If you Google “number of cells in a human body” the top hit is an oddly specific 37.2 trillion. Other estimates go up to 50 trillion.
When I do the math, I used a volume of a body = 0.07m^3 assuming an average mass of 70 kg.
The average cell size diameter is 20 x10E-6 micrometers.
If I assume a cell is spherical, I calculate the number of cells at about 20 trillion and if I assume it is a cube, then I get 8 trillion.
What mistake am I making?
With your username, did you know that
I’d not trust “average cell size diameter” unless it’s strongly weighted toward red blood cells.
The cell estimate is for human cells. Bacterial cells in the human body are estimated at 30 to 100 trillion, not even counting archaea. Frightening.
This article has a chart listing the size of different kinds of human cells. There is a huge variation, from 30 µm[sup]3[/sup] for sperm cells to 4,000,000 µm[sup]3[/sup] for oocytes. There’s also a nice histogram of cell volume vs. frequency, but it’s for a mouse rather than a human. Still I’d expect the human histogram to look roughly similar. For the mouse, almost all cells are between 500 and 2500 µm[sup]3[/sup], with the peak around 1000. That’s pretty far below the OP’s estimate of a 20 µm diameter average, which would have a volume of about 4100 µm[sup]3[/sup].
As others have noted, the size of cells varies widely. In addition, there is a lot of “I heard it so it must be true” urban legendeering in physiology where some estimate becomes widely accepted even if it has little basis in fact. And example of this was the almost universal assumption that the human brain contained 100 billion neurons, and that all brain neurons are essentially identical with the only distinction being their organization, and that humans have brains that are oversized by neuron count as compared with body mass (thus having a high encephalization quotient). Suzana Herculano-Houzel questioned this assumption and found that the estimate of number of neurons had no solid basis, and embarked upon a project to measure the number and size of neurons in different part of the human and other mammalian brains by the innovative process of turning brains into “soup” (dissolving the cell membranes) and filtering for the surviving nuclei, discovering that there were 86 billion neurons which is in line with the brain to body mass ratio of other primates and different from other orders of mammals. She discovered that the key difference is the number of neurons int eh cerebral cortex and a diet consisting of more protein than other primates which which nutrients are made more bioavailable by mechanical processing and heating (cooking). See The Human Advantage: How Our Brains Became Remarkable.
Also, most spheres are not even roughly spherical, and none are cubical, notwithstanding all of the volume of the human body that is not constructed of living cells, e.g. the extracellular matrix and hosted microorganisms. Trying to make a scaling estimate based upon an averaged cell volume and packing of spheres or cubes is going to easily result in order of magnitude errors in estimation.
Stranger
In my science class, we calculate the size of cheek epithelial cells and consistently get around 50 micrometers as a diameter. This forms the basis for the calculations in the OP but I reduced this size because I knew that blood cells were 8 micrometers and I knew there were a lot of them.
When someone quotes that there are 30 trillion blood cells, I am skeptical. When I do the math myself, I don’t come near to that.
A fact they never tell you about in geometry class.
Why is it frightening? We are in symbiosis with our non-human microbes. Together, we and they form a habitat, and without them, we would die (but at least our armpits would not smell as bad in death).
That should have been “…most cells are not even roughly spherical…”
There is the old joke about assuming a horse is a sphere to make the math eaiser which probably applies here, but in general physiology and molecular biology are a lot more complex, and less well understood than you’d guess from just reading textbooks on the topics.
Stranger
Armpits smell bad when you’re alive, and sweating. After death, it’s your intestines (and their contents) that smell the worst, especially when the gases start to burst them open.
Whether a cell is a cube or a sphere makes a factor of 2 difference in the end result.
If cells are packed more loosely or if there is a lot of extra cellular material occupying the volume, there would be even less cells.
I don’t see how there could be “order of magnitude errors”.
The number of blood cells are probably the easiest thing to determine empirically, since it can be calculated from a blood test. Adult males have on average between 4.35-5.65 trillion cells/L. Average blood volume is about 5 L, so around 25 trillion cells would sound about right.
But you did come near to that. 20 trillion isn’t really very different from 37 trillion at this level of estimation, especially when one of the inputs to your measurement is a diameter that’s being converted to a volume (thus amplifying any error in your diameter estimate).
Yes. When I calculate a red blood cell as a flat cylinder rather than a sphere, I get 20 trillion red bloods cells alone!
I just needed to get some expert input and ponder the problem for a bit.