The risks of consuming excess polyunsaturated fats have piqued my curiosity as of late. Like most people in North America, I have overconsumed polyunsaturated fats in a way that is not consistent with our historical intake of those fats. I know that my body dislikes having excessive amounts of any one nutrient, because that entails having smaller amounts of other necessary nutrients. I have no idea if I’m overreacting, but I do dislike the reactive oxygen species that arise when PUFAs burn in chain reactions. It might be behind autoimmune flare-ups and inflammation. More research is needed about this, of course, but it is still a reasonable possibility.

In the meantime, how better to satiate my curiosity about these fat stores than to model how my body will slowly get rid of my excess polyunsaturated fat build-up?

Simplifications

Like in any simple mathematical model, I’m not going to try to create an exact reproduction of reality. Instead, I’m after a general idea of how polyunsaturated fats will evolve with time. To achieve this model, I will assume that:

1. Not all calories are treated equally by the body. The human body has ketone and insulin pathways, and each can behave separately. For simplicity’s sake, I will assume that the body prioritizes the burning of any carbohydrates and protein consumed throughout the day. Therefore, PUFAs will only burn if there is fat consumption.
2. No PUFAS are left unabsorbed through an incomplete digestion. This may well be, of course, an over-simplification. Feel free to adapt the model to reduce inputs accordingly. Assuming full absorption will provide the least optimistic result.
3. The subject’s calorie intake remains constant, and so does his muscle mass. There is no gain or loss of fat in this model, and calories in equal calories out.

Now that I’ve written these simplifications, an old math joke comes to mind, where a mathematics professor had to calculate the mass of a cow without weighing it. The answer went along the lines of, Let us assume that the cow is a perfect sphere…. This model is just for fun’s sake. Don’t assume that you’ll get any useful answers out of it.

Defining PUFA Inputs and Outputs

The evolution of PUFAs in the human body depends on what we consume, and what we burn.

Inputs

1. \(C\) is the subject’s calorie intake.
2. \(k_c\) is the ratio of carbohydrate consumption based on the calorie intake (and its range is 0-1).
3. \(k_{PUFA}\) is the ratio of PUFA consumption based on the non-carbohydrate calorie intake.

Discarding any carbohydrate burning in the fat-based model, the subject’s PUFA stores will increase by the following amount:

\[ dC * (1 - k_c) * k_{PUFA} \]

Outputs

1. \(C\) is the subject’s calorie intake.
2. \(F\) is the subject’s total fat
3. \(P\) is the subject’s total PUFAs.

Again, outputs discard the burning of carbohydrates in the fat-based model. The subject’s PUFA stores will decrease by the following amount:

\[ dC * (1 - k_c) * \frac{P}{F} \]

Putting it Together

Now that we’ve calculated both inputs and outputs, it’s easy to create an equation that calculates \(dP\):

\[ dP = dC * (1 - k_c) \left( k_{PUFA} - \frac{P}{F} \right) \]

Or, using a more familiar layout, it would look like the following first-order linear differential equation:

\[ P'(C) + \frac{1 - k_c}{F}P(C) = k_{PUFA}(1 - k_c) \]

The solution to the above equation is:

\[ P(C) = \frac{F^2}{k_c - 1} + k_1e^{F(k_ck_{PUFA})/(F-k_{PUFA})} + FP \]

A Numeric Solution

Let’s calculate a specific result (in the international system) to see the results better:

1. The subject’s initial fat storage, \(F\), is about 10 kilograms. There are about 9 calories in a gram of fat, so that makes it about 90,000 calories of fat.
2. The subject’s initial PUFA stores, \(P\), are about 4 kilograms after eating a high-PUFA diet. This would be about 36,000 calories of fat.
3. The subject is eating 2,000 calories a day.
4. The subject is eating \(k_c\), about 50% of his total calorie intake as carbohydrates.
5. The subject is eating \(k_{PUFA}\), about 7% of his fat as PUFAs.

This would lead to the following solution:

\[ P(C) = k_1e^{-5.55556*10^{-6}x} + 6300 \]

Substituting \(P(0) = 36000 \) into the equation, it’s easy to calculate the constant \(k_1\). Unless I’ve made a mistake, this should look like the following:

\[ P(C) = 29700e^{-5.55556*10^{-6}x} + 6300 \]

Evolution of PUFA stores in the human body

Let’s take a look at the evolution of PUFA stores in 30-day intervals:

And, for visual people, here’s a graph:

As you can see in this model, as time advances, the percentage of polyunsaturated fat when compared to total fat in the human body declines and tends to match the dietary amount that is being consumed. This model was never designed to be accurate, of course, but it does show an interesting behavior in the evolution of these fats.

If I am not mistaken, the half-life of polyunsaturated fats in the human body is about 6 months. In this case, the decline is faster. But then again, I did assume a very low intake of polyunsaturated fats for the model and the numeric solution. Most people do not keep their polyunsaturated fats below 5% of their total daily calorie consumption.

NB: I have not double-checked my method or my answers. Most recent grads who have taken fancy math classes will probably remember this better than me. I’m only doing this for fun, not to model reality. If you find a mistake in my method or calculations, make sure to tell me. Thanks!