Notebook

Gaussian Models¶

Gaussian models that are typically used to described extra-axonal diffusion. Briefly, we have:

Ball model: Isotropic Gaussian diffusion with diffusion coefficients $\lambda_{iso}$.
Zeppelin Model: Axially symmetric Gaussian tensor, oriented along orientation $\mu$ with parallel and perpendicular diffusivity $\lambda_\parallel$ and $\lambda_\perp$.
Restricted zeppeling model: Axially symmetric Gaussian tensor with time-dependent perpendicular diffusion to account for diffusion restricted in the extra-axonal space between axons. Instead of $\lambda_\perp$, this model has $\lambda_{inf}$ and characteristic coefficient $A$.

Ball: G1¶

the ``Ball'' model (Behrens et al. 2013) models the totallity of all extra-axonal diffusion as a Tensor with isotropic diffusivity $\lambda_{\textrm{iso}}$ as

\begin{equation} E_{\textrm{iso}}(b,\lambda_{\textrm{iso}})=\exp(-b\lambda_{\textrm{iso}}). \end{equation}

In current models, the Ball is usually only used to describe the CSF and/or grey matter compartment of the tissue, where the isotropic diffusion assumption is reasonably valid(Alexander et al. 2010, Jeurissen et al. 2014, Tariq et al. 2016).

In [1]:

from dmipy.signal_models import gaussian_models
from dmipy.core.acquisition_scheme import acquisition_scheme_from_bvalues
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

samples = 100
bvalues = np.linspace(0, 1e9, samples)
bvectors = np.tile([0, 0, 1], (samples, 1))
delta = 1e-2
Delta = 3e-2
scheme = acquisition_scheme_from_bvalues(bvalues, bvectors, delta, Delta)

ball = gaussian_models.G1Ball()

for lambda_iso in [1e-9, 2e-9, 3e-9]:
    plt.plot(bvalues, ball(scheme, lambda_iso=lambda_iso),
             label='lambda_iso=' + str(lambda_iso * 1e9) + 'e-9 m^2/s')
plt.legend()
plt.xlabel('b-value [s/m^2]', fontsize=13)
plt.ylabel('Signal Attenuation', fontsize=13)
plt.title('Signal Attenuation Ball', fontsize=15);

The higher the diffusivity the faster the signal attenuates.

Zeppelin: G2¶

Hindered extra-axonal diffusion, i.e. diffusion of particles in-between axons, is often modeled as an anisotropic, axially symmetric Gaussian, also known as a ``Zeppelin'' (Panagiotaki et al. 2012). Using DTI notation, a Zeppelin with $\lambda_\parallel=\lambda_1$, $\lambda_\perp=\lambda_2=\lambda_3$ and $\lambda_\parallel>\lambda_\perp$ is given as

\begin{equation} E_h(b,\textbf{n},\lambda_\parallel,\lambda_\perp)=\exp(-b\textbf{n}^T(\textbf{R}\textbf{D}^h_{\textrm{diag}}\textbf{R}^T)\textbf{n})\quad\textrm{with}\quad \textbf{D}^h_{\textrm{diag}}= \begin{pmatrix} \lambda_\parallel & 0 & 0 \\ 0 & \lambda_\perp & 0 \\ 0 & 0 & \lambda_\perp \end{pmatrix}. \end{equation}

In [2]:

# We set the parallel diffusivity larger than the perpendicular diffusivity
zeppelin = gaussian_models.G2Zeppelin(lambda_par=1.7e-9, lambda_perp=0.8e-9)
E_zeppelin_par = zeppelin(scheme, mu=[0, 0])
E_zeppelin_perp = zeppelin(scheme, mu=[np.pi / 2, 0.])

plt.plot(bvalues, E_zeppelin_par, label='E_parallel')
plt.plot(bvalues, E_zeppelin_perp, label='E_perpendicular')
plt.legend()
plt.xlabel('b-value [s/m^2]', fontsize=13)
plt.ylabel('Signal Attenuation', fontsize=13)
plt.title('Signal Attenuation Zeppelin', fontsize=15);

Temporal Zeppelin: G3¶

In DTI, the signal attenuation decays like a Gaussian over q-value and like an expontial over diffusion time $\tau$. However, recent works argue that hindered diffusion is actually slower-than-exponential over $\tau$ due to how the external axon boundaries still restrict diffusing particles(Novikov et al. 2014). To account for this, (Burcaw et al. 2015) proposed a modification to the Zeppelin as

\begin{equation} \textbf{D}_{\textrm{diag}}^{\textrm{r}}= \begin{pmatrix} \lambda_\parallel & 0 & 0 \\ 0 & \lambda_\perp^{\textrm{r}} & 0 \\ 0 & 0 & \lambda_\perp^{\textrm{r}} \end{pmatrix}\quad\textrm{with}\quad\lambda_\perp^{\textrm{r}}=D_{\infty}+\frac{A\ln(\Delta/\delta)+3/2}{\Delta-\delta/3} \end{equation}

where perpendicular diffusivity $\lambda_\perp^{\textrm{r}}$ is now time-dependent with $D_{\infty}$ the bulk diffusion constant and $A$ is a characteristic coefficient for extra-axonal hindrance. Notice that when $A=0$ then G3 simplifies to G2.

In [3]:

# we just sample the (restricted) perpendicular direction now
restricted_zeppelin = gaussian_models.G3TemporalZeppelin(
    mu=[np.pi / 2, 0.], lambda_par=1.7e-9, lambda_inf = 1e-9)

for A in [0, 2e-12, 4e-12]:
    plt.plot(bvalues, restricted_zeppelin(scheme, A=A), label='A={} $\mu m^2$'.format(A*1e12))
plt.legend()
plt.xlabel('b-value [s/m^2]', fontsize=13)
plt.ylabel('Signal Attenuation', fontsize=13)
plt.title('Signal Attenuation Restricted Zeppelin', fontsize=15);

Notice that for larger characteristic coefficients the signal attenuates faster.

References¶

Behrens, Timothy EJ, et al. "Characterization and propagation of uncertainty in diffusion‐weighted MR imaging." Magnetic resonance in medicine 50.5 (2003): 1077-1088.
Alexander, Daniel C., et al. "Orientationally invariant indices of axon diameter and density from diffusion MRI." Neuroimage 52.4 (2010): 1374-1389.
Jeurissen, Ben, et al. "Multi-tissue constrained spherical deconvolution for improved analysis of multi-shell diffusion MRI data." NeuroImage 103 (2014): 411-426.
Tariq, Maira, et al. "Bingham–noddi: Mapping anisotropic orientation dispersion of neurites using diffusion mri." NeuroImage 133 (2016): 207-223.
Panagiotaki, Eleftheria, et al. "Compartment models of the diffusion MR signal in brain white matter: a taxonomy and comparison." Neuroimage 59.3 (2012): 2241-2254.
Novikov, Dmitry S., et al. "Revealing mesoscopic structural universality with diffusion." Proceedings of the National Academy of Sciences 111.14 (2014): 5088-5093.
Burcaw, Lauren M., Els Fieremans, and Dmitry S. Novikov. "Mesoscopic structure of neuronal tracts from time-dependent diffusion." NeuroImage 114 (2015): 18-37.