Title page
HbA1c values calculated from blood glucose levels
using truncated Fourier series and implementation
in standard SQL database language
WILHELM TEMSCH,1 ANTON LUGER,2 MICHAELA RIEDL3
1
Section of Medical Information and Retrieval Systems
Core Unit for Medical Statistics and Informatics
Medical University of Vienna, Austria
2
Division of Endocrinology and Metabolism
General Hospital of Vienna, Austria
3
Department of Medicine III
General Hospital of Vienna, Austria
Corresponding author:
Wilhelm Temsch
Core Unit for Medical Statistics and Informatics
Section of Medical Information and Retrieval Systems
Medical University of Vienna
Spitalgasse 23, BT88, room 88.04.807
1090 Vienna, Austria
Postal address:
Besondere Einrichtung für Medizinische Statistik und Informatik
An der Medizinischen Universität Wien
Bauteil 88, Ebene 4 Zimmer 807
Spitalgasse 23
1090 Vienna, Austria
+43 1 40400 6691 (Phone)
+43 1 40400 6697 (Fax)
+43 676 5429863 (Mobile)
Email: Wilhelm.Temsch@meduniwien.ac.at
Summary
Objectives: This article presents a mathematical model to calculate HbA1c values based on
self-measured blood glucose and past HbA1c levels, thereby enabling patients to monitor diabetes therapy between scheduled checkups. This method could help physicians to make treatment decisions if implemented in a system where glucose data are transferred to a remote
server. Systems of this type would yield results without involving any additional cost or effort. The proposed model is both faster and more reliable than chemical analysis, and there is
no need for extra measurements because available values are used exclusively. The method,
however, cannot replace HbA1c measurements; past HbA1c values are needed to gauge the
method. But it provides a useful complement to actual measurement for the trend between
checkups.
Methods: The mathematical model of HbA1c formation was developed based on biochemical
principles. Unlike an existing HbA1c formula [1], the new model respects the decreasing contribution of older glucose levels to current HbA1c values. Convolution is the mathematical
operation by which glucose profiles are transformed to HbA1c profiles. This requires Fourier
series. A php program with embedded SQL statements was written to perform Fourier transform, convolution and inverse Fourier transform. Regression analysis was used to gauge results with previous HbA1c values. Trigonometric, time-cast and sum functions were used exclusively in a total of merely 12 SQL statements, such that the method can be readily implemented in any SQL database.
Results: The predicted HbA1c values thus obtained were in accordance with measured values. They also matched the results of the HbA1c formula in the elevated range. By contrast,
the formula was too “optimistic” in the range of better glycemic control. Individual analysis
of two subjects improved the accuracy of values and reflected the bias introduced by different
glucometers and individual measurement habits.
Conclusions: Individual analysis has some advantages but requires a substantial body of data,
which can be best obtained by implementing the mathematical model such that glucose values
are transferred to a remote server [2].
Key words: diabetes mellitus · glycosylated hemoglobin A · Fourier analysis · SQL database ·
regression analysis
Introduction
Diabetes therapy is mainly about restoring natural glycemic control [3-8] to avoid long-term
consequences such as impaired vision, kidney failure or angiopathy. Ways to prevent these
vascular diseases include dietary measures, oral drugs or insulin substitution. The objective is
always to keep blood glucose levels essentially within the normal range. HbA1c values give
an indication of mean blood glucose levels over the previous 2 or 3 months and therefore are a
suitable parameter to monitor the success of self-performed glycemic control.
HbA1c has been an established parameter in diabetes therapy for over 25 years [9-12]. The
designation indicates hemoglobin molecules that have formed a permanent chemical bond
with glucose molecules. HbA1c values reflect the percentage of hemoglobin having undergone this type of reaction, which is so significant that HbA1c constitutes the backbone of diabetes checkups alongside risk factors such as body weight, blood pressure or blood glucose
values.
Several theories have been presented to explain how hyperglycemia induces the late complications typically associated with diabetes. One of them is the hypothesis of advanced glycation and product formation [13]. According to this theory, glucose molecules react with an
amino group included in all protein molecules to form what is known as aldimine or Schiff
base. Enzymes are not required for this fast reaction. The resultant aldamine is very unstable
and never reaches significant concentrations. Chemical equilibrium will quickly develop. The
bulk of newly formed aldimine will quickly decay back into glucose and protein and is very
slowly but irreversibly rearranged to form stable fructoseamine. Defense mechanisms identify
the protein molecule as being damaged and will attack it by oxidation. This will very slowly
lead up to a configuration in which other bindings sites for protein molecule develop, thereby
giving rise to unwanted cross-linking of different protein molecules.
Collagen affected in this way could result in the formation of new chemical bonds between
collagen molecules. Thus the collagen would lose its elasticity and keep blood vessels, being
a major constituent of their inner surface, in a contracted state. Unwanted bonds of this type
could form while vessels are contracted and prevent their re-expansion. Cross-links could also
form between collagen and complement, which is part of the immune system and destroys
bacteria by binding to them with the help of antibodies under physiological conditions. Erroneous binding and activation of complement at the surface of blood vessels will result in inflammation and contraction. Vessels are thereby reduced in diameter and will no longer be
able to supply oxygen to tissues, such that the tissue is deprived and eventually dies. Eyesight
deteriorates if this happens to the retina, and affected limbs may sooner or later require amputation [13].
The first stable intermediate of this chain is fructoseamine. Hemoglobin is also subject to
aldimine condensation. HbA1c is the fructoseamine emerging from this aldimine [14] and
therefore does not just supply information on blood glucose levels of the previos 2 or 3
months but indeed reflects the destructive action of elevated blood glucose levels.
3
Objectives
Self-management is becoming increasingly important in diabetes therapy. This is a positive
development because greater flexibility will improve quality of life and hence patient acceptance. In addition, personal responsibility contributes to the success of therapy. [5, 8, 15-18].
Conventional insulin therapy (CT) involves a defined regimen of insulin and food intake to be
strictly observed by the patient. A matching regimen is found heuristically in hospital but is
usually no longer adequate on disease progression. In that case, the existing regimen needs to
be replaced by a new one offering greater flexibility while also requiring a higher degree of
personal responsibility.
Patients on functional insulin therapy (FIT) are allowed to eat as much as they like anytime
but have to adjust their insulin doses based on individually derived algorithms. They need to
measure their blood glucose values several times a day and adjust them with short-acting insulin if required. Pump therapy offers a similar degree of flexibility to FIT. Patients have a
certain leeway to adjust therapy on noticing disease progression. Both treatment modalities
(FIT and pump) can only work with responsible patients. Being responsible also means being
interested in the success of therapy and in the development of HbA1c values between scheduled checkups. This interest is also reflected by commercially available HbA1c kits for home
use.
HbA1c values estimated on the basis of current blood glucose and past HbA1c levels are not
actually identical to the HbA1c values present in erythrocytes. However, a good approximation supplied by an appropriate mathematical model would have some advantages over true
measurement. There would be no need for taking additional blood samples as blood glucose
values are measured to determine the required insulin doses anyhow. Values could thus be
calculated at the push of a button. This would be more convenient than performing measurements based on a kit. It would also eliminate the cost and hassle of acquiring the kit. In addition, it may not be possible to ensure the required level of quality control when chemical tests
are performed at home.
The main objective of this article is to provide a mathematical model for HbA1c estimation
(or, more general, risk estimation, like [19, 20]) based on data available to the patient. The
model we have developed is an inexpensive, convenient and reliable tool providing valuable
information for self-management that would enable diabetics to consult a physician earlier
than the scheduled checkup if appropriate.
As a nice spin-off, the HbA1c values obtained through this model will reflect average blood
glucose levels of the previous 2 or 3 months as a logical consequence of the erythrocyte life
span (around 4 months). The model also demonstrates that recent blood glucose values are
overemphasized compared to older ones.
It is also demonstrated that the data collected by diabetics contains a large amount of valuable
information. For example, insulin demand varies with daytime, which is partially due to circadian rhythms of hormone secretion [21, 22]. Mathematical tools can help to extract such
hidden information, thereby enabling patients to further optimize their therapy and lifestyle.
Fourier transform has figured prominently in previous articles [23, 24].
4
Financial implications also play an important role. Complications of diabetes will involve
substantial costs for hemodialysis and patient care. Some of these costs could be avoided by
tools offering more effective self-management.
Methods
In this section we present a mathematical model of HbA1c formation based on biochemistry,
convolution and a triangle-shaped convolution kernel. First intermediate result is an appropriately weighted mean of the glucose values of the recent 4 months. Fourier transform is
adapted to the specific requirements of the problem and implemented in SQL language. Regression analysis was performed with SAS 9.1 software.
Model of HbA1c formation: HbA1c as such indicates a value that has been measured rather
than mathematically derived (or predicted) from blood glucose values. One formula to calculate HbA1c from blood glucose values does already exist [1]. A mean blood glucose level of
130 mg per 100 ml (= 7.2 mmol/l) would be equivalent to 6.5% HbA1c. Any additional
10 mg per 100 ml (= 0.56 mmol/l) translate to an additional 0.3% HbA1c. Lower glucose values will reduce the HbA1c. These considerations result in the following formula:
HbA1c=2.6+0.03*G[mg/100ml] and 2,6+0.54*G[mmol/l]
HbA1c levels change slowly. Even radical changes in glycemic control would take several
weeks to show. Therefore it suffices perfectly to obtain blood glucose values daily or even
weekly for HbA1c estimation. The above formula is fairly useable especially in wellcontrolled diabetes involving constant HbA1c levels over several years. Some glucometers
and software tools for diabetes management rely on this formula to predict HbA1c values.
Despite being a valuable tool, the formula is liable to create a wrong impression. While diabetics who actively maintain glycemic control will manage to keep their HbA1c constant,
they are more likely to keep track of these values in the first place, no matter if they are calculated or measured. Newly diagnosed patients are, however, concerned with many other things
and may therefore not be able to focus on models as abstract as predicting HbA1c values in
this early phase. Some of them will only become interested in such models after their HbA1c
values have stabilized.
Unlike the HbA1c formula, the model presented below respects the gradually decreasing contribution of older glucose values. For this purpose, a triangle-shaped function is used that
reaches zero 120 days in the past, corresponding to the life span of erythrocytes. The HbA1c
formula, by contrast, utilizes a moving mean value derived from the previous 2 to 3 months.
Complexity of the model is not a valid argument. Only about 12 SQL queries were needed to
perform the required Fourier analysis for this project. Once the program has been written and
implemented in a database, it can be executed as conveniently and inexpensively as a simple
query. The model also respects the underlying biochemistry and thus would outperform the
formula in the presence of variable HbA1c levels.
The processes on which the mathematical model is based are described in detail elsewhere
[14]. HbA1c formation starts out by condensation of hemoglobin (H) and glucose (G) to
aldimine (denoted G = H). A double bond between those protein and glucose molecules forms
quickly. Equilibrium between formation and decay will soon develop since G = H will also
decay quickly. The law of mass action applies:
5
[G = H ]
= const
[G ][H]
For details on the law of mass action, the reader is referred to appropriate textbooks of chemistry [25]. The letters in brackets indicate glucose, hemoglobin and aldimine concentrations.
The initial fraction will ideally yield a constant value under identical conditions (e.g. pH or
temperature). This is the equation converted to give [G = H]:
[G = H] = const. × [H] × [G]
Strictly speaking, [G = H] forms at the expense of [H]. In other words, total hemoglobin is reduced by the amount of aldimine forming. The percentage of hemoglobin bound in aldimine
condensed state is roughly 0.5% under normal (non-diabetic) conditions. Therefore the following rule applies if [H0] denotes the whole of hemoglobin:
[H] = 0.995 [H0]
In a seriously hyperglycaemic situation with 540 mg/100 ml (30 mmol/l) glucose, the aldimine level would rise to 3% of total hemoglobin:
[H] = 0.97 [H0]
[H] is still fairly close to [H0] even in this situation. Glucose values that are significantly
higher cannot contribute to HbA1c for long because immediate glucose-lowering treatment
will be required once ketoacidosis develops. Thus [H] will not fall below 97% of total hemoglobin under physiological conditions.
Large amounts of already glycosylated hemoglobin will imply a decrease in [H]. Old erythrocytes can have 10% HbA1c or even more, but they only account for a small fraction of all
erythrocytes present. Their effect is therefore negligible, and we can assume that [H] still applies except for a constant not exceeding a few percent. A good approximation is obtained by
stating that aldimine levels are proportional to glucose concentrations:
[G = H] ~ [G]
The second step in HbA1c formation is that the newly formed aldimine is rearranged to form
stable fructoseamine, which is identical to the HbA1c molecule. The kinetics involved here
differ greatly from the kinetics of aldimine condensation. Conversion is so slow that equilibrium will never develop since erythrocytes will die before that point is reached. This process
is almost irreversible.
Speed of chemical reaction equals change of concentration per time unit expressed by time
derivative:
∂[HbA1c]
= [G = H] × const.
∂t
Thus the reaction speed of HbA1c formation is proportional to the concentration of its immediate precursor aldimine. Since we found that [G = H] was proportional to glucose level [G]
by defining [H] as constant, it follows that:
∂[HbA1c]
~ [G ]
∂t
6
Thus the speed of HbA1c formation is proportional to glucose concentration by a constant yet
to be determined. A single erythrocyte born with no HbA1c is exposed to blood glucose, such
that HbA1c will increase:
T
T
0
0
[HbA1c(T )] = ∫ const. × [G(t)]dt ~ ∫ [G(t)]dt
where [HbA1c(T)] is the HbA1c content of an erythrocyte with age T. [G(t)] indicates blood
glucose values at time t, taking all values between 0 and T. Erythrocytes have an average life
span of 120 days, such that erythrocytes with T > 120 days become increasingly rare. If the
survival function of erythrocytes is defined as g, then the number of cells surviving after time
span T will be 1,000,000*g(T) if the initial number was 1 million cells. There are two possible ways of reading this—either as N0g(T) indicating the number of newly formed erythrocytes (N0) left after time span T, or as the number of erythrocytes with age T in a blood sample being proportional to g(T). The latter view is required for the model at hand. Let T1 be the
time of blood sampling and T a point of time preceding T1. Erythrocytes built at T had a time
span T1−T to accumulate the HbA1c level observed in the sample. Since the age of this fraction of erythrocytes is T1−T, their number is proportional to g(T1−T). This is the contribution
to total HbA1c:
T1
~ g (T1 − T ) ∫ G (t )dt
T
Total HbA1c is the integral starting from an appropriately selected time T0:
T1
T1
T0
T
~ ∫ g (T1 − T )∫ G (t )dtdT
T
Conversion rules for double integrals are applied, assuming that H (T ) := ∫ g ( s )ds (then
0
H(0)=0):
⇒
T1 T1
T1
T1
t
∫ ∫ g (T − T )G(t )dtdT = ∫ Κ = ∫ G (t ) ∫ g (T − T )dTdt = ∫ G(t ) H (T − T ) |
1
1
T0 ≤T ≤t ≤T1
T0 T
T0
1
t
T =T0
dt
T0
T0
H (T1 − T ) |Tt =T0 = H (T1 − t ) − H (T1 − T0 )
H(T) is the integral of erythrocyte life spans up to T, indicating the fraction of erythrocytes not
older than T in blood samples containing erythrocytes of all ages (in portions determined by
the age curve g). H becomes constant if T significantly exceeds 120 days, meaning that no
erythrocytes of that age are detectable in the sample. In mathematical terms, this is expressed
as H(T) equaling H(∞). The point of T0 should be selected far enough in the past for H(T1−T0)
to equal H(∞). If K(T1−t) equals H(T1−t)−H(∞), then K equals 0 for t (the interval from the selected time in the past to the present) because H(T1−t) equals H(∞) too. K(T) indicates the
fraction of erythrocytes older than T. This is the resultant double integral:
T1
∫ G(t ) K (T
1
− t )dt
T0
7
The next step is to extend the integral from − to + infinity:
+∞
T0
T1
−∞
−∞
T0
∫Κ = ∫Κ +
∫
∞
+∫
T1
The first one of the three integrals occurs in a range where K(T) is consistently zero. The third
integral reflects the contribution of erythrocytes (and glucose values) not yet formed. An effect of such future values must be ruled out for the model to be meaningful. Therefore the first
and third integrals must be zero:
T1
∫ G(t ) K (T
1
T0
∞
− t )dt = ∫ G (t ) K (T1 − t )dt =: G ∗ K (T1 )
−∞
This expression is the convolution of G with K. Note that K is not the erythrocyte age curve.
From the viewpoint of HbA1c formation, K overemphasizes recent blood glucose levels, such
that today’s glucose level would affect virtually all erythrocytes present. A glucose value of 1
month previously has left its mark on roughly one-quarter fewer erythrocytes—i.e. those not
yet built. We think of each erythrocyte as featuring a calendar, each sheet giving a day’s mean
blood glucose level. Total HbA1c is the integral over all sheets of all erythrocytes. If an erythrocyte dies, all of its sheets are lost as well. Erythrocytes in a uniformly distributed mix are 2
months old on average. This explains a phenomenon of which experienced diabetics are well
aware, namely that HbA1c values are essentially in accordance with the blood glucose values
of the previous 2 months even though the average life span of erythrocytes is 4 months.
Fourier transform and convolution theorem: Calculation of predicted HbA1c values is
done in two steps: First, an appropriately weighted mean of glucose, denoted GΔ, is calculated. This is the mean of the recent four months with decreasing importance (weight) of older
values in a triangle shaped weighing function. This step requires convolution. In the second
step regression analysis of GΔ plotted against HbA1c values from the laboratory will give the
predicted HbA1c values.
GΔ values are the convolution of glucose profiles with a convolution kernel K. The convolution theorem is a very convenient tool to obtain G * K:
GΔ = G ∗ K = F −1 (Gˆ • Kˆ )
Thus the Fourier coefficients of G and K are simply multiplied for each frequency and subsequently for each inversion of the Fourier transform (reconstruction). Proof of the convolution
theorem can be found in any textbook of analysis [26]. The reader is also referred to section
3.1, No. 30, of a relevant thesis [27]. From the convolution theorem it follows that
G*K = K*G.
Fourier transform of a glucose profile is a typical irregular sampling problem; the intervals of
blood glucose sampling are not always strictly periodic. Relevant tools and solutions to address this problem are described in detail elsewhere [28, 29]. Since Discrete Fourier Transform (DFT), the most popular tool for Fourier transform in information technology, requires
constant sampling intervals, the author decided for an approach based on Fourier series. As
opposed to DFT, which simply maps a vector with N values to another N-vector, Fourier series is a mapping form a continuous function defined on a compact (i.e. finite length) interval
to an infinite series of Fourier coefficients. The time interval from diabetes onset to now is a
compact interval and sampling points are connected by appropriate interpolation. Fourier series actually decomposes the profile into slowly to fast changing profiles. HbA1c is a slowly
8
changing parameter, so fast changes are suppressed in HbA1c formation. A resolution of
weekly intervals is sufficiently fine if we are only interested in HbA1c. This corresponds to a
maximum frequency in the range of 0.5 week–1 according to Shannon’s theorem [27], chapter
5.3. Corresponding Fourier coefficients can be neglected for this purpose and the series can be
truncated.
Linear interpolation of sampling points will generate frequencies above an hour−1 due to the
corners, but these can be neglected, too in case of HbA1c formation. But if, for example, circadian phenomena are explored by Fourier series, the corners play a role and the author
should consider spline or polynomial interpolation rather than linear.
As opposed to DFT the analytic integration is needed. The kth coefficient is calculated
Ck = ∫
now
onset
time Tn)
G (t )e −ikt dt . For the linear interpolated glucose profile with N sampling points Gn (at
∫ G(t )e
−ikt
N −1 Tn +1
dt = ∑
∫ (A
n =1 Tn
n
+ tBn )e −ikt dt (An,Bn are the parameters for G between Tn and
⎛ 1 it ⎞ −ikt
Tn+1). The antiderivative of te −ikt is ∫ te −ikt dt = ⎜ 2 + ⎟ e ⇒
k⎠
⎝k
N −1 Tn +1
⎛
i
⎛ 1 it ⎞ ⎞ −ikt Tn +1
C k = ∫ G (t )e −ikt dt = ∑ ∫ ( An + tBn )e −ikt dt = ∑ ⎜⎜ An + Bn ⎜ 2 + ⎟ ⎟⎟e
t =Tn
k ⎠⎠
⎝k
n =1 Tn
⎝ k
Fourier transform is intended for complex numbers. Blood glucose profile and erythrocyte age
curve are real-valued functions. This can reduce the amount of calculation required for the
Fourier transform by 50%. Real-valued functions will transform to conjugate complexes [27],
chapter 2.2, No. 20. Functions with N samples will map to Fourier transforms with N entries,
indexed −½N…+½N if N is an even number or −½(N-1)…0…+½(N-1) if N is an odd number.
For a real-valued function, the Fourier transform at −k can be derived from the value at +k as
follows:
fˆ (−k ) = fˆ (k )
Only real values should be returned because HbA1c is also real valued. This is the Fourier coefficient for the kth frequency:
(
Hˆ (k ) = a k + ibk Hˆ (k ) = Kˆ (k )Gˆ (k )
)
The Fourier coefficients are complex numbers. Note that Hˆ (−k )e − ikt is exactly the conjugate
complex of Hˆ (k )e ikt . The following equation is for inverse transform and for indices k and –k:
Hˆ (k )e ikt + Hˆ (−k )e − ikt = 2(a k cos kt − bk sin kt ) = 2 Re( Hˆ (k )e ikt )
The next step is to completely reconstruct and evaluate the processed profile H(t) at time t:
H (t ) =
=
1
2π
∑
N /2
k =− N / 2
{
}
1 ˆ
N /2
Hˆ (k )e ikt =
H (0) + ∑k =1 ( Hˆ (k )e ikt + Hˆ (−k )e −ikt ) =
2π
Hˆ (0) 1
+ ∑ Re( Hˆ (k )e ikt )
2π
π
Fourier series are based on periodic functions. The time over which the blood glucose profile
has been monitored is treated as one period of a periodic function. Some effects can only be
understood if we think of the blood glucose profile as a periodic function and perform convo9
lution on it. The average life span of erythrocytes is 120 days. Thus current blood glucose values contribute to calculated HbA1c values from the first 4 months of the observation interval
even those 4 months were several years ago. A similar problem exists in principle for current
values, as the documented interval ends at the present moment. If it is considered a periodic
function, then the value of the next time unit (whether the next day, hour or even second) is
identical to the value measured at baseline several years ago.
Even if we knew those glucose values of the next time unit, they obviously could not contribute to any physiological parameters yet. Since the erythrocyte age curve does not extend into
the future, it is irrelevant for a calculated HbA1c value reflecting the current situation how
blood glucose levels will develop in the future. However, at least 120 days need to be anticipated in the early phase of the observation interval. Any HbA1c values calculated in that
phase must be discarded as nonsensical.
Finally regression analysis of the GΔ−values obtained by this procedure and recent HbA1c values will lead to predicted HbA1c values. This method compensates constant bias, like measuring habits or glucometer bias. A separate regression analysis is recommended in case of a glucometer change. Programmers, that want to use DFT of their favourite software package, have
to build daily means of the glucose profile. Alternatively an approach based on (daily) fasting
glucose values can be made, but changes in medication with long acting insulin could modify
the relation between fasting and mean glucose and should be treated like a glucometer change.
Daily values provide the equidistant step width needed for DFT. For HbA1c formation even
weekly means are sufficient.
Results
Glucose data of 11 diabetic subjects were available for analysis. All timestamps of a specific
individual were shifted by a defined period to avoid secondary identification attributes. It has
been noted that this model requires glucose values over at least 120 days (4 months) for each
newly calculated HbA1c value. Other HbA1c values are ignored. Each HbA1c record analyzed was based on a mean of 582 glucose records.
Table 1: Patient IDs, observation intervals and glucose/HbA1c records
ID
Observation interval
Number of records
(MM/DD/YYYY)
Glucose
HbA1c
1
03/12/2002
04/30/2004
2520
5
2
06/25/2003
08/29/2004
2493
4
3
01/19/2004
08/01/2004
1008
1
4
11/24/2003
09/08/2004
2514
2
5
04/19/2004
08/08/2004
446
1
6
12/22/2003
09/19/2004
1316
2
7
04/05/2004
08/08/2004
966
1
8
05/09/2003
05/13/2004
2276
3
10
9
01/12/2004
07/11/2004
1062
1
10
04/30/2000
05/05/2004
4414
8
11
12/26/2001
04/26/2004
5529
11
Table 2: An appropriate mean glucose value (GΔ) was calculated for each of the 39 HbA1c records by convolution with the triangle-shaped weight function.
ID
Date
HbA1c
G∆
ID
Date
HbA1c
G∆
1
09/25/2002
6.7
123
9
07/11/2004
8.3
136
1
02/03/2003
6.3
120
10
10/11/2000
6.9
109
1
06/02/2003
6.6
129
10
06/13/2001
7.1
116
1
10/06/2003
7.1
132
10
12/05/2001
7.4
127
1
02/16/2004
6.6
123
10
11/27/2002
7.4
121
2
12/17/2003
7.1
129
10
05/07/2003
7.5
120
2
03/09/2004
7.7
127
10
09/17/2003
6.9
119
2
07/21/2004
7.4
132
10
01/14/2004
7.4
118
2
08/29/2004
7.2
134
10
04/21/2004
7.4
114
3
08/05/2004
7.1
124
11
08/08/2002
7.9
157
4
04/15/2004
7.1
114
11
09/05/2002
7.9
167
4
09/08/2004
6.7
104
11
10/22/2002
7.7
156
5
08/19/2004
6.3
127
11
11/28/2002
7.2
150
6
06/17/2004
7.5
143
11
01/29/2003
6.8
134
6
09/19/2004
7.7
142
11
03/19/2003
6.4
126
7
08/08/2004
8.4
195
11
05/14/2003
7.1
127
8
10/06/2003
6.2
102
11
06/23/2003
6.4
117
8
01/19/2004
6.5
104
11
09/03/2003
6.4
121
8
04/19/2004
6.2
111
11
01/14/2004
7.1
129
11
03/05/2004
6.6
138
The overall correlation coefficient between calculated and measured HbA1c values was
0.6645 (39 pairs; p < 0.0001). One patient had to be excluded because of a significant mismatch between HbA1c (9.4) and GΔ (134).
The regression line of this analysis can be read as a linear relationship between GΔ and
HbA1c, its parameters being 4.4 (intercept) and 0.0209 (slope). The established HbA1c for11
mula [1] also has a linear context, the corresponding values being 2.6 and 0.03. Comparison
between both models yielded a difference in the range of better glycemic control. While
GΔ = 130mg/100ml (7.2 mmol/l) is equivalent to 7.11% HbA1c in our own model, the formula yields 6.5% HbA1c [1]. The two lines meet at 8.53% HbA1c and 198 mg/100 ml
(10.99 mmol/l), meaning that both approaches will produce the same HbA1c values in this
range.
For a constant profile, an effect of convolution with K is not expected. A (weighted) mean of
equal values will always give the same number regardless of how they are weighted. This was
verified by substituting a value of 100 for all 5529 glucose values of subject 11 based on all
timestamps available.
Table 3: Test with timestamp data from subject 11 and a constant blood glucose value
08/08/2002
100
09/05/2002
100
10/22/2002
99.999999999999
11/28/2002
100
01/29/2003
100
03/19/2003
100
05/14/2003
99.999999999999
06/23/2003
100
09/03/2003
99.999999999999
01/14/2004
100
03/05/2004
99.999999999998
The results are summarized in table 3. The differences on the order of 10–12 are due to the fact
that the computer ran up against its limits of numerical precision. These figures may vary depending on which computer or database system is used.
The next question concerned the reliability of HbA1c values thus predicted. The value from
the regression line is the mean of a Gaussian-like distribution. Due to statistical fluctuation,
the result is a range of values and probabilities (i.e. 95% CL intervals) rather than a singular
value. Based on regression analysis a value from the “smoothened” glucose profile GΔ yields
an interval carrying a 95% probability that it covers the actual HbA1c. The confidence range
of a value thus predicted depends on its position on the regression line [30]. The confidence
interval is narrower in the cluster of points than in areas where points are isolated or absent.
There is a difference between CL and CLI intervals in regression analysis. CLI refers to an
individual predicted value while CL refers to a point along the regression line. The regression
line is subject to statistical fluctuation as a whole. CL becomes smaller than CLI as more
points are available, since this will increasingly enhance the precision of the regression line.
12
Table 4: Compressed overview of CL and CLI intervals for HbA1c values ranging from 6.5 to
8.5. The CL intervals are narrower in this overview as 27 of 40 HbA1c values are around 7.
HbA1c
CL lower
CL upper
CLI lower
CLI upper
Points
6.5
6.25
6.75
5.59
7.41
5
7
6.86
7.14
6.12
7.88
27
7.5
7.28
7.72
6.60
8.40
6
8
7.67
8.33
7.07
8.93
1
8.5
7.96
9.04
7.48
9.52
1
The SAS output only returned CL and CLI for measured (i.e. actually present) HbA1c values.
Since these values were not straightforward integers or half-integers, the figures in table 4 were obtained by intrapolation.
In some individuals, the relationship between blood glucose profile and HbA1c may be distorted due to physiological conditions or specific habits in glucose measuring. Thus better results may be obtained via individual correlations for all subjects than by blanket analysis.
Table 5: Regression lines and estimates for subjects 10 and 11 based on predicted HbA1c.
Regression lines are almost parallel to those obtained with the HbA1c formula from G∆
Subject 10
Subject 11
G∆
Predicted
Formula
G∆
Predicted
Formula
109
7.03
5.87
157
7.63
7.31
116
7.20
6.08
167
7.95
7.61
127
7.47
6.41
156
7.60
7.28
121
7.32
6.23
150
7.41
7.10
120
7.30
6.20
134
6.91
6.62
119
7.27
6.17
126
6.66
6.38
118
7.25
6.14
127
6.69
6.41
114
7.15
6.02
117
6.37
6.11
121
6.50
6.23
129
6.75
6.47
138
7.03
6.74
All blood glucose data of subject 11 were electronically downloaded from the glucometer and
stored in a database, such that any bias related to selective logging is excluded. The logs of all
other subjects in this article were manually written. The results of CL and CLI analysis for
subject 11 do not significantly differ from the overall picture even though the intervals are
slightly larger due to a smaller number of regression points. Remarkably, the regression line
for predicted HbA1c is almost parallel to the line for the HbA1c formula [1], the latter being
13
0,25% HbA1c lower than the former. The overall difference is two to three times that value in
the range of better glycemic control.
The CL figures obtained for subject 10 do not significantly differ from the overall picture either. Again, the regression line and the line obtained for the HbA1c formula were almost parallel. Both were 1.11% HbA1c apart, which was a larger distance than in subject 11. Again
the lower values were obtained with the formula. The values available for other subjects were
too scanty for individual analysis.
Discussion
The convolution kernel for HbA1c prediction is derived from integration of the erythrocyte
age curve, which would have to be rectangular for a perfect triangle, meaning that all erythrocyte would have to die precisely after 120 days. Integration does succeed in rendering the final segment of the age curve less critical. A significant fraction of erythrocytes dying in
month 4 would imply that an equal fraction will become older than 4 months. The kernel
would then be deformed within a range contributing the 16th part of the entire integral.
Extension: The question arises how values calculated in the past can simplify a system that is
based on continuously collected data. It is certainly not useful to store the Fourier coefficients
because the base frequencies will change over extended observation intervals. It may be possible to reuse some values when new intervals happen to be an integral multiple of the old
ones. If the observation interval is doubled, odd-numbered frequencies need to be calculated
from scratch and even-numbered ones for half the new interval, such that the computation
load is only reduced by 25%. Extending the observation interval by a factor of “k” will allow
1/k² of the total calculation to be reused. Everything else needs to be recalculated.
Considerable machine power can be saved by recalculating not the entire observation interval.
Convolution does not depend on the set of frequencies contained in the Fourier series. Thus
the old results for GΔ should not change. To avoid unwanted effects of periodization, the departure point of the new observation interval should precede the oldest one of the new HbA1c
values by 4 months. Only the immediate result of convolution (i.e. the smoothened profile GΔ)
is invariant. Used in regression analysis, the new values can change predicted HbA1c values
in the past as well. Therefore all HbA1c values need to undergo regression analysis, although
this will not affect the glucose records, which outnumber them by a factor of 800.
Routine application: On first sight, this approach to obtaining HbA1c values may seem impractical for routine application. Manual editing of glucose data as performed in this project is
obviously a laborious task (Table 1). In addition, most glucose records are not archived for the
120 days needed to calculate a predicted HbA1c value. More than ten HbA1c measurements
were available for only one patient analyzed in this article. However, projects exist in which
records of diabetics are being transferred to servers and remain stored there as part of a lifelong health record. Predicted HbA1c values can be obtained at virtually no cost by utilizing
systems of this kind. Some projects focus on transmission technology [31-33] and may even
supply feedback to the patient [31].
The Austrian ELGA project [34] deals with diabetes as a representative example of a life-long
disease and therefore includes private data on glucose, insulin and diet.
14
Predicted HbA1c values in diabetes care: Implementing the presented model in diabetes cares would first require diabetics to transmit their blood glucose values to a server. HbA1c values would then be routinely calculated on a daily or weekly basis. If they deteriorate, a message to the effect that HbA1c is increasing would be sent to the patient by SMS or e-mail, including a recommendation to consult a physician. Care should be taken about ethical aspects
[35] .
A similar approach is indicated for mathematical models analyzing the circadian rhythm of
insulin therapy. The patient could be given relatively safe advice, such as to apply base insulin
2 hours earlier.
Another useful application of the mathematical model would be to anticipate critical situations. Cases that need attention would be filtered out by automated routines and be brought to
the attention of diabetologists who would then advise those patients by e-mail or call them for
a visit if necessary.
In the long term, automated routines of this type should be beneficial for the healthcare system because some level of monitoring between scheduled check-ups will surely reduce costs
by requiring fewer visits to the ward. However, this phase must be preceded by a test period
to verify the reliability of the system.
Linear regression analysis: The CL and CLI intervals compiled in Table 4 illustrate the
great reliability of predicted HbA1c values by ”compressed“ statistics using groups of individual pairs. Figure 1 is a graphical representation including all 39 records.
Figure1: CL and CLI corridors with predicted values (of regression analysis) plotted against
measured values (asterisks) for all individual pairs.
15
10
9
CLI upper
7,88
8,0
8
CL upper
7,5
7,14
7
CL lower
7,00
6,5
6,86
CLI lower
6,12
6
5
6
7
8
The CL interval in Figure 1 is significantly narrower where observation points are densely
clustered and opens trumpet-style as the frequency of observations decreases in the presence
of increasing HbA1c values.
The CLI interval in Figure 1 simply indicates what we get by calculating a HbA1c value with
the proposed method. Figure 1 shows that 1 of 39 points is outside of the CLI corridor, reflecting a 95% probability (i.e. in 19 out of 20 cases) that the value returned from the laboratory for a simultaneously obtained blood sample will be inside this CLI interval.
As already noted, the CL corridor is the area covered by 95% of all regression lines. Thus
there is a 95% probability that the true regression line will be inside the CL corridor. The true
regression line is reflected more closely as more points along its course are available. The CL
corridor will then become narrower and ultimately (if a large number of points are available)
form a straight line rather than a band with flaring ends. While personal glucometer measurements will normally involve 20% fluctuation, a mean number of 582 glucose values as
processed to single GΔ values in the present study is very likely to compensate for this fluctuation. A constant mean level of deviation will naturally remain and will cause parallel translation of regression lines when individual analysis is performed (Table 5). In blanket analysis,
both the deviations introduced by different glucometers and the bias introduced by individual
measurement habits will cause some fluctuation of GΔ values. However, any fluctuation of
glucose values is not expected to be significant in individual analysis.
Lack of a linear association between mean glucose (GΔ) and HbA1c values might conceivably
be a source of fluctuation in linear regression. The HbA1c content of erythrocytes is linearly
proportional to speed of HbA1c formation and duration of exposure to the glycosylating environment. This duration is constant as the life span of erythrocytes is essentially constant. It is
reasonable to infer from the above model of HbA1c formation that its speed is almost linearly
proportional to glucose concentrations. Therefore, since GΔ is a mean glucose level respecting
16
the model of HbA1c formation, a linear association between HbA1c content and GΔ does exist
through formation speed.
Ultimately, the HbA1c values coming from the laboratory are the main source of fluctuation.
In blanket analysis, some fluctuation will also be present on the glucose side as different glucometers and measuring habits enter the equation. For a small number of pairs, the regression
line itself is only obtained within the probability limits of a 95% CL corridor. In a sense, the
difference between CL and CLI is essentially due to statistical fluctuation of HbA1c measurements. The figures would suggest inaccuracies in the range of 0.5% for HbA1c values
coming from the laboratory.
Figure 2: Regression line and line produced by the HbA1c formula. The formula seems too
“optimistic” in the range of better glycemic control. According to the CL corridor, this deviation can be due to statistical fluctuation only for HbA1c > 7.5%.
10
9
CL upper
8
CL lower
7
HbA1c Formula
6
5
100
120
140
160
180
200
220
240
Since the mathematical model is naturally based on glucose values from the logbook, Figure 2
could be read as indicating that true glucose levels are higher than would appear from the logbook. However, this does not imply that the true situation has been intentionally misrepresented. Both functional insulin therapy (FIT) and conventional therapy (CT) usually require
patients to measure their blood glucose levels before meals to adjust insulin doses or food intake. Glucose levels will then remain elevated for a few hours after meals, but these levels are
not normally measured or logged, simply because there are no appropriate columns in the
logbook. In addition, some diabetics may avoid measuring when they know the values will be
high. The reason why better matches are obtained at higher HbA1c levels could be that the
glucose profile may be less stable. Low glucose values before meals would then be less specific due to a higher degree of general variability, as would be increased values after meals.
17
The data for two subjects analyzed individually yielded regression lines that were remarkably
parallel to the lines obtained with the established HbA1c formula. This suggests that the inherent bias in glucose profiles, while varying from one diabetic to the next, is fairly constant
for the same individual over a range of HbA1c values.
Conclusions
The model is best implemented in a system that will collect glucose data of patients and allow
them to be transferred to a database. Today a high level of automation is required as healthcare systems are becoming inordinately expensive. Predicted HbA1c values and other modes
of analyzing blood glucose profiles are no exception but should be automated as well. Feedback supplied to the patients should also be automated unless glucose values exceed a predefined range continuously or repeatedly. A diabetologist should be alerted if a critical situation
occurs, which the proposed system can help to identify [36]. The system cannot prescribe or
change therapy [35] but it can give recommendations involving little risk such as to see a diabetologist.
To some extent, the CL corridor is better suited for monitoring requirements than CLI. Predicted HbA1c values would only be regarded as undergoing change in diabetes care if the CL
limits are exceeded. Some physicians may want to have a safety margin in making statements.
Depending on the specific clinical situation, it may be appropriate to base statements of this
kind not on predicted values along the regression line but on the (upper or lower) CL.
Additional information could be obtained by performing regression analysis for each diabetic
individually. If the proposed system of calculated HbA1c has been implemented in a database
system and enough HbA1c/GΔ pairs of a patient have been collected, then the current status of
glycemic control is indicated by maximum clustering of points. Incipient change can be detected more readily because it is reasonable that predicted values located within that cluster
should be more accurate. Values located outside of the cluster may still indicate change, even
tough with less precision. By reflecting deterioration or improvement, the mathematical
model can be a valuable aid in clinical decision-making.
CL intervals can thus help to distinguish change from mere fluctuation. As a practical example, diabetics with good glycemic control might want to notice incipient change to take appropriate measures. Patients with poor control might want to try different regimens and may
identify the most appropriate strategy by the greater accuracy of predicted HbA1c values in
the higher range.
Individual habits of glucose measurement appear to have a significant effect by introducing
bias. A way to conserve values that would not normally be logged is by automated data transfer from the glucometer to a remote server. In this way, enough data could be accumulated to
obtain meaningful results by individual regression analysis and to eliminate some bias-related
fluctuation.
Theoretical implications of the present study include the triangle-shaped convolution kernel
and the fact that a Fourier transform and Fourier series were implemented in standard SQL
language with trigonometric functions.
18
Acknowledgements
Special thanks are due to Harald Heinzl for his very useful and informative discussion of statistical data analysis. I am indebted to Julian Eigenbauer for manually typing most of the glucose data from hardcopy prints.
19
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21
Table 1: Patient IDs, observation intervals and glucose/HbA1c records
ID
Observation interval
Number of records
(MM/DD/YYYY)
Glucose
HbA1c
1
03/12/2002
04/30/2004
2520
5
2
06/25/2003
08/29/2004
2493
4
3
01/19/2004
08/01/2004
1008
1
4
11/24/2003
09/08/2004
2514
2
5
04/19/2004
08/08/2004
446
1
6
12/22/2003
09/19/2004
1316
2
7
04/05/2004
08/08/2004
966
1
8
05/09/2003
05/13/2004
2276
3
9
01/12/2004
07/11/2004
1062
1
10
04/30/2000
05/05/2004
4414
8
11
12/26/2001
04/26/2004
5529
11
22
Table 2: An appropriate mean glucose value (GΔ) was calculated for each of the 39 HbA1c records by convolution with the triangle-shaped weight function.
ID
Date
HbA1c
G∆
ID
Date
HbA1c
G∆
1
09/25/2002
6.7
123
9
07/11/2004
8.3
136
1
02/03/2003
6.3
120
10
10/11/2000
6.9
109
1
06/02/2003
6.6
129
10
06/13/2001
7.1
116
1
10/06/2003
7.1
132
10
12/05/2001
7.4
127
1
02/16/2004
6.6
123
10
11/27/2002
7.4
121
2
12/17/2003
7.1
129
10
05/07/2003
7.5
120
2
03/09/2004
7.7
127
10
09/17/2003
6.9
119
2
07/21/2004
7.4
132
10
01/14/2004
7.4
118
2
08/29/2004
7.2
134
10
04/21/2004
7.4
114
3
08/05/2004
7.1
124
11
08/08/2002
7.9
157
4
04/15/2004
7.1
114
11
09/05/2002
7.9
167
4
09/08/2004
6.7
104
11
10/22/2002
7.7
156
5
08/19/2004
6.3
127
11
11/28/2002
7.2
150
6
06/17/2004
7.5
143
11
01/29/2003
6.8
134
6
09/19/2004
7.7
142
11
03/19/2003
6.4
126
7
08/08/2004
8.4
195
11
05/14/2003
7.1
127
8
10/06/2003
6.2
102
11
06/23/2003
6.4
117
8
01/19/2004
6.5
104
11
09/03/2003
6.4
121
8
04/19/2004
6.2
111
11
01/14/2004
7.1
129
11
03/05/2004
6.6
138
23
Table 3: Test with timestamp data from subject 11 and a constant blood glucose value
08/08/2002
100
09/05/2002
100
10/22/2002
99.999999999999
11/28/2002
100
01/29/2003
100
03/19/2003
100
05/14/2003
99.999999999999
06/23/2003
100
09/03/2003
99.999999999999
01/14/2004
100
03/05/2004
99.999999999998
24
Table 4: Compressed overview of CL and CLI intervals for HbA1c values ranging from 6.5 to
8.5. The CL intervals are narrower in this overview as 27 of 40 HbA1c values are around 7.
HbA1c
CL lower
CL upper
CLI lower
CLI upper
Points
6.5
6.25
6.75
5.59
7.41
5
7
6.86
7.14
6.12
7.88
27
7.5
7.28
7.72
6.60
8.40
6
8
7.67
8.33
7.07
8.93
1
8.5
7.96
9.04
7.48
9.52
1
25
Table 5: Regression lines and estimates for subjects 10 and 11 based on predicted HbA1c.
Regression lines are almost parallel to those obtained with the HbA1c formula from G∆
Subject 10
Subject 11
G∆
Predicted
Formula
G∆
Predicted
Formula
109
7.03
5.87
157
7.63
7.31
116
7.20
6.08
167
7.95
7.61
127
7.47
6.41
156
7.60
7.28
121
7.32
6.23
150
7.41
7.10
120
7.30
6.20
134
6.91
6.62
119
7.27
6.17
126
6.66
6.38
118
7.25
6.14
127
6.69
6.41
114
7.15
6.02
117
6.37
6.11
121
6.50
6.23
129
6.75
6.47
138
7.03
6.74
26
Figure1: CL and CLI corridors with predicted values (of regression analysis) plotted against
measured values (asterisks) for all individual pairs.
27
Figure 2: Regression line and line produced by the HbA1c formula. The formula seems too
“optimistic” in the range of better glycemic control. According to the CL corridor, this deviation can be due to statistical fluctuation only for HbA1c > 7.5%.
28