Page 186 - Glucose Monitoring Devices

188 CHAPTER 9 Calibration of CGM systems

which is independent of and usually denser than U s (here a uniform 1-min step is
used). The virtual grid allows us to obtain a denser profile, which can be more easily
matched with SMBG samples (see Step 3). Moreover, S v starts from
t 1 100 min, to allow initial condition transient to vanish, so that the reconstruc-
tion of u B ðtÞ is not altered in the window of interest L.
Once all variables have been defined, having U s and U v both uniform, with
U s 4U v , the following matrix equation can be written as follows:
u ¼ H$u þ w (9.23)
IðLÞ BðLÞ
where H is the n N matrix obtained downsampling the N N transfer matrix Hv
of the BG-to-IG system, maintaining only the rows correspondent to sampling
instants in S s . As vector u BðLÞ contains samples of a BG profile, which is a biolog-
ical signal expected to have a certain smoothness, a double integrated white noise
2
model [41] of unknown variance l is chosen to describe entries of u . Thus,
BðLÞ
its covariance matrix is
T 1
2
S ¼ l F F (9.24)
u B ðLÞ
with F N N Toeplitz lower-triangular matrix having ½1; 2; 1; 0; .; 0ŠT as the
first column. Assuming that u and w are uncorrelated, the following quadratic
BðLÞ
optimization problem corresponds to the linear minimum error variance Bayesian
estimate of u :
BðLÞ
T
T
b u ¼ argmin u Hu R 1 u u þ gu T F Fu
BðLÞ IðLÞ BðLÞ IðLÞ BðLÞ BðLÞ BðLÞ
u BðLÞ
(9.25)
2
where parameter g ¼ , estimated by maximum likelihood [41] represents the
s
2
regularization term that balances the data fit with the smoothness of the estimated
l
profile. The optimization problem of Eq. (9.25) admits a closed-form solution,
expressed as follows:
T 1 T 1 T 1
b u ¼ H R H þ gF F H R u (9.26)
BðLÞ IðLÞ
For every SMBG measurement in vector u B (see Eq. 9.20), the BG profile b u ,
BðLÞ
which depends on parameter vector p , is estimated inside the window L that
k
contains the time instant at which the SMBG sample t j ; j ¼ 1; :::; i is acquired.
Step 3: match between estimated BG and available SMBG
For each SMBG sample in vector u B , acquired at time t j ; j ¼ 1; .; i; the corre-
sponding estimated value of b u at time t j is considered, by exploiting the vector
BðLÞ
b u B ðp Þ used in Eq. (9.21):
k
T
k
b u B ðp Þ¼ b u B ðt 1 ; p Þ; .; b u B ðt i 1 ; p Þ; b u B ðt i ; p Þ (9.27)
k
k
k