In MRI (Magnetic Resonance Imaging), gradient coils fulfil an important role to generate a uniform gradient magnetic field over a domain of interest (DOI). As a consequence, a number of numerical methods to analyse and design gradient coils have been developed so far. The design methods are classified to (I) discrete-coils approach and (II) current-density approach [[1], [2], [3]].

The second approach (II) optimises the distribution of current in conductors in a prescribed spatial configuration and includes the target field approach [4], the equivalent magnetic dipole method [5], and so on. This approach is more elaborated and thus powerful than the first approach as mentioned below. As a drawback, it is computationally more expensive because a certain mesh-based method (typically, finite element method [6] and boundary element method [7]) is necessary to solve the Ampere's law (or its equivalence) in terms of the vector potential [8]. Moreover, the current-density methods are an indirect approach, which optimises indirect parameters such as stream function [5]. Since the present study is categorised to the first approach, individual methods for the second approach are left to the review articles [1,2].

The first approach (I) includes the designs of the Helmholtz pair coil [8] (which is for generating a uniform field), the Maxwell pair coil [2], the Golay-type gradient coil [9], and so on. As a development of Golay [9], Roméo et al. [10] proposed to create a desired magnetic field with axially symmetric coils by considering the properties of the higher terms of the field expansion in terms of the spherical harmonics. These studies are combinatorial designs rather than mathematical optimisations of discrete coils.

Another subgroup in (I) relies various optimisation techniques to find optimised parameters regarding geometry of gradient coils. In this subgroup, it is commonly assumed that steady currents run through thin coils or wires. Thus, the (steady-state) magnetic field can be computed by the Biot–Savart's law. Hence, since the magnetic field is represented analytically (as in Eq. (1)), the derivative (gradient) of a given objective function, which is usually related to the magnetic field, can be calculated with respect to a certain parameter regarding the geometrical configuration of the target coil. This enables to apply a variety of gradient-based optimisation methods to the underlying optimisation problems in order to obtain a uniform gradient magnetic field of interest. Many investigations consider the parameters specific to a given configuration of coils [[11], [12], [13], [14], [15], [16], [17], [18]].

On contrast to the above parametric optimisations focusing on a specific type of coils [11,[14], [15], [16], [17], [18]], a more flexible and general-purpose methodology is to approximate a coil with small portions, which are called current/wire elements, and then find their best positions. For this regard, Brey et al. [19] applied the CGD method to optimise locations and/or currents of wire elements. The gradient necessary for the CGD method can be calculated by differentiating the magnetic field with respect to the (cylindrical) coordinates of the elements. Likewise, Lu et al. [20] optimised the positions of wire elements with the momentum-weighted CGD algorithm to take care of the issues of the CGD methods. A possible drawback of optimising the positions of wire elements is to handle a vast number of design variables as the number of such elements increases. This inefficiency is conceivably a serious issue if a coil was complicated in shape because it could consist of many wire elements.

As a more efficient method, this study proposes to approximate a coil with a closed B-spline curve. It is reasonable to choose the control points (CPs) as the design variables. If the degree of B-spline functions is one, the resulting closed B-spline curve is piecewise-linear, which is equivalent to the existing wire elements [19,20]. Otherwise, when a CP is moved, the closed B-spline curve can deform smoothly and locally near the CP. This feature of B-spline functions, which can be generalised to NURBS functions, has been utilised to design surfaces (rather than lines) in shape optimisations using mesh-based PDE solvers such as the isogeometric analysis [21]. With regard to the MRI, the surface shape of the pole piece in a permanent magnet MRI system was designed with using 2D B-spline functions by Ryu et al. [6], although this study is classified to (II).

When expressing a coil with a closed B-spline curve, it will be revealed that the shape sensitivity of a certain objective function, which is related to the quantities for the magnetic field, can be calculated with respect to each CP. To this end, the shape derivative, which is the Eulerian derivative [22] of an objective function with respect to the translation in an arbitrary direction at any point on the coil, is derived from the Biot–Savart's law by considering the perturbation of the point. The derived shape sensitivities are integrated to a general-purpose nonlinear-programming solver to optimise the positions of the design variables, i.e. CPs. As far as the author knows, this kind of NURBS- or B-spline-based optimisation method has been unreported not only for designing gradient coils but also for the other coils/wires.

In what follows, Section 2 constructs the B-spline-based shape optimisation method to find a gradient coil such that ∂Bz∂z, i.e. the gradient in the z-direction of the z-component of the magnetic field B, is close to the desired gradient at given target points. Section 3 rigorously verifies the shape derivative (sensitivity), which is the core of the proposed shape optimisation method. Section 4 solves some shape optimisation problems for z-gradient coils with the proposed coil-design method. Section 5 concludes this study.