9 Modelling of Cerebrospinal Fluid Flow by Computational Fluid Dynamics
221
9.2.2 Spatial Discretization
CFD requires the discretization of the governing equations in space and, if transient
flows are to be studied, also in time. The discretized set of equations is solved in
the CFD software of choice, aspects of which are discussed in Sect. 9.2.4. For
spatial discretization, the domain under investigation must be divided into small
sub-volumes, i.e. a computational grid has to be generated. For the study of CSF
dynamics, the finite volume discretization [9, 31, 32] scheme is most often used,
followed by the finite element method [33, 34]. The lattice Boltzmann method is an
alternative technique that represents the fluid at mesoscale and recovers the NavierStokes equations within limits of low Mach and Knudsen numbers. It has gained
traction in CSF flow modelling due to its good scalability on parallel computing
architectures [35, 36]. While this approach is not based on macroscopic continuum
equations, it still requires discretization of the model domain.
For anatomically accurate modelling of CSF flow, either unstructured grids or an
immersed boundary method [37] needs to be used, as structured grids that follow the
domain contours are very difficult to generate due to the geometric complexity of the
CSF spaces. State-of-the-art grid generation software allows for mostly automatic
unstructured meshing. Higher grid densities are implemented in areas with expected
large velocity gradients, such as at domain walls. This is important, because the
accuracy of the obtained solution depends, among other factors, on the quality of
the computational grid. Since the solution of the original, not discretized set of
governing equations is generally not known and, thus, the error of the discretized
solution cannot be specified, solutions obtained with different grid densities have
to be compared in a procedure termed grid independence or mesh convergence
study. Such grid independence studies will give an estimate of the relative error
introduced by the spatial discretization. If the flow is in transition to turbulence, the
quality of spatial discretization can be assessed using Kolmogorov’s theory [31, 35].
Ultimately, a balance between the accuracy of the results and cost in terms of grid
generation and computational time must be struck.
9.2.3 Boundary and Initial Conditions
Boundary conditions (BC) need to be specified at the domain margins for closure
of the governing equations. CFD models of CSF flow are typically based on MRIderived BC. Unfortunately, it is not possible to obtain absolute pressures via MRI.
This leads to the unsatisfactory situation that only velocity, flow rate or similar BC
can be prescribed in a subject-specific manner, while pressure or impedance BC
that are required to ensure mass conservation in non-compliant domains are generic
or based on generalized lower-order models. The ideal locations for acquisition
of CSF flow BC are typically the aqueduct of Sylvius and the cervical spinal
canal. The geometry of these areas is well defined and flow therein is primarily
221
9.2.2 Spatial Discretization
CFD requires the discretization of the governing equations in space and, if transient
flows are to be studied, also in time. The discretized set of equations is solved in
the CFD software of choice, aspects of which are discussed in Sect. 9.2.4. For
spatial discretization, the domain under investigation must be divided into small
sub-volumes, i.e. a computational grid has to be generated. For the study of CSF
dynamics, the finite volume discretization [9, 31, 32] scheme is most often used,
followed by the finite element method [33, 34]. The lattice Boltzmann method is an
alternative technique that represents the fluid at mesoscale and recovers the NavierStokes equations within limits of low Mach and Knudsen numbers. It has gained
traction in CSF flow modelling due to its good scalability on parallel computing
architectures [35, 36]. While this approach is not based on macroscopic continuum
equations, it still requires discretization of the model domain.
For anatomically accurate modelling of CSF flow, either unstructured grids or an
immersed boundary method [37] needs to be used, as structured grids that follow the
domain contours are very difficult to generate due to the geometric complexity of the
CSF spaces. State-of-the-art grid generation software allows for mostly automatic
unstructured meshing. Higher grid densities are implemented in areas with expected
large velocity gradients, such as at domain walls. This is important, because the
accuracy of the obtained solution depends, among other factors, on the quality of
the computational grid. Since the solution of the original, not discretized set of
governing equations is generally not known and, thus, the error of the discretized
solution cannot be specified, solutions obtained with different grid densities have
to be compared in a procedure termed grid independence or mesh convergence
study. Such grid independence studies will give an estimate of the relative error
introduced by the spatial discretization. If the flow is in transition to turbulence, the
quality of spatial discretization can be assessed using Kolmogorov’s theory [31, 35].
Ultimately, a balance between the accuracy of the results and cost in terms of grid
generation and computational time must be struck.
9.2.3 Boundary and Initial Conditions
Boundary conditions (BC) need to be specified at the domain margins for closure
of the governing equations. CFD models of CSF flow are typically based on MRIderived BC. Unfortunately, it is not possible to obtain absolute pressures via MRI.
This leads to the unsatisfactory situation that only velocity, flow rate or similar BC
can be prescribed in a subject-specific manner, while pressure or impedance BC
that are required to ensure mass conservation in non-compliant domains are generic
or based on generalized lower-order models. The ideal locations for acquisition
of CSF flow BC are typically the aqueduct of Sylvius and the cervical spinal
canal. The geometry of these areas is well defined and flow therein is primarily