Feng Jia^{1}, Sebastian Littin^{1}, stefan kroboth^{1}, Huijun Yu^{1}, Theresa Palm^{2}, Frederik B. Laun^{2}, Mark E. Ladd^{3}, and Maxim Zaitsev^{1}

High performance gradient coils are required to assess the tissue microstructure in human breast in vivo with diffusion-weighted imaging. A deisgn methodology of nonlinear breast gradient coil is proposed to increase resultant gradient strength with the control of field nonlinearity. The method is tested by designing a unilateral breast gradient coil for diffusion weighting. The results are analysis to reveal new insights of coil designs.

The following design requirements of the gradient coil were considered: firstly, the coil has to generate an encoding field with strong gradients at every test point within a region of interest (ROI). Secondly, the coil inductance should be considered during the coil design in order to reduce switching rates. Thirdly, some engineering constraints, such as wire track width, have to be considered for the practical fabrication of the coil. Finally, non-linearity of resultant field needs to be controlled.

Based on the requirements above, an optimization problem is proposed as follows:

$$\min_{\psi}\mbox{ } F, \mbox{ }F:=-\sum_{\vec{x}_i\in{\text{ROI}}}\left|\nabla B_z(\psi,\vec{x}_i)\right|+\alpha_J \lVert\vec{J}(\vec{x})\rVert_{p}$$

$$\text{subject to}\quad\frac{1}{\mu}\int_{\Gamma^{'}}\int_{\Gamma}\frac{|\vec{J}(\vec{x})\vec{J}(\vec{x}^{'}|}{|\vec{x}-\vec{x}^{'}|}d{\Gamma}d\Gamma^{'}\leq W_{\max}\quad\text{and}\quad|B_z(\vec{x}_i)-B_z^*(\vec{x}_i)|\leq\delta\max|B_z^*(\vec{x})|.$$

Here, $$$\psi$$$ denotes the stream function of the electric current
density vector $$$\vec{J}(\vec{x})$$$ where $$$\vec{J}(\vec{x})=
\nabla\times(\psi(\vec{x})\vec{n})$$$ on a current-carrying surface $$$\Gamma$$$(Fig. 1) with a
normal unit vector $$$\vec{n}$$$. The $$$B_z$$$ is the z-component
of the magnetic fields generated by current density $$$\vec{J}$$$ on $$$\Gamma$$$ (Fig. 1) and is calculated using the Biot-Savart law. The $$$B_z^*$$$ is a target magnetic field, which is the
linear z-gradient field of 50 mT/m gradient strength in the examples. The
points $$$\vec{x}_i=(x_i, y_i, z_i),
i=1, …, m$$$, denote the coordinate vectors of $$$m$$$ test points
in the ROI, $$$\mu$$$ indicates the
vacuum permeability, $$$W_{\max}$$$ denotes a specified maximal magnetic
energy of the designed coil which is used to control the coil inductance and δ is a parameter to tune the non-linearity of resultant $$$B_z$$$.
The objective function *F* of the problem (1) consists of two terms. The
first one is related to the gradient strength of the encoding fields and the other
is applied to control the minimal width of wire tracks of the designed coil by
the p-norm (||·||_{p}) method^{2}. One positive constant $$$\alpha_J$$$ is to balance the tradeoff among them.

In order to assess different coil designs, a performance metric is defined as follows:

$$\beta:=\frac{\sum_{\vec{x}_i\in{\text{ROI}}}\lvert\nabla B_z(\psi,\vec{x}_i)\rvert}{m\sqrt{W_{\max}}}$$

Here, the figure of merit β is intended to measure the coil efficiency based on the average of the gradient magnitude of the encoding field over the ROI. From equation of $$$\beta$$$, it is clear that an ideal high-performance coil should have a high $$$\beta$$$. All numerical examples were solved with the function fmincon in MATLAB (The MathWorks. Natick, USA).

1. van der Velden T, van Houtum Q, Gosselink M, et al., Characterization of a breast gradient insert coil at 7 tesla with field cameras. ISMRM. 2016;p3543.

2. Jia F, Littin S, Layton KJ et al., Design of a shielded coil element of a matrix gradient coil. J Magn Reson. 2017;281:217-228.

3. Littin S, Jia F, Layton KJ et al.,Development and implementation of an 84-channel matrix gradient coil. Magn Reson Med. 2017;doi: 10.1002/mrm.26700.

4. Hennig J, Welz AM, Schultz G, et al., Parallel imaging in non-bijective, curvilinear magnetic field gradients: a concept study. MAGMA. 2008; 21: 5–14.

Figure 1: Geometries of a current-carrying surface Г and a
half ellipsoid shaped ROI. The
surface has a shape of cylinder with a hemispherical end and the radius of
107.5 mm and the height of 173 mm. Three principle semi-axes of the ROI have the length of 67.5 mm, 113 mm and 67.5
mm along three coordinate directions, respectively.

Figure 2: The effect of different values of $$$\delta$$$. Here, Wmax
is set to be equal to the magnetic energy (0.1838 [joule],) for the initial
stream function, which is obtained using the design method for a target linear
encoding field in [2].

Figure 3: The resultant field gradient distributions
when $$$\delta$$$ = 5%(a) and 20%(b).