The maths before the scalpal

Leszek Wierzchleyski

6 December 2023

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When an aneurysm forms on a patient’s blood vessel, a critical decision has to be made: should the aneurysm be surgically repaired or left alone? The primary concern is the aneurysm rupturing, as this can cause very serious health complications and possibly death. However, surgical intervention is risky and can cause blood haemorrhaging as well as a stroke or brain damage (if the aneurysm is on the brain). So, the million pound question: to repair or not to repair?

Mathematical and computational modelling can help us answer this question. By understanding the flow dynamics of the blood in and around the aneurysm we can make more informed decisions on whether or not surgical procedures should be carried out.

Patient-specific data can be collected through medical imaging techniques such as CT scans (which use a large number of X-rays from different angles to build a computational picture of blood vessels and blood flow), PET scans (which use positron beams to generate computational images) and MRIs (which collect images by causing protons in the body to align with a magnetic field). This data can then be fed into a mathematical model in order to predict the likely outcome of surgical intervention or leaving the aneurysm alone.

A CT scanner. Image: Tomáš Vendiš, CC BY-SA 3.0

An MRI scanner

First we need to consider some general blood flow rheology (ie fluid characteristics). Blood is a non-Newtonian fluid because its viscosity depends on the rate of shear. Imagine a bottle of tomato ketchup: when you tip it upside down the ketchup probably won’t flow out; but if you tap the side of the ketchup bottle, it suddenly begins to flow.

This is because ketchup is a shear-thinning fluid: the shear force you apply when you tap the side of the bottle reduces the viscosity of the ketchup and allows it to flow. Blood is also a shear-thinning fluid. Broadly speaking, the shear-thinning nature of blood is due to the fact that red blood cells tend to cluster together at low shear rates and spread out at high shear rates.

The graphs below show the flow profiles of a Newtonian fluid (eg water) flowing in a pipe and a shear thinning fluid (eg blood) flowing through a vessel.

Flow profiles of water in a pipe…

…and blood in a vessel

We can clearly see a difference: the flow front is more truncated in the case of blood, which indicates a shear-thinning fluid. For fluids like blood and water, fluid shear stress obeys the relation
$$\tau = K\dot{\gamma}^n,$$
where $\tau$ is the shear stress, $K$ is a constant called the flow consistency index, $\dot{\gamma}$ is the shear rate, and $n$ is the flow behaviour index. This is called the power-law fluid equation. Meanwhile, the effective viscosity for these fluids is given by
$$\mu_{\text{eff}} = K\dot{\gamma}^{n-1}.$$
For $n=1$, shear stress and shear rate are in direct proportion—if this relationship holds, the fluid is Newtonian, and we see this means it has a constant viscosity. For values of $n<1$, we have a shear-thinning fluid---increased shear means increased $\dot{\gamma}$, which causes the viscosity to decrease. In the case of a shear-thickening fluid (meaning viscosity increases with shear) such as corn starch, we would have $n>1$.

After a fancy derivation (based on conservation of momentum), we can use the power-law fluid equation to approximate the velocity of the flow at a specific point in the vessel
$$v = \left(\frac{\tau_\text{wall}}{KR}\right)^{1/n}\frac{R^{1+1/n}-r^{1+1/n}}{1+1/n},$$ where $\tau_\text{wall}$ is the shear stress at the wall, $R$ is the radius of the blood vessel and $r$ is the radial position of a given fluid particle.

This is how we get the profiles shown on the left. We can see from this formula that in the case of a shear-thinning fluid ($n<1$) the $1/n$ terms will be greater than 1, meaning that there will be smaller variation in fluid velocity at smaller radial positions and greater variation at higher radial values. This is what leads to the truncated profile.

The main takeaway from this is that in general, blood flow profiles are relatively simple. This is because most blood vessels are very narrow and so viscous forces will dominate over internal forces leading to a laminar (non-chaotic) flow. But this is not the case in an aneurysm.

In an aneurysm, we often see turbulent flows instead of laminar ones. Turbulent flows are chaotic in the sense that the velocity vectors throughout the fluid point in many different directions. The reason that aneurysms facilitate turbulent flows is that there is an abrupt change in cross-sectional area between the blood vessel and the aneurysm attached to it.

Turbulence creates two problems. First, there is the issue that turbulent flows are harder to analyse and predict. Secondly, the turbulence can cause an increase in pressure and shear stress on the aneurysm, leading to an increased chance of rupture.

A simulation of blood flow in an aneurysm might look like this

The blood flow around an aneurysm can be simulated. The flow regime within the aneurysm is more complex than the flow regime in other parts of the blood vessel. To quantify the risk of rupture, we can investigate the wall shear stress—the force exerted by the fluid on the wall. Unfortunately for us, turbulent flows are not nearly as straightforward as laminar ones and we can no longer use our power-law fluid equation.

Instead, we can estimate the wall shear stress from simulations—an approach known as computational fluid dynamics. We can integrate the quantity calculated over time in order to find a time averaged wall shear stress,
$$\overline{\tau_\text{wall}}=\frac{1}{T}\int^T_{0}\left\lvert \tau_\text{wall}\right\rvert\;\mathrm{d}t,$$
where $T$ is the duration of a cardiac cycle.

If the value of the time averaged shear stress is high (especially when localised) then risk of rupture is increased through mechanical weakening of the vessel wall. However, low wall shear stress can be bad news as well. Low wall shear stress promotes disorganisation of the endothelium (the inner lining of blood vessels) which in turn increases this risk of rupture. This is another reason why the problem is so complex as both high and low wall shear stress can increase the likelihood of rupture. It seems that the distribution of wall shear stress is important, with localised areas of high stress along with larger areas of low stress increasing the likelihood of rupture.

It should be noted that cyclic stressing (stretching and compressing) of the aneurysm wall can weaken it. We therefore need a way to quantify the cyclic wall shear stress. We can do this using a quantity called the oscillatory shear index, which is a measure of the fluctuation of low and high shear stress defined by
$$\text{OSI}=\frac{1}{2}\left(1-\frac{\left\lvert\int^T_{0}\tau_\text{wall}\;\mathrm{d}t\right\rvert}{\int^T_{0}\left\lvert \tau_\text{wall}\right\rvert\;\mathrm{d}t}\right).$$
It has been suggested that when the OSI is high, the vessel walls will be weakened over time, further increasing the likelihood of rupture.

Researchers have proposed that the direction of the wall shear stress may be important and should be part of the computations performed. We can measure the wall stress in all directions and then take the divergence in order to measure the effect of shear stress direction. The divergence quantifies whether the stress generally acts ‘outwards’ or ‘inwards’.

General process workflow of predictive EVAR modelling from model generation to the postprocessing of the simulation results

Now we need to consider whether this divergence has a positive value or a negative value and what this means. A positive value indicates stretching of the vessel wall while a negative value indicates compression. We can think of these regions as two propagating waves. Finding the wave centres of the positive and negative divergence regions can also be beneficial. If these centres are close together, then the stretching and compressing effects will compete with each other which can lead to further damage to the endothelium.

What we have shown here is that by approximating a flow we can calculate a number of different parameters which can help surgeons make key decisions.

So far we have talked about studying the risks associated with leaving an aneurysm alone. On the other side of the coin: once we have decided that intervention is necessary, what can mathematics tell us about surgical procedures?

For this we will consider an aneurysm repair procedure called endovascular aortic repair (EVAR). A key part of this procedure is the installation of a stent graft (a fabric tube supported by a metal mesh which can reinforce the aneurysm wall). This stent can lead to long term complications such as occlusion of blood flow (if a kink forms in the graft) or the graft itself can be fractured. This means that the design of the graft as well as its position is vital. Do you want to take a guess at how we optimise these things? That’s right! More computational modelling!

Through the use of medical imaging, fluid dynamics and solid–fluid interaction, we can simulate the surgical process ahead of time and help surgeons to minimise complications, both in the operating theatre and in the long term.

Leszek Wierzchleyski

Leszek is a mathematical biologist who is interested in how mathematical and computational methods can be applied to medicine. His interests include rowing, rugby and classical music.

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