Mathematical modelling of anterior cruciate ligament reconstruction surgery to improve surgical techniques

Lead Research Organisation: University of Manchester
Department Name: Mathematics


The anterior cruciate ligament (ACL) is the most frequently injured knee ligament, and is one of the structures most commonly injured in sport. Due to the fact that it does not heal naturally, the standard treatment for a ruptured ACL is surgical reconstruction, to which there are several approaches, the most common being patellar tendon (PT) and hamstring tendon (HT) autograft, where the patient's own tendon is extracted and grafted into the location of the ruptured ACL. There is currently no consensus with respect to the choice between these two grafts. Previous decisions on which graft material to use have been empirically based; we aim to develop a theory to provide a justification for this choice based on the differing mechanics of the graft materials. The first aim of this project will be to use mathematical modelling techniques to determine which of the graft choices behaves physically most like a healthy ACL.

It is known that the ACL, PT and HT all behave differently when stretched, some ligaments and tendons are stiffer than others. We will attempt to explain these differences using mathematical models which incorporate the internal structure of each material. There have been attempts to model the ACL mathematically in the past and to model ligament behaviour in general; however, to the the author's knowledge, there has not been an attempt to compare the ACL, PT and HT all within the same framework.

Ligaments and tendons have an extremely hierarchical structure. Their main subunit is a group of fibres called a fascicle, which is made of thinner fibres called fibrils arranged in a crimped pattern. The diameter of fascicles is typically in the range of 50-300 micrometers and that of fibrils is 50-500 nanometers. We hypothesise that it is the arrangement of these fascicles and fibrils that gives tendons and ligaments their differing mechanical properties.

There is also currently no consensus with regards to the techniques used in ACL reconstruction surgery. The single-bundle technique consists of a single graft from the tibial insertion location (footprint) of the ACL to the femural footprint, whereas the double-bundle technique involves fixing two separate grafts in order to simulate the two functional bundles of the ACL. A recent review of the single-bundle and double-bundle reconstruction techniques was inconclusive, although there was some limited evidence that double-bundle ACL reconstruction has some superior results in objective measurements of knee stability and protection against repeat ACL rupture or a new meniscal injury. An analysis of the physical differences between these two methods would help the medical community to come to a consensus with regards to the best ACL reconstruction technique. Therefore, the second objective of this project will be to mathematically compare the single-bundle and double-bundle reconstruction techniques in order to determine whether one of the techniques leads to areas with higher stress concentrations, for example. The modelled grafts will be compared to a model of a healthy ACL in order to determine which reconstruction technique best matches the original anatomical function of the ACL.

The final objective of the project is to determine the optimal techniques for ACL graft placement. In order to attach the graft, tunnels are drilled through the tibia and femur and the grafts are fixated with screws. We shall vary the angle of the tunnels and the area of the insertion site in our models in order to determine the optimal angle and location and will also focus on the mechanics of the grafts at the screw sites. Finally, we will determine the optimal graft tension and will investigate whether an optimal amount of tension and twist can be applied in order to tune the graft tendons to behave more like a healthy ACL.

Planned Impact

1) Firstly, we consider the national importance of our proposed project. According to the NHS website, every year in the UK, there are about 30 ACL injuries for every 100,000 people and ACL injuries account for around 40% of sports injuries. If the ACL is torn, the knee becomes very unstable and loses its full range of movement, which can make it difficult to perform certain movements, such as turning on the spot and may make some sports impossible to play. In a society that is in the midst of an obesity epidemic, it is clearly essential that as many people as possible are physically able to partake in sports. It is, therefore, extremely important that surgery aimed at allowing people to be active is successful.

It is also obviously important that the incidence of rerupture following surgery is minimised. In a recent study by Frobell et al, four ACL reconstruction patients suffered a rerupture within five years of surgery out of a randomised trial of 121 people. This rupture rate is much higher than that in the general population, therefore, a reconstructed ACL is clearly not as reliable as a healthy one. The understanding gained through the proposed project will allow surgeons to optimise their ACL reconstruction
techniques, and hence increase surgical success rates.

2) The subject lends itself well to public engagement, for which funds are requested as part of this proposal. The fact that abstract mathematical techniques can be applied to real-world problems which have the potential to improve surgical techniques is fascinating to a general audience, and comparisons between experimental and theoretical/numerical results demonstrate the power of mathematical modelling in this case.

3) Finally, the UK solid mechanics and biomechanics communities will benefit from the training of a postdoctoral research fellow with a unique interdisciplinary outlook. By working alongside an exceptional multi-disciplinary team, I will have access to scientific expertise of outstanding quality in both breadth and depth. This will help to develop the UK's expertise and the output will be a highly-skilled individual suitable for employment in academia or industry.


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Craddock RJ (2018) Extracellular matrix fragmentation in young, healthy cartilaginous tissues. in European cells & materials

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Gower A (2017) A New Restriction for Initially Stressed Elastic Solids in The Quarterly Journal of Mechanics and Applied Mathematics

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Pearce, P. (2016) Maths in medicine: How to survive a science fair in Mathematics Today

Description Two new mathematical models have been developed that can be used to predict how the mechanical behaviour of a ligament or tendon depends upon the geometrical arrangement of the microscopic fibres that it is made from. These models have been used to explain the mechanical differences between positional and energy-storing tendons. Improvements have also been made on contrast agent staining methods for X-ray computed tomography of ligaments and tendons.
Exploitation Route It may be possible for tissue engineers to use the models described above to explain why surgically reconstructed ligaments and tendons do not always fulfil their mechanical requirements as well as healthy ones and to design replacement tendons with microstructures tailored to their specific functions. This could lead to improvements in surgical techniques and reduced graft failure rates.
Sectors Healthcare

Description Collaboration with Hazel Screen 
Organisation Queen Mary University of London
Department Institute of Bioengineering
Country United Kingdom 
Sector Academic/University 
PI Contribution Hazel Screen and I collaborated on a study about the structure-function relationship in energy storing and positional tendons. I mathematically modelled data provided by her lab.
Collaborator Contribution Hazel Screen's lab provided experimental data for our study.
Impact Publication DOI: 10.1098/rsif.2017.0261
Start Year 2016
Description SET for Britain 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? Yes
Geographic Reach National
Primary Audience Policymakers/politicians
Results and Impact I presented a poster about my research to MPs in Westminster. The event raised awareness amongst MPs about the value of funding science, technology, engineering and mathematics research, and showed them some examples of the impact of the funding.

Ongoing awareness of the value of scientific research is likely to encourage MPs to continue funding. The direct impact of this specific event is difficult to quantify; however, many of the participants mentioned that they had not been aware of the vast array of applications of mathematics research prior to the event and were impressed by what they had seen.
Year(s) Of Engagement Activity 2015
Description School visits (Manchester) 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Schools
Results and Impact I give an annual talk about my research to Year 11 students at Parrs Wood High School as an example of a real-world application of mathematics, so far approximately 90 students have attended in total. The talks are enthusiastically received and interesting discussions with students often follow. One of the teachers at the school has reported that several students have decided to study mathematics at A-level and university as a direct result of my talks.

The school reported that several students said that they would be more likely to study mathematics at university after my talk.
Year(s) Of Engagement Activity 2015,2016