A number of topics from continuum mechanics are presented that are useful in developing mathematical models of brain neuromechanics. Part I reviews basic concepts used to describe deformation or distortion of brain tissue, strains, stresses, their connection and material stiffness. Part II presents concepts from viscoelasticity used to describe the time dependent response of brain tissue such as stress relaxation, creep, response to sinusoidal loading, energy dissipation, characteristic response times and their alteration due to non-mechanical influences. Part III describes recent modeling approaches that can be used to describe microstructural changes in materials due to large deformation, disease or other non-mechanical sources.

The knowledge of in vivo human brain deformation within the skull is essential in understanding brain injury mechanisms. Such measurements have become possible only in recent years thanks to the advancement of magnetic resonance imaging (MRI) technique. In this paper, we first study in vivo human brain deformation under mild impact induced by a 2cm head drop using tagged MRI and the harmonic phase (HARP) imaging analysis technique originally developed for cardiac motion analysis. A finite element (FE) simulation of mild impact is then carried out using a patient-specific 3-D head model. A reasonably good correlation is found between the predicted deformation field from FE modelling and the results from MRI-based assessment. It is found that the maximum deformations occur within a few milliseconds following the impact, which is during the first oscillation of the brain within the skull, with the maximum displacements of 2-3 mm and the maximum strains of 5-10%. To our knowledge, this study is the first attempt where the deformation field obtained by MRI-based assessment is correlated with the prediction of a corresponding FE model, and it is also the first validation of a FE brain injury model on in vivo human brain deformation data.

Cerebrospinal fluid (CSF) pulsations have been proposed as a possible causative mechanism for the ventricular enlargement that characterizes the neurological condition known as hydrocephalus. This paper summarizes recent work by the authors to anaylze the effect of CSF pulsations on brain tissue to determine if they are mechanically capable of enlarging the cerebral ventricles. First a poroelastic model is presented to analyze the interactions that occur between the fluid and porous solid constituents of brain tissue due to CSF pulsations. A viscoelastic model is then presented to analyze the effects of the fluid pulsations on the solid brain tissue. The combined results indicate that CSF pulsations in a healthy brain are incapable of causing tissue damage and thus the ventricular enlargement observed in hydrocephalus. Therefore they cannot be the primary cause of this condition. Finally, a hyper-viscoelastic model is presented and used to demonstrate that small long-term transmantle pressure gradients may be a possible cause of communicating hydrocephalus in infants.

In an infusion test the apparent rate of cerebrospinal fluid (CSF) production is temporarily increased through injection of fluid directly into the CSF system with the result that CSF pressure rises, in theory to a new plateau average, and the change in pressure level gives a measure of resistance to CSF outflow and the rate of approach to the plateau gives information about cerebral compliance. In the first part of this paper we give details of a two-fluid (blood and CSF) spherically symmetric poroelastic model that can simulate an infusion test which includes oscillations in blood pressure. This model has been applied to clinical data where the infusion rate and arterial blood pressure are input and an oscillatory CSF pressure is computed along with spatial parenchyma displacement, strain and local changes in CSF content. In the later part of this paper, the poroelastic model is simplified by spatial integration resulting in a one-compartment model that includes blood pressure oscillations but which, when they are ignored, reduces to a well known one-compartment model. When the arterial pressure pulsations are included, their interaction with a non-linear compliance results in solutions that have to be interpreted very carefully to predict parameter values.

Perhaps the greatest paradox in the hydrocephalus field is the failure of researchers to consistently measure transmantle pressure gradients (ventricle to subarachnoid space) in either human or animal models of the communicating form of the disorder. Without such a gradient, conceptualization of how ventricular distention occurs is diffcult. Based on evidence from both a mathematical model [35] and experiments in skin [51], we observed that the intraventricular injection of anti-β_1 integrin antibodies in rat brains results in a reduction of periventricular pressures to values below those monitored in the ventricles. In addition, many of these animals developed hydrocephalus [30]. We conclude that the dissociation of β_1 integrins from the surrounding matrix fibers generates pressure gradients favouring ventricular expansion suggesting a novel mechanism for hydrocephalus development. Several issues, however, need further clarification. If hydrostatic pressure declines in the periventricular tissues then uid absorption must occur. Aquaporin-4 (AQP4) is a likely candidate for this absorption as it is the predominant water channel in the brain. Indeed, when capillary function is negated, periventricular interstitial uid pressures increase after anti-β_1 integrin antibody administration. This suggests that capillary absorption of parenchymal water may play a pivotal role in the generation of pressure gradients in our hydrocephalus model. Focusing on these issues, we present two poroelastic models to investigate the role of intramantle pressure gradients in ventriculomegaly and to determine if integrin-matrix disassociation represents a complete causative mechanism for hydrocephalus development.

Normal pressure hydrocephalus (NPH) is a neurological condition that occurs in adults usually older than sixty years, characterized by an excessive accumulation of celebrospinal fluid in the brain ventricles in the absence of an elevated intracranial pressure. Although the first description of this disease has been given in 1965 by Hakim and Adams, the progress in improving the diagnosis and treatment outcome of NPH has been slow due mainly to the fact that the causes of NPH continue to remain unknown in most of the cases. The few existing biomechanical models of NPH existing in the literature are based on the bulk flow theory which requires an increased intracranial pressure and thus fail to properly explain the onset of NPH. The aim of the present paper is to formulate the first neuro-mechanical model that will couple the electro-chemical and mechanical properties of the brain. We assume that the brain tissue is a charged hydrated soft tissue made of a solid phase, an interstitial fluid phase and an ion phase with (for now) two monovalent ion species. Using our model, we will show that NPH could be caused by a change in the concentrations of Na^+ and Cl^- in the ventricular cerebrospinal fluid in the absence of an elevated intracranial pressure.

In 2009 the US National Center for Health Statistics showed that cancer is close to becoming the deadliest disease of modern times. Recent years have seen unprecedented advancements in medicine that have contributed to a substantial decrease in the death rates of some serious diseases such as heart disease, stroke, influenza and pneumonia. However, with cancer, equivalent scientific and technological advances have yet to be achieved. Theoretical models capable of explaining the fundamental mechanisms of tumor growth and making reliable predictions are urgently needed. These models can contribute considerably to the design of optimal, personalized therapies that will not only maximize treatment outcomes but also reduce health care costs. Recently [25] we have proposed a non-invasive way of classifying gliomas, primary brain tumors, based on their stiffness. The model uses image mass spectra of proteins present in gliomas and shows that the Young's modulus of a high grade glioma is at least 10kPa higher than the Young's modulus of a low grade glioma. In this paper we will use this model to investigate the effect of mechanics on the growth of gliomas. The proposed mechano-growth model is a non-linear evolution differential equation which is solved analytically using the Adomian method. The time evolution is represented in two ways: (1) using a classical first-order derivative, and (2) using a fractional order derivative. Our results show that the fractional order model captures a very interesting temporal multi-scale effect of tumor transition from low grade (benign) to high grade (malignant) glioma when a certain threshold of mechanical strain is reached in the tissue. For comparison, we also reproduce the results we presented in [25] when linearization is used to solve the evolution equations analytically.