A 3D subject-specific model of the spinal subarachnoid space with anatomically realistic ventral and dorsal spinal cord nerve rootlets

A single, representative healthy, 23-year-old, female Caucasian subject was enrolled in this study. The subject had no previous history of neurological or cardiovascular disorders.

All MRI measurements were obtained with a General Electric 3T scanner (Signa HDxt, software 15.0_M4_0910.a). CSF flow data were collected at three vertebral levels, C2–C3, C7–T1 and T10–T11, using phase-contrast MRI with retrospective electrocardiogram (ECG) gating and 32 cardiac phases [14]. Each slice had a thickness of 5.0 mm and an in-plane resolution of 0.54 × 0.54 mm. Orientation of the slice was made perpendicular to the CSF flow direction and positioned vertically by intersection with a vertebral disk (i.e. C2–C3). A flip angle, TR, TE and VENC was used with a value of 25°, 13.4, 8.26 and 8 cm/s respectively. Detailed information on imaging parameters is provided by Baledent et al. [19].

Oscillatory cardiac-related CSF flow was quantified for the axial locations located at the vertebral disk at the C2–C3, C7–T1 and T10–T11 vertebral levels. As detailed in our previous studies [14, 20], Matlab was used to compute the CSF flow waveform, Q_(t), based on integration of the pixel velocities with Q(t) = ∑A _pixel [V _pixel (t)], where A _pixel is the area of one MRI pixel, V _pixel is the velocity for the corresponding pixel, and Q_(t) is the summation of the flow for each pixel of interest. A smooth distribution of CSF flow along the spine was achieved by interpolating CSF flow between each axial measurement location [21]. Similar to previous studies, the diastolic CSF flow cycle phase was extended in cases when necessary [22]. For correcting eddy current offsets, the cyclic net CSF flow was offset to produce zero net flow over a complete flow cycle [14].

To collect geometric measurements with improved CSF signal, 3D fast imaging employing steady state acquisition (3D FIESTA) was used, and acquisitions were realized with free breathing. The coils used were the HD Neck-Spine Array with 16 Channels for the spine and the 29 element phased array for the upper-neck. Images were collected in three volumes, from the top of the brain to C7, from C5 to T9, and from T9 to S5, with each section containing 140, 104 and 104 sagittal T2-weighted images respectively. The field of view (FOV) size was 30 cm × 30 cm × 7 cm for the craniocervical volume, and 30 cm × 30 cm × 5.25 cm for both the thoracic and lumbosacral volumes. In-plane voxel spacing was 0.547 × 0.547 mm and slice thickness was 1 mm with slice spacing set at 0.499 mm. Echo times (TE) were 1.944, 2.112, 2.100 and repetition times (TR) were 5.348, 5.762, 5.708 for the craniocervical, thoracic, and lumbosacral volumes respectively. Total imaging time for the three levels was ~ 45 min.

The open-source program, ITK-SNAP (Version 3.4.0, University of Pennsylvania, USA) [23], was used to segment the MRI data. Similar to our previous work [24], the cervical, thoracic and lumbar MR image sets were manually segmented in the axial orientation using the semi-automatic contrast-based segmentation tool. The segmented region extended from the foramen magnum to the end of the dural sac. One expert operator completed the segmentation, as our previous study showed strong inter-operator reliability of SSS geometric parameters [24]. A second expert operator reviewed the images to confirm region selection, and in areas of disagreement, discussed in detail with respect to the anatomy. Hyperintensities in the T2-weighted image sets near the epidural space were excluded from the model segmentation (Fig. 1). MRI data were not collected in high-resolution for the entire brain, and thus the cortical and ventricular CSF spaces were not included in the model. After completion, each segmentation was exported as an .STL file with Gaussian smoothing option applied (standard deviation = 0.80 and maximum approximation error = 0.03).

The open source program, Blender (Version 2.77a, Amsterdam, Netherlands), was used for the majority of mesh modifications and all modeling operations in this study. After segmentation, the .STL files generated were imported into Blender. Because of the global reference coordinate set by the MRI, segmentations generated from different image series were automatically registered. However, 3D rigid body translation (~ 5 mm maximum) was required to align each model section due to a small degree of subject movement between the MR image acquisitions. These translations were performed based on a visual best fit.

The following operations were completed to create a lowest-resolution semi-regular surface mesh of the spinal cord and dura while maintaining an accurate representation of the original geometry. After alignment, the triangulated .STL segmentations were converted to quadrilateral meshes using the automatic conversion tool “tris to quads” in Blender. The spinal cord and dural surfaces were separated, and an array of planes was placed along the entire spinal segmentation at a roughly orthogonal orientation to the spinal trajectory. Vertical spacing of these planes was determined by choosing an inter-plane interval (~ 5 mm) that preserved surface contours; this required a minimum of three planes to preserve a change in surface concavity. The circumferential contour of the spinal cord and dura was obtained at each plane using the “intersect (knife)” operation in Blender. The original geometry was then removed. Each surface contour was then vertically extruded ~ 1 mm. Simple circle meshes were place at each contour using the “add circle” command, the “shrink wrap” modifier was then used to form these circles around each profile. The number of vertices in the circles wrapped to the dural and spinal cord profiles was specified to be 55 and 32 respectively. These parameters were determined based on visual inspection of the shrink-wrap fit at the largest profile diameter located at the foramen magnum. Manual adjustment of individual vertices was made to preserve a uniform vertex distribution and surface contour at each slice. To create a continuous quadrilateral mesh of both the spinal cord and dura, the “bridge edge loops” command was used between adjacent contours (Fig. 2).

Manual adjustments were then made by sculpting the remeshed surfaces within the “sculpt mode” workspace in Blender to produce ~ 50% visual interference with the original segmentation surface (Fig. 3). To further improve surface accuracy, a combination of a shrink-wrap and “smooth” modifiers were used simultaneously. Importantly, the “keep above surface” option and “offset” options on the shrink-wrap modifier were used. The values for shrink-wrap offset and smoothing factor in their respective modifier menus must be determined by a trial and error method for each unique mesh until the desired smoothness is justified with overall volume. In this study, values of 0.04 and 0.900 were used for offset and smoothing factor respectively.

The 31 NR pairs, starting from the craniocervical junction, were modeled using the following methodology. For each rootlet, a “circle” mesh was extruded from the SC junction to the dural exit location in Blender. The curvature, radicular line (RL) and descending angle (DA) for each rootlet were determined based on the subject specific segmentation, average cadaveric measurements from the literature and anatomic reference imagery [25‐28] (Fig. 4). The exact method varied by location due to variations in the completeness of the data types; these differences are described below. Note: the 31st nerve root, or coccygeal nerve did not bifurcate into a nerve root pair until after leaving the intrathecal CSF space.

In the left side of the cervical spine, segmentations of the NR were possible to obtain directly from the anatomic MR imaging. These were imported and aligned with the existing model in Blender. A “circle” mesh was extruded along each segmented path and the diameter of this circle was defined as the average NR diameter or thickness from cadaveric measurements for each location. Additionally, in the cervical spine the spinal entry point of each rootlet cylinder was scaled in the cranial direction (~ 150%) along the spinal cord to create a blended transition. Finally, cervical rootlets were mirrored left to right and small adjustments were made to fit them to the correct exit points on the right side of the dura. Mirroring was applied as the NR intersection location at the spinal cord and dura was nearly identical for the left and right side NR.

In the thoracic spine, segmentations were only able to inform NR entry and exit points, and by extension, DA. It is possible that NR points in the thoracic spine were difficult to visualize within this region due to image blurring stemming from respiratory-related tissue motion. NR morphology in the thoracic spine is a steeply descending and tightly packed bundle. Therefore, to reduce unnecessary mesh complexity, a standard NR set was developed as a simplified cylinder with a diameter based on the average NR bundle size in the thoracic region. In addition to this main cylinder, a secondary cylinder was incorporated at the SC entry point to more closely imitate NR branching near the spinal cord. This cylinder extends from just below the primary rootlet entry point to a location approximately one-third of way along the primary rootlet; overall a steeply descending deltoid morphology is created. As in the cervical spine, a blended transition was created at the SC entry point for each NR. This standard NR set was mirrored left to right of the SC and duplicated along the SC for the entire thoracic region.

In the lumbosacral spine, the NR form the cauda equina. High MR image contrast made complete segmentations of this region possible and NR modeling was completed as in the cervical spine. NR were again simplified as a single cylinder of average diameter. Because of this, RLs for this region were not possible to define.

Geometric parameters were calculated along the complete spinal mesh at 1 mm intervals [21]. SSS cross-sectional area, A _cs  = A _d  − A _c  − A _nr , was determined based on cross-sectional area of the NR (A _nr ), SC (A _c ) and dura (A _d ). Hydraulic diameter for internal flow within a tube, D _H  = 4A _cs /P _cs , was determined based on the cross-sectional area and wetted perimeter, P _cs  = P _d  + P _c  + P _nr . Wetted perimeter was computed as the sum of the NR (P _nr ), SC (P _c ) and dura (P _d ) perimeters. Each of these parameters was calculated within a user defined function compiled in ANSYS FLUENT (Ver. 18.1, ANSYS inc, Canonsburg, PA). Note, for geometric analysis, the coccygeal nerve (spinal nerve) was considered to be a part of the spinal cord. 

The hydrodynamic environment at 1 mm slice intervals along the entire spine was assessed by Reynolds number based on peak flow rate, \(\text{Re} = \frac{{Q_{sys} D_{H} }}{{\nu A_{cs} }}\), and Womersley number based on hydraulic diameter. For Reynolds number, Q _sys is the temporal maximum of the local flow at each axial interval along the spine obtained by interpolation from the experimental data and ν is the kinematic viscosity of the fluid. Similar to previous studies, CSF viscosity was assumed to be that of water at body temperature. To evaluate the presence of laminar flow, (Re < 2300), similar to previous studies in CSF and biofluids mechanics, Reynolds number was evaluated at peak systolic flow along the spine. Womersley number, \(\alpha = \frac{{D_{h} }}{2}\sqrt {\omega /\nu }\), where ω is the angular velocity of the volume flow waveform ω = 2π/T, was used to quantify the ratio of unsteady inertial forces to viscous forces. This ratio was previously found to be large relative to viscous forces by Loth et al. [29]. A value greater than 5 for Womersley number indicates transition from parabolic to “m-shaped” velocity profiles for oscillatory flows [30]. CSF pulse wave velocity (PWV) was quantified as an indicator of CSF space compliance. Timing of peak systolic CSF flow rate along the spine was determined based on our previously published method [31]. In brief, a linear fit was computed based on the peak systolic flow rate arrival time with the slope being equivalent to the PWV.

Total intrathecal CSF volume below the foramen magnum was 97.3 cm³ (Table 3). Volumes of the dura mater, spinal cord and 31 NR pairs were 123.0, 19.9 and 5.8 cm³ respectively. The surface areas for the dura mater, spinal cord and NR were 318.5, 112.2 and 232.1 cm² respectively. The average cross-sectional areas of the dura mater, spinal cord and NR were 2.03, 0.33 and 0.10 cm² respectively. The length of the spinal cord down to the conus and spinal dura mater were ~44.8 cm and 60.4 cm respectively. Note, geometric parameters for the spinal cord were computed based on the spinal cord with the coccygeal nerve included as one continuous structure.

Peak-to-peak CSF flow amplitude measured at the C2–C3, C7–C8 and T10–T11 was 4.75, 3.05 and 1.26 cm³/s respectively (Fig. 6a). These were measured at an axial position relative to the model end (foramen magnum) of 4.0, 12.5, and 35.4 cm respectively. Based on the interpolated CSF flow waveform between MRI measurement locations, the maximum peak and mean CSF velocities were present at 38 mm (~ C4–C5, Fig. 7f). Minimum value of peak and mean CSF velocities occurred in the lower lumbar spine and within the thoracic spine from 390 to 410 mm (~ T7–T10, Fig. 7f).

Cerebrospinal fluid flow oscillation had a decreasing magnitude and considerable variation in waveform shape along the spine (Fig. 6a). Spatial temporal distribution of CSF flow rate along the SSS showed that maximum CSF flow rate occurred caudal to C3–C4 at ~ 40 mm (Fig. 6b). CSF pulse wave velocity (PWV) was estimated to be 19.4 cm/s (Fig. 6b).

Average Reynolds and Womersley number was 68.5 and 9.6 respectively. Womersley number ranged from 1.6 to 22.96 (Table 2, Fig. 7d). Maximum Womersley number was present near the foramen magnum (α = 22.96). Womersley number had local minima within the cervical spine and just rostral to the intrathecal sac. Maximum Reynolds number was 174.9 and located at C3–C4.

A variety of software exists to help reconstruct MRI DICOM image files in 3D. Many segmentation software platforms provide automatic segmentation algorithms that can deliver relatively quick visualizations but these segmentations are often not suitable to create 3D models that can be used for CFD modeling or easily exported for 3D printing [33]. In this study, we used the open-source program ITK-SNAP (“The Insight Segmentation and Registration Toolkit”, http://​www.​itk.​org) that supports automatic, semi-automatic and manual approaches. The final model was constructed based on manual segmentation of each slice along the spine by an expert operator previously trained in intrathecal CSF segmentation procedures.

Despite the popularity of CFD studies conducted in the SSS, there is a lack of detailed information on intrathecal segmentation methods based on anatomic MR imaging. The craniocervical junction is highly vascularized with relatively large blood vessels that transverse the region, including the vertebral arteries (3.7 mm diameter for the left vertebral artery and 3.4 mm diameter for the right vertebral artery [34]) and the anterior spinal artery (0.3–1.3 mm diameter [35]). Spinal cord NR can sometimes be seen as dark regions crossing the SSS (Fig. 1d–f). Their length and obliqueness increases progressively moving towards the feet [36]. Denticulate ligaments are located between adjacent sets of NR in the cervical and thoracic spinal cord segments. These structures are too small to be quantified by MRI (thickness of ~ 0.1 mm) but may also appear as slightly darkened regions of SSS on each side of the spinal cord. The CSF on the anterior or posterior side of the spinal cord near the foramen magnum may appear dark in coloration due to flow void artifacts resulting from elevated CSF velocities at this region (and others along the SSS, Fig. 1). Although these regions can appear relatively dark on MR imaging, they should be considered as fluid.

Along the entire spine, the epidural space can appear hyper intense due to the presence of epidural fat (Fig. 1e–f). Care should be taken to not confuse these areas with CSF as it can be difficult to visualize the relatively thin dura mater that separates the two spaces. This ambiguity often confounds automatic segmentation tools and thresholding should be reviewed in detail to ensure accuracy. From our experience, no presently available automated algorithm can avoid over-segmentation of epidural fat, as there can be virtually no border visible between these two regions at many locations along the spine due to MR image resolution limits that do not allow visualization of the relatively thin dura.

The cauda equina begins around the conus medullaris that is located near the lower border of the first lumbar vertebra. This structure is formed by the long rootlets of the lumbar, sacral and coccygeal nerves that run vertically downward to their exit. Similar to the spinal cord NR, ligaments and blood vessels, these small bundles of nerves are not possible to accurately quantify with the current MR image resolution through segmentation alone. In the presented model, they are modeled as curving cylinders as described in our methods with reference to cadaveric studies in the literature and visual interpretation and measurement of NR insertion at the spinal cord and dura.

Although the spinal cord and dura mater were easily visible, smaller structures such as NR were not clearly discernible in the MRI scans used in this study. In our previous study [36], we grossly modeled spinal cord NR as single airfoil shaped structures within the cervical spine only. For the present complete spine model for a healthy subject, we individually modeled the number of nerve rootlets at all vertebral levels (see Fig. 4 for anatomic depiction of nerve rootlets and Table 1 for number of nerve rootlets). The nerve rootlets were each placed with reference to the high-resolution MR imaging, 3D segmented geometry and published cadaveric measurements and images in the literature. Because no single source contained enough information to fully reconstruct the NR geometry, the final model does not strictly adhere to any single set of tabular parameters, but rather, is a best judgment based on the collective information (see Table 1 for parameters). Furthermore, due to limitations in the data as well as the time intensive nature of the modeling process, NR were mirrored left to right along the spinal cord. The duplicate side was subjected to < 3.0 mm translation as necessary to best fit rootlets to the spinal and dural geometry. NR vertical positioning is only referenced by the corresponding vertebral level in the literature. Therefore, vertical positioning was based solely on segmentation data marking SSS entry and exits locations. The resulting model is subject-specific in terms of NR location and orientation, but idealized in terms of the exact structure (Fig. 4).

While the provided model is subject-specific, it can be compared to other MRI-based studies to help understand its similarity to the general population. Overall, the provided model had a SSS volume of 97.34 cm³ and showed a strong similarity with the previous studies cited that, on average, reported SSS volume to be 90.3 cm³ [38‐45]. Table 3 gives a review of studies that used MRI to quantify the volume of anatomical features within the full spine and lumbosacral spine for healthy subjects. In collection, these published studies indicate a decreasing trend in CSF volume with age given by: SSS_volume(ml) = ( − 0.27 × age) + 102 (Fig. 8). The provided model had a volume that was on the higher end of the average reported values, however it was also for a relatively young 23-year-old subject (Table 3). It should be noted that the model was based on high-resolution 0.5 mm isotropic MR images, whereas all cited studies were based on MR images with considerably lower resolution. In addition, many of these studies used axial images with ~ 8 mm slice spacing and a relatively large slice thickness.

The provided subject-specific 3D model was based on a combination of subject-specific MR imaging (Fig. 1) and cadaveric measurements by Bozkurt et al. [25], Zhou et al. [26], Hauck et al. [27] and Lang et al. [28]. The cadaveric studies used to define the NR specifications were selected based on their completeness of information that included spinal cord NR descending angle, radicular line and diameter. As expected, a local enlargement of the spinal cord cross-sectional area was present near the lumbosacral (L2–S2) and cervical (C5–T1) enlargements located near 13 and 40 cm respectively below the foramen magnum (Fig. 7). These locations corresponded to the expected enlargement due to gray matter increase within those regions.

The exact 3D structure of the 31 NR pairs and coccygeal nerve were idealized based on the literature as it was not possible to extract their exact detailed geometry directly from MR imaging. However, it was possible to place each NR pair on a subject-specific basis at the insertion point in the spinal cord and exit point at the dura (details in “Methods”). The resulting model had a total NR volume of 5.8 cm³. This value is similar to that quantified by Hogan et al. (1996) and Martyr et al. (2011) with 7.31 and 9.2 cm³ respectively [38, 46]. The relatively smaller volume in our model is likely due to the smaller size of NR between the L2–S2 levels in comparison to Hogan’s cadaveric measurements [40]. In addition to the noted wide individual variability, Hogan et al. [38] estimated NR volume assuming estimate root lengths from relatively low resolution MRI data. Other studies quantifying cauda equina volume also based their results solely on estimations from MRI segmentations [39, 45‐50].

Total CSF volume in healthy adults has been reported to be ~ 150 mL in many standard medical textbooks [42, 51, 52] and recently published review articles [53, 54]. This value has become ubiquitous within the literature to the point of often not being cited with reference to any empirical study. Methods for CSF volume estimation by relatively crude casting techniques were originally applied [55]. These estimates were later criticized as being prone to significant degree of error [56, 57]. Review of more recent literature using non-invasive MRI-based methods indicates that total CSF volume in healthy adults to range from ~ 250 to 400 cm³ [42, 58‐61]. The difference in CSF volume determined from MRI versus invasive techniques is likely an underlying reason for the discrepancy. The referenced CSF volumetric studies using non-invasive techniques with high-resolution MR imaging may provide a more accurate estimate of total CSF volume. However, invasive measurements provide a lower bound for total CSF volume. More research is needed to fully establish detailed information about CSF volumetric distribution throughout the intracranial cisterns and subarachnoid space of the brain and spine.

At present, all models of the spinal SSS rely on varying degrees of simplification or idealization, often neglecting realistic spinal canal geometry and/or microanatomy. The simplest geometries are coaxial circular annuli employed by Lockey et al. [62], Berkouk et al. [63], Hettiarachchi et al. [64] and Elliott [65] that in some cases also included pathological variations, as well as in Bertram et al. [17] which used an idealized axial distribution for SSS area. Stockman [66] used an elliptical annuli and included microanatomical features, whereas Kuttler [67] modeled an elliptical annulus based on work by Loth et al. [29] who created a SSS from realistic SSS cross sections. The axial distribution of our model spinal cord and dura shows strong similarity to Loth et al. [29], Fig. 3, with a peak SSS area located at the FM and dural sac lumbar enlargement (Fig. 7b). Hsu et al. [40], Pahlavian et al. [36] and Tangen et al. [10, 12] developed CFD models with a subject specific geometry of the SSS reconstructed from MR data. The Pahlavian and Tangen CFD models also included varying degrees of NR detail. Pahlavian idealized NR as smooth airfoil-shaped flat objects and limited the model to the cervical spine. Yiallourou et al. [68] conducted a CFD study to investigate alterations in craniocervical CSF hydrodynamics in healthy controls versus patients with Chiari malformation. In that study, NR were not included in the CFD geometry. The CFD-based velocity profile results were found to lack similarity with in vivo 4D Flow MRI measurements. It was concluded that NR or other relatively small anatomic features are likely needed to accurately reflect CSF velocities within the cervical spine.

The geometric model presented in this study contributes NR microanatomy as discreet rootlets and cauda equina within a complete subject-specific SSS geometry. The model geometry is provided in a downloadable format with the dura, spinal cord and NR as separate files in the .STL (triangular) and .OBJ (quadrilateral) formats (six files in total). This allows modification of each surface separately for modeling purposes. For example, the model could be altered locally to increase the thecal sac volume during upright posture.

The computed parameters for CSF dynamics in terms peak flow rate, mean velocity and Reynolds number (Fig. 7) compare favorably to previous studies. The measured CSF flow rate waveforms (Fig. 6a) had similar magnitude as previous studies in the literature by Loth et al. [29], Linninger et al. [69] and Greitz [70, 71]. For those studies, average value of the peak CSF velocity at C2 vertebral level was ~ 2.5 cm/s. In the present model, peak CSF velocity at C2 vertebral level was 2.16 cm/s (Fig. 7f, towards feet). CSF pulse wave velocity (PWV), was estimated to be 19.4 cm/s in the healthy subject based on feature points of the CSF flow waveform measured along the entire spine (Fig. 6b). This value is lower than those previously reported in the literature that include 4.6 ± 1.7 m/s by Kalata et al. in the cervical spine [31] and ~ 40 m/s by Greitz in a patient [72]. It is difficult to directly compare these results with the present study, as they varied in technique, measurement location and type of subject.

Peak Reynolds number was predicted to be 175 and located within the cervical spine. This value suggests the presence of laminar CSF flow throughout the intrathecal space. However, it should be noted that that the SSS is a highly complex geometry that also contains microscopic structures called arachnoid trabeculae that were not included in the flow calculations. Previous biofluids studies have shown that geometric complexity can allow flow to become partially turbulent at Re > 600 in a stenosis [73], at Re 200–350 in aneurysms [74, 75], in the heart [76] and within CSF in the SSS [77, 78]. More research is needed to define the nature of CSF flow dynamics with respect to turbulence.

Cerebrospinal fluid flow data was collected at three distinct axial locations along the spine for a single subject. Data from these three locations was spatial-temporally interpolated (Fig. 6b) and used in combination with the geometry to quantify axial distribution of CSF dynamics along the spine (Fig. 7). While only representative of the single subject analyzed, the provided parameters give insight into CSF dynamics for a single healthy subject within a complete SC model containing detailed nerve root geometry. For example, the detailed geometry showed that Reynolds number varies significantly along the spine due to the presence of NR (see Fig. 7d Reynolds number variation in cervical spine). Note: validation of numerical models using the provided downloadable CSF flow waveform data should only take into account CSF flow rates measured at the three distinct axial locations (Fig. 6a). Interpolated values are not empirical data to be used for validation purposes.