Home | About | Journals | Submit | Contact Us | Français |

**|**Int J Endocrinol**|**v.2010; 2010**|**PMC2939406

Formats

Article sections

Authors

Related links

Int J Endocrinol. 2010; 2010: 481326.

Published online 2010 August 29. doi: 10.1155/2010/481326

PMCID: PMC2939406

Division of Endocrinology and Metabolism, Bu-Ali Research Institute, Mashhad University of Medical Sciences, Mashhad 9195977178, Iran

*Mahdi Kahrom: Email: gro.cmiri@morhak

Academic Editor: Robert R. Henry

Received 2010 February 8; Accepted 2010 June 13.

Copyright © 2010 Mahdi Kahrom.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article has been cited by other articles in PMC.

HbA_{1c} 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 HbA_{1c} 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 HbA_{1c} which enables analysis of HbA_{1c} behavior and provides emerging new concepts in assessment of diabetes management. Linear relationship of HbA_{1c} and mean plasma glucose along with the kinetic analysis of HbA_{1c} 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 HbA_{1c} changes over time and provides a unique opportunity to address common mistakes and misconceptions in routine application of HbA_{1c} that could have potentially important implications on diabetes control.

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 [2–5]. 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 HbA_{1c} behavior and provides emerging new concepts in assessment of diabetes management.

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]

$$\begin{array}{c}\text{MPG}(\text{mg}/\text{dl})=35.6\times {\text{HbA}}_{1\text{c}}-77.3.\end{array}$$

(1)

or

$$\begin{array}{c}{\text{HbA}}_{1\text{c}}=\frac{1}{35.6}\text{MPG}+2.17.\end{array}$$

(2)

MPG at increasing levels of HbA_{1c} is shown in Table 1 based on DCCT data correlating HbA_{1c} with MPG using 7-point blood glucose profiles along with ADAG data using continuous glucose monitoring systems [5, 9].

Linear regression analysis of MPG versus HbA_{1c}: the Pearson correlation coefficient (*r*) is 0.82; MPG(mg/dl) = 35.6 × HbA_{1c} − 77.3.

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.

Rate of formation of HbA_{1c} simulated from results of prolonged incubation of HbA_{0} with glucose in different concentration.

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:

$$\begin{array}{c}\text{Mean}\hspace{0.17em}\text{Hb}{\text{A}}_{1\text{c}}=\frac{{\sum}_{1}^{120}{\epsilon}_{n}\times \left(1/120\right){\text{RBC}}_{\text{mass}}}{{\text{RBC}}_{\text{mass}}},\end{array}$$

(3)

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

$$\begin{array}{c}{\epsilon}_{2}=2{\epsilon}_{1},\hspace{1em}\hspace{1em}{\epsilon}_{3}=3{\epsilon}_{1},\dots ,\hspace{1em}\hspace{1em}{\epsilon}_{120}=120{\epsilon}_{1}.\end{array}$$

(4)

Hence,

$$\begin{array}{cc}\hfill {\text{HbA}}_{1c}& =\frac{\left(1/120\right){\text{RBC}}_{\text{mass}}\times \left({\epsilon}_{1}+{\epsilon}_{120}\right)\times \left(120/2\right)}{{\text{RBC}}_{\text{mass}}}\hfill \\ \hfill & =\frac{\left({\epsilon}_{1}+{\epsilon}_{120}\right)}{2}.\hfill \end{array}$$

(5)

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:

$$\begin{array}{c}{\text{HbA}}_{1\text{c}}=\frac{\text{MPG}/35.6+2.17}{2}m,\end{array}$$

(6)

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.

Assume that you have visited a diabetic patient with HbA_{1c} = 9% and MPG = 244mg/dl and after adjusting the medications, patient's MPG has fallen to the curve MPG = 137 as visualized by graphic presentation 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

$$\begin{array}{cc}\hfill {\mathrm{\Phi}}_{\text{Upper}}& =\frac{{\text{MPG}}_{1}/35.6+2.17}{2}\left(4-m\right)=\frac{{\text{Hb}}_{1}}{2}\left(4-m\right)\hfill \\ \hfill & =13.5\%.\hfill \end{array}$$

(7)

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

$$\begin{array}{cc}\hfill {\mathrm{\Phi}}_{\text{mean}}& =\frac{{\mathrm{\Phi}}_{\text{Upper}}}{2}=\frac{{\text{MPG}}_{1}/35.6+2.17}{4}\left(4-m\right)\hfill \\ \hfill & =\frac{{\text{Hb}}_{1}}{4}\left(4-m\right)=6.75\text{\%}.\hfill \end{array}$$

(8)

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

$${\mathrm{\Delta}}_{{\mathrm{\Phi}}_{\text{mean}}}=\frac{{\text{MPG}}_{x}/35.6+2.17}{2}m=\frac{{\text{Hb}}_{x}}{2}m=3\%.$$

(9)

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.

$$\begin{array}{cc}\hfill {\mathrm{\Phi}}_{1}& =\frac{{\text{MPG}}_{1}/35.6+2.17}{4}\left(4-m\right)+\frac{{\text{MPG}}_{x}/35.6+2.17}{2}m\hfill \\ \hfill & =\frac{{\text{Hb}}_{1}}{4}\left(4-m\right)+\frac{{\text{Hb}}_{x}}{2}m=9.75\%.\hfill \end{array}$$

(10)

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

$$\begin{array}{c}{\mathrm{\Phi}}_{2}=\frac{{\text{MPG}}_{x}/35.6+2.17}{4}m=\frac{{\text{Hb}}_{x}}{4}m=1.5\text{\%}.\end{array}$$

(11)

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).

$$\begin{array}{cc}\hfill {\text{Hb}}_{\text{mix}}& =\frac{{\mathrm{\Phi}}_{2}\times m+{\mathrm{\Phi}}_{1}\times (4-m)}{4},\hfill \\ \hfill {\text{Hb}}_{\text{mix}}& =\frac{\left(\left({\text{MPG}}_{x}/35.6+2.17\right)/4\right){m}^{2}}{4}\hfill \\ \hfill & \hspace{1em}+\frac{\left(\frac{\frac{{\text{MPG}}_{1}}{35.6}+2.17}{4}\left(4-m\right)+\frac{\frac{{\text{MPG}}_{x}}{35.6}+2.17}{2}m\right)}{4}\hfill \\ \hfill & \hspace{1em}\times \frac{\left(4-m\right)}{4},\hfill \\ \hfill {\text{Hb}}_{\text{mix}}& =\frac{\left({\text{Hb}}_{x}/4\right){m}^{2}+\left(\left({\text{Hb}}_{1}/4\right)\left(4-m\right)+\left({\text{Hb}}_{x}/2\right)m\right)}{4}\hfill \\ \hfill & \hspace{1em}\times \frac{\left(4-m\right)}{4}.\hfill \end{array}$$

(12)

And rearranging gives

$$\begin{array}{c}{\text{Hb}}_{\text{mix}}=\frac{{\text{MPG}}_{x}\left(8m-{m}^{2}\right)+{\text{MPG}}_{1}\left({m}^{2}-8m+16\right)+1236}{570}.\end{array}$$

(13)

or

$$\begin{array}{c}{\text{Hb}}_{\text{mix}}=\frac{{\text{Hb}}_{x}\left(8m-{m}^{2}\right)+{\text{Hb}}_{1}\left({m}^{2}-8m+16\right)}{16}=7.6875\text{\%}.\end{array}$$

(14)

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

Access to the equations governing HbA_{1c} by this comprehensive analysis could have potentially valuable implications on diabetes control. No matter how frequently done, measurement of HbA_{1c} can lead to the desirable mean plasma glucose over previous *m* months and makes all doubts about time wasting over patient's observations, even. Finally, for practical aspects of this model in clinical setting,

$$\begin{array}{c}{\text{MPG}}_{x}=\frac{570\times {\text{Hb}}_{\text{mix}}-{\text{MPG}}_{1}\times \left({m}^{2}-8m+16\right)-1236}{\left(8m-{m}^{2}\right)}.\end{array}$$

(15)

or

$$\begin{array}{c}{\text{Hb}}_{x}=\frac{16\times {\text{Hb}}_{\text{mix}}-{\text{Hb}}_{1}\times \left({m}^{2}-8m+16\right)}{\left(8m-{m}^{2}\right)},\end{array}$$

(16)

where

- Hb
_{1}: initially measured HbA_{1c}, - Hb
_{mix}: measured HbA_{1c}after*m*months, - Hb
_{x}: HbA_{1c}corresponding to the curve on which the patient has moved during previous*m*months, *m*: time interval between measured Hb_{1}and Hb_{mix}in month.

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).

Assume a diabetic patient with Hb_{1} = 12% and MPG_{1} = 350 to whom changing in therapeutic regimens is applied. After two weeks, the rechecked HbA_{1c} is Hb_{mix} = 11%. According to (16), the mean plasma glucose in recent two weeks can be calculated as

$$\begin{array}{c}{\text{Hb}}_{x}=\frac{16\times 11-12\times \left(0.25-4+16\right)}{(4-0.25)}=6.93\%.\end{array}$$

(17)

Hb_{x} = 6.93% represents that the patient is shifted to and moving on the curve MPG = 170 (see (1)) showing a significant improvement in patient's diabetic control. Otherwise, the measured Hb_{mix} = 11% corresponds to the MPG = 315 with a remarkable error and deviation from reality due to a mixture of former and later glycated Hemoglobins.

It is of note that, variation in glycation rates between individuals and also difference in RBC life span especially in hemoglobinopathies are not factored in this model to attenuate intricacy of equations.

Another application of this derived mathematical model is describing the changes in HbA_{1c} with time. For the patient presented in Example 2, (14) takes the form of

$$\begin{array}{c}{\text{Hb}}_{\text{mix}}=\frac{6.93\times \left(8m-{m}^{2}\right)+12\times \left({m}^{2}-8m+16\right)}{16}.\end{array}$$

(18)

and can be plotted as in Figure 4.

Percentage of changes in HbA_{1c} during the time intervals can be expressed as

$$\begin{array}{c}{\mathrm{\Delta}}_{{\text{HbA}}_{1\text{c}}}=\frac{{\text{Hb}}_{1}-{\text{Hb}}_{\text{mix}}(\text{time}-\text{related})}{{\text{Hb}}_{1}-{\text{Hb}}_{x}}\times 100.\end{array}$$

(19)

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 [11–13] 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 via Hb_{mix}, from real MPG calculated by devised equations

$$\begin{array}{c}\text{error}=\frac{{\text{MPG}}_{\text{real}}-{\text{MPG}}_{\text{crude}}}{{\text{MPG}}_{\text{real}}}\times 100.\end{array}$$

(20)

or

$$\begin{array}{cc}\hfill \text{error}& =1-\frac{\left(35.6\times {\text{Hb}}_{\text{mix}}-77.3\right)}{\left(\frac{570\times {\text{Hb}}_{\text{mix}}-{\text{MPG}}_{1}\times \left({m}^{2}-8m+16\right)-1236}{\left(8m-{m}^{2}\right)}\right)}\times 100.\hfill \end{array}$$

(21)

To make the presented 3-variable equation more applicable, it can be used at definite points of time with different values of MPG_{1} and laboratory measured Hb_{mix}. As an instance, error estimation of measured HbA_{1c} for detection of patient's mean plasma glucose in 2 and 3 months intervals is expressed in Tables Tables3 and3 and and4 according4 according to different values of MPG_{1} and Hb_{mix}.

MPG_{1} versus Hb_{mix} showing estimated error of measured HbA_{1c} for detection of patient's MPG in 2-month intervals. Out of range data are ignored.

MPG_{1} versus Hb_{mix} showing estimated error of measured HbA_{1c} for detection of patient's MPG in 3-month intervals.

As presented in Tables Tables3 and3 and and4,4, 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}.

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 time-averaged 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 Figures1 and1 and and2 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) [11–13].

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.

Dr. Mahdi Kahrom has devised this model and written the paper. No conflicts of interest. No funding sources.

1. Service FJ. Correlation between glycemia and glycated hemoglobin. *Comprehensive Therapy*. 1990;16(7):33–40. [PubMed]

2. Crofford OB, Genuth S, Baker L. Diabetes Control and Complications Trial (DCCT): results of feasibility study. *Diabetes Care*. 1987;10(1):1–19. [PubMed]

3. Koenig RJ, Peterson CM, Jones RL, Saudek C, Lehrman M, Cerami A. Correlation of glucose regulation and hemoglobin A1C in diabetes mellitus. *The New England Journal of Medicine*. 1976;295:417–420. [PubMed]

4. Svendsen PA, Lauritzen T, Soegaard U, Nerup J. Glycosylated haemoglobin and steady-state mean blood glucose concentration in type 1 (insulin-dependent) diabetes. *Diabetologia*. 1982;23(5):403–405. [PubMed]

5. Nathan DM, Kuenen J, Borg R, Zheng H, Schoenfeld D, Heine RJ. The A1c-Derived Average Glucose (ADAG) Study Group. Translating the A1C assay into estimated average glucose values. *Diabetes Care*. 2008;31(1):1473–1478. [PMC free article] [PubMed]

6. Davidson MB, Schriger DL, Peters AL, Lorber B. Revisiting the oral glucose tolerance test criterion for the diagnosis of diabetes. *Journal of General Internal Medicine*. 2000;15(8):551–555. [PMC free article] [PubMed]

7. American Diabetes Association: Clinical Practice Recommendations 2001. Standards of medical care for patients with diabetes mellitus. *Diabetes Care*. 2002;24:533–550.

8. Bunn HF, Haney DN, Kamin S, Gabbay K, Gallop P. The biosynthesis of human hemoglobin A1c. *Journal of Clinical Investigation*. 1976;57(6):1652–1659. [PMC free article] [PubMed]

9. Rohlfing CL, Wiedmeyer H-M, Little RR, England JD, Tennill A, Goldstein DE. Defining the relationship between plasma glucose and HbA_{1c}: analysis of glucose profiles and HbA_{1c} in the Diabetes Control and Complications Trial. *Diabetes Care*. 2002;25(2):275–278. [PubMed]

10. Higgins PJ, Bunn HF. Kinetic analysis of the nonenzymatic glycosylation of hemoglobin. *Journal of Biological Chemistry*. 1981;256(10):5204–5208. [PubMed]

11. Tahara Y, Shima K. The response of GHb to stepwise plasma glucose change over time in diabetic patients. *Diabetes Care*. 1993;16(9):1313–1314. [PubMed]

12. Tahara Y, Shima K. Kinetics of HbA_{1c}, glycated albumin, and fructosamine and analysis of their weight functions against preceding plasma glucose level. *Diabetes Care*. 1995;18(4):440–447. [PubMed]

13. Goldstein DE, Little RR, Wiedmeyer HM, England JD, Rohlfing CL. Glycohemoglobin testing in diabetes mellitus: assay methods and clinical interpretation. In: Vasselli JR, Maggio CA, Scriabine A, editors. *Drugs in Development*. Branford, Conn, USA: Neva Press; 1993. pp. 253–267.

14. Beach KW. A theoretical model to predict the behavior of glycosylated hemoglobin levels. *Journal of Theoretical Biology*. 1979;81(3):547–561. [PubMed]

15. Fluckiger R, Winterhalter KH. In vitro synthesis of hemoglobin AIc. *FEBS Letters*. 1976;71(2):356–360. [PubMed]

Articles from International Journal of Endocrinology are provided here courtesy of **Hindawi Publishing Corporation**

PubMed Central Canada is a service of the Canadian Institutes of Health Research (CIHR) working in partnership with the National Research Council's national science library in cooperation with the National Center for Biotechnology Information at the U.S. National Library of Medicine(NCBI/NLM). It includes content provided to the PubMed Central International archive by participating publishers. |