From understanding the effect of aneurysms and what causes strokes to simulating and constructing artificial organs, maths has a huge role to play in developing new medical treatments. But one key part of the human physiology is the study of blood. It’s fairly obvious that blood is key to life – if you bleed too much you die. It has been studied by many eminent figures, from Aristotle who believed blood was required to transport heat around the body to Poiseuille who derived derived a simplified model of mathematical flow in a pipe to describe flow through arteries. We now understand that blood carries oxygen and essential nutrients to our cells, and carries waste products such as urea away to be processed.
Blood, although it might seem like a fluid, is in actual fact much more complex. About 55% percent of blood is made up of what we call blood plasma. This is the fluid portion of blood — water with proteins and various compounds such as carbon dioxide and oxygen dissolved in it. The rest of the blood is made up of platelets, red blood cells and white blood cells. Now bear in mind, the exact make up of blood changes from person to person, and even from artery to capillary to vein as the components of blood diffuse in and out as needed.
Diagram of the make up of blood taken from the Encyclopedia Britannica.
So how do we describe blood mathematically? Blood is effectively a fluid with stuff in it. So is it still a fluid? By the Oxford English Dictionary, a fluid is ‘a substance that has no fixed shape and yields easily to external pressure.’ So the answer is yes, blood is a fluid.
Unfortunately for us mathematicians the story doesn’t actually end there. We typically split fluids into two categories, Newtonian fluids, ie those that we can model using the standard equations for fluid flow (the Navier–Stokes equations) and Non-Newtonian fluids, those that we can’t.
The viscosity of a fluid is a measure of how resistive it against external forces. You can think of it as how easy it is to make the fluid deform. For example water has a low viscosity and honey has a high viscosity. A Newtonian fluid is defined as having a constant viscosity (under normal temperature and pressure conditions) and a Non-Newtonian fluid is one that has a variable viscosity. Now, fluids with variable viscosity are just weird and could have a whole blog post written on them alone. An example of a weird non-Newtonian fluid is a mixture of corn flour and water. If you haven’t had the pleasure of meeting this glop go watch this video. Weird, huh?
So why the tangent about different types of fluid? It’s because here there isn’t an easy answer. Blood is actually a Non-Newtonian fluid — it’s known as a shear-thinning fluid (the opposite to corn flour and water). This is where the fluid becomes less viscous the more pressure is applied to it. What does this actually mean? When we cut a finger, a small number of capillaries in the surface of the skin will also be cut. The pressure in these capillaries will dramatically lessen causing the blood to become more viscous, meaning less blood flows out of the cut allowing a scab to eventually form to completely block the bleeding.
How do we model a Non-Newtonian fluid? The answer is with difficultly. Usually the Navier equation is used (which can be used to derive to the Navier–Stokes equation, you eager beavers!) along with a complex stress relation determined from experimental data and hopeful hypothesis. Essentially the stress relation needs to describe the interaction of the individual particles suspended in the fluid.
However this isn’t always required. The main purpose of modelling is to understand a situation while eliminating as much complexity as possible. In the majority of modelling situations, we can actually treat blood as a Newtonian fluid. Say we want to consider blood flow through the one of the major arteries, the aorta. Here pressure can be assumed to be roughly constant (or in mathematical speak, of the same order of magnitude) so the change in viscosity of the blood will be small enough to approximate the blood as a Newtonian fluid, allowing us to use standard modelling techniques for fluid mechanics. The same applies if we consider flow through a capillary network, through an organ etc.
Understanding how small is small is an important skill in mathematical modelling. Image: Wikipedia user Ellen Levy Finch, usage under CC BY-SA 3.0.
So do mathematicians just ignore what the biology tells us? Well, not quite. It’s a matter of scale. Say we are attempting to model the flow of blood through a cancerous tumour in order to quantify the optimal concentration of cancer drugs. If adding the effect of variable viscosity will change that concentration by a thousandth of a percent but potentially quadruple the complexity of the problem, then it’s fair to assume the viscosity is constant. If we are attempting to model the same drug interacting with the individual components of blood then we cannot assume the blood has the same properties as water and the added complexity makes a real difference between a model being accurate and a model not working.
How do we know if we can make simplifying assumptions or not? Well that’s why we study maths. For me, it is the scales present within biology and the questions of how we model biology that make it so interesting.
This post is part of a series for the UCL Year 12 Maths Research Summer Programme where year 12 students investigate an area of maths under the guidance of a Chalkdust member and PhD student at UCL.The closing event will celebrate the work of the year 12 students and will feature a talk from Prof. Lucie Green, a space scientist and TV & radio presenter. The closing event is on Thursday 12th July and is open to the public. If you would like more information about the programme or to attend the closing event please contact Dr Luciano Rila l.rila@ucl.ac.uk .
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