Computational Methods in Systems Biology

220 N. Paoletti et al.

Nc=20andNp= 200: preliminary experiments suggested that largeNpvalues
and smallNcvalues cause excessive insulin therapy and hypoglycemia.
We design the cost function so as to optimize the following two objectives:

1. Minimize the sum of squared distances between the predicted BG level ̃xG(t+
   k) and a target trajectoryR(t+k):
   d( ̃x(t+k)) =γ(t+k)·( ̃xG(t+k)−R(t+k))^2 (9)
   whereγ(t+k)=γif ̃xG(t+k)<R(t+k) and 1 otherwise. (Remind that
   xG(t)=G(t)=Q 1 (t)/VGin the glucose kinetics subsystem) Parameterγ≥ 1
   allows defining asymmetric cost functions where predicted BG values below
   the target are penalized more than those above the target. Glucose control
   is naturally asymmetric given that hypoglycemia leads to more severe conse-
   quences than (temporary) hyperglycemia, and, as shown in [ 7 ], asymmetric
   costs effectively contribute avoiding hypoglycemia.


2. Minimize step-wise changes in the control input (Διt+k)^2 , whereΔιt+k=
   ιt+k−ιt+k−^1 ,andιt−^1 corresponds to the control input in the previous iter-
   ation, or to the basal insulin rate ̄ιift=0.

In our setup, we fix the target trajectory toR(t+k)=7.8 mmol/L for all time
instants and set penaltyβto 1/50. We set the asymmetric cost penalty toγ=2,
after experimenting with different values (reported in [ 24 ]).

Optimization Algorithm:We solve problem ( 3 ) using non-linear optimization
techniques, where, for a fixed control strategyιt,...,ιt+Nc−^1 , the objective func-
tion value is given in turn as the result of maximizing the objective function over
the uncertainty parameters (and with fixedιt,...,ιt+Nc−^1 ). To solve both min-
imization and maximization problems, we use MATLAB’sfmincon. To reduce
the computational cost of this optimization method, we decrease the number
of decision variables by assuming that, in the prediction model, control inputs
change with period 10 min, and uncertainty parameters with period 30 min.

Hybrid Closed-Loop (HCL) Variant:To compare with our robust MPC approach,
we develop a hybrid closed-loop insulin pumps where only basal insulin is auto-
matically regulated and the patient is responsible for bolus insulin. This reduces
to a MPC that has no knowledge of meals and exercise, and thus, approximates
the behavior of a current state-the-art approved device that requires explicit
meal announcement. In our settings, this is equivalent to fixing the uncertainty
parameters to their default values at rest.
Then the optimization problem of the HCL controller reduces to:

min

ιt,...,ιt+Nc−^1

∑Np

k=1

d(x ̃(t+k)) +β·

N∑c− 1

k=0

(Διt+k)^2 (10)

subject to (4, 5 , 7 ,8) andut+k=(0, 0 ,8) (k=0,...,Np−1).

Note that the constraints on the insulin therapy are the same of the robust
controller ( 4 – 5 ) meaning that the HCL controller is free to synthesize bolus-
like therapy profiles too. This will also serve as the baseline controller in the
evaluation part of Sect. 5.