Foundations of fluid mechanics,

So, what do people, bottles and beakers have in common? At first glance, it might seem easier to describe how they are different. However, we will see that both of the latter can be used as substitutes for people, as (albeit very grossly) simplified examples, if they are looked at in a certain way. Bottles and beakers can be used to create simple models that can help us to understand basic hydrostatic effects within the human body.

The causes of buoyancy stem from the nature of liquids at rest in the gravitational field of the earth. To be more specific: the pressures the liquid is subject to. The scientific subfield dealing with the phenomena that arise from this is known as "hydrostatics".

In physical terms, pressure is defined, very generally, as "the weight exerted on a specific surface area". Pressure p is generated when a physical force F is exerted on the surface of a solid or fluid: in fluids the pressure is exerted evenly p = F/A as long as gravity is not taken into account. A good example of this is a balloon, in which the force F of the gas or water presses against the tension generated by the balloon's skin. In this case, there will be overpressure in the balloon relative to the surrounding atmosphere.

Overpressure (hydrocephalus) can also arise inside the skull, including as a result of the "masses" mentioned above, e.g. excess buildup of CSF, bleeds or a growing tumour. The necessary force is generated by their expansion or growth.

If we then take gravity into account, another special form of pressure will arise due to the weight of the fluid itself. This is known as "hydrostatic pressure". This phenomenon is known as the "basic hydrostatic law" and in physics, is referred to as the "basic hydrostatic equation". This law is essential for understanding why differences in pressure are generated in shunts regardless of the position the body is in. It is also important for understanding why, in the worst case scenario, ventricles can literally be sucked dry, as well how to prevent this from occurring.

To describe this in the simplest possible terms: Every liquid has a mass, and this is subject to a weight within the Earth's gravitational field. Therefore, the load on an imaginary body within the liquid, stemming from the "open surface" (i.e. the boundary between the liquid and the surrounding atmosphere, which is subject to a known amount of (air) pressure) increases in proportion to the depth of the liquid. However, the amount of liquid pressure this generates depends solely on the height of the column of liquid above. This is not affected by the shape of the vessel or the mass of the liquid that is above the area subject to observation.

A good example is a beaker (of any shape) filled with water: unlike in a balloon, the water does not have to be pushed down into the glass by a special force, it only has to be filled up. Naturally, the water's surface is always subject to the ambient pressure, i.e. the atmospheric pressure (which will be approximately 1 bar, depending on the weather). The bottom of the beaker is still subject to the same ambient pressure, but is also subject to the load consisting of the entire weight of the column of water (not the weight of the water in its entirety) above the bottom of the glass. The latter generates additional pressure. This is the hydrostatic pressure mentioned that was mentioned the start. It begins at "zero" on the water's surface, and gets closer to the maximum the deeper the point is under the water (relative to the surface). This is due to the fact that an increasing weight of water is exerting a load on the area in which pressure is being observed.

The following example will demonstrate how pressure ratios in real hydrostatic systems (similar to those the human body or in a shunt, for example) can quickly become complex and confusing, despite the simplicity of the basic law (the basic hydrostatic equation):

In the diagram below there are eight pipes labelled A-H, which are very similar at first glance. However, despite their similarity, they are subject to fundamentally different hydrostatic, and in some cases, hydrodynamic forces (where the water is moving):

A A liquid with px of overpressure was squeezed into the rigid pipe "A". The system was then tightly sealed immediately afterwards. In the rigid pipes, the overpressure is being maintained. If the pipes had been flexible, they would blow up like a balloon due to the px of overpressure. When it is placed in an upright position, the bottom of the pipe is subject to overpressure of px plus the hydrostatic pressure (HP)+ 10 cm.

B Pipe B has an opening, through which it is exposed to the outside air. As a result, there is no longer an overpressure of px. Therefore, by definition, it is 0 at the top and at +10 cmH₂O at the base below.

C Pipe C is open at the bottom and sealed at the top. By definition, the bottom is subject to 0 cmH₂O. The water cannot flow away, as this would create a vacuum above, as has previously been mentioned. However, it will attempt to flow away: the hanging column of water pulls the cap at the top down, generating a negative hydrostatic pressure of -10 cmH₂O, which is also known as suction force.

D Pipe D is identical to C, but in this case the cap is slightly flexible, therefore clearly demonstrating the real existence and effect of the suction. This is important, as "nothing", in the fullest sense of the word, can be seen. There is "no flow" to be seen either, meaning that the existence of suction is a mere assertion.

E Pipe E is open at the top and bottom and therefore just empties out. The driving force behind this process is simply the water, which is able to "fall" down unimpeded.

F In pipe F the hole out to the external air, at which point there is a differential pressure of 0 cmH₂O, is right in the middle. Nothing can flow away, but the hydrostatic pressure at the bottom is now only +5 cmH₂O and the suction at the top is only -5 cmH₂O, due to the lower hydrostatic level.

G In pipe G there is also a flexible cap, to show the real existence and effect of the suction. If the lower cap was also flexible it would bulge outwards due to the positive overpressure.

H In pipe H there is no upper cap, meaning that the liquid flows away, leaving it half-full. Only a positive hydrostatic pressure of + 5 cmH₂O remains at the base.

This section will very specifically look at pipes (C) and (D), which illustrate how the so-called "siphon effect" occurs. This is very important in relation to shunt therapy. As long as the patient is standing up, the catheter full of liquid, which runs from the ventricles (the closed end) to the abdomen (the peritoneum, at the open end), actually constitutes the same type of system. The high level of suction that this produces in the ventricles is physically unavoidable and special implants are required to counteract it.

In essence, this effect is also responsible for the fact that cerebral pressure of a healthy person who is standing up is slightly negative. When in this position, the CSF therefore sinks out of the ventricles down into the dural sac (in the vertebrae). Although the dural sac is not open at the bottom like in pipes C and D, it is quite flexible and is capable of taking in some extra CSF. Luckily, there are also a few other physiological mechanisms to ensure that "physiological" (healthy) cerebral pressure does not drop to -100 cmH₂O when a person stands up, instead only dropping to around -5 to -15 cmH₂O.

If one of the pipes were to fall over (or if a person lies down) the height of the column of water would suddenly increase. The difference in pressure between two different places (such as the neck and base of the bottle) will become level within the Earth's gravitational field as soon as they are at the same hydrostatic level.

To summarise: When using a very rough model, we can treat a person as an upright bottle in the Earth's gravitational field, filled with CSF. When this bottle is in an upright position, the liquid pressure of the CSF in the head will be lower than that in the spinal cavity. When the bottle has been closed we cannot determine what the pressure is simply by looking at it.

It is therefore possible to infer what is happening to the pressure of the CSF in the human body when it is leaning over, bending or lying down on the basis of these simple physical thought experiments based on beakers and bottles. The hydrostatic pressure of the CSF in different parts of the human body is not constant, instead "adapting" to the position it is in and the activity it is doing.

The flow manifests itself if cerebral ventricular system is "disturbed" by a shunt: the difference in pressure between the ventricles and the peritoneum allows for a "net flow" of CSF into the peritoneum through the artificial channel, which is regulated by a special differential pressure or hydrostatic valve. According to the continuity equation, when liquid is being continuously produced, it flows more quickly the narrower the channel is. Because of this, the liquid will flow more quickly through a relatively narrow catheter than, for example, a reservoir with a significantly larger flow cross-section. If, for example, 20 ml of CSF is drained per hour, it will flow through the catheter at a constant rate of 5 millimetres per second. However, the speed of the liquid increases in direct proportion to the cross section width what it is flowing through.

Some of you might already be familiar with this effect from canoeing down narrow channels: the narrower the channel is, the faster the canoe floats along with the current.

If then the flow rate through a hydrocephalus valve with a potentially enlarged cross section is slowed down, the tissue particles, proteins and sometimes also blood may interact with and adhere to the interior surfaces of the valve due to long periods of exposure (due, in part to turbulence and bottlenecks at places with an irregular geometry, where the CSF is almost still). This can gradually lead to valve occlusion.

"... found in relation to toxicity and oncogenity. Minor problems existed in relation to CSF proteins and medium difficulties in relation to corpuscles (blood). Surprisingly with 500 mg/dl protein, most valves showed – in short term – no significant alteration except of some sticking proximal slit probes and sometimes Orbis-Sigma valves. The trials of Brydon et al. (1996) could be confirmed. Protein effects are usually overestimated. Corpuscles and blood are more by far at risk for valve occlusions (Aschoff 1995)."
Aschoff², 2019

Understanding of the exact mechanisms of valve occlusions.at a molecular level is important for the long-term success of a shunt, and is therefore currently the subject of scientific research. We still do not know very much at all about the reasons why blockages occur. However, we suspect that taking flow speed into account is probably of greater importance when designing shunt systems.

Noises in the valves

The flow rate through the pump can also be important for reasons other than occlusions. The occurrence of valve noises (which have been reported by some hydrocephalus patients) can also be explained by phenomena related to the flow rate of the CSF.

This can be illustrated with a simple differential pressure valve. For example, with a ball-cone valve, the pressure proximal to the shunt builds up until it can overcome the force of the spring element in the valve. When the force generated by the pressure finally exceeds the spring force, the ball moves, opening the liquid channel, which means the fluid can move.

Specific conditions may cause the valve to close right after opening, due to a sudden pressure drop in the CSF. One of the reasons for this pressure drop is the reduction in pressure on the side subject to overpressure, due to the liquid being drained away. This can take place very quickly, depending on the specific anatomy of the person (e.g. cerebral compliance, split ventricles etc.). There is also another type of pressure drop, which can be explained by what is known as "Bernoulli's equation". This states that the pressure in a moving fluid is lower than that in a fluid that is still. What this actually means for the valve is that the faster CSF moves through the open valve, the bigger the pressure drop that follows and, in turn, the greater the likelihood of the valve closing quickly again will be.

To allow the pressure to immediately start building up again, it is therefore necessary to repeatedly open and close the valve, which also creates audible noises.

This phenomenon can also be proved experimentally, as can be see in the following video: