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This document delves into the fundamental principles of animal growth, exploring the biological processes involved and the practical applications of growth models in livestock production. It examines various mathematical models, including brody, von bertalanffy, gompertz, and richards, highlighting their strengths and limitations in predicting animal growth patterns. The document emphasizes the importance of understanding the factors influencing growth, such as genetics, nutrition, and environmental conditions, and how these factors can be manipulated to optimize production efficiency. It also discusses the use of mixed models to account for both fixed and random effects, providing a more comprehensive analysis of growth data. The document concludes by emphasizing the importance of incorporating growth models into livestock management practices to improve productivity, optimize resource utilization, and enhance animal welfare.
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República de Panamá Universidad Católica Santa María la Antigua Trabajo de Investigación Licenciatura en Ingeniería en Producción Animal Asignatura: Matemáticas I Profesor: Yordys Gonzales Estudiantes:
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Modeling of growth functions applied to animal production ABSTRACT Animal growth is one of the most important aspects when evaluating productivity in beef cattle enterprises and, in some cases, it is used as a selection criterion. However, it must be considered that growth is not exclusively due to genetic factors but also to environmental effects. To measure animal growth, different linear and non-linear mathematical models have been used, chosen based on their goodness of fit and the ease of biological interpretation of their parameters. Recently, mixed models have been used, where their parameters consist of fixed and random effects, representing the expected values and the variance of the first ones, respectively. This allows for evaluating the variability of different curves among individuals in a population, as well as the covariance between the parameters. The most commonly used criteria for selecting the curve that best fits the data are the coefficient of determination and the percentage of significant and atypical curves found for each function. Additionally, other criteria such as the Akaike information criterion and the Bayesian information criterion can also be applied. The objective of this paper is to provide the reader with an application of non- linear and mixed non-linear models in the analysis of animal growth. Key words: animal growth curves, livestock biomodeling curves, statistical models INTRODUCTION Animal growth can be described using mathematical functions that predict the evolution of live weight performance. These functions allow for evaluations of the production level in livestock enterprises, enabling the classification of the productivity of a specific breed for a given region. They also allow for the calculation of the maximum values of average and current growth, helping to determine the optimal slaughter ages to maximize economic benefits. Furthermore, they provide information for planning feeding programs, carrying capacity, and measuring genetic changes from one generation to another related to production levels. The most widely used growth functions are those proposed by Gompertz in 1825, Verhulst in 1838 (known as the logistic model), Brody in 1945, Von Bertalanffy in 1957, and Richards in 1959. Additionally, linear and polynomial functions have been used to predict adult weight or the degree of maturity without needing repeated measurements or waiting for the animal to reach maturity to make production decisions. Lack of knowledge about growth curves and productive parameters of economic interest has limited the implementation of livestock improvement programs that could increase productivity, such as growth rate, maturity rate at different ages, and slaughter age. These factors can be analyzed based on the zootechnical information of the animals, which requires keeping production records. It is necessary to carry out studies to identify the
mathematical functions that best fit local conditions and to know the genetic parameters to select the best animals. Animal growth begins at the prenatal stage with the fertilization of the egg and ends when the organism reaches its adult weight and the species' specific body conformation. Growth is defined as the quantitative increase in body mass, which translates into weight gain over time. Weight gain occurs through three processes: hyperplasia (cell multiplication), hypertrophy (increase in cell size), and metaplasia (cell transformation). Therefore, animal growth is a cellular response to different factors that may be inherent to the animal or external. If the growth process is not affected by any inhibitory factors, the organism typically follows a mechanism of constant cell multiplication, and once sufficient multiplication occurs, hypertrophy takes place. However, it is possible for inhibitory factors to appear and stop the hyperplasia process, thus halting growth. Although different systems develop in parallel, the rate at which they do so is different, following a strict order of development: the nervous system is the first to develop, followed by the skeletal system, then the muscles, and finally the accumulation of adipose tissue. The increase in live weight throughout an animal's life is a complex phenomenon that depends on the animal's genotype, environmental factors such as nutrition, management, health status, and climatic effects, which have a greater impact during the early stages of growth. Some of these factors persist over time and generate a variable effect with age and animal development. Others, on the contrary, may only affect growth over short periods. Genotypic factors influence fetal development and manifest from birth to adulthood. The offspring grow slowly during the first postpartum month, but then enter a phase of rapid growth until they reach puberty, after which the growth rate decreases until stabilization in adulthood. When comparing different groups of animals, it should be considered that they have been subjected to the same conditions and belong to the same genetic group. It is also recommended to create contemporary groups, take measurements at the same time intervals, and include control groups to obtain more accurate results and analyses. Figure 1 shows the evolution of the animal's weight over time, as well as the growth rate. Initially, weight gain is greater than in adulthood, with a concave upward evolution curve. As the individual develops, the growth rate decreases, and a change in curvature is observed, identifying a point of inflection that corresponds to the maximum value of the growth rate curve. From that moment on, growth slows down, causing the weight gain curve to gradually decline and the evolution curve to increase more slowly until growth stops and the individual's weight stabilizes, which mathematically corresponds to the horizontal asymptote.
The rannor command generated random values for sem0, sem1, and sem2 with a mean of zero and variance of one. These values were multiplied by the deviation value and added to the parameter mean. The value in parentheses after the rannor command indicates the seed value (e.g., rannor(5)) (see Table 2). To display the database with the animal information, the following programming was executed: sas CopiarEditar proc print data=group1; run; To characterize the population and the growth curves for each individual, the mean values, standard deviation, and the minimum and maximum values of the response variable at a given time were obtained. Additionally, to visualize the growth curve of the individuals, the following programming was used (see Table 3). The parameters of the function were initially estimated using the least squares method through the proc nlin command, which is used in non-linear regressions. This procedure first requires the initial or starting values to then evaluate the residuals and the sum of squares for each combination of values, thereby determining the best values for carrying out the algorithmic interaction. There are four methods available for this process: Gauss-Newton, Marquardt, gradient (steepest), and multivariate secant (false position). If none of these methods is specified in the programming, the program defaults to Gauss-Newton. Table 3. Programming to characterize the population sas CopiarEditar proc sort; proc means; var y; by t; proc plot; plot y*t; run; In the programming, a range was included in which the parameter could be located. In this case, it was:
This procedure analyzes the population in general without considering the relationship within each individual. To account for this, the by command was applied, allowing analysis by individual and obtaining parameter estimates for each individual. With the means procedure, the means and variances for each parameter in all individuals were obtained. The programming is shown in Table 5. Table 5. Procedure to obtain the means and variances sas CopiarEditar proc sort data=group1; by animal; proc nlin data=group1 noitprint noprint; parms beta0 = 220 to 550 beta1 = 0.7 to 0. beta2 = 0.001 to 0.0019; model y = beta0 * (1 - beta1 * exp(-beta2 * t)); by animal; output out=brody parms=beta0 beta1 beta2 p=y r=resid; proc means n var mean; var beta0 beta1 beta2 sem0 sem1 sem2 resid; run; To check whether the parameters were correlated or not, since the data simulation assumes that they present low or no correlation, a numerical analysis was performed using the proc corr command and a graphical analysis with the proc plot command. The programming used for this is shown in Table 6. Table 6. Procedure to check parameter correlation sas CopiarEditar proc corr data=brody cov; var beta0 beta1 beta2; proc plot data=brody; plot beta0beta1="." beta0beta2="." beta1*beta2="."; run; RESULTS Figure 2 shows the output, which includes the following information:
Figure 12. Correlation between parameters beta0 and beta1 (a), beta0 and beta2 (b), and beta1 and beta2 (c) The above procedures were performed with all fixed parameters. Figure 13 shows the list of mixed models that have been specified, including the database, the dependent variable, the random effect, and the distributions and optimization techniques. These latter aspects do not need to be programmed. Additionally, the dimensions of the database can be observed: the total number of observations, the number of observations per individual, and how many of these have not been used. This output serves to verify that the specified data and the model are running correctly. Figure 13. General information on the database In Figure 14, the list of parameters with their respective starting values can be seen, as well as the Negloglike value, which evaluates the model. Also shown is the number of iterations the system required to find parameter convergence and the list of criteria used to evaluate the model or compare it with others (see Figure 15). These values were later considered for comparison with the values obtained when the database was processed with two random parameters. Likewise, the estimated values for each of the parameters and their respective standard errors are shown. Also displayed are the degrees of freedom, equal to the number of individuals minus the number of random parameters. In the last column (see Figure 14), the optimization gradient can be observed, indicating a sufficiently small value that can be interpreted as a stationary point for each parameter (see Figure 16). By comparing the information obtained with the information in Figure 7, it can be seen that more information is obtained (lower, upper), which can be used to carry out a better analysis of the database. Furthermore, the values of the parameters (s0 and sigma) are obtained. Figure 14. Output with parameter starting values and iteration history for convergence Figure 15. Values for model evaluation, logarithm of maximum likelihood, AIC, and BIC under the mixed non-linear model methodology with one random parameter In Figure 17, the evaluation methods for the model are identified, showing that the BIC value is significantly lower than the calculated value in the single random parameter model (see Figure 15), indicating that the mixed model with two random parameters is a better model. Figure 16. Estimated values of the parameters under the mixed non-linear model methodology with one random parameter Figure 17. Values for model evaluation,
logarithm of maximum likelihood, and AIC and BIC under the mixed non-linear model methodology with two random parameters In Figure 18, it is clearly shown that the estimated parameter values are similar to the values simulated in the group1 database, which indicates that the model is making a good fit. The calculated values for the parameters were:
structures, unbalanced data, and, in some cases, non-normal data, while flexibly modeling complex data structures. Advances in statistical methodologies and improvements in computational tools have substantially enhanced data analysis systems in livestock enterprises. This information can now be used to make projections and plans in production systems. Mixed models are a rigorous methodology when analyzing correlated data, but their implementation requires that the data to be analyzed have a repeated measures structure. In recent years, this method has been used in fields such as:
Modelado de funciones de crecimiento aplicadas a la producción animal RESUMEN El crecimiento animal es uno de los aspectos más importantes al evaluar la productividad en las empresas de ganado de carne y, en algunos casos, se utiliza como criterio de selección. Sin embargo, debe considerarse que el crecimiento no es exclusivamente atribuible a factores genéticos, sino también a efectos ambientales. Para medir el crecimiento animal, se han utilizado diferentes modelos matemáticos lineales y no lineales, elegidos en función de su bondad de ajuste y la facilidad de interpretación biológica de sus parámetros. Recientemente, se han utilizado modelos mixtos, donde sus parámetros consisten en efectos fijos y aleatorios, representando los valores esperados y la varianza de los primeros, respectivamente. Esto permite evaluar la variabilidad de diferentes curvas entre individuos en una población, así como la covarianza entre los parámetros. Los criterios más comúnmente utilizados para seleccionar la curva que mejor se ajuste a los datos son el coeficiente de determinación y el porcentaje de curvas significativas y atípicas encontradas para cada función. Además, se pueden aplicar otros criterios como el criterio de información de Akaike y el criterio de información bayesiano. El objetivo de este artículo es proporcionar al lector una aplicación de modelos no lineales y modelos mixtos no lineales en el análisis del crecimiento animal. Palabras clave: curvas de crecimiento animal, curvas de biomodelado ganadero, modelos estadísticos INTRODUCCIÓN El crecimiento animal puede ser descrito utilizando funciones matemáticas que predicen la evolución del rendimiento en peso vivo. Estas funciones permiten evaluaciones del nivel de producción en las empresas ganaderas, facilitando la clasificación de la productividad de una raza específica para una determinada región. También permiten calcular los valores máximos de crecimiento promedio y actual, ayudando a determinar las edades de sacrificio óptimas para maximizar los beneficios económicos. Además, proporcionan información para la planificación de programas de alimentación, capacidad de carga y medición de cambios genéticos de una generación a otra relacionados con los niveles de producción.
Las funciones de crecimiento más ampliamente utilizadas son las propuestas por Gompertz en 1825, Verhulst en 1838 (conocida como el modelo logístico), Brody en 1945, Von Bertalanffy en 1957 y Richards en 1959. Además, se han utilizado funciones lineales y polinomiales para predecir el peso adulto o el grado de madurez sin necesidad de mediciones repetidas o esperar a que el animal alcance la madurez para tomar decisiones de producción. La falta de conocimiento sobre las curvas de crecimiento y los parámetros productivos de interés económico ha limitado la implementación de programas de mejora ganadera que podrían aumentar la productividad, como la tasa de crecimiento, la tasa de madurez a diferentes edades y la edad de sacrificio. Estos factores pueden ser analizados con base en la información zootécnica de los animales, lo que requiere llevar registros de producción. Es necesario llevar a cabo estudios para identificar las funciones matemáticas que mejor se ajusten a las condiciones locales y conocer los parámetros genéticos para seleccionar los mejores animales. El crecimiento animal comienza en la etapa prenatal con la fertilización del óvulo y termina cuando el organismo alcanza su peso adulto y la conformación corporal específica de la especie. El crecimiento se define como el incremento cuantitativo de la masa corporal, que se traduce en aumento de peso a lo largo del tiempo. El aumento de peso ocurre a través de tres procesos: hiperplasia (multiplicación celular), hipertrofia (aumento del tamaño celular) y metaplasia (transformación celular). Por lo tanto, el crecimiento animal es una respuesta celular a diferentes factores que pueden ser inherentes al animal o externos. Si el proceso de crecimiento no se ve afectado por factores inhibitorios, el organismo tiende a seguir un mecanismo de multiplicación celular constante, y una vez que ocurre una multiplicación suficiente, se lleva a cabo la hipertrofia. Sin embargo, es posible que aparezcan factores inhibitorios y detengan el proceso de hiperplasia, deteniendo así el crecimiento. Aunque diferentes sistemas se desarrollan en paralelo, la tasa a la que lo hacen es diferente, siguiendo un estricto orden de desarrollo: el sistema nervioso es el primero en desarrollarse, seguido por el sistema esquelético, luego los músculos, y finalmente la acumulación de tejido adiposo.
Tipo de Estudio El estudio fue de tipo teórico-experimental, consistiendo en el desarrollo de procedimientos para el análisis de medidas repetidas en el estudio del crecimiento animal. Modelado de Datos Aplicación del Modelo Fijo A través del modelo de Brody, se analizó la información simulada de 500 animales, utilizando como referencia los valores de parámetros encontrados por Fonseca y Aquino (8) en ganado Nelore (hembras y machos). Los valores de los parámetros reportados por los autores se muestran en la Tabla 1. Tabla 1. Parámetros de ajuste con el modelo de Brody en ganado Nelore (machos y hembras). Tabla Parámetro Nelore hembras y machos (grupo 1) β₀ (peso adulto) 370 ± 41 β₁ (parámetro de ajuste) 0.9205 ± 0. β₂ (parámetro de precocidad) 0.00020 ± 0. Fuente: Adaptado de (7). Para desarrollar el modelado, se simuló una base de datos que contenía información sobre el peso de los animales. En la programación, β₀ se definió como sem0, β₁ como sem1 y β₂ como sem2. Además, se utilizó un tiempo total (t) de 2160 días con intervalos de 90 días entre pesajes, y se consideró Y como la variable de respuesta expresada en kg. La programación para generar los datos simulados se llevó a cabo utilizando el procedimiento IML del paquete estadístico SAS para crear la base de datos con la información de los animales, llamada group1 (ver Tabla 2).
Tabla 2. Programación para crear la base de datos de animales para el grupo 1 sas data group1; do animal = 1 to 500; sem0 = (rannor(6) * 41) + 370; sem1 = (rannor(4) * 0.0116) + 0.9205; sem2 = (rannor(5) * 0.0003) + 0.0020; do t = 0 to 2180 by 90; y = sem0 * (1 - sem1 * exp(-sem2 * t)); output; end; end; run; El comando rannor generó valores aleatorios para sem0, sem1 y sem2 con una media de cero y una varianza de uno. Estos valores fueron multiplicados por el valor de desviación y sumados a la media del parámetro. El valor entre paréntesis después del comando rannor indica el valor de la semilla (por ejemplo, rannor(5)) (ver Tabla 2). Para mostrar la base de datos con la información de los animales, se ejecutó la siguiente programación: sas proc print data=grou1; run; Para caracterizar la población y las curvas de crecimiento de cada individuo, se obtuvieron los valores medios, la desviación estándar y los valores mínimos y máximos de la variable de respuesta en un momento dado. Además, para visualizar la curva de crecimiento de los individuos, se utilizó la siguiente programación (ver Tabla 3). Los parámetros de la función fueron inicialmente estimados utilizando el método de mínimos cuadrados a través del comando proc nlin, que se utiliza en regresiones no lineales. Este procedimiento primero requiere los valores iniciales para luego evaluar los residuos y la suma de cuadrados para cada combinación de valores, determinando así los mejores valores para llevar a cabo la interacción algorítmica. Hay cuatro métodos disponibles para este proceso: Gauss-Newton, Marquardt, gradiente (más pronunciado) y secante
