### Biomechanics

Since Galileo, who made major discoveries about engineering, as well as astronomy, large amounts of concepts, special vocabulary, mathematical formulae, and computer programs have been developed to aid in the design and construction of houses, ships, airplanes, dams, bridges, sky-scrapers etc.
A subfield of biology is now developing which applies engineering concepts and methods to the explanation and prediction of anatomical structures. For example, biomechanics is the research field of the current chairman of this department, Bill Kier. My own research could be described as the biomechanics of embryos and organ formation. These subjects have many potential medical applications.

Let's start with some special vocabulary: "Stress" means the combination of tensions and/or pressures within a piece of material, which could be steel, concrete, rubber, collagen, bone or cells.

"Strain" is not a synonym for stress (to an engineer, or to a bioengineer). "Strain" means the percentage of elongation or compression of any material, caused by stresses. For example, if you pull on a piece of rubber, you produce a large strain by means of rather small amounts of stress. Pulling (or pushing) on steel results in not much strain, even for large amounts of stress. Stress, and also strain can be compression, tension, or shear, and all 3 at the same time, or even at the same place (which some people never get used to!). The ratio of amount of stress required to produce a given amount of strain (for example, one percent elongation or compression) is called the "Young's Modulus"
(Named after the person who translated the Rosetta stone, and proved that light consists of waves!)

Rubber has a small Young's modulus; steel has a large Young's modulus; and diamonds have the largest Young's modulus. It is a measure of stiffness or hardness; it is not a measure of strength; brittle materials often have very large Young's moduli; until they crack! Then the Young's modulus drops to zero. Some scientists don't understand that active contraction is not an increase of Young's modulus, which is a measure of a material's resistance to shape change, but contraction is an increase in the force tending to cause a shape change. Anyway, elastic moduli are proportionality constants; but contractions are forces.

The "shear modulus" of a material is the force needed to slide parts of the material past each other (ratio of force to angular change), and the "bulk modulus" is the ratio of compressive force to volume change. If a given material has a large Young's modulus, then its shear modulus and bulk modulus are also likely to be large. But they are independent variables; they can't be calculated from each other. Rubbery materials can have very large bulk moduli, but very low values for the other two elastic moduli. Almost all materials get a little wider when you squeeze them along one axis; and get a little narrower when you stretch them. This ratio is called the Poisson's ratio and is 0.5 for gels, around 0.3 or 0.2 for metals, and close to zero for cork. That's why corks make better stoppers for bottles than rubber does. Thin rubber sheets develop wrinkles when cells stretch or compress them, & some wrinkles form perpendicular to the forces applied, and other wrinkles form parallel to the force. A recent review article complains about this, revealing that its 2 authors don't realize that gels do exactly the equivalent thing, narrowing in axes perpendicular to tension, which makes it tricky to be sure whether a cell is pulling on one direction, or pushing in directions perpendicular to that. Poisson's ratio is named after a French mathematician of the early 1800s, and can be calculated from measurements of the elastic moduli (at least for simple materials).

For small amounts of strain, up to one or even two percent, stress tends to be linearly proportional to strain for many building materials (steel, concrete, even wood). This proportionality tends to become non-linear for larger strains (except for rubber), and at large enough strains becomes non-linear for any material. We could chose to say that the Young's moduli and other elastic moduli change as a result of non-linearity, or we could say they no longer exist. What is the proportionality constant between variables that are not linearly proportional? Hysteresis is another complication; many materials "give" when stretched, squeezed, twisted, etc. so that the restoring forces get smaller during release of imposed forces, as compared with the (larger) forces that had been needed to produce the distortion of length, angle, volume, etc. Then what are the proportionality constants, like Young's modulus? Do they change, or do they cease to be valid concepts for that situation? Specifically, can you still plug their numerical values into mathematical equations that were derived assuming exact linearity of proportionality? It turns out that small non-linearities can cause wildly different behavior of physical systems (of the kind we saw in this course's brief coverage of behavior of newX:=(oldX*(1.00-oldX)*A. Certain non-linearities of elastic proportionality make body tissues more stable; but others can cause brittleness. Strengths of materials cannot be stronger for a substance than the strengths of bonds between their atoms; but they are always much weaker than that: like a hundredth or a thousandth as strong. Metals are good at redistributing stress.

Tensor variables are to vector variables as vectors are to ordinary quantities, which are called scalars.
Scalars (temperature, chemical concentration, osmotic pressure) just have an amount at each location in space. One number can express (measure) all there is to know about a scalar variable at a location, although scalar variables often vary from place to place, and form interesting gradients.
Vectors (electric fields, patterns of flow, and also gradients!) have an amount and also a direction at each location. In three dimensions, it requires three numbers to express a vector quantity. These three numbers can be directional components relative to (arbitrary?) x, y and z axes, or they could include angles.
Variables like stress, strain, Young's modulus, and susceptibility to piezoelectric voltage generation, are all too complicated to be expressed as vectors. They all have too much ability to vary in amount as functions of direction. In the late 1800s, new categories were invented by mathematicians for the purpose of reasoning cumulatively about quantities that have various amounts of directionality. They invented vectors and tensors. Vectors are tensors of the first rank (or of the first order), which is not quite synonymous, but we will be better off regarding first order as the same as first rank, and second order as the same as second rank. Lots more progress at the biological level are going to be needed before that will become worth worrying about. Stress, strain, and curvatures of surfaces all happen to be second order tensors; Elasticity is a fourth order tensor; and piezoelectric properties are third order tensors (& perhaps birefringence is, too) All these tensor variables just listed above are symmetric tensors; they inherently have exactly the same amount in any two exactly opposite directions. Magnetism is a second order anti-symmetric tensor, which makes its value be minus one unit in one direction whenever its value is plus one in the opposite direction; that makes it behave almost like a vector; & textbooks will tell you it IS a vector.

A big obstacle is that mathematicians believe that tensors belong to them & that biologists shouldn't be allowed to use them. To them, tensors are matrixes of numbers that "transform" (meaning vary as you change directional angles) according to certain rules. If you tell a mathematician that stress or stiffness are second order and fourth order tensors, it seems to him that you are talking nonsense; equating rank and order clinches it! They would take away our car keys if they could. If confronted by territorial displays of mathematicians, try the magnetism-is-really-an-antisymmetric-tensor-that-transforms-like-a-vector gambit.
If only we could get them to treat us as "Fellows of another college, instead of undergraduates" as G. H. Hardy said about ancient Greek mathematicians. They might be impressed by this quote. If they don't know who Hardy was, they aren't worth your trouble. This is the Hardy of Hardy-Weinberg, incidentally.

Biomechanical studies of cell stiffness sometimes use micropipettes to suck out a little hemisphere of membrane and cytoplasm, measuring the ratio of suction force to distance of the surface material is distorted. A reverse method is to use either calibrated thin glass rods to punch dents into cell surfaces, or to use piezo-electrically driven "atomic force microscopes", that sweep sharp needles or crystal tips back and forth, pushing into cell surfaces. These methods measure how much force is just barely sufficient to produce how deep a dent or how big a blister. That's fine; we need measurements of physical properties to build foundations for biophysics. Notice that these methods produce scalar quantities. The properties they attempt to measure are tensors: fourth order tensors, no less! But the measurements produce some unknowably weighted average of whatever different directional components the tensor has, and produce a single number per point. Stress could be 20 units in one direction, but only 1 unit in the perpendicular direction, and poking or sucking methods may produce a "measure" of thirteen units. In the case of cell division, what ultimately matters is the ratio of tensions in perpendicular directions. But sucking or poking can only give you the equivalent of a scalar quantity. The result is endless confusion. Many actually believe that they are being more quantitative to reduce measurements to single numbers.

My own special interest is how mechanical and adhesive properties of cells can cause anatomical patterns to form "spontaneously", in the sense of being caused by properties of materials themselves, not because something tells them what to do - thus the mutual animosity with "Positional Information" advocates. I made movies of how cells respond to adhesion gradients & invented the method of culturing cells directly on sheets of rubber, viscous fluids and gels, so that directions and strengths of cell stresses could simultaneously be mapped at many locations simultaneously. Many other scientists now use this method. I also used thin glass microneedles to map locations of cell-to-glass adhesions, and found surprising spatial distributions, which others confirmed by destructive interference of reflected light. These methods show that cancer cells consistently exert weaker contraction forces, and have broad, disorganized adhesions.
Also, it turns out not to be true that cells maximize their areas of adhesions. Adhesions are often tiny.

An important concept in engineering is continuum mechanics, which means calculating (predicting) the spatial patterns in which stress and strain redistribute themselves in a given material, with certain loads, and particular shapes. Special machines are used to stretch, squeeze or bend small pieces of steel, plastic, wood etc. so as to measure their Young's moduli (& other elastic moduli), their breaking strengths, and how much you can strain them before they permanently distort (their "elastic limit"). The measured quantities are then plugged in to equations intended to predict net distributions of stress, based also on the shape of the object, building, airplane wing, engine-block etc. The goal is to predict how much building material will be sufficient to resist loads. The reason things break is almost always because stresses become concentrated into small areas. Consider what happens when you tear a piece of paper, or cut with scissors, or with a knife, or when something cracks. Why is it more difficult to tear a sheet of rubber? An excellent & readable book about this is "The New Science of Strong Materials" by J. E. Gordon.

Mathematical predictions of stress and strain quickly becomes insolubly difficult for equations.
Until recently, the best solution was some kind of analog computer or physical simulation.
One method was to build scale models out of some material that becomes birefringent in proportion to strain. Plexiglass models of cathedrals and leg bone have been made, loads applied, and the model illuminated by polarized light. Gelatin models of dams were made, and observed between Polaroid filters. Birefringence automatically maps out the distribution of strain, and thus predicts stress, telling engineers whether forces will become too concentrated somewhere in the eventual dam, or cathedral etc.
Photoelasticity is the name of this method, and there are many books about it.

A newer method for accomplishing the same goals is by computer simulations: specifically the kind called "finite element" simulations. The key method is to approximate the building, or dam, or bone or muscle as a large network of dozens or hundreds or thousands of small points in space ("nodes"), connected as lots of little triangles or squares. Imagined forces act along the sides of these triangles, pulling then toward each other or pushing them apart. You can also create the effect of pressures acting on all or many nodes. The usual purpose is to predict how pieces of material will change shape as the result of external and internal forces. I emphasize that the purpose of computer simulations is not theoretical, but experimental. By using simulation programs, you find out what ought to happen as a result of all those little pushing and pulling forces. Will the bridge collapse; if so, then where does more steel need to be added so that there is no location where forces will become too strong, too focused.

Embryological questions are somewhat the reverse of engineering. The question is how internal forces need to be distributed within a material so that it will "break", in the sense of changing spontaneously to some new shape and arrangement of parts, and will then resist changing to other shapes. In other words, how can observed anatomical shapes be crated by combinations of smaller-scale properties (cell adhesion, cell contraction, osmotic pressure, anything else you can discover by studying individual cells or tissues).
Computer programs for finite element simulations tend to be expensive, because they are valued by engineers, but people don't play games with them. So a few embryologists have learned enough programming to write their own finite element simulations. Richard Gordon and Wayne Brodland, both Canadians, have written the most professional programs of this kind. Wayne has some on his web site (http://www.civil.uwaterloo.ca/brodland/), that you can learn a lot from watching. Some Russians have also written excellent computer simulations of embryological events. Mere mortals, with zero formal computer training, have also managed to write interesting finite element simulations. You would enjoy programming, if you had a good language.