On this page we will discuss vector calculus and its relationship to a revolutionary medical technique called Extracorporeal Shockwave Lithotripsy (ESWL). ESWL is a technique for treating kidney stones, which are potentially dangerous and often extremely painful. Kidney stones are crystal-like objects which sometimes form in the kidneys, usually of people between the ages of 20 and 40, and often cause urinary obstruction, infection, and bleeding. Approximately 12% of the population will have a urinary tract stone at some time in their lives. It is ironic that the medical term for a kidney stone is a calculus because, in a certain sense, it is calculus that gets rid of them!

ESWL was first used in Germany in 1980 and approved for use in the United States in 1984. The idea is to use intense sound waves (shockwaves) generated outside the body (extracorporeally) to pulverize the kidney stone ("lithos" is Greek for "stone" and "tripsis" is Greek for "breaking") to a sand-like state, so that it can be passed out naturally without surgery. A key issue is the ability to focus the shockwaves so that they damage only the stone and not the body. The device which performs the lithotripsy is called a lithotripter, and a schematic of one type of lithotripter appears below. The view is towards the top of the patient's head as the patient lays face up on a table.

Like many lithotripters, this one uses a half-ellipsoidal reflecting chamber to focus shockwaves which are created when a powerful spark vaporizes water at one focus of the ellipsoid. Part of each wave never hits the reflector, and this part (light blue) spreads out and weakens. However, the part of the wave which hits the reflector (dark blue) converges on the other focus and becomes very intense, causing the stone to crumble. You can see a photo of an actual lithotripter setup here.

So the important thing is the focusing propetry of ellipses and ellipsoids. You may have heard of it before, but probably nobody ever proved it to you, because the typical proof is kind of messy. However, with aid of vector calculus, we can give a very simple and beautiful little proof due to mathematician Harley Flanders. The reflector is an ellipsoid of revolution about the major axis, so we need only consider its profile - an ellipse. Also, we will work with "rays" rather than waves, thinking of each little part of a wave as moving along a reflected ray.

We want to show that if something like a light ray leaves focus F (with position vector p) and strikes the ellipse at point A, then it will be reflected to focus F (with position vector q). Suppose that the ellipse is given by a smooth parameterization r = r(t) where r is the position vector of A and t is time. Then the velocity vector dr/dt and its opposite -dr/dt are parallel to the tangent line at A, and by the law of reflection (angle of incidence = angle of reflection), we must prove that = . Since and are each less than 180 degrees, this is equivalent to showing that cos = cos.

Before giving the proof, let's notice that if w = w(t) is a vector function of time t, then we can use the definition of vector magnitude and the dot product rule to compute

Now here's the proof, which uses the notation in the above figure.

By the standard definition of an ellipse,

Noticing that p and q are also constant (time derivative 0) we can take derivatives of both sides, using equation (*), to get

which is the same as

.

Since (p - r) / ||p - r|| and (q - r) / ||q - r|| are unit vectors, this means that

and hence cos = cos, which is what we wanted to prove.

So what good is this little vector proof? Well, for one thing, it establishes the focusing accuracy of ellpsoidal lithotripters, which is of no small consequence. ESWL is now the treatment of choice for over 80% of stones in the kidney and ureter. Not only is the recovery time 3-4 days, as opposed to 2-3 weeks with conventional surgery, but the mortality rate is less than 0.01%, as opposed to 2-3% for traditional surgery. The best cure for a "calculus" is a little calculus!

Try this link for more information on kidney stones. For detailed information on ESWL, see the article

"Extracorporeal Shockwave Lithotripsy" by Jeff E. Fegan and Glenn M. Preminger, in High Tech Urology: Technologic Innovations and their Clinical Applications, Joseph A. Smith Jr., M.D., F.A.C.S., W. B. Saunders Company, 1992.

The above book is available in the Ruth Lilly Medical Library here at IUPUI. The reference for Professor Harley Flanders' proof of the reflective property of ellipses (and other conics) is

H. Flanders, The optical property of the conics, Amer. Math. Monthly, 75 (1968), 399.

This issue of the American Mathematical Monthly is available in the IUPUI University Library.

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