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Quantification of Lactation Curves for Diagnosis.
This is a web page version of a talk given in the Research Summaries
session of the annual conference of the
American Association of Bovine Practitioners, September 22,
2006 in St. Paul, MN.
© Jim Ehrlich, D.V.M.,
2006
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To see a larger version of a
picture, click on the picture. It will pop up in a separate window.
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I would like to show you a tool kit I have developed
for the analysis of milk production in dairy cows. Milk production is
important, not only because it pays the bills on dairies, but also
because it is an extremely sensitive measure of cow health. Almost
anything we do to milking cows, or any pathological condition,
influences milk production.
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The first version of this talk that I wrote took half
my allotted time before I even got to the point of showing any data.
That didn't seem acceptable, so I am going to skip a lot of technical
background. That background information is important, so while I don't
have time for it here, let me refer you to the MilkBot website
Here, I am reduced to describing MilkBot as a black box
which takes milk production data, and summarizes each lactation in the
data set as a set of parameter values which characterize the important
aspects of the lactation. The data I am showing here comes from a
commercial herd with automatic ID and recording of milk weights. We
dumped weekly average milk weights into the MilkBot black box.
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Here is the observed data from a sample lactation.
Weekly average milk weights are shown as black dots. |
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The MilkBot fitting engine fits the observed data to
the MilkBot model, as shown by the curve here. It does this by
adjusting values for the four parameters, shown in the upper right. |
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Parameter values determine the magnitude and shape of
the curve. |
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How well the model curve fits the data is shown by the
standard error value in the upper left. The fitting engine's task is to
minimize this standard error within certain important constraints. |
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Because of the way the MilkBot model is constructed,
parameter values have inherent meaning, which is explained in detail on
the web site. The Scale parameter is a simple linear scaler. |
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The Ramp value controls how fast production rises after
calving.
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The Persistence value controls how fast production
declines, modeled with first-order decay kinetics. I'm not going to
dwell on these here, except to say that each parameter controls a
single aspect of the curve, so that it is easy to understand what
parameter changes will be associated with any change in shape of the
curve. With practice, it is not hard to predict parameter values simply
by looking at a curve. |
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Talking of "Models" and "Parameters" adds a layer of
abstraction that can be confusing to those not used to thinking in
those terms. To simplify explanation, MilkBot can also calculate some
reduced parameters, as on this slide showing the same lactation. The
values of the previous slide are transformed by simple algebra to the
statement that 305-day production is 23,939 pounds of milk, with a peak
on day 52 and persistence of 396 days. These numbers summarize the
lactation as a whole. You can see, for example, that the model peak at
52 days is not the same as the day of highest observed production.
Other descriptions are possible, and it remains to be seen what is most
useful and most intuitive for users. |
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More complicated transformations are also possible.
For example, this shows an attempt to quantify variation around the
fitted curve. If we assume that the curve represents what we would
expect in a perfectly controlled environment, then the "blips" shown in
blue represent times when there was a health incident or change in the
environment causing production to deviate from normal. Though nothing
is recorded in this cow's health record, it is reasonable to assume
that there are distinct causes for these blips in production. Anomalies
of this sort turn out to be very common, and MilkBot includes tools to
look at them in a statistical context.
Now I would like to show you a series of lactations, randomly selected
from a normal herd. The point is first to show that MilkBot does an
excellent job of calculating plausible curves. Achieving this kind of
stability from a computer algorithm was not easy. Second, I want to
show the variation which occurs in normal herds, because where there
are patterns of variability, there is interesting science to be done.
The next level of research is to find and characterize causes for this
variation. |
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This is a first lactation, almost flat with persistence
of 581 days and low Scale. If you look carefully, you can see that
points in roughly the first and last thirds of the lactation are higher
than the curve, and points in the middle are lower. This could be due
to herd-wide factors such as changes in weather or feeding, or illness
such as a case of mastitis, or possibly that the heifer was started on
BST at about 170 days-in-milk.
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In this third lactation, production is high and quite
consistent, with the exception of a sudden drop from 130 to 110 pounds,
right near the peak. That drop, to me, suggests some undiagnosed
illness. |
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A second lactation of about 29,000 pounds, quite
similar to the previous one. |
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A fourth lactation with an early peak and poor
persistence. Persistence of 196 days is about one standard deviation
below the mean for this herd. |
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Slightly better persistence than the last slide, also
fourth lactation. |
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Second lactation. |
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This second lactation, is marked "sold or died" with
the comment "mastitis". Clearly something happened at around 60 days.
The lactation appears bimodal, ... |
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... so I asked Milkbot to fit the points before 60
days in light blue, and after in pink. We could characterize the
mastitis event in terms of changes to lactation parameters. |
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A fifth lactation with early peak and good
persistence, though scale and total production are just average. |
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Another second lactation that starts a bit higher than
the previous lactation, but is 5,000 pounds lower in 305-day total
production, due to poor persistence.
It is obvious that a single hypothetical "normal
lactation curve" does not exist; that shape of lactation curves varies
materially between individual cows, and that day-to-day variation
around the curve is also highly variable.
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MilkBot quantifies the shape of lactation curves in a
systematic and repeatable way, so we can calculate summary statistics.
For this group of lactations, mean Scale is 113 pounds with standard
deviation of 21. Persistence is 376 days with standard deviation of
174, and so on. These values vary significantly between individuals,
groups, and herds. Why?
Multivariate analysis of things correlated to parameter
values could be a lot of fun.
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Here is a scatter plot of Persistence versus Age at
Calving. Lactations are color-coded by parity, with first lactations
colored red, 2nd lactations green, and 3rd or greater blue. The
expected relationship between parity and Persistence is visible, and
obviously not linear. I find it interesting that within parity groups,
Persistence does not seem to be influenced by age, which you can see by
looking at any single-color dot set . That is, red dots are first calf
heifers, with higher Persistence than the green or blue dots, but
within any one color there doesn't seem to be any correlation of
Persistence with age. |
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This is a double histogram with horizontal layout and
points again colored by parity. Lactations are divided between left and
right histograms on the basis of a treatment given to cows on one side.
This comes from a pilot study for a trial to evaluate a product that we
think may be partially protective against incidents of subacute rumen
acidosis (SARA). It's not easy to detect a small reduction in the
frequency of a common subclinical disease.
One approach we are considering is to look at consistency of production
within individual lactations. Here I have plotted standard error of
fitted lactations in the two groups, and there doesn't seem to be any
difference, though the size of this pilot study was very small. Mean
and standard deviation are shown graphically by the black lines in the
middle zone.
As I mentioned earlier, we are working on more
sophisticated ways to characterize variability in terms of multi-point
events which I call "blips" in production. It's a bit complicated
devising a good algorithm, but I think it is a very promising approach.
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My last chart is even more complicated. It monitors
group average milk production over time, and tries to seperate
controllable factors from fixed factors influencing bulk tank milk. |
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The thin black line, varying from less than 70 to more
than 90 pounds, shows milk per cow per day, bulk tank milk. That total
is broken down into three components. |
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First, the gray shaded area at the bottom represents
cow quality. It changes only slowly as individual cows enter and leave
the milking string. |
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On top of that, brown arrows represent the lactation
stage component. For each lactation, the difference between that day's
expected production and the mean for the entire
lactation is calculated, and these differences are averaged for the
group. When there are many cows near peak production, values will be
positive, as in the middle portion of this chart. When there are a lot
of cows either pre-peak, as on the left side, or post-peak as on the
right side, then the lactation stage effect will be negative. |
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The remainder, portrayed as a green arrow, is the
unexplained difference, or effect of management plus environment. There
is a lot of calculation going on here. |
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In many ways this last component is the most
interesting, yielding an accurately normalized portrait of day-to-day
changes, which may be due to things like feed quality, heat stress, or
irregularities in the milking or feeding routines. The normalized
picture can be quite different from the raw production curve, as it is
in this example.
This chart is complex, so let me re-state how it is
calculated from the viewpoint of a single lactation.
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Here is a sample lactation. I will magnify the
outlined area in my next slide. Black dots are observed milk
production. The gray zone represents average production from zero to
305 days. |
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I have marked two examples. At 25 days-in-milk, which
is pre-peak, milk production is expected to be below the gray lactation
average by about 7 pounds. The brown arrow on the left points down by
seven pounds. Observed production was about 12 pounds below
expectations, so after removing the minus seven lactation stage effect,
the unexplained residual, or "management effect" is minus 5 pounds.
Seven weeks later at 70 days-in-milk, the lactation
stage effect is plus 14 pounds (the brown
arrow on the right). The black dot for observed production is again
about five pounds below the model curve.
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The chart averages the arrows for each lactation in
the group for a each date on the chart. |
