The claim that "for a conservative force, the work done in changing an object's position is path independent" might seem to follow immediately if the M is simply connected. However, recall that simple-connection only guarantees the existence of a continuous homotopy satisfying we seek a piecewise smooth homotopy satisfying those conditions instead.įortunately, the gap in regularity is resolved by the Whitney's approximation theorem. M is called simply connected if and only if for any continuous loop, c: → M there exists a continuous tubular homotopy H: × → M from c to a fixed point p ∈ c that is, Let M ⊆ R n be non-empty and path-connected. The definition of simply connected space follows:ĭefinition 2-2 (simply connected space). In Lemma 2-2, the existence of H satisfying to is crucial the question is whether such a homotopy can be taken for arbitrary loops. Ībove Lemma 2-2 follows from theorem 2–1. G., “ The application of the linear-quadratic dose-effect equation to fractionated and protracted radiotherapy,” Br. K., “ The linear-quadratic model and most other common radiobiological models result in similar predictions of time-dose relationships,” Radiat. T., “ Application of the linear-quadratic model with incomplete repair to radionuclide directed therapy,” Br. L., and Sgouros G., “ Three-dimensional radiobiologic dosimetry: application of radiobiologic modeling to patient-specific 3-dimensional imaging-based internal dosimetry,” J. thesis, University of Duisburg-Essen, 2009. Jentzen W., “ Prätherapeutische 124I-PET∕CT-Speicheldrüsendosimetrie bei der Radiojodtherapie differenzierter Schilddrüsenkarzinome und Untersuchung von wichtigen Einflussfaktoren auf die 124I-Quantifizierung,” Ph.D.V., “ Application of the linear-quadratic model to radioimmunotherapy: Further support for the advantage of longer-lived radionuclides,” J. L., and Sgouros G., “ Extension of the biological effective dose to the MIRD schema and possible implications in radionuclide therapy dosimetry,” Med. P., Jamar F., and Pauwels S., “ Patient-specific dosimetry in predicting renal toxicity with (90)Y-DOTATOC: relevance of kidney volume and dose rate in finding a dose-effect relationship,” J. Barone R., Borson-Chazot F., Valkema R., Walrand S., Chauvin F., Gogou L., Kvols L.We have devised an algorithmic resolution of the Lea–Catcheside G-factor formula to enable BED calculation for any function, including piecewise defined functions. 2, 3 Analytic calculation of the BED requires functional knowledge of the entire fit, which is not possible with a piecewise defined function such as trapezoidal fitting. Recent attempts to extend the BED formalism to more general applications still depend on dose-rate contributions being analytically integrable. In many instances the lack of a clear pattern for fitting leads to a piecewise trapezoidal fit of the dose-rate data for integration to absorbed dose. 1 The difficulty in applying the BED is that formulas only exist for analytic application for exponential based functional fits. The success of the BED-response relationship established in the context of renal toxicity from radiopeptide therapy has given much credence to the formalism. The use of the biological effective dose (BED) has gained favor in dosimetry as it folds the biological effect of dose rate in with the standard absorbed dose.
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