We survey some recent results concerning the so called Categorical Torelli problem. This is to say how one can reconstruct a smooth projective variety up to isomorphism, by using the homological properties of special admissible subcategories of the bounded derived category of coherent sheaves of such a variety. The focus is on Enriques surfaces, prime Fano threefolds and cubic fourfolds.
Background: The treatment of lung lesions with stereotactic body radiation therapy calls for highly conformal dose, which is evaluated by a number of metrics. Lung stereotactic body radiation therapy clinical trials constrain a plans gradient index. The purpose of this work is to describe the dependence of clinically achievable dose gradient on planning target volume.
Methods: Three hundred seventy-four lung stereotactic body radiation therapy treatment plans were retrospectively reviewed and selected for this study. The relationship between R50% and planning target volume size was observed and compared against the RTOG 0915 and 0813 constraints noting minor and major deviations. Then a least squares regression was used to determine the coefficients for a power functional form of the dependence of gradient measure (GM) on planning target volume size.
Results: Of the 317 peripheral lung SBRT plans, 142 exhibited no deviation, 135 exhibited a minor deviation, and 40 exhibited a major deviation according to the RTOG 0915 dosimetric. conformality and dose fall-off constraints. A plot of gradient measure versus planning target volume size for peripheral lesions, excluding RTOG 0915 major deviations, is fit with an power function of GM = 0.564 V0.215.
Conclusions: Using the PTV size and GM relationship we have characterized, treatment plans with PTV < 85 cm3 can be evaluated subjectively to our previously plans, and given a percentile GM. This relationship and evaluation is useful for volumetric modulated arc therapy lung stereotactic body radiation therapy treatment planning and quality control.
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