September  2012, 11(5): 2157-2177. doi: 10.3934/cpaa.2012.11.2157

Neuronal Fiber--tracking via optimal mass transportation

1. 

Signal Processing Laboratory (LTS5), Ecole Polytechnique Federale de Lausanne (EPFL), ELD 232 - CH-1015 Lausanne, Swaziland

2. 

Department of Computer Sciences, University of Verona, Strada Le Grazie, 15 - I-37134 Verona, Italy, Italy, Italy

Received  April 2011 Revised  September 2011 Published  March 2012

Diffusion Magnetic Resonance Imaging (MRI) is used to (non-invasively) study neuronal fibers in the brain white matter. Reconstructing fiber paths from such data (tractography problem) is relevant in particular to study the connectivity between two given cerebral regions. Fiber-tracking models rely on how water molecules diffusion is represented in each MRI voxel. The Diffusion Spectrum Imaging (DSI) technique represents the diffusion as a probability density function (DDF) defined on a set of predefined directions inside each voxel. DSI is able to describe complex tissue configurations (compared e.g. with Diffusion Tensor Imaging), but ignores the actual density of fibers forming bundle trajectories among adjacent voxels, preventing any evaluation of the real physical dimension of these fiber bundles.
By considering the fiber paths between two given areas as geodesics of a suitable well-posed optimal control problem (related to optimal mass transportation) which takes into account the whole information given by the DDF, we are able to provide a quantitative criterion to estimate the connectivity between two given cerebral regions, and to recover the actual distribution of neuronal fibers between them.
Citation: A. Daducci, A. Marigonda, G. Orlandi, R. Posenato. Neuronal Fiber--tracking via optimal mass transportation. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2157-2177. doi: 10.3934/cpaa.2012.11.2157
References:
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Andrei Agrachev and Paul Lee, Optimal transportation under nonholonomic constraints,, Trans. Amer. Math. Soc., 361 (2009), 6019. doi: 10.1090/S0002-9947-09-04813-2. Google Scholar

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D. C. Alexander, P. L. Hubbard, M. G. Hall, E. A. Moore, M. Ptito, G. J. Parker and T. B. Dyrby, Orientationally invariant indices of axon diameter and density from diffusion MRI,, Neuroimage, (2010). doi: 10.1016/j.neuroimage.2010.05.043. Google Scholar

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Luigi Ambrosio, Nicola Gigli and Giuseppe Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'', Second Ed., (2008). Google Scholar

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P. J. Basser, S. Pajevic, C. Pierpaoli, J. Duda and A Aldroubi, In vivo fiber tractography using DT-MR data,, Magn. Reson. Med., 44 (2000), 625. doi: 10.1002/1522-2594(200010)44:4<625::AID- ?MRM17>3.0.CO;2-O. Google Scholar

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J.-D. Benamou, Y. Brenier and K. Guittet, The Monge-Kantorovitch mass transfer and its computational fluid mechanics formulation,, International Journal for Numerical Methods in Fluids, 40 (2002), 21. doi: 10.1002/fld.264. Google Scholar

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Piermarco Cannarsa and Carlo Sinestrari, Convexity properties of the minimum time function,, Calculus of Variations and Partial Differential Equations, 3 (1995), 273. doi: 10.1007/BF01189393. Google Scholar

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J. Coremans, R. Luypaert, F. Verhelle, T. Stadnik and M Osteaux, A method for myelin fiber orientation mapping using diffusion-weighted MR images,, Magn Reson Imaging, 12 (1994), 443. doi: 10.1016/0730-725X(94)92538-0. Google Scholar

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J. Dauguet, S. Peled, V. Berezovskii, T. Delzescaux, S. K. Warfield, R. Born and C. F. Westin, Comparison of fiber tracts derived from in-vivo DTI tractography with 3D histological neural tract tracer reconstruction on a macaque brain,, Neuroimage, 37 (2007), 530. doi: 10.1016/j.neuroimage.2007.04.067. Google Scholar

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[21]

Mikhail Feldman and Robert J. McCann, Monge's transport problem on a Riemannian manifold,, Transactions of the American Mathematical Society, 354 (2002), 1667. doi: 10.1090/S0002-9947-01-02930-0. Google Scholar

[22]

Albert Fathi and Antonio Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians,, Calculus of Variations and Partial Differential Equations, 22 (2005), 185. doi: 10.1007/s00526-004-0271-z. Google Scholar

[23]

P. Fillard and C. Poupon and J. F. Mangin, A novel global tractography algorithm based on an adaptive spin glass model,, Med. Image Comput. Comput. Assist. Interv., 12 (2009), 927. doi: 10.1007/978-3-642-04268-3_114. Google Scholar

[24]

Ola Friman and Gunnar Farneback and Carl-Fredrik Westin, A Bayesian approach for stochastic white matter tractography,, IEEE transactions on medical imaging, 25 (2006), 965. doi: 10.1109/TMI.2006.877093. Google Scholar

[25]

Wilfrid Gangbo and Robert J. McCann, The geometry of optimal transportation,, Acta Mathematica, 177 (1996), 113. doi: 10.1007/BF02392620. Google Scholar

[26]

P. Hagmann, L. Jonasson, P. Maeder, J. Thiran, V. Wedeen and R. Meuli, Understanding diffusion MR imaging techniques: from scalar diffusion-weighted imaging to diffusion tensor imaging and beyond,, Radiographics, 26 (2006). doi: 10.1148/rg.26si065510. Google Scholar

[27]

P. Hagmann, J. P. Thiran, L. Jonasson, P. Vandergheynst, S. Clarke, P. Maeder and R Meuli, DTI mapping of human brain connectivity: statistical fibre tracking and virtual dissection,, Neuroimage, 19 (2003), 545. doi: 10.1016/S1053-8119(03)00142-3. Google Scholar

[28]

Y. Iturria-Medina, E. J. Canales-Rodríguez, L. Melie-García, P. A. Valdés-Hernández, E. Martínez-Montes, Y. Alemán-Gómez and J M Sánchez-Bornot, Characterizing brain anatomical connections using diffusion weighted MRI and graph theory,, Neuroimage, 36 (2007), 645. doi: 10.1016/j.neuroimage.2007.02.012. Google Scholar

[29]

S. Jbabdi, M. W. Woolrich, J. L. Andersson and T. E. Behrens, A Bayesian framework for global tractography,, Neuroimage, 37 (2007), 116. doi: 10.1016/j.neuroimage.2007.04.039. Google Scholar

[30]

B. W. Kreher, I. Mader and V. G. Kiselev, Gibbs tracking: a novel approach for the reconstruction of neuronal pathways,, Magn. Reson. Med., 60 (2008), 953. doi: 10.1002/mrm.21749. Google Scholar

[31]

M. Lazar and A. L. Alexander, An error analysis of white matter tractography methods: synthetic diffusion tensor field simulations,, Neuroimage, 20 (2003), 1140. doi: 10.1016/S1053-8119(03)00277-5. Google Scholar

[32]

M. Lazar, D. M. Weinstein, J. S. Tsuruda, K. M. Hasan, K. Arfanakis, M. E. Meyerand, B. Badie, H. A. Rowley, V. Haughton, A. Field and A. L. Alexander, White matter tractography using diffusion tensor deflection,, Hum. Brain Mapp., 18 (2003), 306. doi: 10.1002/hbm.10102. Google Scholar

[33]

C. P. Lin, V. J. Wedeen, J. H. Chen, C. Yao and W. Y. Tseng, Validation of diffusion spectrum magnetic resonance imaging with manganese-enhanced rat optic tracts and ex vivo phantoms,, Neuroimage, 19 (2003), 482. doi: 10.1016/S1053-8119(03)00154-X. Google Scholar

[34]

Y. Lu, A. Aldroubi, J. C. Gore, A. W. Anderson and Z. Ding, Improved fiber tractography with Bayesian tensor regularization,, Neuroimage, 31 (2006), 1061. doi: 10.1016/j.neuroimage.2006.01.043. Google Scholar

[35]

G. J. Parker, H. A. Haroon and C. A. Wheeler-Kingshott, A framework for a streamline-based probabilistic index of connectivity (PICo) using a structural interpretation of MRI diffusion measurements,, J. Magn. Reson. Imaging, 18 (2003), 242. doi: 10.1002/jmri.10350. Google Scholar

[36]

Eric Pichon, Carl-Fredrik Westin and Allen Tannenbaum, A Hamilton-Jacobi-Bellman approach to high angular resolution diffusion tractography,, in, 3759 (2005), 180. doi: 10.1007/11566465_23. Google Scholar

[37]

R. Tyrrell Rockafellar, "Convex Analysis,'', Princeton Mathematical Series, (1970). Google Scholar

[38]

Antonio Siconolfi, Metric character of Hamilton-Jacobi equations,, Transactions of the American Mathematical Society, 355 (2003), 1987. doi: 10.1090/S0002-9947-03-03237-9. Google Scholar

[39]

Antonio Siconolfi, Errata to: "Metric character of Hamilton-Jacobi equations'' [Trans. Amer. Math. Soc. 355 (2003), no. 5, 1987-2009 (electronic),, Transactions of the American Mathematical Society, 355 (2003). doi: 10.1090/S0002-9947-03-03410-X. Google Scholar

[40]

J. D. Tournier, F. Calamante, M. D. King, D. G. Gadian and A. Connelly, Limitations and requirements of diffusion tensor fiber tracking: an assessment using simulations,, Magn. Reson. Med., 47 (2002), 701. doi: 10.1002/mrm.10116. Google Scholar

[41]

J. D. Tournier, C. H. Yeh, F. Calamante, K. H. Cho, A. Connelly and C. P. Lin, Resolving crossing fibres using constrained spherical deconvolution: validation using diffusion-weighted imaging phantom data,, Neuroimage, 42 (2008), 617. doi: 10.1016/j.neuroimage.2008.05.002. Google Scholar

[42]

Cédric Villani, "Topics in Optimal Transportation,'', Graduate Studies in Mathematics, 58 (2003). Google Scholar

[43]

Cédric Villani, "Optimal Transport,'', Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338 (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar

[44]

U. C. Wieshmann, C. A. Clark, M. R. Symms, F. Franconi, G. J. Barker and S. D. Shorvon, Reduced anisotropy of water diffusion in structural cerebral abnormalities demonstrated with diffusion tensor imaging,, Magn. Reson. Imaging, 17 (1999), 1269. doi: 10.1016/S0730-725X(99)00082-X. Google Scholar

[45]

X. Wu, Q. Xu, L. Xu, J. Zhou, A. W. Anderson and Z. Ding, Genetic white matter fiber tractography with global optimization,, J. Neurosci. Methods, 184 (2009), 375. doi: 10.1016/j.jneumeth.2009.07.032. Google Scholar

[46]

A. Zalesky, DT-MRI fiber tracking: a shortest paths approach,, IEEE Trans Med Imaging, 27 (2008), 1458. doi: 10.1109/TMI.2008.923644. Google Scholar

show all references

References:
[1]

Yves Achdou and Italo Capuzzo-Dolcetta, Mean field games: numerical methods,, SIAM Journal on Numerical Analysis, 48 (2010), 1136. doi: 10.1137/090758477. Google Scholar

[2]

Andrei Agrachev and Paul Lee, Optimal transportation under nonholonomic constraints,, Trans. Amer. Math. Soc., 361 (2009), 6019. doi: 10.1090/S0002-9947-09-04813-2. Google Scholar

[3]

D. C. Alexander, P. L. Hubbard, M. G. Hall, E. A. Moore, M. Ptito, G. J. Parker and T. B. Dyrby, Orientationally invariant indices of axon diameter and density from diffusion MRI,, Neuroimage, (2010). doi: 10.1016/j.neuroimage.2010.05.043. Google Scholar

[4]

Luigi Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields,, in, 1927 (2008), 1. doi: 10.1007/978-3-540-75914-0_1. Google Scholar

[5]

Luigi Ambrosio and Gianluca Crippa, Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields,, in, 5 (2008). doi: 10.1007/978-3-540-76781-7_1. Google Scholar

[6]

Luigi Ambrosio, Nicola Gigli and Giuseppe Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'', Second Ed., (2008). Google Scholar

[7]

Sigurd Angenent, Eric Pichon and Allen Tannenbaum, Mathematical methods in medical image processing,, American Mathematical Society. Bulletin. New Series, 43 (2006), 365. doi: 10.1090/S0273-0979-06-01104-9. Google Scholar

[8]

Y. Assaf, T. Blumenfeld-Katzir, Y. Yovel and P. J. Basser, AxCaliber: a method for measuring axon diameter distribution from diffusion MRI,, Magn. Reson. Med., 59 (2008), 1347. doi: 10.1002/mrm.21577. Google Scholar

[9]

Jean-Pierre Aubin and Hélène Frankowska, "Set-valued Analysis,'', Modern Birkh{\, (2009). Google Scholar

[10]

Martino Bardi and Italo Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,'', Birkh{\, (1997). Google Scholar

[11]

P. J. Basser, S. Pajevic, C. Pierpaoli, J. Duda and A Aldroubi, In vivo fiber tractography using DT-MR data,, Magn. Reson. Med., 44 (2000), 625. doi: 10.1002/1522-2594(200010)44:4<625::AID- ?MRM17>3.0.CO;2-O. Google Scholar

[12]

Jean-David Benamou and Yann Brenier, A numerical method for the optimal time-continuous mass transport problem and related problems,, in, 226 (1999), 1. doi: 10.1090/conm/226. Google Scholar

[13]

J.-D. Benamou, Y. Brenier and K. Guittet, The Monge-Kantorovitch mass transfer and its computational fluid mechanics formulation,, International Journal for Numerical Methods in Fluids, 40 (2002), 21. doi: 10.1002/fld.264. Google Scholar

[14]

Stefano Bianchini and Fabio Cavalletti, The Monge problem for distance cost in geodesic spaces,, to appear in, (2010). Google Scholar

[15]

Piermarco Cannarsa and Carlo Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,'', Progress in Nonlinear Differential Equations and their Applications, 58 (2004). Google Scholar

[16]

Piermarco Cannarsa and Carlo Sinestrari, Convexity properties of the minimum time function,, Calculus of Variations and Partial Differential Equations, 3 (1995), 273. doi: 10.1007/BF01189393. Google Scholar

[17]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,'', Graduate Texts in Mathematics, 178 (1998). Google Scholar

[18]

J. Coremans, R. Luypaert, F. Verhelle, T. Stadnik and M Osteaux, A method for myelin fiber orientation mapping using diffusion-weighted MR images,, Magn Reson Imaging, 12 (1994), 443. doi: 10.1016/0730-725X(94)92538-0. Google Scholar

[19]

J. Dauguet, S. Peled, V. Berezovskii, T. Delzescaux, S. K. Warfield, R. Born and C. F. Westin, Comparison of fiber tracts derived from in-vivo DTI tractography with 3D histological neural tract tracer reconstruction on a macaque brain,, Neuroimage, 37 (2007), 530. doi: 10.1016/j.neuroimage.2007.04.067. Google Scholar

[20]

T. B. Dyrby, L. V. Sogaard, G. J. Parker, D. C. Alexander, N. M. Lind, W. F. Baaré, A. Hay-Schmidt, N. Eriksen, B. Pakkenberg, O. B. Paulson and J. Jelsing, Validation of in vitro probabilistic tractography,, Neuroimage, 37 (2007), 1267. doi: 10.1016/j.neuroimage.2007.06.022. Google Scholar

[21]

Mikhail Feldman and Robert J. McCann, Monge's transport problem on a Riemannian manifold,, Transactions of the American Mathematical Society, 354 (2002), 1667. doi: 10.1090/S0002-9947-01-02930-0. Google Scholar

[22]

Albert Fathi and Antonio Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians,, Calculus of Variations and Partial Differential Equations, 22 (2005), 185. doi: 10.1007/s00526-004-0271-z. Google Scholar

[23]

P. Fillard and C. Poupon and J. F. Mangin, A novel global tractography algorithm based on an adaptive spin glass model,, Med. Image Comput. Comput. Assist. Interv., 12 (2009), 927. doi: 10.1007/978-3-642-04268-3_114. Google Scholar

[24]

Ola Friman and Gunnar Farneback and Carl-Fredrik Westin, A Bayesian approach for stochastic white matter tractography,, IEEE transactions on medical imaging, 25 (2006), 965. doi: 10.1109/TMI.2006.877093. Google Scholar

[25]

Wilfrid Gangbo and Robert J. McCann, The geometry of optimal transportation,, Acta Mathematica, 177 (1996), 113. doi: 10.1007/BF02392620. Google Scholar

[26]

P. Hagmann, L. Jonasson, P. Maeder, J. Thiran, V. Wedeen and R. Meuli, Understanding diffusion MR imaging techniques: from scalar diffusion-weighted imaging to diffusion tensor imaging and beyond,, Radiographics, 26 (2006). doi: 10.1148/rg.26si065510. Google Scholar

[27]

P. Hagmann, J. P. Thiran, L. Jonasson, P. Vandergheynst, S. Clarke, P. Maeder and R Meuli, DTI mapping of human brain connectivity: statistical fibre tracking and virtual dissection,, Neuroimage, 19 (2003), 545. doi: 10.1016/S1053-8119(03)00142-3. Google Scholar

[28]

Y. Iturria-Medina, E. J. Canales-Rodríguez, L. Melie-García, P. A. Valdés-Hernández, E. Martínez-Montes, Y. Alemán-Gómez and J M Sánchez-Bornot, Characterizing brain anatomical connections using diffusion weighted MRI and graph theory,, Neuroimage, 36 (2007), 645. doi: 10.1016/j.neuroimage.2007.02.012. Google Scholar

[29]

S. Jbabdi, M. W. Woolrich, J. L. Andersson and T. E. Behrens, A Bayesian framework for global tractography,, Neuroimage, 37 (2007), 116. doi: 10.1016/j.neuroimage.2007.04.039. Google Scholar

[30]

B. W. Kreher, I. Mader and V. G. Kiselev, Gibbs tracking: a novel approach for the reconstruction of neuronal pathways,, Magn. Reson. Med., 60 (2008), 953. doi: 10.1002/mrm.21749. Google Scholar

[31]

M. Lazar and A. L. Alexander, An error analysis of white matter tractography methods: synthetic diffusion tensor field simulations,, Neuroimage, 20 (2003), 1140. doi: 10.1016/S1053-8119(03)00277-5. Google Scholar

[32]

M. Lazar, D. M. Weinstein, J. S. Tsuruda, K. M. Hasan, K. Arfanakis, M. E. Meyerand, B. Badie, H. A. Rowley, V. Haughton, A. Field and A. L. Alexander, White matter tractography using diffusion tensor deflection,, Hum. Brain Mapp., 18 (2003), 306. doi: 10.1002/hbm.10102. Google Scholar

[33]

C. P. Lin, V. J. Wedeen, J. H. Chen, C. Yao and W. Y. Tseng, Validation of diffusion spectrum magnetic resonance imaging with manganese-enhanced rat optic tracts and ex vivo phantoms,, Neuroimage, 19 (2003), 482. doi: 10.1016/S1053-8119(03)00154-X. Google Scholar

[34]

Y. Lu, A. Aldroubi, J. C. Gore, A. W. Anderson and Z. Ding, Improved fiber tractography with Bayesian tensor regularization,, Neuroimage, 31 (2006), 1061. doi: 10.1016/j.neuroimage.2006.01.043. Google Scholar

[35]

G. J. Parker, H. A. Haroon and C. A. Wheeler-Kingshott, A framework for a streamline-based probabilistic index of connectivity (PICo) using a structural interpretation of MRI diffusion measurements,, J. Magn. Reson. Imaging, 18 (2003), 242. doi: 10.1002/jmri.10350. Google Scholar

[36]

Eric Pichon, Carl-Fredrik Westin and Allen Tannenbaum, A Hamilton-Jacobi-Bellman approach to high angular resolution diffusion tractography,, in, 3759 (2005), 180. doi: 10.1007/11566465_23. Google Scholar

[37]

R. Tyrrell Rockafellar, "Convex Analysis,'', Princeton Mathematical Series, (1970). Google Scholar

[38]

Antonio Siconolfi, Metric character of Hamilton-Jacobi equations,, Transactions of the American Mathematical Society, 355 (2003), 1987. doi: 10.1090/S0002-9947-03-03237-9. Google Scholar

[39]

Antonio Siconolfi, Errata to: "Metric character of Hamilton-Jacobi equations'' [Trans. Amer. Math. Soc. 355 (2003), no. 5, 1987-2009 (electronic),, Transactions of the American Mathematical Society, 355 (2003). doi: 10.1090/S0002-9947-03-03410-X. Google Scholar

[40]

J. D. Tournier, F. Calamante, M. D. King, D. G. Gadian and A. Connelly, Limitations and requirements of diffusion tensor fiber tracking: an assessment using simulations,, Magn. Reson. Med., 47 (2002), 701. doi: 10.1002/mrm.10116. Google Scholar

[41]

J. D. Tournier, C. H. Yeh, F. Calamante, K. H. Cho, A. Connelly and C. P. Lin, Resolving crossing fibres using constrained spherical deconvolution: validation using diffusion-weighted imaging phantom data,, Neuroimage, 42 (2008), 617. doi: 10.1016/j.neuroimage.2008.05.002. Google Scholar

[42]

Cédric Villani, "Topics in Optimal Transportation,'', Graduate Studies in Mathematics, 58 (2003). Google Scholar

[43]

Cédric Villani, "Optimal Transport,'', Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338 (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar

[44]

U. C. Wieshmann, C. A. Clark, M. R. Symms, F. Franconi, G. J. Barker and S. D. Shorvon, Reduced anisotropy of water diffusion in structural cerebral abnormalities demonstrated with diffusion tensor imaging,, Magn. Reson. Imaging, 17 (1999), 1269. doi: 10.1016/S0730-725X(99)00082-X. Google Scholar

[45]

X. Wu, Q. Xu, L. Xu, J. Zhou, A. W. Anderson and Z. Ding, Genetic white matter fiber tractography with global optimization,, J. Neurosci. Methods, 184 (2009), 375. doi: 10.1016/j.jneumeth.2009.07.032. Google Scholar

[46]

A. Zalesky, DT-MRI fiber tracking: a shortest paths approach,, IEEE Trans Med Imaging, 27 (2008), 1458. doi: 10.1109/TMI.2008.923644. Google Scholar

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