Binge Drinking Dynamics

Raymond O'Connor
October 5, 2012

frequent misconception with science is that certain theories or models are either right or wrong. In actuality, an established theory will never be proven completely wrong, though it may be shown to be only part of a more complete picture. Take Newton’s law of universal gravitation [see…/Newton%27s_law_of_universal_gravitation]. First published in 1687, this theory of gravitation was the best working model of gravity until Einstein developed general relativity in 1916 [see…/General_relativity]. Was Newton wrong? Of course not, his theory described gravity as accurately as was possible at the time. More importantly, we still use Newton’s model of gravity as a first guess when dealing with gravity. Only if we require superior accuracy do we resort to Einstein’s general relativity. In many ways, science attempts to construct a model of reality. The goal is to develop the most accurate model as possible, but often it takes many years and many steps to get there. All of the work performed in the past will always be useful, but it is important to remember when a model is applicable.

While the above paragraph is important to understand about science, I really just included it to cover my ass. What I’m about to present is probably horribly inaccurate, totally not validated by any experiment, and possibly dangerous. But that’s fine! We can still generate a bunch of fancy graphs and ridiculous conclusions. I would like to construct a model of binge drinking in two variables. The first is of course a person’s blood alcohol level. The second, and oft overlooked, is stomach capacity. After all, you can’t drink any more if your stomach is full. We’ll also look at optimal drinking strategies. First, however, we need to look at how we can construct such a model.

Scientific Modeling

In this section, I’ll give a quick overview of the basics of scientific mathematical modeling. The basics are, formulate an equation or set of equations, and either solve them analytically or numerically. Solving an equation analytically means “by hand” or “using Mathematica,” and as it so happens, this is hard for most meaningful equations. Not only is it hard, sometimes it’s actually impossible. Fortunately, we can always resort to numerical solutions, which means breaking down the problem so that a computer can solve it.

Differential equations

A differential equation is one that relates the rate of change of a variable with itself. They are extremely important in science and pretty much all of modern physics is described by one or more differential equation. Shit, I have four of them tattooed on my biceps [see]. As such, when formulating a model, a differential equation is usually a good place to start.

As an example, think about how you piss. When your bladder is full, your piss comes out like the dickens! But towards the end it’s all drips and drops. How can we model this? Well, what we’re interested in is urine flow, which is actually a rate of change—the amount of pee pee that leaves your body per second—so we already know we need a differential equation. The rate of urine loss or flow rate, has to be related to the total amount of urine in your bladder. So we know that urine flow is fast when your bladder is full, and slows as it empties, which tells us that the flow rate is directly proportional to bladder contents:

Equation 1


The first term on the left, dV/dt, represents the flow rate. It’s the derivative that gives the differential equation its name. V represents volume, or total piss in your bladder, so dV/dt means the total amount of change in the volume of piss in your bladder per unit time. So we’re saying that this rate is equal to minus the total volume times some constant a. The minus sign is there because the total amount of pee pee is dropping, and the constant is there to make the units work out correctly. Here’s a secret: a lot of science can be done by looking at units, or what type of thing a variable represents. dV/dt has units of volume per time, and V is just volume. That means they can’t be directly equal, since the units are different, so we need a constant to fix that. We know that a has to have units of “per time.” Sounds weird, but the reciprocal of a, 1/_a_ has units of time and is known as a time constant.

We can solve this equation, in fact, this is the first one many undergrads will see in their differential equations class. The solution is

Equation 2


All we don’t know in this equation is e, b, and c. e is 2.718…, an important constant in mathematics, and when raised to an exponent is known as the exponential function. Super important function, not all that important for our purposes. We now have two other constant, b and c. The constant b is necessary because when we take a derivative of something, it destroys any constants, and c is necessary because the derivative of a constant multiplied by a function is equal to that same constant multiplied by the derivative of the function. That’s called linearity! [see] We can also check the solution, by taking its derivative and seeing if we get the right hand side of Equation (1). Since b is a constant (i.e. doesn’t change), it’s rate of change is zero. The derivative of the exponential function is itself, so we get the right answer. Figure 1 gives a plot of the solution.

Piss solution graph
Figure 1: Graph of pee volume versus time.

Impressive huh? We use a few simple assumptions and ended up with a graph of how the volume of piss in your bladder probably behaves. I say “probably” because it’s probably not completely correct, but I’m sure—just like Newton!—that it’s a good first guess. I know that it’s not exactly right for a number of reasons, the first of which is that the exponential function never actually reaches zero, meaning you’d never completely empty your bladder. You can also flex your pee muscles to make your pee go faster, which is not accounted for at all in the model. We also can’t be sure that the original equation is exactly right; there could be additional higher order terms for example. We also don’t know what a and b are, and we can only find them by urinating multiple times into a container with a stop watch. Weird, but I’m sure Neal would gladly assist with that.

Numerical solution techniques

The urine example above is the simplest differential equation anyone will likely encounter. Because of that, we were able to solve it analytically, or without the help of a computer. A lot of differential equations are much more complicated and more difficult to solve. Many of the only have analytical solutions for under certain, very specific conditions. Even when we have simple differential equations, sometimes we may have more complicated forcing functions, making a numerical solution more attractive. For example, say we did add in the effect of flexing those pee pee muscles, that would take the form of a forcing function, which we could define arbitrarily.

Let’s say we do just that and add a forcing function to Equation (1). To do that, we can simply add the following term:

Equation 3


The forcing function f_(_t) now changes the rate of your pee arbitrarily. It can be anything: say you really have to pee so you force it out real fast in the beginning, but then you relax at the end and say “aahh” while releasing your muscles. There are methods for solving Equation (3), but it actually gets pretty difficult. We can then just resort to a numerical method.

Numerical methods are really just a way of turning calculus (which differential equations are) into algebra (which a computer can easily solve). For our example, the easiest numerical method is known as a forward Euler approximation [see…/Forward_Euler_method], the details of which are not all that important. Ultimately, what we end up doing is replacing the derivative term dV/dt with a discrete approximation that a computer can handle. Forward Euler makes the approximation:

Equation 4


which, after rearranging, gives the following, time-stepping solution

Equation 5


This is known as a time-stepping solution because if you want to solve for 100 particular points in time, you have to solve the equation 100 times in sequence. h represents the time step size, or the distance between time increments. As an example, let’s say that you take a piss, and put some effort into it for the first 10% of your total piss time. The result we get now looks significantly different than before. Figure 2 shows the results in red, with the forcing function in green, and the original solution in blue. It’s as we would expect, with urine volume dropping much quicker than before.

Forced piss solution graph
Figure 2: Graph of pee volume versus time with forcing.

Now we have everything we need to proceed. Next we’ll look at constructing a model of binge drinking.

A Model of Binge Drinking

Binge drinking does a number on your body. Not only do you get shit-faced drunk, you most likely end up with an upset stomach, have to piss every 2 minutes, and might vom or black/brown out. I wanted to create a basic model of binge drinking that captures those important effects. First, blood alcohol concentration (BAC) gives you your level of stupidity, but is also related to blackouts and vomiting. Second, stomach contents (or how much garbage is in your tummy) is related to that bloated feeling, pissing, and vomiting.

Blood alcohol concentration

Blood alcohol concentration is a measure of the mass of alcohol (typically in grams) per volume of blood in your body (in the US we use deciliters). As everyone knows a BAC of 0.08 g/dL is the legal limit for operating a motor vehicle, but when do you actually reach this level? Without a personal breathalyzer, it can be difficult to know for sure. The rule of thumb is one drink per hour, but who follows that, and what BAC does that actually get you? A model of BAC should be able to help us answer these perplexing questions.

The model I will use was implemented by an undergraduate student at the University of South Florida [see…]. This student used a nice two-stage model in which the first stage gives the amount of alcohol absorbed in the stomach, and the second stage gives the amount of alcohol in the blood. The student also matched his model with experimental data, giving me confidence in its accuracy. The model can be stated as

Equation 6


Equation 7


where A is the mass of alcohol in the stomach in grams, B is the mass of alcohol in the blood, f is the forcing function (basically how much alcohol you are drinking) and k 1 and k 2 are the time constants. In the above manuscript, the author found the time constants to be k 1 =0.11 inverse seconds, and k 2 =0.018 inverse seconds. When I first tried out this model, the results were not matching other sources I found online. I adjusted k 1 to 0.25, lengthening the amount of time it requires the alcohol to be absorbed by the stomach. I attribute this change to the fact that the author was using a person taking a single shot of grain alcohol to calibrate his model. Here, I’m more interested in people slugging beers, which are more dilute. Since beer is more dilute, it makes some sense that it may take longer for the alcohol to be absorbed.

Stomach contents

Another important issue during binge drinking is how uncomfortable your stomach is. A stomach is elastic, so that when unfilled it is a certain volume, but it can grow in size to accommodate more food. As the stomach grows, you become more uncomfortable. First, however, we’ll look at a model of stomach emptying. This model will tell us how much crap is in the stomach and how fast it empties. It can be a little difficult to get this exactly right, since most models are based on food and we’re interested in beer. Beer is mostly water, but does contain some carbohydrates that need to be digested. For now, we’ll just stick with a model that was calibrated on food.

The model I found was published in 1977 and can be stated as

Equation 8


where V is the volume of food in the stomach in mL, C is a constant related to the composition of the meal (basically how much water is in it), p and n are constants to be determined, and v is the volume of the meal in a “resting” stomach [see…]

Bladder contents

Finally, bladder contents is another distress of binge drinking. The model I used here is pretty simple, I assume that any volume that enters your stomach, will eventually be absorbed into your bladder. This assumes that what you ingest is all liquid, and while not strictly true for beer, is a pretty good approximation. Especially for the garbage I typically drink. It also assumes that your stomach contents are instantly transferred to your bladder, which is not true as it takes time for excess liquid to traverse your blood stream, enter your liver and kidneys, and eventually make it to your bladder. But, the majority of this liquid does make it to your bladder, so this model should be a fairly good approximation.

Bladder contents can then be modeled by taking the subtractive term of Equation 8, and simply integrating (summing) it across all time. This approach will quickly kill you since your bladder would fill past capacity and burst, so I will assume that you take a piss at some point. Pissing will simply be modeled as setting your bladder contents to zero after the contents get past a certain threshold. This threshold will be discussed further in the following sections. The model can then be stated as (ignoring pissing for now) as

Equation 9


Results and Discussion

We now have everything we need to conduct some numerical experiments. Don’t get too excited! An important point to mention here is that again, this is a model and doesn’t reflect reality exactly. Once useful thing to do with models is to compare results using different inputs. The absolute numbers may not be accurate (which means that our constants are off), but the relative differences in behavior can tell us a lot. With that, we’ll start with a casual night of drinking.

Normal drinking

I’m going to define normal drinking as a casual night at the bar, consisting of steadily drinking six 12 oz. beers (with a alcohol by volume of 5%, equivalent to one Bud heavy can) spread over three hours. Three hours would cover 9:00 to 12:00, and let’s say you go to bed at 1:00, and get eight hours of sleep. Let’s take a look at the numbers for this scenario.

Stomach contents vs. time
Figure 3: Stomach contents for a normal night of drinking.

BAC vs. time
Figure 4: Blood alcohol content for a normal night of drinking.

Bladder contents vs. time
Figure 5: Bladder contents for a normal night of drinking.

Figures 3, 4, and 5 show your BAC versus time, stomach contents versus time, and bladder contents versus time in cyan. I added a couple of horizontal lines: in Figure 4, the red line indicates a BAC of 0.08, or above the legal limit for driving a car. Apparently, drinking six beers over three hours will not put you above 0.08, which I don’t entirely believe. I feel like if I drank six Bud heavies over three days I’d be smashed. In Figure 3, the green line indicates an average stomach capacity without distention. This means that if you stay below the green line, you’ll probably feel pretty normal. The red line indicates the maximum capacity of a stomach, or the point at which you’ll probably puke. Finally, in Figure 5, the green line indicates the point at which you feel like you have to pee, and the red line indicates the maximum capacity of your bladder. I added a pee threshold to this equation, given as a percentage of the difference between minimum pee feeling and blown bladder. For this example I set it to 20%, so you’ll wait a little bit to pee, but since you’re drinking non-competitively, you’ll probably go to the bathroom like a normal human. The model shows that you’ll have to pee eight times, though once you go to sleep you’ll probably wait longer.

What can we do to improve our boring night? One obvious tweak is to drink faster, but consume the same number of beers over the same time period. In other words, chug a beer, wait 30 minutes, chug a beer, repeat. Figures 3, 4, and 5 show this drinking pattern in blue. As we see, if we chug beers, we can spike our BAC above what we’d get by drinking slowly. It looks like we’ll also recover quicker! Win-win! In fact, the maximum BAC attained by chugging is 0.0066 higher. Unfortunately, we have to trade that for a more full stomach and quicker onset of peeing, but I think it’s worth it.

An analysis of the Beer Drinking Royal Rumble

In 2010, Eli put in a performance unmatched in my, and probably many other’s, experience. He downed 18 beers in 87 minutes, an amazing 4.8 minutes per beer. Let’s take a look at what might have been going on in Eli’s superhuman body. Here, as Eli’s an elite competitor, we’ll set the pee threshold to 100%. Figures 6, 7, and 8, show Eli’s performance. I only plotted the time during the rumble in this case so we can get a clear view of the action. First, his BAC may have been as high as 0.25. Many online sources cite 0.15 as the onset of brownout, and 0.2 to 0.25 clear blackout. Next, Eli’s stomach was most likely filled to capacity, and more. His stomach contents goes well above the red line, indicating likely stomach rupture. Finally, he should have had to piss four times in those 87 minutes, while in reality he pissed zero times. Again: ruptured bladder. Granted, the constants used in our models are probably not correct for this case, and should be calibrated for Eli, but this still ranks among the most amazing drinking performances in history.

Stomach contents vs. time
Figure 6: Eli’s stomach contents at BDRR 4.

BAC vs. time
Figure 7: Eli’s BAC at BDRR 4.

Bladder contents vs. time
Figure 8: Eli’s bladder contents at BDRR 4.


Science is a process of careful study, model development, and subsequent refinement. This process can be applied to anything, no matter how ridiculous. Of course we always strive for accuracy, but even with out absolutely correct numbers, we can compare effects, such as constant drinking versus chugging. Here, we constructed a three-part model describing the most important aspects of a night of heavy drinking (aside from embarrassment and hangover depression). The most important being BAC, though secondary effects, stomach capacity and bladder capacity, certainly affect your night. This information isn’t meant to encourage you to consume six beers in three hours and assume you can drive home. I’m pretty sure that’s wrong. But now we can know for sure that chugging fucking rules. Bring on the shotguns!
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