On Existence of L1L^1-solutions for Coupled Boltzmann Transport Equation and Radiation Therapy Treatment Optimization

The paper considers a linear system of Boltzmann transport equations modelling the evolution of three species of particles, photons, electrons and positrons. The system is coupled because of the collision term (an integral operator). The model is intended especially for dose calculation (forward problem) in radiation therapy. It, however, does not apply to all relevant interactions in its present form. We show under physically relevant assumptions that the system has a unique solution in appropriate ($L^1$-based) spaces and that the solution is non-negative when the data (internal source and inflow boundary source) is non-negative. In order to be self-contained as much as is practically possible, many (basic) results and proofs have been reproduced in the paper. Existence, uniqueness and non-negativity of solutions for the related time-dependent coupled system are also proven. Moreover, we deal with inverse radiation treatment planning problem (inverse problem) as an optimal control problem both for external and internal therapy (in general $L^p$-spaces). Especially, in the case $p=2$ variational equations for an optimal control related to an appropriate differentiable convex object function are verified. Its solution can be used as an initial point for an actual (global) optimization.Comment: Corrected typos. Added a new section 3. Revised the argument of Example 7.

Rolling Manifolds: Intrinsic Formulation and Controllability

In this paper, we consider two cases of rolling of one smooth connected complete Riemannian manifold $(M,g)$ onto another one (\hM,\hg) of equal dimension $n\geq 2$. The rolling problem $(NS)$ corresponds to the situation where there is no relative spin (or twist) of one manifold with respect to the other one. As for the rolling problem $(R)$, there is no relative spin and also no relative slip. Since the manifolds are not assumed to be embedded into an Euclidean space, we provide an intrinsic description of the two constraints "without spinning" and "without slipping" in terms of the Levi-Civita connections $\nabla^{g}$ and \nabla^{\hg}. For that purpose, we recast the two rolling problems within the framework of geometric control and associate to each of them a distribution and a control system. We then investigate the relationships between the two control systems and we address for both of them the issue of complete controllability. For the rolling $(NS)$, the reachable set (from any point) can be described exactly in terms of the holonomy groups of $(M,g)$ and (\hM,\hg) respectively, and thus we achieve a complete understanding of the controllability properties of the corresponding control system. As for the rolling $(R)$, the problem turns out to be more delicate. We first provide basic global properties for the reachable set and investigate the associated Lie bracket structure. In particular, we point out the role played by a curvature tensor defined on the state space, that we call the \emph{rolling curvature}. In the case where one of the manifolds is a space form (let say (\hM,\hg)), we show that it is enough to roll along loops of $(M,g)$ and the resulting orbits carry a structure of principal bundle which preserves the rolling $(R)$ distribution. In the zero curvature case, we deduce that the rolling $(R)$ is completely controllable if and only if the holonomy group of $(M,g)$ is equal to SO(n). In the nonzero curvature case, we prove that the structure group of the principal bundle can be realized as the holonomy group of a connection on $TM\oplus \R$, that we call the rolling connection. We also show, in the case of positive (constant) curvature, that if the rolling connection is reducible, then $(M,g)$ admits, as Riemannian covering, the unit sphere with the metric induced from the Euclidean metric of $\R^{n+1}$. When the two manifolds are three-dimensional, we provide a complete local characterization of the reachable sets when the two manifolds are three-dimensional and, in particular, we identify necessary and sufficient conditions for the existence of a non open orbit. Besides the trivial case where the manifolds $(M,g)$ and (\hM,\hg) are (locally) isometric, we show that (local) non controllability occurs if and only if $(M,g)$ and (\hM,\hg) are either warped products or contact manifolds with additional restrictions that we precisely describe. Finally, we extend the two types of rolling to the case where the manifolds have different dimensions

LÄÄKE ♥ LAPSI : Lääkeoppaat leikki-ikäiselle (1-6v.) lapselle ja hänen vanhemmilleen Lasten ja nuorten sairaalan Sydänosastolle K4

LÄÄKE ♥ LAPSI - Lääkeoppaat leikki-ikäiselle (1-6v.) lapselle ja hänen vanhemmilleen Lasten ja nuorten sairaalan Sydänosastolle K4 Opinnäytetyö on työelämälähtöinen ja se tehtiin HUS:n Lasten ja nuorten sairaalan Sydänosastolle K4. Opinnäytetyö koostuu teoreettisesta ja toiminnallisesta osuudesta. Opinnäytetyön tarkoituksena oli toteuttaa lääkehoidon oppaat leikki-ikäiselle sydänlapselle ja hänen vanhemmilleen. Opinnäytetyönä tehtyjen oppaiden tavoitteena on auttaa leikki-ikäistä sydänlasta ymmärtämään lääkityksen tarkoitus sekä vähentää lääkkeiden antamiseen kohdistuvia pelkoja. Vanhemmille pyritään antamaan tarvittavat tiedot ja apuväline, opas, jotta he voisivat tukea ja ohjata lasta lääkehoidossa. Teoreettisessa osuudessa käsitellään leikki-ikäistä lasta, lasten lääkehoidon erityispiirteitä, sydänsairaan lapsen lääkehoitoa, leikki-ikäisen lapsen pelkoja sekä leikki-ikäisen lapsen ja vanhempien valmistamista lääkehoitoon. Myös oppaan laatimisen teoriaa on käsitelty. Toiminnallinen osuus koostuu kahdesta sydänlapsen lääkehoitoa käsittelevästä oppaasta. Sydänlapsen oma lääkekirja on suunnattu leikki-ikäiselle (1-6v.) lapselle. Lapsen opas on värityskirjamuotoon tehty kuvakirja, jossa lyhyiden tekstien avulla kerrotaan lapselle lääkehoidosta. Lapsen oppaasta tehtiin myös valmiiksi väritetty versio. Sydänlapsen lääkitys - Vanhempien opas on leikki-ikäisen lapsen vanhemmille suunnattu ”tietopaketti” sydänlapsen lääkehoidosta.MEDICINE ♥ CHILD - Medicine guides for play-aged (1-6-year old) child and parents at the Cardiac Ward K4 of the Hospital for Children and Adolescents This thesis is based on working life needs and it was made for Children's Hospital Cardiac Ward K4. The thesis consists of a theoretical and a functional section. The purpose of this thesis was to produce medical guides for a play-aged child with heart defects and her/his parents. The aim of the guides made for this thesis is to help play-aged children with heart defects to understand the purpose of the medication and to decrease the fears being aimed at administering medicine. Parents are about to be given the knowledge and instrument, the guide, that are needed so that they can support and instruct their child in medical care. The theoretical section examines the play-aged child, special characteristics of children’s medical care, medical treatment of a child with heart disease, play-aged child’s fears, and preparing a play-aged child and parents for medical care. Also theory of creating a guide has been handled. The functional section consists of two medical guides dealing with medical care of a child with heart defects. Medicine guide for a child with heart defects is directional for a play-aged (1-6-year old) child. The child’s guide is made as a coloring book formed picture book with short texts about medical care. An already colored version of the child's guide was also made. Parents’ guide is written as “informational package” about medical care of a child with heart defects

Rolling Manifolds of Different Dimensions

If $(M,g)$ and (\hM,\hg) are two smooth connected complete oriented Riemannian manifolds of dimensions $n$ and \hn respectively, we model the rolling of $(M,g)$ onto (\hM,\hg) as a driftless control affine systems describing two possible constraints of motion: the first rolling motion $\Sigma_{NS}$ captures the no-spinning condition only and the second rolling motion $\Sigma_{R}$ corresponds to rolling without spinning nor slipping. Two distributions of dimensions (n + \hn) and $n$, respectively, are then associated to the rolling motions $\Sigma_{NS}$ and $\Sigma_{R}$ respectively. This generalizes the rolling problems considered in \cite{ChitourKokkonen1} where both manifolds had the same dimension. The controllability issue is then addressed for both $\Sigma_{NS}$ and $\Sigma_{R}$ and completely solved for $\Sigma_{NS}$. As regards to $\Sigma_{R}$, basic properties for the reachable sets are provided as well as the complete study of the case (n,\hn)=(3,2) and some sufficient conditions for non-controllability