Minor note; ”...a factor of 4 increase in cell density means that the percentage of bioreactor volume that is now cells would increase from 17.5% to 70%, very close to sphere packing density limits.”
Sphere packing density limits shouldn’t be relevant for cells, which are not rigid—they can deform to squish together, unlike in the sphere packing problem. (The problem of getting nutrients distributed is a different issue, of course.)
Thanks for the comment! The tl;dr is that getting nutrients distributed is a closely related issue of “viscosity,” where sufficiently thickly packed cells will have effectively ~0 fluid flow, and that the theoretical upper limit for cells (~infinite viscosity at ~0.65) is close to and slightly lower than sphere packing limits.
Below is some notes from being nerd-sniped by this comment, no guarantees that it’s at all good or useful. (Note that I certainly didn’t know things to that level of granularity before your comment, so it was helpful for you to do this spot check!)
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I agree that sphere packing is a simplifying assumption, as cells are both deformable and not spheres. However, it’s not a crazy one to start with, and I think the practical bounds are much lower.
The problem of getting nutrients distributed is a different issue, of course
Actually, a closely related issue is viscosity, or how difficult it is for fluids to flow through something. The theoretical upper bound is when viscosity approaches ∞, where you can’t get any nutrients at all.
My impression is that the question of the theoretical upper bound of packing with soft objects but other constraints like shear in a viscous solution has been one that interested CHO scientists and other people for over a century, arguably starting with literally Einstein.
Humbird uses φmax=0.65 as the theoretical upper limit for soft spheres as footnote 35 on page 14, citing a PhD thesis. The thesis lays out the problem pretty cleanly (emphasis mine):
There must come a time, as more and more particles are added, which gives continuous three-dimensional contact throughout the suspension, thus making flow impossible, i.e. the viscosity tends to infinity. This will be also correlated with the existence of the yield stress discussed later.
The volume fraction at which this happens is called the maximum packing fraction # and its value will depend on the arrangement of the particles. Maximum packing fractions thus range from approximately 0.5 to 0.75. The maximum packing fraction, as well as being controlled by the type of packing, is very sensitive to particle-size distribution and particle shape.
The thesis then gives the relevant numbers in pg 81-82:
As in the case of suspensions, we define a maximum packing fraction # (which is usually 0.64 or even 0.74 for solid spheres in a face-centered-cubic crystal), depending on cell elasticity, i.e. their compactness [Quemada 1998]. Due to the presence of soft spherical cells, it is expected that the value of # will be in this range. # is determined using the reduced viscosity plot as a function of ( at a shear rate of 10 2 s −1 , = 0.0013 Pa.s the solvent viscosity). In our case, this data is found to match the well-known equation proposed by Krieger and Dougherty [Krieger 1959], this providing the value # = 0.65 (Fig. 4.5).
The surrounding thesis has a lot of science that I don’t feel confident in evaluating, and math that I think (perhaps arrogantly) that I am capable of evaluating, but (unfortunately) like many other problems I’ve encountered in this project, I chose not to.
Obviously I think there’s a reasonably high chance that numbers from a random PhD thesis might be wrong, though I’m somewhat comforted by other lines of attack (including sphere packing) that bounds it at similar numbers . That said, my best guess is that small disagreements here will not change anybody’s bottom-line cause prioritization of cultured meat, though perhaps it might be sometimes relevant for cultured meat implementation.
What is the packing density of muscle cells in muscle tissue (meat)? Why not use that packing density as an estimate for the maximum possible packing density of muscle cells in a bioreactor?
This is a good question. I don’t know how high the packing density is in muscle tissue, but I assume it’s very high.
Note that the theoretical model of bodies and bioreactors are pretty different, e.g., bodies are much more structured and the problems of creating muscle cells and creating the scaffolding for such cells is coupled within the human body.
AFAIK, almost all attempts to make cultured meat is trying to solve the “create cell slurry” problem first and worry about scaffolding later, nobody (and this might be betraying my ignorance) is trying anything like making artificial fascia that grows contemporaneously with muscle cells in a bioreactor.
I’m not entirely sure why, I assume there are strong technical reasons that I don’t (yet) know for this.
Minor note; ”...a factor of 4 increase in cell density means that the percentage of bioreactor volume that is now cells would increase from 17.5% to 70%, very close to sphere packing density limits.”
Sphere packing density limits shouldn’t be relevant for cells, which are not rigid—they can deform to squish together, unlike in the sphere packing problem. (The problem of getting nutrients distributed is a different issue, of course.)
Thanks for the comment! The tl;dr is that getting nutrients distributed is a closely related issue of “viscosity,” where sufficiently thickly packed cells will have effectively ~0 fluid flow, and that the theoretical upper limit for cells (~infinite viscosity at ~0.65) is close to and slightly lower than sphere packing limits.
Below is some notes from being nerd-sniped by this comment, no guarantees that it’s at all good or useful. (Note that I certainly didn’t know things to that level of granularity before your comment, so it was helpful for you to do this spot check!)
-
I agree that sphere packing is a simplifying assumption, as cells are both deformable and not spheres. However, it’s not a crazy one to start with, and I think the practical bounds are much lower.
Actually, a closely related issue is viscosity, or how difficult it is for fluids to flow through something. The theoretical upper bound is when viscosity approaches ∞, where you can’t get any nutrients at all.
My impression is that the question of the theoretical upper bound of packing with soft objects but other constraints like shear in a viscous solution has been one that interested CHO scientists and other people for over a century, arguably starting with literally Einstein.
Humbird uses φmax=0.65 as the theoretical upper limit for soft spheres as footnote 35 on page 14, citing a PhD thesis. The thesis lays out the problem pretty cleanly (emphasis mine):
The thesis then gives the relevant numbers in pg 81-82:
The surrounding thesis has a lot of science that I don’t feel confident in evaluating, and math that I think (perhaps arrogantly) that I am capable of evaluating, but (unfortunately) like many other problems I’ve encountered in this project, I chose not to.
Obviously I think there’s a reasonably high chance that numbers from a random PhD thesis might be wrong, though I’m somewhat comforted by other lines of attack (including sphere packing) that bounds it at similar numbers . That said, my best guess is that small disagreements here will not change anybody’s bottom-line cause prioritization of cultured meat, though perhaps it might be sometimes relevant for cultured meat implementation.
What is the packing density of muscle cells in muscle tissue (meat)? Why not use that packing density as an estimate for the maximum possible packing density of muscle cells in a bioreactor?
This is a good question. I don’t know how high the packing density is in muscle tissue, but I assume it’s very high.
Note that the theoretical model of bodies and bioreactors are pretty different, e.g., bodies are much more structured and the problems of creating muscle cells and creating the scaffolding for such cells is coupled within the human body.
AFAIK, almost all attempts to make cultured meat is trying to solve the “create cell slurry” problem first and worry about scaffolding later, nobody (and this might be betraying my ignorance) is trying anything like making artificial fascia that grows contemporaneously with muscle cells in a bioreactor.
I’m not entirely sure why, I assume there are strong technical reasons that I don’t (yet) know for this.