Stochastic sensitivity analysis for timing and amplitude of pressure waves in the arterial system.

In the field of computational hemodynamics, sensitivity quantification of pressure and flow wave dynamics has received little attention. This work presents a novel study of the sensitivity of pressure-wave timing and amplitude in the arterial system with respect to arterial stiffness. Arterial pressure and flow waves were simulated with a one-dimensional distributed wave propagation model for compliant arterial networks. Sensitivity analysis of this model was based on a generalized polynomial chaos expansion evaluated by a stochastic collocation method. First-order statistical sensitivity indices were formulated to assess the effect of arterial stiffening on timing and amplitude of the pressure wave and backward-propagating pressure wave in the ascending aorta, at the maximum pressure and inflection point in the systolic phase. Only the stiffness of aortic arteries was found to significantly influence timing and amplitude of the backward-propagating pressure wave, whereas other large arteries in the systemic tree showed marginal impact. Furthermore, the ascending aorta, aortic arch, thoracic aorta, and infrarenal abdominal aorta had the largest influence on amplitude, whereas only the thoracic aorta influenced timing. Our results showed that the non-intrusive polynomial chaos expansion is an efficient method to compute statistical sensitivity measures for wave propagation models. These sensitivities provide new knowledge in the relative importance of arterial stiffness at various locations in the arterial network. Moreover, they will significantly influence clinical data collection and effective composition of the arterial tree for in-silico clinical studies.

CSF pressure and velocity in obstructions of the subarachnoid spaces.

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.

CSF Flow in Chiari I and Syringomyelia from the Perspective of Computational Fluid Dynamics.

Phase contrast MR in patients with the Chiari I malformation demonstrates abnormal CSF flow in the foramen magnum and upper cervical spinal canal, related to abnormal pressure gradients. The purpose of this study was to analyze the role of CSF pressure in the pathogenesis of syringomyelia, with computational models. The spinal cord was modeled as a cylindrical poro-elastic structure with homogenous and isotropic permeability. The permeability was then made heterogeneous and anisotropic to represent the different properties of the central canal, gray and white matter. Fluid with a defined pressure, varying both in time and space, was prescribed in the SAS. Simulations were performed to quantify deformations and fluid movement within the cord. In the simulations with uniform permeability fluid moved into the cord in regions of higher pressure and out of the cord in regions of lower pressure. With permeability differences simulating gray and white matter the pattern was more complex, but similar. Adding the central spinal canal, fluid moved into the cord as in the previous case. However, preferential flow along the central canal hindered fluid from flowing back into the SAS. Pressure gradients in the SAS produce movement of fluid in the spinal cord. Assuming different relative permeability in gray matter, white matter and the central spinal canal, abnormal CSF gradients lead to accumulation of fluid within and adjacent to the spinal cord central canal.

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.

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.