Spencer B Parent^{1}, William Handler^{2}, John Drozd^{2}, and Blaine A Chronik^{1,2}

The
current procedures and guidelines for testing forces on medical devices require
that testing be performed in a MR setting, which is both timely and costly. To reduce test time and cost, a
computational model of magnetic force was developed. Using a test case where an
analytic solution of magnetic force can be applied, it is shown that for
materials with magnetic susceptibility (chi)<10^{5} ppm, a
computational model of magnetic force was correct within 10% error and this
error decreases to less than 1% for chi<10^{4} ppm. Size of object is
shown to have little effect on error.

A background magnetic flux
density with spatial gradient <-0.5, 0, 0.5> was implemented at all
boundaries of a cubic simulation domain using *COMSOL* (version 5.2a).
Within this domain a cylinder of radius 1.27cm, length L and magnetic
susceptibility chi was placed along the z axis, with the center of the base of
the cylinder at isocentre. This positioning allowed sufficient symmetry such
that Fx,Fy=0, simplifying the analytic solution.

Results of magnetic force were
produced for two simulation sets. First, the length of the cylinder was kept
constant at 2.54cm and simulations were run varying the magnetic susceptibility
of the cylinder from 1 to 10^{6} ppm. Next, the magnetic susceptibility
was kept constant at 100ppm and simulations were run varying the length of the
cylinder from 1.27 to 25.4 cm.

Magnetic flux density,
magnetization, mesh element volume, and spatial gradient of the B field data -
all as a function of position - was exported from the simulations and used to
calculate the magnetic force in *Matlab* (version
2015b). This calculation was performed with equation (1) of **figure 1**,
using the simulation data as an input.

In order to validate the
results of the simulations, an analytic model of magnetic force was produced, equation
(2) **figure 1**, and this model was solved for all cases between simulation
sets.

**Figure 2 **shows a
linear increase in both simulated and analytic magnetic force for values of
magnetic susceptibility below 10^{5} ppm. For larger susceptibilities
the simulation begins to underestimate force. **Figure 3 **shows the
percent error between the two data sets. Error is less than 1% for chi<10^{4}
ppm, and less than 10% for chi<10^{5} ppm. The increase in
error is most likely due to the inability for the simulation to compensate for
the increasing effect the magnetization has on the surrounding field as chi increases.
Implant-grade metals (ie. stainless steel 316, cobalt-chromium alloys, titanium
alloys) all have chi less than or on the order of 10^{4} ppm, where error
would be less than 1%.

**Figure 4** shows both the analytic
and simulated force due a cylinder of varying length. The percent error between
the two data sets, shown in **figure 5,** increases slightly as length increases but is less than
0.6% for all lengths.

1. Shellock, F. G., Woods, T. O., & Crues, J. V. (2009). MR Labeling Information for Implants and Devices: Explanation of Terminology. Radiology, 253(1), 26–30. https://doi.org/10.1148/radiol.2531091030

2. ASTM F2052-14: Standard Test Method for Measurement of Magnetically Induced Displacement Force on Medical Devices in the Magnetic Resonance. Pennsylvania, USA; ASTM International, 2014.

It can be shown that magnetic force is given by (1) for a
material of magnetic susceptibility chi, volume V, in a magnetic field B. For
a cylinder of height h, magnetic susceptibility chi, and radius r positioned along
the z-axis in a field of B= <-0.5x,0,0.5z> it can be shown that the magnetic force is
given by (2).

Magnetic
force as a function of magnetic susceptibility for a cylinder of radius 1.27cm,
and length 2.54cm. Simulated and analytic results shown.

Percent error between simulated data and analytic solution
for the magnetic force for a cylinder of length 2.54cm, radius 1.27cm,
and varying chi.

Magnetic force as a function of cylinder length for a cylinder
of radius 1.27cm,
and chi 100ppm. Simulated and analytic results shown.

Percent error between simulated data and analytic solution
for the magnetic force for a cylinder of varying length, radius 1.27cm,
and chi 100ppm.