IT IS the ultimate âyour life in their handsâ moment. The anaesthetist is counting down from 10. You are about to lose the ability to feel, to breathe independently. From the instant you lose consciousness, you are relying almost entirely on that anaesthetist to keep you alive and stop you waking mid-surgery. Almost, because human judgements on how best to regulate the flow of drugs are themselves reliant on mathematical models underlying the monitoring systems that anaesthetists use.
At the heart of those models is calculus, the branch of mathematics that lets us explain and predict how change happens. This ability is absolutely fundamental to science, which calculus has crucially underpinned since its invention in its modern form a little over 300 years ago.
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Now we could be moving to the next level. Conventional calculus has its limits when we try to model complex situations. Patient response in anaesthesia is one â hence why there is always an anaesthetist in the room. But a radical, rapidly evolving form of calculus developed in the past few years is giving us a host of mathematical tools that promise to let us understand the finest details of physical processes with unprecedented precision.
It isnât just drug delivery that could see concrete benefits â it could help us solve all manner of problems, from detecting cancer to preventing the spread of pollution to making more efficient batteries. âI canât count the number of ways in which this can be applied,â says , South Africa, who discovered some of the key maths behind the breakthrough.
The original calculus gives us a way to model systems in which there is change and so make predictions. Take speed of motion. What happens when you press a carâs accelerator pedal? You get an increase in speed, yes, but how fast the car accelerates depends on a number of factors: the speed at which it was already travelling, how powerful the engine is, how much force was applied to the pedal, whether you are going uphill or downhill and so on.
Calculus lets you figure this stuff out. In doing so, it has given us the ability to control many aspects of the material world. Indeed, it is no exaggeration to say that calculus drove the scientific revolution, which in turn wrought the industrial revolution and, ultimately, the modern world.
And yet for all that, calculus has its limitations. Consider differentiation, one of the two main processes performed in calculus. Differentiation tells us the rate of change of something. In a car journey, it can give you the rate of change of the location of the car, something we normally call speed. This is known as a first-order derivative â being the first time you have performed differentiation on the original property. A second-order derivative would be the rate of change of that rate of change. We know this, the rate of change of speed, as acceleration. You can go further still: you could perhaps imagine calculating the rate of change of acceleration too, which would be a third-order derivative.
The problem is that we only ever perform differentiations in whole numbers, or âintegersâ â finding first, second, third or even higher-order derivatives. That gives us no way to derive quantities that exist in between those whole numbers. And yet there is plenty of action in the mathematical spaces in between orders defined by whole numbers. It might not make intuitive sense, as with speed and acceleration, and it is hard to get your head around what is meant by two-thirds of a rate of change of something, for instance. But in mathematical terms, such âfractionalâ calculus should have a lot to offer.
So could there be a way to calculate what is going in these in-between spaces, or what mathematicians call fractional order derivatives? As it happens, the mathematician Gottfried Leibniz, who co-invented calculus, was confronted with this very question by Guillaume de lâHĂ´pital in 1695. Leibniz didnât know the answer, but his reply to his fellow mathematician was prescient. âIt will lead to a paradox, from which one day useful consequences will be drawn,â he said.
It isnât clear whether there really is a paradox, but Liebniz was right about this usefulness. Calculating change in the spaces where conventional calculus canât reach â where things get ever more complex â gives us even greater insight into, and control over, the material world. For example, in systems controlled by feedback from sensors and other instruments, devices relying on standard calculus donât work as well as they might in principle. In systems that control vertical take-off and landing systems in aircraft such as the Harrier jump jet, say, or suppress the vibrations on the wings of commercial aeroplanes, conventional calculus tends to give only approximate solutions, never fully achieving the goal. Fractional calculus, on the other hand, offers fine-tuning.
The same applies to much of what goes on inside your body. Viscoelastic materials, which are ubiquitous in biological systems, have properties somewhere between those of liquids and solids. They resist modelling by ordinary calculus because each phase of matter requires a different form of calculus. You might model the motion of a viscous liquid by a first-order differentiation because viscosity generally depends on speed of movement. Equally, you could model an elastic solid by a second-order differentiation, for instance â relating to acceleration, as when a stretched elastic band is released. Neither is quite right for modelling a material like arterial walls or heart muscle, however, where the behaviour and properties of that material depend on the forces they are experiencing. Put simply, you want something in between the two orders that can model in-between states.
This is where fractional calculus comes in. The first steps towards making it work came long after Leibnizâs enigmatic statement. In 1832, mathematicians Bernhard Riemann and Joseph Liouville worked out a way to do the second of the two operations in calculus, known as integration, in fractions. In standard calculus, integration is the reverse process of differentiation: if you integrate speed over time, for instance, you get the distance covered. Riemann and Liouville showed that just as you can perform repeated integrations on the result of an integration, you can also create a recipe for doing this partially, giving a fractional integral.
The next significant step forward came in 1967, when mathematician Michele Caputo came up with a new way to define a fractional integral and its inverse, the fractional derivative. His work opened the floodgates to further ways of creating fractional integrals: once mathematicians saw Caputoâs innovation, many worked out how to create similar tools. âThere are new definitions being created all the time in recent years,â says Arran Fernandez, who studies the mathematical properties of fractional calculus at Eastern Mediterranean University in Cyprus.
âFractional calculus is different in that it models something akin to memoryâ
There are now myriad mathematical processes known as operators that create fractional integrals. In fact, the mathematical literature is full of fractional operators, each designed for a different task, and all of them useful in some way. âThe only reason we use mathematics is to try to capture and understand nature, but itâs impossible for a single mathematical operator to do that,â says Atangana.
He worked out his own fractional operator in 2015 and sent the result to Dumitru Baleanu at Ăankaya University in Ankara, Turkey, to be checked. Baleanu was impressed. âWhen he read it, he said to me, âyou have opened the doors of heavenâ,â says Atangana. The reason for the excitement was that Baleanu could see that , as it is now known, would prove remarkably useful across a wide range of applications. He wasnât wrong.
That is partly because it offered an alternative to the way fractional integration had been done up to that point. This involved using a âpower lawâ, a mathematical relationship between two factors, where one changes as a power of the other â as its square or a cube, for example. We see power law relationships in all kinds of natural phenomena. Earthquakes are one example: the likelihood of an earthquake of a particular magnitude happening within a particular window of time is related to that magnitude by a power law known as the Gutenberg-Richter law.
The problem with power laws is that they describe processes that have no well-defined beginning and end, and only provide an approximate description of how something is evolving. âThey really only serve as first-order approximations,â says Ernst-Jan Wit at the University of Lugano in Switzerland.
Atanganaâs fractional operator swapped out the power law term for something known as a Mittag-Leffler function, which would give his tool the capacity to account for unexpected changes that would pass unnoticed by conventional calculus â promising more precision, but also more versatility.
Imagine groundwater polluted with toxic waste that is seeping into a farmerâs soil, for instance. It travels slowly and disperses in a diffusive manner, spreading out over a large area. But if it encounters a fracture in the subsoil, everything changes. âWhen water meets a fracture, it will go inside the fracture and start running very fast,â says Atangana.
As well as being faster, the flow will be much more directed. These two different kinds of flow mean the system goes through two phases of very different behaviour, a phenomenon known to mathematicians as âcrossoverâ. That has always dictated the use of entirely different mathematical models, making the calculations too complex. But Atangana found a way to overcome that problem, allowing someone to analyse the entire system as it evolves with just one operator.
âNature is full of things that exhibit crossover,â says Atangana. Mathematicians looking to model the spread of disease, for instance, have turned to his operator in droves. In 2018, Baleanu and colleagues from Turkey, Nigeria and Pakistan showed that the new operator was able to give solutions to a previously insoluble problem: how best to vaccinate when an epidemic involves two strains of a pathogen wreaking havoc on a population.
The best effort, based on standard calculus, had involved solving six extremely complex equations, all patched together. Worse still, they only worked if you could find a combination of the equationsâ variables that made the equations reflect real-world values â a laborious process that had to be done by trial and error. With the Atangana-Baleanu operator in play, the researchers quickly found a solution that gave a closer match to the real-world data on the spread of the strains. It has already proved useful in modelling the spread of covid-19 through some populations.
Total recall
Baleanu and his colleagues put this success down to the fact that fractional operators, including the Atangana-Baleanu operator, can model something akin to memory. Think of the time it takes for a laptop battery to lose all of its charge. If it is an old battery that has been through many charging cycles, its useful life will probably be a lot shorter than if it is new. Standard calculus has no way of taking such memory effects into account, making predictions unrealistic. Models and control systems based on fractional derivatives, on the other hand, can factor this kind of thing into their operation.
The power of this aspect of the fractional operator is perhaps best demonstrated in anaesthesia, where memory is also central to the idea of computer-controlled monitoring and drug delivery. âI can use the memory term to track the drugs and avoid overdosing,â says Dana Copot at Ghent University in Belgium.
The way our body processes drugs depends on their concentration in the bloodstream. It is hard to find ways to keep the concentrations of drugs in the blood at their optimal level. Standard calculus canât model the process accurately: it is too blunt an instrument, and under or overshoots the real-world data. As a result, patients might get too little anaesthetic â causing them to wake or feel pain mid-operation â or too much. But fractional calculus can incorporate a memory of what has already been administered. âWith the memory term, we can capture all the characteristics,â says Copot. This allows researchers to find out exactly what amount of which drug should be administered and when.
So although it might be a while before a computer is administering your anaesthetic, Copot is confident that it could be possible one day. After all, fractional calculus is already starting to enter the consulting room. Copot is part of a collaboration that has used it to develop a system for differentiating asthma from chronic obstructive pulmonary disease. It works by comparing physiological symptoms against those predicted by a model with a range of fractional orders. And many more uses of fractional operators will probably emerge over the coming years. Other medical researchers are suggesting that fractional calculus can enable better models of how cancers spread, and how treatment regimes might affect their growth. Outside medicine, a team at Shandong University in Jinan, China, showed earlier this year that fractional operators can give a better live estimate of an electric carâs range by more accurately modelling all the myriad changing factors that are draining its battery.
Not that fractional calculus will be all-conquering. âThe only reason we use mathematics is to try to capture and understand nature, but nature is complex, and nature is above mathematics,â says Atangana. Yet there is no doubt that fractional calculus will reveal more of the finer details of natureâs glorious messiness than any mathematical tool available to us previously. Leibnizâs 300-year-old prediction has proved as accurate as anyone could have hoped.
