Apparent diffusion coefficient (ADC) measurement in diffusion-weighted imaging (DWI) has been reported to be a helpful biomarker for detection and characterization of lesion. In view of the importance of ADC measurement reproducibility, the aim of this study was to probe the variability of the healthy hepatic ADC values measured at 3 MR scanners from different vendors and with different field strengths, and to investigate the reproducibility of normalized ADC (nADC) value with the spleen as the reference organ. Thirty enrolled healthy volunteers received DWI with GE 1.5T, Siemens 1.5T, and Philips 3.0T magnetic resonance (MR) systems on liver and spleen (session 1) and were imaged again after 10 to 14 days using only GE 1.5T MR and Philips 3.0T MR systems (session 2). Interscan agreement and reproducibility of ADC measurements of liver and the calculated nADC values (ADCliver/ADCspleen) were statistically evaluated between 2 sessions. In session 1, ADC and nADC values of liver were evaluated for the scanner-related variability by 2-way analysis of variance and intraclass correlation coefficients (ICCs). Coefficients of variation (CVs) of ADCs and nADCs of liver were calculated for both 1.5 and 3.0-T MR system. Interscan agreement and reproducibility of ADC measurements of liver and related nADCs between 2 sessions were found to be satisfactory with ICC values of 0.773 to 0.905. In session 1, the liver nADCs obtained from different scanners were consistent (P = 0.112) without any significant difference in multiple comparison (P = 0.117 to >0.99) by using 2-way analysis of variance with post-hoc analysis of Bonferroni method, although the liver ADCs varied significantly (P < 0.001). nADCs measured by 3 scanners were in good interscanner agreements with ICCs of 0.685 to 0.776. The mean CV of nADCs of both 1.5T MR scanners (9.6%) was similar to that of 3.0T MR scanner (8.9%). ADCs measured at 3 MR scanners with different field strengths and vendors could not be compared directly. Normalization of ADCs, however, may provide better reproducibility by overcoming these potential issues.

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We study the Dirichlet eigenvalue problem of homogeneous Hörmander operators △ X = m j=1 X 2 j on a bounded connected open domain containing the origin, where X 1 , X 2 , . . . , X m are linearly independent smooth vector fields in R n satisfying Hörmander's condition and a suitable homogeneity property with respect to a family of non-isotropic dilations. Suppose that Ω is a bounded connected open domain in R n containing the origin, and its boundary ∂Ω is smooth and non-characteristic for X. Combining the subelliptic heat kernel estimates, the resolution of singularities in algebraic geometry and some refined analysis involving convex geometry, we establish the explicit asymptotic behaviourwhere λ k denotes the k-th Dirichlet eigenvalue of △ X on Ω, Q 0 is a positive rational number, and d 0 is a non-negative integer. Furthermore, we also give the optimal bounds of index Q 0 , which depends on the homogeneous dimension associated with vector fields X 1 , X 2 , . . . , X m .

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