A Multiscale Pressure-Volume Model of Cerebrospinal Fluid Dynamics: Application to Hydrocephalus

ABSTRACT Hydrocephalus is a brain disease characterized by abnormalities in the cerebrospinal fluid (CSF) circulation. The treatment is surgical in nature and continues to suffer of poor outcomes. The first mathematical model for studying the CSF pressure-volume relationship in hydrocephalus was proposed by Marmarou in the 1970s. However, the model fails to fully capture the complex CSF dynamics controlled by CSF-brain tissue interactions. In this paper we use fractional calculus to introduce multiscaling effects in Marmarou's model. We solve our fractional order non-linear differential equation analytically using a modified Adomian decomposition method

Poiseuille Flow of a Non-Local Non-Newtonian Fluid with Wall Slip: A First Step in Modeling Cerebral Microaneurysms

Cerebral aneurysms and microaneurysms are abnormal vascular dilatations with high risk of rupture. An aneurysmal rupture could cause permanent disability and even death. Finding and treating aneurysms before their rupture is very difficult since symptoms can be easily attributed mistakenly to other common brain diseases. Mathematical models could highlight possible mechanisms of aneurysmal development and suggest specialized biomarkers for aneurysms. Existing mathematical models of intracranial aneurysms focus on mechanical interactions between blood flow and arteries. However, these models cannot be applied to microaneurysms since the anatomy and physiology at the length scale of cerebral microcirculation are different. In this paper, we propose a mechanism for the formation of microaneurysms that involves the chemo-mechanical coupling of blood and endothelial and neuroglial cells. We model the blood as a non-local non-Newtonian incompressible fluid and solve analytically the Poiseuille flow of such a fluid through an axi-symmetric circular rigid and impermeable pipe in the presence of wall slip. The spatial derivatives of the proposed generalization of the rate of deformation tensor are expressed using Caputo fractional derivatives. The wall slip is represented by the classic Navier law and a generalization of this law involving fractional derivatives. Numerical simulations suggest that hypertension could contribute to microaneurysmal formation

A Mathematical Investigation of Sex Differences in Alzheimer’s Disease

Alzheimer’s disease (AD) is an age-related degenerative disorder of the cerebral neuro-glial-vascular units. Not only are post-menopausal females, especially those who carry the APOE4 gene, at a higher risk of AD than males, but also AD in females appears to progress faster than in aged-matched male patients. Currently, there is no cure for AD. Mathematical models can help us to understand mechanisms of AD onset, progression, and therapies. However, existing models of AD do not account for sex differences. In this paper a mathematical model of AD is proposed that uses variable-order fractional temporal derivatives to describe the temporal evolutions of some relevant cells’ populations and aggregation-prone amyloid-β fibrils. The approach generalizes the model of Puri and Li. The variable fractional order describes variable fading memory due to neuroprotection loss caused by AD progression with age which, in the case of post-menopausal females, is more aggressive because of fast estrogen decrease. Different expressions of the variable fractional order are used for the two sexes and a sharper decreasing function corresponds to the female’s neuroprotection decay. Numerical simulations show that the population of surviving neurons has decreased more in post-menopausal female patients than in males at the same stage of the disease. The results suggest that if a treatment that may include estrogen replacement therapy is applied to female patients, then the loss of neurons slows down at later times. Additionally, the sooner a treatment starts, the better the outcome is

The Impact of Anomalous Diffusion on Action Potentials in Myelinated Neurons

Action potentials in myelinated neurons happen only at specialized locations of the axons known as the nodes of Ranvier. The shapes, timings, and propagation speeds of these action potentials are controlled by biochemical interactions among neurons, glial cells, and the extracellular space. The complexity of brain structure and processes suggests that anomalous diffusion could affect the propagation of action potentials. In this paper, a spatio-temporal fractional cable equation for action potentials propagation in myelinated neurons is proposed. The impact of the ionic anomalous diffusion on the distribution of the membrane potential is investigated using numerical simulations. The results show spatially narrower action potentials at the nodes of Ranvier when using spatial derivatives of the fractional order only and delayed or lack of action potentials when adding a temporal derivative of the fractional order. These findings could reveal the pathological patterns of brain diseases such as epilepsy, multiple sclerosis, and Alzheimer’s disease, which have become more prevalent in the latest years

Prediction of ventricular boundary evolutionin hydrocephalic brain via a combined level set and adaptive finite element mesh warping method

Hydrocephalus is a serious neurological disorder caused by abnormalities in cerebrospinal fluid (CSF) flow, resulting in large brain deformations and neuronal damage. Treatments drain the excess CSF from the ventricles either via a CSF shunt or an endoscopic third ventriculostomy. However, patients’responseto these treatments is poor. Mathematical models of hydrocephalus mechanics couldaid neurosurgeons in hydrocephalus treatment planning.Current models and corresponding computational simulations of hydrocephalus are still in their infancy, despitethis being a disease with serious long-term implications.We propose a novel geometriccomputational approach for tracking the evolution of hydrocephalic brain tissue –ventricular CSF interfacevia the level set method and an adaptive mesh warping technique. In our previous work [1], weevolved the ventricular boundary in 2D CT images which required a backtracking line search for obtaining valid intermediate meshes for use with FEMWARP, a finite element-based mesh warping method. In [2], we automatically evolved the ventricular boundary deformation for 2D CT imagesvia the level set method. To help surgeons determine where to implant a shunt, we also computed the brain ventricle volume evolution for 3D MR images using FEMWARP.In this work, we generalize the results from [1,2] and incorporate adaptive mesh refinement following mesh deformation. Based on the use of the solution gradientof the PDE in the mesh warping approach as an enrichment indicator, the mesh is refinedwhere the solution gradient islargeand is coarsened where it is small.We present computational simulations of the onset and treatment of pediatric hydrocephalus based on 2D CT images which demonstrate the success of our combined level set/adaptive finite element-based mesh warping approach

Computing and Information A MATHEMATICAL INVESTIGATION OF THE ROLE OF INTRACRANIAL PRESSURE PULSATIONS AND SMALL GRADIENTS IN THE PATHOGENESIS OF HYDROCEPHALUS

Abstract. 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