Mathematical Framework for Simulating Diffusion Tensor MR Neural Fiber Bundles A. Leemans, 1 * J. Sijbers, 1 M. Verhoye, 1,2 A. Van der Linden, 2 and D. Van Dyck 1 White matter (WM) fiber tractography (i.e., the reconstruction of the 3D architecture of WM fiber pathways) is known to be an important application of diffusion tensor magnetic resonance imaging (DT-MRI). For the quantitative evaluation of several fiber-tracking properties, such as accuracy, noise sensitivity, and robustness, synthetic ground-truth DT-MRI data are re- quired. Moreover, an accurate simulated phantom is also re- quired for optimization of the user-defined tractography param- eters, and objective comparisons between fiber-tracking algo- rithms. Therefore, in this study a mathematical framework for simulating DT-MRI data, based on the physical properties of WM fiber bundles, is presented. We obtained a model of a WM fiber bundle by parameterizing the various features that char- acterize this bundle. We then evaluated three different synthetic DT-MRI models using experimental data in order to test the proposed methodology, and to determine the optimum model and parameter settings for constructing a realistic simulated DT-MRI phantom. Several examples of how the mathematical framework can be applied to compare fiber-tracking algorithms are presented. Magn Reson Med 53:944 –953, 2005. © 2005 Wiley-Liss, Inc. Key words: diffusion tensor MRI; anisotropy; fiber tracking; simulated phantom; synthetic data; model evaluation Diffusion tensor magnetic resonance imaging (DT-MRI) is currently the only method available to obtain quantitative information about the three-dimensional (3D) anisotropic diffusion of water molecules in biological tissue (1,2). This DT anisotropy reflects the presence of spatially oriented microstructures (e.g., neural fibers in the central nervous system), where the mobility of the diffusing particles is mainly determined by the fiber pathway (3). On the basis of this intrinsic property, which assumes that the orienta- tion of the DT field matches the orientation of the corre- sponding underlying fiber system, DT-MRI has been ap- plied in several studies to infer microstructural character- istics and obtain valuable diagnostic information regarding various neuropathological conditions. Excellent reviews on white matter (WM) and neuropsychiatric dis- eases can be found in Refs. 4 and 5. It is known that ambiguous results are obtained when DT-MRI is used to study regions in which WM fibers cross or multiple fibers merge (e.g., Ref. 6). In such regions, the second-rank DT model is incapable of describing multiple fiber orientations within an individual voxel. To overcome this problem, a number of new techniques have been pro- posed, such as high-angular-resolution diffusion-weighted imaging (HARDI) (7), Q-ball imaging (QBI) (8), diffusion spectrum imaging (DSI) (9), persistent angular structure (PAS) reconstruction (10), and generalized DT imaging (GDTI) (11). Although these recently developed tech- niques can provide more accurate and unambiguous re- sults, for simplicity the focus in this paper is confined to classic DT-MRI. An important application of DT-MRI is the reconstruc- tion of the 3D WM fiber network, which is referred to as DT tracking (DTT) or fiber tractography. This technique is based on the assumption that it can accurately retrieve the spatial information of the underlying fiber network, using the available diffusion information of the corresponding tensor field. DTT provides exciting new opportunities to study the central nervous system (CNS) anatomy, and has generated much enthusiasm, resulting in the development of a large number of fiber-tracking algorithms (12–26). Although qualitative results may be very valuable, the lack of a gold standard still precludes an objective quantitative evaluation of these fiber-tracking algorithms with respect to precision, accuracy, reproducibility, etc. Although his- tology has been used to identify major WM fiber bundles, and can provide complementary anatomical information for DT-MRI (27–29), technical difficulties related to tissue preparation impede a quantitative validation of the 3D WM fiber tract reconstruction. To address this lack of a gold standard, an accurate simulated DT-MRI phantom is necessary to evaluate the numerous criteria that characterize a fiber-tracking algo- rithm. Only with such a phantom can a comparison be- tween different fiber-tracking algorithms yield decisive an- swers regarding accuracy, precision, robustness, reproduc- ibility, etc. In addition, with such a phantom one could study the effect of DT-MRI data processing prior to DT and fiber tract computing (e.g., image co-registration, noise fil- tering, and correction of motion artifacts) quantitatively. For example, fiber tracking requires geometrically regis- tered diffusion-weighted (DW) images. With the use of a simulated DT-MRI phantom, the sensitivity of fiber-track- ing algorithms with respect to the misaligned DW images can be studied. Although the need for an accurate simulated DT-MRI phantom has been emphasized in the literature (e.g., Refs. 13 and 30), only a few tractography-related articles have described a technique for computing a simulated DT-MRI phantom. In Ref. 23, a continuous diffusion vector field is used to describe the tracts, omitting information that is contained in the remaining degrees of freedom (DOF) of 1 Vision Laboratory, Department of Physics, University of Antwerp, Antwerp, Belgium. 2 Bio-Imaging Laboratory, Department of Biomedical Sciences, University of Antwerp, Antwerp, Belgium. Grant sponsors: Institute for the Promotion of Innovation Through Science and Technology; Fund for Scientific Research. *Correspondence to: Alexander Leemans, Vision Laboratory, Dept. of Phys- ics, University of Antwerp (CMI), 171 Groenenborgerlaan, U314, B-2020 Ant- werpen, Belgium. E-mail: alexander.leemans@ua.ac.be Received 15 July 2004; revised 2 November 2004; accepted 5 November 2004. DOI 10.1002/mrm.20418 Published online in Wiley InterScience (www.interscience.wiley.com). Magnetic Resonance in Medicine 53:944 –953 (2005) © 2005 Wiley-Liss, Inc. 944