The MilkBot® Model


A lactation model is simply a means to describe lactation curves, and the differences between lactation curves.  Details of how the model is designed have a critical influence on how useful it can be in various applications.  While typical users will be able to understand MilkBot® parameters without learning the details of the MilkBot® model,  understanding the logic behind the model gives a deeper understanding of the system as a whole.

MilkBot® was devised using a theoretical-mechanistic model. This is pseudo-physiology, meaning we use physiological reasoning without worrying whether all our assumptions are exactly correct. The point is more to have a clear chain of logic in the derivation of the model than to model actual physiology. A simple defined derivation makes it easier to interpret results. We may choose to extend the model as experience accumulates, but predictive efficiency does not require strict physiological accuracy. It seems best to start simply.

We start by assuming that milk is produced by a set of hypothetical factory units in the udder, and that each unit produces milk at a a constant rate. This means we can model production by modeling population dynamics of the factories. "Factories" differ from "mammary secretory epithelium" by a layer of abstraction.

To model population dynamics for our "factories", we postulate two opposing forces of creation and destruction. For most purposes, we model creation as a single event, a bell shaped curve with peak near calving and variable width. We model destruction following standard first order decay kinetics. The number of "factories" present at any time is simply the product of the destruction curve and the integral of the creation event. Multiplying by a scaler (milk per factory) gives the day's projected milk production.

The graph at right shows first order decay kinetics, with a rate constant, k, convertible algebraicly to a half-life, t½. We refer to k as the milkbot.decay parameter and t½ as the milkbot.persistence parameter. Either variant can be used, but in most cases we prefer to use the persistence value, since we feel understanding of "half-life" (expressed in days) will be more intuitive to most users.

The blue curve at right has a persistence value of 350 days (decay = .0020), and the green curve persistence of 250 days (decay = .0028).

Pers

For the construction process, we define a bell shaped construction event associated with calving. The number of factories created on or before a certain day will be the area under the curve to the left of that day, i.e. the integral of the curve. Two independent parameters control the characteristics of the creation event. The width of the curve (corresponding to the standard deviation of a Normal curve), is the parameter milkbot.ramp. The offset of the peak of the curve from day zero (calving) is milkbot.offset.

In the graph at right, the blue curve has a ramp value of 30 days, and the green curve a ramp of 50 days. Offset is the time between the peak of the creation event and calving, which is zero in the examples at right.

Crea

Here the integrals of the same bell-shaped creation curves are plotted, what we call the build curve. This gives the number of "factories" created on or before a certain day.

Creaint

Both creation and destruction processes are going on throughout lactation, and the number of factories present on a particular day is the value on the build curve multiplied by the corresponding value on the persistence curve.

Daily milk production is this times a scale factor representing milk per factory, as shown on the next graph.

Comp

It can be seen that the creation process dominates in early lactation, and the destruction curve dominates later on, resulting in the characteristic rise-to-a-peak-then-drop-off shape of a lactation curve.

A curve's peak can be calculated mathematically from its parameter values, as well as area under the curve at any point. These can be used as reduced parameters.

Complact

Because each parameter has a consistent and predictable effect on the shape of the curve, it is easy to understand what changes in parameter values mean in terms of their effect on milk production. It is also straightforward to hypothesize what effect a given intervention might have on individual parameter values. It should be kept in mind that milk "factories" in this model are an abstraction, and not the equivalent of any known physical structure or physiological process.