Engineering Math Software


Calculating the Square Root of my Fear of Blood

by Luke Westbrook

Let me be clear: I do not do blood. It’s not my thing. In high school, I was taken from the third floor to the first in a wheelchair for a hand wound because the sight of my own blood made me feel so faint. And that, dear readers, is why I am not a doctor.

Blood Matters

But blood is important. Obviously. Understatement of the year. What I mean is that there are a lot of people who need blood transfusions for life. So the Red Cross does what it can to maintain the necessary blood supply through voluntary blood drives. One of my favorite things about working for a company like PTC is that it really is a company that cares. Every month or so, PTC hosts a blood drive at our headquarters in Boston. Thus, I arrived at my impasse: blood (and needles!) threatens the security and composure of the undigested food in my stomach, but I also want to be part of the life-saving effort of the Red Cross.

Fortunately, my conscience won out, and I have been trying to do better at donating blood. I donated for the first time a little over a year ago, and it took another year to gather the courage to donate again last January. In that time, I found out that my blood type is in high demand, so when I went in to donate for my third time last Thursday, I was asked to give a double red blood cell donation, called a Power Red donation. Here’s how it works: a normal whole blood donation is the simple removal of one pint of blood, but with a Power Red donation, one pint is removed and pumped into a centrifuge that separates my red blood cells from my plasma. Then, that plasma, plus some saline, is pumped back into my arm. After that is done, the process is repeated, such that the red blood cells from two pints of blood are collected, but I actually retain the fluids since they are pumped back into me.

The Easter Connection

Okay, here’s the point: This whole process, with the centrifuge, was intriguing to me, so as I was trying to think of an Easter blog post, blood separation came to mind. I know Easter and blood sound like very unrelated topics, but think about it. Easter, Passover, springtime? These are celebrations of new life, dead winter trees blooming into vibrant flowers. And in similar fashion, the efforts of the Red Cross regularly offer new life to people. It’s not painting Easter eggs with Mathcad, but I don’t think it’s too much of a stretch.

All About the Centrifuge

Now that we’re on the same page, let’s talk centrifuges. Undoubtedly, you already know what a centrifuge is and does, and you probably know more than I do about them, but let’s start with the concept. A centrifuge uses centripetal force to separate fluids or particulates of different densities. When traveling in a circle, the denser fluid or suspended particles have greater inertia and therefore require greater force to prevent them from traveling in a straight line. Since radial acceleration during circular motion is given by

Acceleration during circular motion equation

we know that the radial force is directly and linearly proportional to the radius from the axis of rotation. Thus, the denser suspended particles will move toward the outer radius of the test tube or spinning bowl or whatever, where the centripetal force is greatest. This is precisely what happens with blood. The red blood cells are denser than the plasma, and so travel to the outer radius of the centrifuge during the separation process, allowing them to be removed from the plasma before the plasma is returned.

Spinning Math

My investigation of centrifugation brought me to an equation that calculates the sedimentation rate of a spherical particle, which is the speed that a suspended particle travels through the medium. Now, a red blood cell is more of a disc rather than a sphere, but it’s a better assumption than a spherical cow.

Equation for sedimentation rate of a spherical particle

I found the necessary values from various sources. Note that I had to create a new unit:

Creating a new mathematical unti

Also, the equation uses g, the acceleration due to gravity, meaning that it is assuming that gravity is the only active force here. So we can calculate the sedimentation rate of red blood cells without centrifugation:

Equation for calculating sedimentation rate without centrifugation

Those cells are probably needing to travel at least a couple of inches, so that’s not going to work. I’d rather not have to wait a few hours for this. What we can do is simulate a greater “g-force” by spinning the blood. Thus, g will be replaced by ar. What we need, then, is the angular velocity of the centrifuge and the radius of centrifugation. Now, I’m not 100% sure what centrifuge was used, but it looked a lot like the Haemonetics® PCS®2 Plasma Collection System. Looking at the specifications, the centrifuge speed can range from 3000-8000 rpm. So I’ll use the halfway point.

Equation showing halfway point of centrifugation speed

Now, what’s the radius of centrifugation? To be honest, I scoured the internet for that information, but simply could not find it. So unfortunately, I’m going to have to eyeball it. Recalling the size of the centrifuge, I would estimate that the radius was about 3 inches, leaving us with

Equation showing radius of centrifugation

Over 2500 times the acceleration due to gravity! Plugging that into the sedimentation rate equation gives us the following,

Equation adding acceleration into sedimentation rate

which is probably pretty reasonable. If

Checking results with quick comparison to known measures

and the draw speed (per the machine specifications) is 40-100mL/min, it would take a few minutes to retrieve a pint of my blood, and then a minute (or two just to be safe) to separate the plasma. And I was told that the draw time was about 7 minutes, so I would say that we are at least reasonably close to the actual sedimentation rate during the donation process. More defined numbers would definitely help get closer.

Maybe I’ll ask next time I give blood. Or bring a tape measure.

[Oh, and I can’t write this post without encouraging readers to make donating blood part of how you celebrate Easter, Passover and springtime by finding a Blood Drive near you!]

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