Page 182 - Glucose Monitoring Devices

184 CHAPTER 9 Calibration of CGM systems

reference to the general calibration function formulation of Eq. (9.1), this means that
there is some expectation on the parameters P, which is usually expressed as
Gaussian distribution described by its mean and standard deviation. The Bayesian
estimation can be used to estimate the parameters P of a generic calibration function
f by using the available data, uðtÞ, and the prior information on P.
In the following subsections, we will give an example of the implementation of a
calibration algorithm that relies on the Bayesian estimation (see Refs. [47,48] for an
extensive description of the method and relative results).

Description of a Bayesian calibration algorithm
Let us define the calibration algorithm as the series of operations that process the
electrical current signal y I ðtÞ, measured by the CGM sensor, and the SMBG mea-
surements, acquired by the fingerprick device, to obtain the IG profile u I ðtÞ. The
two measurements, y I ðtÞ and SMBG, belong to different physical domains, that is,
the current and the glucose domain respectively, as well as to different physiological
sites. SMBG measures glucose concentration into the blood, while the sensor current
reflects glucose concentration in the interstitium. Thus, to calibrate a CGM sensor, a
model describing the relationship between the two measurements is needed.
Letting u B ðtÞ be the glucose concentration in blood (whose samples are observ-
able by SMBG), a widely established description of the relation between BG and IG
profiles is based on a two-compartment model [21] (see Section Critical aspects
affecting calibration), in which the glucose concentration in the interstitial compart-
ment can be described by:
d
s $ u I ðtÞ¼ u I ðtÞþ u B ðtÞ (9.14)
dt
where s is the equilibrium time-constant between plasma and interstitium. In the
literature, the value of s exhibits significant intersubject variability but, in a given
subject, it is commonly assumed to be time invariant [21,23,49,50]. According to
Eq. (9.14), the IG profile can be interpreted as the output of a first-order linear
dynamic system, having the BG profile as input (see first block in the schematic
diagram of Fig. 9.6), and hðtÞ, as defined in Eq. (9.13), as impulse response.
The low-pass filtering nature of hðtÞ causes u I ðtÞ to be a distorted version of
u B ðtÞ, presenting both amplitude attenuation and phase delay. Following the cascade
of blocks of Fig. 9.6, the IG profile u I ðtÞ is the input of the calibration function which
nf
produces as output a current signal referred to the interstitium, y ðtÞ, which once
corrupted by additive noise wðtÞ finally produces the sensor signal y I ðtÞ. To describe
the transformation between glucose and current profiles, let us consider the
following model:
y I ðtÞ¼½u I ðtÞþ bŠ$sðtÞþ wðtÞ (9.15)
where b is the baseline of the glucose profile and sðtÞ represents the sensitivity of the
sensor, described by the model:
(9.16)
sðtÞ¼ s 1 $aðtÞþ s 2 $bðtÞþ s 3