Developing new mathematical modelling techniques

Using mathematics to treat male infertility might seem rather unlikely, but Birmingham postgraduate student Gemma Cupples has taken a classic mathematical model and extended it to create a way to study the swimming behaviour of sperm that could lead to better conception rates.

Gemma, who recently completed the third year of her PhD, funded by a Biotechnology and Biological Sciences Research Council Industrial CASE award, explains the new system in an academic paper – her first – published in the prestigious Journal of Fluid Mechanics. Entitled ‘Viscous propulsion in active transversely isotropic media’, she and co-authors Dr Rosemary Dyson and Professor Dave Smith, her PhD supervisors, show how the development of British mathematician GI Taylor’s two-dimensional ‘swimming sheet’ – a well-known model of microscale propulsion and pumping – can be applied to a range of problems involving biological fluids, from internal fertilisation to the detection of antibiotic-resistant bacteria in blood.

Gemma currently works with University spinout company Linear Diagnostics, which is developing handheld biosensor devices to quickly and inexpensively detect germs in liquid samples, such as E.coli in water or harmful bacteria in blood or urine. The application of Gemma’s mathematical model optimises the efficacy of the devices, enabling the detection of germs at lower concentrations – leading to safer and more informative results.

The mathematical model developed through this project led to an unexpected application – how sperm swim through the female reproductive tract. Male infertility is a particular focus because of a five-year, Engineering and Physical Sciences Research Council-funded project between the University and Birmingham Women’s Hospital (BWH) called Rapid Sperm Capture, which uses cutting-edge technology to develop new techniques to examine sperm: rapid digital camera imaging and computer-based pattern recognition.

‘Colleagues at BWH are very interested in trying to improve how male fertility problems are diagnosed and treated,’ explains Dave, who is the project lead. ‘About one in six couples are infertile and male fertility accounts for half of those, so it’s a very common problem.’

One of the main issues with sperm is their ability to swim. Of the 100 million or so that start out, as few as 10-100 make it to the egg. This is because they have to fight their way through an obstacle course of mucus, consisting of a meshwork of fibres.

‘At the BWH, they are working to understand the job of sperm, yet diagnosis methods are very poor – essentially, they’ve not changed since the 1950s; they largely involve manually counting swimming sperm – and there are no drugs available for treating sperm problems.’

The research team – made up of mathematicians, bioengineers, computer engineers and clinicians – hopes the project will change that by creating a new system using phase-contrast imaging to observe sperm before analysing them mathematically. Gemma will join the project full-time from next January.

‘What we’re doing is charting what the sperm are doing; trying to work out if they are good or bad swimmers,’ says Dave, who is Professor of Applied Mathematics. ‘We’re also looking at the DNA to examine if it’s packaged securely.’

The aim is to be able to automatically, accurately and repeatedly examine a semen sample, collecting simultaneous data on how cells swim and what their shape is like, in order to identify the few cells with the ‘right stuff’ to navigate the obstacle course of mucus to the egg. This could lead to new equipment that would identity the condition of sperm and any necessary treatment or lifestyle changes.

‘The long-term impact could lead to a better use of resources for treatments such as IVF and hopefully improved success rates,’ says Dave.

‘What Gemma did is that she took the very common model for swimming in a fluid, the Taylor swimming sheet from 1951, which explains how microorganisms can swim in the microscale viscous world, and extended Taylor’s analysis to make a more complicated mathematical model that takes into account the work of these fibres in the fluid. What the paper does is tell you how you can pump these fluids.’

Rosemary, a Senior Lecturer in Applied Mathematics, says the beauty of such research is that it can be applied to many different areas. ‘The underlying maths is the same. We can take something Gemma has done and that will turn out to be the piece of methodology we need for another project that, on the surface, is entirely different. Maths affects so many important areas.’

It is this, says Gemma, that gives her the most satisfaction. ‘You see how things we do really have an impact and make a difference – that mathematics is really useful.’