able coil from an engineering point of view (see figure 3.7(b)). This confirms the choice of λ’s having a two fold effect: one, simply to allow an accurate solution of system (3.25) and two, adding weight to the regularisation term associated with the regularisation parameter. The windings associated with figure 3.7(b) do not, however, produce reductions in acoustic noise, and our problem thus requires further searching for an appropriate regularisation parameter set.
3.9
Discussion and Conclusion
This chapter describes one process of designing a gradient coil to reduce simulated acoustic noise output generated by the Lorentz forces. A Tikhonov regularisation method has been used to overcome the ill-conditioned nature of the inverse prob- lem successfully. The major design requirements of producing a linear field in the DSV and small external magnetic fields to the coil have been met. Reducing the deflection of the coil and hence the simulated acoustic noise has been achieved as a secondary design issue to a small extent. A relatively small drop of 0.5 dB was the optimum achievement (from a base of 134.7 dB) while still satisfying the primary design issues.
A linearised coil deflection system for each orthogonal deflection component (3.6) was solved in its most general form. The nature of system (3.6) suggested nu- merical solutions would be required but an application of the Continuity equation avoided this problem. Closed form solutions were obtained for each coil deflection component and the pressure perturbation. The model presented in this chapter gives a complete discription of the cylindrical deflection components and acoustic effects of a typical gradient coil. A corollary of the results is that large amplitude acoustic noise appears to be a negative but necessary result of a gradient coil that
3.9. DISCUSSION AND CONCLUSION 109
produces a linear field.
The ‘thin-walled’ nature of our gradient coil model is a first step approximation to a genuine gradient coil which typically falls into the ‘thick-walled’ class, see Mechefske and Wang [85]. For our ‘thin-walled’ model shear deformation effects are obviously ignored, however, this should not be the case in general. Mechefske and Wang [85] have demonstrated that at least for low switching frequencies shear deformations can be ignored under a ‘thick-walled’ model.
The validity of the assumptions for the coil deflection model found in chapter 2 are confirmed by the results presented here. The form of the infinite pressure wave is identical to the 1D deflection model pressure wave seen in chapter 2. Similar sound pressure level results were produced in the 3D case. This predicts that a necessary requirement for minimising the simulated acoustic noise output generated by the Lorentz forces is minimising the radial coil deflection component. Also demonstrated was the insignificant effects of applying open boundary con- ditions on the pressure wave. Quantitatively, the finite length pressure wave produced essentially the same simulated noise amplitude as the infinite pressure wave case. The extra mathematical and computational effort required to correct the infinite pressure wave by less than 1% is in this authors opinion not worth while. However, this work may be useful to other researchers employing numerical methods, who may be able to incorporate the open boundary effects more simply into their code.
Limitations to the acoustic noise model presented in chapters 2 and 3 are its ability to be generalised to multiple coil vibration combinations. It is believed that combined vibration effects from multiple coils (e.g. xand y gradients together or gradient and RF coils) complicate the vibrational and acoustic fields significantly.
110 3.9. DISCUSSION AND CONCLUSION
In addition to a more complex acoustic field pattern the coupled vibrations may negatively affect the image acquisition stage as well.
Future work will consider a braced gradient coil following Lin [72] but is not in the scope of this thesis. This paper confirms the braces will only need to be ‘ring’ clamps that force nodes in the radial deformation component to reduce the noise. A model for the coil deflection with forced nodes to model braced positions potentially involves a more complex partial differential equation system which may not allow analytical solutions however.
Chapter 4
Achieving Quiet Coils through Image
Processing
4.1
Introduction
So far, our attempts to reduce the acoustically significant noise have been rela- tively unsuccessful. Only small reductions in noise have been gained while forcing the gradient coil to produce an adequately homogeneous linear field. In this chap- ter we abandon the restriction that the gradient coil need produce a linear field. As long as the field produced by the gradient coil provides unambiguous spatial information about the imaging region (i.e. is 1 : 1 over the imaging region), then its role is not defeated. Haacke et al. [45] suggested that useful images ought to be obtainable with coils that generate non-linear fields, and various techniques for doing so are presented by Jankeet al. [59], Langlois et al. [66] and Bernstein et al. [4] for example. Wang et al. [110] and Gunter et al. [44] made use of a phantom image to calibrate the non-linear distortions of the field, using complex three-dimensional structures from which they could calculate the true location of
112 4.1. INTRODUCTION
a point within the reconstructed image. Other researches such as Hennig et al. [49] have developed a theoretical method to encode using non-cartesian directed gradients. Their paper suggests it is possible to select circular slices whithin the DSV by applying a radially directed gradient field in combination with regular cartesian gradients. Further research into using non-linear gradients has resulted in advanced techniques designed to produce accelerated scanning times. The ar- ticle by Stockmannet al.[100] details a method called ‘O-space imaging’ whereby they assert that a radially directed gradient is more efficient in terms of provid- ing more encoding in the angular direction than in the radial direction and is suitable for the natural cylindrical nature of a typical MRI scanner. The radi- ally directed gradient is used in combination with a standard linear gradient to shift the encoding area off centre to gain resolution over regions where the Z2 gradient would be 0 (i.e. the centre of the DSV). One advantage discussed in the paper is that increasing the ring density (number of Z2 slices) leads to significant improvements in resolution, in contrast to cartesian acquisitions. More recently, Forbes, Brideson and While [31] devised a simple correction technique, also based on the use of a phantom, in which the distortion effects due to field non-linearity can in principle be cancelled away exactly, and this technique appears to show promise in recovering good quality images even in the presence of reasonably strong non-linearity.
In this chapter, a method is presented for designing quiet self-shielded gradient coils. Our model follows that of chapter 2 which has been confirmed as acceptable based on the results of chapter 3. However, in this chapter, the noise reduction becomes the primary goal, with the requirement that the gradient coil produce a linear field being only a secondary concern. The algorithm of Forbes, Brideson and While [31] (referred to hereafter as the FBW algorithm) is then used to ex-