A quasi-incompressible and quasi-inextensible finite element analysis of fibrous soft biological tissues

Biomechanics and Modeling in Mechanobiology - The contribution presents an extension and application of a recently proposed finite element formulation for quasi-inextensible and...     

A sub-domain inverse finite element characterization of hyperelastic membranes including soft tissues

Quantification of the mechanical behavior of hyperelastic membranes in their service configuration, particularly biological tissues, is often challenging because of the complicated geometry, material heterogeneity, and nonlinear behavior under finite strains. Parameter estimation thus requires sophisticated techniques like the inverse finite element method. These techniques can also become difficult to apply, however, if the domain and boundary conditions are complex (e.g. a non-axisymmetric aneurysm). Quantification can alternatively be achieved by applying the inverse finite element method over sub-domains rather than the entire domain. The advantage of this technique, which is consistent with standard experimental practice, is that one can assume homogeneity of the material behavior as well as of the local stress and strain fields. In this paper, we develop a sub-domain inverse finite element method for characterizing the material properties of inflated hyperelastic membranes, including soft tissues. We illustrate the performance of this method for three different classes of materials: neo-Hookean, Mooney Rivlin, and Fung-exponential.     

Finite element dynamic analysis of soft tissues using state-space model

A finite element (FE) model is employed to investigate the dynamic response of soft tissues under external excitations, particularly corresponding to the case of harmonic motion imaging. A solid 3D mixed ‘u–p’ element S8P0 is implemented to capture the near-incompressibility inherent in soft tissues. Two important aspects in structural modelling of these tissues are studied; these are the influence of viscous damping on the dynamic response and, following FE-modelling, a developed state-space formulation that valuates the efficiency of several order reduction methods. It is illustrated that the order of the mathematical model can be significantly reduced, while preserving the accuracy of the observed system dynamics. Thus, the reduced-order state-space representation of soft tissues for general dynamic analysis significantly reduces the computational cost and provides a unitary framework for the ‘forward’ simulation and ‘inverse’ estimation of soft tissues. Moreover, the results suggest that damping in soft-tissue is significant, effectively cancelling the contribution of all but the first few vibration modes.     

A finite nonlinear hyper-viscoelastic model for soft biological tissues

Soft tissues exhibit highly nonlinear rate and time-dependent stress-strain behaviour. Strain and strain rate dependencies are often modelled using a hyperelastic model and a discrete (standard linear solid) or continuous spectrum (quasi-linear) viscoelastic model, respectively. However, these models are unable to properly capture the materials characteristics because hyperelastic models are unsuited for time-dependent events, whereas the common viscoelastic models are insufficient for the nonlinear and finite strain viscoelastic tissue responses. The convolution integral based models can demonstrate a finite viscoelastic response; however, their derivations are not consistent with the laws of thermodynamics. The aim of this work was to develop a three-dimensional finite hyper-viscoelastic model for soft tissues using a thermodynamically consistent approach. In addition, a nonlinear function, dependent on strain and strain rate, was adopted to capture the nonlinear variation of viscosity during a loading process. To demonstrate the efficacy and versatility of this approach, the model was used to recreate the experimental results performed on different types of soft tissues. In all the cases, the simulation results were well matched ($R^2⩾0.99$) with the experimental data.     

Trabecular surface remodeling simulation for cancellous bone using microstructural voxel finite element models

A computational simulation method for three-dimensional trabecular surface remodeling was proposed, using voxel finite element models of cancellous bone, and was applied to the experimental data. In the simulation, the trabecular microstructure was modeled based on digital images, and its morphological changes due to surface movement at the trabecular level were directly expressed by removing/adding the voxel elements from/to the trabecular surface. A remodeling simulation at the single trabecular level under uniaxial compressive loading demonstrated smooth morphological changes even though the trabeculae were modeled with discrete voxel elements. Moreover, the trabecular axis rotated toward the loading direction with increasing stiffness, simulating functional adaptation to the applied load. In the remodeling simulation at the trabecular structural level, a cancellous bone cube was modeled using a digital image obtained by microcomputed tomography (microCT), and was uniaxially compressed. As a result, the apparent stiffness against the applied load increased by remodeling, in which the trabeculae reoriented to the loading direction. In addition, changes in the structural indices of the trabecular architecture coincided qualitatively with previously published experimental observations. Through these studies, it was demonstrated that the newly proposed voxel simulation technique enables us to simulate the trabecular surface remodeling and to compare the results obtained using this technique with the in vivo experimental data in the investigation of the adaptive bone remodeling phenomenon.     

Topology modification for surgical simulation using precomputed finite element models based on linear elasticity

Surgical simulators provide another tool for training and practising surgical procedures, usually restricted to the use of cadavers. Our surgical simulator utilises Finite Element (FE) models based on linear elasticity. It is driven by displacements, as opposed to forces, allowing for realistic simulation of both deformation and haptic response at real-time rates. To achieve demanding computational requirements, the stiffness matrix K, which encompasses the geometrical and physical properties of the object, is precomputed, along with K?1. Common to many surgical procedures is the requirement of cutting tissue. Introducing topology modifications, such as cutting, into these precomputed schemes does however come as a challenge, as the precomputed data needs to be modified, to reflect the new topology. In particular, recomputing K?1 is too costly to be performed during the simulation. Our topology modification method is based upon updating K?1 rather than entirely recomputing the matrix. By integrating condensation, we improve efficiency to allow for interaction with larger models. We can further enhance this by redistributing computational load to improve the system's real-time response. We exemplify our techniques with results from our surgical simulation system.     

A finite element model for protein transport in vivo

Background  

Biological mass transport processes determine the behavior and function of cells, regulate interactions between synthetic agents and recipient targets, and are key elements in the design and use of biosensors. Accurately predicting the outcomes of such processes is crucial to both enhancing our understanding of how these systems function, enabling the design of effective strategies to control their function, and verifying that engineered solutions perform according to plan.     

Quantification of soft tissue balance in total knee arthroplasty using finite element analysis

Unbalanced contact force on the tibial component has been considered a factor leading to loosening of the implant and increased wear of the bearing surface in total knee arthroplasty. Because it has been reported that good alignment cannot guarantee successful clinical outcomes, the soft tissue balance should be checked together with the alignment. Finite element models of patients' lower extremities were developed to analyse the medial and lateral contact force distribution on the tibial insert. The distributions for four out of five patients were not balanced equally, even though the alignment angles were within a clinically acceptable range. Moreover, the distribution was improved by changing soft tissue release and ligament tightening for the specific case. Integration of the biomechanical modelling, image matching and finite element analysis techniques with the patient-specific properties and various dynamic loading would suggest a clinically relevant pre-operative planning for soft tissue balancing.     

Density-based load estimation using two-dimensional finite element models: a parametric study

A parametric investigation was conducted to determine the effects on the load estimation method of varying: (1) the thickness of back-plates used in the two-dimensional finite element models of long bones, (2) the number of columns of nodes in the outer medial and lateral sections of the diaphysis to which the back-plate multipoint constraints are applied and (3) the region of bone used in the optimization procedure of the density-based load estimation technique. The study is performed using two-dimensional finite element models of the proximal femora of a chimpanzee, gorilla, lion and grizzly bear. It is shown that the density-based load estimation can be made more efficient and accurate by restricting the stimulus optimization region to the metaphysis/epiphysis. In addition, a simple method, based on the variation of diaphyseal cortical thickness, is developed for assigning the thickness to the back-plate. It is also shown that the number of columns of nodes used as multipoint constraints does not have a significant effect on the method.     

Density-based load estimation using two-dimensional finite element models: a parametric study

A parametric investigation was conducted to determine the effects on the load estimation method of varying: (1) the thickness of back-plates used in the two-dimensional finite element models of long bones, (2) the number of columns of nodes in the outer medial and lateral sections of the diaphysis to which the back-plate multipoint constraints are applied and (3) the region of bone used in the optimization procedure of the density-based load estimation technique. The study is performed using two-dimensional finite element models of the proximal femora of a chimpanzee, gorilla, lion and grizzly bear. It is shown that the density-based load estimation can be made more efficient and accurate by restricting the stimulus optimization region to the metaphysis/epiphysis. In addition, a simple method, based on the variation of diaphyseal cortical thickness, is developed for assigning the thickness to the back-plate. It is also shown that the number of columns of nodes used as multipoint constraints does not have a significant effect on the method.     

Mechanical characterization of anisotropic planar biological soft tissues using finite indentation: experimental feasibility

Heart valve tissue engineering offers a promising alternative for current treatment and replacement strategies, e.g., synthetic or bioprosthetic heart valves. In vitro mechanical conditioning is an important tool for engineering strong, implantable heart valves. Detailed knowledge of the mechanical properties of the native tissue as well as the developing tissue construct is vital for a better understanding and control of the remodeling processes induced by mechanical conditioning. The nonlinear, anisotropic and inhomogeneous mechanical behavior of heart valve tissue necessitates a mechanical characterization method that is capable of dealing with these complexities. In a recent computational study we showed that one single indentation test, combining force and deformation gradient data, provides sufficient information for local characterization of nonlinear soft anisotropic tissue properties. In the current study this approach is validated in two steps. First, indentation tests with varying indenter sizes are performed on linear elastic PDMS rubbers and compared to tensile tests on the same specimen. For the second step, tissue constructs are engineered using uniaxial or equibiaxial static constrained culture conditions. Digital image correlation (DIC) is used to quantify the anisotropy in the tissue constructs. For both validation steps, material parameters are estimated by inverse fitting of a computational model to the experimental results.     

Fracture prediction for the proximal femur using finite element models: Part I--Linear analysis.

Over 90 percent of the more than 250,000 hip fractures that occur annually in the United States are the result of falls from standing height. Despite this, the stresses associated with femoral fracture from a fall have not been investigated previously. Our objectives were to use three-dimensional finite element models of the proximal femur (with geometries and material properties based directly on quantitative computed tomography) to compare predicted stress distributions for one-legged stance and for a fall to the lateral greater trochanter. We also wished to test the correspondence between model predictions and in vitro strain gage data and failure loads for cadaveric femora subjected to these loading conditions. An additional goal was to use the model predictions to compare the sensitivity of several imaging sites in the proximal femur which are used for the in vivo prediction of hip fracture risk. In this first of two parts, linear finite element models of two unpaired human cadaveric femora were generated. In Part II, the models were extended to include nonlinear material properties for the cortical and trabecular bone. While there was poor correspondence between strain gage data and model predictions, there was excellent agreement between the in vitro failure data and the linear model, especially using a von Mises effective strain failure criterion. Both the onset of structural yielding (within 22 and 4 percent) and the load at fracture (within 8 and 5 percent) were predicted accurately for the two femora tested. For the simulation of one-legged stance, the peak stresses occurred in the primary compressive trabeculae of the subcapital region.(ABSTRACT TRUNCATED AT 250 WORDS)     

Fracture prediction for the proximal femur using finite element models: Part II--Nonlinear analysis.

In Part I we reported the results of linear finite element models of the proximal femur generated using geometric and constitutive data collected with quantitative computed tomography. These models demonstrated excellent agreement with in vitro studies when used to predict ultimate failure loads. In Part II, we report our extension of those finite element models to include nonlinear behavior of the trabecular and cortical bone. A highly nonlinear material law, originally designed for representing concrete, was used for trabecular bone, while a bilinear material law was used for cortical bone. We found excellent agreement between the model predictions and in vitro fracture data for both the onset of bone yielding and bone fracture. For bone yielding, the model predictions were within 2 percent for a load which simulated one-legged stance and 1 percent for a load which simulated a fall. For bone fracture, the model predictions were within 1 percent and 17 percent, respectively. The models also demonstrated different fracture mechanisms for the two different loading configurations. For one-legged stance, failure within the primary compressive trabeculae at the subcapital region occurred first, leading to load transfer and, ultimately, failure of the surrounding cortical shell. However, for a fall, failure of the cortical and trabecular bone occurred simultaneously within the intertrochanteric region. These results support our previous findings that the strength of the subcapital region is primarily due to trabecular bone whereas the strength of the intertrochanteric region is primarily due to cortical bone.     

Use of laser microprobe mass analysis (LAMMA) for localizing multiple elements in soft and hard tissues

The potential of laser microprobe mass analysis (LAMMA) as a sensitive microanalytical technique was explored in applications relevant to nephrology. Aluminum and associated elements, such as iron, were localized in fresh tissue biopsies obtained from uremic patients treatment by chronic hemodialysis. The LAMMA was applied to serum, liver, bone, and parathyroid glands of such patients. In addition, we used LAMMA to evaluate the specificity and sensitivity of routine histochemistry, in particular on human bone sections stained by the aluminon method. The high, multielemental sensitivity and molecular microprobe potential of LAMMA established important advantages over other microchemical methods forin situ analysis at the micron level in histological sections.     

Biphasic finite element modeling of hydrated soft tissue contact using an augmented Lagrangian method

A study of biphasic soft tissues contact is fundamental to understanding the biomechanical behavior of human diarthrodial joints. To date, biphasic-biphasic contact has been developed for idealized geometries and not been accessible for more general geometries. In this paper a finite element formulation is developed for contact of biphasic tissues. The augmented Lagrangian method is used to enforce the continuity of contact traction and fluid pressure across the contact interface, and the resulting method is implemented in the commercial software COMSOL Multiphysics. The accuracy of the implementation is verified using 2D axisymmetric problems, including indentation with a flat-ended indenter, indentation with spherical-ended indenter, and contact of glenohumeral cartilage layers. The biphasic finite element contact formulation and its implementation are shown to be robust and able to handle physiologically relevant problems.     

On modelling large deformations of heterogeneous biological tissues using a mixed finite element formulation

This study addresses the issue of modelling material heterogeneity of incompressible bodies. It is seen that when using a mixed (displacement–pressure) finite element formulation, the basis functions used for pressure field may not be able to capture the nonlinearity of material parameters, resulting in pseudo-residual stresses. This problem can be resolved by modifying the constitutive relation using Flory's decomposition of the deformation gradient. A two-parameter Mooney–Rivlin constitutive relation is used to demonstrate the methodology. It is shown that for incompressible materials, the modification does not alter the mechanical behaviour described by the original constitutive model. In fact, the modified constitutive equation shows a better predictability when compared against analytical solutions. Two strategies of describing the material variation (i.e. linear and step change) are explained, and their solutions are evaluated for an ideal two-material interfacing problem. When compared with the standard tied coupling approach, the step change method exhibited a much better agreement because of its ability to capture abrupt changes of the material properties. The modified equation in conjunction with integration point-based material heterogeneity is then used to simulate the deformations of heterogeneous biological structures to illustrate its applications.     

Automatic control of finite element models for temperature-controlled radiofrequency ablation

Background  

The finite element method (FEM) has been used to simulate cardiac and hepatic radiofrequency (RF) ablation. The FEM allows modeling of complex geometries that cannot be solved by analytical methods or finite difference models. In both hepatic and cardiac RF ablation a common control mode is temperature-controlled mode. Commercial FEM packages don't support automating temperature control. Most researchers manually control the applied power by trial and error to keep the tip temperature of the electrodes constant.     

Homogenized stiffness matrices for mineralized collagen fibrils and lamellar bone using unit cell finite element models

Mineralized collagen fibrils have been usually analyzed like a two-phase composite material where crystals are considered as platelets that constitute the reinforcement phase. Different models have been used to describe the elastic behavior of the material. In this work, it is shown that when Halpin–Tsai equations are applied to estimate elastic constants from typical constituent properties, not all crystal dimensions yield a model that satisfy thermodynamic restrictions. We provide the ranges of platelet dimensions that lead to positive definite stiffness matrices. On the other hand, a finite element model of a mineralized collagen fibril unit cell under periodic boundary conditions is analyzed. By applying six canonical load cases, homogenized stiffness matrices are numerically calculated. Results show a monoclinic behavior of the mineralized collagen fibril. In addition, a 5-layer lamellar structure is also considered where crystals rotate in adjacent layers of a lamella. The stiffness matrix of each layer is calculated applying Lekhnitskii transformations, and a new finite element model under periodic boundary conditions is analyzed to calculate the homogenized 3D anisotropic stiffness matrix of a unit cell of lamellar bone. Results are compared with the rule-of-mixtures showing in general good agreement.     

A penetration-based finite element method for hyperelastic 3D biphasic tissues in contact. Part II: finite element simulations

The penetration method allows for the efficient finite element simulation of contact between soft hydrated biphasic tissues in diarthrodial joints. Efficiency of the method is achieved by separating the intrinsically nonlinear contact problem into a pair of linked biphasic finite element analyses, in which an approximate, spatially and temporally varying contact traction is applied to each of the contacting tissues. In Part I of this study, we extended the penetration method to contact involving nonlinear biphasic tissue layers, and demonstrated how to derive the approximate contact traction boundary conditions. The traction derivation involves time and space dependent natural boundary conditions, and requires special numerical treatment. This paper (Part II) describes how we obtain an efficient nonlinear finite element procedure to solve for the biphasic response of the individual contacting layers. In particular, alternate linearization of the nonlinear weak form, as well as both velocity-pressure, v-p, and displacement-pressure, u-p, mixed formulations are considered. We conclude that the u-p approach, with linearization of both the material law and the deformation gradients, performs best for the problem at hand. The nonlinear biphasic contact solution will be demonstrated for the motion of the glenohumeral joint of the human shoulder joint.