Page 240 - Glucose Monitoring Devices

244 CHAPTER 12 Modeling the CGM measurement error

Therefore the mean sensor reading (s)and themeanerror(ε) for each reference BG
were computed using Eq. (12.2):

Z 600
sðrÞ¼ s$Dðs; rÞ ds; εðrÞ¼ sðrÞ r (12.2)
s¼0
Delay estimation. The premise behind our methodology for delay estimation is
that if the CGM measurements are delayed, then the sensor error will be dependent
on the rate of change of glucose. Therefore, by computing Dðs; rÞ for different
ranges of glucose rate of change (bins) and studying the differences between bins,
we can study the delay. To do so, we needed to estimate the density in particular
bins using Eq. (12.3):

2 2
N j
1 X ððs s iÞ þðr r iÞ Þ 1 if vs i is in bin j
D j ðs; rÞ¼ 2 I j ðvs i Þe 2s 2 ; I j ðvs i Þ¼
2ps N j 0 otherwise
i¼1
(12.3)
To compute a robust rate of change, we used sliding linear regression for consec-
utive 20-min windows.

Estimation and modeling of the sensor error distribution
As discussed earlier, CGMs measure glucose in the interstitial fluid while being used
to assess BG, which creates a delay. Therefore, the distribution of sensor errors could
not be computed directly by using the difference between reference and sensors at
the same time points. As the physiological delay described earlier is not constant
across subjects or within a subject over a long period of time, we synchronized
sensor and reference BG using a first-order diffusion model in the calibration equa-
tion (Eq. 12.4):
1
_
CGM ¼ a$G I þ b ; G I ¼ ðG I G B Þ (12.4)
s
where G I and G B are the interstitial and the blood glucose concentrations, respec-
tively, s is the diffusion time constant, and a and b are the calibration parameters.
Spanning the possible values of s, we applied the same linear regression technique
as before to estimate a and b. The final solution is the set ðs; a; bÞ, which produces
the smallest sum of squares. We do not claim that this procedure, derived from the
study by Steil et al. [17], models perfectly the transport of glucose from the blood
to interstitium nor the functional relationship between electrical current in the sensor
and glucose values. However, in the absence of an identifiable model of such a
process, we followed the parsimony principle in choosing the simplest available
one, that is, a linear transformation added to a first order, gain 1, diffusion.
Once sensor and reference data were synchronized, we computed the differences
between sensor and reference estimates, and the first four central moments of their
empirical distribution: mean, variance, skewness, and kurtosis.