Willmore minimizers with prescribed isoperimetric ratio

Motivated by a simple model for elastic cell membranes, we minimize the Willmore functional among two-dimensional spheres embedded in R^3 with prescribed isoperimetric ratio

Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M. Under the assumption that the sectional curvature of M is strictly positive, we prove the existence of a smoothly immersed sphere minimizing the L^{2} integral of the second fundamental form. Assuming instead that the sectional curvature is less than or equal to 2, and that there exists a point in M with scalar curvature bigger than 6, we obtain a smooth 2-sphere minimizing the integral of 1/4|H|^{2} +1, where H is the mean curvature vector

On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow

In this article, we study the axisymmetric surface diffusion flow (ASD), a fourth-order geometric evolution law. In particular, we prove that ASD generates a real analytic semiflow in the space of (2 + \alpha)-little-H\"older regular surfaces of revolution embedded in R^3 and satisfying periodic boundary conditions. We also give conditions for global existence of solutions and prove that solutions are real analytic in time and space. Further, we investigate the geometric properties of solutions to ASD. Utilizing a connection to axisymmetric surfaces with constant mean curvature, we characterize the equilibria of ASD. Then, focusing on the family of cylinders, we establish results regarding stability, instability and bifurcation behavior, with the radius acting as a bifurcation parameter for the problem.Comment: 37 pages, 6 figures, To Appear in SIAM J. Math. Ana

A classification theorem for Helfrich surfaces

In this paper we study the functional \SW_{\lambda_1,\lambda_2}, which is the the sum of the Willmore energy, $\lambda_1$-weighted surface area, and $\lambda_2$-weighted volume, for surfaces immersed in $\R^3$. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our main result is a complete classification of all smooth immersed critical points of the functional with $\lambda_1\ge0$ and small $L^2$ norm of tracefree curvature. In particular we prove the non-existence of critical points of the functional for which the surface area and enclosed volume are positively weighted.Comment: 21 page

The Conformal Willmore Functional: a Perturbative Approach

The conformal Willmore functional (which is conformal invariant in general Riemannian manifold $(M,g)$) is studied with a perturbative method: the Lyapunov-Schmidt reduction. Existence of critical points is shown in ambient manifolds $(\mathbb{R}^3, g_\epsilon)$ -where $g_\epsilon$ is a metric close and asymptotic to the euclidean one. With the same technique a non existence result is proved in general Riemannian manifolds $(M,g)$ of dimension three.Comment: 34 pages; Journal of Geometric Analysis, on line first 23 September 201

A comparison of methods for adapting 177Lu dose-voxel-kernels to tissue inhomogeneities

Abstract In radionuclide therapies, dosimetry is used for determining patient-individual dose burden. Standard approaches provide whole organ doses only. For assessing dose heterogeneity inside organs, voxel-wise dosimetry based on 3D SPECT/CT imaging could be applied. Often, this is achieved by convolving voxel-wise time-activity-curves with appropriate dose-voxel-kernels (DVK). The DVKs are meant to model dose deposition, and can be more accurate if modelled for the specific tissue type under consideration. In literature, DVKs are often not adapted to these inhomogeneities, or simple approximation schemes are applied. For 26 patients, which had previously undergone a -PSMA or -DOTATOC therapy, decay maps, mass-density maps as well as tissue-type maps were derived from SPECT/CT acquisitions. These were used for a voxel-based dosimetry based on convolution with DVKs (each of size ) obtained by four different DVK methods proposed in literature. The simplest only considers a spatially constant soft-tissue DVK (herein named ‘constant’), while others either take into account only the local density of the center voxel of the DVK (herein named ‘center-voxel’) or scale each voxel linearly according to the proper mass density deduced from the CT image (herein named ‘density’) or considered both the local mass density as well as the direct path between the center voxel and any voxel in its surrounding (herein named ‘percentage’). Deviations between resulting dose values and those from full Monte-Carlo simulations (MC simulations) were compared for selected organs and tissue-types. For each DVK method, inter-patient variability was considerable showing both under- and over-estimation of energy dose compared to the MC result for all tissue densities higher than soft tissue. In kidneys and spleen, ‘constant’ and ‘density’-scaled DVKs achieved estimated doses with smallest deviations to the full MC gold standard (∼ underestimation). For low and high density tissue types such as lung and adipose or bone tissue, alternative DVK methods like ‘center-voxel’- and ‘percentage’- scaled achieved superior results, respectively. Concerning computational load, dose estimation with the DVK method ‘constant’ needs about 1.1 s per patient, center-voxel scaling amounts to 1.2 s, density scaling needs 1.4 s while percentage scaling consumes 860.3 s per patient. In this study encompassing a large patient cohort and four different DVK estimation methods, no single DVK-adaption method was consistently better than any other in case of soft tissue kernels. Hence in such cases the simplest DVK method, labeled ‘constant’, suffices. In case of tumors, often located in tissues of low (lung) or high (bone) density, more sophisticated DVK methods excel. The high inter-patient variability indicates that for evaluating new algorithms, a sufficiently large patient cohort needs to be involved

F18-FDG PET/CT imaging early predicts pathologic complete response to induction chemoimmunotherapy of locally advanced head and neck cancer: preliminary single-center analysis of the checkrad-cd8 trial

Aim In the CheckRad-CD8 trial patients with locally advanced head and neck squamous cell cancer are treated with a single cycle of induction chemo-immunotherapy (ICIT). Patients with pathological complete response (pCR) in the re-biopsy enter radioimmunotherapy. Our goal was to study the value of F-18-FDG PET/CT in the prediction of pCR after induction therapy. Methods Patients treated within the CheckRad-CD8 trial that additionally received FDG- PET/CT imaging at the following two time points were included: 3–14 days before (pre-ICIT) and 21–28 days after (post-ICIT) receiving ICIT. Tracer uptake in primary tumors (PT) and suspicious cervical lymph nodes (LN +) was measured using different quantitative parameters on EANM Research Ltd (EARL) accredited PET reconstructions. In addition, mean FDG uptake levels in lymphatic and hematopoietic organs were examined. Percent decrease (Δ) in FDG uptake was calculated for all parameters. Biopsy of the PT post-ICIT acquired after FDG-PET/CT served as reference. The cohort was divided in patients with pCR and residual tumor (ReTu). Results Thirty-one patients were included. In ROC analysis, ΔSUVmax PT performed best (AUC = 0.89) in predicting pCR (n = 17), with a decline of at least 60% (sensitivity, 0.77; specificity, 0.93). Residual SUVmax PT post-ICIT performed best in predicting ReTu (n = 14), at a cutpoint of 6.0 (AUC = 0.91; sensitivity, 0.86; specificity, 0.88). Combining two quantitative parameters (ΔSUVmax ≥ 50% and SUVmax PT post-ICIT ≤ 6.0) conferred a sensitivity of 0.81 and a specificity of 0.93 for determining pCR. Background activity in lymphatic organs or uptake in suspected cervical lymph node metastases lacked significant predictive value. Conclusion FDG-PET/CT can identify patients with pCR after ICIT via residual FDG uptake levels in primary tumors and the related changes compared to baseline. FDG-uptake in LN + had no predictive value. Trial registry ClinicalTrials.gov identifier: NCT03426657

Three-dimensional Monte Carlo-based voxel-wise tumor dosimetry in patients with neuroendocrine tumors who underwent 177Lu-DOTATOC therapy

Abstract Background Patients with advanced neuroendocrine tumors (NETs) of the midgut are suitable candidates for 177Lu-DOTATOC therapy. Integrated SPECT/CT systems have the potential to help improve the accuracy of patient-specific tumor dosimetry. Dose estimations to target organs are generally performed using the Medical Internal Radiation Dose scheme. We present a novel Monte Carlo-based voxel-wise dosimetry approach to determine organ- and tumor-specific total tumor doses (TTD). Methods A cohort of 14 patients with histologically confirmed metastasized NETs of the midgut (11 men, 3 women, 62.3 ± 11.0 years of age) underwent a total of 39 cycles of 177Lu-DOTATOC therapy (mean 2.8 cycles, SD ± 1 cycle). After the first cycle of therapy, regions of interest were defined manually on the SPECT/CT images for the kidneys, the spleen, and all 198 tracer-positive tumor lesions in the field of view. Four SPECT images, taken at 4 h, 24 h, 48 h and 72 h after injection of the radiopharmaceutical, were used to determine their effective half-lives in the structures of interest. The absorbed doses were calculated by a three-dimensional dosimetry method based on Monte Carlo simulations. TTD was calculated as the sum of all products of single tumor doses with single tumor volumes divided by the sum of all tumor volumes. Results The average dose values per cycle were 3.41 ± 1.28 Gy (1.91–6.22 Gy) for the kidneys, 4.40 ± 2.90 Gy (1.14–11.22 Gy) for the spleen, and 9.70 ± 8.96 Gy (1.47–39.49 Gy) for all 177Lu-DOTATOC-positive tumor lesions. Low- and intermediate-grade tumors (G 1–2) absorbed a higher TTD compared to high-grade tumors (G 3) (signed-rank test, p =  < 0.05). The pre-therapeutic chromogranin A (CgA) value and the TTD correlated significantly (Pearson correlation:  = 0.67, p = 0.01). Higher TTD resulted in a significant decrease of CgA after therapy. Conclusion These results suggest that Monte Carlo-based voxel-wise dosimetry is a very promising tool for predicting the absorbed TTD based on histological and clinical parameters

Existence of Integral mm-Varifolds minimizing ∫∣A∣p\int |A|^p and ∫∣H∣p\int |H|^p, p>mp>m, in Riemannian Manifolds

We prove existence and partial regularity of integral rectifiable $m$-dimensional varifolds minimizing functionals of the type $\int |H|^p$ and $\int |A|^p$ in a given Riemannian $n$-dimensional manifold $(N,g)$, $2\leq mm$, under suitable assumptions on $N$ (in the end of the paper we give many examples of such ambient manifolds). To this aim we introduce the following new tools: some monotonicity formulas for varifolds in $\mathbb{R}^S$ involving $\int |H|^p$, to avoid degeneracy of the minimizer, and a sort of isoperimetric inequality to bound the mass in terms of the mentioned functionals.Comment: 33 pages; this second submission corresponds to the published version of the paper, minor typos are fixe