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Brent Cullimore

I don’t mean “blood and guts” metaphorically. But I don’t mean it literally either. Best to call it analytic “blood and guts.” Plus, we’ll skip the guts for now, so you can relax.

And as you’ll see, I could have called this post “City Streets and Sewer Pipes” instead, but I’ll bet you wouldn’t have clicked that.

First, some disclaimers. To the eternal disappointment of my mother, I am not a medical doctor. My brother took that bullet for me. So while I’m fascinated by evolutionary biology and physiology, I can’t even be trusted to apply a bandage straight. As a result, almost every result you’ll see here is almost surely untrue. In fact, they’re perhaps so far removed from reality that you’ll be able to see that for yourself, whether you disappointed your mother or not.

Final numbers aren’t the point here.

Instead, I hope you find the math problem itself strangely attractive (and yes, that was fractal pun).

Where would you like these corpuscles, ma’am?

You are Evolution. You have come up with metazoans, so congrats on that! But to make big metazoans, you’ve got to oxygenate all those thousands of cells using micron-sized piping ... but without spending half of their caloric intake on pumping energy.

As you’ve done countless times before when you’ve hit problems like this, you rely on a dendritic network. A big aorta splits into arteries, each of which in turn splits into small arteries, which split into arterioles, which finally spit into billions of capillaries that are about 8 microns in diameter.

Conveniently, but not coincidentally, 8 microns is also about the size of those blood cells you came up with. They can now line up in the capillaries like a tiny train, making deliveries and pick-ups without ever stopping. At these small scales, you’re back to the “how to feed and clean a single cell using only gas exchange” problem that you solved so many billions of years earlier.

It is so tempting to toss the used corpuscles and collected waste overboard right here. Not that you care about the unsightly mess that would result from an open system. But you do care about efficiency, so only closed systems for you (well, for circulatory systems anyway). Back goes the blood into venules, which collect into small veins, which collect into veins, which then collect into the single vena cava that flows back to that positive displacement pump you call a heart.


You’re not done yet, Evolution. You still have to decide what size each of these little branches should be, and how many of them there should be at each level. You still need to keep your metazoan's appetite in check, so you will have to minimize the pressure drop it takes to get the required flow to each hungry and (diffusively) pooping little cell.

I know you’re fond of really big animals like blue whales and elephants. Economies of scale and all that. But that just means a bigger heart and more flow rate through that heart. You can’t solve your pressure drop problem by keeping the heart small and making the distribution network huge. Yes, a body with 10 billion capillaries would have half the pressure drop that one with 5 billion capillaries has, but where to put them all?

Instead, you’ve got to split and re-split the blood flow without “wasting” volume … you have to fill each fixed body volume with a circulatory system that nourishes it without wasting pumping power. The volume is a key constraint on how your tree is shaped.

Space Trees

Well, space-filling trees anyway, though that term probably doesn’t make much sense either.

How do you make a branching network like a circulatory system that efficiently fills the volume provided? A fractal pattern, where each level you zoom in on resembles the last level you were looking at, turns out to be an optimal solution.

It also turns out that evolution has had to solve this type of problem over and over. In fact, that’s why this post is called Part 1: lungs are also an example of a space-filling tree, though not of a closed or recirculating system. The fact that velocities get close to sonic in most pulmonary passages during a cough or sneeze is fascinating enough to guarantee that someday I’ll complete a dynamic lung model. (A design requirement for a lung, if you think about it, is to be able to develop enough shear force to clear out the crud every now and then. With the opening politely covered, of course.)

City streets and sewer lines also fall into this sort of pattern, though with intentional design of course.

When systems follow these types of patterns, their growth patterns and growth limits can be described by exponential scaling relationships, as described in Scale by Geoffrey West.

So … he’s gonna model this, isn’t he?

Correct. My headstone will be “He came, he saw, he modeled.” Scratched out below it will read “If not well, at least persistently.” You were warned.

You can delve deeper in this sample problem, but otherwise here’s the nickel summary.

A highly non-geometric system such as this …

… was nontheless drawn in a CAD system like this (using the magic of symmetry-exploiting Duplication Factors):

A volume constraint was added, and pressure drop was minimized. The size of the capillaries was fixed, but the numbers of blood vessels at each level and the size (diameter) of them were optimized.

To preserve symmetry, the number of veins was set to be the same as the number of arteries, the number of venules was the same as the number of arterioles, and so forth. The size of each type of vessel was allowed to vary, however, since the arterial system has much thicker walls (being on the high pressure side) than does the venal system. The "capacitance" annotation in the previous artwork refers to veins' ability to expand or contract ... they are the fluid reservoir of the system. This expansion wasn't modeled, since only steady states were calculated, but the thin-walled nature of the venal system's vessels was included.

I started with a best guess as to the number and size of each type of duct (I just can’t call them vessels any more … that means something different in my usually bloodless world). The numbers below are the number of upstream branches, not the total number. For example, there are 40*60=240 small arteries initially, each artery ending in 60 small arteries. Initially there are 40*60*24000*300 = 17.28 billion capillaries total.

You can see from both the original values and the optimized values that these aren’t true fractals. Self-similarity at all levels goes out the window when you see how many arterioles (and venules) there are compared to the branching at neighboring levels.

I don’t know how to reconcile that discrepancy. It might have to do with how special capillaries are, since arterioles and venules are the ones connected to them. (I also found lists that showed twice as many venules as arterioles, and I don’t know how to reconcile that either, so I’m getting used to the feeling.) Certainly neglecting side branches (branching only at the ends of each duct) is questionable, but I didn’t know how else to characterize the branching sensibly.

I also suspect that the real system is more complex than this four- or five-level hierarchy supposes. Unless you can look at a piece of wood and can confidently tell a “branch” from a “limb” from a “twig,” I’m guessing there is a certain arbitrariness to the naming scheme.

You can see that the optimization favored larger diameters in the venal system than in the arterial system, which is realistic. But that might be the only success I can declare since the final optimized pressure drop was embarrassingly low. I know I skipped the hydrostatic head of gravity, but I’m guessing too many other things got skipped as well. One skipped reality is non-Newtonian flow in the capillaries, where the existence of tiny corpuscle trains mean they are better characterized as rails than ducts.

So I learned to love pie charts, which up until now I thought were just for business people. I learned that you can show pretty colors without admitting that the pie on the right should be very very tiny compared to the pie on the left. Here’s the pre- and post-optimized ways in which each tree “spent” the total pressure drop “budget:”

He came, he modeled, but he sure didn't conquer

Whether this was a success or failure, the math of optimized space-filling trees is both fun and important.

Still, I better stick to a water supply system or sewer system model next time, since I can take my tape measure with me. I need to leave this medical stuff to my brother.