# ABSTRACT:

This study examined the use of the Gompertz, Groot, monomolecular, Richards and two-compartment-logistic mathematical models to investigate the kinetics of in vitro gas production of diets composed of combinations of Gliricidia hay or silage. In addition, the effects of Gliricidia hay or silage inclusion on the in vitro cumulative gas production of these diets were evaluated. Rumen fermentation kinetics were analyzed by the in vitro cumulative gas production methodology. The model parameters were estimated using the Gauss Newton method, with the exception of the Richards model, which was used by Marquardt’s algorithm. Model fit was assessed using the determination coefficient, F test for parameters identity, concordance correlation coefficient, root mean square error of prediction, and decomposition of mean square error of prediction into mean error, systematic bias and random error. The models were compared for accuracy (pairwise mean square error of prediction) and precision (delta Akaike’s information criterion). All model evaluation and comparison statistics were calculated using Model Evaluation System software version 3.2.2. The Groot and Richards models did not differ from each other (P>0.05) and were the most precise and accurate (P<0.05). Therefore, the Groot model was selected due to its better accuracy and precision and easier access to the parameters. The inclusion of Gliricidia silage in the diet resulted in an increase in the time to obtain the maximum rate of degradation and in the time after incubation when half of the asymptotic level was reached. The Groot model is recommended to estimate the average curve. Dietary inclusion of Gliricidia silage alters the gas production curve due to the longer time required for the diet to reach the maximum rate of degradation, this can increase the time the diet remains in rumen and promote a reduction in the consumption.

Key words:
fermentation kinetics; Groot model; hay; non-linear functions; silage

# RESUMO:

Palavras-chave:
cinética de fermentação; feno; funções não-lineares; modelo de Groot; silagem

# INTRODUCTION:

In tropical regions, forage plants change their growth pattern according to seasonal environmental scenarios, creating periods of high and low forage availability (SBRISSIA et al., 2020SBRISSIA, A. F. et al. Unravelling the relationship between a seasonal environment and the dynamics of forage growth in grazed swards. Journal Agronomy and Crop Science, v.206, n.5, p. 630-639, 2020. Available from: <Available from: https://doi.org/10.1111/jac.12402 >. Accessed: Nov. 10, 2020. doi: 10.1111/jac.12402.
https://doi.org/10.1111/jac.12402...
). Thus, rational use of forage resources when they are more widely available allows for better dietary planning, in addition to lessening the impacts of climatic fluctuations throughout the year. The preservation of surplus forage in the form of hay and/or silage increases the efficiency of use of the produced forage (HASELMANN et al., 2020HASELMANN, A. et al. Comparing the effects of silage and hay from similar parent grass forages on organic dairy cows’ feeding behavior, feed intake and performance. Animal Feed Science and Technology , v.267, n.1 p.114560, 2020. Available from: <Available from: https://doi.org/10.1016/j.anifeedsci.2020.114560 >. Accessed: Oct. 10, 2020. doi: 10.1016/j.anifeedsci.2020.114560.
https://doi.org/10.1016/j.anifeedsci.202...
).

Gliricidia (Gliricidia sepium) is a leguminous tree with great potential for use in livestock due to its high protein value (MARTINELE et al., 2014MARTINELE, I. et al. Abundance and diversity of rumen protozoa in lambs fed Gliricidia sepium silage. Revista Brasileira de Zootecnia , v.43, n.8, p.436-439, 2014. Available from: <Available from: http://dx.doi.org/10.1590/S1516-35982014000800006 >. Accessed: May. 15, 2020. doi: 10.1590/S1516-35982014000800006.
http://dx.doi.org/10.1590/S1516-35982014...
; SANTANA et al., 2019SANTANA, J. C. S. et al. Fermentation characteristics, chemical composition and protein fractioning of Gliricídia silage submitted to different fermentation periods. Boletim de Indústria Animal , 76, n.1, p.1-9, 2019. Available from: <Available from: https://doi.org/10.17523/bia.2019.v76.e1436 >. Accessed: Jul. 05, 2020. doi: 10.17523/bia.2019.v76.e1436.
https://doi.org/10.17523/bia.2019.v76.e1...
). Providing this material in preserved form results in production indices similar to those achieved with supplementation with soy (FERNANDES et al., 2020FERNANDES, L. S. et al. Performance of sheep grazing Panicum maximum cv. Massai and supplemented with protein sources during the dry season. South African Journal of Animal Science, v.50, n.1, p.1-8, 2020. Available from: <Available from: http://dx.doi.org/10.4314/sajas.v50i1.1 >. Accessed: May. 21, 2020. doi: 10.4314/sajas.v50i1.1.
http://dx.doi.org/10.4314/sajas.v50i1.1...
). Evaluating this ingredient in ruminant diets is an important step to elucidate its benefits in animal feeding as well as reduce the costs of production systems (SANTANA et al., 2020), since protein concentrates are one of the costliest items of the animal diet.

The in vitro cumulative gas production technique is widely used for gravimetric and metabolic assessments of feed stuffs (VELHO et al., 2014VELHO, J. P. et al. Mathematical models for adjustment of in vitro gas production at different incubation times and kinetics of corn silages. Semina: Ciências Agrárias , v.35, n.4, p.2531-2540, 2014. Available from: <Available from: https://doi.org/10.5433/1679-0359.2014v35n4Supl1p253 >. Accessed: Sep. 30, 2019. doi: 10.5433/1679-0359.2014v35n4Supl1p253.
https://doi.org/10.5433/1679-0359.2014v3...
; SANTANA et al., 2020SANTANA, J. C. S. et al. In vitro digestion characteristics of various combinations of elephant grass hay, Gliricidia hay or silage, soybean meal and corn meal in rations for sheep. Tropical Grasslands-Forrajes Tropicales. 8, n.2, p.147-152, 2020. Available from: <Available from: http://dx.doi.org/10.17138/TGFT(8)147-152 >. Accessed: Jul. 05, 2020. doi: 10.17138/TGFT(8)147-152.
http://dx.doi.org/10.17138/TGFT(8)147-15...
). However, it is important to use the most suitable mathematical model to obtain the fermentation parameters and; consequently, the fitting of gas production curves, which can vary depending on the model used (SANTOS et al., 2019SANTOS, A. L. P. et al. Generation of models from existing models composition: An application to agrarian sciences. Plos One, v.14, n.1, p.e0214778, 2019. Available from: <Available from: https://doi.org/10.1371/journal.pone.0214778 >. Accessed: Jul. 10, 2020. doi: 10.1371/journal.pone.0214778.
https://doi.org/10.1371/journal.pone.021...
). Some researchers have used nonlinear models to study the kinetics of in vitro gas production (FARIAS et al., 2011FARIAS, L. N., V. R. et al. Avaliation of bicompartimental logistic and Gompertz mathematical models to estimate gas production from babassu (Orbignya martiana) meal and pie using the semi-automated in vitro technique. Arquivo Brasileiro de Medicina Veterinária e Zootecnia, v.63, n.1, p.136-142, 2011. Available from: <Available from: https://doi.org/10.1590/S0102-09352011000100021 >. Accessed: Oct. 10, 2020. doi: 10.1590/S0102-09352011000100021.
https://doi.org/10.1590/S0102-0935201100...
; VELHO et al., 2014; TEIXEIRA et al., 2016TEIXEIRA, U. H. G. et al. Mathematical models for estimating the parameters of ruminal degradation kinetics of protein concentrates. Revista Brasileira de Saúde e Produção Animal , v.17, n.1, p.73-85, 2016. Available from: <Available from: http://dx.doi.org/10.1590/S1519-99402016000100008 >. Accessed: Nov. 09, 2020. doi: 10.1590/S1519-99402016000100008.
http://dx.doi.org/10.1590/S1519-99402016...
; GOMES et al., 2017GOMES, M. F. F. et al. In vitro fermentation characteristics of ruminant diets using ethanol extract of brown propolis as a nutritional additive. Revista Brasileira de Zootecnia , v.46, n. 7, p.599-605, 2017. Available from: <Available from: http://dx.doi.org/10.1590/S1806-92902017000700007 >. Accessed: May. 25, 2020. doi: 10.1590/S1806-92902017000700007.
http://dx.doi.org/10.1590/S1806-92902017...
), especially the Gompertz, Richards, Groot and monomolecular models. However, the two-compartment logistic model, proposed by PELL & SCHOFIELD (1993PELL, A.; SCHOFIELD, P. Computerized monitoring of gas production to measure forage digestion in vitro. Journal of Dairy Science, v.76, n.4, p.1063-1073, 1993. Available from: <Available from: https://doi.org/10.3168/jds.S0022-0302(93)77435-4 >. Accessed: Oct. 13, 2020. doi: 10.3168/jds.S0022-0302(93)77435-4.
https://doi.org/10.3168/jds.S0022-0302(9...
), has been widely used to estimate the in vitro gas production of feedstuffs and diets for ruminants (OLIVO et al., 2017OLIVO, P. M. et al. Assessing the nutritional value of agroindustrial co-products and feed through chemical composition, in vitro digestibility, and gas production technique. Acta Scientiarum. Animal Sciences, v.39, n.3, p.289-295, 2017. Available from: <Available from: https://doi.org/10.4025/actascianimsci.v39i3.34024 >. Accessed: Oct. 15, 2020. doi: 10.4025/actascianimsci.v39i3.34024.
https://doi.org/10.4025/actascianimsci.v...
; DIAZ et al., 2018DÍAZ, T. G. et al. In vitro gas production kinetics and digestibility in ruminant diets with diferente levels of cashew nut shell liquid. Semina: Ciências Agrárias, v.39, n.4, p.1669-1682, 2018. Available from: <Available from: https://doi.org/10.5433/1679-0359.2018v39n4p1669 >. Accessed: Apr. 25, 2020. doi: 10.5433/1679-0359.2018v39n4p1669.
https://doi.org/10.5433/1679-0359.2018v3...
; SOUZA et al., 2018SOUZA, A. D. V. et al. Thermal decomposition, chemical composition, in vitro digestibility and gas production and in situ degradability of oilseed residues from the biofuel industry. Animal Science Journal, v.89, n.1, p.79-87, 2018. Available from: <Available from: https://doi.org/10.1111/asj.12889 >. Accessed: Sep. 09, 2020. doi: 10.1111/asj.12889.
https://doi.org/10.1111/asj.12889...
; OLIVEIRA et al., 2017OLIVEIRA, V. S. et al. Ruminal kinetics of tropical forrages submitted or not to irrigation. Boletim de Indústria Animal, v.74, n.3, p.195-204, 2017. Available from: <Available from: hhttps://doi.org/10.17523/bia.v74n3p195 >. Accessed: Apr. 10, 2020. doi:10.17523/bia.v74n3p195.
hhttps://doi.org/10.17523/bia.v74n3p195...
; LEAL et al., 2020LEAL, E. S. et al. Inﬂuence of protodioscin content on digestibility and in vitro degradation kinetics in Urochloa brizantha cultivars. Crop and Pasture Science, v.71, n.3, p.278-284, 2020. Available from: <Available from: https://doi.org/10.1071/CP18357 >. Accessed: May. 25, 2020. doi: 10.1071/CP18357.
https://doi.org/10.1071/CP18357...
, SANTANA et al., 2020). Nevertheless, VELHO et al. (2014) warned that a single model should not be used indiscriminately for all types of feed; rather, it is essential that different models be evaluated for each experimental situation.

Incorporating Gliricidia as hay or silage in the traditional way sheep feed based on elephant grass hay, soybean meal and cornmeal can allow a reduction in the quantities of soybean meal in the feed. Added to this, reducing the proportion of expensive soy bran should reduce the cost. Thus, in vitro studies, allow the elaboration of hypotheses about the possible responses of the animals when submitted to these diets.

On this basis, the present study examined the use of the Gompertz, Groot, monomolecular, Richards and two-compartment-logistic mathematical models to elucidate the kinetics of in vitro gas production of diets composed of varied combinations of Gliricidia hay or silage. In addition, the effects of including Gliricidia in the form of hay or silage on the in vitro gas production curve of these diets were evaluated.

# MATERIALS AND METHODS:

Data of three diets (Table 1) formulated for sheep with an estimated weight gain of 200 g/day and dry matter intake estimated at 3.5% live weight, according to the NRC (2007)NRC. Nutrient requirements of small ruminants. 2ª ed. Washinton: DC. National Academy of Sciences, 2007. 362p., were used. Laboratory analyses were performed at the Laboratories of Animal Nutrition and Rumen Fermentation at the Department of Animal Science (DZO) of the Federal University of Sergipe (UFS), located in Aracaju - SE, Brazil. Methodological details of the making of Gliricidia hay and silage can be reported in SANTANA et al. (2020SANTANA, J. C. S. et al. In vitro digestion characteristics of various combinations of elephant grass hay, Gliricidia hay or silage, soybean meal and corn meal in rations for sheep. Tropical Grasslands-Forrajes Tropicales. 8, n.2, p.147-152, 2020. Available from: <Available from: http://dx.doi.org/10.17138/TGFT(8)147-152 >. Accessed: Jul. 05, 2020. doi: 10.17138/TGFT(8)147-152.
http://dx.doi.org/10.17138/TGFT(8)147-15...
).

Table 1
Ingredient and nutritional compositions of the experimental diets.

Three different races on three different days were run. For each diet, five (repetitions) samples were incubated using rumen fluid collected from three hair sheep provided with ruminal fistula (THEODOROU et al., 1994THEODOROU, M. K. et al. A simple gas production method using a pressure transducer to determine the fermentation kinetics of ruminant feeds. Animal Feed Science and Technology , v.48, v.3-4, p.185-197, 1994. Available from: <Available from: https://doi.org/10.1016/0377-8401(94)90171-6 >. Accessed: Sep. 24, 2020. doi: 10.1016/0377-8401(94)90171-6.
https://doi.org/10.1016/0377-8401(94)901...
). The samples were incubated in glass flasks with a total capacity of 200 mL. The 160-mL incubation solution consisted of 80% (128 mL) buffer-mineral solution and 20% (32 mL) rumen inoculum inserted manually with a graduated syringe into glass vials previously washed and dried in a forced-air oven, in which a 1 g sample of the diets (Table 1) was added.

In addition to the 15 vials containing samples of the diets, another four vials were filled only the inoculum and buffer solution to measure possible pressures that were not related to the diets. The incubation solution was prepared as described by THEODOROU et al. (1994THEODOROU, M. K. et al. A simple gas production method using a pressure transducer to determine the fermentation kinetics of ruminant feeds. Animal Feed Science and Technology , v.48, v.3-4, p.185-197, 1994. Available from: <Available from: https://doi.org/10.1016/0377-8401(94)90171-6 >. Accessed: Sep. 24, 2020. doi: 10.1016/0377-8401(94)90171-6.
https://doi.org/10.1016/0377-8401(94)901...
), using cysteine-HCl as a reducing agent (MOULD et al., 2005MOULD, F. L. et al. 2005. A review and simplification of the in vitro incubation medium. Animal Feed Science and Technology , v.123-124, n.1 p.55-172, 2005. Available from: <Available from: https://doi.org/10.1016/j.anifeedsci.2005.05.002 >. Accessed: Jan. 05, 2020. doi: 10.1016/j.anifeedsci.2005.05.002.
https://doi.org/10.1016/j.anifeedsci.200...
).

After the incubation solution and samples were added, CO2 was manually injected for seven seconds into each flask, which was then closed and placed in a forced-air oven at 39 °C during the 48 h of incubation. Data were recorded automatically by the ANKOM RF Gas Production System. The amount of gas produced (mL/100 mg DM) at each time (1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 24, 36 and 48 hours after incubation) was corrected by subtracting the value of gas produced in the un sampled vials of the total gas obtained in the vials containing the samples. Pressure values were corrected for volume using the following equation: y = - 0.772 + 6.087x - 0.382x2 (R2 = 0.95), where: y is the final gas volume in mL; and x is the gas pressure in kilopascal at the respective times (OLIVEIRA et al., 2020OLIVEIRA, V. S. et al. Regression equation to estimate volume of ruminal gases and correlation between chemical composition and fermentation parameters. Diversitas Journal, v.5, n, 4, p. 3238-3249, 2020. Available from: <Available from: hhttps://doi.org/10.17648/diversitas-journal-v5i4-1205 >. Accessed: Apr. Jan. 14, 2020. doi: 10.17648/diversitas-journal-v5i4-1205.
hhttps://doi.org/10.17648/diversitas-jou...
).

The cumulative gas production generated in the 48 h was subjected to five mathematical models (Table 2). The parameters of the different functions can be interpreted biologically as follows: P(t) is the cumulative production at time t. Parameter “A” is the asymptotic gas production (mL of gas/100 mg DM). In the Gompertz and Richards models, parameter “B” represents the time of colonization (h) of the particle (lag); in the Groot model, it is the time after incubation at which half of the asymptotic level was reached (h); and in the monomolecular model, it is the specific gas production rate (mL of gas/h). In the Gompertz and Richards models, parameter “k” is the specific gas production rate (mL of gas/h); in the Groot model, it is an integration constant that determines the sharpness of the curve. Lastly, parameter “m” defines the inflection point of the curve for the Richards model.

Table 2
Models considered in this study to describe the in vitro gas production curve of diets composed of different combinations of Gliricidia hay or silage.

In the two-compartment logistic model, “vNFC” and “vFC” represent the volume of gas produced from the degradation of non-fibrous and fibrous carbohydrates, respectively; “kdNFC” and “kdFC” are the respective degradation rates of non-fibrous and fibrous carbohydrates; and “L” is the time of colonization (h) of the particle (lag).

The parameters of the Gompertz, Groot, monomolecular and two-compartment logistic models were estimated using the modified Gauss Newton method, by the NLIN procedure of SAS software (SAS University Edition (version 12), Sas Institute Inc. Cary, CA, USA). The maximum number of iterations used was 100. Because of the difficulty in fitting the Richards model by the Gauss Newton method, due to the non-convergence of the iterative process, the Marquardt algorithm was used for fitting iterations, adopting 200 iterations as a maximum number.

The following criteria were used to evaluate the models: determination coefficient (R2) and F test for the identity of the parameters (β0 = 0 and β1 = 1) of the regression of predicted on observed data; concordance correlation coefficient (CCC); root mean square error of prediction (RMSEP); and decomposition of the mean square error of prediction (MSEP) into mean error, systematic bias and random error. The models were compared as to their accuracy by pairwise mean square error of prediction (pMSEP) analysis and for precision by the delta Akaike information criterion (AIC) (TEDESCHI, 2006TEDESCHI, L.O. Assessment of the adequacy of mathematical models. Agricultural Systems, v.89, n.2-3, p.225-247, 2006. Available from: <Available from: https://doi.org/10.1016/j.agsy.2005.11.004 >. Accessed: Sep. 15, 2019. doi: 10.1016/j.agsy.2005.11.004.
https://doi.org/10.1016/j.agsy.2005.11.0...
).

Statistical calculations for the evaluation and comparison of the models were performed using Model Evaluation System version 3.2.2 (http://nutritionmodels.tamu.edu/mes.htm, College Station, Tx, USA; TEDESCHI, 2006TEDESCHI, L.O. Assessment of the adequacy of mathematical models. Agricultural Systems, v.89, n.2-3, p.225-247, 2006. Available from: <Available from: https://doi.org/10.1016/j.agsy.2005.11.004 >. Accessed: Sep. 15, 2019. doi: 10.1016/j.agsy.2005.11.004.
https://doi.org/10.1016/j.agsy.2005.11.0...
). Once the model that best described the average in vitro gas production curve of diets was chosen, the effect of diets on gas production kinetics was evaluated using a dummy variable, which consisted of creating binary variables (0 or 1) to represent and compare experimental treatments, and when one of the treatments received 1, the others received 0 (REGAZZI, 2003REGAZZI, A. J. Test for parameter equality in nonlinear regression models. Revista Ceres, v.50, n.287, p.9-26, 2003. Available from: <Available from: http://www.ceres.ufv.br/ojs/index.php/ceres/article/view/2854/710 >. Accessed: Nov. 10, 2020.
http://www.ceres.ufv.br/ojs/index.php/ce...
).

The variables obtained from the parameters of the chosen model were calculated as described in GROOT et al. (1996GROOT, J. C. J. et al. Multiphasic analysis of gas production kinetics for in vitro fermentation of ruminant feeds. Animal Feed Science and Technology, v.64, n.1, p.77-89, 1996. Available from: <Available from: https://doi.org/10.1016/S0377-8401(96)01012-7 >. Accessed: May. 30, 2020. doi: 10.1016/S0377-8401(96)01012-7.
https://doi.org/10.1016/S0377-8401(96)01...
), namely, Ti (h): time of inflection of the curve; Tmax (h): time at which the rate of degradation is maximum; and Rmax (mL of gas/h): maximum fractional rate of substrate degradation. The Ti, Tmax and Rmax variables were subjected to analysis of variance by the PROC GLM command and means were compared using Tukey’s test in the SAS statistical package (SAS University Edition (version 12), Sas Institute Inc. Cary, CA, USA). The significance level was set at 5% for all statistical analyses.

# RESULTS:

All models showed average cumulative gas production estimates and standard deviation close to the observed data as well as high determination coefficients (above 95%) of the regression of predicted on observed data (Table 3). The Gompertz, Groot, Richards and two-compartment logistics models generated predictions similar (P>0.05) to the observed data (β0 = 0 and β1 = 1), whereas the monomolecular model generated predictions different from the data (P<0.05).

Table 3
Evaluation of the adequacy of models for estimating the in vitro gas production of diets composed of varied combinations of Gliricidia hay or silage.

According to CCC analysis, all models were accurate and precise, as the CCC was greater than 0.95 for all of them, the closer to one the better. The RMSEP analysis revealed that the monomolecular model has a lower power to predict the exact gas production value, with a RMSEP of 2.24 mL of gas/100 mg of DM, whereas the other models showed average RMSEP of almost half of this value.

The decomposition of the MSEP showed that the Gompertz, Groot, Richards and two-compartment logistic models had more than 97% of their deviations attributed to random errors, which does not indicate any mean or systematic deficiency of these models. The monomolecular model, conversely, showed about 23% of the deviations associated with a systematic bias, that is, a multiplicative error in the predicted values. The cumulative gas production curves of the diets as estimated by each model are shown in Figure 1.

Figure 1
Cumulative production curves of control diet gases (elephant grass hay + soybean meal + corn meal), gliricidia hay treatment (elephant grass hay + soybean meal + corn meal + gliricidia hay) and treatment gliricidia silage (elephant grass hay + soybean meal + cornmeal + gliricidia silage), projected from the parameters estimated by each model.

In the comparison between the models regarding accuracy and precision, the Groot and Richards models did not differ from each other and were more precise (P<0.05) and accurate (P<0.05) than the other models. Thus, either one can be used to estimate the in vitro cumulative gas production curve of diets composed of different combinations of Gliricidia hay or silage.

However, the Groot model was selected due to its better accuracy and precision and easier access to the parameters. Like this, according to the Groot model, there was no difference (P>0.05) between the curves of the control diet and the diet containing Gliricidia hay (Table 4 and Figure 2). However, the inclusion of Gliricidia silage in the diet changed the fermentation kinetics, which generated a cumulative gas production curve different from that obtained with other experimental diets (Table 4 and Figure 2).

Table 4
Equations and variables obtained by the Groot model of in vitro gas production of diets composed of varied combinations of Gliricidia hay or silage.

Figure 2
Cumulative production curves of control diet gases (elephant grass hay + soybean meal + corn meal), gliricidia hay treatment (elephant grass hay + soybean meal + corn meal + gliricidia hay) and treatment gliricidia silage (elephant grass hay + soybean meal + cornmeal + gliricidia silage).

There was no effect of diet (P>0.05) on the time of inflection of the curve or the maximum fractional rate of degradation (Table 4). However, the diet containing Gliricidia in the form of silage took longer (P<0.05) to reach the maximum degradation rate.

# DISCUSSION:

With the exception of the monomolecular model, all other models showed minimal differences in the model adequacy evaluation criteria. Thus, according to the adopted evaluation criteria, the Gompertz, Groot, Richards and two-compartment-logistic functions would have similar fits. In this way, the sole evaluation of the models, as routinely done (VELHO et al., 2014VELHO, J. P. et al. Mathematical models for adjustment of in vitro gas production at different incubation times and kinetics of corn silages. Semina: Ciências Agrárias , v.35, n.4, p.2531-2540, 2014. Available from: <Available from: https://doi.org/10.5433/1679-0359.2014v35n4Supl1p253 >. Accessed: Sep. 30, 2019. doi: 10.5433/1679-0359.2014v35n4Supl1p253.
https://doi.org/10.5433/1679-0359.2014v3...
; FARIAS et al., 2011FARIAS, L. N., V. R. et al. Avaliation of bicompartimental logistic and Gompertz mathematical models to estimate gas production from babassu (Orbignya martiana) meal and pie using the semi-automated in vitro technique. Arquivo Brasileiro de Medicina Veterinária e Zootecnia, v.63, n.1, p.136-142, 2011. Available from: <Available from: https://doi.org/10.1590/S0102-09352011000100021 >. Accessed: Oct. 10, 2020. doi: 10.1590/S0102-09352011000100021.
https://doi.org/10.1590/S0102-0935201100...
; GOMES et al., 2017GOMES, M. F. F. et al. In vitro fermentation characteristics of ruminant diets using ethanol extract of brown propolis as a nutritional additive. Revista Brasileira de Zootecnia , v.46, n. 7, p.599-605, 2017. Available from: <Available from: http://dx.doi.org/10.1590/S1806-92902017000700007 >. Accessed: May. 25, 2020. doi: 10.1590/S1806-92902017000700007.
http://dx.doi.org/10.1590/S1806-92902017...
), without a comparison in terms of accuracy and precision, may not be a good option.

The literature describes several variations in the models for the fitting of the in vitro cumulative gas production curve of ruminant diets. FARIAS et al. (2011FARIAS, L. N., V. R. et al. Avaliation of bicompartimental logistic and Gompertz mathematical models to estimate gas production from babassu (Orbignya martiana) meal and pie using the semi-automated in vitro technique. Arquivo Brasileiro de Medicina Veterinária e Zootecnia, v.63, n.1, p.136-142, 2011. Available from: <Available from: https://doi.org/10.1590/S0102-09352011000100021 >. Accessed: Oct. 10, 2020. doi: 10.1590/S0102-09352011000100021.
https://doi.org/10.1590/S0102-0935201100...
) studied mathematical models to evaluate gas production from babassu (Orbignya speciosa) meal and cake and preferred the two-compartment logistic model. VELHO et al. (2014VELHO, J. P. et al. Mathematical models for adjustment of in vitro gas production at different incubation times and kinetics of corn silages. Semina: Ciências Agrárias , v.35, n.4, p.2531-2540, 2014. Available from: <Available from: https://doi.org/10.5433/1679-0359.2014v35n4Supl1p253 >. Accessed: Sep. 30, 2019. doi: 10.5433/1679-0359.2014v35n4Supl1p253.
https://doi.org/10.5433/1679-0359.2014v3...
) chose the Gompertz model to describe the kinetics of in vitro gas production of maize silages. TEIXEIRA et al. (2016TEIXEIRA, U. H. G. et al. Mathematical models for estimating the parameters of ruminal degradation kinetics of protein concentrates. Revista Brasileira de Saúde e Produção Animal , v.17, n.1, p.73-85, 2016. Available from: <Available from: http://dx.doi.org/10.1590/S1519-99402016000100008 >. Accessed: Nov. 09, 2020. doi: 10.1590/S1519-99402016000100008.
http://dx.doi.org/10.1590/S1519-99402016...
) investigated the kinetic parameters of rumen degradation of protein concentrates (soybean meal and cotton cake) and recommended the Richards model.

This divergence regarding the different adjusted models is theoretically understandable, as it depends on the fermentation pattern of the feedstuffs under study. However, those authors (VELHO et al., 2014VELHO, J. P. et al. Mathematical models for adjustment of in vitro gas production at different incubation times and kinetics of corn silages. Semina: Ciências Agrárias , v.35, n.4, p.2531-2540, 2014. Available from: <Available from: https://doi.org/10.5433/1679-0359.2014v35n4Supl1p253 >. Accessed: Sep. 30, 2019. doi: 10.5433/1679-0359.2014v35n4Supl1p253.
https://doi.org/10.5433/1679-0359.2014v3...
; FARIAS et al., 2011FARIAS, L. N., V. R. et al. Avaliation of bicompartimental logistic and Gompertz mathematical models to estimate gas production from babassu (Orbignya martiana) meal and pie using the semi-automated in vitro technique. Arquivo Brasileiro de Medicina Veterinária e Zootecnia, v.63, n.1, p.136-142, 2011. Available from: <Available from: https://doi.org/10.1590/S0102-09352011000100021 >. Accessed: Oct. 10, 2020. doi: 10.1590/S0102-09352011000100021.
https://doi.org/10.1590/S0102-0935201100...
; TEIXEIRA et al., 2016TEIXEIRA, U. H. G. et al. Mathematical models for estimating the parameters of ruminal degradation kinetics of protein concentrates. Revista Brasileira de Saúde e Produção Animal , v.17, n.1, p.73-85, 2016. Available from: <Available from: http://dx.doi.org/10.1590/S1519-99402016000100008 >. Accessed: Nov. 09, 2020. doi: 10.1590/S1519-99402016000100008.
http://dx.doi.org/10.1590/S1519-99402016...
) selected the models based on evaluation criteria (root mean square error of prediction, R2, mean square error, among others) without comparisons to determine whether the differences in these criteria are indeed significant.

In the present study, the models were compared for accuracy and precision, as suggested by TEDESCHI (2006TEDESCHI, L.O. Assessment of the adequacy of mathematical models. Agricultural Systems, v.89, n.2-3, p.225-247, 2006. Available from: <Available from: https://doi.org/10.1016/j.agsy.2005.11.004 >. Accessed: Sep. 15, 2019. doi: 10.1016/j.agsy.2005.11.004.
https://doi.org/10.1016/j.agsy.2005.11.0...
), and the Groot and Richards functions were found to be more precise and accurate than the others, which indicates that both can be used to estimate the in vitro cumulative gas production curve of diets. However, despite the good fit, the Richards model showed convergence problems in the iterative process, requiring the use of the Marquardt algorithm and an increase in the number of iterations, possibly because this model needs to estimate an additional parameter. Other authors have also reported convergence difficulties using the Richards model (ZWIETERING et al., 1990ZWIETERING, M. H. et al. Modeling of the Bacterial Growth Curve. Applied and Environmental Microbiology, v.56, n.6, p.1875-1881, 1990. Available from: <Available from: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC184525/ >. Accessed: May. 10, 2019.
https://www.ncbi.nlm.nih.gov/pmc/article...
; KOPUZLU et al., 2014KOPUZLU, S. et al. Estimation of growth curve characteristics of Hemsin male and female sheep. Journal of Applied Animal Research, v.42, n.2 p.228-232, 2014. Available from: <Available from: https://doi.org/10.1080/09712119.2013.842479 >. Accessed: Jul. 8, 2020. doi: 10.1080/09712119.2013.842479.
https://doi.org/10.1080/09712119.2013.84...
).

In addition, a model with three parameters-as is the case with the Groot model-will exhibit more degrees of freedom in the estimates, which can be important when a curve displays a smaller amount of information (ZWIETERING et al., 1990ZWIETERING, M. H. et al. Modeling of the Bacterial Growth Curve. Applied and Environmental Microbiology, v.56, n.6, p.1875-1881, 1990. Available from: <Available from: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC184525/ >. Accessed: May. 10, 2019.
https://www.ncbi.nlm.nih.gov/pmc/article...
). It is also important that all parameters have biological significance. In this respect, the Groot model has advantages over the other functions in that it does not assume a constant fractional rate of fermentation (GROOT et al., 1996GROOT, J. C. J. et al. Multiphasic analysis of gas production kinetics for in vitro fermentation of ruminant feeds. Animal Feed Science and Technology, v.64, n.1, p.77-89, 1996. Available from: <Available from: https://doi.org/10.1016/S0377-8401(96)01012-7 >. Accessed: May. 30, 2020. doi: 10.1016/S0377-8401(96)01012-7.
https://doi.org/10.1016/S0377-8401(96)01...
).

The kinetics of cumulative gas production depends on a sequence of processes. Immediately after incubation, the feed is partially solubilized (ÍTAVO et al., 2016ÍTAVO, L. C. V. et al. Combinations of non-protein nitrogen sources in supplements for Nellore steers grazing. Revista Brasileira de Saúde e Produção Animal, v.17, n.3, p.448-460, 2016. Available from: <Available from: http://dx.doi.org/10.1590/S1519-99402016000300011 >. Accessed: Jul. 3, 2020. doi: 10.1590/S1519-99402016000300011.
http://dx.doi.org/10.1590/S1519-99402016...
) and compounds with greater solubility are quickly fermented (COSTA et al., 2011COSTA, V. A. C. et al. Intake and rumen dynamics of neutral detergent fiber in grazing cattle supplemented with non-protein nitrogen and, or true protein during the rainy season. Revista Brasileira de Zootecnia, v.40, n.12, p.2805-2814, 2011. Available from: <Available from: https://doi.org/10.1590/S1516-35982011001200027 >. Accessed: May. 20, 2020. doi: 10.1590/S1516-35982011001200027.
https://doi.org/10.1590/S1516-3598201100...
; RIBEIRO et al., 2011RIBEIRO, S. S. et al. Effects of ruminal infusion of a slow-release polymercoated urea or conventional urea on apparent nutrient digestibility, in situ degradability, and rumen parameters in cattle fed low-quality hay. Animal Feed Science and Technology , v.164, n.1-2, p.53- 61, 2011. Available from: <Available from: https://doi.org/10.1016/j.anifeedsci.2010.12.003 >. Accessed: May. 10, 2020. doi: 10.1016/j.anifeedsci.2010.12.003.
https://doi.org/10.1016/j.anifeedsci.201...
). Subsequently, the less soluble parts start to ferment (SILVA et al., 2017SILVA, R. N. P. et al. Ruminal degradability of shell of pods of the lima bean (“Phaseolus lunatus” L.) ammoniated with urea. Revista Brasileira de Saúde e Produção Animal , 18, n.1, p.26-37, 2017. Available from: <Available from: http://dx.doi.org/10.1590/S1519-99402017000100004 >. Accessed: Oct. 12, 2020. doi: 10.1590/S1519-99402017000100004.
http://dx.doi.org/10.1590/S1519-99402017...
). In this way, the fractional rate decreases exponentially throughout the incubation time due to the use of substrate by the ruminal microbiota (PELL & SCHOFIELD, 1993PELL, A.; SCHOFIELD, P. Computerized monitoring of gas production to measure forage digestion in vitro. Journal of Dairy Science, v.76, n.4, p.1063-1073, 1993. Available from: <Available from: https://doi.org/10.3168/jds.S0022-0302(93)77435-4 >. Accessed: Oct. 13, 2020. doi: 10.3168/jds.S0022-0302(93)77435-4.
https://doi.org/10.3168/jds.S0022-0302(9...
). Therefore, models that consider a constant fractional rate do not provide an accurate simulation of the phenomena that occur in the rumen environment. For these reasons, we chose the Groot model to fit the in vitro cumulative gas production curve of the diets.

The similarity between the curves of the control and Gliricidia hay diets fitted by the Groot model may have been due to the greater participation of maize in the diets containing Gliricidia. The inclusion of Gliricidia allowed for a reduction in the proportion of soybean, whereas the proportion of maize was increased, thereby increasing the supply of soluble carbohydrates and rapidly fermentable carbohydrates such as starch for the microbial population.

In contrast, the inclusion of Gliricidia silage generated a different curve from those obtained with the other diets, especially for parameter “k”, which determines the sharpness of the curve. This may have been a reflection of the reduction in soluble carbohydrates during the ensiling process, because these carbohydrates are used as substrate for lactic fermentation in the silo (ZARDIN et al., 2017ZARDIN, P. B. et al. Chemical composition of corn silage produced by scientific studies in Brazil - A meta-analysis. Semina: Ciências Agrárias , v.38, n.1; p.503-512, 2017. Available from: <Available from: https://doi.org/10.5433/1679-0359.2017v38n1p503 >. Accessed: Sep. 23, 2019. doi: 10.5433/1679-0359.2017v38n1p503.
https://doi.org/10.5433/1679-0359.2017v3...
; SANTANA et al., 2019SANTANA, J. C. S. et al. Fermentation characteristics, chemical composition and protein fractioning of Gliricídia silage submitted to different fermentation periods. Boletim de Indústria Animal , 76, n.1, p.1-9, 2019. Available from: <Available from: https://doi.org/10.17523/bia.2019.v76.e1436 >. Accessed: Jul. 05, 2020. doi: 10.17523/bia.2019.v76.e1436.
https://doi.org/10.17523/bia.2019.v76.e1...
). Coupled with this is the higher concentration of lignin and cellulose in the diet with Gliricidia silage.

The higher concentration of lignin and cellulose in the diet with Gliricidia silage also explains the longer time taken to obtain the maximum rate of substrate degradation, in that diet. Lignin works as a mechanical barrier against the action of rumen microorganisms (DÍAZ et al., 2018DÍAZ, T. G. et al. In vitro gas production kinetics and digestibility in ruminant diets with diferente levels of cashew nut shell liquid. Semina: Ciências Agrárias, v.39, n.4, p.1669-1682, 2018. Available from: <Available from: https://doi.org/10.5433/1679-0359.2018v39n4p1669 >. Accessed: Apr. 25, 2020. doi: 10.5433/1679-0359.2018v39n4p1669.
https://doi.org/10.5433/1679-0359.2018v3...
), which can increase the time spent by microorganisms to colonize the particle (OLIVEIRA et al., 2017OLIVEIRA, V. S. et al. Ruminal kinetics of tropical forrages submitted or not to irrigation. Boletim de Indústria Animal, v.74, n.3, p.195-204, 2017. Available from: <Available from: hhttps://doi.org/10.17523/bia.v74n3p195 >. Accessed: Apr. 10, 2020. doi:10.17523/bia.v74n3p195.
hhttps://doi.org/10.17523/bia.v74n3p195...
) and; consequently, the time for microorganisms to reach maximum activity.

These findings may be important for sheep production systems. It is possible to partially replace soybean meal with Gliricidia hay in lamb feeding without impairing ruminal fermentation kinetics. However, additional studies with sheep (in vivo) to determine fed intake and animal performance on these or similar feeds are needed to confirm whether these laboratory (in vitro) findings can be reflected in improved production. Other shrub legumes it can also be used depending on availability.

# CONCLUSION:

The Groot and Richards models best describe the kinetics of in vitro gas production of diets with Gliricidia hay or silage. However, the Groot model is recommended to estimate the average curve due to its ease in obtaining the parameters and providing biological explanation. The inclusion of Gliricidia silage alters the fermentation kinetics of the diet. Therefore, it is recommended to use Gliricidia preserved in the form of hay.

# ACKNOWLEDGEMENTS

The Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Financing Code 001 and the National Council funded this research for Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). Supported by the Universidade Federal de Sergipe and Universidade Federal de Mato Grosso do Sul.

# REFERENCES

• CR-2020-0993.R2

# Publication Dates

• Publication in this collection
07 July 2021
• Date of issue
2021