1. Modelling the motion of lung tumors.
   
2. This is joint work with Juergen Meyer (UC). The aim is to find a mechanical model of the way a lung tumour moves as the patient breathes. The idea is to be able to use measurements of the patient’s abdominal movement to predict the location of the tumour, so that treatment beams of radiation can be narrowed, causing less damage to healthy tissue. Currently, we are focussing our efforts on a spring-dashpot model. The model is three-dimensional and general. Our first publication, talks, and posters have concentrated on the one-dimensional asymptotic limit of these governing equations. Excellent agreement between prediction obtained by optimised numerics and experimental observation was observed. We are now testing and improving the full 3D model, using data from Harvard, Stanford, and Wurzburg.
3. Publications:
4. PL Wilson & J Meyer (2009): A spring-dashpot system for modelling lung tumour motion in radiotherapy, Computational and Mathematical Methods in Medicine, to appear.
5. PL Wilson & J Meyer: A general model of lung tumour motion, submitted to Proceedings of the European Consortium on Mathematics in Industry 2008.
6. PL Wilson & J Meyer: A general model of lung tumour motion, poster presented at ECMI 2008, London, UK, July 2008 - winner of Best Poster award.
7. J Meyer & PL Wilson: A three-dimensional spring-dashpot system to model the correlation between abdominal and lung tumour motion, presented at EPSM-ABEC 2008, Christchurch, NZ, October 2008.
8. Formed part of a paper presented at EPSM-ABEC, Perth, Australia, October 2007.

1. Modelling the lipid bilayer of red blood cells.
2. This is part of a grand challenge in modern science: how to incorporate continuum and discrete information into a unified mathematical description of an object for which both kinds of information are important.
3. The human red blood cell is surrounded by a a lipid bilayer only two molecules thick (and so measured in the nanometres) yet extending laterally for micrometres. This incredibly thin sheet is amazingly flexible and durable, and yet can be thought of not as a sheet at all, but as a fluid. Can one description capture these diverse properties?
4. How do cells sense shear? Shear can activate ion channels and cause other responses on nanometre length scales over which shear cannot be felt. So what mechanism mediates between the length scales?
5. This work in progress with Shu Takagi (University of Tokyo and RIKEN) and Huaxiong Huang (York University, Canada) is based on a new model by Joke Blom and Mark Peletier which we feel we have improved with a more accurate description of the fundamental physics involved.
6. Publications:
7. PL Wilson, H Huang & S Takagi (2009): Hydrophobic effect in a continuum model of the lipid bilayer, Communications in Computational Physics, to appear.
8. PL Wilson, H Huang & S Takagi (2008): University of Canterbury Research Report UCDMS2008/2: arXiv:0802.3932v1
9. PL Wilson, H Huang & S Takagi: The lipid bilayer at the mesoscale: a physical continuum model, submitted to Proceedings of the European Consortium on Mathematics in Industry 2008.
10. PL Wilson, H Huang & S Takagi: The lipid bilayer at the mesoscale: a physical continuum model, presented at ECMI 2008, London, UK, July 2008.
11. AMS-NZMS 2007, Wellington, NZ, December 2007;
12. 19th FEL Symposium, Tokyo, Japan, December 2005.
13. And innumerable other “in-house” talks.

1. Blood flow in the brain, and how important your COW is. My work here is a small part of a large effort at UC based around multiscale modelling and numerical methods exploiting new mathematics and the architecture of our IBM Blue Gene supercomputer, dubbed Blue Fern.
2. One particular project is to model the effects of low partial pressure of oxygen in the blood on the formation of Amyloid Beta Peptides in brain cells, the causative factor of Alzheimer’s disease. My role in this project is as second supervisor to Svava Kristinsdottir (UC). Her primary supervisor is Tim David (UC).
3. No publications of which I am a coauthor to date.

1. Transhuman mathematics.
2. An overview of transhumanism can be found here. Essentially, the idea is that as our technology grows in potency and becomes more interwoven with our lives and our bodies, we had better make an effort to consider the moral, ethical, and philosophical challenges we will face.
3. How does the advent of near-unlimited computer processing power and storage, advanced visualisation, human-computer hybridsation, and extended (perhaps indefinite) lifespans impact on the way we do, teach, and think about mathematics?

1. Modelling the lipid bilayer in two and three dimensions.
2. This is work to extend our new Blom & Peletier-like model from the current one- to two- and three-dimensional versions.
3. We have gone some way down the road to getting a two-dimensional version working numerically. But it appears that only limited information can be extracted from the model. However, some insight may be gained into protein-induced curvature, protein-mediated fusion, and general membrane energetics.
   
   
4. Modelling the red blood cell cytoskeleton.
5. In addition to the integrity and mechanical properties resulting from he lipid bilayer, RBC membranes also contain a membrane-bound scaffolding known as a cytoskeleton (CSK). The RBC CSK is a network of elastic elements (spectrin) joined together at 5- or 6-fold junctions, and joined to the membrane at points along the network. The CSK contributes to the visco-elasticity of the membrane. It also has a poorly-understood role in the aggregation of membrane proteins, either through a “corralling” effect, or perhaps even by an unknown mechanism whose effects would be directly opposite to those of a corral. The CSK may have a role in chemical signalling as well as mechanical resistance to deformation. It is one candidate for localling sensing macroscopic shear forces.
6. We are considering extending previous models of the CSK which treat it as a network of linear springs with experimentally-determined properties of both spring elements and network topology. However, there is still debate over the actual in vivo network properties, and of the form of the spectrin elasticity.
   
   
7. Platelet activation.
8. When the interior wall of a blood vessel is damaged, a sublayer is exposed which begins a chemical reaction local to the damaged area. The products of this chemical reaction bind to platelets suspended in the blood stream. These platelets then begin a dramatic change in shape, extending several pseudopodia, and begin to bind to one another at the ends of these protrusions. This growing clot of platelets also binds to the damaged wall, and repair begins. Much of the mechanical aspects of this process is poorly understood.

1. Far-downstream behaviour of merged turbulent boundary layers in curved pipes.
2. This is to study the possible pseudo-wake structure identified in our paper PL Wilson & FT Smith (2007): The development of the turbulent flow in a bent pipe, J. Fluid Mech. 578, 467-494. doi:10.1017/S0022112007005368
3. This is to be cowork with Frank Smith, FRS (University College London).