Page 306 - Glucose Monitoring Devices

Quantifying plasma insulin concentrations 313

have significant effects on the IOB. Therefore the insulin action curves for IOB cal-
culations, usually involving static models with basal and bolus insulin as inputs
and active insulin as the output, are not accurate enough to be used in an AP control
system. Regardless of the sophistication of the IOB calculation, the information
obtained from insulin action curves is usually an approximation of the active insu-
lin in the body and is not a direct estimate of the concentration of insulin in the
bloodstream.
Accurate estimates of PIC can be obtained by using CGM measurements with
adaptive observers designed for simultaneous state and parameter estimation based
on reliable glucose-insulin models. The glucose-insulin dynamic model can be writ-
ten in the form

¼ fðxðtÞ; uðtÞÞ þ wðtÞ;
dxðtÞ
dt wwN ð0; Q w Þ
y k ¼ hðx k Þþ v k ; vwN ð0; R v Þ
where xðtÞ denotes the vector of state variables including the PIC as a state variable,
hðx k Þ denotes the measurement function with y k as the subcutaneous glucose output
measurement, wðtÞ and v k represent the process and observation noise vectors,
respectively, Q w and R v denote the covariance and variance of the process and mea-
surement noise, respectively. To design a state estimator for the augmented system,
the augmented system should be observable.
Nonlinear observers, such as extended or unscented Kalman filters (UKF) and
moving horizon estimation, can be designed for the estimation of the model states
and parameters. The UKF algorithm can handle the nonlinear dynamics of the
glucose-insulin model, is robust to measurement noise, and can compensate for
deviations and converge to the true value of the augmented states through the
Kalman gain correction term added to the estimation. In the UKF algorithm, the
unscented transformation (UT) method is employed for calculating the statistics
of a random variable that undergoes a nonlinear transformation such as the
glucose-insulin dynamics model. The UT characterizes the mean and covariance
estimates with a minimal set of sample points called sigma points. Let the set of
sigma points at sampling instance k be denoted
X ’
i;k ; i˛f0; .; 2ng
with each point being associated with a corresponding weight w i . In the UKF
approach, both the sigma points and the weights are determined deterministically
via specific criteria and equations. The sigma points are propagated through the
nonlinear state dynamics of the glucose-insulin function, which yields propagated
states X ’ and the corresponding mean representing the prior estimates
’ i;krk 1
X b approximated by the weighted average of the transformed points as
kjk 1j
2n
’ X
X b w i X
kjk 1 ¼ i;kjk 1
i¼0