A band presses where the limb turns

A tube of one size presses the calf harder than the ankle worked up a leg station by station, and at each station it used one law: a band presses with its tension per unit height over the limb’s radius. The essay said plainly where that stops. A shin has a ridge of bone where the radius is small and the pressure concentrates. None of that is computed.

This essay computes it, and the ridge turns out to be the smaller half of the story.

A limb has no radius. It has a shape, and a band wrapped round a shape presses each point of it with the band’s tension times the shape’s curvature at that point. Where the limb bends sharply the band presses hard; where it is flat the band presses nothing at all; and where it is hollow the band does not touch it. The number every compression garment is specified by is what that distribution averages to, and it is the pressure at almost no point on the leg.

Laplace’s law is local #

The law every compression calculation uses is older than its garments. A membrane under tension T that curves with curvature κ pushes on whatever is inside it with a pressure Tκ — the same law that sets the pressure in a soap bubble, the tension in a balloon’s wall and the hoop stress in an inflated cylinder.

On a round limb the curvature is one over the radius everywhere, and Tκ becomes T/r. That is the form a cuff’s pressure was computed in, and it is correct for a round limb. The mistake is only in the next step, which nobody makes on purpose: taking the radius of a real limb to be its circumference over 2π, and quoting the result as the pressure.

The circumference is fine; it is what a tape measure gives and it is what fixes the band’s stretch. It is the division by 2π that assumes a circle.

A band rides the hull, and a tape measure does too #

A band under tension cannot follow a hollow. If the skin between two bones is lower than the straight line joining them, the band takes the straight line, because pulling it into the hollow would need a force the band is not supplying. A band lies on the convex hull of the limb — the shape an elastic band stretched round it would take — and so does a tape measure, which is why a limb’s measured girth is the hull’s perimeter rather than the skin’s.

It is the same geometry a light touch never reaching the crowns found at the scale of a cloth’s surface: a plate pressed lightly on a cloth meets only what stands furthest out of it, and a band pressed round a limb meets only the limb’s outermost points.

That makes the simplest honest model of a limb’s section the convex hull of a few circles: one for each bone or muscle mass the band rides over. It is not a survey of anybody’s leg and it is stated as illustrative. What it has, and what makes every number below exact rather than approximate, is that its boundary is made of only two things: arcs of circles, where the band presses with its tension over that circle’s radius, and straight tangent stretches between them, where it presses nothing.

Force goes by turning, pressure by curvature #

Here is the part of the arithmetic that organises everything else.

On an arc of radius r that turns the band through an angle θ, the pressure is T/r and the arc’s length is rθ, so the force the arc carries per unit height is their product — the tension times the angle, with the radius cancelled. A large gentle arc and a small sharp one that turn the band through the same angle carry the same force. The small one carries it on less skin.

And the angles round a convex shape add to exactly one full turn. So the band’s whole inward force, 2πT per unit height, is shared out among the arcs in proportion to the angle each one turns it through, and each arc’s pressure is its share over its length.

That is also why the round-limb number survives. The band’s total force is 2πT whatever the shape, and its length is the measured girth L, so its mean pressure over its own length is 2πT/L — which is exactly T over the circumference divided by 2π. The round-limb number is exact as an average and it is the pressure nowhere in particular.

The same division of labour runs through the whole of this collection. A thread is gripped where it turns: the capstan’s grip on a yarn round a crossing is set by the angle it wraps, not by the crossing’s radius, and every crossing is a force whose size is the tension times the turn. A band on a leg is the same object as a warp end over a weft, drawn at the scale of a body, and a braided hose is the same object again with the pressure pushing outward.

A calf with a ridge down its front #

Put a ridge on the front of a calf. The model is two circles: a large one behind for the muscle, 53 millimetres in radius, and a small one in front for the bone’s edge, 5.3 millimetres, placed so the hull measures 360 millimetres round — the calf of the illustrative leg. Between them the band runs straight, and those straight stretches are the flat inner face of the shin.

With the band tensioned to give 20 millimetres of mercury on a round limb of this girth:

• the ridge takes 214 millimetres of mercury — 10.7 times the specified number — on 2.4 per cent of the girth;
• the muscle takes 21.4, a little over the specified number, on 69 per cent;
• the two flat faces take nothing, on 29 per cent.

The ridge turns the band through 26 per cent of a full turn and so carries 26 per cent of the force. That is the whole of why it is dangerous: it is not that it carries much force, but that it carries a quarter of it on a fortieth of the skin.

A sharper ridge takes the same force on less skin #

The ridge’s radius is the least certain number in the model, and it turns out to matter in a very particular way.

Sharpen the ridge from 27.7 millimetres of radius to 2.2 and its pressure rises from 41 to 529 millimetres of mercury, while its share of the band’s force barely moves — from 40 per cent to 24. The force is set by the angle and the angle hardly changes, because a ridge sticking out of a calf turns the band through much the same angle whether its edge is sharp or blunt. What changes is the length the force is spread over, which is the radius times the angle.

So the pressure on a bony edge is very nearly the tension times the angle it turns the band through, divided by its radius — and of the three, only the radius is a property of the particular person. Two people with the same calf girth and the same stocking can differ by a factor of five in the pressure on their shins.

An ankle is pressed on its bones and not in its hollows #

The ankle is worse, and differently.

Model it as four circles: the broad front of the ankle, the bone on each side, and the heel tendon behind, placed so that the hollows behind the ankle bones are real hollows. The hull comes out at 220 millimetres round, the ankle of the illustrative leg.

The band presses the heel tendon at 93 millimetres of mercury, the outer ankle bone at 82, the inner at 67 and the broad front at 28. Everything else — 62 per cent of the girth — takes nothing: the hollows behind the ankle bones, which the band bridges without touching, and the flatter stretches between the bones, which it touches without pressing.

Two bones and a tendon carry 72 per cent of the band’s force on 18 per cent of its length. The specified pressure is a statement that the band is tight enough, and it says nothing about where.

Why nobody is surprised in a wound clinic #

That last result will not surprise anybody who has bandaged a leg, and the reason it will not is worth stating, because it shows the model has found something real rather than something peculiar to its circles.

The standard practice when applying compression to a leg is to pad the bony prominences and fill the hollows — cotton wool or foam over the shin’s edge and the ankle bones, and packing in the hollows behind the ankle bones — before the compressing layer goes on. It is taught as limb reshaping: make the leg more nearly a cylinder so that the bandage presses it evenly. Pressure damage over the shin’s edge and the front of the ankle is a recognised hazard of compression, and it is why.

The arithmetic here is that practice in numbers. Padding a ridge enlarges its radius, which spreads the same force over more skin — the same trade a cloth makes along its own bearing curve when a load flattens its highest points into more contact; filling a hollow turns a chord into an arc, which gives it some of the turning and therefore some of the force. Both move the section towards the only shape on which the specified number is the real one.

The flat face is the surprise, not the ridge #

Of the two results the ridge is the one anybody would have guessed; a hard edge under a tight band is obviously a pressure point. The flat face is not.

A flat surface is not pressed by a band at all, however tight. It is touched — the band lies on it — but a band pressing on a straight stretch has no curvature to press with, and all of its tension goes into pulling along the surface rather than into it. On the calf model that is 29 per cent of the girth; on a leg with a broad flat front to the shin it would be more.

That has a consequence for anything a compression garment is supposed to do on that face. If the pressure’s job is to press on the veins and tissue beneath a stretch of skin, the stretch has to be curved outward for the garment to do it, and the flatter a face is the less a band does for it — a result that follows from nothing but the geometry, and that a single specified pressure hides completely.

An oval is gentler, and still not round #

Not every part of a leg is ridges and hollows. The thick of the calf is closer to an oval, wider from side to side than from front to back, and an oval has no chords at all: its curvature changes smoothly and every point is pressed.

Even so, it is not even. On an oval whose long axis is a and short axis b, the curvature runs from a/b² at the ends of the long axis to b/a² at the ends of the short one, so the pressure ranges by the aspect cubed. A calf 1.3 times as wide as it is deep is pressed at 30.0 millimetres of mercury at its sides and 13.7 at its front and back — a factor of 2.2 from a shape that looks, to the eye, only a little flattened.

The cube is what makes it large. A shape that departs from round by thirty per cent departs in pressure by a hundred and twenty.

What this does to the round-limb results #

The cuff and tube essays computed a pressure at each station of a leg, and every one of those numbers survives — as an average over its station’s girth.

The tube that presses the calf harder than the ankle does exactly that on average, and it does it by a factor of two for an elastic tube. But the ankle is where the band’s force concentrates hardest onto the least skin, and the calf is where it spreads most evenly, so the peak pressure is higher at the ankle than at the calf even when the average is lower. On the illustrative sections, a band averaging 20 at the ankle peaks at 93; the same stocking averaging 39 at the calf peaks at about 420 on a ridge of 5.3 millimetres and at 42 on the muscle behind it.

So “graduated compression” — higher at the ankle, lower up the leg — is a statement about averages, and whether a particular stocking is graduated at the skin depends on which point of each station is asked about. On the ridge of the shin it is never graduated at all.

The model named #

The limb’s section is the convex hull of stated circles, in millimetres, scaled so the hull’s perimeter is the stated girth: two circles for the calf, four for the ankle, one for the round control. The band’s tension is set by the pressure it would give on a round limb of that girth, which is how a garment is specified, and the tension is the same all the way round — the band is frictionless on the skin. The pressure at a point is the tension times the curvature there: tension over the circle’s radius on an arc, nothing on a chord. The oval is an ellipse at the stated girth, with its curvature taken from its own formula.

The hull is found by sampling each circle’s boundary at 3,600 points and taking the convex hull of all of them; each edge of the hull is either a step along one circle or a tangent bridge between two. Two things are required of every section, not shown: that the arcs turn the band through exactly one full turn between them, and that the band’s length-weighted mean pressure equals the round-limb number for its girth to within one per cent. Both hold on every section and every ridge radius here, and the second is the claim that the specified number is an average.

What was counted #

Three sections — a round limb, a calf with a ridge, an ankle of four circles — each with its parts’ pressures, their shares of the band’s force and their shares of its length. Twelve ridge radii on the same calf, from about 2 to 28 millimetres. And one oval of aspect 1.3, read at 721 points round its girth. Every circle’s position and radius is stated in the section’s own table and none was adjusted to produce a result.

What the model cannot show #

Tissue is not rigid. A band pressing a ridge at 214 millimetres of mercury flattens the skin and fat over it, which enlarges its effective radius, which lowers the pressure — the same thing padding does, done by the leg. How far the tissue gives depends on how much there is over the bone, which varies from one person to the next more than any other number here. The model gives the pressure on an incompressible section, which is the upper bound.

Friction is left out. A real band grips the skin, so its tension is not the same all the way round — a thread gripped where it turns holds different tensions either side of the turn, and so does a band. That would move force from one side of a prominence to the other; it cannot change the total.

And the sections are illustrative. A real ankle is not four circles, and the numbers on its bones are the numbers for these circles. What does not depend on the circles is the structure: that the specified pressure is the average, that force goes by turning, that a flat face takes none, and that an oval’s range is its aspect cubed.

Who found it, and when #

Laplace’s law for a curved membrane is Laplace’s, from 1806, and its application to bandaging is old enough to be in every textbook of wound care, usually as pressure equals tension over radius with the radius taken as a round limb’s. That bony prominences take more pressure is clinical experience, and the practice of padding them and filling the hollows is standard.

What is done here is to keep the law local. Once the curvature is allowed to vary round the section, the round-limb number becomes an average, force becomes a matter of turning angle, and a flat face receives nothing — three statements the circular version cannot make, all of which the clinic already acts on.

Still open: whether a stepped shape leaves rings on the leg #

The question the tube-of-one-size essay ended on still stands. A machine changes a stocking’s size in steps, course by course, and between two steps a tube is one size over a stretch of leg that is not — a short tube pressing its thicker end harder, with a ring at the top of every step.

This essay adds a second direction to the same question. A step changes the tension; the section changes the curvature; and the pressure at a point is the product of the two. How a stocking’s steps up the leg combine with the ridges and hollows round it, and whether the worst point on a leg is on a ridge at a step or somewhere neither argument would pick out alone, is the calculation both essays point at.