Hypoglycaemia‐free artificial pancreas project

Abstract

Driving blood glycaemia from hyperglycaemia to euglycaemia as fast as possible while avoiding hypoglycaemia is a major problem for decades for type‐1 diabetes and is solved in this study. A control algorithm is designed that guaranties hypoglycaemia avoidance for the first time both from the theory of positive systems point of view and from the most pragmatic clinical practice. The solution consists of a state feedback control law that computes the required hyperglycaemia correction bolus in real‐time to safely steer glycaemia to the target. A rigorous proof is given that shows that the control‐law respects the positivity of the control and of the glucose concentration error: as a result, no hypoglycaemic episode occurs. The so‐called hypo‐free strategy control is tested with all the UVA/Padova T1DM simulator patients (i.e. ten adults, ten adolescents, and ten children) during a fasting‐night scenario and in a hybrid closed‐loop scenario including three meals. The theoretical results are assessed by the simulations on a large cohort of virtual patients and encourage clinical trials.

1 Introduction

Insulin was discovered almost 100 years ago. To date, it is the only medical treatment available for type‐1 diabetes and consists either in multiple daily insulin injections or in a continuous subcutaneous insulin infusion. For this purpose, basal‐bolus schemes are widely used. Bolus advisers are designed to help the patient to compute bolus doses.

The design of a closed loop control of blood glucose (BG) in type‐1 diabetes has been initiated 50 years ago [1]. However, the so‐called artificial pancreas (AP) remains a challenge because the system is subject to severe input and output constraints:

• hypoglycaemia (BG < 70 mg/dl) must be avoided,

• the control has to be positive: once it has been injected, insulin cannot be withdrawn from the organism.

The state‐of‐the‐art algorithms for the AP can roughly be split into two main streams [2]:

• Proportional–integral–derivative (PID) and

• model‐predictive control (MPC).

On one hand, PID does not require any mathematical model and is a simple design strategy. However, it cannot cope with the positivity constraints and the insulin pump has eventually to be switched off. The most advanced product in that direction is certainly the Minimed 670G from Medtronic which received the Food and Drug Administration (FDA) approval on September 2016 [3]. This AP system operates under a PID + insulin feedback control law and includes an algorithm that predicts 30 min in advance the occurrence of low glucose levels. In the latter case, insulin delivery is automatically switched off. This safety process leaves the system in open loop until it recovers from hypoglycaemia. On the other hand, MPC is popular because constraints on the control [4] are dealt with in some numerical optimisation process. MPC allows different cost functions to reduce hyperglycaemia and avoid hypoglycaemia, e.g. an ad hoc asymmetric cost function [5]. MPC is the basis of the Diabeloop project [2] in France and of the Cambridge AP project [6]. For both options, the main open problem for any glycaemia regulation control algorithm is to avoid hypoglycaemia because it can rapidly lead to severe complications as cerebral damage or even death [7].

The contributions and approach in this study are quite different: a positive state feedback control – the so‐called hypo‐free strategy (HFS) – is designed and uses the structure of the system or of its model, so that the closed‐loop system remains in the region that respects the positivity constraints. As it is model based, the choice of an adequate model is argued and the long‐term type‐1 model [8] is used later on. It is rigorously proven that positivity of the control is guaranteed, as well as the avoidance of hypoglycaemic episodes. The theoretical proof makes use of the theory of positively invariant sets. Besides, in silico tests are worked out in a realistic environment: the UVA/Padova simulator is used as it is the only one which got the FDA agreement for pre‐clinical tests [9]. As a matter of fact, the simulation results validate the theoretical results. The new control‐law appears to formalise the dynamic version of the static bolus calculation provided by bolus wizards and thus, it is meaningful for physicians, pump manufacturers, and patients as well. The HFS is individualised as it uses the patient's own insulin therapy parameters such as the correction factor (CF), the patient's specific basal rate and the duration of insulin action (DIA). This feature is a key factor as it does not require cumbersome expert optimisation procedure. The HFS is the automated part of a basal/hybrid closed‐loop and embodies step 4 as described in [10]. This control law gathers clinical practice, AP systems hard constraints and theoretical proof.

The paper is organised as follows: Section 2 argues the requirements of the mathematical model and introduces the HFS control law. In Section 3, the properties of the closed loop as stability and positivity of input/state are demonstrated in the nominal case. A tuning procedure is derived from a robustness analysis against parameter uncertainties. In Section 4, the performance of the control law is assessed through in silico tests using the UVA/Padova T1DM simulator. Finally, conclusions and perspectives are presented.

2 HFS control law

2.1 Difficulty of computing the bolus of insulin

Same food, same injection, at the same time of the day was an early option for type‐1 diabetes treatment but it was not very satisfactory. Functional insulin therapy is an educational programme that helps the patient to compute insulin injections [11]. It defines tools such as the insulin sensitivity factor (ISF), also known as the CF, and the carbohydrate‐to‐insulin ratio (CIR). These tools, empirically estimated from clinical protocols [12], are used to compute insulin boluses depending on BG level, BG target, meal carbohydrates (CHO) content, and the insulin on board (IOB) still active from previous boluses.

• The correction bolus $U_{\text{BG}}$, depending on the patient's CF, BG level and BG target, is

  $$U_{\text{BG}}=\frac{\mathrm{B}{\mathrm{G}}_{\text{level}}-\mathrm{B}{\mathrm{G}}_{\text{target}}}{\text{CF}}$$ (1)
• The meal bolus $U_{\text{Carb}}$ depends on the patient's CIR and the amount of CHO in the meal

  $$U_{\text{Carb}}=\frac{\text{CHO}}{\text{CIR}}$$ (2)

• The IOB is the number of insulin units that are still active in the body. IOB is a function of the DIA and the number of previous boluses. It is established from insulin action curves but is computed in different ways according to the different bolus wizards [13].

Nowadays, glucometers and insulin pumps include a bolus wizard. Physicians inform these calculators with individualised values of CIR, CF, DIA, and BG target according to the time of the day. Thus diabetic patients only have to enter the estimated amount of CHO to obtain insulin dose recommendations as

$$U_{\text{Bol}}=U_{\text{BG}}+U_{\text{Carb}}-\text{IOB}$$ (3)

The injection computed with (3) in clinical practice and by type‐1 diabetics in everyday life is expressed in simple terms of needs $\left(U_{\text{BG}}+U_{\text{Carb}}\right)$ minus the amount of insulin, which is still active (IOB). Nevertheless, patients have difficulties in computing the correct insulin doses because

• CF might vary with the time of the day [14], physical activity [15], stress or illness.

• CIR varies according to meal composition [16].

• Incorrect estimation of DIA induces mismatch in the IOB and insulin injection. As a consequence, hypoglycaemia occurs when DIA is underestimated while overestimation of DIA leads to hyperglycaemia. Determination of individualised DIA remains a critical point [17].

• The amount of CHO might be wrongly estimated.

2.2 IOB at time t

In this section, the nominal state space model of the glucose–insulin dynamics is recalled from [8]. Then it is shown for the first time that the IOB at time t can be computed as a combination of those states.

The long‐term model of the glucose–insulin dynamics for type‐I diabetes in [8] is used in a modified version. The model is not the scope of this study; it is used to design the control law, to derive a rigorous proof of the input and state variables positivity (in Section 3.1) and to compute the functional insulin‐therapy parameters to individualise the control law (in Section 4.2). This model features realistic equilibrium, a long‐term fit with clinical data, and the possibility to compute basal rate, insulin sensitivity, CIR from its parameters identified from clinical data [8]. Some of the assumptions used for this model and argued in [8] have recently been used to update alternative models as in [18]. The model has also been used in several works [19, 20, 21].

Derive the model in the simple case when there is no meal, and let $x_1$ be the BG; $x_2$ and $x_3$ are, respectively, the plasma and subcutaneous compartment insulin rate [U/min]. The input u is the insulin infusion rate [U/min]. ${\theta }_1$ [mg/dl/min] is the net balance between the endogenous glucose production and the insulin independent consumption, ${\theta }_2$ [mg/dl/U] is the ISF and ${\theta }_3$ [min] is the time constant of the insulin subsystem related to the DIA. The model is

$${\dot{x}}_1={\theta }_1-{\theta }_2{x}_2,$$ (4)

$${\dot{x}}_2=-\frac{1}{{\theta }_3}{x}_2+\frac{1}{{\theta }_3}{x}_3,$$ (5)

$${\dot{x}}_3=-\frac{1}{{\theta }_3}{x}_3+\frac{1}{{\theta }_3}u.$$ (6)

Note that all the states $x_i$ and the control variable u represent physiological entities, therefore they are positive variables.

The insulin injection rate u is mostly the sum of a basal rate and boluses (as u is an infusion rate, the bolus $u_{\text{bol}}$, which is an insulin injection is defined as a temporary supplementary rate [U/min].) $u=U_{\text{Bas}}+u_{\text{bol}}$. Thus the states $x_2$ and $x_3$ can also be written as the corresponding sums

$$x_3=x_{3_{\text{Bas}}}+x_{3_{\text{bol}}}=x_{3_{\text{Bas}}}+{\tilde{x}}_3,$$ (7)

$$x_2=x_{2_{\text{Bas}}}+x_{2_{\text{bol}}}=x_{2_{\text{Bas}}}+{\tilde{x}}_2.$$ (8)

When fasting, the correct basal insulin rate is established when glycaemia is maintained constant [12]. The equilibrium values $x_{2_{\text{Bas}}}$ and $x_{3_{\text{Bas}}}$ are thus

$$U_{\text{Bas}}=\frac{{\theta }_1}{{\theta }_2}\Rightarrow x_{3_{\text{Bas}}}=x_{2_{\text{Bas}}}=\frac{{\theta }_1}{{\theta }_2},$$ (9)

and from (4) and (9) the glycaemia dynamics is written as

$$\begin{array}{rl}{\dot{x}}_1& ={\theta }_1-{\theta }_2{x}_2={\theta }_1-{\theta }_2\left(x_{2_{\text{Bas}}}+{\tilde{x}}_2\right),\\ {\dot{x}}_1& ={\theta }_1-{\theta }_2\left(\frac{{\theta }_1}{{\theta }_2}+{\tilde{x}}_2\right),\\ {\dot{x}}_1& =-{\theta }_2{\tilde{x}}_2.\end{array}$$ (10)

A physiological definition of IOB is the amount of insulin from previous boluses that are still active in the body or equivalently, the amount of insulin in the subcutaneous and the plasma compartments from previous boluses. According to the first statement, the state representation and the input $u_{\text{bol}}$, the IOB can be written as

$$\text{IOB}(t)={\int }_0^t\left(u_{\text{bol}}(\tau )-{\tilde{x}}_2(\tau )\right)\mathrm{d}\tau .$$ (11)

Now, merging (5) and (6)

$${\theta }_3\left({\dot{\tilde{x}}}_3+{\dot{\tilde{x}}}_2\right)=u_{\text{bol}}-{\tilde{x}}_2.$$ (12)

Assume that no bolus was made before $t=0$, thus ${\tilde{x}}_3(0)=0$ and ${\tilde{x}}_2(0)=0$. Then with (11) and (12)

$$\begin{array}{rl}\text{IOB}(t)& ={\theta }_3{\int }_0^t\left({\dot{\tilde{x}}}_3(\tau )+{\dot{\tilde{x}}}_2(\tau )\right)\mathrm{d}\tau ,\\ \text{IOB}(t)& ={\theta }_3\left({\tilde{x}}_3(t)+{\tilde{x}}_2(t)\right),\end{array}$$ (13)

which is consistent with the second physiological interpretation for IOB.

2.3 Control law design

The notions above are used to introduce a new control law called HFS. The HFS is based on the hyperglycaemia correction bolus formula (3), i.e. especially meaningful when $U_{\text{Carb}}=0$. Using (1) and (13), ISF = CF =  ${\theta }_2$, the correction bolus ${\tilde{u}}_k(t)$ is computed as

$$\begin{array}{rl}{\tilde{u}}_k(t)& =k\left(U_{\text{BG}}(t)-\text{IOB}(t)\right),\\ {\tilde{u}}_k(t)& =k\left(\frac{x_1(t)-x_{1\text{ref}}}{{\theta }_2}-{\theta }_3\left({\tilde{x}}_3(t)+{\tilde{x}}_2(t)\right)\right)\\ \text{Defining}{\tilde{x}}_1(t)& =x_1(t)-x_{1\text{ref}}\\ {\tilde{u}}_k(t)& =k\left(\frac{{\tilde{x}}_1(t)}{{\theta }_2}-{\theta }_3{\tilde{x}}_2(t)-{\theta }_3{\tilde{x}}_3(t)\right).\end{array}$$ (14)

The global infusion rate $u(t)$ will be the state feedback ${\tilde{u}}_k(t)$ modulating the constant insulin infusion rate $U_{\text{Bas}}$

$$u(t)=U_{\text{Bas}}+{\tilde{u}}_k(t).$$ (15)

A family of HFS controllers, parameterised by k [min^–1], is thus defined. The matrices A , B , and F _k are

$$\mathit{A}=\left(\begin{array}{ccc}0& -{\theta }_2& 0\\ 0& -\frac{1}{{\theta }_3}& \frac{1}{{\theta }_3}\\ 0& 0& -\frac{1}{{\theta }_3}\end{array}\right),\mathit{B}=\left(\begin{array}{c}0\\ 0\\ \frac{1}{{\theta }_3}\end{array}\right),$$

$${\mathit{F}}_k=k\left(\begin{array}{ccc}\frac{1}{{\theta }_2}& -{\theta }_3& -{\theta }_3\end{array}\right).$$

Thus, with (5), (6), (10), (14) and (15), the following closed loop will be studied in Section 3

$$\dot{\tilde{\mathit{x}}}(t)=\mathit{A}\tilde{\mathit{x}}(t)+\mathit{B}{\tilde{u}}_k(t),\tilde{\mathit{x}}(0)={\tilde{\mathit{x}}}_0.$$ (16)

$${\tilde{u}}_k(t)={\mathit{F}}_k\tilde{\mathit{x}},$$ (17)

A major property of this family of controllers is that the total quantity of injected insulin does not depend on k as proven in Section 3

$${\int }_0^{\mathrm{\infty }}{\tilde{u}}_k(\tau )\mathrm{d}\tau ={\int }_0^{\mathrm{\infty }}{\tilde{u}}_1(\tau )\mathrm{d}\tau ={\tilde{u}}_1(0).$$ (18)

The gain k allows stretching this total amount of insulin and may be used as a safety factor to recover from the inaccuracies of the mathematical model. The industrial development of the HFS is protected through the patent [22].

3 HFS properties

In this section, rigorous proofs are provided for the stability and positivity of the closed‐loop trajectories. It is proven that this feedback generates a positive control, which ensures the positivity of ${\tilde{x}}_1$, i.e. in medical terms, this property is a guarantee of no hypoglycaemic episodes. This constraint is considered in the early design of the control. For PID BG regulation, hypoglycaemia is reduced thanks to safety saturations where the insulin pump is just set off, whereas for MPC hypoglycaemia risk is reduced thanks to the cost function.

3.1 Input/state positivity

According to (16) and (17), the closed‐loop system reads

$$\begin{array}{rl}\dot{\stackrel{\mathbf{~}}{\mathit{x}}}(t)& =(\mathit{A}+\mathit{B}{\mathit{F}}_k)\tilde{\mathit{x}}(t)=\tilde{\mathit{A}}\tilde{\mathit{x}}(t),\\ \dot{\stackrel{\mathbf{~}}{\mathit{x}}}(t)& =\left(\begin{array}{ccc}0& -{\theta }_2& 0\\ 0& -\frac{1}{{\theta }_3}& \frac{1}{{\theta }_3}\\ \frac{k}{{\theta }_2{\theta }_3}& -k& -k-\frac{1}{{\theta }_3}\end{array}\right)\tilde{\mathit{x}}(t),\end{array}$$ (19)

The positivity of the input/state trajectories, i.e. ${\tilde{u}}_k(t)\ge 0$ and $\tilde{\mathit{x}}(t)\ge 0$, $\forall t\ge 0$, is proven using the notion of positively invariant sets.

Given a dynamical system $\dot{\mathit{x}}=\mathit{D}\mathit{x}$, $\mathit{x}\in {\mathbb{R}}^n$, and a trajectory $\mathit{x}(t,x_0)$, where ${\mathit{x}}_0$ is the initial condition, a non‐empty set $M\subseteq {\mathbb{R}}^n$ is positively invariant if $\forall {\mathit{x}}_0\in M\mathit{x}(t,x_0)\in M\forall t\ge 0$.

If $\mathit{G}\in {\mathbb{R}}^{r\times n}$ then $M(\mathit{G})$ denotes the polyhedron

$$M(\mathit{G})=\left\{\mathit{x}\in {\mathbb{R}}^n|\mathit{G}\mathit{x}\ge 0\right\}.$$

The polyhedral set $M(\mathit{G})$ is a positively invariant set in the sense of Definition 1 [23], if and only if there exists a Metzler matrix $\mathit{H}\in {\mathbb{R}}^{r\times r}$, i.e. $h_{\text{ij}}\ge 0$ for $i\ne j$, such that

$$\mathit{G}\mathit{D}-\mathit{H}\mathit{G}=0.$$ (20)

Here we state the positivity of the input and the states.

Define the sets $M_1=\{\tilde{\mathit{x}}\in {\mathbb{R}}^3|\tilde{\mathit{x}}\ge 0\}$, and $M_2=\left\{\tilde{\mathit{x}}\in {\mathbb{R}}^3|k\left((1/{\theta }_2){\tilde{x}}_1-{\theta }_3({\tilde{x}}_2+{\tilde{x}}_3)\right)\ge 0\right\}$. The largest polyhedral positively invariant set of system (19) is

$$M=M_1\cap M_2.$$

Consider

$$\mathit{G}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\\ \frac{k}{{\theta }_2}& -k{\theta }_3& -k{\theta }_3\end{array}\right).$$

The goal is to compute the largest positively invariant set in $M(\mathit{G})$. Using Proposition 1, it is easy to show that the whole set $M(\mathit{G})$ is positively invariant; just pick

$$\mathit{H}=\left(\begin{array}{cccc}-\frac{1}{{\theta }_3}& 0& {\theta }_2& \frac{{\theta }_2}{k{\theta }_3}\\ 0& -\frac{1}{{\theta }_3}& \frac{1}{{\theta }_3}& 0\\ 0& 0& -\frac{1}{{\theta }_3}& \frac{1}{{\theta }_3}\\ 0& 0& 0& -k\end{array}\right).$$ (21)

and the result follows. □

In practice, any trajectory starting within this polyhedron is ensured to remain in it. From a medical point of view, the positivity of the output ensures that ${\tilde{x}}_1\ge 0$, i.e. guarantees the exclusion of hypoglycaemia episodes $\left(x_1\ge x_{1_{\text{ref}}},\forall t\ge 0\right)$. Moreover, the positivity of the control stands in agreement with the management of insulin injection.

The eigenvalues of $\tilde{\mathit{A}}$ are easily computed as ${\lambda }_1={\lambda }_2=-1/{\theta }_3$ and ${\lambda }_3=-k$, and the system is stable for any $k>0$. The eigenvalues of the insulinaemia subsystem $\left(-1/{\theta }_3\right)$ are not modified by the control law. Consequently, the performance of the closed‐loop depends on the patient's parameter ${\theta }_3$.

The amount of insulin infused through the control (17) does not depend on the stretching factor k.

To prove Proposition 2, transform system (19) into its Jordan form

$$\dot{\mathit{z}}(t)={\mathit{P}}^{-1}\tilde{\mathit{A}}\mathit{P}\mathit{z}(t)=\mathit{J}\mathit{z}(t),$$ (22)

so that $\tilde{\mathit{x}}=\mathit{P}\mathit{z}$, and

$$\mathit{J}=\left(\begin{array}{ccc}-\frac{1}{{\theta }_3}& 1& 0\\ 0& -\frac{1}{{\theta }_3}& 0\\ 0& 0& -k\end{array}\right).$$ (23)

The transformation matrix P is computed as

$$\mathit{P}=\left(\begin{array}{ccc}{\theta }_2{\theta }_3& {\theta }_2{\theta }_3^2& 1\\ 1& 0& \frac{k}{{\theta }_2}\\ 0& {\theta }_3& \frac{k(1-k{\theta }_3)}{{\theta }_2}\end{array}\right).$$ (24)

In the new coordinates z , the positive orthant is positively invariant as matrix J is Metzler. With P , the state trajectories $\tilde{\mathit{x}}(t)$ are computed from the trajectories $\mathit{z}(t)$

$$\left(\begin{array}{rl}& {\tilde{x}}_1(t)={\theta }_2{\theta }_3{z}_1(t)+{\theta }_2{\theta }_3^2{z}_2(t)+z_3(t)\\ & {\tilde{x}}_2(t)=z_1(t)+\frac{k}{{\theta }_2}{z}_3(t),\\ & {\tilde{x}}_3(t)={\theta }_3{z}_2(t)+\frac{k(1-k{\theta }_3)}{{\theta }_2}{z}_3(t)\end{array}\right.$$ (25)

and the control (17) is computed from (14) and (25) as

$$\begin{array}{rl}{\tilde{u}}_k(t)& =k\left(\frac{1}{{\theta }_2}{\tilde{x}}_1(t)-{\theta }_3({\tilde{x}}_2(t)+{\tilde{x}}_3(t))\right)\\ {\tilde{u}}_k(t)& =\frac{k{(1-k{\theta }_3)}^2}{{\theta }_2}{z}_3(t)\\ {\tilde{u}}_k(t)& =z_3(0)\frac{k{(1-k{\theta }_3)}^2}{{\theta }_2}{\mathrm{e}}^{-kt}\end{array}$$ (26)

From (14) and (26) one gets

$$z_3(t)\frac{{(1-k{\theta }_3)}^2}{{\theta }_2}=\frac{{\tilde{x}}_1(t)}{{\theta }_2}-{\theta }_3({\tilde{x}}_2(t)+{\tilde{x}}_3(t))$$ (27)

Integrate (26)

$$\begin{array}{rl}{\int }_0^{\mathrm{\infty }}{\tilde{u}}_k(\tau )\mathrm{d}\tau & ={\left.-z_3(0)\frac{{(1-k{\theta }_3)}^2}{{\theta }_2}{\mathrm{e}}^{-kt}\right|}_0^{\mathrm{\infty }}\\ {\int }_0^{\mathrm{\infty }}{\tilde{u}}_k(\tau )\mathrm{d}\tau & =z_3(0)\frac{{(1-k{\theta }_3)}^2}{{\theta }_2},\text{with}(27)\\ {\int }_0^{\mathrm{\infty }}{\tilde{u}}_k(\tau )\mathrm{d}\tau & =\frac{{\tilde{x}}_1(0)}{{\theta }_2}-{\theta }_3({\tilde{x}}_2(0)+{\tilde{x}}_3(0))={\tilde{u}}_1(0)\end{array}$$ (28)

As control (26) is an exponential function depending on k, this factor k allows stretching the trajectory ensuring that the same amount of insulin is administered for all $k>0$. Setting k to $\mathrm{\infty }$ will inject the bolus at once whereas when k approaches zero the bolus will be injected in infinite time.

Meals are positive disturbances. Thus, insulin injection will increase as the response to the increase of glycaemia due to these disturbances. Hence, the glycaemia trajectory remains in the positivity invariant set $M(\mathit{G})$. Therefore, the previous positivity results remain valid under a scenario which includes a meal as well.

3.2 Robustness

In this section, a tuning procedure of the parameter k is derived through a robustness analysis with respect to parameters’ uncertainties.

3.2.1 Parameters uncertainties

Robustness with respect to parameters’ uncertainties is studied. Parameters may vary with time during the day [14], due to physical activity [15], stress or illness. The state feedback is

$${\hat{\mathit{F}}}_k=k\left[\frac{1}{{\hat{\theta }}_2}-{\hat{\theta }}_3-{\hat{\theta }}_3\right],$$ (29)

where ${\hat{\theta }}_i$ are estimates of the patient's parameters ${\theta }_i$. Considering the closed loop system matrix $\mathit{A}+\mathit{B}{\hat{\mathit{F}}}_k$, the target loop transfer is given by

$$\begin{array}{rl}\hat{L}& ={\hat{\mathit{F}}}_k(s\mathit{I}-\mathit{A}){}^{-1}\mathit{B}\\ \hat{L}& =k\frac{{\theta }_2+2{\hat{\theta }}_2{\hat{\theta }}_{3}s+{\hat{\theta }}_2{\hat{\theta }}_3{\theta }_3{s}^2}{s{\hat{\theta }}_2{\left(1+s{\theta }_3\right)}^2}.\end{array}$$ (30)

Note that the hypoglycaemia risk is increased if parameters ${\theta }_2$ and/or ${\theta }_3$ are underestimated. To illustrate this, assume that the real patient's correction is ${\theta }_2=70\text{mg}/\text{dl}$ but it is estimated at ${\hat{\theta }}_2=35\text{mg}/\text{dl}$. As a consequence, the hyperglycaemic correction bolus will be computed as the double of the required bolus and hypoglycaemia will occur. In the same way, if the DIA (related to parameter ${\theta }_3$) is underestimated, IOB at the time of the correction bolus will be underestimated and hypoglycaemia will follow.

3.2.2 Tuning procedure

The identification of model (31) from a glycaemic holter (see Fig. 1) provides the parameters with a related precision ${\theta }_i^{\ast }={\hat{\theta }}_i\pm {\mathrm{\Delta }}_{{\theta }_i}$. The parameter k is tuned assuming that parameters ${\theta }_2$ and ${\theta }_3$ are underestimated, which is the worst case scenario. Thus $\hat{L}$ is computed with $\theta =\hat{\theta }+{\mathrm{\Delta }}_{\theta }$. The following criterion is used: in case of underestimates of the parameters, the shape of $\hat{L}$ in Black–Nichols chart must be tangent to the shape of $L_{\text{target}}(s)={\displaystyle \frac{1}{{\displaystyle \frac{s}{{\omega }_n}\left(2m+s/{\omega }_n\right)}}}$ at low frequencies. This leads in closed‐loop response to a second order like with a damping ratio $m=0.7$ and an overshoot of 5% (minimum risk of hypoglycaemia). The loop transfer of the system that will lead to the reference second order is shown in Fig. 2.

Fig. 1.

Identification from a glycaemic holter of a UVA/Padova virtual patient

Fig. 2.

$L_{\text{target}}$

and

$\hat{L}$

for

${\hat{\theta }}_{2,3}=50\%{\theta }_{2,3}$

4 In silico results

4.1 UVA/Padova T1DM simulator

The distributed version of the T1DM simulator was approved by the FDA as a preclinical testing platform for control algorithms. The design and tuning of the HFS control‐law use the simple linear model described in [8], whereas the UVa/Padova simulator is based on the non‐linear model given in [24]. This simulator is an extensive description of the metabolism including a model of the insulin pump and a model of the continuous glucose monitoring (CGM) devices. The latter includes a sampling period of 5 min, a delay and a non‐Gaussian noise [25]. This simulator is employed here to demonstrate the robustness and the safety of the HFS control algorithm. Thus, the tests below include a significant mismatch. All available virtual patients of the simulator have been used in these tests.

4.2 Individualisation procedure

The controller is individualised with standard functional insulin therapy tools. The CF, basal rate, and DIA are provided by the clinical protocol, i.e. thanks to physicians analysing a glycaemic holter with standard clinical data such as CGM, insulin infusion, and CHO count. The glycaemic holter can also be used to derive the parameters ${\theta }_1$, ${\theta }_2$, and ${\theta }_3$ by means of identification as explained in [8]. For all the UVA/Padova virtual patients, a glycaemic holter was simulated in an open loop and used to derive the set of individualising parameters $\{{\theta }_1;{\theta }_2;{\theta }_3\}$ by means of identification (see Fig. 1).

• CF =  ${\theta }_2$

• Basal rate $U_{\text{Bas}}=\left({\theta }_1/{\theta }_2\right)$

• DIA related to ${\theta }_3\left({\theta }_3\simeq 12.5\times \text{DIA}\right)$

The tuning parameter k is set with respect to the tuning procedure described in Section 3.3.

4.3 Fasting scenario

The simulations are conducted on ten adults, ten adolescents, and ten children with a fasting overnight scenario:

• where initial BG at 200, 300 and 400 mg/dl at steady state under basal rate;

• each virtual patient uses a CGM device (Dexcom 70) which introduces delay and noise and has a sampling time of 5 min;

• each virtual patient uses a generic pump;

• the BG target is set to 120 mg/dl; and

• the loop was closed at 00:05.

The scenario was repeated ten times to test robustness against CGM noise.

A standard Luenberger observer was used to estimate the states ${\tilde{x}}_2$ and ${\tilde{x}}_3$.

4.4 Fasting results

The control algorithm is designed to steer the glycaemia safely (without hypoglycaemia) back to euglycaemia. The low BG Index (LBGI), as well as time percentage below the level 70 mg/dl, is used to assess the performance of the control low.

Tables 1, 2–3 show the mean of the performance indices with minimum and maximum values.

Table 1.

Ten UVA/Padova adults, fasting results

Initial BG, mg/dl	200	300	400
$\%\le 70$	0 (0; 0)	0 (0; 0)	0 (0; 0)
LBGI	0.001 (0; 0.01)	0.06 (0; 0.47)	0.23 (0; 1.39)

Table 2.

Ten UVA/Padova adolescent, fasting results

Initial BG, mg/dl	200	300	400
$\%\le 70$	0 (0; 0)	0 (0; 0)	0 (0; 0)
LBGI	0 (0; 0)	0.01 (0; 0.10)	0.05 (0; 0.23)

Table 3.

Ten UVA/Padova children, fasting results

Initial BG, mg/dl	200	300	400
$\%\le 70$	0 (0; 0)	0 (0; 0)	0 (0; 0)
LBGI	0 (0; 0)	0.03 (0; 0.16)	0.17 (0; 0.73)

As demonstrated in Section 2, Tables 1, 2–3 illustrate that hypoglycaemia is totally avoided. Moreover LBGI index is extremely low thus the risk is considered minimal. Fig. 3 displays the BG and the injection $u(t)$ computed by the HFS controller for the adults.

Fig. 3.

All adults from UVA/Padova T1DMS, fasting scenario with HFS controller and initial BG at 400 mg/dl

4.5 Meal scenario

As a proof of concept, the HFS was used in a hybrid 30 h closed‐loop scenario. In a basal/hybrid closed‐loop, the control law compensates for hypo‐ and hyperglycaemia whereas a meal bolus is delivered from carb counting. The algorithm was tested in two three meals scenarios (50 g at 07:00, 70 g at 12:00 and 80 g at 18:00) where CHO intake was wrongly estimated by the patient. The scenarios are as follows:

• an underestimation of the CHO is followed by a manual pre‐meal bolus of $70\%U_{\text{Carb}}$,

• an overestimation of the CHO is followed by a manual pre‐meal bolus of $130\%U_{\text{Carb}}$.

To assess the performance of the controller, the results were compared with an open‐loop (OL) basal‐bolus therapy where CHO intake was wrongly estimated as well. The patient

• underestimates the meal's CHO content and a bolus of $70\%U_{\text{Carb}}$ is injected;

• overestimates the meal's CHO content and a bolus of $130\%U_{\text{Carb}}$ is injected.

In a hybrid closed‐loop, each manual pre‐meal bolus generates a variable reference trajectory computed from model (31), the bolus input $U_{\text{bol}}$ and the estimated meal input $\hat{r}=U_{\text{bol}}\times \widehat{\text{CIR}}$. The tuning of the controller, the pump and CGM device remain the same as in the fasting scenario.

4.6 Meal scenario results

The LBGI, high BG index (HBGI) as well as the time percentage below the level 70 mg/dl, between 70 and 180 mg/dl, above 180 mg/dl, and mean BG are used to assess the performance of the control law.

Tables 4, 5–6 show the mean of performance indices with minimum and maximum values for both OL with basal‐bolus therapy and closed‐loop with HFS controller.

Table 4.

Adults OL versus HFS: Avg (min; max)

$50\%U_{\text{Carb}}$	OL	HFS
% time <70	0 (0; 0)	0 (0; 0)
% time in target	70 (28; 92)	78 (57; 92)
% time >180	30 (7; 71)	22 (8; 43)
LBGI	0.002 (0; 0.02)	0.01 (0; 0.05)
HBGI	6.1 (2.1; 16.8)	4.3 (2; 9.9)
mean BG, mg/dl	160 (136; 215)	149 (135; 181)
$120\%U_{\text{Carb}}$	OL	HFS
% time <70	2.8 (0; 11.2)	0 (0; 0)
% time in target	97 (89; 100)	97 (89; 100)
% time >180	0.3 (0; 3)	2.1 (0; 10.9)
LBGI	0.8 (0; 2)	0.1 (0; 0.3)
HBGI	0.6 (0.2; 1.3)	1 (0.4; 2.3)
mean BG, mg/dl	116 (107; 128)	126 (117; 141)

Table 5.

Adolescents OL versus HFS: Avg (min; max)

$50\%U_{\text{Carb}}$	OL	HFS
% time <70	0 (0; 0)	0 (0; 0)
% time in target	53 (43; 60)	59 (45; 72)
% time >180	47 (40; 57)	41 (28; 55)
LBGI	0 (0; 0)	0.03 (0; 0.16)
HBGI	10.8 (7.8; 16.3)	8.8 (4.9; 15.7)
mean BG, mg/dl	184 (170; 184)	171 (153; 206)
$120\%U_{\text{Carb}}$	OL	HFS
% time <70	9 (0; 33)	0 (0; 0)
% time in target	81 (61; 100)	85 (58; 100)
% time >180	10 (0; 39)	15 (0; 42)
LBGI	2.9 (0.05; 15.4)	0.17 (0; 0.46)
HBGI	2.5 (0.2; 9.2)	3.3 (0.5; 10.3)
mean BG, mg/dl	125 (89; 172)	139 (117; 179)

Table 6.

Children OL versus HFS: Avg (min; max)

$50\%U_{\text{Carb}}$	OL	HFS
% time <70	0 (0; 0)	0 (0; 0)
% time in target	46 (32; 66)	57 (47; 66)
% time >180	54 (34; 68)	42 (34; 53)
LBGI	0 (0; 0)	0.07 (0; 0.24)
HBGI	14.7 (8.3; 21.5)	10.5 (6.6; 18.8)
mean BG, mg/dl	203 (172; 233)	180 (161; 219)
$120\%U_{\text{Carb}}$	OL	HFS
% time <70	3.6 (0; 17.9)	0 (0; 0)
% time in target	82 (70; 99)	82 (64; 97)
% time >180	14 (0; 29)	18 (3; 36)
LBGI	0.8 (0.02; 3.1)	0.09 (0; 0.46)
HBGI	2.7 (0.2; 5.8)	3.6 (1.1; 7.6)
mean BG, mg/dl	132 (98; 151)	143 (132; 165)

The control variability grid analysis (CVGA) plot for all the UVA/Padova adolescents is given in Fig. 4 in OL and Fig. 5 with HFS.

Fig. 4.

CVGA plot in OL for the adolescents of UVA/Padova T1DMS. The diamonds represent the underestimation case and the stars represent the overestimation case

Fig. 5.

CVGA plot with HFS for the adolescents of UVA/Padova T1DMS. The diamonds represent the underestimation case and the stars represent the overestimation case

4.7 Discussion

In all scenarios, with all UVA/Padova virtual patients, hypoglycaemia is totally avoided. This is the main feature of the control law, which solves a major issue in glycaemia regulation. The theoretical results and tuning procedure are confirmed through simulations.

The meal scenario shows the effect of the control law in real life conditions. The algorithm copes with both the underdosing and overdosing of insulin boluses.

In the case of underestimates of the CHO:

• the percentage of time in the target (70–180 mg/dl) increased with the HFS in comparison with the OL;

• the percentage of time above the target (>180 mg/dl) decreased with the HFS in comparison with the OL.

The hyperglycaemia is minimised. Fig. 6 shows that the infusion rate is higher than basal after the bolus which confirms that the algorithm detects that a meal has been underestimated.

Fig. 6.

BG and average insulin injection for UVA/Padova T1DMS adolescents, for underestimated bolus injection with HFS controller

In the case of overestimates of the CHO:

• with HFS hypoglycaemia avoided, whereas many hypoglycaemia episodes occur in OL;

• the percentage of time in the target (70–180 mg/dl) is similar to HFS and OL;

• consequently, the percentage of time above the target (>180 mg/dl) slightly increases with the HFS.

Hypoglycaemia is avoided. Fig. 7 shows that the control algorithm reacts very fast after an overestimated bolus and that basal rate is restored when the algorithm computes that IOB will bring glycaemia safely to target.

Fig. 7.

BG and average insulin injection for UVA/Padova T1DMS adolescents, for overestimated bolus injection with HFS controller

The tuning procedure made the control law conservative. Consequently, hypoglycaemia is totally avoided. The drawback of this tuning procedure is that the control law may be too conservative for some patients. For adolescents 1, 4, and 7, childs 1, 2 and 8, the HBGI is above 10 in the scenario with underestimates of the bolus injection and the time above 180 mg/dl is >25% in the scenario with overestimates of the bolus injection. Thus, for these patients, the tuning parameter k might be increased to improve the performance. An adaptive tuning is under research.

5 Conclusion

Hypoglycaemia avoidance and individualisation of the controller are still open problems in the AP project. In this study, a novel closed‐loop control algorithm is developed. This controller is simply individualised using the individual characteristics of the patient (CF, basal, and DIA). Thus, it is immediately comprehensive to physicians and patients.

The main feature of this control law is to ensure the positivity of trajectories. The authors have rigorously proven that this control law guarantees that the hypoglycaemia never occurs. This property allows the controller to cope with the positivity constraint of the insulin injection.

Robustness is a decisive issue as it ensures that the controller will work safely on the non‐nominal diabetic patient model (i.e. the real patient). A robustness analysis against parameters uncertainties is used to derive a tuning procedure of the single parameter k.

Finally, through in silico tests, the theoretical results are confirmed. The HFS algorithm is tested, as a proof of concept, in a hybrid closed loop in a more realistic scenario including meals where the patient wrongly estimated his pre‐meal boluses (both underestimation and overestimation). In all cases, and for all UVA/Padova virtual patients, hypoglycaemia is totally avoided. This makes the HFS a good candidate for automated basal/hybrid closed‐loop AP device.

The satisfactory in silico results obtained on a large cohort of virtual patients encourage clinical trials. A fully‐automated version is left for further research.

The complete model reads

$$\begin{array}{rl}\left(\begin{array}{c}{\dot{x}}_1(t)\\ {\dot{x}}_2(t)\\ {\dot{x}}_3(t)\\ {\dot{x}}_4(t)\\ {\dot{x}}_5(t)\end{array}\right)& =\left(\begin{array}{ccccc}0& -{\theta }_2& 0& {\theta }_4& 0\\ 0& -\frac{1}{{\theta }_3}& \frac{1}{{\theta }_3}& 0& 0\\ 0& 0& -\frac{1}{{\theta }_3}& 0& 0\\ 0& 0& 0& -\frac{1}{{\theta }_5}& \frac{1}{{\theta }_5}\\ 0& 0& 0& 0& -\frac{1}{{\theta }_5}\end{array}\right)\left(\begin{array}{c}{x}_1(t)\\ x_2(t)\\ x_3(t)\\ x_4(t)\\ x_5(t)\end{array}\right)\\ & +\left(\begin{array}{cc}0& 0\\ 0& 0\\ \frac{1}{{\theta }_3}& 0\\ 0& 0\\ 0& \frac{1}{{\theta }_5}\end{array}\right)\left(\begin{array}{c}u(t)\\ r(t)\end{array}\right)+\left(\begin{array}{c}{\theta }_1\\ 0\\ 0\\ 0\\ 0\end{array}\right)\\ y(t)& =x_1(t)\end{array}$$ (31)

where r is the meal input, $x_4$ and $x_5$ are CHO rates [mg/min] in the duodenum and stomach, respectively. ${\theta }_4$ [dl^–1] is a gain and ${\theta }_5$ [min] is the time constant of the digestion compartment. The CIR can be expressed as $\text{CIR}={\theta }_2/{\theta }_4$.