Material characterization of liver parenchyma using specimen-specific finite element models

Mechanical properties of human hepatic tissues to develop liver-mimicking phantoms for medical applications

Using liver phantoms for mimicking human tissue in clinical training, disease diagnosis, and treatment planning is a common practice. The fabrication material of the liver phantom should exhibit mechanical properties similar to those of the real liver organ in the human body. This tissue-equivalent material is essential for qualitative and quantitative investigation of the liver mechanisms in producing nutrients, excretion of waste metabolites, and tissue deformity at mechanical stimulus. This paper reviews the mechanical properties of human hepatic tissues to develop liver-mimicking phantoms. These properties include viscosity, elasticity, acoustic impedance, sound speed, and attenuation. The advantages and disadvantages of the most common fabrication materials for developing liver tissue-mimicking phantoms are also highlighted. Such phantoms will give a better insight into the real tissue damage during the disease progression and preservation for transplantation. The liver tissue-mimicking phantom will raise the quality assurance of patient diagnostic and treatment precision and offer a definitive clinical trial data collection.

An introduction to the Ogden model in biomechanics: benefits, implementation tools and limitations

Constitutive models are important to biomechanics for two key reasons. First, constitutive modelling is an essential component of characterizing tissues' mechanical properties for informing theoretical and computational models of biomechanical systems. Second, constitutive models can be used as a theoretical framework for extracting and comparing key quantities of interest from material characterization experiments. Over the past five decades, the Ogden model has emerged as a popular constitutive model in soft tissue biomechanics with relevance to both informing theoretical and computational models and to comparing material characterization experiments. The goal of this short review is threefold. First, we will discuss the broad relevance of the Ogden model to soft tissue biomechanics and the general characteristics of soft tissues that are suitable for approximating with the Ogden model. Second, we will highlight exemplary uses of the Ogden model in brain tissue, blood clot and other tissues. Finally, we offer a tutorial on fitting the one-term Ogden model to pure shear experimental data via both an analytical approximation of homogeneous deformation and a finite-element model of the tissue domain. Overall, we anticipate that this short review will serve as a practical introduction to the use of the Ogden model in biomechanics. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

An introduction to the Ogden model in biomechanics: benefits, implementation tools and limitations

Constitutive models are important to biomechanics for two key reasons. First, constitutive modelling is an essential component of characterizing tissues' mechanical properties for informing theoretical and computational models of biomechanical systems. Second, constitutive models can be used as a theoretical framework for extracting and comparing key quantities of interest from material characterization experiments. Over the past five decades, the Ogden model has emerged as a popular constitutive model in soft tissue biomechanics with relevance to both informing theoretical and computational models and to comparing material characterization experiments. The goal of this short review is threefold. First, we will discuss the broad relevance of the Ogden model to soft tissue biomechanics and the general characteristics of soft tissues that are suitable for approximating with the Ogden model. Second, we will highlight exemplary uses of the Ogden model in brain tissue, blood clot and other tissues. Finally, we offer a tutorial on fitting the one-term Ogden model to pure shear experimental data via both an analytical approximation of homogeneous deformation and a finite-element model of the tissue domain. Overall, we anticipate that this short review will serve as a practical introduction to the use of the Ogden model in biomechanics. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

NUMERICAL COUPLING ANALYSIS OF THE INFLUENCE OF BLOOD FLOW ON THE MECHANICAL RESPONSE FOR LIVER

As an important basis for determining the state of the liver, the mechanical responses are associated with many factors, and belong to a complex coupling system. Liver tissue has significantly complicated vascular channels. The vascular diameter, vascular deflection angle and vascular depth are defined as the key characteristic parameters. The influences of these parameters on the mechanical responses were analyzed. On the basis of the real mechanical parameters, the coupled numerical model of blood vessel, blood flow and liver tissue was established. The corresponding mechanical responses are obtained by utilizing the different vascular parameters. The effects of vascular parameters on the differences among the mechanical response difference and high strain modulus were analyzed. It was found that the blood vessels in the central area could reduce the liver mechanical response. The inner diameter parameter had main influences on the regions where the stain was more than 0.1. The mechanical difference is greater with larger inner diameter. The influences of vascular depth are greatest when the vascular depth was in the intermediate value, which would increase the liver mechanical responses. With the increment of vascular deflection angle, the liver mechanical response would also increase, and exceed the mechanical response without blood vessels. The findings after analyzing the influence of vascular parameters will provide a basis for the quantitative studies on the influence of blood vessels.

Is the 0.2%-Strain-Offset Approach Appropriate for Calculating the Yield Stress of Cortical Bone?

The 0.2% strain offset approach is mostly used to calculate the yield stress and serves as an efficient method for cross-lab comparisons of measured material properties. However, it is difficult to accurately determine the yield of the bone. Especially when computational models require accurate material parameters, clarification of the yield point is needed. We tested 24 cortical specimens harvested from six bovine femora in three-point bending mode, and 11 bovine femoral cortical specimens in the tensile mode. The Young's modulus and yield stress for each specimen derived from the specimen-specific finite element (FE) optimization method was regarded as the most ideal constitutive parameter. Then, the strain offset optimization method was used to find the strain offset closest to the ideal yield stress for the 24 specimens. The results showed that the 0 strain offsets underestimated (- 25%) the yield stress in bending and tensile tests, while the 0.2% strain offsets overestimated the yield stress (+ 65%) in three-point bending tests. Instead, the yield stress determined by 0.007 and 0.05% strain offset for bending and tensile loading respectively, can effectively characterize the biomechanical responses of the bone, thereby helping to build an accurate FE model.

Comparison of material models for anterior cruciate ligament in tension: from poroelastic to a novel fibril-reinforced nonlinear composite model

Computational models of the knee joint are useful for evaluating stresses and strains within the joint tissues. However, the outcome of those models is sensitive to the material model and material properties chosen for ligaments, the collagen reinforced tissues connecting bone to bone. The purpose of this study was to investigate different compositionally motivated material models and further to develop a model that can accurately reproduce experimentally measured stress-relaxation data of bovine anterior cruciate ligament (ACL). Tensile testing samples were extracted from ACLs of bovine knee joints (N = 10) and subjected to a three-step stress-relaxation test at the toe region. Data from the experiments was averaged and one average finite element model was generated to replicate the experiment. Poroelastic and different fibril-reinforced poro(visco)elastic material models were applied, and their material parameters were optimized to reproduce the experimental force-time response. Material models with only fluid flow mediated relaxation were not able to capture the stress-relaxation behavior (R² = 0.806, 0.803 and 0.938). The inclusion of the viscoelasticity of the fibrillar network improved the model prediction (R² = 0.978 and 0.976), but the complex stress-relaxation behavior was best captured by a poroelastic model with a nonlinear two-relaxation-time strain-recruited viscoelastic fibrillar network (R² = 0.997). The results suggest that in order to replicate the multi-step stress-relaxation behavior of ACL in tension, the fibrillar network formulation should include the complex nonlinear viscoelastic phenomena.

Hyperelastic Ex Vivo Cervical Tissue Mechanical Characterization

This paper presents the results of the comparison between a proposed Fourth Order Elastic Constants (FOECs) nonlinear model defined in the sense of Landau's theory, and the two most contrasted hyperelastic models in the literature, Mooney-Rivlin, and Ogden models. A mechanical testing protocol is developed to investigate the large-strain response of ex vivo cervical tissue samples in uniaxial tension in its two principal anatomical locations, the epithelial and connective layers. The final aim of this work is to compare the reconstructed shear modulus of the epithelial and connective layers of cervical tissue. According to the obtained results, the nonlinear parameter A from the proposed FOEC model could be an important biomarker in cervical tissue diagnosis. In addition, the calculated shear modulus depended on the anatomical location of the cervical tissue (μepithelial = 1.29 ± 0.15 MPa, and μconnective = 3.60 ± 0.63 MPa).

IDENTIFICATION OF UNIAXIAL DEFORMATION BEHAVIOR AND ITS INITIAL TANGENT MODULUS FOR ATHEROMATOUS INTIMA IN THE HUMAN CAROTID ARTERY AND THORACIC AORTA USING THREE-PARAMETER ISOTROPIC HYPERELASTIC MODELS

Uniaxial stretching tests are used for mechanical identification of small fibrous regions of atheromatous arteries. Material constants in isotropic hyperelastic models are determined to minimize the fitting error for the stress–strain curve. We developed a novel method to better characterize the material constants in typical forms of Yeoh, Ogden, Chuong–Fung (CF) and Gasser–Ogden–Holzapfel (GOH) isotropic hyperelastic models for fibrous caps and normal intimal layers from human carotid artery and thoracic aorta by incorporating Young’s modulus, i.e., the initial tangent modulus of uniaxial stress–strain relationships, as one of three material constants. We derived a unified, isotropic form for the anisotropic exponential-type strain energy density functions of CF and GOH models. The uniaxial stress–strain relationship equations were expanded to Maclaurin series to identify Young’s modulus as a coefficient of the linear term of the strain and to examine the roles of the material constants in the nonlinear function. The remaining two material constants were determined by curvefitting. The incorporation of Young’s modulus into the CF and GOH models gave reasonable curvefitting, with errors [Formula: see text], whereas large errors ([Formula: see text]) were observed in one case for the Yeoh model and in two cases for the Ogden model.

The effect of storage time in saline solution on the material properties of cortical bone tissue

BACKGROUND

The use of saline in preserving bone specimens may affect the mechanical properties of specimens. Yet, the reported effects varied and contradicted to each other, with a lack of investigating constitutive material parameters. Therefore, we quantified the effects of preservation time on the constitutive properties of cortical bone.

METHODS

We collected 120 specimens from the mid-diaphysis of six male bovine femora, which were assigned to five groups, including fresh-frozen for 60 days (-20 °C), storage in saline for 3, 10, 36 and 60 days (25 °C). All specimens underwent quasi-static three-point bending tests with a loading rate of 0.02 mm/s. Using the optimization method combined with specimen-specific finite element models, the Young's modulus, tangent modulus, yield stress, effective plastic strain, yield strain, ultimate stress, and toughness were calculated.

FINDINGS

Saline preservation resulted in a significant decrease of Young's modulus, yield stress, ultimate stress and pre-yield toughness (P < 0.001), and a significant increase of effective plastic strain (P = 0.034). After 10 days of preservation, yield stress and pre-yield toughness decreased -14.9% and -21.4%, respectively, and they continued to decrease with longer preservation time. After 36 days of preservation, Young's modulus and ultimate stress decreased -19.2% and -17.3%, respectively, and continued to decrease with longer preservation time. Our data also showed changes of material properties collected after 3-day saline preservation, while the low statistical power must be considered for this group.

INTERPRETATION

Saline preservation impacts on mechanical properties of cortical bone tissue and the effect is already observable after 3 days.

Combining specimen-specific finite-element models and optimization in cortical-bone material characterization improves prediction accuracy in three-point bending tests

Although the beam theory is widely used for calculating material parameters in three-point bending test, it cannot accurately describe the biomechanical properties of specimens after the yield. Hence, we propose a finite element (FE) based optimization method to obtain accurate bone material parameters from three-point bending test. We tested 80 machined bovine cortical bone specimens at both longitudinal and transverse directions using three-point bending. We then adopted the beam theory and the FE-based optimization method combined with specimen-specific FE models to derive the material parameters of cortical bone. We compared data obtained using these two methods and further evaluated two groups of parameters with three-point bending simulations. Our data indicated that the FE models with material properties from the FE-based optimization method showed best agreements with experimental data for the entire force-displacement responses, including the post-yield region. Using the beam theory, the yield stresses derived from 0.0058% strain offset for the longitudinal specimen and 0.0052% strain offset for the transverse specimen are closer to those derived from the FE-based optimization method, compared to yield stresses calculated without strain offset. In brief, we conclude that the optimization FE method is more appropriate than the traditional beam theory in identifying the material parameters of cortical bone for improving prediction accuracy in three-point bending mode. Given that the beam theory remains as a popular method because of its efficiency, we further provided correction functions to adjust parameters calculated from the beam theory for accurate FE simulation.

The estimation method of friction in unconfined compression tests of liver tissue

In traditional unconfined compression tests, the friction between platform and specimen is often considered negligible or minimized by lubrication or other means. However, friction can affect the estimation of material parameters. The percentage difference in radial deformation was investigated in this study. A novel friction estimation method was established and verified using a finite element method. The proposed method was based on the radial deformation during the compression process. Three different hyperelastic material parameters of liver tissue were applied in the simulations. The hyperelastic parameters H1 were obtained by no-slip compression tests, while the others were extracted from the literature. The results showed that the percentage difference in radial deformation was mainly influenced by the friction coefficient and diameter-to-height ( d/ h) ratio of the specimen in unconfined compression tests. The percentage difference increased as the friction coefficient and d/ h increased. Different d/ h and friction coefficient values were tested to validate the proposed method, and the accuracy was estimated to exceed 86%. An optimization strategy for material parameters in unconfined compression tests was proposed accordingly.

Quantifying the in vivo quasi-static response to loading of sub-dermal tissues in the human buttock using magnetic resonance imaging

BACKGROUND

The design of seating systems to improve comfort and reduce injury would benefit from improved understanding of the deformation and strain patterns in soft tissues, particularly in the gluteal region.

METHODS

Ten healthy men were positioned in a semi-recumbent posture while their pelvic and thigh region was scanned using a wide-bore magnetic resonance imaging (MRI) scanner. Independent measurements of deformation for muscles and fat were taken for the transition from non-weight-bearing to weight-bearing loads in three stages. A weight-bearing load was achieved through having the subject supported by a flat, rigid surface. A non-weight-bearing condition was achieved by removing the support under the left buttock, leaving all soft tissue layers undeformed. An intermediate condition partially relieved the subject's left buttock by lowering the support relative to the pelvis by 20mm, which left the buttock partially deformed. For each of these conditions, the thicknesses of muscle and fat tissues below the ischial tuberosity and the greater trochanter were measured from the MRI data.

FINDINGS

In this dataset, the greatest soft tissue deformation took place below the ischial tuberosity, with muscles (mean=17.7mm, SD=4.8mm) deforming more than fat tissues (mean=4.3mm, SD=5.6mm). Muscles deformed through both steps of the transition from weight-bearing to non-weight-bearing conditions, while subcutaneous fat deformed little after the first transition from non-weight-bearing to partial-weight-bearing. High inter-subject variability in muscle and fat tissue strains was observed.

INTERPRETATION

Our findings highlight the importance of considering inter-subject variability when designing seating systems.

Development and evaluation of a finite element model of the THOR for occupant protection of spaceflight crewmembers

New vehicles are currently being developed to transport humans to space. During the landing phases, crewmembers may be exposed to spinal and frontal loading. To reduce the risk of injuries during these common impact scenarios, the National Aeronautics and Space Administration (NASA) is developing new safety standards for spaceflight. The Test Device for Human Occupant Restraint (THOR) advanced multi-directional anthropomorphic test device (ATD), with the National Highway Traffic Safety Administration modification kit, has been chosen to evaluate occupant spacecraft safety because of its improved biofidelity. NASA tested the THOR ATD at Wright-Patterson Air Force Base (WPAFB) in various impact configurations, including frontal and spinal loading. A computational finite element model (FEM) of the THOR to match these latest modifications was developed in LS-DYNA software. The main goal of this study was to calibrate and validate the THOR FEM for use in future spacecraft safety studies. An optimization-based method was developed to calibrate the material models of the lumbar joints and pelvic flesh. Compression test data were used to calibrate the quasi-static material properties of the pelvic flesh, while whole body THOR ATD kinematic and kinetic responses under spinal and frontal loading conditions were used for dynamic calibration. The performance of the calibrated THOR FEM was evaluated by simulating separate THOR ATD tests with different crash pulses along both spinal and frontal directions. The model response was compared with test data by calculating its correlation score using the CORrelation and Analysis rating system. The biofidelity of the THOR FEM was then evaluated against tests recorded on human volunteers under 3 different frontal and spinal impact pulses. The calibrated THOR FEM responded with high similarity to the THOR ATD in all validation tests. The THOR FEM showed good biofidelity relative to human-volunteer data under spinal loading, but limited biofidelity under frontal loading. This may suggest a need for further improvements in both the THOR ATD and FEM. Overall, results presented in this study provide confidence in the THOR FEM for use in predicting THOR ATD responses for conditions, such as those observed in spacecraft landing, and for use in evaluating THOR ATD biofidelity.

Isotropic incompressible hyperelastic models for modelling the mechanical behaviour of biological tissues: a review

Modelling the mechanical behaviour of biological tissues is of vital importance for clinical applications. It is necessary for surgery simulation, tissue engineering, finite element modelling of soft tissues, etc. The theory of linear elasticity is frequently used to characterise biological tissues; however, the theory of nonlinear elasticity using hyperelastic models, describes accurately the nonlinear tissue response under large strains. The aim of this study is to provide a review of constitutive equations based on the continuum mechanics approach for modelling the rate-independent mechanical behaviour of homogeneous, isotropic and incompressible biological materials. The hyperelastic approach postulates an existence of the strain energy function – a scalar function per unit reference volume, which relates the displacement of the tissue to their corresponding stress values. The most popular form of the strain energy functions as Neo-Hookean, Mooney-Rivlin, Ogden, Yeoh, Fung-Demiray, Veronda-Westmann, Arruda-Boyce, Gent and their modifications are described and discussed considering their ability to analytically characterise the mechanical behaviour of biological tissues. The review provides a complete and detailed analysis of the strain energy functions used for modelling the rate-independent mechanical behaviour of soft biological tissues such as liver, kidney, spleen, brain, breast, etc.

The in vivo plantar soft tissue mechanical property under the metatarsal head: implications of tissues׳ joint-angle dependent response in foot finite element modeling

Material properties of the plantar soft tissue have not been well quantified in vivo (i.e., from life subjects) nor for areas other than the heel pad. This study explored an in vivo investigation of the plantar soft tissue material behavior under the metatarsal head (MTH). We used a novel device collecting indentation data at controlled metatarsophalangeal joint angles. Combined with inverse analysis, tissues׳ joint-angle dependent material properties were identified. The results showed that the soft tissue under MTH exhibited joint-angle dependent material responses, and the computed parameters using the Ogden material model were 51.3% and 30.9% larger in the dorsiflexed than in the neutral positions, respectively. Using derived parameters in subject-specific foot finite element models revealed only those models that used tissues׳ joint-dependent responses could reproduce the known plantar pressure pattern under the MTH. It is suggested that, to further improve specificity of the personalized foot finite element models, quantitative mechanical properties of the tissue inclusive of the effects of metatarsophalangeal joint dorsiflexion are needed.

An inverse method to determine the mechanical properties of the iris in vivo

BACKGROUND

Understanding the mechanical properties of the iris can help to have an insight into the eye diseases with abnormalities of the iris morphology. Material parameters of the iris were simply calculated relying on the ex vivo experiment. However, the mechanical response of the iris in vivo is different from that ex vivo, therefore, a method was put forward to determine the material parameters of the iris using the optimization method in combination with the finite element method based on the in vivo experiment.

MATERIAL AND METHODS

Ocular hypertension was induced by rapid perfusion to the anterior chamber, during perfusion intraocular pressures in the anterior and posterior chamber were record by sensors, images of the anterior segment were captured by the ultrasonic system. The displacement of the characteristic points on the surface of the iris was calculated. A finite element model of the anterior chamber was developed using the ultrasonic image before perfusion, the multi-island genetic algorithm was employed to determine the material parameters of the iris by minimizing the difference between the finite element simulation and the experimental measurements.

RESULTS

Material parameters of the iris in vivo were identified as the iris was taken as a nearly incompressible second-order Ogden solid. Values of the parameters μ1, α1, μ2 and α2 were 0.0861 ± 0.0080 MPa, 54.2546 ± 12.7180, 0.0754 ± 0.0200 MPa, and 48.0716 ± 15.7796 respectively. The stability of the inverse finite element method was verified, the sensitivity of the model parameters was investigated.

CONCLUSION

Material properties of the iris in vivo could be determined using the multi-island genetic algorithm coupled with the finite element method based on the experiment.

A numerical investigation on the variation in hip injury tolerance with occupant posture during frontal collisions

OBJECTIVE

More than half of occupant lower extremity (LEX) injuries during automotive frontal crashes are in the knee-thigh-hip (KTH) complex. The objective of this study is to develop a detailed and biofidelic finite element (FE) occupant LEX model that may improve current understanding of mechanisms and thresholds of KTH injuries.

METHODS

Firstly, the pelvis, thigh-knee-hip, and foot models developed in our previous studies were connected into an occupant lower limb model. Further validations, including posterior cruciate ligament (PCL) stretching, thigh lateral loading, KT, and KTH impact loading were then performed to verify the injury predictability of the model under complex frontal and lateral loading corresponding to automotive impacts. Finally, a sensitivity study was performed with the whole lower limb model to investigate the effect of the hip joint angle to acetabulum injury tolerance in frontal impacts.

RESULTS

The whole lower limb model proved to be stable under severe impacts along the knee, foot, and lateral components. In addition, the biomechanical and injury responses predicted by the model correlated well with the corresponding test data. An increase in hip joint extension angle from -30 to +20° relative to neutral posture showed an increase of 19 to 58 percent hip injury tolerance.

CONCLUSIONS

The stability and biofidelity response of the pelvis-lower limb (PLEX) model indicates its potential application in future frontal and lateral impact FE simulations.

Statistical modeling of human liver incorporating the variations in shape, size, and material properties

The liver is one of the most frequently injured abdominal organs during motor vehicle crashes. Realistic numerical assessments of liver injury risk for the entire occupant population require incorporating inter-subject variations into numerical models. The main objective of this study was to quantify the shape variations of human liver in a seated posture and the statistical distributions of its material properties. Statistical shape analysis was applied to construct shape models of the livers of 15 adult human subjects, recorded in a typical seated (occupant) posture. The principal component analysis was then utilized to obtain the modes of variation, the mean model, and 95% statistical boundary shape models. In addition, a total of 52 tensile tests were performed on the parenchyma of three fresh human livers at four loading rates (0.01, 0.1, 1, and 10 s^-1) to characterize the rate-dependent and failure properties of the human liver. A FE-based optimization approach was employed to identify the material parameters of an Ogden material model for each specimen. The mean material parameters were then determined for each loading rate from the characteristic averages of the stress-strain curves, and a stochastic optimization approach was utilized to determine the standard deviations of the material parameters. Results showed that the first five modes of the human liver shape models account for more than 60% of the overall anatomical variations. The distributions of the material parameters combined with the mean and statistical boundary shape models could be used to develop probabilistic finite element (FE) models, which may help to better understand the variability in biomechanical responses and injuries to the abdominal organs under impact loading.

The influence of the specimen shape and loading conditions on the parameter identification of a viscoelastic brain model

The mechanical properties of brain under various loadings have been reported in the literature over the past 50 years. Step-and-hold tests have often been employed to characterize viscoelastic and nonlinear behavior of brain under high-rate shear deformation; however, the identification of brain material parameters is typically performed by neglecting the initial strain ramp and/or by assuming a uniform strain distribution in the brain samples. Using finite element (FE) simulations of shear tests, this study shows that these simplifications have a significant effect on the identified material properties in the case of cylindrical human brain specimens. Material models optimized using only the stress relaxation curve under predict the shear force during the strain ramp, mainly due to lower values of their instantaneous shear moduli. Similarly, material models optimized using an analytical approach, which assumes a uniform strain distribution, under predict peak shear forces in FE simulations. Reducing the specimen height showed to improve the model prediction, but no improvements were observed for cubic samples with heights similar to cylindrical samples. Models optimized using FE simulations show the closest response to the test data, so a FE-based optimization approach is recommended in future parameter identification studies of brain.