Development of a Finite Element Model of Decompressive Craniectomy

Decompressive craniectomy (DC), an operation whereby part of the skull is removed, is used in the management of patients with brain swelling. While the aim of DC is to reduce intracranial pressure, there is the risk that brain deformation and mechanical strain associated with the operation could damage the brain tissue. The nature and extent of the resulting strain regime is poorly understood at present. Finite element (FE) models of DC can provide insight into this applied strain and hence assist in deciding on the best surgical procedures. However there is uncertainty about how well these models match experimental data, which are difficult to obtain clinically. Hence there is a need to validate any modelling approach outside the clinical setting. This paper develops an axisymmetric FE model of an idealised DC to assess the key features of such an FE model which are needed for an accurate simulation of DC. The FE models are compared with an experimental model using gelatin hydrogel, which has similar poro-viscoelastic material property characteristics to brain tissue. Strain on a central plane of the FE model and the front face of the experimental model, deformation and load relaxation curves are compared between experiment and FE. Results show good agreement between the FE and experimental models, providing confidence in applying the proposed FE modelling approach to DC. Such a model should use material properties appropriate for brain tissue and include a more realistic whole head geometry.     

Constitutive model for brain tissue under finite compression

While advances in computational models of mechanical phenomena have made it possible to simulate dynamically complex problems in biomechanics, accurate material models for soft tissues, particularly brain tissue, have proven to be very challenging. Most studies in the literature on material properties of brain tissue are performed in shear loading and very few tackle the behavior of brain in compression. In this study, a viscoelastic constitutive model of bovine brain tissue under finite step-and-hold uniaxial compression with 10 s(-1) ramp rate and 20 s hold time has been developed. The assumption of quasi-linear viscoelasticity (QLV) was validated for strain levels of up to 35%. A generalized Rivlin model was used for the isochoric part of the deformation and it was shown that at least three terms (C(10), C(01) and C(11)) are needed to accurately capture the material behavior. Furthermore, for the volumetric deformation, a two parameter Ogden model was used and the extent of material incompressibility was studied. The hyperelastic material parameters were determined through extracting and fitting to two isochronous curves (0.06 s and 14 s) approximating the instantaneous and steady-state elastic responses. Viscoelastic relaxation was characterized at five decay rates (100, 10, 1, 0.1, 0 s(-1)) and the results in compression and their extrapolation to tension were compared against previous models.     

The influence of anisotropy on brain injury prediction

Traumatic Brain Injury (TBI) occurs when a mechanical insult produces damage to the brain and disrupts its normal function. Numerical head models are often used as tools to analyze TBIs and to measure injury based on mechanical parameters. However, the reliability of such models depends on the incorporation of an appropriate level of structural detail and accurate representation of the material behavior. Since recent studies have shown that several brain regions are characterized by a marked anisotropy, constitutive equations should account for the orientation-dependence within the brain. Nevertheless, in most of the current models brain tissue is considered as completely isotropic. To study the influence of the anisotropy on the mechanical response of the brain, a head model that incorporates the orientation of neural fibers is used and compared with a fully isotropic model. A simulation of a concussive impact based on a sport accident illustrates that significantly lowered strains in the axonal direction as well as increased maximum principal strains are detected for anisotropic regions of the brain. Thus, the orientation-dependence strongly affects the response of the brain tissue. When anisotropy of the whole brain is taken into account, deformation spreads out and white matter is particularly affected. The introduction of local axonal orientations and fiber distribution into the material model is crucial to reliably address the strains occurring during an impact and should be considered in numerical head models for potentially more accurate predictions of brain injury.     

Design and numerical implementation of a 3-D non-linear viscoelastic constitutive model for brain tissue during impact

Finite Element (FE) head models are often used to understand mechanical response of the head and its contents during impact loading in the head. Current FE models do not account for non-linear viscoelastic material behavior of brain tissue. We developed a new non-linear viscoelastic material model for brain tissue and implemented it in an explicit FE code. To obtain sufficient numerical accuracy for modeling the nearly incompressible brain tissue, deviatoric and volumetric stress contributions are separated. Deviatoric stress is modeled in a non-linear viscoelastic differential form. Volumetric behavior is assumed linearly elastic. Linear viscoelastic material parameters were derived from published data on oscillatory experiments, and from ultrasonic experiments. Additionally, non-linear parameters were derived from stress relaxation (SR) experiments at shear strains up to 20%. The model was tested by simulating the transient phase in the SR experiments not used in parameter determination (strains up to 20%, strain rates up to 8s(-1)). Both time- and strain-dependent behavior were predicted accurately (R2>0.96) for strain and strain rates applied. However, the stress was overestimated systematically by approximately 31% independent of strain(rate) applied. This is probably caused by limitations of the experimental data at hand.     

Mechanical properties of brain tissue in-vivo: experiment and computer simulation

Realistic computer simulation of neurosurgical procedures requires incorporation of the mechanical properties of brain tissue in the mathematical model. Possible applications of computer simulation of neurosurgery include non-rigid registration, virtual reality training and operation planning systems and robotic devices to perform minimally invasive brain surgery. A number of constitutive models of brain tissue, both single-phase and bi-phasic, have been proposed in recent years. The major deficiency of most of them, however, is the fact that they were identified using experimental data obtained in vitro and there is no certainty whether they can be applied in the realistic in vivo setting. In this paper we attempt to show that previously proposed by us hyper-viscoelastic constitutive model of brain tissue can be applied to simulating surgical procedures. An in vivo indentation experiment is described. The force-displacement curve for the loading speed typical for surgical procedures is concave upward containing no linear portion from which a meaningful elastic modulus might be determined. In order to properly analyse experimental data, a three-dimensional, non-linear finite element model of the brain was developed. Magnetic resonance imaging techniques were used to obtain geometric information needed for the model. The shape of the force-displacement curve obtained using the numerical solution was very similar to the experimental one. The predicted forces were about 31% lower than those recorded during the experiment. Having in mind that the coefficients in the model had been identified based on experimental data obtained in vitro, and large variability of mechanical properties of biological tissues, such agreement can be considered as very good. By appropriately increasing material parameters describing instantaneous stiffness of the tissue one is able, without changing the structure of the model, to reproduce experimental curve almost perfectly. Numerical studies showed also that the linear, viscoelastic model of brain tissue is not appropriate for the modelling brain tissue deformation even for moderate strains.     

Metabolic and functional studies on post-mortem human brain

The evidence that samples of human brain tissue obtained at autopsy may be used as starting material for the isolation of cellular and subcellular preparations which exhibit metabolic and functional activity when incubated in vitro has been reviewed. Supporting evidence has been found in data from model experiments which used animal brain as the source material. Active preparations have been obtained after considerable (up to 24 h) post mortem delays. Such findings are less surprising when the post mortem stability of key tissue components (enzymes, receptors, nucleic acids) and the retention of cellular integrity are examined. The data from these fields have been reviewed and their relevance to functional studies assessed. Studies which use human autopsy material must consider many additional sources of variation not found in experiments with animal brain and the major problems are briefly discussed. It is argued that functional experiments present few, if any, difficulties not already inherent in static analyses of autopsy material and some procedures which help to minimise these difficulties are outlined. Experimentation in this area is greatly aided by the finding that metabolically and functionally active preparations may be obtained from frozen tissue pieces. Dynamic studies provide a new approach for testing hypotheses of the mechanisms underlying human brain disorders and for studying the actions of neuroactive drugs in man.     

Inverse identification of region-specific hyperelastic material parameters for human brain tissue

The identification of material parameters accurately describing the region-dependent mechanical behavior of human brain tissue is crucial for computational models used to assist, e.g., the development of safety equipment like helmets or the planning and execution of brain surgery. While the division of the human brain into different anatomical regions is well established, knowledge about regions with distinct mechanical properties remains limited. Here, we establish an inverse parameter identification scheme using a hyperelastic Ogden model and experimental data from multi-modal testing of tissue from 19 anatomical human brain regions to identify mechanically distinct regions and provide the corresponding material parameters. We assign the 19 anatomical regions to nine governing regions based on similar parameters and microstructures. Statistical analyses confirm differences between the regions and indicate that at least the corpus callosum and the corona radiata should be assigned different material parameters in computational models of the human brain. We provide a total of four parameter sets based on the two initial Poisson’s ratios of 0.45 and 0.49 as well as the pre- and unconditioned experimental responses, respectively. Our results highlight the close interrelation between the Poisson’s ratio and the remaining model parameters. The identified parameters will contribute to more precise computational models enabling spatially resolved predictions of the stress and strain states in human brains under complex mechanical loading conditions.

     

A nonlinear biphasic model of flow-controlled infusion in brain: Fluid transport and tissue deformation analyses

A biphasic nonlinear mathematical model is proposed for the concomitant fluid transport and tissue deformation that occurs during constant flow rate infusions into brain tissue. The model takes into account material and geometrical nonlinearities, a hydraulic conductivity dependent on strain, and nonlinear boundary conditions at the infusion cavity. The biphasic equations were implemented in a custom written code assuming spherical symmetry and using an updated Lagrangian finite element algorithm. Results of the model showed that both, geometric and material nonlinearities play an important role in the physics of infusions, yielding important differences from infinitesimal analyses. Geometrical nonlinearities were mainly due to the significant enlargement of the infusion cavity, while variations of the parameters that describe the degree of nonlinearity of the stress–strain curve yielded significant differences in all distributions. For example, a parameter set showing stiffening under tension yielded maximum values of radial displacement and porosity not localized at the infusion cavity. On the other hand, a parameter set showing softening under tension yielded a slight decrease in the fluid velocity for a three-fold increase in the flow rate, which can be explained by the substantial increase of the infusion cavity, not considered in linear analyses. This study strongly suggests that significant enlargement of the infusion cavity is a real phenomenon during infusions that may produce collateral damage to brain tissue. Our results indicate that more experimental tests have to be undertaken in order to determine material nonlinearities of brain tissue over a range of strains. With better understanding of these nonlinear effects, clinicians may be able to develop protocols that can minimize the damage to surrounding tissue.     

Physical model simulations of brain injury in the primate

Diffuse brain injuries resulting from non-impact rotational acceleration are investigated with the aid of physical models of the skull-brain structure. These models provide a unique insight into the relationship between the kinematics of head motion and the associated deformation of the surrogate brain material. Human and baboon skulls filled with optically transparent surrogate brain tissue are subjected to lateral rotations like those shown to produce diffuse injury to the deep white matter in the brain of the baboon. High-speed cinematography captures the deformations of the grids embedded within the surrogate brain tissue during the applied load. The overall deformation pattern is compared to the pathological portrait of diffuse brain injury as determined from animal studies and autopsy reports. Shear strain and pathology spatial distributions mirror each other. Load levels and resulting surrogate brain tissue deformations are related from one species to the other. Increased primate brain mass magnified the strain amplified without significantly altering the spatial distribution. An empirically-derived value for a critical shear strain associated with the onset of severe diffuse axonal injury in primates is determined, assuming constitutive similarity between baboon and human brain tissue. The primate skull physical model data and the critical shear strain associated with the threshold for severe diffuse axonal injury were used to scale data obtained from previous studies to man, and thus derive a diffuse axonal injury tolerance for rotational acceleration for humans.     

Postmortem stability of somatostatin in brain tissue

The stability of somatostatin (SS) in brain tissue was studied in human material obtained post-mortem and in the rat. In both human and rat brain, loss of SS was found to occur in tissue frozen to –70°C. In the rat, this loss varied from 26 to 70 percent depending on the type of tissue processing used. These data suggest that, for the study of SS in postmortem brain, use of frozen material should be avoided.     

The mechanical behaviour of brain tissue: large strain response and constitutive modelling

The non-linear mechanical behaviour of porcine brain tissue in large shear deformations is determined. An improved method for rotational shear experiments is used, producing an approximately homogeneous strain field and leading to an enhanced accuracy. Results from oscillatory shear experiments with a strain amplitude of 0.01 and frequencies ranging from 0.04 to 16 Hz are given. The immediate loss of structural integrity, due to large deformations, influencing the mechanical behaviour of brain tissue, at the time scale of loading, is investigated. No significant immediate mechanical damage is observed for these shear deformations up to strains of 0.45. Moreover, the material behaviour during complex loading histories (loading-unloading) is investigated. Stress relaxation experiments for strains up to 0.2 and constant strain rate experiments for shear rates from 0.01 to 1 s(-1) and strains up to 0.15 are presented. A new differential viscoelastic model is used to describe the mechanical response of brain tissue. The model is formulated in terms of a large strain viscoelastic framework and considers non-linear viscous deformations in combination with non-linear elastic behaviour. This constitutive model is readily applicable in three-dimensional head models in order to predict the mechanical response of the intra-cranial contents due to an impact.     

The hydromechanics of hydrocephalus: Steady-state solutions for cylindrical geometry

Hydrocephalus is a state in which the circulation of cerebrospinal fluid is disturbed. This fluid, produced within the brain at a constant rate, moves through internal cavities in it (ventricles), then exits through passages so that it may be absorbed by the surrounding membranes (meninges). Failure of fluid to move properly through these passages results in the distention of the passages and the ventricles. Ultimately, this distention causes large displacements and distortion of brain tissue as well as an increase of fluid in the extracellular space of the brain (edema). We use a two-phase model of fluid-saturated material to simulate the steady state of the hydrocephalic brain. Analytic solutions for the displacement of brain tissue and the distribution of edema for the annular regions of an idealized cylindrical geometry and small-strain theory are found. The solutions are used for a large-deformation analysis by superposition of the responses obtained for incrementally increasing loading. The effects of structural and hydraulic differences of white and gray brain matter, and the ependymal lining surrounding the venticles, are examined. The results reproduce the characteristic steady-state distribution of edema seen in hydrocephalus, and are compared with experiment.     

Uptake and esterification of (4-14C) cholesterol after injection into the cerebral ventricles of conscious cats]

The brain tissue had appreciable capacity to take up and to esterify (4-14C) cholesterol from the cats cerebral ventricles. The highest values of radioactive material in the brain tissue were found 30 minutes after intraventricular injection of (4-14C) cholesterol. Thereafter, within a few hours most of the labelled cholesterol disappeared and by the end of 48 h the brain tissue retained only small amounts of it. at the same time, radioactive material was found in the peripheral venous blood, but in very small amounts. The analysis for cholesterol esters showed that about 50 % of (4-14C) cholesterol taken up by the brain tissue had undergone esterification. The rate of esterification of cholesterol within 48 h remained more or less constant, although a significant peak was noted by the end of 24 h.     

Constitutive model of brain tissue suitable for finite element analysis of surgical procedures.

Realistic finite element modelling and simulation of neurosurgical procedures present a formidable challenge. Appropriate, finite deformation, constitutive model of brain tissue is a prerequisite for such development. In this paper, a large deformation, linear, viscoelastic model, suitable for direct use with commercially available finite element software packages such as ABAQUS is constructed. The proposed constitutive equation is of polynomial form with time-dependent coefficients. The model requires four material constants to be identified. The material constants were evaluated based on unconfined compression experiment results. The analytical as well as numerical solutions to the unconfined compression problem are presented. The agreement between the proposed theoretical model and the experiment is good for compression levels reaching 30% and for loading velocities varying over five orders of magnitude. The numerical solution using the finite element method matched the analytical solution very closely.     

Constitutive modelling of brain tissue: Experiment and theory

Recent developments in computer-integrated and robot-aided surgery—in particular, the emergence of automatic surgical tools and robots—as well as advances in virtual reality techniques, call for closer examination of the mechanical properties of very soft tissues (such as brain, liver, kidney, etc.). The ultimate goal of our research into the biomechanics of these tissues is the development of corresponding, realistic mathematical models. This paper contains experimental results of in vitro, uniaxial, unconfined compression of swine brain tissue and discusses a single-phase, non-linear, viscoelastic tissue model. The experimental results obtained for three loading velocities, ranging over five orders of magnitude, are presented. The applied strain rates have been much lower than those applied in previous studies, focused on injury modelling. The stress-strain curves are concave upward for all compression rates containing no linear portion from which a meaningful elastic modulus might be determined. The tissue response stiffened as the loading speed increased, indicating a strong stress-strain rate dependence. The use of the single-phase model is recommended for applications in registration, surgical operation planning and training systems as well as a control system of an image-guided surgical robot. The material constants for the brain tissue are evaluated. Agreement between the proposed theoretical model and experiment is good for compression levels reaching 30% and for loading velocities varying over five orders of magnitude.     

Anisotropic constitutive equations and experimental tensile behavior of brain tissue

The present study deals with the experimental analysis and mechanical modeling of tensile behavior of brain soft tissue. A transversely isotropic hyperelastic model recently proposed by Meaney (2003) is adopted and mathematically studied under uniaxial loading conditions. Material parameter estimates are obtained through tensile tests on porcine brain materials accounting for regional and directional differences. Attention is focused on the short-term response. An extrapolation of tensile test data to the compression range is performed theoretically, to study the effect of the heterogeneity in the tensile/compressive response on the material parameters. Experimental and numerical results highlight the sensitivity of the adopted model to the test direction.     

Age-dependent material properties of the porcine cerebrum: effect on pediatric inertial head injury criteria

During growth and development, the immature central nervous system undergoes rapid alterations in constituents and structure. We hypothesize that these alterations are accompanied by changes in the mechanical properties of brain tissue which, in turn, influence the response of the brain to traumatic inertial loads. Samples of frontal cerebrum from neonatal (2–3 days) and adult pigs were harvested and tested within 3 h post-mortem. The complex shear modulus of the samples was measured in a custom-designed oscillatory shear testing device at engineering shear strain amplitudes of 2.5% or 5% from 20–200 Hz, at 25°C and 100% humidity. In this range, the elastic and viscous components of the complex shear modulus increased significantly with the development of the cerebral region of the brain. Using an idealized model of the developing head, the age-dependent material properties of brain tissue were shown to affect the mechanical response of the brain to inertial loading. This study is a first step toward developing head injury tolerance criteria specifically for the pediatric population.     

A variational constitutive model for soft biological tissues

In this paper, a fully variational constitutive model of soft biological tissues is formulated in the finite strain regime. The model includes Ogden-type hyperelasticity, finite viscosity, deviatoric and volumetric plasticity, rate and microinertia effects. Variational updates are obtained via time discretization and pre-minimization of a suitable objective function with respect to internal variables. Genetic algorithms are used for model parameter identification due to their suitability for non-convex, high dimensional optimization problems. The material behavior predicted by the model is compared to available tests on swine and human brain tissue. The ability of the model to predict a wide range of experimentally observed behavior, including hysteresis, cyclic softening, rate effects, and plastic deformation is demonstrated.     

On the unimportance of constitutive models in computing brain deformation for image-guided surgery

Imaging modalities that can be used intra-operatively do not provide sufficient details to confidently locate the abnormalities and critical healthy areas that have been identified from high-resolution pre-operative scans. However, as we have shown in our previous work, high quality pre-operative images can be warped to the intra-operative position of the brain. This can be achieved by computing deformations within the brain using a biomechanical model. In this paper, using a previously developed patient-specific model of brain undergoing craniotomy-induced shift, we conduct a parametric analysis to investigate in detail the influences of constitutive models of the brain tissue. We conclude that the choice of the brain tissue constitutive model, when used with an appropriate finite deformation solution, does not affect the accuracy of computed displacements, and therefore a simple linear elastic model for the brain tissue is sufficient.