Dilatation parameterization for two-dimensional modeling of nearly incompressible isotropic materials

A new method to model the stress–strain relationship in two dimensions is proposed, which is particularly suited for analyzing nearly incompressible materials, such as soft tissue. In most cases of soft tissue modeling, plane strain is reported to approximate the deformation when an external compression is applied. However, it is subject to limitations when dealing with incompressible materials, e.g., when solving the inverse problem of elasticity. We propose a novel 2D model for the linear stress–strain relationship by describing the out-of-plane strain as a linear combination of the two in-plane strains. As such, the model can be represented in 2D while being able to explain the three-dimensional deformation. We show that in simple cases where the applied force is dominantly in one direction, one can approximate the sum of the three principal strain components in a plane by a scalar multiplied by the out-of-plane strain. 3D finite-element simulations have been performed. The proposed model has been tested under different boundary conditions and material properties. The results show that the model parametrization is affected mostly by the boundary conditions, while being relatively independent of the underlying distribution of Young's modulus. An application to the inverse problem of elasticity is presented where a more accurate estimate is obtained using the proposed dilatation model compared to the plane-stress and plane-strain models.     

Elastic modulus imaging: some exact solutions of the compressible elastography inverse problem

We consider several inverse problems motivated by elastography. Given the (possibly transient) displacement field measured everywhere in an isotropic, compressible, linear elastic solid, and given density rho, determine the Lamé parameters lambda and mu. We consider several special cases of this problem: (a) for mu known a priori, lambda is determined by a single deformation field up to a constant. (b) Conversely, for lambda known a priori, mu is determined by a single deformation field up to a constant. This includes as a special case that for which the term [see text]. (c) Finally, if neither lambda nor mu is known a priori, but Poisson's ratio nu is known, then mu and lambda are determined by a single deformation field up to a constant. This includes as a special case plane stress deformations of an incompressible material. Exact analytical solutions valid for 2D, 3D and transient deformations are given for all cases in terms of quadratures. These are used to show that the inverse problem for mu based on the compressible elasticity equations is unstable in the limit lambda --> infinity. Finally, we use the exact solutions as a basis to compute non-trivial modulus distributions in a simulated example.     

Sparsity regularization in dynamic elastography

We consider the inverse problem of continuum mechanics with the tissue deformation described by a mixed displacement-pressure finite element formulation. The mixed formulation is used to model nearly incompressible materials by simultaneously solving for both elasticity and pressure distributions. To improve numerical conditioning, a common solution to this problem is to use regularization to constrain the solutions of the inverse problem. We present a sparsity regularization technique that uses the discrete cosine transform to transform the elasticity and pressure fields to a sparse domain in which a smaller number of unknowns is required to represent the original field. We evaluate the approach by solving the dynamic elastography problem for synthetic data using such a mixed finite element technique, assuming time harmonic motion, and linear, isotropic and elastic behavior for the tissue. We compare our simulation results to those obtained using the more common Tikhonov regularization. We show that the sparsity regularization is less dependent on boundary conditions, less influenced by noise, requires no parameter tuning and is computationally faster. The algorithm has been tested on magnetic resonance elastography data captured from a CIRS elastography phantom with similar results as the simulation.     

Shear modulus reconstruction in dynamic elastography: time harmonic case

This paper presents a direct inversion approach for reconstructing the elastic shear modulus in soft tissue from dynamic measurements of the interior displacement field during time harmonic excitation. The tissue is assumed to obey the equations of nearly incompressible, linear, isotropic elasto-dynamics in harmonic motion. A finite element discretization of the governing equations is used as a basis, and a procedure is outlined to eliminate the need for boundary conditions in the inverse problem. The hydrostatic stress (pressure) is also reconstructed in the process, and the effect of neglecting this term in the governing equations, which is common practice, is considered. The approach does not require iterations and can be performed on sub-regions of the domain resulting in a computationally efficient method. A sensitivity study is performed to investigate the detectability of abnormal regions of different size and shear modulus contrast from the background. The algorithm is tested on simulated data on a two-dimensional domain, where the data are generated on a very fine mesh to get a near exact solution, then downsampled to a coarser mesh that is similar to the spatial discretization of actual data, and noise is added. Results showing the effect of the hydrostatic stress term and noise are presented. A reconstruction using MR measured experimental data involving a tissue-mimicking phantom is also shown to demonstrate the algorithm.     

Influence of Poisson's ratio on elastographic direct and inverse problems

We consider the displacement of an elastic material under an external compression (axial or almost axial stress). We assume that only one component of the displacement is observed, in the direction of compression (axial displacement), or alternatively, that two components are observed in a plane. These hypotheses are in accordance with an imaging modality, namely ultrasonic elastography. In the case of a homogeneous medium we show that any value of Poisson's ratio allows us to predict the observed value of the axial displacement. When two components of the displacement are measured in a plane, the Poisson's ratio of the plane strain model that predicts the observed displacement is not the same as the tri-dimensional material. These facts are illustrated by numerical experiments in the case of an inhomogeneous medium. We also present results on experimental phantom data, where the inverse problem of reconstructing the Young's modulus is solved assuming different values for Poisson's ratio.     

基于遗传算法反求耳结构弹性模量

本研究通过力学反问题原理,利用已知的位移求解耳结构弹性模量。随机产生遗传算法的初始种群,使用自编的Matlab算法程序,对初始种群进行遗传迭代计算,把已知的目标位移与种群位移的均方差作为目标函数,以目标函数值最小控制迭代进化的方向。通过耳结构的鼓膜凸和镫骨底板2个控制位移以及砧镫关节周围的8个控制位移这两个算例求解正常砧镫关节的弹性模量,并使用耳结构的鼓膜凸和镫骨底板2个控制位移求解病变砧镫关节的弹性模量。结果表明,使用基于遗传算法的反问题方法计算耳结构的弹性模量是可行的,并且具有稳定性和不受结构力学性能影响的特点,相对误差分别为0.05%和0.2%、0.03%,可为临床病变耳提供有效的力学参数。     

Analytic modeling of breast elastography

The elastic moduli of tumors change during their pathological evolution. Elastographic imaging has potential for detecting and characterizing cancers by mapping the stiffness distribution in tissues. In this paper a micromechanics-based analytical method was developed to detect the location, size, and elastic modulus of a tumor mass embedded in a symmetric two-dimensional breast tissue. A closed-form solution for the strain elastograms (forward problem) was derived. A computational algorithm for the inverse problem was developed for the detection, localization, and characterization of a heterogeneous mass embedded in a breast tissue. Numerical examples were presented to evaluate the proposed method's performance. The detectability of a tumor mass was estimated with respect to lesion location, size, and modulus contrast ratio. It was shown that the micromechanics theory provides a powerful tool for the diagnosis of breast cancer.     

Evaluation of an iterative reconstruction method for quantitative elastography

This paper describes an inverse reconstruction technique based on a modified Newton Raphson iterative scheme and the finite element method, which has been developed for computing the spatial distribution of Young's modulus from within soft tissues. Computer simulations were conducted to determine the relative merits of reconstructing tissue elasticity using knowledge of (a) known displacement boundary conditions (DBC), and (b) known stress boundary conditions (SBC). The results demonstrated that computing Young's modulus using knowledge of SBC allows accurate quantification of Young's modulus. However, the quality of the images produced using this reconstruction approach was dependent on the Young's modulus distribution assumed at the start of the reconstruction procedure. Computing Young's modulus from known DBC provided relative estimates of tissue elasticity which, despite the disadvantage of not being able to accurately quantify Young's modulus, formed images that were generally superior in quality to those produced using the known SBC, and were not affected by the trial solution. The results of preliminary experiments on phantoms demonstrated that this reconstruction technique is capable in practice of improving the fidelity of tissue elasticity images, reducing the artefacts otherwise present in strain images, and recovering Young's modulus images that possess excellent spatial and contrast resolution.     

超声弹性成像逆问题求解的进一步研究

针对作者以前提出的基于有限元迭代的组织弹性模量重建即超声弹性成像逆问题求解的方法进行了进一步的研究.包括采用四边形单元代替三角形单元、尝试对组织边界的弹性模量不进行校正、考虑组织边界的弹性模量未知及含有随机噪声的情形.这些工作为组织弹性模量重建方法的实际应用奠定了基础.     

Non-linear modelling of breast tissue.

Previous approaches to modelling the large deformation of breast tissue, as occurs, e.g. in imaging using magnetic resonance imaging or mammography, include using linear elasticity and pseudo-non-linear elasticity, in which case the non-linear deformation is approximated by a series of small linear isotropic deformations, with the (constant) Young's modulus of each linear deformation an exponential function of the total non-linear strain. In this paper, these two approaches are compared to the solution of the full non-linear elastic problem for tissue with an exponential relationship between stress and strain. Having formulated each model and related the coefficients between the models, numerical simulations are performed on a block of incompressible material. These demonstrate that the simpler models may not be appropriate even in the case of modelling deformations of the human breast under gravity.     

超声弹性成像中的逆问题求解方法

超声弹性成像通常只能得到组织在外部压缩作用下的纵向应变分布.超声弹性成像逆问题的求解,即重建组织内部的弹性模量分布,具有重要的意义.本文提出一种利用组织的应力-应变关系和基于有限元分析的迭代计算方法,从组织的纵向应变分布重建出组织的弹性模量分布.对平面应变状态下的均匀组织内含一圆形异物的模型和弹性模量连续过渡的模型分别进行了计算机仿真.结果初步验证了该方法的可行性.     

Evaluation of the adjoint equation based algorithm for elasticity imaging

Recently a new adjoint equation based iterative method was proposed for evaluating the spatial distribution of the elastic modulus of tissue based on the knowledge of its displacement field under a deformation. In this method the original problem was reformulated as a minimization problem, and a gradient-based optimization algorithm was used to solve it. Significant computational savings were realized by utilizing the solution of the adjoint elasticity equations in calculating the gradient. In this paper, we examine the performance of this method with regard to measures which we believe will impact its eventual clinical use. In particular, we evaluate its abilities to (1) resolve geometrically the complex regions of elevated stiffness; (2) to handle noise levels inherent in typical instrumentation; and (3) to generate three-dimensional elasticity images. For our tests we utilize both synthetic and experimental displacement data, and consider both qualitative and quantitative measures of performance. We conclude that the method is robust and accurate, and a good candidate for clinical application because of its computational speed and efficiency.     

不同部位骨盆骨静态弹性模量及其相关生物力学性质的测定

采用生物力学实验的方法研究骨盆骨的轴向刚度、不同部位骨盆骨的压缩及拉伸弹性模量以及强度,从而为骨盆的临床及实验研究提供理论依据.试验采用12具防腐骨盆骨,利用万能材料试验机测量不同载荷下骨盆骨的位移,计算其轴向刚度;之后截取耻骨、坐骨、髋臼骨前柱及后柱、以及骶髂关节两侧骨,制成10 mm×5mm×10mm大小试件,分成12组,利用杠杆引伸仪、万能材料试验机、电阻应变仪上分别进行压缩及拉伸弹性模量以及强度的测定.在静态载荷状态下,500 N的生理载荷时,骨盆刚度值为181.28±21.4 N.mm-1,1500N时,刚度值增加到352.32 N.mm-1;耻骨、坐骨、髂骨、髋臼前、后柱以及骶髂关节两侧骨质的压缩弹性模量依次为26.7、18.4、31.7、21.3、23.1、31.7、29.6 GPa;压缩强度均值依次为64.27、124.26、91.73、94.22、50.39、107.37、84.23 GPa;拉伸弹性模量依次为29.3、20.4、25.4、23.3、21.2、19.3、17.6 GPa;拉伸强度均值依次为132.53、93.26、95.72、74.22、40.39、84.23、64.27 GPa.静态载荷状态下,刚度值随着载荷的增加逐渐加大;坐骨的抗压缩强度最大,而耻骨的抗拉伸强度最大,髋臼前柱抗压缩及拉伸的强度(94.22、74.22)均明显高于髋臼后柱.     

Young's modulus of collagen at slow displacement rates

Collagen is a key structural component of extracellular matrix and its mechanical properties, particularly its stiffness, have been shown to influence cell function. This study explores the mechanical behavior of type I collagen gels at low rates relevant to that of cell motion. The Young's modulus, E, was obtained for collagen samples of concentrations 1.3, 2 and 3 mg/ml at varying crosshead displacement rates: 0.01, 0.1 and 1 mm/min. Local strain measurement in the gage section were used for both the strain and strain rate determination. The power law models for the modulus at these low strain rates show that the values converge as the displacement rate approaches a quasistatic state. This study provides data that was unavailable in the past on the Young's modulus of collagen at rates relevant to the cell.     

Elastic Modulus of the Thin Filament under Normal Conditions and during Heart Failure

We developed an original approach to prepare samples for electron microscopy in electric field allowing calculation of the Youngs modulus of the thin filament in a direction perpendicular to the axis of the filament (longitudinal to the protomer) under normal conditions and during heart failure induced by 10-day toxic and allergic myocarditis. Electric field stretches thin filaments in the transverse direction and the increase in its diameter linearly depends on the applied voltage. The elastic modulus was in inverse proportion to the applied voltage. We found that during heart failure thin filament had an extreme conformation and to a great extent lost its mobility.     

Elastic deformation behaviour of Ti-24Nb-4Zr-7.9Sn for biomedical applications

In this paper, the elastic deformation behaviour of a recently developed beta-type titanium alloy Ti-24Nb-4Zr-7.9Sn (wt.%) that consists of non-toxic elements and is intended for biomedical applications is described. Tensile tests show that this alloy in the as hot-rolled state exhibits peculiar non-linear elastic behaviour with maximum recoverable strain up to 3.3% and incipient Young's modulus of 42GPa. Solution treatment at high temperature has trivial effect on super-elasticity but decreases strength and slightly increases the incipient Young's modulus. Ageing treatment in the (alpha+beta) two-phase field increases both strength and Young's modulus and results in a combination of high strength and relatively low elastic modulus. In spite of the formation of the alpha phase, short time ageing has no effect on super-elasticity, whereas the non-linear elastic behaviour transforms gradually to normal linear elasticity with the increase of ageing time. We suggest sluggish, partially reversible processes of stress-induced phase transformation and/or incipient kink bands as the origin of the above peculiar elastic behaviour.     

急性上肢哑铃训练对颈总动脉硬度和血液动力学的影响

抗阻力训练对于动脉硬度的影响存在着相互矛盾的结果。为研究不同负荷的急性上肢哑铃训练对颈总动脉硬度和血液动力学的影响,选择10名健康男性志愿者,年龄（21 ± 2）岁,分别进行5 kg和7 kg负荷的哑铃上肢曲臂训练各50次,两种负荷中间休息30 min。用彩色超声多普勒记录静息状态及不同哑铃负荷训练后颈总动脉的管径和轴心流速波形,用电子自动血压计同步测量心率和肱动脉血压。基于检测的实验数据,用经典血液动力学理论对颈总动脉硬度、压力-应变弹性模量以及相关血液动力学参数进行分析和计算。结果表明,受试者经过急性上肢哑铃训练后,心率、最大轴心流速、收缩压和最大壁面切应力等血液动力学参数明显增大,而颈总动脉管径、平均流量率、平均血压等参数没有明显变化。此外,在经过5 kg哑铃训练后,最大流量率和平均轴心流速显著增加,最小轴心流速显著减小,颈总动脉硬度、压力-应变弹性模量、舒张压、最小壁面切应力和振荡剪切指数无明显变化;而7 kg哑铃训练后,颈总动脉硬度、压力-应变弹性模量、振荡剪切指数明显增加,舒张压、最小壁面切应力显著降低,平均轴心流速和轴心流速最小值无明显变化。上肢急性哑铃训练会明显改变部分血液动力学参数,在高负荷训练时会急性增加颈总动脉的硬度。     

基于神经网络法反求耳结构弹性模量

目的通过耳结构的位移反算耳结构弹性模量。方法基于Patran软件建立耳结构有限元模型,使用Mat-lab建立计算耳结构反问题的BP神经网络。对耳结构有限元模型进行频率响应分析,得到鼓膜凸和镫骨足板的位移响应;把位移作为BP神经网络的输入、相对应的结构弹性模量作为输出,对网络进行训练。结果利用训练成熟的BP网络反算出耳结构的弹性模量,相对误差非常小。结论反算结果表明,所使用的反问题方法求解耳结构弹性模量是可行的,可为临床提供确定生物结构力学参数简捷有效的方法。     

Passive stiffness of the gastrocnemius muscle in athletes with spastic hemiplegic cerebral palsy

The passive properties of the muscle–tendon unit are regularly assessed in individuals with cerebral palsy (CP). However, no information is available on the passive properties of adult muscle, and whether any differences exist between the paretic and control muscles. Eleven ambulant male athletes with spastic hemiplegic CP (21.2 ± 3.0 years) and controls without neurological impairment (age = 21.8 ± 2.2 years) completed two and one passive stretch session, respectively. During each session, the ankle was passively dorsiflexed until end range of motion (ROM), whilst recording passive ankle angle, torque and gastrocnemius medialis (GM) myotendinous junction (MTJ) displacement. In addition, GM cross-sectional area (CSA) and length were measured. Subsequently, in vivo stress and strain were determined to calculate elastic modulus. Passive stiffness, MTJ displacement and ROM of the paretic GM were not different from the control muscles. However, the elastic modulus of the paretic GM was two times stiffer than the control GM muscles. In conclusion, athletes with CP exhibit absolute passive muscle stiffness similar to the controls; however, the elastic modulus of the CP muscle was significantly greater. Therefore, throughout the same ROM a smaller GM CSA in CP athletes has to dissipate larger relative torque compared to the control muscles, consequently causing the muscle to elongate to the same extent as the non-paretic muscle under stretch.     

Elastic moduli of normal and pathological human breast tissues: an inversion-technique-based investigation of 169 samples

Understanding and quantifying the mechanical properties of breast tissues has been a subject of interest for the past two decades. This has been motivated in part by interest in modelling soft tissue response for surgery planning and virtual-reality-based surgical training. Interpreting elastography images for diagnostic purposes also requires a sound understanding of normal and pathological tissue mechanical properties. Reliable data on tissue elastic properties are very limited and those which are available tend to be inconsistent, in part as a result of measurement methodology. We have developed specialized techniques to measure tissue elasticity of breast normal tissues and tumour specimens and applied them to 169 fresh ex vivo breast tissue samples including fat and fibroglandular tissue as well as a range of benign and malignant breast tumour types. Results show that, under small deformation conditions, the elastic modulus of normal breast fat and fibroglandular tissues are similar while fibroadenomas were approximately twice the stiffness. Fibrocystic disease and malignant tumours exhibited a 3-6-fold increased stiffness with high-grade invasive ductal carcinoma exhibiting up to a 13-fold increase in stiffness compared to fibrogalndular tissue. A statistical analysis showed that differences between the elastic modulus of the majority of those tissues were statistically significant. Implications for the specificity advantages of elastography are reviewed.