Kelvin-Voigt Fractional Derivative 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.

Requisites on material viscoelasticity for exceptional points in passive dynamical systems

Recent progress in non-Hermitian physics and the notion of exceptional point (EP) degeneracies in elastodynamics have led to the development of novel metamaterials for the control of elastic wave propagation, hypersensitive sensors, and actuators. The emergence of EPs in a parity-time symmetric system relies on judiciously engineered balanced gain and loss mechanisms. Creating gain requires complex circuits and amplification mechanisms, making engineering applications challenging. Here, we report strategies to achieve EPs in passive non-Hermitian elastodynamic systems with differential loss derived from viscoelastic materials. We compare different viscoelastic material models and show that the EP emerges only when the frequency-dependent loss-tangent of the viscoelastic material remains nearly constant in the frequency range of operation. This type of loss tangent occurs in materials that undergo stress-relaxation over a broad spectrum of relaxation times, for example, materials that follow the Kelvin–Voigt fractional derivative (KVFD) model. Using dynamic mechanical analysis, we show that a few common viscoelastic elastomers, such as polydimethylsiloxane and polyurethane rubber, follow the KVFD behavior such that the loss tangent becomes almost constant after a particular frequency. The material models we present and the demonstration of the potential of a widely available material system in creating EPs pave the way for developing non-Hermitian metamaterials with hypersensitivity to perturbations or enhanced emissivity.

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.

The viscoelastic characteristics of in-vitro carotid plaque by Kelvin-Voigt fractional derivative modeling

Atherosclerotic plaque with a thin fibrous cap can be ruptured by shear force. Exploiting the mechanical properties of plaques within different histological regions can help to better understand the physical mechanisms of the plaque. The association between the plaque components and viscoelasticity was studied when mapping the viscoelasticity to histological features. Eleven in-vitro carotid plaques were tested with ramp-hold relaxation nanoindentation tests. Viscoelasticity (elastic modulus E0, fluidity α, and viscosity τ) was characterized by Kelvin-Voigt fractional derivative (KVFD) modeling. There is a significant difference (p < 0.001) on E0, α, and τ between the collagen-rich (CR) group and the non-collagen-rich (NCR) group. In the CR group, the elastic modulus E0 was higher but the fluidity α and viscosity τ were lower than those of the NCR group. Receiver operating characteristic (ROC) analysis revealed that combinations of E0 and α can be used as a CR indicator with an area under the curve (AUC) of 0.770. There was a negative correlation between E0 and the percentages of myxoid degeneration (r = −0.160, p < 0.001), necrosis (r = −0.229, p < 0.001) and inflammatory cells (r = −0.130, p < 0.001), and a positive correlation between elasticity E0 and the percentage of foam cells (r = 0.121, p < 0.001). There was a positive correlation between fluidity α and the percentage of necrosis (r = 0.308, p < 0.001). The results confirmed the clinical evidence that the CR group with higher elasticity and lower fluidity has higher resisting ability, whereas the NCR group with lower elasticity and higher fluidity has accompanied with more myxoid degeneration, extracellular lipids and necrosis.

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.

Wave Propagation in a Fractional Viscoelastic Tissue Model: Application to Transluminal Procedures.

In this article, a wave propagation model is presented as the first step in the development of a new type of transluminal procedure for performing elastography. Elastography is a medical imaging modality for mapping the elastic properties of soft tissue. The wave propagation model is based on a Kelvin Voigt Fractional Derivative (KVFD) viscoelastic wave equation, and is numerically solved using a Finite Difference Time Domain (FDTD) method. Fractional rheological models, such as the KVFD, are particularly well suited to model the viscoelastic response of soft tissue in elastography. The transluminal procedure is based on the transmission and detection of shear waves through the luminal wall. Shear waves travelling through the tissue are perturbed after encountering areas of altered elasticity. These perturbations carry information of medical interest that can be extracted by solving the inverse problem. Scattering from prostate tumours is used as an example application to test the model. In silico results demonstrate that shear waves are satisfactorily transmitted through the luminal wall and that echoes, coming from reflected energy at the edges of an area of altered elasticity, which are feasibly detectable by using the transluminal approach. The model here presented provides a useful tool to establish the feasibility of transluminal procedures based on wave propagation and its interaction with the mechanical properties of the tissue outside the lumen.

Solutions to ramp-hold dynamic oscillation indentation tests for assessing the viscoelasticity of hydrogel by Kelvin-Voigt fractional derivative modeling

Indentation type dynamic mechanical analysis (IT-DMA) is an attractive approach to characterizing viscoelastic properties of soft matter under both quasi-static and dynamic loadings. Frequency-dependent viscoelastic properties are usually characterized by storage and loss moduli whereas extracting inherent viscoelastic model parameters from dynamic loadings remains unsolved. In this study, closed-form solutions to dynamic oscillation loadings were first derived and adapted to a variety of excitations for arbitrary shape indenters. A uniform theoretical framework for ramp-hold-oscillation-relaxation/creep indentation tests are established and Kelvin-Voigt fractional derivative (KVFD) solutions are presented to predict the viscoelastic behavior for both quasi-static loadings (creep, relaxation) and dynamic loadings (sinusoidal displacement/force cycles at 5,10,15,20,30 Hz). Four gelatin-cream hydrogel samples were investigated under ramp-hold-oscillation-relaxation and ramp-hold-oscillation-creep testing conditions. Viscoelastic quantities of elasticity E0, fluidity α, and viscosity τ were extracted by fitting the experimental curves to KVFD predictions. The results show that KVFD models can accurately depict the viscoelastic behavior under both dynamic and quasi-static loadings with high goodness of fit (R2>0.936). To assess the proposed method, comparison of the viscoelastic quantities from different experimental protocols were conducted. Statistical results revealed viscoelastic parameters from different loading are in close agreement. For dynamic loadings, significant differences were found for [E0, α] estimates at different frequencies. For different experimental protocols, estimations were consistent in that E0 showed maximum value on Gel5Cream15 sample, and α was increased with increasing cream percentage. Moreover, viscoelastic quantities extracted from dynamic relaxation were correlated with those of the dynamic creep (r = 0.977, p < 0.001 for E0, and r = 0.969, p < 0.001 for α). Viscoelastic quantities from quasi-static relaxation were highly correlated with those from quasi-static creep (r = 0. 910, p < 0.001 for E0, and r = 0.979, p < 0.001 for α). The complex modulus from KVFD estimations was found to be correlated with that of the DMA measurement (for dynamic relaxation, r = 0.603, p < 0.001 for E′ and r = 0.818, p < 0.001 for E″; for dynamic creep, r = 0.756, p < 0.001 for E′ and r = 0.762, p < 0.001 for E″). Cross validation and assessment by alternative DMA method confirmed that KVFD solutions provided a method for relating viscoelastic measurements at load frequencies from quasi-static to dynamic for soft viscoelastic tissue-like materials.

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.

Sodium Alginate Toughening of Gelatin Hydrogels.

Gelatin is a popular material for the creation of tissue phantoms due to its ease-of-use, safety, low relative cost, and its amenability to tuning physical properties through the use of additives. One difficulty that arises when using gelatin, especially in low concentrations, is the brittleness of the material. In this paper, we show that small additions of another common biological polymer, sodium alginate, significantly increase the toughness of gelatin without changing the Young's modulus or other low-strain stress relaxation properties of the material. Samples were characterized using ramp-hold stress relaxation tests. The experimental data from these tests were then fit to the Generalized Maxwell (GM) model, as well as two models based on a fractional calculus approach: the Kelvin-Voigt Fractional Derivative (KVFD) and Fractional Maxwell (FM) models. We found that for our samples, the fractional models provided better fits with fewer parameters, and at strains within the linear elastic region, the linear viscoelastic parameters of the alginate/gelatin and pure gelatin samples were essentially indistinguishable. When the same ramp-hold stress relaxation experiments were run at high strains outside of the linear elastic region, we observed a shift in stress relaxation to shorter time scales with increasing sodium alginate addition, which may be associated with an increase in fluidity within the gelatin matrix. This leads us to believe that sodium alginate acts to enhance the viscosity within the fluidic region of the gelatin matrix, providing additional energy dissipation without raising the modulus of the material. These results are applicable to anyone desiring independent control of the Young's modulus and toughness in preparing tissue phantoms, and suggest that sodium alginate should be added to low-modulus gelatin for use in biological and medical testing applications.

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.

Wave Simulation in Biologic Media Based on the Kelvin-Voigt Fractional-Derivative Stress-Strain Relation

The acoustic behavior of biologic media can be described more realistically using a stress-strain relation based on fractional time derivatives of the strain, since the fractional exponent is an additional fitting parameter. We consider a generalization of the Kelvin-Voigt rheology to the case of rational orders of differentiation, the so-called Kelvin-Voigt fractional-derivative (KVFD) constitutive equation, and introduce a novel modeling method to solve the wave equation by means of the Grünwald-Letnikov approximation and the staggered Fourier pseudospectral method to compute the spatial derivatives. The algorithm can handle complex geometries and general material-property variability. We verify the results by comparison with the analytical solution obtained for wave propagation in homogeneous media. Moreover, we illustrate the use of the algorithm by simulation of wave propagation in normal and cancerous breast tissue.

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.

Fractional derivative models for ultrasonic characterization of polymer and breast tissue viscoelasticity

The viscoelastic response of hydropolymers, which include glandular breast tissues, may be accurately characterized for some applications with as few as 3 rheological parameters by applying the Kelvin-Voigt fractional derivative (KVFD) modeling approach. We describe a technique for ultrasonic imaging of KVFD parameters in media undergoing unconfined, quasi-static, uniaxial compression. We analyze the KVFD parameter values in simulated and experimental echo data acquired from phantoms and show that the KVFD parameters may concisely characterize the viscoelastic properties of hydropolymers. We then interpret the KVFD parameter values for normal and cancerous breast tissues and hypothesize that this modeling approach may ultimately be applied to tumor differentiation.

Tissue elasticity properties as biomarkers for prostate cancer

In this paper we evaluate tissue elasticity as a longstanding but qualitative biomarker for prostate cancer and sonoelastography as an emerging imaging tool for providing qualitative and quantitative measurements of prostate tissue stiffness. A Kelvin-Voigt Fractional Derivative (KVFD) viscoelastic model was used to characterize mechanical stress relaxation data measured from human prostate tissue samples. Mechanical testing results revealed that the viscosity parameter for cancerous prostate tissue is greater than that derived from normal tissue by a factor of approximately 2.4. It was also determined that a significant difference exists between normal and cancerous prostate tissue stiffness (p < 0.01) yielding an average elastic contrast that increases from 2.1 at 0.1 Hz to 2.5 at 150 Hz. Qualitative sonoelastographic results show promise for cancer detection in prostate and may prove to be an effective adjunct imaging technique for biopsy guidance. Elasticity images obtained with quantitative sonoelastography agree with mechanical testing and histological results. Overall, results indicate tissue elasticity is a promising biomarker for prostate cancer.

Quantitative Characterization of Viscoelastic Properties of Human Prostate Correlated with Histology

Quantification of mechanical properties of human prostate tissue is important for developing sonoelastography for prostate cancer detection. In this study, we characterized the frequency-dependent complex Young's modulus of normal and cancerous prostate tissues in vitro by using stress relaxation testing and viscoelastic tissue modeling methods. After radical prostatectomy, small cylindrical tissue samples were acquired in the posterior region of each prostate. A total of 17 samples from eight human prostates were obtained and tested. Stress relaxation tests on prostate samples produced repeatable results that fit a viscoelastic Kelvin-Voigt fractional derivative (KVFD) model (r 2>0.97). For normal ( n = 8) and cancerous ( n = 9) prostate samples, the average magnitudes of the complex Young's moduli (| E*|) were 15.9 ± 5.9 kPa and 40.4 ± 15.7 kPa at 150 Hz, respectively, giving an elastic contrast of 2.6:1. Nine two-sample t-tests indicated that there are significant differences between stiffness of normal and cancerous prostate tissues in the same gland ( p < 0.01). This study contributes to the current limited knowledge on the viscoelastic properties of the human prostate, and the inherent elastic contrast produced by cancer. (E-mail: mazhang@seas.rochester.edu)