A non-linear mixed-effects model to describe the effect of acarbose intake on postprandial glycaemia in a single rat/Um modelo nao linear de efeitos mistos para descrever o efeito da ingestao de acarbose na glicemia pos-prandial em um unico rato.
The metabolic disease diabetes mellitus is among the ten mortality causes in populations worldwide. The high cost of the disease demands that public and private health systems investigate new treatments, intervention programs (Ejtahed et al., 2012; Konig, Kookhan, Schaffner, Deibert, & Berg, 2014; Shen, Obin, & Zhao., 2013; Costa & Longo, 2014) and drugs capable to improve the patients' quality of life. In living beings, the a-amylases are enzymes that catalyze the hydrolysis of polysaccharides, such as starch and glycogen, to yield glucose and maltose (Wang et al., 2008). However, some organic compounds, among them acarbose, may inhibit the activity of these enzymes (Geng & Bai, 2008; Ritz et al., 2012) and thus prevent or attenuate hyperglycemic peaks (Coniff, Shapiro, Seaton, & Bray, 1995; Pereira et al., 2011; Scheen, Magalhaes, Salvatore, & Lefebvre, 1994; Sybuia, Guilhermetti, Mangolim, Bazotte, & Matioli, 2014-2015; Wong & Jenkins, 2007). Acarbose has been widely studied for the treatment of Type 2 diabetes (Ritz et al., 2012; Rosak, Haupt, Walter, & Werner, 2002; Yee & Fong, 1996) as a therapeutic agent added to food or as a drug administered orally (Espin, Garcia-Conesa, & Tomas-Barberan, 2007; Eng Kiat Loo & Huang, 2007).
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.
Non-linear models have been proposed in the literature to obtain a better understanding of diabetes (Ajmera, Swat, Laibe, Le Novere, & Chelliah, 2013; Boutayeb & Chetouani, 2006; Briegel & Tresp, 2002; Caumo, Saccomani, Toffolo, Sparacino, & Cobelli, 1999; Hovorka et al., 2004; Wu, 2005). Nevertheless, many studies with longitudinal data on glycemia only calculate the area under the curve (AUC) to compare treatments (Sybuia et al., 2014-2015; Tai, 1994). In current research, a non-linear mixed-effects model is investigated to determine the impact of acarbose on postprandial glycemia in a single rat after the ingestion of soluble starch. The analysis was carried out with R Statistical Software (R Core Team, 2015) using NLME package (Pinheiro, Bates, Debroy & Sarkar, 2007).
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 Maringa, Maringa, Parana State, Brazil, following international law on the protection of animals. The rat was maintained under constant temperature (22[degrees]C [+ or -] 1[degrees]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.sup.-1] of soluble starch and the other contained the same quantity of starch plus 10 mg [kg.sup.-1] acarbose. Soluble starch was obtained from Merck (Darmstadt, Germany) and acarbose (Glucobay[R]) from Bayer (Sao Paulo, Brazil). The glucose sensor device was obtained from Medtronic (Sao 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.sup.-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.sup.-1] of acarbose by gavage. Immediately the rat received 1 g [kg.sup.-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 first-order 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
[dG.sub.1]/dt = [k.sub.1] t [G.sub.1] (1)
[G.sub.1] is the glucose absorption function; t is time; [k.sub.1] is the constant of proportionality. The solution of this equation leads to
[G.sub.1](t) = [c.sub.1]exp([k.sub.1][t.sup.2]) (2)
[c.sub.1] and [k.sub.1] are constants that represent the intercept and the shape of [G.sub.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
[G.sub.2] (t) = [c.sub.2]exp([k.sub.2][t.sup.2]). (3)
The constants [k.sub.1] and [k.sub.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.sub.j], with i = 1, ..., 6 and j = 1, ..., 14, by [G.sub.ij], the final model is a linear combination of Equations 2 and 3
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)
with the fixed effects [theta] representing the mean values of the parameters [[PHI].sub.i], and the random effects [b.sub.i] [??] N (0, [PSI]) representing the deviations of [beta], considered to be independent among the profiles. The treatment effect is specified in the model by the parameters y, with [x.sub.i] = 0 if the treatment is starch alone and [x.sub.i] = 1 if the treatment is starch with acarbose. The errors [[epsilon].sub.ij] [??] N(0, [[sigma].sup.2]) are considered to be independent of the random effects and for the different i and j rates.
Since the parameters [[PHI].sub.2] and [[PHI].sub.4] must be negative to make biological sense, we re-parameterized the model in terms of [[PHI]'.sub.2]= log (-[[PHI].sub.2]) and [[PHI]'.sub.2] = log (-[[PHI].sub.2]) (Pinheiro & 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, [PSI].
Analyzing the estimates of the random effects with respect to the treatments, we observed a possible systematic pattern of the parameter [[PHI].sub.0]. After fitting the complete model and various reduced models, we chose the model with only [[gamma].sub.0], [[gamma].sub.3] and [[gamma].sub.4] 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 [[PHI].sub.1] was nearly zero (the parameter [[PHI].sub.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 [[PHI].sub.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
[[beta].sub.2] = -exP([[beta]'.sub.0]) and [[PHI].sub.4] = -exP([[PHI]'.sub.4]).
The first two 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 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.sup.-1] (none of the experimental data was below this cutoff). AUC for the treatment with starch alone was 1.877 mg [dl.sup.-1] min. while that for the treatment plus acarbose was 1.330 mg [dl.sup.-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.sup.-1] and the time was 29 min; in the case of the acarbose curve, the concentration was lower, approximately 101.5 mg [dl.sup.-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.sub.1] and [G.sub.2]) and its rate of variation (G) for the two treatments. The behavior of the curve that represents glucose absorption by the blood ([G.sub.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.sup.-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.
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., 2014-2015; 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).
The authors gratefully acknowledge financial support from the Brazilian Coordination for the Upgrading of Higher Education Personnel (CAPES).
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Received on April 16, 2015.
Accepted on July 7, 2015.
Omar Cleo Neves Pereira (1) *, Paulo Vitor da Costa Pereira (1), Mauricio Fumio Sybuia (2), Emerson Barili (1), Rosangela Santana (1) and Isolde Previdelli (1)
(1) Departamento de Estatistica, Universidade Estadual de Maringa, Av. Colombo, 5790, 87020-900, Maringa, Parana, Brazil, (2) Departamento de Farmacia, Universidade Estadual de Maringa, Parana, Brazil.
* Author for correspondence. E-mail: email@example.com
Caption: 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.
Caption: 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).
Caption: Figure 3. Normal Q-Q plot for the residuals (the linearity of the points indicate a good fit to the normal distribution).
Caption: 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.
Caption: Figure 5. Fitted marginal model (G), the two exponential functions that compose it ([G.sub.1] and [G.sub.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. LB Estimate UB [[beta].sub.0] 84.6104 94.7356 104.8608 [[beta].sub.1] -65.5407 -56.9421 -48.3436 [[beta]'.sub.2] -6.4698 -5.9670 -5.4642 [[beta].sub.3] 31.7150 40.9461 50.1772 [[beta]'.sub.4] -7.4754 -7.1238 -6.7721 [[gamma].sub.0] -30.3190 -15.9917 -1.6644 [[gamma].sub.3] 11.4104 17.6471 23.8838 [[gamma].sub.4] -0.3817 0.0645 0.5108 [MATHEMATICAL 4.7892 8.6508 15.6258 EXPRESSION NOT REPRODUCIBLE IN ASCII] [MATHEMATICAL 0.3324 0.6027 1.0929 EXPRESSION NOT REPRODUCIBLE IN ASCII] [MATHEMATICAL 1.5476 3.2901 6.9946 EXPRESSION NOT REPRODUCIBLE IN ASCII] [MATHEMATICAL 0.1115 0.2451 0.5390 EXPRESSION NOT REPRODUCIBLE IN ASCII] [sigma] 1.1269 1.3660 1.6558