Instruções para elaboração do resumo expandido para a 49a

52a Reunião Anual da Sociedade Brasileira de
Zootecnia
Zootecnia: Otimizando Recursos e Potencialidades
Belo Horizonte – MG, 19 a 23 de Julho de 2015
Modelos não lineares para descrever o crescimento de caprinos jovens da raça Alpina 1
Matheus Lima Corrêa Abreu2, Norberto Silva Rocha2, Raphael dos Santos Gomes2, Davi Leal Barbosa3, Antônio
Paulo de Oliveira Neto3, Flavio Henrique Vidal Azevedo2 e Ricardo Augusto Mendonça Vieira4
1
Parte da tese de doutorado do segundo autor.
Doutorando do Programa de Pós-Graduação em Ciência Animal – UENF, Av. Alberto Lamego, 2000, Campos dos Goytacazes, RJ, Brasil.
Bolsista da CAPES. e-mail: [email protected],[email protected], [email protected]. [email protected].
3
Graduando em Zootecnia - UENF, Av. Alberto Lamego, 2000, Campos dos Goytacazes, RJ, Brasil, Bolsista CNPq.
[email protected], [email protected].
4
Professor Associado, LZO – UENF, Av. Alberto Lamego, 2000, Campos dos Goytacazes, RJ, Brasil. [email protected].
2
Resumo: O objetivo foi avaliar diferentes modelos matemáticos para descrever perfis de crescimento de caprinos
jovens castrados. Os caprinos usados no estudo foram da raça Alpina, castrados aos 15 dias. As variáveis mensuradas
foram peso vivo (W), peso de copo vazio (EBW) e peso de carcaça no nascimento e aos 15, 90, 135, 210, 270 e 365
dias de idade. Foram testados o modelo monomolecular ou equação de Brody e modelos sigmoides de Gompertz,
Richards e o modelo generalizado de Michaelis-Menten. Ainda foram testadas quatro funções de variância
(covariância): homogênea, exponencial, assimptótica e escalonada, para cada modelo. A seleção do modelo foi
baseada no critério de informação de Akaike e suas medidas de verossimilhança. Os modelos sigmoides melhor
descreveram os perfis e o modelo de Gompertz associado com as variâncias homogênea e escalonada foi a melhor
escolha em todos os perfis de crescimento, provavelmente pela fase assimtótica mal definida.
Palavras–chave: análise do crescimento, estrutura de variância, modelo de Gompertz
Nonlinear models to describe the growth of Alpine goats kids
Abstract: The goal of the present study was to evaluate different mathematical models to describe the growth profiles
of growing male kids. The kids used in the present were Alpine wethers (castrated at 15 days of age). The variables
measured were live weight (W), empty body weight (EBW), carcass weight at birth and at 15, 90, 135, 210, 270,and
365 days of age. The models tested were the monomolecular or Brody equation; the Gompertz, Richards, and
generalized Michaelis–Menten models of sigmoid growth. In addition, four types of variance functions (covariance)
were tested, namely, homogeneous, exponential, asymptotic, and power of the mean scaling function for each model.
The model selection was based on the Akaike information criterion and its derived likelihood measures. The sigmoid
simple models better described the time profiles, and the Gompertz model associated with homogeneous and scaled
variances was the best choice in all cases of the growth profiles, most likely because of the ill-defined asymptotic
phase.
Keywords: growth analysis, Gompertz model, variance functions
Introduction
Quantitative description of growth evolved from the observation of cumulative sigmoid (S-shaped) behavior.
Growth is a time-dependent process built on the growing of several body parts to form the body proper. Therefore,
S-shaped mathematical functions operate well in the description of these growth processes, and several improvements
have been added to the original sigmoid equations commonly used for the description of growth (López et al., 2000).
The growth rate of goat kids may be an example of these phenomena because their growth rate starts to decrease soon
after the birth, and the severity of weaning shock may affect the growth rate after weaning; these constraints appear
to introduce another inflection point to the growth profile. We have observed that the growth profiles of goat kids
from ordinary dairy herds in Brazil present at least two visually detectable inflection points: one at the transient stage
during the suckling-weaning phase and the other near the pubertal period, despite the close proximity between these
two events in the time profile. Therefore, the goal of this study was to evaluate the use of traditional and biphasic
models to describe the developmental stages of growth as indicated by possible additional inflection points in the
growth profiles of kids and wethers of dairy herds.
Material e Methods
The set was formed by growing Alpine (Chamois) male kids that were sampled from the annual offspring of
a large dairy herd reared in Viçosa (MG, Brazil). Two weeks after birth, the sampled kids were moved to Campos
dos Goytacazes (RJ, Brazil). The kids were slaughtered at birth and at the intended ages of 15, 90, 135, 210, 270, and
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52a Reunião Anual da Sociedade Brasileira de
Zootecnia
Zootecnia: Otimizando Recursos e Potencialidades
Belo Horizonte – MG, 19 a 23 de Julho de 2015
365 days, with two to four animals slaughtered per age. The kids that survived more than 15 days were castrated at
15 days of age. The kids were weaning until at 90 days and received during the suckling phase 1.8 L of a 50:50 cow
milk and milk replacer (lactose content of 250 g/kg DM) mixture split between the morning and afternoon meals. The
kids after weaning received feed to reached a gain of 150 g/day according to NRC (2007). The roughage-toconcentrate ratio in the offered dry matter was approximately 1:3. The kids were weighed systematically to record
body mass (W) to the nearest 0.1 kg. The animals were rendered insensible, bled, flayed, and eviscerated. The carcass
was weighed. The segments of the gastrointestinal tract (GIT) were separated and weighed full, and each segment
was emptied and weighed after being washed and allowed to dry indoors for almost 3 h. The omental and pelvic fat
and the fat around the kidneys was removed from the carcass and weighed as abdominal fat or non-carcass fat. The
weights of all parts were recorded to the nearest 0.005 kg.
The growth profiles of the first experimental set were described quantitatively by nonlinear models. The
common assumption of normality was assumed for all variables. Nonetheless, different structures for the variances
2
were tested. The general form attributed to the nonlinear models was Y∼Normal (𝜇𝑌𝑡 , 𝜎𝑌𝑡
). The following
mathematical functions were as described to μYt:
Eq. 2: 𝜇𝑌𝑡 = 𝑌0 exp[𝑘(1 − 𝑒𝑥𝑝(−𝐷𝑡))⁄𝐷];
Eq. 1: 𝜇𝑌𝑡 = 𝑌𝑓 − 𝑏 exp(−𝑘𝑡);
Eq. 3: 𝜇𝑌𝑡 = 𝑌𝑓 𝑌0 ⁄[𝑌0𝑚 + (𝑌𝑓𝑚 − 𝑌0𝑚 ) exp(−𝑘𝑡)]
(1⁄𝑚)
; Eq. 4: 𝜇𝑌𝑡 = 𝑌0 𝐾 𝑝 + 𝑌𝑓 𝑡 𝑝 ⁄𝐾 𝑝 + 𝑡 𝑝 ;
In these equations, Yf (mass unit) is the asymptotic growth; Y0 (mass unit) is the initial growth at birth; b is a
scale parameter (mass unit); k is the specific growth rate (1/day) defined as the product 1/Y× dY/dt ; D (1/day) is the
fractional rate of decline of k; m ≥ −1 is a general exponent; K (days) is the time at which half of the asymptotic
growth (Yf) is achieved; p is a dimensionless positive constant; and t is the age expressed in days. Eqs. (1)–(4) are the
monomolecular(Brody et al., 1924), the Gompertz (Gompertz, 1825), and the Richards (Richards, 1959) growth
functions, and the generalized Michaelis–Menten growth model (GMM, López et al., 2000), respectively.
The variances (covariance) were modeled according to the following expressions:
Eq. 5: 𝜎𝑌2𝑡 = 𝜎 2 ;
Eq. 6: 𝜎𝑌2𝑡 = 𝜎 2 exp(𝑐𝑡);
Eq. 7: 𝜎𝑌2𝑡 = 𝜎02 + 𝜎𝑏2 [1 − exp(−𝑠𝑡)];
Eq. 8: 𝜎𝑌2𝑡 = 𝜎 2 (𝜇𝑌𝑡 )
2𝜓
A common error variance was assumed (𝜎 2 ) in Eq. (5) and an exponential increase of the residual variance
with time at a specific increaserate (c, 1/day) was assumed in Eq. (6). We assumed an asymptotic increase (𝜎𝑏2 )
of the initial variance (𝜎02 ) with time at a specific decreasing rate (s,1/day) independent from parameters of 𝜇𝑌𝑡 in Eq.
(10); and a variance that is a power (Ψ, dimensionless) function of the expected mean (𝜇𝑌𝑡 )and the residual variance
(𝜎 2 ) as shown in Eq. (11).
The nonlinear models were fitted to the growth profiles by means ofthe NLMIXED procedure of SAS (version
9, SAS System Inc., Cary, NC, USA). The model that presented the highest likelihood probability (wr) in representing
reality we considered the best choice to describe the growth profiles (Burnham and Anderson, 2004). Despite the
situations in which equal likelihood probabilities among models were observed (_r≤ 2.0, ERr≤ 5), the model with the
smaller number of parameters (Θr) was considered the best choice for representing the growth profile.
(𝜎02 )
Results and Discussion
Only the models that achieved Δr≤ 2.0 are listed and only the best choice for each variable presented (Tables
1 and 2). Given the variables studied, Eq. (2) were the best choices in all of the cases for 𝜇𝑌𝑡 and 𝜎𝑌2𝑡 , in 75% of the
cases associated with Eq. (5) and in 25% of the cases associated with Eq. (8). The Gompertz model, Eq. (2), better
described the variables studied; these S-shaped profiles determines a sigmoid profile for W (Table 1), as noted by
Regadas Filho et al. (2014). The growth pattern for variables (namely, W, EBW, Carcass, and AbFat weights) did not
present an evident asymptotic behavior. Equation (2) does not have an explicit parameter representing the asymptote
reached by final growth, i.e.,Yf. Therefore, the absence of Yf might have had an influence over the outstanding
likelihood performance observed for the Gompertz model. Nonetheless, the poor definition of the asymptotic phase
of the growth profiles of some of the variables reported here might be associated with the poor likelihood probability
levels observed for Eqs.(3) and (4).
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52a Reunião Anual da Sociedade Brasileira de
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Zootecnia: Otimizando Recursos e Potencialidades
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Table 1. Likelihood of the models fitted to the growing body and body parts of the Alpine wethers.
Variable
Model-variance structure
Δr ≤2.0
wr
W (kg)
Gompertz-exponential
0.0
0.159
Brody-scaled
0.2
0.144
Gompertz-scaled
0.3
0.137
Gompertz-homogeneous
0.4
0.130
ER
1.0
1.1
1.2
1.2
Θr
5
5
5
4
EBW (kg)
Gompertz-exponential
Gompertz-homogeneous
Gompertz-scaled
Brody-scaled
Brody-exponential
0.0
0.1
0.3
0.8
1.6
0.171
0.163
0.147
0.115
0.077
1.0
1.1
1.2
1.5
2.2
5
4
5
5
5
Carcass (kg)
Gompertz-homogeneous
Gompertz-exponential
Gompertz-assymptotic
Gompertz-scaled
0.0
1.7
1.8
2.0
0.237
0.101
0.096
0.087
1.0
2.3
2.5
2.0
4
5
6
5
AbFat (g)
Gompertz-scaled
0.0
0.709
1.0
W, body weight; EBW, empty body weight; Carcass, carcass weight; AbFat, abdominal fat weight.
5
Table 2. Parameter estimates of the most suited models for each one of the variables studied in mimicking
growth profiles of Alpine wethers.
Variance (𝜎̂𝑌2𝑡 )
Mean (𝜇𝑌𝑡 )
Variablea
Modelb
W (kg)
G-hg
5.4 exp(0.016(1 − exp(− 0.007t))/0.007)
2.32
EBW (kg)
G-hg
5.20 exp(0.016(1 − exp(− 0.007t))/0.007)
2.22
Carcass (kg)
G-hg
3.4 exp(0.013(1 − exp(− 0.006t))/0.006)
1.62
−1 2
(2×10 ) (µ̂𝑌𝑡 )2×0.934
AbFat (g)
G-s
65 exp(0.029(1 − exp(− 0.006t))/0.006)
a
W, body weight; EBW, empty body weight; Carcass, carcass weight; AbFat, abdominal fat weight.
b
G-hg, Gompertz model with homogeneous variance; G-s, Gompertz model with scaled variance.
Conclusions
Models based on the traditional assumption of a single inflection point (e.g., Gompertz), or even with no
inflection point (e.g., Brody), will present greater likelihoods. Furthermore, not only the inflection point but also the
growth phases (initial, exponential, and asymptotic) that characterize the S-shaped profile need to be clearly
distinguishable. It is also important to emphasize that proper modeling of the variances (covariance) is needed because
this aspect is relevant to the accurate prediction and quantification of uncertainties about the growth of the body and
body parts. These aspects are relevant for an accurate prediction of the body composition and composition of gain of
kids and wethers.
References
Brody, S.; Turner, C.W.; Ragsdale, A.C. 1924. The relation between the initial rise and the subsequent decline of
milk secretion following parturition. J. Gen. Physiol. 6:541–545.
Burnham, K.P.; Anderson, D.R. 2004. Multimodel inference: under-standing AIC and BIC in model selection. Sociol.
Methods Res. 33:261–304
Gompertz, B. 1825. On the nature of the function expressive of the law of human mortality, and on a new mode of
determining the value of life contingencies. Philos. Trans. R. Soc. 115:513–583.
López, S.; France, J.; Gerrits, W.J.J.; Dhanoa, M.S.; Humphries, D.J.; Dijkstra, J. 2000. A generalized Michaelis–
Menten equation for the analysis of growth. J. Anim. Sci. 78:1816–1828.
Regadas Filho, J.G.L.; Tedeschi, L.O.; Rodrigues, M.T.; Brito, L.F.; Oliveira, T.S. 2014. Comparison of growth
curves of two genotypes of dairy goats using nonlinear mixed models. J. Agric. Sci. 152:829–842.
Richards, F.J. 1959. A flexible growth function for empirical use. J. Exp.Bot. 10:290–300.
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