Abstract

Background: Glucose variability is increasingly regarded as a risk factor for diabetes-related complications. The mean amplitude of glycemic excursions (MAGE) is a common measure of glucose variability. While its original definition requires manual calculations, there exist several software programs for automated calculations which have shown varying degrees of agreement. The objective of this study was to develop an automated algorithm for MAGE that would achieve high accuracy against the reference standard of manual calculations and benchmark it against existing alternatives.

Methods: Manual calculations of MAGE were performed on 45 publicly available CGM profiles from a diverse set of patients (i.e., Type 1 diabetes, Type 2 diabetes, and those without diabetes) following the definition described in Service et al. [3]. An automated algorithm for MAGE calculation was developed to identify peaks and nadirs of glycemic excursions based on the crosses of a short and long moving average of the glucose profile, and implemented in version 3+ of R package iglu. The accuracy of the algorithm was evaluated relative to manual calculations using a five-fold cross-validation and compared to other publicly available MAGE calculators, as listed in Table 2.

Results: The newly developed algorithm had the median error of 1.4% relative to manual calculations (iglu v4.1+). The median errors of the six other approaches including AGATA, EasyGV, cgmanalysis, naive iglu algorithm, rGV, and cgmquantify are 9.8%, 11.3%, 20.3%, 42.0%, 78.4%, and 78.1%, respectively.

Conclusions: The newly developed algorithm eliminates the need for tedious manual MAGE calculations and approximates the manual derivation better than existing approaches.

Introduction

Glucose variability is increasingly regarded as a risk factor for diabetes-related complications beyond HbA1c [1, 2]. The mean amplitude of glycemic excursions (MAGE) is a commonly used measure of glucose variability. As defined by Service et al. [3], MAGE is the arithmetic mean of the amplitude (i.e. height) of glucose excursions that are greater than the standard deviation of the glucose values. MAGE is calculated manually by visually inspecting the glucose profiles. However, this manual approach to calculating MAGE is error-prone and impractical for large data sets of continuous glucose monitors (CGMs). Thus, an automated algorithm for MAGE calculation that approximates the manual reference is needed.

Several software programs for automated MAGE calculations have been developed, and these implementations can be divided into two categories: those that only return the numeric MAGE values, and those that additionally provide a visual display of the calculations [4]. Examples in the first category include the Excel®-based workbook EasyGV [5], online GlyCulator2 [6], R package cgmanalysis [7], early versions of R package iglu [8], and Python package cgmquantify [9]. Because these software programs do not output a visual display of the performed calculations, the user has no means to inspect the calculations and confirm accuracy. Examples in the second category include programs developed by Fritzsche [10] and Baghurst [11]. However, we found the software developed by Fritzsche are not compatible with newer operating systems (e.g., Windows 10) to allow importing of CGM data. Additionally, Baghurst does not provide a downloadable implementation of his algorithm. However, according to documentation for cgmanalysis [7] and cgmquantify [9], these packages provide free implementations of Baghurst’s algorithm, albeit they may not fully reflect the original implementation.

The comparison of GlyCulator, EasyGV, Fritzsche, and Baghurst algorithms demonstrates varying degrees of agreement across MAGE algorithms [12]. While the aforementioned algorithms were compared to each other, they were not compared with the manual reference standard.

Thus, the objective of current study is to establish an approach of calculating MAGE that is user-friendly and approximates well the results obtained manually. Therefore, these goals are defined as follows.

1. To computationally emulate the manual method in Service et al. [3] with high accuracy based on the automated detection of peaks and nadirs tailored specifically towards CGM data

2. To provide a freely-available implementation with a visual display

To evaluate accuracy, the newly developed algorithm was compared with the manual derivations along with the other computational approaches.

Methods

CGM Data and Manual MAGE Calculation

Publicly-available CGM data were obtained from [13], from which 45 CGM traces were selected to represent a diverse set of patients (Type 1 diabetes, Type 2 diabetes, and those without diabetes), see Table 1. Each CGM trace corresponds to one day of measurements (24 h), where a day is defined from midnight to midnight as it is customary to calculate MAGE daily. Manual MAGE calculations were performed on each of the 45 CGM traces following the original description of MAGE calculations by Service et al. [3] by two of the authors, who were unaware of the outcome of the proposed algorithm’s outputs.

Table 1: Summary of the 45 CGM traces used for manual calculations. The first three datasets have been obtained as described in [13]. The last dataset is available within R package iglu [8].

Existing MAGE Calculators

We benchmarked the proposed algorithm against a variety of open and closed-source tools. To the best of our knowledge, the list in Table 2 comprises a relatively exhaustive list of existing packages to compute MAGE on CGM data as of May 15, 2024.

Table 2: An exhaustive list (to the best of our knowledge) of existing packages and algorithms to compute MAGE as of May 15, 2024. We also specify the version and release date if we benchmark the proposed MAGE algorithm against it, or the reason we were unable to.

The Algorithm

The original MAGE description in Service et al. [3] requires (i) identification of turning points (e.g., peaks and nadirs) for calculation of glucose excursions; (ii) filtering those turning points to only keep relevant glucose excursions and calculating the mean of those excursion amplitudes (i.e., heights).

To the best of our knowledge, our implementations for both parts (i) and (ii) are unique.

1. Identification of peaks and nadirs. The proposed algorithm uses the crosses of a short and long moving average to identify time intervals where a peak or nadir may exist. Let α be the window size for the short moving average and 𝛽 be the window size for the long moving average, such that α < β. Since the shorter moving average is more affected by local variation in the glucose values than the longer moving average, a peak or nadir (i.e., a turning point, tp_i) must exist on the intervals that are bounded by the crosses of the two moving averages cp_i and cp_i + 1 as demonstrated in Figure 1, with the exception of “whiplash”. (Note: the first and last glucose values are always treated as crossing points [Appendix: Why are the first and last data points always treated as crossing points?].) To solve the issue of “whiplash”, where the true peak/nadir falls outside the interval bounded by cp_i and cp_i + 1 for back-to-back sinusoidal excursions [Appendix: What is whiplash?], we make the assumption that peaks and nadirs alternate (peak, nadir, peak, nadir, etc.) or vice versa and calculate subsequent peaks or nadirs sequentially as the maximum or minimum glucose value, respectively, between the index of tp_i − 1 (left boundary, exclusive) and cp_i + 1 (right boundary, inclusive) for i = 2. The left boundary is cp_i for the base case of i = 1.

   A distinct advantage of using moving averages to identify the turning points is that they inherently smooth out local fluctuations and noise within larger trends in a glucose profile. The amount of smoothing can be tuned by varying window size parameters α and β. By default, the algorithm uses α = 5 and β = 32 as these values maximized the algorithm’s accuracy on the manual MAGE calculations, where α ∈ [1, 7] and β ∈ [16, 38].

2. Eliminating Excursions and Calculation of Mean Amplitudes. An excursion is defined by three turning points E_i, j, k = (tp_i, tp_j, tp_k) s.t. i < j < k , where the amplitude of the excursion is calculated as follows: a(j, i) = |tp_j − tp_i| and |E_i, j, k| = (|E_left|, |E_right|) = (a(j, i), a(k, j)). An “acceptable” excursion, as defined by Service et al. [3], requires the amplitudes of both the left-side and right-side half-excursions to exceed the standard deviation of the entire CGM trace. Computationally, we devised the procedure below to calculate acceptable excursions, given a set of alternating turning points. To prevent double counting excursions, only ascending (MAGE+) or descending (MAGE–) half-excursion amplitudes are averaged (i.e., |E_left| or |E_right|, respectively), where ascending means tp_j is a peak and vice versa. When excursions are symmetric, MAGE+ = MAGE–, however in general they may disagree [11]. By default, our implementation follows the description in Service et al. [3], and returns MAGE+ or MAGE– depending on which type of excursion first crosses the one SD threshold; however, the user can specify either if desired.

   Procedure: Starting from i = 1, j = 2 (note: R is 1-based), check whether the amplitude exceeds the one standard deviation threshold, a(j, i) = a(2, 1) > sd. If false, the algorithm will calculate max  {a(3, 1), a(3, 2)} > sd and so on until a large enough amplitude is found or the CGM trace ends. Note: we use the maximum to enable “accumulation”, an optimization that allows an earlier larger half-excursion to absorb a later smaller half-excursion. Once an acceptable left-side half-excursion is found, the boundaries i and j are recorded and the same procedure is used to find an acceptable right-side excursion, where j is updated if a(k, i) >  = a(j, i). We record E_left once E_right > sd or the CGM trace ends.

Handling Gaps

The algorithm outlined above assumes that the input CGM trace should be treated contiguously, with MAGE calculated over the entire trace and turning points alternate (peak, nadir, peak, nadir, etc.) or vice versa. However, this can lead to suboptimal results if large gaps are present in the trace. While small gaps can be interpolated before MAGE calculation (starting in iglu v3.3.0), CGM values separated by large gaps should be treated as separate segments, with MAGE calculated on each segment independently. By default, MAGE released in iglu v3.5.1+ segments the CGM trace by gaps larger than max_gap = 180 minutes and calculates MAGE via the above algorithm on each segment [Appendix: [Why set a default of 180 minutes for max_gap?]]. The user can then return the MAGE values for each segment (MAGE_i) or a one-number summary for the variability of the entire CGM trace (MAGE_overall). MAGE for the entire trace is reported as an average of the segment MAGE values, weighted by segment length ($MAGE_{overall} = \Sigma \frac{n_i}{N}MAGE_i$, where n_i is the length of segment i and N = Σn_i). The plot displays segments with a solid, black vertical line for the left-boundary and a dashed, black vertical line for the right boundary.

Implementation
The newly developed algorithm for MAGE calculation can be found from version 3 onward in R package iglu [17] as well as the accompanying GUI via Shiny App [8] - both are free and open-source. Implementation in a script-based programming language like R allows the creation of reproducible scripts for all data processing and MAGE calculation steps, and the availability of a point-and-click GUI meets the needs of researchers with limited programming experience. By default, the algorithm uses α = 5 and β = 32, however these choices can be adjusted by the user.

# load package and example data
library(iglu)

# mage calculation using default parameters
mage(example_data_5_subject)
#> # A tibble: 5 × 2
#> # Rowwise: 
#>   id         MAGE
#>   <fct>     <dbl>
#> 1 Subject 1  72.4
#> 2 Subject 2 118. 
#> 3 Subject 3 116. 
#> 4 Subject 4  70.9
#> 5 Subject 5 142.

# Adjust default window sizes in mage calculation
mage(example_data_5_subject, short_ma = 3, long_ma = 30)
#> # A tibble: 5 × 2
#> # Rowwise: 
#>   id         MAGE
#>   <fct>     <dbl>
#> 1 Subject 1  72.3
#> 2 Subject 2 118. 
#> 3 Subject 3 116. 
#> 4 Subject 4  70.9
#> 5 Subject 5 142.

In addition to providing a numerical MAGE value, the software allows the creation of glucose trace plots that highlight the selected peaks and nadirs used for MAGE calculation.

# subset data to better see identified peaks and nadirs
fig1data <- example_data_1_subject[1:200, ]
# plot using mage function with plot = TRUE and show_ma = TRUE 
mage(fig1data, plot = TRUE, show_ma = TRUE, title = "Glucose Trace - Subject 1")

Figure 1: Glucose trace of Subject 2 with Type 2 diabetes from Broll et al. [8], the solid green line is a short MA with α = 5 and the solid orange line is a long MA with β = 32. The peaks and nadirs occur between the crosses of short and long moving averages. By default, the algorithm calculates MAGE+ as in this profile the first segment to exceed one SD threshold is ascending. The provided graphical display in iglu software [17] automatically highlights only those peaks (red) and nadirs (blue) that are used for final calculation.

Additionally, the plot will highlight any gaps in the data due to missing readings. If these gaps are shorter than max_gap = 180, they will be part of the same segment.

# subset data to better show gaps
fig2data <- example_data_1_subject[200:400, ]
# plot using mage function with plot = TRUE
mage(fig2data, plot = TRUE, title = "Glucose Trace with Gaps - Subject 1")

Figure 2. Glucose trace of Subject 5 with Type 2 diabetes from Broll et al. [8] with regions of missing glucose readings automatically highlighted using iglu software [17].

Assessment of Accuracy
The accuracy of the proposed algorithm was evaluated against 45 manual calculations. For each CGM trace, the accuracy was measured as the relative percent difference in the automatically calculated MAGE value and the manual MAGE value, that is: $\frac{|MAGE_{algorithm} - MAGE_{manual}|}{MAGE_{manual}}\cdot 100\%$ .

To find optimal combination of window sizes, we considered α ∈ [1, 7] and β ∈ [16, 38], and recorded the mean errors across 45 CGM traces for each pair of (α, β) values. The optimal (α, β) pair was determined as the one that leads to the minimal in-sample error. Since the obtained minimal error is a biased assessment of accuracy as the optimal (α, β) were selected based on the same 45 traces, five-fold cross-validation was also performed to gain an unbiased assessment of accuracy. The 45 CGM traces were randomly split into five groups of nine samples each. The best (α, β) combination was then chosen using four out of the five groups, and used to evaluate the accuracy on the remaining group. The best combination was defined as the one that gave the lowest percent error compared to manual calculations. This out-of-sample evaluation was performed on all five groups.

The accuracy of EasyGV [5], cgmanalysis [7], cgmquantify [9], rGV, AGATA, and the MAGE algorithm implemented in the earlier version of iglu [8] was also determined against the manual reference derivation. The latter is a naive simplification of original MAGE definition, as it calculates MAGE as the mean of absolute differences between glucose values and the glucose mean that are larger than one standard deviations (and thus does not attempt to explicitly identify peaks and nadirs). We use iglu_ma to denote the proposed algorithm based on moving averages, and iglu_naive to denote the naive algorithm used in earlier versions of the software. We were unable to evaluate GlyCulator2 [6] because the software requires CGM trace to have at least 576 time points, and thus returns NA when applied to 24-hour CGM data. We were also unable to evaluate Fritzsche [10] as importing the CGM data into the provided user interface on Windows 10 lead to errors.

Assessment of Parameter Frequency Robustness

We also test if the optimal long and short window parameters chosen on the 5-minute CGM data - i.e., (α_best, β_best) - are robust to the frequency of the meter / CGM data (e.g., 1-, 5-, 10-, and 15-minute). To do this, we use the iglu::CGMS2DayByDay function to change the raw data’s frequency to 1-, 10-, and 15-minute intervals, and automatically imputes glucose values for gaps less than 45 minutes. We then rerun the hyperparameter tuning code to discover the best (α, β) selected on the new data and report the mean, median, and IQR of errors. Note that this is not a perfect comparison: information is lost when downsampling the 5-minute CGM trace to 10- and 15-minutes but the manual calculations compared against are performed on 5-minute data. Thus, we expect to see a slightly degraded performance on the 10- and 15-minute data simply due to this design limitation and not necessarily an algorithm limitation.

Results

The optimal choice of window sizes (α, β) for the proposed algorithm on the 5-minute CGM data was determined to be (5, 32) as this combination minimized the average error relative to 45 manual calculations. Figure 3 shows a heatmap of the percent average error (across 45 traces) as a function of α and β, where the darkest regions represent the most favorable combinations of α and β. (Note: while there was a tie, we chose hyperparameters in the middle of the dark region.) The minimum corresponds to a mean average error of 8.9% with a much smaller median percent error of 1.4% and [Q1, Q3] = [0.5%, 7.1%]. As expected, the unbiased out-of-sample cross-validated errors are slightly larger, but still relatively low: 9.1% mean with the median of 10.0% and inter-quartile range of [9.4%, 11.8%].

Figure 3. Heatmap of average errors over 45 CGM traces of automatically calculated MAGE values via proposed algorithm relative to manual MAGE values. The short moving average window size α ∈ [1, 7] and the long moving average window size β ∈ [16, 38]. The darker regions correspond to smaller errors, with the minimal error of % achieved at α = 5, β = 32.

Compared to the other available automated algorithms, the next best results were achieved by EasyGV, which had an average error of 23.4%, the median error of 11.3% and the IQR of [6.0%, 22.3%]. In constrast, cgmanalysis, iglu_naive, and cgmquantify all had significantly higher errors, with median errors (and interquartile range) of 20.3% [8.6%, 32.1%], 42.0% [32.6%, 48.3%], and 78.4% [30.8%, 131.9%], respectively. Figure 4 shows boxplots of relative errors of all algorithms on 45 CGM traces. Figure 5 and Table 3 show a correlation matrix of all algorithms on the 45 CGM traces. (Note: see the “Important Notes on rGV” in the Discussion below.)

Table 3. Correlation matrix of various MAGE calculators’ outputs on the 45 CGM traces.

Figure 4. Comparison of relative errors of different automatic algorithms for MAGE calculation relative to manual MAGE values on 45 CGM traces. The proposed method is implemented as ma (moving average) algorithm in R package iglu [17]; best corresponds to the errors with α = 5, β = 32; cv corresponds to the out-of-sample errors based on 5- fold-cross-validation for α, β.

Figure 5. Heatmap of correlation matrix of various MAGE calculators’ outputs on the 45 CGM traces.

Table 4. Correlation matrix of various MAGE calculators’ outputs on the 45 CGM traces and the mean and standard deviations of the 45 CGM traces.

For the proposed algorithm, both in-sample errors with the best choice of window sizes (α, β) and out-of-sample cross-validated errors are displayed. The corresponding numerical summaries are summarized in Table 2.

Table 5: Numerical summaries of relative errors of automatic MAGE calculators on 45 CGM traces compared to manual calculations. The proposed method is implemented as ma (moving average) algorithm in R package iglu [17]; best corresponds to the errors with α = 5, β = 32; cv corresponds to the out-of-sample errors based on 5-fold-cross-validation for α, β.

Table 6: Numerical summaries of relative errors of the proposed MAGE algorithm on 45 CGM traces compared to manual calculations at varying frequencies of the underlying data. The relatively constant mean error rate demonstrates the proposed method and selected hyperparameters (α = 5, β = 32) are robust to the frequency of the meter.

Discussion

There are several software programs for automatic MAGE calculation, however, they show varying agreement [12]. In the current study, a new automated algorithm for MAGE was developed based on moving averages which demonstrated the least amount of error relative to manual reference when compared to other automated algorithms.

The newly developed algorithm approximated the manual results better than EasyGV, cgmanalysis, iglu_naive, and cgmquantify. The largest discrepancy was noted for cgmanalysis and cgmquantify both of which implemented Baghurst’s algorithm [11]. It is unclear whether the deviation from manual reference is due to the underlying algorithm or the specific of implementations of the algorithm. The differences in accuracy among the MAGE algorithms points to the necessity of a visual display of the underlying calculations with identified peaks and nadirs. While such visual display is implemented for the proposed MAGE algorithm in version 3 of R package iglu [17], this functionality is absent from EasyGV, cgmanalysis, and cgmquantify, making it more difficult for the user to verify the accuracy.

The correlation between our proposed algorithm (with Service definition) and manual calculations per Service is 0.93 when compared across the same 45 CGM traces. This is the largest correlation value across the implementations we have compared (0.92 for AGATA, 0.91 for EasyGV, 0.90 for older iglu MAGE algorithm, 0.82 for CGManalysis and 0.66 for CGMQuantify), which is consistent with our proposed approach showing the lowest error rate relative to manual calculations. When comparing correlations between our approach and existing methods, we find correlation of 0.98 with EasyGV, 0.96 with older iglu MAGE algorithm, 0.92 with AGATA, 0.85 with CGManalysis and 0.7 with CGMquantify.

As can be seen in Table 4, there is a relatively high correlation between MAGE and standard deviation of the CGM trace. This has been noted before [33] and has led some to question the importance of using MAGE over the simpler standard deviation. Nevertheless, further research is needed to see if MAGE is better correlated with clinical outcomes (e.g., morbidity or mortality) than the simple standard deviation.

A major challenge for any automated MAGE algorithm is separation of true peaks and nadirs in a larger trend from the smaller glucose fluctuations. A particular advantage of the proposed algorithm is the use of moving averages which inherently smooth out local fluctuations within a larger trend. The amount of smoothing can be tuned by varying the window sizes of the short and long moving averages, and we found that the algorithm is quite robust to the choice of window sizes, with high accuracy in a large range (Figure 3). Additionally, the proposed optimal short and long moving averages are robust to the frequency of the CGM meter, as shown by the similar mean errors of the MAGE algorithm despite the meter frequency in Table 4. One disadvantage is that the moving average is a lagging indicator, and thus is undefined in the beginning of the time period (e.g., a moving average of window size 20 is undefined for the first 19 time points). This could skew the calculations of MAGE if the first times points contain a lot of variation in subject’s glucose levels. To adjust for this, the algorithm replaces the undefined moving average values at the first time points with the first calculated moving average value (e.g., a moving average of window size 20 will be first calculated at 20th time point, and that same value will be assigned to the first 19 time point to be later used in determining the crosses of short and long moving averages). This adjustment can be seen for the long moving average in Figure 1, where its value is flat for the first 32 points. Given the high accuracy of the proposed algorithm compared to manual calculations, this adjustment is adequate, however it is possible that it may not be adequate for some CGM profiles. In practice, we recommend to inspect the visual display of MAGE calculations provided by iglu [17] as in Figures 1 and 2 for additional accuracy validation. Another challenge for MAGE calculations is the presence of asymmetric excursions. To prevent double counting, typically only ascending (MAGE+) or descending (MAGE–) excursion amplitudes are averaged. When excursions are symmetric, MAGE+ = MAGE–, and the accuracy is not affected irrespective of the choice of direction. However, when excursions are asymmetric, the direction matters as the two disagree. By default, our algorithm follows the description in Service et al. [3], and calculates MAGE+ or MAGE– depending on which type of excursion first crosses the one SD threshold. The same decision is made in [12]. However, Baghurst [11] argues that it is more appropriate to use $MAGE_{avg} = \frac{1}{2}(MAGE_{plus}+MAGE_{minus})$. Our implementation allows the user to change the default settings, and explicitly calculate MAGE+, MAGE–, MAGEavg, or MAGEmax, which makes our algorithm’s implementation more flexible compared to existing MAGE software.

Conclusion

We present a new algorithm for MAGE calculation that significantly outperforms existing algorithms. The implementation of this algorithm is free and open-source, it is available in the R package iglu version 3 [17] as well as in the accompanying GUI via Shiny App at https://irinagain.shinyapps.io/shiny_iglu/. Furthermore, the visual displays of the CGM traces, exact values for manual MAGE and the code to reproduce all analyses, along with instructions on how to download each CGM trace, are publicly available at https://github.com/Nathaniel-Fernandes/mage_algorithm_data. The public nature of CGM traces, manual MAGE values, and the algorithm allow additional validation of presented results by a community at large.

Funding Sources: NSF CAREER Award DMS-2044823 to IG

Acknowledgements: The source of subset of the data is the T1D Exchange, but the analyses, content and conclusions presented herein are solely the responsibility of the authors and have not been reviewed or approved by the T1D Exchange.

Disclosures: None

Appendix

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