The relationship between fasting plasma glucose and HbA1c duringintensive periods of glucose control in antidiabetic therapy

Amlan Barua a,n, Jhankar Acharya b, Saroj Ghaskadbi b, Pranay Goel c

a Department of Mathematics, Indian Institute of Science Education and Research Pune, Pune 411008, Indiab Department of Zoology, University of Pune, Pune 41107, Indiac Mathematics and Biology, Indian Institute of Science Education and Research Pune, Pune 411008, India

H I G H L I G H T S

� We examine FPG and HbA1c of type 2 diabetic patients 4 and 8 weeks after starting treatment.� We determine linear and nonlinear regression between FPG and HbA1c .� HbA1c reaches quasi-equilibrium after about 8 weeks of glucose control.� Four-week HbA1c measurements are in excess by about 0.7 mmol/mol.� Regression curves can be used to estimate glycemic status even during intensive glucose control.

a r t i c l e i n f o

Article history:Received 9 June 2014Received in revised form30 July 2014Accepted 12 August 2014Available online 23 August 2014

Keywords:Type 2 diabetes mellitusGlycated hemoglobinNonlinear regression analysis

a b s t r a c t

Objective: HbA1c measurements are typically less variable than fasting plasma glucose (FPG) fordiagnosing diabetes, and for assessment of progress on glucose control therapy. However HbA1c reachessteady-state relative to average plasma glucose over about 120 days. HbA1c thus overestimates averageFPG during first three months of starting therapy in newly diagnosed diabetic patients, and care needs tobe exercised in interpreting HbA1c measurements during this period. At steady-state excellent regressionexists between HbA1c and FPG. We hypothesize that this regression can also be used to obtain reliableestimates of HbA1c relative to FPG at 4 and 8 weeks following the onset of therapy.

Materials and methods: We collected FPG and HbA1c data of type 2 diabetic patients over the first8 weeks of starting antidiabetic treatment. We fit linear and nonlinear regression models to steady-statedata, and estimated how much measured HbA1c deviates at 4 and 8 weeks from these theoreticalrelations.

Results: If measured HbA1c is decremented by 0.7% (8 mmol/mol) at 4 weeks and 0.3% (3 mmol/mol)at 8 weeks, this corrected HbA1c is a better predictor of the corresponding FPG. Using hyperbolicregression, corrections to HbA1c are 0.5 and 0.1% (5 and 1 mmol/mol), respectively.

Conclusion: With the corrections proposed here, HbA1c measurements can be better interpreted inthe early weeks of antidiabetic treatment.

& 2014 Elsevier Ltd. All rights reserved.

1. Clinical significance

In current clinical thinking HbA1c is thought to be useful as astable measure of average FPG only over 3–6 months. Our dataanalysis shows that HbA1c can be an excellent predictor of FPG in a

much shorter period, in particular, in spite of any glucose changesthat may be taking place due to antidiabetic treatment. Mathematicalmodeling shows that a hyperbolic relation provides an excellentcorrespondence (to within 0.5 mmol/mol) between HbA1c and FPGwithin 4 weeks of standard therapy. These results thus haveimplications for the personalization of first-line antidiabetic therapy.

2. Introduction

Type 2 diabetes mellitus (T2DM) is a rapidly growing epidemicin large parts of the world, especially in India and southeast Asia,

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Journal of Theoretical Biology

http://dx.doi.org/10.1016/j.jtbi.2014.08.0200022-5193/& 2014 Elsevier Ltd. All rights reserved.

Abbreviation: AG, average glucose; PG, plasma glucose; FPG, fasting plasmaglucose; T2DM, type 2 Diabetes mellitus; OGTT, oral glucose tolerance test; MBG,mean blood glucose; RBC, red blood cell; ADA, American Diabetes Association;WHO, World Health Organization; DCCT, Diabetes Control and Complications Trial

n Corresponding author.E-mail address: amlan.barua@iiserpune.ac.in (A. Barua).

Journal of Theoretical Biology 363 (2014) 158–163

but, despite the urgency, there is no single consensus regarding itsdiagnosis or treatment. The most widely adopted tests for bothdiagnosis and treatment are those recommended by the AmericanDiabetes Association (ADA), published and updated annually asthe Standards of Medical Care in Diabetes (American DiabetesAssociation, 2014). The oldest, traditional test for diabetes is basedon measuring fasting plasma glucose (FPG). In recent years,however, the ADA has included the measurement of glycosylatedhemoglobin (HbA1c) as one of its accepted tests; in contrast, TheWorld Health Organization (WHO) does not acknowledge HbA1c

among its recommendations. Specifically, the ADA lists the follow-ing four criteria to diagnose diabetes: (i) HbA1c Z6:5%, (ii) FPGZ126 mg=dL (7.0 mmol/L), (iii) 2-h plasma glucose (PG)Z200 mg=dL (11.1 mmol/L) in an oral glucose tolerance test(OGTT), or (iv) a random plasma glucose Z200 mg=dL. In additionto their use in diagnosis, glucocentric measurements are alsodepended upon for antidiabetic therapy. The objective of treat-ment in diabetic patients is to ensure a glycemic reduction as closeto healthy levels as possible. Glucose control targets are deter-mined by the physician keeping in mind factors such as theduration of the disease, life expectancy, risk of hypoglycemia andother complications (American Diabetes Association, 2014).

The HbA1c test has now become a preferred method in manycircumstances (Bonora and Tuomilehto, 2011; Davidson, 2004;Bennett et al., 2007; Nomura et al., 2012). In India, for example, thetest is relatively easily administered, especially in urban centers.HbA1c has the advantage of a much lower variability, while FPGmeasurements need to be repeated, typically three times, toreduce variance and establish confidence (Ollerton et al., 1999).It is therefore not surprising that FPG tests tend to be impractical.On the other hand, the drawback of the HbA1c test is that HbA1c isassumed to reach steady-state relative to average plasma glucoseonly over a timescale associated with the lifespan of a redblood cell, about 120 days, hence HbA1c measurements are typi-cally thought to reflect an average glucose over the previous40–60 days. In particular, when newly diagnosed diabetic patientsundergo glucose control therapy HbA1c is not considered useful forthe initial couple of months. Similarly, clinical studies that alterglucose cannot expect to interpret HbA1c values in the usualmanner while the control is active (and for some time after).However, to the best of our knowledge, these estimates have notbeen directly verified quantitatively. It is therefore imperative toreexamine the question: How soon after beginning therapy innewly diagnosed diabetics can HbA1c be used to obtain meaningfulestimates of the corresponding FPG? In other words, the lowervariability of HbA1c makes it attractive to ask if it can be directlyuseful even during glucose control by antidiabetic treatment.

Good physiological models do not currently exist to directlyestimate FPG from HbA1c measurements. Several factors interferewith the accuracy of HbA1c measurements including geneticmakeup, fetal hemoglobin and chemically modified derivativesof hemoglobin, i.e. carbamylated hemoglobin found in patientswith renal problems, and interpretation of HbA1c results may beaffected by factors such as acute blood loss or hemolytic anemia,which reduces the life period of RBCs (NGSP, 2010). Variousauthors have attempted to mathematically model the glycosyla-tion of hemoglobin (Higgins and Bunn, 1981; Beach, 1979;Osterman-Golkar and Vesper, 2006; Hamren et al., 2008; DeWinter et al., 2006; Uehlinger et al., 1992; Mortensen et al.,1984; Mortensen and Vølund, 1988; Moller et al., 2013). Higginsand Bunn (1981) were among the earliest to investigate thekinetics of glycosylation of hemoglobin. Based on their work,Beach (1979) studied an age-structured dynamic model of glyco-sylation, and derived an analytical expression for the variation oftotal HbA1c with time. Notably, their study assumed a fixederythrocyte life span. Other biochemical models of glycosylation

can be found in Mortensen et al. (1984), Mortensen and Vølund(1988) where various refinements of Beach (1979) were consid-ered. Recently, Osterman-Golkar and Vesper (2006) proposed adiscrete time model considering the incremental increase of HbA1c

and its loss due to multiple factors like erythrocyte clearance andchemical instability of HbA1c. The authors note, however, that theirresults are sensitive to the parameters in their model. Other recentmodels exploring glucose–HbA1c relation take into account variouseffects of RBC turnover rates: a homogeneous RBC lifespan isconsidered in Uehlinger et al. (1992), in De Winter et al. (2006) theauthors construct a random destruction model, and in Hamrenet al. (2008) a transit-compartment model. A very recent study byMoller et al. (2013) suggests that accurate results can still beobtained at the population level even with ignoring differences inRBC lifespan. A natural solution, therefore, to the difficulty inobtaining a physiologically precise relationship between hemo-globin and its glycosylation in vivo, therefore, is epidemiological,that is, to determine the regression relation between the twovariables from apopulation study. Such correlation studiesbetween glucose and HbA1c are clearly important because of theirclinical relevance.

Considerable work has shown that various measures of bloodglucose correlate very well with glycated hemoglobin when bloodglucose is relatively stable over a period of about three months inboth type 1 (Svendsen et al., 1982; Rohlfing et al., 2002; Derr et al.,2003; McCarter et al., 2006; Nathan et al., 1984, 2008) and type2 diabetes, as well as in control subjects. Svendsen et al. (1982)have shown a curvilinear, power-law relationship exists between a5-week mean plasma glucose (MPG) and HbA1c when HbA1c

values are relatively unchanged over two years (n¼15; type1 diabetic patients). Rohlfing et al. (2002) examined type 1 diabeticpatients from the Diabetes Control and Complications Trial (DCCT)to show that a linear regression correlates 7-point MPG and HbA1c

(n¼1439). Nathan et al. (2008) computed an average glucose (AG)based on self-monitored interstitial glucose monitoring in type 1(n¼268) and type 2 (n¼159) diabetic patients to show that HbA1c

is linearly correlated with AG over the previous three months(n¼80 nondiabetic subjects). Derr et al. (2003) investigated theeffects of glycemic variation and mean blood glucose (MBG) onHbA1c in type 1 and type 2 diabetic patients to show that HbA1c isa linear predictor of MBG over the previous 90 days. The steady-state relationship between FPG and HbA1c has been investigatedextensively in type 2 diabetic patients (Paisey et al., 1980; Avignonet al., 1997; Bouma et al., 1999). Paisey et al. (1980) have shownthat a mean of at least three fasting capillary blood glucose havegood correlation with HbA1c (n¼106). Avignon et al. (1997)compared plasma glucose collected at different times during theday (prebreakfast, prelunch, postlunch, extended postlunch) witha single measurement of HbA1c (n¼66) taken within 10 days of theglucose data collection. Multivariate regression showed that post-breakfast glucose correlated better with HbA1c than prebreakfastglucose, a result that confirmed previous findings of Gonen et al.(1977). On the other hand, Bouma et al. (1999) (n¼1020 patients)have argued that FPG values do not reliably predict HbA1c,especially when FPG is continuously changing as a result of anti-diabetic therapy. They studied diabetic patients undergoing ther-apy with either oral hypoglycemic agents or diet control; theyconcluded that only 66% of patients with good HbA1c values couldbe identified through their FPG values. In particular, Bouma et al.recommended that both HbA1c and FPG need to be measuredevery 3 months during glycemic control.

The FPG-HbA1c steady-state curves discussed above are not apriori valid over early periods of antidiabetic treatment. Theproblem can be restated as the following: While repeated FPGmeasurements are a true reflection of the actual glucose statusduring this time, an HbA1c measurement is biased by an earlier

A. Barua et al. / Journal of Theoretical Biology 363 (2014) 158–163 159

period when glucose was high, and therefore overestimates the“actual” HbA1c. The rationale of our study is: the extent to whichmeasured HbA1c deviates from the steady-state FPG-HbA1c rela-tionship needs to be quantified. We note that such estimates arenot readily available in the literature.

Here we examine the relation between FPG and HbA1c usingtwo regression models: a linear model and a nonlinear, hyperbolicmodel that is motivated in part by the biochemistry of glycosyla-tion. We compare model predictions of HbA1c against patient data.In particular, we are interested in asking whether it is possible toreconcile measured FPG and HbA1c during the first several weeksof antidiabetic treatment.

3. Methods

Trial design and patients. We examined newly diagnosedtype 2 diabetic patients (n¼51) attending the Diabetes Unit,KEM Hospital, Pune, India, with FPG concentration greater than126 mg/dl (6.9 mM) and healthy non-diabetic subjects (n¼50)with FPG concentration of r126 mg=dl (6.9 mM), as describedpreviously (Acharya et al., 2014). Fasting blood samples werecollected at baseline (0-week) and at 4 and 8 weeks from non-diabetic subjects as well as diabetic patients. During the studyperiod diabetic patients were put on anti-diabetic therapy tocontrol hyperglycemia as required. Pregnant women, chronicsmokers, individuals with excessive alcohol intake or a history ofa recent (o6 months) cardiovascular event or with symptomaticheart disease (NYHA Class III, IV) and those with clinical infection,inflammatory or malignant disease were excluded from the study.Medical history of all individuals was noted, and each subjectunderwent a standardized physical examination for the measure-ment of body weight, height, waist-to-hip ratio, blood pressureand electrocardiogram (ECG). The study protocol was approved bythe Institutional Ethics Committee, KEM Hospital and ResearchCentre, Pune. An informed consent was obtained in writing fromall individuals upon explaining the nature of the study and itspurpose.

Sample preparation. Blood samples were centrifuged at 4000 rpm(2500 g) for 10 min to separate the plasma. Plasma glucose concen-trations were measured by glucose oxidase method using standardkits on an autoanalyser (Hitachi 902, Japan). HbA1c was measured byusing an HPLC cation exchange column on D-10 HbA1c analyser(Bio-Rad Laboratories, Hercules, CA). The D-10 HbA1c Program iscertified by the NGSP, and produces high-precision results.

Nonlinear model of the relation between FPG and HbA1c. Thekinetics of glycosylation were first obtained by Higgins and Bunn(1981), and later others (Beach, 1979; Osterman-Golkar andVesper, 2006; Svacina et al., 1990; Hamren et al., 2008; Molleret al., 2013). Here we construct a model modified from theHiggins–Bunn scheme and derive its steady-state solutions.

During glycosylation, the aldehyde of glucose binds with theamino groups of HbA and is transformed to an intermediate Schiffbase, preA1c, which undergoes a further change to a more stableketoamine HbA1c. We modify the kinetics proposed by Higgins andBunn (1981) by incorporating a production of hemoglobin, at aconstant rate α (mmol/hr), and decay rates for all three inter-mediate species. The decay term is assumed proportional to theconcentrations of the specie with rate γ (h�1). The reactionschematic is thus

HbA↓α

↓γþGlucose⇋

kB

k1preA1c

↓γ

⟶k2 HbA1c

↓γ

; ð1Þ

where k1 (mM�1h�1) is the rate constant of the first reaction inthe forward direction, kB (h�1) is the rate constant of the reverse

reaction. The first reaction is rapid; the second reaction is some-what slow and occurs mostly in the forward direction with a rateconstant is k2 (h�1).

The steady state FPG–HbA1c solution to this model is

½HbA1c� ¼v½Glucose�

Kþ½Glucose�; ð2Þ

with

v¼ k2αγðk2þγÞ; K ¼ γðkbþk2þγÞ

k1k2þk1γ: ð3Þ

Eq. (2) is the basis of the hyperbolic FPG–HbA1c relationship weuse for nonlinear regression analysis.

Statistical analysis. Two regression analyses, one with a linearmodel and another with a nonlinear, hyperbolic relation werecarried out on the FPG–HbA1c data. The nonlinear regressionequation was adapted from Eq. (2), that is,

½HbA1c� ¼v½FPG�

Kþ½FPG�: ð4Þ

To estimate parameters of the nonlinear fit, linear regression wascarried out between 1/HbA1c and 1/FPG. Goodness-of-fit wasmeasured via the coefficient of determination, r2. Regressionanalyses were carried out in MATLAB.

HbA1c correction to 4- and 8-week samples. We propose correct-ing each measured HbA1c value in such a manner that upontransforming to equivalent FPG values through the regressioncurves we get a distribution that is statistically consistent (up tothe first moment) with the measured FPG distribution. In otherwords, we want

f �1○gðHbAi1cÞ ¼ FPGi ð5Þ

where HbAi1c is the HbA1c value of ith subject, FPGi is the

corresponding FPG value. We choose to apply a constant shift toeach HbA1c uniformly, that is, gðHbAi

1cÞ ¼HbAi1c�β. Given a (linear

or hyperbolic) regression curve HbA1c ¼ f ðFPGÞ, an HbA1c valueinverted through f �1 yields a unique FPG value. Estimates of βwere obtained numerically.

4. Results

A total of 51 diabetic patients and 50 nondiabetic controlsubjects were included in the analysis. FPG and HbA1c data weremeasured from diabetic patients at the beginning of treatment,i.e. at 0 weeks, at 4 weeks following the start of therapy, and at8 weeks. A summary of the FPG and HbA1c of diabetic patients ispresented in Table 1. Data was measured from nondiabetic sub-jects (n¼50) once, and then again after 8 weeks (n¼49).

4.1. Linear and nonlinear regression between HbA1c and FPG

We tested two regression models of the relationship betweenFPG and HbA1c: linear regression, and a nonlinear, hyperbolicregression model, Eq. (4).

Diabetic patients before treatment and non-diabetic subjectsrepresent a “steady-state” group, that is, FPG and HbA1c arepresumed stable over the previous three months. Using linearregression to fit this data we obtained the relation HbA1c¼0.64FPG þ 2.60, with a coefficient of determination, r2¼0.81. We thentested to what extent can linear regression, in addition to thesteady-state data, also explain FPG and HbA1c data of diabeticpatients at 4 weeks, and at 8 weeks. In each of the two cases thecorrelation weakens, r2¼0.76. Thus, linear regression betweensteady-state FPG and HbA1c cannot adequately describe datafollowing intensive glucose control, i.e. at neither 4 nor 8 weeks.

A. Barua et al. / Journal of Theoretical Biology 363 (2014) 158–163160

A similar result was observed with the nonlinear regressionmodel. The steady-state data is described by HbA1c¼26.32 FPG/(18.20 þ FPG) with r2¼0.84; when the diabetic patient data at4 or 8 weeks is included r2 values drop somewhat, to 0.825 and0.81 respectively.

Although neither the linear nor nonlinear model is able to fullyrepresent 4-week and 8-week data as well as the steady-statedata, the nonlinear model performs slightly better of the two; thehyperbolic model has a somewhat higher coefficient of determi-nation than the linear model in each case.

4.2. Regression models underestimate HbA1c by less than 0.7%(8 mmol/mol) at 4 weeks

HbA1c sampled at 4 and 8 weeks cannot be expected to haverelaxed to steady-state; this is reflected in the mean HbA1c lyingabove the regression curves in Fig. 1. We used FPG measured at4 and 8 weeks to predict corresponding HbA1c from the linear andhyperbolic models. At 4 weeks sampled mean HbA1c is 8.7 71.4%(72 715 mmol/mol) and mean FPG is 8.3 72.3 mM. For this FPGdata, linear regression predicts HbA1c to be 8.0% (64 mmol/mol)while nonlinear regression predicts an HbA1c of 8.2% (66 mmol/mol). At 8 weeks sampled mean HbA1c is 7.7% (61 mmol/mol) andmean FPG is 7.5 mM; mean HbA1c predicted by linear and non-linear regression are 7.4 and 7.6% (57 and 60 mmol/mol), respec-tively. The differences of predicted HbA1c from sampled HbA1c,together with p-values testing for the significance, are summar-ized in Table 2.

At 4 weeks the linear model underestimates sampled HbA1c by0.7 % (8 mmol/mol), while the hyperbolic model underestimates itsomewhat less, 0.5% (5 mmol/mol) (significance level, p¼0.06). By8 weeks the linear model underestimates HbA1c by less than 0.3(3 mmol/mol)% (at a significance level of 0.15), while the nonlinearmodel prediction is correct to within 0.1% (1 mmol/mol). We notethat in our study we do not have sufficient power (at an 80% level)to determine if these differences in mean HbA1c at 8 weeks aresignificant.

4.3. Corrections to measured HbA1c that reflect actual FPG at 4 and8 weeks

A single blood sample collected at 4 weeks provides an HbA1c

reading and one FPG reading. If more than one FPG measurementsare collected, say over three successive days, the average of thosemeasurements will better reflect actual FPG. If HbA1c is used toestimate FPG from regression models, it will reflect a value higherthan actual FPG (because HbA1c sampled at 4 weeks is under-estimated by the regression curves).

To address this inconsistency we propose the following: Wecorrect each HbA1c value at 4 weeks in such a manner that theregression models then predict FPG that is statistically consistentwith measured FPG (see Section 3). The correction factor isdependent on the regression model used; we first describe thisusing the nonlinear regression model. Each HbA1c reading mea-sured at 4 weeks, when decreased by 0.5% (5 mmol/mol) andinverted via the hyperbolic regression curve yields a correspondingFPG value; the mean of FPG values obtained in this manner agreeswith 4-week sampled FPG. In other words, a 4-week HbA1c samplemust be corrected by decreasing it 0.5% (5 mmol/mol) in order toobtain a good estimate of actual FPG at that stage. In a similarmanner, if linear regression is used, the HbA1c correction factor is0.7% (8 mmol/mol).

A similar estimate can be obtained for 8-week HbA1c sampling.We propose an 8-week HbA1c correction of 0.1% (1 mmol/mol) forthe hyperbolic model and 0.3% (3 mmol/mol) for the linear model.

To validate the proposed correction factor we compared theFPG distribution obtained from adjusting the HbA1c values andinverting through the linear and nonlinear regression curves, withsampled FPG. The distributions agreed in the mean (as expected),and s.d.'s were comparable as well.

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

FPG mM

HB

A 1c %

Hyperboliclinearnon diabeticcontrols (n=99)0−week (n=51)4−week (n=51) 8−week (n=51)

0−week

4−week

8−week

non diabetics

Fig. 1. Linear and nonlinear regression fits of FPG and HbA1c measured from newlydiagnosed diabetic patients (0-week data) and non-diabetic control subjects. Theequation of the straight line is HbA1c¼0.64 FPG þ 2.60, r2¼0.81. The nonlinearcurve is given by HbA1c¼26.32 FPG/(18.20 þ FPG), r2¼0.84. In contrast to linearregression, hyperbolic regression has the attractive physiological property that thecurve passes through the origin, and the data is somewhat better explained by it aswell. The 4- and 8-week data of the patients lie systematically above theequilibrium regression curves.

Table 2Comparison between HbA1c sampled at 4 and 8 weeks of treatment and HbA1c

predicted by the linear and nonlinear models from sampled FPG. Sampled HbA1c

(mean 7s.d.) at 4 and 8 weeks are 8.7 71.4% (72 715 mmol/mol) and 7.7 71.0%(61 711 mmol/mol), respectively. Differences in the mean HbA1c predicted bymodels and sampled data are as shown. The corresponding p-value for a two-sidedt-test to determine significance of difference in the means is shown alongside inbrackets.

Model 4-week HbA1c 8-week HbA1c

Linear – 0.7% (8 mmol/mol, p¼0.014) – 0.3% (3 mmol/mol, p¼0.15)Hyperbolic – 0.5% (5 mmol/mol, p¼0.06) – 0.1% (1 mmol/mol, p¼0.54)

Table 1Summary statistics of FPG (mM) and HbA1c (%; mmol/mol in parenthesis) of newly diagnosed diabetic patients over 2 months of treatment.

Timepoint FPG HbA1c

Mean 7s.d. Median Min. Max. Mean 7s.d. Median Min. Max.

0-week 10.6 73.4 10.1 5.0 18.4 9.8 72.1 (84 723) 9.6 (81) 6.0 (42) 16.0 (151)4-week 8.3 72.3 8.0 4.5 18.2 8.7 71.4 (72 715) 8.7 (72) 6.2 (44) 12.6 (114)8-week 7.5 71.7 7.1 4.5 12.0 7.7 71.0 (61 711) 7.7 (61) 6.1 (43) 9.8 (84)

A. Barua et al. / Journal of Theoretical Biology 363 (2014) 158–163 161

5. Discussion

In type 2 diabetes FPG and HbA1c correlate very reliably witheach other, although establishing a precise relationship betweenthe two is difficult. Nevertheless, clinical practice invariablyfollows the ADA recommendation of measuring FPG or HbA1c todiagnose and treat diabetes (American Diabetes Association, 2014).In some cases it is easy to measure FPG directly, such as when self-monitoring using a glucometer; however, daily variability isconfounded with the limited accuracy of that test. A high-qualityHbA1c measurement is therefore desirable. Here we have proposedthat HbA1c measurements alone are sufficient to provide anaccurate estimate of FPG as early as within 4 weeks of startingantidiabetic therapy.

We prescribe the following: An HbA1c measurement collectedat 4 weeks of antidiabetic treatment must be decreased by 0.7%(8 mmol/mol), and if required, this corrected HbA1c can be used,using the steady-state regression curve, to obtain an excellentestimate of the FPG at 4 weeks. By 8 weeks the corrections toHbA1c are as low as 0.3% (3 mmol/mol). We note that the 8-weekcorrections may not be significant (p¼0.15; Table 2) because ourstudy is limited in its power to detect this small a difference.We can refine these corrections further still: We propose that anonlinear, hyperbolic relation between FPG and HbA1c is evenmore accurate in relating the two. If the nonlinear regressionmodel is used HbA1c needs to be corrected by 0.5% at 4 weeks, andbarely needs to be corrected at 8 weeks. We recommend theadoption of these new, refined values of HbA1c in clinical practiceto enable the use of HbA1c measurement in antidiabetic treatmentas early as within one month.

Apart from the clinical relevance of these results, our analysishas implications for understanding the physiology of glycation ofhemoglobin as well. Both, theoretical considerations based on theslow timescales of glycation of hemoglobin as well as experimentalstudies have showed that FPG–HbA1c correlations are best effectivein those circumstances when average glucose changes relativelyslowly compared to the timescale of HbA1c equilibration (Paisey etal., 1980; Bouma et al., 1999). In other words, when diabetic patientsare placed on antidiabetic therapy glucose changes continuouslyand HbA1c cannot be expected to be in a good equilibrium relativeto FPG, especially at the start of treatment. As a rule of thumb thisperiod is usually taken to be roughly the lifetime of a red blood cell,about 120 days. Sometimes the HbA1c equilibration period is takento be closer to 3 months: Clinical recommendations, for example,assert that HbA1c monitored at 3-month intervals reflect plasmaglucose consistently. We followed HbA1c and FPG of newly diag-nosed diabetic patients that had just began therapy over thesubsequent two months, sampled at 4 and 8 weeks. We validatethe following picture. As therapy is initiated glucose begins tochange rapidly hence measured HbA1c overestimates actual FPG forseveral weeks after. By eight weeks, however, HbA1c and FPG are inexcellent quasi-equilibrium. This implies that the effective timescaleof glucose control is longer than 2 months, while HbA1c equilibriumhas a relaxation constant considerably less than 120 days (hence8 weeks are sufficient for HbA1c to attain quasi-equilibrium withglucose). Our analysis therefore predicts that the steady-stateregression curves are excellent estimators of the FPG–HbA1c rela-tionship beyond 8 weeks even while glucose continues to change. Inother words, clinical data echoes what is expected from thephysiology described in the kinetic model of glycosylation: Anabrupt change in plasma glucose leads to a temporary misalign-ment of FPG and HbA1c measurements; after about 8 weeks, quasi-equilibrium is established and FPG and HbA1c evolve together inclose agreement.

We stress that a good kinetic model does not currently exist.The biochemical model (1) extends that originally due to Higgins

and Bunn (1981), and is built on minimal assumptions. Inparticular, the steady state equation of (2) was derived under theassumption that the turnover rate of hemoglobin, α, is a constant.However, it is well known that the half-life of RBCs, which containhemoglobin, is not constant: It varies with numerous factors likepatient age, sex, and medical conditions such as inflammation.A more comprehensive kinetic model holds the promise of beingable to explain individual deviations from the regression curve,perhaps in terms of pathophysiological differences betweenpatients. Our preliminary investigation has shown that while itis possible to obtain good estimates of the kinetic parameters inthe model described in the Methods (see reaction (1)), results fromthe model are no better (and at times, worse) than the regressionanalysis that we have presented. Another major limitation whichprevents a detailed kinetic analysis is that we do not have asufficiently good model of glucose changes during the course oftreatment; although glucose changes are roughly linear in the firstcouple of weeks, it remains to be investigated to what extentpatients respond to glucose control, and over what timescales.Kinetic models of plasma glucose, hemoglobin and HbA1c havebeen investigated by several authors since Higgins and Bunn(1981), Beach (1979), Osterman-Golkar and Vesper (2006),Svacina et al. (1990), Hamren et al. (2008), and many of thequestions raised there continue to be important in developing asatisfactory understanding of the factors that influence glycationof hemoglobin in diabetes.

Finally, we discuss the potential generalization of these ideas toother populations. While the methodology we have employed isvalid in general for use on other cohorts, we expect that theresults, including the correction factors we obtain, will vary withthe characteristics of each group. We have considered a populationthat represents a “typical” Indian patient on antidiabetic treatmentwith relatively few co-existing morbidities. However it is impor-tant to note that HbA1c varies significantly across populationsdepending on various factors. Thus, any study that is interested ininvestigating HbA1c correction factors relative to specific featuresof a population must take into account epidemiological considera-tions that include racial backgrounds (Herman et al., 2009;Herman and Cohen, 2012; Venkataraman et al., 2012), sex, age(Lu et al., 2010) and RBC life span (Cohen et al., 2008). Further, thecorrection factors can also be influenced by the specific antidia-betic therapy used, because each particular combination of drugswill influence the HbA1c trajectory (Sherifali et al., 2010; Espositoet al., 2012). Our work thus represents a first step towardsdeveloping correction factors that enable a rapid interpretationof HbA1c in early therapy. Further work is needed to understandthe underlying sources of variation in correction factors acrossdifferent populations.

5.1. Conclusion

Both FPG and HbA1c are now widely used to diagnosetype 2 diabetes mellitus and to assess the progress of antidiabetictreatment. In late-stage treatment the two variables are very wellcorrelated. However, HbA1c is not used in the early phase ofantidiabetic therapy (12 weeks or earlier), during which timeglucose changes continuously. The FPG–HbA1c relation in early-stage treatment remains poorly investigated, although it is highlyrelevant clinically.

Using data obtained from Indian diabetic patients, we investi-gated an FPG–HbA1c relation valid in the initial stage of antidia-betic treatment. We studied linear regression as well as anonlinear regression model derived on the basis of the biochem-istry of glycosylation of HbA1c, to determine a relationshipbetween FPG and HbA1c valid during the first two months oftherapy. We provide an estimate of the correction that should be

A. Barua et al. / Journal of Theoretical Biology 363 (2014) 158–163162

applied to measured HbA1c values in order to avoid over-estimating the true glycemic status of the patient in early-stagetherapy. We conclude that if measured HbA1c is decremented by0.7% (8 mmol/mol) at 4 weeks and 0.3% (3 mmol/mol) at 8 weeks,this corrected HbA1c is a good predictor of the corresponding FPGthrough the steady-state linear regression curve.

On the basis of the corrections proposed here HbA1c measure-ments can be better interpreted in the early weeks of antidiabetictreatment, both clinically as well as in epidemiological studies.

Conflict of interest

The authors declare no potential conflict of interest related tothis article.

Author contributions

A.B., J.A., P.G., and S.G. wrote the manuscript. J.A. and S.G.designed and carried out the experiment. A.B. and P.G. carried outmodeling and statistical analyses.

Acknowledgments

We would like to thank Dr. C.S. Yajnik for his help withcoordinating patient data collection at KEM Hospital, Pune. Weappreciate the help of Ms. Pallavi Yajnik for her help in recruitmentof the diabetic patients for this study. We are grateful to thevolunteers and diabetic patients for their whole-hearted partici-pation during the study. We would like to thank Prof. J.V.Deshpande for useful discussions regarding the statistical analyses,and for his comments on the manuscript. This work was finan-cially supported by Department of Biotechnology, Governmentof India. J.A. was supported by University Grants Commission,Government of India.

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