Boundary conditions
Clearance of tracer after SAH with Neurapheresis therapy and lumbar drain was studied under the following boundary conditions (Fig.
2a). The model system was oriented at 30° to horizontal to mimic a typical patient position within a hospital bed. Fluid was aspirated and returned under Neurapheresis therapy at 2.0 and 1.8 mL/min. The 0.2 mL/min difference between return and aspiration matched the drain rate applied under lumbar drain (see details below). Constant CSF production from the choroid plexus within the left and right ventricles was specified to be 0.1 mL/min (0.2 mL/min total) at the CSF production channel entrance. For the lumbar drain simulation, a drainage rate of 0.2 mL/min was specified, as this is a nominal value typically observed in the literature for that procedure [
34,
35].
To represent CSF pulsation around the brain and spinal cord, an oscillatory velocity inlet boundary condition was defined at the model caudal opening using a User Defined Function. The exact waveform was derived from the C2–C3 CSF flow rate waveform (Fig.
2b) obtained from phase-contrast MR imaging of the healthy 23-year-old female subject [
36]. The angular frequency of the waveform was ω = 6.98 s
−1. A zero-pressure outlet boundary condition was defined at the cranial opening. No-slip boundary conditions were imposed at the model walls (dural, pial and intraventricular spaces) with the walls modeled as stationary. CSF was modeled as incompressible with a density of 998.3 kg/m
3 and viscosity of 0.89 mpa s. Tracer density and viscosity was assumed to be identical to CSF.
Multiphase model
The 3D CFD computations used the ANSYS multi-phase fluid model to track the dispersion of a tracer within the CSF, with tracer volume fraction given by
$$\frac{\partial }{\partial t}\left( {\alpha_{k} \rho_{k} } \right) + \nabla \cdot \left( {\alpha_{k} \rho_{k} \vec{\upsilon }_{m} } \right) = - \nabla \cdot \left( {\alpha_{k} \rho_{k} \vec{\upsilon }_{dr \cdot k} } \right)$$
(3)
where
\(q\) is the bulk fluid phase,
\(\rho_{k}\) and
\(\alpha_{k}\) are the phase density and volume fraction of phase
\(k\),
\(\vec{\upsilon }_{m} = \frac{{\sum\nolimits_{k = 1}^{n} {\alpha_{k} \rho_{k} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {u}_{k} } }}{{\rho_{m} }}\) is the mass-averaged velocity,
\(\rho_{m}\) is the mixture density, and
\(\vec{\upsilon }_{{{\text{dr}},k}}\) is the drift velocity for phase
\(k\), with phase k = 1 being the CSF and k = 2 for the tracer. We assumed the relative velocity (slip velocity) between phase
k and the bulk fluid to be zero. Thus, the drift velocity was considered to be zero.
Based on clinical SAH observations and previous research, we assumed a baseline tracer concentration of 10% (
\(\alpha_{2}\) = 0.1) throughout the model domain. Tangen et al. [
37] showed that blood debris is evenly dispersed throughout the spinal SAS within the first hour post-SAH. For the present study, we expected at least 1-h of time to have passed before a lumbar drain or Neurapheresis therapy system could be applied post-SAH. The lumbar puncture is a common clinical emergency procedure used to aid in SAH diagnosis [
38]. Medical doctors have observed the CSF samples from these SAH patients to be colored with a light pink shade (xanthochromia) [
39], indicating that blood has spread throughout the CSF system down to the lumbar spine, justifying the evenly-mixed tracer concentration used in this model as a reasonable approximation. Steady-streaming velocities were determined based on the average of 10 CSF flow cycles with transient effects eliminated by removal of the first flow cycle. Similar to Kuttler et al. [
40], the fixed velocity field (“frozen flow field”) was used to solve the volume fraction equation for spatial–temporal tracer concentration. The axial distribution of tracer concentration,
\(\alpha (z)\), for 3 mm thick slices along the z-axis was computed by:
$$\alpha (z) = \frac{{\sum\nolimits_{slice} {\left| {\alpha_{k} (z)} \right|V(z)} }}{{\sum\nolimits_{slice} {V(z)} }}$$
(4)
where
\(V\) is the cell volume and summations computed for all cells within each 3 mm thick cross-section. Spatial–temporal distribution of tracer concentration was plotted over 24-h for Neurapheresis therapy and lumbar drain.
To determine concentration profiles over 24-h after SAH, we determined the solute transport due to the steady-streaming CSF velocity field. Molecular diffusion of large molecules within the CSF (MW ~ 150 kDa) is much smaller than steady-streaming and oscillatory CSF velocities [
40]. However, shear and mixing across the cross section has the potential to greatly increase the effective diffusivity in the spinal SAS [
41]. Tangen et al. [
37] found that molecular diffusion had a negligible impact on tracer spread within an idealized geometry representing the spinal and cranial SAS. Kurtcuoglu et al. [
42] also neglected diffusion in their model. To make the computational effort reasonable, molecular diffusion of the tracer was not included in the current study. However, as described in the following paragraphs, the potential impact of neglecting diffusion was estimated.
To help understand the relative importance of diffusive versus advective mass transport, the Sherwood number (\(Sh\)) was calculated and \(Sh = \frac{h}{{{\mathbf{D}}/L}}\) provides the ratio of convective mass transport, \(h\,\), to the effective diffusive mass transport, \({\mathbf{D}}/L\), where \(L\,\) is a characteristic length and \({\mathbf{D}}\) is the effective diffusivity including shear-augmented dispersion. \(h\,\) was computed based on the mean cross-sectional velocity at peak systolic (h = 0.26 m/s and 2.4 m/s for cortical and spinal SAS, respectively). \(L\,\) was assumed to be the minimum gap width between the shells in the cortical SAS (~ 2 mm) and mean of the hydraulic diameter for the spinal SAS (5.87 mm).
To see if the tracer is a good similitude for hemoglobin clearance,
\({\mathbf{D}}\) was calculated for both tracer and hemoglobin using an order-of-magnitude model by Sharp et al. [
41]:
$${\mathbf{D}} = \left( {1 + R_{max} } \right)D$$
(5)
$$R_{max} = P^{2} Sc/\alpha^{2}$$
(6)
$$\alpha^{2} = \beta^{2} /Sc$$
(7)
$$\beta^{2} = (L/2)^{2} \omega /D$$
(8)
where
\(R_{max}\) is the maximum enhancements with optimal mixing,
\(P\) is the characteristic non-dimensional pressure gradient (
\(P\) ~ 152.6 [
41]),
\(\alpha\) is Womersley number and
\(\beta\) is oscillatory Peclet numbers. Schmidt number (
\(Sc\, = \frac{\nu }{D}\)) was described as the ratio of momentum diffusivity of water at room temperature (
\(\nu\) = 0.89 E−06 m
2/s), to molecular diffusion coefficient (
\(D\)). The molecular diffusivity of the fluorescein tracer is
\(D\) = 4.25 E−10 m
2/s [
43,
44] and hemoglobin is
\(D\) = 10.2 E−11 m
2/s [
45].
Solver settings
Simulations were carried out using the PISO Scheme (pressure-implicit with splitting of operators) to solve the flow equations with second-order upwind for momentum discretization, PRESTO! (PREssure STaggering Options) for pressure discretization, and first-order upwind for volume fraction discretization. Default values were used for under relaxation factors. The implicit formulation was used for volume fraction parameters and a dispersed model was used for phase interface modeling. The convergence criteria for velocity, continuity, momentum, and phase volume fraction were set to 1E−06 with time step = 0.1 s and maximum of 100 iterations per time-step. Total simulation time was 14 days to compute 24-h of real-world time in parallel mode with 126 GB RAM and 38 processors. Time required to solve the fixed flow field (11 flow cycles) was 36 h.
Numerical sensitivity studies
Axial distribution of average tracer concentration at 1 h was verified by numerical sensitivity studies for time-step size and mesh resolution. Results were computed for a “coarse”, “medium”, and “fine” mesh with wall prism layers. For the medium mesh, time-step sensitivity was then checked with time-step resolution given by fractions of the cardiac cycle, T = 0.9 (s), for T/18, T/9, T/5. Based on these results, a “Medium” mesh with time step = T/9 was carried forward for computation of final results.