Mathematical modeling for digestible protein in animal feeds for tilapia

The objective of this study was to formulate mathematical models to estimate digestible protein in some animal feeds for tilapia. Literature results of the proximate composition of crude protein, ether extract, and mineral matter, as well as digestible protein obtained in biological assays, were used. The data were subjected to multiple linear stepwise backward regression. Path analysis was performed to measure the direct and indirect effects of each independent variable on the dependent one. To validate the model, the experience used data from independent studies and values obtained from a digestibility trial with juvenile Nile tilapia testing five meat and bone meals, using the Guelph feces collecting system and chromium oxide (III) as an indicator. The obtained model used to estimate digestible protein values (DP) of animal origin is: 953 . 0 R CP; 1.203 15 . 04 2 ) kg (g DP 2 -1 . The path coefficients showed a high direct positive effect (0.900) of crude protein on the digestible protein content. The mineral matter content has an indirect negative effect on protein digestibility (-0.710), reducing the crude protein content and quality.


Introduction
Tilapia is one of the most promising species for aquaculture, due to its rapid growth in intensive farming.Feed is the most expensive component in tilapia farming, representing over 50% of the operating costs (El Sayed, 2006).
The way fish use protein varies by species, and is influenced by feeding habits (Dabrowski & Portella, 2006).Thus, the values of digestible protein in each ingredient are important to develop well-balanced and sustainable diets (Pond et al., 2005).Reliable data on nutrient digestibility are critical to evaluate the inclusion potential of a dietary ingredient, to elaborate low-cost diets, and to minimize the environmental impact of livestock production (Vandenberg & De La Noüe, 2001).
Digestibility values are obtained based on in vivo fecal collection, a methodology routinely used in animal nutrition studies in digestibility trials.In terms of practical conditions, it is costly and difficult to subject every raw material batch to digestibility trials.
Due to the possibility of obtaining the values of crude protein, ether extract, and mineral matter contents by lowcost chemical analysis, and their use in regression equations, the estimation of digestible protein values can be of great practical application (Sakomura & Rostagno, 2007).They may also be an important tool in complementing the biological assays, which depend on a more complex, expensive and prolonged methodology.Mathematical modeling has been widely used to estimate digestible lipids (Sales, 2009;Hua & Bureau, 2009a), available phosphorus (Hua & Bureau, 2006), carbohydrates (Hua & Bureau, 2009b), and protein (Sales, 2008).The development of individual models, both for feed and species will make it possible to obtain data applicable to new situations and physiological features of fish.
The objective of this study was to develop mathematical models to estimate the digestible protein for animal feedstuff for tilapia and to validate them with data from a biological digestibility trial, using meat and bone meal as standard feed and independent studies from the literature.

Material and Methods
Chemical composition and digestible protein, data from some ingredients of animal origin were collected from scientific papers published between 2002 and 2008, mostly obtained for Nile tilapia.The search was conducted on the Scopus and ISI Web of Science databases.
Articles that contained values of dry matter (DM), crude protein (CP), ether extract (EE), mineral matter (MM), and digestible protein (DP) of fish meal, shrimp meal, meat and bone meal, and poultry by products meal were used.At the end of the selection, eight articles were obtained, which resulted in the database described below (Figure 1).For standardization, the data on chemical composition and protein digestibility were expressed in dry matter values.
All data were analyzed by multiple linear regression, testing models with and without intercept, respectively: β 0 value was obtained according to the follow equation: where: Yi = apparent digestible protein (ADP) of the ingredients obtained in a digestibility test; β 0 = β 0 = intercept; X i1 , X i2 X i3 = feed chemical composition variables, respectively, crude protein, ether extract, and mineral matter.
The backward stepwise method was used to remove insignificant independent variables (P<0.05).To determinate which equation best estimates digestible protein, linear equation with intercept and without intercept, the root mean squared error (RMSE) of each one was used.Path analysis was performed to measure the direct and indirect effects of each independent variable on the dependent.
To validate the models, a digestibility assay was conducted at the Universidade Estadual de Maringá -Fish Culture "Estação de Piscicultura de Floriano" -UEM/CODAPAR, from January to February 2010 A practical reference diet was formulated to contain approximately 320 g kg -1 of crude protein, 3120 kcal of digestible energy, 34.0 g kg -1 of crude fiber and 5.0 g kg -1 of phosphorus (Table 1).
Meat and bone meal with different protein levels (Table 2) were used as standard feed to validate the equations and replaced 30% of the reference diet.
In the preparation of test diets, after grinding, weighing, and mixing of ingredients, water was added at 60 °C at a ratio of 25% of the total weight of the diet.The mixture was pelleted in a meat mill and dried in a forced ventilation oven (55 °C) for 48 h.
The apparent digestibility coefficients (ADC) of crude protein were determined by the indirect method using chromic oxide III (5 g kg -1 ) as an inert indicator.Conical fiberglass tanks were used for fecal collection. 1 Vitamin C: calcite salt, active principle ascorbic 2 acid-42%-monophosphate.
Each test diet was assessed in triplicate for five days, each tank was considered a collection repetition.Before feces collection, fish were adapted to the conical tanks, handling, and fed pellet diets for seven days.For each new ingredient, the feces were discarded in the first three days to avoid contamination with the previous diet.At the end of each sampling period, the feces were dried in a forced ventilation oven at 55 °C (48 h) and milled and analyzed according to the methodology described by the AOAC (1990).The gross energy was determined by an adiabatic bomb calorimeter (Parr Instrument Company, Moline, IL, USA).
The chromic oxide content of diets and feces were determined according to Bremer Neto et al. (2005).
The apparent digestibility coefficients for crude protein were calculated according to the equations described by Pezzato et al. (2002).The differences between the digestible protein of the meat and bone meals were determined by analysis of variance (ANOVA; P<0.05), significant values were submitted to linear regression.
The T test of Student was applied to investigate the differences between the mean obtained values from the digestibility trial and the estimated values.The performance of determined prediction equations were evaluated by linear regression analysis between predicted (y) and obtained (x) values, adapted from Sales (2008).The values used in the validation procedure were obtained in the digestibility assay and from four independent studies (Figure 2).All calculations were performed in the statistical package SAS 9.1.3.

Results and Discussion
The standard model for estimating digestible protein, obtained by multiple linear regression, was highly significant (P<0.0001) and had a coefficient of determination R 2 = 0.953 (Figure 3) and accurately described the observations of the dataset (Figure 4).Sales (2008a) Dabrowiski & Portella (2006), different species of fish have different digestive metabolisms, which depend on the feeding habits; therefore, this biological factor should be taken into consideration in the development of mathematical models.
Analyzing the equations, one can notice that the root mean squared error of the equation with intercept was lower than the equation without intercept; therefore, the equation with intercept is the best option to estimate digestible protein (Table 3).
The value of an equation intercept creates a trend in the estimated values, since it is obtained from the average values of the dependent and independent variables (Bhujel, 2008).Thus, the use of prediction models with a constant is not indicated when the values of the chemical composition of the feed used to obtain the model are in a short range.On the other hand, the equations without an intercept can Source: Present digestibility trial; Vásquez-Torres et al., 2010;Gonçalves et al., 2009;Meurer et al., 2003;Sampaio et al., 2001.Figure 2 -Validating dataset consisting of data from 5 studies.
Crude protein (g kg -1 ) Apparent digestibility coefficient (%) protein on the digestible protein content (Figure 5).The mineral matter negatively affected the digestibility of protein by reducing the crude protein content and/or increasing the collagen content, an indigestible protein from the bone matrix (Butolo, 2010), shown by the indirect effect (-0.7101).Protein is the most costly nutrient in diets for domestic animals (Wilson, 2002), and the use of low-quality animal feeds, such as toasted blood meal and feather meal results in high nitrogen excretion into the aquatic environment (Pezzato et al., 2002;Sampaio et al., 2001).This process can reduce water quality, with possible excessive proliferation of algae or microorganisms, both of which are either directly or indirectly harmful to the integrity of the fish.The formulation of diets closely adjusted to the nutritional needs of the species, made from high-quality and properly processed ingredients are the main tools for reducing the negative impacts of aquaculture waste (Cyrino et al., 2010).
Differences (P<0.05) were observed for the ADC of crude protein of feeds (Table 5).Thus, as the composition of the reference diet may influence the results, the feed processing, fecal collecting method, and nutrient levels used for determining the feed ADC are important factors in determining the biological value of each feed, which may reduce the accuracy of the estimated data when the linear coefficient is distant from zero (Rencher & Schaalje, 2008).
There was an increase (P<0.05) in crude protein (CP) digestibility coefficients of meat and bone meal.Comparing these values with others from the literature, proximity is observed between the ADC of 434.80 g kg -1 CP in meat and bone meal and that studied by Pezzato et al. (2002), who obtained an ADC of 731.90 g kg -1 , using a meat and bone meal composition similar to the one used in the present study.The 463.80 g kg -1 CP meat and bone meal had a higher ADC than that obtained by Guimarães et al. (2008), who determined a 78.4% ADC, despite the proximity to the crude protein value.The current study was conducted with a meat and bone meal containing less mineral matter (323.00 g kg -1 ) than that used by Guimarães et al (2008), 411.3 g kg -1 .When evaluating alternative feeds for Australian silver perch (Bidyanus bidyanus) using two meat and bone meals with 492.00 g kg -1 and 543.00 g kg -1 crude protein, Allan et al. (2000) obtained ADC values of crude protein of 715.00 and 739.00 g kg -1 respectively, which were lower than the values obtained with tilapia, in the present study, when considering the chemical composition of the tested ingredients.
The T test did not determine differences between the values obtained and estimated for the meat and bone meals.Furthermore, the linear regression between the observed and estimated values demonstrates highly significant (P<0.0001)linear relationship, an intercept close to zero, a slope close to one, and a high determination coefficient (Figure 6).
In practical terms, the use of mathematical models to estimate the digestible protein values is an important tool, since it is common to buy feed of different chemical compositions, and it would be difficult to carry out digestibility trials for all of them.Observed apparent digestible protein (g kg -1 ) Estimated apparent digestible protein (g kg -1 ) Because of the lower cost of meat and bone meal in some countries, as compared with fish meal, it has been widely used as a source of energy, protein (amino acids), minerals, and vitamins.However, its protein, fat, and mineral composition is highly variable, which even affects the nutritional value of other feeds in the diet.
Multiple linear models are useful to estimate the values of digestible protein, using the chemical composition values of ingredients of animal origin.The values estimated by mathematical models are very close to those obtained by independent experiments.
Digestibility trial data are consistent with those in the literature and confirm the effectiveness of the models.

Conclusions
It is possible to estimate the digestible protein in animal ingredients of tilapia feed through the prediction models.The developed model is

Figure 1 -
Figure 1 -Modeling dataset consisting of data from 8 studies.
(n) = apparent digestibility coefficient; I d = % of chromic oxide in the diet; I f = % of chromic oxide in feces, N d = nutrients in the diet, N f = nutrients in feces.(ing) = apparent digestibility coefficient of the ingredients, ADC (td) = apparent digestibility coefficient of the test diet, ADC (rd) = apparent digestibility coefficient of the reference diet, b = percentage of the reference diet, a = percentage of test ingredient.

Figure 4 -)Figure 3 -
Figure 4 -Comparison of observed and estimated digestible protein content (g/kg) models of the diets from dataset.

Figure 5 -
Figure 5 -Path analysis between chemical composition and digestible protein content of animal ingredients for tilapia.

Figure 6 -
Figure 6 -Comparison of observed and estimated apparent digestible models of protein values from independent studies and digestibility trial.

Table 1 -
Percentage composition of reference diet DM -dry matter; CP -crude protein; EE -ether extract; MM -mineral matter; GE -gross energy.

Table 2 -
Chemical composition of meat and bone meal with different levels of crude protein

Table 3 -
Linear models used to estimate crude protein and digestible energy of ingredients of animal origin for tilapia CP -crude protein; EE -ether extract; MM -mineral matter; R 2 -determination coefficient.

Table 4 -
Path coefficients between the variables of chemical composition and contents of digestible protein and energy

Table 5 -
Apparent digestibility coefficients of meat and bone meal with different levels of crude protein for Nile tilapia DP -digestible protein, CP -crude protein, EE -ether extract; MM -mineral matter.