Combined compression and elongation experiments and non-linear modelling of liver tissue for surgical simulation

Uniaxial stress-strain data were obtained from in vitro experiments on 20 porcine livers for compressions, elongations and cycles of compression and then elongation. There were about 70 cylindrical samples, with diameter 7 mm and varying height (4–11 mm). The combined compression and elongation test provide a unified framework for both compression and elongation for applications such as computer-aided surgical simulation. It enable the zero stress state of the experimental liver sample to be precisely determined. A new equation that combined both logarithmic and polynomial strain energy forms was proposed in modelling these experimental data. The assumption of incompressibility was justified from a preliminary Poisson's ratio for elongation and compression at 0.43±0.16 and 0.47±0.15, respectively. This equation provided a good fit for the observed mechanical properties of liver during compression-elongation cycles and for separate compressions or elongations. The root mean square errors were 91.92±17.43 Pa, 57.55±13.23 Pa and 29.78±17.67 Pa, respectively. In comparison with existing strain energy functions, this combined model was the better constitutive equation. Application of this theoretical model to small liver samples and other tissues demonstrated its suitability as the material model of choice for soft tissue.     

Constitutive Modeling of Liver Tissue: Experiment and Theory

Realistic surgical simulation requires incorporation of the mechanical properties of soft tissue in mathematical models. In actual deformation of soft-tissue during surgical intervention, the tissue is subject to tension, compression, and shear. Therefore, characterization and modeling of soft-tissue in all these three deformation modes are necessary. In this paper we applied two types of pure shear test, unconfined compression and uniaxial tension test to characterize porcine liver tissue. Digital image correlation technique was used to accurately measure the tissue deformation field. Due to gravity and its effect on the soft tissue, a maximum stretching band was observed from the relative strain field on sample undergoing tension and pure shear test. The zero strain state was identified according to the position of this maximum stretching band. Two new constitutive models based on combined exponential/logarithmic and Ogden strain energy were proposed. The models are capable to represent the observed non-linear stress–strain relation of liver tissue for full range of tension and compression and also the general response of pure shear.     

A mesostructurally-based anisotropic continuum model for biological soft tissues--decoupled invariant formulation

Characterising and modelling the mechanical behaviour of biological soft tissues is an essential step in the development of predictive computational models to assist research for a wide range of applications in medicine, biology, tissue engineering, pharmaceutics, consumer goods, cosmetics, transport or military. It is therefore critical to develop constitutive models that can capture particular rheological mechanisms operating at specific length scales so that these models are adapted for their intended applications. Here, a novel mesoscopically-based decoupled invariant-based continuum constitutive framework for transversely isotropic and orthotropic biological soft tissues is developed. A notable feature of the formulation is the full decoupling of shear interactions. The constitutive model is based on a combination of the framework proposed by Lu and Zhang [Lu, J., Zhang, L., 2005. Physically motivated invariant formulation for transversely isotropic hyperelasticity. International Journal of Solids and Structures 42, 6015-6031] and the entropic mechanics of tropocollagen molecules and collagen assemblies. One of the key aspects of the formulation is to use physically-based nanoscopic quantities that could be extracted from experiments and/or atomistic/molecular dynamics simulations to inform the macroscopic constitutive behaviour. This effectively couples the material properties at different levels of the multi-scale hierarchical structure of collagenous tissues. The orthotropic hyperelastic model was shown to reproduce very well the experimental multi-axial properties of rabbit skin. A new insight into the shear response of a skin sample subjected to a simulated indentation test was obtained using numerical direct sensitivity analyses.     

Directional Differences in the Biaxial Material Properties of Fascia Lata and the Implications for Fascia Function

Fascia is a highly organized collagenous tissue that is ubiquitous in the body, but whose function is not well understood. Because fascia has a sheet-like structure attaching to muscles and bones at multiple sites, it is exposed to different states of multi- or biaxial strain. In order to measure how biaxial strain affects fascia material behavior, planar biaxial tests with strain control were performed on longitudinal and transversely oriented samples of goat fascia lata (FL). Cruciform samples were cycled to multiple strain levels while the perpendicular direction was held at a constant strain. Structural differences among FL layers were examined using histology and SEM. Results show that FL stiffness, hysteresis, and strain energy density are greater in the longitudinal vs. transverse direction. Increased stiffness in the longitudinal layer is likely due to its greater thickness and greater average fibril diameter compared to the transverse layer(s). Perpendicular strain did not affect FL material behavior. Differential loading in the longitudinal vs. transverse directions may lead to structural changes, enhancing the ability of the longitudinal FL to transmit force, store energy, or stabilize the limb during locomotion. The relative compliance of the transverse fibers may allow expansion of underlying muscles when they contract.     

生物软组织超弹性及夹持过程仿真分析

目的确定生物软组织的超弹性本构方程,并在此基础上研究生物组织夹持过程的力学响应规律。方法以新鲜的猪肝脏组织为研究对象进行破坏性单轴拉伸实验,并在ABAQUS中对单轴拉伸实验过程进行仿真,通过对比仿真结果与试验数据确定猪肝的超弹性本构方程。以此为基础分别选用尖齿形及波浪齿形夹头对组织的夹持过程进行有限元仿真。结果采用4阶Ogden模型开展拉伸实验的仿真结果与试验数据吻合度较高。组织夹持仿真结果表明,采用尖齿形夹头更容易产生应力集中。结论可以采用4阶Ogden模型描述猪肝的超弹性,并确定相关参数。采用尖齿形夹头更容易造成组织夹持损伤,且组织应力与夹持进给量基本成线性关系。研究结果为手术钳头的设计提供参考。     

A constitutive model of the posterior cruciate ligament

Further to our previous work on the development of a general constitutive framework for transversely isotropic viscohyperelasticity (Limbert, G, Middleton, J. A transversely isotropic viscohyperelastic material. Application to the modelling of biological soft connective tissues. Int J Solids Struct 2004;41(15):4237-60.), we propose a phenomenological constitutive law to describe the anisotropic viscohyperelastic behaviour of the human posterior cruciate ligament (PCL) at high strain rates. The mechanical formulation is based on the definition of a Helmholtz free energy function containing a hyperelastic and a viscous potential. The equations are valid for arbitrary kinematics and satisfy elemental thermodynamic principles. Identification of the constitutive model with experimental data obtained from human PCL specimens was performed and showed the ability of the model to capture accurately the mechanical characteristics of the PCL at various strain rates. Influence of the isotropic and directional viscous stress responses on the global mechanical response are discussed in connection with the modelling hypotheses. This work was motivated by the need to provide an accurate constitutive model of the PCL to be used in finite element analyses of human occupants in car crash simulations. Besides uniaxial tests along the natural fibre orientation of the PCL, additional tests such as equibiaxial, strip biaxial compression-tension and shear tests were also performed in order to assess the physical response of the model in different loading situations. It was found that the model performed as well in these conditions.     

Dynamic compressive response of bovine liver tissues

This study aims to experimentally determine the strain rate effects on the compressive stress-strain behavior of bovine liver tissues. Fresh liver tissues were used to make specimens for mechanical loading. Experiments at quasi-static strain rates were conducted at 0.01 and 0.1?s(-1). Intermediate-rate experiments were performed at 1, 10, and 100?s(-1). High strain rate (1000, 2000, and 3000?s(-1)) experiments were conducted using a Kolsky bar modified for soft material characterization. A hollow transmission bar with semi-conductor strain gages was used to sense the weak forces from the soft specimens. Quartz-crystal force transducers were used to monitor valid testing conditions on the tissue specimens. The experiment results show that the compressive stress-strain response of the liver tissue is non-linear and highly rate-sensitive, especially when the strain rate is in the Kolsky bar range. The tissue stiffens significantly with increasing strain rate. The responses from liver tissues along and perpendicular to the liver surface were consistent, indicating isotropic behavior.     

In vitro technique in estimation of passive mechanical properties of bovine heart part II. Constitutive relation and finite element analysis

A pseudo-elastic constitutive equation describing the mechanical properties of bovine myocardium was developed. The myocardium was modeled as a hyperelastic transversely isotropic material with a minimum viscoelastic loses. The material parameters for the proposed constitutive equations were determined using GA regression technique. In this work, the development of a constitutive equation based on principal stretch ratios is explained. The predictive capability of proposed model was compared against the experimental data obtained from part one. Finally, the constitutive equations were implemented into a commercial finite element program and the results of the mathematical model and FEM were compared with the experimental data.     

虹膜组织力学特性及瞳孔阻滞力定量研究的方法学探索

目的:寻找一种能对虹膜本构关系以及原发性闭角型青光眼瞳孔阻滞力进行定量研究的方法,为闭角型青光眼的早期诊断及治疗方法的选择和治疗效果的评价提供一种定量标准.方法:根据瞳孔阻滞产生机制,设计瞳孔阻滞力仿真方法;依据力学原理和自定义的虹膜面应变、面积模量等建立虹膜的本构关系.结果:设计了一种不破坏虹膜正常功能和形态学结构,对虹膜组织力学特性及瞳孔阻滞力进行定量研究的方法.     

A Nonlinear Constitutive Model for Stress Relaxation in Ligaments and Tendons

A novel constitutive model that describes stress relaxation in transversely isotropic soft collagenous tissues such as ligaments and tendons is presented. The model is formulated within the nonlinear integral representation framework proposed by Pipkin and Rogers (J. Mech. Phys. Solids. 16:59?C72, 1968). It represents a departure from existing models in biomechanics since it describes not only the strain dependent stress relaxation behavior of collagenous tissues but also their finite strains and transverse isotropy. Axial stress?Cstretch data and stress relaxation data at different axial stretches are collected on rat tail tendon fascicles in order to compute the model parameters. Toward this end, the rat tail tendon fascicles are assumed to be incompressible and undergo an isochoric axisymmetric deformation. A comparison with the experimental data proves that, unlike the quasi-linear viscoelastic model (Fung, Biomechanics: Mechanics of Living Tissues. Springer, New York, 1993) the constitutive law can capture the observed nonlinearities in the stress relaxation response of rat tail tendon fascicles.     

基于可行方向法的脂肪组织本构模型参数反求优化

目的研究脂肪组织在中等应变率下本构模型及其参数反求。方法基于脂肪组织力学性能实验,通过有限元方法重构脂肪组织压缩实验,并对常见表征脂肪组织的本构模型进行参数筛选。结合最优化方法中的可行方向法(method of feasible direction,MFD),进行中应变率下脂肪组织本构模型相关参数的反求。结果中应变率(260 s~(-1))下黏弹性本构模型相比Ogden本构模型更适合表征脂肪组织的力学响应,并反求得到适用于仿真的本构模型参数。结论中等应变率下黏弹性本构模型更适合表征脂肪组织力学响应。研究结果为汽车碰撞有限元仿真中探究人体脂肪组织对人体损伤的影响提供参考。     

Soft tissue modelling through autowaves for surgery simulation

Modelling of soft tissue deformation is of great importance to virtual reality based surgery simulation. This paper presents a new methodology for simulation of soft tissue deformation by drawing an analogy between autowaves and soft tissue deformation. The potential energy stored in a soft tissue as a result of a deformation caused by an external force is propagated among mass points of the soft tissue by non-linear autowaves. The novelty of the methodology is that (i) autowave techniques are established to describe the potential energy distribution of a deformation for extrapolating internal forces, and (ii) non-linear materials are modelled with non-linear autowaves other than geometric non-linearity. Integration with a haptic device has been achieved to simulate soft tissue deformation with force feedback. The proposed methodology not only deals with large-range deformations, but also accommodates isotropic, anisotropic and inhomogeneous materials by simply changing diffusion coefficients.     

Characterization of the Highly Nonlinear and Anisotropic Vascular Tissues from Experimental Inflation Data: A Validation Study Toward the Use of Clinical Data for <Emphasis Type="Italic">In-Vivo</Emphasis> Modeling and Analysis

We study whether an inverse modeling approach is applicable for characterizing vascular tissue subjected to various levels of internal pressure and axial stretch that approximate in-vivo conditions. To compensate for the limitation of axial-displacement/pressure/diameter data typical of clinical data, which does not provide information about axial force, we propose to constrain the ratio of axial to circumferential elastic moduli to a typical range. Vessel wall constitutive behavior is modeled with a transversely isotropic hyperelastic equation that accounts for dispersed collagen fibers. A single-layer and a bi-layer approximation to vessel ultrastructure are examined, as is the possibility of obtaining the fiber orientation as part of the optimization. Characterization is validated against independent pipette-aspiration biaxial data on the same samples. It was found that the single-layer model based on homogeneous wall assumption could not reproduce the validation data. In contrast, the constrained bi-layer model was in excellent agreement with both types of experimental data. Due to covariance, estimations of fiber angle were slightly outside of the normal range, which can be resolved by predefining the angles to normal values. Our approach is relatively invariant to a constant or a variable axial response. We believe that it is suitable for in-vivo characterization.     

Strain rate-dependent viscohyperelastic constitutive modeling of bovine liver tissue

The mechanical response of most soft tissue is considered to be viscohyperelastic, making the development of accurate constitutive models a challenging task. In this article, we present a constitutive model for bovine liver tissue that utilizes a viscous dissipation potential, and use it to model the response of bovine liver tissue at strain rates ranging from 0.001 to 0.04 s⁻¹. On the material modeling front of this study, the free energy is assumed to depend on the right Cauchy–Green deformation tensor, whereas a separate rate-dependent viscous potential is posited to characterize viscoelasticity. This viscous dissipation component is a function of the time rate of change of the right Cauchy–Green deformation tensor. On the experimental front, no-slip uniaxial compression experiments are conducted on bovine liver tissue at various strain rates. A numerical correction approach is used to account for the no-slip edge conditions, and the constitutive model is fit to the resulting corrected stress–strain data. The complete derivation of the material model, its implementation in the finite element software package ABAQUS, and a validation study are presented in this article. The results show that bovine liver tissue exhibits a strong strain-rate dependence even at the low strain rates considered here and that the proposed constitutive model is able to accurately describe this response.     

Development of in vivo constitutive models for liver: application to surgical simulation

Advancements in real-time surgical simulation techniques have provided the ability to utilize more complex nonlinear constitutive models for biological tissues which result in increased haptic and graphic accuracy. When developing such a model, verification is necessary to determine the accuracy of the force response as well as the magnitude of tissue deformation for tool–tissue interactions. In this study, we present an experimental device which provides the ability to obtain force–displacement information as well as surface deformation of porcine liver for in vivo probing tasks. In addition, the system is capable of accurately determining the geometry of the liver specimen. These combined attributes provide the context required to simulate the experiment with accurate boundary conditions, whereby the only variable in the analysis is the material properties of the liver specimen. During the simulation, effects of settling due to gravity have been taken into account by a technique which incorporates the proper internal stress conditions in the model without altering the geometry. Initially, an Ogden model developed from ex vivo tension and compression experimentation is run through the simulation to determine the efficacy of utilizing an ex vivo model for simulation of in vivo probing tasks on porcine liver. Subsequently, a method for improving upon the ex vivo model was developed using different hyperelastic models such that increased accuracy could be achieved for the force characteristics compared to the displacement characteristics, since changes in the force variation would be more perceptible to a user in the simulation environment, while maintaining a high correlation with the surface displacement data. Furthermore, this study also presents the probing simulation which includes the capsule surrounding the liver.     

基于超弹性模型的动脉血管数值计算

目的：实验研究表明。血管在周向与轴向两种单轴向拉伸作用下表现出不同的力学特性,本文通过对血管单轴拉伸的数值计算,给出分别适用于周向和轴向荷载的模拟方法。方法：基于超弹性本构模型对轴向和周向两种单轴拉伸作用下血管的应力一应变关系进行数值计算,并结合血管组织结构特点及模型适用范围对结果进行分析,同时通过数值计算对Holzapfel．Gasser-Ogden模型中的各向异性参数对结果的影响展开讨论。结果：计算结果显示单一使用各向同性超弹性应变势函数无法准确完整的模拟两种情况下的单轴拉伸实验,周向拉伸采用各向同性超弹性本构模型的数值结果较好的吻合实验,而轴向拉伸宜采用Holzapfel-Gasser-Ogden模型。Holzapfel。Gasser-Ogden模型中各向异性参数1描述血管中两组增强纤维主方向的分散程度,y值越大即纤维平均主方向与轴向加载方向夹角越小,在外荷载作用下越容易使得纤维旋转到荷载方向;参数K描述血管中每组增强纤维主方向上纤维的分散程度。K值越大,纤维在基体中分散越广泛,材料性子越接近纤维,宏观表现越硬。结论：本文基于超弹性本构模型对轴向和周向两种单轴拉伸作用下血管的应力应变关系进行数值计算,提出分别用多项式形式的各向同性超弹性本构模型数值计算周向荷载作用下应力应变关系、Holzapfel-Gasser-Ogden各向异性超弹性本构模型数值模拟轴向荷载下力学性质,数值结果与实验吻合较好,为心血管系统的数值模拟提供指导,对血管系统的力学机制和临床研究具有重要意义。     

Mechanical characterization of the softening behavior of human vaginal tissue

The mechanical properties of vaginal tissue need to be characterized to perform accurate simulations of prolapse and other pelvic disorders that commonly affect women. This is also a fundamental step towards the improvement of therapeutic techniques such as surgery. In this paper, the softening behavior or Mullins effect of vaginal tissue is studied by proposing an appropriate constitutive model. This effect is an important factor after the birth, since vaginal tissue has been supporting a high load distribution and therefore does not recover its original behavior. Due to the anisotropy of the tissue, the mechanical testing of vaginal tissue, consists in loading-unloading uniaxial tension tests performed along the longitudinal and transverse axes of the vagina. A directional pseudo-elastic model was used to reproduce the inelastic behavior of the tissue. The obtained results may be helpful in the design of surgical procedures with autologous tissue or smart prostheses. A good qualitative agreement has been found between the numerical and experimental results for the vaginal tissue examples, indicating that the constitutive softening model can capture the typical stress-strain behavior observed in this kind of fibrous soft tissue.     

Noninvasive Determination of Ligament Strain with Deformable Image Registration

Ligament function and propensity for injury are directly related to regional stresses and strains. However, noninvasive techniques for measurement of strain are currently limited. This study validated the use of Hyperelastic Warping, a deformable image registration technique, for noninvasive strain measurement in the human medial collateral ligament using direct comparisons with optical measurements. Hyperelastic Warping determines the deformation map that aligns consecutive images of a deforming material, allowing calculation of strain. Diffeomorphic deformations are ensured by representing the deformable image as a hyperelastic material. Ten cadaveric knees were subjected to six loading scenarios each. Tissue deformation was documented with magnetic resonance imaging (MRI) and video-based experimental measurements. MRI datasets were analyzed using Hyperelastic Warping, representing the medial collateral ligament (MCL) with a hexahedral finite element (FE) model projected to a manually segmented ligament surface. The material behavior was transversely isotropic hyperelastic. Warping predictions of fiber stretch were strongly correlated with experimentally measured strains (R ² = 0.81). Both sets of measurements were in agreement with previous ex vivo studies. Warping predictions of fiber stretch were insensitive to bulk:shear modulus ratio, fiber stiffness, and shear modulus in the range of +2.5SD to −1.0SD. Correlations degraded when the shear modulus was decreased to 2.5SD below the mean (R ² = 0.56), and when an isotropic constitutive model was substituted for the transversely isotropic model (R ² = 0.65). MCL strains in the transitional region near the joint line, where the material behavior and material symmetry are more complex, showed the most sensitivity to changes in shear modulus. These results demonstrate that Hyperelastic Warping requires the use of a constitutive model that reflects the material symmetry, but not subject-specific material properties for accurate strain predictions for this application. Hyperelastic Warping represents a powerful technique for noninvasive strain measurement of musculoskeletal tissues and has many advantages over other image-based strain measurement techniques.     

Importance of the superficial tissue layer for the indentation stiffness of articular cartilage

Indentation testing is a widely used technique for nondestructive mechanical analysis of articular cartilage. Although cartilage shows an inhomogeneous, layered structure with anisotropic mechanical properties, most theoretical indentation models assume material homogeneity and isotropy. In the present study, quantitative polarized light microscopy (PLM) measurements from canine cartilage were utilized to characterize thickness and structure of the superficial, collageneous tissue layer as well as to reveal its relation to experimental indentation measurements. In addition to experimental analyses, a layered, transversely isotropic finite element (FE) model was developed and the effect of superficial (tangential) tissue layer with high elastic modulus in the direction parallel to articular surface on the indentation response was studied. The experimental indentation stiffness was positively correlated with the relative thickness of the superficial cartilage layer. Also the optical retardation, which reflects the degree of parallel organization of collagen fibrils as well as collagen content, was related to indentation stiffness. FE results indicated effective stiffening of articular cartilage under indentation due to high transverse modulus of the superficial layer. The present results suggest that indentation testing is an efficient technique for the characterization of the superficial degeneration of articular cartilage.     

Multiaxial mechanical properties and constitutive modeling of human adipose tissue: A basis for preoperative simulations in plastic and reconstructive surgery

A preoperative simulation of soft tissue deformations during plastic and reconstructive surgery is desirable to support the surgeon’s planning and to improve surgical outcomes. The current development of constitutive adipose tissue models, for the implementation in multilayer computational frameworks for the simulation of human soft tissue deformations, has proved difficult because knowledge of the required mechanical parameters of fat tissue is limited. Therefore, for the first time, human abdominal adipose tissues were mechanically investigated by biaxial tensile and triaxial shear tests. The results of this study suggest that human abdominal adipose tissues under quasi-static and dynamic multiaxial loadings can be characterized as a nonlinear, anisotropic and viscoelastic soft biological material. The nonlinear and anisotropic features are consequences of the material’s collagenous microstructure. The aligned collagenous septa observed in histological investigations causes the anisotropy of the tissue. A hyperelastic model used in this study was appropriate to represent the quasi-static multiaxial mechanical behavior of fat tissue. The constitutive parameters are intended to serve as a basis for soft tissue simulations using the finite element method, which is an apparent method for obtaining promising results in the field of plastic and reconstructive surgery.