The variability of blood glucose in type 1diabetes

Stefan Du Rietz  (e-mail: sdr (at) this domain)

How to show the BG variability

In order to get a quantitative image of collected blood glucose (BG) readings, one must produce a relevant graph. A convenient option is a plot of the cumulative probability distribution: the fraction (y) of the sample being smaller than a certain value of the measured variable (x). Fig. 1 shows such a graph of normally distributed data with mean = 0 and standard deviation SD = 1. The fraction is shown as percentage (percentile).

Figure 1. An example of a cumulative probability distribution of normally distributed data

The centre white line in the grey area indicates the arithmetic mean. The other two white lines indicate mean ± SD, where the points A and B are situated, and the vertical outer edges of the grey area indicate mean ± 2 SD. Consequently, the width of the grey area includes about 95% of the data.

To the left, the curve is close to zero, i.e. there are very few data below -3. At x = -2 a few percent of the data have shown up and the curve has begun to rise. At point A (when x = -1, i.e. mean - SD) the curve has risen to 16%, indicating that 16% of the data are below -1. At x = 0, i.e. the mean, the curve has risen to 50%, indicating that 50% of the data are below 0 or that the remainder (100-50 = 50 %) are above 0. In other words, the mean is equal to the median. At point B (when x = 1, i.e. mean + SD) the curve has risen to 84%, indicating that 84% of the data are below 1 or that the remainder (100-84 = 16 %) are above 1. Consequently, 68% (84-16 %) of the data are between -1 and 1. In this way it is always easy to get the percentage of data within a certain range.

An example of actual BG readings

Fig. 2 shows a similar graph of all the regular (8/day) BG readings of the author (SDR) during his last 22 months of multiple daily injections (MDI) therapy. All the readings are from the same meter (One Touch Profile).

Figure 2. Cumulative probability distribution during MDI of all BG readings of SDR (n = 5276):
mean = 5.51 mmol/l  SD = 2.11 mmol/l

Obviously, the probability distribution of BG readings is not normal and symmetric, but very skew. The arithmetic mean is not equal to the median (the 50:th percentile) and there are too many readings above mean + 2 SD (about 5 %) but very few readings below mean - 2 SD. Therefore, conventional parametric statistical analyses cannot be made, which was stated by Kovatchev et al. in 1997 [1]. This has, however, been greatly overlooked in diabetes research: the paper by Kovatchev et al. has so far (2002) only been cited three times by other articles, and then by its own authors! Consequently, in almost all studies, the comparisons of statistical parameters, e.g. mean and SD, of BG readings are erroneous!

However, after logarithmic transformation BG readings become normally distributed. This is due to the multiplicative nature of endocrine and metabolic processes and is in accordance with most biological parameters [2]. The log-transformed BG readings of SDR are shown in fig. 3. Note the symmetric distribution and the nonlinear BG scale (compressed towards higher BG-values).

Figure 3. Cumulative probability distribution during MDI of all regular BG readings of SDR (n = 5276):
GM = 5.12 mmol/l  FV = 1.47  FV⁴ = 4.64

When statistical parameters have been computed on the logarithmic data, a back transformation returns the result expressed in original BG units (fig. 3). Because addition in the logarithmic domain is equivalent to multiplication in the linear domain, the following statements are valid:

1. The arithmetic mean of log-transformed BG readings is equivalent to the geometric mean (GM) of original readings. In the graphs, the centre white border in the grey area indicates GM. Compare this to the median value which is the 50th percentile: there are as many smaller as larger readings. If the statistical model were perfect, GM would be identical with the median.
2. The standard deviation of log-transformed readings is equivalent to a dimensionless factor for original readings, denominated factor of variation (FV) in documents by the author.
3. FV⁴ (the factor of variation multiplied by itself 4 times) is the ratio of the BG-value corresponding to mean + 2 SD in the log-transformed readings to the BG-value corresponding to mean - 2 SD in the log-transformed readings. This ratio, which is indicated in the graphs by the total width of the grey area, is an estimate of the variation range in 95% of the readings.

The graphs can promote new concepts

In recent years, several papers have proposed strict avoidance of hypoglycaemia in order not to introduce "hypoglycaemia unawareness" [3] (see also my discussion of neuroglycopenia). The belief seems to be that it is possible to keep all BG readings above 4 mmol/l and still maintain HbA_1c<7.0% (upper normal limit 5.5%).

Fig. 3 above shows the distribution of all the actual regular BG readings of the author (SDR) during MDI with a geometric mean of 5.12 mmol/l and therefore a HbA_1c within the normal range (on average a few tenths below the upper normal limit). Due to the large BG variation (FV⁴ = 4.64), this low mean brings about a substantial fraction (25.7%) of BG readings below 4 mmol/l.

Because SDR has already tried to minimize the variation, he could only avoid any single reading below 4 mmol/l by increasing the average, which implies moving the curve to the right so that it does not begin to rise until the BG-value is 4 mmol/l. This is equivalent to multiplying all the readings by a factor of 2.88 which is the ratio of 4 mmol/l to the lowest actual BG reading, 1.39 mmol/l. Fig. 4 shows the result of this operation.

Figure 4. Imaginary cumulative probability distribution during MDI of all BG readings of SDR (n = 5276) with no readings below 4 mmol/l.
GM = 14.8 mmol/l  FV = 1.48  FV⁴ = 4.82

Note the large increase in GM (from 5.1 to 14.8 mmol/l) and the majority of BG readings being seriously hyperglycaemic (84% above 10 mmol/l). It goes without saying that the consequences in terms of complications must be devastating. Yet, the variability of the BG readings of SDR is probably lower than that of most individuals with longstanding type 1 diabetes. The common but inappropriate estimate of BG variability, the standard deviation of all blood-glucose values (SDBG), is 2.1 mmol/l in SDR vs. a mean SDBG of 3.9 ± 1.0 mmol/l in a cohort of one hundred patients [4]. [5]

• The Y-axis indicates percentile.
• The X-axis indicates BG readings (whole blood in mmol/l) on a logarithmic scale.
• The total width of the grey area is an estimate of the variation range including 95% of the readings.
• The centre white line in the grey area indicates GM and the other two white lines indicate GM / FV and GM × FV, respectively.

References

1. Kovatchev BP, Cox DJ, Gonder-Frederick LA, Clarke W: "Symmetrization of the blood glucose measurement scale and its applications." Diabetes Care 20(11): 1655-8, 1997. PubMed
2. Zhang CL, Popp FA: "Log-normal distribution of physiological parameters and the coherence of biological systems." Med Hypotheses 43(1): 11-6, 1994. PubMed
3. Bolli GB: "How to ameliorate the problem of hypoglycemia in intensive as well as nonintensive treatment of type 1 diabetes." Diabetes Care 22(3) Suppl 2: B43-52, 1999. PubMed  Full text
4. Moberg E, Kollind M, Lins PE, Adamson U: "Estimation of blood-glucose variability in patients with insulin-dependent diabetes mellitus." Scand J Clin Lab Invest 53(5): 507-14, 1993. PubMed
5. Derr R, Garrett E, Stacy GA, Saudek CD: "Is HbA1c Affected by Glycemic Instability?" Diabetes Care 26(10): 2728-33, 2003. PubMed  Full text

© 2002-2024 Stefan Du Rietz.  Last modified: 2022-06-21 00:41 (MEST).