Page 222 - Glucose Monitoring Devices

The retrofitting algorithm 225

For what concerns S q , the choice
2 3 2
0:25 12:5 0 0
6 7
12:5 100 0 0
6 7
6 7
(11.11)
2 6
7
S q ¼ g
q6 7
6 0 0 10 0 7
4 5
0 0 0 3
has proven effective, where the constant g can be learned on a validation dataset or
q
fixed to g ¼ 0:1. With regard to the additive, possibly nonwhite, random noise
q
wðtÞ, at difference with [16] where a further decomposition and analysis of the
spectral properties of wðtÞ for the Dexcom SEVEN PLUS sensor has been proposed,
here we simply capture the essential error intersamples correlation with an autore-
gressive (AR) process of order 1:
(11.12)
wðt þ 1Þ¼ awðtÞþ eðtÞ
2
2
with a ¼ 0:87 and eðtÞ a white noise with VarðeðtÞÞ ¼ 26:6 [mg /dL ], ct.The
same statistical description can be used for other commercial sensors.
Step B: constrained regularized deconvolution
Step B simultaneously compensates for the distortion introduced by the blood-
to-interstitium glucose transportation and reduces measurement noise affecting
CGM. In fact, employing regularization [18], it prevents noise amplification due
to ill-conditioning [19] and implements a noise filtering leveraging on the physiolog-
ical prior-knowledge on the BG profile smoothness. Adding the constraints allows
using the available accurate BG reference information.
Step B takes as input the recalibrated CGM (produced by Step A) and the
outliers-checked BG reference data (produced by the preprocessing step) and returns
as output a reconstructed quasi-continuous BG profile, retrofitted BG from now on.
To formulate the deconvolution problem, let us introduce bg ðtÞ, the ℝ M the
r
r
vector of all M blood glucose concentrations to be reconstructed. bg ðtÞ is the vecto-
rial representation of the quasi-continuous reconstructed profile, uniformly sampled
with an arbitrary small sampling time T BG , from time t start to time t end :
r
bg ¼½bgðt start Þ; bgðt start þ T BG Þ; .; bgðt end ފ: (11.13)
For sake of clarity, it is assumed from now on that T BG ¼ 1 [min] and t start ¼ 1
[min], so that the BG profile to be estimated is a quasi-continuous signal and the
blood glucose concentration at time t i , bgðt i Þ, is stored in the ith entry of the vector
r
bg .
M
r
The superscript “r” on the ℝ vector bg differentiates the vector to be estimated
m
from the ℝ vector BG containing reference measurements available, m << M.In
the ideal case of perfect reference measurement,
r
bg ¼ C$bg ; (11.14)