Investigating the Effects of Cerebral Aqueduct Blockage on Hydrocephalus Symptoms

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Contributors: Ibrahim Hassan, Siddharth Dhadwal, Parm Ghai, Yan Wah Jemima Chong

Supervisor: Prof. Edmond Young 

1.0. Introduction

Cerebrospinal fluid (CSF) is the fluid that flows through the ventricular system, which surrounds the brain to provide mechanical protection (cushioning) from physical trauma and maintains hemostasis by supplying nutrients and removing waste [1]. The CSF in the ventricular system of the brain is of most interest to this report as it can be the source of headaches. Particularly, the cerebral aqueduct (aqueduct of Sylvius), pictured in Fig. 1.1, is of interest as it is the focus of analysis in this report.

The pressure that CSF exerts on the cranial vault and the brain is known as intracranial pressure (ICP) [3]. The pressure exerted on the ventricular system also falls under ICP [4].  

Hydrocephalus is a neurological condition characterized by the excessive accumulation of CSF within the ventricles of the brain [5]. Disruptions in ICP and CSF flow cause headache because they lead to increased pressure within the skull, which stretches and irritates pain-sensitive structures such as the dura, venous sinuses, and cranial nerves [6]. When CSF outflow is obstructed or venous sinus pressure rises, CSF accumulates and ICP increases, producing mechanical distortion of tissues and venous congestion. This pressure can also compress the optic nerve sheath and meninges, contributing to retro-orbital or diffuse head pain [6]. Reduced intracranial compliance further amplifies these effects, meaning even small changes in CSF flow and ventricular volume cause large increases in ICP and consequently, headaches [6]. This condition can be congenital or acquired, including from infections, tumors or hemorrhage [7], affecting both children and adults. Hydrocephalus symptoms include headache, vomiting, nausea, and disturbance of consciousness [8].

There are two main types of hydrocephalus, communicating and non-communicating [9]. The communicating variant is a result of the obstruction of the flow of CSF in subarachnoid space or at the site of absorption [9]. The non-communicating case is characterized by the blockage of CSF flow in narrow passages connecting the various ventricles, most commonly in the cerebral aqueduct as it is the narrowest part of the ventricular system, with a cross sectional area of only 0.2–1.8 mm2  in adults [9] [10]. Aqueductal stenosis is a condition characterized by the narrowing or obstruction of the cerebral aqueduct [11]. Due to its narrow dimensions, small changes can have a big impact on CSF flow [12]. Restricted flow due to stenosis in the aqueduct will lead to accumulation of CSF fluid in upstream ventricles, leading to enlargement of both lateral ventricles and the 3rd ventricle, which consequently will result in headache and other elevated ICP symptoms [13][14]. Aqueduct stenosis is the cause of 10% hydrocephalus cases in adults [15]. 

Current diagnosis methods include CT and MRI imaging to observe abnormal dilation of upstream ventricles. The lateral and 3rd ventricle can be observed to have a 30-50% dilation [16]. With a continuous and constant CSF production rate, restricted outflow due to blockage in the aqueduct can lead to CSF buildup, increasing pressure gradients in the ventricular system. The normal range for intracranial pressure (ICP) is 8-15 mmHg [17], where treatment usually begins at an elevated pressure of 15 mmHg, and the upper limit for treatment initiation is about 20-25 mmHg [18].  

As current diagnosis methods majorly rely on medical imaging, our interest lies in whether it is possible to estimate quantitative metrics to diagnose stenosis in the cerebral aqueduct. ICP is typically measured within the ventricles and not the aqueduct itself. There is no data on local pressure in the aqueduct with effects of stenosis due to measurement restrictions. Therefore, in this project we will use elevated ICP to estimate radius reduced, and subsequent pressure changes in the aqueduct. The project focuses on the aqueduct stenosis as the sole cause of increased ICP. This report aims to quantify how narrowing of the cerebral aqueduct alters CSF pressure distribution and flow dynamics, and how these changes may contribute to headache symptoms. These approaches allow for a comprehensive examination of CSF behavior, helping to interpret fluid mechanics changes that contribute to pathological conditions and informing potential diagnostic and therapeutic strategies.

2.0 Methods

This project employs two complementary analysis methods to investigate these mechanisms: mathematical modeling and computational fluid dynamics (CFD). Mathematical models provide a broader analytical view of intracranial pressure behavior by representing CSF flow with idealized equations. In this project, the analysis uses the Hagen–Poiseuille flow dynamics, which characterize laminar flow through narrow channels and directly relate pressure gradients, viscosity, and geometry to flow rate. CFD simulations offer detailed visualization and quantification of CSF flow and pressure distributions within physiological boundaries. The ANSYS Fluent simulations numerically solve the Navier–Stokes and continuity equations on a discretized domain using the Finite Volume Method, enabling accurate reconstruction of complex flow fields (see the detailed breakdown of the methods in Sections 3.0 and 4.0).

2.1. Dimensions and parameters

We are simplifying the geometry to an angled venturi tube to simulate stenosis at one location within the cerebral aqueduct. Existing literature simulating the CSF flow in the cerebral aqueduct treats it as a straight cylinder with no angle [19]. Therefore, introducing an angle within our simulations can provide a more physiologically realistic representation of the aqueduct and reveal flow behaviours not present in straight line tubs. 

We consider steady flow and ignore the pulsatile nature of the CSF flow in the cerebral aqueduct because of a difficulty in setting up the simulation and lack of high performance computing availability.

The cerebral aqueduct is 11 mm in length and has a 1.5 mm diameter [20]. The wall thickness is 0.125 mm which results in an internal diameter of 1.25 mm [20]. The mean angle between the cranial and caudal part of the aqueduct was 28° in males and 26° in females [21]. The report is using 28° because it is the most extreme angle away from a straight tube, allowing for investigation of the upper bound of the effects due the curvature of the aqueduct. 

The important properties for CSF in this report are its viscosity and density. CSF has a dynamic viscosity of 0.7-1 mPa⋅s and a density of 1.003-1.008 g/cm^3 [22] [23]. For a healthy individual, the mean velocity of CSF through the aqueduct is taken to be approximately 3.1 cm/s [24]. 

3.0. Mathematical Modelling - Steady State Poiseuille Flow

The CSF is treated as an incompressible Newtonian fluid, allowing the use of classical viscous flow equations. The flow through the aqueduct is assumed to be axisymmetric, and the aqueduct walls are considered rigid, impermeable, and subject to a no-slip boundary condition. Finally, the CSF flow is taken to be fully developed, steady, and laminar, which enables the use of analytical solutions for viscous flow in cylindrical conduits. 

The assumption of laminar flow can be confirmed with Reynolds number Re=  ρvL/μ,

where  ρ = fluid density [kg/m^3]

v = fluid velocity [m/s]

L = characteristic linear dimension, i.e. cerebral aqueduct diameter [m]

μ = dynamic viscosity [m^2/s]

In normal conditions with no blockage, buildup or stenosis,

Re=  ((1005kg/m^3)(0.031m/s)(1.25〖×10〗^(-3) m))/0.8mPa= 47.90

As Re<<2300, flow is laminar. 

With the above assumptions, the Hagen Poiseuille flow equation can be used to express volumetric flow rate in a tube as: 

Q=(Δpπr^4)/L8μ

For effects of stenosis to pressure drop, the above equation can be re-arranged:

 Δp=QR_Ω=Q 8μL/(πr^4 )=u⋅8μL/r^2  

Where R_Ω represents hydraulic resistance, 

u = fluid velocity [m/s]

μ = dynamic viscosity [m^2/s]

L = Length of tube [m]

r = radius of tube [m]

Q = fluid velocity × area [m^3/s]

The pressure drop along the aqueduct in a healthy individual is as follows:

Δp=(0.031m/s)⋅(8(8×10^(-4)  m^2/s)(0.011))/〖(6.25×10^(-4))〗^2 =5.587Pa

 Assuming the downstream pressure at cerebral aqueduct is proportional to the upstream ventricular pressures, it is possible to estimate the ratio of difference between normal and elevated pressures in the aqueduct is similar to ratio of pressure difference measured in the ventricular space 

(Δp_(narrowed aqueduct))/(Δp_(normal  aqueduct) )≈p_(high ICP)/p_(normal  ICP)   

In the Hagen Poisuelle flow equation, with the same fluid properties, the only variable is the radius term. Therefore, the above equation can be re-written as:

p_(high ICP)/p_(normal  ICP) ≈〖r_stenosis〗^4/〖r_normal〗^4 

Taking the midpoint value of the normal ICP to be 11.5 mmHg, and 15 mmHg as the elevated ICP due to stenosis [17] [18]:

r_(stenosis )=r_(normal ) 〖(11.5mmHg/15mmHg)〗^(1/4)

r_(stenosis )= 0.585mm

With this estimation, a -6.43% reduction in aqueduct radius would initiate treatment. The upper limit of 25 mmHg results in r_(stenosis )= 0.515 mm, indicating a -17.65% decrease in adequate diameter would most likely require treatment. 

Using the Hagen Poiseuille flow equation, the pressure drop along the aqueduct stenosis can be found:

〖Δp〗_(stenosis,r=0.585mm)=(0.031m/s)⋅(8(8×10^(-4)  m^2/s)(0.011))/〖(5.85×10^(-4))〗^2 =6.377Pa

〖Δp〗_(stenosis,r=0.585mm)=(0.031m/s)⋅(8(8×10^(-4)  m^2/s)(0.011))/〖(5.15×10^(-4))〗^2 =8.228Pa

CSF flow is driven by arterial pulsations and the cardiac cycle, therefore, flow is actually pulsatile and not steady [25] . The womersley number can determine if the ratio of pulsation frequency is greater than viscous effects. The equation for Womersley number is:

α=R√(ω/υ)

Where α = Womersley number (dimensionless) 

R = radius [m]

ω = frequency of oscillation [rad/s]

υ = kinematic viscosity [m2/s]

As the frequency of oscillation is the same as the heart rate, ω=6.28 rad/s. For a cerebral aqueduct with a normal radius, 

α=(6.25×10^(-4))√(6.28/((0.8mPa⋅s/1005kg/m^3)))=1.755

As α>1, it means that flow is inertia dominated, with a plug-flow like velocity profile. 

4.0. Computational fluid dynamics (CFD) Analysis

Although it is known that CSF flow is pulsatile in the previous section, CFD analysis will assume a steady-state flow due to limiting cThis section presents the CFD method used to simulate cerebrospinal fluid (CSF) flow through the cerebral aqueduct under normal and stenotic conditions. The objective is to quantify how narrowing of the aqueduct alters pressure distribution and flow behavior. All simulations were performed using ANSYS Fluent, using laminar, incompressible and steady-state flow assumptions.

4.1. Ansys Setup

4.1.1. Ansys Workbench Setup

ANSYS Workbench was used to organize the simulation workflow. Each case (normal, stenosis (1), and stenosis (2)) was simulated using an identical workflow to ensure consistent comparison of the end results.

4.1.2. Geometry

The straight section allows the flow to fully develop before entering the aqueduct inlet, ensuring precise CFD results. By the time the fluid passes from the third ventricle into the aqueduct, it has achieved a fully developed, parabolic velocity profile consistent with Hagen–Poiseuille flow in cylindrical conduits. The common straight section in all geometries is shown in Fig. 4.1.2.1. (a), (b), (c).

The areas of analysis relevant to our project (for which pressure profiles are plotted) are as shown in Fig. 4.1.2.2. (a), (b), (c).

4.1.2.1. Normal aqueduct

The baseline geometry consisted of a cylindrical tube with inlet and outlet diameters equal to 1.25 mm, with an angle of 28 degrees [21] from the normal of the inlet to the normal of the outlet.

4.1.2.2. Stenosis 1

Stenosis 1 introduces a mild narrowing to approximate early-stage aqueductal stenosis, with an inlet diameter of 1.25 mm, a neck diameter of 1.17 mm (a 6.4% diameter reduction, as calculated using mathematical modelling, Section 3), and an outlet diameter of 1.25 mm.

4.1.2.3. Stenosis 2

Stenosis 2 represents a more severe narrowing. Here, again the inlet and outlet diameters were taken as 1.25 mm, however the neck diameter was set to 1.03 mm (a 17.6 % reduction in diameter as calculated using mathematical modelling, Section 3).

4.1.3. Mesh

All geometries were meshed using ANSYS Meshing, with an element size of 0.09 mm for all cases. Inflation layers were applied along the aqueduct walls to ensure proper resolution of the boundary-layer flow.

4.1.3.1. Normal Aqueduct

The normal aqueduct displayed a uniform, high-quality tetrahedral mesh with effective boundary layer resolution. This ensured smooth prediction of  pressure fields along the length of the aqueduct.

4.1.3.2. Stenosis 1

Using the same procedure, the stenosis 1 aqueduct displayed a uniform, high-quality tetrahedral mesh with effective boundary layer resolution. This ensured smooth prediction of pressure fields along the length of the aqueduct.

4.1.3.3. Stenosis 2

Continuing to use the same procedure, the stenosis 2 aqueduct displayed a uniform, high-quality tetrahedral mesh with effective boundary layer resolution. This ensured smooth prediction of pressure fields along the length of the aqueduct.

4.2. Setup

4.2.1. Solver Settings

This section shows the solver configuration used for all the CFD cases.

4.2.2. Material Properties

The material used was cerebrospinal fluid (CSF), with a density of 1005 kg/m3 and a dynamic viscosity of 0.8 mPa⋅s [22][23]. These material properties were input based on experimental CSF measurements.

4.2.3. Boundary Conditions

An inlet velocity of 0.031 m/s was applied at the start of the extension [24]. The outlet was assigned a gauge pressure of 0 Pa to simplify the solution, as backflow behaviour was not relevant to the objectives of this analysis, and all walls were modelled with a no-slip boundary condition. A fully developed, parabolic velocity profile was applied at the inlet of the analysis domain by shifting the uniform inlet velocity condition to the entrance of the extension as fully developed velocity-driven conditions more accurately reproduce clinical CSF volumetric flow rates in small ducts compared to pressure-driven boundary conditions.

4.2.4. Residuals

4.2.4.1. Normal Aqueduct

Hybrid initialization from the inlet was used. Simulations were run until residuals were less than 1 x 10-4 for all monitored variables. Convergence was achieved in approximately 45 iterations in all cases.

4.2.4.2. Stenosis 1

4.2.4.3. Stenosis 2

4.3. Results

4.3.1. Pressure Profile on Aqueduct Walls

This section presents the CFD-computed pressure distributions along the aqueduct walls for the normal aqueduct, the aqueduct with "stenosis (1)", and "stenosis (2)". The goal is to quantify the effect of geometric narrowing on the CSF’s outermost layer pressure profile and flow behavior using ANSYS Fluent. All pressure values reported correspond to the maximum static pressure along the aqueduct walls within the defined analysis region (see Section 4.1.2).

4.3.1.1. Normal Aqueduct

For the normal, unobstructed aqueduct, the maximum static pressure predicted by Fluent (CFD facet maximum) was 5.711 Pa in Figure 4.3.1.1. The pressure field along the wall shows a smooth, almost linear decline along the length of the aqueduct. This behavior matches expectations for laminar, fully developed Poiseuille flow in a cylindrical tube, as predicted by the analytical model in Section 3.

We observed that the pressure is highest at the inlet and decreases steadily toward the outlet. No localized pressure spikes or abrupt changes appear, confirming that mesh quality and geometry transitions were smooth.

4.3.1.2. Stenosis 1

For the mildly narrowed aqueduct (≈ 6.4% radius reduction), the maximum static pressure predicted by Fluent (CFD facet maximum) was 6.573 Pa in Figure 4.3.1.2. Compared to the normal case, the pressure gradient steepens as the fluid approaches the narrowed region.

The pressure drop across the stenotic region is higher than along the same length in the normal geometry. The color transition from the green/yellow region upstream to the blue downstream clearly illustrates a pressure drop across the stenosis.

4.3.1.3. Stenosis 2

For the more severe stenosis (≈ 17.6% radius reduction), the CFD wall pressure drop was 8.645 Pa in Figure 4.3.1.3. This case experiences the highest pressure gradients among the three geometries. The narrowed region causes a pronounced pressure drop centered around the neck. The pressure drop across the neck is significantly larger than in Stenosis 1, highlighting the strong sensitivity of flow resistance to radius changes. Downstream of the stenosis, pressure is lower than that of the normal or mild stenosis cases at comparable positions.

5.0. Conclusion

5.1 Discussion

The difference between the mathematical modelling and CFD modelling results was relatively small, although the discrepancy tended to increase as the radius decreased which is most likely due to the simplifying assumptions in the analytical model, whereas the CFD simulation provides a higher-fidelity representation of the flow.

A paper on modelling stenosis caused by tumour growth reports wall pressures of approximately 5.4 Pa, compared with roughly 2.5 Pa under healthy conditions [26]. These values suggest that our results fall within the correct order of magnitude, though direct comparison for the stenosis case is difficult because the tumour-growth model incorporates additional physiological factors. Moreover, the referenced model simulates the entire ventricular system, whereas our work focuses specifically on the cerebral aqueduct. However, it is to be noted that the simplifications used in our models are only sufficient for getting a rough estimate of the pressure. 

Correlating our results to pain is not the most straightforward as there is no numerical method to measure pain. However, insights can be gathered based on the change in ICP, as pain in hydrocephalus is caused by an increased ICP, thus it can be reasoned that larger ICP values result in greater pain. Our results suggest that there is a considerable amount of pain on sensitive tissues within the brain, especially in the stenosis 2 case, as there is roughly a 1.5 times increase from the healthy ICP. Using the healthy ICP value from earlier mentioned paper of 2.5 Pa as the healthy reference would indicate that our stenosis 1 and 2 ICP would likely result in debilitating head pain as the pressure more than doubles and triples in each stenosis case respectively. 

5.2 Suggested Improvements

Firstly, when evaluating the simulation for cerebrospinal fluid dynamics, modelling pulsatile flow is more physiologically representative, as proved in the mathematical modelling section where the Womersley Number is >1 for a normal aqueduct. However, it requires substantially greater computational effort while offering minimal improvement in result precision. The time point selected for this study corresponds to the maximum inflow velocity and therefore, the peak static pressure within the aqueduct. The simulations in this report were limited by the computational power of a Engineering Computing Facility computer available to students at UofT. 

Secondly, the entire ventricular system should be included in the model to allow direct comparison of upstream pressures and capture the interactions between connected ventricles. 

Thirdly, a key requirement is the ability to simulate deformation of the cerebral aqueduct, since its compliance influences resistance and flow patterns. Vessel compliance has been loosely modelled due the limitations of availability of information on the geometry under stenosis.

[1] K. Margetis, “Physiology, Cerebral Spinal Fluid,” StatPearls [Internet]., https://www.ncbi.nlm.nih.gov/books/NBK519007/ (accessed Sep. 18, 2025). 

[2] M. Mortazavi et al., “The ventricular system of the brain: a comprehensive review of its history, anatomy, histology, embryology, and surgical considerations,” Child’s Nervous System, vol. 30, no. 1, pp. 19–35, 2013, doi: https://doi.org/10.1007/s00381-013-2321-3.

[3] U. Kawoos, X. Meng, M.-R. Tofighi, and A. Rosen, “Too Much Pressure: Wireless Intracranial Pressure Monitoring and Its Application in Traumatic Brain Injuries,” IEEE Microwave Magazine, vol. 16, no. 2, pp. 39–53, Mar. 2015, doi: https://doi.org/10.1109/mmm.2014.2377585.

[4] D. M. Olson et al., “Differentiate the Source and Site of Intracranial Pressure Measurements Using More Precise Nomenclature,” Neurocritical Care, vol. 30, no. 2, pp. 239–243, Sep. 2018, doi: https://doi.org/10.1007/s12028-018-0613-x.

[5] M. V. D´Amato and R. Rehder, “Hydrocephalus,” Springer eBooks, pp. 41–49, Jan. 2022, doi: https://doi.org/10.1007/978-3-030-80522-7_3.

[6] S. B. Hladky and M. A. Barrand, “Regulation of brain fluid volumes and pressures: Basic principles, intracranial hypertension, ventriculomegaly and hydrocephalus,” Fluids and Barriers of the CNS, vol. 21, no. 1, Jul. 2024. doi:10.1186/s12987-024-00532-w 

[7] L. C. Daggubati and H. A. Akbari, “Aqueductal stenosis,” Elsevier eBooks, pp. 401–414, Jan. 2023, doi: https://doi.org/10.1016/b978-0-12-819507-9.00022-3.

[8] J. A. Cowan, M. J. McGirt, G. Woodworth, D. Rigamonti, and M. A. Williams, “The syndrome of hydrocephalus in young and middle-aged adults (SHYMA),” Neurological Research, vol. 27, no. 5, pp. 540–547, Jul. 2005, doi: https://doi.org/10.1179/016164105x17242.

[9] F. C. G. Pinto and F. D. R. M. Ureña, “Hydrocephalus Basic Concepts and Initial Management,” Fundamentals of Neurosurgery, pp. 147–160, 2019, doi: https://doi.org/10.1007/978-3-030-17649-5_10.

[10] O. García-González, J. Nicolás Mireles-Cano, P. Silva-Cerecedo, and F. Rueda-Franco, “The Aqueduct,” Child s Nervous System, vol. 30, no. 7, pp. 1147–1149, May 2014, doi: https://doi.org/10.1007/s00381-014-2431-6.

[11] P. Frassanito, B. Goker, and C. Di Rocco, “Aqueductal Stenosis and Hydrocephalus,” Textbook of Pediatric Neurosurgery, pp. 501–519, 2020, doi: https://doi.org/10.1007/978-3-319-72168-2_20.

[12] J. M. Rubino and J. P. Hogg, “Neuroanatomy, cerebral aqueduct (Sylvian),” StatPearls [Internet]., https://www.ncbi.nlm.nih.gov/books/NBK540988/#:~:text=Clinical%20Significance,obliteration%20of%20the%20aqueductal%20conduit (accessed Sep. 18, 2025). 

[13] S. J. Ahmad, R. L. Zampolin, A. L. Brook, A. J. Kobets, and D. J. Altschul, “A case of hydrocephalus confounded by suprasellar arachnoid cyst and concomitant reversible cerebral vasoconstriction syndrome,” Surgical Neurology International, vol. 13, pp. 331–331, Jul. 2022, doi: https://doi.org/10.25259/sni_313_2022.

[14] A. K. Garg, A. Suri, B. S. Sharma, S. A. Shamim, and C. S. Bal, “Changes in cerebral perfusion hormone profile and cerebrospinal fluid flow across the third ventriculostomy after endoscopic third ventriculostomy in patients with aqueductal stenosis: a prospective study,” Journal of Neurosurgery: Pediatrics, vol. 3, no. 1, pp. 29–36, Jan. 2009, doi: https://doi.org/10.3171/2008.10.peds08148.

[15] M. Tisell, “How should primary aqueductal stenosis in adults be treated? - A review,” Acta Neurologica Scandinavica, vol. 111, no. 3, pp. 145–153, Mar. 2005, doi: https://doi.org/10.1111/j.1600-0404.2005.00379.x.

[16] M. Yazbeck and Abed Alrazzak Kerhani, “Aqueductal Stenosis: Pathophysiology, Diagnosis, and Treatment Modalities,” IntechOpen eBooks, May 2025, doi: https://doi.org/10.5772/intechopen.1010710.

[17] L. N. Telano and S. Baker, “Physiology, Cerebral Spinal Fluid (CSF),” PubMed, Jul. 04, 2023. https://www.ncbi.nlm.nih.gov/books/NBK519007/

[18] J. Ghajar, “Traumatic brain injury,” The Lancet, vol. 356, no. 9233, pp. 923–929, Sep. 2000, doi: https://doi.org/10.1016/s0140-6736(00)02689-1.

[19] L. Fin and R. Grebe, “Three Dimensional Modeling of the Cerebrospinal Fluid Dynamics and Brain Interactions in the Aqueduct of Sylvius,” Computer Methods in Biomechanics & Biomedical Engineering, vol. 6, no. 3, pp. 163–170, Jun. 2003, doi: https://doi.org/10.1080/1025584031000097933.

[20] J. Apura, J. Tiago, A. Bugalho de Moura, J. A. Lourenço, and A. Sequeira, “The effect of ventricular volume increase in the amplitude of intracranial pressure,” Computer Methods in Biomechanics and Biomedical Engineering, vol. 22, no. 9, pp. 889–900, 2019, doi: 10.1080/10255842.2019.1587413.

[21] Veeramani Raveendranath, P. K. Dash, Thangaraj Kavitha, and K. Nagarajan, “Magnetic Resonance Morphometry of Normal Cerebral Aqueduct in South Indian Population,” Indian Journal of Neurosurgery, vol. 11, no. 03, pp. 248–252, Nov. 2021, doi: https://doi.org/10.1055/s-0041-1726602.

[22] I. G. Bloomfield, I. H. Johnston, and L. E. Bilston, “Effects of Proteins, Blood Cells and Glucose on the Viscosity of Cerebrospinal Fluid,” Pediatric Neurosurgery, vol. 28, no. 5, pp. 246–251, 1998, doi: https://doi.org/10.1159/000028659.

[23] T. Huff, Prasanna Tadi, and M. Varacallo, “Neuroanatomy, Cerebrospinal Fluid,” Nih.gov, May 2019. https://www.ncbi.nlm.nih.gov/books/NBK470578/

[24] H. Elsafty, A. ELAggan, M. Yousef, and M. Badawy, “Cerebrospinal fluid flowmetry using phase-contrast MRI technique and its clinical applications,” Tanta Medical Journal, vol. 46, no. 2, p. 121, 2018, doi: https://doi.org/10.4103/tmj.tmj_55_17.

[25] H. Mestre et al., “Flow of cerebrospinal fluid is driven by arterial pulsations and is reduced in hypertension,” Nature Communications, vol. 9, no. 1, Nov. 2018, doi: https://doi.org/10.1038/s41467-018-07318-3.

[26] U. U. Haq et al., “Computational modeling and simulation of stenosis of the cerebral aqueduct due to brain tumor,” Engineering Applications of Computational Fluid Mechanics, vol. 16, no. 1, pp. 1018–1030, Apr. 2022, doi: https://doi.org/10.1080/19942060.2022.2056511.