Xiao Fan Ding^{1,2}, William B. Handler^{1}, and Blaine A. Chronik^{1,2}

With the prevalence of medical implants and MRI both on the rise around the world; patients, device manufacturers, and medical professionals alike should know how implants interact with the MR environment. One such interaction is the possibility of torque on an implant due to interaction with the main field. The current methods for measuring induced torque are published by ASTM International. However, although methods are available, their accuracy and precision have yet to be properly studied. This abstract investigates the measurement uncertainties of the two methods for measuring magnetically induced torque published in the test standard, ASTM F2213-17.

In the torsional spring method, a device is placed on a platform suspended on top and bottom by torsion springs. When there is a torque, the device rotates, and a deflection angle is measured. The sources of error are the instrument reading error for finding angular deflection and the error in the torsion spring constant. The dominant source of error for this method would likely be in measuring the deflection angle since the requirement to perform this method is a protractor with 1° increments.

In the pulley method a device is placed on rotatable
surface, as shown in **Figure 1**,
connected to a force sensor by a thread. The force sensor is slowly pulled away
and the surface rotates a full 360°. The sources
of error are the measurement errors from a force sensor and measurement of the
radius of the apparatus. There is also an error in the ‘stickiness’ of the
surface, the torque required to overcome static friction.

To find the error in static friction, measurements are
made using the apparatus in **Figure 1**.
As was in the pulley method, a lightweight thread was extended from
the cylindrical support. Differing however, the thread was
placed over a pulley and attached to a mass. The surface was divided into
twelve 30° sections. For each, the mass suspended was incrementally increased until the weight was enough to set the
surface in motion. The method of changing the mass was to place
staples into a small basket. Staples were chosen since individually, the mass
is insignificant, but multiple staples have a noticeable change in a tangible
quantity.

For a function, $$$x$$$, such that $$$x=f(u_1,u_2,...,u_n)$$$, the uncertainty, $$$\delta_x$$$, based on known sources of error is propagated by ^{[5]},

$$\delta_x^2=\delta_{u_1}^2\left(\frac{\partial x}{\partial u_1}\right)^2+\delta_{u_2}^2\left(\frac{\partial x}{\partial u_2}\right)^2+...+\delta_{u_n}^2\left(\frac{\partial x}{\partial u_n}\right)^2$$

The equation above is true when all independent variables are uncorrelated. An expression for measurement uncertainty of each methods can be calculated and propagated to see how uncertainty changes with torque.

The propagation of torque and measurement uncertainty starting from the smallest observed torque, 1.574 mNm is shown in **Figure 4**.The calibration report a real force sensor ^{[6]}, listed a capacity of 1.11 N, a resolution of 4.46E-4 N, and $$$\delta_F=1.668\times10^{-3}\ \mathrm{N}$$$. Measurements using a digital caliper of the **Figure 1 **apparatus yields $$$R=50.72\ \mathrm{mm}$$$ and $$$\delta_R=0.01\ \mathrm{mm}$$$. The torsion spring method requires a spring constant, $$$k$$$, capable of measuring at least 1° intervals which yields $$$\delta_\theta=0.5^\circ$$$. For $$$\tau=1.574\ \mathrm{mNm}$$$, $$$k$$$ comes to be $$$\tau=1.574\times10^{-3}\ \mathrm{Nm}$$$ with an arbitrarily chosen $$$\delta_k$$$.

From **Figure 4**, as was anticipated, the dominant source of error in the torsion spring method was the instrument uncertainty of the protractor. When $$$\delta_k$$$ becomes negligible,$$$\delta_\tau$$$ is constant at $$$\delta_\tau=k\delta_\theta\sqrt{2}$$$. The dominant source of error in the pulley method is the systematic error of coming from static friction. For comparable measurements of torque, the pulley method, while considering all sources of error, fares better than the torsion spring method.

[1] R.
Kalin and M. S. Staton, “Current clinical issues for MRI scanning of pacemaker
and defibrillator patients.,” *J. Am. Med.
Assoc.*, 2011.

[2] Canadian Agency for Drugs and Technologies in Health (CADTH), “The Canadian medical Imaging Inventory, 2017” 2017

[3] Organisation for Economic Co-operation and Development (OECD), “Health at a Glance 2017” 2017

[4] ASTM International, “F2213-17 Standard Test Method for Measurement of Magnetically Induced Torque on Medical Devices in the Magnetic Resonance Environment”

[5] P.
R. Bevington and D. K. Robinson, *Data
Reduction and Error Analysis for the Physical Sciences*, Third Edition.
McGraw-Hill Higher Education. 2002.

[6] Mark-10 Corporation, "MR03-025 Force Sensor"

**Figure 4**: Propagation of errors as a
function of calculated torque based on the error propagation equation. For the torsion
spring method, uncertainty was propagated from 1.574 mNm until the torque calculated
at 25°. With an uncertainty expression
of $$$\delta_\tau=k\Delta\theta\sqrt{(\frac{\delta_k}{k})^2+2(\frac{\delta_\theta}{\Delta\theta})^2}$$$. $$$\delta_k$$$, which at first is negligible, becomes a more dominant source of error with greater angular deflection. For the pulley method, uncertainty was propagated until the torque calculated at 1.11 N with an expression of $$$\delta_\tau=R(F-F_f)\sqrt{(\frac{\delta_R}{R})^2+2(\frac{\delta_F}{F-F_f})^2+(\frac{\delta_{F_s}}{R(F-F_f)})^2}$$$. With greater force readout, the effect of $$$\delta_{F_f}$$$ diminishes.