A non-linear mixed-effects model to describe the effect of acarbose intake on postprandial glycaemia in a single rat

A non-linear mixed-effects model is proposed to assess the impact of acarbose over time on postprandial glycaemia in a single rat. The model is based on two compartments, one representing the entry of glucose in the blood and the other its exit. The rat was submitted to two treatments: ingestion of starch and ingestion of starch plus acarbose. The model showed great suitability, with inferences on the behavior of glucose levels in response to treatments and supplying a richer description than just the area under the curve. The marginal curves for the two treatments are similar during the first moments; however, after reaching the peak of glucose concentration, they progressively became separate due to acarbose treatment and reached the initial levels more quickly. The proposed model, albeit with a single sample unit, showed similar results to those with larger samples; in other words, acarbose significantly attenuates glycaemia after ingestion of starch.

Several researchers have been trying to understand better the behavior of the disease and the effects of new treatments.However, besides economic factors, the policy of committees for ethics on studies involving animals and humans, which generally recommend the use of smaller numbers in samples should be taken into account.This fact makes it hard to obtain sufficient data to reach statistically robust results.Hence, it is important to propose methods that would accommodate inherent characteristics of research in health and biology.

Material and methods
Current assay was performed with a single adult male Wistar rat.The experimental protocol was approved by the Committee for Ethics of the State University of Maringá, Maringá, Paraná State, Brazil, following international law on the protection of animals.The rat was maintained under constant temperature (22°C ± 1°C) with automatically controlled photoperiod (12h light/12h dark).
The rat received two treatments, or rather, one was given by gavage at 1 g kg -1 of soluble starch and the other contained the same quantity of starch plus 10 mg kg -1 acarbose.Soluble starch was obtained from Merck (Darmstadt, Germany) and acarbose (Glucobay ® ) from Bayer (São Paulo, Brazil).The glucose sensor device was obtained from Medtronic (São Paulo, Brazil).
One day prior to the experiment, the animal was made to fast for 12 h (8:00 p.m.-8:00 a.m.) to discard any interference of intestinal absorption of glucose.At 8:00 a.m., the rat received 1 g kg -1 soluble starch by gavage.During 65 minutes after this application, the glucose concentrations in the rat's blood were recorded at every five minutes (until 09:05 a.m.).Three hours after the administration of starch (at 11:00 a.m.), the rat received 10 mg kg -1 of acarbose by gavage.Immediately the rat received 1 g kg -1 of soluble starch by gavage and glucose levels were recorded every 5 minutes until 12:05 p.m.At 13:30 p.m., the animal was given free access to water and food, and at 20:00 p.m., the animal fasted for 12 h once more.The procedure was repeated for three consecutive days.
The sequence adopted for treatments guaranteed that no residual acarbose influenced the results.Moreover, although the rat is a nocturnal animal, the long period of fasting made it eat in the morning (no significant weight loss was registered during the three days of experiment).
Glucose was measured by a real-time continuous glucose monitoring system (RT-CGMS) technique (Woderer et al., 2007;Carrara et al., 2012;Tavoni et al., 2013).The RT-CGMS is a portable device that requires insertion of a glucose sensor in the animal's subcutaneous tissue.RT-CGMS evaluates glucose levels every 10 sec and the results obtained every 5 as the average sum of 30 glucose concentration rates were sent by radio to a computer for analysis.
Figure 1 shows data on postprandial glucose concentration over time, grouped by day and by treatment.One may observe that all profiles have a clear pattern, with the glucose level in the blood rising quickly at the start followed by a gradual decline.Although as a rule baseline and stabilization of glucose concentration depends on the individual condition, the pattern of glucose levels in current assay is related to the moment when the rat received the treatments.
In pharmacokinetic models, the human body is usually represented as a system of compartments, in which the drug is transferred according to a firstorder or zero-order kinetic equation (Gibaldi & Perrier, 1982).The drug's concentration in the different compartments and over time is determined by a system of differential equations whose solution may be expressed as a linear combination of exponential functions.Similarly, the model used in this study represents the changes in postprandial glucose over time as a process with two compartments: the first representing the absorption of glucose in the blood and the second its elimination.
The ordinary differential equation (ODE) for each compartment represents the variation of postprandial glucose as being proportional to the time since administration and to the amount of glucose at that instant.For example, in the absorption period, the glucose concentration in the blood monotonically increases with time, or rather, its rate of variation grows from zero point (when no absorption has yet occurred) and then declines until it returns to zero again (when the absorption period ends).Therefore, ODE may be written as where: G 1 is the glucose absorption function; t is time; k 1 is the constant of proportionality.The solution of this equation leads to where: c 1 and k 1 are constants that represent the intercept and the shape of G 1 respectively.The elimination process starts immediately after ingestion of the starch and lasts until the glucose level reaches normal rates.Analogously to the absorption period, the glucose elimination function is given by (3) The constants k 1 and k 2 correspond respectively to absorption and elimination rates.
Unlike pharmacokinetic models, the final model must also have an intercept, since the human body tries to maintain the glucose level fluctuating around a constant rate.Thus, denoting the glucose concentration for profile i at time t j , with i = 1, …, 6 and j = 1, …, 14, by G ij , the final model is a linear combination of Equations 2 and 3 ) , 0, 0 in which with the fixed effects β representing the mean values of the parameters Ф i , and the random effects ( ) Ψ representing the deviations of β, considered to be independent among the profiles.The treatment effect is specified in the model by the parameters y, with x i = 0 if the treatment is starch alone and x i = 1 if the treatment is starch with acarbose.The errors 2 (0, ) ij N ε σ are considered to be independent of the random effects and for the different i and j rates.
Since the parameters Ф 2 and Ф 4 must be negative to make biological sense, we re-parameterized the model in terms of  & Bates, 2000).Hence, the model does not have any restrictions with regard to the parameters.

Results and discussion
Since the number of profiles was very near the number of random effects in the model, we were unable to use a positive defined matrix with all the possible covariances (Pinheiro & Bates, 2000).Therefore, we initially used a diagonal matrix with all the parameters to specify the structure of the covariances of the random effects, Ψ. Analyzing the estimates of the random effects with respect to the treatments, we observed a possible systematic pattern of the parameter Ф 0 .After fitting the complete model and various reduced models, we chose the model with only 0 γ , 3 γ and 4 γ by calculating AIC (Akaike, 1974) and BIC (Schwarz, 1978) rates and applying the likelihood ratio test.Employing this model, the estimated standard deviation of the random effect for Ф 1 was nearly zero (the parameter Ф 0 accommodated all the variability of the intercept of the first exponential equation).After testing some structures for the random effects, we chose the diagonal matrix without effect for Ф 1 .The estimated rates and 95% confidence intervals for the fixed effects and for the standard deviations of the random effects are reported in Table 1.Recall that Figure 3 shows a quantile-quantile (Q-Q) graph for the assumption of normal distribution of the residuals.The linearity of the points suggests no serious violation of this assumption.
Another evaluation of the model's adequacy is provided by comparing the individual profiles (observed rates) and the conditional profiles (obtained when the estimates of the random effects are used) and marginal profiles (corresponding to the fixed effects), as presented in Figure 4.Note that the conditional predictions are near the observed concentrations, indicating that the model provides a good representation of the data.The area under the curve (AUC) is a common measure to compare glucose curves.To calculate AUC, we integrated the marginal models estimated for the two treatments in the interval between 0 and 65 min, and subtracted from this rate the area below 70 mg dl -1 (none of the experimental data was below this cutoff).AUC for the treatment with starch alone was 1.877 mg dl -1 min.while that for the treatment plus acarbose was 1.330 mg dl -1 min., approximately 29% smaller.
Another comparative method is to calculate the maximum estimated glucose concentration and report the time the rate is reached; also to find the levels and times of the first and second inflection points, which represent the maximum absorption and elimination, respectively.For the starch curve, the maximum concentration was 108.9 mg dl -1 and the time was 29 min; in the case of the acarbose curve, the concentration was lower, approximately 101.5 mg dl -1 , at a shorter time, at 25 min.As expected, the maximum absorption of acarbose and starch occurred at similar times (around 12 minutes), but the maximum elimination of starch and acarbose occurred at 41 and 38 min.respectively.
Figure 5 presents the fitted marginal model G, the two exponential functions that compose it (G 1 and G 2 ) and its rate of variation (G ' ) for the two treatments.The behavior of the curve that represents glucose absorption by the blood (G 1 ) is equal for the two treatments, increasing from negative values and asymptotically approaching zero.However, the elimination process is substantially different.Thus, the variation rate of glucose concentration changes over the entire period between the treatments, observed in the area under the curve of G ' .The positive area is greater for starch, implying a higher glucose concentration in the blood, while the negative area is greater for acarbose, implying that the glucose concentration declined more quickly.
The process of absorption and elimination of glucose in the blood is dynamic.The human body maintains the homeostasis of glucose levels in the blood using insulin and glucagon.Even during long fasting periods, the glucose levels do not decline drastically and glucose absorption and elimination rates in the blood are kept relatively stable.However, after eating food rich in carbohydrates, the alteration of the absorption and elimination rates raises the level of glucose in the blood.When the absorption process ends, the elimination persists longer until the glucose concentration reaches its reference value again.In this study, we estimated the reference values at 94.74 and 78.74 mg dl -1 for the treatment with starch and acarbose, respectively.The 65-minute duration was only sufficient for the glucose level to return to rates close to the initial ones in the case of acarbose.

Conclusion
The use of animals for scientific purposes has many advantages.However, due to internal pressures on the scientific community to optimize resources and to external pressures from animal protection groups, the number of animals for research should be minimized.Hence, the need to work with few samples in health and biological sciences is growing, prompting statisticians to improve their methods.In current study, with only one experimental unit (a single rat), it was possible to obtain results similar to those of other studies which reported the effect of acarbose on glycaemia carried out with larger samples (Coniff et al., 1995;Pereira et al., 2011;Ritz et al., 2012;Rosak et al., 2002;Scheen et al., 1994;Sybuia et al., 2014Sybuia et al., -2015;;Wong & Jenkins, 2007;Yee & Fong, 1996).
Further, the modified two-compartment model could be applied to a variety of metabolic processes in which the same pattern is observed.The model seems ideal to describe phenomena that may be represented by the entry and the exit of a substance from a homeostatic system (a system with dynamic equilibrium).

Figure 1 .
Figure 1.Glycemic curves after oral administration of starch and soluble starch plus acarbose after three days of experiment in a single rat fasted for 12 hours.Glucose levels were recorded every five minutes with the use of RT-CGMS.
diagnostic graphs in Figure 2 (of the standardized residuals versus the estimated values and the observed values versus the estimated values) do not indicate large deviations from the proposed nonlinear model.

Figure 2 .Figure 3 .
Figure 2. Diagnostic graphs.The left graph plots the standardized residuals versus the fitted rates, while the right panel shows the observed rates versus the fitted ones (the straight line represents a perfect fit).

Figure 4 .
Figure 4. Scatter plot for the observed glucose levels after oral administration of starch (left panel) and soluble starch plus acarbose (right panel), together with the conditional (thin lines) and marginal (thick lines) profiles.Each kind of point or line represents a different profile.

Figure 5 .
Figure 5. Fitted marginal model (G), the two exponential functions that compose it (G 1 and G 2 ) and its rate of variation (G ' ) for the treatment with starch (left panel) and the treatment with soluble starch plus acarbose (right panel).

Table 1 .
Estimates, lower and upper bounds (LB and UB) for the model's parameters.