Mechano-acoustic determination of Young's modulus of articular cartilage

The compressive stiffness of an elastic material is traditionally characterized by its Young's modulus. Young's modulus of articular cartilage can be directly measured using unconfined compression geometry by assuming the cartilage to be homogeneous and isotropic. In isotropic materials, Young's modulus can also be determined acoustically by the measurement of sound speed and density of the material. In the present study, acoustic and mechanical techniques, feasible for in vivo measurements, were investigated to quantify the static and dynamic compressive stiffness of bovine articular cartilage in situ. Ultrasound reflection from the cartilage surface, as well as the dynamic modulus were determined with the recently developed ultrasound indentation instrument and compared with the reference mechanical and ultrasound speed measurements in unconfined compression (n=72). In addition, the applicability of manual creep measurements with the ultrasound indentation instrument was evaluated both experimentally and numerically. Our experimental results indicated that the sound speed could predict 47% and 53% of the variation in the Young's modulus and dynamic modulus of cartilage, respectively. The dynamic modulus, as determined manually with the ultrasound indentation instrument, showed significant linear correlations with the reference Young's modulus (r(2)=0.445, p<0.01, n=70) and dynamic modulus (r(2)=0.779, p<0.01, n=70) of the cartilage. Numerical analyses indicated that the creep measurements, conducted manually with the ultrasound indentation instrument, were sensitive to changes in Young's modulus and permeability of the tissue, and were significantly influenced by the tissue thickness. We conclude that acoustic parameters, i.e. ultrasound speed and reflection, are indicative to the intrinsic mechanical properties of the articular cartilage. Ultrasound indentation instrument, when further developed, provides an applicable tool for the in vivo detection of cartilage mechano-acoustic properties. These techniques could promote the diagnostics of osteoarthrosis.     

Determination of Young's modulus for nanofibrillated cellulose multilayer thin films using buckling mechanics

The Young's modulus of multilayer films containing nanofibrillated cellulose (NFC) and polyethyleneimine (PEI) was determined using the strain-induced elastic buckling instability for mechanical measurements (SIEBIMM) technique. (1) Multilayer films were built up on polydimethylsiloxane substrates using electrostatic layer-by-layer assembly. At 50% relative humidity, SIEBIMM gave a constant Young's modulus of 1.5 ± 0.2 GPa for 35-75 nm thick films. Conversely, in vacuum, the Young's modulus was 10 times larger, at 17.2 ± 1.2 GPa. A slight decrease in buckling wavelength with increasing strain was observed by scanning electron microscopy with in situ compression, and above 10% strain, extensive cracking parallel to the compressive direction occurred. We conclude that whereas PEI acts as a "glue" to hold multiple layers of NFC together, it prevents full development of hydrogen bonding and specific fibril-fibril interactions, and at high humidity, its hygroscopic nature decreases the elastic modulus when compared with pure NFC films.     

Young's modulus varies with differential orientation of keratin in feathers

Feathers are composed of a structure that, whilst being very light, is able to withstand the large aerodynamic forces exerted upon them during flight. To explore the contribution of molecular orientation to feather keratin mechanical properties, we have examined the nanoscopic organisation of the keratin molecules by X-ray diffraction techniques and have confirmed a link between this and the Young's modulus of the feather rachis. Our results indicate that along the rachis length, from calamus to tip, the keratin molecules become more aligned than at the calamus before returning to a state of higher mis-orientation towards the tip of the rachis. We have also confirmed the general trend of increasing Young's modulus with distance along the rachis. Furthermore, we report a distinct difference in the patterns of orientation of beta-keratin in the feathers of flying and flightless birds. The trend for increased modulus along the feathers of volant birds is absent in the flightless ostrich.     

Modeling and characterization of glucose-sensitive hydrogel: Effect of Young's modulus

A multiphysics model concerning the diffusion and enzyme reaction simultaneously is developed in this paper to characterize the equilibrium behavior of the glucose-sensitive hydrogel, which is called the multi-effect-coupling glucose-stimulus (MECglu) model. The responsive behavior of the hydrogel in the chemo-electro-mechanical coupled energy domains is modeled by the nonlinear coupled partial differential equations. They include the Nernst–Planck equations for the diffusion of mobile species and the enzyme reaction catalyzed by the glucose oxidase and the catalase, the Poisson equation for electric potential, and the mechanical equilibrium equation for finite deformation of the glucose-oxidase-loaded pH-sensitive hydrogel. Numerical simulations demonstrate that the MECglu model can consist well with the published experiment for the practical physiological glucose concentration ranging from 0 to 16.5 mM (300 mg/ml). The effect of Young's modulus of the hydrogel is investigated on the distributive concentrations of reacting and diffusive species and the deformation of the glucose-sensitive hydrogels.     

Variation in Young's modulus along the length of a rat vibrissa

Rats use specialized tactile hairs on their snout, called vibrissae (whiskers), to explore their surroundings. Vibrissae have no sensors along their length, but instead transmit mechanical information to receptors embedded in the follicle at the vibrissa base. The transmission of mechanical information along the vibrissa, and thus the tactile information ultimately received by the nervous system, depends critically on the mechanical properties of the vibrissa. In particular, transmission depends on the bending stiffness of the vibrissa, defined as the product of the area moment of inertia and Young's modulus. To date, Young's modulus of the rat vibrissa has not been measured in a uniaxial tensile test. We performed tensile tests on 22 vibrissae cut into two halves: a tip-segment and a base-segment. The average Young's modulus across all segments was 3.34±1.48GPa. The average modulus of a tip-segment was 3.96±1.60GPa, and the average modulus of a base-segment was 2.90±1.25GPa. Thus, on average, tip-segments had a higher Young's modulus than base-segments. High-resolution images of vibrissae were taken to seek structural correlates of this trend. The fraction of the cross-sectional area occupied by the vibrissa cuticle was found to increase along the vibrissa length, and may be responsible for the increase in Young's modulus near the tip.     

Fascin and VASP synergistically increase the Young's modulus of actin comet tails

Cell motility is locally achieved by the interplay between lamellipodia and filopodia at the protrusion sites. The actin cytoskeleton rearranges from a highly branched short filamentous network to well aligned elongated bundles from lamellipodia to filopodia, respectively. This process is governed predominantly by actin binding proteins, VASP and fascin, at the leading edge of migratory cells. Here we use an Arp2/3-complex dependent bead motility assay to study the effect of fascin both on its own and in the presence of VASP. The Young's modulus of phalloidin stabilized comets grown in the presence of fascin increased linearly with concentration above a 0.5 μM threshold. Inversely, the initial velocity of the comets decreased linearly with fascin concentration above the same threshold. Interestingly, VASP and fascin together increased the Young's modulus of the comets compared to those grown in the presence of only one of the two proteins. This effect was most remarkable at low concentration, below 0.5 and 0.15 μM for fascin and VASP, respectively. Our results showed that fascin and VASP work cooperatively to enhance the Young's modulus of the actin network within the comets.     

Effective Young's modulus of bacterial and microfibrillated cellulose fibrils in fibrous networks

The deformation micromechanics of bacterial cellulose (BC) and microfibrillated cellulose (MFC) networks have been investigated using Raman spectroscopy. The Raman spectra of both BC and MFC networks exhibit a band initially located at ≈ 1095 cm(-1). We have used the intensity of this band as a function of rotation angle of the specimens to study the cellulose fibril orientation in BC and MFC networks. We have also used the change in this peak's wavenumber position with applied tensile deformation to probe the stress-transfer behavior of these cellulosic materials. The intensity of this Raman band did not change significantly with rotation angle, indicating an in-plane 2D network of fibrils with uniform random orientation; conversely, a highly oriented flax fiber exhibited a marked change in intensity with rotation angle. Experimental data and theoretical analysis shows that the Raman band shift rate arising from deformation of networks under tension is dependent on the angles between the axis of fibrils, the strain axis, the incident laser polarization direction, and the back scattered polarization configurations. From this analysis, the effective moduli of single fibrils of BC and MFC in the networks were estimated to be in the ranges of 79-88 and 29-36 GPa, respectively. It is shown also that for the model to fit the data it is necessary to use a negative Poisson's ratio for MFC networks and BC networks. Discussion of this in-plane "auxetic" behavior is given.     

The underestimation of Young's modulus in compressive testing of cancellous bone specimens

In order to determine the accuracy of measurements of Young's modulus of cancellous bone by conventional compression testing, two independent strain measurements were made simultaneously during non-destructive uniaxial compression to 0.8% strain of rectangular specimens (n = 18). Strain was measured by an extensometer attached to the compression anvils close to the specimen and by an optical system covering the central half of the specimens. Mean Young's modulus determined by the extensometer technique was 689 MPa, but was 871 MPa when determined by the optical technique (mean difference = 182 MPa, SED = 50 MPa, p less than 0.002). Uneven strain distribution due to lack of support of cut vertical trabeculae at the anvil-specimen interface is believed to be causing the underestimation of Young's modulus measured by the extensometer technique. The influence of friction at the specimen-anvil interface was studied by performing a finite element analysis. It is concluded that Young's modulus of specimens of the chosen geometry on average is underestimated by about 20% by conventional compressing testing. The underestimation seems not to be dependent upon specimen density.     

Estimation of the Young's modulus of articular cartilage using an arthroscopic indentation instrument and ultrasonic measurement of tissue thickness

We evaluated whether the use of cartilage thickness measurement would improve the ability of the arthroscopic indentation technique to estimate the intrinsic stiffness of articular cartilage. First, cartilage thickness and ultrasound reflection from the surface of bovine humeral head were registered in situ using a high-frequency ultrasound probe. Subsequently, cartilage was indented in situ at the sites of the ultrasound measurements using arthroscopic instruments with plane-ended and spherical-ended indenters. Finally, full-thickness cartilage disks (n=30) were extracted from the indented sites (thickness=799-1654microm) and the equilibrium Young's modulus was determined with a material testing device in unconfined compression geometry. We applied analytical and numerical indentation models for the theoretical correction of experimental indentation measurements. An aspect-ratio (the ratio of indenter radius to cartilage thickness) correction improved the correlation of the indenter force with the equilibrium Young's modulus from r(2)=0.488 to r(2)=0.642-0.648 (n=30) for the plane-ended indenter (diameter=1.000mm, height=0.300mm) and from r(2)=0.654 to r(2)=0.684-0.692 (n=30) for the spherical-ended indenter (diameter=0.500mm, height=0.100mm), depending on the indentation model used for the correction. The linear correlation between the ultrasound reflection and the Young's modulus was r(2)=0.400 (n=30). These results suggest that with large indenters, knowledge of the cartilage thickness improves the reliability of the indentation measurements, especially in pathological situations where cartilage thickness may be significantly lower than normal. Ultrasound measurements also provide diagnostically important information about cartilage thickness as well as knowledge of the integrity of the superficial zone of cartilage.     

Anisotropic behavior of the pH dependence part of Young's modulus in a lysozyme triclinic crystal

A numerical simulation of the Young's modulus pH dependence in a triclinic crystal of hen egg white lysozyme is performed. On the basis of calculation of electrostatic interactions between the molecules in the crystal at different pH values, we obtain the Young's modulus quantity as a function of pH. The analysis is carried out along the main crystal axes as well as along the [011] direction. In the last case, a good agreement between our calculations of Young's modulus pH dependence and the experimental data [V. N. Morosov and T. Ya. Morozova (1981) Biopolymers, Vol. 20, pp. 451–467] is obtained. The calculations show a strong anisotropy of both the absolute value of the electrostatic part of Young's modulus and its pH dependence along different directions. A detailed analysis of the electrostatic potential inside the crystal, as well as the charged group distribution in the contact areas, permit us to find the reason for the anisotropy. It is caused by a nonuniform distribution of the charged residues in the lysozyme molecule and especially by the titration behavior of certain acidic groups. It is also shown that Young's modulus depends on the ionic strength, being anisotropic. The reason is that the cavities are not uniformly distributed within the crystal. They are located predominantly along the c axis. © 1995 John Wiley & Sons, Inc.     

The effect of porosity and mineral content on the Young's modulus of elasticity of compact bone

The Young's modulus of elasticity, the calcium content and the volume fraction (1-porosity) of 23 tension specimens and 80 bending specimens, taken from compact bone of 18 species of mammal, bird and reptile, were determined. There was a strong positive relationship between Young's modulus and both calcium content and volume fraction. A power law model fits the data better than a linear model. Young's modulus has a roughly cubic relationship with both calcium content and volume fraction. Over 80% of the total variation in Young's modulus in this data set is explained by these two variables.     

An extended relationship for the characterization of Young's modulus and Poisson's ratio of tunable polyacrylamide gels

Substrates with tunable mechanical properties are crucial for the study of cellular processes, and polyacrylamide gels (PAGs) are frequently used in this context. Several experimental techniques have been proposed to obtain the mechanical properties of PAGs. However, the range of the considered Poisson's ratio values remains quite large and no attempt has been made to propose an analytical relationship allowing the estimation of PAG Young's modulus when both bis-acrylamide and acrylamide concentrations are known. In order to complete the actual knowledge on the mechanical properties of PAGs, we took benefit of our original method based on the micropipette aspiration technique (Boudou et al., J. Biomech. 2006) for characterizing gels made with concentrations in the range 0.02% < or =[Bis]< or =0.20% and 3% < or =[Acry]< or =10%. We found that the PAGs Young's modulus varies nonlinearly with the acrylamide amount. Moreover, our study validates the quasi-incompressibility hypothesis usually made in studies using PAGs (mean Poisson's ratio of 0.480+/-0.012). More generally, and in agreement with data published by other groups, we propose an original nonlinear mathematical relationship allowing the computation of Young's modulus of PAG for any given acrylamide and bis-acrylamide amounts taken in the range of values we considered.     

Quantifying dynamic mechanical properties of human placenta tissue using optimization techniques with specimen-specific finite-element models

Motor-vehicle crashes are the leading cause of fetal deaths resulting from maternal trauma in the United States, and placental abruption is the most common cause of these deaths. To minimize this injury, new assessment tools, such as crash-test dummies and computational models of pregnant women, are needed to evaluate vehicle restraint systems with respect to reducing the risk of placental abruption. Developing these models requires accurate material properties for tissues in the pregnant abdomen under dynamic loading conditions that can occur in crashes. A method has been developed for determining dynamic material properties of human soft tissues that combines results from uniaxial tensile tests, specimen-specific finite-element models based on laser scans that accurately capture non-uniform tissue-specimen geometry, and optimization techniques. The current study applies this method to characterizing material properties of placental tissue. For 21 placenta specimens tested at a strain rate of 12/s, the mean failure strain is 0.472±0.097 and the mean failure stress is 34.80±12.62 kPa. A first-order Ogden material model with ground-state shear modulus (μ) of 23.97±5.52 kPa and exponent (α₁) of 3.66±1.90 best fits the test results. The new method provides a nearly 40% error reduction (p<0.001) compared to traditional curve-fitting methods by considering detailed specimen geometry, loading conditions, and dynamic effects from high-speed loading. The proposed method can be applied to determine mechanical properties of other soft biological tissues.     

Modelling environmental variation in Young's modulus for Pinus radiata and implications for determination of critical buckling height

• Background and Aims Although density-specific stiffness,E/, (where E is Young's modulus and is wood density) is oftenassumed constant by the elastic similarity model, and in determinationof critical buckling height (H_crit), few studies have testedthis assumption within species. Here this assumption is testedfor Pinus radiata growing across an environmental gradient,and theory is combined with data to develop a model of Young'smodulus. • Methods Analyses use an extensive series of environmentalplots covering the range of climatic and edaphic conditionsover which P. radiata is grown in New Zealand. Reduced majoraxis regression was used to determine scaling exponents betweenlog–log plots of H_crit vs. groundline diameter (D), andE/ vs. D. Path analysis was used to identify significant directand indirect (through stem slenderness) edaphic and climaticinfluences on E. • Key Results Density-specific stiffness exhibited 3-foldvariation. As E/ scaled positively with D, the exponent of 0·95between H_crit and D exceeded the assumed value of 0·67under constant E/. The final path analysis model included meanair temperature in early autumn (T_aut) and slenderness as significant(P < 0·05) positive direct influences on E. Tree leafarea index and T_aut were indirectly associated with E throughtheir significant (P < 0·05) positive direct relationshipwith stem slenderness. Young's modulus was most sensitive toT_aut, followed by stem slenderness then leaf area index, andthe final model explained 76 % of the variance in E. • Conclusions The findings suggest that within speciesE/ variation may influence H_crit and the scaling exponent betweenD and H_crit so important in assumptions regarding allometricrelationships. The model presented may provide a useful meansof determining variation in E, E/ and H_crit across environmentalgradients.     

Quantum chemical determination of Young's modulus of lignin. Calculations on a beta-O-4' model compound

The calculation of Young's modulus of lignin has been examined by subjecting a dimeric model compound to strain, coupled with the determination of energy and stress. The computational results, derived from quantum chemical calculations, are in agreement with available experimental results. Changes in geometry indicate that modifications in dihedral angles occur in response to linear strain. At larger levels of strain, bond rupture is evidenced by abrupt changes in energy, structure, and charge. Based on the current calculations, the bond scission may be occurring through a homolytic reaction between aliphatic carbon atoms. These results may have implications in the reactivity of lignin especially when subjected to processing methods that place large mechanical forces on the structure.     

Obtaining source current density related to irregularly structured electromagnetic target field inside human body using hybrid inverse/FDTD method

Inverse method is inherently suitable for calculating the distribution of source current density related with an irregularly structured electromagnetic target field. However, the present form of inverse method cannot calculate complex field–tissue interactions. A novel hybrid inverse/finite-difference time domain (FDTD) method that can calculate the complex field–tissue interactions for the inverse design of source current density related with an irregularly structured electromagnetic target field is proposed. A Huygens’ equivalent surface is established as a bridge to combine the inverse and FDTD method. Distribution of the radiofrequency (RF) magnetic field on the Huygens’ equivalent surface is obtained using the FDTD method by considering the complex field–tissue interactions within the human body model. The obtained magnetic field distributed on the Huygens’ equivalent surface is regarded as the next target. The current density on the designated source surface is derived using the inverse method. The homogeneity of target magnetic field and specific energy absorption rate are calculated to verify the proposed method.     

Validation of a voxel-based FE method for prediction of the uniaxial apparent modulus of human trabecular bone using macroscopic mechanical tests and nanoindentation

Assessment of the mechanical properties of trabecular bone is of major biological and clinical importance for the investigation of bone diseases, fractures and their treatments. Finite element (FE) methods are getting increasingly popular for quantifying the elastic and failure properties of trabecular bone. In particular, voxel-based FE methods have been previously used to calculate the effective elastic properties of trabecular microstructures. However, in most studies, bone tissue moduli were assumed or back-calculated to match the apparent elastic moduli from experiments, which often lead to surprisingly low values when compared to nanoindentation results. In this study, voxel-based FE analysis of trabecular bone is combined with physical measures of volume fraction, micro-CT (microCT) reconstructions, uniaxial mechanical tests and specimen-specific nanoindentation tests for proper validation of the method. Cylindrical specimens of cancellous bone were extracted from human femurs and their volume fraction determined with Archimede's method. Uniaxial apparent modulus of the specimens was measured with an improved tension-compression testing protocol that minimizes boundary artefacts. Their microCT reconstructions were segmented to match the measured bone volume fraction and used to create full-size voxel models with 30-45 microm element size. For each specimen, linear isotropic elastic material properties were defined based on specific nanoindentation measurements of its embedded bone tissue. Linear FE analyses were finally performed to simulate the uniaxial mechanical tests. Additional parametric analyses were performed to evaluate the potential errors on the predicted apparent modulus arising from variations in segmentation threshold, tissue modulus, and the use of 125-mm(3) cubic sub-regions. The results demonstrate an excellent correspondence between experimental measures and FE predictions of uniaxial apparent modulus. In conclusion, the adopted voxel-based FE approach is found to be a robust method to predict the linear elastic properties of human cancellous bone, provided segmentation of the microCT reconstructions is carefully calibrated, tissue modulus is known a priori and the entire region of interest is included in the analysis.     

Solving inverse problems for biological models using the collage method for differential equations

In the first part of this paper we show how inverse problems for differential equations can be solved using the so-called collage method. Inverse problems can be solved by minimizing the collage distance in an appropriate metric space. We then provide several numerical examples in mathematical biology. We consider applications of this approach to the following areas: population dynamics, mRNA and protein concentration, bacteria and amoeba cells interaction, tumor growth.