Adjustment of basal insulin infusion rate in T1DM by hybrid PSO

Abstract

Basal insulin infusion rate which should be adjusted to increase or decrease insulin delivery with the varying blood sugar level plays a key role in type 1 diabetes mellitus (T1DM) patients for maintaining the blood glucose level approximately steady within reference range in order to avoid the complications developed from diabetes. This paper proposes an effective hybrid particle swarm optimization (HPSO) algorithm for solving the basal insulin infusion rate problem. In HPSO, bad experience lesson learning scheme and local search based on chaotic dynamics are proposed to make a good balance between global exploration and local exploitation. Simulation results based on a set of well-known optimization benchmark instances and the basal insulin infusion rate adjustment problem for T1DM demonstrate the effectiveness of the proposed HPSO. In silico tests on standard virtual subject via HPSO show that under nominal condition the blood glucose concentrations could be kept within a range of 80–150 mg/dL within less than 5 days; meanwhile, in case of random variations in meal timings within $\pm$60 min or meal amounts within $\pm$75 % deviation from the nominal values, the blood glucose concentrations could be kept within the safe regions.

Appendix

Five well-known benchmark problems used for the purpose of comparison are:

• 1. Sphere model

  $$\begin{aligned} f_1 (x)=\sum \limits _{i=1}^{30} x_i^2,\quad x\in [-100,100]^{30} \end{aligned}$$

  Its global minimum is equal to 0.

• 2. Schwefel’ Problem 2.22

  $$\begin{aligned} f_2 (x)=\sum _{i=1}^{30} |x_i|+\prod \limits _{i=1}^{30} |x_i|,\quad x\in [-10,10]^{30} \end{aligned}$$

  Its global minimum is equal to 0.

• 3. Schwefel’ Problem 1.2

  $$\begin{aligned} f_3 (x)=\sum _{i=1}^{30} \left( \sum _{i=1}^i x_i\right) ^2,\quad x\in [-100,100]^{30} \end{aligned}$$

  Its global minimum is equal to 0.

• 4. Schwefel’ Problem 2.21

  $$\begin{aligned} f_4 (x)=\hbox {MAX}\{|x_i|,\ 1\le i\le 30\},\quad x\in [-100,100]^{30} \end{aligned}$$

  Its global minimum is equal to 0.

• 5. Generalized Rosenbrock’s function

  $$\begin{aligned} f_5 (x)&= \sum \limits _{i=1}^{30} {\left[ {100(x_{i+1} -x_i^2 )^2+(x_i -1)^2} \right] },\\&\quad x\in [-30,30]^{30} \end{aligned}$$

  Its global minimum is equal to 0.

• 6. Step function

  $$\begin{aligned} f_6 (x)=\sum \limits _{i=1}^{30} (\lfloor x_i +0.5 \rfloor )^2,\quad x\in [-100,100]^{30} \end{aligned}$$

  Its global minimum is equal to 0.

• 7. Quadric noise function

  $$\begin{aligned} f_7 (x)=\sum \limits _{i=1}^{30} {ix_i^4 +\hbox {random}[0,1]},\quad x\in [-1.28,1.28]^{30} \end{aligned}$$

  Its global minimum is equal to 0.

• 8. Generalized Schwefel’s Problem 2.26

  $$\begin{aligned} f_8 (x)=-\sum \limits _{i=1}^{30} {\left( {x_i \sin \sqrt{\left| {x_i} \right| }}\right) },\quad x\in [-500,500]^{30} \end{aligned}$$

  Its global minimum is equal to $-12{,}569.5.$

• 9. Generalized Rastrigin’s function

  $$\begin{aligned} f_9 (x)&= \sum \limits _{i=1}^{30} {\left[ {x_i^2 -10\cos (2\pi x_i )+10} \right] },\\&\quad x\in [-5.12,5.12]^{30} \end{aligned}$$

  Its global minimum is equal to 0.

• 10. Ackley’s function

  $$\begin{aligned} f_{10} (x)&= -20\exp \left( -0.2\sqrt{1/30\sum _{i=1}^{30}{x_i^2}}\right) \nonumber \\&\quad -\,\exp \left( 1/30\sum _{i=1}^{30} {\cos (2\pi x_i)}\right) +20+e \\&\quad \quad x\in \left[ {-32,32} \right] ^{30} \end{aligned}$$

  Its global minimum is equal to 0.

• 11. Generalized Griewank’s function

  $$\begin{aligned} f_{11} (x)&= \frac{1}{4{,}000}\sum \limits _{i=1}^{30} {x_i^2 } -\prod \limits _{i=1}^{30} \cos \left( \frac{x_i}{\sqrt{i}}\right) +1,\\&\quad x\in [-600,600]^{30} \end{aligned}$$

  Its global minimum is equal to 0.

• 12. Generalized Penalized function

  $$\begin{aligned}&f_{12} (x)=\frac{\pi }{30}\left\{ 10\sin ^2(\pi y_1)+ \sum \limits _{i=1}^{29} (y_i-1)^2\right. \\&\qquad \qquad \quad \left. \times [1+10\sin ^2(\pi y_{i+1} )]+(y_{30}-1)^2 \right\} \\&\qquad \qquad \quad \,+\sum \limits _{i=1}^{30} {u(x_i,10,100,4)}, \\&\qquad y_i =1+\frac{(x_i +1)}{4}, \\&\qquad u(x_i,a,k,m)=\left\{ {\begin{array}{l@{\quad }l} k(x_i -a)^m&{} x_i >a \\ 0&{} -a\le x_i \le a \\ k(-x_i-a)^m&{} x_i<-a \\ \end{array}} \right. \\&\qquad \qquad \qquad \quad \qquad \qquad x\in [-50,50]^{30} \end{aligned}$$

  Its global minimum is equal to 0.

• 13. Generalized penalized function

  $$\begin{aligned}&f_{13} (x)=0.1\left\{ \sin ^2(3\pi x_1)+\sum \limits _{i=1}^{29} (x_i-1)^2\right. \\&\qquad \qquad \quad \times \,[1+\sin ^2(3\pi x_{i+1})]+(x_{30}-1)^2\\&\qquad \qquad \quad \times \, \left. [1+\sin ^2(2\pi x_{30})]\right\} \\&\qquad \qquad \quad +\,\sum \limits _{i=1}^{30} {u(x_i,5,100,4)}, \\&\qquad u(x_i,a,k,m)=\left\{ {\begin{array}{l@{\quad }l} k(x_i-a)^m&{}\quad x_i <a \\ 0&{}\quad -a\le x_i \le a \\ k(-x_i-a)^m&{}\quad x_i \ge a \\ \end{array}} \right. \\&\qquad \qquad \qquad \qquad \qquad \, x\in [-50,50]^{30} \end{aligned}$$

  Its global minimum is equal to 0.