# Genomineerden 2011

**Immersed boundary method for pulsatile flow in cerebral aneurysms**

*Julia Mikhal (Universiteit Twente); Waaier 2, vr 10:15-10:35*

We present a numerical method for simulation of blood flow inside the human brain. The focus is on cerebral aneurysms that may form on some vessels. The precise blood flow and the forces on the vessel wall are computed. This contributes to our understanding of possible long-term rupture of aneurysms from an analysis of the short-time pulsatile flow. The computational model is based on the incompressible Navier-Stokes equations in 3D. Flow in complex aneurysm geometries is represented with the use of a volume-penalizing Immersed Boundary (IB) method. The main feature of our IB method is the so-called masking function which equals “0” inside the flow domain while in solid parts it takes the value “1”. This technique allows a fast and relatively simple definition of any geometry. The flow inside the defined geometry is simulated on the basis of a skew-symmetric finite-volume discretization and explicit time-stepping. We compute the blood flow at various physiologically relevant flow speeds and for several types of pulsatile forcing of the flow. The model aneurysm consists of a curved vessel with a spherical cavity attached to it. Time-dependent flow and the evolution of the forces on the vessel wall are analyzed for a basic sinusoidal forcing and a parameterized realistic cardiac cycle. For relatively slow flows the flow forcing pattern dominates the forces on the walls. Upon increasing the flow-speed we observe a lively unsteady flow with more and more vortical structures arising inside the curved vessel and in the spherical cavity of the aneurysm. At these faster flows the pulsatile forcing pattern is less pronounced and the flow is dominated by its intrinsic Navier- Stokes unsteadiness. The simulations confirm that, within a physiologically relevant range of flow speeds, strong transitions in flow behavior and in force levels develop inside the aneurysm cavity, which may contribute to the long-term risk of aneurysm rupture.

**Representations of finite groups on ordered vector spaces**

*Marten Wortel (Universiteit Leiden); Waaier 2, vr 10:35-10:55*

In this talk we study representations of groups as positive automorphisms of ordered vector spaces, and the theory of ordered decompositions of such representations, which is not at all well developed, not even for finite groups. We will lay down the basic notions of positive representations and present our results for finite groups. We will explain how the concept of ordered decompositions leads to the notion of band-irreducible representations, and indicate why it is not evident that a bandirreducible representation of a finite group is finite-dimensional. Nevertheless, this is in fact the case, and furthermore, we can completely determine the structure of finite-dimensional positive representations of a finite group.

**Cost Reductions in Inventory Systems by the Use of Lateral Transshipments**

*Sandra van Wijk (Technische Universiteit Eindhoven); Waaier 2, vr 10:55-11:15*

We consider an inventory system with multiple stock points. When a demand occurs at a stock point and there is a part on stock, the demand can be immediately satisfied. Otherwise, one may apply a lateral transshipment, in which case a part is shipped from a nearby stock point if it has a part on stock. The demand is lost otherwise. Both options have a cost, and the optimal choice may depend on the stock levels of all stock points in the system. For inventory problems with lateral transshipments, currently only limited insights are available on optimal policy structures. It is known that a lot of costs can be saved via lateral transshipments; see in particular Kranenburg [2006], who showed a cost reduction of 50% for a spare parts inventory control problem at ASML, compared to the situation without lateral transshipments. But, there is a lack of insights into when exactly costs can be saved via lateral transshipments. This depends on the parameters settings, such as the costs for a lateral transshipment.

For this, we studied a system with two stock points, for which we completely characterized the structure of the optimal policy. We derived conditions under which simple, easy to implement, policies are always optimal. Furthermore, we identified the parameter settings under which one can gain most from lateral transshipments. For more than two stock points, we characterized the optimal policy structure for a system where only one stock point issues lateral transshipments. Moreover, we constructed an approximation algorithm for the general multi- location setting, determining the performance when executing a given policy. This can also be used for the optimization of parameters within a given class of policies, as from an implementation point of view, a simple parameterized policy may be much more attractive than an overall optimal policy. Furthermore, we investigated the gap between the optimal policy within a given class of parameterized policies and the overall optimal policy.

**A population-based model to describe geometrical uncertainties in radiotherapy: applied to prostate cases**

*Eka Budiarto (Technische Universiteit Delft) Waaier 2, vr 11:15-11:35*

Cancer patients receive radiotherapy treatments in a number of sessions or fractions. Local motions and deformations of organs between treatment fractions introduce geometrical uncertainties into radiotherapy. These un- certainties can decrease the quality of the treatment, since these can lead to underdosage in the tumour tissue, making the treatment less effective, or overdosage in the surrounding healthy tissue, damaging the healthy organs. A practical method to fully include these uncertainties is still lacking. This research proposes a model based on principal component analysis to describe the patient specific local probability distributions of voxel motions (to be used later in inverse treatment planning). As usually only a very limited number of data for a new patient is available, the analysis is extended to use population data, taken from patients with similar conditions. One of the challenges is to filter out the effects of variations in the shapes of the organ over the population, and get only the general movements and deformations. The assumption that general movements and deformations of a specific organ are similar despite the shape variations is justified retrospectively. A proof of principle of the method for deformations of the prostate and the seminal vesicles is presented.

**Controlling plants**

*Harsh Vinjamoor (Rijksuniversiteit Groningen) Waaier 2, vr 11:35-11:55*

Several physical systems such as ships, chemical plants, surgical robots etc. play an important role in everyday life. We call such systems “plants”. In the above plants some variables are available to the user. The rudder angle in ships, the flow of chemicals in reactors, the knob for the desired position of a robot tip etc. are variables that are manipulated by the user. The user in turn wants the plant to behave as desired. “Desirable” can have different meanings depending on the application. For a chemical plant it can mean that a certain product acquires the desired concentration quickly. For a surgical robot it can mean that the robot tip moves without vibrations. Many of these plants and desired behaviours can be represented by differential equation models. Our aim is to study these models and modify them so that they have a desired behaviour. We influence the behaviour of the plant by attaching another system to it; this other system is called a “controller”. We present a general scheme for constructing such a controller and discuss its applications for various classes of plants and desired behaviours.

**Bayesian approach to inverse problems**

*Bartek Knapik (Vrije Universiteit Amsterdam) Waaier 2, vr 11:55-12:15*

In this talk we study a Bayesian approach to estimating a parameter *μ *from an observation *Y *following the model

The unknown parameter *μ *is an element of a separable Hilbert space *H*1, and is mapped into another Hilbert space *H*2 by a known, compact, injective, linear operator *K*: *H*1 *à H*2. The image *Kμ *is perturbed by unobserved, scaled Gaussian white noise *Z*. The inverse problem of estimating *μ *been studied by both statisticians and numerical mathematicians, but rarely from a theoretical Bayesian perspective. In order to make inference about *μ *we put a Gaussian process prior on *μ*. Our interest is in studying the properties of the posterior distribution in two aspects of inverse problems – recovering the full parameter *μ *and recovering linear functionals of *μ*.

This talk provides the main results of our work that can be compared with well known frequentist theory. Both in nonparametric and linear functional case, we show the rate of the contraction of the posterior distribution around the truth, and we investigate frequentist coverage of Bayesian credible sets. The additional result is Bernstein-von Mises phenomenon for linear functionals of *μ*, which under suitable conditions on the linear functional and the prior shows that the posterior for the linear functional of the truth is centered at the frequentist maximum likelihood estimator.

In particular, we explain how the behavior of the posterior depends on the regularity of the element *μ*, the regularity of the prior, and the ill-posedness of the operator *K*, which is defined by its spectral properties. We illustrate the results by simulations and pictures in the particular example that *K *is the Volterra operator.