Intravoxel Dephasing due to Wave Gradients

This article assumes prior knowledge of Wave-CAIPI[1]. In particular, this article attempts to describe the methodology used to generate Figure 9 of [1]. For simplicity, assume a two-dimensional problem. $x$ and $y$ will denote the readout and phase encode directions, and $k_x$ and $k_y$ will denote the respective k-space directions. Let $m$ be the underlying image and $w$ be the Wave-PSF as described in Wave-CAIPI[1].

Consider the voxel corresponding to the center of the image (at $x = 0$ mm and $y = 0$ mm). The sinusoid gradients induce a spreading along the readout direction as a function of $y$. This can be modelled as convolution as follows:

$$v(x, y=0) = m(x, y=0) \ast_x w(x, y=0)$$

Here, $v$ denotes the acquired data in the $(x, y)$ domain. The subscript in $(\ast_x)$ implies this convolution is applied only along the $x$ direction.

In reality, the underlying image $m$ exists as a continuum and the wave gradients induce a spreading within the voxel itself. Assuming the image resolution is at 1 mm isotropic resolution, the voxel at $(0, 0)$ is in effect the sum of all the contributions within the continuum $[-0.5, 0.5] \text{mm} \times [-0.5, 0.5] \text{mm}$. This summing effect can be modelled as,

$$v(x, y=0) = \int_{y = -0.5 \text{mm}}^{0.5 \text{mm}} \left[m(x, y) \ast_x w(x, y) \right]dy$$

Expanding the above,

$$v(x, y=0) = \int_{y = -0.5 \text{mm}}^{0.5 \text{mm}} \int_t m(t, y) w(x - t, y) dydt$$

Now comes the main assumption:

$$m(t, y) \approx m(t, 0) \text{ for } y \in (-0.5, 0.5)$$

With this in place, the previous equation decouples to the following:

$$v(x, y=0) = \int_t m(t, 0) \left[\int_{y = -0.5 \text{mm}}^{0.5 \text{mm}} w(x - t, y) dy\right]dt$$

Let $W(x, y)$ be defined as,

$$W(x, y) = \int_{y = -0.5 \text{mm}}^{0.5 \text{mm}} w(x, y) dy$$

With using $W$, the equation looks once again like the original convolution model.

$$v(x, y) = m(x, y) \ast_x W(x, y)$$

Looking at the PSF $W(x, y)$ at the center voxel (as in Figure 9 of [1]) which models the intravoxel dephasing within a voxel, we see that wave gradients induce a small side lobe.

References.

Bilgic B, Gagoski BA, Cauley SF, et al. Wave-CAIPI for highly accelerated 3D imaging. Magn Reson Med. 2015;73(6):2152-2162. doi:10.1002/mrm.25347