Kelvin-Voigt Fractional Derivative Model Research Articles

Viscoelastic Characteristics in Mouse Model of Hepatic Steatosis With Inflammation by Kelvin-Voigt Fractional Derivative Modeling.

The aim of the work described here was to measure the characteristics of viscoelasticity and fluidity in a mouse model of hepatic steatosis and inflammation using a nano-indentation test and the Kelvin-Voigt fractional derivative (KVFD) model and to explore the viscoelasticity and fluidity characteristics in mice with different degrees of hepatic steatosis with inflammation. Twenty-five ApoE mice were randomly divided into an experimental high-fat diet group (n=15) and an ordinary-food control group (n=10), then subdivided into four subgroups based on pathological degree of hepatic steatosis: S0 (normal), S1 (mild), S2 (moderate) and S3 (severe). The 25 liver specimens from these mice were evaluated by a slope-keeping relaxation nano-indentation test. Elasticity (E0) was significantly higher in the S3 group than in the S1 and S2 groups, while fluidity (α) and viscosity (τ) were significantly lower in S3 than in S1 and S2 (all p values < 0.05). The following cutoff values for the diagnosis of hepatic steatosis >33% with inflammation were also determined: E0 > 85.01 Pa (area under the curve [AUC]: 0.917, 95% confidence interval [CI]: 0.735-0.989), α ≤ 0.38 (AUC: 0.885, 95% CI: 0.695-0.977),\ and τ ≤ 3.92 (AUC: 0.813, 95% CI: 0.607-0.939). Increases in the degree of hepatic steatosis with inflammation in mice paralleled gradual increases in the stiffness of the liver and gradual decreases in the fluidity and viscosity of the liver.

Mechanical validation of viscoelastic parameters for different interface pressures using the Kelvin-Voigt fractional derivative model.

The knowledge of the biomechanical properties of tissues is useful for different applications such as disease diagnosis and treatment monitoring. Reverberant Shear Wave Elastography (RSWE) is an approach that has reduced the restrictions on wave generation to characterize the shear wave velocity over a range of frequencies. This approach is based on the generation of a reverberant field that is generated by the reflections of waves from inhomogeneities and tissue boundaries that exist in the tissue. The Kelvin-Voigt Fractional Derivative model is commonly used to characterize elasticity and viscosity of soft tissue when using shear wave ultrasound elatography. These viscoelastic characteristics can be then validated using mechanical measurements (MM) such as stress relaxation. During RSWE acquisition, the effect of interface pressure, induced by pushing the probe on the skin through the gel pad, on the viscous and elastic characteristics of tissue can be investigated. However, the effect of interface pressure on the validity of the extracted viscous and elastic characteristics was not investigated before. Therefore, the purpose of this study was to compare the estimation of the viscoelastic parameters at different thickness of gel pad against the viscoelastic characteristics obtained from MM. The experiments were conducted in a tissue-mimicking phantom. The results confirm that the relaxed elastic constant (μ0) can be depreciated. In addition, a higher congruence was found in the viscous parameter (ηα) estimated at 6 and 7 mm. On the other hand, a difference in the order of fractional derivative (α) was found.

Comprehensive experimental assessments of rheological models’ performance in elastography of soft tissues

Elastography researchers have utilized several rheological models to characterize soft tissue viscoelasticity over the past thirty years. Due to the frequency-dependent behavior of viscoelastic parameters as well as the different techniques and frequencies employed in various studies of soft tissues, rheological models have value in standardizing disparate techniques via explicit mathematical representations. However, the important question remains: which of the several available models should be considered for widespread adoption within a theoretical framework? We address this by evaluating the performance of three well established rheological models to characterize ex vivo bovine liver tissues: the Kelvin-Voigt (KV) model as a 2-parameter model, and the standard linear solid (SLS) and Kelvin-Voigt fractional derivative (KVFD) models as 3-parameter models. The assessments were based on the analysis of time domain behavior (using stress relaxation tests) and frequency domain behavior (by measuring shear wave speed (SWS) dispersion). SWS was measured over a wide range of frequency from 1 Hz to 1 kHz using three different tests: (i) harmonic shear tests using a rheometer, (ii) reverberant shear wave (RSW) ultrasound elastography scans, and (iii) RSW optical coherence elastography scans, with each test targeting a distinct frequency range. Our results demonstrated that the KVFD model produces the only mutually consistent rendering of time and frequency domain data for liver. Furthermore, it reduces to a 2-parameter model for liver (correspondingly to a 2-parameter “spring-pot” or power-law model for SWS dispersion) and provides the most accurate predictions of the material viscoelastic behavior in time (>98% accuracy) and frequency (>96% accuracy) domains. Statement of SignificanceRheological models are applied in quantifying tissues viscoelastic properties. This study is unique in presenting comprehensive assessments of rheological models:•We employed experimental data in both the frequency domain (shear wave speed (SWS) vs. frequency) and time domain (stress relaxation) to assess rheological models’ performances.•SWS were acquired over a wide frequency range, 1 Hz to 1 kHz, by three independent techniques.•Using the frequency domain analysis, we evaluated how well each model can predict measured time domain behaviors (and vice versa).•This presents wide-ranging experimental proofs as the most comprehensive study of its type in terms of the number of experiments, frequency range, and conjoined assessments of time and frequency domains behaviors, demonstrating the most appropriate rheological model for soft tissues.

Characterization of tissue mimicking phantoms and soft tissues using ultrasound shear wave elastography and power-law models

Characterization of mechanical properties of tissue has become an important area of study over the last two decades. Shear wave elastography (SWE) is being used in many clinical applications to characterize, and multiple mechanical properties are being reported. To more comprehensively characterize soft tissues, we have focused on frequency-domain methods for evaluating viscoelasticity. We measure the shear wave velocity and attenuation and the variation with frequency. We will present results with the Kelvin-Voigt fractional derivative model and general power laws to characterize data from SWE experiments in tissue-mimicking phantoms and exvivo and in vivo tissue. Tissue mimicking phantoms from CIRS, inc., that have varying amounts of viscoelasticity will be investigated. Measurements from ex vivo and in vivo liver and kidney will also be used for this study. We will examine the range of the power-law exponents associated with these samples and the different models to provide understanding of the range of these parameters and how well tissue mimicking phantoms realize the parameters measured in soft tissues.

Fluidity and elasticity form a concise set of viscoelastic biomarkers for breast cancer diagnosis based on Kelvin-Voigt fractional derivative modeling.

Cancer progression involves biomechanical changes within transformed cells and the surrounding extracellular matrix (ECM). The viscoelastic features of fluidity and elasticity that are based on a novel Kelvin-Voigt fractional derivative (KVFD) model were found capable of discriminating normal, benign and malignant breast biopsy tissues on the cellular scale. The improved specificity of KVFD model parameters derives from greater accuracy of fitting the entire approaching force-indentation measurement curve ([Formula: see text] > 0.99) compared with traditional elastic models ([Formula: see text] < 0.86). Moreover, model parameters can be interpreted in terms of histopathological features. First, statistical comparisons reveal there are significant differences (p < 0.001) in elasticity E0, fluidity [Formula: see text], and viscosity [Formula: see text] among healthy, benign, and malignant groups. Malignant breast tissues show low-value, broad-distributions in E0 and with high fluidity [Formula: see text] as compared with healthy and benign tissues. Second, histograms of E0 and [Formula: see text] provide distinctive features by fitting to Gaussian mixture (GM) models. The histograms of E0 and [Formula: see text] are best fit by two kernels GM for malignant tissues, indicating that the cells are soft but with high fluidity and the ECM is stiff but with low fluidity. However, the data suggest one-kernel GM model for benign tissue and a patched uniform distribution for healthy tissue. Third, using fluidity [Formula: see text] as the test statistic, the area under the receiver operator characteristiccurve (AUC) is 0.701 ± 0.012 (p < 0.0001) for control versus malignant and 0.706 ± 0.013 (p < 0.0001) for benign versus malignant group. Variations in tissue fluidity and elasticity offer a concise set of viscoelastic biomarkers that correlate well with histopathological features.

ANALYSIS OF FORCED VIBRATIONS OF NONLINEAR PLATES IN A VISCOELASTIC MEDIUM UNDER THE CONDITIONS OF THE DIFFERENT COMBINATIONAL INTERNAL

In the present paper, the force driven dynamic response of a nonlinear plate embedded in a viscoelastic medium, damping features of which are described by the Kelvin-Voigt fractional derivative model, is studied. The motion of the plate is described by three coupled nonlinear differential equations with due account for the fact that the plate is being under the conditions of the internal combinational resonance accompanied by the external resonance, resulting in the interaction of three modes corresponding to the mutually orthogonal displacements. A comparative analysis of numerical calculations for the cases of free and forced vibrations has been carried out.

Analysis of thin nonlinear plates’ forced vibrations in a viscoelastic medium under the conditions of the additive combinational internal resonance

The force driven dynamic response of a nonlinear plate embedded in a viscoelastic medium, damping features described by the Kelvin-Voigt fractional derivative model, is studied in the present paper. The motion of the plate is described by three coupled nonlinear differential equations with due account for the fact that the plate is being under the conditions of the internal combinational resonance accompanied by the external resonance, resulting in three modes corresponding interaction to the mutually orthogonal displacements. The displacement functions are determined in terms of linear vibrations’ eigenfunctions. The solution of motion nonlinear governing equations was obtained by the utilization of the multiple scales method, so doing the amplitude functions are expanded into power series in terms of the small parameter and depend on different time scales. The influence of viscosity on the energy exchange mechanism between interacting modes was analyzed. A comparative analysis of numerical calculations for the free and forced vibrations cases was carried out.

Modeling ramp-hold indentation measurements based on Kelvin–Voigt fractional derivative model

Interpretation of experimental data from micro- and nano-scale indentation testing is highly dependent on the constitutive model selected to relate measurements to mechanical properties. The Kelvin–Voigt fractional derivative model (KVFD) offers a compact set of viscoelastic features appropriate for characterizing soft biological materials. This paper provides a set of KVFD solutions for converting indentation testing data acquired for different geometries and scales into viscoelastic properties of soft materials. These solutions, which are mostly in closed-form, apply to ramp-hold relaxation, load-unload and ramp-load creep-testing protocols. We report on applications of these model solutions to macro- and nano-indentation testing of hydrogels, gastric cancer cells and ex vivo breast tissue samples using an atomic force microscope (AFM). We also applied KVFD models to clinical ultrasonic breast data using a compression plate as required for elasticity imaging. Together the results show that KVFD models fit a broad range of experimental data with a correlation coefficient typically R2 > 0.99. For hydrogel samples, estimation of KVFD model parameters from test data using spherical indentation versus plate compression as well as ramp relaxation versus load-unload compression all agree within one standard deviation. Results from measurements made using macro- and nano-scale indentation agree in trend. For gastric cell and ex vivo breast tissue measurements, KVFD moduli are, respectively, 1/3–1/2 and 1/6 of the elasticity modulus found from the Sneddon model. In vivo breast tissue measurements yield model parameters consistent with literature results. The consistency of results found for a broad range of experimental parameters suggest the KVFD model is a reliable tool for exploring intrinsic features of the cell/tissue microenvironments.

Assessing composition and structure of soft biphasic media from Kelvin–Voigt fractional derivative model parameters4

Kelvin–Voigt fractional derivative (KVFD) model parameters have been used to describe viscoelastic properties of soft tissues. However, translating model parameters into a concise set of intrinsic mechanical properties related to tissue composition and structure remains challenging. This paper begins by exploring these relationships using a biphasic emulsion materials with known composition. Mechanical properties are measured by analyzing data from two indentation techniques—ramp-stress relaxation and load-unload hysteresis tests. Material composition is predictably correlated with viscoelastic model parameters. Model parameters estimated from the tests reveal that elastic modulus E0 closely approximates the shear modulus for pure gelatin. Fractional-order parameter α and time constant τ vary monotonically with the volume fraction of the material’s fluid component. α characterizes medium fluidity and the rate of energy dissipation, and τ is a viscous time constant. Numerical simulations suggest that the viscous coefficient η is proportional to the energy lost during quasi-static force-displacement cycles, EA. The slope of EA versus η is determined by α and the applied indentation ramp time Tr. Experimental measurements from phantom and ex vivo liver data show close agreement with theoretical predictions of the relation. The relative error is less than 20% for emulsions 22% for liver. We find that KVFD model parameters form a concise features space for biphasic medium characterization that described time-varying mechanical properties.

Ramp-hold relaxation solutions for the KVFD model applied to soft viscoelastic media

The standard step-hold load-relaxation profile can yield variable estimates of mechanical properties due to the difficulty in achieving a step strain experimentally. A ramp-hold profile overcomes this limitation if appropriate model functions can be derived. Utilizing Boltzmann hereditary integral operators for two indentation geometries, analytical ramp solutions for load-relaxation were developed based on the Kelvin–Voigt fractional derivative (KVFD) model. The results identify three model parameters for characterizing viscoelastic behavior from a single model curve fit to the data: the elastic modulus E0, fractional-order parameter α, and relaxation time constant . The quantitative nature of the analysis was validated through measurements on gelatin emulsion samples exhibiting viscoelastic behavior. KVFD-model-based solutions provide mathematically simple and experimentally flexible descriptions of load-relaxation behavior for a range of viscoelastic properties and experimental conditions; e.g. one closed-form solution can fit the ramp and the hold phases of the relaxation time series. Experiments show that the solution for a spherical indenter and plate compressor each fit well to the corresponding experimental relaxation curves with a coefficient of determination R2 > 0.98. Parameters obtained from the spherical-tip indentation and plate-compression geometries agree within one standard deviation, confirming that the ramp solution based KVFD model yields consistent measurements for characterizing viscoelastic materials.

Measurement of Viscoelastic Properties of Polyacrylamide-Based Tissue-Mimicking Phantoms for Ultrasound Elastography Applications

Many ailments and/or malfunctions of the body have been observed to change the viscous behavior and elastic properties of biological soft tissues. The technique of elastography has evolved to image such properties. The clinical evidence gathered during studies involving elastography to identify cancerous lesions is very promising. However, the quantification of the resolution and specificity of elastography is best achieved under a controlled study using tissue-mimicking phantoms. One challenge is to reproduce viscoelastic behavior in phantoms as observed in biological tissues. In this paper, polyacrylamide gel based tissue-mimicking phantoms have been developed to experimentally study the role of viscoelastic properties in a controlled manner. To measure the Young's modulus, the phantoms were subjected to linear loading, and the stress-strain relationship is deduced therefrom. It is seen that the phantoms show hysteresis behavior. The viscoelastic properties of these phantoms were measured by subjecting the samples to cyclic loading. Normal forces during this process of loading were also measured as a measure of sample elasticity. To emulate the normal and pathological lesions, samples were prepared with varying concentration of monomer and studied. Three models, namely, Maxwell, Kelvin-Voigt (KV), and Kelvin-Voigt fractional derivative (KVFD), were chosen to fit the experimental data. Of these, the KVFD model was found to be best fitting for the experimental data obtained. Results indicate that stiffer samples exhibit large variations in the storage modulus when the precompression levels are altered.