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Spherical Mean Representation of any Compartment Model¶

The idea of Spherical Mean models is to fit the spherical mean of a model to the spherical mean of the data, rather than fitting the separate DWIs per shell. As such, spherical mean models have no concept of orientation, dispersion or crossings. (Kaden et al. 2015) proposed the spherical mean technique by providing the analytic expression of the Zeppelin model. Later, (Kaden et al. 2016) combined the spherical mean model of a Zeppelin and Stick model to create his multi-compartment spherical mean technique.

Taking a step forward, Dmipy allows to estimate the spherical mean representation of any Compartment Model, and combine them in a MultiCompartmentSphericalMeanModel in the same way as a standard MultiCompartmentModel. The spherical mean for a model is estimated as follows:

If available, the spherical mean for a model is estimated using an analytic expression (e.g. for C1Stick, G2Zeppelin and G3RestrictedZeppelin).
For round models such as G1Ball or restricted sphere models, the spherical mean is just a value for the given shell parameters.
For cylinder models without analytic expressions, we estimate the spherical mean using the its rotational_harmonics_representation. In short, the spherical mean of any spherical function (if sufficiently expanded) is equal to $c_{m=0}^{l=0} / (2 \sqrt{\pi})$, where $c_{m=0}^{l=0}$ is the zeroth order spherical harmonics coefficient.

A model's spherical mean for a given acquisition scheme can be called using model.spherical_mean(acquisition_scheme, **parameters). To illustrate, we show the composition of the well-known Wu-Minn HCP acquisition scheme

In [1]:

# as an example we estimate the spherical mean of the Gaussian phase cylinder model.
from dmipy.data.saved_acquisition_schemes import wu_minn_hcp_acquisition_scheme
scheme = wu_minn_hcp_acquisition_scheme()
scheme.print_acquisition_info

Acquisition scheme summary

total number of measurements: 288
number of b0 measurements: 18
number of DWI shells: 3

shell_index |# of DWIs |bvalue [s/mm^2] |gradient strength [mT/m] |delta [ms] |Delta[ms] |TE[ms]
0           |18        |0               |0                        |10.6       |43.1      |N/A  
1           |90        |1000            |56                       |10.6       |43.1      |N/A  
2           |90        |2000            |79                       |10.6       |43.1      |N/A  
3           |90        |3000            |97                       |10.6       |43.1      |N/A

It can be seen that this scheme has 4 measured shells (including the b0 measurements). This means that the spherical mean for any of dmipy's models will also only have 4 values. To illustrate, we estimate the spherical mean of the Gaussian phase cylinder approximation (which does not have an analytical representation):

In [2]:

from dmipy.signal_models import cylinder_models
cylinder = cylinder_models.C4CylinderGaussianPhaseApproximation()

diameter = 2e-6  # diameter of 2 microns
lambda_par = 1.7e-9  # parallel intra-axonal diffusivity
cylinder.spherical_mean(scheme, diameter=diameter, lambda_par=lambda_par)

Out[2]:

array([ 1.        ,  0.63540094,  0.47616228,  0.3917526 ])

Notice that for the spherical mean respresentation it is not necessary to give the model a value for its orientation 'mu'. This is because a spherical mean model has no concept of orientation (as it has been integrated out of the model representation). Currently, the dmipy models that have a spherical mean representation are the following:

cylinder_models.C1Stick
cylinder_models.C2CylinderStejskalTannerApproximation
cylinder_models.C3CylinderCallaghanApproximation
cylinder_models.C4CylinderGaussianPhaseApproximation
gaussian_models.G1Ball
gaussian_models.G2Zeppelin
gaussian_models.G3TemporalZeppelin
sphere_models.S1Dot
sphere_models.S2SphereStejskalTannerApproximation
sphere_models.S4SphereGaussianPhaseApproximation

In addition, the spherical mean is also available for any GammaDistributed model.

Constructing a MultiCompartmentSphericalMeanModel¶

A MultiCompartmentModel for spherical mean representations of models is done by calling MultiCompartmentSphericalMeanModel with the same parameters as the MultiCompartmentModel. For example, making a spherical mean representation of the Multi Compartment Microscopic Diffusion Imaging (MC-MDI) (Kaden et al. 2016) model is done as follows:

In [3]:

from dmipy.signal_models import cylinder_models, gaussian_models
from dmipy.core.modeling_framework import MultiCompartmentSphericalMeanModel
stick = cylinder_models.C1Stick()
zeppelin = gaussian_models.G2Zeppelin()
mc_mdi_model = MultiCompartmentSphericalMeanModel(
    models=[stick, zeppelin])
mc_mdi_model.parameter_names

Out[3]:

['G2Zeppelin_1_lambda_perp',
 'C1Stick_1_lambda_par',
 'G2Zeppelin_1_lambda_par',
 'partial_volume_0',
 'partial_volume_1']

Notice that the spherical mean representation has no orientation parameters 'mu'. Adding tortuosity and equality constraints on the model parameters is done the same as before:

In [4]:

mc_mdi_model.set_tortuous_parameter('G2Zeppelin_1_lambda_perp',
                                    'C1Stick_1_lambda_par',
                                    'partial_volume_0',
                                    'partial_volume_1')
mc_mdi_model.set_equal_parameter('G2Zeppelin_1_lambda_par',
                                 'C1Stick_1_lambda_par')
mc_mdi_model.parameter_names

Out[4]:

['G2Zeppelin_1_lambda_par', 'partial_volume_0', 'partial_volume_1']

With this the spherical mean model can be fitted to dMRI data as mc_mdi_model.fit(scheme, data). The application of this model on HCP data is given in the MC-MDI model example.

References¶

Kaden, Enrico, Frithjof Kruggel, and Daniel C. Alexander. "Quantitative mapping of the per‐axon diffusion coefficients in brain white matter." Magnetic resonance in medicine 75.4 (2016): 1752-1763.
Kaden, Enrico, et al. "Multi-compartment microscopic diffusion imaging." NeuroImage 139 (2016): 346-359.