Page 173 - Glucose Monitoring Devices

Calibration of minimally invasive CGM sensors 175

The choice of the calibration function fð $Þ is critical. It has to be invertible, and
it has to precisely describe the relationship between the electrical current signal and
glucose concentration, which can be, in the most general case, nonlinear and time
variant (in this case, time t would be, explicitly, an input of fð $Þ). Moreover, the
choice of using either the electrical current or the BG measurements as an indepen-
dent variable in the calibration model may affect the calibration performance [13].
The most common and simplest calibration model adopted by manufacturers of
CGM systems is a first-order time-independent linear function [14e17], with param-
eters P ¼½s; bŠ, where s and b are called as sensor sensitivity and baseline (or
offset), respectively. In this case, the model of the measurements reported in
Eq. (9.1) in a general form turns into
yðtÞ¼ fðP; uðtÞÞ þ wðtÞ¼ s$uðtÞþ b þ wðtÞ (9.3)
The numerical determination of the model parameters b s and b is, thus, required.
b
For such a scope, if for instance two BG references uðt 1 Þ and uðt 2 Þ are available at
times t 1 and t 2 , knowing the electrical current values given by the sensor at the same
time instants, yðt 1 Þ and yðt 2 Þ, the so-called two-point calibration can be performed
[18], which allows the estimation of sensitivity, b s, and baseline, b, from the two
b
measured pairs as follows:
8
yðt 2 Þ yðt 1 Þ
>
> b s ¼
>
<
uðt 2 Þ uðt 1 Þ
(9.4)
> yðt 2 Þ yðt 1 Þ
>
> b
: b ¼ yðt 2 Þ $uðt 2 Þ
uðt 2 Þ uðt 1 Þ
In general, when multiple pairs of electrical current and BG samples are avail-
able at times t i ði ¼ 1; 2; .; NÞ, as shown in Fig. 9.2, a linear regression is
used to fit the sensitivity and baseline to the data. In particular, including the
measurement noise wðt i Þ, the model of the measurements becomes
yðt i Þ¼ s$uðt i Þþ b þ wðt i Þ (9.5)
and the numerical determination of model parameters is done by minimizing the
residual sum of squares, that is, the differences between the measurements, yðt i Þ,
and the model predictions, b yðt i Þ¼ s$uðt i Þþ b:
N N
h i
X 2 X 2
b s; b ¼ argmin wðt i Þ ¼ argmin ðyðt i Þ b yðt i ÞÞ (9.6)
b
s;b i¼1 s;b i¼1
Finally, the calibrated glucose profile b uðtÞ is obtained from the measured current
signal yðtÞ and the estimated calibration parameters b s and b by inverting the calibra-
b
tion function:
yðtÞ b b
b uðtÞ¼ (9.7)
b s