Non-ideal Behaviors of Magnetically Driven Screws in Soft Tissue

Arthur W. Mahoney, Nathan D. Nelson, Erin M. Parsons, and Jake J. Abbott

Abstract— This paper considers the requirements for sta-ble control of simple devices that consist of a permanentmagnet rigidly attached to a screw operating in soft tissue.Placing these devices in a rotating magnetic field causes thedevice’s permanent magnet to rotate synchronously with theapplied field, making the attached screw generate forwardpropulsion. Although there has been significant research inthe area of magnetic screws, this paper describes magneticphenomena that have not been addressed in prior work, yetwill be critical in the design of future steering and controlstrategies for medical applications. Using a time-averaged modelof magnetic torque, we analyze the magnetic torque presentwhen intentionally steering the screw or due to naturallyoccurring disturbances in the screw’s heading. We predict andexperimentally demonstrate the existence of magnetic torquesthat simultaneously act to stabilize and destabilize magneticscrews operating in tissue. We find that there exists an appliedfield rotation speed, above which screws may become unstable,and we find that the interaction between the stabilizing anddestabilizing magnetic torques may cause potentially undesiredbut predictable behavior while steering.

I. INTRODUCTION

Untethered micro- and mesoscale actuated devices (e.g.,microrobots) have become an active area of research in thefield of minimally invasive medicine due to their potentialfor locomotion and manipulation in hard-to-reach areas of thehuman body [1]. Many applications of interest will requiresuch devices to locomote through soft tissue. These mayinclude radioactive seed positioning for prostate brachyther-apy, positioning an untethered device within the brain forhyperthermia treatments, and navigating through the vitreoushumor of the eye to perform therapy of the retina such astargeted drug delivery.

One form of untethered device that is showing promisefor medical applications are those actuated using forcesand torques produced by magnetic fields. The helical mi-croswimmer, which consists of a rigid helix attached to amagnetic element designed to mimic the motion of bacterialflagella, is one such device [2]–[4]. When placed in a rotatingmagnetic field, a torque is applied to the device through themagnetic head, which causes the attached helix to rotate andproduce fluidic propulsion. Helical microswimmers operatewell in the low-Reynolds-number regime where viscous dragdominates inertia [5]. Because the helix of the microswim-mer does not displace the medium around the magnetichead, helical microswimmers are not well suited for moving

This material is based upon work supported by the National ScienceFoundation under Grant Nos. 0952718 and 0654414.

A. W. Mahoney is with the School of Computing, and N. D. Nelson,E. M. Parsons, and J. J. Abbott are with the Department of MechanicalEngineering, University of Utah, UT 84112, USA.

email: art.mahoney@utah.edu

through soft tissue. Instead, similar devices that consist ofa magnetic element attached to a solid screw, rather than ahelix, have been proposed for use in tissue [6]–[11].

In [6], the authors introduce a device of 11.5 mm inlength consisting of a 7.5 mm-long diametrically magnetizedpermanent magnet attached to a 4.0 mm-long brass woodscrew. While operating their devices in agar gel to simulatetissue, the authors show that their device’s forward velocityis linearly related to the rotation frequency of the appliedfield, and they demonstrate that their device can be usedin real tissue by driving it through a 25 mm-thick sampleof bovine tissue. In [7], the authors demonstrate that bypitching the rotation axis of the rotating applied field out ofalignment with the principal axis of the screw, the devicescan be made to steer toward a desired direction in agargel. In their experiments, they find that the smallest turningradius can be obtained with their shortest device when theapplied field rotates at 0.1 Hz and the rotation axis of theapplied field is kept pitched away from the principal axis oftheir screw by 60 toward the direction of the desired turn.The authors show that the average magnetic torque causingtheir device to turn is maximized in this configuration,and that applying torque in this manner generates turnsof highest curvature. Other research has explored variousscrew-tip cutting mechanisms [8], trailing a wire [9], localhyperthermia treatment using inductive heating [10], andusing screw devices as pH sensors [11].

This paper continues on the work of [6] and [7] towardstable open-loop control of untethered screw devices insoft tissue. In this study, we numerically analyze the time-averaged magnetic torque that causes the screw to pitch inthe direction of the applied field’s axis of rotation, whichis present when intentionally steering the screw or due tonaturally occurring disturbances in the screw’s heading. Weshow that not only is there a stabilizing magnetic torquethat causes the screw to realign with the rotation axis ofthe applied field as desired, but there also always existsa competing undesired destabilizing magnetic torque thatsimultaneously causes the screw to turn perpendicular to thedesired direction of alignment. We demonstrate that thereexists a rotation speed of the applied field above which thedestabilizing torque will always be greater in magnitude thanthe stabilizing torque. This can cause the screw to becomeunstable and diverge away from the desired trajectory. Thisphenomenon is not documented in prior literature and mustbe taken into account when designed future steering andcontrol strategies for medical applications.

2012 IEEE/RSJ International Conference onIntelligent Robots and SystemsOctober 7-12, 2012. Vilamoura, Algarve, Portugal

978-1-4673-1735-1/12/S31.00 ©2012 IEEE 3559

B

m Ω

Ψ

x

y

z

φ

θ

pitc

h

yaw

Fig. 1. The principal axis of the untethered screw device is constrainedto lie parallel to the z device axis making the dipole moment of the screwm lie in the x-y screw plane. The applied magnetic field (magnetic fluxdensity) B rotates around and is perpendicular to the rotation axis Ω. Thescrew is steered in space by pitching Ω away from the screw’s principalaxis by the angle ψ toward the desired direction of steering.

II. MODELING MAGNETIC CONTROL

Let the applied magnetic field B (magnetic flux density)rotate with angular velocity Ω, whose direction and magni-tude define the applied field’s axis of rotation and speed, andwhere B is perpendicular to Ω. Define a device coordinateframe with axes x,y,z as shown in Fig. 1, where the z axisis parallel to the screw’s principal axis, the x axis lies in theplane spanned by the z axis and Ω, and the y axis lies in thedirection of z × x. This breaks down when z is parallel toΩ, however, this paper is concerned with the case when z isnot parallel to Ω. For simplicity, we assume that the screw’sprincipal axis lies parallel to the z device axis. We define ψto measure the angle between z and Ω, which we refer toas the “pitch angle.”

The instantaneous magnetic torque τ produced on a dipolemoment m (i.e., the small permanent magnet attached to thescrew) by B at the location of m causes m to align withB and is expressed by τ = m × B. We have chosen toparameterize m and B with the angles, φ and θ, respectively,which results in the representations

B = |B|

cosψ cos θsin θ

− sinψ cos θ

(1)

m = |m|

cosφsinφ

0

(2)

and the resulting magnetic torque τ is given by

τ = |m||B|

sinψ cos θ sinφ− sinψ cos θ cosφ

sin θ cosφ− cosψ cos θ sinφ

(3)

We denote the components of τ that lie parallel to the xand y device axis to be the destabilizing τx and stabilizing

τ y torques, respectively. The component of τ in the devicez direction, τ z , causes the screw to roll and is referredto as the propulsion torque. When Ω is aligned with theprincipal axis of the screw (i.e., ψ = 0), the magnetictorque is purely composed of the propulsion torque, andthe components representing the stabilizing and destabilizingmagnetic torques have zero magnitude. The result is a lineartrajectory in the direction of the screw’s principal axis.

The screw is steered through space by pitching the rotationaxis Ω of the applied field in the direction of the desiredturn. Without loss of generality, Ω is assumed to be rotatedabout the y device axis. When Ω is not aligned with theprincipal axis of the screw, the stabilizing τ y and destabiliz-ing τx magnetic torques cause the screw to align with anddeviate from Ω, respectively. In Newtonian fluids, the screwresponds by aligning its principal axis with Ω over time [2].In viscoelastic environments such as soft tissue, however,the stabilizing and destabilizing magnetic torques introducecomplex steering behavior.

Rather than model the complex behavior, we have chosento simplify analysis by examining the average stabilizingand destabilizing torques over one revolution of the appliedfield, the assumption being that the tendency of the screwto change its heading in the stabilizing or destabilizingdirections during one applied field revolution will be relatedto the averaged stabilizing or destabilizing torques during thesame field revolution, while operating in tissue.

The model used to average the stabilizing and destabi-lizing torque is derived by constraining the screw as if itwere placed within a lumen such that the stabilizing anddestabilizing torques have no effect on the screw’s headingand the screw only rotates about its principal axis (i.e., φ istime-varying but ψ is constant). The model is simplified inthis way because, in practice, the screw rotates much fasterabout its principal axis than it changes its heading whenoperating in tissue. When averaging over one revolution ofthe applied field, even with the presence of the stabilizingand destabilizing torques, the actual change in heading dueto these torques is small and will not significantly alter theanalysis of the averaged torques themselves.

The dominating motion during one applied field revolutionwill be the screw’s rotation around its principal axis. This ro-tation is modeled with the assumption that any inertial effectsare dominated by linear drag caused by friction between thescrew and the tissue, making the inertial torque negligible.The net torque acting upon the screw along its principal axisconsists of the magnetic torque generated by the applied fieldand a linear damping torque. This assumption is often madewhen analyzing helical swimmers in low-Reynolds-regimefluids where fluidic drag dominates inertia [5]. Aside fromtorque, forces that act on the screw in tissue include the forcedue to gravity and any forces acting on the screw due to themechanics of the tissue itself. Due to the force acting onthe screw from the tissue, the screw typically does not sinkunder the force of gravity as it would in Newtonian fluids[3] and we assume its effect is negligible. The full effect ofthe forces caused by tissue mechanics remains to be studied.

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The linear damping torque about the screw’s principal axisis modeled with the coefficient c and is |τ f | = cφ, where φis the angular velocity of the screw about the same axis. Themagnetic torque applied about the screw’s principal axis isτ z , whose magnitude is found in (3). Since the net torqueis zero, the differential equation governing the rotationaldynamics of the screw about its principal axis, for someconstant pitch angle ψ and constant field rotation speed |Ω|,can be found as

φ =|m||B|c

(sin θ(t) cosφ(t)− cosψ cos θ(t) sinφ(t)

)(4)

which can be solved numerically given initial conditions φ(0)and θ(0), since θ(t) = |Ω|t is known for all t.

The solution φ(t) of (4) describes the screw’s rotationalresponse to the rotating field for the given pitch angle ψand field rotation speed |Ω|. There exists a field rotationspeed |Ω|, above which the screw’s rotation cannot remainsynchronized with the rotating field. This speed, known asthe “step-out” speed, is a function of the linear damping termc, the field magnitude |B|, the screw dipole moment |m|, thepitch angle ψ, and is given by |m||B| cos(ψ)/c. When |Ω| islarger than the step-out frequency, the screw tends to behavechaotically and control is limited. When |Ω| is less than step-out, the screw remains synchronized with the rotating field,although the magnetic torque τ tends to oscillate.

After solving (4) numerically to produce φ(t) for a givenpitch angle ψ and field rotation speed |Ω|, the magnitude ofthe torque about the y axis and the x axis averaged over oneturn of the applied field is produced by integrating the x andy components of (3) using the numerical solutions for φ(t)and θ(t) to obtain φ(θ), which is used in the integrations

τx =|m||B|

2πsinψ

∫ 2π(k+1)

2πk

cos θ sin(φ(θ)

)dθ (5)

τ y = −|m||B|2π

sinψ

∫ 2π(k+1)

2πk

cos θ cos(φ(θ)

)dθ (6)

where k ∈ Z+ is selected to reject the screw’s transientresponse due to the initial conditions when solving (4).Although we have observed the initial transient response tobe insignificant after two rotations of the applied field, wechose k = 9 as a conservative lower limit of integration withinitial conditions φ(0) = 0 and θ(0) = 90. Averagingthe magnetic torque after the transient response simplifiesanalyses by removing fluctuation caused by the oscillationin τ . Neglecting the transient response, the values obtainedfor τx and τ y only change with the pitch angle ψ and fieldrotation speed |Ω| used to obtain the solution of (4).

We find that τx and τ y can be normalized by scalingthem by the maximum possible total magnetic torque |m||B|,and they can be enumerated over the pitch angle ψ andthe normalized field rotation speed, which is defined as theratio of |Ω| to the screw’s step-out speed when ψ = 0.Fig. 2(a) and Fig. 2(b) show non-dimensional contour plotsof normalized τx and τ y with ψ ∈ [0, 90). Because thescrew’s step-out speed varies with ψ, and we are interestedin the screw’s behavior when the screw is synchronized with

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

0.34

0.29

0.24

0.19

0.140.090.04

Step-out Speed

Max Stab. TorqueMax Torque Ridge

Normalized Applied Field Speed,|m||B||Ω|c

Pitc

h A

ngle

, Ψ

(de

g)

(a) Average scaled stabilizing torque, τy/|m||B|. The maximumstabilizing torque occurs at a normalized rotation speed of 0.13 andpitch angle 60.0. The curve “Max Torque Ridge” describes the pitchangle necessary to maximize stabilizing torque for each normalizedapplied field rotation speed.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

Normalized Applied Field Speed,

0.05

0.10

0.15

0.20

0.25

0.30

|m||B||Ω|c

Step-out SpeedMax Stab. Torque

Pitc

h A

ngle

, Ψ

(de

g)

(b) Average scaled destabilizing torque, τx/|m||B|.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

0.33 0.66 1.00 1.33 1.66 2.00

Step-out SpeedMax Stab. Torque

Normalized Applied Field Speed,|m||B||Ω|c

Unstable

Pitc

h A

ngle

, Ψ

(de

g)

(c) Ratio of the average destabilizing torque to average stabilizingtorque, τx/τy . The region where the screw becomes unstable islabeled and shaded.

Fig. 2. The average stabilizing torque (a), average destabilizing torque(b), and the ratio between them (c), as a function of the pitch angle andnormalized applied field rotation speed. The point where the maximumstabilizing torque occurs is shown in all three contour plots for reference.

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5.6mm

1.4mm

(a) (b) (c)

x z

y

Fig. 3. The Helmholtz coils (a), the untethered screw device (b), andthe container (c) used for experiments presented in this paper. The screwdevice is 5.6 mm long and 1.4 mm in diameter and consists of a 1 mm cubepermanent magnet rigidly attached to brass screw with 0.87 mm pitch. Thestatic world coordinate frame used in experiments is shown in (c).

the rotating field, the normalized field rotation speed wasranged so as not to exceed step-out for each value of ψ. Thestep-out speed is shown as a bold, dashed line. When Ω ispitched down (i.e., ψ < 0), the results of Fig. 2(a) and Fig.2(b) change sign but Fig. 2(c) is unchanged.

Fig. 2(a) shows the average stabilizing torque that causesthe screw to turn toward Ω increases with ψ up to thecurve labeled “Max Torque Ridge” for any field rotationspeed with the global scaled maximum of 0.35 occurringat normalized rotation speed of 0.13 and pitch angle of60.0 (which corresponds to the same result found by [7]).As the average stabilizing torque increases, Fig. 2(b) showsthat the average destabilizing torque tends to increase aswell, and can become larger than the stabilizing torque.The region of Fig. 2(c) where the ratio between the averagedestabilizing and stabilizing torques is larger than one cor-responds to configurations where the average destabilizingtorque is larger than the average stabilizing torque. Whenoperating in this region, the magnetic torque will producemore rotation of the screw away from than toward Ω. Ifthe screw’s heading is deviated from Ω, due to naturallyoccurring disturbances, then the screw may become unstablewhen the average destabilizing torque is large. Operating thescrew so that the average stabilizing torque is larger than theaverage destabilizing torque will enable the screw to rejectdisturbances, realigning faster when the stabilizing torquedominates the destabilizing torque.

From Fig. 2(b), the averaged destabilizing torque is onlyzero when ψ = 0 or |Ω| = 0. This means that while steeringthe screw by increasing or decreasing ψ, there will alwaysbe a destabilizing torque present that causes the screw toyaw in the direction perpendicular to the desired turn. Whenthe desired turn direction is up in Fig. 1 (i.e., ψ > 0), thedestabilizing torque will cause the screw to yaw around thepositive x device axis according to the right-hand rule. Whenthe desired turn direction is down in Fig. 1 (i.e., ψ < 0), thedestabilizing torque creates yaw around the negative x deviceaxis. Although the screw will always have the tendency toyaw, its effect can be reduced by operating in regions ofFig. 2(b) where the average destabilizing torque is small (i.e.at small angles of ψ and/or at slow speeds).

III. EXPERIMENTAL RESULTS

The experimental setup used to generate controlled mag-netic fields consists of three nested sets of Helmholtz coils(described in detail in [3] and shown in Fig. 3(a)) arrangedorthogonally such that the magnetic field in the center of thecoils can be assigned arbitrarily, with each Helmholtz paircorresponding to one basis direction of the magnetic field.Each set of Helmholtz coils generates a magnetic field thatis approximately uniform in the center of the workspace,with a maximum field strength of 10 mT. Computer visionfor screw tracking was performed using a Basler A602FCcamera fitted with a Computar MLH-10X macro zoom lenswith the image plane parallel to the x-z plane of a staticworld coordinate frame, depicted in Fig. 3(c).

Similarly to [6] and [7], agar gel was used to simulatesoft tissue for our experiments. Agar gel is commonly usedin research as an inexpensive and semi-transparent tissuephantom. For example, agar gel has been used to simulate thevitreous humor of the eye [12], brain tissue [13], and humanbreast tissue [14]. The agar gel used herein consisted of0.5 wt. % culinary agar to unfiltered water, which we found tohandle well for our initial experiments. For a comparison, ithas been found that 0.6 wt. % agar closely models propertiesof human brain tissue [13]. Our agar gel was cut intoblocks, fitting a 45 mm×16 mm×38 mm acrylic container,and positioned in the common center of the Helmholtz coils.The screw used in this study is 5.6 mm long and 1.4 mm indiameter and consists of a 1 mm cube Grade-N50 permanentmagnet rigidly attached to a brass wood screw with 0.87 mmpitch. During experimentation, the step-out frequency of thescrew at ψ = 0 ranged from 7.3 Hz to 8.0 Hz and the fieldstrength |B| was set to 10.0 mT.

The screw’s ability to reject disturbances in orientationis demonstrated in the results presented in Fig. 4, whichshows several screw trajectories as viewed from the imagingsystem with Ω aligned with the world’s z axis throughoutthe duration of each run. Four typical trajectories whenoperating at normalized applied field rotation speeds of 0.20and 0.85 are shown on the left side of Fig. 4(a) and Fig. 4(b),respectively, along with the points where each trajectory exitsthe agar sample plotted on the right. At normalized speed0.20, the average stabilizing torque is sufficient to overcomethe average destabilizing torque and return the orientationof the screw to Ω resulting in a tight clustering of exitpoints near the opposite position of the entry point. Withnormalized speed of 0.85, the average destabilizing torqueovercomes the stabilizing torque and perpetually increasesthe magnitude of naturally occurring disturbances resultingin unstable behavior and large variation in the position of theexit points as shown. It is important to note, however, thatthe screw only naturally rejects disturbances in heading, notposition. The position of the screw may drift off the desiredtrajectory without feedback control.

We demonstrate the screw’s tendency to yaw while steer-ing by pitching Ω an angle of ψ measured from the screw’sprincipal axis as seen from the vision system. As the screw

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5 10 15 20 25 30 35 40 455

10

15

20

25

0 4 8

x, m

m

z, mm y, mm

Ω

12 16

(a) Normalized applied field speed 0.20

5 10 15 20 25 30 35 40 455

10

15

20

25

0 4 8z, mm

x, m

m

y, mm

Ω

12 16

(b) Normalized applied field speed 0.85

Fig. 4. Trajectories of the screw at two applied field rotation speeds inagar gel (0.5 wt. %) with Ω parallel to the z world axis. The exit points ofeach trajectory are shown on the right side of the figure. Each trajectory isobtained using the computer vision system. Each exit point is measured byhand.

(a) ψ = 20° (b) ψ = 40°

(c) ψ = -20°

Top

0s 10s 20s 30s

Front

0s 10s 20s30s

Front

0s 9s 18s27s

Top

0s 9s 18s27s

(d) ψ = -40°

Top

0s 9s 18s27s

Front

0s 9s 18s27s

Top

0s 9s 18s27s

Front

9s0s 18s 27s

Fig. 5. Front and top views of screw trajectories demonstrating the screw’stendency to yaw while turning. The angle between the screw’s principal axisand Ω (as viewed from the x-z world plane) is maintained at ψ = ±20

and ±40. The field is rotated at a normalized applied field rotation speedof 0.20 in all four cases.

turns, Ω was kept in the x-z world plane, but manuallyadjusted to maintain a constant ψ using visual feedbackwith the assistance of guidelines drawn on realtime video

0 100 200 300 400 500 600t = 0 s

t = 600 s5

15

25

35

Time, s

Ang

le, d

eg

(a) Normalized applied field speed 0.001 and ψ = 10.0

t = 15.5 s

t = 0 s

Applied Field Revolutions (#)

Hor

izon

tal A

ngle

(de

g)

5

15

25

35

10 20 300

(b) Normalized applied field speed 0.3 and ψ = 10.0

Fig. 6. Angle of the screw’s principal axis measured from horizontal asa function of the number of applied field revolutions while rotating at anormalized rotation speed of 0.001 (a) and 0.3 (b). The right side showsimages of the screw in the agar at the beginning and end of the experiment.

of the screw presented to the user. Fig. 5 shows a front view(in the x-z world plane) and a top view (in the y-z worldplane, taken with a Canon PowerShot G10) of the resultingtrajectories with ψ = 20 and 40 in Figs. 5(a) and 5(b), andψ = −20 and −40 in Figs. 5(c) and 5(d), respectively. Inall four cases, the applied field is rotated at a normalizedrotation speed of 0.20 and each trajectory stops when thescrew either collides with the container or exits the top of theagar sample. When ψ > 0, Fig. 2 predicts that the stabilizingtorque will cause the screw to turn in the direction of Ωbut the destabilizing torque will cause the screw to rotateabout the x axis of the device frame. In the world frame, thescrew will tend to simultaneously turn about the y world axisand the z world axis as shown in Figs. 5(a) and 5(b). Whenψ < 0, the screw will tend to simultaneously turn about the−y world axis and the −z world axis as shown in Figs. 5(c)and 5(d). This tendency to yaw will need to be taken intoaccount while navigating untethered screws devices.

In Sec. II, we applied the assumption that the screw’sprincipal axis is constrained during one complete rotationof the applied field. This tends to be a valid assumptionwhen driving the screw through tissue because the headingof the principal axis typically does not change significantlyover one revolution of the applied field. This assumptionbreaks down when rotating the applied field very slowly. Atslow rotation speeds, the instantaneous magnetic torque (3)causes the screw’s heading to vary significantly throughoutone cycle. This is demonstrated in Fig. 6, which shows themeasured heading of the screw (obtained using the computervision system) as a function of time when driving the screwwith a normalized applied field rotation speed of 0.001and 0.3, with the pitch angle ψ the same in both cases.When rotating the applied field slowly, the screw’s headingtends to oscillate at a magnitude of 7 throughout one

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applied field revolution, whereas the screw’s heading tends tooscillate at a magnitude of 1 when rotating the applied fieldrapidly. Without any explicit understanding of the screw-tissue interaction, the drop in magnitude may indicate thatthe oscillations of the screw exceed the inherent bandwidth ofthe tissue. If the oscillation of the screw is undesired (e.g.,due to potential tissue damage), then the screw should bedriven at faster rotation speeds, which may come at the costof increasing the undesired yaw while steering.

IV. DISCUSSION

Various screw geometries have been proposed for drivingthrough soft tissue. In [6], the authors propose a similardevice to that used in our experiments, where a permanentmagnet is rigidly attached to a single screw; in [7] and[8], the authors control a screw-like device consisting ofa helical thread wrapped around a cylindrical permanentmagnet (diametrically magnetized); and in [11], the authorspropose a device consisting of screws rigidly attached toboth ends of a cylindrical permanent magnet. The resultspresented in Sec. II are purely an analysis of the interactionbetween the screw’s dipole moment and the applied magneticfield, therefore, it applies to screws of any geometry. Becausethe magnetic torque applied to the screw is a pure torque (i.e.,not an immediate result of an applied force), its affect on thescrew (as a free body) is invariant to where it is applied on thedevice itself. As a result, the screw used herein would exhibitthe same behavior if the cube magnet were positioned in themiddle of the screw as it does with the magnet positioned atthe rear. We do expect, however, that the screw’s length andthe density of the tissue will influence the magnitude of thebehavior demonstrated herein.

The screw used herein is 5.6 mm long. It is likely thatthe screw will need to be scaled down for many medicalapplications. We do not yet have a good understanding ofscrew-tissue interaction. However, we can approximate therole of friction as the screw is scaled if we model thescrew as if it were a cylinder and if we assume that thefriction between the tissue and the screw is proportional tothe surface area. Applying these assumptions, the surfacearea of the screw is A = 2πdl, where d and l are thescrew’s diameter and length. If d and l are scaled by afactor s, then the surface area becomes A = 2πdls2 and thusscales with s2. If we scale the dimensions of the permanentmagnet rigidly attached to the screw by the same factor s,then the magnet’s volume scales with s3. The magnitudeof the screw’s dipole moment m is proportional to themagnet’s volume causing m, and subsequently the appliedmagnetic torque τ (given by (3)), to also scale with s3. Thisimplies that as the screw is scaled down, the surface frictiondecreases at a slower rate than the magnetic torque and theapplied magnetic field magnitude |B| may need to increaseto generate effectively the same behavior presented herein.Fully understanding how the screw will scale to smallersizes will require the screw-tissue interaction to be betterunderstood. This will be the subject of future work.

V. CONCLUSION

In this study, we have analyzed the stabilizing and desta-bilizing magnetic torques present when intentionally steeringan untethered screw device or due to naturally occurringdisturbances in the screw’s heading. We have shown theo-retically and experimentally that there exists an applied fieldrotation speed, above which untethered screws devices willbecome unstable. We have also demonstrated that complexinteraction between the stabilizing and destabilizing mag-netic torques can cause a screw device to turn away fromdesired directions of steering in a predictable way. Thesephenomena are not reported in prior literature and they mustbe taken into account when designing future steering andclosed-loop control strategies for medical applications.

ACKNOWLEDGMENT

The authors wish to thank Jeremy Greer and Dr. EberhardBamberg for fabricating the screw pictured in Fig. 3.

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