Multi-objective blood glucose control for type 1 diabetes

Abstract

For people with type 1 diabetes, automatic controllers aim to maintain the blood glucose concentration within the desired range of 60–120 mg/dL by infusing the appropriate amount of insulin in the presence of meal and exercise disturbances. Blood glucose concentration outside the desired range can be harmful to an individual’s health but concentration below 60 mg/dL, a state known as hypoglycemia, is considered to be more harmful than the concentration above 120 mg/dL, a state known as hyperglycemia. In this paper, two techniques to address this issue within a multi-parametric model based control framework are presented. The first technique introduces asymmetry into the objective function to penalize the deviation towards hypoglycemia more than the deviation towards hyperglycemia. The second technique is based upon placing higher priority on satisfaction of constraints on hypoglycemia than on satisfaction of constraints on hyperglycemia. The performance of both the control techniques is analyzed and compared in the presence of disturbances.

Appendix

This appendix presents an algorithm for the solution of multi-parametric Mixed Integer Linear Programs (mp-MILP) of the following form [9]:

$$ \begin{aligned}&J(x(t)) = \mathop {\min }\limits_{{\pi_{c} ,\pi_{d} }} \phi_{1}^{\text{T}} \pi_{c} + \phi_{2}^{\text{T}} \pi_{d} \\&s.t.\quad G_{c} \pi_{c} + G_{d} \pi_{d} \le S + Fx(t)\end{aligned} $$

(A.1)

where ϕ ₁ and ϕ ₂ are constant vectors, π _c and π _d are vectors of continuous and discrete variables, respectively, and G _c , G _d , S and F are constant vectors and matrices of appropriate dimensions.

An initial feasible π_d is obtained by solving the following MILP:

$$ \begin{aligned}&\mathop {\min }\limits_{{\pi_{c} ,\pi_{d} ,x(t)}} \phi_{1}^{\text{T}} \pi_{c} + \phi_{2}^{\text{T}} \pi_{d} \\&s.t.\quad G_{c} \pi_{c} + G_{d} \pi_{d} \le S + Fx(t)\end{aligned} $$

(A.2)

where x(t) is treated as a vector of free variables to find a starting feasible integer solution. Let the solution of (A.2) be given by $\pi_{d} = \bar{\pi }_{d} .$

Fix $\pi_{d} = \bar{\pi }_{d}$ in (A.1) to obtain a multi-parametric LP problem of the following form:

$$ \begin{aligned}&\hat{J}(x(t)) = \mathop {\min }\limits_{{\pi_{c} }} \phi_{1}^{\text{T}} \pi_{c} + \phi_{2}^{\text{T}} \bar{\pi }_{d} \\&s.t.\quad G_{c} \pi_{c} + G_{d} \bar{\pi }_{d} \le S + Fx(t)\end{aligned} $$

(A.3)

The solution of the multi-parametric LP [14] subproblem in (A.3) which represents a parametric upper bound on the final solution is given by (i) a set of parametric profiles, $\hat{J}( {x( t )} )^{i} ,$ and the corresponding critical regions, CR ⁱ, and (ii) a set of infeasible regions where $\hat{J}( {x( t )} )^{i} = \infty .$

For each critical region, CR ⁱ, obtained from the solution of the multi-parametric LP subproblem in (A.3), an MILP subproblem is formulated as follows:

$$\begin{aligned}&\mathop {\min }\limits_{{\pi_{c} ,\pi_{d} ,x(t)}} \phi_{1}^{\text{T}} \pi_{c} + \phi_{2}^{\text{T}} \pi_{d} \\ &s.t.\quad G_{c} \pi_{c} + G_{d} \pi_{d} \le S + Fx(t) \\&\phi_{1}^{\text{T}} \pi_{c} + \phi_{2}^{\text{T}} \pi_{d} \le \hat{J}(x(t))^{i} \\ &\pi_{d} \ne \bar{\pi }_{d} \\&x \in {\text{CR}}^{i} \end{aligned} $$

(A.4)

The integer solution,$\pi_{d} = \bar{\pi }_{d}^{1} ,$ and the corresponding CRs, obtained from the solution of (A.4), are then recycled back to the multi-parametric LP subproblem—to obtain another set of parametric profiles. Note that the integer cut, $\pi_{d} \ne \bar{\pi }_{d} ,$ and the parametric cut, $\phi_{1}^{\text{T}} \pi_{c} + \phi_{2}^{\text{T}} \pi_{d} \le \hat{J}( {x( t )} )^{i} ,$ are accumulated at every iteration.

If there is no feasible solution to the MILP subproblem (A.4) in a CRⁱ, that region is excluded from further consideration and the current upper bound in that region represents the final solution.

The set of parametric solutions corresponding to an integer solution, $\pi_{d} = \bar{\pi }_{d} ,$ which represents the current upper bound are then compared to the parametric solutions corresponding to another integer solution, $\pi_{d} = \bar{\pi }_{d}^{1} ,$ in the corresponding CRs in order to obtain the lower of the two parametric solutions and update the upper bound. This is achieved by using a procedure presented by Acevedo and Pistikopoulos [1]. Based upon the above theoretical developments, the steps of the algorithm are stated in Table 2.

Table 2 Solution Steps of the mp-MILP Algorithm