Estimation of CSF flow resistance in the upper cervical spine.

Chiari I patients have increased CSF velocities in the foramen magnum due hypothetically to increased pressure gradients or reduced flow resistance. We calculated flow resistance in the cervical spinal canal in a group of subjects with and without the Chiari malformation. Eight subjects including healthy volunteers and Chiari I patients were studied. From 3D high resolution MR images of the cervical spine mathematical models of the subarachnoid spaces were created by means of standard programs for segmentation and discretization. Oscillatory flow through the subarachnoid space was simulated. Cross-sectional area of the subarachnoid space was computed at each level from C1 through C4 and the length of this spinal canal segment was measured. Peak caudad CSF flow velocity at each level was plotted against cross-section area. CSF volumetric flux and resistance were calculated for each subject. The correlation between velocity and resistance was calculated. In all subjects, peak velocities increased progressively from C1 to C4 by 0.6 to 0.7 cm/s per level. Spinal canal areas diminished from C1 to C5 in each subject at a rate of -0.25 to -0.29 cm(2) per level. Resistance averaged 4.3 pascal/ml/s in the eight subjects; 3.8 pascal/ml/s in patients with tonsilar herniation and 6.0 pascal/ml/s in volunteers. Velocity correlated inversely with resistance (R(2) = 0.6). CSF velocities correlated inversely with the flow resistance in the upper cervical spinal canal. Resistance tends to be lower in Chiari I patients than in healthy volunteers.

Simulating CSF flow dynamics in the normal and the Chiari I subarachnoid space during rest and exertion.

BACKGROUND AND PURPOSE: CSF fluid dynamics in healthy subjects and patients with Chiari I have been characterized during rest with phase-contrast MR imaging and CFD. CSF flow velocities and pressures in the nonresting state have not been adequately characterized. We used computer simulations to study CSF dynamics during increased heart rates in the normal and Chiari I subarachnoid space. MATERIALS AND METHODS: Cyclic CSF flow was simulated for multiple cycles in idealized 3D models of the subarachnoid space for normal and Chiari I malformation subarachnoid spaces, with flow cycles corresponding to 80 or 120 heart beats per minute. Flow velocities and pressures were computed by the Navier-Stokes equations. Synchronous bidirectional flow and flow patterns were displayed in Star-CD and inspected visually. Peak velocities and pressure differences in the 2 models were compared for the 2-cycle frequencies. RESULTS: Elevating the cycle rate from 80 to 120 cpm increased peak superior-inferior pressure gradients (top-bottom) by just 0.01% in the normal model and 2% in the Chiari model. Corresponding average pressure gradients increased by 92% and 100%, respectively. In addition, in both models, the range of synchronous bidirectional flow velocities increased. Systolic velocities had smaller increases with faster cycling. For each cycle rate, peak and average pressure gradients in the Chiari model were greater than in the normal model by 11%-16%. CONCLUSIONS: Raising the cycle rate from 80 to 120 cpm increased superior-inferior average pressure gradients and the range of synchronous bidirectional flow velocities in the normal and Chiari I models.

Patient-specific 3D simulation of cyclic CSF flow at the craniocervical region.

BACKGROUND AND PURPOSE: Flow simulations in patient-specific models of the subarachnoid space characterize CSF flow in more detail than MR flow imaging. We extended previous simulation studies by including cyclic CSF flow and patient-specific models in multiple patients with Chiari I. We compared simulation results with MR flow measurements. MATERIALS AND METHODS: Volumetric high resolution image sets acquired in 7 patients with Chiari I, 3 patients who had previous craniovertebral decompression, and 3 controls were segmented and converted to mathematical models of the subarachnoid space. CSF flow velocities and pressures were calculated with high spatial and temporal resolution during simulated oscillatory flow in each model with the Navier-Stokes equations. Pressures, velocities, and bidirectional flow were compared in the groups (with Student t test). Peak velocities in the simulations were compared with peak velocities measured in vivo with PCMR. RESULTS: Flow visualization for patients and volunteers demonstrated nonuniform reversing patterns resembling those observed with PCMR. Velocities in the 13 subjects were greater between C2 and C5 than in the foramen magnum. Chiari patients had significantly greater peak systolic and diastolic velocities, synchronous bidirectional flow, and pressure gradients than controls. Peak velocities measured in PCMR correlated significantly (P = .003; regression analysis) despite differences between them. CONCLUSIONS: In simulations of CSF, patients with Chiari I had significantly greater peak systolic and diastolic velocities, synchronous bidirectional flow, and pressure gradients than controls.

Effect of tonsillar herniation on cyclic CSF flow studied with computational flow analysis.

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.

Characterization of cyclic CSF flow in the foramen magnum and upper cervical spinal canal with MR flow imaging and computational fluid dynamics.

CSF flow has been shown to exhibit complex patterns in MR images in both healthy subjects and in patients with Chiari I. Abnormal CSF flow oscillations, according to prevailing opinion, cause syringomyelia and other clinical manifestations that affect some patients with the Chiari I malformation. For this article, we reviewed the literature on PC MR of CSF flow, collected the published CFD studies relevant to CSF flow, and performed flow simulations. PC MR creates cine and still images of CSF flow and measurements of flow velocities. CFD, a technique used to compute flow and pressure in liquid systems, simulates the CSF flow patterns that occur in a specific geometry or anatomy of the SAS and a specific volume of flow. Published PC MR studies show greater peak CSF velocities and more complex flow patterns in patients with Chiari I than in healthy subjects, with synchronous bidirectional flow one of the characteristic markers of pathologic flow. In mathematic models of the SAS created from high-resolution MR images, CFD displays complex CSF flow patterns similar to those shown in PC MR in patients. CFD shows that the pressure and flow patterns vary from level to level in the upper spinal canal and differ between patients with Chiari and healthy volunteers. In models in which elasticity and motion are incorporated, CFD displays CSF pressure waves in the SAS. PC MR and CFD studies to date demonstrate significant alterations of CSF flow and pressure patterns in patients with Chiari I. CSF flow has nonlaminar complex spatial and temporal variations and associated pressure waves and pressure gradients. Additional simulations of CSF flow supplemented by PC MR will lead to better measures for distinguishing pathologic flow abnormalities that cause syringomyelia, headaches, and other clinical manifestations in Chiari I malformations.

CSF flow dynamics at the craniovertebral junction studied with an idealized model of the subarachnoid space and computational flow analysis.

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.

Numerical solution of the bidomain equations.

Knowledge of cardiac electrophysiology is efficiently formulated in terms of mathematical models. However, most of these models are very complex and thus defeat direct mathematical reasoning founded on classical and analytical considerations. This is particularly so for the celebrated bidomain model that was developed almost 40 years ago for the concurrent analysis of extra- and intracellular electrical activity. Numerical simulations based on this model represent an indispensable tool for studying electrophysiology. However, complex mathematical models, steep gradients in the solutions and complicated geometries lead to extremely challenging computational problems. The greatest achievement in scientific computing over the past 50 years has been to enable the solving of linear systems of algebraic equations that arise from discretizations of partial differential equations in an optimal manner, i.e. such that the central processing unit (CPU) effort increases linearly with the number of computational nodes. Over the past decade, such optimal methods have been introduced in the simulation of electrophysiology. This development, together with the development of affordable parallel computers, has enabled the solution of the bidomain model combined with accurate cellular models, on geometries resembling a human heart. However, in spite of recent progress, the full potential of modern computational methods has yet to be exploited for the solution of the bidomain model. This paper reviews the development of numerical methods for solving the bidomain model. However, the field is huge and we thus restrict our focus to developments that have been made since the year 2000.