Page 184 - Glucose Monitoring Devices

186 CHAPTER 9 Calibration of CGM systems

where u B is the i 1 vector containing the SMBG samples acquired at calibration
times t j ; j ¼ 1; .; i ði ¼ 1; .; MÞ:
T
u B ¼½u B ðt 1 Þ; .; u B ðt i 1Þ; u B ðt i ފ (9.20)
b u B is the i 1 vector (function of the model parameters), obtained transforming the
i 1 vector y , containing y I ðt j Þ, j ¼ 1; :::; i, into BG values. Both the BG-to-IG ki-
I
netics and the calibration model are considered in the transformation (as discussed
later). The i 1 vector w represents the error. Note that the length i of the vectors
increases of one unit each time a new SMBG is acquired for calibration, as all pre-
vious measurements are anyway considered.
The unknown parameters vector p is estimated by exploiting the data contained
in u B and y in addition to some a priori knowledge on the distribution of p, derived
I
from a data training set (details in Section Implementation). In particular, the a priori
distribution of the parameters vector p has mean m and covariance matrix S p . The
p
error vector w is assumed to contain white noise samples, uncorrelated from p, with
zero mean and diagonal covariance matrix S w . The error variance is assumed con-
2
2
stant over time, that is, S w ¼ s $ I and s is estimated from the training set (details
w
w
in Section Implementation).
The Bayesian maximum a posteriori estimate of p is obtained by solving the
following optimization problem:
T 1
T
1
b p ¼ argmin½u B b u B ðpފ S w ½u B b u B ðpފ þ m p S p m p (9.21)
p
p
p
which, given the presence of nonlinearities, does not have a closed-form solution.
Thus, the estimate b p is found by looking iteratively into the parameter space. The
iterative procedure, schematically described by the diagram of Fig. 9.7, is summa-
rized in five steps:
• (i) initialization of parameter vector p;
• (ii) estimation of IG profile b u I ðtÞ according to the calibration model of Eq.
(9.15);
• (iii) estimation of BG profile, b u B ðtÞ accounting for the distortion introduced by
the BG-to-IG kinetics, through nonparametric deconvolution;
• (iv) matching between SMBG measurements and b u B ðtÞ;
• (v) update of the parameter vector for the next iteration.
The following subsections describe individually each of these five steps.

Step 0: parameter initialization
At the first iteration of the ith calibration (i ¼ 1; :::; M) the parameter vector is
initialized to the mean value of the prior distribution, p ¼ m p
0
Step 1: use of calibration model
At each iteration k (k ¼ 0; 1; :::; Niter), the interstitial glucose u I ðtÞ is estimated
from the current signal y I ðtÞ by inverting Eq. (9.15), which depends on the parameter
vector p :
k
y I ðtÞ
b u I ðt; p Þ¼ b (9.22)
k
sðt; s 2 ; s 3 Þ