- Research
- Open Access

# Barrier dysfunction or drainage reduction: differentiating causes of CSF protein increase

- Mahdi Asgari
^{1, 2}, - Diane A. de Zélicourt
^{1}and - Vartan Kurtcuoglu
^{1, 2, 3}Email authorView ORCID ID profile

**14**:14

https://doi.org/10.1186/s12987-017-0063-4

© The Author(s) 2017

**Received:**24 February 2017**Accepted:**9 May 2017**Published:**18 May 2017

## Abstract

### Background

Cerebrospinal fluid (CSF) protein analysis is an important element in the diagnostic chain for various central nervous system (CNS) pathologies. Among multiple existing approaches to interpreting measured protein levels, the Reiber diagram is particularly robust with respect to physiologic inter-individual variability, as it uses multiple subject-specific anchoring values. Beyond reliable identification of abnormal protein levels, the Reiber diagram has the potential to elucidate their pathophysiologic origin. In particular, both reduction of CSF drainage from the cranio-spinal space as well as blood–CNS barrier dysfunction have been suggested ρas possible causes of increased concentration of blood-derived proteins. However, there is disagreement on which of the two is the true cause.

### Methods

We designed two computational models to investigate the mechanisms governing protein distribution in the spinal CSF. With a one-dimensional model, we evaluated the distribution of albumin and immunoglobulin G (IgG), accounting for protein transport rates across blood–CNS barriers, CSF dynamics (including both dispersion induced by CSF pulsations and advection by mean CSF flow) and CSF drainage. Dispersion coefficients were determined a priori by computing the axisymmetric three-dimensional CSF dynamics and solute transport in a representative segment of the spinal canal.

### Results

Our models reproduce the empirically determined hyperbolic relation between albumin and IgG quotients. They indicate that variation in CSF drainage would yield a linear rather than the expected hyperbolic profile. In contrast, modelled barrier dysfunction reproduces the experimentally observed relation.

### Conclusions

High levels of albumin identified in the Reiber diagram are more likely to originate from a barrier dysfunction than from a reduction in CSF drainage. Our in silico experiments further support the hypothesis of decreasing spinal CSF drainage in rostro-caudal direction and emphasize the physiological importance of pulsation-driven dispersion for the transport of large molecules in the CSF.

## Keywords

- Spinal Canal
- Dispersion Coefficient
- Barrier Permeability
- Spinal Subarachnoid Space
- Albumin Quotient

## Background

Despite continued advances in non-invasive medical imaging, cerebrospinal fluid (CSF) analysis in general and CSF protein analysis in particular have remained important tools for the diagnosis of various disorders of the central nervous system (CNS) [1]. Yet while it is accepted that abnormal changes in CSF protein content are indicative of pathological conditions, the reasons leading to the measured protein concentrations are often a matter of debate [2].

While some proteins found in the CSF are synthesized within the CNS (choroid plexus, brain and spine) or the meninges, most of them originate in the blood serum under normal conditions [2–4]. They pass through blood-CNS barriers (either the blood–brain barrier, BBB, or blood-CSF barrier, BCSFB) into CNS fluids [5]. Equilibrium between the rate-limited influx of serum derived proteins through these barriers and their efflux with CSF drainage determines the protein content of the CSF [6]. Changes in the concentrations of these proteins may thus reflect alterations in either (1) serum protein levels, (2) intrathecal protein synthesis [7], (3) barrier properties [8], or (4) CSF dynamics and drainage [2].

Of the four possible causes for changes in CSF protein concentration listed above, the Reiber diagram corrects for variations in serum protein levels and identifies intrathecal protein synthesis (see Fig. 1a). However, it cannot distinguish between changes in CNS barrier properties and changes in CSF dynamics and drainage, both of which have been hypothesized as possible causes for abnormal albumin quotients [2, 8, 9]. In this study, we have employed a set of computational tools to test these two competing hypotheses.

To this end, we have analysed how changes in barrier function, CSF drainage rates and pulsatility translate to changes of albumin and IgG quotients in the Reiber diagram, where IgG was chosen from the family of immunoglobulins arbitrarily as a common biomarker for inflammatory neurological disorders [10]. Our models reproduce the empirical mathematical relationship between the two quotients given by Reiber, quantify the effect of CSF pulsation on protein distribution and show that barrier dysfunction rather than decreased cerebrospinal fluid drainage is the likely cause of abnormally high albumin values in the Reiber diagram. Our results further emphasize the pathophysiological importance of dispersion, CSF drainage and blood-CNS barrier permeability for the transport of large molecules in the spinal subarachnoid space.

## Methods

### Three-dimensional model of protein dispersion induced by CSF pulsation

Dispersion as the combined effect of diffusion and advection by pulsatile fluid motion with zero net flow is the governing mechanism for the faster transport of solutes in the CSF compared to pure diffusion [11–14]. To determine dispersion coefficients of albumin and IgG along the spine, we first solve the axisymmetric three-dimensional Navier–Stokes equations and associated advection–diffusion equation for protein transport in a segment of the spinal canal.

#### Model characteristics

Model parameters

Parameter | Value | References |
---|---|---|

| ||

In the cortical subarachnoid space | 29.4 | [27] |

In the ventricular space | 7.6 | [27] |

In the spinal space | 4.8 | [27] |

| ||

Ventricular space | 30 | |

Cortical subarachnoid space | 90 | |

Spinal subarachnoid space | 30 | |

| ||

Pore radius, r | 19.4 | [6] |

Albumin hydrodynamic radius, \(a_{Al}\) | 3.58 | [6] |

Immunoglobulin G hydrodynamic radius, \(a_{IgG}\) | 5.34 | [6] |

| ||

CSF total production and drainage rate, F [ml/day] | 500 | [30] |

CSF pulsation | ||

CSF pulsation amplitude in the cervical region [mm/s] | 10 | [25] |

CSF pulsation amplitude in the lumbar region [mm/s] | 0 | [24] |

CSF pulsation time period [s] | 0.8 | [25] |

| ||

Density, \(\rho\) [kg/m | 1000 | |

Viscosity, \(\mu\) [Pa s] | 0.001 | |

| ||

Porosity, \(\varepsilon\) | 0.99 | [21] |

Permeability in the longitudinal direction, K | 1.45 · 10 | [21] |

Permeability in the radial direction, K | 2.36 · 10 | [21] |

| ||

Albumin concentration in the lumbar CSF [mg/ml] | 0.363 | [27] |

Albumin CSF/blood quotient in the lumbar space | 0.002 | [31] |

Albumin quotient ratio (lumbar to cisternal) | 2 | [27] |

Albumin quotient ratio (cortical subarachnoid space to cisternal) | 3 | [27] |

| ||

Spinal cord diameter | 10 | |

Spinal subarachnoid space thickness, w | 4 | |

Spinal segment length | 100 | |

Spine length between cistern and lumbar space | 700 | |

Protein properties [m | ||

Albumin diffusion coefficient, D | 6 · 10 | |

Immunoglobulin G diffusion coefficient, D | 2.4 · 10 |

The model domain is treated as porous, with permeability and porosity metrics according to literature values for the subarachnoid space [18]. A velocity (flow) boundary condition derived from MRI measurements of spinal CSF [19] is imposed at the inlet boundary (proximal site), while a constant pressure boundary condition is imposed at the outlet (distal site). Both the inner and outer boundaries of the spinal canal are treated as impermeable walls with zero slip and zero solute flux conditions. Constant solute concentration is imposed at the axial boundaries.

#### Solution methodology

Equations (1) to (3) are discretized using an implicit Euler scheme for the temporal derivatives and central differencing for the first and second order spatial derivatives. All calculations are conducted with a time step size of 10^{−4} s and spatial resolution of 100 μm in both axial and radial directions. Grid and time-step independence were confirmed.

#### Evaluation of the dispersion coefficient

### One dimensional model of protein distribution in the spinal CSF

_{i}, represents the influx of serum proteins into the CSF, while the sink term, S

_{o}, represents protein efflux due to CSF drainage [23]. The dimensions of the domain are reported in Table 1.

#### Evaluation of the dispersion coefficient D*

The dispersion coefficient depends on both the solute considered and the amplitude of the CSF pulsations. The latter has been shown to increase from zero in the lumbar space [24] to a maximum of about 10 mm/s in the cervical region [25]. Accordingly, we applied our three-dimensional model to characterize the dispersion coefficients of albumin and IgG for CSF pulsation amplitudes ranging between 0 and 10 mm/s. The corresponding dispersion values are reported in results section. Expectedly, dispersion equals to diffusion for the pulsation amplitude of zero (i.e. in the lumbar space) and increases for the higher pulsation amplitudes, reaching a maximum for 10 mm/s velocity (i.e. in the cervical space). Since there is an almost linear relation between the imposed velocity and calculated dispersion coefficient, we consider a linear increase of the dispersion coefficient from \(D_{\hbox{min} }^{*}\) equal to the pure diffusion coefficient in the lumbar space to a value of \(D_{\hbox{max} }^{*}\) in the cervical region.

#### Evaluation of the source term

_{b}stands for the diffusive permeability of the blood-CNS barriers for the protein under consideration and \(C_{blood}\) is the serum protein concentration. The permeability of the barrier to albumin molecules in different regions of the CSF compartments has been measured with radioactive studies [27]. However, it is not known how this permeability might change due to barrier opening. In order to model such permeability variations in pathological situations, we use the membrane pore model described in [6], which was demonstrated to accurately capture barrier permeability for different proteins. In this model, permeability depends on the ratio of protein size to pore size:

#### Evaluation of the sink term

*F*is the CSF drainage rate. The total CSF turn-over rate has been estimated to 500 ml/day in humans [30]. However, the distribution of the corresponding drainage between cranial and spinal compartments is not fully known [30], let alone its distribution along the spinal axis. To address this issue, we leverage available data on the spatial distribution of albumin concentrations at steady state, namely the known relative concentrations of albumin in the cisterns, lumbar and cortical subarachnoid spaces, and reported albumin concentration gradients along the spinal subarachnoid space.

_{Al}, in a given CSF compartment [6]:

_{Al}is known, namely the cisterns, cortical or spinal subarachnoid spaces, P

_{bc}stands for the barrier permeability in that compartment and \(\overline{{F_{c} }}\) for the mean CSF drainage rate to be determined. The corresponding results are reported in Table 3. The obtained mean drainage characteristics for the spinal compartment, \(\overline{{F_{spinal} }}\), are then employed as baseline for other tested scenarios.

Having calculated the mean CSF drainage rate for the spinal compartment, we determine its local value by making use of reported albumin concentration gradients along the neuraxis. Due to the low CSF turnover rate, sequential sampling of CSF through a lumbar puncture allows one to sequentially access CSF portions from the lumbar, thoracic and finally cervical subarachnoid spaces. Using this method, a decrease of Q_{Al} was observed from the first 0–3 ml of CSF to the last 27–30 ml of CSF obtained by lumbar puncture [31]. Having an opposite gradient in CSF drainage has been hypothesized as the most probable mechanism for these changing CSF protein concentrations [6]. Accordingly, we assume spinal CSF drainage to increase linearly from zero at x = 0 in the lumbar sac (end of lumbar region) to twice \(\overline{{F_{spinal} }}\) in the cervical region, thereby ensuring that the average spinal drainage matches the above determined value, \(\overline{{F_{spinal} }}\). Note that only at exactly x = 0 is CSF drainage zero, but that integrated over a segment, for example along the lumbar region, there is CSF drainage.

#### Solution method

Equation (5) for solute transport is discretized using finite differences in Matlab with a forward Euler time stepping scheme and second order central differences for the spatial second derivatives. Neumann boundary conditions of zero flux for concentrations are imposed on the proximal end of the cervical region and the distal end of the lumbar space. These zero flux boundary conditions are reasonable due to the closed end of the lumbar and the steady-state equilibrium between protein influx and efflux in the lumped compartment of cranial space. The equation is solved with a time-step size of 6 s and a spatial resolution of 3.5 mm, with confirmed time-step and grid independence.

### The Reiber diagram

## Results

### Transport of the molecules in the spinal canal

^{−11}and 2.4 · 10

^{−11}m

^{2}/s, respectively. A peak CSF velocity of 10 mm/s was considered as Ref. [25]. The resulting dispersion coefficients are summarized in Table 2. Since the CSF pulsation amplitude reduces along the spinal canal towards the lumbar space [24], we also calculated the dispersion coefficient for lower velocities. Puy et al. showed that CSF pulsations can change in pathological situations [33], demonstrating an up to four fold increase in amplitude. To evaluate the impact of such pathological variations on protein distribution, we also calculated dispersion coefficients for accordingly increased velocities. We observed an almost linear increase in the dispersion coefficients with increasing velocity amplitude.

Calculated protein dispersion coefficients

Molecule | Diffusion coefficient (m | Maximum CSF velocity (mm/s) | Dispersion coefficient (m |
---|---|---|---|

Immunoglobulin G | 2.4 · 10 | 10 | 4.0 · 10 |

Albumin | 6.0 · 10 | 2.5 | 2.8 · 10 |

5 | 2.2 · 10 | ||

10 | 6.0 · 10 | ||

20 | 1.3 · 10 | ||

40 | 2.7 · 10 |

### Distribution of albumin and IgG in the spinal CSF: baseline condition

CSF drainage distribution and albumin quotients in different CSF compartments

| |

Cortical region | 82% |

Spinal region | 18% |

| |

Lumbar region | 0.002 |

Cortical subarachnoid space | 0.003 |

Cistern | 0.001 |

### Impact of CSF pulsation amplitude change on protein distribution

### Impact of barrier dysfunction and CSF drainage on protein quotients

Figure 6b illustrates the effect of change in barrier permeability for three different constant CSF drainage rates. The center (dashed) curve corresponds to nominal drainage, while the upper and lower solid curves correspond to 30% increased and decreased drainage rates, respectively. All three curves are hyperbolic. We used the upper and lower curves to calculate representations of the population variation coefficient, obtaining values of 0.48, 0.44 and 0.4 for albumin quotients of 0.001, 0.002 and 0.003, respectively. Note that the population variation coefficient determined by Reiber based on patient data is constant over a range of albumin quotients.

Figure 6c illustrates the effect of change in barrier permeability for three different baseline IgG permeabilities, reflecting the variation of the barrier permeability to IgG to different extent than for albumin as shown by Seyfert et al. [34]. The center (dashed) curve corresponds to nominal baseline IgG permeability, while the upper and lower solid curves correspond to 30% increased and decreased baseline IgG permeability, respectively. The representation of the population variation coefficient is in this case 0.6 for all albumin quotients.

## Discussion

The biochemical analysis of the cerebrospinal fluid is an important diagnostic tool for pathologies of the CNS. For example, changes in CSF immunoglobulin content can be indicative of inflammatory reactions in the brain. To account for inter-individual and normal intra-individual variability, it is advantageous to assess relative rather than absolute values of protein concentration as done in the Reiber diagram. While the Reiber diagram can indicate intrathecal synthesis of proteins, it is debated whether higher than normal readings of relative albumin concentrations are indicative of CNS barrier dysfunction or reduction in CSF drainage. Here we have employed a set of computational models to assess which one of these two changes is the more likely cause of increased albumin concentration in CSF relative to that in the blood plasma.

The Reiber diagram features a hyperbolic relationship between albumin quotient and, for example, IgG quotient, where ‘quotient’ refers to the concentration of the respective protein in CSF relative to its concentration in blood plasma. Reiber derived this empirical relationship from measurements in a large set of patients in which intrathecal synthesis of the protein of interest could be excluded. He hypothesized that this non-linear relationship was caused by inter-patient variability in CSF drainage rates [32]. However, as shown in Fig. 6a, our models indicate that variations in the rate of CSF drainage would yield a linear relationship between the quotients rather than the experimentally determined hyperbolic one. Reiber also calculated the variation coefficient for his patient database and found it to be constant for a large range of albumin quotients. Our calculations show that the variation coefficient does not stay constant for different baseline CSF drainage values (Fig. 6b), indicating that inter-patient variability in CSF drainage alone may not result in the protein quotient relationship observed by Reiber. One should thus not, without further case-dependent evidence, attribute abnormally high albumin quotients identified in the Reiber diagram to reduced CSF drainage.

Others have attributed increased albumin quotients to blood-CNS barrier dysfunction. Indeed, as shown in Fig. 6a, variation in barrier permeability leads to the expected hyperbolic relationship between protein quotients. This is further confirmed by a constant population variation coefficient as illustrated in panel (c) for different baseline IgG permeabilities. Consequently, high albumin quotients identified in the Reiber diagram may be seen as indicative of a CNS barrier dysfunction.

Our calculations of the distribution of CSF efflux indicate 18% drainage in the spinal compartment and 82% drainage in the cranial compartment. This distribution matches well with the measurements of Marmarou et al. [35] in cats, where absorption in the spinal space accounted for 16% of the total CSF drainage and the cranial space contributed 84%. Similar results were obtained by Gehlen et al. using a lumped parameter model of coupled cardiovascular and CSF dynamics [36]. Albumin quotients calculated based on this drainage distribution are within the range of values obtained experimentally in healthy subjects [31].

Seyfert et al. measured albumin and immunoglobulin concentration gradients in the spinal CSF by sequential CSF sampling through lumbar puncture. They showed a decreasing protein concentration profile from lumbar to cervical space [31]. It was hypothesized that this concentration gradient results from the variation of CSF drainage along the spine [6]. Our calculations show that the hypothesized drainage gradient along the spinal canal with minimum drainage rate in the lumbar space would, indeed, result in a longitudinal concentration gradient for albumin and IgG (Fig. 4). Therefore, our results support the existence of rostro-caudally decreasing spinal CSF drainage.

Puy et al. correlated the magnitude of CSF pulsation with protein distribution in different CSF compartments [33]. We calculated the dispersion rate of albumin in the spinal CSF for different pulsation amplitudes as reported in Table 2, and employed these values in our global protein distribution model. Increased CSF pulsation diminishes the longitudinal concentration gradient in the spinal canal, while reduced pulsation intensifies it (Fig. 5). These results are in line with the measurements of Puy et al. [33]. Therefore, changed CSF dynamics in pathologies such as hydrocephalus and Chiari malformation could have an impact on protein distribution in the spinal canal.

The two computational models developed in this study have the following main limitations: First and foremost, we have simplified the spinal canal anatomy substantially to a 3D axisymmetric annular conduit and a 1D representation, respectively, considering the spinal subarachnoid space as a porous medium. Both the macroscopic anatomy as well as the microanatomy of the CSF spaces as defined by, e.g. arachnoid trabeculae, could play an important role in fluid and solute dynamics. Neglecting the microanatomy can lead to discrepancies between computed and measured metrics of spinal CSF dynamics [19]. In our models, the effect of microstructures is approximated by the introduction of anisotropic permeability of the porous medium representing the spinal subarachnoid space.

The second main limitation pertains to the issue of parameter uncertainty. For instance, we have considered the overall CSF drainage rate to be equal to the estimated value of CSF production, which itself is only known approximatively [30]. We have dealt with parameter uncertainty by performing sensitivity analyses, which show that our main conclusions are robust with respect to reasonable variations of the model parameters. Concretely, we have shown that the hyperbolic protein quotient function in the Reiber diagram that results from variation in barrier permeability does not depend on baseline CSF drainage (Fig. 6b) or IgG permeability values (Fig. 6c). We have also made sure that the population variation coefficient does not only stay constant for a 30% change in IgG baseline permeability (Fig. 6c), but also for much larger and smaller changes (up to 100% change). Finally, we checked that the derived dispersion coefficients do not depend on the computational domain length and hydraulic conductivity of the domain.

## Declarations

### Authors’ contributions

MA implemented the computational model and performed the calculations. DAZ supervised model implementation and calculations. VK conceived the study and directed the research. All authors analyzed the data and wrote the manuscript. All authors read and approved the final manuscript.

### Acknowledgements

We gratefully acknowledge the financial support provided by the Swiss National Science Foundation through Grant 200021_147193 CINDY, Marie Heim-Vögtlin fellowship PMPDP2_151255 and NCCR Kindey.CH.

### Competing interests

The authors declare that they have no competing interests.

### Availability of data and materials

All data generated or analyzed during this study are included in this published article.

### Funding

The presented study was financially supported by the Swiss National Science Foundation through Grant 200021_147193 CINDY, Marie Heim-Vögtlin fellowship PMPDP2_151255 and NCCR Kindey.CH.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.

## Authors’ Affiliations

## References

- Frankfort SV, Tulner LR, van Campen JP, Verbeek MM, Jansen RW, et al. Amyloid beta protein and tau in cerebrospinal fluid and plasma as biomarkers for dementia: a review of recent literature. Curr Clin Pharmacol. 2008;3(2):123–31.View ArticlePubMedGoogle Scholar
- Reiber H. Proteins in cerebrospinal fluid and blood: barriers, CSF flow rate and source-related dynamics. Restor Neurol Neurosci. 2003;21(3–4):79–96.PubMedGoogle Scholar
- Reiber H, Padilla-Docal B, Jensenius JC, Dorta-Contreras AJ. Mannan-binding lectin in cerebrospinal fluid: a leptomeningeal protein. Fluids Barriers CNS. 2012;9(1):17.View ArticlePubMedPubMed CentralGoogle Scholar
- Rosen H, Sunnerhagen KS, Herlitz J, Blomstrand C, Rosengren L. Serum levels of the brain-derived proteins S-100 and NSE predict long-term outcome after cardiac arrest. Resuscitation. 2001;49(2):183–91.View ArticlePubMedGoogle Scholar
- Engelhardt B, Sorokin L. The blood-brain and the blood-cerebrospinal fluid barriers: function and dysfunction. Semin Immunopathol. 2009;31(4):497–511.View ArticlePubMedGoogle Scholar
- Rapoport SI. Passage of proteins from blood to cerebrospinal fluid. Neurobiology of cerebrospinal fluid 2. New York: Springer; 1983. p. 233–45.Google Scholar
- Winfield JB, Shaw M, Silverman LM, Eisenberg RA, Wilson HA 3rd, et al. Intrathecal IgG synthesis and blood-brain barrier impairment in patients with systemic lupus erythematosus and central nervous system dysfunction. Am J Med. 1983;74(5):837–44.View ArticlePubMedGoogle Scholar
- Sharief MK, Ciardi M, Thompson EJ. Blood-brain barrier damage in patients with bacterial meningitis: association with tumor necrosis factor-alpha but not interleukin-1 beta. J Infect Dis. 1992;166(2):350–8.View ArticlePubMedGoogle Scholar
- Zetterberg H, Jakobsson J, Redsater M, Andreasson U, Palsson E, et al. Blood-cerebrospinal fluid barrier dysfunction in patients with bipolar disorder in relation to antipsychotic treatment. Psychiatry Res. 2014;217(3):143–6.View ArticlePubMedGoogle Scholar
- Akaishi T, Narikawa K, Suzuki Y, Mitsuzawa S, Tsukita K, et al. Importance of the quotient of albumin, quotient of immunoglobulin G and Reibergram in inflammatory neurological disorders with disease-specific patterns of blood–brain barrier permeability. Neurol Clin Neurosci. 2015;3(3):94–100.View ArticleGoogle Scholar
- Asgari M, de Zélicourt D, Kurtcuoglu V. Glymphatic solute transport does not require bulk flow. Sci Rep. 2016;6:38635.View ArticlePubMedPubMed CentralGoogle Scholar
- Kurtcuoglu V, Soellinger M, Summers P, Poulikakos D, Boesiger P. Mixing and modes of mass transfer in the third cerebral ventricle: a computational analysis. J Biomech. 2007;129(5):695–702.View ArticleGoogle Scholar
- Hettiarachchi HD, Hsu Y, Harris TJ Jr, Penn R, Linninger AA. The effect of pulsatile flow on intrathecal drug delivery in the spinal canal. Ann Biomed Eng. 2011;39(10):2592–602.View ArticlePubMedGoogle Scholar
- Siyahhan B, Knobloch V, de Zelicourt D, Asgari M, Schmid Daners M, et al. Flow induced by ependymal cilia dominates near-wall cerebrospinal fluid dynamics in the lateral ventricles. J R Soc Interface. 2014;11(94):20131189.View ArticlePubMedPubMed CentralGoogle Scholar
- Panjabi MM, Oxland T, Takata K, Goel V, Duranceau J, et al. Articular facets of the human spine. Quantitative three-dimensional anatomy. Spine (Phila Pa 1976). 1993;18(10):1298–310.View ArticleGoogle Scholar
- Panjabi MM, Takata K, Goel V, Federico D, Oxland T, et al. Thoracic human vertebrae. Quantitative three-dimensional anatomy. Spine (Phila Pa 1976). 1991;16(8):888–901.View ArticleGoogle Scholar
- Zaaroor M, Kosa G, Peri-Eran A, Maharil I, Shoham M, et al. Morphological study of the spinal canal content for subarachnoid endoscopy. Minim Invasive Neurosurg. 2006;49(4):220–6.View ArticlePubMedGoogle Scholar
- Gupta S, Soellinger M, Grzybowski DM, Boesiger P, Biddiscombe J, et al. Cerebrospinal fluid dynamics in the human cranial subarachnoid space: an overlooked mediator of cerebral disease. I. Computational model. J R Soc Interface. 2010;7(49):1195–204.View ArticlePubMedPubMed CentralGoogle Scholar
- Yiallourou TI, Kroger JR, Stergiopulos N, Maintz D, Martin BA, et al. Comparison of 4D phase-contrast MRI flow measurements to computational fluid dynamics simulations of cerebrospinal fluid motion in the cervical spine. PLoS ONE. 2012;7(12):e52284.View ArticlePubMedPubMed CentralGoogle Scholar
- Jasak H. OpenFOAM: open source CFD in research and industry. Int J Nav Arch Ocean. 2009;1(2):89–94.Google Scholar
- Gupta S, Soellinger M, Boesiger P, Poulikakos D, Kurtcuoglu V. Three-dimensional computational modeling of subject-specific cerebrospinal fluid flow in the subarachnoid space. J Biomech Eng. 2009;131(2):021010.View ArticlePubMedGoogle Scholar
- Pineda SM, Diaz G, Coimbra CFM. Approximation of transient 1D conduction in a finite domain using parametric fractional derivatives. J Heat Trans-T Asme. 2011;133(7):071301.View ArticleGoogle Scholar
- Abbott NJ. Evidence for bulk flow of brain interstitial fluid: significance for physiology and pathology. Neurochem Int. 2004;45(4):545–52.View ArticlePubMedGoogle Scholar
- Schellinger D, LeBihan D, Rajan SS, Cammarata CA, Patronas NJ, et al. MR of slow CSF flow in the spine. AJNR Am J Neuroradiol. 1992;13(5):1393–403.PubMedGoogle Scholar
- Pahlavian SH, Bunck AC, Loth F, Tubbs RS, Yiallourou T, et al. Characterization of the discrepancies between four-dimensional phase-contrast magnetic resonance imaging and in silico simulations of cerebrospinal fluid dynamics. J Biomech Eng. 2015;137(5):051002.View ArticleGoogle Scholar
- Poduslo JF, Curran GL, Wengenack TM, Malester B, Duff K. Permeability of proteins at the blood–brain barrier in the normal adult mouse and double transgenic mouse model of Alzheimer’s disease. Neurobiol Dis. 2001;8(4):555–67.View ArticlePubMedGoogle Scholar
- Cutler RW, Murray JE, Cornick LR. Variations in protein permeability in different regions of the cerebrospinal fluid. Exp Neurol. 1970;28(2):257–65.View ArticlePubMedGoogle Scholar
- Bechter K, Schmitz B. Cerebrospinal fluid outflow along lumbar nerves and possible relevance for pain research: case report and review. Croat Med J. 2014;55(4):399–404.View ArticlePubMedPubMed CentralGoogle Scholar
- Bechter K, Benveniste H. Quinckes’ pioneering 19th centuries CSF studies may inform 21th centuries research. Neurol Psychiatry Brain Re. 2015;21(2):79.View ArticleGoogle Scholar
- Brinker T, Stopa E, Morrison J, Klinge P. A new look at cerebrospinal fluid circulation. Fluids Barriers CNS. 2014;11:10.View ArticlePubMedPubMed CentralGoogle Scholar
- Seyfert S, Faulstich A. Is the blood-CSF barrier altered in disease? Acta Neurol Scand. 2003;108(4):252–6.View ArticlePubMedGoogle Scholar
- Reiber H. Flow rate of cerebrospinal fluid (CSF)—a concept common to normal blood-CSF barrier function and to dysfunction in neurological diseases. J Neurol Sci. 1994;122(2):189–203.View ArticlePubMedGoogle Scholar
- Puy V, Zmudka-Attier J, Capel C, Bouzerar R, Serot J-M, et al. Interactions between flow oscillations and biochemical parameters in the cerebrospinal fluid. Front Aging Neurosci. 2016;8:154. doi:10.3389/fnagi.2016.00154.View ArticlePubMedPubMed CentralGoogle Scholar
- Seyfert S, Quill S, Faulstich A. Variation of barrier permeability for albumin and immunoglobulin G influx into cerebrospinal fluid. Clin Chem Lab Med. 2009;47(8):955–8.View ArticlePubMedGoogle Scholar
- Marmarou A, Shulman K, LaMorgese J. Compartmental analysis of compliance and outflow resistance of the cerebrospinal fluid system. J Neurosurg. 1975;43(5):523–34.View ArticlePubMedGoogle Scholar
- Gehlen M, Kurtcuoglu V, Daners MS. Patient specific hardware-in-the-loop testing of cerebrospinal fluid shunt systems. IEEE Trans Biomed Eng. 2016;63(2):348–58.View ArticlePubMedGoogle Scholar