A Scaling Method and its Applications to Problems in Fractional Dimensional Space

A scaling method is proposed to find (1) the volume and the surface area of a generalized hypersphere in a fractional dimensional space and (2) the solid angle at a point for the same space. It is demonstrated that the total dimension of the fractional space can be obtained by summing the dimension of the fractional line element along each axis. The regularization condition is defined for functions depending on more than one variable. This condition is applied (1) to find a closed form expression for the fractional Gaussian integral, (2) to establish a relationship between a fractional dimensional space and a fractional integral, (3) to develop the Bochner theorem, and (4) to obtain an expression for the fractional integral of the Mittag–Leffler function. Some possible extensions of this work are also discussed

Solutions of a particle with fractional δ\delta-potential in a fractional dimensional space

A Fourier transformation in a fractional dimensional space of order \la (0<\la\leq 1) is defined to solve the Schr\"odinger equation with Riesz fractional derivatives of order \a. This new method is applied for a particle in a fractional $\delta$-potential well defined by V(x) =- \gamma\delta^{\la}(x), where $\gamma>0$ and \delta^{\la}(x) is the fractional Dirac delta function. A complete solutions for the energy values and the wave functions are obtained in terms of the Fox H-functions. It is demonstrated that the eigen solutions are exist if 0< \la<\a. The results for \la= 1 and \a=2 are in exact agreement with those presented in the standard quantum mechanics

Computer-Based Intensity Measurement Assists Pathologists in Scoring Phosphatase and Tensin Homolog Immunohistochemistry - Clinical Associations in NSCLC Patients of the European Thoracic Oncology Platform Lungscape Cohort.

Phosphatase and tensin homolog (PTEN) loss is frequently observed in NSCLC and associated with both phosphoinositide 3-kinase activation and tumoral immunosuppression. PTEN immunohistochemistry is a valuable readout, but lacks standardized staining protocol and cutoff value. After an external quality assessment using SP218, 138G6 and 6H2.1 anti-PTEN antibodies, scored on webbook and tissue microarray, the European Thoracic Oncology Platform cohort samples (n = 2245 NSCLC patients, 8980 tissue microarray cores) were stained with SP218. All cores were H-scored by pathologists and by computerized pixel-based intensity measurements calibrated by pathologists. All three antibodies differentiated six PTEN+ versus six PTEN- cases on external quality assessment. For 138G6 and SP218, high sensitivity and specificity was found for all H-score threshold values including prospectively defined 0, calculated 8 (pathologists), and calculated 5 (computer). High concordance among pathologists in setting computer-based intensities and between pathologists and computer in H-scoring was observed. Because of over-integration of the human eye, pixel-based computer H-scores were overall 54% lower. For all cutoff values, PTEN- was associated with smoking history, squamous cell histology, and higher tumor stage (p &lt; 0.001). In adenocarcinomas, PTEN- was associated with poor survival. Calibration of immunoreactivity intensities by pathologists following computerized H-score measurements has the potential to improve reproducibility and homogeneity of biomarker detection regarding epitope validation in multicenter studies

Computer-Based Intensity Measurement Assists Pathologists in Scoring Phosphatase and Tensin Homolog Immunohistochemistry — Clinical Associations in NSCLC Patients of the European Thoracic Oncology Platform Lungscape Cohort

Introduction: Phosphatase and tensin homolog (PTEN) loss is frequently observed in NSCLC and associated with both phosphoinositide 3-kinase activation and tumoral immunosuppression. PTEN immunohistochemistry is a valuable readout, but lacks standardized staining protocol and cutoff value. Methods: After an external quality assessment using SP218, 138G6 and 6H2.1 anti-PTEN antibodies, scored on webbook and tissue microarray, the European Thoracic Oncology Platform cohort samples (n = 2245 NSCLC patients, 8980 tissue microarray cores) were stained with SP218. All cores were H-scored by pathologists and by computerized pixel-based intensity measurements calibrated by pathologists. Results: All three antibodies differentiated six PTEN+ versus six PTEN- cases on external quality assessment. For 138G6 and SP218, high sensitivity and specificity was found for all H-score threshold values including prospectively defined 0, calculated 8 (pathologists), and calculated 5 (computer). High concordance among pathologists in setting computer-based intensities and between pathologists and computer in H-scoring was observed. Because of over-integration of the human eye, pixel-based computer H-scores were overall 54% lower. For all cutoff values, PTEN- was associated with smoking history, squamous cell histology, and higher tumor stage (p &lt; 0.001). In adenocarcinomas, PTEN- was associated with poor survival. Conclusion: Calibration of immunoreactivity intensities by pathologists following computerized H-score measurements has the potential to improve reproducibility and homogeneity of biomarker detection regarding epitope validation in multicenter studies. © 2018 International Association for the Study of Lung Cance

Programmed death-ligand 1 expression influenced by tissue sample size. Scoring based on tissue microarrays' and cross-validation with resections, in patients with, stage I-III, non-small cell lung carcinoma of the European Thoracic Oncology Platform Lungscape cohort

PD-L1, as assessed by immunohistochemistry, is a predictive biomarker for immuno-oncology treatment in lung cancer. Different scoring methods have been used to assess its status, resulting in a wide range of positivity rates. We use the European Thoracic Oncology Platform Lungscape non-small cell lung carcinoma cohort to explore this issue. PD-L1 expression was assessed via immunohistochemistry on tissue microarrays (up to four cores per case), using the DAKO 28-8 immunohistochemistry assay, following a two-round external quality assessment procedure. All samples were analyzed under the same protocol. Cross-validation of scoring between tissue microarray and whole sections was performed in 10% randomly selected samples. Cutoff points considered: >= 1, 50 (primarily), and 25%. At the two external quality assessment rounds, tissue microarray scoring agreement rates between pathologists were: 73% and 81%. There were 2008 cases with valid immunohistochemistry tissue microarray results (50% all cores evaluable). Concordant cases at 1, 25, and 50% were: 85, 91, and 93%. Tissue microarray core results were identical for 70% of cases. Sensitivity of the tissue microarray method for 1, 25, and 50% was: 80, 78, and 79% (specificity: 90, 95, 98%). Complete agreement between tissue microarrays and whole sections was achieved for 60% of the cases. Highest sensitivity rates for 1% and 50% cutoffs were detected for higher number of cores. Underestimation of PD-L1 expression on small samples is more common than overestimation. We demonstrated that classification of PD-L1 on small biopsy samples does not represent the overall expression of PD-L1 in all non-small cell cancer carcinoma cases, although the majority of cases are 'correctly' classified. In future studies, sampling more and larger biopsies, recording the biopsy size and tumor load may permit further refinement, increasing predictive accuracy.Pathogenesis and treatment of chronic pulmonary disease