Validation and application of an intervertebral disc finite element model utilizing independently constructed tissue-level constitutive formulations that are nonlinear,anisotropic, and time-dependent

Finite element (FE) models are advantageous in the study of intervertebral disc mechanics as the stress–strain distributions can be determined throughout the tissue and the applied loading and material properties can be controlled and modified. However, the complicated nature of the disc presents a challenge in developing an accurate and predictive disc model, which has led to limitations in FE geometry, material constitutive models and properties, and model validation. The objective of this study was to develop a new FE model of the intervertebral disc, to validate the model?s nonlinear and time-dependent responses without tuning or calibration, and to evaluate the effect of changes in nucleus pulposus (NP), cartilaginous endplate (CEP), and annulus fibrosus (AF) material properties on the disc mechanical response. The new FE disc model utilized an analytically-based geometry. The model was created from the mean shape of human L4/L5 discs, measured from high-resolution 3D MR images and averaged using signed distance functions. Structural hyperelastic constitutive models were used in conjunction with biphasic-swelling theory to obtain material properties from recent tissue tests in confined compression and uniaxial tension. The FE disc model predictions fit within the experimental range (mean±95% confidence interval) of the disc?s nonlinear response for compressive slow loading ramp, creep, and stress-relaxation simulations. Changes in NP and CEP properties affected the neutral-zone displacement but had little effect on the final stiffness during slow-ramp compression loading. These results highlight the need to validate FE models using the disc?s full nonlinear response in multiple loading scenarios.     

A volumetric model for growth of arterial walls with arbitrary geometry and loads

Stress and deformation in arterial wall tissue are factors which may influence significantly its response and evolution. In this work we develop models based on nonlinear elasticity and finite element numerical solutions for the mechanical behaviour and the remodelling of the soft tissue of arteries, including anisotropy induced by the presence of collagen fibres. Remodelling and growth in particular constitute important features in order to interpret stenosis and atherosclerosis. The main object of this work is to model accurately volumetric growth, induced by fluid shear stress in the intima and local wall stress in arteries with patient-specific geometry and loads. The model is implemented in a nonlinear finite element setting which may be applied to realistic 3D geometries obtained from in vivo measurements. The capabilities of this method are demonstrated in several examples. Firstly a stenotic process on an idealised geometry induced by a non-uniform shear stress distribution is considered. Following the growth of a right coronary artery from an in vivo reconstructed geometry is presented. Finally, experimental measurements for growth under hypertension for rat carotid arteries are modelled.     

Identification of in vivo material and geometric parameters of a human aorta: toward patient-specific modeling of abdominal aortic aneurysm

Recent advances in computational modeling of vascular adaptations and the need for their extension to patient-specific modeling have introduced new challenges to the path toward abdominal aortic aneurysm modeling. First, the fundamental assumption in adaptation models, namely the existence of vascular homeostasis in normal vessels, is not easy to implement in a vessel model built from medical images. Second, subjecting the vessel wall model to the normal pressure often makes the configuration deviate from the original geometry obtained from medical images. To address those technical challenges, in this work, we propose a two-step optimization approach; first, we estimate constitutive parameters of a healthy human aorta intrinsic to the material by using biaxial test data and a weighted nonlinear least-squares parameter estimation method; second, we estimate the distributions of wall thickness and anisotropy using a 2-D parameterization of the vessel wall surface and a global approximation scheme integrated within an optimization routine. A direct search method is implemented to solve the optimization problem. The numerical optimization method results in a considerable improvement in both satisfying homeostatic condition and minimizing the deviation of geometry from the original shape based on in vivo images. Finally, the utility of the proposed technique for patient-specific modeling is demonstrated in a simulation of an abdominal aortic aneurysm enlargement.     

Long bone torsion: I. Effects of heterogeneity, anisotropy and geometric irregularity

The influences of heterogeneity, anisotropy and geometric irregularity on the unrestrained, linearly elastic torsional response of long bones are assessed. Longitudinal geometric variations contribute insignificantly to the torsional response for typical long bone geometries. Anisotropy, heterogeneity and transverse geometric irregularity significantly influence the torsional response. A procedure is discussed which uses an approximate means to characterize both heterogeneity and anisotropy in predicting the torsional response. The accuracy of circular and elliptical annulus models of the bone cross-sectional geometry are assessed by comparing the stress predictions of these simple models to those of finite element models of the bone geometry.     

A finite element model of stress-mediated vascular adaptation: application to abdominal aortic aneurysms

Despite rapid expansion of our knowledge of vascular adaptation, developing patient-specific models of diseased arteries is still an open problem. In this study, we extend existing finite element models of stress-mediated growth and remodelling of arteries to incorporate a medical image-based geometry of a healthy aorta and, then, simulate abdominal aortic aneurysm. Degradation of elastin initiates a local dilatation of the aorta while stress-mediated turnover of collagen and smooth muscle compensates the loss of elastin. Stress distributions and expansion rates during the aneurysm growth are studied for multiple spatial distribution functions of elastin degradation and kinetic parameters. Temporal variations of the degradation function are also investigated with either direct time-dependent degradation or stretch-induced degradation as possible biochemical and biomechanical mechanisms for elastin degradation. The results show that this computational model has the capability to capture the complexities of aneurysm progression due to variations of geometry, extent of damage and stress-mediated turnover as a step towards patient-specific modelling.     

Soft tissue modelling for applications in virtual surgery and surgical robotics

Soft tissue modelling has gained a great deal of importance, for a large part due to its application in surgical training simulators for minimally invasive surgery (MIS). This article provides a structured overview of different continuum-mechanical models that have been developed over the years. It aims at facilitating model choice for specific soft tissue modelling applications. According to the complexity of the model, different features of soft biological tissue will be incorporated, i.e. nonlinearity, viscoelasticity, anisotropy, heterogeneity and finally, tissue damage during deformation. A brief summary of experimental methods for material characterisation and an introduction to methods for geometric modelling are also provided. The overview is non-exhaustive, focusing on the most important general models and models with specific biological applications. A trade-off in complexity must be made for enabling real-time simulation, but still maintaining realistic representation of the organ deformation. Depending on the organ and tissue types, different models with emphasis on certain features will prove to be more appropriate, meaning the optimal model choice is organ and tissue-dependent.     

A finite element model of stress-mediated vascular adaptation: application to abdominal aortic aneurysms

Despite rapid expansion of our knowledge of vascular adaptation, developing patient-specific models of diseased arteries is still an open problem. In this study, we extend existing finite element models of stress-mediated growth and remodelling of arteries to incorporate a medical image-based geometry of a healthy aorta and, then, simulate abdominal aortic aneurysm. Degradation of elastin initiates a local dilatation of the aorta while stress-mediated turnover of collagen and smooth muscle compensates the loss of elastin. Stress distributions and expansion rates during the aneurysm growth are studied for multiple spatial distribution functions of elastin degradation and kinetic parameters. Temporal variations of the degradation function are also investigated with either direct time-dependent degradation or stretch-induced degradation as possible biochemical and biomechanical mechanisms for elastin degradation. The results show that this computational model has the capability to capture the complexities of aneurysm progression due to variations of geometry, extent of damage and stress-mediated turnover as a step towards patient-specific modelling.     

Soft tissue modelling for applications in virtual surgery and surgical robotics

Soft tissue modelling has gained a great deal of importance, for a large part due to its application in surgical training simulators for minimally invasive surgery (MIS). This article provides a structured overview of different continuum-mechanical models that have been developed over the years. It aims at facilitating model choice for specific soft tissue modelling applications. According to the complexity of the model, different features of soft biological tissue will be incorporated, i.e. nonlinearity, viscoelasticity, anisotropy, heterogeneity and finally, tissue damage during deformation. A brief summary of experimental methods for material characterisation and an introduction to methods for geometric modelling are also provided.

The overview is non-exhaustive, focusing on the most important general models and models with specific biological applications. A trade-off in complexity must be made for enabling real-time simulation, but still maintaining realistic representation of the organ deformation. Depending on the organ and tissue types, different models with emphasis on certain features will prove to be more appropriate, meaning the optimal model choice is organ and tissue-dependent.     

A novel methodology for generating 3D finite element models of the hip from 2D radiographs

Finite element (FE) modelling has been proposed as a tool for estimating fracture risk and patient-specific FE models are commonly based on computed tomography (CT). Here, we present a novel method to automatically create personalised 3D models from standard 2D hip radiographs. A set of geometrical parameters of the femur were determined from seven a–p hip radiographs and compared to the 3D femoral shape obtained from CT as training material; the error in reconstructing the 3D model from the 2D radiographs was assessed. Using the geometry parameters as the input, the 3D shape of another 21 femora was built and meshed, separating a cortical and trabecular compartment. The material properties were derived from the homogeneity index assessed by texture analysis of the radiographs, with focus on the principal tensile and compressive trabecular systems. The ability of these FE models to predict failure load as determined by experimental biomechanical testing was evaluated and compared to the predictive ability of DXA. The average reconstruction error of the 3D models was 1.77 mm (±1.17 mm), with the error being smallest in the femoral head and neck, and greatest in the trochanter. The correlation of the FE predicted failure load with the experimental failure load was r²=64% for the reconstruction FE model, which was significantly better (p<0.05) than that for DXA (r²=24%). This novel method for automatically constructing a patient-specific 3D finite element model from standard 2D radiographs shows encouraging results in estimating patient-specific failure loads.     

Does clinical data quality affect fluid-structure interaction simulations of patient-specific stenotic aortic valve models?

Numerical models are increasingly used in the cardiovascular field to reproduce, study and improve devices and clinical treatments. The recent literature involves a number of patient-specific models replicating the transcatheter aortic valve implantation procedure, a minimally invasive treatment for high-risk patients with aortic diseases. The representation of the actual patient’s condition with truthful anatomy, materials and working conditions is the first step toward the simulation of the clinical procedure.The aim of this work is to quantify how the quality of routine clinical data, from which the patient-specific models are built, affects the outputs of the numerical models representing the pathological condition of stenotic aortic valve.Seven fluid–structure interaction (FSI) simulations were performed, completed with a sensitivity analysis on patient-specific reconstructed geometries and boundary conditions. The structural parts of the models consisted of the aortic root, native tri-leaflets valve and calcifications. Ventricular and aortic pressure curves were applied to the fluid domain.The differences between clinical data and numerical results for the aortic valve area were less than 2% but reached 12% when boundary conditions and geometries were changed. The difference in the aortic stenosis jet velocity between measured and simulated values was less than 11% reaching 27% when the geometry was changed. The CT slice thickness was found to be the most sensitive parameter on the presented FSI numerical model.In conclusion, the results showed that the segmentation and reconstruction phases need to be carefully performed to obtain a truthful patient-specific domain to be used in FSI analyses.     

Numerical discretization-based estimation methods for ordinary differential equation models via penalized spline smoothing with applications in biomedical research

Differential equations are extensively used for modeling dynamics of physical processes in many scientific fields such as engineering, physics, and biomedical sciences. Parameter estimation of differential equation models is a challenging problem because of high computational cost and high-dimensional parameter space. In this article, we propose a novel class of methods for estimating parameters in ordinary differential equation (ODE) models, which is motivated by HIV dynamics modeling. The new methods exploit the form of numerical discretization algorithms for an ODE solver to formulate estimating equations. First, a penalized-spline approach is employed to estimate the state variables and the estimated state variables are then plugged in a discretization formula of an ODE solver to obtain the ODE parameter estimates via a regression approach. We consider three different order of discretization methods, Euler's method, trapezoidal rule, and Runge-Kutta method. A higher-order numerical algorithm reduces numerical error in the approximation of the derivative, which produces a more accurate estimate, but its computational cost is higher. To balance the computational cost and estimation accuracy, we demonstrate, via simulation studies, that the trapezoidal discretization-based estimate is the best and is recommended for practical use. The asymptotic properties for the proposed numerical discretization-based estimators are established. Comparisons between the proposed methods and existing methods show a clear benefit of the proposed methods in regards to the trade-off between computational cost and estimation accuracy. We apply the proposed methods t an HIV study to further illustrate the usefulness of the proposed approaches.     

The micromechanical environment of intervertebral disc cells: effect of matrix anisotropy and cell geometry predicted by a linear model

Cells of the intervertebral disc exhibit spatial variations in phenotype and morphology that may be related to differences in their local mechanical environments. In this study, the stresses, strains, and dilatations in and around cells of the intervertebral disc were studied with an analytical model of the cell as a mechanical inclusion embedded in a transversely isotropic matrix. In response to tensile loading of the matrix, the local mechanical environment of the cell differed among the anatomic regions of the disc and was strongly influenced by changes in both matrix anisotropy and parameters of cell geometry. The results of this study suggest that the local cellular mechanical environment may play a role in determining both cell morphology in situ and the inhomogeneous response to mechanical loading observed in cells of the disc.     

Finite element simulation of spinal deformities correction by in situ contouring technique

Biomechanical models have been proposed in order to simulate the surgical correction of spinal deformities. With these models, different surgical correction techniques have been examined: distraction and rod rotation. The purpose of this study was to simulate another surgical correction technique: the in situ contouring technique. In this way, a comprehensive three-dimensional Finite Element (FE) model with patient-specific geometry and patient-specific mechanical properties was used. The simulation of the surgery took into account elasto-plastic behavior of the rod and multiple moments loading and unloading representing the surgical maneuvers. The simulations of two clinical cases of hyperkyphosis and scoliosis were coherent with the surgeon's experience. Moreover, the results of simulation were compared to post-operative 3D measurements. The mean differences were under 5 degrees for vertebral rotations and 5 mm for spinal lines. These simulations open the way for future predictive tools for surgical planning.     

Determinants of biventricular cardiac function: a mathematical model study on geometry and myofiber orientation

In patient-specific mathematical models of cardiac electromechanics, usually a patient-specific geometry and a generic myofiber orientation field are used as input, upon which myocardial tissue properties are tuned to clinical data. It remains unclear to what extent deviations in myofiber orientation and geometry between model and patient influence model predictions on cardiac function. Therefore, we evaluated the sensitivity of cardiac function for geometry and myofiber orientation in a biventricular (BiV) finite element model of cardiac mechanics. Starting out from a reference geometry in which myofiber orientation had no transmural component, two new geometries were defined with either a 27 % decrease in LV short- to long-axis ratio, or a 16 % decrease of RV length, but identical LV and RV cavity and wall volumes. These variations in geometry caused differences in both local myofiber and global pump work below 6 %. Variation of fiber orientation was induced through adaptive myofiber reorientation that caused an average change in fiber orientation of \({\sim }8^\circ \) predominantly through the formation of a component in transmural direction. Reorientation caused a considerable increase in local myofiber work \(({\sim }18\,\%)\) and in global pump work \(({\sim }17\,\%)\) in all three geometries, while differences between geometries were below 5 %. The findings suggest that implementing a realistic myofiber orientation is at least as important as defining a patient-specific geometry. The model for remodeling of myofiber orientation seems a useful approach to estimate myofiber orientation in the absence of accurate patient-specific information.

     

Development of a patient-specific anatomical foot model from structured light scan data

The use of anatomically accurate finite element (FE) models of the human foot in research studies has increased rapidly in recent years. Uses for FE foot models include advancing knowledge of orthotic design, shoe design, ankle–foot orthoses, pathomechanics, locomotion, plantar pressure, tissue mechanics, plantar fasciitis, joint stress and surgical interventions. Similar applications but for clinical use on a per-patient basis would also be on the rise if it were not for the high costs associated with developing patient-specific anatomical foot models. High costs arise primarily from the expense and challenges of acquiring anatomical data via magnetic resonance imaging (MRI) or computed tomography (CT) and reconstructing the three-dimensional models. The proposed solution morphs detailed anatomy from skin surface geometry and anatomical landmarks of a generic foot model (developed from CT or MRI) to surface geometry and anatomical landmarks acquired from an inexpensive structured light scan of a foot. The method yields a patient-specific anatomical foot model at a fraction of the cost of standard methods. Average error for bone surfaces was 2.53 mm for the six experiments completed. Highest accuracy occurred in the mid-foot and lowest in the forefoot due to the small, irregular bones of the toes. The method must be validated in the intended application to determine if the resulting errors are acceptable.     

Dependence of mechanical behavior of the murine tail disc on regional material properties: a parametric finite element study

In vivo rodent tail models are becoming more widely used for exploring the role of mechanical loading on the initiation and progression of intervertebral disc degeneration. Historically, finite element models (FEMs) have been useful for predicting disc mechanics in humans. However, differences in geometry and tissue properties may limit the predictive utility of these models for rodent discs. Clearly, models that are specific for rodent tail discs and accurately simulate the disc's transient mechanical behavior would serve as important tools for clarifying disc mechanics in these animal models. An FEM was developed based on the structure, geometry, and scale of the mouse tail disc. Importantly, two sources of time-dependent mechanical behavior were incorporated: viscoelasticity of the matrix, and fluid permeation. In addition, a novel strain-dependent swelling pressure was implemented through the introduction of a dilatational stress in nuclear elements. The model was then validated against data from quasi-static tension-compression and compressive creep experiments performed previously using mouse tail discs. Finally, sensitivity analyses were performed in which material parameters of each disc subregion were individually varied. During disc compression, matrix consolidation was observed to occur preferentially at the periphery of the nucleus pulposus. Sensitivity analyses revealed that disc mechanics was greatly influenced by changes in nucleus pulposus material properties, but rather insensitive to variations in any of the endplate properties. Moreover, three key features of the model-nuclear swelling pressure, lamellar collagen viscoelasticity, and interstitial fluid permeation-were found to be critical for accurate simulation of disc mechanics. In particular, collagen viscoelasticity dominated the transient behavior of the disc during the initial 2200 s of creep loading, while fluid permeation governed disc deformation thereafter. The FEM developed in this study exhibited excellent agreement with transient creep behavior of intact mouse tail motion segments. Notably, the model was able to produce spatial variations in nucleus pulposus matrix consolidation that are consistent with previous observations in nuclear cell morphology made in mouse discs using confocal microscopy. Results of this study emphasize the need for including nucleus swelling pressure, collagen viscoelasticity, and fluid permeation when simulating transient changes in matrix and fluid stress/strain. Sensitivity analyses suggest that further characterization of nucleus pulposus material properties should be pursued, due to its significance in steady-state and transient disc mechanical response.     

Patient-specific modeling of dyssynchronous heart failure: A case study

The development and clinical use of patient-specific models of the heart is now a feasible goal. Models have the potential to aid in diagnosis and support decision-making in clinical cardiology. Several groups are now working on developing multi-scale models of the heart for understanding therapeutic mechanisms and better predicting clinical outcomes of interventions such as cardiac resynchronization therapy. Here we describe the methodology for generating a patient-specific model of the failing heart with a myocardial infarct and left ventricular bundle branch block. We discuss some of the remaining challenges in developing reliable patient-specific models of cardiac electromechanical activity, and identify some of the main areas for focusing future research efforts. Key challenges include: efficiently generating accurate patient-specific geometric meshes and mapping regional myofiber architecture to them; modeling electrical activation patterns based on cellular alterations in human heart failure, and estimating regional tissue conductivities based on clinically available electrocardiographic recordings; estimating unloaded ventricular reference geometry and material properties for biomechanical simulations; and parameterizing systemic models of circulatory dynamics from available hemodynamic measurements.     

Finite element simulation of spinal deformities correction by in situ contouring technique

Biomechanical models have been proposed in order to simulate the surgical correction of spinal deformities. With these models, different surgical correction techniques have been examined: distraction and rod rotation. The purpose of this study was to simulate another surgical correction technique: the in situ contouring technique. In this way, a comprehensive three-dimensional Finite Element (FE) model with patient-specific geometry and patient-specific mechanical properties was used. The simulation of the surgery took into account elasto–plastic behavior of the rod and multiple moments loading and unloading representing the surgical maneuvers. The simulations of two clinical cases of hyperkyphosis and scoliosis were coherent with the surgeon's experience. Moreover, the results of simulation were compared to post-operative 3D measurements. The mean differences were under 5° for vertebral rotations and 5 mm for spinal lines. These simulations open the way for future predictive tools for surgical planning.