Parya Momayyez
McGill University

### Introduction!

The focus of my PhD research has been on the problem of reconstructing anatomical connectivity in the brain using the information provided by diffusion-MRI measurements on the local orientation of white matter fibre tracts. We started by extending a 2D model of curve completion to 3D to provide a probabilistic representation of the most likely 3D curves connecting two end points, while being tangent to their associated orientation vectors. The algorithm is based on a 3D directional random walk, under which particles tend to travel in their present direction of heading, but with a slight change of orientation at each time step. The change in orientation is controlled by two consecutive orientation deviations, which affect the spherical components of the particleâ€™s state. Using this directional random walk as a prior model of completion, 3D stochastic completion fields are obtained by the product of a source field and a sink field. The former represents the probability distribution of particles which start from the source while following a directional random walk and the latter stands for the probability distribution of particles which reach a sink state starting from every position and orientation in the underlying 5D space. We demonstrated that the maximum likelihood curves obtained under such a model represent those curves which minimize a weighted sum of curvature squared plus torsion squared plus length.

We then moved on to formulate a computational model in order to estimate 3D stochastic completion fields in a local, parallel and efficient network. To do so, the Fokker-Planck equation, which is a partial differential equation describing the evolution of the probability density function of particles following a directional random walk, was used. A rotation invariant estimation is achieved by using spherical harmonics to compute the probability function in the spherical domain, while spatial diffusion terms were also added to make the advection error in the x, y and z directions isotropic. We also suggested an adaptation of the stochastic completion field algorithm to problems where dense orientation data arises, as is the case for diffusion-MRI data. The adaptation allows for improved utilization of the local orientation information available at each voxel. We then formulated a more general model for a directional random walk where diffusion is considered on the entire statespace. Additionally, a variation of the model is suggested where the 3D orientation change is drawn from a single distribution, i.e., a 3D Brownian distribution.

Finally, the application of the proposed model for diffusion-MRI based connectivity measurements was explored. It is explained how different model parameters, such as angular drift and diffusion terms and the lifetime scale can be automatically set based on the local diffusion data. We based the reconstruction of white matter connectivity patterns on the measured local fibre orientations provided by diffusion-MRI data and have exploited their consistency across voxels in the selection of completion fields parameters. But the parameter selection process can be modified to incorporate other relevant measures, should they be available. The possibility of incorporating noise measures into the computed connectivity measures was also discussed. Experiments on phantom and synthetic data demonstrated, both quantitatively and qualitatively, the potential of our novel algorithm both for reconstructing diffusion-MRI based anatomical connectivity measures and for providing a probabilistic view of the underlying fibre trajectories. We also investigated the performance of the stochastic completion field algorithm by looking at the white matter connections in the human brain between cortical regions involved in language processing.

### Publications!

#### Diffusion MRI Related

Parya MomayyezSiahkal and Kaleem Siddiqi. "3D Stochastic Completion Fields for Mapping Connectivity in Diffusion MRI". IEEE Transactions on Pattern Analysis and Machine Intelligence, to appear as a regular paper, 2012.

P. MomayyezSiahkal and K. Siddiqi. "Rotation Invariant Completion Fields for Mapping Diffusion MRI Connectivity". Proceedings of Information Processing in Medical Imaging, (IPMI'11), 2011.

J. Campbell, P. MamayyezSiahkal, P. Savadjiev, I. R. Leppert, K. Siddiqi, and G. B. Pike. "A new comprehensive framework for probabilistic tractography of fanning fibres". Proceedings of the International Society for Magnetic Resonance in Medicine (ISMRM'12), 2011.

P. Momayyez and K. Siddiqi. "Probabilistic Anatomical Connectivity Using Completion Fields". International Conference On Medical Image Computing and Computer Assisted Intervention (MICCAI'10), 2010.

P. Momayyez and K. Siddiqi. "Probabilistic connectivity in fibre tractography". Proceedings of the International Society for Magnetic Resonance in Medicine (ISMRM'10), 2010.

Parya Momayyez, Kaleem Siddiqi. "3D Stochastic Completion Fields for Fiber Tractography". IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA'09), 2009.