Mathematical models for the tumour control probability (TCP) are used to estimate the expected success of radiation treatment protocols of cancer. There are several TCP models in the literature, from the simplest (Poissonian TCP) to the well-advanced stochastic birth-death processes. Simple and complex models often make the same predictions. Hence, here, we present a systematic study where we compare six of these TCP models: the Poisson TCP, the Zaider-Minerbo TCP, a Monte Carlo TCP and their corresponding cell cycle (two-compartment) models. Several clinical non-uniform treatment protocols for prostate cancer are employed to evaluate these models. These include fractionated external beam radiotherapies, and high and low dose rate brachytherapies. We find that in realistic treatment scenarios, all one-compartment models and all two-compartment models give basically the same results. A difference occurs between one-compartment and two-compartment models due to reduced radiosensitivity of quiescent cells.We find that care must be taken for the right choice of parameters, such as the radiosensitivities α and β and the hazard function h. Typically, different hazard functions are used for fractionated treatment (fractionated survival fraction) and for brachytherapies (Lea-Catcheside protraction factor). We were able to combine these two approaches into one 'effective' hazard function. Based on our results, we can recommend the use of the Poissonian TCP for everyday treatment planning. More complicated models should only be used when absolutely necessary.
The cell cycle can be understood as the sequestration of cells in the quiescent compartment, where they are less sensitive to radiation. We suggest that our model can be used in combination with synchronization methods to optimize treatment timing.
Adversarial attacks for image classification are small perturbations to images that are designed to cause misclassification by a model. Adversarial attacks formally correspond to an optimization problem: find a minimum norm image perturbation, constrained to cause misclassification. A number of effective attacks have been developed. However, to date, no gradient-based attacks have used best practices from the optimization literature to solve this constrained minimization problem. We design a new untargeted attack, based on these best practices, using the well-regarded logarithmic barrier method.On average, our attack distance is similar or better than all state-of-the-art attacks on benchmark datasets (MNIST, CIFAR10, ImageNet-1K). In addition, our method performs significantly better on the most challenging images, those which normally require larger perturbations for misclassification. We employ the LogBarrier attack on several adversarially defended models, and show that it adversarially perturbs all images more efficiently than other attacks: the distance needed to perturb all images is significantly smaller with the LogBarrier attack than with other state-of-the-art attacks.
We approximate the homogenization of fully nonlinear, convex, uniformly elliptic Partial Differential Equations in the periodic setting, using a variational formula for the optimal invariant measure, which may be derived via Legendre-Fenchel duality. The variational formula expresses H(Q) as an average of the operator against the optimal invariant measure, generalizing the linear case. Several nontrivial analytic formulas for H(Q) are obtained. These formulas are compared to numerical simulations, using both PDE and variational methods. We also perform a numerical study of convergence rates for homogenization in the periodic and random setting and compare these to theoretical results.
We are interested in the shape of the homogenized operator F (Q) for PDEs which have the structure of a nonlinear Pucci operator. A typical operator is H a 1 ,a 2 (Q, x) = a 1 (x)λ min (Q)+a 2 (x)λmax(Q). Linearization of the operator leads to a non-divergence form homogenization problem, which can be solved by averaging against the invariant measure. We estimate the error obtained by linearization based on semi-concavity estimates on the nonlinear operator. These estimates show that away from high curvature regions, the linearization can be accurate. Numerical results show that for many values of Q, the linearization is highly accurate, and that even near corners, the error can be small (a few percent) even for relatively wide ranges of the coefficients.
<p>The Swarm constellation provides information on both along- and across-track magnetic field gradients. Spatial changes of the magnetic vector field elements are described by a magnetic field gradient tensor, whose elements and their uncertainties can be estimated using the Virtual Observatory (VO) concept, whereby data within a cylinder centred on the VO with axis perpendicular to the Earth&#8217;s surface are reduced to a central point at satellite altitude. Recent experiments have shown that analysing data collected over a 4 month window provides the best compromise between reducing bias from the way the satellite orbits sample each VO cylinder and preserving information on temporal changes of the field, and that the data provide spatial information sufficient to resolve 300 non-overlapping VOs. We invert annual first differences of the 5 independent gradient tensor elements (providing estimates of secular variation, SV, gradients) at these 300 VOs over the Swarm era for advective velocity at the core-mantle boundary, forcing the flow to have minimal acceleration while providing an adequate fit to the data. We obtain flows similar to those from previous SV inversions but purely from the gradient information. The resolution of the SV gradients is higher than that of the SV itself, resulting in a ~30% increase in the number of effective flow parameters; this is thought to be because the gradients are less affected by long period external signals that are difficult to remove from the data, resulting in an improved signal to noise ratio. Although very little temporal change in the flow is required to reproduce even rapid changes in the magnetic field, we are able to isolate some robust flow changes, in particular regarding changes in the azimuthal flow acceleration, associated with the geomagnetic impulse in the Pacific region in around 2016.</p>
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