Ravi Namani

Measurements of mechanical anisotropy in brain tissue and implications for transversely isotropic material models of white matter

White matter in the brain is structurally anisotropic, consisting largely of bundles of aligned, myelin-sheathed axonal fibers. White matter is believed to be mechanically anisotropic as well. Specifically, transverse isotropy is expected locally, with the plane of isotropy normal to the local mean fiber direction. Suitable material models involve strain energy density functions that depend on the I4 and I5 pseudo-invariants of the Cauchy-Green strain tensor to account for the effects of relatively stiff fibers. The pseudo-invariant I4 is the square of the stretch ratio in the fiber direction; I5 contains contributions of shear strain in planes parallel to the fiber axis. Most, if not all, published models of white matter depend on I4 but not on I5. Here, we explore the small strain limits of these models in the context of experimental measurements that probe these dependencies. Models in which strain energy depends on I4 but not I5 can capture differences in Youngs (tensile) moduli, but will not exhibit differences in shear moduli for loading parallel and normal to the mean direction of axons. We show experimentally, using a combination of shear and asymmetric indentation tests, that white matter does exhibit such differences in both tensile and shear moduli. Indentation tests were interpreted through inverse fitting of finite element models in the limit of small strains. Results highlight that: (1) hyperelastic models of transversely isotropic tissues such as white matter should include contributions of both the I4 and I5 strain pseudo-invariants; and (2) behavior in the small strain regime can usefully guide the choice and initial parameterization of more general material models of white matter.

Anisotropic Mechanical Properties of Magnetically Aligned Fibrin Gels Measured by Magnetic Resonance Elastography

The anisotropic mechanical properties of magnetically aligned fibrin gels were measured by magnetic resonance elastography (MRE) and by a standard mechanical test: unconfined compression. Soft anisotropic biomaterials are notoriously difficult to characterize, especially in vivo. MRE is well-suited for efficient, non-invasive, and non-destructive assessment of shear modulus. Direction-dependent differences in shear modulus were found to be statistically significant for gels polymerized at magnetic fields of 11.7 and 4.7 T compared to control gels. Mechanical anisotropy was greater in the gels polymerized at the higher magnetic field. These observations were consistent with results from unconfined compression tests. Analysis of confocal microscopy images of gels showed measurable alignment of fibrils in gels polymerized at 11.7 T. This study provides direct, quantitative measurements of the anisotropy in mechanical properties that accompanies fibril alignment in fibrin gels.

Elastic Characterization of Transversely Isotropic Soft Materials by Dynamic Shear and Asymmetric Indentation

The mechanical characterization of soft anisotropic materials is a fundamental challenge because of difficulties in applying mechanical loads to soft matter and the need to combine information from multiple tests. A method to characterize the linear elastic properties of transversely isotropic soft materials is proposed, based on the combination of dynamic shear testing (DST) and asymmetric indentation. The procedure was demonstrated by characterizing a nearly incompressible transversely isotropic soft material. A soft gel with controlled anisotropy was obtained by polymerizing a mixture of fibrinogen and thrombin solutions in a high field magnet (B = 11.7 T); fibrils in the resulting gel were predominantly aligned parallel to the magnetic field. Aligned fibrin gels were subject to dynamic (20-40 Hz) shear deformation in two orthogonal directions. The shear storage modulus was 1.08 ± 0. 42 kPa (mean ± std. dev.) for shear in a plane parallel to the dominant fiber direction, and 0.58 ± 0.21 kPa for shear in the plane of isotropy. Gels were indented by a rectangular tip of a large aspect ratio, aligned either parallel or perpendicular to the normal to the plane of transverse isotropy. Aligned fibrin gels appeared stiffer when indented with the long axis of a rectangular tip perpendicular to the dominant fiber direction. Three-dimensional numerical simulations of asymmetric indentation were used to determine the relationship between direction-dependent differences in indentation stiffness and material parameters. This approach enables the estimation of a complete set of parameters for an incompressible, transversely isotropic, linear elastic material.

Poroviscoelastic cartilage properties in the mouse from indentation

A method for fitting parameters in a poroviscoelastic (PVE) model of articular cartilage in the mouse is presented. Indentation is performed using two different sized indenters and then these data are fitted using a PVE finite element program and parameter extraction algorithm. Data from a smaller indenter, a 15 mum diameter flat-ended 60 deg cone, is first used to fit the viscoelastic (VE) parameters, on the basis that for this tip size the gel diffusion time (approximate time constant of the poroelastic (PE) response) is of the order of 0.1 s, so that the PE response is negligible. These parameters are then used to fit the data from a second 170 mum diameter flat-ended 60 deg cone for the PE parameters, using the VE parameters extracted from the data from the 15 mum tip. Data from tests on five different mouse tibial plateaus are presented and fitted. Parameter variation studies for the larger indenter show that for this case the VE and PE time responses overlap in time, necessitating the use of both models.

Reliability of structure tensors in representing soft tissues structure

Structure tensors have been applied as descriptors of tissue morphology for constitutive modeling. Here the reliability of these tensors in representing tissues structure is investigated by model simulations of a few examples of generated and measured planar fiber orientation distributions. Reliability was evaluated by comparing the data with the orientation density distribution recovered from the structure tensor representation, and with a orientation density recovered from an alternative representation by Fourier series of spherical harmonics. The results show that except for the case of uniform or close to uniform orientation distributions, the distributions recovered from the structure tensor fit the data poorly. On the other hand, orientation distributions recovered from Fourier series of spherical harmonics converge to the data distributions provided sufficient terms are included in the truncated series. These results suggest that the structure tensor is a reliable descriptor of tissue structure only under very limited cases.

Shear wave propagation in anisotropic soft tissues and gels

The propagation of shear waves in soft tissue can be visualized by magnetic resonance elastography (MRE) to characterize tissue mechanical properties. Dynamic deformation of brain tissue arising from shear wave propagation may underlie the pathology of blast-induced traumatic brain injury. White matter in the brain, like other biological materials, exhibits a transversely isotropic structure, due to the arrangement of parallel fibers. Appropriate mathematical models and well-characterized experimental systems are needed to understand wave propagation in these structures. In this paper we review the theory behind waves in anisotropic, soft materials, including small-amplitude waves superimposed on finite deformation of a nonlinear hyperelastic material. Some predictions of this theory are confirmed in experimental studies of a soft material with controlled anisotropy: magnetically-aligned fibrin gel.

Inverse finite element analysis of indentation tests to determine hyperelastic parameters of soft-tissue layers

Indentation is being used increasingly to probe the mechanical behaviour of cells, the extra-cellular matrix, and organs. Multiple hyperelastic parameters have been extracted from indentation data, but their accuracy has not been evaluated. Consequently, this paper examines whether two non-linear hyperelastic parameters or even a single generalized elastic modulus E can be extracted accurately from atomic force microscopy and nanoindentation force–depth data. Tissue was modelled as incompressible isotropic Mooney–Rivlin (MR), polynomial (POLY), or exponential (EXP) layers. For each model, n = 9 sets of the two hyperelastic parameters (all with R2 > 0.99) were extracted from an artificial ‘benchmark’ by running our inverse finite-element-based extraction algorithm multiple times. Error between the mean extracted E and the known ‘benchmark’ value was 5.5 per cent for an MR layer, 3.6 per cent for a POLY layer, and 20.6 per cent for an EXP layer. Errors in at least one of the two hyperelastic parameters reached 75 per cent for the MR layer, 33 per cent for the POLY layer, and 400 per cent for the EXP layer, indicating that indentation has limited potential to extract two or more hyperelastic parameters accurately. However, the generalized elastic modulus extracted from indentation could be used as a quantitative measure to evaluate the influence of altered processing conditions, pathology, etc.

Measurement of Brain Biomechanics in vivo by Magnetic Resonance Imaging

Computer models of head-brain biomechanics offer enormous potential for improved understanding and prevention of traumatic brain injury (TBI). However existing computer models remain controversial because their predictions have yet to be rigorously compared to measured biomechanical data. The nonlinear, anisotropic, viscoelastic, heterogeneous character of brain tissue, and the intricate connections between the brain and skull, all complicate modeling efforts. In order to make progress toward the goal of accurate simulation of TBI, experimental techniques to address these issues must be developed. In this paper we describe two magnetic resonance (MR) imaging techniques to characterize brain deformation, estimate brain material properties, and illuminate the boundary conditions between brain and skull. MR tagging is used to estimate displacement and strain fields in response to rigid-body acceleration of the skull, and MR elastography is used to visualize shear wave propagation induced by oscillatory loading at the surface of the skull.

Application of the Virtual Fields Method to Magnetic Resonance Elastography data

This paper deals with the application of the Virtual Fields Method to the identification of the shear modulus of a gel from Magnetic Resonance Elastography data. Volume deformation fields in the cube were recorded at different times during the harmonic loading and the full harmonic response has been reconstructed using Fast Fourier Transform. Strains were then obtained by direct spatial differentiation, without any smoothing. The VFM was then applied with inertial forces balancing out elastic forces, without including the loading force which was not measured here. It has been shown that the choice of the virtual field is critical with such a spatial wave deformation field. A wide range of spatially harmonic virtual fields has been tested at different times within the loading period. The identified shear modulus has been shown to be consistent and to correlate with the value obtained from a simplified approach based on the shear wave solution. This is a feasibility study, it will be extended in the future to heterogeneous materials with a more thorough procedure to build up relevant virtual fields.

Integrative model of coronary flow in anatomically based vasculature under myogenic, shear, and metabolic regulation

Coronary blood flow is regulated to match the oxygen demand of myocytes in the heart wall. Flow regulation is essential to meet the wide range of cardiac workload. The blood flows through a complex coronary vasculature of elastic vessels having nonlinear wall properties, under transmural heterogeneous myocardial extravascular loading. To date, there is no fully integrative flow analysis that incorporates global and local passive and flow control determinants. Here, we provide an integrative model of coronary flow regulation that considers the realistic asymmetric morphology of the coronary network, the dynamic myocardial loading on the vessels embedded in it, and the combined effects of local myogenic effect, local shear regulation, and conducted metabolic control driven by venous O2 saturation level. The model predicts autoregulation (approximately constant flow over a wide range of coronary perfusion pressures), reduced heterogeneity of regulated flow, and presence of flow reserve, in agreement with experimental observations. Furthermore, the model shows that the metabolic and myogenic regulations play a primary role, whereas shear has a secondary one. Regulation was found to have a significant effect on the flow except under extreme (high and low) inlet pressures and metabolic demand. Novel outcomes of the model are that cyclic myocardial loading on coronary vessels enhances the coronary flow reserve except under low inlet perfusion pressure, increases the pressure range of effective autoregulation, and reduces the network flow in the absence of metabolic regulation. Collectively, these findings demonstrate the utility of the present biophysical model, which can be used to unravel the underlying mechanisms of coronary physiopathology.