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According to some theories, obstruction of CSF flow produces a pressure drop in the subarachnoid space in accordance with the Bernoulli theorem that explains the development of syringomyelia below the obstruction. However, Bernoulli's principle applies to inviscid stationary flow unlike CSF flow. Therefore, we performed a series of computational experiments to investigate the relationship between pressure drop, flow velocities, and obstructions under physiologic conditions. We created geometric models with dimensions approximating the spinal subarachnoid space with varying degrees of obstruction. Pressures and velocities for constant and oscillatory flow of a viscid fluid were calculated with the Navier-Stokes equations. Pressure and velocity along the length of the models were also calculated by the Bernoulli equation and compared with the results from the Navier-Stokes equations. In the models, fluid velocities and pressure gradients were approximately inversely proportional to the percentage of the channel that remained open. Pressure gradients increased minimally with 35% obstruction and with factors 1.4, 2.2 and 5.0 respectively with 60, 75 and 85% obstruction. Bernoulli's law underestimated pressure changes by at least a factor 2 and predicted a pressure increase downstream of the obstruction, which does not occur. For oscillatory flow the phase difference between pressure maxima and velocity maxima changed with the degree of obstruction. Inertia and viscosity which are not factored into the Bernoulli equation affect CSF flow. Obstruction of CSF flow in the cervical spinal canal increases pressure gradients and velocities and decreases the phase lag between pressure and velocity. [ABSTRACT FROM AUTHOR]

This article examines a 1D incompressible two-fluid model including artificial tensor diffusion. The aim is to obtain a formulation that provides convergent numerical solutions for all flow conditions within the stratified and the stratified wavy flow regime. With appropriate simplifications, the two-fluid model reduces to one momentum balance, one mass conservation and two algebraic equations. It has previously been established that a formulation that is well posed in possessing exclusively real characteristics can be obtained by including an axial diffusion term in the momentum balance. In this article, however, we demonstrate that this is not sufficient to obtain a system suitable for numerical simulations. Although the unbounded growth rates of the standard two-fluid model are eliminated, linear stability theory predicts that infinitesimal wavelengths still experience finite growth. This entails that grid refinement always will result in new unstable wavelengths being resolved. On the other hand, if artificial axial diffusion is added to both the mass and the momentum equations as suggested here, a cut-off wavelength is established below which all wavelengths are stable. Thus, a numerically converging model is formed, which retains the long-wavelength properties of the standard two-fluid model. The conclusions of the mathematical analysis are substantiated by numerical simulations of 1D gravity waves. [ABSTRACT FROM PUBLISHER]

An improved method for calculating smooth derivatives of multilinear finite element fields is presented. The method is also applicable to finite difference schemes. A simple one-dimensional model problem is discussed analytically, and the new method shows superior behaviour compared to existing popular smoothing techniques. Two- and three-dimensional problems involving isoparametric elements are tested, and the results show improvements over tile widely used global least-squares method, with lumped mass matrix, especially at the boundaries and for non-uniform meshes. [ABSTRACT FROM AUTHOR]

Background and Purpose: The Chiari I malformation, characterized by tonsils extending below the foramen magnum, has increased CSF velocities compared with those in healthy subjects. Measuring the effect of tonsillar herniation on CSF flow in humans is confounded by interindividual variation. The goal of this study was to determine the effect of herniated tonsils on flow velocity and pressure dynamics by using 3D computational models., Materials and Methods: A previously described 3D mathematic model of the normal subarachnoid space was modified by extending the tonsils inferiorly. The chamber created was compared with the anatomy of the subarachnoid space. Pressures and velocities were calculated by CFA methods for sinusoidal flow of a Newtonian fluid. Results were displayed as 2D color-coded plots and 3D animations. Pressure gradients and flow velocities were compared with those in the normal model. Velocity distributions were also compared with those in clinical images of CSF flow., Results: The model represented grossly the subarachnoid space of a patient with Chiari I malformation. Fluid flow patterns in the Chiari model were complex, with jets in some locations and stagnant flow in others. Flow jets, synchronous bidirectional flow, and pressure gradients were greater in the Chiari model than in the normal model. The distribution of flow velocities in the model corresponded well with those observed in clinical images of CSF flow in patients with Chiari I., Conclusions: Tonsillar herniation per se increases the pressure gradients and the complexity of flow patterns associated with oscillatory CSF flow.

Background and Purpose: How CSF flow varies with the anatomy of the subarachnoid space has not been sufficiently well studied. The goal of this study was to develop an idealized 3D computational model of the subarachnoid space and then to use this model to study the detailed spatiotemporal effects of anatomic variations on CSF pressures and velocities., Materials and Methods: We created a geometric model with a computer-assisted design program. The model contained a central structure for the brain and spinal cord axis and a second surrounding structure for the peripheral borders of the subarachnoid space. Model dimensions were adjusted to capture the main characteristics of the normal human posterior fossa and cervical spinal anatomy. CSF flow was modeled as water with a sinusoidal flow pattern in time. Velocities and pressures during craniocaudal and caudocranial flow were calculated with computational fluid dynamics (CFD) software. Simulated flow was compared with published phase-contrast MR imaging measurements of CSF flow in healthy human subjects., Results: The model contained geometric characteristics of the posterior fossa and spinal canal. Flow velocities varied with the time in the cycle and location in space. Flow velocities had spatial variations that resembled those in healthy human subjects. Reynolds numbers were moderate, showing a laminar flow regime. Pressure varied uniformly along the long axis of the model during craniocaudal and caudocranial flow., Conclusions: In an idealized geometric approximation of the human subarachnoid space, CSF velocities and pressures can be studied in spatiotemporal detail with mathematic models.