Cytoskeletal Mechanics in Pressure-Overload Cardiac Hypertrophy
We have shown that the cellular contractile dysfunction characteristic of pressure-overload cardiac hypertrophy results not from an abnormality intrinsic to the myofilament portion of the cardiocyte cytoskeleton but rather from an increased density of the microtubule component of the extramyofilament portion of the cardiocyte cytoskeleton. To determine how, in physical terms, this increased microtubule density mechanically overloads the contractile apparatus at the cellular level, we measured cytoskeletal stiffness and apparent viscosity in isolated cardiocytes via magnetic twisting cytometry, a technique by which magnetically induced force is applied directly to the cytoskeleton through integrin-coupled ferromagnetic beads coated with Arg-Gly-Asp (RGD) peptide. Measurements were made in two groups of cardiocytes from cats with right ventricular (RV) hypertrophy induced by pulmonary artery banding: (1) those from the pressure-overloaded RV and (2) those from the normally loaded same-animal control left ventricle (LV). Cytoskeletal stiffness increased almost twofold, from 8.53±0.77 dyne/cm2 in the normally loaded LV cardiocytes to 16.46±1.32 dyne/cm2 in the hypertrophied RV cardiocytes. Cytoskeletal apparent viscosity increased almost fourfold, from 20.97±1.92 poise in the normally loaded LV cardiocytes to 87.85±6.95 poise in the hypertrophied RV cardiocytes. In addition to these baseline data showing differing stiffness and, especially, apparent viscosity in the two groups of cardiocytes, microtubule depolymerization by colchicine was found to return both the stiffness and the apparent viscosity of the pressure overload–hypertrophied RV cells fully to normal. Conversely, microtubule hyperpolymerization by taxol increased the stiffness and apparent viscosity values of normally loaded LV cardiocytes to the abnormal values given above for pressure-hypertrophied RV cardiocytes. Thus, increased microtubule density constitutes primarily a viscous load on the cardiocyte contractile apparatus in pressure-overload cardiac hypertrophy.
It has been clear for some time that myocardium hypertrophying in response to a substantial, sustained increase in systolic pressure loading is characterized by linked contractile and energetic abnormalities, wherein mechanical output per unit muscle mass is decreased while energy utilization is paradoxically increased.1 We have now determined that the contractile defect of the pressure overload–hypertrophied feline RV cardiac muscle cell, or cardiocyte, is accounted for to a remarkable degree by increased microtubule density, with normal contractile function being restored when the microtubules are depolymerized.2 3 4 A major residual question, however, is the nature of the mechanical and metabolic loads imposed during contraction by increased microtubule density in the hypertrophied cardiocyte. Since we have also found in both normal and hypertrophied cells that the mechanical coupling between sarcomere and sarcolemma is unaltered by microtubule depolymerization, even in the presence of an extracellular load,3 we would suggest the hypothesis that the microtubule component of the cytoskeleton imposes an intracellular load that physically resists sarcomere shortening rather than altering the structure or function of the sarcomere itself. An expected consequence of this impediment to sarcomere motion would be an increase in cardiac energy requirements for a given amount of mechanical output. Thus, the mechanical and energetic abnormalities in pressure-overload cardiac hypertrophy may be causally linked via increased internal loading of the contractile cytoskeleton, such that mechanical energy generated within the sarcomeres is lost through an immediate conversion of mechanical energy to heat energy, ie, an increase in apparent viscosity, instead of being conserved within cellular elastic elements as potential energy.
Although a number of different techniques have been used to apply mechanical stress to the cell surface, all of these earlier approaches have resulted in deformation either of the whole cell or of a relatively large area of the cell membrane in a relatively nonspecific manner. Thus, these approaches cannot be used to probe cytoskeletal properties directly via transmembrane mechanical coupling, especially when the goal is to focus in a striated muscle cell on interactions among the extramyofilament cytoskeletal polymer systems, ie, the microfilaments, microtubules, and intermediate filaments. However, magnetic twisting cytometry, a newly developed method in which known stresses are applied directly to these interconnected cytoskeletal elements via specific transmembrane integrin receptors,5 has now allowed us to characterize cytoskeletal stiffness and apparent viscosity in both normal and pressure overload–hypertrophied feline RV cardiocytes.
Materials and Methods
Pressure-overload hypertrophy of the RV was induced in cats by partially occluding the pulmonary artery with a 3.2-mm (internal diameter) band, just as we have described before.1 Since a new steady state of feline RV mass is reached by 2 weeks after this procedure,4 we chose a postoperative recovery period of 2 to 4 weeks before final study. All operative procedures were carried out under full surgical anesthesia, consisting initially of ketamine HCl (11 mg/kg IM) and acepromazine maleate (0.3 mg/kg IM); after intubation and respiratory support, the cats were given nitrous oxide (0.5 L/min) and 3% isoflurane (1.5 L/min) supplemented with 100% O2. All procedures and the care of the cats were in accordance with institutional guidelines.
Hemodynamic Studies and Cardiocyte Isolation
At the time of final study, the cats were anesthetized as above; right heart and systemic arterial pressures were then obtained as before.1 The methods that were next used to obtain Ca2+-tolerant quiescent cardiocytes from the RV and LV of each cat have been described in detail in our previous work.6 7 After obtaining the RV and LV weights, cardiocytes were isolated separately from each ventricle and then maintained for 1 hour before plating at 37°C in a collagenase-free Krebs-Henseleit buffer of the following composition (mmol/L): NaCl 140.0, KCl 4.8, MgSO4 2.4, CaCl2 1.8, NaH2PO4 1.2, NaHCO3 4.0, HEPES 12.0, and glucose 12.5 at pH 7.4. Cardiocytes were plated in 1.8 mmol/L Ca2+ DMEM at a density of 3×104 cells per well on 96-well plastic dishes (Removawells, Immulon II, Dynatech) coated with 2 μg/cm2 of laminin (Upstate Biotechnology) for 1 hour and then rinsed three times with DMEM to remove contracted cells before starting the magnetic twisting cytometry experiments. The cardiocytes, which now consisted of 95% rod-shaped cells in both the RV and the LV samples, were quiescent unless electrically stimulated to contract.
Magnetic Twisting Cytometry
Detailed descriptions of the magnetic twisting cytometry technique have been published recently.5 8 9 10 In brief, ferromagnetic microbeads (4.5-μm diameter, Spherotech) were precoated with a synthetic RGD peptide (Peptite 2000, Telios) in 1-ethyl-3-(3-dimethylaminopropyl)carbodiimide (Sigma) containing 0.1 mol/L sodium phosphate buffer (pH 5.0) and stored overnight at 4°C to facilitate peptide conjugation with the beads.9 This peptide is a specific ligand for integrin receptors.5 A ligand coating-density of 50 μg/mL peptide per milligram beads was chosen, since this is in excess of the saturating concentration for promoting maximal bead-to-cell binding.10
The RGD-coated beads were washed in PBS and quenched in serum-free medium containing 1% BSA before addition to the cells. The beads were added to each well at 80 μg per well (four to eight beads per cell) for 15 minutes in DMEM; unbound beads were washed away with 1% BSA/DMEM. Fig 1A⇓ illustrates the experimental preparation, a quiescent cardiocyte to which ferromagnetic beads are bound. As before, when using this peptide coating-density, we could not detect breakage of bonds between the beads and the cell surface receptors during mechanical stress application.5
The cardiocyte-containing well was placed into the magnetic twisting cytometer and maintained at 37°C (Fig 1B⇑). The entire well was rotated around the vertical axis at 10 Hz to minimize any disturbances from external magnetic fields and hence increase the signal-to-noise ratio. A very brief (10-microsecond) but strong (1000-gauss) homogeneous magnetic pulse was then applied to magnetize all surface-bound beads in the horizontal direction (Fig 1B⇑). At this point, the magnetic field vector generated by the beads in the horizontal direction (⇀B) was measured with an in-line magnetometer. A typical readout from the magnetic twisting cytometer is shown in Fig 1C⇑. In the absence of an applied twisting field, the magnetic signal from the beads exhibited only a small, gradual remanent field decay due to thermal motion and membrane movements (relaxation curve).11 Then a twisting torque (twist on) was generated 20 seconds after the initial 1000-gauss magnetizing pulse by applying a much weaker but sustained (1-minute) vertical homogeneous 30-gauss magnetic field (⇀H) that was not strong enough either to remagnetize the beads or to spatially displace them. Rather, the beads tended to rotate toward the vertical direction in order to align with the newly applied field lines, much as a compass needle reorients to align with magnetic north (Fig 1B⇑). In this manner, a controlled torque could be applied directly to the specific integrin cell surface receptors to which the beads were bound. In the absence of any interconnection with the cytoskeleton, there would be no resistance to this shear stress; thus, the beads would rotate 90° immediately, such that the magnitude of the magnetic vector measured in the horizontal direction would drop from a maximum level to zero with an infinitely steep slope. Therefore, we could measure the mechanical response of the cytoskeleton to stresses that were applied directly to specific transmembrane receptors by calculating changes in the rate and degree of bead rotation, or angular strain, on the basis of the ratio of the remanent magnetic fields before and after application of the magnetic twist (see “Appendix”).
The torque of the applied twisting field ⇀H shown in Fig 1B⇑ (and thus the rotational stress) is proportional to the magnitude of the twisting field, bead magnetization, and the cosine of the angle between the twisting field vector and the bead magnetic moment.8 Strain was quantified as the degree of bead rotation in response to stress application. Stiffness, shown in Fig 1C⇑ as inversely related to the decrease in the remanent field after the twisting field was applied, was calculated as the ratio of stress to strain and was used as a measure of the resistance to mechanical deformation. As also shown in Fig 1C⇑, we measured elastic energy stored in the cytoskeleton by turning off the twisting torque (twist off) for 1 minute and quantifying the ability of the beads to rotate back toward the initial undisturbed state. Apparent viscosity, shown in Fig 1C⇑ as inversely related to the slope of remanent field recovery after the twisting field was removed, was determined by measuring the time constant of recovery and multiplying it times stiffness. The magnetic torque, or stress applied to the ferromagnetic beads, was calibrated as before in a polydimethylsiloxane solution of known viscosity, and the average apparent stress (ie, torque per unit bead volume) was determined as a function of the applied twisting field.10 Although the foregoing is intended to provide a conceptual basis for these experiments, the “Appendix” sets forth the basis for the actual measurements.
Cytoskeletal mechanics were characterized in both hypertrophied cardiocytes from pressure-overloaded RVs and control cardiocytes from normally loaded LVs from the same cats. Given the time required for the following protocols, we elected to define cytoskeletal properties in one to three wells of RV and of LV cardiocytes from each cat, since baseline cytoskeletal properties were stable within this time frame. The experimental variables used included microtubule depolymerization, in superfusates containing either physiological or markedly reduced external Ca2+ levels, and microtubule hyperpolymerization. To define the effects of microtubule depolymerization, colchicine at a final concentration of 10−6 mol/L was added to the chamber containing either hypertrophied or normal cardiocytes, and cytoskeletal properties were studied by magnetic twisting cytometry over the ensuing hour, by which time cardiocyte microtubules are fully depolymerized.3 To characterize the effects of microtubule hyperpolymerization, normal cardiocytes were exposed to taxol at a final concentration of 10−5 mol/L, and cytoskeletal properties were studied over the ensuing 3 hours, by which time a stable level of cardiocyte microtubule hyperpolymerization is reached.3
Magnetic cytometry was used to characterize the cytoskeletal properties of quiescent cardiocytes. In order to relate this information, especially that pertinent to increased microtubule density in hypertrophied cardiocytes, to mechanical influences that might obtain during active cardiocyte contraction, we used our laser diffraction system,12 which uses the pattern cast by transilluminated sarcomeres to measure sarcomere length, to define the dependence of microtubule effects on the rate and extent of sarcomere motion in electrically stimulated, hypertrophied RV cardiocytes. For this purpose, isolated pressure-hypertrophied RV cardiocytes were placed in the Krebs-Henseleit buffer for 1 hour either with or without 10−6 mol/L colchicine and then stimulated to contract at frequencies ranging from 0.10 to 0.83 Hz. After sarcomere shortening had reached a steady state at each stimulation frequency, the extent and velocity of sarcomere motion during contraction were determined by laser diffraction measurements of sarcomere length using a sampling rate of 1 kHz.
Cell Adhesion Study
If the attachment affinity of the peptide-coated beads differed between RV and LV cardiocytes, the response of each cell to the twisting torque might differ independent of changes in cytoskeletal properties. To evaluate this possibility, the affinity of cardiocytes for extracellular matrix protein ligands was measured in three cats with RV pressure-overload hypertrophy produced by pulmonary artery banding for 2 weeks. Multiwell dishes (Falcon) were precoated with either laminin (Upstate Biotechnology) or fibronectin (Sigma) at concentrations of 0, 10, 20, and 50 mg/mL. Normal LV and hypertrophied RV cardiocytes were then plated in DMEM onto these dishes at a concentration of 150 000 cells per well and allowed to attach for 60 minutes. Unattached cells were removed by gently washing the wells three times with buffer. The number of attached cells was determined by counting all cells in each well. Data are presented as cells/mm3.
The mean±SEM values are shown for each group of data. Differences in selected measures were evaluated via either a paired or unpaired Student's t test, as appropriate, with a significant difference said to exist at the level specified for each set of data. Where stated, group means were first compared by a one-way or two-way ANOVA, and if a difference was found, then each experimental mean was compared with that of the control group by the appropriate post hoc test as individually specified.
Characteristics of the Experimental Model
The major features of the surgical model used in the present study are summarized in the Table⇓. In the PA band group, RV systolic pressure was increased almost threefold. The ratios of RV weight to body weight and to tibial length were each increased significantly in the PA band cats, with RV mass almost doubling. Body weight was similar in each group, and the ratio of LV weight to body weight did not differ between control and surgical animals. There was no evidence of right heart failure, either in terms of increases in RV end-diastolic pressure or the ratio of liver weight to body weight or in terms of the presence of ascites and/or pleural effusion in any cat at the time of study.
Cardiocyte Mechanics: Effects of Microtubule Perturbations
We have shown that microtubule depolymerization, whether effected by low temperature or by colchicine, restores the initially abnormal contractile performance of the pressure-hypertrophied feline RV cardiocyte to normal after hypertrophy is complete.2 3 4 Fig 2A⇓ shows first at baseline the changes in the remanent magnetic field versus time after application of a twisting force to the integrin-attached beads on hypertrophied RV cardiocytes with intact microtubules. One hour after the addition of 10−6 mol/L colchicine, both the initially reduced field displacement and the prolonged time course of field recovery for the hypertrophied RV cardiocytes returned to levels comparable to those for normal cardiocytes at baseline (colchicine [Fig 2A⇓] versus baseline [Fig 4A⇓]). Fig 2B⇓ provides summary data for cytoskeletal viscosity in normally loaded LV cardiocytes and pressure-hypertrophied RV cardiocytes using the experimental protocol shown in Fig 2A⇓, and Fig 2C⇓ provides concurrently obtained summary data for cytoskeletal stiffness. All cells were sampled sequentially at the indicated times after drug exposure. Initially, there were statistically significant differences between the RV and LV cardiocytes in both viscosity and stiffness; however, both of these differences were abolished with microtubule depolymerization. Of interest, the difference in viscosity between the RV and LV cardiocytes before microtubule depolymerization (a 4.2-fold difference) was considerably greater than the initial difference in stiffness (a 1.9-fold difference).
The constitutive mechanical properties of a striated muscle cell, even when quiescent, must be a composite function of contributions from both the myofilament and the extramyofilament components of the cytoskeleton. Indeed, there is clear evidence for Ca2+- and energy-dependent crossbridge interactions in quiescent cardiocytes14 and diastolic cardiac muscle.15 16 Since the goal of the present study was to examine the mechanical properties of microtubules as an element of the extramyofilament cytoskeleton, a particular effort was made to do so under circumstances in which the contribution of any activated myofilaments, even in these quiescent cells, to the net mechanical properties of the cytoskeleton had been minimized. Thus, in order to block actin-myosin crossbridge interactions, we prepared a Ca2+-free DMEM superfusate containing 0.1 mmol/L EGTA and 10 mmol/L 2,3-butanedione monoxime. These agents had no discernible effect herein on cardiocyte morphology but have been shown to inhibit myofilament activation in cardiac muscle.17 18 Fig 3A⇓ provides summary data for cytoskeletal viscosity in both normally loaded LV cardiocytes and pressure-hypertrophied RV cardiocytes, and Fig 3B⇓ provides concurrently obtained summary data for cytoskeletal stiffness. All cells were sampled sequentially at the indicated times after drug exposure. Just as in the normal Ca2+ superfusate (Fig 2⇑), there were initial statistically significant differences between the RV and LV cardiocytes in both viscosity and stiffness; however, both of these differences were once again abolished with microtubule depolymerization. Of interest here, although the absolute values for both viscosity and stiffness were reduced at all time points for both groups of cardiocytes compared with the values obtained in the normal Ca2+ superfusate (Fig 2⇑), perhaps attributable in part to reduced Ca2+ leakage into cells with abnormal sarcolemmal integrity, the difference in viscosity between the RV and LV cardiocytes before microtubule depolymerization (a 3.7-fold difference) remained considerably greater than the initial difference in stiffness (a 1.4-fold difference), and both of these ratios are similar to those seen in the normal Ca2+ superfusate.
Thus, there are increases in viscosity and stiffness, larger in the former case, in hypertrophied cardiocytes that apparently are attributable to the microtubule component of the extramyofilament cytoskeleton. If an excess of microtubules is indeed responsible for these abnormalities of hypertrophied cardiocytes, the same abnormalities should appear in normal cardiocytes if excessive microtubule polymerization is caused to occur in these cells. Taxol is a diterpene that lowers the critical concentration of αβ-tubulin heterodimers required to form microtubules19 ; thus, it acts in a manner closely analogous to the demonstrated effect2 3 4 of an extending stress load on the αβ-tubulin heterodimer/microtubule system of cardiocytes exposed to a pressure overload. In this context, it is of interest that addition of 10−5 mol/L taxol to the superfusate of normal cardiocytes caused a decrease in field displacement and a prolongation of the time course of field recovery to levels comparable to those of hypertrophied RV cardiocytes at baseline (taxol [Fig 4A⇓] versus baseline [Fig 2A⇑]).
Cardiocyte Mechanics: Effects of Contraction Frequency
The magnetic twisting data showed that for a given mechanical shear load, the time constant of remanent field recovery after loading is about twice as great for hypertrophied RV cells as it is for normal LV cells (Fig 2A⇑). The addition of colchicine to the hypertrophied RV cells returned this time constant to normal (colchicine [Fig 2A⇑] versus baseline [Fig 4A⇑]). If this effect is based primarily on an abnormality in viscosity as opposed to stiffness of the extramyofilament cytoskeleton in hypertrophied RV cardiocytes, one would expect it to become more apparent as the rate of sarcomere motion during contraction increases. In cardiac muscle at relatively low stimulation rates, there is a stimulation frequency-dependent increase in the rate of sarcomere motion during contraction. Therefore, we would predict that as stimulation frequency and thus the rate of sarcomere motion during contraction increase, any viscous extramyofilament cytoskeletal impediment to sarcomere motion would become engaged primarily at higher stimulation frequencies. The data in Fig 5⇓ show only a partial restoration of contractile function after microtubule depolymerization at low stimulation frequencies, perhaps due to the abnormal Ca2+ metabolism seen in this model,1 but a full restoration at higher stimulation frequencies, thus bearing out this prediction. Thus, we conclude that at physiological contraction frequencies, ie, >0.4 Hz with attendant sarcomere shortening velocities >2.5 μm/s, increased microtubule density in the hypertrophied cardiocyte tends to interfere with normal sarcomere mechanics primarily via a viscous impediment to sarcomere motion.
Cell Adhesion Study
Our primary goal of characterizing the mechanical effects of microtubule alterations was found not to be critically dependent on initial binding of the RGD peptide–coated beads to the cardiocytes. Nevertheless, to further validate comparisons in terms of cytoskeletal properties between normal and hypertrophied cardiocytes, we wished to determine whether the binding affinity of integrin-mediated coupling of ferromagnetic beads to the cardiocyte integrin receptors was comparable in the two cell types. Cell surface integrin receptors mediate cell adhesion to extracellular matrix molecules, such as laminin and fibronectin.20 Therefore, we elected to evaluate the adhesion of normal and hypertrophied cardiocytes to these two proteins when used as substrates as an indirect index of the cardiocyte-bead binding affinity. The data in Fig 6⇓ show that adhesion of normal and hypertrophied cardiocytes to these two protein substrates was indeed comparable at all protein concentrations tested.
In normal cardiocytes, microtubules are concentrated mainly in the perinuclear regions, but they have also been observed to invest the myofibrils and traverse their surfaces at various angles.21 22 23 24 In cardiac hypertrophy, microtubules undergo significant changes both in concentration2 3 4 and in regional distribution.22 23 24 25 We have previously found that microtubules are selectively and persistently increased in myocardium hypertrophying in response to pressure overloading, that in RV cardiocytes from the PA band model used in the present study this alteration is responsible for the reduced extent and velocity of sarcomere shortening in externally unloaded cells and for the reduced extent of sarcomere and cellular shortening in externally loaded cells, that these contractile defects can be reproduced in normal cells when microtubule hyperpolymerization is induced by a chemical as opposed to a pathophysiological stimulus, and that these contractile defects are specific to the microtubule component of the cytoskeleton.2 3 4 As yet unknown, however, was the specific effect in biophysical terms of the increased microtubule density characteristic of cardiocytes hypertrophying in response to a pressure overload.
This data set, which we generated in an effort to define this mechanical effect, comprises the initial characterization of the constitutive properties of the extramyofilament cytoskeleton both of striated muscle cells in general and of normal and pressure overload–hypertrophied cardiocytes in particular. In hypertrophied RV cardiocytes, both the apparent viscosity and the stiffness of the cytoskeleton were found to be greater than in normal LV cardiocytes, and the increase in apparent viscosity was considerably more pronounced than the increase in stiffness. Microtubule depolymerization in hypertrophied cardiocytes, either with or without the opportunity for diastolic myofilament activation, normalized both viscosity and stiffness. Furthermore, microtubule hyperpolymerization in normal cardiocytes caused cytoskeletal viscosity and stiffness to mimic those of hypertrophied cells. These findings were not due to differences between the two groups of cells in terms of their affinity for extracellular matrix protein ligands. Finally, when these observations in quiescent cardiocytes were extended to contracting cardiocytes, the microtubule-based impediment to sarcomere motion was found to be exhibited only at higher sarcomere shortening rates.
It is important to note that the magnetic twisting cytometry method used in the present study is very different from conventional methods used for probing cellular constitutive properties in which the cells are stretched along their principal axis. In the present study, rotational shear stresses were applied to the cytoskeleton via specific integrin receptors. Thus, it would be expected that the stiffness of the extramyofilament cytoskeletal network measured here would not necessarily reflect either the stiffness of the individual filaments in a one-to-one fashion or that of the cell as a whole. Indeed, at least in striated muscle, extramyofilament cytoskeletal stiffness may or may not predict cellular stiffness, since despite the cytoskeletal stiffness increase reported in the present study for pressure-hypertrophied cardiocytes, we do not find a change in overall cellular stiffness.26 Our data are instead consistent with the idea represented by the “tensegrity” model,8 wherein it is the structural arrangements of the cytoskeletal filaments rather than their absolute quantity that determines cytomechanical properties and thus the effects observed in our own work on cardiocyte contraction. Furthermore, the damping parameter that we used in the present study, ie, apparent viscosity, does not represent intracellular fluid viscosity. Rather, the changes in apparent viscosity observed in hypertrophied cardiocytes appear to represent structural damping attributable to the microtubule component of the cytoskeleton. In these terms, apparent viscosity might well reflect a process of intracellular frictional dissipation that impedes cellular shortening. Hence, one might expect this impeding effect, as observed here experimentally, to become more pronounced at higher rates of sarcomere motion, and one would also expect on this basis the loss of energetic efficiency during contraction seen in our earlier work with this model.1
Thus, as hypothesized to be the case initially, the present study demonstrates that the microtubule-dependent biophysical effect is one of imposing a primarily viscous intracellular load on the shortening sarcomeres. Again, as suggested initially, this single molecular defect could well be explicative not only of abnormal sarcomere and cellular motion during contraction but also of the linked contractile and energetic abnormalities characteristic of pressure-overload cardiac hypertrophy.
Selected Abbreviations and Acronyms
|LV||=||left ventricle (ventricular)|
|PA band||=||pulmonary artery band|
|RV||=||right ventricle (ventricular)|
As explained for a more general case elsewhere,8 magnetic twisting cytometry was used here to apply a measurable sheer stress (σ) to ferromagnetic beads bound to the cardiocyte transmembrane integrin receptors and, hence, to the cytoskeleton. This stress results in a quantifiable rotation, or angular strain (ε), of the ferromagnetic beads, which is measured as a change in the remanent magnet field (⇀B) by an in-line magnetometer. From these measurements of σ, ε, and ⇀B, both stiffness (E) and viscosity (η) can be calculated.
As shown in panels A and B of Fig 7⇓, the beads are magnetized at t0. After the beads are magnetized, there is a small but reproducible spontaneous decay, or relaxation, of their remanent magnetic field, shown as ⇀B1. Following a 20-second equilibration period after bead magnetization, an external 30-gauss twisting field (⇀H) is applied at t1. As a result, the beads rotate away from the magnetizing horizontal field toward the vertical, ie, the direction of the external twisting field, and the remanent magnetic field decreases (⇀B2t1). Sixty seconds after application of the external twisting field, at t2, this field is discontinued (⇀B2t2). The beads then untwist and rotate back toward the horizontal, and there is an increase, or recovery, of the remanent magnetic field. However, 60 seconds after the external twisting field is discontinued, at t3, the beads have not fully untwisted to the horizontal, such that the remanent field does not fully recover. As a result of this plastic deformation, ⇀B2t3≠⇀B1t3.
Stiffness is the ratio of σ at t2 and ε at t2:
Stress is the product of c, an experimentally derived calibration factor (dyne·cm−2·gauss−1), the external twisting field in gauss (⇀H), and the angular displacement produced by the twisting field (B2t2/B1t2). For each bead size, c is determined in a known viscosity standard with no elastic opposing resistance; for a given external twisting field, c relates observed angular displacement (B2t2/B1t2) to the effective stress applied to the microbead and had a value of 0.51 in these experiments. ⇀B2t2 is the remanent magnetic field at t2, 60 seconds after ⇀H is applied, or 80 seconds after magnetization, and ⇀B1t2 is the remanent magnetic field at t2, when ⇀H is never applied and there is spontaneous remanent field relaxation. That is,
If application of the twisting field did not cause any degree of plastic deformation, strain at t2 would be due entirely to elastic deformation and would occur when the remanent field reached its steady-state nadir at time t2. However, as our experiments showed, application of a twisting field did result in plastic deformation. Therefore, the residual angular displacement at t3 (εt3, the plastic yielding angle) must be subtracted from the angular displacement that is caused by purely elastic deformation at t2 (εt2). Stiffness is then calculated using the corrected angular strain, which in turn is calculated as the angular displacement at t2 (φ1) minus the remaining angular displacement (φ3) resulting from plastic deformation at t3, 60 seconds after the twisting field is discontinued. Thus,
Angular displacement is calculated by examining changes in the remanent magnetic field. As shown in Fig 7C⇑, the angle φ in a triangle with one right angle can be calculated from the ratio of the hypotenuse (⇀B2) and the base (⇀B1) as follows:
By analogy, in Fig 7B⇑,
Viscosity is calculated by examining the time course of recovery of the remanent magnetic field from t2 to t3 after discontinuing the twisting field. The time course of this change in angular strain can be characterized as follows:where τ is the time constant of recovery of angular strain. This time constant is dependent on both the viscosity (η) and the stiffness (E) of the cardiocyte cytoskeleton. Since τ is equal to the ratio of η/E, η can be calculated from E and τ, and angular strain at any point in time is calculated as cos−1(B2/B1).
In considering why in mathematical terms τ=η/E, let us stipulate that Tmag, the torque applied by the twisting field, is equal in magnitude but directionally opposite to the torque created by the viscosity (Tvis) plus the torque created by the elasticity of the cytoskeleton (Telas). That is,Thus, when Tmag is turned off,Rearranging Equation 10,Integrating both sides of Equation 11,This then yields:Taking the antilog of both sides of Equation 13,
This study was supported by grants HL-37196 (Dr Cooper), HL-48788 (Drs Cooper and Zile), HL-46491 (Dr Ingber), and HL-33009 (Dr Wang) from the National Heart, Lung, and Blood Institute and by research funds from the Department of Veterans Affairs (Drs Cooper and Zile). The authors thank Mary Barnes and Tommy Gallien for their excellent technical assistance.
Reprint requests to George Cooper, IV, MD, Cardiology Section, VA Medical Center, 109 Bee St, Charleston, SC 29401-5799. E-mail george email@example.com
- Received September 6, 1996.
- Accepted November 21, 1996.
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