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It is shown that the continuum level density (CLD) at unbound energies can be calculated with the complex scaling method (CSM), in which the energy spectra of bound states, resonances and continuum states are obtained in terms of $L^2$ basis functions. In this method, the extended completeness relation is applied to the calculation of the Green functions, and the continuum-state part is approximately expressed in terms of discretized complex scaled continuum solutions. The obtained result is compared with the CLD calculated exactly from the scattering phase shift. The discretization in the CSM is shown to give a very good description of continuum states. We discuss how the scattering phase shifts can inversely be calculated from the discretized CLD using a basis function technique in the CSM.Comment: 14 pages, 9 figures, To be published in Progress of Theoretical Physic

BackgroundWe performed a dosimetric comparison of spot-scanning proton therapy (SSPT) and intensity-modulated radiation therapy (IMRT) for hepatocellular carcinoma (HCC) to investigate the impact of tumor size on the risk of radiation induced liver disease (RILD).MethodsA number of alternative plans were generated for 10 patients with HCC. The gross tumor volumes (GTV) varied from 20.1 to 2194.5 cm3. Assuming all GTVs were spherical, the nominal diameter was calculated and ranged from 3.4 to 16.1 cm. The prescription dose was 60 Gy for IMRT or 60 cobalt Gy-equivalents for SSPT with 95% planning target volume (PTV) coverage. Using IMRT and SSPT techniques, extensive comparative planning was conducted. All plans were evaluated by the risk of RILD estimated using the Lyman-normal-tissue complication probability model.ResultsFor IMRT the risk of RILD increased drastically between 6.3–7.8 cm nominal diameter of GTV. When the nominal diameter of GTV was more than 6.3 cm, the average risk of RILD was 94.5% for IMRT and 6.2% for SSPT.ConclusionsRegarding the risk of RILD, HCC can be more safely treated with SSPT, especially if its nominal diameter is more than 6.3 cm.

The results suggested that by properly adjusting irradiation control parameters, gated proton spot-scanning beam therapy can be robust to target motion. This is an important first step toward establishing treatment plans in real patient geometry.