Low-tech mechanical measurements

# Introduction

There are many reasons to study the mechanical properties of small, soft materials. For instance, mechanical measurements can help one study the physics of tissue movements during development, or how cells feel and respond to their mechanical environment.

Fortunately, there are several methods that someone, even a hobbyist without a big research budget, or much technical experience, could use to measure mechanical properties of materials such as cells, embryos, and soft tissues. These cheap, low-tech methods can be particularly helpful for biomechanical measurements on small soft things, since these are often difficult to work with using more high-tech devices.

# Hertz contact:

 Fig. 1. Hertz contact method with a ball. Young's modulus (E) is the ratio of tensile stress (σ) to tensile strain (ε) in elastic materials. For a non-adhesive ball resting on a thick layer of elastic material, the Young's modulus of the layer is approximately: (1) $$E=3(1 - ν^2)f/(4d^{3/2}r^{1/2}).$$ Where d is indentation depth, f is the weight of the ball (in the medium), r is ball radius, and ν is the Poisson ratio of the material. The Poisson ratio relates strain in one direction to strain in another direction. Often one can pick a reasonable guess for the Poisson ratio (e.g. 1/2).

This technique is particularly simple for sheets of elastic materials1. For this technique one places a sphere of known density (perhaps something like the ball from the end of a ball-point pen) onto a surface, and then measures how deep the indentation is2. Cell biologists have used this to calibrate the stiffness of gels when looking at cells' ability to sense the stiffness of their environment [1].

# Micropipette aspiration:

 Fig. 2. Micropipette aspiration applied to a sea urchin blastula-stage embryo (Lytechinus variegatus). Left: Image through the microscope with the embryo partially sucked into the micropipette. The arrowhead indicates tissue pulling into the micropipette in response to a 100 Pa suction.3 Right: The estimate of the creep compliance (which describes how soft a material is as a function of time after a stress is applied) based on the length of the tissue pulled into the micropipette (arrowhead) as a function of time since suction began.

This technique is quite useful for measuring the mechanics of cells and soft tissues, going back at least to Mitchison and Swann's (1954) work with sea urchin embryos[2], but is still widely used today[3][4]. The basic approach is to suck on the sample of interest through a narrow channel (typically a micropipette, but not necessarily). One measures how far the sample sucks into the tube at a given pressure and time. The method itself can be extremely simple. The tube has to be thin and clear, with a clean, flat end at the sample surface (which also has to be clean, non-porous, and flat enough to form a seal). The pressure can be controlled just by raising and lowering a column of water continuous with the end of the tube. One can, with appropriate analysis techniques and careful experimental design, get lots of information out of the experiment[5][6][7][3][4]: the viscoelastic properties of the material, or whether it behaves as a liquid with surface tension, and much more. However, even very simple approaches can tell one a lot about the mechanics of a system[2][9][8].

# Bending beams:

 Fig. 3. Bending beams for measuring mechanical properties. Starting with a long thin elastic beam (e.g. a glass rod; top), one can measure the stiffness of the beam by measuring its deflection when known weights (e.g. pieces of bent wire) are hung from the tip, with the beam oriented horizontally, so it bends with the added weight (middle). When one then presses some material or structure with unknown properties (lower three panels) by moving the base of the beam (purple arrow), one can press against something with unknown mechanical properties. One can calculate the force (F) applied to the material from the beam's stiffness, k, and the tip deflection, x. From the displacement of the material (d) and applied force, one can calculate properties (e.g. stiffness, elastic modulus, etc.). There are several possible configurations. Using just the beam, one can press against another long, beam-like structure to measure its stiffness. By gluing a flat plate to the tip of the rod, one can do compression tests on blocks or spheres of material. By gluing a short rod perpendicular to the beam's tip, one can indent a material. Note that for the 2nd and 3rd configuration, one would want to do the calibration after gluing the plate or rod to the beam.

This is not so much a single technique as a trick that makes measuring very small forces possible with low-tech, simple equipment. Examples include measuring the stiffness of the frog embryonic notochord[10] or sea urchin embryo[11]. Slightly more sophisticated approaches have also been used for other embryonic tissues[12], and the same basic principle is used in Atomic Force Microscopy. The end of a long thin elastic beam will deflect in a simple manner. If F is a force pushing the tip of the beam to the side, the deflection (x) is given by k*x=F, where k is the spring constant.

If you know k, then you can press on some object with the tip of the beam (again pressing from the side, not straight on), and measure x to calculate the force (F) on the object. By measuring how the object deforms (d) in response to that force, you can calculate its stiffness (from the same equation, or something more appropriate to its geometry). Alternatively, if the object pushes on the tip of the beam, it will deflect the beam's tip, so you can measure the force the object exerts on the beam.

One can make beams that can be used to measure the stiffness of small, soft materials (e.g. biological tissues, cells, gels, etc) by heating a thin glass rod or fiber and pulling it out to the desired length and thickness. Depending on the stiffness one wants, one can also use the glass fiber from a fiber optic cable. The convenient features of glass rods and fibers for making measurements of small forces are that they are stiff and elastic (spring-like), one can easily make them, and one can make them very, very sensitive. All one needs to make them more sensitive is to make the rod longer or thinner (so k goes down). Then one can use the fiber to press on something else. The key thing is to calibrate the fiber so that you get k. One approach is to hang known weights off its tip (e.g. a skinny metal wire[11] or plastic sphere and measure the deflection of the tip [10]).

# Literature Cited

1. Lo CM, Wang HB, Dembo M, Wang YL (2000) Cell movement is guided by the rigidity of the substrate. Biophys J 79: 144-152
2. Mitchison JM, Swann MM (1954) The Mechanical Properties of the Cell Surface: I. The Cell Elastimeter. J Exp Biol 31: 443-460.[http://jeb.biologists.org/content/31/3/443.full.pdf+html?sid=c8ecd497-dcb9-4311-a99c-65cc4d458f51]
3. Zhou EH, Quek ST, Lim CT (2010) Power-law rheology analysis of cells undergoing micropipette aspiration. Biomechanics and Modeling in Mechanobiology 9: 563-572;DOI 10.1007/s10237-010-0197-7.[http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Citation&list_uids=20179987]
4. von Dassow M, Strother JA, Davidson LA (2010) Surprisingly simple mechanical behavior of a complex embryonic tissue. PLoS One 5:e15359; DOI:10.1371/journal.pone.0015359.[http://www.ncbi.nlm.nih.gov/pubmed/21203396]
5. Sato M, Theret DP, Wheeler LT, Ohshima N, Nerem RM (1990) Application of the micropipette technique to the measurement of cultured porcine aortic endothelial cell viscoelastic properties. J Biomech Eng 112: 263-268.[http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Citation&list_uids=2214707]
6. Evans E, Yeung A (1989) Apparent viscosity and cortical tension of blood granulocytes determined by micropipet aspiration. Biophysical Journal 56: 151-160.[http://www.sciencedirect.com/science/article/B94RW-4V8RSPC-G/2/8329a7079d4ef6beb05c4fb197956cf3]
7. Hochmuth RM (2000) Micropipette aspiration of living cells. J Biomech 33: 15-22.[http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Citation&list_uids=10609514]
8. von Dassow M, Davidson LA (2009) Natural variation in embryo mechanics: gastrulation in Xenopus laevis is highly robust to variation in tissue stiffness. Dev Dyn 238: 2-18;DOI:10.1002/dvdy.21809.[http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Citation&list_uids=19097119]
9. Takata H, Kominami T (2001) Ectoderm exerts the driving force for gastrulation in the sand dollar Scaphechinus mirabilis. Development Growth & Differentiation 43: 265-274.
10. Adams DS, Keller R, Koehl MA (1990) The mechanics of notochord elongation, straightening and stiffening in the embryo of Xenopus laevis. Development 110: 115-130.[http://dev.biologists.org/content/110/1/115.abstract]
11. Davidson LA, Oster GF, Keller RE, Koehl MA (1999) Measurements of mechanical properties of the blastula wall reveal which hypothesized mechanisms of primary invagination are physically plausible in the sea urchin Strongylocentrotus purpuratus. Dev Biol 209: 221-238
12. Zamir EA, Taber LA (2004) On the effects of residual stress in microindentation tests of soft tissue structures. J Biomech Eng 126: 276-283.[http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Citation&list_uids=15179859]