the cardiac myocyte is a key experimental system for exploring the mechanical properties of the diseased and healthy heart. Compared with molecular or myofibril studies, the contractile protein apparatus is arrayed in a physiologically relevant orientation within a living cell. Furthermore, use of myocytes avoids problems inherent to multicellular preparations, including heterogeneity of cell types (34), diffusion-limited extracellular spaces (10, 24), nonuniform shortening of sarcomeres during isometric contraction, and the mechanical influence of the extracellular matrix (25) and allows clear optical interrogation of sarcomere length (SL).

The dynamic and passive mechanical properties of muscle tissue have been explored with many force and length stimuli, including steps, ramps, sinusoids, and pseudorandom binary sequences (for review see Ref. 13). To demonstrate the capabilities of our instrument, we used stochastic system identification techniques to measure the dynamic stiffness of a passive ventricular myocyte. Because, to the authors' knowledge, measurement of the dynamic stiffness of an intact mammalian myocyte has not been published, it is possible that features of the observed transfer function in muscle tissue are influenced by complications inherent to multicellular experiments. In addition, the myocyte is a clear division for anatomically based finite-element models of the heart (15). Stochastic system identification provides a framework for generating thorough linear and nonlinear models of individual cells under various conditions that could abstract the underlying molecular biology of the cell, balancing computational and scientific complexity.

Protocols for isolating millions of functional, primary cardiac muscle cells from ventricular tissue are readily available (1, 2, 8). Myocytes are typically used for experimentation within 4–6 h of isolation. However, with currently available instrumentation, it is difficult for a single researcher to successfully explore the mechanical characteristics of more than one or two myocytes under loaded conditions within this 4- to 6-h period. An array that could perform these measurements on 10–100 cells in parallel would be advantageous from ethical and scientific perspectives.

To provide a suitable basis for an instrument array, the core elements of the mechanical testing device (actuator, force sensor, and attachment strategy) should be modular, compact, and inexpensive. However, the majority of instruments used to measure the mechanical properties of cardiac muscle cells are not designed to meet these criteria. Currently available instruments typically use commercial displacement actuators that are expensive and bulky (when associated electronics are considered) to stretch living muscle cells. These instruments often apply cantilevers as force transducers. For example, researchers have used optical fibers, suction pipettes, glass needles, steel foil (32), microfabricated polysilicone beams (23, 31), and 12-μm wire (18). The position sensor used to detect the deflection of the cantilever is the key element of this type of force transducer, and the compromise between resolution and cost is critical for an array application.

Finally, the attachment of myocytes to the force sensor and actuator remains a significant challenge and is limited by the fragility of the sarcolemma and the need to support large forces during contraction (for review see Ref. 9). Of the many attachment methodologies attempted, the etched carbon fiber approach, developed by Le Guennec et al. (22) and adapted by Yasuda et al. (36), was perhaps the most suitable for an array application because of its simplicity and reasonable attachment strength. However, the cells are bound on only one side, introducing the possibility of shear.

Here we describe the design and development of a modular instrument and novel attachment mechanism that can be used to explore the mechanics of single muscle cells and is suitable for use in an instrument array for high-throughput science. We present the mechanical and electrical characteristics of the device and demonstrate its functionality by providing the first measure of the passive dynamic stiffness of a single intact ventricular myocyte at varied SLs.

DESIGN CONSIDERATIONS

The core of the instrument (Fig. 1), which is conceptually similar to the design of Iwazumi (18), utilizes two Lorentz force actuators that can simultaneously be used as force sensors. A permandur motor structure, which concentrates a magnetic field in two air gaps, was manufactured using a wire electrical discharge machine (EDM; Charmilles, Lincolnshire, IL) and a five-axis machining center (HAAS Automation, Oxnard, CA). Stainless steel cantilevers (with a rectangular section removed from the center to form a loop) serve as the actuator and force sensor. The long arms of the cantilever loop pass through the air gap magnetic field, providing the basis for actuation. A bench-top deposition system (Para Tech Coating, Aliso Viejo, CA) was used to coat the motor structure in 3 μm of parylene to ensure biological compatibility and provide electrical insulation. Optical position sensors reflect from the back surface of each cantilever, their photodiode current is amplified, and the resulting position signal can be used to control the current source driving the actuator. All electronics necessary to amplify position signals and drive currents through the cantilevers are built into the module (Fig. 2). Circuit design and board layout utilized PSpice and Allegro packaged within Cadence Design Systems software (Cadence Design Systems, San Jose, CA), and double-sided circuit boards were manufactured using a board engraver (LPKF Laser & Electronics, Garbsen, Germany).

Data Acquisition, Signal Processing, and Control

Data acquisition, signal processing, and digital control were implemented using a data acquisition card (model 6052E, National Instruments, Austin, TX) and custom Visual Basic.Net (VB.Net) code. Four analog inputs, two position signals and two current signals, were each sampled at 2 kHz, and two analog outputs set the current through the cantilevers. Signal-processing algorithms were implemented in VB.Net and Matlab to perform digital control, power spectral estimation, and stochastic system identification to estimate samples of the impulse response or Markov parameters between two data sets (typically a force input and position output).

Stochastic system identification.

A typical experiment to probe the impulse response of a system (a cantilever or myocyte) applied a 100-s, filtered, zero-mean, unit-variance, Gaussian white noise input sampled at 2 kHz. The filter was chosen to scale and distribute the power applied to the system over the appropriate measurement bandwidth. To find the impulse response, 2,048-point auto- and cross-correlation estimates were made of the input (x) and output (y) and used to populate the following equation (19)

where R_XX is a Toeplitz matrix of autocorrelation coefficients, R_XY is a vector of cross-correlation estimates, h_XY is a vector of samples of the impulse response, and n = 2,048 in this case. Equation 1 was solved for h_XY by inversion of the Toeplitz matrix using a Levinson recursion algorithm implemented in Matlab.

The quality of the system identification was assessed by calculating the variance accounted for or coherence squared, coh²(jω), using the following equation (19)

where S_XY(jω), S_XX(jω), and S_YY(jω) are the cross, input, and output power spectral densities, respectively. Conceptually, coh²(jω) will be less than unity at a particular frequency if there is noise or interference in the data at that frequency (e.g., sensor noise or extraneous inputs, such as mechanical interference) and/or nonlinearities in the system under test (e.g., harmonics).

Characterizing a sample.

A detailed description of the modeling strategy used to characterize a sample is beyond the scope of this work. Briefly, static, modal, and harmonic finite-element analyses in Ansys (Ansys, Canonsburg, PA) were used to assess the effect of loading the cantilevers with a sample with material properties similar to a myocyte. For frequencies up to the first resonance, an invertible lumped-parameter model (Fig. 3 and Eq. 3) of the dynamic stiffness of cantilevers 1 2, H₁(s) and H₂(s), and the muscle cell, H_M(s), was adequate to describe the dynamics of the system. The multi-input–multi-output lumped-parameter model used the forces applied to the cantilevers as inputs and the displacements of the tips of the cantilevers as outputs

To explore the properties of a myocyte suspended between the cantilevers, we applied a 100-s force signal (F₁) to cantilever 1 (Fig. 3) sampled at 2 kHz. The applied current was typically band limited between 0.1 and 100 Hz and scaled to produce a standard deviation of position and force of <1 μm and <2 μN, respectively. The positions of the tips of cantilevers 1 and 2 (x₁ and x₂) were recorded, and the transfer function X₁/X₂(jω) (the Fourier transform of h_X2X1) was estimated using Eq. 1. The dynamic stiffness of the sample could then be estimated by rearranging the second row of Eq. 4a to give where H_M(jω) and H₂(jω) represent the dynamic stiffness of the sample and cantilever 2, respectively.

SL estimation.

A digital microscope was custom-built using a ×10, ×20, or ×50 lens (M-Plan Apo, Mitutoyo America, Aurora, IL), an infinity-corrected tube lens (Edmund Optics, Barrington, NJ), and a digital video camera (model DFW-SX900 CMOS, Sony, Tokyo, Japan). It was used to capture 1,280 × 960 pixel images of the myocyte at 7 frames/s. The spatial Fourier transform of an image of a 2-μm graticule was used to find the sensitivity of the digital microscope (4.6 pixels/μm). To estimate SL, the dominant spatial period (found using the spatial Fourier transform) of each row of a user-defined region of the myocyte image was averaged. Each row was zero padded to reduce the bin spacing of the transform and band-pass filtered to remove extraneous information.

In good preparations, we found that the SL varied across the width of the cell by approximately ±0.02 μm (± the standard deviation of the distribution of SL estimates from each row). The variance of the SL estimation was used as an indication of the quality of the cell under experimentation. For example, if the sarcomeres responded unevenly to stretch across the width of a cell, the myocyte was discarded. It was difficult to determine the SL in the immediate vicinity of the attachment region because of the curved surface of the borosilicate glass capillary beneath the cell (Fig. 4).

Position Sensor

The position sensor was a critical element of the design, inasmuch as its resolution limited the minimum detectable force and position. A commercially available confocal sensor consisting of a light-emitting diode, a split lens, and a photodiode (model HEDS-1300, Agilent, Palo Alto, CA) was selected for its simplicity, robustness, low cost, and ease of alignment (3). These characteristics made it ideal for use in an instrument array.

A low-noise transimpedance amplifier was designed to ensure that the position resolution was limited by the shot noise of the photodiode within the sensor (model HEDS-1300). Circuit noise and interference were quantified at several operating points (current levels through the photodiode) by estimation of the power spectra of the position signal output. In the present device iteration with the cantilevers fully immersed in physiological saline, the shot noise of the circuit was limited between 10 Hz and 10 kHz with a typical noise floor of 0.75 nm/Hz and sensitivity of ∼20 ×10³ V/m.

Force Sensor

The cantilevers, which were typically 5 mm long × 1 mm wide × 0.025 mm thick, were cut from 25-μm annealed stainless steel 316 foil using the wire EDM. Stainless steel 316 was used for its high conductivity, low cost, reasonable reflectivity, and low magnetic susceptibility (provided it was annealed after it was cold worked). Our original choice of stainless steel 304 was abandoned when it was found that the cantilevers became magnetized over time by the strong magnetic field within the motor structure.

Early designs used straight sections of wire or ribbon fixed at both ends, which simplified the motor structure, inasmuch as only one air gap with a uniform magnetic field was required. However, the ratio of stiffness to first resonant frequency for a uniform beam fixed at both ends is about eight times that of a beam fixed only at one end (27). Therefore, a cantilever provides better force resolution (lower stiffness) for a given measurement bandwidth (set by the resonant frequency). This motivated the design of a novel cantilever actuator with a rectangular section removed from its center to create a current loop (Fig. 1).

It is desirable for the force sensor to be significantly stiffer than the myocyte, so that the loading effects of the myocyte can be ignored. However, primarily because of limitations in position sensor resolution, this is difficult to achieve. To appreciate this problem, consider the position resolution required to measure the stiffness of muscle tissue. Rearranging Eq. 4 and treating the muscle and cantilevers as simple springs, the relationship between the displacement applied to one side of the cell, Δx₁ (the actuator), and the displacement of the other cantilever, Δx₂ (the force sensor), is approximately

where k_m and k₂ represent magnitude of the stiffness of the muscle and force sensor, respectively. The displacement, Δx₁, is often limited to avoid nonlinear effects in the muscle tissue, and the position sensor resolution provides a lower bound on Δx₂. The combination of these constraints sets the maximum stiffness, k₂, of the force sensor. We have found that female guinea pig ventricular myocytes have stiffness between 0.04 and 0.4 N/m at 0 Hz, depending on cell length and contraction state. The cantilevers were designed with a stiffness of 2 N/m. In the worst case, when H_m = 0.04 N/m, only 2% of the displacement applied by the actuator will couple through the cell to the force sensor.

Motor Design and Actuation

The primary goal of the mechanical structure of the motor was maximization of the coupling between current and force by concentration of the magnetic flux in the air gaps though which the arms of the cantilevers passed. However, it was also desirable that the structure be fully submersible in physiological saline (to avoid the effects of surface tension and evaporation on the cantilevers), biologically inert, and electrically insulated and, also, provide means for transmission microscopy of the cell while it was attached to the cantilevers (necessary for direct optical measurement of SL). The design illustrated in Fig. 1 met these objectives. The core of the device was the cantilever assembly consisting of the two cantilevers, each attached to 100-μm-thick glass rectangles separated by a 130-μm-diameter multimode optical fiber. The cantilever assembly was glued onto one-half of the motor structure, and the remaining elements of the motor were held together by magnetic attraction. Small aluminum spacers ensured that the structure was stable and supported two stainless steel tubes that were inserted into fluidic inlets at the base of the structure.

Magnetic design.

Because of its high saturation flux density, permandur was used as the base material for the motor structure. Nickel-plated, sintered, neodynium-iron-boron magnets (Dexter, Elk Grove Village, IL) were used for their high residual magnetic flux density. The main structure of the motor was designed using the computer-aided design software Solid Edge (UGS, Plano, TX) and manufactured using the wire EDM and five-axis machining center. Three-dimensional magnetic finite-element analysis (Ansys) and iterative redesign were used to optimize the motor structure by identifying regions of saturation or peak flux density in the permandur field guides and flux leakage outside the air gap. The simulation used ∼106,000 eight-node elements (Solid96, Ansys). The magnets were modeled with isotropic relative magnetic permeability: μ_r = 1.044 and H_c = 0.6 × 10⁶ A/m. The permandur was modeled as an isotropic, nonlinear material, with a 10-point linearly interpolated function used to relate magnetic flux density (B) to magnetic field intensity (H). Thermal effects and the magnetic fields produced by passing current through the cantilevers were ignored in this study.

Fluidics.

The motor structure was designed to be inserted into a glass tube (15 mm wide × 6 mm long × 12 mm high), which could then be sealed at the base using Compound 111 (Dow Corning, Midland, MI). This approach allowed the entire motor to be submerged during tests. Solutions were applied through two inlets in the base of the motor structure or through the open top of the glass tube. A small section of a coverslip was suspended over the bath to ensure a flat surface for imaging and to allow the stable formation of a positive meniscus above the glass tube. A small nozzle inserted into the meniscus was used to pump fluid from the bath and was adjusted to minimize mechanical disruption of the surface. The temperature of the fluid in the bath was recorded during experiments using a resistive temperature detector.

Actuation.

The motor structure was mounted directly on a circuit board, and the ends of the cantilevers were soldered to two independent, low-noise current sources, each capable of providing up to ±0.4 A. The resistance of the cantilevers was ∼4 Ω, and 1 μW of power was dissipated into the solution during a typical measurement of the dynamic stiffness of a myocyte (σ_force ∼ 1 μN). The current-force relationship applied to each cantilever (referred to the tip) was approximately the same, typically between 1 × 10⁻³ and 1.1 × 10⁻³ N/A, corresponding to an air gap magnetic flux density of ∼ 1.0 T. The motor could provide peak forces of ±400 μN and peak displacements of approximately ±200 μm. The Lorentz force driving the actuation was distributed along the cantilever arms starting ∼1.5 mm below their top edge. The estimation of cantilever and myocyte stiffness assumes that force is applied to the tip of the cantilevers at the point of cell attachment. The system was calibrated such that all forces were expressed as an equivalent force at this point.

System Calibration

For calibration of the position sensors in the apparatus before each test, the digital microscope was focused on the top edge of each cantilever as it was being driven by a 0.5-Hz sinusoidal current. Custom edge-detection code written in Matlab was used to track the displacement. The position sensor sensitivity was typically ∼20 × 10³ V/m when the cantilevers where immersed in physiological saline.

The force-current relationship was calibrated using a combination of modeling and stochastic system identification. Briefly, a finite-element model of the cantilever structure in a vacuum was generated using Ansys. Eleven simulations were run over 3- to 6-mm lengths. The results of the simulation were used to express the change in the first resonant frequency and stiffness at 0 Hz (estimated by application of a 1-mN force distributed across the top edge) as functions of cantilever length.

Stochastic system identification was then used to measure the mechanical transfer function of each cantilever between 0.1 Hz and 10 kHz. Nonlinear least-squares estimation was used to fit the estimated Markov parameters to a second-order mechanical impulse response. Comparison of the empirical resonant frequency with the finite-element model results provided an estimate of the stiffness of the cantilever. Then the displacement resulting from a known current was measured using the digital microscope to determine the current-force relationship.

Myocyte Isolation, Loading, and Attachment

Isolation.

Standard enzymatic techniques (4, 29) were used to isolate ventricular myocytes from ∼2- to 3-mo-old Dunkin-Hartley female guinea pigs: The experimental protocol was approved by the Institutional Animal Care and Use Committee at the Massachusetts Institute of Technology and was conducted in accordance with the National Institute of Health Guide for the Use and Care of Laboratory Animals. Briefly, animals were anesthetized with Telazol (50 mg/kg; equal parts by weight of tiltetamine-HCl and zolazepam-HCl) and then decapitated, and the heart was quickly excised and placed in heparin-Tyrode solution (140 mM NaCl, 5.4 mM KCl, 1 mM MgCl₂, 1.8 mM CaCl₂, 5 mM HEPES, 11 mM glucose, and 500 IU/L heparin). The heart was mounted on a Langendorff apparatus and perfused via the aorta with heparin-Tyrode-EGTA solution (140 mM NaCl, 5.4 mM KCl, 1 mM MgCl₂, 5 mM HEPES, 11 mM glucose, 1 mM EGTA, and 500 IU/L heparin) and high-K⁺ solution (mM: 4 NaCl, 10 KCl, 1 MgCl₂, 0.025 CaCl₂, 130 K-glutamate, 5 HEPES, and 11 glucose) and then collagenase solution (high-K⁺ solution + 1 g/l collagenase). The ventricles were cut from the heart, and the myocytes were mechanically dissociated in high-K⁺ solution through a series of steps. Cell suspensions were spun in a centrifuge at 300 rpm for 1 min, and the pellet was suspended in BSA-Tyrode solution (normal Tyrode solution with 1 g/l BSA and 620 IU/L trypsin inhibitor). Myocytes were used for experimentation within 4–6 h of isolation.

Loading.

The motor structure was immersed in oxygenated Tyrode solution, and dilute cell suspension (10 μl) was placed on a custom stage directly above the cantilevers. The cells were illuminated using the optical fiber passing through the center of the motor (Fig. 1B) and imaged using the digital microscope. The myocytes were stimulated by 5- to 10-V, 0.2-Hz, 10-ms pulses of 100-μm-diameter platinum electrodes. Given the configuration of the electrodes and the presence of the metallic cantilevers, it was difficult to accurately estimate the electric field strength applied to the cells. A cell was selected for experimentation if it contracted >5% under electric field stimulation and its sarcomere pattern was ordered (with average resting SL = 1.7–1.9 μm). A pulled borosilicate glass capillary with a ∼2-μm-diameter tip was gently touched against the cell and allowed to attach for ∼20 s. The cell was then lifted from the inspection stage and lowered onto the cantilevers.

Attachment.

Several different strategies and materials were explored to promote cell attachment. The attachment characteristics of isolated cells were highly dependent on the collagenase-protease blend used to isolate them. We achieved the best results with a novel clamp structure on each capillary (Fig. 4) and the C8176 collagenase blend (Sigma-Aldrich, St. Louis, MO). The cell was clamped between a square-cross-section borosilicate glass capillary and an etched carbon fiber (kindly provided by Prof. Peter Kohl, University of Oxford) (22). Although the fibers used in this study are not commercially available, it is possible that the fibers of Yasuda et al. (36) could be applied in a similar manner. The carbon fiber was glued at one end to the top surface of the glass capillary. For attachment of a cell, the carbon fiber was bent upward, and the cell was gently placed between it and the glass capillary. The carbon fiber was then slowly lowered onto the cell. For adjustment of the clamp force, the distance between the cell and the point where the carbon fiber was glued to the glass was varied. Very good attachment could be achieved with mild deformation of the cell.

The attachment mechanism could support significant forces and typically released the cell only while it was dying. When passive myocytes were slowly stretched to SL of 2.4 μm, 2- to 3-μN forces were applied. Twitch forces as high as 4.3 μN were measured without detachment (at diastolic SL ∼ 2.2 μm). However, in experiments in which twitch force vs. SL was explored using 0.2-Hz, 10-ms electric field pulses, spontaneous contractions leading to cell death often began when the cells were producing ∼2 μN of force at SL of 2.0–2.1 μm. We hypothesize that the rapid generation of force during the twitch response damaged the cells. After cell death, debris on the etched carbon fiber could be removed using the fine tip of the loading capillary. The attachment apparatus was typically replaced after 10 experimental days.

In the present system, the process of loading and attaching an individual myocyte required practice. However, once familiar with the approach, we were able to experiment on three to five cells per 4-h experimental period (during which the cells remained viable). Once the myocyte was in position and the clamp was lowered, attachment (defined as the ability to apply displacements to the cell) was achieved ∼90% of the time. The primary limitation was accidental cell death during cell alignment and clamp closure. The probability of success for the combined process of loading and attaching a well-aligned cell with ordered sarcomeres that responded strongly to an applied electric field was ∼20%.

EXPERIMENTAL VALIDATION

To demonstrate the viability of cells attached to the cantilevers, the twitch force produced by a myocyte was measured. A cell was selected and attached to the top of the cantilevers and then allowed to stabilize for 5 min in 28°C oxygenated Tyrode solution with 1.8 mM Ca²⁺. An electrical stimulus was applied (10 ms, 5- to 10-V pulses at 0.2 Hz), and the cell was stretched to an SL of 2.0 μm. Five sequential twitches were recorded, and force was measured (Fig. 5).

To explore the functionality of the instrument, we measured the passive dynamic stiffness of a single ventricular myocyte. A cell was again selected and attached to the top of the cantilevers and allowed to stabilize for 5 min in 28°C oxygenated Tyrode solution with 1.8 mM Ca²⁺. After the resting SL (1.81 μm) was measured, mild tension was applied (producing ∼5% strain). After the pretensioning procedure, the cell had a SL of 1.9 μm and was 21 μm wide and 86 μm long (measured between the attachment points).

To measure the dynamic stiffness, we applied a 10⁶-point displacement signal with Gaussian distribution and 1-μm standard deviation (±4 μm peak) to the actuator cantilever. The signal was sampled at 2 kHz and shaped using a custom-made digital filter with a cutoff between 40 and 60 Hz. Displacements of the actuator and force sensor cantilevers were recorded. The cell was then stretched to SL of 2.14 and 2.37 μm, and the measurement was repeated. Good agreement between the relative increase in SL and the increase in cell length (96 and 105 μm) for these stretches demonstrates that the attachment mechanism was successfully transmitting motion of the cantilevers to displacement of the contractile apparatus within the cell.

Equations 5 and 2 were used to estimate the dynamic stiffness of the myocyte (H_M(jω) in Eq. 5) and coh²(jω) of the measurement at 215 frequencies linearly separated by 0.31 Hz (Fig. 6). The dynamic stiffness of the force sensor cantilever (H₂(jω) in Eq. 5) was measured independently before the cell was attached and after it was removed using a 10⁶-point stochastic displacement signal scaled to produce a standard deviation of 10 μm [to improve the signal-to-noise ratio (SNR)]. The stiffness of the cell was converted to a modulus by optical measurement of cell width and with the assumptions that the cross section of the cell was elliptical and that the minor axis of the cross section was one-third of the major axis (12, 33). To demonstrate the increase in dynamic modulus with SL, the magnitude of the dynamic stiffness at 10 frequency points between 9 and 12 Hz was averaged, and the results are plotted vs. SL in Fig. 7.

The 1-μm standard deviation (±4 μm peak) displacements applied to the actuator cantilever would be large enough to disrupt any strongly bound cross bridges. The peak displacements are considerably above the noise floor of the position sensor. However, a band-limited, stochastically varying signal distributes power across many frequencies. Furthermore, at short SL, when H_m(jω) at 0 Hz was ∼0.04 N/m, only 2% of the applied displacement would couple through the cell to the force sensor cantilever. The power spectra of the measured displacement of the actuator and force sensor cantilevers at SL of 1.9 and 2.37 μm are presented Fig. 8. The noise floor or power spectrum of the position signal when the force sensor cantilever was stationary is also given in Fig. 8. The SNR of the force sensor displacement was ∼2.4 at 30 Hz (Fig. 8A); at a longer SL (Fig. 8B), the SNR is 4.4 at 30 Hz, as the muscle cell is considerably stiffer and the displacement of the force sensor is larger (as suggested by Eq. 6).

Also, displacement power applied to the actuator cantilever was deliberately increased for frequencies <10 Hz. The increase in displacement was required because of significant low-frequency position sensor noise that rose above the shot noise limit at ∼10–20 Hz (Fig. 8). Without frequency shaping, larger displacements would have been needed across all frequencies to maintain the same coh²(jω) at low frequencies (significantly increasing the standard deviation of the applied displacement).

DISCUSSION

We have described the design and development of a modular instrument for exploring the mechanics of intact mammalian myocytes. The instrument utilized a novel motor design and attachment strategy that allowed cells to be stretched to SL >2.4 μm. To verify the viability of the myocytes in the device, a representative series of myocyte contractions were captured. To demonstrate the operation of the instrument, stochastic system identification techniques were used to measure the dynamic stiffness of a myocyte in 28°C 1.8 mM Ca²⁺-Tyrode solution at 1–60 Hz at three SLs. To the authors' knowledge, this represents the first measurement of the dynamic stiffness of a single intact mammalian cardiac myocyte.

A frequency-shaped, 100-s noise sequence with a Gaussian amplitude probability distribution was used to perform the stochastic system identification and generate estimates of the dynamic stiffness of a single cell at 215 distinct frequencies. As discussed by Rossmanith (28), the use of broad-band signals, such as shaped white noise, instead of sinusoids, is advantageous, because it can greatly reduce the time required to measure the complex modulus at low frequencies and simplify nonlinear analysis. The standard deviation of the stimuli used in this study was ∼1 μm (1% strain), and the peak-to-peak variation in displacement was ±4 μm. These displacements were large enough to disrupt any strongly bound cross bridges (for review see Ref. 16).

As illustrated in Fig. 8, the larger displacements were necessary to ensure that displacement of the force sensor was above the noise floor. The coh²(jω) in Fig. 7 is less than unity, indicating the presence of noise in the data or nonlinearities in the system. The SNR improves as the cell is stretched, and it is likely, in part, responsible for the parallel increase in coh²(jω). This hypothesis is further supported at all SL by the joint reduction in SNR and coh²(jω) at frequencies <3 Hz, that occurs as the position sensor noise rises faster than the applied displacement (Fig. 8). The position sensor in this instrument is ideal for an array application because of its low cost, robustness, and ease of alignment. We are considering design modifications to boost the resolution of the sensor by an order of magnitude, which would allow an equivalent reduction in the applied displacement.

Activated muscle is highly nonlinear, as demonstrated by the classic step experiments of Huxley and Simmons (17). Several researchers observed nonlinearities in the form of harmonics when sinusoids with amplitude >0.02–0.2% of muscle length were applied to activated rabbit psoas muscle or glycerinated flight muscle (7, 20). Although we measured passive mechanical properties in this proof-of-principle experiment, it is possible that a nonlinear response of the muscle was also responsible for the reduction in coh²(jω). The larger-amplitude displacements at low frequencies due to input noise shaping might have further contributed to the lower coh²(jω) at these frequencies.

The passive dynamic stiffness measured in this study can be compared with that measured by other researchers using intact cardiac muscle tissue. Shibata et al. (30) measured the passive dynamic stiffness of intact papillary muscle from rabbits at 24°C with 2.5 mM external Ca²⁺ at eight frequencies between 0.05 and 30 Hz. Their results show the same gradual increase in stiffness and phase we observed, and the magnitude of the dynamic modulus was on a similar order (∼80 kPa at the muscle length that produced peak force). Using sinusoids with amplitude equal to 4.06% of muscle length over a temperature range of 5–37°C, Pinto and Fung (11, 26) observed the same gradual increase in the magnitude of the passive dynamic stiffness of rabbit papillary muscle at 0.01–100 Hz.

Finally, Kirton et al. (21) conducted a detailed analysis of the effect of 2,3-butanedione monoxime (BDM) and external Ca²⁺ concentration on the dynamic stiffness of intact passive rat trabecula at 20°C. They also measured the same gradual increase in magnitude and phase when 20 mM BDM was applied. However, at 1.25 mM external Ca²⁺ without BDM, they observed significant changes in phase as a function of frequency, possibly suggesting that the muscle tissue was mildly activated. Although our external Ca²⁺ concentration was higher than that used by Kirton et al., we did not observe significant changes in phase in this study. The phase response observed by Kirton et al. has been associated with the cycling of cross bridges and the generation of force. It is possible that the larger displacements used in our proof-of-principle experiment disrupted strongly bound cross bridges and, hence, muscle activation.

The stress-strain (or, equivalently, force-SL) relationship of passive cardiac muscle is highly nonlinear and time varying and demonstrates significant viscoelasticity. Step or ramp disturbances are the most common means to probe the dynamics of this relationship. For example, researchers measure the force required to apply large ramp displacements of varied magnitude to a muscle sample at constant velocity using a stretch-release protocol (5, 14, 35). The dynamic stiffness results presented in Fig. 5 were captured using relatively small-amplitude, 100-s, zero-mean displacements centered around three SLs. However, they can be compared with results gathered by stretch-release protocols, for example, the work of Helmes et al. (14), by recognizing that 1) the gradient of the stress-strain relationship is constant for small changes in muscle length around a given SL (i.e., for small displacements, the muscle has approximately constant stiffness), and 2) Helmes et al. found that when a series of displacement ramps were applied to muscle tissue, the time-varying directional hysteresis in the stress-strain relationship disappeared (see Fig. 5A of Ref. 14). With these points in mind, the modulus of the dynamic stiffness at a particular frequency can be converted to a stiffness that represents the gradient of the release curve in a passive force-muscle length relationship. The increase in dynamic modulus with frequency is analogous to the increase in the gradient of the force-SL relationship with increasing ramp velocity (see Fig. 6 in Ref. 14).

Although they are uncommon in cardiac muscle research, band-limited stochastic inputs excite more features of a system than ramp or step signals. As such, they have the potential, when combined with stochastic system identification techniques, to produce time-varying, nonlinear models that could be used to predict the response of passive muscle to steps and ramps. The results presented here represent a step toward that goal.

The novel motor structure described here has several important advantages. 1) The unique means of actuation of the cantilever, analogous to a single loop of a conductor in a voice coil motor, ensures that the actuator/force sensor has very low moving mass. 2) This design conveys flexibility, inasmuch as the dimensions of the cantilevers can be adjusted to tune the trade-off between bandwidth (resonant frequency) and force sensitivity (stiffness). 3) By designing the motor system such that it could be inserted into a glass tube and fully immersed, we avoid interference due to surface tension and evaporation. 4) The motor structure was well suited to feedback control of cantilever position.

Another novel aspect of this work was the attachment methodology. The approach combined a mild clamping force with the inherent attachment of intact myocytes to borosilicate glass and etched carbon fiber. This technique is still far from the ideal attachment to an intact myocyte that would couple forces through natural protein structures such as the intercalated disks or costameres (for review see Ref. 6). However, the strategy was repeatable, gentle enough to avoid damage to the cells, and strong enough to support significant forces and, also, successfully coupled external displacements to the contractile apparatus. Furthermore, the cells were firmly attached immediately after clamping, an important property for rapid parallel testing.

The primary goal of this work was to design a system that was capable of measuring the mechanical properties of individual cells and was also appropriate for use in an instrument array. To be suitable for an array, the major components of the design, including the actuators, position sensors, fluidics, and data acquisition system, must be compact and inexpensive. The majority of these requirements were satisfied by the present design. It is also necessary to simplify as much as possible the challenge of working with single myocytes. An average of 5 min are needed to select and load a cell onto the cantilevers. The loading time is limited by the manual manipulation of the cells.

In conclusion, we have described the design and development of a modular instrument that has successfully been used to make the first measurement of the passive dynamic stiffness of intact ventricular myocytes. The system utilized a novel motor structure and attachment mechanism and represents a first step toward an instrument array for high-throughput single-cell muscle physiology.

GRANTS

This research was supported by, or supported in part by, the US Army through the Institute for Soldier Nanotechnologies, under Contract DAAD-19-02-D-0002 with the US Army Research Office. The content does not necessarily reflect the position of the Government, and no official endorsement should be inferred.

FOOTNOTES

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