Clustered ventilation defects are a hallmark of asthma, typically seen via imaging studies during asthma attacks. The mechanisms underlying the formation of these clusters is of great interest in understanding asthma. Because the clusters vary from event to event, many researchers believe they occur due to dynamic, rather than structural, causes. To study the formation of these clusters, we formulate and analyze a lattice-based model of the lung, considering both the role of airway bistability and a mechanism for organizing the spatial structure. Within this model we show how and why the homogeneous ventilation solution becomes unstable, and under what circumstances the resulting heterogeneous solution is a clustered solution. The size of the resulting clusters is shown to arise from structure of the eigenvalues and eigenvectors of the system, admitting not only clustered solutions but also (aphysical) checkerboard solutions. We also consider the breathing efficiency of clustered solutions in severely constricted lungs, showing that stabilizing the homogeneous solution may be advantageous in some circumstances. Extensions to hexagonal and cubic lattices are also studied.

We present a shape-oriented data assimilation strategy suitable for front-tracking tumor growth problems. A general hyperbolic/elliptic tumor growth model is presented as well as the available observations corresponding to the location of the tumor front over time extracted from medical imaging as MRI or CT scans. We provide sufficient conditions allowing to design a state observer by proving the convergence of the observer model to the target solution, for exact parameters. In particular, the similarity measure chosen to compare observations and simulation of tumor contour is presented. A specific joint stateparameter correction with a Luenberger observer correcting the state and a reduced-order Kalman filter correcting the parameters is introduced and studied. We then illustrate and assess our proposed observer method with synthetic problems. Our numerical trials show that state estimation is very effective with the proposed Luenberger observer, but specific strategies are needed to accurately perform parameter estimation in a clinical context. We then propose strategies to deal with the fact that data is very sparse in time and that the initial distribution of the proliferation rate is unknown. The results on synthetic data are very promising and work is ongoing to apply our strategy on clinical cases.