Towards microstructure-informed material models for human brain tissue.

Emerging evidence suggests that the mechanical behavior of the brain plays a critical role in development, disease, and aging. Recent studies have begun to characterize the mechanical behavior of gray and white matter tissue and to identify sets of material models that best reproduce the stress-strain behavior of different brain regions. Yet, these models are mainly phenomenological in nature, their parameters often lack clear physical interpretation, and they fail to correlate the mechanical behavior to the underlying microstructural composition. Here we make a first attempt towards identifying general relations between microstructure and mechanics with the ultimate goal to develop microstructurally motivated constitutive equations for human brain tissue. Using histological staining, we analyze the microstructure of brain specimens from different anatomical regions, the cortex, basal ganglia, corona radiata, and corpus callosum, and identify the regional stiffness and viscosity under multiple loading conditions, simple shear, compression, and tension. Strikingly, our study reveals a negative correlation between cell count and stiffness, a positive correlation between myelin content and stiffness, and a negative correlation between proteoglycan content and stiffness. Additionally, our analysis shows a positive correlation between lipid and proteoglycan content and viscosity. We demonstrate how understanding the microstructural origin of the macroscopic behavior of the brain can help us design microstructure-informed material models for human brain tissue that inherently capture regional heterogeneities. This study represents an important step towards using brain tissue stiffness and viscosity as early diagnostic markers for clinical conditions including chronic traumatic encephalopathy, Alzheimer's and Parkinson's disease, or multiple sclerosis. STATEMENT OF SIGNIFICANCE: The complex and heterogeneous mechanical properties of brain tissue play a critical role for brain function. To understand and predict how brain tissue properties vary in space and time, it will be key to link the mechanical behavior to the underlying microstructural composition. Here we use histological staining to quantify area fractions of microstructural components of mechanically tested specimens and evaluate their individual contributions to the nonlinear macroscopic mechanical response. We further propose a microstructure-informed material model for human brain tissue that inherently captures regional heterogeneities. The current work provides unprecedented insights into the biomechanics of human brain tissue, which are highly relevant to develop refined computational models for brain tissue behavior or to advance neural tissue engineering.

Micromechanically-motivated analysis of fibrous tissue.

Collagen fibers are the main load bearing component in fibrous tissues. Systematic analyses of their structure and orientation are thus crucial for the development of material models that enable to predict the mechanical tissue response. To this end, biaxial tests at different stretch ratios were performed on two tissue samples of the medial layer extracted from a human aorta. The tissues were loaded in the circumferential and axial directions simultaneously. We develop here a micromechanical model which is based on structural parameters of collagen fibers that were extracted from second-harmonic generation images of the two samples. The tissue is modeled as a periodic six-layered laminate in which the individual layers are treated as periodic fibrous structures with one family of fibers. We make use of the Hill-Mandel theory in the context of periodic homogenization to determine the overall mechanical tissue response. Both the analytical and numerical models are able to capture the overall mechanical response of the two tissue samples using a straightforward representation of the tissue structure together with a limited set of material parameters. Up to 10% of strains the model captures the almost linear response of both tissue samples. Beyond that stretch level the stiffening of the tissues becomes more evident, especially in the circumferential direction. In cases where the axial stretch is larger than the circumferential stretch the predictions are somewhat stiffer, while a very good agreement is obtained when the circumferential stretch is dominant. The stiffening of one tissue sample was substantially larger than the other, implying that higher-order stiffening mechanisms may kick in at larger strains. Our sensitivity analyses reveal that the parameters of the material model and the fiber dispersion have a minor effect on the tissue response. The novel modeling approach has the potential to reduce the need of time-consuming experimental data of the mechanical behavior of fibrous tissues.

Multiscale FEM simulations of cross-linked actin network embedded in cytosol with the focus on the filament orientation.

The present contribution focuses on the application of the multiscale finite element method to the modeling of actin networks that are embedded in the cytosol. These cell components are of particular importance with regard to the cell response to external stimuli. The homogenization strategy chosen uses the Hill-Mandel macrohomogeneity condition for bridging 2 scales: the macroscopic scale that is related to the cell level and the microscopic scale related to the representative volume element. For the modeling of filaments, the Holzapfel-Ogden ß-model is applied. It provides a relationship between the tensile force and the caused stretches, serves as the basis for the derivation of the stress and elasticity tensors, and enables a novel finite element implementation. The elements with the neo-Hookean constitutive law are applied for the simulation of the cytosol. The results presented corroborate the main advantage of the concept, namely, its flexibility with regard to the choice of the representative volume element as well as of macroscopic tests. The focus is particularly placed on the study of the filament orientation and of its influence on the effective behavior.

Viscoelastic parameter identification of human brain tissue.

Understanding the constitutive behavior of the human brain is critical to interpret the physical environment during neurodevelopment, neurosurgery, and neurodegeneration. A wide variety of constitutive models has been proposed to characterize the brain at different temporal and spatial scales. Yet, their model parameters are typically calibrated with a single loading mode and fail to predict the behavior under arbitrary loading conditions. Here we used a finite viscoelastic Ogden model with six material parameters-an elastic stiffness, two viscoelastic stiffnesses, a nonlinearity parameter, and two viscous time constants-to model the characteristic nonlinearity, conditioning, hysteresis and tension-compression asymmetry of the human brain. We calibrated the model under shear, shear relaxation, compression, compression relaxation, and tension for four different regions of the human brain, the cortex, basal ganglia, corona radiata, and corpus callosum. Strikingly, unconditioned gray matter with 0.36kPa and white matter with 0.35kPa were equally stiff, whereas conditioned gray matter with 0.52kPa was three times stiffer than white matter with 0.18kPa. While both unconditioned viscous time constants were larger in gray than in white matter, both conditioned constants were smaller. These rheological differences suggest a different porosity between both tissues and explain-at least in part-the ongoing controversy between reported stiffness differences in gray and white matter. Our unconditioned and conditioned parameter sets are readily available for finite element simulations with commercial software packages that feature Ogden type models at finite deformations. As such, our results have direct implications on improving the accuracy of human brain simulations in health and disease.

Rheological characterization of human brain tissue.

The rheology of ultrasoft materials like the human brain is highly sensitive to regional and temporal variations and to the type of loading. While recent experiments have shaped our understanding of the time-independent, hyperelastic response of human brain tissue, its time-dependent behavior under various loading conditions remains insufficiently understood. Here we combine cyclic and relaxation testing under multiple loading conditions, shear, compression, and tension, to understand the rheology of four different regions of the human brain, the cortex, the basal ganglia, the corona radiata, and the corpus callosum. We establish a family of finite viscoelastic Ogden-type models and calibrate their parameters simultaneously for all loading conditions. We show that the model with only one viscoelastic mode and a constant viscosity captures the essential features of brain tissue: nonlinearity, pre-conditioning, hysteresis, and tension-compression asymmetry. With stiffnesses and time constants of µ∞=0.7kPa, µ1=2.0kPa, and τ1=9.7s in the gray matter cortex and µ∞=0.3kPa, µ1=0.9kPa and τ1=14.9s in the white matter corona radiata combined with negative parameters α∞ and α1, this five-parameter model naturally accounts for pre-conditioning and tissue softening. Increasing the number of viscoelastic modes improves the agreement between model and experiment, especially across the entire relaxation regime. Strikingly, two cycles of pre-conditioning decrease the gray matter stiffness by up to a factor three, while the white matter stiffness remains almost identical. These new insights allow us to better understand the rheology of different brain regions under mixed loading conditions. Our family of finite viscoelastic Ogden-type models for human brain tissue is simple to integrate into standard nonlinear finite element packages. Our simultaneous parameter identification of multiple loading modes can inform computational simulations under physiological conditions, especially at low to moderate strain rates. Understanding the rheology of the human brain will allow us to more accurately model the behavior of the brain during development and disease and predict outcomes of neurosurgical procedures. STATEMENT OF SIGNIFICANCE: While recent experiments have shaped our understanding of the time-independent, hyperelastic response of human brain tissue, its time-dependent behavior at finite strains and under various loading conditions remains insufficiently understood. In this manuscript, we characterize the rheology of human brain tissue through a family of finite viscoelastic Ogdentype models and identify their parameters for multiple loading modes in four different regions of the brain. We show that even the simplest model of this family, with only one viscoelastic mode and five material parameters, naturally captures the essential features of brain tissue: its characteristic nonlinearity, pre-conditioning, hysteresis, and tension-compression asymmetry. For the first time, we simultaneously identify a single parameter set for shear, compression, tension, shear relaxation, and compression relaxation loading. This parameter set is significant for computational simulations under physiological conditions, where loading is naturally of mixed mode nature. Understanding the rheology of the human brain will help us predict neurosurgical procedures, inform brain injury criteria, and improve the design of protective devices.

Mechanical characterization of human brain tissue.

Mechanics are increasingly recognized to play an important role in modulating brain form and function. Computational simulations are a powerful tool to predict the mechanical behavior of the human brain in health and disease. The success of these simulations depends critically on the underlying constitutive model and on the reliable identification of its material parameters. Thus, there is an urgent need to thoroughly characterize the mechanical behavior of brain tissue and to identify mathematical models that capture the tissue response under arbitrary loading conditions. However, most constitutive models have only been calibrated for a single loading mode. Here, we perform a sequence of multiple loading modes on the same human brain specimen - simple shear in two orthogonal directions, compression, and tension - and characterize the loading-mode specific regional and directional behavior. We complement these three individual tests by combined multiaxial compression/tension-shear tests and discuss effects of conditioning and hysteresis. To explore to which extent the macrostructural response is a result of the underlying microstructural architecture, we supplement our biomechanical tests with diffusion tensor imaging and histology. We show that the heterogeneous microstructure leads to a regional but not directional dependence of the mechanical properties. Our experiments confirm that human brain tissue is nonlinear and viscoelastic, with a pronounced compression-tension asymmetry. Using our measurements, we compare the performance of five common constitutive models, neo-Hookean, Mooney-Rivlin, Demiray, Gent, and Ogden, and show that only the isotropic modified one-term Ogden model is capable of representing the hyperelastic behavior under combined shear, compression, and tension loadings: with a shear modulus of 0.4-1.4kPa and a negative nonlinearity parameter it captures the compression-tension asymmetry and the increase in shear stress under superimposed compression but not tension. Our results demonstrate that material parameters identified for a single loading mode fail to predict the response under arbitrary loading conditions. Our systematic characterization of human brain tissue will lead to more accurate computational simulations, which will allow us to determine criteria for injury, to develop smart protection systems, and to predict brain development and disease progression. STATEMENT OF SIGNIFICANCE: There is a pressing need to characterize the mechanical behavior of human brain tissue under multiple loading conditions, and to identify constitutive models that are able to capture the tissue response under these conditions. We perform a sequence of experimental tests on the same brain specimen to characterize the regional and directional behavior, and we supplement our tests with DTI and histology to explore to which extent the macrostructural response is a result of the underlying microstructure. Results demonstrate that human brain tissue is nonlinear and viscoelastic, with a pronounced compression-tension asymmetry, and we show that the multiaxial data can best be captured by a modified version of the one-term Ogden model.

Stress softening and permanent deformation in human aortas: Continuum and computational modeling with application to arterial clamping.

Inelastic phenomena such as stress softening and unrecoverable inelastic deformations induced by supra-physiological loading have been observed experimentally in soft tissues such as arteries. These phenomena need to be accounted for in constitutive models of arterial tissues so that computational models can properly predict the outcome of interventional procedures such as arterial clamping and balloon angioplasty that involve non-physiological tissue loading. Motivated by experimental data, a novel pseudo-elastic damage model is proposed to describe discontinuous softening and permanent deformation in arterial tissues. The model is fitted to experimental data and specific material parameters for 9 abdominal and 14 thoracic aortas are provided. Furthermore, the model was implemented in a finite element code and numerically analyzed with respect to experimental tests, i.e. cyclic uniaxial tension in circumferential and longitudinal directions. Results showed that the model is able to capture specific features including anisotropy, nonlinearity, and damage-induced inelastic phenomena, i.e. stress softening and permanent deformation. Finite element results of a more complex boundary-value problem, i.e. aortic clamping considering the three aortic layers, residual stress, non-symmetric blood pressure after clamping, and patient-specific data are also presented.

Diameter-related variations of geometrical, mechanical, and mass fraction data in the anterior portion of abdominal aortic aneurysms.

OBJECTIVE: Maximum aortic diameter is an important measure in rupture prediction of abdominal aortic aneurysms (AAAs). Analyzing the variations of geometrical, material, and biochemical properties with increased AAA diameters advances understanding of the effect of lesion enlargement on patient specific vascular properties. METHODS: 96 AAA samples were harvested during open surgical aneurysm repair. Geometrical factors such as the maximum intraluminal thrombus (ILT) thickness, wall thickness, and AAA expansion rate were measured. Biaxial extension and peeling tests were performed to characterize the biaxial mechanical responses and to quantify the dissection properties of aneurysmal tissue. Mass fraction analysis quantified the dry weight percentages of elastin and collagen within the AAA wall. Linear regression models were used to correlate geometrical, mechanical, and mass fraction data with maximum AAA diameter. RESULTS: Both ILT thickness and AAA expansion rate increased and were positively correlated with maximum AAA diameter, while there was a slight increase in wall thickness for AAAs with a larger maximum diameter. For the biaxial mechanical responses, mean peak stretches and maximum tangential moduli in the circumferential and longitudinal axes did not correlate with maximum AAA diameters. However, the quantified energy to propagate tissue dissections within intima-media composites showed a significant inverse correlation with maximum AAA diameter. Elastin content decreased significantly with increasing AAA diameter. CONCLUSION: Larger AAA diameters are associated with thicker ILTs, higher AAA expansion rates, and pronounced elastin loss, and may also lead to a higher propensity for tissue dissection and aneurysm rupture.

Uniaxial tensile testing approaches for characterisation of atherosclerotic plaques.

The pathological changes associated with the development of atherosclerotic plaques within arterial vessels result in significant alterations to the mechanical properties of the diseased arterial wall. There are several methods available to characterise the mechanical behaviour of atherosclerotic plaque tissue, and it is the aim of this paper to review the use of uniaxial mechanical testing. In the case of atherosclerotic plaques, there are nine studies that employ uniaxial testing to characterise mechanical behaviour. A primary concern regarding this limited cohort of published studies is the wide range of testing techniques that are employed. These differing techniques have resulted in a large variance in the reported data making comparison of the mechanical behaviour of plaques from different vasculatures, and even the same vasculature, difficult and sometimes impossible. In order to address this issue, this paper proposes a more standardised protocol for uniaxial testing of diseased arterial tissue that allows for better comparisons and firmer conclusions to be drawn between studies. To develop such a protocol, this paper reviews the acquisition and storage of the tissue, the testing approaches, the post-processing techniques and the stress-strain measures employed by each of the nine studies. Future trends are also outlined to establish the role that uniaxial testing can play in the future of arterial plaque mechanical characterisation.

High-quality conforming hexahedral meshes of patient-specific abdominal aortic aneurysms including their intraluminal thrombi.

In order to perform finite element (FE) analyses of patient-specific abdominal aortic aneurysms, geometries derived from medical images must be meshed with suitable elements. We propose a semi-automatic method for generating conforming hexahedral meshes directly from contours segmented from medical images. Magnetic resonance images are generated using a protocol developed to give the abdominal aorta high contrast against the surrounding soft tissue. These data allow us to distinguish between the different structures of interest. We build novel quadrilateral meshes for each surface of the sectioned geometry and generate conforming hexahedral meshes by combining the quadrilateral meshes. The three-layered morphology of both the arterial wall and thrombus is incorporated using parameters determined from experiments. We demonstrate the quality of our patient-specific meshes using the element Scaled Jacobian. The method efficiently generates high-quality elements suitable for FE analysis, even in the bifurcation region of the aorta into the iliac arteries. For example, hexahedral meshes of up to 125,000 elements are generated in less than 130 s, with 94.8 % of elements well suited for FE analysis. We provide novel input for simulations by independently meshing both the arterial wall and intraluminal thrombus of the aneurysm, and their respective layered morphologies.

Variations of dissection properties and mass fractions with thrombus age in human abdominal aortic aneurysms.

INTRODUCTION: Thrombus ages, defined as four relative age phases, are related to different compositions of the intraluminal thrombus (ILT) in the abdominal aortic aneurysm (AAA) (Tong et al., 2011b). Experimental studies indicate a correlation between the relative thrombus age and the strength of the thrombus-covered wall. METHODS: On 32 AAA samples we performed peeling tests with the aim to dissect the material (i) through the ILT thickness, (ii) within the individual ILT layers and (iii) within the aneurysm wall underneath the thrombus by using two extension rates (1mm/min, 1mm/s). Histological investigations and mass fraction analysis were performed to characterize the dissected morphology, to determine the relative thrombus age, and to quantify dry weight percentages of elastin and collagen in the AAA wall. RESULTS: A remarkably lower dissection energy was needed to dissect within the individual ILT layers and through the thicknesses of old thrombi. With increasing ILT age the dissection energy of the underlying intima-media composite continuously decreased and the anisotropic dissection properties for that composite vanished. The quantified dissection properties were rate dependent for both tissue types (ILT and wall). Histology showed that single fibrin fibers or smaller protein clots within the ILT generate smooth dissected surfaces during the peeling. There was a notable decrease in mass fraction of elastin within the thrombus-covered intima-media composite with ILT age, whereas no significant change was found for that of collagen. CONCLUSIONS: These findings suggest that intraluminal thrombus aging leads to a higher propensity of dissection for the ILT and the intima-media composite of the aneurysmal wall.

A finite element-based constrained mixture implementation for arterial growth, remodeling, and adaptation: theory and numerical verification.

We implemented a constrained mixture model of arterial growth and remodeling in a nonlinear finite element framework to facilitate numerical analyses of diverse cases of arterial adaptation and maladaptation, including disease progression, resulting in complex evolving geometries and compositions. This model enables hypothesis testing by predicting consequences of postulated characteristics of cell and matrix turnover, including evolving quantities and orientations of fibrillar constituents and nonhomogenous degradation of elastin or loss of smooth muscle function. The nonlinear finite element formulation is general within the context of arterial mechanics, but we restricted our present numerical verification to cylindrical geometries to allow comparisons with prior results for two special cases: uniform transmural changes in mass and differential growth and remodeling within a two-layered cylindrical model of the human aorta. The present finite element model recovers the results of these simplified semi-inverse analyses with good agreement.

Gender differences in biomechanical properties, thrombus age, mass fraction and clinical factors of abdominal aortic aneurysms.

OBJECTIVE: The main purpose of the present study is the investigation of gender differences in the biomechanical properties, thrombus age, mass fraction and key clinical factors of abdominal aortic aneurysms (AAAs). MATERIALS AND METHODS: A total of 90 AAA samples (78 males and 12 females) were harvested from open surgical aneurysm repairs. Biaxial extension and peeling tests were performed to characterise the biaxial mechanical responses and to determine dissection properties of both the intraluminal thrombi (ILTs) and the thrombus-covered walls. Relative thrombus age was determined by characterising the ILT histological microstructure. Mass fraction analyses quantified dry weight percentages of elastin and collagen within the AAA walls. Moreover, we statistically compared clinical factors between male and female. RESULTS: The luminal layers of the female thrombi and the female AAA walls showed a significantly lower tissue stiffness (modulus) in the longitudinal direction when compared to males. Gender differences were also shown in the dissection properties of the intima-media composite within the AAA walls, in which a statistically significantly lower energy to propagate a dissection was quantified for females than for males. Moreover, 82% of female thrombi were relatively older (ILT age phases III and IV), twice that of male thrombi (43%). A pronounced lower elastin content was identified for the intima-media composites of male AAA walls, whereas female AAA walls had significantly lower dry weight percentages of collagen. Regarding clinical factors, nicotine pack years, serum creatinine and AAA expansion rate were found to be much higher for male patients. CONCLUSION: These findings may help to explain higher risks for AAA growth in males and the ruptures of smaller-sized AAAs in females.

Modeling sample/patient-specific structural and diffusional responses of cartilage using DT-MRI.

We propose a new 3D biphasic constitutive model designed to incorporate structural data on the sample/patient-specific collagen fiber network. The finite strain model focuses on the load-bearing morphology, that is, an incompressible, poroelastic solid matrix, reinforced by an inhomogeneous, dispersed fiber fabric, saturated with an incompressible fluid at constant electrolytic conditions residing in strain-dependent pores of the collagen-proteoglycan solid matrix. In addition, the fiber network of the solid influences the fluid permeability and an intrafibrillar portion that cannot be 'squeezed out' from the tissue. We implement the model into a finite element code. To demonstrate the utility of our proposed modeling approach, we test two hypotheses by simulating an indentation experiment for a human tissue sample. The simulations use ultra-high field diffusion tensor magnetic resonance imaging that was performed on the tissue sample. We test the following hypotheses: (i) the through-thickness structural arrangement of the collagen fiber network adjusts fluid permeation to maintain fluid pressure (Biomech. Model. Mechanobiol. 7:367-378, 2008); and (ii) the inhomogeneity of mechanical properties through the cartilage thickness acts to maintain fluid pressure at the articular surface (J. Biomech. Eng. 125:569-577, 2003). For the tissue sample investigated, both through-thickness inhomogeneities of the collagen fiber distribution and of the material properties serve to influence the interstitial fluid pressure distribution and maintain fluid pressure underneath the indenter at the cartilage surface. Tissue inhomogeneity appears to have a larger effect on fluid pressure retention in this tissue sample and on the advantageous pressure distribution.

Constrained Mixture Models as Tools for Testing Competing Hypotheses in Arterial Biomechanics: A Brief Survey.

Hypothesis testing via numerical models has emerged as a powerful tool which permits the verification of theoretical frameworks against canonical experimental and clinical observations. Cleverly designed computational experiments also inspire new methodologies by elucidating important biological processes and restricting parametric spaces. Constrained mixture models of arterial growth and remodeling (G&R) can facilitate the design of computational experiments which can bypass technical limitations in the laboratory, by considering illustrative special cases. The resulting data may then inform the design of focused experimental techniques and the development of improved theories. This work is a survey of computational hypothesis-testing studies, which exploit the unique abilities offered by the constrained mixture theory of arterial G&R. Specifically, we explore the core hypotheses integrated in these models, review their basic mathematical conceptualizations, and recapitulate their most salient and illuminating findings. We then assess how a decade's worth of constrained mixture models have contributed to a lucid, emerging picture of G&R mechanisms.

Remodeling of intramural thrombus and collagen in an Ang-II infusion ApoE-/- model of dissecting aortic aneurysms.

Fibrillar collagen endows the normal aortic wall with significant stiffness and strength and similarly plays important roles in many disease processes. For example, because of the marked loss of elastic fibers and functional smooth cells in aortic aneurysms, collagen plays a particularly important role in controlling the dilatation of these lesions and governing their rupture potential. Recent findings suggest further that collagen remodeling may also be fundamental to the intramural healing of arterial or aneurysmal dissections. To explore this possibility further, we identified and correlated regions of intramural thrombus and newly synthesized fibrillar collagen in a well-established mouse model of dissecting aortic aneurysms. Our findings suggest that intramural thrombus that is isolated from free-flowing blood creates a permissive environment for the synthesis of fibrillar collagen that, albeit initially less dense and organized, could protect that region of the dissected wall from subsequent expansion of the dissection or rupture. Moreover, alpha-smooth muscle actin positive cells appeared to be responsible for the newly produced collagen, which co-localized with significant production of glycosaminoglycans.

Mechanics, mechanobiology, and modeling of human abdominal aorta and aneurysms.

Biomechanical factors play fundamental roles in the natural history of abdominal aortic aneurysms (AAAs) and their responses to treatment. Advances during the past two decades have increased our understanding of the mechanics and biology of the human abdominal aorta and AAAs, yet there remains a pressing need for considerable new data and resulting patient-specific computational models that can better describe the current status of a lesion and better predict the evolution of lesion geometry, composition, and material properties and thereby improve interventional planning. In this paper, we briefly review data on the structure and function of the human abdominal aorta and aneurysmal wall, past models of the mechanics, and recent growth and remodeling models. We conclude by identifying open problems that we hope will motivate studies to improve our computational modeling and thus general understanding of AAAs.

Perspectives on biological growth and remodeling.

The continuum mechanical treatment of biological growth and remodeling has attracted considerable attention over the past fifteen years. Many aspects of these problems are now well-understood, yet there remain areas in need of significant development from the standpoint of experiments, theory, and computation. In this perspective paper we review the state of the field and highlight open questions, challenges, and avenues for further development.