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. 2021 Feb 2;99(2):skab030. doi: 10.1093/jas/skab030

A dynamic mammary gland model describing colostrum immunoglobulin transfer and milk production in lactating sows

Uffe Krogh 1, Hélène Quesnel 1, Nathalie Le Floch 1, Aude Simongiovanni 2, Jaap van Milgen 1,✉
PMCID: PMC7887553  PMID: 33527135

Abstract

The physiology of the sow mammary gland is qualitatively well described and understood. However, the quantitative effect of various biological mechanisms contributing to the synthesis of colostrum and milk is lacking and more complicated to obtain. The objective of this study was to integrate physiological and empirical knowledge of the production of colostrum and milk in a dynamic model of a single sow mammary gland to understand and quantify parameters controlling mammary gland output. In 1983, Heather Neal and John Thornley published a model of the mammary gland in cattle, which was used as a starting point for the development of this model. The original cattle model was reparameterized, modified, and extended to describe the production of milk by the sow mammary gland during lactation and the prepartum production of colostrum as the combined output of immunoglobulins (Ig) and milk. Initially, the model was reparameterized to simulate milk synthesis potential of a single gland by considering biological characteristics and empirical estimations of sows and piglets. Secondly, the model was modified to simulate more accurately the responses to changes in milk removal rates. This was done by linking the ejectable milk storage capacity to the number of secretory cells rather than being constant throughout lactation. Finally, the model was extended to include the prepartum synthesis of milk and the kinetics of Ig into and out of the mammary gland. A progressive capacity of secretory cells to synthesize milk was used to differentiate the time between the onset of milk synthesis and Ig transfer. Changes in maximum milk removal rate, duration of milk ejection, and nursing interval exerted a great impact on the modeled milk output. Changes by ±60% in one of these parameters were capable of increasing milk output by 28% to 39% during the first 4 wk in lactation compared with the reference parameterization. This suggests that the ability of the piglet to remove milk from the gland exerts a key control on milk synthesis during lactation. Modeling colostrum as the combined output of Ig and milk allowed to represent the rapid decline in Ig concentration observed during the first hours after farrowing. In conclusion, biological and empirical knowledge was integrated into a model of the sow mammary gland and constitutes a simple approach to explore in which conditions and to what extent individual parameters influence Ig kinetics and milk production.

Keywords: conceptual modeling, lactation, milk synthesis, nursing behavior, physiology, sows

Introduction

Experimental studies investigating mammary gland physiology and development have brought valuable insights into basic biological mechanisms of sow colostrum and milk production. These studies have been compiled from various perspectives (Farmer, 2019, Hurley, 2019, Quesnel and Farmer, 2019). In addition, empirical relationships between piglet growth and output of colostrum, nutrients, and milk have been established and used to predict the mammary output of colostrum and milk (Noblet and Etienne, 1989; Hansen et al., 2012; Theil et al., 2014). The integration of biological mechanisms with empirical knowledge of the sow mammary gland into a dynamic model offers a great potential to explore, understand, and quantify the production of colostrum and milk in sows. Such knowledge further constitutes the basis to understand mammary gland nutrient metabolism and is a fundamental step toward a more accurate nutrient supply in sows. Neal and Thornley (1983) proposed a semi-mechanistic model that was able to represent a range of different lactation curves in cattle based on mammary gland biology combined with empirical-based knowledge of mammary gland output. The biological similarities among lactating mammals (McManaman and Neville, 2003) qualify the cattle model by Neal and Thornley (1983) as a sound starting point for the development of a sow mammary gland model. However, differences between cattle and sows in biology, livestock production aspects (i.e., retrieve milk or provide it to piglets), and characteristics of colostrum and milk production have to be considered. In contrast to cattle, the sow mammary glands are emptied around 30 times per day (Špinka and Illmann, 2015). Moreover, one or two piglets in a litter typically are removed or die before weaning (Muns et al., 2016), and the existence of teat ownership, where one piglet nurses the same gland at successive nursings (Špinka and Illmann, 2015), illustrates the relevance to model individual glands rather than the whole udder. Finally, ingestion of colostrum is essential for the piglets to supply energy for short-term survival as well as immunoglobulins (Ig) for health protection and long-term survival (Quesnel et al., 2012). Consequently, the overall aim was to integrate physiological and empirical knowledge on colostrum and milk production in sows in a model of a single sow mammary gland to better understand under which conditions and to what extent the piglets and the sow drive the production of colostrum and milk. In this initial effort to integrate empirical and mechanistic knowledge in a model of the sow mammary gland, simplicity was prioritized above biological detail and the resulting complexity. The objective was to simulate the transfer of Ig and the production of milk in the sow mammary gland based on the model framework developed for lactating cattle by Neal and Thornley (1983).

Materials and Methods

The conversion of the cattle model developed by Neal and Thornley (1983) into a sow model simulating the kinetics of Ig transfer and milk production in the sow mammary gland was implemented in a three-step procedure described below. In the first step, the original cattle model was reparameterized to simulate the milk synthesis potential of the sow mammary gland (model version I). In the second step, the original model structure was modified to simulate more accurately how changes in milk removal affect cell number and the quantity of milk that can be excreted by the gland during a milk ejection episode (model version II). In the third step, the model was extended to include the prepartum milk synthesis, the prepartum storage of colostrum Ig, and the postpartum excretion of the stored Ig (model version III). Colostrum was modeled as the combined secretion of Ig and milk from the gland. A description of the general model structure is given below followed by a detailed description of the individual steps of model development. Table 1 illustrates the changes made in the three steps of the model development, converting the original cattle model into a model of a single sow mammary gland. This sow mammary gland model focuses on the essential characteristics of colostrum and milk production and, consequently, simplicity was highly prioritized as a trade-off with biological detail. A 30% and 60% increase and decrease of individual input parameters of the model compared with the reference parameterization were used to explore the behavior of the model (Table 2). Although a 60% change may not be of biological relevance for all parameters, this approach was chosen to illustrate the impact of individual parameters on model responses. Such a characterization of the model becomes useful for the understanding of the quantitative aspects of sow lactation when it is placed in a biologically relevant context.

Table 1.

The main characteristics of the three steps in the model development1

Description and changes from the original model Related rates/pools
Model version I: reparameterization
Adjusting parameter values to characteristics of the sow mammary gland and milk output
• D0 milk synthesis Initial cell number
• Peak lactation milk synthesis capacity Milk synthesis rate
• Storage capacity of ejectable milk Milk synthesis rate
• Gland filling rate kinetics Milk synthesis rate
• Time to regulate cell number to changes in milk removal Variable cell loss rate
• Nursing interval Milk removal rate
• Duration of milk ejection Milk removal rate
• Piglet ability to remove milk from the gland Milk removal rate
• Persistency of milk synthesis Hormone decay rate
Model version II: structure modification
Simulating how varying milk removal rates influence ejectable milk and secretory cells
• Milk storage capacity is proportional to cell number Milk synthesis rate
• Proportion of milk in gland regulates cell number Variable cell loss rate
Model version III: model extension
Simulating the synthesis of milk and transfer of Ig before farrowing and the removal of Ig after farrowing
• Prepartum hormone Hormone excretion rate
• Prepartum cell differentiation level Milk synthesis rate
• Storage of Ig in the gland Prepartum Ig transfer rate
• Removal of stored Ig after farrowing Postpartum Ig removal rate
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1Model version I constitutes a reparameterized version of the original model by Neal and Thornley (1983) accounting for differences between cattle and sows. Model version II constitutes a modification to simulate the impact of milk removal rate on gland ejectable milk and secretory cell number. Model version III constitutes an extension to simulate the synthesis of milk and transfer of Ig before farrowing as well as the removal of Ig after farrowing.

Table 2.

Percent change (+/–) in milk output from day 0 to 28 after farrowing and the maximum variable cell loss rate relative to the reference (Ref) output as the response to changes in individual input parameters from the Ref parameterization (–60%, –30%, +30%, and +60%)

Ref. input Unit –60% –30% 30% 60%
Milk output from day 0 to 28, kg/d (Ref. output: 31.8 kg/d) Kg/d % Change of Ref.
 Hormone decay rate kH 2.45 %/d 1 0 0 −1
 Maximum cell proliferation rate vm 1 cell divisions/d −49 −16 8 21
 Fractional basal cell loss rate ks 10 %/d 9 5 −6 −14
 Fractional variable cell loss rate ksM 58 %/d 2 1 −1 −1
 Michaelis–Menten constant of variable cell loss rate Mh-p 0.9 g/g −71 −17 4 4
 Time period of average kr 2 d 0 0 0 0
 Maximum milk synthesis rate kM 0.21 g milk/1,000 cells/min −44 −9 −2 −7
 Ejectable milk storage capacity Mm-p 11 g milk/1,000 cells −60 −17 7 11
 Maximum milk removal rate rm 150 g milk/min −62 −29 20 28
 Nursing interval t interval 50 min 39 30 −25 −41
 Milk ejection duration t duration 22.5 s −69 −33 22 29
Maximum variable cell loss rate, cells/min (Ref. output: 0.137 cells/min) Cells/min % Change of Ref.
 Hormone decay rate kH 2.45 %/d 13 7 −7 −13
 Maximum cell proliferation rate vm 1 cell divisions/d −99 −80 110 231
 Fractional basal cell loss rate ks 10 %/d 197 86 −57 −86
 Fractional variable cell loss rate ksM 58 %/d −36 −15 11 20
 Michaelis–Menten constant of variable cell loss rate Mh-p 0.9 g/g 239 121 −79 −97
 Time period of average kr 2 d −4 −2 2 4
 Maximum milk synthesis rate kM 0.21 g milk/1,000 cells/min −100 −91 53 84
 Ejectable milk storage capacity Mm-p 11 g milk/1,000 cells 89 7 −3 −6
 Maximum milk removal rate rm 150 g milk/min 214 99 −73 −96
 Nursing interval t interval 50 min −99 −91 54 90
 Milk ejection duration t duration 22.5 s 242 110 −78 −97
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General model structure

The overall structure of the final sow model is illustrated in Figure 1 and includes the model structure developed by Neal and Thornley (1983) as well as the modifications and extensions to the original model structure. The model consists of four pools representing the systemic level of a nonspecific lactation hormone (H, arbitrary units), the secretory cell number in a single gland (C, number of cells), the quantity of ejectable milk in a single gland (M, g), and the quantity of Ig in a single gland (Ig, g). These pools are determined by the balance between the input and output rates of the respective pool. The rates are determined by constants and/or the pool levels (Figure 1). The lactation hormone pool level is controlled by the prepartum hormone excretion rate (EH, units/min) and the postpartum hormone decay rate (DH, units/min). The lactation hormone works as a stimulator of the cell proliferation rate (Pc, number of cells/min) into the secretory cell pool (C) constituting a single mammary gland. These secretory cells are lost, partly by a basal cell loss rate (Lc-basal, number of cells/min) mimicking a finite lifetime of the secretory cells, and partly by a variable cell loss rate (Lc-var, number of cells/min) mimicking the adaptation of cell apoptosis to changes in milk removal. The variable cell loss rate is modeled to be positively related to the average quantity of milk in the gland across a specified time period (Mav-p, g/g; i.e., a negative feedback mechanism) expressed as the ratio between the quantity of ejectable milk in the gland and the overall storage capacity of milk in the gland. A similar mechanism was included in the original model by Neal and Thornley (1983) to mimic the regulation of mammary gland cell number. This is in line with the observed impact of milking frequency on the number of secretory cells in ruminants (Boutinaud et al., 2004). The milk synthesis rate (SM, g/min) is positively related to the number of secretory cells (C) and inhibited by the quantity of ejectable milk (M) relative to the storage capacity of milk in the gland (i.e., negative feedback mechanism). The same mechanism controls the prepartum Ig transfer rate (TIg, g/min) into the pool of total Ig in the gland (Ig). The frequency of gland emptying (nursing interval), the duration of the active emptying process (milk ejection duration), and the capacity of the piglet to drink milk (maximum milk removal rate) constitute the most important model constants determining the milk removal rate (RM, g/min) and postpartum Ig removal rate (RIg, g/min) as illustrated by the piglet in Figure 1.

Figure 1.

Figure 1.

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A Forrester diagram illustrating the basal structure of the sow mammary gland model. Pools are indicated by solid boxes. Fluxes/rates are indicated by solid arrows and controlled by valves, which are regulated by factors as indicated by dashed arrows. The time average of the milk proportion in the gland (Mav-p, g/g) used to control variable cell loss rate (i.e., auxiliary variable) is indicated by a solid circle. Modifications and extensions to the original model structure by Neal and Thornley (1983) are indicated with shaded area. The shaded dashed arrow linking the lactation hormone pool (H, units) and the milk synthesis rate (SM, g/min) represents the introduction of a progressing increase in cell differentiation level until farrowing (i.e., increasing capacity of a cell to synthesize milk).

Model development

The different steps in the process of model development (Table 1) are described below, while the numerical values of parameters and initial pool values for the different versions of the model as well as the equations used to calculate rates and pool sizes are given in E-Supplementary S1.

Model version I: reparameterization

The reparameterization of the original model by Neal and Thornley (1983) was based on biological differences between cattle and sows and on empirical estimations of the milk synthesis potential of a single sow mammary gland. In sows, mammary growth and milk synthesis by a single gland adapt to the milk removal by the piglet nursing this gland (Farmer and Hurley, 2015) as illustrated by King et al. (1997), who showed that milk yield was greater in sows that nursed 2-wk-old piglets immediately after farrowing than in sows nursing newborn piglets. Thus, the capacity of the piglet to remove milk from the gland is a key factor determining the milk synthesis of that gland. The nursing interval and the duration of milk ejection also affect milk removal, and these traits are identical for all glands and piglets in a litter (Špinka and Illmann, 2015). Consequently, the first stage in the conversion of the original model into a sow mammary gland model aimed to simulate the maximum capacity of a single mammary gland to synthesize milk.

In contrast to dairy cows, the milk production of sows is not easily measured. Thus, the piglet growth rate was used as a proxy to estimate milk intake of individual piglets, and the maximal milk synthesis capacity was estimated based on piglet growth of the 5% piglets with the greatest growth rate between day 0 and 28 of lactation in the studies by Krogh et al. (2016, 2017). The growth rate of the 5% fastest growing piglets constituted approximately 400 g/d during the second to fourth week of lactation (i.e., peak lactation). We assumed that the growth potential was fulfilled for these piglets and that milk intake was the limiting factor for growth. In support of this, growth rates of up to 349 g/d were achieved during the fourth week of age in piglets with ad libitum access to milk replacers in the study of Zijlstra et al. (1996) Piglet growth rate was converted into piglet milk intake based on the energy required for energy retention and heat production of suckling piglets (Noblet and Etienne, 1987) combined with the milk energy concentration, which was calculated based on standard energy values and measured concentrations of fat, protein, and lactose in milk (Krogh et al., 2016, 2017). Based on these calculations, a growth rate capacity of 400 g/d corresponds to a milk output from the gland of approximately 1,800 g milk/d and a total milk synthesis of around 40 kg between farrowing and day 28 of lactation (i.e., liquid milk including water and dry matter). The colostrum intake was estimated using the prediction model of Theil et al. (2014). The mammary gland output on day 0 (i.e., colostrum) was assumed to be approximately 550 g, based on the colostrum intake of the 5% piglets with the greatest growth rate during lactation. A detailed description of the underlying data used to estimate colostrum and milk synthesis capacity is given in E-Supplementary S2. These empirical estimations of the maximum capacity of the mammary gland to synthesize colostrum and milk were used to reparameterize the original model combined with sow-specific characteristics as described later.

Piglets in a litter simultaneously remove milk from their respective gland for a duration of approximately 20 s in intervals of around 50 min during the lactation period (Špinka and Illmann, 2015). Thus, with an average nursing interval of approximately 50 min between nursings (i.e., 28.8 nursings/d) and a daily milk synthesis capacity of the gland of around 1,800 g/d, an average of 63 g milk (i.e., 1,800/28.8) is released from the gland during the 20 s of milk ejection. Furthermore, it was assumed that the gland was completely filled in 100 min and 80% filled in 50 min as indicated by Spinka et al. (1997). The filling of the gland was assumed to follow a curvilinear time relationship as indicated by the association between milk output and time between milking in dairy cows (Klopčič et al., 2013). Based on this, the ejectable milk storage capacity of a gland was estimated to constitute approximately 79 g (i.e., 63/0.80). Accordingly, the numerical values of the duration of milk ejection (tduration: ≈20 s), nursing interval (tinterval: 50 min), and ejectable milk storage capacity (Mm: 79 g) were directly applied in model version I to describe the sow-specific parameters of the milk removal rate (RM, g/min). In addition to these sow-specific parameters, the piglet-specific parameter, maximum milk removal rate (rm, g/min), was adjusted to ensure that the gland was almost emptied at each nursing event to reach a daily output of approximately 1,800 g milk/d. The use of Michaelis–Menten kinetics to describe milk removal rate (i.e., exponential decrease in the amount of ejectable milk in the gland during milk ejection) and a relatively short duration of milk ejection at each nursing event (tduration: ≈20 s) resulted in an incomplete removal of ejectable milk (maximum of 5 g of ejectable milk in the gland immediately after milk ejection). Moreover, the three Michaelis–Menten constants (Mh, KR, and kM) in the original model of Neal and Thornley (1983) that were associated with the ejectable milk storage capacity were adjusted to constitute the same percentage of ejectable milk storage capacity (Mm, g) as in the original cattle model. These three Michaelis–Menten constants are involved in the determination of variable cell loss rate, milk synthesis rate, and milk removal rate.

Mammary gland cell number adapts to changes in milk removal by the piglets (Kim et al., 2000). In the original cattle model, the variable cell loss rate was controlled by the average quantity of total milk in the udder across a 21-d period. For sows, Theil et al. (2005) showed that glands that were not emptied during the first 3 d after farrowing underwent rapid regression and were not able to synthesize milk to support piglet survival during lactation. They also observed that glands that were not suckled for 1 d after farrowing produced less milk than regularly suckled glands as indicated by a greater weight gain of piglets nursing the regularly suckled glands. Consequently, the average quantity of total milk in the gland was reparameterized to constitute a 2-d average instead of the 21-d average applied in the original model. As a result of the shorter time average of the quantity of total milk in the gland (i.e., 2 vs. 21 d), the variable cell loss rate becomes more rapidly activated when the gland is not emptied.

The final stage of the model reparameterization was based on the empirical estimation of the gland capacity to synthesize milk. The initial number of secretory cells was reparameterized to simulate an output of approximately 550 g milk on the first day in lactation (day 0). The fractional hormone decay rate constant was reparameterized to simulate the anticipated persistency of milk synthesis capacity in sows. Finally, the maximum milk synthesis rate constant, with great impact on milk synthesis rate (SM), was reparameterized so that the quantity of ejectable milk in the gland immediately before gland emptying reached approximately 80% of storage capacity during lactation.

Model version II: structure modifications

The reparameterization of the original cattle model leading to model version I of the sow mammary gland aimed to simulate the milk synthesis to fulfill the growth potential of suckling piglets, with the milk removal rate (RM) as the main driving force for milk synthesis of the corresponding gland. The aim of model version II was to simulate the gland response to changes in piglet demand based on biological observations and empirical estimations.

The empirical estimation of piglet milk intake from day 0 to 28 in lactation showed that the variation in peak milk output ranged from approximately 500 g milk/d for the 5% slowest growing piglets to 1,800 g/d for the 5% fastest growing piglets. Moreover, around 1 to 2 piglets in an average litter (i.e., 10% to 15% per litter) die during the lactation period terminating milk output for the gland (Muns et al., 2016). Milk that is no longer removed from the gland results in a decrease in cell number of this gland (Hurley, 2001), and weaning (i.e., complete termination of milk removal) has been observed to cause a reduction in mammary parenchymal mass of two-thirds within a week after weaning (Ford et al., 2003). Additionally, Kim et al. (2000) found that piglet gain (proxy of milk intake) and mammary gland size were positively correlated. These biological observations and empirical estimations as well as the fact that milk is mainly stored in the alveoli of the sow mammary gland (Farmer and Hurley, 2015) indicate that the number of secretory cells and the quantity of ejectable milk in the mammary gland adapt to the milk demand of the piglet. In this context, weaning was assumed to have a similar effect on the mammary gland as piglet death. Consequently, the model was modified to express storage capacity relative to the number of secretory cells instead of being constant throughout lactation as assumed in the original cattle model (Neal and Thornley, 1983). A consequence of this modification was that the averaged quantity of milk in the gland and the two Michaelis–Menten constants associated with the variable cell loss rate and milk synthesis rate were expressed in grams of milk relative to the total gland storage capacity (g milk) to correctly match the units of the rates and pools in the model. Finally, the numerical value of the fractional variable cell loss rate constant was calibrated based on the empirical observations presented above, to reflect the reduction in secretory cell number as a response to termination of milk removal.

Model version III: model extension

The model was extended to also simulate the output of colostrum, modeled as the combined output of Ig and milk from the gland. Consequently, the original model was extended to simulate the prepartum transfer of Ig and synthesis of milk in the gland, their storage, and the postpartum secretion of the resulting colostrum. This extension was based on the integration of biological mechanisms and empirical observations as described below.

Mammary gland size increases substantially during the last 2 wk of gestation (Hurley et al., 1991), with some transfer of colostrum components (including Ig) in the alveoli occurring from around 10 d before farrowing (Kensinger et al., 1986). Palombo et al. (2018) observed large increases in the number of expressed genes between 6 and 2 d before farrowing with even greater increases observed between day 2 before farrowing and day 1 of lactation. This likely suggests an acceleration of cell differentiation a few days before farrowing. In support of this, lactose synthesis appears to be initiated around 1 d before farrowing as indicated by an increased lactose concentration in sow plasma. Indeed lactose diffuses from the alveoli to plasma due to the loose integrity between alveolar cells during this stage (Hartmann et al., 1984). In contrast to lactose, Ig are not synthesized in the gland but by plasma cells and are transferred into the alveoli of the gland from around 10 d before farrowing. This is supported by the decrease in plasma concentrations of IgG in sows (i.e., the main Ig in colostrum) occurring from around day 106 in gestation (Huang et al., 1992; Feyera et al., 2019). Moreover, the Ig molecules are less likely to passively return to plasma due to their large molecular size. Taken together, these studies illustrate major changes in mammary gland cell number from around 10 d before farrowing, with differences in time between the onset of Ig transfer and of lactose synthesis. These physiological changes were used as the basis to extend the original model to simulate the prepartum Ig transfer and the milk synthesis representing colostrum production (i.e., Ig + milk).

The original model by Neal and Thornley (1983) started at parturition. In model version III, the time period was extended to initiate 10 d before farrowing. The lactation hormone was modeled with a prepartum excretion rate (EH, g/min) starting 10 d before farrowing with an initial basal hormone level and a rapid increase in the excretion rate around 3 d before parturition. The simulation of the hormone excretion rate before farrowing was inspired by the observed changes in reproductive and lactogenic hormone levels during late gestation (Foisnet et al., 2010; Vanklompenberg et al., 2013) and parameterized to attain the same level at farrowing (time zero) as in the original model. For simplicity, the equations used to describe the impact of hormone level during the prepartum period on mammary cell proliferation were kept identical to the original cattle model.

The prepartum milk synthesis rate is controlled by the number of secretory cells using the same structure as for the postpartum milk synthesis in the original model by Neal and Thornley (1983). However, in contrast to the original cattle model, the prepartum capacity of the secretory cells to synthesize milk (i.e., differentiation level) is modeled to increase during the prepartum period obtaining a full differentiation level at farrowing as indicated by Palombo et al. (2018) and Hurley et al. (1991). The shape of the cell differentiation curve is determined by a basal differentiation level and modeled to rapidly increase around 1 d before farrowing based on the sudden increase in plasma lactose observed by Hartmann et al. (1984). The increase in differentiation level is controlled by the prepartum increase in hormone level in combination with a steepness constant. In line with the original model, the secretory cells were assumed to be fully differentiated after farrowing. The prepartum development in differentiation level was included as a mechanism to model the biological differences between milk synthesis and Ig transfer during the prepartum period. The prepartum Ig transfer rate is modeled so that Ig are exclusively transferred into the alveoli before farrowing using the same model structure as for the milk synthesis rate (i.e., controlled by secretory cell number) but parameterized differently. The total quantity of Ig in the gland is exclusively removed after parturition at a rate proportional to the milk removal rate.

Model implementation

The model was implemented with the Vensim 7.3.5 modeling software (Ventana Systems Inc., Harvard, MA, USA) using the Runge–Kutta method (fourth order) numerical integration procedure with adaptive step size control and a maximum integration step of 0.125 min.

Results

Impact of model changes

The decay rate of the lactation hormone level after farrowing (DH, units/min; Figure 2A) was the same (shape and values) in all model versions and identical to that of Neal and Thornley (1983). The extension made in model version III included that the lactation hormone started 10 d before farrowing at a low basal level (20 units) to reach a specified level of 1,000 units at farrowing and with at a rapid increase starting around 3 d before farrowing (Figure 2A). The constant milk storage capacity (applied in model version I) resulted in a large quantity of ejectable milk in the gland with an average of 76 g before nursing and 57 g after nursing when the maximum milk removal rate was low (i.e., 60 g/min; Figure 2B, dashed lines). The introduction of a dynamic milk storage capacity (model versions II and III), which was controlled by the number of secretory cells, reduced the average quantity of ejectable milk in the gland compared with model version I, especially at a low milk removal rate. The average levels of ejectable milk in the gland were 29 g immediately before nursing and 15 g immediately after nursing when the maximum milk removal rate was low (i.e., 60 g/min; Figure 2B, dotted lines). The model extension in model version III resulted in an increasing quantity of ejectable milk in the gland from 0 to 20 g during the 10 d before farrowing, with the most rapid increase starting 1 d before farrowing (Figure 2B). The introduction of a dynamic milk storage capacity in model versions II and III reduced the daily milk output by an average of approximately 100 g/d (≈8%) compared with model version I for the reference parameterization mimicking milk synthesis of gland nursed by an average piglet (Figure 2C). The quantity of Ig in the gland (Ig) before farrowing increased from 10 d before farrowing to a maximum at 24 g at farrowing (Figure 2D). The curve shape of the quantity of Ig in the gland was modeled to slowly increase during the initial phase (day −10 to −3) followed by a slightly more rapid increase (day −3 to farrow; Figure 2D). After farrowing, the quantity of Ig in the gland declined exponentially with almost no Ig remaining in the gland 24 h after the onset of nursing (0.8 g Ig at 24 h; Figure 2D).

Figure 2.

Figure 2.

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Model scenarios illustrating the main similarities and differences between model versions I, II, and III with regard to the lactation hormone level (A), ejectable milk in the gland (B), daily milk output (C), and total Ig in the gland (D). (A) The lactation hormone curve after farrowing was identical for all model versions (I, II, and III), while a prepartum hormone release was exclusively included in model version III. (B) The storage capacity of ejectable milk was constant throughout lactation in the model version I (dashed lines) but linked to the secretory cell number in model versions II and III (dotted lines). This difference influenced the quantity of ejectable milk in the gland and was particularly noticeable, where the model was parameterized to mimic a slow-growing piglet modeled as a piglet with a low milk removal rate (i.e., rm = 60 g/min). The upper dotted and upper dashed lines indicate the quantity of milk in the gland immediately before milk ejection, while the lower dotted and lower dashed lines indicate the quantity of milk in the gland immediately after milk ejection. The quantity of milk in the gland before farrowing was included in model version III (solid line). (C) Daily milk output in model version I (dashed line; constant storage capacity) and in model versions II and III (solid line; milk storage linked to cell number) using the reference parameterization mimicking the average piglet. (D) Kinetics of the total quantity of Ig in the gland included in model version III using the reference parameterization mimicking the average piglet.

Behavior of the final model (model version III)

The reference parameterization of model version III was used to simulate the mammary gland nursed by an average piglet. The impact of a 30% and 60% increase and decrease in numerical values of individual model parameters (compared with the reference situation) on milk output, variable cell loss rate, secretory cell number, and quantity of Ig in the gland at farrowing and 24 h after farrowing was simulated to describe the behavior of the model (Tables 2 and 3). Milk output during the first 28 d of lactation was shown to increase by 28% and 29% when the maximum milk removal rate (rm, g/min) and the milk ejection duration (tduration, s) were increased by 60% relative to the reference level, respectively (Table 2). Milk output during the first 28 d of lactation increased by 39% when nursing interval (tinterval, min) was reduced by 60% relative to the reference level (Table 2). The quantity of Ig in the gland at farrowing was directly related to the maximum cell proliferation rate (vm, cell divisions/d) with a 60% increase in Ig at farrowing as a response to a 60% increase in the maximum cell proliferation rate (Table 3). The quantity of Ig in the gland at farrowing increased by 29% and 9% when the maximum Ig transfer rate (kIg, g Ig/1,000 cells/min) and Ig storage capacity (Igm-p, g/1,000 cells) were increased by 60%, respectively. Similarly, the quantity of Ig in the gland at 24 h after farrowing increased by 67% and 18% when the maximum Ig transfer rate (kIg, g Ig/1,000 cells/min) and Ig storage capacity (Igm-p, g/1,000 cells) were increased by 60%, respectively.

Table 3.

Percent change (+/–) in the maximum number of secretory cells and the quantity of Ig in the gland at farrowing and 24 h after farrowing relative to the reference (Ref) output as the response to changes in individual input parameters from the Ref parameterization (–60%, –30%, +30%, and +60%)

Ref. input Unit –60% –30% 30% 60%
Maximum number of secretory cells, cells (Ref. output: 5,959 cells) Cells % Change of Ref output
 Hormone decay rate kH 2.45 %/d 2 1 −1 −2
 Maximum cell proliferation rate vm 1 cell divisions/d −52 −20 14 26
 Fractional basal cell loss rate ks 10 %/d 22 11 −11 −23
 Fractional variable cell loss rate ksM 58 %/d 7 3 −2 −4
 Michaelis–Menten constant of variable cell loss rate Mh-p 0.9 g/g −77 −28 16 20
 Time period of average kr 2 d −1 −1 1 2
 Maximum milk synthesis rate kM 0.21 g milk/1,000 cells/min 21 19 −14 −24
 Ejectable milk storage capacity Mm-p 11 g milk/1,000 cells −33 −5 3 5
 Maximum milk removal rate rm 150 g milk/min −42 −22 15 20
 Nursing interval t interval 50 min 21 19 −14 −23
 Milk ejection duration t duration 22.5 s −46 −24 16 20
Ig in gland at farrowing, g (Ref. output: 24 g) g % change of Ref output
 Maximum cell proliferation rate vm 1 cell divisions/d −60 −30 30 60
 Fractional basal cell loss rate ks 10 %/d 15 7 −6 −12
 Maximum Ig transfer rate kIg 0.0042 g Ig/1,000 cells/min −55 −25 17 29
 Total Ig storage capacity Igm-p 21 g/1,000 cells −41 −14 6 9
Ig in gland at 24 h, g (Ref. output: 0.8 g) g % change of Ref output
 Maximum cell proliferation rate vm 1 cell divisions/d 186 86 −50 −75
 Fractional basal cell loss rate ks 10 %/d −29 −14 14 28
 Maximum milk removal rate rm 150 g milk/min 252 24 −4 −5
 Nursing interval t interval 50 Min −33 −29 40 194
 Milk ejection duration t duration 22.5 s 411 32 −5 −6
 Maximum Ig transfer rate kIg 0.0042 g Ig/1,000 cells/min −73 −40 37 67
 Total Ig storage capacity Igm-p 21 g/1,000 cells −59 −24 12 18
 Fractional Ig removal rate eIg-p 0.2 g Ig/g milk 903 243 −72 −92
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The impact of sudden changes in piglet suckling behavior (compared with the reference situation) on the variable cell loss rate, secretory cell number, and daily milk output is shown in Figure 3. A temporary reduction of 60% in the maximum milk removal rate between days 10 and 14 in lactation (i.e., maximum milk removal rate of 60 g/min) was used to simulate the mammary gland nursed by a piglet being sick from day 10 to 14 in lactation. A complete lack of milk removal from day 10 in lactation and onward (i.e., maximum milk removal rate of 0 g/min) was used to simulate the death of a piglet. Both the temporary reduction and the complete lack of milk removal stimulated the variable cell loss rate, which peaked at a level 5.0-fold greater (short-term reduction) and 6.5-fold greater (non-suckled gland) than the peak variable cell loss rate of the reference gland (Figure 3A). Simulation of a sick piglet resulted in a 31% reduction of secretory cell number and a 58% reduction in daily milk output (Figure 3B and C). When the maximum milk removal rate returned to the level of the average piglet on day 14, cell number and milk output returned to the levels of the average piglet, which were reached on day 27 in lactation (Figure 3B and C). Simulation of removing the pig from the litter resulted, naturally, in the complete cessation of milk output (Figure 3C) from the gland and a 66% reduction of secretory cell number on day 17 in lactation as compared with the level on day 10 (Figure 3B).

Figure 3.

Figure 3.

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Impact of change in piglet suckling behavior on variable cell loss rate (A), secretory cell number (B), and daily milk output (C). A temporary reduction of 60% in the maximum milk removal rate parameter (i.e., 60 g/min) between days 10 and 14 in lactation was used to simulate a piglet being sick between days 10 and 14 in lactation (Sick). A complete stop of milk removal from day 10 and onward was used to simulate a dead piglet (Non-suckling). These two scenarios were compared with the reference parameterization of model version III (Reference).

The impact of maximum milk removal rate constant and nursing interval on the dynamic response of variable cell loss rate, daily milk output, and ejectable milk in the gland on day 10 (M, g) is shown in Figures 4 and 5. A 60% decrease in the maximum milk removal rate constant increased the maximum variable cell loss rate by 214%, while the 60% increase almost eliminated the variable cell loss rate (Figure 4A). From a level at 1,247 g/d in the reference situation, maximum milk output was reduced to 463 g/d when maximum milk removal rate constant was decreased by 60%, whereas it reached 1,769 g/d when maximum milk removal rate constant was increased by 60% (Figure 4B). The quantity of ejectable milk in the gland on day 10 was 15, 42, and 50 g greater immediately before nursing than immediately after when the maximum milk removal rate was decreased by 60%, kept at reference level, and increased by 60%, respectively (Figure 4C). A 60% increase in the nursing interval (tinterval, g/min) increased the maximum variable cell loss rate by 90%, while the 60% decrease almost eliminated the variable cell loss rate (Figure 5A). A 60% increase in nursing interval decreased the maximum milk output to 664 g/d from 1,247 g/d in the reference situation, whereas it reached 1,904 g/d when the nursing interval was decreased by 60% (Figure 5B). The quantity of ejectable milk in the gland on day 10 was 21, 42, and 39 g greater immediately before nursing than immediately after when the nursing interval was decreased by 60%, kept at reference level, and increased by 60%, respectively (Figure 5C). A 60% reduction and 60% increase in nursing interval corresponded to 72 and 18 nursing events during a 24-h period, respectively.

Figure 4.

Figure 4.

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Impact of maximum milk removal rate (rm, g/min) on variable cell loss rate (A), daily milk output (B), and the quantity of ejectable milk in the gland on day 10 in lactation during a 180-min time period (C). A 60% reduction (–60%; i.e., rm = 60 g/min) and a 60% increase (+60%; i.e., rm = 240 g/min) in the maximum milk removal rate were compared with the reference parameterization of model version III (Reference; i.e., rm=150 g/min).

Figure 5.

Figure 5.

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Impact of nursing interval (tinterval, min) on variable cell loss rate (A), daily milk output (B), and the quantity of ejectable milk in the gland on day 10 in lactation during a 180-min time period (C). A 60% reduction (–60%; i.e., tinterval = 20 min) and a 60% increase (+60%; i.e., tinterval = 80 min) in the nursing interval were compared with the reference parameterization of model version III (Reference; i.e., tinterval = 50 min).

Discussion

There are indications that the capacity for piglet growth is greater than the capacity of the sow to produce milk. Ad libitum feeding of milk replacer to newborn piglets (0 to 4 wk of age) has indeed been shown to increase piglet growth rates to levels exceeding the growth rates of sow-nursed piglets (Harrell et al., 1993; Zijlstra et al., 1996). On the other hand, milk production is controlled by piglet suckling demand and nursing behavior in general as indicated by a 30% increase in litter growth rate by allowing two groups of six piglets to alternately suckle the sow in 30 min intervals compared with a sow nursing six piglets with no intervention (Auldist et al., 2000). This paradox emphasizes that piglet growth is a consequence of a complex balance between piglet demand and sow capacity to synthesize colostrum and milk. Prediction models have been developed to estimate colostrum and milk yields in sows based on piglet gain (Noblet and Etienne, 1989; Hansen et al., 2012; Theil et al., 2014). Such prediction models are helpful to quantify nutrient output. Kim (1999) developed a mammary gland model, in which protein and energy intake of the sow was used to predict mammary gland growth (i.e., milk synthesis capacity). This model is useful to better understand some of the mechanisms driving the capacity of the sow to synthesize milk. A different approach was taken by Pettigrew et al. (1992), who developed a dynamic model of energy and protein metabolism of lactating sows, in which nutrient metabolism was integrated based on nutrient availability and a time-dependent nutrient demand for milk production. These different models have provided valuable insights into the quantitative and mechanistic aspects of sow lactation. However, an integration of the different aspects of these models may be useful to understand and describe under which conditions and to what extent the piglets and the sow drive the production of colostrum and milk. The approach of the present mammary gland model allows considering piglet characteristics that influence the demand for milk as well as sow characteristics with impact on milk synthesis capacity of a single gland. In this context, the mammary gland model developed for cattle by Neal and Thornley (1983) provided a simple and suitable model framework and a sound starting point for investigating these relationships and their impact on the sow mammary gland.

Milk removal rate

The capacity of the piglet to remove milk from the gland during the short duration of milk ejection is reflected in the model by the maximum milk removal rate constant and represents a direct demand for milk by the piglet. The maximum milk removal rate was used to investigate to which extent the individual piglet influences the output of milk by the mammary gland. Changes in the piglet demand simulated by varying the maximum milk removal rate resulted in milk outputs of around 1,800 g/d at peak lactation (Figure 4). This indicates that the ability of the piglet to remove milk from the gland exerts a control on milk synthesis during lactation. In support of this, data from Krogh et al. (2016, 2017) showed that the maximum observed daily gain (as a proxy of milk intake) increased linearly from 300 g/d in piglets weighing less than 900 g at birth to almost 500 g/d in piglets weighing more than 1,900 g at birth (E-Supplementary S3). A positive correlation between stomach capacity and piglet body weight at birth as observed by Lynegaard et al. (2020) may at least partly explain and support this relationship. The duration of milk ejection and the nursing interval are also central parameters of quantitative importance for the removal of milk and Ig from the gland (Table 2). The nursing behavior of sows implies that the duration of milk ejection and the nursing interval apply to all piglets within a litter (Špinka and Illmann, 2015). However, more knowledge about interactions between nursing interval, duration of milk removal, and the capacity of the piglet to remove milk from the gland is needed to more accurately quantify and understand the key mechanism controlling milk synthesis under different conditions.

Milk synthesis capacity

The effect of the milk removal rate on milk synthesis capacity was modeled as an adjustment of the variable cell loss rate (i.e., adaptation of secretory cell number) and is controlled by the averaged proportion of milk in the gland during a 2-d period. The reduction in the time period for calculating the average quantity of milk in the gland from a 21-d average in the original cattle model to a 2-d average in the sow model allowed to model short-term responses to changes in specific parameters of interest for sow lactation. Indeed, this regulation reflects a central biological mechanism, in which the regression of the mammary gland functions as a response to milk stasis in the mammary gland of sows (Theil et al., 2005). This regulation was shown to have a substantial impact on the model response as illustrated by the increase in variable cell loss rate as a response to a reduction in maximum milk removal rate (Figure 4A) and to an increase in the nursing interval (Figure 5A). This mechanism was also present in the structure of the original model by Neal and Thornley (1983) but less responsive due to the parametrization of the cattle model.

The milk synthesis rate is also directly and increasingly inhibited as the total capacity of the gland to store milk is approached (Figure 1) causing the quantity of milk in the gland to increase in a curvilinear manner (Figure 5B). This immediate regulation mechanism reflects the quadratic response of milk output to differences in milking interval in dairy cows (Klopčič et al., 2013) and in sows (Spinka et al., 1997). The model simulations illustrate that this immediate regulation mechanism may be of quantitative significance in some conditions. Indeed, milk output was reduced by an increase in the nursing interval from 20 to 50 min (Figure 5B). This reduction was partly driven by the immediate regulation mechanism as indicated by the quantity of milk in the gland (Figure 5C) and partly from the negative feedback mechanism stimulating variable cell loss rate (Figure 5A).

A complete cessation of milk removal from day 10 in lactation and onward was simulated to mimic piglet removal, death, or weaning. This cessation caused a dramatic increase in variable cell loss rate and a reduction in total secretory cell number by 66% during a 7-d period, in agreement with the two-third reduction in mammary gland size 1 wk after weaning as reported by Ford et al. (2003). Furthermore, the short-term reduction in the milk removal parameter to simulate a sick piglet (Figure 3) suggested that a lower cell number following a 4-d period with a low milk removal rate resulted in a carry-over effect where milk output was reduced for around 2 wk after the piglet recovered. This carry-over effect on milk output constituted around one-third of the total reduction in milk output associated with the sick piglet. The ability of the model to adjust cell number and secretory capacity to the level of milk removal is in line with the observed flexibility of the ruminant mammary gland capable of adjusting cell number according to changes in milking frequency (Boutinaud et al., 2004). However, more empirical data on the sow mammary gland response to changes in milk removal are needed to accurately quantify and understand the mechanisms controlling milk synthesis under different conditions.

Dynamic milk storage capacity

The inclusion of a link between mammary cell number and the milk storage capacity of the gland resulted in a dynamic capacity of the gland to store milk during the lactation period (model versions II and III), which indirectly affected the quantity of ejectable milk and the milk synthesis capacity of the gland. The link between cell number and storage capacity is particularly relevant in modeling the sow mammary gland where milk is only stored in the alveoli and ducts in close association with the milk-synthesizing cells (Kim et al., 2000; Farmer and Hurley, 2015). In dairy cows, milk stored in alveoli and ducts constitutes only 60% to 85% of total milk volume, depending on milking interval, while cisternal storage constitutes 15% to 40% (Ayadi et al., 2003). A link between cell number and the storage capacity of the gland will be helpful to apply this modeling framework in a nutrient-based model simulating mammary gland nutrient uptake, output, and metabolism. Indeed, mammary nutrient uptake is influenced by the quantity of milk in the gland as illustrated by an abrupt change from a net uptake to a net release of amino acids by the mammary glands as a consequence of weaning (i.e., no removal of milk; Trottier et al., 1997).

Limitations of extrapolating from one gland to the whole udder level

The mammary gland model was developed to simulate milk outputs of around 1,800 g/d at peak lactation and average milk output of almost 1,400 g/d between days 0 and 28 from glands suckled by piglets with a high milk demand. Extrapolating this milk output to a sow nursing 14 piglets in a litter would correspond to 25 kg milk/d at peak lactation and a total litter weight of around 160 kg on day 28 of lactation. However, such a high milk yield may not be achieved, even for the best-performing sows (Hojgaard et al., 2019). Thus, on the sow level, the variation in performance is likely explained by different factors related to characteristics of the sow and the piglets in the litter. Indeed, variation in piglet birth weight and stomach capacity, as discussed above, and the variation in mammary gland weight at farrowing observed by Kim et al. (2000) may explain differences in milk synthesis capacity among glands in the sow udder. Also, the total supply of nutrients for milk synthesis may be a limiting factor, although this is not considered in the present model. In this context, the physical upper limit for feed intake (Thingnes et al., 2012) and the capacity to mobilize body reserves (Schenkel et al., 2010) may limit the supply of nutrients to support such a great milk yield. For comparison, the total energy requirement to support a milk production of 25 kg/d corresponds to an energy intake of approximately 190 MJ or around 14 kg of standard lactation diet each day, assuming a milk energy concentration of 5.0 MJ/kg, a conversion efficiency of feed to milk of 78%, and that the requirement for sow maintenance constitutes 30 MJ metabolizable energy (Theil et al., 2004). The maximum intake of energy may reach up to 150 MJ metabolizable energy/d (i.e., 11 kg feed/d) at peak lactation as indicated by Thingnes et al. (2012). The supply of energy from mobilized body reserves may constitute a maximum of 42.5% body lipids and 27.4% of the level at farrowing in body protein in extreme cases as presented by Schenkel et al. (2010). Thus, in a 250-kg sow with a composition of 23% lipids and 16% proteins, this corresponds to the daily mobilization of around 43 MJ or approximately 3 kg of feed. These estimations suggest that energy and likely also other nutrients become limiting to sustain a milk yield of 25 kg/d in many cases and at least in conditions where sow nutrient intake and mobilization of body reserves are not maximized and the dietary composition is not completely balanced with the demand. This illustrates the importance of understanding to what extent the piglets and the sow are the drivers of milk production under various nutritional and physiological conditions. Finally, this indicates that interactions between individual glands in the same udder should be considered to describe colostrum and milk production on the udder level. These relationships between the supply, uptake, and metabolism of nutrients in the mammary gland are complex and require separate studies to describe and understand the nutritional impact on milk synthesis in sows.

Colostrum synthesis (Ig + milk)

To our knowledge, this is the first model that considers the production of colostrum as the combination of Ig and milk in two dynamic pools. This approach allows considering the different characteristics of colostrum and milk. For example, the quantity of Ig in the gland at farrowing is by definition not affected by suckling behavior and consequently modeled exclusively as a sow characteristic. The Ig in the gland are transferred from the blood during the last 10 d of pregnancy and after farrowing, while the quantitative transfer of Ig is considered to be of the minor quantitative importance of the transfer of Ig (Feyera et al., 2019). The maximum quantity of Ig in the gland (at farrowing) was parameterized in the model to reach a maximum quantity of 24 g at farrowing (Figure 2). This was empirically estimated based on the intake curve of newborn piglets (Le Dividich et al., 1997), the average colostrum intake of sow nursed piglets (Krogh et al., 2015), and the composition of Ig in colostrum during the first 24 h after farrowing (Hurley, 2015). In contrast to Ig transfer, milk synthesis increases rapidly as a response to milk removal. Consequently, the continuous synthesis and removal of milk cause a dilution of the Ig components in milk after farrowing. This approach implied that colostrum is defined as the combination of different components (Ig + milk), which are separately controlled rather than being defined as a distinct secretion from the mammary gland in a specific time period relative to farrowing (Theil et al., 2014). This modeling approach allowed to represent the rapid decline in Ig concentration during the first hours after farrowing (Hurley, 2015). Moreover, the impact of milk removal rate on the quantity of Ig removed from the gland during the first hours after farrowing suggests that the change in Ig concentration from a gland between farrowing and day 1 may be used as an indicator of the suckling intensity of that gland.

The removal rate of Ig and milk from the gland is mainly controlled by nursing, frequency, duration, and intensity of milk removal by the piglet. However, in contrast to the established lactation, the nursing behavior of newborn piglets is characterized by the existence of nonsynchronous nursing events and by the piglets suckling different glands during the initial hours after birth (De Passille and Rushen, 1989). These differences in nursing behavior during the first hours after birth are not considered in the current version of the model. Consequently, the colostrum output should be used to investigate the overall and general regulation of colostrum production but requires further development and more knowledge to understand how the output of Ig and milk is affected by litter characteristics on a short time scale during the first 24 h after the onset of farrowing.

Conclusions

To our knowledge, this is the first model to simulate the production of colostrum and milk in sows using physiological aspects to describe the balance between the capacity of the sow and the demand by the piglets. The model simulations suggest that the piglet demand for milk represents the majority of the variation in milk output from individual glands. The approach of representing colostrum as the combined output of Ig and milk illustrates the gradual transition between colostrum to milk synthesis; however, more knowledge is required to consider the nursing behavior of piglets during this period. The model constitutes a simple approach to investigate and potentially predict mammary output based on relatively simple and (some) measurable parameters. Finally, the model may serve as framework to further investigate the impact of nutrient supply, uptake, and metabolism on milk synthesis.

Supplementary Material

skab030_suppl_Supplementary_Material
Click here for additional data file. (500.6KB, pdf)

Acknowledgments

The financial support for this project was provided through a collaborative project between INRAE, Institut Agro, Saint-Gilles, France, and Ajinomoto Animal Nutrition Europe, Paris, France.

Glossary

Abbreviation

Ig

immunoglobulins

Conflict of interest statement

The authors declare no conflict of interest.

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