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The Physics of How Microbubbles Sort Cells

January 2016 Technical

The Physics of How Microbubbles Sort Cells

I’m going to devote this first post to floatation, which is the driving principle behind our cell sorting technology. It’s hard to imagine many other physical phenomena that humans have more of an intuition for than floating and sinking. The exact same rules of floating and sinking apply to a stem cell being lifted from a sample on the back of an Akadeum microbubble. In principal, it is as simple as pair of water wings that keeps a 2 year old afloat.

Understanding these rules lets you anticipate important performance features of separation systems like ours. It allows for questions to be answered like

  • How long will it take for my cells to get to the top of the tube? (Not long)
  • Can microbubbles lift a spheroid? (Probably)
  • How many microbubbles do you need to float a large cell? (One)

The answer to these questions may have a big impact on the success of your cell separations, and so floatation is where we will start.

Why microbubbles float and what determines how fast they rise

Archimedes

Archimedes of Syracuse

Archimedes of Syracuse gets credit for figuring out the means of determining whether an object in a liquid will sink or float. You can read more about that here: Archimedes’ Principle.  In short, if the density of an object is less than the density of the liquid it displaces, the object will float. Otherwise, the object will be either neutrally buoyant or will sink.

Sir George Gabriel Stokes

Sir George Gabriel Stokes

For us, whether an object floats or sinks is not enough.  Rather, we want to know things like how quickly a microbubble will move through a user’s sample, especially if that bubble is carrying a cellular payload.

For that question, we turn to Sir George Gabriel Stokes.  Stokes is one of the fathers of modern fluid dynamics, which may in part explain his stern expression.1 Relevant to microbubbles, his law states that the features that matter are these:

  1. The difference in density between the bubble and the surrounding fluid. The greater the density difference, the faster the upward velocity;
  2. The radius of the bubble. At a given density, the larger the bubble, the faster it rises;
  3. The gravitational or centrifugal force applied. For bubbles separating cells on the benchtop, only gravity acts on the bubbles.  By placing the sample in a centrifuge, g can be increased many fold, for good or for ill depending on your application.

These three elements can all be controlled experimentally through your choice of microbubble and your use of a centrifuge.  For completeness’ sake, a fourth feature, the sample viscosity also is important but is not readily modified by most users (including us).

As for our currently available products, terminal velocity under gravity alone (i.e., no centrifugation) produces a theoretical rising velocity of ~ 1 mm/min.  In actual practice, higher velocities can be achieved through mechanisms I’ll discuss in a future post.


Nuts and Bolts:

Stokes’ Law addresses a special (but highly relevant) circumstance in which fluid is moving around a sphere in a nonturbulent way.  As a user of cells, you might be able to violate the nonturbulent assumption, but you’d probably significantly damage your cells in the process.  Under this assumption, the buoyant force experienced by the microbubble, which we’ll call FB, can be expressed as follows:

Formula - Buoyant Force of Microbubble

where ρp is the density of the microbubbles, in kg/m3, ρf is the density of water, g is the acceleration due to gravity or centrifugation, in m/s2,  and r is the radius of the microbubble in meter.2 To predict the rate of rise, the buoyancy is not enough – you also need to include the drag experienced by the bead as well:

Formula - Drag experienced by Bead

where μ is the dynamic viscosity of water, typically ~ 10-4 Pa-s, and V is the velocity of the beads in m/s.   The terminal velocity, which is the highest speed with which a particle of a given density can rise in a fluid of given viscosity, can be reached algebraically by setting FB = F (the condition under which any additional buoyancy of the bead is completely counteracted by additional drag) by solving for V :

Formula - Solve for Velocity

This equation sums up a lot of what you’ll need to have a working intuition about our microbubbles, specifically that the velocity at which bubbles will rise in your sample is:

  • Linearly related to the difference in density between the bubble and the surrounding fluid,
  • Linearly related to the gravitational (or centrifugal) force applied to the sample, and
  • Proportional to the square of the bubble radius, meaning that small changes in bubble dimension can have outsize effects on rise time.

The other physical feature in these equations, the viscosity of the sample, is not for most users a controllable feature, but completeness’ sake it does impact microbubble velocity.

For our microbubbles, in the absence of an attached cell, the typical maximum rising velocity is just shy of 1 mm/minute.    In practice, for reasons that we’ll explore in future posts, the observed velocity is substantially more.


1 There are plenty of more details.  If you’re interested in understanding more of the details, check out the Wikipedia entry on Stokes Flow.

2 Usually, we’ll stick to meter-kilogram-second and SI units.  For a great reason to stay in one measurement system, see here: Mars Climate Orbiter