Algorithm 1018: FaVeST—Fast Vector Spherical Harmonic Transforms

Vector spherical harmonics on the unit sphere of ℝ 3 have broad applications in geophysics, quantum mechanics, and astrophysics. In the representation of a tangent vector field, one needs to evaluate the expansion and the Fourier coefficients of vector spherical harmonics. In this article, we develop fast algorithms (FaVeST) for vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has a computational cost proportional to N log √ N for N number of evaluation points. The adjoint FaVeST, which evaluates a linear combination of vector spherical harmonics with a degree up to ⊡ M for M evaluation points, has cost proportional to M log √ M . Numerical examples of simulated tangent fields illustrate the accuracy, efficiency, and stability of FaVeST.

Axillary lymph node coverage of tangent radiotherapy: Comparison of supine versus prone positioning.

200 Background: Recent randomized evidence has demonstrated low rates of axillary nodal failure in patients with one to two positive lymph nodes who receive local therapy with sentinel lymph node biopsy and simple tangential radiotherapy alone. Decreasing utilization of complete axillary nodal dissections has potential implications for radiotherapy treatment delivery design. We sought to compare the inadvertent coverage of the axillary lymph nodal regions between supine versus prone position for whole breast radiotherapy using simple tangent fields. Methods: Twenty patients with breast cancer who had previously received whole breast radiotherapy with simple tangent fields were randomly selected. Patients were selected such that 10 patients had received treatment in the supine position and 10 patients in the prone position. Axillary lymph node levels I-III were contoured by a single physician according to the RTOG breast contouring atlas. The previously generated radiotherapy plans, each to deliver a prescribed dose between 42.56 and 50Gy, were dosimetrically compared to assess differences in coverage of the nodal volumes. Results: In both positions, dose to each of the axillary nodal regions was low. For level I, the volume receiving 95% of the prescribed dose (V95) was 34.7% for supine positioning and 1.6% for prone positioning. All other analyzed volumes, specifically V25 (71.5% vs 32.6%), V50 (63.3% vs 28.2%), and V75 (57.6% vs 24.1%), were 50% greater for supine compared to prone positioning. Level II coverage was less with V95 of 6% and 0.1% respectively. Similarly V25 (32.8% vs 5.7%), V50 (25.7% vs 3.8%), and V75 (20.3% vs 2.1%) were less compared to level one and greater in the supine position. Level III coverage was less than 10% for both positions at all measured volumes of V95, V75, V50, and V25. Conclusions: Delivery of radiotherapy using simple tangents inadequately covers all axillary nodal levels. Coverage is greater in the supine position with very little inadvertent coverage in the prone position. The risk of nodal recurrence should be carefully considered when deciding to use a simple tangential field design. Additional factors such as dose to the lung and heart should also be considered for positioning.