Modeling Lactation and Optimizing Milking

Chapter 6

Abstract

Mathematical Modeling of

Parametric Models

Mahdi Bouallegue and Naceur M'Hamdi

Lactation Curves: A Review of

The mathematical representation of milk production against time represents one of the most successful applications of mathematical modeling in agriculture. Models provide summary information, which is useful in making management and breeding decisions. Several empirical mathematical functions have been proposed to describe the lactation curve of dairy cattle differing in mathematical properties, in the number of parameters and in their degree of relationships with the main features of a typical lactation pattern, such as peak yield, time at peak and persistency. This review gives an overview of the parametric models used to fit of lactation curves in dairy cattle. Parametric models are those that found large application to fit the lactation curves, basically due to their limited mathematical complexity, and their abilities to fit a large kind of curves. Models to describe the lactation curve have been classified into two main groups: linear and nonlinear models. Nonlinear parametric functions have represented the preferred tools for modeling lactation curves with the main aim of predicting yields and parameters describing the shape of the curve in addition to important parameters such as peak yield and persistency.

Nonlinear models need iterative techniques to be solved. Different iterative

Keywords: mathematical modeling, parametric models, nonlinear regression,

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

goodness of fit, shape of lactation curve

1. Introduction

95

methods frequently employed in nonlinear regression models are Marquardt, Newton, Gauss and Dud. Wood model was the most popular parametric model with the largest application can be found in the immediate and easy understanding of relationships between its parameters and main curvatures of the lactation pattern.

#### Chapter 6

## Mathematical Modeling of Lactation Curves: A Review of Parametric Models

Mahdi Bouallegue and Naceur M'Hamdi

#### Abstract

The mathematical representation of milk production against time represents one of the most successful applications of mathematical modeling in agriculture. Models provide summary information, which is useful in making management and breeding decisions. Several empirical mathematical functions have been proposed to describe the lactation curve of dairy cattle differing in mathematical properties, in the number of parameters and in their degree of relationships with the main features of a typical lactation pattern, such as peak yield, time at peak and persistency. This review gives an overview of the parametric models used to fit of lactation curves in dairy cattle. Parametric models are those that found large application to fit the lactation curves, basically due to their limited mathematical complexity, and their abilities to fit a large kind of curves. Models to describe the lactation curve have been classified into two main groups: linear and nonlinear models. Nonlinear parametric functions have represented the preferred tools for modeling lactation curves with the main aim of predicting yields and parameters describing the shape of the curve in addition to important parameters such as peak yield and persistency. Nonlinear models need iterative techniques to be solved. Different iterative methods frequently employed in nonlinear regression models are Marquardt, Newton, Gauss and Dud. Wood model was the most popular parametric model with the largest application can be found in the immediate and easy understanding of relationships between its parameters and main curvatures of the lactation pattern.

Keywords: mathematical modeling, parametric models, nonlinear regression, goodness of fit, shape of lactation curve

#### 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

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

Mathematical Modeling of Lactation Curves: A Review of Parametric Models

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 lactation defined 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

Total production obtained 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

Initial production (y 0) estimated by the average of the productions of the 4th,

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

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 ym and tp are obtained, respectively, as ordinate and abscissa of the point where the derivative of the function of adjust-

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

Lactation curve for dairy cattle (observed and adjusted yields) [17]. With y0 = initial yield; ym = yield at the

peak of production; tp = peak time; th = time of mid-lactation; and tf = time of end of lactation.

to maintain milk production after the peak. Assuming uniform nutrition and

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

mathematical function over the interval of lactation duration.

dt is equal to 0.

[15, 16].

prolonged lactations beyond 305 days.

DOI: http://dx.doi.org/10.5772/intechopen.90253

5th, and 6th postpartum days [13].

production (y0).

ment of the curve dy tð Þ

observed.

Figure 1.

97

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 welldefined 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

#### Mathematical Modeling of Lactation Curves: A Review of Parametric Models DOI: http://dx.doi.org/10.5772/intechopen.90253

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 lactation defined 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 production obtained 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 ym and tp are obtained, respectively, as ordinate and abscissa of the point where the derivative of the function of adjust-

#### ment of the curve dy tð Þ dt is 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

#### Figure 1.

Lactation curve for dairy cattle (observed and adjusted yields) [17]. With y0 = initial yield; ym = yield at the peak of production; tp = peak time; th = time of mid-lactation; and tf = time of end of lactation.

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 welldefined 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

Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…

flexible and general, such as polynomials or splines [12].

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

2. Description of the standard lactation curve

96

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].

Another advantage of parametric models is that they summarize distribution characteristics through a small number of parameters (in the majority of cases, three parameters). Rekaya et al. [21] highlighted that a function with a minimum number of parameters and a significant biological interpretation is the most desirable. Although increasing the number of parameters in the model improves the quality of fit for some functions, interpreting the parameters of the most complex models is difficult, and, in many cases, it is impossible to connect them with the

Mathematical Modeling of Lactation Curves: A Review of Parametric Models

Among the parametric models, lactation curves adjusted by exponential functions or integrating an exponential component into the model formula were widely used. The first attempt to develop a mathematical model to describe the lactation curve dates back to 1923. Brody et al. [1] used an exponential function in the

This model has been adjusted to monthly lactation yields. Its expression highlights the scaling factor for adjusting production to initial level a and the c parameter associated with the descending phase of the lactation curve and highlighted as a measure of persistency. Although this model is a good attempt to describe the descending phase of the lactation curve, it does not model the growth rate in the ascending phase to reach the peak of lactation. In order to overcome this limitation, Brody et al. [22] presented an improved version of their model, which takes into account the increase in milk yield until production maximum by incorporating an

While this is a great improvement over the first model, Cobby and Le Du [23] reported that this model underestimates milk yield in the middle of lactation and overestimates milk yield at the end of lactation. This model was followed by an exponential parabolic function introduced by Sikka [24] for modeling milk yield:

This model provided a good fit for lactation curves of primiparous cows, but it was less effective for multiparous cows, resulting in a bell-shaped curve that does not respect the regularity of the lactation curve around peak production. Then, numerous proposals were made to improve these aspects. Fischer [25] attempted to improve model 3 by replacing the exponential decline integrated in this model by

This model underestimated the maximum milk yield and also resulted in a relatively early estimate of the peak date [26]. The individuality of this model is that the ratio a/b estimates the duration of lactation. Vujicic and Bacic [27] attempted to

Yt ¼ a exp ð Þ �ct (2)

Yt ¼ a exp ð Þ� �bt a exp ð Þ �ct (3)

Yt <sup>¼</sup> <sup>a</sup> exp bt � ct<sup>2</sup> (4)

Yt ¼ a � bt � a exp ð Þ �ct (5)

Yt <sup>¼</sup> t c�<sup>a</sup> exp ð Þ �ct (6)

classic characteristics of the lactation curve [13].

DOI: http://dx.doi.org/10.5772/intechopen.90253

exponential decline function into the model:

means of a linear decline:

modify model 2:

99

3.3 Exponential functions

following form:

#### 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].

#### 3.1 Empirical models

An empirical model has a theory that refers only to the level of reality for which the phenomenon being considered is expressed. These so-called empirical models (models based exclusively on experience and observation and not on theory), whether linear or not linear, represent the huge majority of studies published in the bibliography [12]. In fact, empirical models have found great application in different fields of animal science, mainly because of their limited mathematical complexity. Most of them permit to estimate certain classic characteristics of the curve (date and level of the peak of production, persistency, and total production). Many empirical mathematical functions have been proposed to describe the lactation curve of dairy cattle [16, 20]. These functions differ in their mathematical properties, the number of parameters, and their degree of relationship to the main characteristics of a typical lactation structure, such as yield at peak time, persistency, and total yield. Nonlinear parametric models have been a preferred tool for modeling mean curves of homogeneous animal groups [12].

#### 3.2 Parametric curves

To describe the temporal evolution of milk secretion (lactation curve), scientists generally use parametric curves where the variation over time is modeled using linear or nonlinear functions. Most of the mathematical functions proposed to fit lactation curves in dairy cattle are primarily aimed at describing the phenomenon of milk secretion. Their basic assumption is that lactation is characterized by a continuous and deterministic component with an increasing phase to a maximum, followed by a decreasing slope [12]. The mathematical tool used in this approach can be represented by an analytic function of general time:

$$Y = f(t) + \varepsilon \tag{1}$$

where Y is often the daily output obtained on the day of control t; f (t) is a continuous function, differentiable over the interval represented by the duration of lactation; and ε is the random residual.

The use of parametric models has several advantages. Indeed, these models allow algebraically to calculate characteristic parameters of the curve. For example, the yield and the date of the peak of production are obtained, respectively, as ordinate and abscissa of the point where the first derivative of the function df tð Þ dt is equal to 0.

Production between two dates is obtained by calculating the integral of f(t) over this time interval. The total production also corresponds to the integral of the lactation curve over the duration of the lactation.

Mathematical Modeling of Lactation Curves: A Review of Parametric Models DOI: http://dx.doi.org/10.5772/intechopen.90253

Another advantage of parametric models is that they summarize distribution characteristics through a small number of parameters (in the majority of cases, three parameters). Rekaya et al. [21] highlighted that a function with a minimum number of parameters and a significant biological interpretation is the most desirable. Although increasing the number of parameters in the model improves the quality of fit for some functions, interpreting the parameters of the most complex models is difficult, and, in many cases, it is impossible to connect them with the classic characteristics of the lactation curve [13].

#### 3.3 Exponential functions

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].

Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…

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

An empirical model has a theory that refers only to the level of reality for which the phenomenon being considered is expressed. These so-called empirical models (models based exclusively on experience and observation and not on theory), whether linear or not linear, represent the huge majority of studies published in the bibliography [12]. In fact, empirical models have found great application in different fields of animal science, mainly because of their limited mathematical complexity. Most of them permit to estimate certain classic characteristics of the curve (date and level of the peak of production, persistency, and total production). Many empirical mathematical functions have been proposed to describe the lactation curve of dairy cattle [16, 20]. These functions differ in their mathematical properties, the number of parameters, and their degree of relationship to the main characteristics of a typical lactation structure, such as yield at peak time, persistency, and total yield. Nonlinear parametric models have been a preferred tool for model-

To describe the temporal evolution of milk secretion (lactation curve), scientists generally use parametric curves where the variation over time is modeled using linear or nonlinear functions. Most of the mathematical functions proposed to fit lactation curves in dairy cattle are primarily aimed at describing the phenomenon of milk secretion. Their basic assumption is that lactation is characterized by a continuous and deterministic component with an increasing phase to a maximum, followed by a decreasing slope [12]. The mathematical tool used in this approach

where Y is often the daily output obtained on the day of control t; f (t) is a continuous function, differentiable over the interval represented by the duration of

The use of parametric models has several advantages. Indeed, these models allow algebraically to calculate characteristic parameters of the curve. For example, the yield and the date of the peak of production are obtained, respectively, as ordinate and abscissa of the point where the first derivative of the function df tð Þ

Production between two dates is obtained by calculating the integral of f(t) over

this time interval. The total production also corresponds to the integral of the

Y ¼ f tð Þþ ε (1)

dt is

because they have a simple biological or economic interpretation [19].

3. Mathematical modeling of lactation curve

ing mean curves of homogeneous animal groups [12].

can be represented by an analytic function of general time:

lactation; and ε is the random residual.

lactation curve over the duration of the lactation.

equal to 0.

98

3.1 Empirical models

3.2 Parametric curves

Among the parametric models, lactation curves adjusted by exponential functions or integrating an exponential component into the model formula were widely used. The first attempt to develop a mathematical model to describe the lactation curve dates back to 1923. Brody et al. [1] used an exponential function in the following form:

$$Y\_t = a \exp\left(-ct\right) \tag{2}$$

This model has been adjusted to monthly lactation yields. Its expression highlights the scaling factor for adjusting production to initial level a and the c parameter associated with the descending phase of the lactation curve and highlighted as a measure of persistency. Although this model is a good attempt to describe the descending phase of the lactation curve, it does not model the growth rate in the ascending phase to reach the peak of lactation. In order to overcome this limitation, Brody et al. [22] presented an improved version of their model, which takes into account the increase in milk yield until production maximum by incorporating an exponential decline function into the model:

$$Y\_t = a \exp\left(-bt\right) - a \exp\left(-ct\right) \tag{3}$$

While this is a great improvement over the first model, Cobby and Le Du [23] reported that this model underestimates milk yield in the middle of lactation and overestimates milk yield at the end of lactation. This model was followed by an exponential parabolic function introduced by Sikka [24] for modeling milk yield:

$$Y\_t = a \exp\left(bt - ct^2\right) \tag{4}$$

This model provided a good fit for lactation curves of primiparous cows, but it was less effective for multiparous cows, resulting in a bell-shaped curve that does not respect the regularity of the lactation curve around peak production. Then, numerous proposals were made to improve these aspects. Fischer [25] attempted to improve model 3 by replacing the exponential decline integrated in this model by means of a linear decline:

$$Y\_t = a - bt - a \, \exp\left(-ct\right) \tag{5}$$

This model underestimated the maximum milk yield and also resulted in a relatively early estimate of the peak date [26]. The individuality of this model is that the ratio a/b estimates the duration of lactation. Vujicic and Bacic [27] attempted to modify model 2:

$$Y\_t = t \ c^{-a} \ \exp\left(-ct\right) \tag{6}$$

This model seems to be the first attempt to develop a model that varies both directly and exponentially over time. The disadvantage of this model is that it does not consider the initial evolution to the peak of production. Since the abovementioned exponential models do not correctly translate the ascendant phase of the lactation curve, Wood [28] proposed to adjust the entire curve by an incomplete gamma-type function:

$$Y\_t = at^b \exp\left(-ct\right) \tag{7}$$

Despite the limits reported, the Wood model remains the most used function for modeling lactation curves [35]. In addition, it has been used to describe traits other than milk yield, such as fatty acids [37], and it has also been used for adjustment of extended lactations [34, 36]. Many other models have been reported in the literature especially after the appearance of critics of Wood's model. To anticipate these restrictions, many authors have proposed improvements to Wood's model in order to increase its flexibility. As a result, several derivatives of Wood's function have appeared while noting improvements in the modeling of the lactation curve. Beever et al. [20] summarizes the improvements made by these derivatives in greater flexibility in modeling curve shapes and in improving the mathematical properties of the model by decreasing the correlation between model parameters [32]. Cobby and Le Du [23] reported that milk yield after the peak of production declines at a constant rate and proposed a modification of Wood's gamma function, substituting

b ) and

the power function (tb) for an asymptotic function of the type exp. (t

Mathematical Modeling of Lactation Curves: A Review of Parametric Models

Gaines resulting in a model of the form:

DOI: http://dx.doi.org/10.5772/intechopen.90253

resulting in the following model:

model equal to 1:

101

replacing the first exponential function of the equation obtained by the function of

The advantage of this model over that of Wood [28] is that declines in milk production are modeled exponentially [16]. This model allows a better adjustment of the initial phase of the lactation curve with a good estimate of peak production [26]. Rowlands et al. [26] compared models 3, 7, and 8 and concluded that model 8 describes the initial evolution of milk yield up to 5 weeks better than model 7. They also observed that models 7 and 8 slightly underestimated the initial yield and model 3 slightly overestimated the peak of lactation. Model 7 provided the best position of the peak yield date. According to Olori et al. [38], this model has a parameter that cannot be estimated by linear regression, and this limits its use in practice. Dhanoa [32] proposed a slightly different writing of Wood's model:

Such model reduces the correlations between the parameters of the curve in many cases. In addition, this model provides, with parameter b, a direct estimate of the peak lactation date. In an attempt to include a season effect in the Wood model, Goodall [39] proposed the introduction of a categorical variable D which takes the value 0 from the period October to March and the value 1 from April to September,

where D estimates the effect of season. This change takes into account the

Another modification of Wood's function was attempted by Jenkins and Ferrell [40] (1984) through placing the exponent of t, which is the value of b in Wood's

This model has its limits since curve fitting results have shown that evolution to maximum yield is relatively slow [41]. Based on the model of Cobby and Le Du [23], Wilmink [42] proposed a linear model to describe the lactation curve:

quantitative estimation of the effect of seasonal changes.

Yt ¼ a � b t � a exp ð Þ �c t (8)

Yt <sup>¼</sup> atb c exp ð Þ �c t (9)

Yt <sup>¼</sup> at<sup>b</sup> exp ð Þ �c t <sup>þ</sup> dD (10)

Yt ¼ at exp ð Þ �c t (11)

The incomplete gamma function is probably the most popular parametric model of the lactation curve. It generates the standard form of the lactation curve as the product of a constant, a power function, and an exponential decline function [12]. The disjunction of the Wood equation in its components emphasizes the direct relation of its parameters with the main elements of the shape of the lactation curve. In this expression, the power function tb permits to integrate the ascending phase of the lactation curve, whereas the exponential term accounts for the downward phase. For these reasons, Wood [28] interpreted parameters b and c as indices of growth intensity and output decline, respectively; the function tb exp (�ct) thus appears as the form factor of the curve [13]. Parameter a is then the scale factor of the level of production, which Wood [28] associates with the average production level of the beginning of lactation. Wood [29] tried to justify the use of model 7 from a physiological point of view, but does not interpret parameters b and c from a practical biological point of view. Parameter b is an index of a cow's capacity for the relevant use of energy to produce milk, but mathematically according to Wood [29], parameter b represents the rate of growth of production to yield maximal, and parameter c alternatively represents the rate of decline after the peak of lactation. Cobby and Le Du [23] argued that these interpretations of parameters b and c are fairly simplified and may be erroneous. Wood [30, 31] sought to interpret parameters a, b, and c as a function of the energy flows in the body. According to its approach, the parameter would translate a potential of production which is the function of the intrinsic capacity of secretion of the udder, the level of the mobilizable reserves, and the capacities of ingestion and digestion of the animal. On the other hand, the expression tb is explained by the fact that all the secretory cells would not be functional immediately after calving. According to Masselin et al. [13], the last aspect should rather be interpreted as the growth of the body's ability to ensure gluconeogenesis to synthesize lactose, which is the main element determining the exit of water from milk. Finally, parameter c would integrate the progressive reduction of the contribution of body reserves to the milk secretion and the exponential decline of the number of secretory cells. Several critics have been reported in Model 7. Cobby and Le Du [23] reported an overestimation of early lactation production and underestimation of peak lactation. Dhanoa [32] reported problems of strong correlations between the estimated parameters. Another reproach has been given to the Wood model that by construction, calving day production is constrained to be zero. This is not real in most mammal species. However, Tozer, and Huffaker [33] have indicated that a cow produces colostrum just after calving, which has no economic value, so considering zero milk yield after calving is not a significant problem. Rowlands et al. [26] reported that the Wood curve does not provide good consistency with the data collected on higher dairy cows at the peak of lactation. According to Macciotta et al. [12], Wood's model limitations are an overestimation of milk production per day in the first part of the curve and an underestimation around and after the peak of lactation. These weaknesses have been reported by some authors [34, 35] primarily because of the multiplicative structure, and the model is characterized by higher correlation between its parameters (ranging from 0.70 to 0.90, [12]) which results in higher sensitivity to data distribution [2].

Mathematical Modeling of Lactation Curves: A Review of Parametric Models DOI: http://dx.doi.org/10.5772/intechopen.90253

Despite the limits reported, the Wood model remains the most used function for modeling lactation curves [35]. In addition, it has been used to describe traits other than milk yield, such as fatty acids [37], and it has also been used for adjustment of extended lactations [34, 36]. Many other models have been reported in the literature especially after the appearance of critics of Wood's model. To anticipate these restrictions, many authors have proposed improvements to Wood's model in order to increase its flexibility. As a result, several derivatives of Wood's function have appeared while noting improvements in the modeling of the lactation curve. Beever et al. [20] summarizes the improvements made by these derivatives in greater flexibility in modeling curve shapes and in improving the mathematical properties of the model by decreasing the correlation between model parameters [32]. Cobby and Le Du [23] reported that milk yield after the peak of production declines at a constant rate and proposed a modification of Wood's gamma function, substituting the power function (tb) for an asymptotic function of the type exp. (t b ) and replacing the first exponential function of the equation obtained by the function of Gaines resulting in a model of the form:

$$Y\_t = a - bt - a \exp\left(-ct\right) \tag{8}$$

The advantage of this model over that of Wood [28] is that declines in milk production are modeled exponentially [16]. This model allows a better adjustment of the initial phase of the lactation curve with a good estimate of peak production [26]. Rowlands et al. [26] compared models 3, 7, and 8 and concluded that model 8 describes the initial evolution of milk yield up to 5 weeks better than model 7. They also observed that models 7 and 8 slightly underestimated the initial yield and model 3 slightly overestimated the peak of lactation. Model 7 provided the best position of the peak yield date. According to Olori et al. [38], this model has a parameter that cannot be estimated by linear regression, and this limits its use in practice. Dhanoa [32] proposed a slightly different writing of Wood's model:

$$Y\_t = at^{b\cdot c} \exp\left(-ct\right) \tag{9}$$

Such model reduces the correlations between the parameters of the curve in many cases. In addition, this model provides, with parameter b, a direct estimate of the peak lactation date. In an attempt to include a season effect in the Wood model, Goodall [39] proposed the introduction of a categorical variable D which takes the value 0 from the period October to March and the value 1 from April to September, resulting in the following model:

$$Y\_t = at^b \exp\left(-ct + dD\right) \tag{10}$$

where D estimates the effect of season. This change takes into account the quantitative estimation of the effect of seasonal changes.

Another modification of Wood's function was attempted by Jenkins and Ferrell [40] (1984) through placing the exponent of t, which is the value of b in Wood's model equal to 1:

$$Y\_t = at \exp\left(-ct\right) \tag{11}$$

This model has its limits since curve fitting results have shown that evolution to maximum yield is relatively slow [41]. Based on the model of Cobby and Le Du [23], Wilmink [42] proposed a linear model to describe the lactation curve:

This model seems to be the first attempt to develop a model that varies both directly and exponentially over time. The disadvantage of this model is that it does

abovementioned exponential models do not correctly translate the ascendant phase of the lactation curve, Wood [28] proposed to adjust the entire curve by an incom-

The incomplete gamma function is probably the most popular parametric model of the lactation curve. It generates the standard form of the lactation curve as the product of a constant, a power function, and an exponential decline function [12]. The disjunction of the Wood equation in its components emphasizes the direct relation of its parameters with the main elements of the shape of the lactation curve. In this expression, the power function tb permits to integrate the ascending phase of the lactation curve, whereas the exponential term accounts for the downward phase. For these reasons, Wood [28] interpreted parameters b and c as indices of growth intensity and output decline, respectively; the function tb exp (�ct) thus appears as the form factor of the curve [13]. Parameter a is then the scale factor of the level of production, which Wood [28] associates with the average production level of the beginning of lactation. Wood [29] tried to justify the use of model 7 from a physiological point of view, but does not interpret parameters b and c from a practical biological point of view. Parameter b is an index of a cow's capacity for the relevant use of energy to produce milk, but mathematically according to Wood [29], parameter b represents the rate of growth of production to yield maximal, and parameter c alternatively represents the rate of decline after the peak of lactation. Cobby and Le Du [23] argued that these interpretations of parameters b and c are fairly simplified and may be erroneous. Wood [30, 31] sought to interpret parameters a, b, and c as a function of the energy flows in the body. According to its approach, the parameter would translate a potential of production which is the function of the intrinsic capacity of secretion of the udder, the level of the mobilizable reserves, and the capacities of ingestion and digestion of the animal. On the other hand, the expression tb is explained by the fact

that all the secretory cells would not be functional immediately after calving. According to Masselin et al. [13], the last aspect should rather be interpreted as the growth of the body's ability to ensure gluconeogenesis to synthesize lactose, which is the main element determining the exit of water from milk. Finally, parameter c would integrate the progressive reduction of the contribution of body reserves to the milk secretion and the exponential decline of the number of secretory cells. Several critics have been reported in Model 7. Cobby and Le Du [23] reported an overestimation of early lactation production and underestimation of peak lactation. Dhanoa [32] reported problems of strong correlations between the estimated parameters. Another reproach has been given to the Wood model that by construction, calving day production is constrained to be zero. This is not real in most mammal species. However, Tozer, and Huffaker [33] have indicated that a cow produces colostrum just after calving, which has no economic value, so considering zero milk yield after calving is not a significant problem. Rowlands et al. [26] reported that the Wood curve does not provide good consistency with the data collected on higher dairy cows at the peak of lactation. According to Macciotta et al. [12], Wood's model limitations are an overestimation of milk production per day in the first part of the curve and an underestimation around and after the peak of lactation. These weaknesses have been reported by some authors [34, 35] primarily because of the multiplicative structure, and the model is characterized by higher correlation between its parameters (ranging from 0.70 to 0.90, [12]) which results in higher sensitivity to data distribution [2].

Yt <sup>¼</sup> atb exp ð Þ �c t (7)

not consider the initial evolution to the peak of production. Since the

Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…

plete gamma-type function:

100

Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…

$$Y\_t = a + b \exp\left(-kt\right) + ct \tag{12}$$

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

Mathematical Modeling of Lactation Curves: A Review of Parametric Models

DOI: http://dx.doi.org/10.5772/intechopen.90253

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 model-

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

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

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))<sup>2</sup> to the second model of Singh and Gopal

, the same author observed that the function 18 gives a better fit than

Yt <sup>¼</sup> <sup>t</sup>=<sup>a</sup> <sup>þ</sup> bt <sup>þ</sup> ct<sup>2</sup> (18)

Yt ¼ a � b t þ c Log tð Þ (19)

Yt <sup>¼</sup> <sup>a</sup> <sup>þ</sup> b t <sup>þ</sup> c t<sup>2</sup> <sup>þ</sup> d Log tð Þ (20)

yt <sup>¼</sup> <sup>a</sup> <sup>þ</sup> bX <sup>þ</sup> cX<sup>2</sup> <sup>þ</sup> <sup>f</sup> log 1ð Þþ <sup>=</sup><sup>X</sup> <sup>e</sup>ð Þ log 1ð Þ <sup>=</sup><sup>X</sup> <sup>2</sup> (21)

ing lactation curves. Day t production is described as follows:

proposed two new models: linear cum log model:

[54] and proposed the use of a five-variable linear model:

and quadratic cum log model:

method.

coefficient R<sup>2</sup>

103

According to Wilmink [42], parameter k is related to the peak of lactation time and usually constitutes a fixed value, derived from a preliminary analysis of average production, implying that the model has only three parameters to estimate. Brotherstone et al. [4] emphasized the importance of parameter k in the ability of the Wilmink model to model daily production in early lactation. In an attempt to overcome underestimation of peak yield and overestimation of yield in the decreasing phase of lactation that results from the use of the Wood model, Cappio-Borlino et al. [43] proposed a nonlinear modification of the Wood model in the following form:

$$Y\_t = \mathbf{a} \cdot \mathbf{t}^{\mathbf{b} \cdot \exp\left(-\mathbf{c}t\right)} \tag{13}$$

while this proposal is more complex than Wood's equation, this model reduces the extent of underestimation at the beginning of lactation and overestimation at the final stage of lactation. Franci et al. [44] have successfully used to adjust the lactation curves of dairy ewes, characterized by a rapid increase in milk yield to the peak of lactation.

#### 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:

$$Y\_t = a - bt\tag{14}$$

In this model, parameter a is an estimator of initial production, and b was 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:

$$y\_t = a + bt + ct^2 \tag{15}$$

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

$$y\_t = a + bt - ct^2 \tag{16}$$

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

$$y\_t = a + bt + ct^2 + dt^3 \tag{17}$$

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 d which have no biological or zootechnical sense [13]. Dag et al. [49] reported that the cubic model best matched

Mathematical Modeling of Lactation Curves: A Review of Parametric Models DOI: http://dx.doi.org/10.5772/intechopen.90253

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:

$$Y\_t = t/a + bt + ct^2\tag{18}$$

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 R<sup>2</sup> , 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:

$$Y\_t = a - bt + c \, \text{Log}(t) \tag{19}$$

and quadratic cum log model:

$$Y\_t = a + bt + c\,t^2 + d\,\text{Log}(t)\tag{20}$$

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))<sup>2</sup> to the second model of Singh and Gopal [54] and proposed the use of a five-variable linear model:

$$y\_t = a + bX + cX^2 + f \log \left( \mathbf{1} / X \right) + e \left( \log \left( \mathbf{1} / X \right) \right)^2 \tag{21}$$

Yt ¼ a þ b exp ð Þþ �k t ct (12)

Yt <sup>¼</sup> a tb exp ð Þ �ct (13)

Yt ¼ a � bt (14)

yt <sup>¼</sup> <sup>a</sup> <sup>þ</sup> bt <sup>þ</sup> ct<sup>2</sup> (15)

yt <sup>¼</sup> <sup>a</sup> <sup>þ</sup> bt � ct<sup>2</sup> (16)

yt <sup>¼</sup> <sup>a</sup> <sup>þ</sup> bt <sup>þ</sup> ct<sup>2</sup> <sup>þ</sup> dt<sup>3</sup> (17)

According to Wilmink [42], parameter k is related to the peak of lactation time and usually constitutes a fixed value, derived from a preliminary analysis of average

while this proposal is more complex than Wood's equation, this model reduces the extent of underestimation at the beginning of lactation and overestimation at the final stage of lactation. Franci et al. [44] have successfully used to adjust the lactation curves of dairy ewes, characterized by a rapid increase in milk yield to the

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

In this model, parameter a is an estimator of initial production, and b was 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

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

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 d which have no biological or zootechnical sense [13]. Dag et al. [49] reported that the cubic model best matched

production, implying that the model has only three parameters to estimate. Brotherstone et al. [4] emphasized the importance of parameter k in the ability of the Wilmink model to model daily production in early lactation. In an attempt to overcome underestimation of peak yield and overestimation of yield in the decreasing phase of lactation that results from the use of the Wood model, Cappio-Borlino et al. [43] proposed a nonlinear modification of the Wood model

Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…

in the following form:

peak of lactation.

the persistency:

buffalo in first lactations:

degree polynomial:

102

4. Linear adjustment models

where X = t / length of lactation, a is a parameter associated with the peak of production, and f and e are associated with the upward part of the production curve and b and c with the descending part.

Yt ¼ a Log b t ð Þ exp ð Þ �c t (27) Yt ¼ a Log b t ð Þ= cosh ð Þ c t (28) Yt ¼ a arctan ð Þ b t exp ð Þ �ct (29)

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

Mathematical Modeling of Lactation Curves: A Review of Parametric Models

DOI: http://dx.doi.org/10.5772/intechopen.90253

ai bi <sup>1</sup> � tanh <sup>2</sup>

occurs at time ci. The duration of each phase is associated with 2b�<sup>1</sup>

where n is the number of lactation phases considered and tanh is the hyperbolic tangent function. For each phase i, the maximum yield is equal to ai and bi and

sents 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

ð Þ bið Þ t � ci � � � � (30)

<sup>i</sup> which repre-

Yt <sup>¼</sup> <sup>X</sup><sup>n</sup> i¼1

quantitative level of production.

105

5.1 Methods applied for adjusting the lactation curve

Nonlinear and linear estimation methods have been used to adjust lactation curves, where the method employed is determined by the nature of the model to be

lactation:

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 [57–59]. 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:

$$Y\_t = a + b \log\left(305/t\right) + ct \tag{22}$$

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

$$y\_t = a + bt^{1/2} + c \log(t) \tag{23}$$

This model can be considered as inspired by model 19 suggested by Singh and Gopal [54], substituting time t for the square root of t in 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 t <sup>b</sup>,1 � exp ð Þ �<sup>t</sup> , Log tð Þ, and arctan ð Þ<sup>t</sup> and decreasing functions such as, and, where arc-tan and cosh, respectively, refer to the arc-tangent trigonometric function and the hyperbolic cosine function. These reflections gave birth to the following six models:

$$Y\_t = at^b / \cosh\left(ct\right) \tag{24}$$

$$Y\_t = a(1 - \exp\left(-bt\right)) / \cosh\left(ct\right) \tag{25}$$

$$Y\_t = a \arctan\left(bt\right) / \cosh\left(ct\right) \tag{26}$$

Mathematical Modeling of Lactation Curves: A Review of Parametric Models DOI: http://dx.doi.org/10.5772/intechopen.90253

$$Y\_t = a \operatorname{Log}(bt) \exp\left(-ct\right) \tag{27}$$

$$Y\_t = a \operatorname{Log}(bt) / \cosh\left(ct\right) \tag{28}$$

$$Y\_t = a \arctan\left(bt\right) \exp\left(-ct\right) \tag{29}$$

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:

$$Y\_t = \sum\_{i=1}^{n} \left\{ a\_i b\_i \left[ 1 - \tanh^2(b\_i (t - c\_i)) \right] \right\} \tag{30}$$

where n is the number of lactation phases considered and tanh is the hyperbolic tangent function. For each phase i, the maximum yield is equal to ai and bi and occurs at time ci. The duration of each phase is associated with 2b�<sup>1</sup> <sup>i</sup> which 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.

#### 5.1 Methods applied for adjusting the lactation curve

Nonlinear and linear estimation methods have been used to adjust lactation curves, where the method employed is determined by the nature of the model to be

where X = t / length of lactation, a is a parameter associated with the peak of production, and f and e are associated with the upward part of the production curve

Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…

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 [57–59]. 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

Yt ¼ a þ b log 305 ð Þþ =t ct (22)

yt <sup>¼</sup> <sup>a</sup> <sup>þ</sup> bt<sup>1</sup>=<sup>2</sup> <sup>þ</sup> c log tð Þ (23)

= cosh ð Þ ct (24)

Yt ¼ að1 � exp ð Þ �b t = cosh ð Þ ct (25) Yt ¼ a arctan ð Þ b t = cosh ð Þ ct (26)

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

model. However, this model tends to underestimate peak yield, while

For a competitive model, Papajcsik and Bodero [66] have introduced trigonometric functions and combined functions with increasing variation such as

Yt <sup>¼</sup> at<sup>b</sup>

<sup>b</sup>,1 � exp ð Þ �<sup>t</sup> , Log tð Þ, and arctan ð Þ<sup>t</sup> and decreasing functions such as, and, where arc-tan and cosh, respectively, refer to the arc-tangent trigonometric function and the hyperbolic cosine function. These reflections gave birth to the following six

This model can be considered as inspired by model 19 suggested by Singh and Gopal [54], substituting time t for the square root of t in the second term of the

overestimating post-peak yield [38]. Catillo et al. [65] reported that this model was effective in estimating lactation curves and milk production characteristics of

and b and c with the descending part.

a logarithmic model:

Italian water buffaloes.

5. Other models

t

models:

104

used. Some models can be developed using a nonlinear and linear estimation at the same time, such as the polynomial regression model of Ali and Schaeffer [55]. Others can be transformed into linear models. Wood [28] has already noted that the gamma function can be converted into a simple linear regression model by performing a logarithmic transformation to determine the initial values of the parameters a, b, and c by means of a least square's estimation. Congleton and Everett [69] reported that the adjustment of the incomplete gamma function with linear regression after a logarithmic transformation has the advantage of requiring a minimum of computational machine time. Due to the large number of lactations analyzed, parameter estimation is obtained by linear regression rather than an iterative nonlinear technique. In linear models the parameters are linear functions of the lactation day, and the least square estimates of the parameters can always be obtained analytically by a simple linear regression. On the other hand, nonlinear models such as the Wilmink function can only be solved by numerical methods following iterative optimization procedures. Adjusting data with a nonlinear regression model has specific advantages. Indeed, nonlinear models are often derived on the basis of biological and/or physical considerations. Thus, the parameters of a nonlinear model usually have direct interpretation in terms of the processes studied. In addition, the main advantage of nonlinear regression compared to other curve fitting procedures is the wide range of functions that can be adjusted. The objective in nonlinear regression is to obtain estimates of the parameters that minimize the residual effects, measured as the sum of the squares of the distances of the data points on the curve [17].

Different iterative methods such as Marquardt, Gauss-Newton, and Does not Use Derivatives (DUD) are frequently used in nonlinear regression models. The simplest method, known as the DUD, does not require the specification of the partial derivatives with respect to the parameters of the mathematical model. It is an algorithm that brings the derivatives closer by differences. However, it is important to note that this algorithm does not give good estimates especially when

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

Lactation curves for milk production, percent fat, and percent milk protein (Wood, 1976).

the parameters are strongly correlated. Another method commonly used in nonlinear regression is the iterative Gauss-Newton method also available in SAS. The algorithm of this technique is based on Taylor series development near the initial parameter values [67]. Generally, Marquardt's nonlinear regression was the most commonly used method to adjust lactation curves using nonlinear models [34]. The Marquardt method, which follows an intermediate path between the Gauss-Newton (Taylor series method) and Newton (second derivative) methods, was often considered better to achieve convergence when the parameter estimates

Mathematical Modeling of Lactation Curves: A Review of Parametric Models

were strongly correlated [34].

Figure 2.

107

6. Lactation curve of milk constituents

DOI: http://dx.doi.org/10.5772/intechopen.90253

To estimate the parameters following an iterative procedure, it is necessary to have those initial values, which will be subjected to successive iterations. These initial values are adjusted, and the sum of the squares of the residues is reduced significantly to each iteration. The process of estimating the parameters continues until a convergence criterion is met, accepting that from this point on, a negligible or no improvement in data fit is possible [17]. A major difficulty of this procedure is to enter the initial values of the model parameters. If the pre-estimates are not correct, the process may not converge to the minimum of the sum of error squares. Moreover, it is not always possible to know if the process converges toward the best estimate of the minimum of the sum of error squares [70]. The initial values should be reasonably close to the estimates of parameters that are still unknown if the optimization procedure cannot converge. The consequence of a bad choice of the initial parameters is the obtaining of low values of the coefficient of determination, the standard errors that are high [71], and consequently a poor quality of adjustment of the model to data. Fadel [71] discussed a technique for identifying appropriate estimates of initial parameter values using the nonlinear procedure of the statistical analysis system (SAS, PROC NLIN). This technique was illustrated via a segmented nonlinear model with four parameters to estimate (b1, b2, b3, and b4), frequently used for the modeling of fiber digestion as a function of fermentation time (t):

$$\begin{cases} \mathbf{f\_1(t)} = \mathbf{b1} + \mathbf{b4}, \text{si } \mathbf{t} \le \mathbf{b3} \\ \mathbf{f\_2(t)} = \mathbf{b1} \exp\left[ \mathbf{-b2} \left( \mathbf{t} \cdot \mathbf{b3} \right) \right] + \mathbf{b4}, \text{si } \mathbf{t} \ge \mathbf{b3} \end{cases} \tag{31}$$

The principle of this technique consists in using a network of values for b3 (example of 2 to 6 per unit of 0.1) with fixed estimates of the parameters b1, b2, and b4. The SAS program generates a set of data sets for each proposed value. Then b1, b2, and b4 will be calculated for each estimate of b3 of the proposed network. The combination of the parameters estimated from the solution with the smallest value of the sum of the squares of the residuals will be used as initial values for the final analysis [71].

Mathematical Modeling of Lactation Curves: A Review of Parametric Models DOI: http://dx.doi.org/10.5772/intechopen.90253

Different iterative methods such as Marquardt, Gauss-Newton, and Does not Use Derivatives (DUD) are frequently used in nonlinear regression models. The simplest method, known as the DUD, does not require the specification of the partial derivatives with respect to the parameters of the mathematical model. It is an algorithm that brings the derivatives closer by differences. However, it is important to note that this algorithm does not give good estimates especially when the parameters are strongly correlated. Another method commonly used in nonlinear regression is the iterative Gauss-Newton method also available in SAS. The algorithm of this technique is based on Taylor series development near the initial parameter values [67]. Generally, Marquardt's nonlinear regression was the most commonly used method to adjust lactation curves using nonlinear models [34]. The Marquardt method, which follows an intermediate path between the Gauss-Newton (Taylor series method) and Newton (second derivative) methods, was often considered better to achieve convergence when the parameter estimates were strongly correlated [34].

#### 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

Figure 2. Lactation curves for milk production, percent fat, and percent milk protein (Wood, 1976).

used. Some models can be developed using a nonlinear and linear estimation at the same time, such as the polynomial regression model of Ali and Schaeffer [55]. Others can be transformed into linear models. Wood [28] has already noted that the

To estimate the parameters following an iterative procedure, it is necessary to have those initial values, which will be subjected to successive iterations. These initial values are adjusted, and the sum of the squares of the residues is reduced significantly to each iteration. The process of estimating the parameters continues until a convergence criterion is met, accepting that from this point on, a negligible or no improvement in data fit is possible [17]. A major difficulty of this procedure is to enter the initial values of the model parameters. If the pre-estimates are not correct, the process may not converge to the minimum of the sum of error squares. Moreover, it is not always possible to know if the process converges toward the best estimate of the minimum of the sum of error squares [70]. The initial values should be reasonably close to the estimates of parameters that are still unknown if the optimization procedure cannot converge. The consequence of a bad choice of the initial parameters is the obtaining of low values of the coefficient of determination, the standard errors that are high [71], and consequently a poor quality of adjustment of the model to data. Fadel [71] discussed a technique for identifying appropriate estimates of initial parameter values using the nonlinear procedure of the statistical analysis system (SAS, PROC NLIN). This technique was illustrated via a segmented nonlinear model with four parameters to estimate (b1, b2, b3, and b4), frequently used for the model-

ing of fiber digestion as a function of fermentation time (t):

analysis [71].

106

f <sup>1</sup>ðÞ¼ t b1 þ b4, si t≤ b3

<sup>f</sup> <sup>2</sup>ðÞ¼ <sup>t</sup> b1 exp ½ �þ ‐b2 tð Þ ‐b3 b4, si t>b3

The principle of this technique consists in using a network of values for b3 (example of 2 to 6 per unit of 0.1) with fixed estimates of the parameters b1, b2, and b4. The SAS program generates a set of data sets for each proposed value. Then b1, b2, and b4 will be calculated for each estimate of b3 of the proposed network. The combination of the parameters estimated from the solution with the smallest value of the sum of the squares of the residuals will be used as initial values for the final

(31)

gamma function can be converted into a simple linear regression model by performing a logarithmic transformation to determine the initial values of the parameters a, b, and c by means of a least square's estimation. Congleton and Everett [69] reported that the adjustment of the incomplete gamma function with linear regression after a logarithmic transformation has the advantage of requiring a minimum of computational machine time. Due to the large number of lactations analyzed, parameter estimation is obtained by linear regression rather than an iterative nonlinear technique. In linear models the parameters are linear functions of the lactation day, and the least square estimates of the parameters can always be obtained analytically by a simple linear regression. On the other hand, nonlinear models such as the Wilmink function can only be solved by numerical methods following iterative optimization procedures. Adjusting data with a nonlinear regression model has specific advantages. Indeed, nonlinear models are often derived on the basis of biological and/or physical considerations. Thus, the parameters of a nonlinear model usually have direct interpretation in terms of the processes studied. In addition, the main advantage of nonlinear regression compared to other curve fitting procedures is the wide range of functions that can be adjusted. The objective in nonlinear regression is to obtain estimates of the parameters that minimize the residual effects, measured as the sum of the squares of the distances of

Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…

the data points on the curve [17].

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.

Author details

Mahdi Bouallegue<sup>1</sup>

109

\* and Naceur M'Hamdi<sup>2</sup>

2 National Agronomic Institute of Tunisia, Tunis, Tunisia

provided the original work is properly cited.

\*Address all correspondence to: mahdibouallegue@yahoo.fr

1 Higher Institute of Agronomy of Chott-Meriem, Sousse, Tunisia

Mathematical Modeling of Lactation Curves: A Review of Parametric Models

DOI: http://dx.doi.org/10.5772/intechopen.90253

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Mathematical Modeling of Lactation Curves: A Review of Parametric Models DOI: http://dx.doi.org/10.5772/intechopen.90253

### Author details

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

Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…

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

function of Wood.

7. Conclusion

lactation curves parameters.

108

Mahdi Bouallegue<sup>1</sup> \* and Naceur M'Hamdi<sup>2</sup>


\*Address all correspondence to: mahdibouallegue@yahoo.fr

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### References

[1] Brody S, Turner CW, Ragsdale AC. The rate of decline of milk secretion with the advance of the period lactation. The Journal of General Physiology. 1923; 5:442-444

[2] Silvestre AM, Petim-Batista F, Colaco J. The accuracy of seven mathematical functions in modeling dairy cattle lactation curves based on test-day records from varying sample schemes. Journal of Dairy Science. 2006; 89:1813-1821

[3] Farhangfar H, Rowilnson P, Willis MB. Estimation of lactation curve parameters for Iranian Holstein dairy cows using nonlinear models. In: British Society of Animal Science Proceedings. 109p. 2000

[4] Rekik B, Gara A, Hamouda M, Hammami H. Fitting lactation curves of dairy cattle in different types of herds in Tunisia. Livestock Production Science. 2003;83:309-315

[5] Wood PDP. A note on the estimation of total lactation yield from production on a single day. Animal Production. 1974;19:393-396

[6] De Vries A. Economic value of pregnancy in dairy cattle. Journal of Dairy Science. 2006;89:3876-3885

[7] Gipson TA, Grossman M. Diphasic analysis of lactation curves in dairy goats. Journal of Dairy Science. 1989;72: 1035-1044

[8] Tekerli M, Akinci Z, Dogan I, Akcan A. Factors affecting the shape of lactation curves of Holstein cows from the Baliksir province of Turkey. Journal of Dairy Science. 2000;83:1381-1386

[9] Gengler N. Persistency of lactation yields: A review. Interbull Bulletin. 1996;12:97-102

[10] France J, Thornley JHM. Mathematical Models in Agriculture. London, United Kingdom: Butterworths; 1984

[19] Leclerc H, Duclos D, Barbat A, Druet T, Ducrocq V. Environmental effects on lactation curves included in a test-day model genetic evaluation.

DOI: http://dx.doi.org/10.5772/intechopen.90253

Mathematical Modeling of Lactation Curves: A Review of Parametric Models

[28] Wood PDP. Algebraic model of the lactation curve in cattle. Nature. 1967;

[29] Wood PDP. A note on seasonal fluctuations in milk production. Animal

[30] Wood PDP. The biometry of lactation. The Journal of Agricultural

[31] Wood PDP. A simple model of lactation curves for milk yield, food requirements, and body weight. Animal

[32] Dhanoa MS. A note on an alternative form of the lactation model of Wood. Animal Production. 1981;32:349-351

[34] Dematawewa CMB, Pearson RE, Vanraden PM. Modeling extended lactation of Holsteins. Journal of Dairy

[35] Dijkstra J, Lopez S, Bannink A, Dhanoa MS, Kebreab E, Odongo NE, et al. Evaluation of a mechanistic lactation model using cow, goat and sheep data. The Journal of Agricultural

[36] Steri R, Dimauro C, Canavesi EF, Nicolazzi EL, Macciotta NPP. Analysis

[37] Craninx M, Steen A, Van Laar H, Van Nespen T, Martín-Tereso J, De Baets B, et al. Effect of lactation stage on the odd- and branched-chain milk fatty acids of dairy cattle under grazing and indoor conditions. Journal of Dairy

of lactation shapes in extended lactations. Animal. 2012;6(10):

Science. 2008;91:2662-2677

Production. 1972;15:89-92

Science. 1977;88:333-339

Production. 1979;28:55-63

[33] Tozer PR, Huffaker RG. Mathematical equations to describe lactation curves for Holstein-Friesian cows in New South Wales. Australian Journal of Agricultural Research. 1999;

Science. 2007;90:3924-3936

Science. 2010;148:249-262

50:431-440

1572-1582

216:164-165

[20] Beever DE, Rook AJ, France J, Dhanoa MS, Gill M. A review of empirical and mechanistic models of lactational performance by the dairy cow. Livestock Production Science.

[21] Rekaya R, Carabaño MJ, Toro MA. Bayesian analysis of lactation curves of

[22] Brody S, Turner CW, Ragsdale AC. The relation between the initial rise and the subsequent decline of milk secretion following parturition. The Journal of General Physiology. 1924;6:541

[23] Cobby JM, Le Du YLP. On fitting curves to lactation data. Animal Production. 1978;26:127-133

[24] Sikka LC. A study of lactation as affected by heredity and environment. The Journal of Dairy Research. 1950;17:

Württemberg spotted mountain cows on the shape of the lactation curve and how it may be influenced by non-genetic factors. Züchtungskunde. 1958;30:296-304

[26] Rowlands GJ, Slucey S, Russel AM. A comparison of different models of the lactation curve in dairy cattle. Animal

[25] Fischer A. Research with

Production. 1982;35:135-144

[27] Vujicic I, Bacic B. 1961. [New equation of the lactation curve] Novi Sad. Ann. Sci. Agric. No. 5. In: Schultz AA, editor. 1974. Factors Affecting the Shape of the Lactation Curve and Its Mathematical Description. MS Thesis, University of Wisconsin, Madison

Holstein-Friesian cattle using a nonlinear model. Journal of Dairy Science. 2000;83:2691-2701

Animal. 2008;2:344-353

1991;29:115-130

231-252

111

[11] Guo Z, Swalve HH. Comparison of different lactation curve sub-models in test day models. Interbull Bulletin. 1997; 16:75-79

[12] Macciotta NPP, Dimauro C, Rassu SPG, Steri R, Pulina G. The mathematical description of lactation curves in dairy cattle. Italian Journal of Animal Science. 2011;10(4):213-223

[13] Masselin S, Sauvant D, Chapoutot P, Milan D. Adjustment models for the lactation curves. Annales de Zootechnie. 1987;36:171-206

[14] Knight CH, Wilde CJ. Mammary cell changes during pregnancy and lactation. Livestock Production Science. 1993;35:3-19

[15] Stanton TL, Jones LR, Everett RW, Kachman SD. Estimating milk, fat and protein lactation curves with a test day model. Journal of Dairy Science. 1992; 75:1691-1700

[16] Sherchand L, Mcnew RW, Kellogg DW, Johnson ZB. Selection of a mathematical model to generate lactation curves using daily milk yields of Holstein cows. Journal of Dairy Science. 1995;78:2507-2513

[17] Lopez S. Non-linear functions in animal nutrition. In: France JE, Kebreab E, editors. Mathematical Modeling in Animal Nutrition. Oxfordshire: CABI; 2008. 574p

[18] Gaines WL. The Energy Basis of Measuring Milk Yield in Dairy Cows. Illinois Agricultural Experiment Station. Bulletin No 308. Urbana: University of Illinois; 1928. pp. 403-438

Mathematical Modeling of Lactation Curves: A Review of Parametric Models DOI: http://dx.doi.org/10.5772/intechopen.90253

[19] Leclerc H, Duclos D, Barbat A, Druet T, Ducrocq V. Environmental effects on lactation curves included in a test-day model genetic evaluation. Animal. 2008;2:344-353

References

5:442-444

89:1813-1821

109p. 2000

2003;83:309-315

1974;19:393-396

1035-1044

1996;12:97-102

110

[1] Brody S, Turner CW, Ragsdale AC. The rate of decline of milk secretion with the advance of the period lactation. The Journal of General Physiology. 1923; [10] France J, Thornley JHM.

London, United Kingdom: Butterworths; 1984

16:75-79

Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…

1987;36:171-206

1993;35:3-19

75:1691-1700

Mathematical Models in Agriculture.

[11] Guo Z, Swalve HH. Comparison of different lactation curve sub-models in test day models. Interbull Bulletin. 1997;

[13] Masselin S, Sauvant D, Chapoutot P, Milan D. Adjustment models for the lactation curves. Annales de Zootechnie.

[14] Knight CH, Wilde CJ. Mammary cell changes during pregnancy and lactation. Livestock Production Science.

[15] Stanton TL, Jones LR, Everett RW, Kachman SD. Estimating milk, fat and protein lactation curves with a test day model. Journal of Dairy Science. 1992;

Kellogg DW, Johnson ZB. Selection of a

[16] Sherchand L, Mcnew RW,

mathematical model to generate lactation curves using daily milk yields of Holstein cows. Journal of Dairy Science. 1995;78:2507-2513

[17] Lopez S. Non-linear functions in animal nutrition. In: France JE, Kebreab E, editors. Mathematical Modeling in Animal Nutrition. Oxfordshire: CABI; 2008. 574p

[18] Gaines WL. The Energy Basis of Measuring Milk Yield in Dairy Cows. Illinois Agricultural Experiment Station. Bulletin No 308. Urbana: University of

Illinois; 1928. pp. 403-438

[12] Macciotta NPP, Dimauro C, Rassu SPG, Steri R, Pulina G. The mathematical description of lactation curves in dairy cattle. Italian Journal of Animal Science. 2011;10(4):213-223

[2] Silvestre AM, Petim-Batista F, Colaco J. The accuracy of seven mathematical functions in modeling dairy cattle lactation curves based on test-day records from varying sample schemes. Journal of Dairy Science. 2006;

[3] Farhangfar H, Rowilnson P,

[4] Rekik B, Gara A, Hamouda M, Hammami H. Fitting lactation curves of dairy cattle in different types of herds in Tunisia. Livestock Production Science.

[5] Wood PDP. A note on the estimation of total lactation yield from production on a single day. Animal Production.

[6] De Vries A. Economic value of pregnancy in dairy cattle. Journal of Dairy Science. 2006;89:3876-3885

[7] Gipson TA, Grossman M. Diphasic analysis of lactation curves in dairy goats. Journal of Dairy Science. 1989;72:

[8] Tekerli M, Akinci Z, Dogan I, Akcan A. Factors affecting the shape of lactation curves of Holstein cows from the Baliksir province of Turkey. Journal of Dairy Science. 2000;83:1381-1386

[9] Gengler N. Persistency of lactation yields: A review. Interbull Bulletin.

Willis MB. Estimation of lactation curve parameters for Iranian Holstein dairy cows using nonlinear models. In: British Society of Animal Science Proceedings.

[20] Beever DE, Rook AJ, France J, Dhanoa MS, Gill M. A review of empirical and mechanistic models of lactational performance by the dairy cow. Livestock Production Science. 1991;29:115-130

[21] Rekaya R, Carabaño MJ, Toro MA. Bayesian analysis of lactation curves of Holstein-Friesian cattle using a nonlinear model. Journal of Dairy Science. 2000;83:2691-2701

[22] Brody S, Turner CW, Ragsdale AC. The relation between the initial rise and the subsequent decline of milk secretion following parturition. The Journal of General Physiology. 1924;6:541

[23] Cobby JM, Le Du YLP. On fitting curves to lactation data. Animal Production. 1978;26:127-133

[24] Sikka LC. A study of lactation as affected by heredity and environment. The Journal of Dairy Research. 1950;17: 231-252

[25] Fischer A. Research with Württemberg spotted mountain cows on the shape of the lactation curve and how it may be influenced by non-genetic factors. Züchtungskunde. 1958;30:296-304

[26] Rowlands GJ, Slucey S, Russel AM. A comparison of different models of the lactation curve in dairy cattle. Animal Production. 1982;35:135-144

[27] Vujicic I, Bacic B. 1961. [New equation of the lactation curve] Novi Sad. Ann. Sci. Agric. No. 5. In: Schultz AA, editor. 1974. Factors Affecting the Shape of the Lactation Curve and Its Mathematical Description. MS Thesis, University of Wisconsin, Madison

[28] Wood PDP. Algebraic model of the lactation curve in cattle. Nature. 1967; 216:164-165

[29] Wood PDP. A note on seasonal fluctuations in milk production. Animal Production. 1972;15:89-92

[30] Wood PDP. The biometry of lactation. The Journal of Agricultural Science. 1977;88:333-339

[31] Wood PDP. A simple model of lactation curves for milk yield, food requirements, and body weight. Animal Production. 1979;28:55-63

[32] Dhanoa MS. A note on an alternative form of the lactation model of Wood. Animal Production. 1981;32:349-351

[33] Tozer PR, Huffaker RG. Mathematical equations to describe lactation curves for Holstein-Friesian cows in New South Wales. Australian Journal of Agricultural Research. 1999; 50:431-440

[34] Dematawewa CMB, Pearson RE, Vanraden PM. Modeling extended lactation of Holsteins. Journal of Dairy Science. 2007;90:3924-3936

[35] Dijkstra J, Lopez S, Bannink A, Dhanoa MS, Kebreab E, Odongo NE, et al. Evaluation of a mechanistic lactation model using cow, goat and sheep data. The Journal of Agricultural Science. 2010;148:249-262

[36] Steri R, Dimauro C, Canavesi EF, Nicolazzi EL, Macciotta NPP. Analysis of lactation shapes in extended lactations. Animal. 2012;6(10): 1572-1582

[37] Craninx M, Steen A, Van Laar H, Van Nespen T, Martín-Tereso J, De Baets B, et al. Effect of lactation stage on the odd- and branched-chain milk fatty acids of dairy cattle under grazing and indoor conditions. Journal of Dairy Science. 2008;91:2662-2677

[38] Olori VE, Brotherstone S, Hill WG, Mcguirk BJ. Fit of standard models of the lactation curve to weekly records of milk production of cows in a single herd. Livestock Production Science. 1999;58: 55-63

[39] Goodall EA. An analysis of seasonality of milk production. Animal Production. 1983;36:69-72

[40] JenkinsTG Ferrell CL. A note on lactation curves of crossbred cows. Animal Production. 1984;39:479-482

[41] Landete-Castillejos T, Gallego L. Technical note: The ability of mathematical models to describe the shape of lactation curves. Journal of Animal Science. 2000;78:3010-3013

[42] Wilmink JBM. Adjustment of testday Milk, fat and protein yield for age, season and stage of lactation. Livestock Production Science. 1987;16:335-348

[43] Cappio-Borlino A, Pulina G, Rossi G. A nonlinear modification of Wood's equation fitted to lactation curves of Sardinian dairy ewes. Small Ruminant Research. 1995;18:75-79

[44] Franci O, Pugliese C, Acciaioli A, Parisi G, Lucifero M. Application of two models to the lactation curve of Massese ewes. Small Ruminant Research. 1999; 31:91-96

[45] Delage J, Leroy AM, Poly J. Une étude sur les courbes de lactation. Annales de Zootechnie. 1953;3:225-267

[46] Ostergaard V. Strategies for concentrate feeding to attain optimum feeding level in high yielding dairy cows. Chap. V - food intake. In: Beretning fra statens husdyrbrugs forsog. Vol. 482. 1979. pp. 37-67

[47] Dave BK. First lactation curve of Indian water buffalo. JNKVV Research Journal. 1971;5:93

[48] Sauvant D, Morand-Fehr P. Clasification des types de courbes de lactation et d'evolution de la composition du lait de la chèvre. Rech. Ovine Caprine. 1975;1:90

[57] Druet T, Jaffrezic F, Boichard D, Ducrocq V. Modeling lactation curves and estimation of genetic parameters for first lactation test-day records of French Holstein cows. Journal of Dairy Science.

DOI: http://dx.doi.org/10.5772/intechopen.90253

Mathematical Modeling of Lactation Curves: A Review of Parametric Models

[65] Catillo G, Macciotta NPP,

Science. 2002;85(5):1298-1306

[67] Grossman M, Koops WJ.

1988;47:201-205

1988;71:1598-1608

101-108

2003. 452 p

87:169-173

2004

Carretta A, Cappio Borlino A. Effects of age and calving eeason on lactation curves of milk production traits in Italian water buffaloes. Journal of Dairy

[66] Papajcsik IA, Bodero J. Modelling lactation curves of Friesan cows in a subtropical climate. Animal Production.

Multiphasic analysis of lactation curves in dairy cattle. Journal of Dairy Science.

[68] Rook AJ, France J, Dhanoa MS. On the mathematical description of lactation curves. The Journal of Agricultural Science. 1993;121:97-102

[69] Congleton WRJ, Everett RW. Error and bias in using the incomplete gamma function to describe lactation curves. Journal of Dairy Science. 1980;63:

[71] Fadel JG. Technical note: Estimating parameters of nonlinear segmented models. Journal of Dairy Science. 2004;

[70] Vohnout KD. Mathematical Modeling for System Analysis in Agricultural Research. Amsterdam, The Netherlands: Elsevier Science;

[72] Silvestre AM, Martins AM, Santos VA, Ginja MM, Colaço JA. Lactation curves for milk, fat and protein in dairy cows: A full approach. Livestock Science. 2009;122:308-313

[73] Eicher R. Evaluation of the metabolic and nutritional situation in dairy herds: Diagnostic use of milk components. In: Proceedings of WBC Congress, Quebec 11–16 July, Canada.

[58] Schaeffer LR. Application of random regression models in animal breeding. Livestock Production Science.

[59] Schaeffer LR, Jamrozik J, Kistemaker GJ, Van Doormaal BJ. Experience with a test-day model. Journal of Dairy Science. 2000;83:

[60] Jamrozik J, Schaeffer LR, Dekkers JCM. Genetic evaluation of dairy cattle using test day yields and random regression model. Journal of Dairy Science. 1997;80:1217-1226

article/viewFile/237/237

2000;66:251-261

2006;45:13-23

p. 443

113

[61] Guo Z, Swalve HH. Modeling of the lactation curve as a sub-model in the evaluation of test day records. In: Paper Presented at the INTERBULL Open Meeting 7–8 September, Prague, Czech Republic. 1995. Available at https:// journal.interbull.org/index.php/ib/

[62] Kettunen A, Mantysaari E, Poso J. Estimation of genetic parameters for daily milk yield of primiparous Ayrshire cows by random regression test day models. Livestock Production Science.

[63] Quinn N, Killen L, Buckley F. Modeling fat and protein concentration curves for Irish dairy cows. Irish Journal of Agricultural and Food Research.

[64] Schaeffer LR, Dekkers JCM. Random regressions in animal models for test-day production in dairy cattle. In: Proc. 5th World Congr. Genet. Appl. Livest. Prod. Vol. XVIII. Guelph; 1994.

2003;86:2480-2490

2004;86:35-45

1135-1144

[49] Dag B, Keskin I, Mikailsoy F. Application of different models to the lactation curves of unimproved awassi ewes in Turkey. South African Journal of Animal Science. 2005;35:238-243

[50] Nelder JA. Inverse polynomials, a useful group of multi-factor response functions. Biometrics. 1966;22:128-141

[51] Yadav MC, Katpatal BG, Kaushik SN. Components of inverse polynomial function of lactation curve, and factors affecting them in Hariana and its Friesian crosses. The Indian Journal of Animal Sciences. 1977;47: 777-781

[52] Batra TR. Comparison of two mathematical models in fitting lactation curves for pureline and crossline dairy cows. Canadian Journal of Animal Science. 1986;66:405-414

[53] Bhat PN, Kumar R, Garg RC. Note on comparative efficiency of various lactation curve functions in Hariana cattle. The Indian Journal of Animal Sciences. 1981;51:102

[54] Singh RP, Gopal R. Lactation curve analysis of buffaloes maintained under village conditions. The Indian Journal of Animal Sciences. 1982;52:1157-1163

[55] Ali TE, Schaeffer LE. Accounting for covariance among test day milk yields in dairy cows. Canadian Journal of Animal Science. 1987;67:637-644

[56] Macciotta NPP, Vicario D, Cappio-Borlino A. Detection of different shapes of lactation curve for milk yield in dairy cattle by empirical mathematical models. Journal of Dairy Science. 2005; 88:1178-1191

Mathematical Modeling of Lactation Curves: A Review of Parametric Models DOI: http://dx.doi.org/10.5772/intechopen.90253

[57] Druet T, Jaffrezic F, Boichard D, Ducrocq V. Modeling lactation curves and estimation of genetic parameters for first lactation test-day records of French Holstein cows. Journal of Dairy Science. 2003;86:2480-2490

[38] Olori VE, Brotherstone S, Hill WG, Mcguirk BJ. Fit of standard models of the lactation curve to weekly records of milk production of cows in a single herd. Livestock Production Science. 1999;58:

Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…

[48] Sauvant D, Morand-Fehr P. Clasification des types de courbes de

[49] Dag B, Keskin I, Mikailsoy F. Application of different models to the lactation curves of unimproved awassi ewes in Turkey. South African Journal of Animal Science. 2005;35:238-243

composition du lait de la chèvre. Rech.

[50] Nelder JA. Inverse polynomials, a useful group of multi-factor response functions. Biometrics. 1966;22:128-141

Kaushik SN. Components of inverse polynomial function of lactation curve, and factors affecting them in Hariana and its Friesian crosses. The Indian Journal of Animal Sciences. 1977;47:

[52] Batra TR. Comparison of two mathematical models in fitting lactation curves for pureline and crossline dairy cows. Canadian Journal of Animal

[53] Bhat PN, Kumar R, Garg RC. Note on comparative efficiency of various lactation curve functions in Hariana cattle. The Indian Journal of Animal

[54] Singh RP, Gopal R. Lactation curve analysis of buffaloes maintained under village conditions. The Indian Journal of Animal Sciences. 1982;52:1157-1163

[55] Ali TE, Schaeffer LE. Accounting for covariance among test day milk yields in dairy cows. Canadian Journal of Animal

[56] Macciotta NPP, Vicario D, Cappio-Borlino A. Detection of different shapes of lactation curve for milk yield in dairy cattle by empirical mathematical models. Journal of Dairy Science. 2005;

Science. 1986;66:405-414

Sciences. 1981;51:102

Science. 1987;67:637-644

88:1178-1191

[51] Yadav MC, Katpatal BG,

777-781

lactation et d'evolution de la

Ovine Caprine. 1975;1:90

[39] Goodall EA. An analysis of

Production. 1983;36:69-72

seasonality of milk production. Animal

[40] JenkinsTG Ferrell CL. A note on lactation curves of crossbred cows. Animal Production. 1984;39:479-482

[41] Landete-Castillejos T, Gallego L. Technical note: The ability of mathematical models to describe the shape of lactation curves. Journal of Animal Science. 2000;78:3010-3013

[42] Wilmink JBM. Adjustment of testday Milk, fat and protein yield for age, season and stage of lactation. Livestock Production Science. 1987;16:335-348

[43] Cappio-Borlino A, Pulina G, Rossi G. A nonlinear modification of Wood's equation fitted to lactation curves of Sardinian dairy ewes. Small Ruminant Research. 1995;18:75-79

[44] Franci O, Pugliese C, Acciaioli A, Parisi G, Lucifero M. Application of two models to the lactation curve of Massese ewes. Small Ruminant Research. 1999;

[45] Delage J, Leroy AM, Poly J. Une étude sur les courbes de lactation. Annales de Zootechnie. 1953;3:225-267

[46] Ostergaard V. Strategies for concentrate feeding to attain optimum feeding level in high yielding dairy cows. Chap. V - food intake. In: Beretning fra statens husdyrbrugs forsog. Vol. 482. 1979. pp. 37-67

[47] Dave BK. First lactation curve of Indian water buffalo. JNKVV Research

Journal. 1971;5:93

112

31:91-96

55-63

[58] Schaeffer LR. Application of random regression models in animal breeding. Livestock Production Science. 2004;86:35-45

[59] Schaeffer LR, Jamrozik J, Kistemaker GJ, Van Doormaal BJ. Experience with a test-day model. Journal of Dairy Science. 2000;83: 1135-1144

[60] Jamrozik J, Schaeffer LR, Dekkers JCM. Genetic evaluation of dairy cattle using test day yields and random regression model. Journal of Dairy Science. 1997;80:1217-1226

[61] Guo Z, Swalve HH. Modeling of the lactation curve as a sub-model in the evaluation of test day records. In: Paper Presented at the INTERBULL Open Meeting 7–8 September, Prague, Czech Republic. 1995. Available at https:// journal.interbull.org/index.php/ib/ article/viewFile/237/237

[62] Kettunen A, Mantysaari E, Poso J. Estimation of genetic parameters for daily milk yield of primiparous Ayrshire cows by random regression test day models. Livestock Production Science. 2000;66:251-261

[63] Quinn N, Killen L, Buckley F. Modeling fat and protein concentration curves for Irish dairy cows. Irish Journal of Agricultural and Food Research. 2006;45:13-23

[64] Schaeffer LR, Dekkers JCM. Random regressions in animal models for test-day production in dairy cattle. In: Proc. 5th World Congr. Genet. Appl. Livest. Prod. Vol. XVIII. Guelph; 1994. p. 443

[65] Catillo G, Macciotta NPP, Carretta A, Cappio Borlino A. Effects of age and calving eeason on lactation curves of milk production traits in Italian water buffaloes. Journal of Dairy Science. 2002;85(5):1298-1306

[66] Papajcsik IA, Bodero J. Modelling lactation curves of Friesan cows in a subtropical climate. Animal Production. 1988;47:201-205

[67] Grossman M, Koops WJ. Multiphasic analysis of lactation curves in dairy cattle. Journal of Dairy Science. 1988;71:1598-1608

[68] Rook AJ, France J, Dhanoa MS. On the mathematical description of lactation curves. The Journal of Agricultural Science. 1993;121:97-102

[69] Congleton WRJ, Everett RW. Error and bias in using the incomplete gamma function to describe lactation curves. Journal of Dairy Science. 1980;63: 101-108

[70] Vohnout KD. Mathematical Modeling for System Analysis in Agricultural Research. Amsterdam, The Netherlands: Elsevier Science; 2003. 452 p

[71] Fadel JG. Technical note: Estimating parameters of nonlinear segmented models. Journal of Dairy Science. 2004; 87:169-173

[72] Silvestre AM, Martins AM, Santos VA, Ginja MM, Colaço JA. Lactation curves for milk, fat and protein in dairy cows: A full approach. Livestock Science. 2009;122:308-313

[73] Eicher R. Evaluation of the metabolic and nutritional situation in dairy herds: Diagnostic use of milk components. In: Proceedings of WBC Congress, Quebec 11–16 July, Canada. 2004

**115**

**Chapter 7**

*Moez Ayadi*

**Abstract**

system.

profitability

**1. Introduction**

facilitate the storage of the milk produced [2–7].

Optimization of Milking

Frequency in Dairy Ruminants

To make a decision on the number of milkings per day for each ruminant is a key factor to optimize the use of a machine milking. Currently, this decision is mainly taken from yield and stage of lactation data, but no udder capacity is taken into account. Milk is stored in the udder in the alveolar and cisternal compartments. Milk partitioning in the udder varied widely according to species, breed, lactation stage, parity, and milking interval. The increase in milking frequency has improved milk production in dairy ruminants. However, this practice reduces the milk composition, fertility, and productive life. To avoid increasing the number of milkings per day and reducing milk losses, a strategy based on the selection of ruminants with large udder cistern to store a large quantity of milk was adopted. Animals with great cisterns tolerate extended milking intervals and are milked faster with simplified routines. Ultrasonography will be a useful tool to measure udder cistern and to predict high-yielding animals. In practice, we propose to use the evaluation of udder cistern area, as helping criteria of udder milk storage capacity, establishing the optimal milking frequencies for each ruminant according to the production

**Keywords:** udder morphology, milking frequency, cisternal capacity, ultrasound,

The problems faced by farmers vary according to the region of production, the breed, the breeding, the feeding system, and the environmental conditions. Indeed, data in Europe show that milk production is surplus, but it is in deficit in Africa and South America [1]. The breeding programs for dairy animals have led to an increase in the quantity of milk. The reasons of this increase in milk yield include udder size, connective tissue mass, and secretory tissue. In fact, a hypertrophy of the secretory tissue of the udder is accompanied by a large milk production that can only be expressed phenotypically when the volume of the udder cistern is important to

In practice, because of the selection pressure exerted on the morphology of the udder, for example, cows with a high milk production must be milked at least three times a day. So, the increase in milking frequency has improved milk production in cows (15–20%, [8]). However, this practice reduces the fat matter, protein, fertility, and productive life of dairy cows [9]. It should be noted that the increase in the number of milkings per day is not accepted by farmers who are looking for farming practices to reduce the number of milkings per week and improve the quality of life on the farm [6].

[74] Reist M, Erdin D, Von Euw D, Tschuemperlin K, Leuenberger H, Chilliard Y, et al. Estimation of energy balance at the individual and herd level using blood and milk traits in highyielding dairy cows. Journal of Dairy Science. 2002;85:3314-3327

[75] Loeffler SH, Devries MJ, Schukken YH. The effects of time of disease occurrence, milk yield and body condition on fertility of dairy cows. Journal of Dairy Science. 1999;82: 589-2600

#### **Chapter 7**

[74] Reist M, Erdin D, Von Euw D, Tschuemperlin K, Leuenberger H, Chilliard Y, et al. Estimation of energy balance at the individual and herd level

Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…

using blood and milk traits in highyielding dairy cows. Journal of Dairy Science. 2002;85:3314-3327

[75] Loeffler SH, Devries MJ,

589-2600

114

Schukken YH. The effects of time of disease occurrence, milk yield and body condition on fertility of dairy cows. Journal of Dairy Science. 1999;82:

## Optimization of Milking Frequency in Dairy Ruminants

*Moez Ayadi*

#### **Abstract**

To make a decision on the number of milkings per day for each ruminant is a key factor to optimize the use of a machine milking. Currently, this decision is mainly taken from yield and stage of lactation data, but no udder capacity is taken into account. Milk is stored in the udder in the alveolar and cisternal compartments. Milk partitioning in the udder varied widely according to species, breed, lactation stage, parity, and milking interval. The increase in milking frequency has improved milk production in dairy ruminants. However, this practice reduces the milk composition, fertility, and productive life. To avoid increasing the number of milkings per day and reducing milk losses, a strategy based on the selection of ruminants with large udder cistern to store a large quantity of milk was adopted. Animals with great cisterns tolerate extended milking intervals and are milked faster with simplified routines. Ultrasonography will be a useful tool to measure udder cistern and to predict high-yielding animals. In practice, we propose to use the evaluation of udder cistern area, as helping criteria of udder milk storage capacity, establishing the optimal milking frequencies for each ruminant according to the production system.

**Keywords:** udder morphology, milking frequency, cisternal capacity, ultrasound, profitability

#### **1. Introduction**

The problems faced by farmers vary according to the region of production, the breed, the breeding, the feeding system, and the environmental conditions. Indeed, data in Europe show that milk production is surplus, but it is in deficit in Africa and South America [1]. The breeding programs for dairy animals have led to an increase in the quantity of milk. The reasons of this increase in milk yield include udder size, connective tissue mass, and secretory tissue. In fact, a hypertrophy of the secretory tissue of the udder is accompanied by a large milk production that can only be expressed phenotypically when the volume of the udder cistern is important to facilitate the storage of the milk produced [2–7].

In practice, because of the selection pressure exerted on the morphology of the udder, for example, cows with a high milk production must be milked at least three times a day. So, the increase in milking frequency has improved milk production in cows (15–20%, [8]). However, this practice reduces the fat matter, protein, fertility, and productive life of dairy cows [9]. It should be noted that the increase in the number of milkings per day is not accepted by farmers who are looking for farming practices to reduce the number of milkings per week and improve the quality of life on the farm [6].

To avoid increasing the number of milkings per day and reducing milk losses, a strategy based on the selection of ruminants with large udder cistern to store a large quantity of milk was adopted [2, 3]. Therefore, noninvasive in vivo imaging techniques to measure udder storage capacity have been developed [10–14, 5].

Indeed, the study of the internal morphology of the udder in ruminants will know an important development soon. Scientific advances such as embryo transplantation and cloning can contribute to increased uniformity of livestock. Therefore, with this new orientation, it is interesting to take into account, in addition to the external volume of the udder, the internal size of the udder, the capacity of distension of the cells, and the kinetics of udder filling to ensure better adaptation of the udders to the number of milkings (conventional mechanical milking and robotic) and the different milk production systems (extensive, intensive) to maximize farmer's income.

#### **2. Morphology of the mammary gland in dairy ruminants**

#### **2.1 Udder morphology and milk production**

The purpose of the use of morphological traits in a dairy breeding scheme is to improve the functional longevity of animals by reducing the frequency of reforms and facilitate the adaptation of animals to milking, mechanics, as well as the work of the breeder. In fact, the study of mammary morphometry in dairy animals permits identifying correlations between some morphological traits and milk production as well as the possibility of mechanical milking.

Several authors have studied the external morphological characteristics of the udder in ruminants for performance evaluation and mechanical milking skills [15–18]. The importance of these morphological measurements of the cow's udder has been accentuated by the interest in the application of the system of mechanical milking by robot [19]. In dairy cows, the large udders are usually the ones that give the most milk. Moreover, the correlations between the estimated volume of the udder and the milk production can vary from 0.60 to 0.82 depending on the breed [20]. According to the same authors, the teats must be implanted vertically, and the distance between them must never be less than 6 cm. Wide teats are associated with udder health risks and with the quality of produced milk [21]. In cattle, positive correlations have been confirmed between the distances of the teats and the teat diameter and milk yield of cows [15]. The same researchers have shown that some features of cows' mammary morphology may be associated with a risk of higher mastitis such as low teats and wide teats, as they may increase the risk of injury and the entry of pathogens inside the udder.

In ewes, the criteria of the size and shape of the udder and teats are positively correlated with milk production [21] and milk flow [22]. The presence of developed udder implies a good ability to withstand long intervals between milking and even the practice of single milking (one milking per day). Indeed, the removal of one milking per day indicates a greater loss of milk in breeds with smaller cistern udders [4]. In addition, the udders with rather horizontal teats and inserted high relative to the base of the cisterns are associated with the increase in the fraction of milk drip, requiring a manual intervention of massage and udder movement to collect the milk not extracted by the machine.

In Murciano-Granadina goats, positive correlations between teat length and milk flow (r = 0.55) and between teat surface (r = 0.47–0.58) and residual milk were reported [23]. In the Saanen goats, a positive correlation (r = 0.65) was found between the circumference of the udder and daily milk production.

**117**

*Optimization of Milking Frequency in Dairy Ruminants DOI: http://dx.doi.org/10.5772/intechopen.87303*

rous (3.40 cm) than multiparous (6.10 cm).

so-called udder simple cistern (case of ruminants).

linked to stimulation of the udder when introducing the cannula).

important to determine the appropriate intervals between milkings.

**2.2 Storage of milk in the udder**

traits.

Unlike other dairy animals, there has been little work on the study of mammary morphology in camels [24–29]. A good udder in camels is characterized by welldeveloped and symmetrical neighborhoods with vertically implanted, uniform, and well-spaced teats [30]. In the same context, [28] reported that the length and depth of the udder in camels are of the same order of magnitude as those indicated for cows [6] and for buffalos [31]. In camels, the teats placed very close to each other and sometimes fused are frequent. Juhasz and Nagy [32] showed a great heterogeneity in the morphology of the udder and teats in camels. They defined at least five different forms of teats. Ayadi et al. [28] found that daily milk production is positively correlated with teat distance (r = 0.61), udder depth (r = 0.29), and breast vein diameter (r = 0.34). The conformation of the udder in camels varies considerably according to the breed, the age, and the stage of lactation. Indeed, [25] reported that camels' teat length varies with parity with shorter teats in primipa-

Recently, for the development of a dromedary selection program according to the udder and teat typology adapted to mechanical milking, [33] proposed a 5-point linear scoring template for evaluating the udder of dairy camels based on five main

A functional mammary gland is an exocrine gland consisting of a tubuloalveolar epithelial tissue and a stroma. In order to fulfill its production function, the udder is richly vascularized and innervated. There are two main categories of udders: the so-called udders composed without cistern (case of rodents and primates) and the

In the ruminant udder, it is possible to distinguish an alveolar compartment (alveoli and fine channels) from an interconnected cistern compartment (large canals and cistern). The volumes of milk accumulated in each of these anatomical compartments can be measured accurately. Milk was first evaluated by draining the milk accumulated in the cistern by insertion of a cannula into the teat canal; the alveolar milk is then recovered by milking followed or not by an injection of oxytocin [2]. The use of a cannula to drain milk has been widely used. By using this technique, the volume of milk may be overestimated (in addition to milk, a fraction of alveolar milk can be recovered by endogenous oxytocin secretion conditioned or

Intravenous injection of oxytocin receptor blocking agent (Atosiban), which inhibits milk ejection, has been developed [34, 35]. Milking after injection of

In cows, the volume of milk contained in the alveolar compartment is preponderant since it represents between 70 and 80% of the total quantity of milk 12 hours after milking [2, 3, 5]. This fraction is about 50–75% in ewes [37, 38, 4] and reaches even 80–90% in goats [39]. The cisternal milk represented 3–19% of total milk in camels [28, 12, 26] and 5% in buffaloes [40]. These proportions may vary depending on the breed of animals but also on the stage of lactation. In addition, the volume of milk stored in the cistern increases during lactation because of the decrease in secretory tissue during lactation [41]. Likewise, the volume of milk

Atosiban permits to collect the cistern milk; then oxytocin injection reverses the effect and the alveolar milk can be collected. The use of Atosiban is therefore recommended in ruminants [36]. Moreover, noninvasive in vivo imaging techniques (ultrasonography) have been used to measure cistern udder storage capacity [10, 11, 5, 13]. Certainly, whatever the measurement method, knowledge of the distribution of milk in the udder, as well as the kinetics of its filling according to the species, is particularly

*Optimization of Milking Frequency in Dairy Ruminants DOI: http://dx.doi.org/10.5772/intechopen.87303*

*Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…*

niques to measure udder storage capacity have been developed [10–14, 5]. Indeed, the study of the internal morphology of the udder in ruminants will know an important development soon. Scientific advances such as embryo transplantation and cloning can contribute to increased uniformity of livestock. Therefore, with this new orientation, it is interesting to take into account, in addition to the external volume of the udder, the internal size of the udder, the capacity of distension of the cells, and the kinetics of udder filling to ensure better adaptation of the udders to the number of milkings (conventional mechanical milking and robotic) and the different milk production systems (extensive, intensive) to

**2. Morphology of the mammary gland in dairy ruminants**

**2.1 Udder morphology and milk production**

tion as well as the possibility of mechanical milking.

the entry of pathogens inside the udder.

not extracted by the machine.

maximize farmer's income.

To avoid increasing the number of milkings per day and reducing milk losses, a strategy based on the selection of ruminants with large udder cistern to store a large quantity of milk was adopted [2, 3]. Therefore, noninvasive in vivo imaging tech-

The purpose of the use of morphological traits in a dairy breeding scheme is to improve the functional longevity of animals by reducing the frequency of reforms and facilitate the adaptation of animals to milking, mechanics, as well as the work of the breeder. In fact, the study of mammary morphometry in dairy animals permits identifying correlations between some morphological traits and milk produc-

Several authors have studied the external morphological characteristics of the udder in ruminants for performance evaluation and mechanical milking skills [15–18]. The importance of these morphological measurements of the cow's udder has been accentuated by the interest in the application of the system of mechanical milking by robot [19]. In dairy cows, the large udders are usually the ones that give the most milk. Moreover, the correlations between the estimated volume of the udder and the milk production can vary from 0.60 to 0.82 depending on the breed [20]. According to the same authors, the teats must be implanted vertically, and the distance between them must never be less than 6 cm. Wide teats are associated with udder health risks and with the quality of produced milk [21]. In cattle, positive correlations have been confirmed between the distances of the teats and the teat diameter and milk yield of cows [15]. The same researchers have shown that some features of cows' mammary morphology may be associated with a risk of higher mastitis such as low teats and wide teats, as they may increase the risk of injury and

In ewes, the criteria of the size and shape of the udder and teats are positively correlated with milk production [21] and milk flow [22]. The presence of developed udder implies a good ability to withstand long intervals between milking and even the practice of single milking (one milking per day). Indeed, the removal of one milking per day indicates a greater loss of milk in breeds with smaller cistern udders [4]. In addition, the udders with rather horizontal teats and inserted high relative to the base of the cisterns are associated with the increase in the fraction of milk drip, requiring a manual intervention of massage and udder movement to collect the milk

In Murciano-Granadina goats, positive correlations between teat length and milk flow (r = 0.55) and between teat surface (r = 0.47–0.58) and residual milk were reported [23]. In the Saanen goats, a positive correlation (r = 0.65) was found

between the circumference of the udder and daily milk production.

**116**

Unlike other dairy animals, there has been little work on the study of mammary morphology in camels [24–29]. A good udder in camels is characterized by welldeveloped and symmetrical neighborhoods with vertically implanted, uniform, and well-spaced teats [30]. In the same context, [28] reported that the length and depth of the udder in camels are of the same order of magnitude as those indicated for cows [6] and for buffalos [31]. In camels, the teats placed very close to each other and sometimes fused are frequent. Juhasz and Nagy [32] showed a great heterogeneity in the morphology of the udder and teats in camels. They defined at least five different forms of teats. Ayadi et al. [28] found that daily milk production is positively correlated with teat distance (r = 0.61), udder depth (r = 0.29), and breast vein diameter (r = 0.34). The conformation of the udder in camels varies considerably according to the breed, the age, and the stage of lactation. Indeed, [25] reported that camels' teat length varies with parity with shorter teats in primiparous (3.40 cm) than multiparous (6.10 cm).

Recently, for the development of a dromedary selection program according to the udder and teat typology adapted to mechanical milking, [33] proposed a 5-point linear scoring template for evaluating the udder of dairy camels based on five main traits.

#### **2.2 Storage of milk in the udder**

A functional mammary gland is an exocrine gland consisting of a tubuloalveolar epithelial tissue and a stroma. In order to fulfill its production function, the udder is richly vascularized and innervated. There are two main categories of udders: the so-called udders composed without cistern (case of rodents and primates) and the so-called udder simple cistern (case of ruminants).

In the ruminant udder, it is possible to distinguish an alveolar compartment (alveoli and fine channels) from an interconnected cistern compartment (large canals and cistern). The volumes of milk accumulated in each of these anatomical compartments can be measured accurately. Milk was first evaluated by draining the milk accumulated in the cistern by insertion of a cannula into the teat canal; the alveolar milk is then recovered by milking followed or not by an injection of oxytocin [2]. The use of a cannula to drain milk has been widely used. By using this technique, the volume of milk may be overestimated (in addition to milk, a fraction of alveolar milk can be recovered by endogenous oxytocin secretion conditioned or linked to stimulation of the udder when introducing the cannula).

Intravenous injection of oxytocin receptor blocking agent (Atosiban), which inhibits milk ejection, has been developed [34, 35]. Milking after injection of Atosiban permits to collect the cistern milk; then oxytocin injection reverses the effect and the alveolar milk can be collected. The use of Atosiban is therefore recommended in ruminants [36]. Moreover, noninvasive in vivo imaging techniques (ultrasonography) have been used to measure cistern udder storage capacity [10, 11, 5, 13]. Certainly, whatever the measurement method, knowledge of the distribution of milk in the udder, as well as the kinetics of its filling according to the species, is particularly important to determine the appropriate intervals between milkings.

In cows, the volume of milk contained in the alveolar compartment is preponderant since it represents between 70 and 80% of the total quantity of milk 12 hours after milking [2, 3, 5]. This fraction is about 50–75% in ewes [37, 38, 4] and reaches even 80–90% in goats [39]. The cisternal milk represented 3–19% of total milk in camels [28, 12, 26] and 5% in buffaloes [40]. These proportions may vary depending on the breed of animals but also on the stage of lactation. In addition, the volume of milk stored in the cistern increases during lactation because of the decrease in secretory tissue during lactation [41]. Likewise, the volume of milk

is higher in multiparas [41]. This is due to the immaturity of the development of cisterns in primiparas [42]. Studies on milk accumulation in the udder after milking have been conducted in dairy ruminants.

Recently, [43] proposed a 6-point linear scoring template for evaluating the cisternal size of the udder of dairy cows (0 = absent cistern; 6 = very large cistern), evaluated by ultrasound according to the methodology of [5]. This classification optimizes the milking frequency according to the stage of lactation and the production system.

#### **3. Milking frequency**

Milk production (quantity and quality of milk) is regulated at different levels: by genetic factors, diet, various endocrines, and environmental controls. One of the levers for acting on the metabolic and secretory activity of the udder is the frequency of milking. Generally, cows are milked twice a day with milking intervals ranging from 8 to 16 hours, though studies have been conducted to determine animal milking management systems that combine maximization of quantitative and qualitative production with reduced work constraints. Research showed that for a frequency of two milkings per day, a 12–12 interval would be beneficial for highproducing cows (3–5% gain over a 10–14 interval) [37], suggesting the appearance of a brake on secretion beyond a certain time limit. To determine this limit, several studies place it between 10 and 18 hours depending on the animals [44]. These differences could be due to inter-individual variations and could also be related to anatomical features of the udder.

In fact, animals with large udder cistern produce more milk and withstand relatively longer intervals between milkings than animals with a small udder cistern, which cannot transfer their alveolar milk and in which a brake on the secretion is set up faster. Such an observation has been verified in cows [2, 5], ewes [37, 13], goats [39], and camels [45, 12]. Therefore, it has been shown that when milk can flow continuously from the udder, milk production increases [44].

#### **3.1 Decrease in milking frequency**

The consequences of reducing the number of milkings on ruminant milk production have been studied by many authors. Certainly, the passage from two milkings to a single milking per day leads to a loss of milk production from 10 to 50% in cows [42].

Short-term (1 week) trials of mid-lactation Friesian and Jersey cows from two milkings to one daily milking reported milk yield decreases ranging from 10 to 25% [2]. The responses would depend on the stage of lactation since the loss of production would be more pronounced for animals in early lactation than for animals at the end of lactation (−38 vs. −28%) [44]. This can be related to the anatomy of the gland since it is known that the proportion of milk stored in the cistern varies with relation to the stage of lactation in cows [46, 41]. In ewes, switching to one milking per day causes a decrease in production from 15 to 35% [47–49]. The lowest losses are reported in Sardi ewes, known for their high capacity for storage and high production capacity, while the largest losses are observed in pre-Alpine ewes with small tanks and low production.

In goats, one milking per day leads to production decrease from 6 to 35% compared to two milkings per day [50]. As in cows and sheep, race and stage of lactation have an effect, which can be related to the storage capacity of the udder. Undeniably, the largest losses are recorded at the beginning of lactation [39], and the lowest losses

**119**

*Optimization of Milking Frequency in Dairy Ruminants DOI: http://dx.doi.org/10.5772/intechopen.87303*

lactation [50, 51].

*3.1.1 Alveolar distension*

traffic of the constituents of milk.

*3.1.2 Decrease in udder blood flow*

hypothesis has not been confirmed in goats [58].

*3.1.3 Increase of tight junction's permeability*

are observed in Canary goats, with very large cisterns. Overall, a decrease in milking frequency causes production losses depending on the animal's storage capacity. Finally, it seems that this milking practice increases the concentration of somatic cells in milk [44], the increase being more marked as the number of cells in the milk at the beginning of the experiment is important. In dairy sheep, switching to one milking per day does not significantly modify the composition of milk [47], whereas

in goats, an increase in fat matter and casein concentrations is reported [39].

In dairy ruminants, as time after milking increased, there is (i) an increase in alveolar distension, (ii) a decrease in udder blood flow, (iii) an increase in tight junction's permeability, and (vi) an accumulation of putative feedback inhibitor of

The first signals of local regulation of mammary gland activity are probably the degree and duration of alveolar distension. Studies by [52] have shown that the amount of alveolar milk in goats is low compared to animals with a high volume of residual milk. Despite the size of the alveolar compartment of the udder of cows reaches its maximum around 16 hours after milking, the longer the interval increases beyond 16 hours, the more the cells are filled with milk. In fact, the increase in pressure following the accumulation of milk throughout the mammary ducts generally leads to an inhibition of the secretion of milk [44]. According to [53], the dilation of the mammary alveoli is accompanied by a decrease in prolactin concentrations when the milking frequency is reduced. Furthermore, the increase of the intra-alveolar pressure causes the compression of the mammary epithelial cells (CEMs) altering the activity of their cytoskeletons and thus the intracellular

Dairy production and mammary blood flow are positively correlated throughout lactation, with the synthesis of 1 L of milk requiring the passage of approximately 300–500 L of blood regardless of the ruminant species [54]. The increase in intramammary pressure (IMP) related to milk accumulation decreases the mammary blood flow (−10% after 24 hours in cows) [55] and −50% after 36 hours in goats [56]. The availability of hormones and nutrients would be reduced in the gland, thus decreasing the rate of secretion. This decrease in mammary blood flow could also be related to the activation of the sympathetic nervous system by the accumulation of milk [57]. Draining more frequently would therefore avoid a decline in blood flow, which could be a limiting factor for milk production, although the latter

A regulating mechanism involved in the practice of a single milking per day acts on the tight junctions, leading to an increase in the alveolar permeability. Really, the change in the chemical composition of milk during the practice of daily milking can be attributed to an increase in the serum in milk, as a result of changing the permeability of tight junctions. Furthermore, the increased permeability of the tight junction is achieved at around 17–18 hours of milking in cows [51], 19–20 hours in sheep [13], beyond 21 hours in goats [39], and 16 hours in camels [45]. Indeed, the change of the permeability of the mammary epithelium membrane during the practice of a single daily milking suggests a rapid increase in the concentration of lactose in the

#### *Optimization of Milking Frequency in Dairy Ruminants DOI: http://dx.doi.org/10.5772/intechopen.87303*

are observed in Canary goats, with very large cisterns. Overall, a decrease in milking frequency causes production losses depending on the animal's storage capacity. Finally, it seems that this milking practice increases the concentration of somatic cells in milk [44], the increase being more marked as the number of cells in the milk at the beginning of the experiment is important. In dairy sheep, switching to one milking per day does not significantly modify the composition of milk [47], whereas in goats, an increase in fat matter and casein concentrations is reported [39].

In dairy ruminants, as time after milking increased, there is (i) an increase in alveolar distension, (ii) a decrease in udder blood flow, (iii) an increase in tight junction's permeability, and (vi) an accumulation of putative feedback inhibitor of lactation [50, 51].

#### *3.1.1 Alveolar distension*

*Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…*

have been conducted in dairy ruminants.

tion system.

**3. Milking frequency**

anatomical features of the udder.

**3.1 Decrease in milking frequency**

small tanks and low production.

50% in cows [42].

is higher in multiparas [41]. This is due to the immaturity of the development of cisterns in primiparas [42]. Studies on milk accumulation in the udder after milking

Recently, [43] proposed a 6-point linear scoring template for evaluating the cisternal size of the udder of dairy cows (0 = absent cistern; 6 = very large cistern), evaluated by ultrasound according to the methodology of [5]. This classification optimizes the milking frequency according to the stage of lactation and the produc-

Milk production (quantity and quality of milk) is regulated at different levels: by genetic factors, diet, various endocrines, and environmental controls. One of the levers for acting on the metabolic and secretory activity of the udder is the frequency of milking. Generally, cows are milked twice a day with milking intervals ranging from 8 to 16 hours, though studies have been conducted to determine animal milking management systems that combine maximization of quantitative and qualitative production with reduced work constraints. Research showed that for a frequency of two milkings per day, a 12–12 interval would be beneficial for highproducing cows (3–5% gain over a 10–14 interval) [37], suggesting the appearance of a brake on secretion beyond a certain time limit. To determine this limit, several studies place it between 10 and 18 hours depending on the animals [44]. These differences could be due to inter-individual variations and could also be related to

In fact, animals with large udder cistern produce more milk and withstand relatively longer intervals between milkings than animals with a small udder cistern, which cannot transfer their alveolar milk and in which a brake on the secretion is set up faster. Such an observation has been verified in cows [2, 5], ewes [37, 13], goats [39], and camels [45, 12]. Therefore, it has been shown that when milk can flow

The consequences of reducing the number of milkings on ruminant milk production have been studied by many authors. Certainly, the passage from two milkings to a single milking per day leads to a loss of milk production from 10 to

Short-term (1 week) trials of mid-lactation Friesian and Jersey cows from two milkings to one daily milking reported milk yield decreases ranging from 10 to 25% [2]. The responses would depend on the stage of lactation since the loss of production would be more pronounced for animals in early lactation than for animals at the end of lactation (−38 vs. −28%) [44]. This can be related to the anatomy of the gland since it is known that the proportion of milk stored in the cistern varies with relation to the stage of lactation in cows [46, 41]. In ewes, switching to one milking per day causes a decrease in production from 15 to 35% [47–49]. The lowest losses are reported in Sardi ewes, known for their high capacity for storage and high production capacity, while the largest losses are observed in pre-Alpine ewes with

In goats, one milking per day leads to production decrease from 6 to 35% compared to two milkings per day [50]. As in cows and sheep, race and stage of lactation have an effect, which can be related to the storage capacity of the udder. Undeniably, the largest losses are recorded at the beginning of lactation [39], and the lowest losses

continuously from the udder, milk production increases [44].

**118**

The first signals of local regulation of mammary gland activity are probably the degree and duration of alveolar distension. Studies by [52] have shown that the amount of alveolar milk in goats is low compared to animals with a high volume of residual milk. Despite the size of the alveolar compartment of the udder of cows reaches its maximum around 16 hours after milking, the longer the interval increases beyond 16 hours, the more the cells are filled with milk. In fact, the increase in pressure following the accumulation of milk throughout the mammary ducts generally leads to an inhibition of the secretion of milk [44]. According to [53], the dilation of the mammary alveoli is accompanied by a decrease in prolactin concentrations when the milking frequency is reduced. Furthermore, the increase of the intra-alveolar pressure causes the compression of the mammary epithelial cells (CEMs) altering the activity of their cytoskeletons and thus the intracellular traffic of the constituents of milk.

#### *3.1.2 Decrease in udder blood flow*

Dairy production and mammary blood flow are positively correlated throughout lactation, with the synthesis of 1 L of milk requiring the passage of approximately 300–500 L of blood regardless of the ruminant species [54]. The increase in intramammary pressure (IMP) related to milk accumulation decreases the mammary blood flow (−10% after 24 hours in cows) [55] and −50% after 36 hours in goats [56]. The availability of hormones and nutrients would be reduced in the gland, thus decreasing the rate of secretion. This decrease in mammary blood flow could also be related to the activation of the sympathetic nervous system by the accumulation of milk [57]. Draining more frequently would therefore avoid a decline in blood flow, which could be a limiting factor for milk production, although the latter hypothesis has not been confirmed in goats [58].

#### *3.1.3 Increase of tight junction's permeability*

A regulating mechanism involved in the practice of a single milking per day acts on the tight junctions, leading to an increase in the alveolar permeability. Really, the change in the chemical composition of milk during the practice of daily milking can be attributed to an increase in the serum in milk, as a result of changing the permeability of tight junctions. Furthermore, the increased permeability of the tight junction is achieved at around 17–18 hours of milking in cows [51], 19–20 hours in sheep [13], beyond 21 hours in goats [39], and 16 hours in camels [45]. Indeed, the change of the permeability of the mammary epithelium membrane during the practice of a single daily milking suggests a rapid increase in the concentration of lactose in the

blood plasma and increased serum protein in milk and content of milk in Cl and Na and a reduction in lactose and K [59].

#### *3.1.4 Accumulation of the feedback inhibitor of lactation*

The causes of the decrease in milk production for daily single milking are not well known. Indeed, in dairy cows, it has been shown, reduction of the milking frequency in one quarter of the mammary gland and not in the other quarters, that the quantity of milk in the treated unit only once a day decreased [60]. The same results were observed in sheep and goats [61]. In addition, incomplete emptying of the udder causes a decrease in production [62]. In order to prevent engorgement, the mammary gland has a feedback mechanism on milk synthesis; it produces a glycoprotein that inhibits its synthesis. Therefore, frequent emptying reduces the amount of this inhibiting factor in contact with the CEMs. This local chemical factor with inhibitory activity on milk secretion, called feedback inhibitor of lactation (FIL) or lactation inhibitor factor (LIF), is a low-molecular-weight protein (7.6 kDa), which has been shown in goats [63]. The FIL reduces the secretion rate of milk in vitro [62] and in vivo [63] when in contact with the alveolar epithelium. It works by inhibiting the constitutive secretion of proteins by CEMs by reversible blocking of the early stages of the biosynthesis-secretion pathway. In addition, the FIL would also inhibit lactose synthesis. Finally, FIL would regulate the number of cell tissue by triggering apoptosis [54]. Indeed, incomplete milking or milking removals would allow an accumulation of the FIL in contact with the CEMs, which would explain the reductions in milk production described above.

Recently, serotonin (5-HT) has been proposed to be an autocrine/paracrine regulator of lactation in the mouse, humans, and more recently in the bovine. The enzymatic machinery necessary for 5-HT biosynthesis has been detected in the mammary epithelium [64]. Other researches support the concept that serotonin (5-HT) is a feedback inhibitor of lactation in the bovine [65].

#### **3.2 Increase in milking frequency**

Increasing milking frequency in dairy cattle to more than two milkings per day has resulted in an increase in milk production without any negative effect on the health of the animal. There are various reasons for the practice of three milkings a day, namely, increase in the time of use of the milking machine, the size of the herd, and milk production. In fact, milking three times a day results in an increase in milk production from 3 to 39% compared to two times in dairy cows [66], 15–35% in ewes [47], 10–20% in goats [67, 62, 39], and 4–13% in camels [68, 45]. The response to increased milking frequency would be greater in high-producing, primiparous, and late-lactating cows [66, 69]. Erdman and Varner [70] in their review of 40 comparative studies of increased milking frequency reported that switching from 2 to 3 milkings per day resulted in a stable increase in milk production in terms of quantity (3.5 kg/day) and not by a proportional increase.

In studies on the milking robot, it has been shown that cows, when given free time, are milked on average between 2.7 and 3.9 times a day [71]. In addition, when a rate of four milkings per day is applied for 4 weeks, a production increase of 14% is observed [2]. However, switching to six milkings per day for 2 days only increases production by 10–15% [8]. Such observations suggest that an interval between milkings of 6–8 hours is physiologically ideal for the animal and that there would be no advantage in increasing the rate of milking above four milkings per day in cows [53], as in ewes [47]. An increase in milking frequency would therefore allow improved persistence of production in goats and cows [2, 51].

**121**

and camels [45].

**4. Conclusions**

ruminants.

*Optimization of Milking Frequency in Dairy Ruminants DOI: http://dx.doi.org/10.5772/intechopen.87303*

cows increases the California mastitis test (CMT) score.

each ruminant according to the production system.

There are contradictions in the literature regarding the effects of switching to three milkings per day. For some, changes in milk composition at three milkings per day would be insignificant, while others report a decrease in milk fat compared with cows milked twice a day [8]. For some, this decrease would be greater for primiparous cows, while for others, the decrease would be greater for multiparous cows [66]. At the lactation scale, [69] noted a slight decrease in protein and casein concentrations in milk, enough to reduce cheese yield by 1.5%. Somatic cell milk content is used as an indicator of the microbiological quality of milk. Indeed, the number of somatic cells decreases when milking frequency increases [71]. On the other hand, [66] report that switching from two to three milkings per day in dairy

The increase of milking frequency could lead to an increased release of lactogenic hormones which will stimulate the synthetic activities of the CEMs and allow a better persistence of the lactation. These hormones may, in addition to their metabolic effects, increase the number of secretory cells and thus increase the volume of milk secreted [72]. Indeed, even if it is admitted in ruminants that the CEMs deteriorate and that their number decreases progressively with the advancement of lactation, by triggering apoptosis [54], the activity of synthesis of remaining cells is unchanged [72]. This decrease in the number of cells would be modulated by the frequency of milking. The increase of milking frequency causes cellular hypertrophy followed by an increase in the number of CEMs by proliferation of new cells. In addition, an increase in enzyme activities, reflecting their potential for synthesis, is observed in response to an increase in milking frequency in goats [62], cows [2],

Deciding about the number of milkings per day for each ruminant is a key factor in optimizing the use of mechanical milking. Currently, this decision is primarily based on the production level and stage of lactation data, but no udder capacity is taken into account. Therefore, it is recommended to use ruminants with large cisterns in order to minimize the effect of hydrostatic pressure on the cells and consequently reduce production losses. In practice, we propose to use the evaluation of udder cistern area by ultrasonography as a criterion to estimate milk storage capacity in the udder in order to establish the appropriate milking frequency for

Research opportunities are open to broaden and consolidate this study. Indeed,

the work on the heritability and the repeatability of this character "glandular cistern" is essential in order to incorporate it into the breeding programs for dairy

#### *Optimization of Milking Frequency in Dairy Ruminants DOI: http://dx.doi.org/10.5772/intechopen.87303*

*Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…*

and a reduction in lactose and K [59].

tions in milk production described above.

**3.2 Increase in milking frequency**

(5-HT) is a feedback inhibitor of lactation in the bovine [65].

quantity (3.5 kg/day) and not by a proportional increase.

improved persistence of production in goats and cows [2, 51].

*3.1.4 Accumulation of the feedback inhibitor of lactation*

blood plasma and increased serum protein in milk and content of milk in Cl and Na

The causes of the decrease in milk production for daily single milking are not well known. Indeed, in dairy cows, it has been shown, reduction of the milking frequency in one quarter of the mammary gland and not in the other quarters, that the quantity of milk in the treated unit only once a day decreased [60]. The same results were observed in sheep and goats [61]. In addition, incomplete emptying of the udder causes a decrease in production [62]. In order to prevent engorgement, the mammary gland has a feedback mechanism on milk synthesis; it produces a glycoprotein that inhibits its synthesis. Therefore, frequent emptying reduces the amount of this inhibiting factor in contact with the CEMs. This local chemical factor with inhibitory activity on milk secretion, called feedback inhibitor of lactation (FIL) or lactation inhibitor factor (LIF), is a low-molecular-weight protein (7.6 kDa), which has been shown in goats [63]. The FIL reduces the secretion rate of milk in vitro [62] and in vivo [63] when in contact with the alveolar epithelium. It works by inhibiting the constitutive secretion of proteins by CEMs by reversible blocking of the early stages of the biosynthesis-secretion pathway. In addition, the FIL would also inhibit lactose synthesis. Finally, FIL would regulate the number of cell tissue by triggering apoptosis [54]. Indeed, incomplete milking or milking removals would allow an accumulation of the FIL in contact with the CEMs, which would explain the reduc-

Recently, serotonin (5-HT) has been proposed to be an autocrine/paracrine regulator of lactation in the mouse, humans, and more recently in the bovine. The enzymatic machinery necessary for 5-HT biosynthesis has been detected in the mammary epithelium [64]. Other researches support the concept that serotonin

Increasing milking frequency in dairy cattle to more than two milkings per day has resulted in an increase in milk production without any negative effect on the health of the animal. There are various reasons for the practice of three milkings a day, namely, increase in the time of use of the milking machine, the size of the herd, and milk production. In fact, milking three times a day results in an increase in milk production from 3 to 39% compared to two times in dairy cows [66], 15–35% in ewes [47], 10–20% in goats [67, 62, 39], and 4–13% in camels [68, 45]. The response to increased milking frequency would be greater in high-producing, primiparous, and late-lactating cows [66, 69]. Erdman and Varner [70] in their review of 40 comparative studies of increased milking frequency reported that switching from 2 to 3 milkings per day resulted in a stable increase in milk production in terms of

In studies on the milking robot, it has been shown that cows, when given free time, are milked on average between 2.7 and 3.9 times a day [71]. In addition, when a rate of four milkings per day is applied for 4 weeks, a production increase of 14% is observed [2]. However, switching to six milkings per day for 2 days only increases production by 10–15% [8]. Such observations suggest that an interval between milkings of 6–8 hours is physiologically ideal for the animal and that there would be no advantage in increasing the rate of milking above four milkings per day in cows [53], as in ewes [47]. An increase in milking frequency would therefore allow

**120**

There are contradictions in the literature regarding the effects of switching to three milkings per day. For some, changes in milk composition at three milkings per day would be insignificant, while others report a decrease in milk fat compared with cows milked twice a day [8]. For some, this decrease would be greater for primiparous cows, while for others, the decrease would be greater for multiparous cows [66]. At the lactation scale, [69] noted a slight decrease in protein and casein concentrations in milk, enough to reduce cheese yield by 1.5%. Somatic cell milk content is used as an indicator of the microbiological quality of milk. Indeed, the number of somatic cells decreases when milking frequency increases [71]. On the other hand, [66] report that switching from two to three milkings per day in dairy cows increases the California mastitis test (CMT) score.

The increase of milking frequency could lead to an increased release of lactogenic hormones which will stimulate the synthetic activities of the CEMs and allow a better persistence of the lactation. These hormones may, in addition to their metabolic effects, increase the number of secretory cells and thus increase the volume of milk secreted [72]. Indeed, even if it is admitted in ruminants that the CEMs deteriorate and that their number decreases progressively with the advancement of lactation, by triggering apoptosis [54], the activity of synthesis of remaining cells is unchanged [72]. This decrease in the number of cells would be modulated by the frequency of milking. The increase of milking frequency causes cellular hypertrophy followed by an increase in the number of CEMs by proliferation of new cells. In addition, an increase in enzyme activities, reflecting their potential for synthesis, is observed in response to an increase in milking frequency in goats [62], cows [2], and camels [45].

#### **4. Conclusions**

Deciding about the number of milkings per day for each ruminant is a key factor in optimizing the use of mechanical milking. Currently, this decision is primarily based on the production level and stage of lactation data, but no udder capacity is taken into account. Therefore, it is recommended to use ruminants with large cisterns in order to minimize the effect of hydrostatic pressure on the cells and consequently reduce production losses. In practice, we propose to use the evaluation of udder cistern area by ultrasonography as a criterion to estimate milk storage capacity in the udder in order to establish the appropriate milking frequency for each ruminant according to the production system.

Research opportunities are open to broaden and consolidate this study. Indeed, the work on the heritability and the repeatability of this character "glandular cistern" is essential in order to incorporate it into the breeding programs for dairy ruminants.

*Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…*

#### **Author details**

Moez Ayadi

Département de Biotechnologie Animale, Institut Supérieur de Biotechnologie de Beja, Université de Jendouba, Tunisie

\*Address all correspondence to: moez\_ayadi2@yahoo.fr

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**123**

pp. 46-64

*Optimization of Milking Frequency in Dairy Ruminants DOI: http://dx.doi.org/10.5772/intechopen.87303*

> [8] Eslamizad M, Dehghan-Banadaky M, Rezayazdi K, Moradi-Shahrbabak M. Effects of 6 times daily milking during early versus full lactation of Holstein cows on milk production and blood metabolites. Journal of Dairy Science. 2010;**93**(9):4054-4061

[9] Smith JW, Ely LO, Graves WM, Gilson WM. Effect of milking interval in DHI performance measures. Journal of Dairy Science. 2002;**85**:3526-3533

[10] Ruberte J, Carretero A, Fernandez M, Navarro M, Caja G, Kirchner F, et al. Ultrasound mammography in the lactating ewe and its correspondence to anatomical section. Small Ruminant

Research. 1994;**13**:199-204

pp. 91-93

[11] Caja G, Such X, Ruberte J, Carretero A, Navarro M. The use of ultrasonography in the study of mammary gland cisterns during lactation in sheep. In: Proceedings of the International Symposium on Milking and Milk Production of Dairy Sheep and Goats. Wageningen, The Netherlands: Wageningen Pers; 1999.

[12] Caja G, Salama OA, Fathy A, El-Sayed H, Salama AAK. Milk partitioning and accumulation in the camel udder according to time elapsed after milking. In: Proceedings of the 62nd Annual Meeting of EAAP. Stavanger; 2011. p. 363

[13] Castillo V, Such X, Caja G, Salama AA, Albanell E, Casals R. Changes in alveolar and cisternal compartments induced by milking interval in the udder of dairy ewes. Journal of Dairy Science.

[14] Rovai M, Caja G, Such X. Evaluation of udder cisterns and effects on milk yield of dairy ewes. Journal of Dairy

2008;**91**:3403-3411

Science. 2008;**91**:4622-4629

[1] FAOSTAT. Production. Live Animals. Food and Agriculture Organization of the United Nations. 2016. Available from http://faostat3.fao.org/faostat-gateway/ go/to/download/Q/QA/E [Accessed:

[2] Knight CH, Hirst D, Dewhurst RJ. Milk accumulation and distribution in the bovine udder during the interval between milkings. The Journal of Dairy

[3] Davis SR, Farr VC, Copeman PJA, Carruthers VR, Knight CH, Stelwagen K. Partitioning of milk accumulation between cisternal and alveolar compartments of the bovine udder: Relationship to production loss during once daily milking. The Journal of Dairy

[4] Castillo V, Such X, Caja G, Casals R, Salama AAK, Albanell E. Long- and short-term effects of omitting two weekend milkings on the lactational performance and mammary tight junction permeability of dairy ewes. Journal of Dairy Science.

[5] Ayadi M, Caja G, Such X, Knight CH. Use of ultrasonography to estimate cistern size and milk storage at different milking intervals in the udder of dairy cows. The Journal of Dairy Research.

[6] Ayadi M, Caja G, Such X, Knight CH. Effect of omitting one milking weekly on lactational performances and morphological udder changes in dairy cows. Journal of Dairy Science.

[7] Marnet PG. Milking procedures and facilities. In: Park YW, Haenlein GFW, editors. Milk and Dairy Products in Human Nutrition: Production, Composition and Health. 1st ed.

Nowega: John wiley and Sons Ltd.; 2013.

Research. 1994;**61**:167-177

Research. 1998;**65**:1-8

2009;**92**:3684-3695

2003;**70**:1-7

2003;**86**:2352-2358

**References**

July 15, 2017]

*Optimization of Milking Frequency in Dairy Ruminants DOI: http://dx.doi.org/10.5772/intechopen.87303*

#### **References**

*Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…*

Département de Biotechnologie Animale, Institut Supérieur de Biotechnologie de

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

**122**

**Author details**

Beja, Université de Jendouba, Tunisie

provided the original work is properly cited.

\*Address all correspondence to: moez\_ayadi2@yahoo.fr

Moez Ayadi

[1] FAOSTAT. Production. Live Animals. Food and Agriculture Organization of the United Nations. 2016. Available from http://faostat3.fao.org/faostat-gateway/ go/to/download/Q/QA/E [Accessed: July 15, 2017]

[2] Knight CH, Hirst D, Dewhurst RJ. Milk accumulation and distribution in the bovine udder during the interval between milkings. The Journal of Dairy Research. 1994;**61**:167-177

[3] Davis SR, Farr VC, Copeman PJA, Carruthers VR, Knight CH, Stelwagen K. Partitioning of milk accumulation between cisternal and alveolar compartments of the bovine udder: Relationship to production loss during once daily milking. The Journal of Dairy Research. 1998;**65**:1-8

[4] Castillo V, Such X, Caja G, Casals R, Salama AAK, Albanell E. Long- and short-term effects of omitting two weekend milkings on the lactational performance and mammary tight junction permeability of dairy ewes. Journal of Dairy Science. 2009;**92**:3684-3695

[5] Ayadi M, Caja G, Such X, Knight CH. Use of ultrasonography to estimate cistern size and milk storage at different milking intervals in the udder of dairy cows. The Journal of Dairy Research. 2003;**70**:1-7

[6] Ayadi M, Caja G, Such X, Knight CH. Effect of omitting one milking weekly on lactational performances and morphological udder changes in dairy cows. Journal of Dairy Science. 2003;**86**:2352-2358

[7] Marnet PG. Milking procedures and facilities. In: Park YW, Haenlein GFW, editors. Milk and Dairy Products in Human Nutrition: Production, Composition and Health. 1st ed. Nowega: John wiley and Sons Ltd.; 2013. pp. 46-64

[8] Eslamizad M, Dehghan-Banadaky M, Rezayazdi K, Moradi-Shahrbabak M. Effects of 6 times daily milking during early versus full lactation of Holstein cows on milk production and blood metabolites. Journal of Dairy Science. 2010;**93**(9):4054-4061

[9] Smith JW, Ely LO, Graves WM, Gilson WM. Effect of milking interval in DHI performance measures. Journal of Dairy Science. 2002;**85**:3526-3533

[10] Ruberte J, Carretero A, Fernandez M, Navarro M, Caja G, Kirchner F, et al. Ultrasound mammography in the lactating ewe and its correspondence to anatomical section. Small Ruminant Research. 1994;**13**:199-204

[11] Caja G, Such X, Ruberte J, Carretero A, Navarro M. The use of ultrasonography in the study of mammary gland cisterns during lactation in sheep. In: Proceedings of the International Symposium on Milking and Milk Production of Dairy Sheep and Goats. Wageningen, The Netherlands: Wageningen Pers; 1999. pp. 91-93

[12] Caja G, Salama OA, Fathy A, El-Sayed H, Salama AAK. Milk partitioning and accumulation in the camel udder according to time elapsed after milking. In: Proceedings of the 62nd Annual Meeting of EAAP. Stavanger; 2011. p. 363

[13] Castillo V, Such X, Caja G, Salama AA, Albanell E, Casals R. Changes in alveolar and cisternal compartments induced by milking interval in the udder of dairy ewes. Journal of Dairy Science. 2008;**91**:3403-3411

[14] Rovai M, Caja G, Such X. Evaluation of udder cisterns and effects on milk yield of dairy ewes. Journal of Dairy Science. 2008;**91**:4622-4629

[15] Rogers GW, Spenser SB. Relationships among udder and teat morphology and milking characteristics. Journal of Dairy Science. 1991;**74**:4189-4194

[16] Le Du JF, Chevalerie A, Taverna M, Dano Y. Aptitude des vaches à la traite mécanique: Relation avec certaines caractéristiques physiques du trayon. Annales de Zootechnie. 1994;**43**:77-90

[17] Such X, Caja G, Pérez L. Comparison of milking ability between Manchega and Lacaune dairy ewes. In: Proceeding of International Symposium on Milking and Milk Production of Dairy Ewe and Goats EAAP Publication No 95. Wageningen, The Netherlands: Wageningen Pers; 1999. pp. 45-50

[18] Ayadi M, Such X, Ezzehizi N, Zouari M, Najar T, Ben M' Rad M, et al. Relationship between mammary morphology traits and milk yield of Sicilo-Sarde dairy sheep in Tunisia. Small Ruminant Research. 2011;**96**:41-45

[19] Armstrong DV, Daugherty LS. Milking robots in large dairy farms. Computers and Electronics in Agriculture. 1997;**17**:123-128

[20] Labussière J, Richard P. Machine milking. Anatomical, physiological and technological aspects. Literature development. Annales de Zootechnie. 1965;**14**:63-126

[21] Labussière J. Review of physiological and anatomical factors influencing the milking ability of ewes and the organization of milking. Livestock Production Science. 1988;**18**:253-274

[22] Marnet PG, Combaud JF, Dano Y. Relationships between characteristics of the teat and milkability in Lacaune ewes. In: Proceedings of 6th International Symposium on Milking and Milk Production of Dairy

Sheep and Goats; 26 September-1 October 1999. Athens, Greece: EAAP; 1999. pp. 41-44

[23] Peris S, Caja G, Such X. Relationships between udder and milking traits in Murciano-Granadina dairy goats. Small Ruminant Research. 1999;**33**:171-179

[24] Hammadi M, Atigui M, Ayadi M, Barmat A, Belgacem A, Khaldi G, et al. Training period and short time effects of machine milking on milk yield and milk composition in Tunisian Maghrebi camels (*Camelus dromedarius*). Journal of Camel Practice and Research. 2010;**17**:1-7

[25] Eisa MO, Ishag IA, Abu-Nikhaila AM. A note on the relationships between udder morphometric and milk yield of Lahween camel (*Camelus dromedarius*). Livestock Research for Rural Development. 2010;**22**(10):1-11. Available from http://www.lrrd.org/ lrrd22/10/eisa22188.htm [Accessed: December 12, 2017]

[26] Atigui M, Hammadi M, Barmat A, Farhat M, Khorchani T, Marnet PG. First description of milk flow traits in Tunisian dairy dromedary camels under intensive farming system. The Journal of Dairy Research. 2014;**81**:173-182

[27] Nagy P, Juhasz J. Present knowledge and future questions regarding intensive camel milk production. In: Proceeding of 4th International Conference of ISOCARD, "Silk Road Camel: The Camelids, Main Stakes for Sustainable Development"; 8-12 June 2015. Almaty, Kazakhstan: ISOCARD; 2015, 2015. pp. 51-56

[28] Ayadi M, Aljumaah RS, Musaad A, Samara EM, Abelrahman MM, Alshaikh MA, et al. Relationship between udder morphology traits, alveolar and cisternal milk compartments and machine milking performances of dairy camels (*Camelus dromedarius*). Spanish

**125**

*Optimization of Milking Frequency in Dairy Ruminants DOI: http://dx.doi.org/10.5772/intechopen.87303*

> Pharmacokinetics and inhibition of milk ejection in dairy cows. The Journal

[36] Rovai M, Such X, Caja G, Knight CH. Interbreed differences in cisternal and alveolar milk partitioning in the udder according to yield in dairy sheep. Journal of Dairy Science. 2000;**83**(Suppl.1):166 (Abstract)

[38] McKusick BC, Thomas DL, Berger YM, Marnet PG. Effect of milking interval on alveolar versus cisternal milk accumulation and milk production and composition in dairy ewes. Journal of Dairy Science. 2002;**85**:2197-2206

[39] Salama AAK, Caja G, Such X, Peris S, Sorensen A, Knight CH. Changes in cisternal udder compartment induced by milking interval in dairy goats milked once or twice daily. Journal of Dairy Science. 2004;**87**:1181-1187

[40] Thomas CS, Svennersten-Sjaunja K, Bhosrekar MR, Bruckmaier

RM. Mammary cisternal size, cisternal milk and milk ejection in Murrah buffaloes. The Journal of Dairy Research. 2004;**71**:162-168

CH. Changes in cisternal compartment based on stage of lactation and time since milk ejection in the udder of dairy cows. Journal of Dairy Science.

[42] Davis SR, Farr VC, Stelwagen K. Regulation of yield loss and milk composition during once-daily milking: A review. Livestock Production Science.

[43] Molenaar A, Leath SR, Caja G, Henderson HV, Cameron C, Challies M,

[41] Caja G, Ayadi M, Knight

2004;**87**:2409-2415

1999;**59**:77-94

of Dairy Research. 1999;**66**:1-8

[37] Marnet PG, McKusick BC. Regulation of milk ejection and milkability in small ruminants. Livestock Production Science.

2001;**70**:125-133

Journal of Agricultural Research.

[29] Ayadi M, Musaad A, Aljumaah RS, Konuspayeva G, Bengoumi M, Faye B. Machine milking parameters for an efficient and healthy milking dairy camels (*Camelus dromedarius*). Journal of Camel Practice and Research.

[30] Dioli M. Body and udder

Livestock Research for Rural Development. 2010;**22**(20):1-10. Available from: http://www.lrrd.org/ lrrd22/1/pras22020.htm [Accessed:

[32] Juhasz J, Nagy P. Challenges in the development of a large-scale milking system for dromedary camels. In: Proceedings of the WBC/ICAR 2008 Satellite Meeting on Camelid Reproduction; Budapest, Hungary.

[33] Ayadi M, Aljumaah RS, Samara EM, Bernard F, Gerardo C. A proposal for linear assessment scheme for the udder of dairy camels (*Camelus dromedarius*). Tropical Animal Health and Production.

[34] Bruckmaier RM, Rothenanger E, Blum JW. Measurement of mammary gland cistern size and determination of the cisternal milk fraction in dairy cows. Milchwissenschaft. 1994;**49**:543-546

[35] Wellnitz O, Bruckmaier RM, Albrecht C, Blum JW. Atosiban, an oxytocin receptor blocking agent:

October 19, 2014]

2008. pp. 84-87

2016;**48**:927-933

conformation faults. In: Pictorial Guide to Traditional Management, Husbandry and Diseases of the One-Humped Camel. 2nd ed. Spain. Chapters 8 to 13.

[31] Prasad RMV, Sudhakar K, Raghava Rao E, Ramesh Gupta B, Mahender M. Studies on the udder and teat morphology and their relationship with milk yield in Murrah buffaloes.

2013;**1**:790-797

2018;**25**(1):1-7

2013. p. 125

*Optimization of Milking Frequency in Dairy Ruminants DOI: http://dx.doi.org/10.5772/intechopen.87303*

Journal of Agricultural Research. 2013;**1**:790-797

*Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…*

Sheep and Goats; 26 September-1 October 1999. Athens, Greece: EAAP;

[24] Hammadi M, Atigui M, Ayadi M, Barmat A, Belgacem A, Khaldi G, et al. Training period and short time effects of machine milking on milk yield and milk composition in Tunisian Maghrebi camels (*Camelus dromedarius*). Journal of Camel Practice and Research.

[25] Eisa MO, Ishag IA, Abu-Nikhaila AM. A note on the relationships between udder morphometric and milk yield of Lahween camel (*Camelus dromedarius*). Livestock Research for Rural Development. 2010;**22**(10):1-11. Available from http://www.lrrd.org/ lrrd22/10/eisa22188.htm [Accessed:

[26] Atigui M, Hammadi M, Barmat A, Farhat M, Khorchani T, Marnet PG. First

Tunisian dairy dromedary camels under intensive farming system. The Journal of Dairy Research. 2014;**81**:173-182

[27] Nagy P, Juhasz J. Present knowledge and future questions regarding intensive camel milk production. In: Proceeding of 4th International Conference of ISOCARD, "Silk Road Camel: The Camelids, Main Stakes for Sustainable Development"; 8-12 June 2015. Almaty, Kazakhstan: ISOCARD; 2015, 2015.

[28] Ayadi M, Aljumaah RS, Musaad A, Samara EM, Abelrahman MM, Alshaikh

MA, et al. Relationship between udder morphology traits, alveolar and cisternal milk compartments and machine milking performances of dairy camels (*Camelus dromedarius*). Spanish

description of milk flow traits in

[23] Peris S, Caja G, Such X. Relationships between udder and milking traits in Murciano-Granadina dairy goats. Small Ruminant Research.

1999. pp. 41-44

1999;**33**:171-179

2010;**17**:1-7

December 12, 2017]

pp. 51-56

[15] Rogers GW, Spenser SB. Relationships among udder and teat morphology and milking

[17] Such X, Caja G, Pérez L. Comparison of milking ability

1991;**74**:4189-4194

1999. pp. 45-50

2011;**96**:41-45

1965;**14**:63-126

characteristics. Journal of Dairy Science.

[16] Le Du JF, Chevalerie A, Taverna M, Dano Y. Aptitude des vaches à la traite mécanique: Relation avec certaines caractéristiques physiques du trayon. Annales de Zootechnie. 1994;**43**:77-90

between Manchega and Lacaune dairy ewes. In: Proceeding of International Symposium on Milking and Milk Production of Dairy Ewe and Goats EAAP Publication No 95. Wageningen, The Netherlands: Wageningen Pers;

[18] Ayadi M, Such X, Ezzehizi N, Zouari M, Najar T, Ben M' Rad M, et al. Relationship between mammary morphology traits and milk yield of Sicilo-Sarde dairy sheep in Tunisia. Small Ruminant Research.

[19] Armstrong DV, Daugherty LS. Milking robots in large dairy farms. Computers and Electronics in

Agriculture. 1997;**17**:123-128

[20] Labussière J, Richard P. Machine milking. Anatomical, physiological and technological aspects. Literature development. Annales de Zootechnie.

[21] Labussière J. Review of physiological and anatomical factors influencing the milking ability of ewes and the organization of milking. Livestock Production Science. 1988;**18**:253-274

[22] Marnet PG, Combaud JF, Dano Y. Relationships between characteristics

of the teat and milkability in Lacaune ewes. In: Proceedings of 6th International Symposium on Milking and Milk Production of Dairy

**124**

[29] Ayadi M, Musaad A, Aljumaah RS, Konuspayeva G, Bengoumi M, Faye B. Machine milking parameters for an efficient and healthy milking dairy camels (*Camelus dromedarius*). Journal of Camel Practice and Research. 2018;**25**(1):1-7

[30] Dioli M. Body and udder conformation faults. In: Pictorial Guide to Traditional Management, Husbandry and Diseases of the One-Humped Camel. 2nd ed. Spain. Chapters 8 to 13. 2013. p. 125

[31] Prasad RMV, Sudhakar K, Raghava Rao E, Ramesh Gupta B, Mahender M. Studies on the udder and teat morphology and their relationship with milk yield in Murrah buffaloes. Livestock Research for Rural Development. 2010;**22**(20):1-10. Available from: http://www.lrrd.org/ lrrd22/1/pras22020.htm [Accessed: October 19, 2014]

[32] Juhasz J, Nagy P. Challenges in the development of a large-scale milking system for dromedary camels. In: Proceedings of the WBC/ICAR 2008 Satellite Meeting on Camelid Reproduction; Budapest, Hungary. 2008. pp. 84-87

[33] Ayadi M, Aljumaah RS, Samara EM, Bernard F, Gerardo C. A proposal for linear assessment scheme for the udder of dairy camels (*Camelus dromedarius*). Tropical Animal Health and Production. 2016;**48**:927-933

[34] Bruckmaier RM, Rothenanger E, Blum JW. Measurement of mammary gland cistern size and determination of the cisternal milk fraction in dairy cows. Milchwissenschaft. 1994;**49**:543-546

[35] Wellnitz O, Bruckmaier RM, Albrecht C, Blum JW. Atosiban, an oxytocin receptor blocking agent:

Pharmacokinetics and inhibition of milk ejection in dairy cows. The Journal of Dairy Research. 1999;**66**:1-8

[36] Rovai M, Such X, Caja G, Knight CH. Interbreed differences in cisternal and alveolar milk partitioning in the udder according to yield in dairy sheep. Journal of Dairy Science. 2000;**83**(Suppl.1):166 (Abstract)

[37] Marnet PG, McKusick BC. Regulation of milk ejection and milkability in small ruminants. Livestock Production Science. 2001;**70**:125-133

[38] McKusick BC, Thomas DL, Berger YM, Marnet PG. Effect of milking interval on alveolar versus cisternal milk accumulation and milk production and composition in dairy ewes. Journal of Dairy Science. 2002;**85**:2197-2206

[39] Salama AAK, Caja G, Such X, Peris S, Sorensen A, Knight CH. Changes in cisternal udder compartment induced by milking interval in dairy goats milked once or twice daily. Journal of Dairy Science. 2004;**87**:1181-1187

[40] Thomas CS, Svennersten-Sjaunja K, Bhosrekar MR, Bruckmaier RM. Mammary cisternal size, cisternal milk and milk ejection in Murrah buffaloes. The Journal of Dairy Research. 2004;**71**:162-168

[41] Caja G, Ayadi M, Knight CH. Changes in cisternal compartment based on stage of lactation and time since milk ejection in the udder of dairy cows. Journal of Dairy Science. 2004;**87**:2409-2415

[42] Davis SR, Farr VC, Stelwagen K. Regulation of yield loss and milk composition during once-daily milking: A review. Livestock Production Science. 1999;**59**:77-94

[43] Molenaar A, Leath SR, Caja G, Henderson HV, Cameron C, Challies M, et al. Development of ultrasound methodology to measure cow udder cistern storage capacity in the new Zeland pasture-fed context. Proceedings of the New Zealand Society of Animal Production. 2013;**73**:114-116

[44] Stelwagen K, Knight CH, Farr VC, Davis SR, Prosser CG, McFadden TB. Continuous versus single drainage of milk from the bovine mammary gland during a 24 hour period. Experimental Physiology. 1996;**81**:141-149

[45] Ayadi M, Hammadi M, Khorchani T, Barmat A, Atigui M, Caja G. Effects of milking interval and cisternal udder evaluation in Tunisian Maghrebi dairy dromedaries (*Camelus dromedarius* L.). Journal of Dairy Science. 2009;**92**:1452-1459

[46] Dewhurst RJ, Knight CH. Relationship between milk storage characteristics and the short-term response of dairy cows to thricedaily milking. Animal Production. 1994;**58**:181-187

[47] Négrao JA, Marnet PG. Cortisol, adrenalin, noradrenalin and oxytocin release and milk yield during first milkings in primiparous ewes. Small Ruminant Research. 2003;**67**:69-75

[48] Nudda A, Pulina G, Vallabell R, Bencini R, Enne G. Ultrasound technique for measuring mammary cistern size of dairy ewes. The Journal of Dairy Research. 2000;**67**:101-106

[49] Salama AAK, Such X, Caja G, Rovai M, Casals R, Albanell E, et al. Effects of once versus twice daily milking throughout lactation on milk yield and milk composition in dairy goats. Journal of Dairy Science. 2003;**86**:1673-1680

[50] Wilde CJ, Knight CH, Addey CVP, Blatchford DR, Travers M, Bennett CN, et al. Autocrine regulation of mammary cell differentiation. Protoplasma. 1990;**159**:112-117

[51] Stelwagen K, Knight CH. Effect of unilateral once or twice daily milking of cows on milk yield and udder characteristics in early and late lactation. The Journal of Dairy Research. 1997;**64**:487-494

[52] Peaker M, Blatchford DR. Distribution of milk in the goat mammary gland and its relation to the rate and control of milk secretion. The Journal of Dairy Research. 1988;**55**:41-48

[53] Stelwagen K. Effect of milking frequency on mammary functioning and shape of the lactation curve. Journal of Dairy Science. 2001;**84**:E204-E211

[54] Akers RM. Lactation and the Mammary Gland. 1st ed. USA: Iowa State Press; 2002

[55] Guinard-Flament J, Rulquin H. Effect of one daily milking on mammary blood flow (MBF) in dairy cows. Livestock Production Science. 2001;**70**:180 (Abstract)

[56] Stelwagen K, Davis SR, Farr VC, Eichler SJ, Politis I. Effect of once daily milking and concurrent somatotropin on mammary tight junction permeability and yield of cows. Journal of Dairy Science. 1994;**77**:2974-3001

[57] Bruckmaier RM, Wellnitz O, Blum JW. Inhibition of milk ejection in cows by oxytocin receptor blockade, a-adrenergic receptor stimulation and in unfamiliar surroundings. The Journal of Dairy Research. 1997;**64**:315-325

[58] Lacasse P, Prosser CG. Mammary blood flow does not limit milk yield in lactating goats. Journal of Dairy Science. 2003;**86**:2094-2097

[59] Auldist MJ, Prosser CG. Differential effects of short-term once-daily milking

**127**

*Optimization of Milking Frequency in Dairy Ruminants DOI: http://dx.doi.org/10.5772/intechopen.87303*

> [67] Knight CH, Foran D, Wilde CJ. Interactions between autocrine and endocrine control of milk yield; thricedaily milking of bromocriptine-treated goats. Journal of Reproduction and Fertility. 1990:**5**. Abstract. no. 30

[68] Alshaikh MA, Salah MS. Effect of milking interval on secretion rate and composition of camel milk in late lactation. The Journal of Dairy

[69] Klei LR, Lynch JM, Barbano DM, Oltenacu PA, Lednor AJ, Bandler DK. Influence of milking three times a day on milk quality. Journal of Dairy

[70] Erdman RA, Varner M. Fixed yield responses to increased milking frequency. Journal of Dairy Science.

[71] Ipema AH. Integration of robotic milking in dairy housing systems review of cow traffic and milking capacity aspects. Computers and Electronics in

[72] Knight CH, Wilde CJ. Mammary cell changes during pregnancy and lactation. Livestock Production Science.

Research. 1994;**61**:451-456

Science. 1997;**80**:427-436

Agriculture. 1997;**17**:79-94

1995;**78**:1199-1203

1993;**35**:3-19

on milk yield, milk composition and concentration of selected blood metabolites in cows with high or low pasture intake. Proceedings of the New Zealand Society of Animal Production.

[60] Stelwagen K, van Espen DC, Verkerk GA, McFadden HA, Farr VC. Elevated plasma cortisol reduces permeability tight junctions in the lactating bovine mammary epithelium.

The Journal of Endocrinology.

[62] Wilde CJ, Addey CV, Knight CH. Regulation of intracellular casein degradation by secreted milk proteins.

Biochimica et Biophysica Acta.

[64] Stull M, Pai A, Vomachka V, Marshall AJ, Jacob AM, Horseman ND. Mammary gland homeostasis employs serotonergic regulation of epithelial tight junctions. Proceedings of the National Academy of Sciences.

2007;**104**(42):16708-16713

2008;**91**(5):1834-1844

[66] Allen DB, DePeters EJ, Laben RC. Three times a day milking: Effects on milk production, reproductive efficiency and udder health. Journal of Dairy Science. 1986;**69**:1441-1446

[65] Hernandez L, Stiening L, Wheelock CM, Baumgard JB, Parkhurst LH, Collier RJ. Evaluation of serotonin as a feedback inhibitor of lactation in the bovine. Journal of Dairy Science.

[63] Wilde CJ, Addey CVP, Li P, Fernig DG. Programmed cell death in bovine mammary tissue during lactation and involution. Experimental Physiology.

[61] Wilde CJ, Knight CH. Milk yield and mammary function in goats during and after once-daily milking. The Journal of Dairy Research. 1990;**57**:441-447

1998;**58**:41-43

1998;**159**:173-178

1989;**992**:315-319

1997;**82**:943-953

#### *Optimization of Milking Frequency in Dairy Ruminants DOI: http://dx.doi.org/10.5772/intechopen.87303*

on milk yield, milk composition and concentration of selected blood metabolites in cows with high or low pasture intake. Proceedings of the New Zealand Society of Animal Production. 1998;**58**:41-43

*Lactation in Farm Animals - Biology, Physiological Basis, Nutritional Requirements…*

cell differentiation. Protoplasma.

[51] Stelwagen K, Knight CH. Effect of unilateral once or twice daily milking of cows on milk yield and udder characteristics in early and late lactation. The Journal of Dairy

DR. Distribution of milk in the goat mammary gland and its relation to the rate and control of milk secretion. The Journal of Dairy Research.

[53] Stelwagen K. Effect of milking frequency on mammary functioning and shape of the lactation curve. Journal of Dairy Science. 2001;**84**:E204-E211

[54] Akers RM. Lactation and the Mammary Gland. 1st ed. USA: Iowa

[55] Guinard-Flament J, Rulquin H. Effect of one daily milking on mammary blood flow (MBF) in dairy cows. Livestock Production Science.

[56] Stelwagen K, Davis SR, Farr VC, Eichler SJ, Politis I. Effect of once daily milking and concurrent somatotropin

permeability and yield of cows. Journal of Dairy Science. 1994;**77**:2974-3001

[57] Bruckmaier RM, Wellnitz O, Blum JW. Inhibition of milk ejection in cows by oxytocin receptor blockade, a-adrenergic receptor stimulation and in unfamiliar surroundings. The Journal of

Dairy Research. 1997;**64**:315-325

2003;**86**:2094-2097

[58] Lacasse P, Prosser CG. Mammary blood flow does not limit milk yield in lactating goats. Journal of Dairy Science.

[59] Auldist MJ, Prosser CG. Differential effects of short-term once-daily milking

Research. 1997;**64**:487-494

[52] Peaker M, Blatchford

1988;**55**:41-48

State Press; 2002

2001;**70**:180 (Abstract)

on mammary tight junction

1990;**159**:112-117

et al. Development of ultrasound methodology to measure cow udder cistern storage capacity in the new Zeland pasture-fed context. Proceedings of the New Zealand Society of Animal

Production. 2013;**73**:114-116

TB. Continuous versus single drainage of milk from the bovine mammary gland during a 24 hour period. Experimental Physiology.

1996;**81**:141-149

2009;**92**:1452-1459

1994;**58**:181-187

[46] Dewhurst RJ, Knight

[44] Stelwagen K, Knight CH, Farr VC, Davis SR, Prosser CG, McFadden

[45] Ayadi M, Hammadi M, Khorchani T, Barmat A, Atigui M, Caja G. Effects of milking interval and cisternal udder evaluation in Tunisian Maghrebi dairy dromedaries (*Camelus dromedarius* L.). Journal of Dairy Science.

CH. Relationship between milk storage characteristics and the short-term response of dairy cows to thricedaily milking. Animal Production.

[47] Négrao JA, Marnet PG. Cortisol, adrenalin, noradrenalin and oxytocin release and milk yield during first milkings in primiparous ewes. Small Ruminant Research. 2003;**67**:69-75

[48] Nudda A, Pulina G, Vallabell R, Bencini R, Enne G. Ultrasound

mammary cistern size of dairy ewes. The Journal of Dairy Research.

[49] Salama AAK, Such X, Caja G, Rovai M, Casals R, Albanell E, et al. Effects of once versus twice daily milking throughout lactation on milk yield and milk composition in dairy goats. Journal of Dairy Science.

[50] Wilde CJ, Knight CH, Addey CVP, Blatchford DR, Travers M, Bennett CN, et al. Autocrine regulation of mammary

technique for measuring

2000;**67**:101-106

2003;**86**:1673-1680

**126**

[60] Stelwagen K, van Espen DC, Verkerk GA, McFadden HA, Farr VC. Elevated plasma cortisol reduces permeability tight junctions in the lactating bovine mammary epithelium. The Journal of Endocrinology. 1998;**159**:173-178

[61] Wilde CJ, Knight CH. Milk yield and mammary function in goats during and after once-daily milking. The Journal of Dairy Research. 1990;**57**:441-447

[62] Wilde CJ, Addey CV, Knight CH. Regulation of intracellular casein degradation by secreted milk proteins. Biochimica et Biophysica Acta. 1989;**992**:315-319

[63] Wilde CJ, Addey CVP, Li P, Fernig DG. Programmed cell death in bovine mammary tissue during lactation and involution. Experimental Physiology. 1997;**82**:943-953

[64] Stull M, Pai A, Vomachka V, Marshall AJ, Jacob AM, Horseman ND. Mammary gland homeostasis employs serotonergic regulation of epithelial tight junctions. Proceedings of the National Academy of Sciences. 2007;**104**(42):16708-16713

[65] Hernandez L, Stiening L, Wheelock CM, Baumgard JB, Parkhurst LH, Collier RJ. Evaluation of serotonin as a feedback inhibitor of lactation in the bovine. Journal of Dairy Science. 2008;**91**(5):1834-1844

[66] Allen DB, DePeters EJ, Laben RC. Three times a day milking: Effects on milk production, reproductive efficiency and udder health. Journal of Dairy Science. 1986;**69**:1441-1446

[67] Knight CH, Foran D, Wilde CJ. Interactions between autocrine and endocrine control of milk yield; thricedaily milking of bromocriptine-treated goats. Journal of Reproduction and Fertility. 1990:**5**. Abstract. no. 30

[68] Alshaikh MA, Salah MS. Effect of milking interval on secretion rate and composition of camel milk in late lactation. The Journal of Dairy Research. 1994;**61**:451-456

[69] Klei LR, Lynch JM, Barbano DM, Oltenacu PA, Lednor AJ, Bandler DK. Influence of milking three times a day on milk quality. Journal of Dairy Science. 1997;**80**:427-436

[70] Erdman RA, Varner M. Fixed yield responses to increased milking frequency. Journal of Dairy Science. 1995;**78**:1199-1203

[71] Ipema AH. Integration of robotic milking in dairy housing systems review of cow traffic and milking capacity aspects. Computers and Electronics in Agriculture. 1997;**17**:79-94

[72] Knight CH, Wilde CJ. Mammary cell changes during pregnancy and lactation. Livestock Production Science. 1993;**35**:3-19

### *Edited by Naceur M'Hamdi*

Greater knowledge of lactation allows us to alter environmental, nutritional, and milking procedures, or general management to maximize production. This book, focusing on lactation in farm animals (biology, physiological basis, nutritional requirements, and modelization), presents invited papers from internationally recognized scientists. This volume contains seven chapters covering the key topics related to milk production and lactation biology and physiology. The authors show that animals raised on a well-controlled nutrition regimen may have significant enhancement of succeeding lactations. Furthermore, the usefulness of a milk yield prediction system depends upon how accurately it can predict daily milking patterns and its ability to adjust to factors affecting supply. Milk yield prediction models have proven helpful for genetic analysis and for bio-economic modeling. On the whole, this book serves as an inspirational basis for both scientists and farmers.

Published in London, UK © 2020 IntechOpen © ddukang / iStock

Lactation in Farm Animals -

Biology, Physiological Basis, Nutritional Requirements, and Modelization

IntechOpen Book Series

Veterinary Medicine and Science, Volume 3

Lactation in Farm Animals

Biology, Physiological Basis, Nutritional

Requirements, and Modelization

*Edited by Naceur M'Hamdi*