9(S)-HODE br ARTICLE IN PRESS br D CEUS
ARTICLE IN PRESS
3-D CEUS for Prediction of Prostate Cancer R. R. WILDEBOER et al. 3
were sufficient to resolve clinically relevant contrast-agent kinetics (Schalk et al. 2015b). To prevent noise and motion artefacts from obscuring the recordings, we performed signal spatiotemporal subspace selection through singular value decomposition (Demene et al. 2015) after spatial filtering (performed block-wise, isolating the first four signal compo-nents) and spatial downsampling, as described by Schalk et al. (2015b) for the model-fit and system identification approaches.
The 1-D formulation of the convective-dispersion eqn
(1) gives rise to the LDRW model (Mischi 2003; Mischi et al. 2003; Strouthos et al. 2010), that is, the shape of TICs was considered to be the result of a UCA dispersion pro-cess in a flowing carrier fluid (i.e., the blood flow). Assum-ing that the conversion from the video quantization levels (Q) to the actual UCA concentration was known and given by a logarithmic 9(S)-HODE based on the dynamic range
In this formulation, t0 is the theoretical injection time, a is the area under the TIC, k is a dispersion-related parameter and m is the mean transit time. These parame-ters were estimated by fitting the actual TIC by this model through a mean squared error minimization over the time interval [T0, Tend] fa^; ^k; m^g ¼ mina;k;m Tend QLDRW ðtÞ QðtÞ 2 !:ð3Þ
The minimization procedure initializes with the naı¨ve (i.e., recirculation-free) maximum-likelihood estimates of m and k (Kuenen et al. 2014); and a^ = 10 max [Q(t)] was empirically chosen as a good initial guess for the area under the TIC curve. To avoid arte-facts caused by recirculation, the fitting procedure was subsequently carried out only from 8 s after injection to 12 s after reaching the peak intensity (PI) in the five-element moving-average filtered TIC.
In addition to the LDRW variables, the spatiotem-poral correlation (r), spectral coherence (r) or mutual information (MI) between the voxel of interest and its neighbors in a shell-shaped kernel were considered to be a measure of convective dominance over the dispersion. The shell-shaped kernel included all pixels between a minimum and maximum distance from the center pixel, defined by the imaging system resolution and desired resolution of the parametric analysis, respectively. For the 3-D implementation of these analyses, we referred to the previously published works on the subject (Schalk et al. 2015a, 2015b, 2018). Taking into account the reso-lution of the imaging system employed in this study, the analysis was carried out with an inner kernel radius of 1.0 mm and an outer kernel radius of 2.5 mm.
System identification analysis
The transition of concentration evolution from voxel to voxel could also be considered governed by a convective-dispersive system. As such, eqn (1) in 1-D gave rise to the following Green’s function representing the system’s spatiotemporal impulse response mapping the TIC at the center voxel to that of another voxel, n, at distance Ln (Mischi 2003; van Sloun et al. 2017b) w nð j n ; v; D Þ ¼ p4pDt
provided that v and D are locally constant and homoge-neous. H(t) is the Heaviside step function.
Based on the TICs measured at the center voxel (c0) and voxel n(cn), the local values of v and D could be esti-mated by system identification. In contrast to the Wiener filter maximum-likelihood implementation in 2-D, we used a direct model-based least-squares minimization procedure (van Sloun et al. 2017b),