Calculating the Mean Amplitude of Glycemic Excursions from

Research ArticleCalculating the Mean Amplitude of Glycemic Excursions fromContinuous Glucose Data Using an Open-Code ProgrammableAlgorithm Based on the Integer Nonlinear Method

Xuefei Yu1 Liangzhuo Lin1 Jie Shen2 Zhi Chen2 Jun Jian3

Bin Li1 and Sherman Xuegang Xin 4

1Southern Medical University Guangzhou China2TheThird Affiliated Hospital of Southern Medical University Guangzhou China3Guangzhou Power Supply Bureau Co Ltd Huadu Branch Guangzhou China4School of Medicine South China University of Technology Guangzhou 520006 China

Correspondence should be addressed to Sherman Xuegang Xin xinxgscuteducn

Received 20 September 2017 Revised 18 January 2018 Accepted 7 February 2018 Published 8 March 2018

Academic Editor Jesus Pico

Copyright copy 2018 Xuefei Yu et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The mean amplitude of glycemic excursions (MAGE) is an essential index for glycemic variability assessment which is treated as akey reference for blood glucose controlling at clinic However the traditional ldquoruler and pencilrdquo manual method for the calculationof MAGE is time-consuming and prone to error due to the huge data size making the development of robust computer-aidedprogram an urgent requirement Although several software products are available instead of manual calculation poor agreementamong them is reported Therefore more studies are required in this field In this paper we developed a mathematical algorithmbased on integer nonlinear programming Following the proposedmathematical method an open-code computer program namedMAGECAA v10 was developed and validated The results of the statistical analysis indicated that the developed program wasrobust compared to the manual method The agreement among the developed program and currently available popular softwareis satisfied indicating that the worry about the disagreement among different software products is not necessary The open-codeprogrammable algorithm is an extra resource for those peers who are interested in the related study on methodology in the future

1 Introduction

Clinical researches have suggested that high glycemic vari-ability may cause more serious damage to the body thanhigh level stable blood glucose [1] which relates to thedevelopment of diabetic complication [2ndash6] and the increaseof mortality in critically serious patients without diabetes[7 8] In such circumstances how to quantitatively evaluatethe glycemic variability in diabetic blood glucose monitoris essential for the clinical diagnosis and treatment Severalindexes have been proposed for the quantitative evaluationof the glycemic variability such as MBG SDBG IQR LAGEM-value CV J-index IGC GRADEMODD LBGI HBGIADRR TI LI PGS CONGA and MAGE [9ndash16] Presentlyclinicians and researchers trend to choose MAGE as thepreferred index [17 18] making it a ldquopopular standardrdquo

in the quantitative evaluation of the within-day glycemicvariability

MAGE is an arithmetic average of either the upward ordownward of all glycemic excursions exceeding the threshold(standard deviation of blood glucose (SDBG) obtained fromall blood glucose concentrations within 24-hour period) thedirection of the calculation is determined by the first count-able excursion [19ndash21] Following the original definition byService et al [19] clinicians and researchers applied the ldquorulerand pencilrdquo graphical approach to calculateMAGEHoweverthis kind of manual approach is time-consuming and error-prone when dealing with huge amount of data typically char-acterized by 288 observations in 5 minutes apart over a 24-hour period generated from continuous glucose monitoring(CGM)Thus the development of computer-aided programsto calculate MAGE becomes an urgently needed task

HindawiComputational and Mathematical Methods in MedicineVolume 2018 Article ID 6286893 9 pageshttpsdoiorg10115520186286893

2 Computational and Mathematical Methods in Medicine

To address the urgent need above several automaticprograms have been developed [22ndash25] making great con-tribution to the automatic calculation of MAGE However itwas pointed out that the agreement among them is poor [26]Therefore more studies are required to explore the automaticcalculation of MAGE Currently available programs can beroughly divided into two groups One group has detaileddescriptions about algorithm with graphical display such asthe automated algorithm described by Baghurst [22] and thecomputer software described by Fritzsche et al (Fritzsche)[23] The other group only provides executable softwarefor automatic calculation without detailed descriptions ofthe algorithms used in the software such as the web-basedapplication ldquoGlyCulatorrdquo [24] and the Excel-based workbookldquoEasyGVrdquo [25] However to the best of our knowledgeall of these methods did not provide programmable opencodes which are important resources required for peers toimplement more studies on methodology in this field

Therefore in this report we developed a mathematicalalgorithm based on integer nonlinear programming methodFollowing the proposed mathematical method a computer-aided program named MAGECAA v10 was developed Thecode of our program is open if peers are interested init please contact xuefeiyusmueducn for downloadingTo validate the developed program comparison study wasimplemented using blood glucose CGM datasets obtainedfrom T1D T2D and gestational diabetes patients againstthe manual method (MAGEo) and other currently popularsoftware products

2 Materials and Methods

21 The Proposed Mathematical Algorithm Let 119891(119905) (119905 =1199051 1199052 119905119871) represent the discrete blood glucose valuesobtained in CGM then 119891 is a discrete function defined intime set 1199051 1199052 119905119871 Let SDBG be the standard deviationof 119891(119905) (119905 = 1199051 1199052 119905119871) A graph depicting the glycemicvariability can be formed by connecting all the discrete valuesof function 119891 When the difference of a peak and an adjacentnadir exceeds SDBG the corresponding peak is labeled as avalid peak The key point for the calculation of MAGE is tocorrectly count the valid peak or nadir

Amplitude is the difference of functional values in a peakand a nadir of the function graph A valid amplitude islabeled and countedwhen it is bigger than SDBGTo computeMAGE these valid amplitudes of function 119891 should be firstlysearched From themathematical perspective a peak or nadirshould be an extreme point of the function These extremepoints related to valid amplitudes are called valid extremepoints The MAGE computation problem can be solved bycalculating all valid extreme points

Suppose that 1199051198971 1199051198972 119905119897119873 is the sequence of all extremepoints of function 119891 We know that the local maximumpoints and local minimum points of function 119891 shouldbe arranged as follows interleaved between a maximumpoint and a minimum point For simplicity 1199051198971 1199051198972 119905119897119873is denoted as 1 2 119873 and 1198991 1198992 119899119870 is an arbitraryvalid subsequence of sequence 1 2 119873 These local

maximumpoints and localminimumpoints can be staggeredin subsequence 1198991 1198992 119899119870 Then

1 le 1198991 lt 1198992 lt sdot sdot sdot lt 119899119870 le 119873(minus1)119899119896+1minus119899119896 = minus1 119896 = 1 2 119870 minus 1 (1)

where 2 le 119870 le 119873 This equation ensures that the selectedextreme points are staggered

Thus the above MAGE computation problem can betransformed to an integer nonlinear programming (INLP)problem

argmax1198701198991 1198992119899119870

119885119870 (1198991 1198992 119899119870)

= argmax1198701198991 1198992119899119870

119870minus1sum119896=1

10038161003816100381610038161003816119891 (119905119899119896+1) minus 119891 (119905119899119896)10038161003816100381610038161003816 (2)

subject to 10038161003816100381610038161003816119891 (119905119899119896+1) minus 119891 (119905119899119896)10038161003816100381610038161003816 ge SDBG119896 = 1 2 119870 minus 1

1 le 1198991 lt 1198992 lt sdot sdot sdot lt 119899119870 le 119873(minus1)119899119896+1minus119899119896 = minus1 119896 = 1 2 119870 minus 1

(3)

According to the principle of INLP problem function(2) should have an optimum solution 119899lowast1 119899lowast2 119899lowast119870 and anoptimumvalue 119885lowast119870with respect to a constant119870 it should alsohave an optimum value 119885lowast119870minus1 for constant119870 minus 1 and

119885lowast119870minus1 le 119885lowast119870 (4)

Suppose that 119870lowast is the maximal 119870 with respect tothe optimal solution of function (2) and then the optimalsolution 119899lowast1 119899lowast2 119899lowast119870lowast represents the valid extreme pointsto compute MAGE

If the extreme point 119899lowast1 is the local minimum point then

MAGE+ = 1[119870lowast2][119870lowast2]sum119896=1

10038161003816100381610038161003816119891119899(119896minus1)lowast2+2 minus 119891119899(119896minus1)lowast2+1 10038161003816100381610038161003816

MAGEminus = 1[(119870lowast minus 1) 2][(119870lowastminus1)2]sum119896=2

10038161003816100381610038161003816119891119899(119896minus1)lowast2+1 minus 119891119899(119896minus1)lowast2 10038161003816100381610038161003816 MAGE = MAGE+MAGEa = MAGE+ +MAGEminus2

(5)

If the extreme point 119899lowast1 is the local maximum point then

MAGE+ = 1[(119870lowast minus 1) 2][(119870lowastminus1)2]sum119896=2

10038161003816100381610038161003816119891119899(119896minus1)lowast2+1 minus 119891119899(119896minus1)lowast2 10038161003816100381610038161003816

MAGEminus = 1[119870lowast2][119870lowast2]sum119896=1

10038161003816100381610038161003816119891119899(119896minus1)lowast2+2 minus 119891119899(119896minus1)lowast2+1 10038161003816100381610038161003816 MAGE = MAGEminusMAGEa = MAGE+ +MAGEminus2

(6)

Computational and Mathematical Methods in Medicine 3

Now our object is to get the optimal solution 119899lowast1 119899lowast2 119899lowast119870lowast of function (2) If the INLP problem is solvedby enumeration algorithm the amounts of different subse-quences 1198991 1198992 119899119870 should be (119873119870 ) For 3 le 119870 le 119873 theamounts of all subsequences are

(119873119873) + ( 119873119873 minus 1) + sdot sdot sdot + (1198733) asymp 2119873 (7)

It is difficult to directly solve function (2) when 119873 islarge A faster optimization algorithm is usually used to solvethe above INLP problem For example a penalty function

algorithm [27 28] was used here to transform the above INLPproblem (2) to an unconstrained optimization problem asfollows

Set

119892119896 (1198991 1198992 119899119870) = SDBG minus 10038161003816100381610038161003816119891119899119896+1 minus 119891119899119896 10038161003816100381610038161003816 ℎ119896 (1198991 1198992 119899119870) = (minus1)119899119896+1minus119899119896 + 1

119896 = 1 2 119870 minus 1(8)

The new unconstrained integer optimization problembecomes

argmin11989911198992 119899119870

119884119870 (1198991 1198992 119899119870) = argmin1198701198991 1198992119899119870

[minus119870minus1sum119896=1

10038161003816100381610038161003816119891119899119896+1 minus 119891119899119896 10038161003816100381610038161003816 +119870minus1sum119896=1

120583max 0 1198921198962 + 119870minus1sum119896=1

120582ℎ2119896] (9)

where 3 le 119870 le 119873 and120583119896 and 120582119896 (119896 = 1 119870minus1) are penaltycoefficients and tend to be +infin

When the optimum value 119884lowast119870 gt 0 problem (9) has nooptimal solution

Differential evolution (DE) algorithm a faster opti-mization algorithm proposed by Storn and Price [29] isa simple but powerful population-based stochastic searchtechnique to solve global optimization problems over con-tinuous domains Many researchers modified DE algorithmto improve its performance when it was applied to a specificproblem [30ndash33] The idea of a modified DE algorithmproposed by Lin [34] to solve mixed-integer nonlinearprogramming problem is used here

DE searches for a global optimal point in an 119899-dimensional hyperspace Let 119878 be the 119870-dimensional searchspace of the INLP problem under consideration The DEevolves a population ofNP 119899-dimensional individual vectorsthat is solution candidatesN119894119870 = (1198991198941 119899119894119870) isin 119878 119894 = 1 NP from one generation to the next The evolution beginswith a randomly initialized population of 119899-dimensionalinteger parameter vectors in space 1 2 119873119870 In eachvector integer parameters are sorted in an ascending orderEach vector forms a candidate solution to the unconstrainedoptimization problem At each generation119866 DE employs themutation and crossover operations to produce a trial vectorU119866119894119870 for each individual vector N119866119894119870 also called target vectorin the current population

The details of the employment of the DE algorithm tosolve the above INLP problem are as follows from (a) to (d)

(a) InitializationA randomly initialized population is createdto cover the entire search space uniformly as in the followingform

N0119894119870 = (1 1 1)+ NINT [(1205881198941 1205881198942 120588119894119870) times (119873 minus 1)] (10)

where 120588119894119895 is a random number in the range [0 1] and NINT[B]is expressed as the nearest integer vector to real vector B

(b) Mutation Operation For each target vectorN119866119894119870 at genera-tion 119866 randomly sample three other individuals N1198661199031119870 N

1198661199032119870

and N1198661199033119870 from the same generation where 1199031 1199032 and 1199033 arerandom and mutually different integers generated over therange [1119873119875] which should be different from the currenttrial vectorrsquos index 119894 Then an associated mutant vector V119866119894119870 =(V1198661198941 V119866119894119870) can be generated by using strategy

V119866119894119870 = N1198661199031119870 +NINT [119865 times (N1198661199032119870 minusN1198661199033119870)] (11)

where 119865 is a factor in [0 1] for scaling differential vectors(c) Crossover Operation The crossover operation is appliedto each pair of the generated mutant vector V119866119894119870 and itscorresponding target vector N119866119894119870 to generate a trial vectorU119866119894119870 = (1199061198661198941 1199061198661198942 119906119866119894119870)119906119866119894119895 =

V119866119894119895 rand [0 1] le CR or 119895 = 119895rand119899119866119894119895 otherwise

119895 = 1 2 119870(12)

where CR isin [0 1] is a crossover constant that is determinedby users 119895rand is a randomly chosen index in [1 119870] whichensures that 119906119866119894119895 gets at least one parameter from V119866119894119895 Theinteger parameters of target vector N119866119894119870 are also sorted fromsmall to big

(d) Selection OperationThe trial vectorU119866119894119870 is compared to itscorresponding target vectorN119866119894119870 using the greedy criterion todecide whether a member of generation 119866 + 1 existed or notIf vectorU119866119894119870 yields a smaller cost function value 119884(U119866119894119870) than119884(N119866119894119870) then N119866+1119894119870 is set to U119866119894119870 otherwise the old vector isretained The operation is expressed as follows

N119866+1119894119870 = U119866119894119870 if 119884 (U119866119894119870) lt 119884 (N119866119894119870) N119866119894119870 otherwise (13)


Start

Input CGMdataset

Display the amount of the datasetcalculate the mean and the SDBG of the dataset

Identify all extreme points and set the quantity of them as N

The maximum amplitude of this dataset is less than SDBG so there is no MAGE

Generate a population of NP (depends onK) K-dimensional individual vectors fromthose N extreme points randomly (K startas 3 from 3 to N)

Mutate each vector of those K-dimensionalindividual vectors to a new vectormake a selection on the vector before and after the mutation to form a new population of NP K-dimensional individual vectors

End

Select and record the optimum vector

which minimizes Y the most

Identical to the initiallyset mutation number

Calculate the parameters MAGE+ MAGEminus MAGE and MAGEa based on the found optimum vector

Display the parameters and plots of these valid extreme points

End

No

Yes

No

Yes

Record minimum Y

Iff(t)max minus f(t)min lt SDBG

Figure 1 Flow of the algorithm of the program of MAGECAA v10

The above (b) to (d) steps are repeated till the evolutiontimes arrived to certain number (general 119866 = 200) fromthe last evolutionary generation the individual vector N119866119894119870with the smallest value of the objective function 119884 is theoptimal solution of the problemof the current119870-dimensionalextremepoint combinationThe algorithmwill search119870 from3 to 119873 till the objective function 119884 obtains the minimum

value so as to obtain the optimal extreme point combination119899lowast1 119899lowast2 119899lowast119870lowast and then use formulas (5)sim(6) to calculatethe MAGE value Figure 1 shows the entire algorithm

22TheDeveloped ProgramBased on the ProposedMathemat-ical Method Based on the proposed mathematical methoda computer automated program named MAGECAA v10


t

t minus 1

t + 1

(a)

t

t minus 1

t + 1

(b)

Figure 2 (a) shows turning points 119905 and 119905 + 1 to be equal in values (not being extreme points) but there is an amplitude from peak point 119905 minus 1to nadir point 119905 (b) shows the combination of points 119905 and 119905 + 1 as one point 119905 + 1 or remove point 119905 then let point 119905 + 1 be a local maximumpoint

was developed The MAGE calculation program can bedescribed as a process that selects valid extreme points froma time-ordered set of glucose concentrations whose adjacentdifferences are all greater than the threshold (typically 1 SDBGobtained from24-hour period blood glucose concentrations)It can be summarized as finding the optimum vector combi-nation solution of the valid extreme points and using INLPto establish the mathematic method which can be solved bydifferential evolution (DE) algorithm Once all valid extremepoints of countable excursions have been identified theMAGE is determined by MAGE+ or by MAGEminus dependingon the direction of the first countable excursion For morein-depth understanding it depends on the first valid extremepoint of the vector combination because that point indicatesthe direction In addition the average of both MAGE+ andMAGEminus designated as MAGEa is also calculated MAGE-CAA v10 is based on INLP and has several different outputsSDBG MAGE+ MAGEminus MAGE MAGEa and so forthBesides it also needs plot to show all valid extreme pointsjoined by straight lines MATLAB (MathWorks USA) ischosen as the programming environment accordingly Gen-erally some data points could not be extreme point accordingto the mathematical definition like points shown in Figure 2which shows turning points 119905 and 119905 + 1 which are equal invalues (not being extreme points) but there is an amplitudefrom peak point 119905 minus 1 to nadir point 119905 so we combine points119905 and 119905 + 1 as one point 119905 + 1 or remove point 119905 and then letpoint 119905 + 1 be a local maximum point If the points appear tobe opposite then point 119905+1 should be a localminimumpoint

MAGECAA v10 consists of the followingmajormodules(1) import CGM data and calculate the SDBG as the thresh-old (2) identify all extreme points (3) find the optimumvector combination solution of valid extreme points and (4)display the calculated parameters and plots

23 Collection of CGM Data The CGM datasets obtainedby using CGMS Gold (Medtronic USA) collected fromclinical treatment are used to evaluate the proposed programAll CGM datasets were provided by the Third AffiliatedHospital of Southern Medical University Only complete24-hour CGM data were selected for comparison studyAll patients have provided their written informed consent5 CGM recordings from 3 T1D patients 116 CGM mea-surements contributed by 58 T2D patients and 127 CGM

measurements based on gestational diabetes patients havebeen collected Outpatients had been treated with eitherdiet oral hypoglycemic agents oral hypoglycemic agents plusinsulin or insulin alone depending on their glycemic control

24 Data Analysis The validation of MAGECAA v10 wasimplemented by comparison againstMAGEo andMAGEc Adoctorwho has beenwell trained in using the originalmanualmethod to analyze CGM data was invited from Departmentof Endocrinology of theThird AffiliatedHospital of SouthernMedical University and he did not know the effect ofMAGE-CAA v10 Our research team analyzed the complete patientpopulation (119899 = 248)CGMmeasurements usingMAGECAAv10 and two other popular software products EasyGV andFritzschewhereas the doctor performed amanual analysis ona randomly drawn sample (119899 = 60) As calculator Fritzscheallows the analyzer to choose whether or not to consider thefirst andor the final glucose value of the CGM trace as astart or end point of a glucose excursion it will result in fourdifferentMAGE values per CGM trace To keep following theoriginal description of the calculation of MAGE and for thecomparison of the different software products the first andfinal glucose values were both taken into account

25 Statistical Analyses Spearmanrsquos correlation analysis wasapplied in evaluating the relationship between the MAGEvalues obtained by different methods with respect to thesame patients Bland-Altman plots were used to represent theagreement of the methods [35 36] 119875 lt 001 was consideredsignificant

3 Results

Figure 3 presents the screenshot of the graphical user inter-face of the developed software named MAGECAA v10

31 Comparison of MAGECAA v10 with the Original Man-ual Method As shown in Figure 4 Spearmanrsquos correlationanalysis identified that there is a highly significant linearcorrelation between MAGEc and MAGEo (119903 = 0998 resp119875 lt 001) and the mean difference found in Bland-Altmanplot was minus003 plusmn 021 which was statistically significantlysmall This result indicated that the developed program wasrobust compared with the manual method


Table 1 The correlation coefficient and difference of the results separately calculated using MAGEc Fritzsche and EasyGV software

MAGEc Fritzsche

119903 mean plusmn SD 119903 mean plusmn SD(mmolesLminus1) (mmolesLminus1)

MAGEc mdash mdash mdash mdashFritzsche 0987 014 plusmn 055 mdash mdashEasyGV 0926 052 plusmn 117 0926 038 plusmn 112Note 119875 lt 001

3

4

5

6

7

8

9

10

11

Bloo

d gl

ucos

e con

cent

ratio

n (m

mol

esL

)

2 4 6 8 10 12 14 16 18 20 22 240Time (hours)

Figure 3 Screenshot of the graphical user interface of the MAGE-CAA v10 based on the MATLAB5 programming environment forthe mean amplitude of glycemic excursions (MAGE) calculationAfter computation 24-hour continuous glucose monitoring profilesare shown in the plots with all valid extreme points joined by straightlines Besides the calculation of MAGE it also calculates the stan-dard deviation of blood glucose (SDBG) the average of all upwardvalid excursions (MAGE+) the average of all downward valid excur-sions (MAGEminus) and the average of all valid excursions (MAGEa)The CGM data were collected from patients with type 1 diabetesAfter calculation the results are as follows SDBG = 163mmolesLMAGE+ = 443mmolesL MAGEminus = 463mmolesL MAGEc =443mmolesL (the first account excursion is from nadir to peak)and MAGEa = 453mmolesL

32 Agreement ofMAGECAA v10 with Fritzsche and EasyGVTo evaluate the agreement of MAGECAA v10 with two cur-rently available popular software products that is Fritzscheand EasyGV we did pairwise comparison of MAGE amongthem Table 1 shows the correlation coefficients for MAGEcalculation between the calculators which ranged from 0926to 0987 (119875 lt 001 for all) Figure 5 shows the Bland-Altmanplot among the three software products the dashed lines rep-resent the 95 confidence limit of the differences between thetwo methods and the solid line demonstrates that the meandifference between the methods is close to 0 indicating thatlittle difference existed among themThemean differences are014 plusmn 055 (MAGEc versus Fritzsche) 052 plusmn 117 (MAGEcversus EasyGV) and 038 plusmn 112 (Fritzsche versus EasyGV)showing the good agreement between these three softwareproducts

33 Comparison of MAGEc with MAGEa In the originaldefinition of MAGE its direction of the calculation is deter-mined by the first countable excursion SoMAGEc=MAGE+or MAGEc = MAGEminus it is somewhat arbitrary and ignoreshalf of the valid excursions However MAGEa represents theaverage of MAGE+ and MAGEminus it does not consider thedirection thus involving all the valid excursions

To explore whether MAGEa may be a more usefulindex than MAGEc which depends on MAGE+ or MAGEminusthe relationship between MAGEc and MAGEa was testedby using all the 248 CGM measurements via Spearmanrsquoscorrelation analysis and Bland-Altman analysis As shown inFigure 6 the correlation coefficient was 119903 = 0993 (119875 lt 001)and themean difference was minus005plusmn048The results showedthat little difference was observed between usingMAGEa andMAGEc indicating that MAGEa may be a useful index forevaluating glycemic variability

4 Discussions

We developed a computer-aided open-code program namedMAGECAA v10 based on INLP algorithm for automaticcalculation of MAGE Compared with the existing methodsthe proposed novel method turns to search the optimalsolution of the combination of extreme points from overallCGM measurements instead of searching adjacent extremepoints from the beginning to the end step by step As forthe computational time if used for one person the proposedmethod is comparable with currently available methods ifused for batch calculation the proposed method is morepowerful The programmable open codes are useful for thestudy of methodology for automatic calculation of MAGEThe comparison study usingMAGECAA v10 against manualmethod indicated that the agreement is satisfiedThepairwisecomparison study between MAGECAA v10 and two otheravailable software products Fritzsche and EasyGV basedon Spearmanrsquos correlation analysis and Bland-Altman plotsdemonstrated that the agreements between them met therequirement Our study showed that the worry about thedisagreement among the currently available popular softwareproducts is not necessary which is different from the proposalby Sechterberger et al [26] The reason may be the increasedamount of CGM data used in our research

TheMAGEvalue depends onMAGE+ orMAGEminus follow-ing the direction of the first accountable glucose excursionConsidering the fact that only one direction of glucose excur-sion is adopted in current popular software unavoidablyresulting in the omission of the other directions of valid


0

4

8

12

16

20M

AGEc

(mm

oles

L)

4 8 12 16 200MAGEo (mmolesL)

(a)

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

MAG

Eo minus

MAG

Ec (m

mol

esL

)

4 8 12 16 200Average of MAGEo and MAGEc (mmolesL)

(b)Figure 4 (a) Correlation of the mean amplitude of glycemic excursions values (MAGE) obtained from the proposed program (MAGEc)and the original manual methods (MAGEo) The data of 60 continuous glucose monitoring (CGM) measurements were randomly chosenfrom all collected CGM data (b) Bland-Altman plot shows the difference between MAGEc and MAGEo on 119910-axis and the mean of the twocomputed indices on 119909-axis The dashed lines represent the 95 confidence limit of the differences between the two methods and the solidline indicates that the mean difference between the methods is close to 0

minus4minus3minus2minus1

0123456

MAG

Ec minus

Frit

zsch

e (m

mol

esL

)

4 8 12 16 200Average of MAGEc andFritzsche (mmolesL)

(a)


0123456

MAG

Ec minus

Eas

yGV

(mm

oles

L)

4 8 12 16 200

Average of MAGEc andEasyGV (mmolesL)

(b)


0123456

Fritz

sche

minus E

asyG

V (m

mol

esL

)

4 8 12 16 200Average of Fritzsche and

EasyGV (mmolesL)

(c)Figure 5 Bland-Altman plots showing the mean difference between MAGEc EasyGV and Fritzsche when applied to the same continuousglucose monitoring datasets

0

4

8

12

16

20

MAG

Ea (m

mol

esL

)

4 8 12 16 200MAGEc (mmolesL)

(a)

4 8 12 16 200Average of MAGEc and MAGEa (mmolesL)

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

MAG

Ec minus

MAG

Ea (m

mol

esL

)

(b)Figure 6 (a) Correlation between the mean amplitudes of glycemic excursions computerized (MAGEc) and the average of all valid glycemicexcursions (MAGEa) for the total collected 248 continuous glucose monitoring measurements (b) Bland-Altman plot shows the meandifference between MAGEc and MAGEa


excursions we implemented an extra experiment in whichdata from both directions of the valid glycemic excursions areutilized MAGEa was used to represent the mean of MAGE+and MAGEminus for the calculation of MAGE As shown by ourdata a close linear correlation between MAGEa and MAGEcwas observed indicating that the difference betweenMAGE+and MAGEminus is significantly small Therefore we proposedthatMAGEamight be another suitable parameter to quantifyglycemic variability

To conclude an open-code software program namedMAGECAA v10 for automatic calculation of MAGE basedon a mathematical algorithm has been proposed and eval-uated The programmable open codes are useful for furthermethodology study in the future The comparison studyindicated that the agreement among the proposed softwareand existing software is satisfied and the worry about thedisagreement among currently available different popularsoftware products is not necessary

Abbreviations

MAGE Mean amplitude of glycemic excursionsSDBG Standard deviation of blood glucoseCGM Continuous glucose monitoringFritzsche Fritzsche et alrsquos softwareINLP Integer nonlinear programmingMAGEo Calculated values obtained from the

original manual methodMAGEc Calculated values obtained from

MAGECAA v10MAGE+ The average of all upward valid glycemic

excursionsMAGEminus The average of all downward valid

glycemic excursionsMAGEa The average of all downward validated

glycemic excursionsDE Differential evolution

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This study was partially supported by the National KeyResearch and Development Program of China (nos2016YFC0100800 and 2016YFC0100801) the National Nat-ural Science Foundation of China (Grants nos 61671229 and61528102) Science and Technology Program of GuangdongChina (nos 2015B020214006 and 2016A050502026)GuangdongNatural Science Foundation (no 2015A030313234) andScience and Technology Program of Guangzhou China (no201704020091)

References

[1] LMonnier EMas C Ginet et al ldquoActivation of oxidative stressby acute glucose fluctuations compared with sustained chronic

hyperglycemia in patients with type 2 diabetesrdquo The Journal ofthe AmericanMedical Association vol 295 no 14 pp 1681ndash16872006

[2] I B Hirsch and M Brownlee ldquoThe effect of glucose variabilityon the risk of microvascular complications in type 1 diabetesResponse to Kilpatrick et al and Bolli [9]rdquo Diabetes Care vol30 no 1 pp 186-187 2007

[3] J Bragd et al ldquoCan glycemic variability as calculated formblood glucose self-monitoring predict the development ofcomplications in type 1 diabetes over a decaderdquo Diabetes ampMetabolism vol 34 no 6 pp 612ndash616 2008

[4] J Skrha et al ldquoGlucose variability HbA1c and microvascularcomplicationsrdquo Reviews in Endocrine amp Metabolic Disordersvol 17 no 1 pp 1ndash8 2016

[5] M Prazny et al ldquoShort-term study and long-term glycemicvariability and its relationship to microvascular complicationsof diabetesrdquo Vnitrni Lekarstvi vol 62 no Supply 4 85 pages2016

[6] M Caprnda ldquoGlycemic variability and Vascular Complicationin Patients with Type 2 Diabetes Mellitusrdquo Folia Med vol 59no 3 pp 270ndash278 2017

[7] S Eslami Z Taherzadeh M J Schultz and A Abu-HannaldquoGlucose variability measures and their effect on mortality asystematic reviewrdquo Intensive Care Medicine vol 37 no 4 pp583ndash593 2011

[8] J Hermanides T M Vriesendorp R J Bosman D F ZandstraJ B Hoekstra and J H Devries ldquoGlucose variability is associ-ated with intensive care unit mortalityrdquo Critical Care Medicinevol 38 no 3 pp 838ndash842 2010

[9] J Schlichtkrull O Munck and M Jersild ldquoThe M-Value anIndex of Blood-sugar Control in Diabeticsrdquo Journal of InternalMedicine vol 177 no 1 pp 95ndash102 1965

[10] J M Wojcicki ldquoMathematical descriptions of the glucosecontrol in diabetes therapy Analysis of the Schlichtkrull rsquoMrsquo-valuerdquo Hormone and Metabolic Research vol 27 no 1 pp 1ndash51995

[11] J Wojcicki ldquoldquoJrdquo-Index A New Proposition of the Assessmentof Current Glucose Control in Diabetic PatientsrdquoHormone andMetabolic Research vol 27 no 01 pp 41-42 1995

[12] B P Kovatchev D J Cox L A Gonder-Frederick and WClarke ldquoSymmetrization of the blood glucose measurementscale and its applicationsrdquo Diabetes Care vol 20 no 11 pp1655ndash1658 1997

[13] N R Hill P C Hindmarsh R J Stevens I M Stratton J CLevy and D R Matthews ldquoA method for assessing quality ofcontrol from glucose profilesrdquo Diabetic Medicine vol 24 no 7pp 753ndash758 2007

[14] B P Kovatchev E Otto D Cox L Gonder-Frederick andW Clarke ldquoEvaluation of a new measure of blood glucosevariability in diabetesrdquo Diabetes Care vol 29 no 11 pp 2433ndash2438 2006

[15] D Rodbard T Bailey L Jovanovic H Zisser R Kaplanand S K Garg ldquoImproved quality of glycemic control andreduced glycemic variability with use of continuous glucosemonitoringrdquo Diabetes Technology ampTherapeutics vol 11 no 11pp 717ndash723 2009

[16] I B Hirsch A K Balo K Sayer A Garcia B A Buckinghamand T A Peyser ldquoA simple composite metric for the assessmentof glycemic status from continuous glucose monitoring dataImplications for clinical practice and the artificial pancreasrdquoDiabetes Technology ampTherapeutics vol 19 pp S38ndashS48 2017


[17] L Monnier C Colette and D R Owens ldquoGlycemic variabilityThe third component of the dysglycemia in diabetes Is itimportant how to measure itrdquo Journal of Diabetes Science andTechnology vol 2 no 6 pp 1094ndash1100 2008

[18] D Rodbard ldquoNew and improved methods to characterizeglycemic variability using continuous glucose monitoringrdquoDiabetes Technology amp Therapeutics vol 11 no 9 pp 551ndash5652009

[19] F J Service G D Molnar J W Rosevear E Ackerman LC Gatewood and W F Taylor ldquoMean amplitude of glycemicexcursions a measure of diabetic instabilityrdquo Diabetes vol 19no 9 pp 644ndash655 1970

[20] F J Service and R L Nelson ldquoCharacteristics of glycemicstabilityrdquo Diabetes Care vol 3 no 1 pp 58ndash62 1980

[21] F J Service P C OrsquoBrien and R A Rizza ldquoMeasurements ofglucose controlrdquo Diabetes Care vol 10 no 2 pp 225ndash237 1987

[22] P A Baghurst ldquoCalculating the mean amplitude of glycemicexcursion from continuous glucose monitoring data An auto-mated algorithmrdquo Diabetes Technology amp Therapeutics vol 13no 3 pp 296ndash302 2011

[23] G Fritzsche K-D Kohnert P Heinke L Vogt and ESalzsieder ldquoThe use of a computer program to calculate themean amplitude of glycemic excursionsrdquo Diabetes TechnologyampTherapeutics vol 13 no 3 pp 319ndash325 2011

[24] D Czerwoniuk W Fendler L Walenciak and W MlynarskildquoGlyCulator A glycemic variability calculation tool for contin-uous glucose monitoring datardquo Journal of Diabetes Science andTechnology vol 5 no 2 pp 447ndash451 2011

[25] N R Hill et al ldquoNormal Reference Range for Mean TissueGlucose and Glycemic Variability Derived from ContinuousGlucose Monitoring for Subjects Without Diabetes in DifferentEthnic GroupsrdquoDiabetes Technology ampTherapeutics vol 13 no9 pp 921ndash928 2011

[26] M K Sechterberger Y M Luijf and J H Devries ldquoPooragreement of computerized calculators for mean amplitude ofglycemic excursionsrdquo Diabetes Technology amp Therapeutics vol16 no 2 pp 72ndash75 2014

[27] M Anjidani and S Effati ldquoSteepest descent method for solvingzero-one nonlinear programming problemsrdquo Applied Mathe-matics and Computation vol 193 no 1 pp 197ndash202 2007

[28] M Kılınc et al ldquoExploiting integrality in the global optimiza-tion of mixed-integer nonlinear programming problems withBARONrdquo Optimization Methods amp Software pp 1ndash23 2017

[29] R Storn and K Price ldquoDifferential Evolution - A Simple andEfficient Heuristic for Global Optimization over ContinuousSpacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[30] S Kukkonen and C A Coello Coello ldquoGeneralized differentialevolution for numerical and evolutionary optimizationrdquo Studiesin Computational Intelligence vol 663 pp 253ndash279 2017

[31] P Rakshit ldquoUncertainty Management in Differential EvolutionInduced Multiobjective Optimization in Presence of Measure-ment Noiserdquo IEEE Transactions on Systems Man CyberneticsSystems vol 44 no 7 pp 922ndash937 2017

[32] S Das A Konar and U K Chakraborty ldquoTwo improveddifferential evolution schemes for faster global searchrdquo inProceedings of the Genetic and Evolutionary Computation Con-ference (GECCO rsquo05) pp 991ndash998 June 2005

[33] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (CEC rsquo05)vol 2 pp 1785ndash1791 IEEE September 2005

[34] Y C Lin ldquoA mixed-coding scheme of evolutionary algrithmsto solve mixed-integer nonlinear programming problemsrdquoComputer Mathematics with Applications vol 47 no 8-9 pp1295ndash1307 2004

[35] JM Bland andDGAltman ldquoMeasuring agreement inmethodcomparison studiesrdquo Statistical Methods in Medical Researchvol 8 no 2 pp 135ndash160 1999

[36] D Giavarina ldquoUnderstanding Bland Altman analysisrdquo Bio-chemia Medica vol 25 no 2 pp 141ndash151 2015

Stem Cells International

Hindawiwwwhindawicom Volume 2018


MEDIATORSINFLAMMATION

of

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Immunology ResearchHindawiwwwhindawicom Volume 2018

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine


Behavioural Neurology

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary andAlternative Medicine

Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom


To address the urgent need above several automaticprograms have been developed [22ndash25] making great con-tribution to the automatic calculation of MAGE However itwas pointed out that the agreement among them is poor [26]Therefore more studies are required to explore the automaticcalculation of MAGE Currently available programs can beroughly divided into two groups One group has detaileddescriptions about algorithm with graphical display such asthe automated algorithm described by Baghurst [22] and thecomputer software described by Fritzsche et al (Fritzsche)[23] The other group only provides executable softwarefor automatic calculation without detailed descriptions ofthe algorithms used in the software such as the web-basedapplication ldquoGlyCulatorrdquo [24] and the Excel-based workbookldquoEasyGVrdquo [25] However to the best of our knowledgeall of these methods did not provide programmable opencodes which are important resources required for peers toimplement more studies on methodology in this field

Therefore in this report we developed a mathematicalalgorithm based on integer nonlinear programming methodFollowing the proposed mathematical method a computer-aided program named MAGECAA v10 was developed Thecode of our program is open if peers are interested init please contact xuefeiyusmueducn for downloadingTo validate the developed program comparison study wasimplemented using blood glucose CGM datasets obtainedfrom T1D T2D and gestational diabetes patients againstthe manual method (MAGEo) and other currently popularsoftware products

2 Materials and Methods

21 The Proposed Mathematical Algorithm Let 119891(119905) (119905 =1199051 1199052 119905119871) represent the discrete blood glucose valuesobtained in CGM then 119891 is a discrete function defined intime set 1199051 1199052 119905119871 Let SDBG be the standard deviationof 119891(119905) (119905 = 1199051 1199052 119905119871) A graph depicting the glycemicvariability can be formed by connecting all the discrete valuesof function 119891 When the difference of a peak and an adjacentnadir exceeds SDBG the corresponding peak is labeled as avalid peak The key point for the calculation of MAGE is tocorrectly count the valid peak or nadir

Amplitude is the difference of functional values in a peakand a nadir of the function graph A valid amplitude islabeled and countedwhen it is bigger than SDBGTo computeMAGE these valid amplitudes of function 119891 should be firstlysearched From themathematical perspective a peak or nadirshould be an extreme point of the function These extremepoints related to valid amplitudes are called valid extremepoints The MAGE computation problem can be solved bycalculating all valid extreme points

Suppose that 1199051198971 1199051198972 119905119897119873 is the sequence of all extremepoints of function 119891 We know that the local maximumpoints and local minimum points of function 119891 shouldbe arranged as follows interleaved between a maximumpoint and a minimum point For simplicity 1199051198971 1199051198972 119905119897119873is denoted as 1 2 119873 and 1198991 1198992 119899119870 is an arbitraryvalid subsequence of sequence 1 2 119873 These local

maximumpoints and localminimumpoints can be staggeredin subsequence 1198991 1198992 119899119870 Then

1 le 1198991 lt 1198992 lt sdot sdot sdot lt 119899119870 le 119873(minus1)119899119896+1minus119899119896 = minus1 119896 = 1 2 119870 minus 1 (1)

where 2 le 119870 le 119873 This equation ensures that the selectedextreme points are staggered

Thus the above MAGE computation problem can betransformed to an integer nonlinear programming (INLP)problem

argmax1198701198991 1198992119899119870

119885119870 (1198991 1198992 119899119870)

= argmax1198701198991 1198992119899119870

119870minus1sum119896=1

10038161003816100381610038161003816119891 (119905119899119896+1) minus 119891 (119905119899119896)10038161003816100381610038161003816 (2)

subject to 10038161003816100381610038161003816119891 (119905119899119896+1) minus 119891 (119905119899119896)10038161003816100381610038161003816 ge SDBG119896 = 1 2 119870 minus 1

1 le 1198991 lt 1198992 lt sdot sdot sdot lt 119899119870 le 119873(minus1)119899119896+1minus119899119896 = minus1 119896 = 1 2 119870 minus 1

(3)

According to the principle of INLP problem function(2) should have an optimum solution 119899lowast1 119899lowast2 119899lowast119870 and anoptimumvalue 119885lowast119870with respect to a constant119870 it should alsohave an optimum value 119885lowast119870minus1 for constant119870 minus 1 and

119885lowast119870minus1 le 119885lowast119870 (4)

Suppose that 119870lowast is the maximal 119870 with respect tothe optimal solution of function (2) and then the optimalsolution 119899lowast1 119899lowast2 119899lowast119870lowast represents the valid extreme pointsto compute MAGE

If the extreme point 119899lowast1 is the local minimum point then

MAGE+ = 1[119870lowast2][119870lowast2]sum119896=1

10038161003816100381610038161003816119891119899(119896minus1)lowast2+2 minus 119891119899(119896minus1)lowast2+1 10038161003816100381610038161003816

MAGEminus = 1[(119870lowast minus 1) 2][(119870lowastminus1)2]sum119896=2

10038161003816100381610038161003816119891119899(119896minus1)lowast2+1 minus 119891119899(119896minus1)lowast2 10038161003816100381610038161003816 MAGE = MAGE+MAGEa = MAGE+ +MAGEminus2

(5)

If the extreme point 119899lowast1 is the local maximum point then

MAGE+ = 1[(119870lowast minus 1) 2][(119870lowastminus1)2]sum119896=2

10038161003816100381610038161003816119891119899(119896minus1)lowast2+1 minus 119891119899(119896minus1)lowast2 10038161003816100381610038161003816

MAGEminus = 1[119870lowast2][119870lowast2]sum119896=1

10038161003816100381610038161003816119891119899(119896minus1)lowast2+2 minus 119891119899(119896minus1)lowast2+1 10038161003816100381610038161003816 MAGE = MAGEminusMAGEa = MAGE+ +MAGEminus2

(6)






Set


119896 = 1 2 119870 minus 1(8)


argmin11989911198992 119899119870

119884119870 (1198991 1198992 119899119870) = argmin1198701198991 1198992119899119870


10038161003816100381610038161003816119891119899119896+1 minus 119891119899119896 10038161003816100381610038161003816 +119870minus1sum119896=1

120583max 0 1198921198962 + 119870minus1sum119896=1

120582ℎ2119896] (9)










1198661199032119870





119895 = 1 2 119870(12)





Start

Input CGMdataset






End






End

No

Yes

No

Yes

Record minimum Y







t

t minus 1

t + 1

(a)

t

t minus 1

t + 1

(b)








3 Results





MAGEc Fritzsche



3

4

5

6

7

8

9

10

11

Bloo

d gl

ucos

e con

cent

ratio

n (m

mol

esL

)

2 4 6 8 10 12 14 16 18 20 22 240Time (hours)





4 Discussions




0

4

8

12

16

20M

AGEc

(mm

oles

L)


(a)

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

MAG

Eo minus

MAG

Ec (m

mol

esL

)




0123456

MAG

Ec minus

Frit

zsch

e (m

mol

esL

)


(a)


0123456

MAG

Ec minus

Eas

yGV

(mm

oles

L)

4 8 12 16 200


(b)


0123456

Fritz

sche

minus E

asyG

V (m

mol

esL

)


EasyGV (mmolesL)


0

4

8

12

16

20

MAG

Ea (m

mol

esL

)


(a)


minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

MAG

Ec minus

MAG

Ea (m

mol

esL

)





Abbreviations









Acknowledgments


References











































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of
























Set


119896 = 1 2 119870 minus 1(8)


argmin11989911198992 119899119870

119884119870 (1198991 1198992 119899119870) = argmin1198701198991 1198992119899119870


10038161003816100381610038161003816119891119899119896+1 minus 119891119899119896 10038161003816100381610038161003816 +119870minus1sum119896=1

120583max 0 1198921198962 + 119870minus1sum119896=1

120582ℎ2119896] (9)










1198661199032119870





119895 = 1 2 119870(12)





Start

Input CGMdataset






End






End

No

Yes

No

Yes

Record minimum Y







t

t minus 1

t + 1

(a)

t

t minus 1

t + 1

(b)








3 Results





MAGEc Fritzsche



3

4

5

6

7

8

9

10

11

Bloo

d gl

ucos

e con

cent

ratio

n (m

mol

esL

)

2 4 6 8 10 12 14 16 18 20 22 240Time (hours)





4 Discussions




0

4

8

12

16

20M

AGEc

(mm

oles

L)


(a)

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

MAG

Eo minus

MAG

Ec (m

mol

esL

)




0123456

MAG

Ec minus

Frit

zsch

e (m

mol

esL

)


(a)


0123456

MAG

Ec minus

Eas

yGV

(mm

oles

L)

4 8 12 16 200


(b)


0123456

Fritz

sche

minus E

asyG

V (m

mol

esL

)


EasyGV (mmolesL)


0

4

8

12

16

20

MAG

Ea (m

mol

esL

)


(a)


minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

MAG

Ec minus

MAG

Ea (m

mol

esL

)





Abbreviations









Acknowledgments


References











































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















Start

Input CGMdataset






End






End

No

Yes

No

Yes

Record minimum Y







t

t minus 1

t + 1

(a)

t

t minus 1

t + 1

(b)








3 Results





MAGEc Fritzsche



3

4

5

6

7

8

9

10

11

Bloo

d gl

ucos

e con

cent

ratio

n (m

mol

esL

)

2 4 6 8 10 12 14 16 18 20 22 240Time (hours)





4 Discussions




0

4

8

12

16

20M

AGEc

(mm

oles

L)


(a)

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

MAG

Eo minus

MAG

Ec (m

mol

esL

)




0123456

MAG

Ec minus

Frit

zsch

e (m

mol

esL

)


(a)


0123456

MAG

Ec minus

Eas

yGV

(mm

oles

L)

4 8 12 16 200


(b)


0123456

Fritz

sche

minus E

asyG

V (m

mol

esL

)


EasyGV (mmolesL)


0

4

8

12

16

20

MAG

Ea (m

mol

esL

)


(a)


minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

MAG

Ec minus

MAG

Ea (m

mol

esL

)





Abbreviations









Acknowledgments


References











































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















t

t minus 1

t + 1

(a)

t

t minus 1

t + 1

(b)








3 Results





MAGEc Fritzsche



3

4

5

6

7

8

9

10

11

Bloo

d gl

ucos

e con

cent

ratio

n (m

mol

esL

)

2 4 6 8 10 12 14 16 18 20 22 240Time (hours)





4 Discussions




0

4

8

12

16

20M

AGEc

(mm

oles

L)


(a)

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

MAG

Eo minus

MAG

Ec (m

mol

esL

)




0123456

MAG

Ec minus

Frit

zsch

e (m

mol

esL

)


(a)


0123456

MAG

Ec minus

Eas

yGV

(mm

oles

L)

4 8 12 16 200


(b)


0123456

Fritz

sche

minus E

asyG

V (m

mol

esL

)


EasyGV (mmolesL)


0

4

8

12

16

20

MAG

Ea (m

mol

esL

)


(a)


minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

MAG

Ec minus

MAG

Ea (m

mol

esL

)





Abbreviations









Acknowledgments


References











































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of





















MAGEc Fritzsche



3

4

5

6

7

8

9

10

11

Bloo

d gl

ucos

e con

cent

ratio

n (m

mol

esL

)

2 4 6 8 10 12 14 16 18 20 22 240Time (hours)





4 Discussions




0

4

8

12

16

20M

AGEc

(mm

oles

L)


(a)

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

MAG

Eo minus

MAG

Ec (m

mol

esL

)




0123456

MAG

Ec minus

Frit

zsch

e (m

mol

esL

)


(a)


0123456

MAG

Ec minus

Eas

yGV

(mm

oles

L)

4 8 12 16 200


(b)


0123456

Fritz

sche

minus E

asyG

V (m

mol

esL

)


EasyGV (mmolesL)


0

4

8

12

16

20

MAG

Ea (m

mol

esL

)


(a)


minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

MAG

Ec minus

MAG

Ea (m

mol

esL

)





Abbreviations









Acknowledgments


References











































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















0

4

8

12

16

20M

AGEc

(mm

oles

L)


(a)

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

MAG

Eo minus

MAG

Ec (m

mol

esL

)




0123456

MAG

Ec minus

Frit

zsch

e (m

mol

esL

)


(a)


0123456

MAG

Ec minus

Eas

yGV

(mm

oles

L)

4 8 12 16 200


(b)


0123456

Fritz

sche

minus E

asyG

V (m

mol

esL

)


EasyGV (mmolesL)


0

4

8

12

16

20

MAG

Ea (m

mol

esL

)


(a)


minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

MAG

Ec minus

MAG

Ea (m

mol

esL

)





Abbreviations









Acknowledgments


References











































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of






















Abbreviations









Acknowledgments


References











































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of












































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of























of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















Documents

Calculating the Mean Amplitude of Glycemic Excursions from