Measuring the linear and nonlinear elastic properties of brain tissue with shear waves and inverse analysis

Photoacoustic viscoelasticity assessment of prefrontal cortex and cerebellum in normal and prenatal valproic acid-exposed rats

Mechanical properties of brain tissues are from principal features from different points of view; diagnosis, the performance of the brain and neurological disorders. Particularly viscoelastic properties of the brain tissues are determinative. In this study based on a proposed accurate and non-invasive method, we have measured the viscoelastic properties of prefrontal cortex and cerebellum, two important brain regions involved in motor learning and pathophysiology of autism spectrum disorder (ASD). In this regard, using photoacoustic systems, viscoelastic properties of tissues from the cerebellum and prefrontal cortex of normal and prenatal VPA (Valproic acid)-exposed (i.e. autistic-like) offspring rats are measured. Results of our study show that the cerebellums of normal tissues are stiffer than the tissue obtained from autistic-like rats, while the viscoelasticity of the prefrontal cortex of normal tissues is higher than that of autistic ones. The proposed method for the measurement of viscoelastic properties of the brain tissue has the potential not only for the fundamental studies but as a diagnosis technique.

An atlas of the heterogeneous viscoelastic brain with local power-law attenuation synthesised using Prony-series

This review addresses the acute need to acknowledge the mechanical heterogeneity of brain matter and to accurately calibrate its local viscoelastic material properties accordingly. Specifically, it is important to compile the existing and disparate literature on attenuation power-laws and dispersion to make progress in wave physics of brain matter, a field of research that has the potential to explain the mechanisms at play in diffuse axonal injury and mild traumatic brain injury in general. Currently, viscous effects in the brain are modelled using Prony-series, i.e., a sum of decaying exponentials at different relaxation times. Here we collect and synthesise the Prony-series coefficients appearing in the literature for twelve regions: brainstem, basal ganglia, cerebellum, corona radiata, corpus callosum, cortex, dentate gyrus, hippocampus, thalamus, grey matter, white matter, homogeneous brain, and for eight different mammals: pig, rat, human, mouse, cow, sheep, monkey and dog. Using this data, we compute the fractional-exponent attenuation power-laws for different tissues of the brain, the corresponding dispersion laws resulting from causality, and the averaged Prony-series coefficients. STATEMENT OF SIGNIFICANCE: Traumatic brain injuries are considered a silent epidemic and finite element methods (FEMs) are used in modelling brain deformation, requiring access to viscoelastic properties of brain. To the best of our knowledge, this work presents 1) the first multi-frequency viscoelastic atlas of the heterogeneous brain, 2) the first review focusing on viscoelastic modelling in both FEMs and experimental works, 3) the first attempt to conglomerate the disparate existing literature on the viscoelastic modelling of the brain and 4) the largest collection of viscoelastic parameters for the brain (212 different Prony-series spanning 12 different tissues and 8 different animal surrogates). Furthermore, this work presents the first brain atlas of attenuation power-laws essential for modelling shear waves in brain.

Effects of Different Types of Electric Fields on Mechanical Properties and Microstructure of Ex Vivo Porcine Brain Tissues

Electrotherapy plays a crucial role in regulating neuronal activity. Nevertheless, the relevant therapeutic mechanisms are still unclear; thus, the effects of electric fields on brain tissue's mechanical properties and microstructure need to be explored. In this study, focusing on the changes in mechanical properties and microstructure of ex vivo porcine brain tissues under different types of electric fields, directional and alternating electric fields (frequencies of 5, 20, 50, and 80 Hz, respectively) integrate with a custom-designed indentation device. The experimental results showed that for the ex vivo brain tissue, the directional electric field (DEF) can reduce the elastic properties of brain tissue. Simultaneously, the DEF can increase the cell spacing and reduce the proteoglycan content. The transmission electron microscope (TEM) analysis observed that the DEF can reduce the integrity of the plasma membrane, the endoplasmic reticulum's stress response, and the myelin lamella's separation. The alternating electric field (AEF) can accelerate the stress relaxation process of brain tissue and change the time-dependent mechanical properties of brain tissue. Meanwhile, with the increase in frequency, the cell spacing decreased, and the proteoglycan content gradually approached the control group without electric fields. TEM analysis observed that with the increase in frequency, the integrity of the plasma membrane increases, and the separation of the myelin lamella gradually disappears. Understanding the changes in the mechanical properties and microstructure of brain tissue under AEF and DEF enables a preliminary exploration of the therapeutic mechanism of electrotherapy. Simultaneously, the essential data was provided to support the development of embedded electrodes. In addition, the ex vivo experiments build a solid foundation for future in vivo experiments.

Linear, weakly nonlinear and fully nonlinear models for soft tissues: which ones provide the most reliable estimations of the stiffness?

Benign and malignant lesions in tissues or organs can be detected by elastographic investigations in which pathological regions are spotted from local alterations of the stiffness. As is known, the shear modulus provides a measure of the stiffness of an elastic material. Based on the classical theory of linear elasticity, an elastogram yields estimations of the linear shear modulus from measurements of the speed of small-amplitude transverse waves propagating in the medium tested. In this paper, we show that the estimation of the shear modulus can be improved significantly by employing the fourth-order weakly nonlinear theory of elasticity (FOE), and indicate how the stiffness can be assessed more precisely with the use of FOE. We discuss also why FOE provides more reliable results than the fully nonlinear theory of elasticity. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

The effects of electric fields on the mechanical properties and microstructure of ex vivo porcine brain tissues

As a popular tool for regulating the physiological conditions of the brain and treating brain diseases, electrotherapy has become increasingly mature in clinical applications. However, the mechanical properties and microstructure of the brain that change with weak electric fields are often overlooked. Thus, the mechanical behaviors of the brain tissue, which play a critical role in modulating the brain form and brain function, need to be taken into account. Herein, the direct current electric fields were combined with a customized indentation device and simultaneously focused on the changes in the mechanical properties and microstructure of ex vivo porcine brain tissues under electric fields. The experimental results showed that the electric fields reduced the shear modulus and viscosity and increased the relaxation rate of ex vivo porcine brain tissues. Moreover, electric fields polarized the cell bodies and reduced proteoglycan content in the cortex. The TEM observation confirmed that the electric fields deepened the degree of endoplasmic reticulum expansion and decreased the structural integrity of the cell membrane and myelin sheath. This study confirmed the effect of electric fields on ex vivo brain tissues; concurrently, it created comparable space in microscopic structure/compositions and mechanical parameters for future deeper brain experiments under stress-electric field coupling.

Nonlinear Characterization of Tissue Viscoelasticity With Acoustoelastic Attenuation of Shear Waves

Shear-wave elastography (SWE) measures shear-wave speed (SWS), which is related to the underlying shear modulus of soft tissue. SWS in soft tissue changes depending on the amount of external strain that soft tissue is subjected to due to the acoustoelastic (AE) phenomenon. In the literature, variations of SWS as a function of applied uniaxial strain were used for nonlinear characterization, assuming soft tissues to be elastic, although soft tissues are indeed viscoelastic in nature. Hence, nonlinear characterization using SWS alone is insufficient. In this work, we use SWS together with shear-wave attenuation (SWA) during incremental quasi-static compressions in order to derive biomechanical characterization based on the AE theory in terms of well-defined storage and loss moduli. As part of this study, we also quantify the effect of applied strain on measurements of SWS and SWA since such confounding effects need to be taken into account when using SWS and/or SWA, e.g., for staging a disease state, while such effects can also serve as an additional imaging biomarker. Our results from tissue-mimicking phantoms with varying oil percentages and ex vivo porcine liver experiments demonstrate the feasibility of our proposed methods. In both experiments, SWA was observed to decrease with applied strain. For 10% compression in ex vivo livers, shear-wave attenuation decreased, on average, by 28% (93 Np/m), while SWS increased, on average, by 20% (0.26 m/s).

Measured Hyperelastic Properties of Cervical Tissue with Shear-Wave Elastography

The nonlinear mechanical behaviour of cervical tissue causes unpredictable changes in measured elastograms when pressure is applied. These uncontrolled variables prevent the reliable measurement of tissue elasticity in a clinical setting. Measuring the nonlinear properties of tissue is difficult due to the need for both shear modulus and strain to be taken simultaneously. A simulation-based method is proposed in this paper to resolve this. This study describes the nonlinear behaviour of cervical tissue using the hyperelastic material models of Demiray–Fung and Veronda–Westmann. Elastograms from 33 low-risk patients between 18 and 22 weeks gestation were obtained. The average measured properties of the hyperelastic material models are: Demiray–Fung—A1α = 2.07 (1.65–2.58) kPa, α = 6.74 (4.07–19.55); Veronda–Westmann—C1C2 = 4.12 (3.24–5.04) kPa, C2 = 4.86 (2.86–14.28). The Demiray–Fung and Veronda–Westmann models performed similarly in fitting to the elastograms with an average root mean square deviation of 0.41 and 0.47 ms−1, respectively. The use of hyperelastic material models to calibrate shear-wave speed measurements improved the consistency of measurements. This method could be applied in a large-scale clinical setting but requires updated models and higher data resolution.

Mechanical properties of brain tissue based on microstructure

Research on the mechanical properties of brain tissue has gradually deepened recently. Two indentation protocols were used here to characterize the mechanical properties of cortical tissues. Further, histological staining was used to explore the correlation between the mechanical properties and microstructure on the basis of the density of cell nuclei and proteoglycan content. No significant difference was observed in transient contact stiffness between the cerebral cortex and cerebellar cortex at the depth interval of 0-600 μm under the cortical surface; however, the average shear modulus of the cerebral cortex was higher than that of the cerebellar cortex. The cerebral cortex responded more quickly to the change in load and released stress more thoroughly than the cerebellar cortex. In addition, the density of cell nuclei was related to both the transient contact stiffness and second time constant of cortical tissues. Proteoglycan content had a more significant impact on the shear modulus, second time constant, and stress relaxation rate of cortical tissues. Exploring mechanical properties thoroughly will provide more detailed mechanical information for future brain chip implantation. Alternatively, linking the mechanical properties of cortical tissues to the microstructure can provide basic data for the design and manufacture of substitute materials for brain tissue.

A Shift in Tissue Stiffness During Hippocampal Maturation Correlates to the Pattern of Neurogenesis and Composition of the Extracellular Matrix

Aging changes the mechanical properties of brain tissue, such as stiffness. It has been proposed that the maintenance and differentiation of neural stem cells (NSCs) are regulated in accordance with extracellular stiffness. Neurogenesis is observed in restricted niches, including the dentate gyrus (DG) of the hippocampus, throughout mammalian lifetimes. However, profiles of tissue stiffness in the DG in comparison with the activity of NSCs from the neonatal to the matured brain have rarely been addressed so far. Here, we first applied ultrasound-based shear-wave elasticity imaging (SWEI) in living animals to assess shear modulus as in vivo brain stiffness. To complement the assay, atomic force microscopy (AFM) was utilized to determine the Young’s modulus in the hippocampus as region-specific stiffness in the brain slice. The results revealed that stiffness in the granule cell layer (GCL) and the hilus, including the subgranular zone (SGZ), increased during hippocampal maturation. We then quantified NSCs and immature neural cells in the DG with differentiation markers, and verified an overall decrease of NSCs and proliferative/immature neural cells along stages, showing that a specific profile is dependent on the subregion. Subsequently, we evaluated the amount of chondroitin sulfate proteoglycans (CSPGs), the major extracellular matrix (ECM) components in the premature brain by CS-56 immunoreactivity. We observed differential signal levels of CSPGs by hippocampal subregions, which became weaker during maturation. To address the contribution of the ECM in determining tissue stiffness, we manipulated the function of CSPGs by enzymatic digestion or supplementation with chondroitin sulfate, which resulted in an increase or decrease of stiffness in the DG, respectively. Our results illustrate that stiffness in the hippocampus shifts due to the composition of ECM, which may affect postnatal neurogenesis by altering the mechanical environment of the NSC niche.

Acousto-elasticity of transversely isotropic incompressible soft tissues: characterization of skeletal striated muscle

Using shear wave elastography, we measure the changes in the wave speed with the stress produced by a striated muscle during isometric voluntary contraction. To isolate the behaviour of an individual muscle from complementary or antagonistic actions of adjacent muscles, we select theflexor digiti minimimuscle, whose sole function is to extend the little finger. To link the wave speed to the stiffness, we develop an acousto-elastic theory for shear waves in homogeneous, transversely isotropic, incompressible solids subject to an uniaxial stress. We then provide measurements of the apparent shear elastic modulus along, and transversely to, the fibre axis for six healthy human volunteers of different age and sex. The results display a great variety across the six subjects. We find that the slope of the apparent shear elastic modulus along the fibre direction changes inversely to the maximum voluntary contraction (MVC) produced by the volunteer. We propose an interpretation of our results by introducing the S (slow) or F (fast) nature of the fibres, which harden the muscle differently and accordingly, produce different MVCs. A natural follow-up on this study is to apply the method to patients with musculoskeletal disorders or neurodegenerative diseases.

Nonlinear plane waves in saturated porous media with incompressible constituents

We consider the propagation of nonlinear plane waves in porous media within the framework of the Biot–Coussy biphasic mixture theory. The tortuosity effect is included in the model, and both constituents are assumed incompressible (Yeoh-type elastic skeleton, and saturating fluid). In this case, the linear dispersive waves governed by Biot’s theory are either of compression or shear-wave type, and nonlinear waves can be classified in a similar way. In the special case of a neo-Hookean skeleton, we derive the explicit expressions for the characteristic wave speeds, leading to the hyperbolicity condition. The sound speeds for a Yeoh skeleton are estimated using a perturbation approach. Then we arrive at the evolution equation for the amplitude of acceleration waves. In general, it is governed by a Bernoulli equation. With the present constitutive assumptions, we find that longitudinal jump amplitudes follow a nonlinear evolution, while transverse jump amplitudes evolve in an almost linearly degenerate fashion.

An ultrasonic method to measure stress without calibration: The angled shear wave method

Measuring stress levels in loaded structures is crucial to assess and monitor structure health and to predict the length of remaining structural life. Many ultrasonic methods are able to accurately predict in-plane stresses inside a controlled laboratory environment but struggle to be robust outside, in a real-world setting. That is because these methods rely either on knowing beforehand the material constants (which are difficult to acquire) or require significant calibration for each specimen. This paper presents an ultrasonic method to evaluate the in-plane stress in situ directly, without knowing any material constants. The method is simple in principle, as it only requires measuring the speed of two angled shear waves. It is based on a formula that is exact for incompressible solids, such as soft gels or tissues, and is approximately true for compressible "hard" solids, such as steel and other metals. The formula is validated by finite element simulations, showing that it displays excellent accuracy, with a small error on the order of 1%.

Why Are Viscosity and Nonlinearity Bound to Make an Impact in Clinical Elastographic Diagnosis?

The adoption of multiscale approaches by the biomechanical community has caused a major improvement in quality in the mechanical characterization of soft tissues. The recent developments in elastography techniques are enabling in vivo and non-invasive quantification of tissues' mechanical properties. Elastic changes in a tissue are associated with a broad spectrum of pathologies, which stems from the tissue microstructure, histology and biochemistry. This knowledge is combined with research evidence to provide a powerful diagnostic range of highly prevalent pathologies, from birth and labor disorders (prematurity, induction failures, etc.), to solid tumors (e.g., prostate, cervix, breast, melanoma) and liver fibrosis, just to name a few. This review aims to elucidate the potential of viscous and nonlinear elastic parameters as conceivable diagnostic mechanical biomarkers. First, by providing an insight into the classic role of soft tissue microstructure in linear elasticity; secondly, by understanding how viscosity and nonlinearity could enhance the current diagnosis in elastography; and finally, by compounding preliminary investigations of those elastography parameters within different technologies. In conclusion, evidence of the diagnostic capability of elastic parameters beyond linear stiffness is gaining momentum as a result of the technological and imaging developments in the field of biomechanics.

Analyzing acoustoelastic effect of shear wave elastography data for perfused and hydrated soft tissues using a macromolecular network inspired model

Shear wave elastography (SWE) has enhanced our ability to non-invasively make in vivo measurements of tissue elastic properties of animal and human tissues. Recently, researchers have taken advantages of acoustoelasticity in SWE to extract nonlinear elastic properties from soft biological tissues. However, most investigations of the acoustoelastic effects of SWE data (AE-SWE) rely on classic hyperelastic models for rubber-like (dry) materials. In this paper, we focus solely on understanding acoustoelasticity in soft hydrated tissues using SWE data and propose a straightforward approach to modeling the constitutive behavior of soft tissue that has a direct microstructural/macromolecular interpretation. Our approach incorporates two constitutive features relevant to biological tissues into AE-SWE: static dilation of the medium associated with nonstructural components (e.g. tissue hydration and perfusion) and finite extensibility derived from an ideal network of biological filaments. We evaluated the proposed method using data from an in-house tissue-mimicking phantom experiment, and ex vivo and in vivo AE-SWE data available in the SWE literature. In conclusion, predictions made by our approach agreed well with measurements obtained from phantom, ex vivo and in vivo tissue experiments.

Development of Phantom Material that Resembles Compression Properties of Human Brain Tissue for Training Models

There is a need to quantify and reproduce the mechanical behavior of brain tissue for a variety of applications from designing proper training models for surgeons to enabling research on the effectiveness of personal protective gear, such as football helmets. The mechanical response of several candidate phantom materials, including hydrogels and emulsions, was characterized and compared to porcine brain tissue under similar strains and strain rates. Some candidate materials were selected since their compositions were similar to brain tissue, such as emulsions that mimic the high content of lipids. Others, like silicone, were included since these are currently used as phantom materials. The mechanical response of the emulsion was closer to that of the native porcine brain tissue than the other candidates. The emulsions, created by addition of oil to a hydrogel, were able to withstand compressive strain greater than 40%. The addition of lipids in the emulsions also prevented the syneresis typically seen with hydrogel materials. This allowed the emulsion material to undergo freeze-thaw cycles with no significant change in their mechanical properties.

Spectral Quantification of Nonlinear Elasticity Using Acoustoelasticity and Shear-Wave Dispersion

Tissue biomechanical properties are known to be sensitive to pathological changes. Accordingly, various techniques have been developed to estimate tissue mechanical properties. Shear-wave elastography (SWE) measures shear-wave speed (SWS) in tissues, which can be related to shear modulus. Although viscosity or stress-strain nonlinearity may act as confounder of SWE, their explicit characterization may also provide additional information about tissue composition as a contrast modality. Viscosity can be related to frequency dispersion of SWS, which can be characterized using multi-frequency measurements, herein called spectral SWE (SSWE). Additionally, nonlinear shear modulus can be quantified and parameterized based on SWS changes with respect to applied stress, a phenomenon called acoustoelasticity (AE). In this work, we characterize the nonlinear parameters of tissue as a function of excitation frequency by utilizing both AE and SSWE together. For this, we apply incremental amounts of quasi-static stress on a medium, while imaging and quantifying SWS dispersion via SSWE. Results from phantom and ex vivo porcine liver experiments demonstrate the feasibility of measuring frequency-dependent nonlinear parameters using the proposed method. SWS propagation in porcine liver tissue was observed to change from 1.8 m/s at 100 Hz to 3.3 m/s at 700 Hz, while increasing by approximately 25% from a strain of 0% to 12% across these frequencies.

Generalization of the Zabolotskaya equation to all incompressible isotropic elastic solids

We study elastic shear waves of small but finite amplitude, composed of an anti-plane shear motion and a general in-plane motion. We use a multiple scales expansion to derive an asymptotic system of coupled nonlinear equations describing their propagation in all isotropic incompressible nonlinear elastic solids, generalizing the scalar Zabolotskaya equation of compressible nonlinear elasticity. We show that for a general isotropic incompressible solid, the coupling between anti-plane and in-plane motions cannot be undone and thus conclude that linear polarization is impossible for general nonlinear two-dimensional shear waves. We then use the equations to study the evolution of a nonlinear Gaussian beam in a soft solid: we show that a pure (linearly polarized) shear beam source generates only odd harmonics, but that introducing a slight in-plane noise in the source signal leads to a second harmonic, of the same magnitude as the fifth harmonic, a phenomenon recently observed experimentally. Finally, we present examples of some special shear motions with linear polarization.

Ex Vivo Evaluation of Mouse Brain Elasticity Using High-Frequency Ultrasound Elastography

OBJECTIVE

Most neurodegenerative diseases are highly linked with aging. The mechanical properties of the brain should be determined for predicting and diagnosing age-related brain diseases. A preclinical animal study is crucial for neurological disease research. However, estimation of the elasticity properties of different regions of mouse brains remains difficult because of the size of the brain. In this paper, high-frequency ultrasound elastography (HFUSE) based on shear wave imaging was proposed for mapping the stiffness of the mouse brain at different ages ex vivo.

METHODS

For HFUSE, a 40-MHz ultrasound array transducer with an ultrafast ultrasound imaging system was used in this paper. The accuracy and resolution during HFUSE were determined through a mechanical testing system and by conducting phantom experiments.

RESULTS

In the experiments, the error in the elastic modulus measurement was approximately 10% on average, and the axial resolution was 248 μm. Animal testing was conducted using mice that were 4 (young aged) and 11 (middle aged) months old. The elasticity distributions of the cortex and hippocampus in the mouse brains were obtained through HFUSE.

CONCLUSION

The average shear moduli of the cortex and hippocampus were 3.84 and 2.33 kPa for the 4-month-old mice and 3.77 and 1.94 kPa for the 11-month-old mice, respectively. No statistical difference was observed in the cortex stiffness of mice of different ages. However, the hippocampus significantly softened with aging.

A comparison of hyperelastic constitutive models applicable to shear wave elastography (SWE) data in tissue-mimicking materials

Shear wave elastography (SWE) techniques have received substantial attention in recent years. Strong experimental data in SWE suggest that shear wave speed changes significantly due to the known acoustoelastic effect (AE). This presents both challenges and opportunities toward in vivo characterization of biological soft tissues. In this work, under the framework of continuum mechanics, we model a tissue-mimicking material as a homogeneous, isotropic, incompressible, hyperelastic material. Our primary objective is to quantitatively and qualitatively compare experimentally measured acoustoelastic data with model-predicted outcomes using multiple strain energy functions. Our analysis indicated that the classic Neo-Hookean and Mooney-Rivlin models are inadequate for modeling the AE in tissue-mimicking materials. However, a subclass of strain energy functions containing both high-order/exponential term(s) and second-order invariant dependence showed good agreement with experimental data. Based on data investigated, we also found that discrepancies may exist between parameters inversely estimated from uniaxial compression and SWE data. Overall, our findings may improve our understanding of clinical SWE results.

Indentation response in porcine brain under electric fields

Electric fields in the environment can have profound effects on brain function and behavior. In clinical practice, some noninvasive/microinvasive therapies with electrical fields such as transcranial electrical stimulation (tES), deep brain stimulation (DBS), and electroconvulsive therapy (ECT) have emerged as powerful tools for the treatment of neuropsychiatric disorders and neuromodulation. Nonetheless, currently, most studies focus on the mechanisms and effects of therapies and do not to address the mechanical properties of brain tissue under electric fields. Thus, the mechanical behavior of brain tissue, which plays an important role in modulating both brain form and brain function, should be given attention. The present study addresses this paucity by presenting, for the first time, the mechanical properties of brain tissue under various intensities of direct current electric field (0, 2, 5, 10, 20, and 50 V) using a custom-designed indentation device. Prior to brain indentation, validation tests were performed in different hydrogels to ensure that there was no interference in the electric fields from the indentation device. Subsequently, the load trace data obtained from the indentation-relaxation tests was fitted to both linear elastic and viscoelastic models to characterize the sensitivity of the mechanical behavior of the brain tissue to the electric fields. The brain tissue was found to be softened at a higher electric field level and less viscous, and substantially responded more quickly with an increase in electric field. The explanations for the above behaviors were further discussed based on the analysis of the resistance and thermal responses during the testing process. Understanding the effect of electric fields on brain tissue at the mechanical level can provide a better understanding of the mechanisms of some therapies, which may be beneficial to guide therapy protocols.

Identification of the visco-hyperelastic properties of brain white matter based on the combination of inverse method and experiment

To fully understand the brain injury mechanism and develop effective protective approaches, an accurate constitutive model of brain tissue is firstly required. Generally, the brain tissue is regarded as a kind of viscoelastic material and is simply used in the simulation of brain injury. In fact, the brain tissue has the behavior of the visco-hyperelastic property. Therefore, this paper presents an effective computational inverse method to determine the material parameters of visco-hyperelastic constitutive model of brain white matter through compression experiments. First, with the help of 3D hand scanner, 3D geometries of brain white matter specimens are obtained to make it possible to establish the accurate simulation models of the specific specimens. Then, the global sensitivity analysis is adopted to evaluate the importance of the material parameters and further determine the parameters which may be identified. Subsequently, based on the genetic algorithm, the optimal material parameters of brain white matter can be identified by minimizing the match error between the experimental and simulated responses. Finally, by comparing the experiment and simulation results on the other specific specimen, and the simulation results with the material parameters from the references, respectively, the accuracy and reliability of the constitutive model parameters of brain white matter are demonstrated. Graphical abstract The main flowchart of the computational inverse technique for determining the material parameters of specimen-specific on brain white matter. Generalization: Combining the computational inverse method and unconfined uniaxial compression experiment of the specific specimen, an effective identification method is presented to accurately determine the hyperelastic and viscoelastic parameters of brain white matter in this paper.

Fourier-Domain Shift Matching: A Robust Time-of-Flight Approach for Shear Wave Speed Estimation

Our primary objective of this work was to design and test a new time-of-flight (TOF) method that allows measurements of shear wave speed (SWS) following impulsive excitation in soft tissues. Particularly, under the assumption of the local plane shear wave, this work named the Fourier-domain shift matching (FDSM) method, estimates SWS by aligning a series of shear waveforms either temporally or spatially using a solution space deduced by characteristic curves of the well-known 1-D wave equation. The proposed SWS estimation method was tested using computer-simulated data, and tissue-mimicking phantom and ex vivo tissue experiments. Its performance was then compared with three other known TOF methods: lateral time-to-peak (TTP) method with robust random sampling consensus (RANSAC) fitting method, Radon sum transformation method, and a modified cross correlation method. Hereafter, these three TOF methods are referred to as the TTP-RANSAC, Radon sum, and X-corr methods, respectively. In addition to an adapted form of the 2-D Fourier transform (2-D FT)-based method in which the (group) SWS was approximated by averaging phase SWS values was considered for comparison. Based on data evaluated, we found that the overall performance of the above-mentioned temporal implementation of the proposed FDSM method was most similar to the established Radon sum method (correlation = 0.99, scale factor = 1.03, and mean difference = 0.07 m/s), and the 2-D FT (correlation = 0.98, scale factor = 1.00, and mean difference = 0.10 m/s) at high signal quality. However, results obtained from the 2-D FT method diverged (correlation = 0.201) from these of the proposed temporal implementation in the presence of diminished signal quality, whereas the agreement between the Radon sum approach and the proposed temporal implementation largely remained the same (correlation = 0.98).

Simulation of nonlinear transient elastography: finite element model for the propagation of shear waves in homogeneous soft tissues

In this study, visco-hyperelastic Landau's model, which is widely used in acoustical physic field, is introduced into a finite element formulation. It is designed to model the nonlinear behaviour of finite amplitude shear waves in soft solids, typically, in biological tissues. This law is used in finite element models based on elastography, experiments reported in Jacob et al, the simulations results show a good agreement with the experimental study: It is observed in both that a plane shear wave generates only odd harmonics and a nonplane wave generates both odd and even harmonics in the spectral domain. In the second part, a parametric study is performed to analyse the influence of different factors on the generation of odd harmonics of plane wave. A quantitative relation is fitted between the odd harmonic amplitudes and the non-linear elastic parameter of Landau's model, which provides a practical guideline to identify the non-linearity of homogeneous tissues using elastography experiment.

Assessing the mechanical properties of anisotropic soft tissues using guided wave elastography: Inverse method and numerical experiments

Determining the mechanical properties of soft biological tissues can be of great importance. For example, the microstructures of many soft tissues, such as those of the human Achilles tendon, have been identified as typical anisotropic materials. This paper proposes an inverse approach that uses guided wave elastography to determine the anisotropic elastic and hyperelastic parameters of thin-walled transversely isotropic biological soft tissues. This approach was developed from the theoretical solutions for the dispersion relations of guided waves, which were derived based on a constitutive model suitable for describing the deformation behavior of such tissues. The properties of these solutions were investigated; in particular, sensitivity to data errors was addressed by introducing the concept of the condition number. To further validate the proposed inverse approach, the guided wave elastography of thin-walled transversely isotropic soft tissues was investigated using numerical experiments. The results indicated that the four constitutive parameters (other than the tensile modulus along the direction of the fibers, E_L) could be determined with a good level of accuracy using this method.

ECM Mechano-Sensing Regulates Cytoskeleton Assembly and Receptor-Mediated Endocytosis of Nanoparticles

It is possible to create sophisticated and target-specific devices for nanomedicine thanks to technological advances in the engineering of nanomaterials. When on target, these nanocarriers often have to be internalized by cells in order to accomplish their diagnostic or therapeutic function. Therefore, the control of such uptake mechanism by active targeting strategy has today become the new challenge in nanoparticle designing. It is also well-known that cells are able to sense and respond to the local physical environment and that the substrate stiffness, and not only the nanoparticle design, influences the cellular internalization mechanisms. In this frame, our work reports on the cyclic relationship among substrate stiffness, cell cytoskeleton assembly and internalization mechanism. Nanoparticles uptake has been investigated in terms of the mechanics of cell environment, the resulting cytoskeleton activity and the opportunity of activate molecular specific molecular pathways during the internalization process. To this aim, the surface of 100 nm polystyrene nanoparticles was decorated with a tripeptide (RGD and a scrambled version as a control), which was able to activate an internalization pathway directly correlated to the dynamics of the cell cytoskeleton, in turn, directly correlated to the elastic modulus of the substrates. We found that the substrate stiffness modulates the uptake of nanoparticles by regulating structural parameters of bEnd.3 cells as spreading, volume, focal adhesion, and mechanics. In fact, the nanoparticles were internalized in larger amounts both when decorated with RGD, which activated an internalization pathway directly correlated to the cell cytoskeleton, and when cells resided on stiffer material that, in turn, promoted the formation of a more structured cytoskeleton. This evidence indicates the directive role of the mechanical environment on cellular uptake of nanoparticles, contributing new insights to the rational design and development of novel nanocarrier systems.

Mechanics of ultrasound elastography

Ultrasound elastography enables in vivo measurement of the mechanical properties of living soft tissues in a non-destructive and non-invasive manner and has attracted considerable interest for clinical use in recent years. Continuum mechanics plays an essential role in understanding and improving ultrasound-based elastography methods and is the main focus of this review. In particular, the mechanics theories involved in both static and dynamic elastography methods are surveyed. They may help understand the challenges in and opportunities for the practical applications of various ultrasound elastography methods to characterize the linear elastic, viscoelastic, anisotropic elastic and hyperelastic properties of both bulk and thin-walled soft materials, especially the in vivo characterization of biological soft tissues.

A viscoelastic analysis of the P56 mouse brain under large-deformation dynamic indentation

The brain is a complex organ made up of many different functional and structural regions consisting of different types of cells such as neurons and glia, as well as complex anatomical geometries. It is hypothesized that the different regions of the brain exhibit significantly different mechanical properties which may be attributed to the diversity of cells within individual brain regions. The regional viscoelastic properties of P56 mouse brain tissue, up to 70μm displacement, are presented and discussed in the context of traumatic brain injury, particularly how the different regions of the brain respond to mechanical loads. Force-relaxation data obtained from micro-indentation measurements were fit to both linear and quasi-linear viscoelastic models to determine the time and frequency domain viscoelastic response of the pons, cortex, medulla oblongata, cerebellum, and thalamus. The damping ratio of each region was also determined. Each region was found to have a unique mechanical response to the applied displacement, with the pons and thalamus exhibiting the largest and smallest force-response, respectively. All brain regions appear to have an optimal frequency for the dissipation of energies which lies between 1 and 10Hz.

STATEMENT OF SIGNIFICANCE

We present the first mechanical characterization of the viscoelastic response for different regions of mouse brain. Force-relaxation tests are performed under large strain dynamic micro-indentation, and viscoelastic models are used subsequently, providing time-dependent mechanical properties of brain tissue under loading conditions comparable to what is experienced in TBI. The unique mechanical properties of different brain regions are highlighted, with substantial variations in the viscoelastic properties and damping ratio of each region. Cortex and pons were the stiffest regions, while the thalamus and medulla were most compliant. The cerebellum and thalamus had highest damping ratio values and those of the medulla were lowest. The reported material parameters can be implemented into finite element computer models of the mouse to investigate the effects of trauma on individual brain regions.

Characterizing white matter tissue in large strain via asymmetric indentation and inverse finite element modeling

Characterizing the mechanical properties of white matter is important to understand and model brain development and injury. With embedded aligned axonal fibers, white matter is typically modeled as a transversely isotropic material. However, most studies characterize the white matter tissue using models with a single anisotropic invariant or in a small-strain regime. In this study, we combined a single experimental procedure - asymmetric indentation - with inverse finite element (FE) modeling to estimate the nearly incompressible transversely isotropic material parameters of white matter. A minimal form comprising three parameters was employed to simulate indentation responses in the large-strain regime. The parameters were estimated using a global optimization procedure based on a genetic algorithm (GA). Experimental data from two indentation configurations of porcine white matter, parallel and perpendicular to the axonal fiber direction, were utilized to estimate model parameters. Results in this study confirmed a strong mechanical anisotropy of white matter in large strain. Further, our results suggested that both indentation configurations are needed to estimate the parameters with sufficient accuracy, and that the indenter-sample friction is important. Finally, we also showed that the estimated parameters were consistent with those previously obtained via a trial-and-error forward FE method in the small-strain regime. These findings are useful in modeling and parameterization of white matter, especially under large deformation, and demonstrate the potential of the proposed asymmetric indentation technique to characterize other soft biological tissues with transversely isotropic properties.

Numerical Simulation of Focused Shock Shear Waves in Soft Solids and a Two-Dimensional Nonlinear Homogeneous Model of the Brain

Shear waves that propagate in soft solids, such as the brain, are strongly nonlinear and can develop into shock waves in less than one wavelength. We hypothesize that these shear shock waves could be responsible for certain types of traumatic brain injuries (TBI) and that the spherical geometry of the skull bone could focus shear waves deep in the brain, generating diffuse axonal injuries. Theoretical models and numerical methods that describe nonlinear polarized shear waves in soft solids such as the brain are presented. They include the cubic nonlinearities that are characteristic of soft solids and the specific types of nonclassical attenuation and dispersion observed in soft tissues and the brain. The numerical methods are validated with analytical solutions, where possible, and with self-similar scaling laws where no known solutions exist. Initial conditions based on a human head X-ray microtomography (CT) were used to simulate focused shear shock waves in the brain. Three regimes are investigated with shock wave formation distances of 2.54 m, 0.018 m, and 0.0064 m. We demonstrate that under realistic loading scenarios, with nonlinear properties consistent with measurements in the brain, and when the shock wave propagation distance and focal distance coincide, nonlinear propagation can easily overcome attenuation to generate shear shocks deep inside the brain. Due to these effects, the accelerations in the focal are larger by a factor of 15 compared to acceleration at the skull surface. These results suggest that shock wave focusing could be responsible for diffuse axonal injuries.