Synopsis

The following work presents a novel, generally applicable framework (MODERN) which takes, as input, statistics from histology and automatically generates biologically plausible numerical phantoms whose morphological features are optimized to match the input statistics. As a proof of concept, MODERN is used to generate three-dimensional geometrical meshes representing a bundle configuration of axons which can be immediately integrated into Monte Carlo simulations for diffusion MRI acquisitions. The obtained features of the optimised phantoms are shown to match those from the input values. The statistics used were obtained from axonal segmentations from synchrotron imaging.

Introduction

Methods

Samples from the perfusion fixed splenium of a 32-month-old female Vervet monkey brain were stained with 0.5% osmium tetroxide(OsO4) to give contrast to the myelin. These were embedded in EPON and imaged with x-ray phase contrast tomography(PCT) at beamline ID16A of the European Synchrotron Research Facility(ESRF) in Grenoble, France. The PCI experiments produced 75nm isotropic resolution 3D-volumes, with a field-of-view(FOV) of 150µm in all dimensions. Fifty-four axons were semi-manually segmented using ITK-Snap[6], and the axonal centerlines were generated using a layered surface segmentation method developed at the DTU and the DRCMR. In MODERN, we utilize three statistics commonly used in characterization of vascularities(which can similarly be applied to axons): the tortuosity$$$(tt)$$$ and the local/global mean angular deviations$$$(\theta^l$$$and $$$\theta^g$$$resp.). Each metric was computed using the centerlines of each segmented axon. The metrics are defined as follows:$$tt=\sum_{i=0}^{\#S}\frac{|S_{\#S}-S_1|}{|S_i-S_{i-1}|}$$ $$\theta^I=\sum_{i=2}^{\#S-1}\frac{ang((S_{i+}-S_i ), (S_i -S_{i-1}))}{\#S}$$ $$\theta^g=\sum_{i=2}^{\#S}\frac{ang((S_{\#S}-S_1), (S_i-S_{i-1}))}{\#S}$$

Where$$$ang(v,w)$$$ computes the angle between two vectors,$$$\#S$$$ is the number of segments, and$$$S_i$$$ denotes the i-th point of a given axon skeleton. MODERN steps: The workflow of MODERN requires four steps as is shown in Figure1.The input parameters are the desired values for the ICVF, the number of cylinders to generate, the tortuosity, and the mean local/global deviations. 1)Gamma distributed radii fitting. Taking as input the number of cylinders, the parameters of a gamma distribution ($$$\alpha$$$ $$$\beta$$$), and the desired intra-cellular volume fraction(ICVF), a set of cylinders is sampled and positioned inside a circular boundary to satisfy the desired ICVF until a user set stopping criterion is reached(Figure1-a). 2)Initial bundle configuration. An initial configuration of regular, undulating trajectories is chosen such that the obtained tortuosity is close to the objective value (Figure1-b). 3)DE Optimization. The full set of trajectories is then optimized using a multi-objective cost function defined as follows:$$E({\cup}S)=\lambda_1(tt'-tt({\cup}S))^2+\lambda_2(\theta^{l'}-\theta^{l}({\cup}S))^2+\lambda_3(\theta^{g'}-\theta^{g}({\cup}S))^2+\lambda_4{\tau}({\cup}S,lr)$$

Where $$${\cup}S$$$ is the full set of generated trajectories,$$$tt'$$$,$$$\theta^{l'}$$$,$$$\theta^{g'}$$$ are the desired (input) values to achieve, and$$$tt(\cup S)$$$,$$$\theta^{l}({\cup}S),$$$,$$$\theta^{g'}({\cup}S)$$$ are the mean values of the descriptors computed from the full set of trajectories. Finally, $$$\tau({\cup}S,lr)$$$ is a penalization term that checks that no axon control point is outside a limit radius ($$$lr$$$) defined by the initial configuration and that there are no intersections between trajectories. The objective function is then optimized using a bespoke implementation of differential evolution which saves any set of parameters that fulfil the no overlapping condition$$$(\tau=0)$$$ and is close to the objective value by a given threshold(Figure1-c). 4)Post-processing and added perturbations. The resulting trajectories are meshed and decimated. Optionally, local perturbations can be added to each strand to add local complexity along the trajectory, in contrast with the usual cylindrical representation of the axons(Figure1-d).

Results and Discussion

Conclusion

Acknowledgements

Capital Region Research Foundation (grant number: A5657)

Thomas Yu is supported by the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie project TRABIT (agreement No 765148).

References

1. Hall M, Alexander DC. Convergence and parameter choice for Monte-Carlo simulations of diffusion MRI data. (2009).

2. Novikov, Dmitry S. and Fieremans, Els and Jensen, Jens H. and Helpern, Joseph A. Random walks with barriers. Nature Publishing Group. (2011)

3. Nedjati-Gilani et. al. Machine learning based compartment models with permeability for white matter (2017)

4. Close, Thomas G. and Tournier, Jacques Donald and Calamante, Fernando and Johnston, Leigh A. and Mareels, Iven and Connelly, Alan. A software tool to generate simulated white matter structures for the assessment of fibre-tracking algorithm. (2009)

5. Ginsburger, Kevin and Poupon, Fabrice and Beaujoin, Justine and Estournet, Delphine and Matuschke, Felix and Mangin, Jean-Francois and Axer, Markus and Poupon, Cyril. Improving the Realism of White Matter Numerical Phantoms: A Step toward a Better Understanding of the Influence of Structural Disorders in Diffusion MRI. (2018)

6. Yushkevich, Paul A. and Piven, Joseph and Hazlett, Heather Cody and Smith, Rachel Gimpel and Ho, Sean and Gee, James C. and Gerig, Guido. User-guided 3D active contour segmentation of anatomical structures: Significantly improved efficiency and reliability. (2006)