Fitting of two-tensor models without ad hoc assumptions to detect crossing fibers using clinical DWI data.

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.

[1]  Khader M Hasan,et al.  Computation of the fractional anisotropy and mean diffusivity maps without tensor decoding and diagonalization: Theoretical analysis and validation , 2003, Magnetic resonance in medicine.

[2]  Alan Connelly,et al.  Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution , 2004, NeuroImage.

[3]  Jerry L. Prince,et al.  Resolution of crossing fibers with constrained compressed sensing using diffusion tensor MRI , 2012, NeuroImage.

[4]  Arnold Neumaier,et al.  Global Optimization by Multilevel Coordinate Search , 1999, J. Glob. Optim..

[5]  Jeremy D. Schmahmann,et al.  Diffusion spectrum magnetic resonance imaging (DSI) tractography of crossing fibers , 2008, NeuroImage.

[6]  Richard Frayne,et al.  Effect of b value on contrast during diffusion‐weighted magnetic resonance imaging assessment of acute ischemic stroke , 2002, Journal of magnetic resonance imaging : JMRI.

[7]  Saša Singer,et al.  Efficient Implementation of the Nelder–Mead Search Algorithm , 2004 .

[8]  Matt Hall,et al.  Resolving axon fiber crossings at clinical b-values: an evaluation study. , 2011, Medical physics.

[9]  Stephan E Maier,et al.  Biexponential analysis of diffusion-related signal decay in normal human cortical and deep gray matter. , 2008, Magnetic resonance imaging.

[10]  Khader M Hasan,et al.  A framework for quality control and parameter optimization in diffusion tensor imaging: theoretical analysis and validation. , 2007, Magnetic resonance imaging.

[11]  Dorothee P Auer,et al.  A regularized two‐tensor model fit to low angular resolution diffusion images using basis directions , 2008, Journal of magnetic resonance imaging : JMRI.

[12]  Carl-Fredrik Westin,et al.  Geometrically constrained two-tensor model for crossing tracts in DWI. , 2006, Magnetic resonance imaging.

[13]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[14]  D. Tuch.  Q‐ball imaging , 2004, Magnetic resonance in medicine.

[15]  O. Abe,et al.  Quantitative diffusion tensor analysis using multiple tensor ellipsoids model and tensor field interpolation at fiber crossing. , 2008, Academic radiology.

[16]  Klaus Hahn,et al.  A review of diffusion tensor magnetic resonance imaging computational methods and software tools , 2011, Comput. Biol. Medicine.

[17]  N. Makris,et al.  High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity , 2002, Magnetic resonance in medicine.

[18]  S. Haker,et al.  On high b diffusion imaging in the human brain: ruminations and experimental insights. , 2009, Magnetic resonance imaging.

[19]  Per Christian Hansen,et al.  REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems , 1994, Numerical Algorithms.

[20]  George T. Gilber.  Positive definite matrices and Sylvester's criterion , 1991 .

[21]  N. D. de Souza,et al.  Diffusion‐weighted imaging of the prostate and rectal wall: comparison of biexponential and monoexponential modelled diffusion and associated perfusion coefficients , 2009, NMR in biomedicine.

[22]  Mark A. Lukas,et al.  Comparing parameter choice methods for regularization of ill-posed problems , 2011, Math. Comput. Simul..

[23]  Arash Kamali,et al.  Mapping the human brain white matter tracts relative to cortical and deep gray matter using diffusion tensor imaging at high spatial resolution. , 2009, Magnetic resonance imaging.

[24]  Nadim Joni Shah,et al.  Non-Gaussian diffusion in human brain tissue at high b-factors as examined by a combined diffusion kurtosis and biexponential diffusion tensor analysis , 2011, NeuroImage.

[25]  James Hardy Wilkinson,et al.  The Least Squares Problem and Pseudo-Inverses , 1970, Comput. J..