An Innovative Mathematical Model: A Key to the Riddle of HbA1c

HbA1c is a standard clinical assessment of glycemia and the basis of most data relating glycemic control to complications. While daily blood glucose testing gives a picture of day-to-day fluctuations, the HbA1c test offers an overview of how well glucose has been controlled over the past 4 months. I devised an innovative mathematical model to describe novel equations governing HbA1c which enables analysis of HbA1c behavior and provides emerging new concepts in assessment of diabetes management. Linear relationship of HbA1c and mean plasma glucose along with the kinetic analysis of HbA1c formation has been used as the basic suppositions to construct this model. The main application of this devised model is prediction of mean plasma glucose at any desired point in time after a change in therapy and with great certainty. This model also appraises the pattern of HbA1c changes over time and provides a unique opportunity to address common mistakes and misconceptions in routine application of HbA1c that could have potentially important implications on diabetes control.


Introduction
Maintenance of blood glucose levels as close as possible to the nondiabetic range over time is an important goal in the current management of patients with diabetes. Assessment of a patient's diabetes management can be accomplished by directly analyzing the pattern of multiple blood glucose samples drawn over time [1]. However, a high degree of cooperation is required on the part of the patient to collect a sufficient number of blood glucose samples that adequately represent typical diurnal glucose patterns. Once collected, statistical analysis is then necessary to assess the central tendency and variability of glucose levels. As an alternative, a patient's HbA 1c level can be easily and conveniently determined from a single blood sample.
A large number of studies have shown that HbA 1c is strongly associated with the preceding mean plasma glucose (MPG) over previous weeks and months [2][3][4][5]. Based on the statistical relation of HbA 1c and MPG, HbA 1c is widely used as a clinical estimation of MPG, and it has been proposed as a diagnostic criterion for diabetes, as well [6]. HbA 1c has, therefore, become a standard assessment of glycemia [7] and a standard part of diabetes management.
One of the most important limitations of HbA 1c is that it is not applicable in short intervals. Erythrocyte life span in normal conditions averages ∼120 days, and the glycation of hemoglobin (Hb) is a continuous, nonenzymatic, relatively slow and nearly irreversible process [8] that means change in effects of previous glycation on Hb takes several weeks to months to occur. To permit a much clearer assessing of diabetes management, it is generally recommended that the HbA 1c assay be used every 2-3 months. Ideally, if measured each 120 days (4 months) it gives a precise estimation of MPG over preceding 4 months, reliably comparable to previous HbA 1c value. If measurement is taken earlier than erythrocyte life span intervals (4 months), because of existing previously glycated Hbs which have not reached end of their lives, the estimated MPG would be affected by previous plasma glucose levels. This would be an important issue, and if measured following a significant variations in plasma glucose during changes in patient's diabetes control or medication, then it would end to a remarkable error in estimation of MPG.
I devised an innovative mathematical model to describe novel equations governing HbA 1c which enables analysis of 2 International Journal of Endocrinology HbA 1c behavior and provides emerging new concepts in assessment of diabetes management.

Suppositions and Theory
Quarterly HbA 1c and corresponding seven-point capillary blood glucose profiles obtained in the DCCT have been analyzed to define the relationship between HbA 1c and MPG. HbA 1c is linearly related to MPG based on linear regression analysis weighted by the number of observations per subject (Figure 1), producing a relationship of [9] MPG mg/dl = 35.6 × HbA 1c − 77.3. (1) or HbA 1c = 1 35. 6 MPG + 2.17.
MPG at increasing levels of HbA 1c is shown in Table 1 based on DCCT data correlating HbA 1c with MPG using 7point blood glucose profiles along with ADAG data using continuous glucose monitoring systems [5,9]. The kinetic analysis of HbA 1c formation depicted in Figure 2 shows the linear relationship between HbA 1c formation rate and time, with the slope proportional to the MPG [10]. The higher the blood glucose is, the faster HbA 1c will be formed, resulting in higher HbA 1c levels. It also demonstrates the distribution of HbA 1c amount in erythrocytes with different ages. For instance, it is elicited from the curve MPG = 137, that the HbA 1c rate in newly born RBCs is 0%, and in RBCs with 60 and 120 days old, 6 and 12%, respectively.
It can be corroborated that the mean HbA 1c in a collection of erythrocytes with different ages and HbA 1c rates is the median point or arithmetic mean of the upper and lower limits of the curve. Erythrocyte life span in normal conditions is about 120 days and the level of HbA 1c at any point in time is contributed to by all circulating erythrocytes, from the oldest (120 days old) to the youngest. Since the rate of RBC formation is equal to its degradation, the percentage of RBC count in a single day is 1/120 of total RBC mass. Hence, 1/120 of RBC collection are one day old, 1/120 are two days old, and likewise 1/120 are 120 days old. The mean value of HbA 1c in a collection of RBCs with different ages can be calculated by averaging of HbA 1c rate in each RBC as follows: Mean HbA 1c = 120 1 ε n × (1/120)RBC mass RBC mass , where ε n is HbA 1c rate in RBCs with n days old and RBC mass is total number of RBCs in the body. Since glycation of Hb according to Figure 2 follows a linear pattern, it is expected that  Hence, Accordingly, the mean value of HbA 1c in a collection of RBCs would be the arithmetic mean of upper and lower limits of the curve. The mathematical relationship between data leading to the curves depicted in Figure 2 can be correlated by the following formula: where MPG contributes the mean plasma glucose in which Hb glycation is progressing and m is the variation of time in month.
We will now map this model into mathematical expressions and start with an example.  in Figure 3. After one month you are interested in calculating the HbA 1c which is the mixture of previously and newly glycated Hbs and is estimated to be in range of 6% to 9%.
As described in Figure 3, after passing one month of changes in MPG, the erythrocytes with 3 to 4 months old will reach the end of their lives and destroy themselves. The remaining RBCs have HbA 1c ranging from 0 to 13.5%. This upper extreme can be calculated as As described before (by (5)) the mean value of HbA 1c in this group of RBCs is arithmetic mean of upper and lower extremes of the curve, that is Over the past one month, these previously glycated RBCs undergo new glycation on the curve MPG = 137 to convey the prior mean HbA 1c to a newly higher point. This displacement of mean HbA 1c point on the second curve over m months can be written as The sum of contributions (8) and (9) represents the cumulative mean HbA 1c in this group of RBCs with former and later glycation on two different curves.
Furthermore, during the past one month, second group of RBCs have been newly formed and undergone glycation on the new curve (MPG = 137) with mean HbA 1c of Following all above steps, we can find the final desired HbA 1c by averaging equations Φ 1 and Φ 2 considering their coefficients according to available RBCs in each group (3 : 1, three months versus one month).
Therefore, the estimated HbA 1c after one month would be 7.6875%. Although this is the answer to our initial riddle, this value (Hb mix ) is a simply measurable variable by laboratory assays. In fact, our unknown desirable variable in this setting would be MPG x and Hb x representative of the second curve in which glycation occurs over recent months. And this is 4 International Journal of Endocrinology or It is of note that, the final equation (16) is independent of presumed equations correlating HbA 1c with MPG such as DCCT data ( (1) and (2)) or other data such as Nathan's et al. [5]. The calculated Hb x has a capability to be converted to the corresponding MPG using any of mentioned HbA 1c -MPG relationships (Table 1). and can be plotted as in Figure 4. Percentage of changes in HbA 1c during the time intervals can be expressed as and is presented in Table 2.
The calculated changes of HbA 1c over time derived from devised mathematical model are in full quantitative agreement with previous clinical studies [11][12][13] showing that plasma glucose levels in the preceding 30 days contribute ∼50% to the final results, and PG levels from 90-120 days earlier contribute only ∼10%.
As briefly described, without applying the presented equations, early measurement of HbA 1c will end to a crude and erroneous estimation of patient's MPG. How frequently should it be checked is a great controversy among authorities, but the general trend and recommendation vary from 2 to 3 months.
Additional application of our mathematical model is calculation of emerged error at any desired time intervals, defined as deviation of the crude estimation of MPG derived   Tables 3 and 4, estimated error emerged in different values of MPG 1 and Hb mix ranges from −50% to +20% for 2-month interval and −26% to +5% for 3-month interval. Negative and positive errors contribute to overestimation and underestimation of patient's MPG, respectively. The higher the difference between Hb 1 and Hb mix is, the bigger the error emerged from crude estimation of patient's MPG via laboratory measured Hb mix .

Discussion
Hemoglobin is continuously glycated during the 120-day life span of erythrocyte such that the cumulative amount of HbA 1c in an erythrocyte is directly proportional to the timeaveraged concentration of glucose within the erythrocyte [8,10,14,15]. Glycated hemoglobins provide an index of the patient's average blood glucose concentration over a long time period. This index is not affected by short-term fluctuations in blood sugar (hour to hour) and hence gives a relatively precise reflection of the state of blood glucose control in diabetes.
To introduce novel applications and new concepts about HbA 1c , an innovative mathematical simulation was analytically modeled to describe the HbA 1c behavior and process of events. The basic suppositions are cited from available equations expressed in Figures 1 and 2 [9,10]. The devised model is used to predict the mean plasma glucose at any desired point in time with great certainty. By using derived formulas, it does not take 120 days to detect a clinically meaningful and reliable value for HbA 1c and MPG over preceding months.
In addition to the presented application, I specifically was interested in assessing the pattern of HbA 1c changes over time and calculation of emerged error during crude estimation of MPG from Hb mix . As described in Table 2 and Figure 4, change in HbA 1c shows a prompt fall upon institution of rigorous diabetic control. This finding can be readily explained by decay of older erythrocytes with highest rates of glycated Hb, as demonstrated in Figure 3. This model refutes the explanation that recent PG levels (i.e., 3-4 weeks earlier) contribute considerably more to the level of HbA 1c than do long past PG levels (i.e., 3-4 months earlier) [11][12][13].
According to calculated error for 2-and 3-month interval and its explained logic, without employment of the devised model, HbA 1c should be used with caution as a surrogate measure of MPG because it may significantly under or overestimate patient's MPG.
The tests currently in use for diagnosis are the fasting plasma glucose test and the less common oral glucose tolerance test. However, these tests can be inaccurate if a person has eaten recently or is sick. Advantages of the HbA 1c test are that it can be given at any time and, because it reflects blood glucose levels over a longer period, it is not unduly influenced by events on the day of the test. This devised model also makes HbA 1c more befitting and useful for being a main part of guidelines on using the HbA 1c test as a diagnostic tool for diabetes. However, a consensus statement is necessary because right now there is no agreement on what HbA 1c level would constitute a diagnosis of diabetes.