Nonlinear model of human descending thoracic aortic segments with residual stresses

The nonlinear static deformation of human descending thoracic aortic segments is investigated. The aorta segments are modeled as straight axisymmetric circular cylindrical shells with three hyperelastic anisotropic layers and residual stresses by using an advanced nonlinear shell theory with higher-order thickness deformation not available in commercial finite element codes. The residual stresses are evaluated in the closed configuration in an original way making use of the multiplicative decomposition. The model was initially validated through comparison with published numerical and experimental data for artery and aorta segments. Then, two different cases of healthy thoracic descending aorta segments were numerically simulated. Material data and residual stresses used in the models came from published layer-specific experiments for human aortas. The material model adopted in the study is the mechanically based Gasser–Ogden–Holzapfel, which takes into account collagen fiber dispersion. Numerical results present a difference between systolic and diastolic inner radii close to the data available in literature from in vivo measurements for the corresponding age groups. Constant length of the aortic segment between systolic and diastolic pressures was obtained for the material model that takes the dispersion of the fiber orientations into account.     

Axial prestretch and circumferential distensibility in biomechanics of abdominal aorta

Elastic arteries are significantly prestretched in an axial direction. This property minimises axial deformations during pressure cycle. Ageing-induced changes in arterial biomechanics, among others, are manifested via a marked decrease in the prestretch. Although this fact is well known, little attention has been paid to the effect of decreased prestretch on mechanical response. Our study presents the results of an analytical simulation of the inflation–extension behaviour of the human abdominal aorta treated as nonlinear, anisotropic, prestrained thin-walled as well as thick-walled tube with closed ends. The constitutive parameters and geometries for 17 aortas adopted from the literature were supplemented with initial axial prestretches obtained from the statistics of 365 autopsy measurements. For each aorta, the inflation–extension response was calculated three times, with the expected value of the initial prestretch and with the upper and lower confidence limit of the initial prestretch derived from the statistics. This approach enabled age-related trends to be evaluated bearing in mind the uncertainty in the prestretch. Despite significantly decreased longitudinal prestretch with age, the biomechanical response of human abdominal aorta changes substantially depending on the initial axial stretch was used. In particular, substituting the upper limit of initial prestretch gave mechanical responses which can be characterised by (1) low variation in axial stretch and (2) high circumferential distensibility during pressurisation, in contrast to the responses obtained for their weakly prestretched counterparts. The simulation also suggested the significant effect of the axial prestretch on the variation of axial stress in the pressure cycle. Finally, the obtained results are in accordance with the hypothesis that circumferential-to-axial stiffness ratio is the quantity relatively constant within this cycle.     

Wall shear in pulsatile flow

This paper deals with a mathematical attempt to determine the wall shear during normal flow of blood in the ascending and the descending thoracic aorta. A simple model is used, but the results obtained are in agreement with published experimental results for the descending thoracic aorta. It is suggested that the degree of fluctuation in the pressure gradient at a given station is the major factor in determining the level of wall shear at that point.     

Estimating material parameters of human skin in vivo

An accurate mathematical representation of the mechanical behaviour of human skin is essential when simulating deformations occurring in the skin during body movements or clinical procedures. In this study constitutive stress–strain relationships based on experimental data from human skin in vivo were obtained. A series of multiaxial loading experiments were performed on the forearms of four age- and gender matched subjects. The tissue geometry, together with recorded displacements and boundary forces, were combined in an analysis using finite element modelling. A non-linear optimization technique was developed to estimate values for the material parameters of a previously published constitutive law, describing the stress–strain relationship as a non-linear anisotropic membrane. Ten sets of material parameters where estimated from the experiments, showing considerable differences in mechanical behaviour both between individual subjects as well as mirrored body locations on a single subject. The accuracy of applications that simulate large deformations of human skin could be improved by using the parameters found from the in vivo experiments as described in this study.     

Development and validation of a human biomechanical model for rib fracture and thorax injuries in blunt impact

From 1990 to approximately 50,000–120,000 people die annually of road traffic accidents in China. Traffic accidents are the main cause of death of Chinese adults aged 15–45 years. This study aimed to determine the biomechanical response and injury tolerance of the human body in traffic accidents. The subject was a 35-year-old male with a height of 170 cm, weight of 70 kg and Chinese characteristics at the 50th percentile. Geometry was generated by computed tomography and magnetic resonance imaging. A human-body biomechanical model was then developed. The model featured in great detail the main anatomical characteristics of skeletal tissues, soft tissues and internal organs, including the head, neck, shoulder, thoracic cage, abdomen, spine, pelvis, pleurae and lungs, heart, aorta, arms, legs, and other muscle tissues and skeletons. The material properties of all tissues in the human body model were obtained from the literature. Material properties were developed in the LS-DYNA code to simulate the mechanical behaviour of the biological tissues in the human body. The model was validated against cadaver responses to frontal and side impact. The predicted model response reasonably agreed with the experimental data, and the model can further be used to evaluate thoracic injury in real-world crashes. We believe that the transportation industry can use numerical models in the future to simultaneously reduce physical testing and improve automotive safety.     

Material properties of the placenta under dynamic loading conditions

Trauma during pregnancy especially occurring during car crashes leads to many foetal losses. Numerical modelling is widely used in car occupant safety issue and injury mechanisms analysis and is particularly adapted to the pregnant woman. Material modelling of the gravid uterus tissues is crucial for injury risk evaluation especially for the abruption placentae which is widely assumed as the leading cause of foetal loss. Experimental studies on placenta behaviour in tension are reported in the literature, but none in compression to the authors' knowledge. This lack of data is addressed in this study. To complement the already available experimental literature data on the placenta mechanical behaviour and characterise it in a compression loading condition, 80 indentation tests on fresh placentae are presented. Hyperelastic like mean experimental stress versus strain and corridors are exposed. The results of the experimental placenta indentations compared with the tensile literature results tend to show a quasi-symmetrical behaviour of the tissue. An inverse analysis using simple finite element models has permitted to propose parameters for an Ogden material model for the placenta which exhibits a realistic behaviour in both tension and compression.     

Mathematical modelling of intra-aortic balloon pump

Background: Ischemic heart diseases now afflict thousands of Iranians and are the major cause of death in many industrialised countries. Mathematical modelling of an intra-aortic balloon pump (IABP) could provide a better understanding of its performance and help to represent blood flow and pressure in systemic arteries before and after inserting the pump.

Methods: A mathematical modelling of the whole cardiovascular system was formulated using MATLAB software. The block diagram of the model consists of 43 compartments. All the anatomical data was extracted from the physiological references. In the next stage, myocardial infarction (MI) was induced in the model by decreasing the contractility of the left ventricle. The IABP was mathematically modelled and inserted in the model in the thoracic aorta I artery just before the descending aorta. The effects of IABP on MI were studied using the mathematical model.

Results: The normal operation of the cardiovascular system was studied firstly. The pressure–time graphs of the ventricles, atriums, aorta, pulmonary system, capillaries and arterioles were obtained. The volume–time curve of the left ventricle was also presented. The pressure–time curves of the left ventricle and thoracic aorta I were obtained for normal, MI, and inserted IABP conditions. Model verification was performed by comparing the simulation results with the clinical observations reported in the literature.

Conclusions: IABP can be described by a theoretical model. Our model representing the cardiovascular system is capable of showing the effects of different pathologies such as MI and we have shown that MI effects can be reduced using IABP in accordance with the modelling results. The mathematical model should serve as a useful tool to simulate and better understand cardiovascular operation in normal and pathological conditions.     

Mechanical properties of the coronary vasculature: indirect evaluation.

Information on the mechanical properties of the coronary vascular bed can be obtained indirectly by modelling the vascular system. This indirect approach, unlike 'in vitro' measurements, allows to take into account the vasomotor conditions of the circulatory district as well as the effect of the surrounding embedding tissue on the vascular performance. An experimental manoeuvre of sudden occlusion and subsequent release of the thoracic descendent aorta on 5 anaesthetized dogs with open pericardium induces a step-like variation in the coronary perfusion pressure and the occurrence of oscillations in the mean coronary flow. Such a behaviour can be described using a second-order model ('windkessel'+inductance, which takes into account blood inertia in the large vessels). The value of the coefficients entering the equations have been obtained with a 'best-fit' procedure (minimum of the chi-squared variable) on the haemodynamical data. Coefficient variations are in agreement with the direct estimation of the myocardial compliance and volume, measured by Ultrasound Echocardiographic imaging (4-chamber projection mode).     

A transversely isotropic hyperelastic constitutive model of the PDL. Analytical and computational aspects

This study describes the development of a constitutive law for the modelling of the periodontal ligament (PDL) and its practical implementation into a commercial finite element code. The constitutive equations encompass the essential mechanical features of this biological soft tissue: non-linear behaviour, large deformations, anisotropy, distinct behaviour in tension and compression and the fibrous characteristics. The approach is based on the theory of continuum fibre-reinforced composites at finite strain where a compressible transversely isotropic hyperelastic strain energy function is defined. This strain energy density function is further split into volumetric and deviatoric contributions separating the bulk and shear responses of the material. Explicit expressions of the stress tensors in the material and spatial configurations are first established followed by original expressions of the elasticity tensors in the material and spatial configurations. As a simple application of the constitutive model, two finite element analyses simulating the mechanical behaviour of the PDL are performed. The results highlight the significance of integrating the fibrous architecture of the PDL as this feature is shown to be responsible for the complex strain distribution observed.     

Pulse wave velocities in the aorta

This paper is an analytical study on the pulse wave velocities in the aorta. In conformity to a physiologic state of loading, the distensibility of the vessel wall has been accounted for. The wall material is treated by using the theory of large elastic deformations. The orthotropicity of wall tissues and the effect of the surrounding tissues have been incorporated in the analysis which is based on the use of the strain energy function suggested by Vaishnavet al. Numerical values of the wave velocities of the canine middle descending thoracic aorta are computed by using the derived analytical expressions.     

A Transversely Isotropic Hyperelastic Constitutive Model of the PDL. Analytical and Computational Aspects

This study describes the development of a constitutive law for the modelling of the periodontal ligament (PDL) and its practical implementation into a commercial finite element code. The constitutive equations encompass the essential mechanical features of this biological soft tissue: non-linear behaviour, large deformations, anisotropy, distinct behaviour in tension and compression and the fibrous characteristics. The approach is based on the theory of continuum fibre-reinforced composites at finite strain where a compressible transversely isotropic hyperelastic strain energy function is defined. This strain energy density function is further split into volumetric and deviatoric contributions separating the bulk and shear responses of the material. Explicit expressions of the stress tensors in the material and spatial configurations are first established followed by original expressions of the elasticity tensors in the material and spatial configurations. As a simple application of the constitutive model, two finite element analyses simulating the mechanical behaviour of the PDL are performed. The results highlight the significance of integrating the fibrous architecture of the PDL as this feature is shown to be responsible for the complex strain distribution observed.     

Unsteady and three-dimensional simulation of blood flow in the human aortic arch

A three-dimensional and pulsatile blood flow in a human aortic arch and its three major branches has been studied numerically for a peak Reynolds number of 2500 and a frequency (or Womersley) parameter of 10. The simulation geometry was derived from the three-dimensional reconstruction of a series of two-dimensional slices obtained in vivo using CAT scan imaging on a human aorta. The numerical simulations were obtained using a projection method, and a finite-volume formulation of the Navier-Stokes equations was used on a system of overset grids. Our results demonstrate that the primary flow velocity is skewed towards the inner aortic wall in the ascending aorta, but this skewness shifts to the outer wall in the descending thoracic aorta. Within the arch branches, the flow velocities were skewed to the distal walls with flow reversal along the proximal walls. Extensive secondary flow motion was observed in the aorta, and the structure of these secondary flows was influenced considerably by the presence of the branches. Within the aorta, wall shear stresses were highly dynamic, but were generally high along the outer wall in the vicinity of the branches and low along the inner wall, particularly in the descending thoracic aorta. Within the branches, the shear stresses were considerably higher along the distal walls than along the proximal walls. Wall pressure was low along the inner aortic wall and high around the branches and along the outer wall in the ascending thoracic aorta. Comparison of our numerical results with the localization of early atherosclerotic lesions broadly suggests preferential development of these lesions in regions of extrema (either maxima or minima) in wall shear stress and pressure.     

Theoretical Study of Stress-Modulated Growth in the Aorta

A theoretical model is presented for stress-modulated growth in the aorta. The model consists of a pseudoelastic tube composed of two layers representing the intima/media and the adventitia. Finite volumetric growth is included by letting the time-rate of change of the zero-stress dimensions of each volume element depends linearly on the local stresses. After analyzing the model, we examine its fundamental growth response under changes in loads, material properties, and growth parameters. The behaviour of the model is quite sensitive to changes in material nonlinearity and in the coefficients of the growth law. Next, growth of the aorta is simulated during development and maturity. For an appropriate choice of the parameters, the model exhibits patterns of growth that agree qualitatively with known characteristics of aortic growth. Comparison of model results with published experimental data during hypertension in the rat shows good agreement in the time course of the vessel radii and residual strain. Finally, the implications of the results are discussed in the context of deducing a general mechanical growth law for soft tissues. The proposed model should be useful in studies to determine the biomechanical factor that regulates growth.     

In vivo characterization of the mechanical properties of human skin derived from MRI and indentation techniques

The human skin is an exceedingly complex and multi-layered material. This paper aims to introduce the application of the finite element analysis (FEA) to the in vivo characterization of the non-linear mechanical behaviour of three human skin layers. Indentation tests combined with magnetic resonance imaging (MRI) technique have been performed on the left dorsal forearm of a young man in order to reveal the mechanical behaviour of all skin layers. Using MRI images processing and a pre and post processor allows to make numerically individualized 2D model which consists of three skin layers and the muscles. FEA has been applied to simulate indentation tests. Neo-Hookean slightly compressible material model of two material constants (C(10), K) has been used to model the mechanical behaviour of the three skin layers and the muscles. The identification of material model parameters was done by applying Levenberg-Marquardt algorithm (LMA). Our methodology of identification provides a range of values for each constant. Range of values of different material properties of epidermis, dermis, hypodermis are respectively, C10(E)=0.12+/-0.06 MPa, C10(D)=1.11+/-0.09 MPa, C10(H)=0.42+/-0.05 KPa, K(E)=5.45+/-1.7 MPa, K(D)=29.6+/-1,28 MPa, K(H)=36.0+/-0.9 KPa.     

An in vivo parameter identification method for arteries: numerical validation for the human abdominal aorta

A method for identifying mechanical properties of arterial tissue in vivo is proposed in this paper and it is numerically validated for the human abdominal aorta. Supplied with pressure-radius data, the method determines six parameters representing relevant mechanical properties of an artery. In order to validate the method, 22 finite element arteries are created using published data for the human abdominal aorta. With these in silico abdominal aortas, which serve as mock experiments with exactly known material properties and boundary conditions, pressure-radius data sets are generated and the mechanical properties are identified using the proposed parameter identification method. By comparing the identified and pre-defined parameters, the method is quantitatively validated. For healthy abdominal aortas, the parameters show good agreement for the material constant associated with elastin and the radius of the stress-free state over a large range of values. Slightly larger discrepancies occur for the material constants associated with collagen, and the largest relative difference is obtained for the in situ axial prestretch. For pathological abdominal aortas incorrect parameters are identified, but the identification method reveals the presence of diseased aortas. The numerical validation indicates that the proposed parameter identification method is able to identify adequate parameters for healthy abdominal aortas and reveals pathological aortas from in vivo-like data.     

Determination of material parameters of the two-dimensional Holzapfel–Weizsäcker type model based on uniaxial extension data of arterial walls

Soft tissues are anisotropic materials yet a majority of mechanical property tests have been uniaxial, which often failed to recapitulate the tensile response in other directions. This paper aims to study the feasibility of determining material parameters of anisotropic tissues by uniaxial extension with a minimal loss of anisotropic information. We assumed that by preselecting a certain constitutive model, we could give the constitutive parameters based on uniaxial extension data from orthogonal strip samples. In our study, the Holzapfel–Weizsäcker type strain energy density function (H–W model) was used to determine the material parameters of arterial walls from two fresh donation bodies. The key points we applied were the relationships between strain components in uniaxial tensile tests and the methods of stochastic optimisation. Further numerical experiments were taken. The estimate–effect ratio, defined by the number of data with the precision of estimation less than 0.5% over whole size of data, was calculated to demonstrate the feasibility of our method. The material parameters for Chinese aorta and pulmonary artery were given with the maximum root mean square (RMS) errors 0.042, and the minimal estimate–effect ratio in numerical experiments was 90.79%. Our results suggest that the constitutive parameters of arterial walls can be determined from uniaxial extension data, given the passive mechanical behaviour governed by H–W model. This method may apply to other tissues using different constitutive models.     

A new constitutive model for multi-layered collagenous tissues

Collagenous tissues such as the aneurysmal wall or the aorta are multi-layered structures with the mean fibre alignments distinguishing one layer from another. A constitutive representation of the multiple collagen layers is not yet developed, and hence the aim of the present study. The proposed model is based on the constitutive theory of finite elasticity and is characterized by an anisotropic strain-energy function which takes the material structure into account. The passive tissue behaviour is modelled and the related mechanical response is assumed to be dominated by elastin and collagen. While elastin is modelled by the neo-Hookean material the constitutive response of collagen is assumed to be transversely isotropic for each individual layer and based on an exponential function. The proposed constitutive function is polyconvex which ensures material stability. The model has five independent material parameters, each of which has a clear physical interpretation: the initial stiffnesses of the collagen fabric in the two principal directions, the shear modulus pertaining to the non-collagenous matrix material, a parameter describing the level of nonlinearity of the collagen fabric, and the angle between the principal directions of the collagen fabric and the reference coordinate system. An extension-inflation test of the adventitia of a human femoral artery is simulated by means of the finite element method and an error function is minimized by adjusting the material parameters yielding a good agreement between the model and the experimental data.