)+m_1{I}_l+k_{a1}{I}_{sc1}+k_{a2}{I}_{sc2},$$ (7)

where $I_p$ and $I_l$ are insulin in the plasma and liver compartments, $I_{sc1}$ and $I_{sc2}$ are insulin mass in the two subcutaneous compartments from delivered insulin and described using a linear model [26], and $m_{\mathrm{*}}$ and $k_{a\mathrm{*}}$ are the virtual subject-specific parameters. The $U_{id}$ and $EGP$ can be described as:

$$U_{id}(t)=\frac{\left[V_{m0}+V_{mx}X(t)\left(1+r_{1}risk\right)\right]G_t(t)}{K_{m0}+G_t(t)},$$ (8)

$$EGP(t)=k_{p1}-k_{p2}{G}_p(t)-k_{p3}{X}^L(t)+\xi X^H(t),$$ (9)

where $X^L$ is the delayed insulin action on EGP described using a chain of two compartments from $I_p$ [27], $X^H$ is the delayed glucagon action on EGP, X is insulin in the remote compartment described using a linear model [27], $k_{p^{\mathrm{*}}},\xi$ are virtual subject-specific parameters, and $V_{m0},V_{mx},K_{m0}$ are virtual subject-specific nonlinear (Michaelis–Menten) model parameters. The risk term captures the risk function to model increase in $U_{id}$ during hypoglycemia and $r_1$ is a model parameter [26].

Finally, the $Ra$ term is proportional to the nonlinear emptying rate of the stomach compartment [29]

$$Ra(t)\propto k_{\mathit{min}}+\frac{k_{\mathit{max}}-k_{\mathit{min}}}{2}\left(\mathit{tanh}\left(Q_{\mathit{sto}}\right)\right),$$ (10)

where $Q_{\mathit{sto}}$ is meal mass in the stomach compartment and $k_{m\mathrm{*}}$ are the virtual subject-specific parameters.

In this work, prolonged hyperglycemia is induced by changing the following three sets of parameters while no other changes are made to the simulator:

1. Insulin sensitivity: $V_{mx}$ and $k_{p3}$ are decreased to limit the effect of insulin to increase utilization $U_{id}$ in (8) and decrease production $EGP$ in (9), respectively,

2. Glucose effectiveness: defined as the effect of glucose to intrinsically limit it’s own utilization at basal insulin, $V_{m0}$ in (8) and $k_{p2}$ in (9) are decreased to limit the independent effect of glucose in $U_{id}$ and $EGP$, respectively,

3. Meal rate of absorption: $k_{min}$ and $k_{max}$ in (10) are decreased to limit the rate of meal appearance and extend the meal response.

Fig. 1 shows trajectories from open-loop simulation for 9 hours (sampled at 1 minute) with 60 gram meals and optimal feedforward boluses based on nominal parameters. The simulations were performed for nominal parameters and for the aforementioned parameters multiplied by a factor of 0.75 to induce resistance. The parameter combinations induce a significance degree of resistance as observed by the difference in glucose trajectories due to insufficient basal and bolus insulin. The IOB decay is not changed in either of the two scenarios.

Fig. 1:

Comparison of trajectories from a cohort of 10 virtual subjects under nominal scenario (blue) and under resistance scenario (red) shown as mean response (solid line and dashdot line, respectively) and one standard deviation (shaded regions). (A) plasma glucose $G_p$ in mg/dL, (B) plasma glucose ROC ${\dot{G}}_p$ in mg/dL/min, (C) normalized IOB ($IOBn$, unitless) approximated as $I_{sc1}+I_{sc2}$ with basal insulin removed and normalization as per (19), and (D) plane of mean glucose ROC with one standard deviation versus mean normalized IOB.

Under nominal scenario, the trajectory starts and stays at the origin in the glucose ROC-IOB plane due to the correct nominal basal insulin. It jumps to the right along the IOB axis due to the feedforward bolus, after that the glucose ROC increases due to the meal, reaches its ROC peak, and eventually crosses the zero ROC threshold, which corresponds to the meal peak in the time domain, reaches the ROC trough that is much smaller than the peak, after that the trajectory converges back to the origin. In contrast, for the case of induced resistance, the trajectory drifts away from the origin along the ROC axis at the beginning as the nominal basal insulin is not sufficient. As with the nominal case, the trajectory jumps to the right from the same bolus, after that, however, it shows a slower glucose ROC and lower ROC peak compared to the nominal case, crosses the zero ROC threshold much later indicating delayed meal peak, and with a muted ROC trough converges back to the origin. These aspects of trajectories in a region of glucose ROC-IOB plane are discussed in the context of controller design in the ensuing section.

III. Glucose ROC and IOB Jointly Weighted Zone MPC

A. Notation

Let $\mathbb{R}$ denote set of real numbers, ${\mathbb{R}}_{>0}$ denote set of positive real numbers, ${\mathbb{Z}}_{a:b}$ denote a finite set of integers $\{a,\dots ,b\}$, and ${\mathbb{Z}}_{+}$ denote set of positive integers, all of appropriate dimensions. In the zone MPC formulation, variables for each sample instance are indexed using variable k in the subscript and typeset using normal math font, such as state $x_k$ and input $u_k$, while those defined in the optimization problem are indexed using variable i in the subscript and distinguished by using sans serif math fonts, such as predicted state ${\mathrm{x}}_i$ and decision variable ${\mathrm{u}}_i$.

B. Designing the Adaptive Weight Q

An MPC cost with glucose ROC or velocity $v$ weighting was introduced in [23], [30] as:

$$\sum _{i=1}^{N_y}\left(Q\left({\text{v}}_i\right){\hat{\text{z}}}_i^2+{\check{\text{z}}}_i^2\right),$$ (11)

where v is the predicted velocity and $\hat{\text{z}}$ is the predicted glucose deviation from the upper zone boundary weighted by $Q\left(v\right)\in (0,1]$ for negative velocities up to 1 mg/dL/min using a cosine function [23], $\check{\text{z}}$ is deviation below the lower zone boundary, and $N_y\in {\mathbb{Z}}_{+}$ is the prediction horizon in this simplified representation. The velocity weighting in (11) avoids hypoglycemia by reducing insulin delivery smoothly to basal when the velocity is increasingly negative, in particular, following a meal peak. Experiments with $Q\left(v\right)>1$ did not yield a desirable control law [23].

A feature of prolonged hyperglycemia is eventual low glucose ROC with not enough IOB (Fig. 1). Thus, any increase in weight cannot be made only as a function of glucose ROC without the knowledge of IOB. The muted convergence of open-loop resistance scenario trajectory in the region of $v-IOBn$ plane, as shown in Fig. 1(D), can be used to inform design of a closed-loop weighting scheme to systematically increase insulin delivery when the trajectory enters the low $v-IOBn$ region. The timely addition of insulin could move the resistance scenario trajectory closer to the nominal scenario trajectory. We postulate the following three properties:

P1 The weight is turned off for sufficiently negative $v$,

P2 The weight is kept at baseline for sufficiently positive $v$,

P3 Otherwise, the weight is modulated around baseline as a function of $v$ and IOB.

The design properties of the weighting function are based on primacy of glucose ROC over IOB due to key factors such as asymmetric risk of hypoglycemia over hyperglycemia, relatively short prediction horizon of 45 minutes compared to the natural glucose dynamics, and glucose ROC can be reliably estimated from CGM while IOB can only be estimated from recorded deliveries with no approved sensing device. While [P1] is a property retained from the previous controller, [P2] is designed to limit aggressive response without reliable IOB measurement, and the modulation in [P3] is designed to be not dependent of glucose level and provide desired performance with and without a feedforward bolus, as discussed later.

The asymmetric weighting Q as a function of $v$ (mg/dL/min) and $IOBn$ (unitless) is shown in Fig. 2. In particular, higher weight in the low $v-IOBn$ plane, as shown in Fig. 2(D), is designed to nudge glucose out of prolonged hyperglycemia by the priming the controller to increase insulin delivery when the trajectory enters the aforementioned region, also shown in Fig. 1(D), while backing off when $v$ or $IOBn$ move out of the region. The joint surface is constructed through interpolation of sample points from newly designed univariate weighting function $Q\left(v\right)$ when $IOBn=0,Q\left(IOBn\right)$ when $v=0$, and other joint information at select points as described in Appendix I. Polynomial equations of second degree are fit to the surfaces partitioned by magnitude of $v$ based on design thresholds as shown in (12) where $ϵ>0$ for a well-posed problem as described later, and $Q(v,IOBn)\in \left[ϵ3\right]$ with a baseline value of $Q=1$. The parameters $p_{\mathrm{*}}$ and $q_{\mathrm{*}}$ are reported in Table IV in Appendix I.

Fig. 2:

(A). The asymmetric weighting Q as a function of glucose ROC $(v$, mg/dL/min) and normalized IOB ($IOBn$, unitless). The projection plots in each of the two dimensions are: (B) Q versus $v$ plane, (C) Q versus $IOBn$ plane, and (D) $v$ versus $IOBn$ plane. It is worth noting that $IOBn$ was truncated at 0.15 as described in Appendix I-B.

TABLE IV:

Estimated parameters of the weighting surface.

$p_{00}$	$p_{10}$	$p_{01}$	$p_{20}$	$p_{11}$	$p_{02}$
3.204	1.733	−15.545	−0.223	−10.156	−1.917
$q_{00}$	$q_{10}$	$q_{01}$	$q_{20}$	$q_{11}$	$q_{02}$
3.142	−1.141	−15.309	−0.159	10.238	−1.363

$$\begin{gathered} Q=\{\begin{array}{cc}ϵ\hfill & \hfill \text{for}v<-1.5\\ p_{00}+p_{10}v+p_{01}IOBn \\ +p_{20}{v}^2+p_{11}vIOBn \\ +p_{02}IOB{n}^2& \hfill \text{for}-1.5\le v<0\\ q_{00}+q_{10}v+q_{01}IOBn \\ +q_{20}{v}^2+q_{11}vIOBn \\ +q_{02}IOB{n}^2& \hfill \text{for}0\le v\le 1.5\\ 1\hfill & \hfill \text{for}v>\mathrm{1.5.}\end{array} \end{gathered}$$ (12)

The adaptation of Q (over time) has been designed to have the following salient characteristics:

• Modulation of Q. Within the region defined by $\left|v\right|\le 1.5$ mg/dL/min and $0\le IOBn\le 0.15$ in the $v-IOBn$ plane, the effect of $v$ on Q is modulated by $IOBn$, and vice-versa. Thus, for a given $v$, the controller is more aggressive when $IOBn$ is low and similarly for a given $IOBn$, the controller is more aggressive when the magnitude of $v$ is low. Any additional insulin over baseline weighting will result in an increase in $v$ or $IOBn$ that will automatically inhibit further increase of Q.

• Glucose level. By design, Q is independent from the glucose level $G_p$. In particular, Q will be highest at the origin of $v-IOBn$ plane, which during the nominal condition corresponds to the glucose value within the zone boundaries. However, the actual cost incurred in that case will be zero as the zone excursion variables $\hat{z},\check{z}$ will be zero. This enables the controller to aggressively correct drifts from the upper zone boundary and may reduce offsets more effectively.

• Feedforward insulin. Following feedforward insulin, such as during a meal or for correcting prolonged hyperglycemia, the weight will change to a value on the right half of the $v-IOBn$ plane, but never go below $Q=1$ as long as $v$ is positive. This implies that controller is primed to correct for meal related hyperglycemia while the IOB from bolus has an effect. Further, the nominal threshold for $IOBn=0.15$ (Appendix I) is tuned such that Q may not increase > 1 around meal peak when $v$ will be low, when additional insulin may be not required following an optimal bolus, but interjects later more aggressively when additional insulin may be required such as in case of an extended meal disturbance response [25].

• Feedback insulin only. With feedback insulin only in response to rising glucose from a meal, the weight will gradually change to a value in the top of half of the $v-IOBn$ plane as $IOBn$ increases from the delivered insulin, but never goes below $Q=1$ as long as $v$ is positive. Since the feedback action gradually increases the IOB, $Q>1$ is likely in the beginning of the meal excursion before reaching the peak, and hence the controller may be more assertive than the baseline weight.

• Velocity weighting only. By construction, the joint weighting $Q(v,IOBn)$ is functionally equivalent to the velocity weighting $Q\left(v\right)$ in [23] when $IOBn=0.15$.

C. MPC Problem

The proposed zone MPC design is based on a linear control-relevant model of insulin-glucose dynamics, as described in Appendix II–A, subject to input constraints, as described in Appendix II-B, with zones boundaries as described in Appendix II-C. Extending the description for (11), let the control horizon be $N_u\in {\mathbb{Z}}_{1:N_y}$, and the weights $Q,\hat{D},\hat{R},\check{R}\in {\mathbb{R}}_{>0}$ of appropriate dimensions, and the decision variable input u, predicted state x, predicted glucose y, predicted glucose velocity $\mathrm{v}$, and predicted glucose excursion z. At each k, with updated state $x_k$, a constrained finite horizon optimization problem is solved to determine the optimal control sequence ${\text{u}}^{*}$:

$$\left\{{\text{u}}_0^{*},\dots ,{\text{u}}_{N_{\text{u}}-1}^{*}\right\}=\text{arg min}J\left(x_k,\left\{{\text{u}}_0,\dots ,{\text{u}}_{N_{\text{u}}-1}\right\}\right),$$ (13)

where the cost function $J(\cdot ,\cdot )$ is defined as:

$$J(\cdot ,\cdot )=\sum _{i=1}^{N_y}\left(Q\left({\text{v}}_i,IOB{n}_k\right){\hat{\text{z}}}_i^2+{\check{\text{z}}}_i^2+\hat{D}{\hat{\text{v}}}_i^2\right)$$ (14a)

$$+\sum _{i=0}^{N_u-1}\left(\hat{R}{\hat{\text{u}}}_i^2+\check{R}{\check{\text{u}}}_i^2\right)$$ (14b)

subject to

$${\text{x}}_0=x_k,$$ (15a)

$$\begin{array}{cc}{\text{x}}_{i+1}=A{\text{x}}_i+B{\text{u}}_i& \forall i\in {\mathbb{Z}}_{0:N_y-1},\end{array}$$ (15b)

$$\begin{array}{cc}{\text{y}}_i=C_y{\text{x}}_i,{\text{v}}_i=C_v{\text{x}}_i& \forall i\in {\mathbb{Z}}_{0:N_y},\end{array}$$ (15c)

$$\begin{array}{cc}{\text{u}}_i\in \left[{\check{\lambda }}_{k+i}{\hat{\lambda }}_{k+i}\right]& \forall i\in {\mathbb{Z}}_{0:N_u-1},\end{array}$$ (15d)

$$\begin{array}{cc}{\text{u}}_i=0& \forall i\in {\mathbb{Z}}_{N_u:N_y-1},\end{array}$$ (15e)

$$\begin{array}{cc}{\hat{\text{u}}}_i=\text{max}\left\{{\text{u}}_i,0\right\}& \forall i\in {\mathbb{Z}}_{0:N_u-1},\end{array}$$ (15f)

$$\begin{array}{cc}{\check{\text{u}}}_i=\text{min}\left\{{\text{u}}_i,0\right\}& \forall i\in {\mathbb{Z}}_{0:N_u-1},\end{array}$$ (15g)

$$\begin{array}{cc}{\text{z}}_i=Z\left({\text{y}}_i,k+i\right)& \forall i\in {\mathbb{Z}}_{0:N_y},\end{array}$$ (15h)

$$\begin{array}{cc}{\hat{\text{z}}}_i=\text{max}\left\{{\text{z}}_i,0\right\}& \forall i\in {\mathbb{Z}}_{0:N_y},\end{array}$$ (15i)

$$\begin{array}{cc}{\check{\text{z}}}_i=\text{min}\left\{{\text{z}}_i,0\right\}& \forall i\in {\mathbb{Z}}_{0:N_y},\end{array}$$ (15j)

$$\begin{array}{cc}{\hat{\text{v}}}_i=\text{max}\left\{{\text{v}}_i,0\right\}& \forall i\in {\mathbb{Z}}_{0:N_y},\end{array}$$ (15k)

where $IOBn$ is the normalized IOB as described in Appendix I. Consequently, the insulin microbolus $u_{\mu \mathrm{b}}$ to be delivered is calculated as:

$$u_{\mu \text{b},k}={\text{u}}_0^{*}+u_{\text{basal},k},$$ (16)

where $u_{\text{basal},k}$ is person-specific time-varying basal in microboluses (U/5-min). In addition to the terms described in (11), the positive velocity penalty term in (14a) $\hat{D}{\stackrel{\mathrm{ˆ}}{\mathrm{v}}}_i^2$ [23], which is active in glucose range [140 180] mg/dL, and asymmetric penalty (14b) on deviation above basal $\hat{\text{u}}$ and deviation below basal $\check{\text{u}}$ [31], are unmodified in this work.

The controller parameters were tuned using the 10 adult virtual subject cohort of the UVA/Padova metabolic simulator and were fixed for the cohort at the following values: $N_y=9,N_u=5,\hat{R}=6500,\check{R}=100,\hat{D}=1000$, and $Q(v,IOBn)$ as described in Section III-B with $ϵ={10}^{-6}$. The model predicted v and $IOBn$ results in changing Q over the prediction horizon. Using auxiliary variables to bound the zone excursion z, velocity v and input u, for a given sequence of $Q\left({\mathrm{v}}_i,IOB{n}_k\right)\forall i\in {\mathbb{Z}}_{1:N_y}$, the optimization problem in (13) is a convex quadratic program. Since the $Q\left({\text{v}}_i,IOB{n}_k\right)$ is itself a function of calculated control sequence ${\text{u}}^{*}$, the solution has to be obtained iteratively [23]. Using an initial assumption of $\mathrm{v}$ and $IOBn$ using current state $x_k$ and $\mathrm{u}=0$, an initial weight column vector value is used to calculate ${\text{u}}^{*}$ that is then fed back to recalculate Q. The convergence criteria for terminating iterations is set to a maximum absolute difference between successively recalculated Q column vector values to be less than 10⁻⁶ or if the iteration exceeds a set limit of ten.

In (14a), the value of $IOB{n}_k$ is a function of index k and not i. In other words, the IOB value stays constant throughout the prediction horizon. Similarly, the IOB constraint (30) is also implemented in the same fashion. There are two main reasons for this: first, large changes in the IOB typically only occur due to a feedforward bolus that is unknown to the controller over the prediction horizon and second, the IOB will not change significantly over the shorter prediction horizon relative to the IOB decay curve. A longer horizon may require incorporation of $IOB{n}_i$ in the calculation of the control sequence. Notwithstanding, at each k, $IOB{n}_k$ is updated using the historical insulin delivery U.

Due to limitations with subcutaneous insulin absorption and size of meal disturbance, the feedback control is usually augmented with a user-requested feedforward insulin bolus. In this work, a bolus calculator uses an estimate of meal carbohydrate size M (g) and person-specific time-varying insulin-to-carbohydrate ratio ${\mathrm{C}\mathrm{R}}_k$ (g/U) to calculate a feedforward bolus $u_{\mathrm{f}\mathrm{f}\mathrm{b}}$ as:

$$u_{\text{ffb},k}=\{\begin{array}{rr}\hfill \frac{\text{M}}{{\text{CR}}_k}& \hfill \text{if}{G}_{\text{CGM},k}\ge 120\text{mg}/\text{dL},\\ \hfill 0.8\frac{\text{M}}{{\text{CR}}_k}& \text{otherwise},\hfill \end{array}$$ (17)

where the bolus size is reduced by 20% if the current glucose level is below a threshold as a precaution. Additionally, a correction bolus up to 2 U is added to $u_{\mathrm{f}\mathrm{f}\mathrm{b}}$ to correct elevated glucose up to $G_{\text{CGM }}=150$ mg/dL [23]. The total insulin delivered $u_{\mathrm{I}\mathrm{N},k}$ is a sum of microboluses and feedforward boluses:

$$u_{\text{IN},k}=u_{\mu \text{b},k}+u_{\text{ffb},k}.$$ (18)

Although tuning for a number of design parameters is fixed, it is worth noting that the person-specific variables, namely total daily insulin, basal rate, correction factor and insulin-to-carbohydrate ratio in addition to the adaptive weight Q personalizes the closed-loop insulin delivery.

IV. Numerical Simulations

The numerical simulations were performed on an independent cohort of 100 adult virtual subjects from the UVA/Padova metabolic simulator [26]. Each virtual subject’s simulation was repeated ten times to produce variability from measurement noise, as described in (4), and meal disturbance, as noted later, thus resulting in a total of 1000 data points for any given comparison. The virtual subjects contain a range of parameters to represent real-life variability including those on the limits of plausibility. The algorithm verification data was presented to the FDA for transition to a clinical study. A subset of those simulation scenarios have been expanded upon for this paper to highlight utility of the proposed formulation.

The simulations compare the proposed zone MPC controller with glucose ROC and IOB joint weighting (QvIOB) to a reference zone MPC controller with velocity weighting only (Qv) [23] with day zones boundaries of [90 120] mg/dL and overnight zone boundaries of [100 120] mg/dL [11], [32]. The two controllers differed in only two aspects: the use of weighting function Q and l ∈ [2 8] in the reference Qv controller. The simulations allows for a head-to-head comparison with a clinically-validated controller.

The comparison of the two controllers was performed for the following two scenarios:

1. Scenario A: the three set of simulator parameters, discussed in Section II, were modified by a factor of 0.75 to induce prolonged hyperglycemia,

2. Scenario B: nominal simulator parameters.

For Scenario A, the correction factor ${\mathrm{C}\mathrm{F}}_k$ was also scaled by a factor of 0.5 for both controllers so that safety constraint was sufficiently relaxed commensurate with the degree of induced resistance to allow a fair comparison of the effect of weight adaptation, whereas for Scenario B, no such modification was made. The simulation started at 6 am and three meals were provided over a total of 24 hours. Each meal amount was sampled from a normal distribution while the meal time was sampled from an uniform distribution over span of ±1 hour from a reference value. Specifically, the three meals had a mean of 50 g (breakfast), 75 g (lunch) and 75 g (dinner) with a standard deviation of 10 g. Thus, the average total daily carbohydrate intake provided in this scenario was 50 + 75 + 75 = 200 g. The meal timings were from a reference value of 9 am (breakfast), 1 pm (lunch) and 7 pm (dinner).

Within each of the two scenarios, four different test conditions compared the two controllers under different level of prior insulin from feedforward boluses calculated using the nominal insulin-to-carbohydrate ratio and mismatches in meal size information:

1. no bolus (feedback insulin only),

2. nominal bolus,

3. underestimation of meal bolus by 25%,

4. overestimation of meal bolus by 25%.

For each simulation, the noisy glucose measurements $G_{\text{CGM }}$ were used as the measured signal for feedback control while the glucose outcomes are calculated using plasma glucose $G_{\mathrm{p}}$, both sampled at five minutes. The controller performances are compared using outcomes based on glucose and insulin data calculated for the complete day and night period (24 hours) and for the overnight period (12 am to 6 am, 6 hours). The results are summarized in the figures as empirical cumulative distribution function (CDF) of glucose, where better control will move the CDF curve steeper and to the left in the desired glucose range, and as control-variability grid analysis (CVGA) scatter plot of maximum and minimum glucose with pre-specified glucose zones [33]. In the CVGA plot, a point corresponds to one simulation (each virtual subject has 10 points) where better control will keep the points within zones A and B on the lower left. The lower bound of CVGA plot was relaxed from the original value of 110 mg/dL [33] to 125 mg/dL in order to visualize more points for the induced resistance scenario.

In the tables, the glucose outcomes summarized the distribution of glucose using mean and standard deviation (SD) and using percent time in range, below and above glucose thresholds of clinical interest [34]. The low blood glucose index (LBGI) and high blood glucose index (HBGI) quantify the glycemic risks as implemented in [35] and the number of CVGA data points in the pre-specified zones were used to highlight the regulation trade-off. The insulin outcomes summarized the sum of all insulin $\left({\sum }_k u_{\mathrm{I}\mathrm{N},k}\right)$, sum of microbolus insulin $\left({\sum }_k u_{\mu \mathrm{b},k}\right)$, sum of feedforward insulin $\left({\sum }_k u_{\mathrm{f}\mathrm{f}\mathrm{b},k}\right)$, and sum of microbolus insulin deviation from nominal basal greater than zero $\left({\sum }_k \left(u_{\mu \mathrm{b},k}-u_{\text{basal},k}\right)={\sum }_k \Delta u_{\mu \mathrm{b},k}>0\right)$ highlighting insulin increased above basal to limit hyperglycemia, and less than and equal to zero $\left({\sum }_k \left(u_{\mu \mathrm{b},k}-u_{\text{basal},k}\right)={\sum }_k \Delta u_{\mu \mathrm{b},k}\le 0\right)$ highlighting insulin reduced below basal to limit hypoglycemia. Due to asymmertic insulin penalty weights $\hat{R}\gg \check{R}$, insulin microbolus less than basal are mostly zero, i.e. complete pump suspension [31]. Thus, ${\sum }_k \Delta u_{\mu \mathrm{b},k}\le 0$ is proportional to the length of pump suspension multiplied by the virtual subject’s basal rate.

With repeated data for each virtual subject, the p-value was calculated using linear mixed-effects models with random intercept for the null hypothesis of no fixed effect difference between the two controllers. Models were fit using the maximum likelihood method and the degree of freedom using the Satterthwaite approximation. No p-values could be calculated for the CVGA regions for data summarized as scalar counts at the cohort level. The estimate and p-value for distribution of Q was calculated using a intercept only fixed effect with random intercept where the p-value was calculated for the null hypothesis of $Q=1$ using an F-test. A threshold of p = 0.001 was used to highlight an effect.

A. Induced Resistance Scenario

For Scenario A with induced resistance, the QvIOB controller consistently improved glucose outcomes versus the Qv controller, for all test conditions, by gradually increasing insulin delivery without incurring additional time in hypoglycemia (Table I). The difference between controllers were larger with less prior insulin, in particular for the overnight period, when the controllers had more leeway to act. For the no bolus test condition and complete period, the changes in mean glucose and percent time in 70 – 180 mg/dL were 183.5 mg/dL versus 192.2 mg/dL, p < 0.001 and 53.5% versus 48.9%, p < 0.001, respectively with lower HBGI (9.9 versus 11.5, p < 0.001) and no change in LBGI (p = 0.158) through increased insulin above basal (19.5 U versus 17.7 U, p < 0.001) or approximately 10% more. Similarly, for the overnight period, percent time in 70 – 140 mg/dL improved by 18% (70.9% versus 52.9%, p < 0.001 though increased insulin above basal (2.6 U versus 2.2 U, p < 0.001) or approximately 18% more. The choice of nominal ${\mathrm{C}\mathrm{F}}_k$ would have resulted in smaller effect sizes due to control action saturating from the safety constraint (30) (data not shown).

TABLE I:

Outcomes during induced resistance Scenario A comparing the proposed controller (QvIOB) versus the reference controller (Qv) for two selected test conditions and grouped by the two time periods.

Day and Night
	Nominal bolus			No bolus
	QvIOB	Qv	p-value	QvIOB	Qv	p-value
Mean glucose	154.6	158.1	< 0.001†	183.5	192.2	< 0.001†
SD glucose	26.5	26.2	0.017	47.5	48.9	< 0.001†
LBGI	0.0	0.0	0.206	0.0	0.0	0.158
HBGI	4.4	4.8	< 0.001†	9.9	11.5	< 0.001†
% < 54	0.0	0.0	0.614	0.0	0.0	0.882
% < 70	0.0	0.0	0.660	0.0	0.0	0.659
% ∈ [70, 140]	36.2	30.6	< 0.001†	27.6	20.6	< 0.001†
% ∈ [70, 180]	81.2	78.7	< 0.001†	53.5	48.9	< 0.001†
% > 180	18.8	21.3	< 0.001†	46.5	51.1	< 0.001†
% > 300	0.1	0.1	0.540	3.1	4.5	< 0.001†
${\sum }_k{u}_{\mathrm{IN},k}$	51.0	50.0	< 0.001†	45.5	43.9	< 0.001†
${\sum }_k{u}_{\mu b,k}$	36.0	34.7	< 0.001†	45.5	43.9	< 0.001†
${\sum }_k{u}_{\text{ffb},k}$	15.0	15.3	< 0.001†	0.0	0.0	—
${\sum }_k\Delta u_{\mu b,k}>0$	11.0	9.6	< 0.001†	19.5	17.7	< 0.001†
${\sum }_k\Delta u_{\mu b,k}\le 0$	−2.6	−2.5	< 0.001†	−1.6	−1.5	< 0.001†
CVGA zone: A	84	57	—	0	0	—
CVGA zone: B	905	933	—	767	696	—
CVGA zone: C	5	7	—	231	302	—
CVGA zone: D	5	3	—	2	2	—
CVGA zone: E	0	0	—	0	0	—
Overnight
Mean glucose	129.0	133.7	< 0.001†	136.6	145.3	< 0.001†
SD glucose	10.4	10.4	0.852	16.6	17.4	0.004
LBGI	0.0	0.0	0.485	0.0	0.0	0.303
HBGI	1.0	1.4	< 0.001†	2.0	3.0	< 0.001†
% < 54	0.0	0.0	0.943	0.0	0.0	0.882
% < 70	0.0	0.0	0.978	0.0	0.0	0.914
% ∈ [70, 140]	83.0	70.8	< 0.001†	70.9	52.9	< 0.001†
% ∈ [70, 180]	99.3	99.2	0.284	93.3	89.4	< 0.001†
% > 180	0.6	0.8	0.248	6.7	10.5	< 0.001†
% > 300	0.0	0.0	—	0.0	0.1	0.020
${\sum }_k{u}_{\mathrm{IN},k}$	8.6	8.3	< 0.001†	8.9	8.7	< 0.001†
${\sum }_k{u}_{\mu b,k}$	8.6	8.3	< 0.001†	8.9	8.7	< 0.001†
${\sum }_k{u}_{\text{ffb},k}$	0.0	0.0	—	0.0	0.0	—
${\sum }_k\Delta u_{\mu b,k}>0$	2.5	2.0	< 0.001†	2.6	2.2	< 0.001†
${\sum }_k\Delta u_{\mu b,k}\le 0$	−1.2	−1.0	< 0.001†	−1.0	−0.8	< 0.001†
CVGA zone: A	933	930	—	620	536	—
CVGA zone: B	65	69	—	377	454	—
CVGA zone: C	1	1	—	3	10	—
CVGA zone: D	0	0	—	0	0	—
CVGA zone: E	0	0	—	0	0	—

Data are summarized as mean and the significant effects are highlighted using †. The percent time threshold are in mg/dL. The unabridged table is available at https://doi.org/10.5281/zenodo.8034034.

Fig. 3 compares the CDF and CVGA plots for the no bolus test condition for both time periods. The QvIOB controller mean CDF stayed on the left and did not cross the Qv controller mean CDF highlighting the consistent improvement with larger differences for the overnight period. The modified CVGA plots indicates a decreased maximum as well as minimum glucose due to CVGA points moving from the zone C to the zone B, and from zone B to zone A for all test conditions with larger shifts for the complete time period (Table I). The number of CVGA points in zone D and zone E were comparable between the controllers. Finally, small differences in the size of feedforward bolus insulin were due to meal variability and bolus calculation in (17) as a function of glucose.

Fig. 3:

Comparison of glycemic performance for Scenario A and no bolus test condition for both time periods. The (A) empirical CDF plot compares mean for QvIOB (blue, solid line) versus Qv (red, dashed-dot line) with one standard deviation shown as the shaded region. The (B) CVGA plot, with lower bound relaxed to 125 mg/dL, compares the CVGA data mean (vertical and horizontal line with keys matching (A)) and one standard deviation shown as ellipses where the individual Qv controller CVGA points (red squares) moved from the left and top region to the interior as QvIOB controller CVGA points (blue circles).

B. Nominal Scenario

For Scenario B with nominal parameters, the improvement in glucose outcomes were again consistent for all test conditions but more modest than Scenario A due to nominal insulin requirements (Table II). For the case of nominal bolus test condition, the mean glucose was lower (136.2 mg/dL versus (136.2 mg/dL versus 137.9 mg/dL, p < 0.001) but no significant change in glucose SD (30.4 mg/dL versus 30.2 mg/dL, p = 0.151). Similarly, for the overnight period, percent time in the tighter glucose range 70 – 140 mg/dL was higher (97.1% versus 95.7%, p < 0.001) and no change in percent time < 70 mg/dL (0.2% versus 0.1%, p = 0.389) through increased insulin above basal (1.7 U versus 1.3 U, p < 0.001) or approximately 30% more.

TABLE II:

Outcomes during nominal Scenario B comparing the proposed controller (QvIOB) versus the reference controller (Qv) for two selected test conditions and grouped by the two time periods.

Day and Night
	Nominal bolus			No bolus
	QvIOB	Qv	p-value	QvIOB	Qv	p-value
Mean glucose	136.2	137.9	< 0.001†	156.0	161.3	< 0.001†
SD glucose	30.4	30.2	0.151	48.9	49.8	< 0.001†
LBGI	0.2	0.1	< 0.001†	0.2	0.1	< 0.001†
HBGI	2.6	2.7	< 0.001†	6.0	6.7	< 0.001†
% < 54	0.0	0.0	0.745	0.1	0.1	0.065
% < 70	0.2	0.2	0.376	0.5	0.3	< 0.001†
% ∈ [70, 140]	64.5	63.2	< 0.001†	52.3	49.1	< 0.001†
% ∈ [70, 180]	88.6	88.0	< 0.001†	69.3	66.6	< 0.001†
% > 180	11.2	11.9	< 0.001†	30.2	33.1	< 0.001†
% > 300	0.1	0.1	0.986	1.2	1.6	< 0.001†
${\sum }_k{u}_{\mathrm{IN},k}$	40.4	39.9	< 0.001†	36.7	35.6	< 0.001†
${\sum }_k{u}_{\mu b,k}$	27.5	26.9	< 0.001†	36.7	35.6	< 0.001†
${\sum }_k{u}_{\text{ffb},k}$	12.9	13.0	0.156	0.0	0.0	—
${\sum }_k\Delta u_{\mu b,k}>0$	5.7	4.8	< 0.001†	13.6	12.0	< 0.001†
${\sum }_k\Delta u_{\mu b,k}\le 0$	−5.8	−5.4	< 0.001†	−4.6	−4.0	< 0.001†
CVGA zone: A	164	153	—	1	0	—
CVGA zone: B	792	808	—	819	804	—
CVGA zone: C	3	4	—	91	116	—
CVGA zone: D	40	34	—	63	62	—
CVGA zone: E	1	1	—	26	18	—
Overnight
Mean glucose	115.8	118.7	< 0.001†	116.7	121.1	< 0.001†
SD glucose	10.3	10.3	0.848	11.3	11.5	0.381
LBGI	0.2	0.2	< 0.001†	0.3	0.2	< 0.001†
HBGI	0.3	0.4	< 0.001†	0.3	0.6	< 0.001†
% < 54	0.0	0.0	0.542	0.2	0.1	0.255
% < 70	0.2	0.1	0.389	0.6	0.4	0.028
% ∈ [70, 140]	97.1	95.7	< 0.001†	95.1	92.1	< 0.001†
% ∈ [70, 180]	99.7	99.8	0.780	99.2	99.1	0.442
% > 180	0.1	0.1	0.514	0.3	0.6	0.001
% > 300	0.0	0.0	—	0.0	0.0	—
${\sum }_k{u}_{\mathrm{IN},k}$	7.0	6.8	< 0.001†	7.0	6.8	< 0.001†
${\sum }_k{u}_{\mu b,k}$	7.0	6.8	< 0.001†	7.0	6.8	< 0.001†
${\sum }_k{u}_{\text{ffb},k}$	0.0	0.0	—	0.0	0.0	—
${\sum }_k\Delta u_{\mu b,k}>0$	1.7	1.3	< 0.001†	1.7	1.2	< 0.001†
${\sum }_k\Delta u_{\mu b,k}\le 0$	−2.0	−1.7	< 0.001†	−1.9	−1.7	< 0.001†
CVGA zone: A	789	831	—	797	824	—
CVGA zone: B	198	160	—	168	152	—
CVGA zone: C	13	9	—	35	24	—
CVGA zone: D	0	0	—	0	0	—
CVGA zone: E	0	0	—	0	0	—

Data are summarized as mean and the significant effects are highlighted using †. The percent time threshold are in mg/dL. The unabridged table is available at https://doi.org/10.5281/zenodo.8034034.

Fig. 4 compares the CDF and CVGA plots for the no bolus test condition for both time periods. The QvIOB controller mean CDF stayed on the left and did not cross the Qv controller mean CDF highlighting small but consistent improvement. The CVGA plot indicates a decrease in maximum as well as minimum glucose with most of the CVGA points in zone B and zone A. However, there was a small increase in CVGA points in the zone D and zone E during the complete period and in the lower zone C during the overnight period indicating the QvIOB controller was more aggressive during nominal scenario for a few virtual subjects, but this was deemed acceptable for clinical use. Consequently, the amount of insulin below basal was slightly higher (−4.6 U versus −4.0 U, p < 0.001 for the no bolus and complete period, or approximately 15% more) indicating that longer pump suspensions were needed.

Fig. 4:

Comparison of glycemic performance for Scenario B and no bolus test condition for both time periods. The plots have the same key as Fig. 3 with CVGA lower bound at 110 mg/dL.

V. Clinical Validation

The proposed controller was evaluated as part of three separate multicentre clinical studies using the iAPS platform running on an unlocked smartphone [36]. These were a two-week outpatient study in 10 adults with in-clinic induced stress (NCT04142229) [8], a three-month outpatient study in 35 adults with data-driven adaptation supervisory layer (NCT04436796) [37], and a 48-hour simulated outpatient study in 11 adult pregnant women (NCT04492566) [38]. In these studies, participants announced meals to the system where the closed-loop arm outperformed or was at least as good as the standard-of-care arm. Table III compares the glucose time in range outcomes from the aforementioned clinical studies as well other recent studies utilizing different versions of zone MPC controller in adults, adolescents and children, and pregnant women cohorts.

TABLE III:

Comparison of glucose time-in-range outcomes from seven recent clinical studies utilizing different versions of zone MPC controller.

					Day and Night			Overnight
#	Study	Controller	T1D Cohort	n (Duration)	[70 180]	> 180	< 70	[70 180]	> 180	< 70
1	Forlenza et al. [39]•	[23]	Adult	19 (2 weeks)	71.6	24.9	1.3	73.7	25.4	0.7
2	Pinsker et al. [40]⋄	[20], [41]	Adult	15 (48 hours)	88.0	10.5	1.5	88.1	7.5	1.1
3	Deshpande et al. [32]	[23]*	Pediatric	20 (48 hours)	63.5	34.7	1.8	61.9	37	1.1
4	Deshpande et al. [11]	[23]*	Adult	10 (2 weeks)	74.0	23.7	2.3	70.4	26.9	2.8
5	Kaur et al. [8]	Current	Adult	10 (2 weeks)	74.4	24.8	0.8	78.3	21	0.7
6	Pinsker et al. [37]	Current†	Adult	35 (13 weeks)	69	28	2.1	70	28	1.9
7	Ozaslan et al. [38]	Current, [42]‡	Pregnant women	11 (48 hours)	90.1	4.9	5	88.1	9.6	2.2

^•: data are reported as median, other study data are reported as mean;

^⋄: setpoint MPC with a modified bolus calculator;

^*updated zone boundaries as described in Section IV and [11];

^†: with a data-driven adaptation supervisory layer;

^‡: updated tuning including decreased zones boundaries of [80 110] mg/dL during day and [80 100] mg/dL during overnight targeting glucose in the range [63 140] mg/dL as described in [42].

It is not straightforward to perform a head-to-head comparison for outcomes from different studies due to difference in durations of the study, clinical protocol and level of supervision (simulated outpatient versus at-home studies), inherent variability of the cohort, and level of adherence to treatment. Overall, the proposed controller use in outpatient at-home studies was safe with no adverse events. The percent time in 70–180 mg/dL for the three listed outpatient studies were close or above the clinical recommendations of > 70% [34] for both the day and night period as well as for the overnight period. In the pregnancy with T1D study [38], given much tighter pregnancy-specific target range of [63 140] mg/dL, true indication for hypoglycemia is captured by percent time < 63 mg/dL, which was 2% in this study again satisfying the clinical recommendations [34]. It is worth noting that the time in range during the overnight hours with the QvIOB controller in [8] was lower than the 48 hours simulated outpatient study [40], which used set point MPC and a modified feedforward bolus calculator. Compared with other outpatient studies in adults, the QvIOB controller had a comparable day and night and overnight performance while was the only controller used in pregnant women cohort [38]. In the ensuing subsections, we highlight performance of the QvIOB controller during clinical validation.

A. Empirical distribution of Q During Outpatient Use

Fig. 5 compares the empirical distributions of the weight Q from all 13 participants who completed the outpatient study with induced stress [8]. In order to highlight the adaptation of Q under different glucose values, the distribution is evaluated for data partitioned by it, starting with for all $G_{\mathrm{C}\mathrm{G}\mathrm{M}}$, for glucose above clinical hyperglycemia threshold $G_{\mathrm{C}\mathrm{G}\mathrm{M}}>180$ mg/dL and below that threshold $G_{\mathrm{C}\mathrm{G}\mathrm{M}}\le 180$ mg/dL, and specifically for glucose in $G_{\mathrm{C}\mathrm{G}\mathrm{M}}\in$ [120 180] mg/dL to highlight incurred cost (14a) and (14b) due to non-zero zone excursion $\hat{\text{z}}$ from the upper zone boundary of 120 mg/dL. The data distributions are shown for both time periods.

Fig. 5:

Comparison of histograms of Q during two-week clinical study with induced stress [8] for (A) day and night and (B) overnight time period. Each period shows subplots grouped by different four different glucose $G_{\text{CGM }}$ thresholds where the data is plotted for $i=1$. The red dashed lines represents the mean value.

The controller weight adaptation was more aggressive during the overnight hours $(Q=2(p<0.001\left)\right)$ versus the complete period $(Q=1.5(p<0.001\left)\right)$. With feedforward boluses during the day, the ability or need for the controller to increase the weight was limited. For $G_{\mathrm{C}\mathrm{G}\mathrm{M}}>180\mathrm{m}\mathrm{g}/\mathrm{d}\mathrm{L},Q$ was similar to one for both overnight period and the complete period $\left(1.1\right(p=0.586\left)\right)$ and $\left(0.9\right(p=0.011\left)\right)$ indicating that the $IOBn$ value was saturated. On the other hand, for $G_{\text{CGM}}\le 180$ mg/dL, and in particular for glucose in 120–180 mg/dL range, Q was larger than one, with overnight period showing larger values than the complete period (2.2 (p < 0.001)) and (1.7 (p < 0.001)), and (2.0 (p < 0.001)) and (1.5 (p < 0.001)), respectively.

It is worth noting that in the empirical distributions, particularly during the complete period for all glucose values, there were three distinct concentration of values: $Q=0$ for negative and falling $v,Q=1$ when $v$ and $IOBn$ were high such as following a meal, and $Q=3$ for condition of both low $v$ and $IOBn$ as described in Section III-B. Finally, increases in Q may not have necessarily resulted in more insulin delivery as either the cost may not have increased if $\hat{\text{z}}$ was zero, or the input constraint (29) may have limited insulin delivery. To contrast with the reference Qv controller [23], the Q values in that case can only change between 0 and 1.

B. Adaptation of Q During Induced Stress

Fig. 6 demonstrates the adaptation of Q during prolonged hyperglycemia from induced pharmacological stress. The adult participant underwent supervised stress session using oral hydrocortisone (total of 80 mg in three separate doses over 8 hours) resulting in higher insulin requirement over the nominal case [8]. The resulting prolonged hyperglycemia (subplot 1) was observable from the start of the stress session from 7 am until the next day at 12 pm, and especially during the overnight period. During the day and the evening, due to feedforward boluses (subplot 2) from large meals (subplot 1), ${\hat{\lambda }}_{\text{IOB},k}$ did not allow for much increased microboluses (subplot 2) although Q (subplot 3) was greater than one in some time segments. During the overnight period, with low $v$ (subplot 4) and $IOBn$ (subplot 5), the QvIOB controller increased controller microboluses up to 1–2.5 times the basal (subplot 2) with $Q>1$ resulting in glucose near the zone boundary by 6 am. It is worth noting that during the stress session, with $IOBn$ as large as 0.8, the controller was able to prevent hypoglycemia following the dinner with zero microboluses as well as following large positive rise in $v$ and $IOBn>0$ such as during breakfast, kept $Q=1$ to limit overdelivery. Note that the insulin delivery would have been higher, especially during the overnight period, if ${\hat{\lambda }}_{\mathrm{I}\mathrm{O}\mathrm{B},k}$ were relaxed instead of use of unmodified profiles. Thus, there was a significant disturbance effect that was not timely rejected by the controller.

Fig. 6:

Adaptation of Q by the QvIOB controller during an induced stress session for an adult participant [8]. The first subplot shows CGM (left) and meal (right, triangular marker) with zone boundaries as per Appendix II-C, the second subplot shows controller microbolus (left) and feedforward boluses (right, triangular marker), followed by subplots for Q, $v$ and $IOBn$. Each marker • denotes a controller instance k. For each k, the Q and $v$ trajectories only show value for $i=1$. The overnight period is highlighted by a red dashed rectangle.

C. Advisory Mode Simulation

To demonstrate the benefit of the proposed design if it were used in other clinical studies, data from an adolescent participant using the Qv controller [32] was fed through the QvIOB controller to retrospectively compare the two insulin deliveries. Fig. 7 demonstrates advisory mode simulations [43] during an instance of overnight prolonged hyperglycemia (subplot 1) following a large meal of 116 grams (subplot 1) and an accompanying bolus of 18.5 U around 6 pm (subplot 2). The controller microboluses (subplot 2) from the Qv controller with a maximum of $Q=1$ (subplot 3) were not sufficient to limit hyperglycemia where the constraints (29) were not a limitation since this participant had small a ${\mathrm{C}\mathrm{F}}_k$ (or nominally high insulin requirements). Starting at approximately midnight following the relaxation of ${\hat{\lambda }}_{\mathrm{I}\mathrm{O}\mathrm{B},k}$ after a meal and with both $v$ (subplot 4) and $IOBn>0$ (subplot 5) being low, the QvIOB controller could adapt $Q>1$ to increase insulin microboluses (subplot 2). In the advisory mode, given the same inputs and a fixed output, a valid comparison can only be made for the same time instance k, and hence it bears emphasizing that although the simulations are informed by data, they are hypothetical comparisons that cannot account for other factors such as changing output due to suggested input from the new controller. On comparing with the nominal basal rate (subplot 2), at each k for the overnight period, the QvIOB controller suggested insulin varied from 1–4 times the basal in comparison to the clinical data that varied from 1 – 2.5 times basal, suggesting if the QvIOB controller were used in this study, it could have limited hyperglycemia. It is worth noting that the QvIOB controller response to falling glucose levels, following the meal around 7 pm, was similar to the clinical data highlighting that it retained the design to prevent hypoglycemia.

Fig. 7:

Advisory mode simulations comparing QvIOB controller (blue) with clinical data from Qv controller (red) for an adolescent participant [32]. The subplots have same key as Fig. 6. Each marker • denotes a controller instance k. Glucose and hence glucose ROC values are identical in both comparisons. For each k, the Q and $v$ trajectories only show value for $i=1$.

VI. Conclusion

A zone MPC formulation with adaptive weighting for short time scales using a joint function of predicted glucose ROC and normalized IOB was proposed to address prolonged hyperglycemia while retaining the feature to avoid hypoglycemia. The QvIOB controller was implemented on a smartphone-based platform, the algorithm was verified via numerical simulations, which included scenario with an increase in insulin requirements, and validated in three FDA approved clinical studies. The continuous adaptation scheme using the glucose ROC-IOB plane was capable of decreasing mean glucose and increasing time in 70 – 180 mg/dL, with significant increase in mean $Q>1$ for several situations, both in simulations and during real-life use that highlights its utility and safety.

The other zone MPC formulations with adaptive weights on the input [20], [21] may not elicit appropriate control response under conditions of prolonged hyperglycemia due to the trust index based on recent prediction history being not high enough [20] and smaller control penalties weight due to low $v$ [21]. The QvIOB controller, in those conditions, can adjust glucose using adaptive weight based on feedback and recorded deliveries independently of glucose magnitude, and hence does not require ad hoc rules based on glucose levels or explicit user intervention. This also resulted in improved performance over the previous Qv controller for the nominal scenario simulations while with a minor increase in hypoglycemic risk and longer pump suspensions. Thus, improvements during the induced resistance scenario were not achieved just by an overly aggressive controller but one that can adjudicate when to intervene, and thus performs equitably during both resistance and nominal scenarios.

Broadly speaking, other MPC formulations with various adaptation and personalization schemes, discussed in the Introduction, particularly when addressing glucose regulation during physiological conditions that either have unreliable measurements of underlying variables or limited model-based description or both, such as during stress, could also exploit additional patterns in the plane of measured physiological states.

Several design parameters in MPC, such as weights in the cost function, are typically constant after being tuned to achieve the desired performance [12]. On the other hand, the tuned controller deployed to units with large variability, such as due to inter-person variability in this application, may require retuning that may not be always possible or safe after deployment. The aspect of varying weights systematically using feedback from a function of measured variables could make the controller more adaptive to each unit’s characteristics while using a traditional MPC framework.

A few limitations of the QvIOB controller are the timing of gradual increase in insulin microboluses only for low $v$ and $IOBn$ may have limited effect on reducing meal peak and glucose variability (glucose standard deviation). The adaptive weights with dependence on fixed thresholds, in particular $IOBn$, may have limited ability to adapt to large changes in insulin requirements leading to an offset. In the future, the weighting surface should be tailored to a person’s changing requirements by learning from data including from trajectories in the glucose ROC-IOB plane.

Appendix I

Constructing the adaptive weight Q

A. Normalization

Given significant difference in insulin requirements between persons, such as by daily caloric (and carbohydrate) intake and lifestyles, the IOB value has to be normalized. People with T1D have record of their open-loop total daily insulin (TDI), which is the sum of all insulin delivered over 24 hours, consisting of basal (total daily basal (TDBa)) and feedforward insulin boluses (total daily boluses (TDBo)). As IOB is calculated in deviation of basal and due to day-by-variability in TDBo, the normalization is performed using long term average of TDI and TDBa as:

$$IOB{n}_k=\frac{IO{B}_k}{\text{TDI}-\text{TDBa}}.$$ (19)

Similarly, the magnitude of glucose ROC varies between persons and in the young children particularly (data not shown); however, the variations are much smaller and not straightforward to normalize, and hence model-based $v$ are directly used.

B. Sample Points and Univariate Weighting Functions

The thresholds for $v$ and $IOBn$, and the maximum value of Q were tuned using the 10 adult virtual subject cohort of the UVA/Padova metabolic simulator and clinical data. Starting from the left of $v$ axis in Fig. 8(A), sufficiently large negative threshold was set at $v=-1.5$ mg/dL/min for weight $Q=ϵ>0$, and the threshold for Q = 3 was set at $v=\pm 0.25$ mg/dL/min, and the sufficiently large positive threshold was set at $v=1.5$ mg/dL/min for weight Q = 1. The threshold for $IOBn$ was calculated assuming a 50 – 50 split between TDBa and TDBo. Considering three large meals in a day, $IOBn$ would peak at 0.33. A threshold of around half of that $IOBn=0.15$ was chosen for Q > 1 as shown in Fig. 8(B). In the code, any $IOBn>0.15$ was truncated at 0.15. In order to create a joint surface, in addition to the joint points associated for $Q\left(v\right)$ when $IOBn=0$ and $Q\left(IOBn\right)$ when $v=0$, both constructed using linear equations connecting the specified thresholds as shown in Fig. 8, additional link points were provided as $(v,IOBn,Q)$ triplet: (−3, 0.15, ϵ),(−1.5, 0.15, ϵ) and (3, 0.15, 1) to cover the extremities of $v$ for the upper threshold of $IOBn$.

Fig. 8:

Sample points used for interpolation shown for (A) $Q\left(v\right)$ and (B) $Q\left(IOBn\right)$.

C. Surface Fitting

The sample points were interpolated using a triangulation-based natural neighbor interpolation to create the surface shown in Fig. 2. The surfaces between $\left|v\right|\le 1.5$ mg/dL/min and partitioned by $v=0$ were fit by a second degree polynomial function using least squares. The estimated parameters are reported in Table IV where the R-squared value was greater than 0.98 for both surfaces.

Finally, the authors also experimented with scaling Q with singular variable defined as a direct interaction between $v$ and $IOBn$ as $Q(v\times IOBn)$ but it did not yield as flexible weighting as the joint function proposed in this work.

Appendix II

MPC Problem Setup: Model, Constraints and Zones

A. Insulin-Glucose Model

The model predictions are based on a linear three-state control-relevant personalized model proposed in [44]. The model was identified using simulated data from 10 UVA/Padova simulator virtual subjects. At steady-state and in open-loop without any meal disturbance, insulin bolus of 1 U, 2 U after 24 hours, and 3 U after 48 hours were given and data was sampled at five minute intervals. Input insulin deviations u and output glucose deviation y around the equilibrium are defined as:

$$u=u_{\text{IN}}-u_{\text{basal}},y=G_p-y_s,$$ (20)

where $u_{\mathrm{I}\mathrm{N}}$ is delivered insulin, $u_{\text{basal}}$ is virtual subject-specific basal insulin converted from U/h to microboluses (U/5-min), and $y_s=110$ mg/dL is steady-state glucose reference. An output-error structure [45] of third order with two sample delay was identified that achieved a high fit [44]. Informed by the identified models

$$M\left(q^{-1}\right)=\frac{K{q}^{-3}}{\left(1-p_1{q}^{-1}\right){\left(1-p_2{q}^{-1}\right)}^2}$$ (21)

was used to capture control-relevant dynamics around the bandwidth by underestimating the phase and overestimating the gain compared to the identified models by fixing the poles at $p_1=0.98$ and $p_2=0.965$ while the gain parameter K was personalized using the “1800 rule” based on virtual subject’s total daily insulin (TDI > 0) with unit scaling: $K=-\frac{1800\times 90\times \left(1-p_1\right){\left(1-p_2\right)}^2}{\text{TDI}}$ [31], [44], where $q^{-1}$ is the backward shift operator. A state-space realization of M [23], [31] with an additional output of glucose velocity $v$ used in this work is:

$$x_{k+1}=A{x}_k+B{u}_k,$$ (22)

$$y_k=C_y{x}_k,v_k=C_v{x}_k,$$ (23)

where $A\in {\mathbb{R}}^{3\times 3},B\in {\mathbb{R}}^{3\times 1},C_y\in {\mathbb{R}}^{1\times 3}$ and $C_v\in {\mathbb{R}}^{1\times 3}$ are defined as:

$$A :=\left[\begin{array}{ccc}{p}_1+2p_2& -2p_1{p}_2-p_2^2& p_1{p}_2^2\\ 1& 0& 0\\ 0& 1& 0\end{array}\right],$$ (24)

$$B:=-\frac{1800\times 90\times \left(1-p_1\right){\left(1-p_2\right)}^2}{\mathbf{\text{TDI}}}{\left[\mathrm{1\; 0\; 0}\right]}^{\top },$$ (25)

$$C_y:=\left[\mathrm{0\; 0\; 1}\right],C_v:=\left[\mathrm{0.1\; 0}-0.1\right].$$ (26)

It follows from the state-space realization that the three current state elements are related to predicted glucose as: $y_k=x_{3,k},y_{k+1}=x_{2,k},y_{k+2}=x_{1,k}$. Given a new CGM measurement every five minutes, $v_k$ is calculated as difference of the first and the third state element over 10 minutes resulting in predicted glucose ROC in mg/dL/min. Finally, at each $k$ with new glucose measurement $G_{\mathrm{C}\mathrm{G}\mathrm{M},k}$, the state $x_k$ is updated by an observer as:

$$x_k=A{x}_{k-1}+B{u}_{k-1}+L\left(\left(G_{\text{CGM},k}-y_s\right)-C{x}_k\right),$$ (27)

where the gain L is calculated by solving a dual of the state feedback problem where the weights were tuned using the UVA/Padova metabolic simulator to achieve an acceptable noise rejection performance [31].

B. Calculation of IOB and Input Constraints

The IOB values are calculated directly using a convolution of pre-specified decay curves ${\mathrm{\Theta }}_l$ with decay time $l$ in hours [23], [46] and recent insulin delivery history $U$ offset from basal as:

$$IO{B}_k=\sum _{\tau =1}^m{\text{Θ}}_{\tau }^l{U}_{k-\tau },$$ (28)

where $m=8\times \frac{60}{5}=96$ covers a history of eight hours, which is sufficient for physiological effect of insulin. The decay curve $\mathrm{\Theta }$ is parameterized using a cosine function for smooth switching between curves of different length [23]. Curve length of $l=5$ hours is used for calculation of $IOBn$ in (12).

The input $u_k$ is constrained by scalar time-varying bounds:

$$u_k\in \left[{\check{\lambda }}_k{\hat{\lambda }}_k\right].$$ (29)

The lower bound ${\check{\lambda }}_k\le 0$ enforces the nonnegativity requirement for insulin delivery and is equal to the negative basal ${\check{\lambda }}_k=-u_{\text{basal },k}$. The upper bound ${\hat{\lambda }}_k\ge 0$ is calculated using two conditions. First, the insulin delivery is limited by an amplitude constraints ${\hat{\lambda }}_{\mathrm{U}\mathrm{B},k}$ equal to four times $u_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{a},k}$ from 10 pm to 4 am, and equal to 1 U for other times of the day. Second, the insulin delivery is upper bounded by an IOB constraint to prevent overdelivery as:

$${\hat{\lambda }}_{\text{IOB},k}=\text{max}\left\{\frac{G_{\text{CGM},k}-y_s}{{\text{CF}}_k}-{\overline{IOB}}_k,0\right\},$$ (30)

$${\overline{IOB}}_k=\sum _{\tau }{\text{Θ}}_{\tau }^4{U}_{k-\tau }^{\text{ffb}}+\sum _{\tau }{\text{Θ}}_{\tau }^{[\mathrm{2\; 6}],G_{\text{CGM}}}{U}_{k-\tau }^{\mu \text{b}},$$ (31)

where ${\mathrm{C}\mathrm{F}}_k$ is the person-specific time-varying correction factor (mg/dL/U) defined as the amount of change in glucose level due to an unit insulin bolus, which is indicative of insulin sensitivity, and ${\overline{IOB}}_k$ is a sum of IOB from feedforward boluses $U_k^{\mathrm{f}\mathrm{f}\mathrm{b}}$ using a fixed curve ${\mathrm{\Theta }}^4$ and IOB from the controller microboluses $U_k^{\mu \mathrm{b}}$ using a varying curve [46] ${\mathrm{\Theta }}^{\left[\mathrm{2\; 6}\right],G_{\mathrm{C}\mathrm{G}\mathrm{M}}}$ as described in Appendix A.4 of [23] based on $G_{\text{CGM}}\in$ [120 300] $\mathrm{m}\mathrm{g}/\mathrm{d}\mathrm{L}$ and $l\in \left[2\mathrm{ }6\right]$. The upper bound is selected as:

$${\hat{\lambda }}_k=\text{min}\left\{{\hat{\lambda }}_{\text{UB},k},{\hat{\lambda }}_{\text{IOB},k}\right\}.$$ (32)

C. Zones

The controller cost function penalized glucose excursion ${\text{z}}_k$ from the time-varying zone scalar boundaries $\left[\begin{array}{ll}{\check{\xi }}_k& {\hat{\xi }}_k\end{array}\right]$ calculated using a zone excursion function $Z$ as:

$${\text{z}}_k=Z(y,k):=\underset{\alpha \in ℝ}{\text{arg min}}\left\{{\alpha }^2\mid y+y_s-\alpha \in \left[{\check{\xi }}_k,{\hat{\xi }}_k\right]\right\},$$ (33)

that results in ${\text{z}}_k$ calculated from the nearest zone boundary, and is zero if glucose is within the zone boundaries. The day zone boundary values were [90 120] mg/dL from 6 am till 10 pm and the overnight zone boundary values were [100 120] mg/dL from midnight to 4 am with smooth transition between the diurnal zone boundaries [31].