Wood (1967) reparameterized the gamma probability density function to yield a nonlinear equation that mimics records over the lactation course of dairy cows with a satisfactory fit ( Ferreira et al., 2015 ). The equation can take the form of the following expression:

The parameter μ _t represents an expected value, that is, a population mean about random lactation records over time ( t , days or weeks) from one cow or, preferably, from a random sample of cows taken from a larger population. Parameters α , β , and λ are constants, and μ _t is a nonlinear function. Isolated, λ is a fractional rate of decline expressed as reciprocal of time units ( t ⁻¹), and a formal differential equation ( Dijkstra et al., 2005 ) reveals that the specific fractional rate is time-dependent, actually:

Equation 2 represents a time trajectory instead of a single parameter that otherwise would offer a direct biological interpretation for both theoretical and practical purposes. In fact, λ is the asymptotic fractional rate for κ t ( l i m t → ∞   κ t = λ ) . Therefore, Wood's model presents some attributes revealed by analytic geometry that may be of use for animal scientists ( Figure 1 ).

We choose equation 1 because of its flexibility and feasibility to mimic lactation records as demonstrated by several authors ( Dijkstra et al., 2010 ; Ferreira et al., 2015 ; López et al., 2015 ; Ahmed et al., 2019 ). The geometric properties of equation 1 are nonlinear functions of its parameters. The time to peak milk production ( t _p ) and peak milk production ( ( μ t p ) ) have been already described ( Wood, 1967 ):

These two attributes are time-independent. The inflection point ( t _i ) of equation 1 is one of two possible solutions (i.e., roots) obtained by equating the second derivative of equation 1 to zero. These two solutions are also time-independent:

Only the root t _i > t _p is valid. All parameters must be subjected to the same constraints described by Wood (1967) . As a corollary of the parametric space for β , that is, β ≤ 1, lactation records with ascending and descending phases about peak milk yield are those with a constrained parametric space, i.e. 0 < β ≤ 1; whereas those records that show only a descending phase starting from the onset of lactation at parturition are those for which the parametric space is constrained to β ≤ 0 (concave profiles). We remind the reader that equation 1 resumes to the exponential model for β = 0. The cumulative or total milk production ( TMμ _t ) is the area under the curve used to describe the lactation records. Here, we provide an analytical solution that yields the definite integral as follows:

Numerically, the upper limit ( u ) of the integral was set at 305 d, and the lower limit ( l ) can be a small value greater than zero (10⁻⁶) for numerical integration. The incomplete gamma function is Γ [ β   +   1 ,   λ t ]   =   Γ [ β   +   1 ]   –   γ [ β   +   1 ,   λ t ] . The gamma function Γ [ β + 1 ] = ∫ 0 ∞ t β exp(− t ) dt is a numerical result from a definite integral, and the lower incomplete gamma function is given by γ [ β + 1 ,   λ t ] = ∫ 0 ∞ v β exp(− λv ) dv . The average milk production was computed as μ ¯ 0 , 305 = T M μ t / 305 .

The geometry of the function depicted in Figure 1 can yield a fractional rate constant of decline with reference to peak milk yield. The dashed descending line tangent to the inflection point after the time for peak milk yield is the geometric place defined by equation 7a :

in which ( t ', μ t ′ ) is the point where equation 7a crosses the tangent line to the point ( t _p , μ t p ), and ( d μ t / d t ) t i is the value of the first derivative of equation (1) at t _i ( Figure 1a ). The tangent to ( t _p , μ t p ) is parallel to the abscissa axis, i.e., μ t ′ = μ t p , and the abscissa coordinate t ' can be isolated in equation 7a and solved for t = t _i . Finally, if we divide both sides of equation 7a by μ t p and manipulate signs for κ _d because ( d μ t / d t ) t i < 0 , for not changing the trend of equation 7a , we have the persistency trend over time ( P _t ) as follows:

Therefore, the specific rate constant of decline after peak milk yield arbitrarily becomes the positive value, as follows:

The new parameter κ _d can be d⁻¹ or w⁻¹ and dependent on t _i ( equation 5 ) as an intrinsic attribute of equation 1 . We can relate the fractional rate described by equation 8 to the phenomenon of stromal apoptosis in the mammary gland ( Stefanon et al., 2002 ; Dijkstra et al., 1997 , 2010 ). The relative rate of decline midway between peak milk yield and the arbitrary end point of lactation ( t _f = 305) is an additional rate computed from the parameters of Wood's equation, namely r d = 2 β λ ( β − λ t f ) − 1 − λ ( Dijkstra et al., 2010 ).

Another critical factor of the lactation might be the rate of DNA proliferation in the mammary gland, which would be responsible for the rise in milk production after parturition ( Dijkstra et al., 2010 ). We can geometrically generate a point estimator for an ascending rate of milk production by calculating the average milk production during the ascending phase until peak milk yield, as follows:

for ∫ t p 305 α t β exp ( − λ t ) d t = − α λ − ( β + 1 ) ( Γ [ β + 1 , 305 λ ] − Γ [ β + 1 , λ t p ] ) . By computing the tangent of the δ angle ( Figure 1b ), as equation 10 , we have:

and the following ratio yields a constant, fractional ascendant rate of milk production given by equation 11 .

This rate can be d⁻¹ or w⁻¹. The rate κ _α can be associated with the process of cell proliferation until peak milk yield ( Dijkstra et al., 1997 , 2010 ). The constraint 0 < β ≤ 1 must hold for a valid κ _α ; nonetheless, even though estimates for the fixed parameter β are negative, there are possible occurrences of positive values from the random cow effect over parameter β that may result in typical ascending-peak-descending (convex-shaped) lactation profiles.

Brody and colleagues proposed a model to mimic the standard course of milk flow of dairy cows, that is, the rise of milk flow after parturition and its steady decline after peak milk yield ( Brody et al., 1924 ). Their model describes a biphasic phenomenon, in which the rise follows a monomolecular change and, simultaneously, an exponential decay follows the peak milk yield. Mathematically, the final equation has the form:

The mean μ _t has the same meaning as previously described, but θ ₁ and θ ₂ are scale parameters, κ ₁ (d⁻¹ or w⁻¹) is the characteristic constant of the decline of milk production after peak milk yield (“after the second month” in those authors' words), and κ ₂ (d⁻¹ or w⁻¹) is the characteristic diminishing constant of the rising course of lactation. It is possible that equation 12 mimics concave-shaped lactation profiles, but this will depend on final parametric estimates: if θ ₁ > 0 and θ ₂ ≤ 0, the equation resumes to an exponential decay. All derivations applicable to Wood's equation also apply to the Brody's equation.

There is an elegant mechanistic equation based on the pool size of DNA in the cell population of the mammary gland during lactation. This model roots on cell proliferation and death and mimics the typical convex-shaped lactation profile as follows:

Parameter μ ₀ (kg · d⁻¹ or kg · w⁻¹) is the theoretical initial milk production at parturition, θ _T (d⁻¹ or w⁻¹) represents the specific rate of cell proliferation at parturition, κ ₃ (d⁻¹ or w⁻¹) is a decay parameter, and κ ₄ (d⁻¹ or w⁻¹) is the specific rate of cell death. Dijkstra et al. (1997) showed several attributes of equation 13 as general nonlinear functions of its parameters.

The gamma probability density function (PDF) models time-to-event data ( Mood et al., 1974 ; Stroup, 2013 ). Because Wood (1967) reparametrized the gamma density function as previously mentioned, we believe that it is worthy evaluating this model in its original form because the properties of the gamma PDF may be of use to describe lactation records. We can compute the mean and variance based on this lactation function. The model can have the following form:

The mean μ _t represents the lactation course, and α and r are scale and shape parameters, respectively. Parameter λ (d⁻¹ or w⁻¹) has the same meaning as previously described. The constraints α ₀ > 0, r > 0, and λ > 0 must hold. The equivalence between equations 1 and 14 is trivial, because of β = r − 1, α = α ₀ λ ^r / Γ [ r ], and λ is the parameter in common. One can use the same rationale for Wood's model to obtain the nonlinear parametric functions of interest.

We used the nlme function of R to fit equations 1 , 12 , 13 , and 14 to the lactation records. The gnls and nlme functions belong to the nlme package ( Pinheiro et al., 2017 ). The stochastic version of the models was set as follows:

The record y t i j k corresponds to the milk production rate (kg · d⁻¹) for the k -th cow of the i -th breed, during the j -th parity order, and recorded at time t . The basic nonlinear function that fits y t i j k is g ( Θ m ,   t i j k ) = μ t i j k , ∀ μ _t related to equations 1 , 12 , 13 , and 14 . Parameter Θ _m represents the vector of parameters of the m -th model. However, prior to the fit of the model equations, we had to group lactation records by the random cow and cow × parity order intersecting effects by using the groupedData function of nlme. The resulting formula for the grouped data was y ~ Time|cow/cpo, i.e., y is daily milk yield, time is t , cow and cpo are the random cow and cow × parity order (cpo) intersection. Breed ( B ), parity order ( PO ), and their interaction ( BPO ) were the fixed effects associated with each parameter of μ _t for each model evaluated.

We modeled the variance associated with the error term ( e t i j k ) according to four conditions:

Parameters represent the residual variance ( σ ²), the power ( ψ ) that scales the absolute mean | g ( Θ _m , t_ijk )|, and the assumed positive correlation parameter (0 < ϕ < 1). The model fitted with the variance function in equation 17 challenges the fitted one based on the traditional homoscedastic assumption described by equation 16 . Parameter ω is the intercept of the VarConstPower function ( equation 18 ), which allowed modeling the variances when | g ( Θ _m , t_ijk )| = 0 or equal to a baseline or background value, and when the variance scales to the mean for | g ( Θ _m , t_ijk )| > 0 or greater than a baseline value ( Pinheiro and Bates, 2000 ; Rohem Júnior et al., 2020 ; Vieira et al., 2020 ). The fit of the model with the first-order autoregressive correlation structure given by equation 19 challenges the traditional assumption of independent time records on the same subject ( Vonesh, 2012 ); this was accomplished by fitting the corCAR1 function of nlme package ( Pinheiro and Bates, 2000 ). We also checked for possible diagonal and symmetric covariance structures for random effects by using the pdDiag (diagonal, D) and pdSymm (unstructured, U) covariance classes available in the R software ( Pinheiro and Bates, 2000 ).

The models were initially fitted with the gnls function with no random effects introduced. Thence, by using the nlme function, we gradually introduced random effects (cow and cow × parity order) in the different fixed parameters contained in g ( Θ _m , t_ijk ) creating several model versions. An example is the introduction of the random effects over parameter λ in Wood's equation, in sequence over parameters λ and β , and so on for all equations studied, which formed the different model versions. In addition, we assessed the quality of fit of the model solutions by using the I-T approach ( Buckland et al., 1997 ; Burnham and Anderson, 2004 , 2014 ; Burnham et al., 2011a , b ; Sober, 2002 ). This methodology is based on the corrected Akaike Information Criterion as AICc_m ( Akaike, 1974 ; Sugiura, 1978 ; Hurvich and Tsai, 1989 ; Cavanaugh, 1997 ), and the derived measures Δ _m ( m -th Akaike differences), p_m ( m -th model probabilities), and ER_m ( m -th evidence ratios) computed for each m −th feasible model version fitted. We discarded solutions that yielded heavy-tailed 0.95 confidence intervals (0.95CI) for variance components of the pdDiag or pdSymm structures, as well as for parameters of the correlation structure, the power-of-the-mean function, or both ( Pinheiro and Bates, 2000 ).

Only one solution satisfied all criteria for model selection ( Table 2 ). Therefore, we obtained the random effects as outputs of the nlme function and computed the nonlinear geometric functions of the parameters. In sequence, we fitted the values for each cow and cpo for all BPO , which generated new random variables to be analyzed by using the GLIMMIX procedure (SAS University Edition, SAS Systems Inc., Cary, NC, USA). We checked the patterns of the Pearson residuals to evaluate the quality of fit of the model and the assumed probability density function. The model definition was as follows:

The linear predictor is η_ijk and the Greek uppercase letters represent the i -th breed ( B_i ), the j -th parity order ( P_j ), and their interaction ( BP_ij ). We assumed that variables could be y_ijk | c_k ~ Gamma( μ_ijk , Φ ) or y_ijk | c_k ~ Normal( μ_ijk , σ ²), and the random effect of cow is identically and independently distributed as c_k ~ Normal(0, σ c 2 ). The Greek capital Φ is a scale parameter, and the link functions used were as follows ( Stroup, 2013 ): the identity link or η_ijk = μ_ijk was used for the Normal distribution, whereas the inverse and log links, namely η i j k = μ i j k − 1 or η_ijk = log ( μ_ijk ), were used for the Gamma PDF. Because the Normal distribution and the Gamma distribution produced equal residual patterns, the alternative Gamma PDF was chosen as the best solution. Therefore, the Gamma was used as a generalizing distribution for variables in the domain (0, ∞).

The Akaike criterion ( Akaike, 1974 ) corrected for small samples ( Sugiura, 1978 ; Hurvich and Tsai, 1989 ; Cavanaugh, 1997 ), namely AICc_m , has been reported as one of the most important measures to evaluate the predictive power of mathematical models in a multiple hypotheses framework ( Buckland et al., 1997 ; Sober, 2002 ; Burnham and Anderson, 2004 ). The AICc_m and its derived measures constitute the I-T approach, which allows the comparison of the quality of fit of different models and, whenever necessary, establishes parameters for model averaging. If one takes the smallest AICc_m value to choose a model over the others, does an incomplete procedure within the I-T framework ( Burnham and Anderson, 2004 ; Vieira et al., 2018 ; 2020 ). The smallest AICc_m value indicates the lower information loss due to the fit of a given model among the feasible models evaluated. However, if one assumes that a given model is the best solution and ignores the others, depending on the p_m for the chosen model, its uncertainty can be large. This is why we assumed that a low uncertainty for a given model is 1 − p_m < 0.10 ( Burnham and Anderson, 2004 ; Vieira et al., 2018 ).

The frequentist statistical tools available nowadays allow the association of random factors to the fixed parameters of nonlinear models and the fit of the resultant models by maximum or restricted maximum likelihood. In addition, one can challenge the traditional assumptions of homoscedasticity and independence of the errors with these tools based on the theories of nonlinear and generalized linear mixed-effects models ( Pinheiro and Bates, 2000 ; Vonesh, 2012 ; Stroup, 2013 ). In our study, the use of the available statistical tools (nlme from R and GLIMMIX of SAS) provided considerable improvements in model fitting and prediction. Based on the I-T approach, we observed that Wood's equation was superior in terms of quality of fit and the best choice to represent the diversity in shape and form of the lactation profiles, given the data. Nonetheless, many studies use the R ² to compare models, but this measure is not sensitive to differences among nonlinear models, which make comparisons useless because of the difficulty to demonstrate differences. Therefore, models that actually differ based on I-T may be considered equal if one bases on R ² or, alternatively, its adjusted version for model selection ( Spiess and Neumeyer, 2010 ; Hossein-Zadeh, 2016 ).

In this study, we challenged the ability of some models to mimic actual lactation records over time. Although the abundant dataset (37810 time-records) might have favored both accuracy and precision of the estimates, considerable variation may have arisen due to different ethnic types, parity orders, low number of cows for some breed-parity order interactions, and incomplete or irregular lactation records to characterize an entire lactation (0-305 days). These sources of variation might have constrained some of the results and increased the difficulty of obtaining valid, convergent, and robust solutions for some models. Nonetheless, the dataset offered the opportunity for Wood's model to exhibit its flexible nature to fit both convex and concave lactation profiles. Concave profiles are rather frequent than one might expect; these concave profiles are characteristic of animals that did not present peak milk yield, and equation 1 mimicked those profiles when other models ( equations 12 - 14 ) failed. Within the framework of nonlinear mixed-effects models, Wood's model even was able to fit lactation profiles with incomplete records, which are needed to characterize an entire lactation trend ( Table 3 and Figure 2 , panel e).

Several authors faced difficulties with fitting models to atypical lactation profiles, that is, those profiles that deviate from the standard lactation convex-shaped trend ( Rekik and Ben Gara, 2004 ; Macciotta et al., 2005 ; López et al., 2015 ). Therefore, the greater flexibility of equation 1 provides the necessary accuracy and precision to the selection of cows based on ranking their performances as parametric attributes of the lactation profiles such as peak milk yield, mean milk production, ascending and descending rates, and more importantly, the total milk production per lactation.

There was a positive agreement between random β and total milk production estimates. Therefore, the use of the peak milk yield as a parameter for selection may result in the exclusion of cows that may produce high quantities of milk, even though the shape parameter β ≤ 0 for a given breed. Our findings favored the use of all lactation profiles from all cows. They avoided an equivocated exclusion of animals before the fit of the model, i.e., all animals were evaluated regardless of the occurrence of a peak milk yield or not ( Rekik and Ben Gara, 2004 ). The poor performance of the other models ( equations 12 − 14 ) may be explained by the no prior arbitrary exclusion of concave lactation records, as well as incomplete and irregular lactation profiles from our dataset. The advantage of the nonlinear mixed-effects models relies exactly upon the joint analysis of all lactation records employing a matrix of random effects, which encompasses the variation from the random sample of cows about the fixed parameters.

There are some reports that present a poor performance for equation 1 if compared with equations 12 and 13 . However, they used the nonlinear least-squares method of estimation to fit the models, did not consider random factors over the fixed parameters, nor did challenged the independence of residuals and homogeneity of variances ( Brody et al., 1924 ; Dijkstra et al., 1997 ; Hossein-Zadeh, 2016 ). Generally, convex-shaped lactation records are selected because they resemble the typical, standard lactation profile ( Morant and Gnanasakthy, 1989 ; Dijkstra et al., 1997 ; Macciotta et al., 2005 ). Nonetheless, the final shape and form of the nonlinear model depend on its parametric estimates ( Wood, 1967 ; Grossman and Koops, 1988 ; Morant and Gnanasakthy, 1989 ; Dijkstra et al., 1997 ; Macciotta et al., 2005 ). The sign and magnitude of the β estimate determine if the curve is flatter, presents a sharp peak, or decline monotonically at the onset of lactation at parturition ( Congleton Jr. and Everett, 1980 ; Morant and Gnanasakthy, 1989 ). In addition, it is important to note that the inflection point in the descending phase is part of a typical convex-shaped lactation profile ( Druet et al., 2003 ).

Congleton Jr. and Everett (1980) and Macciotta et al. (2006) believed that the lack of points at the ascending phase do not allow the proper characterization of the typical convex-shaped lactation profiles with a characteristic peak milk yield. Nonetheless, our dataset contains concave-shaped lactation records with several time points, as the better-characterized convex-shaped profiles do. This is why the accommodation of the random factor to parameter β resulted in the superiority of equation 1 to represent the data, because final random estimates can be either 0 < β ≤ 1 and β ≤ 0. Because of the factors mentioned previously, we believe that those variations occurred when cows presented the peak milk yield on the day of parturition or nearly after parturition. These types of lactation profiles occurred for groups BPO ₄₁ and BPO ₄₃, which might be associated with the poor dairy temperament of Zebu breeds and its crossbreeds with dairy breeds ( Bangar and Verma, 2017 ). Sometimes, even cows that exhibited a typical convex-shaped lactation profile in a previous lactation fail to present a peak milk yield in a further lactation, because of environmental factors such as metabolic disorders, mastitis, other diseases, and nutritional or other management errors ( Wood, 1968 , 1970 , 1972 , 1976 , 1980 ; Macciotta et al., 2005 ; Hossein-Zadeh, 2016 ; Ahmed et al., 2019 ). Nevertheless, the cows that presented typical convex-shaped lactation profiles were those that produced more milk in the entire course of lactation than their counterparts did.

The ascending phase of equation 1 is represented by κ_α and can be associated with the proliferation of secretory cells of the epithelial parenchyma in the mammary gland of cows ( Capuco et al., 2001 ), dairy does ( Knight and Peaker, 1984 ), and mice and rats ( Dijkstra et al., 1997 ). The DNA content of the mammary gland is richer in the ascending phase, which results in greater RNA transcription and the consequent translation into enzymes responsible for the synthesis of milk components such as galactosyl transferase, fatty acid synthase, and acetyl-CoA carboxylase ( Knight and Peaker, 1984 ; Boutinaud et al., 2004 ). The continuous milk production increase may be associated with the net number of secretory cells that result from the processes of cell mitosis or differentiation ( Capuco et al., 2001 ; Dijkstra et al., 1997 , 2010 ). Homeorhetic mechanisms orchestrate changes in the mammary gland and other metabolic organs of the animals to provide the machinery and nutrients for milk synthesis by the secretory tissue. The net results are the changes observed for milk yields in the course of the lactation, including changes in other variables that dairy animals go through during the dry and transition periods ( Bauman and Currie, 1980 ; Bauman et al., 1999 ; Bauman, 2000 ).

The smaller estimates for κ_α of primiparous cows within each breed observed in our study may be explained by the incomplete development of their mammary glands. First parity order cows are younger and still grow after the first calving; therefore, there is a competition for nutrients between the processes of growth, gestation, and the subsequent lactation, which delays the growth of the mammary gland and its full secretory potential observed in a mature cow ( Bauman and Currie, 1980 ; Bauman et al., 1999 ; Bauman, 2000 ). Indeed, the mammary glands of cows ( Capuco et al., 2001 ) and dairy does ( Knight and Peaker, 1984 ) continue to grow until and after the first parturition. The volume of secretory tissue in the mammary gland of second parity order dairy does is greater than that of the first parity order does, but the greater differentiation in the quantity of secretory parenchyma occurs during pregnancy ( Fowler et al., 1990 ; Knight and Wilde, 1993 ). Those biological facts may explain the greater κ_α estimates for multiparous cows and, consequently, their greater peak milk yields estimated in our study.

The peak milk yield can be used as a parametric reference for phenotypic selection of cows because of the good positive agreement between total milk production and peak milk yield in dairy cattle ( Grossman and Koops, 1988 ; Tekerli et al., 2000 ; Hossein-Zadeh, 2014 ; López et al., 2015 ) and buffaloes ( Hossein-Zadeh, 2016 ).

The descending phase occurs soon after the peak milk yield ( Figure 1 ) ( Wood, 1967 , 1976 ; Knight and Peaker, 1984 ; Dijkstra et al., 1997 , 2010 ; López et al., 2015 ). We used the inflection point at the descending phase to obtain an estimator for κ_d . This specific constant rate can be an index of persistency: the greater the absolute values, the lower the lactation persistence, i.e., the sharper the decline after peak milk yield. Nonetheless, those cows that presented lactation profiles with greater κ_α estimates also presented greater arbitrarily positive κ_d estimates (smaller persistency). We observed a negative Pearson correlation between these two specific rates, which clearly demonstrated the relationship. Wood (1969) observed that as cows grow older, they start with greater milk yield records at parturition, but their rate of decline in milk production reduces as lactation advances post-peak milk yield (persistency reduces = larger absolute κ_d ). Our results confirm that primiparous cows present flatter lactational profiles, which means a greater persistency if compared with their older counterparts ( Grossman and Koops, 1988 ; Bauman et al., 1999 ; Cobuci and Costa, 2012 ; Hossein-Zadeh, 2014 , 2016 ; López et al., 2015 ). Therefore, the geometrization of equation 1 yielded parameter estimates that agreed with the biological trends reported in the literature.

The milk production decline post-peak yield may be associated to the reduction in the numbers of secretory cells in the mammary glands of cows ( Capuco et al., 1997 ; Cobuci and Costa, 2012 ), dairy does ( Knight and Peaker, 1984 ; Quarrie et al., 1994 ; Dijkstra et al., 1997 ), and small rodents ( Walker et al., 1989 ; Strange et al., 1992 ; Dijkstra et al., 1997 ). The number of epithelial secretory cells recorded at the peak milk yield can be reduced by 40% until the end of lactation in cows ( Capuco et al., 2001 ) and by 30 ( Knight and Peaker, 1984 ) to 40% in dairy does ( Fowler et al., 1990 ). Capuco et al. (2001) reported that the number of secretory cells results from the balance between cell proliferation and death, i.e., the quantity of secretory tissue reduces as the cell death rate overcomes the cell proliferation rate. This resultant cell loss occurs by some sort of a programmed mechanism for cell death also called cell apoptosis ( Walker et al., 1989 ; Strange et al., 1992 ; Quarrie et al., 1994 ; Stefanon et al., 2002 ). The first report about apoptosis was for no lactating mice and rats during the period of mammary gland involution ( Walker et al., 1989 ; Strange et al., 1992 ). Nonetheless, Knight and Peaker (1984) detected the reduction in the number of cells of the secretory tissue after the eighth week of lactation in Saanen does. However, the association of this reduction in the number of secretory cells to the phenomenon of lactational apoptosis in ruminants was introduced by Quarrie et al. (1994) . Therefore, from the estimation of κ_α and κ_d , we can infer quantitatively about the anatomic and physiological changes during pregnancy and mainly after parturition. We can also speculate, based on our correlated κ_α and κ_d estimates, that the faster the cell proliferation occurs, the sooner the process of cell apoptosis prevails in the secretory tissue of the mammary gland. In addition, we can use those parametric estimates to predict the impact of long-term environmental constraints over milk production trends ( Wood, 1972 ), e.g., the attenuation or worsening of the effects of the oxidative stress over the secretory capacity of the mammary gland, and the consequent result in milk production ( Bernabucci et al., 2002 , 2005 ). Other solutions may be of use for short-term effects that dramatically alter the lactation trends ( Ahmed et al., 2019 ).

We can make a direct algebraic comparison, for instance, between equations 1 and 13 . If we take the root for t_i > t_p ( equation 21 ) among other possible root solutions to d ² μ_t / dt ² = 0 for μ_t from equation 13 ^1,2 , the time at the inflection point would be as follows:

Thence, we can apply the same rationale for κ_d as described by equations 7b and 8 , with the properties described by Dijkstra et al. (1997) . In sequence, we can also obtain an equivalent specific ratio as described by equation 11 , because

In equation 22 , t p = κ 3 − 1 log ( κ 4 − 1 θ T ) , μ t p = μ 0 ( κ 4 / θ T ) κ 4 / κ 3 exp ( k 3 − 1 ( θ T − κ 4 ) ) , and the definite integral can be solved numerically. Unfortunately, the model described by equation 13 did not fit our dataset with the same flexibility as Wood's model did, and the comparison was only on a theoretical basis.