Sharp error estimates for spline approximation: explicit constants, nn-widths, and eigenfunction convergence

In this paper we provide a priori error estimates in standard Sobolev (semi-)norms for approximation in spline spaces of maximal smoothness on arbitrary grids. The error estimates are expressed in terms of a power of the maximal grid spacing, an appropriate derivative of the function to be approximated, and an explicit constant which is, in many cases, sharp. Some of these error estimates also hold in proper spline subspaces, which additionally enjoy inverse inequalities. Furthermore, we address spline approximation of eigenfunctions of a large class of differential operators, with a particular focus on the special case of periodic splines. The results of this paper can be used to theoretically explain the benefits of spline approximation under $k$-refinement by isogeometric discretization methods. They also form a theoretical foundation for the outperformance of smooth spline discretizations of eigenvalue problems that has been numerically observed in the literature, and for optimality of geometric multigrid solvers in the isogeometric analysis context.Comment: 31 pages, 2 figures. Fixed a typo. Article published in M3A

Whole breast radiotherapy in prone and supine position: is there a place for multi-beam IMRT?

Background: Early stage breast cancer patients are long-term survivors and finding techniques that may lower acute and late radiotherapy-induced toxicity is crucial. We compared dosimetry of wedged tangential fields (W-TF), tangential field intensity-modulated radiotherapy (TF-IMRT) and multi-beam IMRT (MB-IMRT) in prone and supine positions for whole-breast irradiation (WBI). Methods: MB-IMRT, TF-IMRT and W-TF treatment plans in prone and supine positions were generated for 18 unselected breast cancer patients. The median prescription dose to the optimized planning target volume (PTVoptim) was 50 Gy in 25 fractions. Dose-volume parameters and indices of conformity were calculated for the PTVoptim and organs-at-risk. Results: Prone MB-IMRT achieved (p= 600 cc heart dose was consistently lower in prone position; while for patients with smaller breasts heart dose metrics were comparable or worse compared to supine MB-IMRT. Doses to the contralateral breast were similar regardless of position or technique. Dosimetry of prone MB-IMRT and prone TF-IMRT differed slightly. Conclusions: MB-IMRT is the treatment of choice in supine position. Prone IMRT is superior to any supine treatment for right-sided breast cancer patients and left-sided breast cancer patients with larger breasts by obtaining better conformity indices, target dose distribution and sparing of the organs-at-risk. The influence of treatment techniques in prone position is less pronounced; moreover dosimetric differences between TF-IMRT and MB-IMRT are rather small

Reproducibility of deep inspiration breath hold for prone left-sided whole breast irradiation

Background: Investigating reproducibility and instability of deep inspiration breath hold (DIBH) in the prone position to reduce heart dose for left-sided whole breast irradiation. Methods: Thirty patients were included and underwent 2 prone DIBH CT-scans during simulation. Overlap indices were calculated for the ipsilateral breast, heart and lungs to evaluate the anatomical reproducibility of the DIBH maneuver. The breathing motion of 21 patients treated with prone DIBH were registered using magnetic probes. These breathing curves were investigated to gain data on intra-fraction reproducibility and instability of the different DIBH cycles during treatment. Results: Overlap index was 0.98 for the ipsilateral breast and 0.96 for heart and both lungs between the 2 prone DIBH-scans. The magnetic sensors reported population amplitudes of 2.8 +/- 1.3 mm for shallow breathing and 11.7 +/- 4.7 mm for DIBH, an intra-fraction standard deviation of 1.0 +/- 0.4 mm for DIBH, an intra-breath hold instability of 1.0 +/- 0.6 mm and a treatment time of 300 +/- 69 s. Conclusion: Prone DIBH can be accurately clinically implemented with acceptable reproducibility and instability

The relation between patient discomfort and uncompensated forces of a patient support device for breast and regional lymph node radiotherapy

Although many authors stated that a user-centred design approach in medical device development has added values, the most common research approach within healthcare is evidence-based medicine, which tend to focus on functional data rather than patient wellbeing and comfort. End user comfort is well addressed in literature for commercial products such as seats and hand tools but no data was found for medical devices. A commercial patient support device for breast radiotherapy was analysed and a relation was found between discomfort and uncompensated internal body forces. Derived from CT-images, simplified patient free-body diagrams were analysed and pain and comfort evaluated. Subsequently, a new patient position was established and prototypes were developed. Patient comfort- and prototype optimization was done through iterative prototyping. With this approach, we were able to compensate all internal body forces and establish a force neutral patient free-body diagram. This resulted in comfortable patient positioning and favourable medical results

Potential benefits of crawl position for prone radiation therapy in breast cancer

Purpose: To investigate crawl position with the arm at the treated side alongside the body and at the opposite side above the head for prone treatment in patients requiring breast and regional lymph node irradiation. Methods: Patient support devices for crawl position were built for CT simulation and treatment. An asymmetric fork design resulted from an iterative process of prototype construction and testing. The fork's large horn supports the hemi-thorax, shoulder, and elevated arm at the nontreated side and the head. The short, narrow horn supports the arm at the treated side. Between both horns, the treated breast and its regional lymph nodes are exposed. Endpoints were pain, comfort, set-up precision, beam access to the breast and lymph nodes, and plan dose metrics. Pain and comfort were tested by volunteers (n = 9); set-up precision, beam access, and plan dose metrics were tested by means of a patient study (n = 10). The AIOTM (Orfit, Wijnegem, Belgium) prone breastboard (AIOTM) was used as a reference regarding comfort and set-up precision. Results: Pain at the sternum, the ipsilateral shoulder, upper arm, and neck was lower in crawl position than with bilateral arm elevation on AIOTM. Comfort and setup precision were better on the crawl prototype than on AIOTM. In crawl position, beam directions in the coronal and near-sagittal planes have access to the breast or regional lymph nodes without traversing device components. Plan comparison between supine and crawl positions showed better dose homogeneity for the breast and lymph node targets and dose reductions to all organs at risk for crawl position. Conclusions: Radiation therapy for breast and regional lymph nodes in crawl position is feasible. Good comfort and set-up precision were demonstrated. Planning results support the hypothesis that breast and regional lymph nodes can be treated in crawl position with less dose to organs at risk and equal or better dose distribution in the target volumes than in supine position. The crawl technique is a candidate methodology for further investigation for patients requiring breast and regional lymph node irradiation

Powell-Sabin B-splines and unstructured standard T-splines for the solution of the Kirchhoff-Love plate theory exploiting Bézier extraction

The equations that govern Kirchhoff–Love plate theory are solved using quadratic Powell–Sabin B-splines and unstructured standard T-splines. Bézier extraction is exploited to make the formulation computationally efficient. Because quadratic Powell–Sabin B-splines result in inline image-continuous shape functions, they are of sufficiently high continuity to capture Kirchhoff–Love plate theory when cast in a weak form. Unlike non-uniform rational B-splines (NURBS), which are commonly used in isogeometric analysis, Powell–Sabin B-splines do not necessarily capture the geometry exactly. However, the fact that they are defined on triangles instead of on quadrilaterals increases their flexibility in meshing and can make them competitive with respect to NURBS, as no bending strip method for joined NURBS patches is needed. This paper further illustrates how unstructured T-splines can be modified such that they are inline image-continuous around extraordinary points, and that the blending functions fulfil the partition of unity property. The performance of quadratic NURBS, unstructured T-splines, Powell–Sabin B-splines and NURBS-to-NURPS (non-uniform rational Powell–Sabin B-splines, which are obtained by a transformation from a NURBS patch) is compared in a study of a circular plat

Computation of multi-degree B-splines

Multi-degree splines are smooth piecewise-polynomial functions where the pieces can have different degrees. We describe a simple algorithmic construction of a set of basis functions for the space of multi-degree splines, with similar properties to standard B-splines. These basis functions are called multi-degree B-splines (or MDB-splines). The construction relies on an extraction operator that represents all MDB-splines as linear combinations of local B-splines of different degrees. This enables the use of existing efficient algorithms for B-spline evaluations and refinements in the context of multi-degree splines. A Matlab implementation is provided to illustrate the computation and use of MDB-splines

Whole breast and regional nodal irradiation in prone versus supine position in left sided breast cancer

Background: Prone whole breast irradiation (WBI) leads to reduced heart and lung doses in breast cancer patients receiving adjuvant radiotherapy. In this feasibility trial, we investigated the prone position for whole breast + lymph node irradiation (WB + LNI). Methods: A new support device was developed for optimal target coverage, on which patients are positioned in a position resembling a phase from the crawl swimming technique (prone crawl position). Five left sided breast cancer patients were included and simulated in supine and prone position. For each patient, a treatment plan was made in prone and supine position for WB + LNI to the whole axilla and the unoperated part of the axilla. Patients served as their own controls for comparing dosimetry of target volumes and organs at risk (OAR) in prone versus in supine position. Results: Target volume coverage differed only slightly between prone and supine position. Doses were significantly reduced (P < 0.05) in prone position for ipsilateral lung (Dmean, D2, V5, V10, V20, V30), contralateral lung (Dmean, D2), contralateral breast (Dmean, D2 and for total axillary WB + LNI also V5), thyroid (Dmean, D2, V5, V10, V20, V30), oesophagus (Dmean and for partial axillary WB + LNI also D2 and V5), skin (D2 and for partial axillary WB + LNI V105 and V107). There were no significant differences for heart and humeral head doses. Conclusions: Prone crawl position in WB + LNI allows for good breast and nodal target coverage with better sparing of ipsilateral lung, thyroid, contralateral breast, contralateral lung and oesophagus when compared to supine position. There is no difference in heart and humeral head doses