Jonas Lynge Olesen1,2 and Sune Nørhøj Jespersen1,2
1Center of Functionally Integrative Neuroscience (CFIN) and MINDLab, Department of Clinical Medicine, Aarhus University, Aarhus, Denmark, 2Department of physics and Astronomy, Aarhus University, Aarhus, Denmark
Synopsis
A main idea contained in the standard model of diffusion is to model neurons with zero-width sticks. A resulting signature is the prediction that in the large b limit, the isotropically averaged signal scales as $$$1/\sqrt{b}$$$ which has been verified in white matter but not gray matter. This has multiple proposed causes including dendrite curvature and branching. Here, we report on Monte Carlo simulations in 3D reconstructed neurons and find that branching and curvature do not break the power law scaling. On the other hand,the soma is found to limit the regime in which stick scaling is observed.
Introduction
The
widely used ”standard model” of diffusion in the brain is a multicompartment
Gaussian view of tissue as consisting of intra-neurite and extra-neurite water,
with the intra-neuritic part behaving as cylinders with effectively zero
diameter1-4. This gives rise to the prediction of $$$1/\sqrt{b}$$$ scaling of the isotropically averaged signal at large b-values,
which has been verified in human brain white matter5-6. In contrast, this power law was not observed
in gray matter6, which has been
suggested to be caused by effects such as water exchange across dendrites, cell
soma contributions, or dendrite curvature and branching6-7. Here, we use Monte Carlo simulations of
diffusion in 3D reconstructed neurons from NeuroMorpho.Org to investigate
whether power law scaling can be observed even in the presence of cell bodies
and dendrite deviations from sticks, such as e.g. curvature and branching.Methods
We consider
diffusion in pyramidal cells which are one of the main types of neurons in the
cortex8. The neurons are from human brain
and obtained from the NeuroMorpho.Org database9 under reconstructions from the Allen
Brain Atlas10. In these data, the cell structures
are represented by connected nodes with associated radii. From these, we
approximated the cell geometry by assigned to each node a corresponding sphere
and to each node connection a truncated cone making the dendrite diameter change
linearly between neighbours (see fig. 1). The cell soma is modeled as a single
sphere with a diameter chosen to reproduce the soma surface area estimated from
microscopy as reported in the NeuroMorpho database. Finally, the axons of the cells were incompletely characterized and therefore disregarded.
The dMRI
signal was generated by Monte Carlo simulating one million particles in five
neurons each with a number of particles in proportion its relative volume. Trajectories
were generated by updating the particle positions with normally distributed steps
at each time point with a time resolution resulting in a rms displacement of 0.13
mum (on the order of 1/10 of typical dendrite diameters). After each step, it
is evaluated whether each particle remains within the cell structure and if
not, the violating particle’s step is rolled back and repeated until a valid
position is achieved.Results
Figure 2 shows the
signal as a function of diffusion weighting relative to the intrinsic
diffusivity $$$bD_0$$$. This was set to the free diffusivity of water at body
temperature (3 mum^2/ms12) but the results can be generalized to any
diffusivity by modifying the sequence timings proportionally. Spheres with
twice the soma diameters centred at each soma are used to divide the particles
into approximate somatic and dendritic subpopulations whose signals can then be
studied separately.
Firstly, it is seen
that the dendritic part of the signal agrees well with the stick power-law
behaviour within range of practically relevant diffusion weightings ($$$bD_0$$$<100) even though exponential damping is clearly observed at extreme weightings.
Staying within practical range, figure 3 shows the apparent exponent for
different sequence timings which comes as close as $$$\sim$$$0.53 to the stick prediction
of 0.5 for experimentally reasonable timings. This demonstrates that the stick
power law is not broken by realistic undulations and branching.
With increased
sequence timing, the somatic contribution gains increased importance. Indeed,
while the dendritic signal follows the power law, the full signal exhibits more
complicated behaviour due to the somatic contribution. To recover the power law
in the full signal, one must either use sufficient weighting to suppress the
somas which is impractical because it requires very high weighting which in
turn increases the power-law deviation of the dendritic signal. An interesting alternative appears for large pulse separations, an intermediate b-value range exists where the
somatic signal changes slowly and effectively behaves as an immobile water
fraction. As shown in fig. 4 and 5, it is conceivable that this might be
exploited to recover the stick scaling at moderate diffusion weighting.Conclusion
Using Monte
Carlo simulations in pyramidal cells, we demonstrate that realistic branching and
undulations do not break the stick power law scaling. The presence of soma on
the other hand complicates the signal behaviour but it is possible that the
power law can be recovered at an intermediate b-value range with appropriate
sequence timings.Acknowledgements
No acknowledgement found.References
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