Effect of Tonsillar Herniation on Cyclic CSF Flow Studied with Computational Flow Analysis

S.O. Linge; V. Haughton; A.E. Løvgren; K.A. Mardal; A. Helgeland; H.P. Langtangen

doi:10.3174/ajnr.A2496

Abstract

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.

Abbreviations

AP: anteroposterior
CFA: computational flow analysis
LR: left-right
PCMR: phase-contrast MR imaging
Pmax: maximum pressure
Pmin: minimum pressure
SI: superior-inferior

CSF flows alternately downward and upward through the foramen magnum during the cardiac cycle because of the expansion and contraction of the brain during cardiac systole and diastole. The oscillatory CSF flow during the cardiac cycle has a complex pattern with some regions of relatively high velocity and others with relatively low velocity. The flow patterns have been demonstrated by means of axial PCMR.¹ Pressure gradients associated with the cranial and caudal flow have been demonstrated in an animal model in which tonsillar herniation was simulated with a balloon catheter.²

CSF flow is incompletely characterized by PCMR. The images show velocities at the levels selected for study but not throughout a volume. Furthermore, PCMR images do not show CSF pressure gradients. The cyclic CSF flow through the craniocervical region has recently been investigated in an idealized 3D model of the normal subarachnoid space by means of CFA.³ The model demonstrates flow and pressure throughout the selected space, with greater spatial and temporal resolution than PCMR. The flow characteristics in the model were found to correspond well with flow patterns demonstrated by PCMR in healthy adult human subjects. This model allows an investigator to measure flow changes resulting from changes in 1 single anatomic or physiologic parameter.

In the Chiari I malformation, CSF flows with higher velocities than in normal subjects,^1,4,5 and pressures are increased.⁵ The hyperdynamic CSF flow and altered pressure dynamics in the Chiari malformation hypothetically have a role in the pathogenesis of syringomyelia,^6,7 motor and sensory abnormalities, headache, and so forth associated with the Chiari I malformation.⁵ The effect of tonsillar herniation on CSF flow dynamics, therefore, needs further elaboration. A technique for investigating the effect of changing anatomy on CSF flow is the application of CFA to suitable models. The parameters affecting CSF flow are better controlled in such an experiment than in measurements in patient groups because of significant interindividual variation. In this study, we used a mathematic model of the subarachnoid space to measure the effect of tonsillar herniation on CSF velocities and pressures in the craniocervical region during a complete cardiac cycle.

Materials and Methods

Geometric Model of the Subarachnoid Space

We used a geometric model³ of the subarachnoid space in which the brain and the spinal cord were modeled respectively as a cone-shaped structure and a tube. These were located inside another conical structure and tube so that the space between the 2 formed an idealized model of the cranial and upper cervical subarachnoid space. With the Star-CD graphics program (CD-Adapco, Melville, New York),⁸ 2 cones were added to the inferior surface of the brain model, to simulate tonsils extending into the posterior portion of the upper cervical spinal canal. The “tonsils” had a length of 3.5 cm and a diameter of 0.5 cm at foramen magnum so that they narrowed the posterior subarachnoid space (Fig 1A). In all other aspects, the model with herniated tonsils was identical to the previously described idealized model of the normal subarachnoid space (Fig 1B). No structure to represent the tentorium or channels corresponding to the fourth ventricle or its foramina was included in the model. The region that corresponded to the craniovertebral junction was located midway between the top and bottom of the model, extending 20 cm superoinferiorly. The cross-sectional geometry of the superior and inferior ends of the model was extended 2 cm as they were in the normal model so that inflow and outflow boundary conditions could be specified more efficiently. The walls of the model were assumed rigid, nonpermeable, and immobile. Sagittal and axial sections of the model were inspected by a neuroradiologist who determined if the model had the gross appearance of the subarachnoid space in a Chiari I malformation.

Fig 1.

The idealized model (A) of a subarachnoid space with herniated tonsils and the original model (B) with normally positioned tonsils. The models are shown with sagittal MR images illustrating a typical normal individual and a typical Chiari I malformation. For the normal model, a midline sagittal plane is shown and for the tonsillar herniation case, its paramedian through the nearest tonsil. A white reference line indicates the midpoint of the model at the correlate of the craniovertebral junction. The extensions to the model to facilitate the description of boundary conditions are demonstrated by the red lines. Anterior is to the reader's left and posterior to the right.

Simulation of Flow and Pressure in the Model

The geometry of the Chiari model was converted in ICEM CFD⁹ to a hexahedral computational mesh with 615 680 nodes. Each computational cell had the smallest distance between nodes ranging from 0.15 to 1.25 mm. The higher resolutions were used in places with higher geometric complexity. Standard CFA methods available in Star-CD were used to calculate velocities and pressures during several cycles of oscillatory flow through the model. The boundary conditions and fluid characteristics that were used previously for the normal model³ were also used for the simulation of flow in the Chiari model. We assumed a pressure at the top of the model corresponding to 20 cm of water. We assumed a plug-shaped inflow velocity profile varying sinusoidally at 1 cycle per second. The magnitude of peak flow was set at 17 mL/s, which corresponds to the flow rate selected empirically for the normal model to produce peak velocities of 2–3 cm/s near the cranio-occipital junction. Flow was assumed laminar in all of flow space. No-slip (zero-velocity) conditions were specified at the walls. The flowing fluid was assumed to have the properties of water. With the finite volume method in Star-CD, the Navier-Stokes equations were solved numerically to give a fluid velocity vector and a scalar pressure at each computational point in the grid at millisecond intervals. Data from the simulations were postprocessed in Star-CD and Matlab¹⁰ to display color-coded velocity and pressure data in 2D sagittal and axial sections and to display the velocity vector cinematically. Simulations were initiated with zero-flow velocity, but a steady periodic state was reached already in the third flow cycle, allowing this cycle to be used for our investigations. To confirm that grid resolution was sufficient, we repeated simulations with another grid having 35% fewer computational points.

Plots of the velocity vectors in sagittal planes and in successive axial planes were inspected for evidence of flow jets, stagnant flow, and synchronous bidirectional flow (ie, flow simultaneously in both cranial and caudal directions). The location in the subarachnoid space and the time in the flow cycle for the peak velocities and pressures were determined by inspection. Velocity characteristics in the Chiari model and the normal model were compared.

Animation

Animations were created with the volume-rendering tool VoluViz.¹¹ The animations use particle-tracking display to demonstrate the direction of flow.¹² They have color scales to demonstrate the magnitude of flow. The animations were inspected interactively to describe the flow patterns.

Velocities as a Function of Level

Components of the velocity vector in the SI, LR, and AP directions were plotted for the time of maximal inferior and maximal superior flow (t = 0.25 and 0.75 seconds) and for change in flow direction (t = 0.00 and 0.50 seconds) for both the Chiari and the normal model. Changes in velocity vectors along the spinal column were assessed. Differences between the Chiari model and the normal model were noted.

Pressure Gradients

Pressure gradients were assessed by inspection of sections made in sagittal and axial planes through the models. Pressure differences (top-bottom) were tabulated for 4 phases of the cycle (t = 0.00, 0.25, 0.50, and 0.75 seconds). Pressures in the Chiari model and the normal model were compared.

Validation of Chiari Model

A neuroradiologist (the second author) compared axial displays of flow velocity distributions at the craniovertebral junction in the Chiari model with velocities in axial PCMR images from patients with Chiari I. The model was considered validated if visual inspection revealed similar distribution and magnitudes of velocities in the tonsillar herniation model and in the PCMR images from patients.

Reynolds Number Calculation

Reynolds number (Re) is defined as

where u_s is the mean fluid velocity in m/s, L is the characteristic length (hydraulic diameter) in m, and v is the kinematic fluid viscosity in m²/s, for which we used a value of 0.700 ·10⁻⁶ m²/s (water at 37°C). The numbers were calculated for maximal fluid velocity and characteristic length.

Results

Geometric Model of the Subarachnoid Space

The geometric model of the Chiari I malformation had diminished volumes of the craniovertebral junction and upper posterior cervical subarachnoid space compared with the normal idealized model. In sagittal and axial sections, the geometric model with inferiorly displaced tonsils was judged by the neuroradiologist to approximate the subarachnoid space in the Chiari I malformation (Fig 1).

Simulations of Flow and Pressure in the Chiari Model

The simulations showed velocity patterns changing in space and with the phase of the cardiac cycle (Figs 2 and 3). Sagittal images showed larger velocities in the spinal canal than in the cranial vault (Fig 2). Axial images showed regions of localized above-average velocities (“jets”) in cardiac phases with caudal flow. These jets had velocities of 5.7 cm/s at peak caudal flow (Fig 3A), more than twice the velocity in the normal model (Fig 3B). The AP and LR components of the velocity vectors (in-plane or x and y components) showed the same patterns (Fig 4 A, -B). At the peak cranial volume flow rate, velocity components in the superior direction had no clear jet pattern. However, the transverse velocity components had clear jet patterns (Fig 4C, -D). These jet patterns were seen for both in-plane velocity components in any axial plane chosen for t = 0.00, 0.25, 0.50, or 0.75 seconds.

Fig 2.

Sagittal plane sections showing flow velocities in the Chiari model at the time when the caudal volume flow rate is maximal (t = 0.25 seconds, A) and when cranial volume flow rate is maximal (t = 0.75 seconds, B). C and D, For comparison, corresponding sections in the normal model are shown. Velocity scales in are the same as in A and B, respectively. The plane selected for the Chiari model is paramidline to demonstrate the extension of the tonsils into the spinal canal.

Fig 3.

Axial sections show flow velocity at the craniovertebral junction in the Chiari model at the time when caudal volume flow rate is maximal (t = 0.25 seconds). B, For comparison, a corresponding section for the original model (without tonsil herniation) is shown at an equivalent time in the cycle. Note that the distribution of peak velocities in both models is similar and that in the presence of tonsillar herniation, velocities are more than double those in absence of tonsillar herniation.

Fig 4.

Axial sections showing the LR (A) and AP components (B) of the flow velocity vector at the craniovertebral junction in the Chiari model at the time when the caudal volume flow rate is maximal (t = 0.25 seconds). C and D, Similar velocity components are also shown, when cranial volume flow rate is maximal (t = 0.75 seconds). Velocity scales are the same as in A and B, respectively.

Results were essentially identical when simulations were run on the coarser grid. For example, in the axial section from the craniovertebral junction at the time when the caudal flow rate was maximal (Fig 3A), minimum and maximum velocities differed by <1 mm/s.