Stephan E Maier^{1,2}

A simplified approach is presented to obtain consistent tissue ADC data, irrespective of diffusion protocol, location within field of view and MR system. Examples of application in prostate diffusion scans are presented. This could greatly facilitate the establishment of suitable cut-off values to differentiate aggressive from indolent disease in prostate cancer, but as detailed in the abstract would entail several other substantial benefits that are universally applicable to diffusion imaging in tissues.

Due to the non-monoexponential signal decay in tissues [1], basic monoexponential ADC computation with different b_{max} settings yields different results (Fig. 1). There is no universally agreed upon setting for b_{max}, since it typically is adjusted for optimal lesion conspicuity and for optimal SNR, i.e, factors that are also influenced by field strength, echo time, voxel geometry and coil setup. Thus, unless b_{max} falls within a narrow range, ADC values among sites are not comparable. Another system factor, are gradient non-linearities, that bring about a diffusion encoding strength that is location dependent. Since this non-linearity is associated with a local change of b_{max}, tissue ADCs, unlike simple liquid diffusion, cannot readily be corrected.

We have fitted various models to high-b diffusion data sets of the
prostate [2]. For each model we investigated the ability of the resulting
fitting parameters to differentiate suspected tumor areas from normal
tissue and how they correlate with Gleason score. We observed that once
parameters of a model are linearly combined, all models, including the
mono-exponential model exhibited the same ability for differentiation.
Moreover, for the kurtosis model with $$$S = S_0 · \exp(−bADC_K +
b^2ADC_K^2K/6)$$$ [3], ADC_{K} or the kurtosis parameter *K* alone permitted equally good differentiation, which is explained by the high covariance of the parameters (see Figs. 2 and 3).

Jensen et al [4] have analyzed the behavior of the kurtosis model for different diffusion scenarios. In particular, they analyzed a one-dimensional model with barriers for short diffusion times. Rearrangement of the formulas given in [4] shows that $$$K = (1 − ADC_K/ADC_f ) · 6/5$$$, where ADC_{f} is the diffusion coefficient of free diffusing water. Thus kurtosis *K* depends linearly on ADC_{K}, and reaches 6/5 for completely restricted diffusion, where ADC_{K}=0 and 0 for ADC_{K}=ADC_{f} , i.e., 3 μm^{2}/ms at 37 C.

The fact that ADC_{K} and *K* are highly correlated, opens up the possibility to only measure one value and then compute the other value with a known and preferably standardized correlation function. It can be assumed that this correlation holds as long as diffusion times are not too long and complete exchange occurs, in which case *K* would tend towards zero, a scenario not supported by our data. It also holds as long as that we do not study tissues with micro-anisotropy [5], such as gray matter (see Fig. 3). But the assumption of correlation is probably valid for all abdominal tissues, most tumor tissues, and even white matter.

The proposed approach was tested in data obtained during multi-parametric prostate MR exams at 3 Tesla. An endo-rectal coil was used to obtain echo-planar diffusion-weighted images for 15 evenly spaced b-factors between 0 and 3500 s/mm^{2}. A quadratic correlation function between *K* and ADC_{K} was obtained with existing ROI data [2] through non-linear fitting and subsequently applied to subsets of the multi-b data.

1) Mulkern RV, Barnes AS, Haker SJ, Hung YP, Rybicki FJ, Maier SE, Tempany CM. Biexponential characterization of prostate tissue water diffusion decay curves over an extended b-factor range. Magn Reson Imaging. 2006;24(5):563–8.

2) Langkilde F, Kobus T, Fedorov A, Dunne R, Tempany C, Mulkern RV, Maier SE.Evaluation of fitting models for prostate tissue characterization using extended-range b-factor diffusion-weighted imaging. Magn Reson Med. 2018;79(4):2346–2358.

3) Jensen JH, Helpern JA, Ramani A, Lu H, Kaczynski K. Diffusional kurtosis imaging: the quantification of non-Gaussian water diffusion by means of magnetic resonance imaging. Magn. Reson. Med. 2005;53: 1432–1440.

4) Jensen JH and Helpern JA. MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR Biomed. 2010; 23: 698–710.

5) Szczepankiewicz F, Lasic S, van Westen D, Sundgren PC, Englund E, Westin CF , Stalberg F, Latt J, Topgaard D, and Nilsson M. Quantification of microscopic diffusion anisotropy disentangles effects of orientation dispersion from microstructure: applications in healthy volunteers and in brain tumors. Neuroimage, 104:241–52, 2015.

6) Kitajima K, Kaji Y, Kuroda K, and Sugimura K. High b-value diffusion-weighted imaging in normal and malignant peripheral zone tissue of the prostate: effect of signal-to-noise ratio. Magn Reson Med Sci, 7(2):93–9, 2008.

7) Kim CK, Park BK, and Kim B. High-b-value diffusion-weighted imaging at 3 T to detect prostate cancer: comparisons between b values of 1,000 and 2,000 s/mm^{2}. Am J Roentgenol, 194(1):W33–7, 2010.

ADC maps of prostate diffusion scans obtained with an endo-rectal coil. The ADCs computed from different diffusion scans with non-identical b_{max} look different. This precludes a meaningful comparison among protocols and systems. Also, recommendations for suitable cut-off values to differentiate aggressive from indolent disease in prostate cancer cannot be established.

Kurtosis *K* vs kurtosis diffusion coefficient ADC_{K}, as plotted here for 40 ROI measurements from prostate transition zones (TZ) of 27 patients, are highly correlated [2].

Scatter plot *K* vs ADC_{K} of brain pixels,
prostate pixels and all 152 prostate pheripheral and transition zone ROIs measured in study [2]. The
brain data looks similar to data shown in Jensen’s original kurtosis
paper [3]). The area with low K and low ADC_{K} (separated by
gray line), which does not contain any prostate data, was verified to stem only from gray
matter areas, which are characterized by micro-anisotropy [5].

Fits with predetermined correlation between ADC_{K} and *K*. Subsets of data were used for fitting with the proposed approach, i.e., a b_{max}=700 s/mm^{2} set with 4 b-factors, a b_{max}=1633 s/mm^{2} set with 8 b-factors and a b_{max}=2567 s/mm^{2} set with 12 b-factors. Resulting ADC_{K}
maps look virtually the same (compare with Fig 1). Diffusion-weighted
images (DWI) measured and extrapolated from b_{max}=700 at b=1633 and 2567
s/mm^{2} look alike, whereas diffusion-weighted images that result from basic
mono-exponential fits look very different [6,7].