Mathematical Modeling of Lactation Curves: A Review of Parametric Models

1. Introduction

The lactation curves provide useful information for selection programs and for developing suitable management decisions and production strategies at the farm level. So, the modeling of the lactation curve is not a new research topic, the first reference of a model of lactation curve is attributed to Brody et al. [1]. Because of computational difficulties, and the limitations of computer means, early models of lactation curves based on simple logarithmic transformations of exponentials, polynomials, and other linear functions were developed [2]. Mathematical models are able to predict milk yields. The application of these models on the first lactations data can provide important predictive information. Indeed, predicting the evolution of milk production at the individual or herd level is a powerful tool for managing herd performance. Linear and nonlinear methods were used to estimate the parameters of production peak and inclining and declining phases of milk production during lactation [3]. The incomplete gamma function or Wood function proved relatively powerful in fitting the observed daily milk yield [4]. According to Wood [5], knowledge of the parameters of lactation curves can predict total production from a single control, regarding the number of controls available for prediction. In recent years lactation curve fitting functions have been implemented in dairy farm management software [6]. Moreover, lactation curve modeling is a tool for monitoring individual yields, feeding planning, early detection of diseases before clinical signs appear, and selecting animals for breeding [7]. Another frequently advanced interest in curve production modeling is the measure of persistency. The selection of animals with higher persistency (low decrease of production during the second phase of lactation) is interesting. Thus, cows showing higher yield at the peak of production followed by rapid decline are undesirable and will be easily detected and identified using the adjusted lactation curve. The cost of milk production depends largely on the lactation persistency. The unexpected drop in production after the peak increases the cost of production, because of production inequitably along the lactation [8]. Economically, cows with flat lactation curves are more persistent and produce milk at lower cost [9]. Indeed, the incidence of metabolic and reproductive disorders, arising from the physiological stress of high milk production, would be lowered. The animal may have a more stable diet, favoring in particular the proportion of fodder in the ration and thus reducing production costs [8]. Thanks to these interests and utilities of the lactation curve, the choice of one or more appropriate mathematical functions capable of effectively describing the evolution of milk production throughout lactation is a crucial point. Thus, the interest of the study of the lactation curve is reflected by a multiplicity of mathematical models. Really, the mathematical representation of milk production during the lactation period is one of the most successful applications of mathematical modeling in agriculture [10]. The choice of a model as well as the quantity and the quality of the information necessary for its estimation must, therefore, be reasoned according to the desired use. For example, for studies of the effect of environmental factors on the shape of the lactation curve and the estimation of the classical parameters of the curve by adjusting the data sets classified by groups of animals according to a well-defined factor, a simple model with fewer parameters can meet these objectives. The selection of a mathematical function must, therefore, be based on the ease of parameter estimation, its versatility (possible modeling of the different constituents of the milk and not just the quantity of milk), and the quality of the adjustment. Guo and Swalve [11] recommend the use of the model with the lowest possible number of parameters. The availability of data collected through individual lactation and the development of genetic evaluation methods based on elementary controls, as well as the evolution of the specific requirements of the dairy cattle industry, have oriented the interest of modelers toward more linear functions flexible and general, such as polynomials or splines [12].

2. Description of the standard lactation curve

Milk production evolves during lactation following a cycle that is similar in all dairy females and usually characterized by two different phases: an ascending phase from parturition to peak production (the maximum production) and a downward phase, from this peak to the dry period. The slope of this phase represents the persistency of lactation [13]. Knight and Wilde [14] explain that this phenomenon is related to the exponential increase in the volume of secretory cells, during gestation due to the phenomenon of hyperplasia (proliferation of cells) and between calving and the peak of lactation by hypertrophy (intensification of their activity). The descending phase of lactation is the longest during which the milk secretion gradually decreases until dry up. This second phase is explained by the involution of secretory cells but especially by the fall of their numbers. The phenotypic expression of these biological processes is represented by standard or typical form (Figure 1) of the lactation curve, obtained by plotting on the abscissa the time elapsed since calving and on the ordinate the corresponding daily production [15, 16].

The general appearance of this curve is relatively constant between many domestic species. The lactation curve can be characterized by a number of parameters [13]:

The length of lactationdefined by the subscale interval. In most studies concerning lactation curve modeling, this duration is standardized at an interval of 5–305 days. Animals with lactation times greater than 305 days are considered lactations of 305 days [12]. Currently in most countries, several cows have prolonged lactations beyond 305 days.

Total productionobtained by combining daily milk production. Total milk yield is considered to be of high economic importance [18]. Total production is also the area under the lactation curve, which mathematically translates as the integral of a mathematical function over the interval of lactation duration.

Initial production (y 0)estimated by the average of the productions of the 4th, 5th, and 6th postpartum days [13].

The growth rate in the ascending phase. This phase ranges from calving to maximum production. Masselin et al. [13] expresses the growth rate of the increasing phase by the difference between the maximum (ym) and the initial production (y0).

The maximum daily production (ym) and the date at which this maximum (tp) is observed.

The peak of production is the highest yield of lactation, and its date is expressed in week (Wood, 1967) or in day. When a mathematical model of adjustment of the lactation curve is available, parameters y_mand t_pare obtained, respectively, as ordinate and abscissa of the point where the derivative of the function of adjustment of the curve dytdtis equal to 0.

The persistency of production in decreasing phase.It is most often identified with a measure of decay of production in a period of time. It is defined as the ability of a cow to maintain milk production after the peak. Assuming uniform nutrition and management conditions, the post-peak decline rate is calculated as the proportion of the decline in milk yield from the previous month, usually ranging from 4 to 9% [14].

3. Mathematical modeling of lactation curve

The interest of the lactation curve is reflected by a variety of mathematical models proposed to describe or predict it. These models are appreciated and used because they have a simple biological or economic interpretation [19].

4. Linear adjustment models

Dairy production can be considered as a linear combination of the time since calving [13]. Gaines [18] developed a simple linear first-degree model to measure the persistency:

In this model, parameter ais an estimator of initial production, and bwas proposed as a direct measure of absolute persistency and assumed to be constant during lactation. This model has been compared to other proposals [45], and it has been used to compare feeding strategies of dairy cows with their performance [46].

After the application of this simple linear model, several researchers have attempted to adapt the parabolic or quadratic model whose general form is as follows:

Dave [47] used a quadratic form for modeling the lactation curve of water buffalo in first lactations:

Sauvant and Fehr [48] sought to adjust lactation curves of dairy goats by a third-degree polynomial:

In this model, the interest of the presence of the term of degree 3 with respect to the parabolic model is the introduction of an asymmetry in the curve. The limits of this model are at the level of its parameters b, c, and dwhich have no biological or zootechnical sense [13]. Dag et al. [49] reported that the cubic model best matched the data collected in Awassi ewes and allowed for an appropriate description of the shape of the lactation curve. These authors also indicated that the total milk yield estimated by the cubic model was similar to that obtained by the Fleischmann method.

Other higher-degree polynomials have been used to model milk production. With these models, the parameters remain simple to estimate. However, interpretation and biological significance remain a major difficulty. In addition, the adjustments obtained by some authors are satisfactory, but as indicated by Masselin et al. [13], they cover a portion of the lactation curve. Nelder [50] suggested that if biological responses over time were to be modeled with quadratic regression, then it was better to first perform an inverse data transformation. However, a polynomial of the second order is unbounded and tends to be symmetrical with respect to its stationary point, while the characteristic lactation curve is not symmetrical with respect to the asymptote. To obtain an asymmetry, it will be necessary to estimate more parameters. An inverse quadratic polynomial is bounded and generally nonnegative and has no integrated symmetry. As a result of these suggestions, Nelder [50] proposed a polynomial function (called the inverse function) to adjust the response curves in multifactorial experiments, particularly in the context of modeling lactation curves. Day t production is described as follows:

Yadav et al. [51] were the first to value this design to model the lactation curve of dairy cattle and found it appropriate. According to Batra [52], this model gave a good fit for low-yielding lactations with an early lactation peak. Based on the coefficient R², the same author observed that the function 18 gives a better fit than the gamma function when weekly milk control data were used. This function was also greater to parabolic and exponential Wood models for modeling average lactation curves using weekly data from Hariana cows [53]. Olori et al. [38] showed that model 18 underestimates the milk yield around the peak and overestimates it immediately afterwards. An inverse transformation of data as proposed by Nelder [50] to obtain the properties will allow to model the lactation curve with a quadratic polynomial. Singh and Gopal [54] increased the number of parameters by including the term Log (t)as an additional co-variable. The introduction of the logarithm breaks the symmetry of the parabolic model [13]. Therefore, these authors have proposed two new models: linear cum log model:

and quadratic cum log model:

These authors indicated that these models were superior to the Wood models and the inverse polynomial when fitted to the bi-weekly controlled performance data. At the same time as one of the limits, these models are undefined at t = 0, because Log (t) = ∞ [16], although these models have not been widely applied. They have contributed as support for the development of other models. Ali and Schaeffer [55] added the term e(Log(t))² to the second model of Singh and Gopal [54] and proposed the use of a five-variable linear model:

where X = t / length of lactation, ais a parameter associated with the peak of production, and fand eare associated with the upward part of the production curve and band cwith the descending part.

Linear model has a greater number of coefficients that allow the adjustment of large forms, while its parameters do not have a technical and biological meaning [12]. Two mathematical models (Ali and Schaeffer and Wilmink) have been used successfully to adjust individual curves [2, 56]. Both models have also been implemented in earlier versions of random regression models [57, 58, 59]. A potential problem is raised by the author authors of model 21, and it is that the parameters of the regression model are strongly correlated, which can strongly limit its use. However, the results reported in several studies using this model are contradictory. According to Jamrozik et al. [60], this model gives results very similar to those obtained with the Wilmink model, despite the fact that the Ali and Schaeffer model includes additional parameters. Guo and Swalve [11, 61] have found that this model is less efficient than others, notably that of Wilmink. Concerning the limits of this model, Macciotta et al. [56] found very strong correlations in absolute values (from 0.85 to 0.99) of lactation curve coefficients with a standard form [62]. The correlations obtained by these authors are much higher than those reported by Ali and Schaeffer [55]. Olori et al. [38] compared different functions and showed that the function of Ali and Schaeffer was slightly better than that of Wilmink. Quinn et al. [63] reported that this model is inappropriate for the description of milk component lactation curve (percentages of fat and protein). Schaeffer and Dekkers [64] proposed to adjust the lactation curves by a logarithmic model:

Guo and Swalve [11] introduced the mixed logarithmic model:

This model can be considered as inspired by model 19 suggested by Singh and Gopal [54], substituting time tfor the square root of tin the second term of the model. However, this model tends to underestimate peak yield, while overestimating post-peak yield [38]. Catillo et al. [65] reported that this model was effective in estimating lactation curves and milk production characteristics of Italian water buffaloes.

5. Other models

For a competitive model, Papajcsik and Bodero [66] have introduced trigonometric functions and combined functions with increasing variation such as tb,1−exp−t, Logt, and arctantand decreasing functions such as, and, where arc-tanand cosh, respectively, refer to the arc-tangenttrigonometric function and the hyperbolic cosine function. These reflections gave birth to the following six models:

The authors compared the effectiveness of 20 different mathematical models, including these six models, and concluded that model 24 and the Wood model better fit the data of the Holstein cows. Although, this study is cited in most of the following studies as an advantage for Wood’s function over the model proposed, but we note that this work directs the thinking of modelers to the possibility of the use of complex mathematical models and particularly of trigonometric mathematical functions. An approach was introduced by Grossman and Koops [67] who proposed that lactation could be visualized as a multiphase biological process, that is, visualizing lactation as a two-step process divided in a slant until the peak of lactation is established as the first phase and the progressive decrease in production after the peak as a second phase. The suggested multiphasic logistic function determines the total milk yield by obtaining the sum of the yield resulting from each phase of lactation:

where nis the number of lactation phases considered and tanhis the hyperbolic tangent function. For each phase i, the maximum yield is equal to aiand biand occurs at time ci. The duration of each phase is associated with 2bi−1which represents the time required to obtain 75% of the total asymptotic production during this phase. This model has been applied in two-phase or three-phase model, with better adjustment resulting from the three-phase model with a low correlation between residues. For a two-phase model, the first phase could be considered the peak phase because of its proximity to the general spike and its short duration. Likewise, the second phase must be studied because it corresponds to the phase of “persistency.” This model has been criticized by Rook et al. [68] who reported lack of justification for lactation to be visualized as a multiphase process. Despite the wide range of models available to adjust lactation curves, the situation cannot be considered satisfactory because of the importance of the remarks that can be made to most of the proposed models. In this regard, an almost general reproach of the comparison of the adjustment quality of these models is the insufficiency of the evaluation of the adjustment’s quality. Indeed, the coefficient of determination, which is only a very partial element of this evaluation, is one of the main parameters used for this purpose. Recent studies incorporate residue variation, and the diversification of the results in this respect according to the data used prevents the publication of standard criteria for comparing the quality of the residues of fit models. Olori et al. [38] adjusted data from a single, uniformly managed herd to five mathematical models. They concluded that the relevance of the empirical models of the lactation curve does not depend on the mathematical form of function but on the biological nature of lactation. We notice that the adjusted character remains in most cases the raw milk production. So, it seemed logical to compare the quality of the models for the quantitative level of production.

6. Lactation curve of milk constituents

Lactation curves relating milk yield and milk constituents would be considered for influencing different of lactation curves. However, it is necessary to consider factors that influence milk yield and milk constituents, for example, different genetic, effect of climate, and nutrition. These factors will cause changes in milk compositions and milk yield which may be the data for prediction of the lactation curve and lactation persistency [72]. This link has already been studied at the phenotypic and genetic level. The genetic correlations of milk yield with negative percent of fat and protein were negative (0.25 and 0.27, respectively), and the same sign was observed at the phenotypic level (0.28 and 0.39, respectively) [72]. The ordinary description of milk secretion refers to the appearance of changes in milk composition during lactation, i.e., the decrease in milk yield is accompanied by an increase in fat and protein contents. Milk composition can be used as a diagnostic and monitoring tool in nutritional assessment [73]. Several studies have shown a correlation between energy levels and milk composition using different traits such as fat/protein ratio (FPR), protein/fat ratio, fat/lactose ratio [74]. Higher FPR values are associated with a decrease in dry matter intake and an increase in fat mobilization on the negative energy balance phase after calving [73]. Thus, FPR changes in milk may be an indication of a cow’s ability to adapt to the requirements of milk production and postpartum reproductive efficiency [75]. The richness of the milk (fat and protein contents) follows an inverse curve to that of the milk secretion, mainly because of the effect of the dilution. It decreases rapidly during the first weeks, stabilizes at a minimal level (nadir point), and rises again due to less dilution. The relative composition of milk constituents changes profoundly during the first days after parturition. The concentration of immunoglobulins decreases rapidly after parturition in favor of caseins. The nadir point of the fat concentration curve is reached approximately 3 weeks after peak lactation of milk yield, while that of protein is established near the peak of lactation [15]. As a result, milk fat and protein are often modeled with the same functions as those used to model milk production, provided that they can take a convex form. Most of the models generated for the description of the lactation curve focused only on milk yield, although the first reference found on the adjustment of lactation curve adapted to milk composition was that of Wood [28] who studied its seasonal variation. Figure 2 illustrates the shape of the lactation curve of milk yield and its fat and protein composition, expressed as a percentage and adjusted by the incomplete gamma function of Wood.

7. Conclusion

Mathematical models allow the lactation curve to be represented in terms of a set of parameters to be estimated. Various models have been used to study the lactation in dairy cattle. Each function has advantages and disadvantages. Parametric models such as Wood and Wilmink models have several advantages. Indeed, parametric models offer the possibility to calculate primary and secondary parameters, which have a biological meaning and are therefore easy to interpret. These parameters are key elements to study the effect of the environment factors on the shape of the lactation curves. Recently the increased availability of records per individual lactations and the genetic evaluation based on test day records has oriented the interest of modelers toward general linear functions, as polynomials or splines. But these functions present some computational difficulties especially at the level of the lactation curves parameters.