Pattern formation and travelling waves in a multiphase moving boundary model of tumour growth.

We employ the multiphase, moving boundary model of Byrne et al. (2003, Appl. Math. Lett., 16, 567-573) that describes the evolution of a motile, viscous tumour cell phase and an inviscid extracellular liquid phase. This model comprises two partial differential equations that govern the cell volume fraction and the cell velocity, together with a moving boundary condition for the tumour edge, and here we characterize and analyse its travelling-wave and pattern-forming behaviour. Numerical simulations of the model indicate that patterned solutions can be obtained, which correspond to multiple regions of high cell density separated by regions of low cell density. In other parameter regimes, solutions of the model can develop into a forward- or backward-moving travelling wave, corresponding to tumour growth or extinction, respectively. A travelling-wave analysis allows us to find the corresponding wave speed, as well as criteria for the growth or extinction of the tumour. Furthermore, a stability analysis of these travelling-wave solutions provides us with criteria for the occurrence of patterned solutions. Finally, we discuss how the initial cell distribution, as well as parameters related to cellular motion and cell-liquid drag, control the qualitative features of patterned solutions.

Time-dependent manufacturing processes lead to a new class of inverse problems.

The control of time-dependent, energy beam manufacturing processes has been achieved in the past through trial-and-error approaches. We identify key research gaps and generic challenges related to inverse problems for these processes that require a multidisciplinary problem-solving approach to tackle them. The generic problems that we identify have a wide range of applications in the algorithmic control of modern manufacturing processes.

New models for energy beam machining enable accurate generation of free forms.

We demonstrate that, despite differences in their nature, many energy beam controlled-depth machining processes (for example, waterjet, pulsed laser, focused ion beam) can be modeled using the same mathematical framework-a partial differential evolution equation that requires only simple calibrations to capture the physics of each process. The inverse problem can be solved efficiently through the numerical solution of the adjoint problem and leads to beam paths that generate prescribed three-dimensional features with minimal error. The viability of this modeling approach has been demonstrated by generating accurate free-form surfaces using three processes that operate at very different length scales and with different physical principles for material removal: waterjet, pulsed laser, and focused ion beam machining. Our approach can be used to accurately machine materials that are hard to process by other means for scalable applications in a wide variety of industries.