Labile haemoglobin as a glycaemic biomarker for patient-specific monitoring of diabetes: mathematical modelling approach

Abstract

Diabetes mellitus constitutes a major health problem and its clinical presentation and progression may vary considerably. A number of standardized diagnostic and monitoring tests are currently used for diabetes. They are based on measuring either plasma glucose, glycated haemoglobin or both. Their main goal is to assess the average blood glucose concentration. There are several sources of interference that can lead to discordances between measured plasma glucose and glycated haemoglobin levels. These include haemoglobinopathies, conditions associated with increased red blood cell turnover or the administration of some therapies, to name a few. Therefore, there is a need to provide new diagnostic tools for diabetes that employ clinically accessible biomarkers which, at the same time, can offer additional information allowing us to detect possible conflicting cases and to yield more reliable evaluations of the average blood glucose level concentration. We put forward a biomathematical model to describe the kinetics of two patient-specific glycaemic biomarkers to track the emergence and evolution of diabetes: glycated haemoglobin and its labile fraction. Our method incorporates erythrocyte age distribution and utilizes a large cohort of clinical data from blood tests to support its usefulness for diabetes monitoring.

1. Introduction

Diabetes mellitus encompasses a group of metabolic disorders characterized by high blood sugar concentration and irregular insulin levels. It has been recognized as one of the major global health emergencies of the twenty-first century. Nearly, 9% of the adult population worldwide has diabetes [1–3]. The incidence of this condition keeps growing, on a global scale, due to a combination of factors that include an increasing intake of sugar and saturated fats and less physical activity. Thus, in the next decades the associated mortality and expenditure are expected to rise and significantly impact on society. Therefore, there is a need to improve the current tools employed in the management and monitoring both of diabetic patients and those at high risk of developing it.

In diabetic patients, sustained periods of hyperglycaemia ultimately lead to micro- and macrovascular complications. These complications often involve ischaemic heart disease, nerve damage, retinopathy, kidney disease and limb loss. Monitoring of glycaemic levels is thus essential for better control and to adjust treatment. Classically, monitoring of diabetic patients' blood sugar levels has been performed through plasma glucose tests. Those measurements, however, depend on many factors [4]. This is why other methods providing better information on the average glucose levels have been developed, most notably the measurement of HbA_1c, also known as glycated haemoglobin. HbA_1c levels do not directly yield an estimate of blood glucose, instead they rely on determining the amount of glycated haemoglobin in the red blood cells (RBC) which occurs as a consequence of the cell's continuous exposure to glucose over its lifetime. The use of glycated haemoglobin as a biomarker to assess patient glycaemic levels over a preceding period of eight to 12 weeks has had a major clinical impact on the management of diabetes. It has provided a convenient tool to relate glycaemia and the emergence of diabetic complications [5–7].

The fraction of glycated haemoglobin is considered the gold standard in the clinic for monitoring diabetes mellitus, particularly in type 2 diabetes. A well-known conversion table between the HbA_1c fraction and the average blood glucose level concentration, the so-called ADAG formula [8], is widely employed in the clinic. One of the advantages in the use of HbA_1c for the screening and diagnosis of diabetes is that the patient does not need to be fasting when the blood samples are extracted. It displays a much smaller intra-individual biological variability with respect to that of plasma glucose. Also, it is not influenced by sudden glycaemic variations such as those that occur, for instance, under stress. HbA_1c is correlated with the average glucose plasma levels over a time frame of two to three months [9]. Furthermore, drugs known to affect glucose metabolism have a moderate impact on HbA_1c levels.

There exist, however, limitations in the use of HbA_1c. It may exhibit in vivo effects due to physiological conditions such as pregnancy [10], age [11] and genetic determinants [12]. It may depend also on pathological conditions such as iron deficiency [13], haemolytic anaemia, renal failure, type 1 diabetes under rapid development, or recent blood losses/transfusions, to name a few. In addition, it may have analytical interference with hyperbilirubinaemia, elevated serum triglycerides, increased white blood cell counts and the presence of other haemoglobin variants [14–16].

HbA_1c constitutes an essentially permanent (irreversible) form of haemoglobin and results from a non-enzymatic glycation reaction during the exposure of free haemoglobin HbA₀ to blood glucose. Remarkably, besides the HbA_1c fraction, there exists a transient (reversible) form, known as haemoglobin A_1d aldimine fraction, HbA_1d, exhibiting much faster kinetics. Both HbA_1c and HbA_1d are routinely measured by chromatography. However, while the HbA_1c component has turned out to be widely employed for monitoring metabolic control, the HbA_1d fraction is considered to be a disturbing element in blood analysis. Only a few authors have recognized the potential clinical utility of the labile fraction as a biomarker indicating clinical situations that alter glycation in the previous hours before acquiring the blood sample [17–21].

Quantifying the interplay of both HbA_1c and HbA_1d and their combined potential use for an improved assessment of average glucose levels in patients and for early detection of diabetes and haemoglobinopathies has not been explored so far. Most mathematical models developed previously on processes related to diabetes have studied the dynamics of the glucose–insulin feedback system. The general goal of those studies has been to describe glucose homeostatic control, to analyse experimental data, to identify and quantify relevant biophysical parameters, to design clinical trials and to evaluate diabetes prevention or disease modification therapies. These models ranged from those taking into account pancreatic insulin production, with a coarser/finer level of detail over cellular and subcellular scales, to short-term organ/tissue models accounting for the intravenous and oral glucose tolerance tests as well as for the euglycaemic hyperinsulinaemic clamp, up to total-body, long-term diabetes models aiming to represent disease progression in terms of b-cell population dynamics over a long period of years (see [22] and references therein). More recent models [23–26] did include the dynamics of HbA_1c and were aimed at estimating average glucose levels in patients. To our knowledge, the role of HbA_1d as a potential biomarker complementing HbA_1c for blood glucose assessment has not been studied mathematically.

Here, we develop and validate a mathematical model describing the dynamics of both forms of haemoglobin, HbA_1c and HbA_1d, in response to temporal changes of blood glucose. Our approach describes the processes of HbA_1c and HbA_1d formation inside erythrocytes by means of chemical reaction kinetic differential equations and incorporates the age structure of RBC populations. Our goal is to improve the accuracy of blood glucose estimation and to study the potential application of HbA_1d in combination with the standard glycated haemoglobin HbA_1c.

2. Material and methods

2.1. Age-structured model for quantifying haemoglobin glycation

To account for the glycation kinetics of HbA_1c and HbA_1d, our mathematical model incorporates the ageing process of RBCs. In healthy human adults, RBCs survive in the circulatory system up to about 110–120 days after their release from the bone marrow [27]. During the lifespan of RBCs, haemoglobin becomes increasingly glycated at a rate that depends on the blood plasma glucose concentration. Inside RBCs, the process of haemoglobin glycation essentially takes place in two steps. In the first step, condensation of a glucose molecule with an amino group of free haemoglobin HbA₀ gives rise to a Schiff's base (aldimine), producing labile haemoglobin HbA_1d. This first reaction product is unstable and dissociates [28]. Thus, if the level of glucose drops for a sustained period of time of at least 2 h, the concentration of HbA_1d decreases. Otherwise, such glucose fluctuations do not produce any significant effect on HbA_1d. In the second step, a stable ketoamine form is produced with a much slower reaction rate by means of an Amadori rearrangement of the aldimine linkage, resulting in haemoglobin HbA_1c. Once HbA_1c is formed, it irreversibly remains in the RBC. Clearance of HbA_1c does not occur until the death of RBCs. Figure 1 summarizes the glycation process of haemoglobin inside RBCs.

Figure 1.

Scheme of the glycation kinetics of free haemoglobin HbA₀ inside RBCs resulting in the formation of the irreversible HbA_1c form through the intermediate HbA_1d. As RBCs age, the glycation level of their haemoglobin content increases.

Let G(t) denote the glucose concentration in blood plasma and consider an RBC which, at time t, has an age a. As each human RBC typically contains of the order of 2.7 × 10⁸ haemoglobin molecules, its glycation may be assumed to be a continuous process during the lifetime of an RBC. Let , and represent the respective concentrations of free (HbA₀), labile (HbA_1d) and glycated (HbA_1c) haemoglobin in an RBC of age a at time t. If the RBCs are subjected to a spatially uniform bath of free glucose, the partial differential equations describing the interplay of , , and G(t) read as

2.1a

2.1b

and

2.1c

Equations (2.1) constitute a continuous time model for the three different haemoglobin forms accounting for the age-structure of RBCs. The first term on the right-hand side of equation (2.1a) describes the condensation of glucose with the free (or available) haemoglobin concentration with a rate constant k_FL. The importing of glucose by RBCs is mediated by the GLUT family of membrane transporters and is much faster than the other intervening processes. Thus, the glucose concentration rapidly equilibrates between the serum and the RBC cytoplasm. The reverse reaction where glucose is released from HbA_1d takes place with a rate constant k_LF. The last term in equation (2.1a) accounts for the death rate of RBCs, with a characteristic time τ(a) that, in general, depends on their age. Similarly, equations (2.1b) and (2.1c) encompass the dynamics of the labile and glycated haemoglobin concentrations. The term describes the Amadori rearrangement of the aldimine linkage with a rate constant k_LS giving rise to HbA_1c. As this is an irreversible process, there is no transformation term of HbA_1c back into HbA_1d.

2.2. Derivation of equations for the labile and glycated haemoglobin fractions

Our first main goal is to derive a closed set of differential equations that can use patient-specific data. Thus, it is necessary to focus on global time-varying quantities having no explicit dependence on age. This can be achieved by integrating equations (2.1) over the age a ∈ [0, ∞) of the RBCs. Let us define the time-dependent functions

2.2a

2.2b

2.2c

accounting for the amounts of free, labile and glycated haemoglobin at time t, respectively. Using equations (2.1) and (2.2), we obtain the equations

2.3a

2.3b

2.3c

where the functions , and represent the concentration of free, labile and glycated haemoglobin for RBCs of zero age, respectively. We have resorted to the first mean value theorem for definite integrals involving the death rate of RBCs in order to write

2.4a

2.4b

2.4c

with τ_F, τ_L and τ_S being characteristic lifetimes.

Equations (2.3) can be further simplified by using the fact that together with τ_F ≃ τ_L ≃ τ_S ≡ τ_RBC, where τ_RBC is interpreted as the mean lifetime of RBCs. We next note that the total concentration H_RBC of haemoglobin inside an RBC does not vary significantly during its lifespan. Therefore, H_F(t) + H_L(t) + H_S(t) = H_RBC is taken as a constant. Under these assumptions, equations (2.3) become

2.5a

2.5b

2.5c

By adding up the three equations in (2.5), it follows that does not depend on time, which is to be expected under homeostatic conditions.

As H_RBC is a constant, we may define the following normalized fractions:

2.6a

2.6b

2.6c

which are amenable to direct comparison with clinical data. Given that h_F = 1 − h_L − h_S and using equations (2.5), it follows that the kinetics of the labile and glycated haemoglobin fractions is governed by the two following non-autonomous linear ordinary differential equations (ODEs):

2.7a

and

2.7b

Equations (2.7) comprise our general theoretical framework to describe the dynamics of both the labile and stable forms of haemoglobin if the glucose profile G(t) is known. The solutions to equations (2.7) can be exactly expressed in terms of quadratures (see appendix A). Moreover, when the temporal profile of G(t) is a T-periodic function it is possible to prove, using Floquet theory (see appendix B), that the solutions of system (2.7) are periodic and bounded. From a more clinical perspective, the set of equations (2.7) will allow us to uncover whether the labile haemoglobin can provide additional patient-specific information to complement the one obtained from glycated haemoglobin.

2.3. Patients

The study group consisted of 11 505 adult patients from Health Management Area Campo de Gibraltar (AGSCG) whose data were collected between January 2014 and June 2015. We excluded patients with chronic renal disease. There were 4229 non-diabetic patients and 7276 patients diagnosed with diabetes. Table 1 shows a summary of the main patients characteristics. Both groups (diabetic and non-diabetic) were matched for sex and age. Each row displays both the range of values (between brackets) and the mean of these values.

Table 1.

Description of the patient population included in this study (total of 11 505 patients). Shown are the percentage of non-diabetic and diabetic patients, their distribution by sex and age, the fasting plasma glucose (FPG) values and the glycated and labile haemoglobin percentages. The mean of the groups per age, FPG, HbA_1c and HbA_1d is shown together with the range of values observed in the data.

	non-diabetic N = 4229	diabetic N = 7276
female sex	58.2%	49.1%
age (in years)	57 (44–68)	65 (54–73)
FPG mg dl−1	100 (91–109)	137 (118–164)
HbA1c (%)	5.6 (5.4–5.9)	6.9 (6.3–7.8)
HbA1d (%)	1.7 (1.6–1.8)	1.9 (1.8–2.1)

2.4. Virtual patients

For our computations, we have used two classes of virtual patients. In the first class, we have employed the clinically measured FPG to create a glucose profile, as described below. For the second class, fully synthetic glucose functions were constructed similar to those registered via continuous glucose monitoring systems. The levels of glycated and labile haemoglobin were obtained according to our proposed model (2.7).

2.5. Glycated haemoglobin, labile haemoglobin and plasma glucose measurements

HbA_1c and HbA_1d fractions were measured using the ADAMS HGA-8160 and HA-8180 high performance liquid chromatography (HPLC) methods (A. Menarini Diagnostics, Barcelona. Spain), in diabetic mode and calibrated according to the International Federation of Clinical Chemistry (IFCC) Reference Measurement Procedure. Fasting plasma glucose (FPG) was measured using the enzymatic hexokinase UV method (Beckman Coulter AUR). The values of HbA_1d were obtained in a non-standard percentage, while those of HbA_1c were obtained in a standard percentage. Moreover, the numerical values for the HbA_1c and HbA_1d fractions returned by the measuring system, after processing, were expressed with a precision of 0.1%.

2.6. Sensitivity analysis and estimation of the kinetic constants

Our model equations (2.7) contain four kinetic constants: k_FL, k_LF, k_LS and τ_RBC. The values of k_FL, k_LF and k_LS were previously estimated in [29] in the context of an in vitro study of the chemical reaction that takes place during the glycation process. As these constants may display variations between in vivo and in vitro conditions, we estimated them from our patient database by resorting to classical methods and sensitivity analysis. The first goal here is to determine the most influential constants of model (2.7) in order to simplify the estimation problem. The values of these constants will be estimated using the least-squares method and fitting our model equations (2.7) to the data collected for h_s and h_l.

To study the influence of the kinetic constants on the ODEs solutions and to identify possible strong dependences, in addition to improve their estimation, a sensitivity analysis was performed. We resorted to variance-based methods described in [30,31] using Sobol's indexing. Variance-based sensitivity analysis works within a probabilistic framework, it decomposes the variance of the output of the model or system into fractions which can be attributed to inputs or sets of inputs. Using a priori information of the kinetic constants, distribution functions are assumed. Sobol's indexing is a computation method for those measures of sensitivity. In the method of Sobol each sensitivity effect, expressed through a conditional variance, is computed by evaluating a multidimensional integral via a Monte Carlo method.

We calculated the total sensitivity index function, denoted by TSI_i(t), to quantify the relative impact of the variability in a given constant i on the model output. The values of this index indicate the parameter's influence on the model's response. If the index is close to zero, then the parameter is non-influential.

Uniform probability distributions, over an interval [a, b], were employed for k_FL and k_LS around the values determined in [29]. For k_LF, we assumed a normal distribution with a mean equal to the kinetic value in [29], whereas a truncated normal distribution with a mean of 52 days for τ_RBC was used. The details for the corresponding distributions and their parameters for each of the kinetic constants are shown in table 2.

Table 2.

Parameters for the probability distributions used for the estimation of each of the kinetic constants k_FL, k_LS, k_LF and τ_RBC.

kinetic constants	distribution	parameters
kFL	uniform	a = 10−5, b = 6 × 10−5
kLS	uniform	a = 0.0025, b = 0.0085
kLF	normal	μ = 0.33, σ = 0.05
τRBC	truncated normal	μ = 52, σ = 25

The sensitivity analysis of the ODE solutions is depicted in figure 2. For the labile fraction h_L, we observed that the constants k_LS and τ_RBC did not show any substantial dependence on the solution. The other two constants, k_FL and k_LF, did have a noticeable effect on the variance of the solutions. For the glycated fraction h_S, we noted that k_FL and k_LS exhibit the largest dependence. Therefore, the mean lifetime of RBCs, τ_RBC, did not display in all the examined cases a significant effect on the solutions for h_L and h_S. By contrast, it was found that the rate constant k_FL, which controls the condensation of glucose with free haemoglobin, is the most sensitive of all the intervening kinetic constants in our model.

Figure 2.

Total sensitivity index function TSI(t) for the kinetic constants of the model corresponding to: (a) h_L and (b) h_S. Each curve represents (dark blue) k_FL, (green) k_LF, (red) k_LS and (cyan) τ_RBC.

Once the sensitivity analysis of the four kinetic constants was carried out, we assessed their values from our patient database, for which we had available labile and glycated haemoglobin, together with blood glucose under fasting conditions. Our estimation procedure was iterative. We first fixed k_LS and τ_RBC and obtained the values of k_FL and k_LF by fitting h_L with the corresponding patient data. Subsequently, we estimated k_LS and τ_RBC by fitting h_S with the corresponding patient data while fixing the previously calculated values for k_FL and k_LF. This process was performed eight times for each group of patients (diabetic and non-diabetic), by taking random samples of 100 individuals. In all cases, the system (2.7) was solved for each patient using a triangular periodic glucose function of the form

2.8

where G₀ is the fasting plasma glucose and G_m is the peak value for glucose. This function, and its 8 h shifts, reach their maxima at times 2 h, 10 h, 18 h, respectively, whereas the minima occur at 6 h, 14 h and 22 h and last for 2 h. The average glucose obtained from the ADAG formula [8]

2.9

and FPG were used to build the function glucose for each patient. In the formula (2.9), AG denotes the average blood glucose.

The calculated means from the fittings were henceforth taken as estimates of the kinetic constants, which are displayed in table 3. Remarkably, these constants are in the range of those previously found in the literature.

Table 3.

Estimated kinetic constants and their comparison with values from the literature.

constant (units)	value from literature	estimation	(CI 90%)
kFL (dl mg−1 h−1 )	1.66 × 10−5 [29]	5.2 × 10−5	(5.1 × 10−5, 5.4 × 10−5)
kLF (h−1)	0.33 [29]	0.31	(0.30, 0.32)
kLS (h−1)	0.0055 [29]	0.0025	(0.0025, 0.0026)
τRBC (days)	52 [27]	53	(53, 54)

Figure 3 shows the best fitting, employing a least-squares method, for HbA_1c versus HbA_1d. Our aim at this point was to make a large comparison between real and virtual patient data. The real measurements of HbA_1c and HbA_1d fractions were extracted from a random sample of 1000 patients. The model (2.7), with the constants from table 3, was applied to obtain the HbA_1c and HbA_1d fractions for 1000 virtual patients. Each virtual patient is a replica of a real one, taking into account the available data. Non-diabetic patients were chosen due to their stable glucose levels, which justifies such periodic glucose profiles for the simulation of their virtual analogues.

Figure 3.

Linear regression of HbA_1d versus HbA_1c. The blue line (HbA_1d = 0.23 HbA_1c + 0.42) and the red line (HbA_1d = 0.23 HbA_1c + 0.41) fit, respectively, patient data (blue points) and the results from virtual patients (red crosses) using our model equations (2.7). The values for the kinetic constants used in the simulations are given in table 3.

In table 4, we collect the mean and the standard deviations calculated from a linear regression between our model and the patient data. Note that the relative errors between the clinical data and the outcome of our mathematical model are as small as 1% and 0.6% for the means of the labile and glycated haemoglobin, respectively.

Table 4.

Comparison between the clinical data and the results obtained for virtual patients simulated from our model equations (2.7) for the mean and standard deviation of HbA_1d and HbA_1c, respectively.

	math model mean	clinical data mean
HbA1d (%)	1.6189	1.6365
HbA1c (%)	5.2690	5.3029
	math model std	clinical data std
HbA1d (%)	0.1181	0.1026
HbA1c (%)	0.2373	0.1890

3. Results

The main motivation of our study is to reveal what additional and clinically relevant information can be obtained from labile haemoglobin that cannot be extracted from glycated haemoglobin data alone. To further understand the labile haemoglobin dynamics, we performed a series of numerical simulations using virtual patients for which glucose time profiles were constructed as those registered via continuous glucose monitoring systems.

3.1. Labile haemoglobin evolves similarly to glucose but with a time shift

First, we wanted to identify the relevant timescales for labile and glycated haemoglobin evolution according to the temporal profile of glucose using our mathematical model (2.7). Labile haemogloblin was observed to vary on substantially shorter timescales than glycated haemoglobin, as expected from its reaction kinetics. Figure 4 shows the glucose concentration, together with the labile and glycated haemoglobin fractions, corresponding to a snapshot of a virtual healthy patient having stable levels of glucose. HbA_1d follows glucose variations but with a few hours of delay. By contrast, glycated haemoglobin remains constant during the same time window. This relationship between labile haemoglobin and glucose could be useful to assess a patient's average glucose level by performing just one joint measurement of G(t) and HbA_1d. Such combined information would allow the clinician to have access to an extended temporal frame of a few hours for glucose variations and to detect possible anomalies that may not be captured by means of only isolated glucose measurements (as e.g. with FPG).

Figure 4.

Simulated labile haemoglobin (red curve) and glycated haemoglobin (black curve inset) profiles taking as an input the periodic glucose function of a healthy patient (blue curve).

3.2. Labile haemoglobin can detect episodes of hyperglycaemia and hypoglycaemia

As glycated haemoglobin averages over very long timescales, it cannot detect short episodes of hyperglycaemia and hypoglycaemia. We wanted to test if the labile fraction could be used to indicate the presence of such episodes, that is a question of practical clinical relevance. Figure 5 illustrates simulations of our model equations (2.7) for the evolution of the labile fraction and the glucose of virtual patients experiencing episodes of hyperglycaemia. In figure 5a, a glucose function with a short event of hyperglycaemia around hour 30 occurs. Two hours later, a peak in the labile haemoglobin profile is apparent.

Figure 5.

Simulated labile haemoglobin (red curve) and glycated haemoglobin (black curve inset) profiles taking as input glucose function displaying (a) short and (b) large hyperglycaemia episodes. (a) The short hyperglycaemia event occurs around hour 30. (b) The long hyperglycaemia event begins at hour 28 and lasts for about 8 h.

Similarly, figure 5b shows a considerable longer episode of hyperglycaemia. While glucose maintains a maximum plateau, labile haemoglobin keeps increasing and reaches even higher values than those shown in figure 5a, despite the fact that in both cases the glucose peaks had the same amplitude. Note that labile haemoglobin lags behind glucose and its decrease is slower than that of glucose. Furthermore, our simulations confirm that glycated haemoglobin does not exhibit any relevant change in any of the two cases (see insets in figure 5). Thus, labile haemoglobin not only reveals elevations in glucose levels, but could also provide information on the duration of such events.

Analogous results are obtained when analysing the scenario of a virtual patient experiencing an episode of hypoglycaemia. Figure 6 shows a drop in the labile haemoglobin fraction, both for (a) short and (b) long events. When the glucose fall persists for a longer time, the labile haemoglobin displays a substantial decrease. Again, we find a relationship between glucose and its accumulated average over a time interval of a few hours. Meanwhile, glycated haemoglobin does not present any significant change in any of the two examined cases (see insets in figure 6).

Figure 6.

Simulated labile haemoglobin (red curve) and glycated haemoglobin (black curve inset) profiles taking as an input a glucose function displaying (a) short and (b) large hypoglycaemia episodes. (a) The short hypoglycaemia event occurs around hour 26. (b) The long hypoglycaemia event begins at hour 26 and lasts for about 6 h.

3.3. Labile haemoglobin can provide an advanced biomarker of response to treatments

We also explored whether the labile haemoglobin fraction may be useful to monitor the response of a diabetic patient under treatment. To this end, we simulated two virtual patients having identical temporal profiles for the glucose except for the presence of a hyperglycaemic event happening during the night period. Figure 7 shows a comparison of the two virtual patients. In this scenario, one of the patients experiences a sudden increase in the glucose level that lasts for about 8 h until it returns to the same values as the other patient. Both patients are then subjected to blood tests at 8.00 (corresponding to hour 32 in figure 7a). Interestingly, using a standard blood test, the measured glucose and glycated haemoglobin values would be expected to be undistinguishable in the two examined cases. However, the resulting labile haemoglobin turns out to be, in contrast, appreciably different. The patient that underwent the hyperglycaemic episode would exhibit a larger labile haemoglobin fraction (figure 7b). As both virtual patients were using the same medication, even though one of them suffered an hyperglycaemic event, this suggests that including the labile haemoglobin fraction routinely in current standard blood tests could be of help to track if a treatment is effective for a patient or some possible complications may occur.

Figure 7.

(a) Two glucose profiles without (blue dotted curve) and with (red solid curve) an episode of hyperglycaemia. (b) Changes in the labile haemoglobin fraction level in response to the profiles shown in (a). In both panels, vertical black dotted lines mark the delay time between both curves (see text for more details).

To further measure the relationship between labile haemoglobin and the average glucose, we computed the linear correlation coefficient between both quantities. We simulated 100 glucose profiles for 72 h and generated their corresponding labile fractions. We then calculated the linear correlation coefficient between labile haemoglobin and average glucose during the previous T hours, for each glucose profile. Figure 8a depicts the mean and the standard deviation of these coefficients, taking into account the average glucose up to the previous 12 h. There is a strong correlation between the labile haemoglobin and the average glucose in the interval 1 < T < 9 h, with a maximum occurring at T = 4 h. The linear correlation coefficients are stable and have low standard deviations regardless of which particular glucose profiles are used. Figure 8b,c shows a linear relationship between the labile haemoglobin and the average glucose during the previous 4 and 8 h, respectively. It is apparent that the best fit of the data is achieved at T = 4 h, but a good result is also observed at T = 8 h.

Figure 8.

(a) Linear correlation coefficient between the labile haemoglobin fraction and the average glucose during the previous T hours. The red dots represent the average of the correlation coefficient for 100 different glucose profiles together with standard deviation. The labile haemoglobin corresponding to each glucose profile was calculated using the mathematical model (2.7). Values above 0.85 (blue dotted line) indicate a high correlation. Linear dependence between the labile haemoglobin and the average glucose during the previous (b) T = 4 (AG₄ = 72.7 h_L − 16.0) and (c) T = 8 (AG₈ = 49.4 h_L + 19.5) hours.

4. Discussion and conclusion

The development of quantitative approaches to predict the levels of glucose metabolism has remained an important area of research in biomedicine during the last few decades [32]. The advent of new monitoring technologies for in vivo, such as continuous glucose measuring devices [33], has motivated more comprehensive mathematical models and efficient algorithms to process huge amounts of real time data [34–36]. While extremely useful both to track special groups of patients in clinical trials and for validating new hypothesis and mechanisms, that technology is not routinely available to all patients. Therefore, methods are required that can assess glucose levels employing means that are widely accessible to most patients worldwide.

This work is a proof of principle that the inclusion of HbA_1d, in addition to HbA_1c and FPG, provides a relevant combination of biomarkers to track the glycaemic levels in blood at different timescales. Supported by biomathematical modelling, such a combination may achieve a major improvement towards more robust and accurate patient-specific monitoring of diabetes and other related conditions (e.g. stress hyperglycaemia). To the best of our knowledge, this is the first time that labile haemoglobin has been embodied into a mathematical model as a biomarker to detect episodes of hyperglycaemia and hypoglycaemia. We also stress that the use of HbA_1d as a biomarker does not represent any extra cost. Its fraction is measured by chromatography in parallel with HbA_1c, thus making it easily implementable in practice.

Our mathematical model captures the relationship between labile haemoglobin and blood glucose levels and is able to predict their temporal evolution in virtual patients. The underlying kinetics of the model has been validated using a very large database of over 11 000 patients (non-diabetic and diabetic). This has enabled us to characterize the key parameters and to reveal how labile haemoglobin varies when compared with blood glucose levels. Interestingly, this biomarker is not affected by minor alterations such as food intake or nervous states that could cause fluctuations in glucose during sample acquisition.

We have also shown that labile haemoglobin can detect episodes of hyperglycaemia and hypoglycaemia in patients during the preceding hours before sample acquisition, even when their glucose levels may have been restored to normal values. Our study gives evidence that measuring HbA_1d after a substantial increase/decrease of glycaemia has the potential to inform about changes in the average glucose and to detect such episodes during the subsequent 8 h after their irruption. This fact can be of vital importance to control diabetic patients and adjust their treatments. Nowadays, two to three months are needed to evaluate the response to treatment changes through HbA_1c. According to our proposal, using HbA_1d would be expected to significantly shorten the evaluation time and decision making in specific patients in a limited amount of days as, for example, in hospital admissions.

Finally, our model is interesting from a purely mathematical point of view; it consists of a system of linear non-autonomous ODEs where the glucose function drives the internal dynamics of the two haemoglobin forms. We have found the exact solutions in terms of quadratures, which should allow not only the use of qualitative but also quantitative approaches to identify new scenarios relevant from a clinical perspective, as those considered in the present work. Moreover, although we have briefly examined, via a Floquet theory, the existence of periodic solutions when the glucose function is itself a T-periodic function, further studies are still required in the more general case of a bounded and quasi-periodic glucose function.

In conclusion, we have put forward a simple mathematical model for the dynamics of the labile and glycated haemoglobin forms under the effect of time-varying blood glucose. Using a large patient database, we have identified the key kinetic constants of the model enabling us to show that the labile haemoglobin provides information of the averaged glucose in a timescale of about 2–8 h. That is, in an intermediate temporal window between the volatile fast-fluctuating blood glucose levels and the very slowly varying glycated haemoglobin, which currently constitute the golden standard for the diagnosis and management of diabetes in the clinic. We have substantiated that HbA_1d could be very helpful as a complementary biomarker of blood glucose concentration and HbA_1c. By using a widely available technology in the clinic at no extra cost, this is a reliable method of monitoring the rapid occurrence of glycaemic disorders that escape detection when using only blood glucose concentration and HbA_1c. Hence, this method offers a novel biomarker for both diagnosis and early treatment response follow-up.

Appendix A. Exact solutions of the model equations

The system (2.7) is a set of linear non-autonomous ODEs that can be reduced to quadratures. First, we formally integrate equation (2.7b) to yield

A 1

where h_S(0) is the initial value for the glycated haemoglobin. Let α_L= k_LF + k_LS + 1/τ_RBC. By plugging (A 1) into equation (2.7a) and differentiating with respect to t, we get the second order differential equation

A 2

Equation (A 2) can be rewritten in the equivalent form

A 3

By setting

equation (A 3) can be expressed as

A 4

Equation (A 4) has the formal solution

A 5

where

is the initial condition for f(t), with h_L(0) being the initial value for the labile haemoglobin. To find f(0), we make use of equation (2.7a) which gives f(0) = k_FL(1 − h_S(0)). Using this fact, together with (A 1), and substituting into (A 5), we get

A 6

whose solution is

A 7

We have thus obtained the exact (coupled) solutions, in terms of quadratures involving the glucose G(t), of h_S(t) and h_L(t), given by the formulae (A 1) and (A 7), respectively.

Appendix B. Periodic solutions for system (2.7)

In this appendix, we briefly study the existence of periodic solutions of the linear system (2.7) where G(t) is assumed to be a T-periodic function. By the Massera Theorem [37], equations (2.7) possess T-periodic solutions if and only if they have bounded solutions on the positive half line [0, +∞). Therefore, if we prove that a solution of system (2.7) is bounded, we will ensure the existence of a periodic solution.

Let us consider a fundamental matrix solution Φ for (2.7). Thus, the monodromy matrix is defined as

The eigenvalues of the monodromy matrix are called characteristic multipliers or Floquet multipliers λ of the system, and they verify that λ = e^μT. The complex number μ is called the Floquet exponent [38]. Then, if the characteristic multipliers λ of the homogeneous system associated with (2.7) all have modulus less than one, then the solution of the system is bounded. Moreover, if the characteristic multipliers all have modulus equal to one and if the algebraic multiplicity equals the geometric multiplicity of each characteristic multiplier with modulus one, then the solution is also bounded (see [39]). Thus, a solution of system (2.7) is bounded if

where m_a(λ) and m_g(λ) denote the algebraic and the geometric multiplicities, respectively.

As it was not possible to calculate in closed-form the characteristic multipliers associated to (2.7), we resorted to a numerical analysis of the monodromy matrix in order to obtain the eigenvalues of this matrix. The glucose function family G(t) = a − bcos(π/4t), with a ∈ [50, 200], b ∈ [0, 100], a + b ≤ 250 and a − b ≥ 50 was taken to cover the maximum number of biologically relevant cases. This yielded a family of periodic systems.

Our results showed that the corresponding eigenvalues λ₁ and λ₂ for the different monodromy matrices, upon variation of a and b in the ranges above-mentioned, have moduli less than or equal to one: |λ₁| < 0.88 and |λ₂| ≤ 1. Hence, we conclude that there exist solutions bounded and periodic for system (2.7) when the glucose function G(t) is a T-periodic function.

Ethics

This study was approved by the Research Ethics Commission in the province of Cádiz (Spain) and was conducted within the terms of the Helsinki Statement.

Data accessibility

This article has no additional data.

Authors' contributions

G.F.C., M.R.D. and V.M.P.-G. conceived and design the work. G.F.C. and V.M.P.-G. proposed the mathematical model. O.L.-T. and J.B.-B. performed the numerical calculations with contributions from G.F.C. and V.M.P.-G. J.E.-S. collected the clinical data. G.F.C., M.R.D., J.E.-S. and V.M.P.-G. analysed the clinical data. G.F.C., O.L.-T., J.B.-B. and V.M.P.-G. wrote the paper with contributions from M.R.D., A.M.-D. and J.E.-S. All authors read and approved the manuscript.

Competing interests

We declare we have no competing interests.

Funding

This work was supported by the Spanish Ministerio de Economía y Competitividad/FEDER MTM2015-71200-R.