Method for characterizing viscoelasticity of human gluteal tissue

Characterizing compressive transient large deformation properties of biological tissue is becoming increasingly important in impact biomechanics and rehabilitation engineering, which includes devices interfacing with the human body and virtual surgical guidance simulation. Individual mechanical in vivo behaviour, specifically of human gluteal adipose and passive skeletal muscle tissue compressed with finite strain, has, however, been sparsely characterised. Employing a combined experimental and numerical approach, a method is presented to investigate the time-dependent properties of in vivo gluteal adipose and passive skeletal muscle tissue. Specifically, displacement-controlled ramp-and-hold indentation relaxation tests were performed and documented with magnetic resonance imaging. A time domain quasi-linear viscoelasticity (QLV) formulation with Prony series valid for finite strains was used in conjunction with a hyperelastic model formulation for soft tissue constitutive model parameter identification and calibration of the relaxation test data. A finite element model of the indentation region was employed. Strong non-linear elastic but linear viscoelastic tissue material behaviour at finite strains was apparent for both adipose and passive skeletal muscle mechanical properties with orthogonal skin and transversal muscle fibre loading. Using a force-equilibrium assumption, the employed material model was well suited to fit the experimental data and derive viscoelastic model parameters by inverse finite element parameter estimation. An individual characterisation of in vivo gluteal adipose and muscle tissue could thus be established. Initial shear moduli were calculated from the long-term parameters for human gluteal skin/fat: G(∞,S/F)=1850 Pa and for cross-fibre gluteal muscle tissue: G(∞,M)=881 Pa. Instantaneous shear moduli were found at the employed ramp speed: G(0,S/F)=1920 Pa and G(0,M)=1032 Pa.     

Determination of Poisson's ratio of articular cartilage by indentation using different-sized indenters

Articular cartilage is often characterized as an isotropic elastic material with no interstitial fluid flow during instantaneous and equilibrium conditions, and indentation testing commonly used to deduce material properties of Young's modulus and Poisson's ratio. Since only one elastic parameter can be deduced from a single indentation test, some other test method is often used to allow separate measurement of both parameters. In this study, a new method is introduced by which the two material parameters can be obtained using indentation tests alone, without requiring a secondary different type of test. This feature makes the method more suitable for testing small samples in situ. The method takes advantages of the finite layer effect. By indenting the sample twice with different-sized indenters, a nonlinear equation with the Poisson's ratio as the only unknown can be formed and Poisson's ratio obtained by solving the nonlinear equation. The method was validated by comparing the predicted Poisson's ratio for urethane rubber with the manufacturer's supplied value, and comparing the predicted Young's modulus for urethane rubber and an elastic foam material with modulii measured by unconfined compression. Anisotropic and nonhomogeneous finite-element (FE) models of the indentation were developed to aid in data interpretation. Applying the method to bovine patellar cartilage, the tissue Young's modulus was found to be 1.79 +/- 0.59 MPa in instantaneous response and 0.45 +/- 0.26 MPa in equilibrium, and the Poisson's ratio 0.503 +/- 0.028 and 0.463 +/- 0.073 in instantaneous and equilibrium, respectively. The equilibrium Poisson's ratio obtained in our work was substantially higher than those derived from biphasic indentation theory and those optically measured in an unconfined compression test. The finite element model results and examination of viscoelastic-biphasic models suggest this could be due to viscoelastic, inhomogeneity, and anisotropy effects.     

Characterization of human passive muscles for impact loads using genetic algorithm and inverse finite element methods

The objective of this study is to identify the dynamic material properties of human passive muscle tissues for the strain rates relevant to automobile crashes. A novel methodology involving genetic algorithm (GA) and finite element method is implemented to estimate the material parameters by inverse mapping the impact test data. Isolated unconfined impact tests for average strain rates ranging from 136 s⁻¹ to 262 s⁻¹ are performed on muscle tissues. Passive muscle tissues are modelled as isotropic, linear and viscoelastic material using three-element Zener model available in PAMCRASH^TM explicit finite element software. In the GA based identification process, fitness values are calculated by comparing the estimated finite element forces with the measured experimental forces. Linear viscoelastic material parameters (bulk modulus, short term shear modulus and long term shear modulus) are thus identified at strain rates 136 s⁻¹, 183 s⁻¹ and 262 s⁻¹ for modelling muscles. Extracted optimal parameters from this study are comparable with reported parameters in literature. Bulk modulus and short term shear modulus are found to be more influential in predicting the stress-strain response than long term shear modulus for the considered strain rates. Variations within the set of parameters identified at different strain rates indicate the need for new or improved material model, which is capable of capturing the strain rate dependency of passive muscle response with single set of material parameters for wide range of strain rates.     

Creep indentation of single cells

An apparatus for creep indentation of individual adherent cells was designed, developed, and experimentally validated. The creep cytoindentation apparatus (CCA) can perform stress-controlled experiments and measure the corresponding deformation of single anchorage-dependent cells. The apparatus can resolve forces on the order of 1 nN and cellular deformations on the order of 0.1 micron. Experiments were conducted on bovine articular chondrocytes using loads on the order of 10 nN. The experimentally observed viscoelastic behavior of these cells was modeled using the punch problem and standard linear solid. The punch problem yielded a Young's modulus of 1.11 +/- 0.48 kPa. The standard linear solid model yielded an instantaneous elastic modulus of 8.00 +/- 4.41 kPa, a relaxed modulus of 1.09 +/- 0.54 kPa, an apparent viscosity of 1.50 +/- 0.92 kPa-s, and a time constant of 1.32 +/- 0.65 s. To our knowledge, this is the first time that stress-controlled indentation testing has been applied at the single cell level. This methodology represents a new tool in understanding the mechanical nature of anchorage-dependent cells and mechanotransductional pathways.     

In vivo characterization of the mechanical properties of human skin derived from MRI and indentation techniques

The human skin is an exceedingly complex and multi-layered material. This paper aims to introduce the application of the finite element analysis (FEA) to the in vivo characterization of the non-linear mechanical behaviour of three human skin layers. Indentation tests combined with magnetic resonance imaging (MRI) technique have been performed on the left dorsal forearm of a young man in order to reveal the mechanical behaviour of all skin layers. Using MRI images processing and a pre and post processor allows to make numerically individualized 2D model which consists of three skin layers and the muscles. FEA has been applied to simulate indentation tests. Neo-Hookean slightly compressible material model of two material constants (C(10), K) has been used to model the mechanical behaviour of the three skin layers and the muscles. The identification of material model parameters was done by applying Levenberg-Marquardt algorithm (LMA). Our methodology of identification provides a range of values for each constant. Range of values of different material properties of epidermis, dermis, hypodermis are respectively, C10(E)=0.12+/-0.06 MPa, C10(D)=1.11+/-0.09 MPa, C10(H)=0.42+/-0.05 KPa, K(E)=5.45+/-1.7 MPa, K(D)=29.6+/-1,28 MPa, K(H)=36.0+/-0.9 KPa.     

Viscoelastic properties of single attached cells under compression

The viscoelastic properties of single, attached C2C12 myoblasts were measured using a recently developed cell loading device. The device allows global compression of an attached cell, while simultaneously measuring the associated forces. The viscoelastic properties were examined by performing a series of dynamic experiments over two frequency decades (0.1-10 Hz) and at a range of axial strains (approximately 10-40%). Confocal laser scanning microscopy was used to visualize the cell during these experiments. To analyze the experimentally obtained force-deformation curves, a nonlinear viscoelastic model was developed. The nonlinear viscoelastic model was able to describe the complete series of dynamic experiments using only a single set of parameters, yielding an elastic modulus of 2120 +/- 900 Pa for the elastic spring, an elastic modulus of 1960 +/- 1350 for the nonlinear spring, and a relaxation time constant of 0.3 +/- 0.12 s. To our knowledge, it is the first time that the global viscoelastic properties of attached cells have been quantified over such a wide range of strains. Furthermore, the experiments were performed under optimal environmental conditions and the results are, therefore, believed to reflect the viscoelastic mechanical behavior of cells, such as would be present in vivo.     

Passive and active mechanical properties of the superficial and deep digital flexor muscles in the forelimbs of anesthetized Thoroughbred horses

The superficial (SDF) and deep digital flexor (DDF) muscles are critical for equine forelimb locomotion. Knowledge of their mechanical properties will enhance our understanding of limb biomechanics. Muscle contractile properties derived from architectural-based algorithms may overestimate real forces and underestimate shortening capacity because of simplistic assumptions regarding muscle architecture. Therefore, passive and active (=total - passive) force-length properties of the SDF and DDF muscles were measured directly in vivo. Muscles from the right forelimbs of four Thoroughbred horses were evaluated during general anesthesia. Limbs were fixed to an external frame with the muscle attached to a linear actuator and load cell. Each muscle was stretched from an unloaded state to a range of prefixed lengths, then stimulated while held at that length. The total force did not exceed 4000 N, the limit for the clamping device. The SDF and DDF muscles produced 716+/-192 and 1577+/-203 N maximum active isometric force (F(max)), had ascending force-length ranges (R(asc)) of 5.1+/-0.2 and 9.1+/-0.4 cm, and had passive stiffnesses of 1186+/-104 and 1132+/-51 N/cm, respectively. The values measured for F(max) were much smaller than predicted based on conservative estimates of muscle specific tension and muscle physiological cross-sectional area. R(asc) were much larger than predicted based on muscle fiber length estimates. These data suggest that accurate prediction of the active mechanical behavior of architecturally complex muscles such as the equine DDF and SDF requires more sophisticated algorithms.     

Static versus dynamic gerbil tympanic membrane elasticity: derivation of the complex modulus

An accurate estimation of tympanic membrane stiffness is important for realistic modelling of middle ear mechanics. Tympanic membrane stiffness has been investigated extensively under either quasi-static or dynamic loading conditions. It is known that biological tissues are sensitive to strain rate. Therefore, in this work, the mechanical behaviour of the tympanic membrane was studied under both quasi-static and dynamic loading conditions. Experiments were performed on the pars tensa of four gerbil tympanic membranes. A custom-built indentation apparatus was used to perform in situ tissue indentations and testing was done applying both quasi-static and dynamic sinusoidal indentations up to 8.2?Hz. The unloaded shape of the tympanic membrane was measured and used to create specimen-specific finite element models to simulate the experiments. The frequency dependent Young's modulus of each specimen was then estimated by an inverse analysis in which the error between experimental and simulated indentation data was optimised for each indentation frequency separately. Using an 8?μm central region thickness, we found Young's moduli between 71 and 106?MPa (n = 4) at 0.2?Hz indentation frequency. A standard linear viscoelastic model and a viscoelastic model with a continuous relaxation spectrum were used to derive a complex modulus in the frequency domain. Due to experimental limitations, the indentation frequency upper limit was 8.2?Hz. The average relative modulus increase in this domain was 14% and the increase was the strongest below 6?Hz.     

Models of skeletal muscle to explain the increase in passive stiffness in desmin knockout muscle

Absence of desmin in skeletal muscle was found to induce an increase in passive stiffness. The present study aimed at developing rheological models of passive muscle to explain this stiffening. Models were elaborated by using experimental data depicting muscle viscoelastic behaviour. The experimental protocol included stepwise extension tests applied on control and desmin knockout soleus muscles from mice. Linear and non-linear models were composed of elastic and viscous elements. They were constructed with the aim at taking the presence or absence of desmin into account by simulating desmin as an elastic element. Furthermore, associated adaptation of connective tissues in absence of desmin was modelled as an additional elastic element. Differences in passive behaviour induced by absence of desmin were predicted by using a linear model and a non-linear one. The non-linear model was selected because: (1) it is able to predict experimental viscoelastic kinetics accounting for the increase in passive stiffness in muscles lacking desmin, (2) its design is consistent with morphological data, and (3) stiffness characteristics of its elements are in accordance with the literature. Finally, this modelling approach demonstrates that both absence of desmin and adaptation of connective tissue are required to explain the increase in passive stiffness in desmin knockout muscles.     

The length dependence of muscle active force: considerations for parallel elastic properties.

The purpose of this study was to choose between two popular models of skeletal muscle: one with the parallel elastic component in parallel with both the contractile element and the series elastic component (model A), and the other in which it is in parallel with only the contractile element (model B). Passive and total forces were obtained at a variety of muscle lengths for the medial gastrocnemius muscle in anesthetized rats. Passive force was measured before the contraction (passive A) or was estimated for the fascicle length at which peak total force occurred (passive B). Fascicle length was measured with sonomicrometry. Active force was calculated by subtracting passive (A or B) force from peak total force at each fascicle or muscle length. Optimal length, that fascicle length at which active force is maximized, was 13.1 +/- 1.2 mm when passive A was subtracted and 14.0 +/- 1.1 mm with passive B (P < 0.01). Furthermore, the relationship between double-pulse contraction force and length was broader when calculated with passive B than with passive A. When the muscle was held at a long length, passive force decreased due to stress relaxation. This was accompanied by no change in fascicle length at the peak of the contraction and only a small corresponding decrease in peak total force. There is no explanation for the apparent increase in active force that would be obtained when subtracting passive A from the peak total force. Therefore, to calculate active force, it is appropriate to subtract passive force measured at the fascicle length corresponding to the length at which peak total force occurs, rather than passive force measured at the length at which the contraction begins.     

Surface viscoelastic effects in cell cleavage

The effect of passive surface traction on the cleavage of cells is incorporated in the cytokinesis hydrodynamic model of Zinemanas and Nir [Biomechanics of cell Division, pp. 281-305, Plenum Press, New York (1987)]. Different rheological behaviours were examined to model the surface tensions which arise due to the passive deformation of the cortex: a Mooney-Rivlin material, an STZC material and a viscoelastic material. The calculations show that passive surface tensions may play a significant role in determining the local surface deformations as well as in the modulation of the surface forces. Varying the rheological model has limited effect on the overall deformations. The latter appear to be affected mostly by contractile filament interactions.     

Viscoelastic properties of isolated rat colon smooth muscle cells

The measurement of the biomechanical properties of gastrointestinal smooth muscle cells is important for the basic understanding of digestive function and the interaction of muscle cells with the matrix. Externally applied forces will deform the cells depending upon their mechanical properties. Hence, the evoked response mediated through stretch-sensitive ion-channels in the smooth muscle cell membrane will depend upon membrane properties and the magnitude of the external force. The aim of this study was to test the hypothesis that gastrointestinal smooth muscle cells behave in a viscoelastic manner. Smooth muscle cells were dissociated from the muscle layers of the descending colon. The viscoelastic properties of the isolated cells were characterized by measuring the mechanical deflection response of the cell membrane to a negative pressure of 1cm H(2)O applied across the cell through a micropipette and fitting the response to a theoretical viscoelastic solid model. The viscoelastic mechanical constants of the isolated cells (N=9) were found to be as follows: k(1)=19.99+/-2.86 Pa, k(2)=7.19+/-1.21 Pa, mu=25.36+/-6.14 Pas and tau=4.84+/-0.95 s. This study represents, to the best of our knowledge, the first quantitative mechanical properties of isolated living smooth muscle cells from the gastrointestinal tract. The mechanical properties determined in this study will be of use in future analytical and numerical smooth muscle cell models to better predict the mechanism between the magnitude of mechanical stimuli, mechanosensitivity and the evoked afferent responses.     

Model and in vivo studies on human trunk load partitioning and stability in isometric forward flexions

To resolve the trunk redundancy to determine muscle forces, spinal loads, and stability margin in isometric forward flexion tasks, combined in vivo-numerical model studies was undertaken. It was hypothesized that the passive resistance of both the ligamentous spine and the trunk musculature plays a crucial role in equilibrium and stability of the system. Fifteen healthy males performed free isometric trunk flexions of approximately 40 degrees and approximately 65 degrees +/- loads in hands while kinematics by skin markers and EMG activity of trunk muscles by surface electrodes were measured. A novel kinematics-based approach along with a nonlinear finite element model were iteratively used to calculate muscle forces and internal loads under prescribed measured postures and loads considered in vivo. Stability margin was investigated using nonlinear, linear buckling, and perturbation analyses under various postures, loads and alterations in ligamentous stiffness. Flexion postures significantly increased activity in extensor muscles when compared with standing postures while no significant change was detected in between flexed postures. Compression at the L5-S1 substantially increased from 570 and 771 N in upright posture, respectively, for +/-180 N, to 1912 and 3308 N at approximately 40 degrees flexion, and furthermore to 2332 and 3850 N at approximately 65 degrees flexion. Passive ligamentous/muscle components resisted up to 77% of the net moment. In flexion postures, the spinal stability substantially improved due both to greater passive stiffness and extensor muscle activities so that, under 180 N, no muscle stiffness was required to maintain stability. The co-activity of abdominal muscles and the muscle stiffness were of lesser concern to maintain stability in forward flexion tasks as compared with upright tasks. An injury to the passive system, on one hand, required a substantial compensatory increase in active muscle forces which further increased passive loads and, hence, the risk of injury and fatigue. On the other hand, it deteriorated the system stability which in turn could require greater additional muscle activation. This chain of events would place the entire trunk active-passive system at higher risks of injury, fatigue and instability.     

The passive properties of muscle fibers are velocity dependent

The passive properties of skeletal muscle play an important role in muscle function. While the passive quasi-static elastic properties of muscle fibers have been well characterized, the dynamic visco-elastic passive behavior of fibers has garnered less attention. In particular, it is unclear how the visco-elastic properties are influenced by lengthening velocity, in particular for the range of physiologically relevant velocities. The goals of this work were to: (i) measure the effects of lengthening velocity on the peak stresses within single muscle fibers to determine how passive behavior changes over a range of physiologically relevant lengthening rates (0.1–10L_o/s), and (ii) develop a mathematical model of fiber viscoelasticity based on these measurements. We found that passive properties depend on strain rate, in particular at the low loading rates (0.1–3L_o/s), and that the measured behavior can be predicted across a range of loading rates and time histories with a quasi-linear viscoelastic model. In the future, these results can be used to determine the impact of viscoelastic behavior on intramuscular stresses and forces during a variety of dynamic movements.     

Mapping motor neuron activity to overt behavior in the leech

As an initial step in constructing a quantitative biomechanical model of the medicinal leech (Hirudo medicinalis), we determined the passive properties of its body wall over the physiological range of dimensions. The major results of this study were:

1. The ellipsoidal cross section of resting leeches is maintained by tonic muscle activation as well as forces inherent in the structure of the body wall (i.e., residual stress).
2. The forces required for longitudinal and circumferential stretch to maximum physiological dimensions were similar in magnitude. Cutting out pieces of body wall did not affect the passive longitudinal or circumferential properties of body wall away from the edges of the cut.
3. The strain (i.e., the percentage change in dimension) of different body segments when subject to the same force was identical, despite differences in muscle crosssections.
4. Serotonin, a known modulator of tension in leech muscles, affected passive forces at all physiological muscle lengths. This suggests that the longitudinal muscle is responsible for at least part of the passive tension of the body wall.
5. We propose a simple viscoelastic model of the body wall. This model captures the dynamics of the passive responses of the leech body wall to imposed step changes in length. Using steady-state passive tensions predicted by the viscoelastic model we estimate the forces required to maintain the leech at any given length over the physiological range.

     

Elastic properties of an intact and ACL-ruptured knee joint: measurement, mathematical modelling, and haptic rendering

An analytical, dynamic model of the human knee joint has been developed to simulate the unloaded knee joint behaviour in 6 degrees of freedom. It is based on extensive robot-based measurements of the elastic properties of a human cadaver knee joint. The measured data are compared with data from the literature to ensure that a proper database for modelling is used. The analytical modelling of the passive elastic joint properties is done with Local Linear Model Trees. The deduced knee joint model incorporates passive elastic properties of the internal knee joint structures, passive elastic muscle forces, damping forces, gravitational forces, and external forces. There are two sets of parameters, one simulating the movement of the intact knee joint, and a second simulating the knee joint with ruptured anterior cruciate ligament. The dynamic model can be easily processed in real-time. It is implemented in the haptic display of the Munich Knee Joint Simulator (MKS), which enables a person to move a plastic leg driven by a robot manipulator and feel the simulated knee joint force. Orthopaedic physicians judged the performance of the dynamic knee joint model by executing physical knee joint tests at the MKS.     

Elbow and wrist joint contact forces during occupational pick and place activities

A three-dimensional, mathematical model of the elbow and wrist joints, including 15 muscle units, 3 ligaments and 4 joint forces, has been developed. A new strain gauge transducer has been developed to measure functional grip forces. The device measures radial forces divided into six components and forces of up to 250N per segment can be measured with an accuracy of +/-1%. Ten normal volunteers were asked to complete four tasks representing occupational activities, during which time their grip force was monitored. Together with kinematic information from the six-camera Vicon data, the moment effect of these loads at the joints was calculated. These external moments are assumed to be balanced by the internal moments, generated by the muscles, passive soft tissue and bone contact. The effectiveness of the body's internal structures in generating joint moments was assessed by studying the geometry of a simplified model of the structures, where information about the lines of action and moment arms of muscles, tendons and ligaments is contained. The assumption of equilibrium between these external and internal joint moments allows formulation of a set of equations from which muscle and joint forces can be calculated. A two stage, linear optimisation routine minimising the overall muscle stress and the sum of the joint forces has been used to overcome the force-sharing problem. Humero-ulnar forces of up to 1600N, humero-radial forces of up to 800N and wrist joint forces of up to 2800N were found for moderate level activity. The model was validated by comparison with other studies.     

Changes in shear wave propagation within skeletal muscle during active and passive force generation

Muscle force can be generated actively through changes in neural excitation, and passively through externally imposed changes in muscle length. Disease and injury can disrupt force generation, but it can be challenging to separate passive from active contributions to these changes. Ultrasound elastography is a promising tool for characterizing the mechanical properties of muscles and the forces that they generate. Most prior work using ultrasound elastography in muscle has focused on the group velocity of shear waves, which increases with increasing muscle force. Few studies have quantified the phase velocity, which depends on the viscoelastic properties of muscle. Since passive and active forces within muscle involve different structures for force transmission, we hypothesized that measures of phase velocity could detect changes in shear wave propagation during active and passive conditions that cannot be detected when considering only group velocity. We measured phase and group velocity in the human biceps brachii during active and passive force generation and quantified the differences in estimates of shear elasticity obtained from each of these measurements. We found that measures of group velocity consistently overestimate the shear elasticity of muscle. We used a Voigt model to characterize the phase velocity and found that the estimated time constant for the Voigt model provided a way to distinguish between passive and active force generation. Our results demonstrate that shear wave elastography can be used to distinguish between passive and active force generation when it is used to characterize the phase velocity of shear waves propagating in muscle.     

Characterization of muscle belly elastic properties during passive stretching using transient elastography

Passive muscle stretching can be used in vivo to assess the viscoelastic properties of the entire musculo-articular complex, but does not allow the specific determination of the muscle or tendon viscoelasticity. In this respect, the local muscle hardness (LMH) of the gastrocnemius medialis (GM) belly was measured during a passive ankle stretching of 10 subjects using transient elastography. A Biodex isokinetic dynamometer was used to stretch ankle plantar flexors, to measure ankle angle, and the passive torque developed by the ankle joint in resistance to the stretch. Results show that the LMH increased during the stretching protocol, with an averaged ratio between maximal LMH and minimal LMH of 2.62+/-0.46. Furthermore, LMH-passive torque relationships were nicely fitted using a linear model with mean correlation coefficients (R(2)) of 0.828+/-0.099. A good reproducibility was found for the maximal passive torque (ICC=0.976, SEM=2.9Nm, CV=5.5%) and the y-intercept of the LMH-passive torque relationship (ICC=0.893, SEM=105Pa, CV=7.8%). However, the reproducibility was low for the slope of this relationship (ICC=0.631, SEM=10.35m(-2), CV=60.4%). The y-intercept of the LMH-passive torque relationship was not significantly changed after 10min of static stretching. This result confirms the finding of a previous study indicating that changes in passive torque following static stretching could be explained by an acute increase in muscle length without any changes in musculo-articular intrinsic mechanical properties.     

Extraction of quasi-linear viscoelastic parameters for lower limb soft tissues from manual indentation experiment.

A manual indentation protocol was established to assess the quasi-linear viscoelastic (QLV) properties of lower limb soft tissues. The QLV parameters were extracted using a curve-fitting procedure on the experimental indentation data. The load-indentation responses were obtained using an ultrasound indentation apparatus with a hand-held pen-sized probe. Limb soft tissues at four sites of eight normal young subjects were tested in three body postures. Four QLV model parameters were extracted from the experimental data. The initial modulus E0 ranged from 0.22 kPa to 58.4 kPa. The nonlinear factor E1 ranged from 21.7 kPa to 547 kPa. The time constant tau ranged from 0.05 s to 8.93 s. The time-dependent materials parameter alpha ranged from 0.029 to 0.277. Large variations of the parameters were noted among subjects, sites, and postures.