Medico

More Mathematics into Medicine!
P. Deuflhard∗ O. D¨ssel† A. K. Louis‡ S. Zachow§ , o , , March 5, 2009

Abstract This article presents three success stories that show how the coaction of mathematics and medicine has pushed a development towards patient specific models on the basis of modern medical imaging and “virtual labs”, which, in the near future, will play an increasingly important role.Thereby the interests of medicine and mathematics seem to be consonant: either discipline wants the results fast and reliably. As for the medical side, this means that the necessary computations must run in shortest possible times on a local PC in the clinics and that their results must be accurate and resilient enough so that they can serve as a basis for medical decisions. As for themathematical side, this means that highest level requirements for the efficiency of the applied algorithms and the numerical and visualization software have to be met. Yet there is still a long way to go, until anatomically correct and medically useful individual functional models for the essential body parts and for the most frequent diseases will be at hand. This will only be possible, if more mathematicsenters into medicine.

∗ Zuse

Institute Berlin (ZIB) and Freie Universit¨t Berlin, Institute for Mathematics a Karlsruhe (TH), Institut f¨ r Biomedizinische Technik u ‡ Universit¨t des Saarlandes, Saarbr¨ cken, Institute for Mathematics a u § Zuse Institute Berlin (ZIB), Research Group Medical Planning
† Universit¨t a

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Mathematics in Medical Imaging
History of a successWhen Konrad R¨ntgen discovered the “X-rays” in 1895 in W¨rzburg, Germany, o u he opened a window to non-invasive insight into the human body. The attenuation of X-rays is strongly dependent on the tissue through which they travel, for example, bone attenuates more than fat. The thus produced shadow images deliver visual information about the interior of the body. The disadvantage is that densetissue like bones masks the tissue along the same path of the Xrays, whence the application was restricted to those parts of the body with few bones. Examinations of the brain were, due to the surrounding skull, virtually impossible without the dangerous addition of contrast agents. The conceptual and technical breakthrough was achieved in the sixties: Nobel laureates Allan Cormack and GodfreyHounsfield suggested an imaging system, where shadow images of an imaginary cross section through the body and for many directions are registered simultaneously. Each of these shadow images contributes only little information. The exploitation of the fact that all of these images cover the same part of the body allowed for images of the interior of the body with previously unknown resolution. This was thestarting point of a long success story. Computed Tomography (CT). Computed tomography, also known as computer assisted tomography (CAT), is the name of an imaging system, where the role of mathematics is evident. With the large number of collected data the calculation of the desired image information can be achieved only by means of powerful computers. Fundamental for the use of computers is thedevelopment of efficient algorithms, based on a precise mathematical model of the complex connection between measured data and image information to be determined. The measured data, i.e. the attenuation of the X-ray intensity after travelling through the body, is related to the X-ray attenuation coefficient, interpreted mostly as the density of the tissue, along the path of the rays. In mathematicalterms, the observed attenuation is related to the line integral of the X-ray attenuation coefficient along the ray path. The arising integral equation is, in the 2D case, named as Radon transform, after the Austrian mathematician Johann Radon. In the first commercial scanners, Godfrey Hounsfield solved this integral equation by standard discretization methods. He projected the solution on a pixel...