Vibrometry as a noncontact alternative to dynamic and viscoelastic mechanical testing in cartilage

Physiological loading of knee cartilage is highly dynamic and may contribute to the progression of osteoarthritis. Thus, an understanding of cartilage's dynamic mechanical properties is crucial in cartilage research. In this study, vibrometry was used as a fast (2 h), noncontact and novel alternative to the slower (30 h), traditional mechanical and biochemical assays for characterization of cartilage from the condyle, patella, trochlear groove and meniscus. Finite-element models predicted tissue resonant frequencies and bending modes, which strongly correlated with experiments (R2 = 0.93). Vibrometry-based viscoelastic properties significantly correlated with moduli from stress relaxation and creep tests, with correlation strengths reaching up to 0.78. Loss modulus also strongly correlated with glycosoaminoglycan (GAG) content. Dynamic properties measured by vibrometry significantly differed among various knee cartilages, ranging between 6.1 and 56.4 MPa. Interestingly, meniscus viscoelastic properties suggest that contrary to common belief, it may lack shock absorption abilities; instead, condylar hyaline cartilage may be a better shock absorber. These data demonstrate for the first time that vibrometry is a noncontact approach to dynamic mechanical characterization of hyaline and fibrocartilage cartilage with concrete relationships to standard quasi-static mechanical testing and biochemical composition. Thus, with a single tool, vibrometry greatly facilitates meeting multiple regulatory recommendations for mechanical characterization of cartilage replacements.


Introduction
Cartilage is a dynamically loaded tissue. Typical activities, such as walking, can generate large stresses between 1 and 6 MPa but may even peak up to 12 MPa [1]. These stresses are applied cyclically at frequencies ranging from 0.01 to 2 Hz, typically resulting in up to 4 million loading cycles per year [1,2]. Several factors, such as cyclical mechanical fatigue, excessive strains and rapid strain rates, are often implicated with the onset of osteoarthritis, a condition afflicting approximately 14% of adults [3]. Under cyclical compression, cartilage's biphasic and dynamic mechanical properties protect against these potentially damaging factors via interstitial fluid pressurization [1]. Despite this arguably hostile mechanical environment, cartilage actually relies on cyclic loading for processes such as solute transport and matrix protein synthesis [4]. Given its relevance to daily activities and role in maintaining tissue homeostasis, cartilage dynamic mechanical properties are a crucial component to the field's ongoing research endeavours and to those seeking to replace or regenerate cartilage's function in a dynamic environment.
The importance of cartilage dynamics has not gone unnoted. Cyclic loading has been used to enhance the functional properties of both native and tissue-engineered cartilages [5][6][7]. Also, for translational purposes, the Food and Drug Administration (FDA) specifically recommends that both static and dynamic mechanical properties of any knee cartilage repair or replacement product be ascertained [8]. Quasi-static mechanical measurements using viscoelastic testing conditions, such as stress relaxation and creep indentation, are routinely employed to evaluate cartilage and its replacements [9]. Dynamic mechanical testing is often performed under confined compression, unconfined compression or indentation to obtain a dynamic modulus and, occasionally, viscoelastic properties [10,11]. However, most of these mechanical testing modalities are destructive, which either limits the pursuit of certain investigations or requires a large amount of testable tissue. It is certain that a noncontact form of dynamic mechanical testing would garner much attention among cartilage researchers.
Laser Doppler vibrometry has been extensively used in various engineering applications, such as for the assessment of structural stability and rotor vibrations and for the fabrication of micro-electro-mechanical systems devices [12]. One of the first biological applications of vibrometry was in auditory research, specifically measuring vibrations of the tympanic membrane and cochlea [13,14]. Vibrometry has slowly been adopted by others studying biological specimens, such as single cells [15], arteries [16], skin [17] and hydrogels [18]. A major advantage of vibrometry showcased in these works is the ability to perform noncontact and noninvasive measurements. Given this key characteristic, vibrometry is well poised to become a valuable measurement modality in the biological sciences.
The objective of this work was to investigate the use of vibrometry as a noncontact dynamic mechanical testing modality in comparison to quasi-static mechanical and biochemical assays for articular cartilage characterization. Other forms of dynamic testing have shown that dynamic mechanical properties correlate well with quasi-static properties in many biological materials and are typically greater than their quasistatic counterparts [19][20][21][22]. This study focuses on the cartilages of the knee, which are known to have differing mechanical properties [23]. We hypothesized that vibrometer-based measurements would recapitulate the relationship between dynamic and quasi-static mechanical properties found through other testing modalities. We further hypothesized that a dynamic structure-function relationship could be ascertained by studying dynamic mechanical properties along with biochemical composition. Lastly, we hypothesized that noncontact laser vibrometry can measure differences in the dynamic modulus and viscoelastic properties of cartilage explants from the condyle, patella, trochlear groove and meniscus. Inclusion of the meniscus, a fibrocartilage, is important for validating vibrometry-based measurements on cartilage tissues with differing mechanical, structural and biochemical properties from hyaline cartilage. The results from this work will enable cartilage researchers to examine native and engineered tissue dynamics without damage, which could facilitate time-course studies surrounding development, disease and repair. These studies may further propel the characterization of cartilage restoration therapies and determine their ability to withstand the dynamic mechanical demands of the native cartilage environment.

Tissue harvest
Six bovine knee joints from 2-to 4-week-old calves were obtained 48 h post-mortem (Research 87 Inc., Boylston, MA). Using a biopsy punch, 5 mm plugs of cartilaginous tissue were obtained from four different locations: femoral condyle (medial and lateral load-bearing regions), trochlear groove, patella and meniscus (figure 1a). All punches were trimmed down to a height of 1.5-2 mm for consistency, maintaining the superficial zone intact and removing the calcified zone. Meniscus punches remained thicker in order to avoid compressive deformation and uneven cutting. The punches were then washed in phosphate buffered saline (PBS) and stored at −20°C. Before testing, all samples were thawed at room temperature in fresh PBS for 1 h. These tissue punches were used for all of the mechanical testing modalities: vibrometry, stress relaxation and creep. The cartilage explants were allowed to recover and equilibrate between tests. It is noteworthy that the geometrical requirements for the samples are imposed by the stress relaxation and creep tests and are not necessary for vibrometry.

Vibrometry
Tissue deformation due to axial vibrations was measured using a PSV-500-B Xtra Laser Scanning Vibrometer (Polytec). As shown in figure 2a, the tissue punch was placed on top of a piezoelectric actuator (P-885.3 PICMA). The punch's bottom surface was secured with wax. The piezo's motion was controlled by the vibrometer's internal function generator. The vibrometer lens was focused on the entire sample surface. Using the high-contrast display option in the PSV 9.4 software (Polytec), a grid with approximately 20 points covering both the sample and piezo was used to align the laser with the live video display. A second higher-density grid (100-150 points) over the same area was then defined to establish the measurement points. It is important to note that measurements were taken of both the sample and the piezo. The function generator was used to generate a chirp signal ranging from 10 to 100 000 Hz to identify resonant frequencies. This was followed by a 500 Hz sinusoidal excitation. Further details on the acquisition settings can be found in electronic supplementary material, table S1. The samples were not damaged and then used to perform destructive mechanical testing.

Vibrometry analysis
First and second resonant frequencies were found by identifying peaks in the sample velocity data across the frequency sweep (figure 2b) using the PSV 9.4 software. For this analysis, data from the piezo were only used to confirm that the piezo's resonant frequencies do not interfere with those of the sample. To differentiate from peaks due to rigid body motion, velocity and deformation were plotted as two-dimensional surfaces to verify the presence of a bending mode at resonance. The dynamic modulus, G*, of each sample was determined based on the American Society for Testing and Materials International (ASTM International) guidelines for calculating dynamic mechanical properties for disc-shaped specimens via vibration [24,25]. Briefly, for each resonant frequency, a dynamic modulus was calculated, ,

ð2:1Þ
where i is an index referencing either the first or second resonant frequency, f i are the resonant frequencies, d is the diameter, m is the mass and t is the thickness. Poisson's ratio, ν, is a function of the ratio of the second to first resonant frequencies ( f 2 /f 1 ) and the ratio of sample thickness to radius (t/r). In turn, the geometric factors, K i , are functions of ν and t/r. Functions for ν and K i were fit to previous empirical data [24,25] using MATLAB (electronic supplementary material, figure S1) and used to determine values for the present study. The dynamic moduli for both frequencies were averaged to give a single G* value per sample. The viscoelastic properties were determined using the data from sinusoidal excitation (figure 2c). Using the PSV 9.4 software, an inverse Fourier transform was used to convert data from the frequency domain to the time domain. In this analysis, data from both the piezo and sample were necessary to measure the phase shift, δ, between the piezo's and tissue's response to the reference signal. The tangent of the phase shift is known to be the ratio of viscous to elastic contributions [26], also known as the loss and storage moduli, respectively. Using G* and δ, storage and loss moduli were calculated as follows: and where G 0 is the storage modulus and G 00 is the loss modulus. It is important to note that since none of the expressions used to calculate G*, G 0 or G 00 depends on the displacement or velocity signal amplitudes, there is no need to normalize the tissue response by the piezo data.

Stress relaxation
Unconfined stress relaxation compressive testing was performed on the 5 mm diameter tissue punches using an Instron model 5565 (Instron, Canton, MA, USA). The samples, which were previously used for nondestructive vibrometry, were placed in a PBS bath at room temperature. Sample heights were determined by lowering a platen that contacts the entire surface until a load change of 0.2 N was detected, and samples were preconditioned at 5% compressive strain for 15 cycles. Stress relaxation tests were performed as follows: samples were compressed to 10% strain at   royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 18: 20210765 a rate of 1% sample height/s, held for 200 s, and then compressed to 20% strain, held for 450 s. The instantaneous modulus (E i ), relaxation modulus (E r ) and coefficient of viscosity of the 10% and 20% strain curves were determined by using the standard linear solid viscoelastic model to fit the data in MATLAB (Math-Works) [27]. The relationships between the instantaneous and relaxation moduli and the component spring elements (E 1 and E 2 ) are shown in figure 3. The viscoelastic model's springs and dashpot can also be directly related to G 0 and G 00 as shown here: where ω is the frequency and η is the coefficient of viscosity.

Creep indentation
A compression indentation apparatus was used to assess creep and recovery deformation of the cartilaginous tissues [28]. Briefly, the 5 mm punches were photographed and glued to a cylindrical sample holder. Indentation was performed using a 1 mm flat-ended, porous indenter tip, perpendicular to the surface at the centre of the sample. The sample surface is assumed to be a semi-infinite half space, which allows the single measurement point to be representative of the whole sample. A 0.5 g tare weight was applied at samples' indentation site, and a variable weight (7.5-10 g for patella and condyle, 4-7.5 g for trochlear and 2-4 g for meniscus) was used to achieve an approximately 10% strain. Sample height was digitally measured using ImageJ (National Institutes of Health). The aggregate modulus (H A ), shear modulus (µ s ) and permeability were obtained using a semi-analytical, semi-numerical, biphasic model followed by finite-element optimization [29]. The solution process begins with the calculation of μ s at equilibrium (t = ∞) for a specified set of ν values ranging from 0 to 0.499 using where u is the displacement, h is the sample height, a is the radius, P 0 is the load and κ is a function of a/h and ν known as the geometric scaling function [30]. With the exception of ν and κ, all parameters are obtained experimentally. This is followed by the use of a set of specified time (t) values in the expression below: u[log 10 (t), S]=h ¼ G[log 10 (t); S, P 0 =(2m s a 2 ), a=h, n s ], ð2:5aÞ and where S is known as the shift factor, shifting the solution along thelogarithmic time axis, and G is the inverse Laplace transform of the creep function. This produces a family of solutions in the (ν, t) plane that can be represented by a bicubic function whose parameters are numerically obtained via curve fitting. This algorithm is described in detail elsewhere [31]. Lastly, these parameters were used as initial guesses for a biphasic model finite-element optimization algorithm that fits the entirety of the creep curve, including the initial rise toward equilibrium [32].

Validation through finite-element modelling
Abaqus/CAE 2018 (Dassault Systèmes) was used to validate our approach for determining the dynamic modulus of cartilage based on resonance. One representative sample from each group was used to define geometry, mass, G* and permeability for the models. These values were all directly measured from the methods described above. An encastre boundary condition was imposed on the bottom surface. The models were meshed with 10 344 C3D8R elements. Given the small deformations of the experiment, linear elasticity was assumed. In addition to G* and permeability, the solid phase of the porous elastic model was also defined by a void ratio, set to 0.5, based on the literature [33,34]. The fluid phase was characterized by the specific weight of the wetting liquid and also based on previous cartilage studies [20,35]. All parameters are listed in electronic supplementary material, table S2. A linear perturbation frequency analysis was performed to find all the resonant frequencies found within 0-20 kHz and their corresponding bending modes. The frequency range was based on the experimental frequency sweeps showing that all first and second resonant frequencies occurred below 20 kHz.

Biochemical assays
Cartilage tissue samples (approx. 2-3 mg) were weighed before and after lyophilization. The lyophilized samples were digested in 125 µg ml −1 papain (Sigma) + 5 mM N-acetyl-L-cysteine + (Sigma) 5 mM ethylenediaminetetraacetic acid (EDTA) in phosphate buffer pH 6.5 for 18 h at 60°C. DNA content was measured with a Picogreen assay (ThermoFisher Scientific). Glycosoaminoglycan (GAG) content was measured using a Blyscan glycosaminoglycan assay kit (Biocolor, Newtownabbey, Northern Ireland). Total collagen content was measured using a modified chloramine-T hydroxyproline assay using a bovine Sircol collagen standard (Biocolor) [36]. DNA, GAG and collagen content were measured on a GENios microplate reader (Tecan). For the quantification of pyridinoline (PYR) cross-links, separate tissue samples (approx. 1 mg wet weight (WW)) were weighed, lyophilized and acid digested for 12 h in 6 N HCl. After evaporation, the dried hydrolysate was resuspended in a 75%/25% (v/v) solution of 0.1% formic acid and acetonitrile. Samples were measured through mass spectrometry using a PYR standard (BOC Sciences) and a Cogent diamond hydride 2.0 HPLC column (Avantor, VWR) with a flow rate of 400 µl min −1 on an ACQUITY UPLC I-Class core system with an ACQUITY Qda quadrupole mass spectrometer (Waters) [37].

Statistics and accessibility
All data are reported as average ± standard deviation. Two-way and one-way analysis of variances (ANOVAs) with Tukey's post-hoc test (GraphPad Prism 8.1.1) were used to determine statistical significance. Measured and modelled resonant frequencies were found to be normally distributed and were correlated using the Pearson correlation test. Correlations between dynamic vibrometer measures and quasi-static and biochemical measures were performed using the Spearman's ρ-test. Correlation strengths, ρ, are generally considered strong if greater than or equal to 0.70, moderate if between 0.40 and 0.69 and weak if less than 0.40 [38]. This study's datasets can be found on the Dryad Digital Repository [39].

Tissue sample physical properties
Tissue punches were obtained from the condyle, patella, trochlear groove and meniscus. Although the punches from the other groups were consistent in size, meniscus punches had a significantly smaller diameter (figure 1c). Compared to the condyle, patella and trochlear groove, meniscus explant diameters were 11.9%, 11.5% and 8.7% smaller, respectively. The meniscus punches also had a significantly greater thickness and WW compared to all other groups. The meniscus samples showed a 44.2%, 53.8% and 53.4% increased  royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 18: 20210765 thickness and 39.0%, 43.0% and 53.4% increased WW when compared to the condyle, patella and trochlear groove, respectively. Dimensional and weight data were used to normalize data and obtain mechanical properties.

Vibrometry
Vibrational spectra were obtained for measurement points on the sample and piezo, as depicted in figure 4a. Representative data are shown in figure 4b. In order to distinguish between rigid body motion and resonance, bending mode shapes were identified for each group's first and second resonant frequencies (figure 5 and electronic supplementary material, videos S1 and S2). Mode shapes are often described by nodal lines or circles (electronic supplementary material, figure S2) [40]. For example, a mode with one nodal line, but no circles, is denoted by (0, 1). Points along nodal lines and circles are relatively stable at resonance compared to other areas. The first resonant frequency's bending mode, (0, 2), was characterized by displacements that were symmetrical around two orthogonal nodal diameters in-plane with the sample, and was known as a saddle mode [24]. This shape was consistent across all groups. The second resonant frequency for the condyle, patella and trochlear groove was characterized by symmetry around three nodal diameters, (0, 3). The second resonant frequency for the meniscus differed from the rest, consisting of a single nodal circle, (1, 0). This mode is often referred to as a breathing mode. Videos of the bending modes at resonance can be found in the electronic supplementary material.

Resonant frequencies and dynamic moduli
First resonant frequencies for the condyle, patella and trochlear groove were 5352 ± 2011 Hz, 7823 ± 2682 Hz and 8768 ± 1957 Hz, respectively (figure 6a). An increasing trend was observed, with the trochlear groove being significantly greater than the condyle. The first resonant frequency for the meniscus was 2708 ± 65 Hz, significantly lower than all the other groups. The trends observed with the first resonant frequencies were repeated with the second resonant frequencies. Figure 6b shows the dynamic moduli, calculated based on resonant frequencies. The relationship among the groups was maintained. Dynamic moduli were 27.4 ± 14.3 MPa, 46.7 ± 11.0 MPa, 56.4 ± 29.9 and 6.1 ± 0.5 MPa for the condyle, patella, trochlear groove and meniscus, respectively.

Viscoelastic properties
In addition to the significant differences in dynamic modulus, there were also considerable differences among the groups' phase shifts, δ. The measured phase shifts were 7.0 ± 1.4°, 3.8 ± 2.0°, 2.1 ± 1.0°and 8.2 ± 0.2°for the condyle, patella, trochlear groove and meniscus, respectively (figure 7a). The tangents of the phase shifts for the patella and trochlear groove were 46% and 70% lower compared to the condyle, respectively. Although this did not result in significant differences in G 0 among these three groups (figure 7b), the differences in their G 00 did reach statistical significance. The loss moduli (figure 7c) for the condyle, patella and trochlear groove were 4.0 ± 0.3 MPa, 3.1 ± 1.6 MPa and 1.6 ± 0.2 MPa, respectively. The meniscus followed a different pattern from the hyaline cartilage groups (G 00 = 0.9 ± 0.1 MPa). Although the tangent of the phase was comparable to the condyle's, the meniscus had a substantially lower G 0 and G 00 compared to the condyle, reaching statistical significance in the case of the loss modulus.

Quasi-static mechanics
Representative curves and model fits for stress relaxation and creep tests are shown in figure 8. Stress relaxation was performed at 10% and 20% strain. There were no statistically significant differences among the groups for either instantaneous modulus (figure 9a) or viscosity (not shown) at either strain. The relaxation modulus (figure 9b) for the meniscus at 20% strain was significantly lower than the condyle's by 83%. Both aggregate and shear moduli (figure 9c) from creep indentation tests were significantly different among the groups. For example, the trochlear groove and meniscus aggregate moduli were reduced by 44% and 87% compared to the 1 s t r e s o n a n t f r e q u e n c y royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 18: 20210765 condyle, respectively. Several quasi-static measures followed similar trends to those obtained through vibrometry. Instantaneous modulus correlated with G 0 , while relaxation, aggregate and shear moduli correlated with G 00 .

Finite-element validation
The experimental (figure 5) and computationally predicted (figure 10a) bending modes were strikingly similar for the condyle, patella and trochlear groove. The predicted displacements for the meniscus were small and thus scaled differently (figure 10a-insets). For the meniscus, the predicted first and second resonant frequency modes were characterized by one and four nodal diameters, respectively. Taken together, the measured and modelled resonant frequencies significantly and strongly correlated across all tissues (R 2 = 0.93, figure 10b) with a slope of approximately 1, providing strong validation of the experimental approach.

Biochemical composition
Hydration of the tissue samples was relatively similar among the condyle, patella and trochlear groove, averaging around 79% (figure 11a). The meniscus was significantly less hydrated at 71 ± 4%. With regards to collagen/WW (figure 11b), the condyle's collagen content was significantly lower than all other groups at only 7.6 ± 0.6%. The patella (12.9 ± 2.3%) and trochlear groove (11.6 ± 0.8%) were not statistically different from each other, but both had significantly less collagen than the meniscus (24.6 ± 3.7%). The meniscus had over three times more collagen content than was present in the condyle. PYR/ WW (figure 11c) followed the same trend as collagen: 916.1 ± 48.2 ng mg −1 , 1363 ± 217.3 ng mg −1 , 1208 ± 233.0 ng mg −1 and 3355 ± 697.0 ng mg −1 for the condyle, patella, trochlear groove and meniscus, respectively. The inverse trend was observed with GAG content (figure 11d). The condyle's GAG content (4.8 ± 0.5%) was more than eight times that found in the meniscus (0.6 ± 0.2%). The GAG content found in the patella and trochlear groove was 2.6 ± 1.1% and 2.7 ± 0.9%, respectively.

Comparison between measurement techniques
Several correlations between dynamic mechanical properties and quasi-static properties and between dynamic mechanical properties and biochemical composition were found (figure 12). The dynamic modulus, G*, was most strongly and significantly correlated with the 10% and 20% instantaneous moduli from stress relaxation (ρ ≥ 0.65, p = 0.001). No associations were found between G* and creep indentation parameters or between G* and biochemical components. The storage modulus, G 0 , which was proportional to G*, was solely correlated to the instantaneous moduli. The loss modulus, G 00 , trended with the relaxation moduli, reaching statistical significance with the values from the 20% strain stress relaxation tests. G 00 was more strongly related to the properties obtained from creep indentation. For example, the correlation between G 00 and shear modulus had a strength of 0.78 (p = 0.002). Similarly, G 00 was also strongly correlated with GAG content (ρ = 0.72, p = 0.002).

Discussion
Physiological loading of knee articular cartilage is highly dynamic. This loading environment is necessary for nutrient transport and mechanical signalling [4], but may also result in tissue fatigue and the eventual onset of osteoarthritis [3]. Therefore, as recommended by the FDA [8], mechanical characterization of cartilage replacements should include both static and dynamic testing. Most dynamic tests used in cartilage research are destructive, often limiting the scope of a study due to tissue scarcity. Given the success of vibrometry in engineering and even a few biological applications, this study's objective was to investigate the use of vibrometry as a noncontact method for dynamic mechanical testing of cartilage. Our first hypothesis was that dynamic measurements from vibrometry would correlate with those obtained from quasi-static mechanical testing, as shown in previous studies using other dynamic mechanical testing modalities [20]. The second hypothesis was that vibrometry and  Figure 7. Phase shifts (δ) were measured to determine viscoelastic properties. (a) The tangent of δ represents the ratio of viscous to elastic contributions to the tissue's mechanics. This is significantly greater for the condyle and meniscus than the patella and trochlear groove. The storage (b) and loss (c) moduli trend differently, but in both cases, the meniscus has the lowest values. Error bars represent standard deviation and n = 6. biochemical assays would elucidate cartilage's dynamic structure-function relationship. Lastly, the third hypothesis was that vibrometry would detect the differences in the dynamic mechanical properties among four regions of the knee, consisting of both hyaline and fibrocartilage. All three hypotheses were upheld. Vibrometry allowed for the measurement of phase shifts and calculation of storage and loss moduli, which correlated well with several standard quasi-static mechanical properties. Additionally, the loss moduli correlated strongly with GAG content, the tissue constituent that is typically related to compressive properties [41]. Moreover, dynamic properties not only differed among cartilage groups but also agreed with previously published work using non-vibrometry methods. These data show that the sole use of vibrometry can yield data obtained from three distinct assays. We found that vibrometry is an efficient and noncontact testing modality for native cartilage dynamic and viscoelastic testing and is well suited for providing necessary data in the regulatory process for knee cartilage replacement products.
This study showed that vibrometry is an important advancement towards noncontact dynamic mechanical testing of cartilage. Several strong correlations between traditional destructive assays used in cartilage research and dynamic mechanical properties obtained from noncontact vibrometry were found (figure 12). In this work, cartilage's viscoelasticity was described by the standard linear solid model, consisting of two elastic springs and a viscous dashpot (figure 3). Although the storage modulus (G 0 ) is associated with the relaxation modulus (equation (2.3a) and figure 3), it is also dependent on the spring-related 10% and 20% instantaneous moduli obtained from stress relaxation. Given the multiple relationships between G 0 and all of the standard linear solid's viscoelastic parameters, it was not surprising to find significant, albeit muted, correlations. With regard to biochemical components, there were no correlations with G 0 . The loss moduli (G 00 ), associated with viscous behaviour, were correlated with measures from all the assays performed in this work. Previously described mathematical relationships between G 00 and the mechanical components of the standard linear solid model imply that G 00 should correlate with the Maxwell portion of the standard linear solid (figure 3) [26]. The present data's correlations between G 00 and the 10% and 20% relaxation moduli were unexpected (statistical significance determined by p = 0.052 for 10% and p = 0.031 for 20%), given that the Maxwell spring depends on both the instantaneous and relaxation moduli (E 1 = E i − E r ), but the single spring, E 2 , is equivalent to the relaxation modulus. Since G 00 only depends on E 1 , its relationship may be related to other factors not contained within the standard linear solid's model. In this and other studies performed by royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 18: 20210765 our group, we have frequently found a similar relationship between the relaxation moduli from stress relaxation and aggregate moduli from creep experiments (data not shown). This is further reinforced by the correlations between G 00 and the aggregate and shear moduli from creep indentation, as these are also measures of the long-term elastic response. It is possible that these surprising associations with G 00 may be due to the high-frequency excitation used in our vibrometry experiments. It is well known that G 0 and G 00 are frequency dependent and may even reach a 'crossover point' at which the relative relationship between these two moduli plateaus and is inverted [42,43]. This phenomenon has been observed in several biomaterials and tissues, such as collagen/hyaluronan gels [44], vocal fold tissue [45] and airway smooth muscle [46]. Therefore, although these particular relationships were not anticipated and merit further investigation, independent verification through both stress relaxation and creep indentation makes a very strong case for their validity. royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 18: 20210765 Notably, the correlations found in this study, especially between G 00 and the creep-related parameters, suggest that quasi-static mechanical properties may predict dynamic properties. Inasmuch as the FDA and ASTM International recommend that both dynamic and viscoelastic properties be reported [8,47], these correlations demonstrate that vibrometry may obtain both dynamic and viscoelastic properties, an important observation in light of the regulatory requirements for cartilage tissue replacements. Although biochemical composition is frequently used to describe cartilage's structure-function relationships, it is often only related to quasi-static mechanical measures. A few studies have investigated the association between cartilage composition and dynamic properties but typically only measure G* without examining its elastic and viscous components [48,49]. In this study, the strongest correlations were found between the loss modulus (G 00 ) and GAG content. This was not unexpected given the relationships between G 00 and the relaxation, aggregate and shear moduli, described above. There were no correlations with collagen content. In the present work, the tissue was only axially perturbed. However, perturbations in different directions (e.g. parallel to the direction of articulation) may provide additional insights. A three-dimensional vibrometry set-up would detect radial and circumferential deformations, which would likely be correlated to collagen content. Further work to understand how dynamic mechanical properties relate to biochemical composition is needed, especially given the strong correlations reported in this study.
Vibrometer-based dynamic testing revealed differences in viscoelastic characteristics among the cartilage groups. G 0 (figure 7b) was similar to the dynamic modulus, G* (figure 6b), for all groups. It has been previously reported that at high frequencies (greater than 40 Hz), cartilage behaves more like an elastic solid with reduced viscous contributions [1]. The small phase shifts (figure 7a) in the present study in which dynamic loading at 500 Hz (figure 2c) was employed support this, thus resulting in a relatively low G 00 compared to the corresponding G 0 value (figure 7c). Phase shifts previously reported at 10 Hz [1] were comparable (approx. 5°) to our measured values, which averaged at 5.2°. However, despite these low phase shift values, there are significant differences in G 00 among the groups. Interestingly, the meniscus, which is often described as a shock absorber, had the lowest G 00 . This is in contrast to condylar cartilage, which has the greatest G 00 . Values reported here are comparable to previously published data for both the condyle [50] and the meniscus [19]. Though the role of the meniscus as a shock absorber has been questioned [51,52], this is the first time that it has been quantitatively established that, using G 00 as the outcome measure, condylar hyaline cartilage has a superior energy dissipation capacity when compared to the meniscus. Shock absorption has been quantified in numerous ways [53,54]; however, in this work, G 00 was selected because it describes viscous energy dissipation independently from elastic energy storage. The vibrometer's ability to reproduce the viscoelastic properties of different knee cartilages is strongly demonstrated in this work and may continue to shed light onto the function of other types of cartilage in future studies.
The dynamic modulus of cartilage is the most basic measure of tissue function under the cyclic loading conditions found in daily locomotive activities. It should be noted that the dynamic moduli measured here (figure 6b) and reported elsewhere [1,20] are significantly greater (greater than 10×) than the moduli measured using quasistatic approaches ( figure 9). This is due to internal fluid pressurization. Similar to quasi-static properties, there are topographical variations in dynamic properties across the different knee cartilages (figure 6). Here, an increasing trend was found from the condyle, patella, to trochlear groove groups; this has also been found by others in bovine tissues [20]. However, it is important to note that studies using human tissue report significantly greater compressive moduli for the condyle compared to the trochlear groove [55,56]. With regards to the meniscus, a separate study using human tissue [57] agreed with the meniscus data presented here. Both the data contained within the present study and independent studies [19,20,57] focusing on individual cartilages show that the meniscus has a lower dynamic compressive modulus when compared to values for the condyle, patella and trochlear groove. The compressive properties of the meniscus are known to be relatively small in comparison to its tensile properties, likely due to compressive load transfer into hoop stresses [19]. The hoop stresses are largely borne by the circumferential fibres found in the meniscus. This structural difference, as compared to the structure of hyaline cartilage, is likely responsible for the meniscus second mode shape and expected given the literature on the effect of structure on the bending modes of composites [58][59][60]. Regardless of structure, the agreement in knee cartilage dynamic moduli obtained through noncontact vibrometry with previous reports using destructive methods is impressive and lends credence to vibrometry's efficacy. Vibrometry's use for mechanical characterization of biological tissue is in its nascent stage, and thus requires further study and improvements to reach its potential. A limitation of this study is the use of high frequencies, beyond what is typically observed physiologically. This was required in our set-up in order to excite the entire specimen without damage. At low frequencies, vibrations merely resulted in translation, not the deformation or bending necessary to identify resonant frequencies. Although vibrometers can measure displacements at low frequencies, the mode of excitation would require increased contact, similar to what is used in destructive mechanical testing such as dynamic mechanical analysis (DMA). Therefore, it is expected that cartilage's viscoelastic properties would differ at lower frequencies typically used by other testing systems [61]. Despite this constraint, others have shown that cartilage's dynamic properties are relatively frequency stable when viscous properties are minimized at approximately 40 Hz and beyond [1,[62][63][64]. This is supported by the similarity in G* values reported here by resonant frequencies up to 15 kHz and those measured at 40 Hz [1]. Furthermore, our vibrometry-based storage and loss moduli measurements obtained for bovine hyaline articular cartilage at 15 kHz are within the same order of magnitude as those reported for human and bovine specimens using dynamic mechanical analysis at 88 Hz [62][63][64]. A related limitation to the translation of this work is the use of a piezoelectric drive to induce vibrations. The piezo did not damage the sample, thus allowing it to be used for other mechanical testing. However, in situ applications will require other means of vibrational excitation, such as ultrasound transducers [65]. A third limitation in this study may be that tissues remained in ambient conditions for the duration of the measurement. Although measurements were relatively short (less than 5 min), drying may be a concern. Given this concern, we have fabricated a fluid-filled sample chamber that nonetheless permits the piezo-mediated vibrations to excite the tissue and does not obstruct measurement. Preliminary data using this chamber (data not shown) are encouraging, yielding comparable results to those shown here and assuaging concerns regarding drying. It is important to note that vibrometry-based measurements for the entire study were completed in approximately 2 h, and provided information that collectively took 30 h to acquire from quasi-static mechanical tests and biochemical assays. This represents a 93% reduction in time. Lastly, we do not expect changes in tissue properties arising from enzymatic degradation or the freeze-thaw cycle in this work given that the joint remained closed prior to testing, cartilage cells are known to very metabolically inactive [66], and that refrigeration is the standard way of storing cartilage tissue [67]. However, this might not be the case when working with large samples, where properties can be affected by testing times; thus, groups working with large tissues and/or requiring a fast turnover will find this decrease in time of utmost importance. Given the benefits and advantages of vibrometry, optimization of this tool for cartilage research is well justified.
This study successfully shows the use of vibrometry in a noncontact manner to measure cartilage's dynamic mechanical properties. These properties were particularly distinctive to higher frequency dynamics, which may be representative of the high strain rates observed in traumatic injury. The dynamic properties reported here showed several correlations with quasi-static and biochemical properties, thus providing important insights into cartilage function under cyclic loading. Vibrometer-based measurements were also precise enough to detect differences in viscoelastic properties among the cartilages of the knee. This work not only solidifies the relationship among quasi-static, viscoelastic and dynamic properties over a short testing time but also presents a practical option for high-throughput nondestructive testing, ideal for applications in cartilage developmental biology, tissue engineering and translational medicine.