Eur Radiol (2012) 22:891–899 893 Fig. 1 Diffusion in the liver was measured in five locations (ROI 1 to 5) with a diameter of 22.3 mm. a Transverse plane of the medial liver with ROI 1 central in the right lobe (segments 5/8), ROI 2 dorso- lateral in the right lobe (segments 6/7) and ROI 3 ventral in the left representing molecular diffusion and fslow is the fraction of molecular diffusion (fslow = 1 - ffast). Equation 1 was fitted by the Nelder-Mead simplex direct search method with bound constraints (Table 1), which performs a constrained non-linear minimisation of the sum of the squared residuals [27, 28]. Finally, the conventional ADC was determined using linear regression analysis of the natural log of signal intensities S versus all seven b-values [2]: S ¼ A expð b ADCÞ where A is the intercept of the signal. Parametric maps 2 ð Þ For one subject, parametric maps of ADC and Dslow were calculated by fitting the diffusion signal of a 4 by 4 pixels area to the mono- and bi-exponential model. The area was then moved 4 pixels until the whole liver matrix (192 × 144) was covered, resulting in a parametric map of 48 × 36. Subsequently, the parametric map was rescaled to the original matrix size by linear interpolation. Influence of blood vessels To illustrate the effects of microperfusion on the mono- and bi-exponential models, an ROI was drawn in one subject and blood vessels were included in the ROI. Then the same ROI was moved slightly until the blood vessels were excluded from the ROI. In both cases ADC, Dslow, Dfast and f were calculated and the curves presented in a figure for comparison. Table 1 Fitting parameters aNumber of parameters Bi-exponential fit (p=3)a ffast (unitless) Dfast (×10−3 mm2/s) Dslow (×10−3 mm2/s) lobe (segments 2/3). b Transverse plane of the cranial liver with ROI 4 dorso-lateral in the right lobe (segment 7) c Transverse plane of the caudal liver with ROI 5 near the gallbladder (segment 5) Simulation study A systematic literature search was performed in order to compare our results with previous findings. We includ- ed multi-region DWI liver studies that determined the ADC for healthy liver tissue for at least 20 patients and published a P-value for the significance of the ADC difference among the regions. Then the results of the reviewed studies were simulated by performing mono- and bi-exponential analyses on our data using the b- values and regions mentioned in the review studies. The ADC, Dslow, Dfast, fslow and ffast resulting from the simulation were then compared with those of the reviewed studies. Statistical analysis Statistical analysis was performed using SPSS (SPSS 18, Chicago, IL, USA). All data were tested for normality using Shapiro–Wilk tests. For normally distributed data (ADC and Dslow) one-way ANOVA tests were used to compare measurements between different liver regions. For non- normally distributed data (Dfast and f) the differences between liver regions were determined using non- parametric Kruskal–Wallis tests. The central tendency and variability of the data were quantified by the median m1=2 and the median deviation s1=2 [29] where s1=2 ¼ medianðjxi m1=2jÞ For each individual fitting procedure the coefficient of correlation (goodness-of-fit) R2 was adjusted for 3 ð Þ Lower-bound Higher-bound Initial guess 1e-8 1e-5 1e-5 1 100 10 0.4 10 1