Stress Relaxation of Porcine Gluteus Muscle Subjected to Sudden

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Eran Linder-Ganz. Amit Gefen ... =G1 ·exp(−t/1)+G2 ·exp(−t/2)+G , which provided good fit, ... showed that only G2 was affected by preconditioning, while GL,.
Stress Relaxation of Porcine Gluteus Muscle Subjected to Sudden Transverse Deformation as Related to Pressure Sore Modeling Avital Palevski Ittai Glaich Sigal Portnoy Eran Linder-Ganz Amit Gefen e-mail: [email protected] Department of Biomedical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel

Computational studies of deep pressure sores (DPS) in skeletal muscles require information on viscoelastic constitutive behavior of muscles, particularly when muscles are loaded transversally as during bone-muscle interaction in sitting and lying immobilized patients. In this study, we measured transient shear moduli G共t兲 of fresh porcine muscles in vitro using the indentation method. We employed a custom-made pneumatic device that allowed rapid 共2000 mm/ s兲 4 mm indentations. We tested 8 gluteus muscles, harvested from 5 adult pigs. Each muscle was indented transversally (perpendicularly to the direction of fibers) at 3 different sites, 7 times per site, to obtain nonpreconditioned (NPC) and preconditioned (PC) G共t兲 data. Short-term 共GS兲 and long-term 共GL兲 shear moduli were obtained directly from experiments. We further fitted measured G共t兲 data to a biexponential equation G共t兲 = G1 · exp共−t / ␶1兲 + G2 · exp共−t / ␶2兲 + G⬁, which provided good fit, visually and in terms of the correlation coefficients. Typically, plateau of the stress relaxation curves (defined as 10% difference from final GL) was evident ⬃20 s after indentation. Short-term shear moduli GS (mean NPC: 8509 Pa, PC: 5711 Pa) were greater than long-term moduli GL (NPC: 609 Pa, PC: 807 Pa) by about an order of magnitude. Statistical analysis of parameters showed that only G2 was affected by preconditioning, while GL, GS, G⬁, ␶1, ␶2, and G1 properties were unaffected. Since DPS develop over time scales of minutes to hours, but most stress relaxation occurs within ⬃20 s, the most relevant property for computational modeling is GL (mean ⬃700 Pa), which is, conveniently, unaffected by preconditioning. 关DOI: 10.1115/1.2264395兴 Keywords: pressure ulcer, decubitus, striated muscle, viscoelastic mechanical properties, indentation

1

Introduction

Deep pressure sores 共DPS兲 are attributed to deficient blood perfusion and lymphatic drainage, to excessive local deformations in tissues, and to ischemia-reperfusion sequences. In immobilized 共bedridden or wheelchair bound兲 patients, DPS are a lifethreatening, costly, and common complication 关1–4兴. The sites Contributed by the Bioengineering Division of ASME for publication in the JOURBIOMECHANICAL ENGINEERING. Manuscript received July 29, 2005; final manuscript received April 9, 2006. Review conducted by Michael Sacks.

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most susceptible to DPS are the regions that make contact with the supporting surface and also contain bony prominences that tend to concentrate mechanical loads and stresses 关1,5兴. The majority of the wounds appear on the lower part of the body, mostly around the sacrum and ischial tuberosities 共38–56%兲 关2,6兴. Prolonged compression of vascularized soft tissues, which obstructs pathways of nutrient supply and waste clearance, is considered the most important mechanical cause for onset of DPS 关1,5兴. Prolonged excessive compression has been shown to cause necrosis of skin, subcutaneous tissues and striated muscle, but muscular tissue appears to be the most sensitive one 关7兴. When DPS are formed in muscle tissue, the clinical classification is typically of a severe injury that is expected to expand towards the skin 关1,5,8兴. Treatment is consequently difficult, and in most cases, imposes surgery to remove necrotic tissues 关5兴. Biomechanical models showed that mechanical properties of the muscle play a dominant role in the prognosis of DPS. Specifically, exposure of striated muscle tissue to intensive and prolonged compression may pathologically alter its microstructure, thereby causing a change in tissue stiffness 关1,8兴. Changes in the mechanical properties of injured muscle tissue affect the mechanical stress and strain distributions around the site of the injury, and may expose additional uninjured regions of muscle tissue to intensified stresses in a positive-feedback mechanism that widens the injured area 关1,8兴. Therefore, understanding the etiology of DPS requires information on internal stress and strain distributions in deep tissues, particularly in the muscles underlying bony prominences during human sitting and lying, such as in the gluteus muscles which envelop the ischial tuberosities in these postures 关1,5,8兴. Computational models employed for this purpose make use of constitutive 共stress-strain兲 relationships to describe tissue behavior. However, a problem in formulating these constitutive equations is the limited availability of experimental data on tissue mechanical properties. Specifically, there are very limited published data on viscoelastic mechanical properties of striated muscle under transverse compression 关8–10兴, i.e., compression of muscle tissue perpendicularly to the direction of muscle fibers, as in the case of DPS biomechanics, where bony prominences 共e.g., the sacrum and ischial tuberosities兲 laterally compress the muscles 共e.g., the longissimus and gluteus, respectively兲. Published data of transverse viscoelastic mechanical properties for skeletal muscles, obtained in vitro 关9兴 and in vivo 关8,10兴, are all from rats, and so, there is paucity of information regarding transverse muscle properties in larger animal models, particularly in the pig, which is commonly used in pressure sore studies 关7,11兴. Another problem is that previous protocols of muscle transverse loading 关8,10兴 included ramp-and-hold tests with relatively long transition periods 共of 1 – 10 s, depending on the size of the step deformation兲, rather than a 共nearly兲 instantaneous step deformation. The response to a rapid step deformation is required to accurately evaluate the viscoelastic response 共particularly in short terms兲 to any general wave shape of deformation in absence of a specific constitutive model, based on Boltzmann’s principle of superposition in linear viscoelasticity, or based on Fung’s quasilinear viscoelastic theory 关12兴. In this study, rapid indentation experiments were conducted using a custom-made pneumatic stress-relaxation testing device that produced minimal transition periods 共as opposed to slow stepmotor-driven systems 关8,10兴兲, thereby measuring more accurate relaxation time courses, particularly in the short-term phase of tissue response. Determining the time course of relaxation of muscle tissue in the transverse direction is significant for DPS biomechanics because recently, it was indicated that pressure sores can develop in muscles on a time scale of minutes-to-hours, not necessarily hours-to-days, as it has long been believed 关8兴. For biomechanical modeling of the development of DPS, it is required to know when the mechanical properties of muscle tissue can be considered stable 共after viscoelastic stress relaxation兲, with respect to the time when a patient was put in the immobilized position

Copyright © 2006 by ASME

Transactions of the ASME

that led to DPS. If significant stress relaxation still continues when DPS onset/progress, this must be considered in the modeling 关1,5,8兴. Nonetheless, we hypothesize that the long-term viscoelastic characteristics 共e.g., the long-term shear modulus兲 are the ones relevant to DPS modeling. Accordingly, the specific aims of this study were 共i兲 experimental characterization of the in vitro shear moduli of fresh porcine gluteus muscle and characterization of the time course of relaxation in response to a 共nearly ideal兲 step deformation in the transverse direction, and 共ii兲 mathematical formulation of this response, in a form that can be used for future analytical and computational models.

2

Materials and Methods

2.1 Experimental Apparatus and Protocol. Muscle tissue is a complex hierarchial structure. At a microscopic scale, muscle tissue contains muscle fibers packed together by endomysium connective tissue, and individual packs are wrapped by perimysium connective tissue. Although this heterogeneous structure generally exhibits anisotropic mechanical behavior, for the purpose of biomechanical measurements at the macroscopic level, it is common to approximate muscle tissue as being a homogenous isotropic material 关1,8兴. This assumption allows a transient shear modulus of muscle tissue during viscoelastic stress relaxation G共t兲 to be calculated, if the muscle is also considered a semi-infinite volume that is indented by a rigid indentor with a circular flat tip 关8,13兴: G共t兲/共1 − ␯兲 = 关F共t兲/␦兴/共2␲d兲

共1兲

where ␯ is the Poisson’s ratio, d is the diameter of the indentor 共taken here is 12 mm兲, ␦ is the depth of indentation 共set as 4 mm兲, and F共t兲 is the measured time-dependent relaxation force. For viscoelastic materials such as muscle tissue, the load F共t兲 decreases with time to an asymptote during the ‘‘hold’’ period of the indentation. Accordingly, for a step indentation, a short-term 共instantaneous兲 shear modulus GS can be calculated from Eq. 共1兲 using the peak, or instantaneous load measured immediately at the start of the “hold” period. A long-term 共asymptotic兲 shear modulus GL can be calculated using the asymptotic load value. Equation 共1兲 was successfully used in previous studies to determine the transient shear modulus G共t兲 and its characteristics GS and GL for many soft tissues 关13兴. In the present study, we employed Eq. 共1兲 to determine the transient shear modulus G共t兲 of fresh porcine skeletal muscle tissue in the transverse direction 共i.e., perpendicular to the direction of muscle fibers兲. In order to solve Eq. 共1兲 for the transient shear modulus G共t兲 of skeletal muscles, we set ␯ = 0.5, considering that muscle tissue contains approximately 75% water and is therefore nearly incompressible 关8兴. Porcine gluteus muscle was selected for this study due to anatomical and physiological similarities to human skeletal muscles 关14兴 and because the pig is a well-accepted model for human pressure sores, particularly those affecting muscles 关7,15兴. Ten fresh muscles 共gluteus medius兲 were harvested from 5 euthanized adult pigs 共weighing 65± 23 kg兲 immediately after the time of death. Equation 共1兲 was shown to provide the most accurate G共t兲 evaluation for indentor radii that equal to or are less than 25% the thickness of the samples being studied 关16兴. Hence, we verified that the thickness of the central third of all our muscle specimens 共where indentations were conducted兲 was at least ⬃4 cm. All muscles were immersed in saline just after dissection, and were transported at 4 ° C to the testing facility, where recording of the relaxation responses began within 20 min from dissection, and was completed within 3 h from the time of euthanasia, to ensure that no rigor mortis occurred 关17,18兴. Muscles were tested in air, after being allowed a few minutes to equilibrate with room temperature, so that muscle temperature was ⬃24° C during measurements. The muscles were kept moist during experiments using a saline spray. In vitro indentation experiments were conducted using a Journal of Biomechanical Engineering

custom-made pneumatic testing device 共Fig. 1兲 that has a swift indentation speed of 2000 mm/ s during the ramp phase 共compared with an indentation speed of 0.25– 1 mm/ s in previous studies 关8,10兴兲. Hence, the ramp phase of 4 mm indentations 共4 mm was set as the constant indentation depth in all experiments兲 lasted 0.002 s. Such swift indentations may induce transient stress waves in the tested specimens. To verify that transient stress waves decayed before influencing measurements of longterm shear moduli—the property hypothesized herein to be the most relevant for pressure sore modeling—we conducted experiments with a viscoelastic polyurethane gel specimen. The size, thickness, and stiffness 共long-term shear modulus ⬃1000 Pa兲 of the gel specimen were similar to those of our muscle specimens. Use of a synthetic phantom, rather than tissue, eliminated the effect of biological variability and allowed isolation of the stress wave effect on long-term shear moduli GL. We conducted 10 trials of swift indentations 共0.002 s risetime of ramp兲, and 10 additional trials in which we loaded the viscoelastic gel quasistatically 共which ensured that no stress waves occurred兲. In all swift indentation tests, stress waves appeared in the polyurethane gel specimen within the first 0.01 s of relaxation, with magnitude, frequency, and decay pattern that were similar to those seen during muscle tissue indentations. However, the long-term shear moduli GL obtained in the quasistatic and swift indentation trials were statistically indistinguishable 共p-value of 0.85 in an unpaired twotail t-test兲. We concluded that transient stress waves did not affect the long-term shear moduli GL in our experimental setup, and hence, the same deformation parameters 共0.002 s risetime of ramp deformation and 4 mm depth of indentation兲 were used for testing muscle specimens. Each muscle was indented 7 times at 3 different sites that were at least 5 cm apart on the surface of the muscle’s central third, in order to obtain both nonpreconditioned 共NPC兲 and preconditioned 共PC兲 viscoelastic mechanical properties. We verified that the rapid indentations driven by the pneumatic device did not cause tears or any other type of visible damage to the muscle surface. At each indentation site 共3 sites per muscle兲, NPC results were obtained from the first indentation run, and PC results were obtained from the last 3 indentation cycles. The last 3 indentations at each site were assigned as PC data because in preliminary muscle tissue testing as well as for the muscles analyzed herein, stress relaxation curves overlapped visually and were shown to be stable for the last 3 indentation cycles 共out of 7兲 by two separate one-way analysis of variance 共ANOVA兲 tests comparing the fifth, sixth, and seventh cycles of GS and GL. These tests showed that both GS and GL properties were statistically indistinguishable across the last 3 preconditioning cycles, indicating arrival at a stable stress relaxation response. The duration of each measurement and the pause between subsequent preconditioning indentations at the same site 共which allowed elastic recovery of tissue兲 was set as 60 s, due to experimental design limitations, which required that all measurements be completed early enough before rigor mortis may onset. However, the relaxation response was observed to plateau well within the 60 s duration. Specifically, G共t兲 reached 10% difference from the final value 共GL兲 after a mean time of ⬃20 s, and a 2–4% difference from the final value 共GL兲 after a mean time of ⬃40 s. In each stress relaxation trial, the indentor, with a circular, flat tip 共diameter 12 mm兲, was placed over the central part of the muscle and carefully lowered to the surface using its system of axial adjustors and elevator 共Fig. 1兲. As soon as indentor-muscle contact was detected, as determined by a threshold reading of 0.1 N from the force transducer attached to the indentor, position was held and the pneumatic piston 共BDALS 10X30, Koganei Co., Japan兲 driving the indentor was armed at a constant air pressure of 400 kPa as measured by a pressure regulator 共to ensure identical conditions across all measurements兲. A stopper plate was set at a distance of 4 mm from the surface of the indentor to ensure that OCTOBER 2006, Vol. 128 / 783

Fig. 1 The pneumatic indentation testing device.

all indentations are to the same 4 mm depth. After release of the pressurized pneumatic piston, force readings were recorded over time for each indentation run using a force transducer attached to the indentor. The force transducer has a response time of less than 5 ␮s, capacity of 20 N, accuracy of ±2.5%, and drift ⬍3% per logarithmic time scale 共Flexiforce A201, Tekscan Co., MA, USA兲. Force signals were amplified using operational amplifier 共LM324N, National Instruments Co., TX, USA兲 and were sampled continuously at a frequency of 1000 Hz for the first 3 s and at 100 Hz afterwards 共using A/D card and LabView 7.0 software, National Instruments Co.兲 to maintain reasonable data file sizes 共Fig. 2兲. Results from two of the muscles and ⬃10% of stress relaxation trials from the other 8 muscles were excluded due to interrupted indentor contact manifested by noisy data. 2.2 Data Analysis. Since DPS develop over time scales of minutes-to-hours 关8兴, but most stress relaxation was shown to occur within ⬃20 s 共as indicated above兲, the most relevant property for DPS modeling studies emerges to be the long-term shear modulus of muscles GL. Considering that once an immobilized person was positioned in a bed or wheelchair, he is not expected to change posture for a long period of time, the NPC modulus, particularly, seems to be relevant for computational modeling of DPS. Nevertheless, we acquired short-term and long-term, NPC and PC shear moduli for completeness. Hence, short-term 共instantaneous兲 shear moduli 共GS兲 and long-term shear moduli 共GL兲, were obtained directly from substituting the experimental force data in Eq. 共1兲. Specifically, we calculated GS from each stress relaxation trial by substituting the peak force, max兵F共t兲其, measured at the instance of deformation. The corresponding GL was 784 / Vol. 128, OCTOBER 2006

calculated by substituting the mean of force data over the terminal 1 s 共100 data points兲 of the plateau phase of the stress relaxation curve 共we defined the beginning of the plateau phase as the time point at which GL reached 10% difference from the final value兲. We then calculated means and standard deviations of experimentally obtained, NPC and PC GS and GL values, as follows. First, to reduce prospective measurement errors in PC GS and GL data, and considering that ANOVA showed stable, repeatable stress relaxations during the last 3 preconditioning cycles 共both GS and GL were statistically indistinguishable across the fifth, sixth, and seventh preconditioning cycles兲, we averaged each parameter across the 3 last indentation runs at each site, and termed the mean values from the 3 last runs as the fully PC GS and GL data. Second, in order to further reduce variability, we averaged the NPC and fully PC GS and GL data 共separately兲 across the 3 indentation sites in each muscle, and hence, obtained the effective NPC and PC GS and GL properties per muscle. Using the Matlab 共version 7兲 software “curve fitting tool,” the experimental transient shear moduli G共t兲 were found to have a good fit to a biexponential equation 关19兴, both visually and in terms of correlation coefficients 共for a total of 43 NPC and fully PC curves: R2 ⬎ 0.9 for 37 curves, R2 ⬎ 0.85 for 5 curves, and R2 ⬎ 0.7 for one curve兲. Experimental G共t兲 data were therefore fitted to a function given by G共t兲 = G1 · exp共− t/␶1兲 + G2 · exp共− t/␶2兲 + G⬁

共2兲

where ␶1 and ␶2 are the short and long relaxation times, G1 , G2 are constants, and G⬁ is the long-term shear modulus from curve fitting. Substituting t = 0 in Eq. 共2兲 yields that the short-term 共or Transactions of the ASME

Fig. 2 Representative transient shear modulus G„t… data obtained for in vitro nonpreconditioned „NPC, dark gray… and preconditioned „PC, seventh indentation run, light gray… indentations of porcine gluteus medius muscle. Lines show biexponential curve fitting „Eq. „2…… to the NPC „thin line… and PC „thick line… experimental data. The first 0.8 s of the stress relaxation are magnified in the central plot to show the transient response. The first 0.05 s are magnified in the top right plot to show the ramp loading phase.

instantaneous兲 shear modulus calculated from curve fitting is G1 + G 2 + G ⬁. The values of G1 , G2 , G⬁ , ␶1 , ␶2, the parameters of the transient relaxation modulus function 共Eq. 共2兲兲, were obtained for the naive NPC indentation and the 3 last indentation runs at each site 共Fig. 2兲. Similarly to the above-described processing of directly measured GS and GL values, in order to obtain mean PC parameter values from curve fitting, we averaged G1 , G2 , G⬁ , ␶1 , ␶2 parameters across the 3 last indentation runs at each site, and referred to the mean values from these last 3 maneuvers as the fully PC data. In order to further reduce variability in G1 , G2 , G⬁ , ␶1 , ␶2 data, we averaged each parameter across the 3 sites per muscle specimen 共separately for the NPC and PC results兲, so that an effective mechanical characterization of the NPC and PC relaxation response and transient shear modulus G共t兲 per muscle was obtained. 2.3 Statistical Testing. All data sets were tested for normal distribution using the Shapiro-Wilk test, and all parameter values were found to be normally distributed with the exception of PC values of ␶1. However, PC ␶1 values were distributed very close to a normal distribution as was evident from the linearity of a normal quantile plot drew for this parameter. Statistical outlier values were removed from the GS , GL , G1 , G2 , G⬁ , ␶1 , ␶2 data sets, based on the results of a Grubb’s test run separately for each parameter 关20兴. We typically removed 1–3 outlier values per parameter. Paired t-tests were then conducted for NPC versus PC values of measured short-term 共GS兲 and long-term 共GL兲 shear moduli across all muscle specimens to determine if these shear moduli are affected by preconditioning. “One-tail” tests were employed since preconditioning a soft tissue in vitro or in situ was shown to consistently decrease the short-term and long-term shear moduli at the preconditioning site 关8,19兴. In all statistical tests, a p-value lower than 0.05 was considered statistically significant. Journal of Biomechanical Engineering

3

Results

Table 1 shows the mean values and standard deviations obtained for all parameters, GS , GL , G1 , G2 , G⬁ , ␶1 , ␶2, during NPC and PC indentations. Our measured variations in mechanical properties, due to biological variability, are similar to those reported in previous studies 关9,17,19兴. Short-term shear moduli GS were in the order of thousands of Pascals, and long-term shear moduli GL were in the order of hundreds of Pascals 共Table 1兲. Hence, stress relaxation was shown to decrease NPC shear moduli 14-fold, and likewise, decrease PC shear moduli sevenfold 共Table 1兲. We studied the effect of preconditioning on measured shortterm shear moduli 共GS兲 and long-term shear moduli 共GL兲 of porcine gluteus muscles by means of paired t-tests. Preconditioning was shown to significantly decrease mean GS 共⬃0.67-fold, Table 1兲 共p = 0.04兲. However, we found that the effect of preconditioning on GL was not statistically significant 共p = 0.32兲. It should be noted that although mean PC GL values appear to be higher than mean NPC GL 共⬃1.3-fold, Table 1兲, individual sites in a muscle always displayed a decrease of GL values with preconditioning 共though not a statistically significant decrease兲, and so, the difference between means of NPC and PC GL should be attributed to biological variability and not to preconditioning. A retrospective power analysis indicated that the chance of detecting a statistically significant effect of preconditioning on GL with our sample size was 62%, and that the minimal detectable difference in testing NPC versus PC GL was 298 Pa 共for “one side” test with power of 90%兲. Accordingly, it is possible that preconditioning might have affected GL to an extent smaller than our above test resolution. Taking this reservation into account, we conclude that GL, aforementioned as the most relevant property for DPS related computational modeling, is conveniently found to be unaffected by preconditioning. OCTOBER 2006, Vol. 128 / 785

PC N = 23 807± 406 NPC N = 17 609± 215 NPC N = 19 8509± 3934

PC N = 24 5711± 1788

GL 共Pa兲 GS 共Pa兲

NPC N = 17 575± 174

PC N = 23 811± 406

4

NPC= nonpreconditioned. PC= preconditioned. b

a

NPC N = 16 0.051± 0.035 PCb N = 24 3922± 2250 NPCa N = 19 4663± 3775

PC N = 19 0.016± 0.019

NPC N = 19 1146± 556

PC N = 24 343± 92

NPC N = 19 8.59± 6.63

PC N = 23 9.01± 6.36

G⬁ 共Pa兲

␶2 共s兲 G2 共Pa兲

␶1 共s兲 G1 共Pa兲

Table 1 Parameter values for porcine gluteus transient shear modulus of the form: G„t… = G1 · exp„−t / ␶1… + G2 · exp„−t / ␶2… + Gⴥ, and experimental values for short-term „GS… and long-term „GL… shear moduli „mean± SD…. N denotes the number of values used to calculate the mean ±SD of each parameter after excluding noisy data and outliers.

The biexponential curve fitting 共Eq. 共2兲兲 provided excellent approximations of the long-term shear modulus G⬁ with respect to measured GL 共error of 6% and 0.5% for the NPC and PC values, respectively, Table 1兲. However, errors in approximating the shortterm modulus GS by G1 + G2 + G⬁ were more substantial 共error of 25% and 11% for the NPC and PC values, respectively, Table 1兲. As already noted, stress relaxation was shown to occur rather quickly, and hence, 90% of the full relaxation 共GL兲 was reached within ⬃20 s 共mean兲 from the instant of deformation 共Fig. 2兲. Relaxation times were more than two orders of magnitude apart, with the short relaxation time constant being less than 0.1 s, and the long time constant being less than 20 s 共Table 1兲. We further notice that the ramp risetime induced by the pneumatic testing device 共0.002 s兲 is substantially shorter than the short relaxation time constant ␶1 共⬃26-fold NPC, ⬃ eightfold PC, Table 1兲, which ensures that information on the short-term phase of stress relaxation was adequately captured 关12兴.

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Discussion

The objective of the present study was to characterize the viscoelastic stress relaxation and shear moduli of porcine gluteus muscles in vitro, in a manner suitable for analytical and computational studies of DPS in humans. Such modeling generally requires that the viscoelastic mechanical properties of muscles be specified, particularly, in the transverse direction 共perpendicularly to muscle fibers兲 to mimic compression of bony prominences in sitting or lying humans. The results provided herein can be employed to describe the time course of stress relaxation of gluteus muscles in the transverse direction, in full detail. In many DPS related studies, however, which deal with prolonged excessive compression of muscle tissue by bony prominences in immobilized patients 关1,5,8兴, it is mainly the long-term shear modulus GL 共or G⬁兲 that is of interest. The present results indicate that stress relaxation in the transverse direction of muscle tissue occurs within less than 1 min, i.e., shorter than the minutes-to-hours time scale required for developing DPS 关8兴. Hence, GL in the transverse direction is indeed the relevant mechanical property when modeling muscle tissue as related to DPS. Previous studies showed that mechanical properties of muscle tissue reach a steady state response, and therefore, become repeatable as a result of preconditioning 关8,21兴. It has been shown herein that for the sample size considered in this study, GL was statistically unaffected by preconditioning, and this may simplify DPS modeling, particularly when it is desirable to take postural changes of patients 共which induce muscle preconditioning兲 into account. It should be noted that the exclusion of statistical outliers, averaging of data, and selection of a one-tailed statistical analysis considerably increased the probability that differences resulting from preconditioning would be detected. However, in spite of that, the effect of preconditioning on long-term moduli was not significant, further supporting the main result of this study. The latter finding, that GL is not affected by preconditioning, is also in accordance with a previous study of soft tissue viscoelasticity, which found that preconditioning does not affect the long-time behavior of human ankle ligament 关22兴. Taken together, our findings therefore indicate that GL of skeletal muscles in the transverse direction should be taken as ⬃700± 300 Pa 共NPC and PC combined兲. This value is in exact agreement with in vivo GL data 共for transverse loading兲 which were previously obtained from gracilis of anesthetized rats 关8兴, and with a literature review suggesting that the long-term shear moduli of human skeletal muscle in the transverse direction should be in the order of 250– 1200 Pa 关8兴. This study had some limitations which should be considered while interpreting the results. The first limitation concerns the assumption that muscle tissue is homogenous. This assumption allowed characterization of an effective viscoelastic mechanical behavior at the continuum level, but not local variants of tissue Transactions of the ASME

mechanical properties. However, the effective viscoelastic properties at the continuum level of muscle are sufficient for most DPS modeling tasks 关1,8兴. The analytical solution of the indentation problem 共Eq. 共1兲兲 further assumes homogenous elastic tissue with a perfectly flat surface and boundaries at an infinite distance from the site of indentation. The latter assumption was closely approached because the indentor’s diameter was small compared to the distance between the indentor and the tissue boundaries. A second limitation is that the G共t兲 experimental data presented here are subject to a time limit of 60 s, the “hold” time of the stress relaxation tests. This time limit was necessary, because experiments had to be carefully limited in time frame to ensure that measurements are taken prior to rigor mortis effects 关17,18兴. Determination of 60 s data capture periods may result in some small overestimation of the long-term 共asymptotic兲 shear modulus, due to approximation of the point at which the stress relaxation test is ended as an infinite time 关8,12,22兴. Considering, for example, the case of a wheelchair bound patient whose posture is changed once every 30 min 关8兴. The mean rate of change in 共PC兲 shear moduli at the terminal second of measurement 共dG共t兲 / dt at t = 60 s兲 was −0.05 Pa/ s, and hence, further decrease in GL due to ongoing stress relaxation of muscles in such patient who is immobilized for 30 min 共1800 s兲 must be smaller than −0.05 Pa/ s · 1800 s = 90 Pa, i.e., maximal ⬃13% decrease from the presently measured ⬃700 Pa value. Considering also the biological variability in GL properties 共ratio of standard deviation over mean ⬃35% for NPC and ⬃50% for PC GL properties兲, long-term shear moduli obtained after a longer relaxation period are expected to be similar to those obtained herein. A third limitation relates to the rapid rate of indentations. We employed swift indentations 共0.002 risetime of ramp deformation兲 to obtain a relaxation response that is as similar as possible to the theoretical “step response.” However, as opposed to theoretical 共ideal兲 step deformations, swift deformations in an experimental system may involve stress waves, some of which are apparent in the magnification of the short-term response in Fig. 2 共top right frame兲. These stress waves were shown to damp out completely within less than 0.5 s from the onset of deformation. Nonetheless, we verified, using phantom experiments, that GL, the relevant property for DPS modeling, is unaffected by these stress waves 共see Methods兲. Unfortunately, we cannot exclude a possibility that the short-term transient constants of the relaxation response ␶1 and G1, which depend on the deformation rate 关12兴, might have been affected by stress waves. Last, the viscoelastic properties reported herein represent passive muscle behavior, and do not consider stiffness due to muscle contractility. However, as this study is relevant to modeling the mechanical conditions in muscles of immobilized 共wheelchair bound and bedridden兲 patients who typically have no neural stimulation of muscles, the passive properties are of interest. In closing, further work in applying various analytical and computational models to the study of DPS can make use of the quantitative formulation of the stress relaxation and shear moduli of porcine gluteal muscle tissue in the transverse direction, which are reported in this study.

Acknowledgment The authors would like to thank Dr. Ehud Willenz from the Neufeld Cardiac Research Institute of Sheba Medical Center, Israel, for handling the animals and conducting the dissections. Dr.

Journal of Biomechanical Engineering

Uri Zaretsky from the Department of Biomedical Engineering at Tel Aviv University is thanked for helping with the design of the pneumatic stress relaxation testing device. This study was partially supported by the Slezak Super Center for Cardiac Research and Biomedical Engineering 共AG兲 and by the Internal Research Fund of Tel Aviv University 共AG兲.

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OCTOBER 2006, Vol. 128 / 787