A spherical rearrangement proof of the stability of a Riesz-type inequality and an application to an isoperimetric type problem

We prove the stability of the ball as global minimizer of an attractive shape functional under volume constraint, by means of mass transportation arguments. The stability exponent is $1/2$ and it is sharp. Moreover, we use such stability result together with the quantitative (possibly fractional) isoperimetric inequality to prove that the ball is a global minimizer of a shape functional involving both an attractive and a repulsive term with a sufficiently large fixed volume and with a suitable (possibly fractional) perimeter penalization.

A fractional Hadamard formula and applications

We derive a shape derivative formula for the family of principal Dirichlet eigenvalues $$\lambda _s(\Omega )$$ λ s ( Ω ) of the fractional Laplacian $$(-\Delta )^s$$ ( - Δ ) s associated with bounded open sets $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N of class $$C^{1,1}$$ C 1 , 1 . This extends, with a help of a new approach, a result in Dalibard and Gérard-Varet (Calc. Var. 19(4):976–1013, 2013) which was restricted to the case $$s=\frac{1}{2}$$ s = 1 2 . As an application, we consider the maximization problem for $$\lambda _s(\Omega )$$ λ s ( Ω ) among annular-shaped domains of fixed volume of the type $$B\setminus \overline{B}'$$ B \ B ¯ ′ , where B is a fixed ball and $$B'$$ B ′ is ball whose position is varied within B. We prove that $$\lambda _s(B\setminus \overline{B}')$$ λ s ( B \ B ¯ ′ ) is maximal when the two balls are concentric. Our approach also allows to derive similar results for the fractional torsional rigidity. More generally, we will characterize one-sided shape derivatives for best constants of a family of subcritical fractional Sobolev embeddings.

Multi-volume hemacytometer

Cell counting has become an essential method for monitoring the viability and proliferation of cells. A hemacytometer is the standard device used to measure cell numbers in most laboratories which are typically automated to increase throughput. The principle of both manual and automated hemacytometers is to calculate cell numbers with a fixed volume within a set measurement range (105 ~ 106 cells/ml). If the cell concentration of the unknown sample is outside the range of the hemacytometer, the sample must be prepared again by increasing or decreasing the cell concentration. We have developed a new hemacytometer that has a multi-volume chamber with 4 different depths containing different volumes (0.1, 0.2, 0.4, 0.8 µl respectively). A multi-volume hemacytometer can measure cell concentration with a maximum of 106 cells/ml to a minimum of 5 × 103 cells/ml. Compared to a typical hemacytometer with a fixed volume of 0.1 µl, the minimum measurable cell concentration of 5 × 103 cells/ml on the multi-volume hemacytometer is twenty times lower. Additionally, the Multi-Volume Cell Counting model (cell concentration calculation with the slope value of cell number in multi-chambers) showed a wide measurement range (5 × 103 ~ 1 × 106 cells/ml) while reducing total cell counting numbers by 62.5% compared to a large volume (0.8 µl-chamber) hemacytometer.

Final results of a phase 1b study of isatuximab short-duration fixed-volume infusion combination therapy for relapsed/refractory multiple myeloma

Part B of this phase 1b study (ClinicalTrials.gov number, NCT02283775) evaluated safety and efficacy of a fixed-volume infusion of isatuximab, an anti-CD38 monoclonal antibody, in combination with pomalidomide and dexamethasone (Pd) in relapsed/refractory multiple myeloma patients. Isatuximab (10 mg/kg weekly for 4 weeks, then every other week) was administered as a fixed-volume infusion of 250 mL (mL/h infusion rate) with standard doses of Pd on 28-day cycles. Patients (N = 47) had a median of three prior treatment lines (range, 1–8). Median duration of exposure was 36.9 weeks and median duration of first, second, and 3+ infusions were 3.7, 1.8, and 1.2 h, respectively. The most common non-hematologic treatment-emergent adverse events were fatigue (63.8%), infusion reactions (IRs), cough, and upper respiratory tract infection (40.4% each). IRs were all grade 2 and occurred only during the first infusion. The overall response rate was 53.2% in all patients (55.5% in response-evaluable population, 60.0% in daratumumab-naïve patients). Efficacy and safety findings were consistent with data from the isatuximab plus Pd infusion schedule in Part A of this study and also from the phase 3 ICARIA-MM study, and these new data confirm the safety, efficacy, and feasibility of fixed-volume infusion of isatuximab.

Multi-Volume Hemacytometer

Cell counting has become an essential method for monitoring the viability and proliferation of cells. A hemacytometer is the standard device used to measure cell numbers in most laboratories which are typically automated to increase throughput. The principle of both manual and automated hemacytometers is to calculate cell numbers with a fixed volume within a set measurement range (105~106 cells/ml). If the cell concentration of the unknown sample is outside the range of the hemacytometer, the sample must be prepared again by increasing or decreasing the cell concentration. We have developed a new hemacytometer that has a multi-volume chamber with 4 different depths containing different volumes (0.1, 0.2, 0.4, 0.8 µl respectively). A multi-volume hemacytometer can measure cell concentration with a maximum of 106 cells/ml to a minimum of 5×103 cells/ml. Compared to a typical hemacytometer with a fixed volume of 0.1 µl, the minimum measurable cell concentration of 5×103 cells/ml on the multi-volume hemacytometer is twenty times lower. Additionally, the Multi-Volume Cell Counting model (cell concentration calculation with the slope value of cell number in multi-chambers) showed a wide measurement range (5×103 ~1×106 cells/ml) while reducing total cell counting numbers by 62.5% compared to a large volume (0.8 µl-chamber) hemacytometer.