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"source": [
"# Spherical Mean Representation of any Compartment Model"
]
},
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"cell_type": "markdown",
"metadata": {},
"source": [
"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.\n",
"\n",
"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:\n",
"- If available, the spherical mean for a model is estimated using an analytic expression (e.g. for C1Stick, G2Zeppelin and G3RestrictedZeppelin).\n",
"- For round models such as G1Ball or restricted sphere models, the spherical mean is just a value for the given shell parameters.\n",
"- 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.\n",
"\n",
"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"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
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{
"name": "stdout",
"output_type": "stream",
"text": [
"Acquisition scheme summary\n",
"\n",
"total number of measurements: 288\n",
"number of b0 measurements: 18\n",
"number of DWI shells: 3\n",
"\n",
"shell_index |# of DWIs |bvalue [s/mm^2] |gradient strength [mT/m] |delta [ms] |Delta[ms] |TE[ms]\n",
"0 |18 |0 |0 |10.6 |43.1 |N/A \n",
"1 |90 |1000 |56 |10.6 |43.1 |N/A \n",
"2 |90 |2000 |79 |10.6 |43.1 |N/A \n",
"3 |90 |3000 |97 |10.6 |43.1 |N/A \n"
]
}
],
"source": [
"# as an example we estimate the spherical mean of the Gaussian phase cylinder model.\n",
"from dmipy.data.saved_acquisition_schemes import wu_minn_hcp_acquisition_scheme\n",
"scheme = wu_minn_hcp_acquisition_scheme()\n",
"scheme.print_acquisition_info"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"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):"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 1. , 0.63540094, 0.47616228, 0.3917526 ])"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from dmipy.signal_models import cylinder_models\n",
"cylinder = cylinder_models.C4CylinderGaussianPhaseApproximation()\n",
"\n",
"diameter = 2e-6 # diameter of 2 microns\n",
"lambda_par = 1.7e-9 # parallel intra-axonal diffusivity\n",
"cylinder.spherical_mean(scheme, diameter=diameter, lambda_par=lambda_par)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"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:\n",
"- cylinder_models.C1Stick\n",
"- cylinder_models.C2CylinderStejskalTannerApproximation\n",
"- cylinder_models.C3CylinderCallaghanApproximation\n",
"- cylinder_models.C4CylinderGaussianPhaseApproximation\n",
"- gaussian_models.G1Ball\n",
"- gaussian_models.G2Zeppelin\n",
"- gaussian_models.G3TemporalZeppelin\n",
"- sphere_models.S1Dot\n",
"- sphere_models.S2SphereStejskalTannerApproximation\n",
"- sphere_models.S4SphereGaussianPhaseApproximation\n",
"\n",
"In addition, the spherical mean is also available for any GammaDistributed model."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Constructing a MultiCompartmentSphericalMeanModel"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"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:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"['G2Zeppelin_1_lambda_perp',\n",
" 'C1Stick_1_lambda_par',\n",
" 'G2Zeppelin_1_lambda_par',\n",
" 'partial_volume_0',\n",
" 'partial_volume_1']"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from dmipy.signal_models import cylinder_models, gaussian_models\n",
"from dmipy.core.modeling_framework import MultiCompartmentSphericalMeanModel\n",
"stick = cylinder_models.C1Stick()\n",
"zeppelin = gaussian_models.G2Zeppelin()\n",
"mc_mdi_model = MultiCompartmentSphericalMeanModel(\n",
" models=[stick, zeppelin])\n",
"mc_mdi_model.parameter_names"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"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:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"['G2Zeppelin_1_lambda_par', 'partial_volume_0', 'partial_volume_1']"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mc_mdi_model.set_tortuous_parameter('G2Zeppelin_1_lambda_perp',\n",
" 'C1Stick_1_lambda_par',\n",
" 'partial_volume_0',\n",
" 'partial_volume_1')\n",
"mc_mdi_model.set_equal_parameter('G2Zeppelin_1_lambda_par',\n",
" 'C1Stick_1_lambda_par')\n",
"mc_mdi_model.parameter_names"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"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](http://nbviewer.jupyter.org/github/AthenaEPI/mipy/blob/master/examples/example_multi_compartment_spherical_mean_technique.ipynb)."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## References\n",
"- 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.\n",
"- Kaden, Enrico, et al. \"Multi-compartment microscopic diffusion imaging.\" NeuroImage 139 (2016): 346-359."
]
}
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