It is shown that for $\gamma$-ray $(>1 MeV)$ therapy the delivered dose can be approximated by the dual attenuated x-ray transform of the filtered beam profiles. The implied treatment geometry is appropriate for the new multileaf collimators. The number of intensity-modulated beams required for conformal radiotherapy is examined using the mathematics of tomographic reconstruction. For a 2D tomotherapy geometry the sampling requirement is at most $(2 \pi r_{max} W_{max} + 5/2)$ beams, where $r_{max}$ and $W_{max}$ are the maximum spatial extent and frequency, respectively, of the radiation dose. We generalize this ``Bow Tie" solution to 3D, suggesting a sufficient beam number given by $(\Delta \omega / 2 \pi W_{max})(2 \pi r_{max} W_{max} + 5/2)^{2}$, where $\Delta \omega$ is the frequency resolution of the beam front modulation delivered by the multilear collimator. The matrix inversion implicit in this bound suggests a beam selection criteria. The beam angles should be chosen such that the SVD inversion to beam profiles is non-singular for the entire configuration. The direct function metric among beam profiles provides another criteria for choosing beam angles. By maximizing the overlap between the sampled and continuous beam profile functions, the intensity (in the $\rho$-metric) is derived and displayed relative to the 3D data set for a ranking of beam orientations.

The formalism above is applied to the derivation and evaluation of radiotherapy plans for brain and prostate tumors based on real patient data from the UMASS Medical School and Massachusetts General Hospital. Dose-volume histograms are examined for evidence of beam number thresholds in conformal treatments.

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1996-1997 Mathematics in High Performance Computing