Phillip L. Wilson

A Mathematical Model of the Metabolic and Perfusion Effects on Cortical Spreading Depression

Cortical spreading depression (CSD) is a slow-moving ionic and metabolic disturbance that propagates in cortical brain tissue. In addition to massive cellular depolarizations, CSD also involves significant changes in perfusion and metabolism—aspects of CSD that had not been modeled and are important to traumatic brain injury, subarachnoid hemorrhage, stroke, and migraine. In this study, we develop a mathematical model for CSD where we focus on modeling the features essential to understanding the implications of neurovascular coupling during CSD. In our model, the sodium-potassium–ATPase, mainly responsible for ionic homeostasis and active during CSD, operates at a rate that is dependent on the supply of oxygen. The supply of oxygen is determined by modeling blood flow through a lumped vascular tree with an effective local vessel radius that is controlled by the extracellular potassium concentration. We show that during CSD, the metabolic demands of the cortex exceed the physiological limits placed on oxygen delivery, regardless of vascular constriction or dilation. However, vasoconstriction and vasodilation play important roles in the propagation of CSD and its recovery. Our model replicates the qualitative and quantitative behavior of CSD—vasoconstriction, oxygen depletion, extracellular potassium elevation, prolonged depolarization—found in experimental studies. We predict faster, longer duration CSD in vivo than in vitro due to the contribution of the vasculature. Our results also help explain some of the variability of CSD between species and even within the same animal. These results have clinical and translational implications, as they allow for more precise in vitro, in vivo, and in silico exploration of a phenomenon broadly relevant to neurological disease.

A spring–dashpot system for modelling lung tumour motion in radiotherapy

A 3D system of springs and dashpots is presented to model the motion of a lung tumour during respiration. The main guiding factor in configuring the system is the spatial relationship between abdominal and lung tumour motion. A coupled, non-dimensional triple of ordinary differential equations models the tumour motion when driven by a 3D breathing signal. Asymptotic analysis is used to reduce the system to a single equation driven by a 3D signal, in the limit of small lateral and transverse tumour motions. A numerical scheme is introduced to solve this equation, and tested over wide parameter ranges. Real clinical data is used as input to the model, and the tumour motion output is in excellent agreement with that obtained by a prototype tumour tracking system, with model parameters obtained by optimization. The fully 3D model has the potential to accurately model the motion of any lung tumour given an abdominal signal as input, with model parameters obtained from an internal optimization routine.

Fluid–body interactions: clashing, skimming, bouncing

Solid–solid and solid–fluid impacts and bouncing are the concern here. A theoretical study is presented on fluid–body interaction in which the motion of the body and the fluid influence each other nonlinearly. There could also be many bodies involved. The clashing refers to solid–solid impacts arising from fluid–body interaction in a channel, while the skimming refers to another area where a thin body impacts obliquely upon a fluid surface. Bouncing usually then follows in both areas. The main new contribution concerns the influences of thickness and camber which lead to a different and more general form of clashing and hence bouncing.

Body-rock or lift-off in flow

Conditions are investigated under which a body lying at rest or rocking on a solid horizontal surface can be removed from the surface by hydrodynamic forces or instead continues rocking. The investigation is motivated by recent observations on Martian dust movement as well as other small- and large-scale applications. The nonlinear theory of fluid-body interaction here has unsteady motion of an inviscid fluid interacting with a moving thin body. Various shapes of body are addressed together with a range of initial conditions. The relevant parameter space is found to be subtle as evolution and shape play substantial roles coupled with scaled mass and gravity effects. Lift-off of the body from the surface generally cannot occur without fluid flow but it can occur either immediately or within a finite time once the fluid flow starts up: parameters for this are found and comparisons are made with Martian observations.

The Lipid Bilayer at the Mesoscale: A Physical Continuum Model

A new model of inter-molecular interactions is introduced into a continuum paradigm for the lipid bilayer membrane. The model promotes the hydrogen bond network responsible for the hydrophobic effect. Physically-realistic numerical bilayers are obtained from the model.

A General Model of Lung Tumour Motion

A limiting factor for the effective delivery of radiotherapy to lung tumours is the tumour motion as the patient breathes. If the tumour position is known at all times then treatment parameters may be adjusted accordingly. We formulate a general approach to model the spatial relationship between an external respiratory signal and the tumour position. The model treats the tumour as a point mass attached to a spring-dashpot system driven by abdominal motion. We present the model and show results of numerical computations based on clinical data.

The development of the turbulent flow in a bent pipe

The three-dimensional incompressible turbulent flow through a slender bent pipe of simple cross-section is analysed, the pipe gradually bending the rapid flow through a substantial angle. The ratio of the relative radius of curvature to the magnitude of the turbulent fluctuations is crucial: analysis of the entry region involving exact solutions of the governing equations shows three different downstream developments, depending on the magnitude of that ratio. The main velocity components are found in each case, and one downstream development studied in detail is when turbulence dominates the flow.The main novel points and results are as follows. (i) The present physical Situation which arises commonly in industrial settings has been little studied previously by theory or experiments. (ii) The working applies for any two-tier mixing-length model. (iii) As a most surprising feature, the fully developed flow far downstream is not unique, being found to depend instead on the global flow behaviour (thus the centreline velocity is not determined simply by the pressure drop, in contrast to the laminar case). (iv) A quite accurate predictive tool based on approximation is suggested for the downstream flow. (v) Crossflow maxima are found to occur very close to the walls, as observed in experiments. (vi) Other comparisons are made with experimental data and prove generally favourable.