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Mathematical Modelling in Lactation: An analysis of hormonal regulation and complex lactation mechanisms

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Abstract

 Lactation is a unique natural mechanism shared amongst mammalian species that serves the primary purpose of providing offspring with the vital nutrition and immune support necessary in the early stages of fetal development.  The process of lactation has been extensively researched and has led to the application of mathematical principles to model the phenomena.  Furthermore, advancements of biotechnology have resulted in the discovery of mathematical functions and models to describe the various biochemical processes and mechanisms of mammalian lactation.  The various stages of the lactation cycle are controlled by the endocrine system of the animal and both reproductive and lactation hormones play a central role in the formation of the mammary glands, production of milk and the eventual release of milk from the mammary glands.  The investigation of the two primary stages of the lactation cycle will be further investigated with reference to the mathematical relationships of hormone levels and trends.  In particular, the first phase of the lactation cycle, mammogenesis, will be explored in order to provide a background on the influence of reproductive hormones on the growth and development of the mammary system.  In addition, the subsequent stage, lactogenesis, will be examined with an emphasis on the relationship between milk production in the mammary glands and various hormone levels during cow gestation and parturition.  As well, the investigation of a study describing the mathematical relationship between milk production and the inhibition of the hormone prolactin will highlight the significance of mathematical models to describe the lactation mechanism. Additionally, the role of positive feedback loops will be detailed with reference to the function of the hormone oxytocin on the mechanism of milk release from the mammary glands. The relationship between oxytocin and milk ejection will be further defined through the examination of the BovCycle model to describe changes in oxytocin levels during the lactation process.  Finally, an exploration of mathematical models and equations regarding ductal displacement and elongation will provide a better understanding of the rate of mammary growth with response to various parameters.

Introduction

The process of mammalian lactation is composed of many complex biochemical processes and mechanisms that can be represented with mathematical models and equations to establish a more concrete understanding of the intricacies of the phenomena. Firstly, comprehending the fundamental role the endocrine system plays in preparation for and during lactation allows further analysis and the discovery of unique relationships between hormone levels and the stages of lactation.  Furthermore, the vital role of placental hormones throughout the stages of lactation exhibits a relationship between the reproductive system and the initiation of lactation in mammals.  Mathematical models allow researchers to quantify and analyze natural mechanisms such as lactation to better understand biological processes. In particular, the most notable relationship that is essential to understanding the lactation process is the presence of a positive feedback loop which is responsible for the regulation of a vital lactation hormone, oxytocin.  In addition, the investigation of ductal elongation in response to various parameters has led to the creation of mathematical models and equations that describe mammary growth rates which is essential to understanding the lactation process for mammalian species.

Mammogenesis

Throughout the course of a mammal’s life, the mammary gland undergoes greater changes in size, composition, structure, and activity than any other tissue or organ (Knight & Peaker,1982).  Mammogenesis is the process that describes the repeated cycles of growth and maturation of the mammary gland.  Furthermore, mammogenesis is established during early fetal development and continues during a mammal’s estrous cycles and pregnancy (Rezaei et al., 2016).  Mammogenesis is a necessary process as mammary glands require proper growth and development to reach their functional state to perform lactation. Most of mammogenesis occurs during gestation, which is the prolonged stage of the reproductive cycle that proceeds conception and during this stage the placental hormones estrogen and progesterone are responsible for mammary growth and maturation (Rezaei et al., 2016). Furthermore, lactation and reproduction are closely related in mammals as mammary development occurs slightly during estrous cycles which initiates “positive allometric growth” of the central mammary structures.  Additionally, greater mammary development occurs while the female is pregnant as vital pregnancy hormones such as estrogen and progesterone also function as the primary stimulators of mammary tissue growth during pregnancy (Rezaei et al., 2016).  In addition to estrogen and progesterone, growth hormone, prolactin and glucocorticoids (particularly cortisol) contribute to mammary gland development during the process of mammogenesis (Hartmann et al., 1996).  In addition to estrogen and progesterone, growth hormone, prolactin, and glucocorticoids (particularly cortisol) contribute to mammary gland development during the process of mammogenesis (Rezaei et al., 2016).

Fig. 1 Relationship Between Stages of Mammogenesis and Udder Size for Dairy Cows (Hartmann et al., 1996)

Mammogenesis varies between mammalian species in terms of the amount of growth and development of the mammary gland required to achieve mammary functionality for that species’ lactation mechanisms.  During gestation, most species undergo the greatest mammary development, but the levels of estrogen and progesterone during mammogenesis vary between species.  For example, in cows, progesterone levels are elevated throughout the entirety of gestation while estrogen levels only increase in the second half of pregnancy (Figure 2) (Rezaei et al., 2016).  In contrast, during mare pregnancy, progesterone reaches its maximum concentration in early gestation and estrogen levels do not increase until right before parturition, or birth, at the end of gestation (Figure 3) (Hormonal Changes in the Mare During Pregnancy, n.d.).

Fig. 2 Estrogen and Progesterone Levels Throughout Cow Gestation Until Parturition (Fora, 2006)
Fig. 3 Estrogen and Progesterone Levels Throughout Mare Gestation (Hormonal Changes in the Mare During Pregnancy, n.d.)

Mammogenesis is a crucial stage of the lactation cycle that is responsible for the sufficient growth and development of mammary glands to their functional state in preparation for lactation (Rezaei et al., 2016).  Various levels of the placental hormones estrogen and progesterone play a role in the development of the mammary system prior to the lactation stage of lactogenesis where milk production occurs (López-Fontana et al., 2012). The transition from mammogenesis to lactogenesis happens in a fluid fashion, as the presence of milk inhibiting reproductive hormones is maintained up until parturition.  

Hormonal Control of Lactogenesis

Following conception, mammogenesis takes the mammary glands through gradual development up until just before parturition, at which point the mammary glands enter a new stage of development known as lactogenesis. As such, lactogenesis is, in a general sense, the stage of lactation in which the mammary glands undergo rapid development in a short period prior to parturition. Lactogenesis can be characterized by two distinct periods: Lactogenesis I and Lactogenesis II.

Lactogenesis I is characterized by a proportionally high serum concentration of the reproductive hormones, notably: estrogens, progesterone, prolactin, and oxytocin (Neville et al., 2002). These hormones stimulate rapid differentiation of the mammary tissues, prepping the body of the gestating mother for the initiation of full lactation.

Fig. 4 Blood serum progesterone concentrations in 10 cows, taken at intervals before and after parturition (Convey, 1974)

Prolactin and oxytocin are two hormones that have very particular application within the greater reproductive cycles, whereas progesterone and estrogens have more general application within the bodies of female mammals. Furthermore, oxytocin and prolactin are both produced in the hypothalamus and pituitary glands within the brain, whereas the progesterone and estrogens that take part in the regulation of lactation are produced in the placenta (Popa, 2021). The differences between these hormones can be seen in their respective blood serum concentrations throughout the different stages of lactation, and as such can be understood as the hormonal regulators of milk synthesis.

Fig. 5 Blood serum estrone and estradiol concentrations during gestation and parturition in cows (Convey, 1974)

 Lactogenesis II is largely triggered by a sharp drop in blood serum levels of estrogens and progesterone (Neville, 2011). This can be observed in figures 4 and 5 and can be explained by the expulsion of the placenta during parturition. Thus, the prolactin and oxytocin levels become disproportionately high in comparison to the estrogen/progesterone levels. As such, it can be hypothesized that prolactin and oxytocin stimulate the production of milk, whereas estrogens/progesterone inhibit the production of milk.

To no surprise, prolactin has been shown to be directly involved in the synthesis of 𝛼-lactalbumin which is directly involved in the synthesis of lactose, a key component of mammalian milk (Ingram et al., 2007). Further studies have shown that the inhibition of prolactin through a drug known as bromocriptine can reduce milk production by 60-70% (in goats). Furthermore, the suppression of genes involved in the synthesis of milk components has shown to invariably drop the concentration of circulating prolactin (also in goats) (Cowie et al., 2011). This could therefore indicate a proportional relationship between the frequency of milking and the production of milk, which has been confirmed through lactation curves and the study of the positive feedback loop characteristic to lactation (figure 6).

Fig. 6 Mean milk yields of three control goats (⬤)and goats treated with increasing doses of bromocriptine (○) (Cowie et al., 2011)

Section 3: Positive feedback loop in lactation

Positive feedback is a mechanism in which a stimulus produces a product which can intrinsically lead to new stimulus. The stimulus amplifies the original action since the more stimulus, the more product it produces. In a positive feedback loop, this cycle continues indefinitely. The complete process of the positive feedback loop is that after a signal is sent by stimulus to a sensor organ which detects the change and produces a nerve impulse, the control center will receive the impulse and send this signal to the effector organ in question. Thus, the effector organ responds to the change caused by stimulus, producing new subsequent stimulus, and so the loop continues (positive feedback, n.d.) (Figure 7).

Fig. 7 Positive feedback loop visualization (positive feedback, n.d.)

Lactation is a prime example of a positive feedback loop in which infant suckling is the stimulus and milk release are the response. The more frequently babies suckle, the more milk ejects, and the increasing amount of milk triggers more suckling behavior (Brodeur-Doucet, 2021). This loop will continue until the feeding infant no longer depends on a maternal diet and the lactation period is finished.

Fig. 8 positive feedback loop of lactation – milk release (positive feedback, n.d.)

The complete positive feedback loop of lactation in milk release process consists of 4 steps (Figure 8). First, when babies suck the nipples, a signal is sent to alveoli tissues in the mammary gland. As a sensor organ, alveoli trigger sensory nerve impulses which travel from the nipples to the brain, which is a control center (World Health Organization, 2009). Next, the hypothalamus and pituitary gland of the brain synthesize the hormone oxytocin. Neurosecretory cells of paraventricular nucleus in the hypothalamus produce oxytocin which travels down the axons to the posterior pituitary (Brunton, 2018; Endocrine System, n.d.) (Figure 9). Then, oxytocin, which is then stored at the axon terminals known as Herring bodies, travels through blood vessels to the mammary gland (Popa, 2021).

Fig. 9 Oxytocin production in the hypothalamus and pituitary gland within the brain (Endocrine System, n.d.)

Circulating oxytocin triggers myoepithelial cell contraction by decreasing the diameter of alveolar lumen. The pressure inside the mammary gland is increased and myoepithelial cells generate a force to push the milk out so that milk can be squeezed from breast alveoli (Scott & Brown, 2013; Haaksma et al., 2011). This is similar to the idea of wringing out a sponge. Finally, milk travels from alveoli tissues through many mammary ducts to nipple pores, ready to be obtained by the feeding young.

Indeed, oxytocin plays an important role in the milk ejection process during lactation, and the amount of milk released is proportional to the oxytocin concentration level. The more oxytocin in the blood, the greater the milk yield (Brunton, 2018; Negrão & Marnet, 2006).

However, oxytocin concentration levels can be affected by the emotions and mood of the feeding mother. If the mother is in pain, or is stressed/anxious, less oxytocin would be produced, leading to difficult milk ejection. On the other hand, if the mother is happy and feels at ease, more oxytocins would be produced, leading to easier milk ejection. Oxytocin begins to release when the lactation period starts (World Health Organization, 2009).

A study by Omari et al. (2019) investigates how the concentration level of oxytocin in dairy cows changes during lactation. It revealed that the concentration level of oxytocin in dairy cows is decreasing throughout the lactation period which can be illustrated by the following equations defined in the adapted “BovCycle” model.

Fig. 10 adapted “BovCycle” model equations to describe change of concentration level of oxytocin (Omari et al. 2019)

Originally, the BovCycle model described in the study was only capable of mathematically modelling the rate of oxytocin synthesis and removal in the blood. However, an adapted BovCycle model was introduced in order to accurately model the changes in oxytocin concentration levels throughout lactation. A new term “Oxylac” was thus added, corresponding to the variable instantaneous concentration level in the blood of cows defined by the exponential equation of figure 10. The term “Oxysyn” corresponds to the rate of oxytocin synthesis and the term “Oxycle” corresponds to the metabolic clearance rate of oxytocin.

 A curve of oxytocin concentration versus lactation time is represented by the Gaussian function (2) in figure 11 with parameter values c1 = 0.0007 and c2 = 1.5. These values are determined by an experiment measuring the oxytocin concentration level in some dairy cows. As seen in figure 11, concentration levels of oxytocin in the cow’s blood are decreasing nonlinearly from the beginning of lactation. After 100 days, oxytocin concentration is nearly 0 which means that cows do not produce milk during late lactation period as much as they produce during early lactation period (Omari et al. 2019).

Fig. 11 Plot of the time-dependent oxytocin concentration level during lactation as defined by eq. (2) (Omari et al. 2019)

Thus, the positive feedback loop can illustrate how suckling behavior is connected to the amount of milk ejected from the mammary gland of mammals. During the lactation period, milk yield varies according to frequency of suckling frequency and concentration level of oxytocin in the blood.

The rate of the ductal elongation

In the sweat glands, the ductal elongation (division of the cell) occurs during puberty regulated by TEBs (Terminal End Bud). TEBs are multi-layered structures that direct the growth of the duct throughout the fat pad (a mass of packed fat molecules). During the process of elongation, TEBS are more specifically responsible for the division of the main ducts (figure 12). However, for division to occur, limited geometric constraints must be respected. Thus, the TEB must have an ideal structure for ductal elongation.

Fig. 12 Pubertal mammary ductal growth dynamics. (Scheele et al., 2017)

We can represent the rate of the ductal elongation and TEB structuring with a mathematical equation. In order to model this structure, we base the equations on the two main compartments, where high levels of proliferation are made to produce the cells that form the mature duct. By taking into account five measurable parameters, we can accurately predict the ductal displacement and elongation. (Paine et al., 2016)

Fig. 13 Mathematical equation of the ductal elongation rate of the inside (lum) and outside (bas) compartiment of the TEB. (Paine et al., 2016)

To represent this model numerically, we must divide a single TEB into eight sections and individually quantify the behaviour of the population of cells within these sections. Thus, by modifying the population equation in adding more parameters (figure 13), we can come up with an equation that will best decribe the ductal elongation of the mammary glands. In the equation, the lambda variable is the ductal elongation rate in the two main compartiment (the inside (lum) and the outside (bas) compartiment). Other variables are taken into account in this equation such as cell number, cell migration, etc (Paine et al., 2016).

Although this equation is very useful to calculate the the ductal elongation rate, it can also be used to know if an animals mammary glands are growing normally or if the animal in question  has an underlying health condition affecting mammary growth. In figure 12, a normal division is represented, whereas in figure 14, the division is characteristic of a cancerous growth. (Scheele et al., 2017)

Fig. 14 Cancerous growth in the sweat glands (Ingthorsson et al., 2015.)

Conclusion

By understanding the various stages of lactation of mammals, researchers are better able to represent the biological phenomena using mathematical models and equations to simplify the complexities of mammalian lactation.  Mammogenesis takes place pre- and post-conception up until parturition. This is a vital stage of the lactation cycle, as it ensures the proper growth and development of the mammary glands for them to be fully functional. This period is characterized by high concentrations of estrogens, progesterone, prolactin and oxytocin which all stimulate mammary tissue differentiation and growth. The exact levels of progesterone and estrogen during gestation may vary species-to-species, but the transition to lactogenesis is common to all.

Lactogenesis (II) takes place following parturition when the placental concentrations of estrogens and progesterone plummet due to the ejection of the placenta. Although the blood concentrations of oxytocin and prolactin drop too, these latter hormones drop significantly less, leading to a large hormonal disproportion in comparison to the blood concentrations prior to parturition. These hormonal changes coincide with the beginning of milk production, leading to the hypothesis that progesterone/estrogens inhibit milk production, whereas prolactin and oxytocin stimulate milk production. Although it is not yet known for certain whether this is the case, certain studies involving bromocriptine, a prolactin inhibitor, seem to confirm the hypothesis.

Furthermore, a proportional relationship between suckling frequency and milk production has been observed, indicating a potential positive feedback loop. A positive feedback loop is when a response to a stimulus creates more stimulus, essentially amplifying the stimulus indefinitely. Indeed, this is the exact mechanism which describes the release of oxytocin during milk suckling by the infant. As the infant suckles, a signal reaches the brain and oxytocin is produced and released into the blood. This circulating oxytocin causes milk ejection through myoepithelial cell contraction, which is ready to be obtained by the young. The cycle repeats itself as the infant continues to suckle and stimulate the nipple/teat.

Finally, extensive analysis regarding ductal displacement and elongation has led to the creation of mathematical models and equations to accurately describe the rate of mammary gland growth with respect to various parameters. Not only can researchers use this mathematical relationship to describe mammary growth but can also provide analysis into underlying health conditions that may impact mammary development and therefore disrupt the natural mechanisms of lactation due to insufficient growth. Mathematical models and equations allow researchers to describe complex biological mechanisms and processes such as lactation in order to better understand and represent the intricacies of natural phenomena.

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