Animal growth evaluation is important to optimize management and feeding practices in livestock farming, as well as for the genetic improvement of species. In this sense, nonlinear statistical models have been adopted to describe the animal growth curve over time, which allows the evaluation of how genetic and environmental factors influence growth patterns. The shape of the growth curve can also be used in breeding programs and in individual animals, whose growth parameters are hereditary and responsive to selection procedures (Fitzhugh Jr, 1976; Wurzinger et al., 2005).

The nonlinear statistical models of Brody, Von Bertalanffy, Logistic, and Gompertz are widely used to describe animal growth curves (De Sousa et al., 2021). In these models (Table 1), parameter A represents the size or asymptotic value, B is an integration constant related to the initial values of the response variable under study and has no well-defined biological interpretation, k corresponds to the growth or maturation rate, and t is time. These parameters are fundamental to understanding the growth patterns observed in different biological contexts (Lôbo et al., 2006).

Table 1

Model	Equation
Brody	Y(t) = A(1 − Be−kt) + εi
Gompertz	Y(t) = Ae−Be−kt + εi
Von Bertalanffy	Y(t) = A(1 − Be−kt)3 + εi
Logistic	Y(t) = A (1 + Be−kt)−1 + εi

Y(t) = observed measure of response variable; A = the size or asymptotic value; B = an integration constant related to the initial values of the response variable; k = the growth or maturation rate; t = time; ε_i = fitting error.

Growth curve studies have been widely developed in beef cattle (Hartati and Putra, 2021; Arré et al., 2019), in chickens and broilers (Yadav et al., 2023; Masoudi and Azarfar, 2017) and in sheep and goats (Sharif et al., 2021; De Sousa et al., 2021), among other species. These studies are based on the use of computational methods to estimate the parameters of nonlinear models and are generally performed using statistical software such as R, SAS, and SPSS, which provide a wide variety of functions with the most diverse purpose and an environment in which users must write or select commands to carry out the study. These instructions must be entered via the command line and must follow their own syntax established by some programming language, which is a challenge for those who have no programming experience. This can lead to delays or even errors in the process of estimating the curve parameters.

In the literature consulted, few initiatives for the development of tools that make it easier to apply nonlinear models to describe animal growth were observed. Reiter and Vorholt (2024) propose a web application that allows a fast and interactive analysis of microbial growth curves. However, the resources and analyses offered by this system are not the most suitable for studying the growth curve of animals, since their proposal is to analyze microbial abundance over time. Enot et al. (2018) and Wang et al. (2017) proposed software for the study of, respectively, the growth of tumors and the child growth.

In this context, this research aims to create and evaluate FiT, a software that facilitates the analysis of animal growth curves, allowing farmers, researchers, and professionals, including animal scientists, veterinarians, extension agents, and related specialists, to quickly visualize and analyze growth curves without the requirement for coding knowledge.

The data used to validate the FiT system were originally proposed in Sarmento et al. (2006), a study that aimed to verify, among the Brody, Von Bertalanffy, Logistic, Gompertz, and Richards models, which one best fits the growth data of Santa Inês sheep. The database used consists of 7271 records of birth weight and weight at 28, 56, 84, 112, 140, 168 and 196 days of age from an experimental herd provided by the Empresa Estadual de Pesquisa Agropecuária da Paraíba (EMEPA-PB), controlled from 1983 to 2000, today called Empresa Paraibana de Pesquisa, Extensão Rural e Regularização Fundiária – EMPAER.

The FiT tool is characterized as a web-based software application developed using Python and R languages in conjunction with the Flask framework. The Python programming language was chosen because of its simplicity, clarity, and wide range of available libraries. Flask, known for its ease of use and flexibility, was adopted to efficiently build the web application structure. R, which is widely recognized as a powerful and established tool in the field of data analysis and statistics, was used to estimate the parameters of nonlinear models through the use of the nlsLM function of the minpack.lm package (Mullen, 2022). The D function of R was also used to calculate the numerical derivatives of the models and to estimate functions of the fitted models. To incorporate these R functions, the project uses the rpy2 package which allows R scripts to be called directly within the Python environment, facilitating advanced statistical analyses and the use of specialized packages available in R, ensuring a robust and adaptable solution to complex data analysis needs.

The FiT software performs the adjustment of classical nonlinear models, such as Brody, Von Bertalanffy, Logistic and Gompertz, whose parameters to be estimated during the adjustment are Â, B̂ and K̂. The convergence and quality of the results obtained are directly affected by the definition of initial values for these parameters, which is a critical and challenging activity. Considering that the explanatory variables (e.g., age or time measurements) and response (e.g., weight or other characteristic of interest) are, respectively, x and y, in FiT, the definition of the initial values of these parameters follows the following logic: A (asymptotic value): it is defined as the mean value of all y that corresponds to the highest x found in the database; B (initial value): the ratio of the mean value of all y corresponding to the lowest x found in the database by the lowest value of x itself; K (growth constant): 0.01, which was the value that presented the best results during the experiments, regardless of the model to be adjusted.

To evaluate the fit of the models, FiT calculates the metrics SSR – sum of squared residuals, SST – total sum of squares, MSE – mean squared error, R2 – coefficient of determination, AIC – Akaike information criterion, and BIC – Bayesian information criterion, in addition to the weights and ages at the inflection point (IP) and asymptotic point (AP). IP corresponds to the point at which the function changes from increasing to decreasing, indicating a deceleration in the growth rate of the variable under study (Sarmento et al., 2006). The moment at which this pace of growth stabilizes is represented by AP (Mischan et al., 2011).

According to Mischan et al. (2011), to determine the values of IP and AP in a growth curve, it is necessary to calculate the successive derivatives of the time-adjusted nonlinear model. The IP represents the value at which the second derivative of the function with respect to time is equal to zero, i.e., Y"(t) = 0, starting from the value zero on the abscissa axis. The AP represents the value at which the fourth derivative of the function with respect to time cancels out, that is, also starting from the value zero on the axis of the abscissas. Y^IV(t) = 0. The searches for the roots of the derivatives in question were carried out using the numerical method of bisection (Burden and Faires, 1985).

The evaluation of the tool was conducted by comparing the results produced by the FiT tool with the results achieved by Sarmento et al. (2006), which used the same database of Santa Inês sheep.

The evaluation of the system was conducted by comparing the results produced by FiT with the results in Sarmento et al. (2006) using the same database of Santa Inês sheep. The results achieved are quite similar (Table 2). Although the values of the parameters and evaluation metrics are slightly different, the order of the models is the same as that found in the reference work when both the MSE and R2 are considered as comparison factors. It was not possible to evaluate the convergence of the results of the Richard model, since it has not yet been implemented in the system.

Table 2

Model	Parameters estimated						MSEr	MSEf	R2r	R2f
	Âr	Âf	B̂r	B̂f	K̂r	K̂f
Bertalanffy	24.8055	24.7747	0.4799	0.4855	0.0159	0.0159	17.2646	17.2624	0.7202	0.7323
Brody	27.4134	27.5391	0.8865	0.8898	0.0094	0.0092	17.3723	17.3825	0.7185	0.7304
Gompertz	24.1653	24.1039	1.8731	1.9070	0.0191	0.0192	17.2688	17.2624	0.7205	0.7323
Logistic	23.1683	23.0721	4.3772	4.5695	0.0287	0.0292	17.4055	17.4047	0.7180	0.7301

The values Â_r, B̂_r, K̂_r, MSE_r and R2_r refer to the results found in Sarmento et al. (2006) and Â_f, B̂_f, K̂_f, MSE_f and R2_f correspond to the results obtained by the FiT.

The perceived difference is slight and can be justified by the adoption of different strategies for parameter estimation and model fitting. In FiT, which uses the nlsLM() package of R, the estimation of the parameters is based on the Levenberg-Marquardt algorithm. This algorithm is a refinement of the Gauss-Newton procedure, adopted in the reference work, which increases the chances of local convergence and avoids the occurrence of divergence (Moré, 1978). Regarding fitting strategy, in Sarmento et al. (2006), the adjustment of the models occurred for each animal, and the values of Â, B̂, and K̂ reflect the mean of these values. In FiT, the fit of the models reflects the average curve for all animals at once.

In the Charts section (Figure 4), FiT exhibits the values of the inflection and asymptotic points considering the Gompertz model, which was chosen by Sarmento et al. (2006) as the best-fit model. The weight of the animal at the inflection point (IPy) and the weight gain at the inflection point (AGRmax) also coincide with the values found in the reference study. The 1st derivative graph shows the absolute growth rates (AGR), which represent the daily weight gain over time. Furthermore, FiT calculates the AP, which is a very important additional information in the study of the growth patterns, but which is rarely presented in studies in the literature. This point represents the moment when the animal’s weight gain stabilizes and can be interpreted as an indication of the moment when the increase in weight tends to stability (Mischan et al., 2011), which can serve as a reference to indicate the moment of commercialization, slaughter, or even adjustments in the feeding strategy, since the animal can deposit more fat from this moment on.

Figure 4

It is noteworthy that the AP is not easily obtained with statistical software, as it requires additional knowledge and programming skills. Therefore, most publications in the animal growth analysis field do not present this estimate, such as Sarmento et al. (2006), even though it is very important to understand growth patterns. On the other hand, IP is a point widely addressed in the literature; however, its identification is usually based on model-specific equations. In FiT, the calculation of IP and AP is based on the identification of important points of the derivatives of the equation, as mentioned above, so it can be applied to all growth models with sigmoidal behavior, provided that there are weight samples from birth to adulthood of the animal.

In addition to the confidence in the results generated by the FiT tool, it should be noted that the user only needs a database in CSV format to carry out the study easily and quickly. Between uploading the database and completing the adjustment of the models, it took approximately 15 seconds. This saves users time and effort, as it avoids the need to invest time in learning how to use tools like R to perform a growth analysis, whether academic or not. However, it is still extremely important that the user has the necessary knowledge to interpret the results of the estimated parameters.

Built on top of modern, powerful, and established technologies like Python and R, FiT is an easy-to-use web application that streamlines the process of animal growth analysis. By adopting FiT, researchers and professionals can reduce the time required to adjust nonlinear statistical models to analyze growth curves from hours to minutes without the requirement for coding knowledge.