Livestock Research for Rural Development 26 (2) 2014  Guide for preparation of papers  LRRD Newsletter  Citation of this paper 
The
objective of this study was to identify the model (Exponential, France,
Gompertz, Logistic and Dualpool logistic models) that best fits the
cumulative gas production curve in ruminant diets consisted of
the substitution of maize with crude glycerol (0, 4, 8 and 12%). Alfalfa hay
(Medicago sativa) was used as roughage and comprised 60% of the diet.
The criteria used
were the graphic analysis of the observed and estimated curves,
graphic analysis of residues,
mean square error,
coefficient of determination (r²), residual mean absolute
deviation (RMAD),
mean percentage error and relative efficiency.
Dualpool logistic, Gompertz
and logistic models showed the highest r^{2 }values, as well as the
lowest mean squared error. The France and Gompertz models had the lowest
mean percentage error. The dualpool logistic model showed the lowest RMAD
and relative efficiency greater than 1.0 in all comparisons, and therefore,
was considered the most efficient of the models evaluated.
Keywords: curve, gas production, in vitro, mathematical models, rumen kinetics
The technique of in vitro gas production may describe the kinetics of food ruminal fermentation, in addition of estimating the degradation rate and extent of the food solubles and insoluble fractions by monitoring gas production rates at determined time intervals (Beuvink and Kogut 1993) and quantifying the residues on the final time point (Pell and Schofield 1993).
The mathematical description of gas production curves allows analyzing data and comparing substrates or characteristics of the fermentation environment, thereby providing useful information relative to composition of the substrate and the fermentation of soluble and slowly fermentable substrate fractions (Groot et al 1996).
The mathematical nonlinear models commonly used to estimate the kinetics of ruminal fermentation are the exponential models, proposed by Ørskov and McDonald (1979), Mertens and Loften (1980), France et al (1993), Beuvink and Kogut (1993) and the Gompertz model, proposed by Lavrencic et al (1997); the single model proposed by Schofield et al (1994); the dualpool logistic models proposed by Schofield et al (1994) and Groot et al (1996). The models may assume one, two or more fermentation compartments and/or phases (Groot et al 1996; Huhtanen et al 2008). Therefore, data analyzed by different models may generate different results, and hence, their interpretation demands caution and common sense (Mello et al 2008).
According to Noguera et al (2004), mathematical models have advantages and disadvantages relative to one another. Estimating variations exist between them depending on the experimental conditions and food evaluated. It is therefore essential, prior assessment of the most appropriate model for suitable interpretation of the nutritional value of feedstuffs for ruminants (Beuvink and Kogut 1993) particularly in diets utilizing alternative foods for substituting the conventional ones.
Crude glycerol could be used as an alternative energy source for ruminant feeding, due to its increasing availability and favorable price as a result of the expansion of the biofuel industry, as well as to the increase in grain maize prices.
The objective of the present study was to identify among nonlinear mathematical models the one that best fitted biologically the invitro cumulative gas production curves in diets based on alfalfa hay and ground maize grain with the addition of increasing levels of crude glycerol in substitution of maize.
The experiment was carried out in the ruminant sector of Laboratory of
Animal Science, and the chemical analyses were performed at the Animal
Nutrition, both belonging to the Department of Animal Science of the School
of Agronomy of the
Federal University of Rio Grande do Sul (UFRGS), Brazil.
Two Texel sheep, with 40kg average body weight and fitted with a rumen fistula, were used as inoculum donor. The animals were kept in a 120m^{2} paddock with a shelter during the entire experiment. Alfalfa hay was daily fed at 3% the animals’ body weight. Mineral salt and water were supplied ad libitum. Before the experiment started, the animals were submitted to a 10day adaptation period to the above described diet. The experimental protocol followed the guidelines of the Ethics Committee on the Use of Animals in Research as Number 18.442 in compliance with Law 11.794.
The experimental treatments consisted of substituting maize with liquid crude glycerol (0, 4, 8 and 12%) in dry matter basis. Alfalfa hay (Medicago sativa) was used as roughage and comprised 60% of the diet. Table 1 shows the nutritional composition of the ingredients of the experimental diets.
Table 1. Dry matter (DM), organic matter (OM), crude protein (CP), neutral detergent fiber (NDF) and acid detergent fiber (ADF), crude energy (CE) and glycerol contents of the ingredients in the experimental diets 

Ingredients 
DM 
OM 
CP 
NDF 
ADF 
CE (MJ/kg) 
Glycerol (%) 
(%) 
……….% DM..……… 

Alfalfa hay 
88.5 
88.3 
19.2 
54.4 
30.6 
18.6 
 
Ground maize 
88.8 
99.1 
8.53 
16.0 
3.46 
18.7 
 
Crude glycerol^{*} 
81.4 
 
0.11 
 
 
14.5 
80.0 
^{ *}methanol content lower than 45 mg/L 
Invitro cumulative gas production was measured using the technique of automatic invitro gas production (ANKOM Gas Production System). One day before the incubation to each 250 ml fermentation bottle, 1g composite sample consisting of 60% alfalfa hay and ground maize was placed. Crude glycerol substituting 0, 4, 8, and 12% of the total maize (40%) on the dry matter basis were individually added to the flasks, according to experimental treatment, and the flasks were then kept in an oven at 39ºC. On incubation day 100 ml of a mixture containing culture medium (Goering and Van Soest 1975), which was previously heated and maintained at 39ºC, and saturated with CO_{2} for approximately 15 minutes before being added to the bottles. After the addition of the mixture, bottles were saturated with CO_{2 }for approximately 30 seconds, immediately closed, and maintained in the incubator at 39ºC until the inoculum was added.
Two hours after the animals received the morning meal, ruminal liquid and part of solid ruminal material were collected. The objective of collecting solid material was to also collect microorganisms attached to the substrate. The collected materials were homogenized in a food processor at a 1:1 ratio (1 part of liquid to 1 part of solid). The homogenized material was then filtered through four layers of gauze and maintained under constant CO_{2} gas flow, always keeping the temperature near 39ºC before and during the addition of the material to the bottles. A volume of 25 ml of inoculum was added to each bottle using a 50 ml glass syringe. The bottles were then placed in an incubator at 39ºC for 48 hours.
Invitro cumulative gas pressure was automatically measured every 10 minutes. The gas production was determined for each sample in four repeated runs. The gas production curves obtained in each run were very reproducible (repeatability estimated by RELM method = 0.79).
In order to transform pressure into volume data, the following equation recommended by the manufacturer was applied: V_{x} =V_{j}P_{psi} x 0.068004084, where: V_{x} = volume at 39ºC in ml; V_{j} = head space of the digestion bottle in ml; P_{psi }= cumulative pressure recovered by the software.
Gas production data were expressed in ml of gas produced per gram of organic matter incubated.
The models were fit to the cumulative gas production data and was presented in Table 2.
The criteria adopted to verify the goodness of fit of the models were the coefficient of determination (r^{2}), mean square error (MSE), residual mean absolute deviation (RMAD), graphic analysis of observed and predicted curves, graphic analysis of residues, mean percentage error (MPE), and relative efficiency (RE).
The MSE was obtained by analysis of variance, using the PROC NLIN of SAS (SAS Inst., Inc., Cary, NC; Version 9.5), dividing the sum of squared errors (SSE) by the number of observations because the models presented different number of parameters to be estimated.
The r² values resulted from the division of the sum of squares of the model (SSM) by the sum of total squares (STS). RMAD was calculated as the sums of difference between the observed and predicted values, divided by the number of observations.
The PROC CORR of SAS (SAS Inst., Inc., Cary, NC; Version 9.5) was used for r², RMAD, graphic analysis of observed and predicted curves and graphic analysis of residues.
The relative efficiency was calculated as where MSE is the mean square error, and RE _{(i/j)} indicates the efficiency of model i relative to model j.
The parameters of the models were estimated using the iterative method of Marquardt inserted in the NLIN procedure of SAS (SAS Inst., Inc., Cary, NC; Version 9.5).
The values of the mean squared error (MSE), coefficient of determination (r^{2}), residual mean absolute deviation (RMAD) and mean prediction error (MPE), obtained from fitting the data of organic matter gas production to the tested models, were submitted to analysis of variance (PROC GLM) and the means were compared by the test of Tukey at the level of 5% error probability.
No difference was observed in the initial hours in the gas production profiles, only differences in total gas production are observed (Figure 1).
The exponential (A) and France (B) models were less effective in gas production fitting to the curves in the beginning of incubation (0 to 5 hours) at all crude glycerol levels as they underestimated gas production. The exponential model yielded negative values, which is biologically impossible (Figure 1). Moreover, these models presented exponential growth during the incubation period, when, in fact, gas production curves are usually sigmoidal, as those produced by the Gompertz (C), logistic (D) and dualpool logistic (E) models. These models presented three phases: an initial phase with no or slow gas production, exponential phase of rapid gas production, and asymptotic with gas production reduction. Different from Gompertz model, the logistic e dualpool logistic models overestimated gas production during the initial period of incubation (0 to 2 hours) at all crude glycerol levels evaluated.
The exponential and France model underestimated gas production in the transition between the end of the exponential phase and the beginning of the asymptotic phase, specifically between 12 and 24 hours of incubation, whereas the logistic model overestimated gas production during this phase. The Gompertz and duopool logistic models best estimate at all stages of gas production.
A:  A: 
B:  B: 
C:  C: 
D:  D: 
E:  E: 
Figure 1. Observed and predicted curves of
cumulative gas production (ml/g incubated OM) and residual
dispersion during the incubation period as determined by the
exponential (A), France (B), Gompertz (C), logistic (D) and dualpool logistic (E) models when increasing crude glycerol levels (0, 4, 8, and 12%) were included in the diet 
At the end of the incubation period (between 45 and 48 hours of incubation), the exponential and France model overestimated gas production for all crude glycerol levels, while the logistic model overestimated gas production for the crude glycerol levels of 0 and 4% and underestimated gas production for the levels of 8 and 12%.
The plot of residues as a function of incubation time shows the goodness of fit provided by each model tested. The widest residual dispersion occurred in the initial incubation times in all evaluated models. The exponential (A) and France (B) models presented similar trends of residual dispersion as a function of incubation time for all crude glycerol levels. The Gompertz (C) and logistic (D) presented wide residual dispersion during the 48 hours of incubation, indicating the goodness of fit of these models was less suitable. The most homogenous residual dispersion was obtained with the dualpool logistic model (E) (Figure 1).
The evaluated parameters were not influenced by dietary crude glycerol inclusion levels, therefore, the differences between the evaluated parameters were considered among the models evaluated.
Considering mean squared error (MSE), the dualpool logistic model presented the lowest residual variance and the exponential model, the highest residual variance (Table 3).
Table 3. Means of the mean squared error (MSE), coefficient of determination (r^{2}), residual mean absolute deviation (RMAD) and mean prediction error (MPE) obtained by fitting organic matter gas production data to the tested models 

Model 
MSE 
r^{2} 
RMAD 
MPE 
France 
12.4^{b} 
0.98^{b} 
2.53^{b} 
0.22^{b} 
Exponential 
24.5^{a} 
0.97^{c} 
4.03^{a} 
15.4^{a} 
Gompertz 
6.33^{bc} 
0.99^{ab} 
1.85^{b} 
2.35^{b} 
Logistic 
8.01^{bc} 
0.99^{ab} 
2.33^{b} 
9.60^{c} 
Dual pool 
3.01^{c} 
0.99^{a} 
1.03^{c} 
8.55^{c} 
Mean 
10.8 
0.98 
2.36 
1.05 
CV (%) 
61.2 
0.97 
33.7 
581 
Means withincolumns followed by different superscripts are statistically different (P<0.05) by the test of Tukey 
The dualpool logistic, Gompertz and logistic models presented the highest r^{2} values (0.99), indicating that these models had the best goodness of fit (Table 3).
When residual mean absolute deviation (RMAD) was evaluated (Table 3), the highest dispersion was obtained with exponential model as compared with the other models, as shown in Figure 1. This would indicate that this model would be the least suitable to describe mean gas production curves. Based on this criterion, the lowest deviation was observed with the dualpool logistic model.
According to MPE results, the mean of observed values were underestimated, except for the exponential model, which overestimated these values. The MPE values closer to zero were obtained with the France and Gompertz models (Table 3).
When the RE value of a model compared with another model is higher than 1.0 (RE > 1.0), this indicates that the first model was more efficient. When the relative efficiency of the models was calculated, the exponential model was shown to be the least efficient, and therefore, the least indicated for fitting data (Table 4). As to the other evaluated models, the France model was only more efficient than the exponential model, whereas the logistic model was more efficient than both the France and exponential models and the Gompertz model was more efficient than the previous three. The most efficient model for fitting data was the dualpool logistic model.
Table 4. Relative efficiency (RE) of the fitted models 

Models^{1} 
Models^{2} 

France 
Exponential 
Gompertz 
Logistic 
Dual pool 

France 
 
0.50 
1.96 
1.55 
4.12 
Exponential 
1.97 
 
3.87 
3.05 
8.13 
Gompertz 
0.50 
0.26 
 
0.79 
2.10 
Logistic 
0.64 
0.33 
1.26 
 
2.66 
Dual pool 
0.24 
0.12 
0.47 
0.37 
 
^{ 1}Model j for the calculation of relative efficiency. ^{2}Model i for the calculation of relative efficiency 
The exponential model assumes that the gas production rate depends only on the substrate available for fermentation after the lag time is achieved (Ørskov and McDonald 1979; McDonald 1981). The France model assumes that the gas production rate is directly proportional to food degradation rate, which varies as a function of incubation time and food colonization time by microorganisms (France et al 1993). Thereby, it is highly flexible in fitting or not the observed data to the sigmoidal curve (Dhanoa et al 2000). However, this was not the case in the present study, and moreover, it generated parameters with unrealistic biological interpretation (initial negative values). The logistic model assumes that the gas production rate is proportional to substrate content and microbial mass (Schofield et al 1994); however, the specific rate is fixed along incubation time and the point of inflection is limited between 0.20 and 0.40 of Vf, by establishing that b = 1 (Brown et al 1976).
The Gompertz and dualpool logistic models presented the most suitable curve shapes until the end of incubation. The Gompertz model assumes that the specific gas production rate is proportional to microbial mass, which in turn depends on the concentration of digestible substrate (Lavrencic et al 1997). However, the fractional rate exponentially decreases along incubation time due to the inactivation of bacteria caused by substrate depletion (Schofield et al 1994), with the point of inflection fixed at 0.37 Vf. In contrast, the dualpool logistic model, from the nutritional perspective, presents more relevant characteristics than the Gompertz model and other evaluated models because it divides total gas production in two fractions (pools) as a function of digestion rates (fast and slow). According to Schofield et al (1994) and Groot et al (1996), multipool models present better goodness off it than the models based on kinetics of the first order; in addition, they are more suitable when a high number of readings is made (Fondevilla and Barrios 2001), such as in the present study that used an automated system programmed to record gas production every 10 minutes.
This is demonstrated when the residual dispersion graphs are analyzed (Figure 1). According to the visual inspection of residual inspection graphs, the dualpool logistic model presented the lowest residual dispersion as a function of incubation time for all glycerol levels, generating the best fitting of data.
The r^{2} values obtained in the present study are consistent with those obtained by Schofield and Pell (1995) when fitting the dualpool logistic model (r^{2}=0.99). It should be mentioned, however, that the r^{2} provides only suggests the association between observed and predicted values, and does not really represent the biological association of the curve. Therefore, the r^{2 } per se is not a suitable criterion to evaluate the goodness of fit of a model because it does not evidence the differences that are clearly detected when curve and residual graphs are analyzed (Mello et al 2008). In some cases, despite presenting marked discrepancies between observed and predicted values, nonlinear models may have high r^{2} values (Ratkowsky 1990).
According to Sarmento et al (2006), low RMAD indicate better goodness of fit. In the present study, the dualpool logistic model presented the lowest deviation value.
Mean prediction error (MPE) indicates if the models overestimated (positive values) or underestimated (negative values) the predicted values relative to the observed values. In this study, the France model presented the lowest bias, i.e., MPE value closer to zero.
The RE of the dualpool logistic model tested in the present study was higher than 1.0 compared with all the other models evaluated, and therefore, it was considered the most efficient.
According with all the criteria adopted to verify the goodness of fit of the models, the dualpool logistic model presents the best goodness of fit to the cumulative gas production in the period of 48 h on diets were maize was substituted by different crude glycerol levels.
The dualpool logistic model presents the best goodness of fit to the cumulative gas production in the period of 48 h on diets with different crude glycerol inclusion levels.
The present study received financial support of CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico – Brazilian Scientific and Technological Development Council).
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Received 24 July 2013; Accepted 26 January 2014; Published 4 February 2014