Fitting of two-tensor models without ad hoc assumptions to detect crossing fibers using clinical DWI data Klaus Hahn a, ⁎, Sergei Prigarin b , Khader M. Hasan c a Institute of Biomathematics and Biometry, Helmholtz Center Munich HMGU, German Research Center for Environmental Health, Ingolstaedter Landstrasse 1, D-85764 Neuherberg, Germany b Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of Russian Academy of Sciences, pr. Lavrentieva 6, Novosibirsk, 630090, Russia c Department of Diagnostic and Interventional Imaging, University of Texas Health Science Center at Houston, Houston, TX 77030, USA abstract article info Article history: Received 8 May 2012 Revised 25 September 2012 Accepted 30 October 2012 Keywords: Fiber crossing Full two-tensor model Ad hoc parameters Clinical DWI data DTI Icosahedral Analysis of crossing fibers is a challenging topic in recent diffusion-weighted imaging (DWI). Resolving crossing fibers is expected to bring major changes to present tractography results based on the standard tensor model. Model free approaches, like Q-ball or diffusion spectrum imaging, as well as multi-tensor models are used to unfold the different diffusion directions mixed in a voxel of DWI data. Due to its seeming simplicity, the two-tensor model (TTM) is applied frequently to provide two positive-definite tensors and the relative population fraction modeling two crossing fiber branches. However, problems with uniqueness and noise instability are apparent. To stabilize the fit, several of the 13 physical parameters are fixed ad hoc, before fitting the model to the data. Our analysis of the TTM aims at fitting procedures where ad hoc parameters are avoided. Revealing sources of instability, we show that the model's inherent ambiguity can be reduced to one scalar parameter which only influences the fraction and the eigenvalues of the TTM, whereas the diffusion directions are not affected. Based on this, two fitting strategies are proposed: the parsimonious strategy detects the main diffusion directions without extra parameter fixation, to determine the eigenvalues and the population fraction an empirically motivated condition must be added. The expensive strategy determines all 13 physical parameters of the TTM by a fit to DWIs alone; no additional assumption is necessary. Ill-posedness of the model in case of noisy data is cured by denoising of the data and by L-curve regularization combined with global minimization performing a least-squares fit of the full model. By model simulations and real data applications, we demonstrate the feasibility of our fitting strategies and achieve convincing results. Using clinically affordable diffusion acquisition paradigms (encoding numbers: 21, 2*15, 2*21) and b values (b =500–1500 s/mm 2 ), this methodology can place the TTM parameters involved in crossing fibers on a more empirical basis than fitting procedures with technical assumptions. © 2013 Elsevier Inc. All rights reserved. 1. Introduction The two-tensor model (TTM) and its extensions to more diffusion compartments offer a simple and intuitive physical model to explore the presence of multiple fiber orientations in a voxel. In diffusion tensor imaging (DTI), the TTM is frequently assigned to slow and fast diffusion components [1,2]. In another context, the TTM models crossing fibers in clinically affordable data sets [3–8]. In general, the crossing of compact white matter fibers cannot be separated by the Gaussian single-tensor model [9]. Model free approaches such as Q-ball imaging [10], spherical deconvolution [11] or diffusion spectrum imaging [12] are also used to obtain crossing fibers by the estimation of fiber orientation distributions. These methods, however, require more scan time than TTMs, as more gradient orientations and more b values at lower signal-to-noise ratios are acquired. An evaluation of fiber crossing using several high angular resolution diffusion imaging methods (HARDI) with 60 gradients, including Q-ball and spherical deconvolution approaches under clinical conditions, was recently presented by Ramirez et al. [13]. Due to its physical relevance and its simple generalization to more compartments in a voxel, the TTM has attracted the attention of several groups. The main problem to overcome was the instability of biexponential model fitting [14]. Tuch et al. [3] tried to fit the TTM using 126 gradients. They constrained the TTM by ad hoc fixing the eigenvalues. Peled et al. [4] and Stamatios et al. [5] also simplified the TTM for fitting. Peled et al. [4] reduced the 13 physical parameters of the full model to 4 free parameters, fixing 9 unknowns by using information from the single-tensor model. Stamatios et al. [5] used Peled's approach and introduced in addition a discrete basis for the two tensors and simplified the determination of the population Magnetic Resonance Imaging 31 (2013) 585–595 ⁎ Corresponding author. Tel.: +49 89 3187 4483; fax: +49 98 3187 3369. E-mail address: hahn@helmholtz-muenchen.de (K. Hahn). 0730-725X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.mri.2012.10.016 Contents lists available at SciVerse ScienceDirect Magnetic Resonance Imaging journal homepage: www.mrijournal.com