Shear wave imaging the active constitutive parameters of living muscles

Shear wave elastography (SWE) of human skeletal muscles allows for measurement of muscle elastic properties in vivo and has important applications in sports medicine and for the diagnosis and treatment of muscle-related diseases. Existing methods of SWE for skeletal muscles rely on the passive constitutive theory and have so far been unable to provide constitutive parameters describing muscle active behavior. In the present paper, we overcome this limitation by proposing a SWE method for quantitative inference of active constitutive parameters of skeletal muscles in vivo. To this end, we investigate the wave motion in a skeletal muscle described by a constitutive model in which muscle active behavior has been defined by an active parameter. An analytical solution relating shear wave velocities to both passive and active material parameters of muscles is derived, based upon which an inverse approach has been developed to evaluate these parameters. To demonstrate the usefulness of the reported method, in vivo experiments were carried out on 10 volunteers to obtain constitutive parameters, particularly those describing active deformation behaviors of living muscles. The results reveal that the active material parameter of skeletal muscles varies with warm-up, fatigue and rest. STATEMENT OF SIGNIFICANCE: Existing shear wave elastography methods are limited to imaging the passive parameters of muscles. This limitation is addressed in the present paper by developing a method to image the active constitutive parameter of living muscles using shear waves. We derived an analytical solution demonstrating the relationship between constitutive parameters of living muscles and shear waves. Relying on the analytical solution, we proposed an inverse method to infer active parameter of skeletal muscles. We performed in vivo experiments to demonstrate the usefulness of the theory and method; the quantitative variation of the active parameter with muscle states such as warm-up, fatigue and rest has been reported for the first time.

Noninvasive measurement of local stress inside soft materials with programmed shear waves

Mechanical stresses across different length scales play a fundamental role in understanding biological systems' functions and engineering soft machines and devices. However, it is challenging to noninvasively probe local mechanical stresses in situ, particularly when the mechanical properties are unknown. We propose an acoustoelastic imaging-based method to infer the local stresses in soft materials by measuring the speeds of shear waves induced by custom-programmed acoustic radiation force. Using an ultrasound transducer to excite and track the shear waves remotely, we demonstrate the application of the method by imaging uniaxial and bending stresses in an isotropic hydrogel and the passive uniaxial stress in a skeletal muscle. These measurements were all done without the knowledge of the constitutive parameters of the materials. The experiments indicate that our method will find broad applications, ranging from health monitoring of soft structures and machines to diagnosing diseases that alter stresses in soft tissues.

Probing the Mechanical Properties of Large Arteries by Measuring Their Deformation In Vivo with Ultrasound

Aging and cardiovascular diseases (CVDs) may alter the microstructures of arteries and hence their mechanical properties. Therefore, the measurement of intrinsic artery mechanical properties in vivo can provide valuable information in understanding aging and CVDs and is of clinical significance. The accuracy of advanced ultrasound imaging techniques in measuring the deformation of large arteries under blood pressure is good. However, the assessment of arterial stiffness in vivo remains a challenge. An inverse method to infer the constitutive parameters of arteries in vivo from the blood pressure-arterial radius relationship (P-r curve) is proposed here. The stability analysis reveals that a key constitutive parameter, b_θ, which measures the circumferential hardening of an artery, can be reliably identified. An in vivo experiment was performed on the common carotid arteries of 41 healthy volunteers (age: 37 ± 17 y). The value of b_θ varies significantly (from 0.55 ± 0.15 for the young group to 0.93 ± 0.29 for the older group, p < 0.01) and is positively correlated with age (r = 0.673, p < 0.01). Furthermore, our theoretical analysis and experimental study have revealed a strong correlation between the clinic-used stiffness index β and b_θ. This study shows that the arterial material parameter b_θ can be measured in vivo, which makes it promising as a new biomarker in the diagnosis of CVDs.

Self-Organization of Tissue Growth by Interfacial Mechanical Interactions in Multilayered Systems

Morphogenesis is a spatially and temporally regulated process involved in various physiological and pathological transformations. In addition to the associated biochemical factors, the physical regulation of morphogenesis has attracted increasing attention. However, the driving force of morphogenesis initiation remains elusive. Here, it is shown that during the growth of multilayered tissues, a morphogenetic process can be self-organized by the progression of compression gradient stemmed from the interfacial mechanical interactions between layers. In tissues with low fluidity, the compression gradient is progressively strengthened during growth and induces stratification by triggering symmetric-to-asymmetric cell division reorientation at the critical tissue size. In tissues with high fluidity, compression gradient is dynamic and induces cell rearrangement leading to 2D in-plane morphogenesis instead of 3D deformation. Morphogenesis can be tuned by manipulating tissue fluidity, cell adhesion forces, and mechanical properties to influence the progression of compression gradient during the development of cultured cell sheets and chicken embryos. Together, the dynamics of compression gradient arising from interfacial mechanical interaction provides a conserved mechanism underlying morphogenesis initiation and size control during tissue growth.

AFM-based indentation method for measuring the relaxation property of living cells

Probing the mechanical properties of cells is critical for understanding their deformation behaviors and biological functions. Although some methods have been proposed to characterize the elastic properties of cells, it is still difficult to measure their time-dependent properties. This paper investigates the use of atomic force microscope (AFM) to determine the reduced relaxation modulus of cells. In principle, AFM is hard to perform an indentation relaxation test that requires a constant indenter displacement during load relaxation, whereas the real AFM indenter displacement usually varies with time during relaxation due to the relatively small bending stiffness of its cantilever. We investigate this issue through a combined theoretical, computational, and experimental effort. A protocol relying on the choice of appropriate cantilever bending stiffness is proposed to perform an AFM-based indentation relaxation test of cells, which enables the measurement of reduced relaxation modulus with high accuracy. This protocol is first validated by performing nanoindentation relaxation tests on a soft material and by comparing the results with those from independent measurements. Then indentation tests of cartilage cells are conducted to demonstrate this method in determining time-dependent properties of living cells. Finally, the change in the viscoelasticity of MCF-7 cells under hyperthermia is investigated.

Mechanical characterization of functionally graded soft materials with ultrasound elastography

Functionally graded soft materials (FGSMs) with microstructures and mechanical properties exhibiting gradients across a spatial volume to satisfy specific functions have received interests in recent years. How to characterize the mechanical properties of these FGSMs in vivo/in situ and/or in a non-destructive manner is a great challenge. This paper investigates the use of ultrasound elastography in the mechanical characterization of FGSMs. An efficient finite-element model was built to calculate the dispersion relation for surface waves in FGSMs. For FGSMs with large elastic gradients, the measured dispersion relation can be used to identify mechanical parameters. In the case where the elastic gradient is smaller than a certain critical value calculated here, our analysis on transient wave motion in FGSMs shows that the group velocities measured at different depths can infer the local mechanical properties. Experiments have been performed on polyvinyl alcohol (PVA) cryogel to demonstrate the usefulness of the method. Our analysis and the results may not only find broad applications in mechanical characterization of FGSMs but also facilitate the use of shear wave elastography in clinics because many diseases change the local elastic properties of soft tissues and lead to different material gradients. This article is part of the theme issue 'Rivlin's legacy in continuum mechanics and applied mathematics'.

Guided wave elastography of layered soft tissues

In vivo mechanical characterization of soft biological tissues has broad applications ranging from disease diagnosis to tissue engineering. Shear wave elastography based on the bulk wave theory has been widely used to measure the mechanical properties of soft tissues. Given that most soft tissues basically have layered structures, the dispersive feature of elastic waves should be considered when the thickness of the interested layer is comparable to or smaller than the wavelength. Bearing this fundamental issue in mind, we propose an ultrasound-based guided wave elastography (GWE) method to characterize the mechanical properties of layered soft tissues. The dispersion relations of guided waves in layered structures were derived first, and its explicit expression was achieved. An inverse approach based on the dispersion relation to characterize the mechanical properties of layered soft tissues was then established. Both finite element analysis (FEA) and phantom experiments were carried out to validate the new method. In vivo experiments on forearm skin demonstrate the usefulness of the present method in characterizing layered soft tissues. STATEMENT OF SIGNIFICANCE: Layered soft tissues and artificial soft materials are ubiquitous in both nature and engineering. Imaging their in vivo/in situ mechanical properties finds important applications and remains a great challenge to date. Here, we propose an ultrasound-based guided wave elastography method to in vivo/in situ characterize the elastic properties of layered soft materials. We validate the method via finite element analysis and phantom experiments and further demonstrate its usefulness in practice by performing in vivo measurements on forearm skins. Given that the dispersive feature of elastic waves in layered soft media is considered in our method, it provides the opportunity to assess the intrinsic elastic properties of an individual layer in a non-destructive manner as shown in our experiments.

An Inverse Method to Determine Arterial Stiffness with Guided Axial Waves

Many cardiovascular diseases can alter arterial stiffness; therefore, measurement of arterial wall stiffness can provide valuable information for both diagnosis of such diseases in the clinic and evaluation of the effectiveness of relevant drugs. However, quantitative assessment of the in vivo elastic properties of arterial walls in a non-invasive manner remains a great challenge. In this study, we found that the elastic modulus of the arterial wall can be extracted from the dispersion curve of the guided axial wave (GAW) measured using the ultrasound elastography method. It is shown that the GAW in the arterial wall can be well described with the Lamb wave (LW) model when the frequency exceeds a critical value f_c, whose explicit form is determined here based on dimensional analysis method and systematic finite-element simulations. Further, an inverse procedure is proposed to determine both f_c and the elastic modulus of the arterial wall. Phantom experiments have been performed to validate the inverse method and illustrate its potential use in the clinic.

An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves

Arterial stiffness is highly correlated with the functions of the artery and may serve as an important diagnostic criterion for some cardiovascular diseases. To date, it remains a challenge to quantitatively assess local arterial stiffness in a non-invasive manner. To address this challenge, we investigated the possibility of determining arterial stiffness using the guided circumferential wave (GCW) induced in the arterial wall by a focused acoustic radiation force. The theoretical model for the dispersion analysis of the GCW is presented, and a finite element model has been established to calculate the dispersion curve. Our results show that under described conditions, the dispersion relations of the GCW are basically independent of the curvature of the arterial wall and can be well-described using the Lamb wave (LW) model. Based on this conclusion, an inverse method is proposed to characterize the elastic modulus of artery. Both numerical experiments and phantom experiments had been performed to validate the proposed method. We show that our method can be applied to the cases in which the artery has local stenosis and/or the geometry of the artery cross-section is irregular; therefore, this method holds great potential for clinical use.

Curvature induced hierarchical wrinkling patterns in soft bilayers

Sinusoidal wrinkling will occur in a planar film-substrate bilayer when the uniaxial compressive strain imposed to the system exceeds a critical value. However, when a core-shell soft cylinder is subjected to axial compression, surface wrinkling patterns may evolve from the sinusoidal mode to the diamond-like mode, depending on the modular ratio and the curvature of the system. Inspired by this phenomenon, we here propose a simple yet robust strategy to fabricate hierarchical wrinkling patterns by controlling the curvature of a film-substrate system. To quantitatively understand the experimental results, a three-dimensional finite element model has been built to track the wrinkling pattern evolution. Furthermore, a phase diagram is provided based on the theoretical analysis and finite element simulations, which may guide the experimental design. In addition, the wetting properties of the surface with hierarchical micropatterns fabricated using the proposed method are investigated. The results show that the hierarchical surface wrinkles lead to anisotropic wetting behavior, which can be tuned by controlling the imposed compressive strain. The tunable anisotropic wetting surface fabricated here may find a broad range of applications such as in the development of sensors, fluidic devices, micro-reactors and biomedical devices.

Controlling elastic wave propagation in a soft bilayer system via wrinkling-induced stress patterns

Compression of a film/substrate bilayer system with different surface/interfacial structures can lead to diverse buckling patterns including sinusoidal wrinkles, ridges, folds, creases and tilted sawteeth wrinkles. In this paper, we show that elastic wave band gaps in the film/substrate bilayer system largely depend on the wrinkling patterns. More interestingly, we find that different wrinkling patterns investigated here can coexist and evolve in one bilayer system and the elastic wave propagation behaviors can be controlled by manipulating the hybrid wrinkling patterns. Our analysis also reveals that the periodic stress pattern plays a dominant role in tuning the bandgap structures in comparison to geometrical patterns caused by surface instability. A careful investigation of the transmission spectra of the composite systems has validated the main findings given by the analysis based on the Bloch wave theory. Potential use of the method and materials reported here to gain wide attenuation frequency ranges and the design of nesting Fibonacci superlattices have been demonstrated.

Buckling of an elastic fiber with finite length in a soft matrix

Elastic fibers embedded in a soft matrix are frequently encountered in nature and engineering across different length scales, ranging from microtubules in cytosol and filament networks to dissociative slender fish bones in muscles and fiber-reinforced soft composites. Fibers may buckle when the composite is subjected to compression; this study investigates this issue through a combination of experiments, finite-element simulations and theoretical analysis. Analysis reveals the important role of the interfacial shear forces and leads to an explicit solution to predict the occurrence of buckling for a slender fiber with finite length. The results reported in this paper will help understand the formation of shapes in some natural systems and provide guidelines for the design of soft biocomposites.

Characterization of the nonlinear elastic properties of soft tissues using the supersonic shear imaging (SSI) technique: inverse method, ex vivo and in vivo experiments

Dynamic elastography has become a new clinical tool in recent years to characterize the elastic properties of soft tissues in vivo, which are important for the disease diagnosis, e.g., the detection of breast and thyroid cancer and liver fibrosis. This paper investigates the supersonic shear imaging (SSI) method commercialized in recent years with the purpose to determine the nonlinear elastic properties based on this promising technique. Particularly, we explore the propagation of the shear wave induced by the acoustic radiation force in a stressed hyperelastic soft tissue described via the Demiray-Fung model. Based on the elastodynamics theory, an analytical solution correlating the wave speed with the hyperelastic parameters of soft tissues is first derived. Then an inverse approach is established to determine the hyperelastic parameters of biological soft tissues based on the measured wave speeds at different stretch ratios. The property of the inverse method, e.g., the existence, uniqueness and stability of the solution, has been investigated. Numerical experiments based on finite element simulations and the experiments conducted on the phantom and pig livers have been employed to validate the new method. Experiments performed on the human breast tissue and human heel fat pads have demonstrated the capability of the proposed method for measuring the in vivo nonlinear elastic properties of soft tissues. Generalization of the inverse analysis to other material models and the implication of the results reported here for clinical diagnosis have been discussed.