Models that rely on an approximate solution of the Bloch-Torrey equation for magnetization with diffusion-induced magnetization transfer under the narrow pulse approximation (NPA):

• Söderman’s model (Söderman and Jönsson 1995);
• Callaghan’s model (Callaghan 1995).

Models that rely on the Gaussian phase approximation (GPA), i.e. that diffusion remains Gaussian but with apparent diffusion coefficient that depends on the cylinder’s radius:

• Neuman’s model (Neuman 1974);
• Van Gelderen’s model (Vangelderen et al. 1994).

Models that rely on the geometric simplification of diffusion between two parallel planes under the NPA:

• Stanisz’s model (Stanisz et al. 1997).

Neuman (1974) and Vangelderen et al. (1994) model the diffusion in the perpendicular plane to the cylinder axis as a two-dimensional zero-mean Gaussian distribution with variance \(2 D_\perp^\text{app} \tau\) where \(D_\perp^\text{app}\) is the apparent radial diffusivity in the intra-axonal space and is defined as a function of the cylinder radius \(R\).

Stanisz et al. (1997) take on a different approach and provide models for the restricted diffusion in cylindrical and spherical geometries. As such, it is useful to model both isotropic restricted diffusion in glial cells and anisotropic restricted diffusion in axons. This is achieved by using in both cases a simple one-dimensional model of restricted diffusion within infinite parallel membranes. This assumption is validated by comparing the signal loss with respect to the model proposed by Neuman (1974). Accordingly, they use the model of signal attenuation induced by water diffusing between flat, impermeable barriers of spacing \(\ell\) proposed in Tanner (1978):

One gets restricted diffusion in a cylinder or in a sphere by setting \(\ell\) to the corresponding radius \(R\). The signal in this model is not a mono-exponential decay and therefore the resulting density of water molecule displacements is not a Gaussian distribution, which is probably a more realistic assumption for water diffusing in a trapped environment. The model however relies on the short pulse duration (SPD) approximation. In the case of restricted diffusion within a cylinder, the diffusion gradient strength \(G\) should also be replaced by \(G \sqrt{1 - \braket{\mathbf{n}, \mathbf{u}}^2}\), where \(\mathbf{u}\) is the cylinder axis.

Neuman (1974) assumes Gaussian diffusion during the gradient pulse and that:

Under these assumptions, Neuman (1974) predicts that the signal attenuation induced by diffusion perpendicular to the cylinder axis reads:

where \(\mathrm{TE}\) is the echo time. Note that, similarly to Soderman’s model, the signal attenuation induced by diffusion perpendicular to the cylinder axis under Neuman’s model is independent from the diffusion time \(\Delta\). However, in contrast to Soderman’s model, the signal depends on the echo time \(\mathrm{TE}\) and the free diffusion coefficient \(D_0\).

Söderman and Jönsson (1995) assumes that the pulse duration \(\delta\) is small enough to neglect the diffusion during the pulse and that the diffusion time \(\Delta\) is long enough to make diffusion restricted within a cylinder of radius \(R\) and free diffusion coefficient \(D_0\). In other terms, they assume that:

Under these assumptions, Söderman and Jönsson (1995) predicts that the signal attenuation induced by diffusion perpendicular to the cylinder axis reads:

where \(J_1\) is the Bessel function of the first kind of order \(1\). Note that the signal attenuation induced by diffusion perpendicular to the cylinder axis is independent from the diffusion time \(\Delta\) and the free diffusion coefficient \(D_0\).

Vangelderen et al. (1994) make no assumptions on the acquisition sequence and the gradient waveform. They assume Gaussian diffusion during the gradient pulse.

Under these assumptions, Vangelderen et al. (1994) predicts that the signal attenuation induced by diffusion perpendicular to the cylinder axis reads:

where \(\alpha_m\) are the roots of the equation:

Callaghan (1995) relaxes the assumption that \(\Delta \gg \frac{R^2}{D_0}\). It still assumes that:

Under these assumptions, Callaghan (1995) predicts that the signal attenuation induced by diffusion perpendicular to the cylinder axis reads:

where \(J_n^\prime\) is the derivative of the Bessel function of the first kind of order \(n\) and \(\beta_{nm}\) are the non-negative arguments at which \(J_n^\prime\) equals zero or equivalently, at which \(J_n\) has a local extremum:

• \(E_r(\mathbf{q}; \mathbf{u}_j)\): the signal attenuation induced by restricted diffusion in an axon bundle with mean orientation \(\mathbf{u}_j\) according to previous theory (4 free parameters);
• \(E_h(\mathbf{q})\): the signal attenuation induced by hindered diffusion in the extra-axonal space modeled as diffusion tensor model \(E_h(\mathbf{q}) = e^{-\mathbf{q}^T \mathbf{D}_h \mathbf{q}}\) (6 free parameters);
• \(E_\mathrm{csf}(\mathbf{q})\): the signal attenuation induced by free diffusion in the CSF, i.e. \(E_\mathrm{csf}(\mathbf{q}) = e^{-D_\mathrm{csf} \|q\|^2}\) (1 free parameter).

• Models have focus either on the axonal density or the axon diameter:

  • Axon density: CHARMED (Assaf and Basser 2005);
  • Axon diameter: AxCaliber (Assaf et al. 2008; Barazany, Basser, and Assaf 2009).

• Axon diameter estimation requires acquisitions with multiple diffusion times (\(\Delta\));

• The estimation of both axonal density and axon diameter is improved if time dependence of the extra-axonal apparent radial diffusivity is taken into account (Burcaw, Fieremans, and Novikov 2015; De Santis, Jones, and Roebroeck 2016).

• \(f_\mathrm{csf} = 0\);
• \(M\) up to \(2\);
• Fixed distribution of axonal radii: \(\{(1.5, 0.0212), (2.5, 0.1072), (3.5, 0.1944), (4.5, 0.2667), (5.5, 0.2150), (6.5, 0.1956) \}\);
• Neuman’s model for restricted diffusion in cylinders.