Axial Stress Provides a Lower Bound on Shear Wave Velocity in Active and Passive Muscle

Ultrasound shear wave elastography can be used to characterize mechanical properties of unstressed tissue by measuring shear wave velocity (SWV), which increases with increasing tissue stiffness. Measurements of SWV have often been assumed to be directly related to the stiffness of muscle. Some have also used measures of SWV to estimate stress, since muscle stiffness and stress covary during active contractions. However, few have considered the direct influence of muscle stress on SWV, independent of the stress-dependent changes in muscle stiffness, even though it is well known that stress alters shear wave propagation. The objective of this study was to determine how well the theoretical dependency of SWV on stress can account for measured changes of SWV in passive and active muscle. Data were collected from six isoflurane-anesthetized cats; three soleus muscles and three medial gastrocnemius muscles. Muscle stress and stiffness were measured directly along with SWV. Measurements were made across a range of passively and actively generated stresses, obtained by varying muscle length and activation, which was controlled by stimulating the sciatic nerve. Our results show that SWV depends primarily on the stress in a passively stretched muscle. In contrast, the SWV in active muscle is higher than would be predicted by considering only stress, presumably due to activation-dependent changes in muscle stiffness. Our results demonstrate that while SWV is sensitive to changes in muscle stress and activation, there is not a unique relationship between SWV and either of these quantities when considered in isolation.


ABSTRACT 23
Ultrasound shear wave elastography can be used to characterize mechanical properties 24 of unstressed tissue by measuring shear wave velocity (SWV), which increases with 25 increasing tissue stiffness. Measurements of SWV have often been assumed to be directly 26 related to the stiffness of muscle. Some have also used measures of SWV to estimate stress, 27 since muscle stiffness and stress covary during active contractions. However, few have 28 considered the direct influence of muscle stress on SWV, independent of the stress-dependent 29 changes in muscle stiffness, even though it is well known that stress alters shear wave 30 propagation. The objective of this study was to determine how well the theoretical 31 dependency of SWV on stress can account for measured changes of SWV in passive and 32 active muscle. Data were collected from six isoflurane-anesthetized cats; three soleus muscles 33 and three medial gastrocnemius muscles. Muscle stress and stiffness were measured directly 34 along with SWV. Measurements were made across a range of passively and actively 35 generated stresses, obtained by varying muscle length and activation, which was controlled 36 by stimulating the sciatic nerve. Our results show that SWV depends primarily on the stress 37 in a passively stretched muscle. In contrast, the SWV in active muscle is higher than would 38 be predicted by considering only stress, presumably due to activation-dependent changes in 39 muscle stiffness. Our results demonstrate that while SWV is sensitive to changes in muscle 40 stress and activation, there is not a unique relationship between SWV and either of these 41 quantities when considered in isolation. Ultrasound shear wave elastography can be used to measure the mechanical properties 46 of biological tissues non-invasively (1-4). In simple, homogeneous materials, shear wave 47 velocity (SWV) provides an accurate estimate of the shear modulus and, through a direct 48 relationship, the tensile or Young's modulus, which is more relevant to how muscles act on 49 the skeleton. The relatively low cost and ease of use of ultrasound systems for elastography 50 has inspired many to use SWV to measure skeletal muscle properties (5, 6). However, 51 skeletal muscle is a complex tissue and the relationship between its material properties and 52 SWV is incompletely understood (7-9). This has led to SWV being used to estimate muscle 53 force (10-12) and Young's modulus (13-15), though it is unclear under which conditions 54 either of these properties can be uniquely determined. 55 Muscle is unique in its ability to generate tension when activated -a process that 56 leads to simultaneous changes in Young's modulus that are thought to arise from increased 57 cross-bridge attachment (16)(17)(18)(19). The tensile load on a muscle changes with passive 58 stretching, and these loads can also alter the Young's modulus of muscle through the 59 nonlinear stress-strain properties of its passive structures (20, 21). The change in muscle 60 tension with activation and stretch complicates the relationship between SWV and material 61 properties since it is well known that the speed with which a shear wave propagates changes 62 with tension (22, 23), or more precisely stress. Traditional beam analysis shows that shear 63 wave propagation depends on stress and material properties, specifically shear modulus (22, 64 24). Recently, Martin et al. (25) demonstrated that the analysis of shear wave propagation in 65 beams provides a reasonable approximation for how they also propagate in tendons. In 66 addition, they found that the speed with which a shear wave propagates along a tendon is 67 primarily determined by stress. Muscle is obviously different from tendon. We previously 68 demonstrated that SWV is sensitive to activation-dependent changes in muscle stiffness even 69 measured with a load cell (Sensotec model 31; Honeywell, Golden Valley) mounted on the 120 motor shaft. The knee and ankle joints were fixed to metal clamps. The whole left hindlimb 121 was submerged in a temperature controlled saline bath (26 o C) to permit ultrasound imaging 122 without the transducer contacting the muscle. For a figure and more detailed information 123 about the recording setup see (26). 124 Nerve stimulation. 125 A bipolar cuff electrode was folded around the sciatic nerve. The electrode was 126 connected to a stimulus isolation unit (Grass stimulator S8800) that was under computer 127 control. We generally used supramaximal currents in our stimulus pulses except for a few 128 low force trials where we deliberately stimulated the muscle at sub-maximal levels. One 129 twitch was evoked before each contraction to enable the muscle to adapt to its current length, 130 minimizing history effects. Five-second trains were needed to allow muscle force to stabilize 131 before measuring SWV. Such long trains created a problem with fatigue. In the soleus we 132 found that 100 Hz for 5s produced substantial high frequency fatigue. This fatigue recovered 133 quickly with rest but still created a 20% change in force within a trial (27). This was avoided 134 in the soleus by using 40 Hz trains, which were still sufficient to maintain tetanic 135 contractions. In the MG fatigue could not be avoided. In a single trial, force often fell by 136 30% from the start of the train until the end. A synchronization pulse from the ultrasound 137 unit allowed the SWV measurements to be associated with the correct force. A 2-min 138 recovery time was allowed between subsequent stimulations. However, even with this rest, 139 force from the MG continued to fall throughout the experiment. This was not a major 140 problem since our goal is to compare stress with SWV. We did not see any evidence that 141 fatigue changed the relationship between force and SWV. 142

Short Range Stiffness 143
A direct measure of muscle stiffness can be obtained by characterizing short-range 144 stiffness (SRS). SRS describes the elastic properties of muscle in response to small, rapid 145 changes in length. Measurements of SRS can be made in an in situ muscle preparation that 146 allows for direct measures of muscle force and length. The SRS of individual muscles scales 147 linearly with actively generated force and is thought to be directly related to the number of 148 attached cross-bridges. We measured SRS with a fast length perturbation before the end of 149 tetanic stimulation (26). The perturbation had a displacement of 2 mm and a speed of 2 m/s. Vermon, Tours, France) was used for all ultrasound elastography. The parameters of the 158 system were set to 1) mode: MSK-foot and ankle, 2) opt: std, 3) persist: no, to maximize the 159 sampling rate and avoid bias due to locked proprietary processing. The research grade 160 software on our system provides shear wave motion images sampled at 8 kHz in sequences of 161 42 frames for each acoustic radiation pulse, recorded simultaneously with a single B-mode 162 ultrasound image. The size of the region of interest was selected to include the belly of the 163 muscle (up to 4x1 cm), and the speed of the compressional sound wave in the tissues was 164 assumed to be 1,540 m/s. A custom-built frame held the transducer in a constant position 165 relative to muscle, with the transducer oriented parallel to the fascicle plane. 166

Data collection. 167
A MATLAB XPC controller (MathWorks, Natick, MA) was used to control the 168 muscle puller, sequence nerve stimulation, and record all data. Acquisition of ultrasound 169 images and collection of the muscle data were synchronized with a trigger pulse from the 170 ultrasound machine. 171 The experiment began by measuring the forces needed to accelerate the puller shaft, 172 membrane, and saline in the bath, using the same step change in displacement used to 173 measure SRS. This was later subtracted from measurements made on the muscle. Next, we 174 measured the length-tension properties of the muscle. This set the operating range for the 175 puller. We always kept the muscle at Lo minus 5mm during rest periods to keep passive force 176 near zero. This minimized the stretch of the tendon-aponeurosis. We then began data 177 collection. The muscle was moved to the test length prior to each 5 s trial. First a passive 178 trial was measured, followed two seconds later an active trial; this time was sufficient for 179 repositioning the probe as needed to optimize image acquisition during muscle stimulation. 180 One B-mode image and one shear wave movie were captured simultaneously for each passive 181 and active condition. The muscle was moved back to rest length for two minutes and the 182 process repeated. We measured the force-SRS-SWV properties at three different lengths: i) 183 medium ( Lo ) to get the maximum active force, ii) short, a length chosen so that active force 184 was about half of the peak force, iii) long, where active force was again half the peak value. 185 At the long length there was substantial passive tension. See Fig 1A for the length tension 186 curve and measured force in the soleus. Most active measurements were made using 187 supramaximal stimulation, but we also used sub-maximal stimulation in some measurements 188 made at Lo. This was to obtain multiple levels of active force at the same level of passive 189 stretch. Finally, we measured passive muscle properties at long lengths. These were 190 conducted at the end of each experiment as the imposed stretch exceeded the physiological 191 length of muscle, eventually resulting in muscle damage. We found that the physical limits of 192 the muscle were typically reached when passive force was equal to the peak active force. 193 Positioning the ultrasound probe during active trials was difficult, particularly for the 194 MG in which fascicles shorten by about 5mm during a tetanic contraction, increasing the 195 angle of pennation (29). By trial-and-error we were able to position the probe to obtain clear 196 images. However, this trial-and-error process led to increased fatigue and tendon stretch 197 while the optimal position was determined. 198

Data Processing 199
In this manuscript we present a simplified model of shear wave propagation, which 200 we refer to as the stress model. We assume muscle is a homogenous, anisotropic, infinitely 201 large material. The model considers a plane wave propagating along the muscle fibres with 202 minimal contribution of the bending moment. If we also assume that the transverse shear 203 modulus of the muscle did not change with stress, the constitutive equation for wave 204 propagation derived from Timoshenko beam theory can be reduced to (25): 205 In this equation, v is the SWV along the muscle fibres in m/s, µo is the shear modulus of 207 unstressed passive muscle in Pa, σ is stress (tension per unit area) in Pa, and ρ is tissue 208 density in kg/m 3 . 209 SWV was estimated offline using displacement movies created by the Aixplorer 210 ultrasound machine, as we detailed previously (26), and summarized here. Displacement of 211 the wave was displayed as a function of time, depth, and distance along the fascicle. There 212 were 5 separate pushes covering the selected area of the movie. The B mode image was used 213 to further select part of the displacement movie that covered only the muscle of interest 214 without any aponeuroses. The displacement movie was then averaged over this depth to 215 create a 2D image where displacement is a function of time and distance along the fascicle. 216 For each push the 2D displacement image was fit with a regression line the slope of which 217 was an estimate of the SWV. 218 We used muscle cross sectional area to compute stress from the measured 219 experimental forces. Cross sectional area at Lo was estimated using the maximum tetanic 220 tension at Lo, assuming the specific tension for muscle is 230 kPa (30, 31). 221 where ao is the cross-sectional area at Lo, po is tetanic tension at Lo, and s is the specific 223 tension. 224 Muscle cross sectional area is not constant but rather changes with passive stretching 225 and activation. The influence of both can be approximated by considering the change in 226 muscle fiber length relative to Lo such that the area at any length is given by: 227 Eq. 3 = 228 Mean fascicle length was estimated using muscle weight and density (1060 kg/m 3 ) to 229 estimate volume, and dividing by area to determine mean fascicle length at Lo, xfo: 230 where w is the weight of the muscle, and σ is the density. 232 When a muscle was passively stretched to long lengths, it produced substantial 233 tension. For example, the MG, stretched to a force equal to po, causes a tendon-aponeurosis 234 stretch of about 5 mm (29, 32). This tendon stretch was estimated and used to correct 235 estimates of fascicle length. The assumed stretch was: 236 where Δxt is change in tendon length due to changing force, k is the stiffness of the tendon-238 aponeurosis, p is the current force and po is the tetanic force at Lo. The stiffness of the tendon-239 aponeurosis, k, was estimated using the model of Cui et al. (30). 240 This approach allowed us to compute the fascicle length, xf , as: 241 where xe is the experimentally measured change in muscle-tendon length from Lo. 243 The distributions of errors resulting from the stress model (Eq. 1) were analysed using 244 a signed-rank test to determine if the mean for active and passive conditions differed from 245 zero. Linear mixed effects models were used to describe these errors as functions of either 246 short range stiffness or muscle stress. In both models, cat was considered as a random factor. 247 Analysis was conducted in MATLAB 2019a. All results are presented as mean ± standard 248 error, unless noted. 249

251
The anatomical parameters from all muscles used in the study are shown in A typical example of SWV versus stress in the cat soleus is shown in Fig. 1. Fig. 1A  255 shows the relation between the conditions imposed in the experiment and the muscle length-256 tension curve. The largest active stresses (about 230 kPa) were obtained at Lo (0 mm). Using 257 sub-maximal stimulus currents, lower stresses (about 120 kPa) were measured at Lo (green 258 points at lower stress). Note that at this length passive stress is near zero. By shortening the 259 muscle, we could stimulate maximally and obtain less stress (yellow points). Stretching the 260 muscle to long lengths and fully activating it also produced less active stress but significant 261 passive stress (blue points). The red points show data when the muscle was passively 262 stretched. Note that this involved some unusually long lengths beyond physiological limits. 263 Fig. 1B shows the same data set with SWV plotted as a function of stress. Force was 264 normalized by the fixed area ao (Eq. 2). The solid black line represents the stress model. Fig.  265 1C shows the same data but now normalized by length-dependent calculations of area, 266 estimated using fascicle length (a in Eq. 3). Normalization has the greatest influence on 267 points measured at long lengths. In this animal, active and passive points measured at long 268 lengths were similar to the stress model. Active short length data points lie above the stress 269 model. 270 Passive data from all cats are shown in Fig 2A. The SWV for passively stretched 271 muscle is described well by the stress model when the estimate of cross-sectional area is 272 corrected to account for changes at different muscle lengths. SWV measurements from both 273 the MG and soleus are close to what is predicted by the stress model. 274 The active trials from all cats are shown in Fig 2B. Note that stresses for some 275 measurements are very high (~2 times po), probably higher than the physiological range of 276 the muscle. This is due to the measurements being obtained at long lengths. Fig. 3 shows the 277 same data, but for each cat shown separately. All 6 muscles show some active points that are 278 substantially above the black line from the stress model. Note that the active trials in MG 279 show large variability, likely due to the large change in pennation angle that occurs with MG 280 activation and the challenges that poses for ultrasound measurements. 281 The error in SWV, that is the experimental value minus the prediction from the stress 282 model, is shown in Fig 4. Fig. 4A shows the distribution or errors for passive (mean=0.20 283 kPa, sdev=1.42) and active conditions (mean=2.72 kPa, sdev=3.12). A MATLAB) showed that the distribution of active errors was non-285 Gaussian (p<.00001). We therefore used a Wilcoxon signed-rank test (signrank, MATLAB) 286 to compare the mean of each distribution to zero. We found that the mean of the passive 287 distribution was not significantly different from zero (p=.99), but that the mean of the active 288 distribution was different (p<.000001). Because the errors were significantly different from 289 zero for active muscles, we attempted to model these residual errors as a function of both 290 muscle stress and the independently measured short-range stiffness (SRS). Fig. 4B shows the 291 error plotted as a function of stress for both active and passive conditions. Fig. 4C shows the 292 same data plotted against SRS. In both cases, data from all animals are plotted. 293 The error in SWV was correlated with our measures of SRS in both active and 294 passive muscle. This was assessed using a linear mixed effects model with SRS and 295 activation state of the muscle (passive or active) as the independent variables and the error 296 difference between the squared value of the measured SWV (SWV 2 ) and that predicted by the 297 stress model as the dependent variable (Fig. 5). SWV 2 was used to yield a linear relationship 298 between our independent and dependent variables (Eq. 1). Cat was considered as a random 299 factor having an independent effect on the average slope and intercept across activation 300 states. The error in the SWV 2 increased significantly with SRS for both passive (p=0.0007) 301 and active (p=0.0011) conditions, though there was no significant difference in these slopes 302 (p=0.47). The errors in the active conditions were offset from those in the passive conditions 303 (∆=48±21 m 2 /s 2 ; p=0.02). The R 2 for the overall model was 0.64 indicating that there remains 304 a significant amount of measurement variance not accounted for by SRS. 305 SRS was strongly correlated with muscle stress in each animal, having an average R 2 306 of 0.97±0.01 for the passive conditions and 0.70±0.05 for the active conditions. For these 307 reasons modeling the SWV 2 as a function of stress (R 2 =0.63) worked almost as well as when 308 it was modeled as a function of stiffness. However, when both stress and SRS were 309 considered as independent variables, only SRS had a significant effect on the error in SWV 2 310 (SRS: p<0.0001; stress: p=0.3). 311

DISCUSSION 313
The aim of this study was to examine the relationship between muscle stress and 314 SWV in passive and active muscle tissues. We applied controlled changes in force by 315 lengthening and activating the soleus and MG muscles of the cat. Our primary hypothesis 316 was that changes in SWV can be predicted by a simple model accounting for muscle stress 317 and knowledge of the SWV for unstressed muscle. Our results show that this stress model 318 fits passive data well but only sets the lower bound for SWV measured in active muscle. 319 Active muscle showed more variability and often had SWV values well above what was 320 predicted by the stress model. These experiments were not designed to determine if the 321 increased SWV was due to increased muscle stiffness, but activation dependent changes in 322 stiffness appears to account for some of the error. 323 We are the first to show that SWV in passive muscle can be predicted by stress. This 324 is important because stress and stiffness covary in muscles. It has proven difficult to 325 decouple the effect of one from the other on human muscle tissue because only indirect 326 measures of force or stiffness are possible (7, 33). Our results are in line with a recent study 327 on tendon tissues where Martin et al. (25) demonstrated that tensional pre-load is the main 328 determinant of shear wave propagation in tendon, with the shear modulus playing a role only 329 at the lowest tensions. Other studies measured SWV in animal muscle ex-vivo (8, 34), 330 showing SWV increased with increasing tension. Eby et al. also measured Young's modulus,331 showing it increased with stress. They suggested Young's modulus accounted for the change 332 in SWV. We believe their observations can, in part, be attributed directly to tension rather 333 than Young's modulus. Our analysis of Eby et al.'s data show the stress model accounts for 334 about half of their measured SWV. The difference between our results and theirs may be due 335 to our use of living muscle tissue, whereas Eby et al's studied excised, dead tissue, which 336 might have a higher stiffness that contributed to their higher SWV for a given stress. Still, 337 the Eby results are consistent with our main conclusion, which is that the stress model sets a 338 lower bound on the SWV within muscle tissue. 339 Our data show the stress model is incomplete for active muscle. It still appears to set 340 a lower bound but underestimates measures SWV versus stress. Others have shown SWV 341 increases in active muscle, but did not have a direct measure of muscle stress (33,35,36). So, 342 we are unaware of direct comparisons that can be made to our experimental results. 343 Understanding shear wave propagation in contracting muscle presents a challenge because 344 active muscle tension and SRS both depend on the number of attached cross-bridges. In a 345 previous attempt to clarify the active stress-stiffness-SWV relationship, we demonstrated that 346 SWV varies with muscle stiffness when tension is held constant (26). We manipulated 347 muscle temperature to elicit changes in muscle stiffness (SRS) without changes in active 348 force. Here, we show that SRS explains part of the error in the stress model, emphasizing the 349 importance of understanding how both muscle stress and stiffness contributed to SWV in 350 active muscle. 351 There are some limitations to our study. We had a surprising amount of variability in 352 our data, particularly for active measurements in the MG. We have not identified the source, 353 but can formulate several hypotheses. First, we expect greater variance in SWV at high 354 velocities because of limited sampling time, but there is no reason to believe the error should 355 be greater than data obtained from passive muscle at the same tension. Second, positioning 356 the ultrasound probe was more difficult in active muscle because of movement of the muscle 357 relative to the probe. This was particularly true for the MG, which had substantial changes in 358 pennation angle during active contractions potentially explaining the increased variability in 359 this muscle. However, it should not lead to an overall increase in SWV compared to passive 360 muscle or the stress model. A third problem seems to be the quality of the displacement 361 movie used to calculate SWV. Our subjective impression was that the clarity of the B-mode 362 images, and displacement movie, were not as good in active muscle, particularly at short 363 muscle lengths. The fascicles did not always stand out as clearly. Combined with the 364 heterogeneity of active stress within the muscle at short lengths or sub-maximal activation, 365 this could be another reason we see so much variability. 366 In summary, we found that a theoretical model of how stress influences SWV 367 predicted our experimental results well for passively stretched muscles. In contrast, this same 368 model significantly underpredicted the SWV in active muscles. These findings were 369 consistent in the soleus and medial gastrocnemius muscles, suggesting our results generalize 370 across muscles with different architectures and fiber types. These findings demonstrate that 371 SWV is sensitive to changes in muscle stress and that a model of how stress influences SWV 372 can be used to predict the SWV in passively stretched living muscles. This same model 373 provides only a lower bound on the SWV in active muscle, presumably due to activation-374 dependent changes in muscle stiffness. Together, our results provide further clarity on the 375 factors influencing shear wave propagation in muscle. relationship was assessed using a linear mixed effects model with SRS and activation state of 518 the muscle (passive or active) as the independent variables and the error difference between 519 the measured SWV2 that predicted by the stress model as the dependent variable. Cat was 520 considered as a random factor having an independent effect on the average slope and 521 intercept across activation states. A linear mixed effects model was used to determine if SRS 522 accounted for some of the error. SWV error squared was treated as the dependent variable, 523 SRS as continuous independent factor, and cat as a random factor. 524 525