Magnetic Tweezers with Magnetic Flux Density Feedback Control

In this work, we present a single-pole magnetic tweezers (MT) device designed for integration with substrate deformation tracking microscopy (DTM) and/or traction force microscopy (TFM) experiments intended to explore extracellular matrix rheology and human epidermal keratinocyte mechanobiology. Assembled from commercially available off-the-shelf electronics hardware and software, the MT device is amenable to replication in the basic biology laboratory. In contrast to conventional solenoid current-controlled MT devices, operation of this instrument is based on real-time feedback control of the magnetic flux density emanating from the blunt end of the needle core using a cascade control scheme and a digital proportional-integral-derivative (PID) controller. Algorithms that compensate for an apparent spatially non-uniform remnant magnetization of the needle core that develops during actuation are implemented into the feedback control scheme. Through optimization of PID gain scheduling, the MT device exhibits magnetization and demagnetization response times of less than 100 ms without overshoot over a wide range of magnetic flux density setpoints. Compared to current-based control, magnetic flux density-based control allows for more accurate and precise magnetic actuation forces by compensating for temperature increases within the needle core due to heat generated by the applied solenoid currents. Near field calibrations validate the ability of the MT device to actuate 4.5 μm-diameter superparamagnetic beads with forces up to 25 nN with maximum relative uncertainties of ±30% for beads positioned between 2.5 and 40 μm from the needle tip.


I. INTRODUCTION
Despite advances in our understanding of the pathophysiology of congenital and acquired human blistering skin diseases, the biophysical mechanisms by which cell-cell and cell-matrix anchoring junctions endow epidermal keratinocytes with such an innate mechanical resilience are not completely understood 1 .
Fortunately, during the past several decades, cell mechanics has been the subject of intense exploration amongst biologists, physicists, and engineers [2][3][4] . As such, numerous methodologies have been developed that can be used to apply mechanical loads and deformations to individual cells via cell-cell and cell-matrix anchoring junction proteins, including magnetic tweezers, optical tweezers, and magnetic twisting cytometry, to name a few [5][6][7][8][9][10] . Moreover, the well-established techniques of deformation tracking microscopy (DTM) 11,12 and cell traction force microscopy (TFM) [13][14][15][16] can be used to quantify substrate deformations and traction stresses present at cell-matrix junctions that individual cells and multicellular sheets use to attach to model biological surfaces 16,17 . As the overarching goal of this work, we set out to demonstrate how a scientific apparatus that integrates magnetic tweezers (MT) and substrate deformation tracking/traction force microscopy (DTM/TFM) can be employed to explore the mechanobiology of cell-cell and cell-matrix anchoring junctions within human epidermal keratinocytes cultured in vitro as a model for investigating skin fragility disorders. To date, the techniques of DTM and TFM have been widely incorporated into the biology laboratory due to the relative simplicity of the experimental setup and existence of open source computational machinery for analyzing image-based data [17][18][19] . Arguably, the implementation of MT devices is much more limited. Towards this end, in this first of two companion publications 20 , we present a detailed discourse on the concept, design, assembly, operation, and calibration of an MT device for ease of replication in the biology laboratory using commercially available off-the-shelf electronics, mechanical hardware, and computer software. In contrast to conventional MT devices that achieve magnetic actuation forces through feedback control of solenoid currents, our MT device is unique in that it is based on feedback control of the magnetic flux density emanating from the needle core. Potential advantages of this control configuration are demonstrated herein.

II. MAGNETIC TWEEZERS -CONCEPT
In theory, operation of a single-pole MT device is based on the principle that an electrical current, , applied to a solenoid coil surrounding a soft ferromagnetic core will induce a magnetic field in the core that emanates from both the blunt end of the core and the needle tip ( Fig. 1(a)) 21 . The magnetic field within the core, # , at any point in time, , is a function of ; the relative permeability of the core, # ; the saturation magnetization of the core, # ()* ; and the coercivity of the core, # . Note that # is closely related to the remnant magnetization of the core, # ,-. , the magnitude of which depends on the material's magnetization history. Here, # ( ) , # ()* ( ) , and # ,-. ( ) can all be regarded as intrinsic material properties of the core that all have a temperature dependence, denoted by the variable, . If a superparamagnetic bead is exposed to the magnetic field present in front of the needle tip, *23 , the MT will impose a magnetic force on the bead, 56 , that is a function of bead diameter, ; the relative permeability of the bead, 8 ; and gradients present in the magnetic field, ∇ *23 ; given by the relationship: where J is the permeability of free space 22 . Here Equation (1) Ideally, the most accurate method for controlling 56 in an MT experiment would be through feedback control of *23 . However, placement of one or more magnetic flux density sensors within the proximity of the needle tip to provide feedback measurements of *23 is not practical for most applications.
As such, conventional MT designs are based on real-time feedback control of to indirectly control *23 , and thus 56 for a given 22,23 . Using image-based particle tracking, it has also been shown possible to monitor in real-time such that closed-loop feedback control of 56 can be achieved by adjusting either or , the latter accomplished through rapid spatial positioning of the needle tip using a motorized micromanipulator 23 . Despite these remarkable advances in MT devices, it is important to recognize that all core materials develop remnant magnetization after exposure to an applied magnetic field. Additionally, temperature increases in the needle core due to heat generated from current flow in the solenoid coil will lead to changes in # ( ), # ()* ( ), and # ,-. ( ). Collectively, thermally-induced fluctuations within the electron structure of the core material can ultimately oppose magnetization by an applied field 24 . All of these effects can lead to erroneous control of 56 if one utilizes only solenoid currents, , to control *23 while disregarding the thermal and magnetic history of the core. To nullify these effects, many research groups have employed various combinations of in-operation degaussing current bursts 23,25,26 , compensation currents based on magnetization history 23 , and/or adjunctive cooling systems [27][28][29] . As an alternative and much less widely adopted approach 30-32 , we demonstrate control of 56 based on feedback control of the magnetic field present at the blunt end of the needle core, MNOP* , or conceptually, For ease of review, a comprehensive list of abbreviations and symbols can be found in the Supplementary Material.

A. Hardware
Much of the design of our single-pole MT device is modeled on the work of Kollmannsberger and Fabry 23 , modified to implement a magnetic flux density-based control system. As detailed in Fig. 2 The MT is constructed from a cylindrical ASTM A753-08 Alloy Type 4 soft ferromagnetic core (Scientific Alloys, Westerly, RI), 4.76 mm in diameter and ~165 mm long, machined with one blunt end orthogonal to the longitudinal axis of the core and the other end tapered at an angle of 7.5° from the longitudinal axis ( Fig. 1(b)). The core was hydrogen annealed at 1175°C for 4 to 6 hours, followed by controlled cooling to 370°C at a rate of 175-315°C/hour (Vac-Met Inc., Warren, MI). After annealing, the tapered tip was hand-sharpened to a needle point with a radius of curvature of ~10 µm using a series of

B. Magnetic flux density-control scheme
As a surrogate for *23 , operation of our MT device is based on control of the magnetic flux density emanating in the axial direction from the blunt end of the needle core, a quantity we define as MNOP* . Our feedback control routine utilizes a cascade configuration with a master or outer loop that uses a 1.0 kHz digital proportional-integral-derivative (PID) controller within the target computer for setpoint control of MNOP* , and a faster slave or inner analog loop operating within the bipolar power supply for setpoint control of . In the slave loop, the bipolar power supply drives to the setpoint corresponding to the analog voltage signal output from the master loop. The bipolar power supply is specifically designed for powering inductive loads with typical response times of ~220 to 280 µs at 1.7 kHz for a 2-mH nominal load. These specifications are similar to those reported for other MT devices 23,24,33 . The master loop uses consecutive 30-sample averages of MNOP* measurements to update analog output control voltages that are subsequently passed to the bipolar power supply (i.e., slave loop) (see Fig. 2).

A. Degaussing and magnetic properties of the needle core
Prior to any initial actuation of the MT device, a degaussing current is applied to the solenoid to demagnetize the needle as part of our standard operating protocol. Specifically, the degaussing routine is composed of 3 consecutive applications of a 60 Hz current waveform, ( ) Q.)R , defined as , where the initial current, J = 4 A, and spans from 0 to 5 s. To facilitate tuning of the PID control parameters for the master loop, we first characterized the magnetic behavior of the HyMu80 core. The longitudinal axis of the needle core was inclined at an angle of 25° from the horizontal (defined as the standard MT device configuration), and a second magnetic flux density sensor (Honeywell SS-495A1) was vertically oriented with the center of its sensing loop positioned ~100 µm lateral to the needle tip within the same horizontal plane. This second sensor was used to measure the magnetic flux density at this location/position, an experimental quantity defined here as *23 . Following degaussing of the core, the current in the solenoid was incrementally looped with the device configured for conventional feedback control of I from 0 A to −4 A, then from −4 A to 0 A, then from 0 A to 4 A, and finally from 4 A back to 0 A over a period of 30 s while recording MNOP* and *23 . The results of this characterization are shown in Fig. 3. The saturation flux density for MNOP* was approximately ±200 Gs with a linear response between ±150 Gs, and minimal hysteresis due to hydrogen annealing of the needle core. *23 was observed to exhibit a much more exaggerated hysteresis loop in response to the applied solenoid current loop, with a remnant magnetic flux density of roughly −1.95 Gs at the end of the 0 A to −4 A to 0 A sequence compared to remnant magnetic flux density of 0.75 Gs measured for MNOP* . Note that the signs of *23 and MNOP* are opposite, consistent with the physical orientation of magnetic field lines that are generated during actuation of the MT device.

B. Actuation response times
To demonstrate the operating characteristics of the MT device under MNOP* -control, a typical OFF/ON waveform sequence of 0.25 Hz is used, as shown in Fig. 4(a). The output of the magnetic flux density sensor following completion of the initial degaussing routine is set as the zero magnetic flux density reference point, or MNOP* __ = 0 Gs. PID gain schedules in LabVIEW TM were empirically derived to achieve the widest range of controllable MNOP*` settings (i.e., ±185 Gs) to maximize the range of magnetic actuation forces, 56 . PID gain settings were also set to minimize response times required to achieve these MNOP* With PID gain scheduling, our MT device was able to reliably control MNOP*` and MNOP* __ setpoints to within ±0.005 Gs (for time-averaged data) for magnetic flux density setpoints ranging from 0 Gs to ±185 Gs.
Collectively, response times over the entire range of MNOP*` setpoints were ≤75 ms (magnetization or loading response times), while response times to achieve MNOP* __ = 0 Gs were ≤100 ms (demagnetization or unloading response times). In comparison to the ~5 ms response times of a recently published MNOP* -controlled MT device 32 and the 1 to 5 ms-response times of most -control MT systems 23,34 , the response time of our MNOP* -controlled device is predictably slower, a consequence of the cascade feedback control scheme and the 1 kHz digital PID control loop. However, we note that with upgraded data acquisition AI/AO hardware, it would be possible to sample MNOP* at higher rates (~90 kS/s) while increasing the frequency of our digital PID control loop (~3 kHz), likely enabling faster magnetization/demagnetization response times. More importantly, with programmable digital PID gain scheduling, our cascade control loop enables optimization of response times without overshoot in flux density setpoints across the entire magnetic induction response of the core material, including within the domain of non-linear magnetic saturation. In contrast, given a specified set of analog circuit components, hardware-based feedback systems for MNOP* -control would be limited to optimized flux density responses within smaller defined domains of the overall magnetic induction response. Consequently, hardware-based feedback systems for MNOP* -control are more likely to exhibit overshoot in magnetic flux density setpoints during operation 32 .

C. Comparison to conventional current-based feedback control
In MNOP* -control mode, our MT instrument inherently achieves the prescribed MNOP* ON and OFF setpoints in a manner that (i) overcomes thermal effects that alter the magnetization properties of the core and (ii) negates the remnant magnetization that develops within the core following extinction of the field that was generated during the ON portion of the actuation sequence. Fig. 5 showcases these two major benefits of magnetic flux density-based feedback control as opposed to current-based feedback control.
Note how over the sequential ON segments of a typical 10-cycle 0/175 Gs OFF/ON 0.125 Hz waveform sequence, MNOP*` is maintained precisely at 175.0 Gs (Fig. 5(a)). However, due to heating of the core from the solenoid currents, the coil currents, , required to achieve this magnetization ON setpoint increase in magnitude with each actuation cycle. Conversely, if a similar 10-cycle OFF/ON waveform is conducted using -control with an ON setpoint of −3.0 A (higher current intentionally selected to illustrate thermal effects), the corresponding setpoints of MNOP*` diminish as cycle number increases ( Fig. 5(b)). Admittedly, in first 80 seconds in the -control MT experiment, the decrease in MNOP*` is small and would be associated with negligible variations in 56 . However, for experiments of a much longer duration or larger currents where heating of the core becomes more pronounced, magnetic flux density-based feedback control would provide more reliable cycle-to-cycle control of 56 . In Fig. 5(c), notice how the solenoid current for MNOP* -control is positive during the 0 Gs OFF setpoints the for cycles #2 through #10 (Fig. 5(c)). In other words, in the MNOP* -control scheme, a small reverse current is automatically generated to null the small remnant magnetization that develops within the core following the magnetization of the ON segment. In contrast, in -control mode, MNOP* OFF setpoints do not return to 0 Gs for cycles #2 through #10 as returns to 0 A (Fig. 5(d)).

D. Non-uniform remnant magnetization of the needle
To investigate whether MNOP* -control can serve as a reproducible surrogate for controlling *23 , we performed a series of experiments in which the MT device was oriented in the standard configuration and then actuated using OFF/ON MNOP* -controlled 0.25-Hz waveform sequences while simultaneously measuring the magnetic flux density, *23 , at a spatial location ~100 µm from the needle tip using a second vertically oriented Hall effect sensor (see Fig. 6(a) and Sec. IV.A). MNOP* __ was set to 0 Gs for all test waveforms, and MNOP*` setpoints spanned from 0 Gs to 185 Gs. A typical sequence for a 0/175 Gs MNOP* OFF/ON waveform is shown in Fig. 6  repeatably ±0.2 Gs over the 10 cycles of operation (see Fig. 7(b)). However, despite precise and accurate control of MNOP* __ setpoints as hallmarked by ∆ MNOP* __ variations of ±5 mGs (see Fig. 7(c)), the magnitude of ∆ *23 __ monotonically increases as a function of the MNOP*` setpoint (see Fig. 7(d)). In other words, even though we had nullified the remnant magnetization present at the blunt end of the core following actuation of the MT device, the needle tip still maintained a finite remnant magnetization at OFF setpoints.
Interestingly, evidence of non-uniform remnant magnetization of the needle core in a MNOP* -controlled MT device was also observed in the stepwise force characterization presented by Kah et al. 32 , where 5.09 µm-diameter superparamagnetic beads were subject to a finite MT force of ~0.23 nN at a bead-tip distance of ~30 µm despite controlling for a MNOP* __ setpoint of 0 Gs.
Spatially non-uniform remnant magnetization was also observed in other MT needle prototypes developed during the course of this work. Specifically, less pronounced differences in remnant magnetization between the needle tip and the blunt end of the core were observed for needles with a 15° taper compared to the 7.5° taper of the needle shown in Fig. 6(a). As such, we speculate that these observations might be attributed to differences in the length scale of magnetic domains within the HyMu80 Ni/Fe alloy relative to the length scales that define the bulk geometry of the core and its transition to the needle tip. Previous numerical modeling of MT devices has suggested that the magnitude of the magnetic field that develops within a soft ferromagnetic core in response to an energized solenoid varies along the longitudinal axis of the core as a function of core radius 33 . Assuming this to be true, it seems only logical to conclude that a spatially non-uniform remnant magnetization would be present upon removal of the applied magnetic field. Given the applied nature of this work, however, the true physical underpinnings of the apparent non-uniform remnant magnetization of the needle core were not further investigated.

E. Compensation for null magnetic force actuation
Even after degaussing the needle core, the MT device was observed to exert forces on superparamagnetic beads suspended in glycerol and positioned in close proximity to the tip, i.e., *23 ≠ 0 Gs. A phenomenon that others have also observed 35 , we postulate that the small magnetic field present at the needle tip following degaussing is due to either a distortion of the earth's magnetic field or a permanent magnetization of the needle tip. Collectively then, based on core characterization and MNOP* -control experiments (see Secs. IV.A and IV.D), we identified two sources of magnetization that could lead to non-zero *23 fields, and thus non-zero 56 , despite controlling MNOP* to output 0 Gs. The first source is referred to as permanent tip magnetization, i.e., the presence of a non-zero *23 despite degaussing of the needle core. The second source is referred to as the remnant tip magnetization, a term used to describe the apparent non-zero *23 that exists despite controlling MNOP* to output 0 Gs following any magnetization of the core. Higher frequency actuation is possible but was not further explored in this work.

V. CALIBRATION
Here, δ denotes the magnitude of the positional vector of the superparamagnetic bead, , calculated as the Euclidean separation distance between the centroid of the bead and the needle tip. Eq. (4) assumes low Reynolds numbers, i.e., equilibrium between 56 (δ) and the drag force that a spherical object experiences under laminar flow conditions. Reynolds numbers for calibrations conducted in silicone oil and glycerol were ~10 −8 and ~10 −7 , respectively. With this method of data reduction, a family of six discrete calibration δ-56 curves were generated, one for each MNOP*` * setpoint (where 56 denotes the scalar magnitude of 56 ).
Near field calibrations conducted with silicone oil are shown in Fig. 10 (see Fig. 1(a)). Arguably, this finding is consistent with the data shown in Fig. 3, where *23 measurements demonstrated much more rapid saturation in magnetic flux density with increasing solenoid currents compared to MNOP* . In other words, values of *23 seem to exhibit little to no change as MNOP* ranges between 150 Gs and 185 Gs. Alternatively, we considered the possibility that the superparamagnetic beads were saturating in response to *23 for MNOP*` * ≥ 150 Gs and thus leading to negligible increases 56 .
However, M-450 beads are not known to exhibit magnetic saturation for applied fields of <1000 Gs, so this possibility seems unlikely 36 . The more plausible physical possibility is that the needle core overall exhibits a spatially non-uniform magnetization in response to an applied field, consistent with our previous observations of an apparent spatially non-uniform remnant magnetization.
Calibrations conducted with silicone oil exhibited minor differences in 56 predictions when subject to small variations in room temperature and relative humidity that were present within the laboratory environment. In contrast, large variations in 56 for the same MNOP*` * flux density setpoint appeared for calibrations conducted in glycerol when deviations in ambient temperature and relative humidity were not accounted for in the assumed value of (see Fig. 11). Due to its hygroscopic nature, a thin film (<1000 µm) of pure glycerol exposed to laboratory conditions rapidly equilibrates with water vapor present in the ambient air forming a water-glycerol mixture. Consequently, the dynamic viscosity of this equilibrium glycerol-water mixture is a function of both relative humidity and temperature [37][38][39] . For example, at 20°C, viscosities of glycerol-water mixtures vary more than 12-fold from 765 cP to 60.1 cP as the relative humidity varies from 10% to 50% 38,40 . Similarly, though of smaller magnitude, A temperature increase from 20° to 30°C reduces the dynamic viscosity by ~50% 39 . Fig. 11 compares a MNOP*` * = 175 Gs calibration conducted in silicone oil at 25 o C to a calibration done in glycerol at 25°C. Here, the calibrations assume a dynamic viscosity for either pure glycerol based on a relative humidity of 0% or a water-glycerol mixture at 25°C that accounts for a relative humidity of 19% present in the ambient air. When humidity is properly accounted for, the silicone and glycerol calibrations demonstrate excellent quantitative agreement.
However, glycerol calibrations that do not account for humidity effects generate erroneous 56 values, with several-fold deviations in predicted magnetic actuation forces in the near field when compared to calibrations conducted in silicone. With this insight, it seems plausible that some of the discrepancy in force capabilities of previously described MT devices might be secondary to glycerol-based calibrations done in the absence of temperature and humidity assessments.
Fits of near field 56 (δ) calibration measurements were done for each of the six MNOP*` * flux density setpoints using the following three-term power law 33 : Fit parameters for each calibration curve are shown in Table I. Fit of the MNOP*` * =25 Gs was poor. Values of J increase with increasing magnitudes of MNOP*` * setpoints. With the exception of the 25 Gs MNOP*` * setpoint, ranged from ~2.3 to 2.9, consistent with the modeling predictions of Bijamov et al. 33 Moreover, we note that the best-fit values of δ J are ~10 µm, a value that is roughly equivalent to the radius of curvature of our needle tip, a finding also observed by Bijamov et al. 33 We attempted to assign a universal fit for all of the data as done by Kollmannsberger and Fabry 23 , but this led to unacceptable uncertainties in 56 (δ) predictions. Consequently, for our MT device, 56 (δ) are evaluated by means of three-parameter power-law fits specific to each of the six calibrated MNOP*` * flux density setpoints. Using a standard approach to uncertainty analysis 41 , defining ∆ 56 (δ) as the uncertainty in 56 (δ) for a 95% confidence level, and assuming ±1 µm for the uncertainty in δ at a 95% confidence level for experiments in which the needle tip and the superparamagnetic bead are both present within the same horizontal imaging plane, the relative uncertainty, ω 56 , defined here as Δ 56 (δ) / 56 (δ), for various MNOP*` * setpoints and for δ spanning from 2.5 µm to 50 µm are plotted in Fig. 12. Excluding the 25 Gs, 50 Gs, and 75 Gs MNOP*` * setpoints, relative uncertainties in 56 (δ) are <30% for 2.5 µm < δ < 40 µm, when using MNOP*` * setpoints ≥100 Gs. Note that small variations of superparamagnetic bead diameter are accounted for within the uncertainties of the power law fit parameters because each calibration curve was based on testing of at least 5 independent beads. Overall, our 56 (δ) uncertainties are quantitatively similar to those reported by Kollmansburger et al. 23

VI. CONCLUSIONS
As motivation for this work, we propose that a magnetic tweezers (MT) device, integrated with substrate deformation tracking microscopy (DTM) and traction force microscopy (TFM), can be used to investigate the biophysical mechanisms of human blistering skin disease. While integration and application of MT with DTM/TFM is presented in a companion paper 20 , here we have presented key findings on the design, assembly, operation, and calibration of an MT device that is amenable to replication in any biology laboratory. Unique to our device, we have employed feedback control of the magnetic flux density emanating from the blunt end of the soft ferromagnetic needle core as an alternative to conventional devices that are based on feedback control of solenoid current. Using a cascade control scheme with PID gain scheduling, loading and unloading response times of our device (<100 ms) are necessarily slower than conventional current-based feedback-controlled MT devices. However, by intentionally avoiding overshoot and by intrinsically compensating for thermal heating effects, magnetic flux density-based feedback control of an MT device achieves more reproducible magnetic fields at the needle tip and are hence less cycle-tocycle variation in magnetic actuation forces. Direct measurements of magnetic fields emanating from the needle tip suggest the existence of a small but finite permanent magnetization of the tip that remains even after degaussing of the needle core. Additionally, our observations suggest that the needle core experiences a spatially non-uniform magnetization in response to the applied field of the solenoid coil, where fields at the tip demonstrate more rapid saturation and increased remnant magnetization compared to fields emanating from the blunt end of the core. Further investigation will be required to determine the physical mechanisms that give rise to non-uniform magnetization of the core. Nevertheless, empirically derived control schemes can be developed to achieve null magnetic actuation forces and robust operation of an MT device based on magnetic flux density control. Rigorous calibration of our MT device has validated its ability to produce magnetic actuation forces up to 25 nN on 4.5 µm-diameter superparamagnetic beads with a maximum relative error of ±30% using magnetic flux density setpoints ≥100 Gs and for beads positioned between 2.5 µm and 40 µm from the needle tip.

VII. SUPPLEMENTARY MATERIAL
A comprehensive list of abbreviations and symbols used in this work, photographs of the magnetic tweezers instrumentation and setup, and far field calibration data can be found in the Supplementary Material.          9. Force calibration of the MT device. A series of DIC images at select timepoints are shown demonstrating movement of a 4.5 µm-diameter superparamagnetic bead suspended in 10,000 cSt silicone oil and subject to actuation with the MT device using a 6-cycle OFF/ON 0.25 Hz MNOP* -control waveform with MNOP*` * = 175 Gs. MNOP* __ control setpoints were programmed to maintain null actuation forces during OFF portions of the calibration sequence. Time stamps correlate with the real-time movie of the calibration sequence that can be found here (Vid 1). During calibration, the bead drifts slightly away from the needle tip during the OFF segments of each actuation cycle, as visualized by the eccentricity between the bead's tracked centroid at the beginning of each OFF segment (black cross) and the bead's tracked outer diameter (OD) at the end of each OFF segment (blue dashed circle). Bead drift is unrelated to magnetic actuation.