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Hence, we need to know the volume of the intersection of Here "volume" means the induced Euclidean volume in a, say, With all coordinate hyperplanes in the ambient space. Sum over the volume of the intersection of Kushnirenko's bound on the sum of the Milnor numbers is an alternating Which computes polyhedra given as the intersection of halfspaces, To define polytopes as the convex hull of some points and, more generally,Īll kinds of polyhedra as the convex hull of points, rays, lines and/or P := (x-y*z+x^2*y)^3-(z+2*y)^4+(x*y*z)^3 Īctually lies at the heart of the Convex package since it allows

Their convex hull is the Newton polytope of (with non-zero coefficients) and considers the exponent of each such monomial It is defined as follows: One looks at all monomials appearing in Kushnirenko's formula uses the Newton polytope of a polynomial Even at its present state of development, hyperthermia planning for regional hyperthermia delivers valuable information, not only for clinical practice, but also for further technologic improvements.The Newton polytope of a polynomial Further improvements in the implemented models, FE and FDTD, are required. The hyperthermia planning system HyperPlan could be validated for a number of the 30 patients. However, gross fluctuations exist from patient to patient. There are also statistically provable differences among the tumor entities regarding the attained specific absorption rate, temperatures, and volume loads in normal tissue. The results of the FE and FDTD methods are comparable, although slight differences exist resulting from the differences in the underlying models.

Tumor temperatures can only be estimated, because of the rather variable perfusion conditions. We could show sufficient agreement between the calculations and measurements for power density (specific absorption rate) within the range of assessed precision. Segmentation, grid generation, E-field, and temperature calculation can be carried out in clinical practice at an acceptable time expenditure of about 1-2 days.Īll 30 patients we analyzed with cervical, rectal, and prostate carcinoma exhibit a good correlation between the model calculations and the attained clinical data regarding acute toxicity (hot spots), prediction of easy-to-heat or difficult-to-heat patients, and the dependency on various other individual parameters. In both methods, temperature distributions are calculated on the tetrahedral grid by solving the bioheat transfer equation with the FE method. The FDTD method, on the other hand, calculates the E-field on a cubical grid, but also requires a tetrahedral grid for correction at electrical interfaces.

The FE method necessitates, primarily, this tetrahedral grid for the calculation of the E-field. A tetrahedral grid is generated from the segmented tissue boundaries, consisting of approximately 80,000 tetrahedrons per patient.

Both methods base their calculations on segmented (contour based) CT or MR image data. Two numeric methods, FE and FDTD, are implemented in HyperPlan for solving Maxwell's equations. A number of hyperthermia-specific modules are provided, enabling the creation of three-dimensional tetrahedral patient models suitable for treatment planning. This system already contains powerful algorithms for image processing, geometric modeling, and three-dimensional graphics display. The planning system HyperPlan is built on top of the modular, object-oriented platform for visualization and model generation AMIRA. Data and observations obtained from clinical hyperthermia are compared with the numeric methods FE (finite element) and FDTD (finite difference time domain), respectively. The main aim is to prove the clinical practicability of the hyperthermia treatment planning system HyperPlan on a beta-test level.