A Gift For the Readers of Intermediate Physics for Medicine and Biology

The holiday season is a time when we often exchange gifts. My gift to the readers of the 4th edition of Intermediate Physics for Medicine and Biology is a new homework problem. It belongs to the chapter on magnetic resonance imaging, and specifically to Section 18.12 (this section about functional MRI has no homework problems associated with it, so the new one fills the gap), but it also draws heavily on Section 8.1 about the magnetic force on a current, sometimes called the Lorentz force. The purpose of the problem is to determine if you can use MRI and the Lorentz force to detect nerve activation.

Section 18.12

Problem 37 1/2 Suppose your median nerve, having a radius of 2 mm, carries a current density of 10 Amps per square meter over a length of 10 millimeters. (Assume all the axons are simultaneously active, so the current density is uniformly distributed throughout the nerve).

a) You are having a magnetic resonance image taken, and the steady uniform magnetic field has a strength of 4 Tesla and is directed perpendicular to the nerve. Calculate the magnitude and direction of the magnetic force on the nerve.

b) Assume the nerve is held in position by an elastic force equal to the product of k and s, where k is the spring constant of 400 Newtons per meter and s is the distance the nerve is displaced from its equilibrium position. Calculate the displacement of the nerve experiencing the force found in part a.

c) Finally, assume that a magnetic field gradient of 36 milliTesla per meter is present, so that when the nerve moves the distance calculated in part b, it enters a region of different magnetic field strength. Calculate the change in magnetic field that the nerve experiences because of its motion. Calculate the change in resonance angular frequency (assuming you are imaging protons). If the gradient and current last for 15 milliseconds, what is the change in phase of the MRI signal?

Where did I come up with this problem? It is based on a paper that Peter Basser and I recently published, titled “Mechanical Model of Neural Tissue Displacement During Lortenz Effect Imaging,” which appeared in the January 2009 issue of the journal Magnetic Resonance in Medicine (Volume 61, Pages 59–64). The mechanical problem is somewhat more complicated than described in part b of the above problem, but the calculated displacement is similar to what Basser and I find for the more accurate calculation. If you solve the new homework problem correctly, you should obtain a very small displacement, implying a phase shift too small to measure with current technology. The conclusion is that nerve action currents are unlikely to be measurable using this method.

Do you want the solution to the problem? Send me an email (roth@oakland.edu) and I will be happy to supply it.

Happy Holidays!