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V. Kurtcuoglu et al.
in axial direction, allowing for the placement of a single transverse measurement
plane for velocity-encoded MRI. Furthermore, there are closed-form solutions of
flow in idealized representations of the aqueduct (e.g. straight elliptic pipe) [38]
and the spinal canal (straight elliptic annulus) [39], allowing for the filtering of
higher frequency noise components in the acquired flow data. This can be done
by first integrating the flow profile in space for each measured point in time,
which yields a volumetric or mass flow curve that contains less noise. In order
to recuperate the spatial velocity field, this flow rate is applied to the respective
idealized geometry with available closed-form solution of the governing equations
[32]. Finally, conformal mapping is performed to map the solution back to the actual
domain [40].
The pulsatile components in CSF flow are caused primarily by the expansion
and contraction of blood vessels in and around the CNS, and by respiration,
which changes central venous pressure and affects pressure conditions in the spinal
compartment. This suggests that the CSF boundaries have to be flexible entities that
can transfer momentum by deformation. Nevertheless, most CFD models assume
rigid CSF domain boundaries, as this simplifies model setup and calculations
tremendously. There are two general approaches by which boundary motion can
be taken into account: either the entire chain of displacements from blood vessels
or the diaphragm to tissue and CSF is modelled, or the displacement of the
boundaries is measured directly. The former approach poses great challenges, the
most serious of which is the lack of material data to model the interaction between
CSF and the surrounding deformable structures. In addition, the complexity of the
vascular network in the brain and limits in the effective resolution of MRI require
simplifications of the vascular model (see Sect. 9.2.1), introducing errors that are
difficult to quantify.
Direct measurement of the motion of CSF space boundaries is not trivial, but
feasible. As early as 1992, Enzmann and Pelc used phase-contrast MRI to measure
brain motion [41]. With displacement-encoded MR imaging, brain displacement
down to 0.01 mm can be measured [42, 43]. This is sufficient for deriving ventricle
wall motion, though it has to be noted that averaging over several cardiac cycles is
necessary. Such averaging may lead to underestimation of the actual displacement if
respiration is not accounted for by, e.g. additional respiratory gating. In contrast to
ventricular wall movement, the motion of the pia mater is likely too small for direct
measurement. The MRI acquisitions result in a discrete displacement map within
which the CSF boundaries do not necessarily coincide with the measured locations.
Hence, spatial interpolation must be performed first to cover the boundaries.
Since the temporal frequency spectrum of blood flow into the brain through the
carotid and vertebral arteries does not extend beyond 10 Hz under normal resting
conditions, a brain motion acquisition interval of 0.05 s within a cardiac cycle is
sufficient to meet the Nyquist-Shannon sampling criterion [44, 45]. However, the
temporal discretization of the governing equations requires much smaller time steps,
necessitating a temporal interpolation of the acquired brain motion data. This can be
performed by Fourier decomposition of the acquired discrete signal and resampling
of the reconstructed continuous signal [32].
V. Kurtcuoglu et al.
in axial direction, allowing for the placement of a single transverse measurement
plane for velocity-encoded MRI. Furthermore, there are closed-form solutions of
flow in idealized representations of the aqueduct (e.g. straight elliptic pipe) [38]
and the spinal canal (straight elliptic annulus) [39], allowing for the filtering of
higher frequency noise components in the acquired flow data. This can be done
by first integrating the flow profile in space for each measured point in time,
which yields a volumetric or mass flow curve that contains less noise. In order
to recuperate the spatial velocity field, this flow rate is applied to the respective
idealized geometry with available closed-form solution of the governing equations
[32]. Finally, conformal mapping is performed to map the solution back to the actual
domain [40].
The pulsatile components in CSF flow are caused primarily by the expansion
and contraction of blood vessels in and around the CNS, and by respiration,
which changes central venous pressure and affects pressure conditions in the spinal
compartment. This suggests that the CSF boundaries have to be flexible entities that
can transfer momentum by deformation. Nevertheless, most CFD models assume
rigid CSF domain boundaries, as this simplifies model setup and calculations
tremendously. There are two general approaches by which boundary motion can
be taken into account: either the entire chain of displacements from blood vessels
or the diaphragm to tissue and CSF is modelled, or the displacement of the
boundaries is measured directly. The former approach poses great challenges, the
most serious of which is the lack of material data to model the interaction between
CSF and the surrounding deformable structures. In addition, the complexity of the
vascular network in the brain and limits in the effective resolution of MRI require
simplifications of the vascular model (see Sect. 9.2.1), introducing errors that are
difficult to quantify.
Direct measurement of the motion of CSF space boundaries is not trivial, but
feasible. As early as 1992, Enzmann and Pelc used phase-contrast MRI to measure
brain motion [41]. With displacement-encoded MR imaging, brain displacement
down to 0.01 mm can be measured [42, 43]. This is sufficient for deriving ventricle
wall motion, though it has to be noted that averaging over several cardiac cycles is
necessary. Such averaging may lead to underestimation of the actual displacement if
respiration is not accounted for by, e.g. additional respiratory gating. In contrast to
ventricular wall movement, the motion of the pia mater is likely too small for direct
measurement. The MRI acquisitions result in a discrete displacement map within
which the CSF boundaries do not necessarily coincide with the measured locations.
Hence, spatial interpolation must be performed first to cover the boundaries.
Since the temporal frequency spectrum of blood flow into the brain through the
carotid and vertebral arteries does not extend beyond 10 Hz under normal resting
conditions, a brain motion acquisition interval of 0.05 s within a cardiac cycle is
sufficient to meet the Nyquist-Shannon sampling criterion [44, 45]. However, the
temporal discretization of the governing equations requires much smaller time steps,
necessitating a temporal interpolation of the acquired brain motion data. This can be
performed by Fourier decomposition of the acquired discrete signal and resampling
of the reconstructed continuous signal [32].