A Multistep Algorithm for Processing and Calibration of Microdialysis ...

DIABETES TECHNOLOGY & THERAPEUTICS Volume 15, Number 10, 2013 ª Mary Ann Liebert, Inc. DOI: 10.1089/dia.2013.0041

ORIGINAL ARTICLE

A Multistep Algorithm for Processing and Calibration of Microdialysis Continuous Glucose Monitoring Data Zeinab Mahmoudi, MSc,1 Mette Dencker Johansen, PhD,1 Jens Sandahl Christiansen, MD, DMSc, FRCPI,2 and Ole Kristian Hejlesen, PhD1,3,4

Abstract Background: The deviation of continuous subcutaneous glucose monitoring (CGM) data from reference blood glucose measurements is substantial, and adequate signal processing is required to reduce the discrepancy between subcutaneous glucose and blood glucose values. The purpose of this study was to develop a multistep algorithm for the processing and calibration of continuous subcutaneous glucose monitoring data with high accuracy and short delay. Algorithm Presentation: The algorithm comprises three steps: rate-limiting filtering, selective smoothing, and robust calibration. Initially, the algorithm detects nonphysiological glucose rate-of-change and corrects it with a weighted local polynomial. Noisy signal parts that require smoothing are then detected based on zero crossing count of the sensor signal first-order differences, and an exponentially weighted moving average smooths the noisy parts of the signal afterward. Finally, calibration is performed using a first-order polynomial as the conversion function, with coefficients being estimated using robust regression with a bi-square weight function. Algorithm Performance: The performance of the algorithm was evaluated on 16 patients with type 1 diabetes mellitus. To compare the algorithm with state-of-the-art CGM data denoising and calibration, the rate-limiting filter and selective smoothing were replaced with an adaptive Kalman filter, and the calibration method was replaced with the calibration algorithm presented in one of the Medtronic (Northridge, CA) CGM patents. The median (mean) of the absolute relative deviation (ARD) of the sensor glucose values processed by the newly developed algorithm from capillary reference blood glucose measurements was 14.8% (22.6%), 10.6% (14.6%), and 8.9% (11.7%) in hypoglycemia, euglycemia, and hyperglycemia, respectively, whereas for the alternative algorithm, the median (mean) was 22.2% (26.9%), 12.1% (15.9%), and 8.8 (11.3%), respectively. The median (mean) ARD in all ranges was 10.3% (14.7%) for the new algorithm and 11.5% (15.8%) for the alternative algorithm. The new algorithm had an average delay of 2.1 min across the patients, and the alternative algorithm had an average delay of 2.9 min. Conclusions: The presented algorithm may increase the accuracy of CGM data. Background

M

anaging diabetes and maintaining good glycemic control is a crucial task for diabetes patients and healthcare professionals, as these factors are of critical importance in the prevention of diabetes complications.1–4 In recent years, continuous glucose monitoring (CGM) has been used to detect glucose variations (e.g., by generation of hypoand hyperglycemic alarms).5–8 The use of CGM enhances patients’ ability to reduce glycosylated hemoglobin while avoiding the risk of hypoglycemia.9 There is abundant information in CGM data; however, because of the substantial deviation of CGM data from blood glucose (BG) levels, both in amplitude and in phase, the in1

Department Department 3 Department 4 Department 2

of of of of

formation is inaccurate and should not be interpreted prior to processing.10–12 The dynamics from BG to CGM data consist of two parts: conversion of blood glucose to interstitial glucose (IG) and conversion of IG to CGM data. IG amplitude is blunted compared with BG level.12,13 The bluntness can cause the deviation of IG from BG to reach 50%.14 In addition, it is speculated that IG is physiologically delayed compared with BG.15–17 Using data from a pig model, Nielsen et al.18 demonstrated that there is no delay between IG excursions and central nervous system glucose excursions; this suggests that the delay between BG and IG may be of lesser clinical importance than initially assumed. However, when the delay produced by signal processing algorithms in CGM devices is added to

Health Science and Technology, Aalborg University, Aalborg, Denmark. Endocrinology and Diabetes, Aarhus University Hospital, Aarhus, Denmark. Health and Nursing Science, University of Agder, Agder, Norway. Computer Science, University of Tromsø, Tromsø, Norway.

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826 the physiological and instrumental delay, the overall delay between CGM data and BG values is substantial and needs to be minimized in order to increase the clinical accuracy of CGM devices. Whereas the subcutaneous CGM sensor samples interstitial fluid, the CGM data do not reflect IG perfectly. The discrepancies between CGM data and IG levels are nonphysiologic, including the delay between CGM data and IG values and the artifacts such as electronic noise, sensor drift, and movement artifacts.12,17,19 The current signal processing algorithms for noise elimination and calibration purposes cause large delays and do not reduce the discrepancies effectively.4,20 Therefore, development of accurate and reliable CGM remains a challenging issue in terms of artifact correction, delay reduction, and calibration improvement. The present study presents a CGM processing algorithm aiming to reduce delay and deviation of CGM data from BG levels. Furthermore, we provide preliminary results from application of the algorithm to real patient data. The newly developed algorithm can be implemented both in real time and retrospectively. Materials and Methods Subjects We included CGM datasets from 32 patients with type 1 diabetes mellitus. Patients were 33.6 – 9.9 (mean – SD) years old, with a body mass index of 23.8 – 3.0 kg/m2 and glycated hemoglobin level of 8 – 1.6%. In total, 21 patients were male. The datasets were randomly selected from a larger population of 135 patients with type 1 diabetes. The data were collected at two centers (Medical Department M, Aarhus University Hospital, Aarhus, Denmark; and Profil Institute for Metabolic Research, Neuss, Germany) that contributed to the clinical in vivo development phase of the SCGM 1 system (Roche Diagnostics, Mannheim, Germany).21 The SCGM 1 is a microdialysis CGM device allowing up to 120 h of data collection. The CGM data obtained from patients are averaged over a 60s interval; therefore, a 1-min signal is available at the sensor site. The duration of data recordings was up to 5 days per patient. Glucose measurements from capillary BG were taken by nurses up to 20 times per day using a built-in BG meter. These measurements were used as reference values for calibration and evaluation of the algorithm. The reference BG values were measured twice to confirm the values. The two values at each measurement instance were averaged before they were used in the algorithm. Data were divided randomly into two sets of equal size: a training set for development of the algorithm and a validation

MAHMOUDI ET AL. set for testing the algorithm. The training dataset comprised 16 patients with a total of 1,701.4 h of 1-min CGM data and 2,208 reference-sensor pairs. The validation dataset also comprised 16 patients with a total of 1,804.8 h of CGM data and 2,640 pairs. Data from each patient were exclusive and formed part of either the training or validation set but not both. CGM signal processing algorithm Figure 1 demonstrates the main elements of the proposed algorithm. The algorithm operates on the raw CGM sensor signal with a sampling frequency of 1 per minute. In the first block, a rate-limiting filter limits the rate-ofchange if a physiological threshold is exceeded. The second block then evaluates if the signal is noisy by counting the number of zero crossings (ZCs) of the signal first-order differences. If the count exceeds a predefined threshold, the signal is considered noisy. Selective smoothing is performed in the same block; only noisy signals are smoothed. Finally, in the third block, a robust linear calibration converts the current measured by the sensor (in nA) to BG levels (in mg/dL) using reference capillary BG measurements. A detailed description of the blocks is provided in the following sections. The algorithm was developed in MATLAB (version 7.12.0 [R2011a]; The MathWorks, Natick, MA). Rate-limiting filtering. To limit the signal rate-of-change causing the sensor glucose (SG) rate to exceed the physiological limit, a rate-limiting filter is applied. We assume that any signal rate exceeding the threshold has a nonphysiological origin (e.g., is caused by sensor artifacts). The physiological limit of BG rate-of-change is considered to be 4 mg/dL/min.22,23 However, the threshold applied to the signal varies according to sensor sensitivity: a sensor with higher sensitivity produces a stronger current than a sensor with lower sensitivity for the same IG value; therefore, the threshold applied should be higher. Sensor sensitivity is recalculated when a new reference BG sample is available. The sensor sensitivity in Eq. 1 is obtained using the newly available reference BG value (BGj) and the 10 interstitial signal (ISIG) values immediately after BGj; j is the sample, and ISIG is the signal fed into the algorithm. The purpose of using the 10 ISIG values after BGj is to compensate for the physiological time delay between ISIG and BG. The feasibility of using ISIG values ahead of BG measurements for delay compensation when pairing between BG and ISIG values is required is discussed in the Calibration section. The BG values for the

FIG. 1. Overall diagram of the algorithm for processing of the continuous glucose monitoring signal. ISIG, interstitial signal; ZC, zero crossing.

CONTINUOUS GLUCOSE MONITORING ALGORITHM

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sensitivity calculation are the same measurements that are used for calibration. Sensor sensitivity ¼

median(ISIGj þ 1 ,    , ISIGj þ 10 ) BGj

(1)

The rate-limiting threshold (bI) applied to the signal is calculated using the corresponding sensor sensitivity and the physiological limit of the BG rate-of-change (b) bI ¼ b · Sensor sentivity

(2)

where bI is in nA/min. If the rate-of-change between two consecutive ISIG values exceeds the rate-limiting threshold, the most recent ISIG value is considered an artifact and thus replaced by a weighted local polynomial regression estimate24: d ¼ b1 þ b2 t ISIG where t is the time at which the artifact occurs,

(3)   b1 b2

are the

d is the corrected artifact. parameters of the regression, and ISIG Parameter estimation is based on the 10 previous valid (not replaced) nearest neighboring points including the weighted current artifact, using a weighted least square fit. wi is the weight of the ith neighboring point and decreases exponentially as the distance from the artifact to be replaced increases: wi ¼ kdi where i ¼ 1, 2, . . . , 10

(4)

  b1 ¼ (X¢WX)  1 X¢WY b2

(5)

W ¼ diagfwi g

(6)

k acts as a forgetting factor, and its value determines the length of the memory of the past data that contribute to the   estimation of bb1 ; d is the distance from the artifact to be 2

replaced. In Eq. 5, X is the design matrix of the regression model, and Y is the matrix of the responses (signal points). If the signal is very noisy, it will be impossible to find 10 valid points of the signal in the near vicinity of the artifact, and the 10 nearest previous points may be located far from the artifact point. In this case, if the forgetting factor is small, the weights wi become so small that the matrix X’WX becomes singular and consequently non-invertible. We choose k = 0.9 to avoid this problem. The resulting signal from the rate-limiting filtering block is called s. Selective smoothing. We present a selective moving average that smooths only the noisy parts of the signal without affecting the remainder. Therefore, it acts as an on/off filter. To detect the noisy parts of the s signal to start smoothing, we count the number of ZCs of the first-order differences (Diff) of the s signal in a moving 5-min segment starting at sample (k – 4) and ending at sample time k: Diff: Sk  3  Sk  4 Sk  2  Sk  3 . . . Sk  Sk  1

(7)

Any data segment with ZC ‡ 1 is considered noisy. By counting the ZCs of the first-order derivative and comparing them with a threshold, we locate the abrupt

changes in the signal. The number of ZCs per segment is proportional to the dominant frequency of the signal segment.25,26 As a result, signals with low-frequency dominant components will have fewer ZCs per signal segment than signals with highfrequency dominant components.25 Taking the derivative of the signal increases the proportion of the energy of high-frequency components to the energy of low-frequency components in the signal spectrum. This makes it easier to detect the noise because the noise resides in the high-frequency component. To selectively smooth noisy parts of the signal, the algorithm applies a weighted moving average of order l: ^(k) ¼ u

w1 s(k) þ w2 s(k  1) þ . . . þ wl s(k  l þ 1) l

+ i ¼ 1 wi

(8)

^(k) is the smoothed signal, i is the sample, and the where u weight wi is exponential: wi = li, where l is a forgetting factor with a value between 0 and 1 that indicates to what degree the ^(k). In the current immemory of past data contributes to u plementation l = 0.7 and l = 50. The output of the smoothing block is called Icgm. Calibration. To convert CGM data from the nA range to mg/dL, the signal is calibrated. The reference BG measurements are considered in 6-h windows; therefore, if several reference BG measurements are present in one window, only the first is used. This procedure yields a maximum of four reference BG values per 24-h calibration interval. Calibration is performed using the four most recent reference BG values and is updated when a new BG measurement is available. Only the reference BG levels within the range of the CGM devices (40 mg/dL £ BG £ 400 mg/dL) are considered valid for calibration. For calibration, reference BG values are paired with sensor values from 10 min ahead to compensate for the physiological lag as is common in other calibration algorithms.27–29 In a real-time situation, whenever a reference BG value is collected, the algorithm starts waiting for the CGM value 10 min ahead. The BG collection time acts as the reference time to start shifting. In a retrospective situation, each reference BG value is paired with the CGM value already available 10 min ahead because all CGM values are accessible. Because the interstitial response to BG variation is via a time constant rather than a transport lag,30 the feasibility of a 10-min shift for physiological delay compensation is not clear. However, with the assumption that the CGM sensor samples IG in steady state or near steady state, the delay compensation achieved by shifting will be reasonable. The paired BG–Icgm pairs form the calibration set. Modification of calibration set. For each calibration set and before calibration, the Pearson product-moment correlation coefficient between reference BG measurements and Icgm values and also the relative SD (RSD) of the reference BG values are calculated. If both measures are below their thresholds, the calibration set is modified. The thresholds for correlation coefficient and RSD used in the current implementation are 0.75 and 28%, respectively: Correlation coefficient ¼

RSD% ¼

cov(BG, Icgm ) rBG rIcgm

SD % mean

(9)

(10)

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MAHMOUDI ET AL.

where cov is the covariance between BG measurements and Icgm values in the calibration set and rBG and rIcgm are SD values of BG measurements and Icgm values in the calibration set. The RSD follows the variation in variance. The calibration set is first modified for low correlation coefficient and then for low RSD. To increase the correlation coefficient between reference BG measurements and Icgm values, BG and Icgm values are sorted in ascending order and re-paired according to the new order. To increase the RSD of reference BG measurements, the highest reference BG value in the calibration set is increased, and the lowest reference BG value is decreased, in a stepwise manner, until the RSD of BG values reaches the threshold. The corresponding Icgm values are modified with the same percentage to keep the sensor sensitivity intact. The procedure of low RSD correction is represented in Eq. 11 and also in Figure 2: 8  BGmaxr ¼ BGmax þ 0:01r · BGmax > > max max > :  maxr Icgm ¼ Icgm  0:01r · Icgm minr

min

function. The mathematical details of the implemented robust calibration are included in the Appendix. Adaptive offset. To increase the sensitivity of the SG signal in the hypoglycemic range, a second-order polynomial c Usually CGM devices overestioffset is subtracted from SG. mate BG in the hypoglycemic range (BG £ 70 mg/dL).31 Therefore, a true hypoglycemic BG value may not be estimated as hypoglycemic by CGM systems. After data analysis, we found that overestimation mostly occurs in the nearc (70 mg=dLpSG c < 85 mg=dL). hypoglycemic range of the SG The adaptive offset allows the SG to be corrected in the nearhypoglycemic range. c The value of the offset is adjusted with the value of SG: c  offset SG ¼ SG

(13)

d2 þ a2  SG c þ a3 Offset ¼ a1  SG

(14)

c < 85, For 70pSG (11)

min

where r ¼ 1, 2, . . . , g, r is the iteration number, and g is the number of iterations for which the RSD of BG values reaches the threshold.

c the offset is 0. For other values of SG, The Appendix contains the mathematical method of a1, a2, and a3 estimation. Comparing the new algorithm with an alternative CGM algorithm

Conversion function The modified Icgm signal is called Icgm and is converted to SG by a linear function: c ¼ m · Icgm þ b SG

(12)

The parameters m and b are estimated from the modified calibration set using robust regression with a bi-square weight

In its general form, the new algorithm consists of two parts: filtering and calibration. We compared the new algorithm with the CGM algorithm presented by Mueller et al.32 in one of the Medtronic CGM patents, which also comprises both filtering and calibration. An adaptive Kalman filter is suggested to filter the current measured by the sensor before the calibration. We chose this algorithm for comparison because the act of the rate-limiting filtering and the selective smoothing together is comparable with the adaptive Kalman filtering in optimization of the trade-off between smoothing and delay. The Kalman filter model introduced by Facchinetti et al.23 was used in the following form: Ik þ 1 ¼ 2Ik  Ik  1 þ Gw wk ISIGk ¼ Ik þ #k

(15)

where I is the signal measured by the sensor if the sensor was ideal (without being affected by noise), ISIG is the raw current measured by the sensor, wk is the process noise (zero mean Gaussian) with the covariance matrix Q, and tk is the measurement noise with the covariance matrix R. Considering I as the first state (x1), I at the previous time step as the second state (x2) and the current measured by the sensor as the system output (y), the above model has the state space form 

      x1 x1 2 1 1 ¼ þ wk x2 k þ 1 x2 k 1 0 0 |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflffl{zfflfflffl} |ffl{zffl} xk þ 1

FIG. 2. Flowchart of the calibration set modification for low relative SD (RSD). BG, blood glucose; Icgm, filtered continuous glucose monitoring current; max, maximum; min, minimum.

/





xk

x1 yk ¼ ½ 1 0  þ #k |fflfflffl{zfflfflffl} x2 k |fflfflffl{zfflfflffl} c xk

Gw

(16)

CONTINUOUS GLUCOSE MONITORING ALGORITHM The states are estimated using predictor–corrector equations: Predictor : ^xkjk  1 ¼ F^xk  1jk  1

(17)

Corrector : ^xkjk ¼ ^xkjk  1 þ Lk (yk  C^xkjk  1 )

(18)

^yk ¼ C^xkjk

(19)

where ^x represents an estimate of the states and the subscript kjk - 1 denotes the estimate at step k based on measurements up to (and including) step k - 1. The Kalman filter output (^ y) is sent to calibration. The variance of the measurement noise (d) was measured with the method suggested by Mueller et al.32:

dk ¼

 k +i ¼ k  l jISIGi  ISIGi  l j 

k

+i ¼ k  l jISIGi  ISIGi  1 j lþ1

l Rk ¼ dk

2 (20)

where k is the time sample and l is the length of the moving segment in which the variance is measured; l was selected to 5 min. The Kalman gain (Lk) is updated in each time step by calculating the state estimate covariance (Pk):

 FPk  1 CT (CPk  1 CT þ Rk )  1 CPk  1 FT Lk ¼ Pk CT (CPk CT þ Rk )  1

(22)

(23)

The initial conditions of P and ^x and also the value of Q are given in the Appendix. In the Medtronic CGM calibration, the linear regression sensitivity ratio (LRSR) is calculated: n

LRSR ¼

+i ¼ 1 [Icgm i BGi ] n

+i ¼ 1 [Icgm 2i ]

(24)

where Icgm is the Kalman filter output and n is the number of BG–Icgm pairs. If LRSR is less than a sensitivity threshold value, then a modified linear regression sensitivity ratio (MLRSR) is calculated using the offset value: then offset ¼ 3

if LRSR < 7,

performance measures: absolute relative deviation (ARD) and delay. The same as calibration, only the reference BG values within the range of 40 mg/dL £ BG £ 400 mg/dL were considered valid to be used for evaluation and BG values were paired with the sensor values from the same time. ARD The ARD was determined for each patient as the deviation of processed SG values from the paired reference BG measurements:   SGi  BGi · 100 ARDi ¼ (27) BGi where i ¼ 1, 2, . . . , n and n is the number of BG–SG pairs in the evaluation datasets. Time delay

(21)

Pk ¼ FPk  1 FT þ Gw QGwT

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The filters used in CGM systems are usually nonlinear, for example, the median filter published in Medtronic CGM patents,32,33 or nonlinear elements are cascaded to linear finite impulse response or infinite impulse response filters, such as the filtering method published in the Dexcom (San Diego, CA) patent.19 The nonlinearity makes it impossible to quantify the delay caused by these filters using group/phase delay analysis. In CGM delay measurement, this explains why researchers mainly used methods such as correlation analysis,34,35 time difference in peak/nadir,17 or other maximum agreement measures such as the root mean square error23 and the most orderly Poincare´ plot.36 We used selective filtering that comprises a nonlinear element in combination with finite impulse response filtering, which makes the whole filtering process nonlinear. The delay caused by the algorithm was evaluated using the Pearson correlation coefficient as a statistical estimator. Correlation analysis is one of the simplest and most widely used methods for time delay estimation in different applications37; for example, Ionescu et al.38 applied cross-correlation analysis for online estimation of time delay between the propofol signal and the bispectral index signal in closed-loop sedation in the intensive care unit. The delay was estimated using an offline method and was defined as the time shift resulting in the maximum correlation coefficient between the processed SG signal and the instrumental delay-compensated raw signal (ISIG in the algorithm). In Eq. 28, SGT is the SG value shifted T min and rSGT , ISIG is the Pearson correlation coefficient between SGT and ISIG. T is increased in 1-min steps from 0 to 30:

then offset ¼ 0

if LRSR q 7,

Delay ¼ argT (rSGT , ISIG )max

(28)

n

MLRSR ¼

+i ¼ 1 [(Icgmi  offset)BGi ] n

+i ¼ 1 [(Icgmi  offset)2 ]

(25) Statistical analysis

Therefore, the calibrated SG value is: SG ¼ (Icgm  offset) · MLRSR

(26)

Statistical analyses were performed in MATLAB (version 7.12.0 [R2011a]). The Wilcoxon signed rank test was used to compare the median of ARD in the new algorithm and the median of ARD in the alternative algorithm.

Performance measures

Results

We compared the processed CGM data (SG) with time-paired reference BG measurements. In the following, we introduce two

The results are given for the validation dataset. Referencesensor pairs used for calibration were not used in accuracy

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MAHMOUDI ET AL. Table 1. Median (Mean) Absolute Relative Deviation Percentage

Number of readings New algorithm Alternative algorithm

Hypoglycemia

Euglycemia

Hyperglycemia

All ranges

157 14.8 (22.6) 22.2 (26.9)

709 10.6 (14.6) 12.1 (15.9)

365 8.9 (11.7) 8.8 (11.3)

1231 10.3 (14.7) 11.5 (15.8)

assessment because they are by definition perfectly accurate in the algorithm. The SCGM 1 sensor causes around 31 min (SD – 2 min) of instrumental delay, which is the time that the sensor requires to respond to the changes in subcutaneous glucose.17,21 The delay was considered fixed when delay compensation was required.21 We also considered the delay fixed throughout the data and compensated for this by shifting the signal 31 min backward before applying the algorithm. Table 1 demonstrates the median ARD for BG–SG pairs for the new algorithm and the alternative algorithm. Hypoglycemia and hyperglycemia were defined as a reference BG level of £70 mg/dL and BG > 180 mg/dL, respectively. Figure 3 depicts the performance of the two algorithms for one of the patients. The average delays produced by the algorithms across the patients were 2.1 and 2.9 min for the new algorithm and the alternative algorithm, respectively. Medians of the delays for both algorithms were 1 min across the patients. Discussion This study describes and evaluates a new algorithm consisting of an automatic selective denoising and a robust calibration for CGM systems. The most noteworthy aspect of the denoising in the algorithm is automatic noise detection and the application of filtering to only the noisy parts of the signal. This lowers the delay introduced to the CGM signal associated with filtering. The strengths of the calibration are its ability to handle spiky self-monitored BG measurements through a robust procedure and its correction of SG in the hypoglycemic range, which is achieved by adding an adaptive offset to the calibration conversion function. The SG values processed by the algorithm had low deviation from capillary reference BG measurements. With regard to Table 1, the new algorithm had a median ARD of 10.3% in ‘‘all ranges,’’ which was lower than the median ARD of the alternative algorithm in ‘‘all ranges’’ (P < 0.001); however, the delays were similar. The difference was more obvious in the hypoglycemic range, where the new algorithm had a 33% lower median ARD (P < 0.01). The source of this improvement lies in the adaptive offset used in the new algorithm to correct sensor readings in the hypoglycemic range. Also, in euglycemia, the new algorithm yielded more promising results with a lower median ARD than the alternative algorithm (P < 0.01). In hyperglycemia, the alternative algorithm seems to outperform the new algorithm; however, the difference was not significant (P > 0.05). The lower performance of the alternative algorithm, besides the calibration part, may arise from the intrinsic quality of the Kalman filter: the dependency of the filter performance on the process noise covariance Q, the measurement noise covariance R, and the predictor model reliability. Q and R are

generally unknown and work as tuning parameters, and the Q/R ratio greatly affects the performance.39 However, in the Kalman filter we implemented, R varies adaptively, and therefore the Q/R ratio is not fixed. Because of the variable glucose dynamics during the monitoring, the performance of the Kalman filter could potentially be enhanced if both R and Q were variable, for example, by considering the intraindividual variability as suggested by Facchinetti et al.40 The Kalman filter adapts smoothing of the CGM signal using the Q/R ratio. If Q/R is relatively small (noisy sensor measurements), the Kalman gain according to Eq. 23 tends to be small; thus, the signal measured by the CGM sensor is less trusted, and the model contributes more heavily than the sensor measurement in updating the states according to Eq. 18. In the application of the Kalman filter for CGM data, the predictor is usually an autoregressive model in which the estimated variables are conditioned by previous information of the variables under study.39 In the case of noisy measurements, the Kalman filter output relies heavily on a first-order autoregressive model, which may substantially deviate from the nature of the IG system and therefore be a source of error. This issue was also raised by Rebrin et al.,30 who argued that in order to benefit from the model-based filtering approaches in CGM, the model should be able to produce more reliable values than the measurements. Facchinetti et al.41 applied the SCGM—a three-module software consisting of denoising, enhancement, and prediction—on the output of the Dexcom SEVEN Plus continuous glucose monitor. This postprocessing reduced the mean ARD from 15.1% to 10.3% as evaluated in patients with type 1 diabetes. Keenan et al.42 investigated the accuracy of the Enlite sensor (Medtronic Diabetes, Northridge, CA), which is a new-generation subcutaneous glucose sensor. Data were calibrated with two different methods: Guardian REAL-Time (RT) (Medtronic Diabetes, Northridge, CA) algorithm and Paradigm Veo (Medtronic Diabetes, Northridge, CA) algorithms using self-monitored BG measurements. The data included patients with type 1 and patients with type 2 diabetes. The mean ARD for the reference plasma glucose–SG pairs was 13.86% for the Veo-calibrated sensor, which was lower than the mean ARD for the Guardian-calibrated sensor. Damiano et al.43 compared three commercial CGM systems: the Navigator (Abbott Diabetes Care, Alameda, CA), the Dexcom SEVEN Plus, and the Guardian RT. The Navigator had the best mean ARD of 11.8%. However, as also stated by the authors, because calibration of the sensors was performed using reference highquality plasma glucose rather than self-monitored BG measurements, the accuracy of the CGM systems could have been overestimated. In another study done by Mazze et al.,44 the Guardian RT and Dexcom STS CGM systems are reported to have mean ARD values of 19.9% and 16.7% and median ARD values of 16.9% and 14.2%, respectively. Barcelo´ Rico et al.11 reported that the GlucoDay (Menarini Diagnostics, Florence,

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FIG. 3. The output signals of the two algorithms for one of the patients. All plots are from the same patient. (A) The raw current measured by the sensor and the filtered signal of the selective smoothing block. (B) Zero crossing count of the signal first-order differences used for noise detection in the new algorithm. The zero crossing measurement is performed on the output of the rate-limiting filtering block (the s signal). (C) The raw current measured by the sensor and the filtered signal of the adaptive Kalman filter. (D) The variance of the measurement noise used in the adaptive Kalman filter in the alternative continuous glucose monitoring algorithm. (E) The calibrated sensor glucose from the new algorithm and the alternative algorithm. SMBG, self-monitored blood glucose.

Italy) had mean and median ARD values of 14.70% and 13.20%, respectively. The new algorithm with mean ARD of 15.1% and median ARD of 10.3% compares well with the above-mentioned results. Weinstein et al.45 indicated that the FreeStyle Navigator CGM had mean and median ARDs of 35.7% and 26.4%, respectively, in the hypoglycemic range. Yielding mean and median ARDs of 22.9% and 15.0% in the hypoglycemic range, our algorithm seems to perform better than this result. The mean delay of the FreeStyle Navigator CGM was 15 min as reported by Garg et al.35; however, in another study by

Kovatchev et al.,36 the mean delay of the Navigator was reported to be 12.5 min. Mazze et al.44 reported that Guardian RT and Dexcom STS CGM systems had mean delays of 21 and 7 min, respectively. In the above-mentioned studies, the reported delays include the summation of the physiological time lag plus the inherent electrochemical sensor delay and the delay caused by CGM filtering. The manufacturerreported delays associated with filtering per se are 8.25 min for the Guardian RT and 3 min for the CGMS Gold (Medtronic Diabetes, Northridge, CA),17 which indicate the

832 average delay of 2 min caused by filtering in our algorithm is quite comparable with delays of the existing CGM algorithms. The most commonly available CGM systems in the market are enzyme-based. This demands the accuracy of the results of our study be compared with the published accuracy of the algorithms implemented on enzyme-based CGM systems. Because the algorithm is developed on microdialysis-based CGM data, a comparison will only be reliable if it can be demonstrated that the statistical quality of the microdialysisbased CGM data is not different from that of the enzymebased CGM data. To address this issue, we performed the Kolmogorov–Smirnov test between the statistical parameters (mean, SD, and variance) of the scaled raw signal of the microdialysis-based sensor and the raw signal of the enzymebased sensor to test the null hypothesis that the parameters have the same distribution. Three Kolmogorov–Smirnov tests were run: between the mean of the microdialysis-based CGM data and the mean of the enzyme-based CGM data, between the SDs of the two types of data, and, finally, between the variances of the two types of data. The P values of all three tests were larger than the significance level of 0.05. We were hence able to retain the null hypothesis and suggest that the algorithm developed and tested on microdialysis-based CGM data may give a similar result if it is evaluated on the enzymebased CGM data. It is therefore meaningful to compare the result obtained in the study with the published accuracy results obtained from enzyme-based CGM data in the literature. Because the microdialysis-based CGM data and enzymebased CGM data were statistically equal, the improvement of the accuracy obtained in our study could be ascribed to the algorithm itself and not to the data. The rate-limiting filter in the algorithm utilizes a threshold based on a physiological maximum BG rate-of-change of 4 mg/ dL/min; however, the rate-of-change threshold is identified and implemented differently in other algorithms.19,32,33 The artifact correction by weighted local polynomial regression introduces a smoother signal than hard limit delimitation by clipping the signal rate to the maximum physiological limit as implemented in other algorithms.17,32,33 The smoother sensor signal produced by the algorithm makes the signal subsequently require a less aggressive filtering and therefore makes it possible to use selective smoothing. The detection of noisy parts of the signal implemented in our algorithm is defined in the time domain. Previous algorithms apply frequency-based noise detection and smooth the signal with a finite impulse response filter in the presence of noise19; however, the time domain-noise detection used in our algorithm is more straightforward and easier to implement because it does not need frequency transformation. A large number of ZCs is an indicator of the presence of high-frequency variations. If the frequency of the variations is larger than the physiological limit of the BG frequency, these variations are caused by noise. The maximum frequency (which is also called band edge frequency) of BG in diabetes patients in daily life with a typical schedule of meals, exercise, sleep, and insulin injection is 0.9 · 10–3 Hz.46 The intrinsic BG dynamics are therefore not faster than approximately 18 min (the inverse of the band edge frequency, the shortest period for physiological variation in BG). This suggests that if the length of the ZC noise detection segment is short enough, the meal-related variations in the

MAHMOUDI ET AL. glucose trend will not mistakenly be detected as noise by the filter. The variations in the glucose trend leading to any ZC of the first-order differences in the 5-min segment used in the algorithm are consequently due to noise, and they are therefore not physiologically meaningful. Crossing the zero line of the first-order differences at least once in the segment should therefore be detected as noise according to the discussion above. This explains the selection of ZC = 1 as the threshold for detection of a noisy signal. By increasing the ZC threshold, the sensitivity of noise detection decreases, and the roughness in the filtered signal consequently rises. Facchinetti et al.47 proposed an online method for detection of three failures in the glucose–insulin pumps system for type 1 diabetes patients who use continuous subcutaneous insulin infusion. The failures are spike in the CGM profile, loss of sensitivity of the glucose sensor, and failure in the insulin delivery pump. The SG level is compared with the predicted CGM value, and if the SG values are not consistent with the predictions, a failure alert is generated. Prediction is performed by a Kalman filter, the inputs of which are the measured CGM level by the sensor and the programmed insulin delivery profile. The CGM spike detection in their work is comparable with the ZC noise detection presented in our algorithm. However, the method proposed by Facchinetti et al.47 is individualized, and the model needs to be identified offline from the previous data. The selective smoothing that uses selective moving average in the algorithm does not introduce unnecessary delay, whereas the state-of-the-art ordinary moving average does.17,23,48 Selective smoothing is comparable with adaptive (for example, Kalman) filtering in optimization of the tradeoff between smoothing and delay.40,49 The robust calibration implemented in the algorithm attenuates the effect of outliers in the calibration set by a weighting function independently of whether the outlier is caused by an erroneous reference BG measurement entered by the patient or by a spike in the Icgm signal. Correction of the calibration set was also recommended previously.27,50 Goode et al.50 suggested performing linear least square regression and updating reference BG measurement by the patient if the R2 value is lower than a predefined threshold, whereas Keenan and Mastrototaro27 presented a method for checking the entered capillary reference BG measurement by several interconnected criteria and demanding that the patient update the capillary measurement if the entered reference value did not meet the criteria. The autocorrection feature of our algorithm reduces the demand for patients to repeat reference BG measurement in the case of imprecisely measured values. The defined thresholds for correlation coefficient, RSD, and the threshold for SG to involve adaptive offset were chosen after several experiments in order to obtain a high performance (low ARD and low delay). The same applies to k in the rate-limiting filter and l in the selective smoothing. More appropriate threshold values could be achieved by using search-based optimization techniques. Although the algorithm can be applied prospectively, it does not compensate for the physiological delay between SG and BG. Nevertheless, real-time delay compensation has been suggested10,28,51 and may be used in combination with the algorithm. The focus of our algorithm is high-quality calibration of CGM data. In other kinds of studies presented in the literature, the focus is, however, on the conversion of

CONTINUOUS GLUCOSE MONITORING ALGORITHM calibrated CGM into valid BG values by compensation for the physiological lag between IG and BG. Two such examples are the studies by Guerra et al.52 and Bequette,48 in which the physiological lag is compensated for through monoexponential dynamics between IG and BG. The estimation of BG from CGM is one step ahead of calibration, when the already calibrated CGM data are available. Indeed, the quality of calibration strongly affects the quality of BG estimation as also mentioned by Bequette.48 Although a 10-min shift is a rough approximation of the physiological lag between IG and BG and does not satisfy BG-to-IG dynamics as is required in BG estimation, the shift would be sufficient for calibration per se. Conclusions The results suggest the ability of the algorithm to enhance the accuracy of CGM; however, more thorough evaluation is required. Such evaluation must address the algorithm performance in various types of CGM devices. Acknowledgments The authors wish to express their appreciation to Roche Diagnostics, Mannheim, Germany, and the Profil Institute for Metabolic Research, Neuss, Germany, for providing the access to the datasets. The data collection was supported by an unrestricted grant from Roche Diagnostics. The research was funded by a PhD scholarship (562/06-21-10045) from Aalborg University, Aalborg, Denmark. Author Disclosure Statement No competing financial interest exists. References 1. Ali MK, Bullard KM, Imperatore G, Barker L, Gregg EW; Division of Diabetes Translation, National Center for Chronic Disease Prevention and Health Promotion: Characteristics associated with poor glycemic control among adults with self-reported diagnosed diabetes—National Health and Nutrition Examination Survey, United States, 2007–2010. MMWR Surveill Summ 2012;61:32–37. 2. Wang PH, Lau J, Chalmers TC: Meta-analysis of effects of intensive blood-glucose control on late complications of type I diabetes. Lancet 1993;341:1306–1309. 3. Alexiadou K, Doupis J: Management of diabetic foot ulcers. Diabetes Ther 2012;3:4. 4. Francescato MP, Geat M, Stel G, Cauci S: Accuracy of a portable glucose meter and of a continuous glucose monitoring device used at home by patients with type 1 diabetes. Clin Chim Acta 2012;413:312–318. 5. Davey RJ, Stevens K, Jones TW, Fournier PA: The effect of short-term use of the guardian RT continuous glucose monitoring system on fear of hypoglycaemia in patients with type 1 diabetes mellitus. Prim Care Diabetes 2012;6: 35–39. 6. Harvey RA, Dassau E, Zisser HC, Bevier W, Seborg DE, Jovanovic L, Doyle FJ 3rd: Clinically relevant hypoglycemia prediction metrics for event mitigation. Diabetes Technol Ther 2012;14:1–9. 7. Iscoe KE, Davey RJ, Fournier PA: Increasing the low-glucose alarm of a continuous glucose monitoring system prevents exercise-induced hypoglycemia without triggering any false alarms. Diabetes Care 2011;34:e109.

833 8. Buckingham B, Beck RW, Tamborlane WV, Xing D, Kollman C, Fiallo-Scharer R, Mauras N, Ruedy KJ, Tansey M, Weinzimer SA, Wysocki T: Continuous glucose monitoring in children with type 1 diabetes. J Pediatr 2007;151: 388–393. 9. Mauras N, Beck R, Xing D, Ruedy K, Buckingham B, Tansey M, White NH, Weinzimer SA, Tamborlane W, Kollman C: A randomized clinical trial to assess the efficacy and safety of real-time continuous glucose monitoring in the management of type 1 diabetes in young children aged 4 to k

(A8)

 4 0 nA2 0 4 

^ x0 ¼

(q) wi

(A13)

Q was considered constant during filtering for each patient. The following values were used as initial conditions for the state covariance and the state vector:

1

0

  Q nA2 R 0

 y0 nA 0

(A14)

(A15)