The design of an NMR excitation scheme, whether selective or nonselective, is essentially the simultaneous inversion of an array of Bloch equations driven by magnetic fields which differ according to well-defined constraints. We find that if relaxation effects are negligible, nearly exact inversion of the Bloch equations is straightforward when performed in a special time-varying frame of reference. Repeated inversions of the Bloch equations for small perturbations provide the basis for arbitrarily large, optimal adjustments of the magnetization response to an applied time-varying magnetic field. Choice of the target response to be sought at each iteration is not trivial if overall adjustments of more than one-half rotation are required. We present the analysis both formally and in geometric terms and show how it leads to a general algorithm for the optimization of NMR excitation schemes. The unprecedented efficiency of the algorithm and its ability to generate novel pulses from distant starting approximations are demonstrated in the optimization of slice-selective pi pulses for inversion and refocusing, and a prefocused slice-selective pi/2 pulse. Other applications are discussed, including use of the algorithm to compensate for instrumental imperfections such as radiofrequency inhomogeneity.

NMR has shown promise as a non-invasive and nonhazardous method of imaging the mobile proton distributtion (mainly water) on a small scale in both animal and human tissues (Mansfield and Maudsley, 1977) and more recently on a much larger whole-body scale (Mansfield, et ah, 1978; Damadian et ah, 1977).In this paper we report the first localization of a breast cancer by this technique in a simple mastectomy specimen from a 47-year-old woman.Following mastectomy the specimen was rapidly transferred from the City Hospital to the Physics Department and the NMR scan started about 1J hours following operation. The specimen measured some 15 cm in the long axis, one end of which was the axillary tail; it had a maximum thickness of about 4 cm at the centre tapering off to about 1 cm at the periphery. The surface of the specimen was marked with waterproof ink for identification purposes and placed flat in a plastic container. The slice thickness scanned included the whole of the breast; the images produced correspond to coronal scan projections of water and fat distributed in the whole breast. METHODThe NMR line scanning technique used here has been largely described elsewhere (Mansfield, et al., 1976). The principle of operation is: mobile protons contained in the water, fat or oil distributed throughout biological material are first aligned in a large static magnetic field. A weak nuclear magnetization results due to the small magnetic moment possessed by each proton or hydrogen nucleus. The weak polarization produced is proportional to the localized proton density (water content, etc.); this is read out line by line across the specimen by applying switched magnetic field gradients to define position within the specimen, and weak radiofrequency (rf) pulses tuned to the nuclear resonance frequency. The rf pulses perturb the nuclear magnetization from alignment and transient nuclear free-induction decay (FID) signals are recorded and Fourier analysed to yield the effective proton density of the specimen along a particular line.The static field used was approximately 1 kG corresponding to a resonance frequency of 4.0 MHz.In addition to distributed content of water and fat etc., the spatial distribution of spin lattice relaxation times T\ (x, y) can be measured. Ti is the time for disturbed protons to realign in the large polarizing field. Changes in T\ reflect differences in mobility of the water and also differences in metabolic activity and dissolved salts contained in the cytoplasmic and extra-cellular water.Immediately after scanning, the tissue was immersed in 20% formalin in which it remained for four days. It was then deep frozen and horizontal slices of about 8 mm thickness prepared and photographed. Paraffin sections of selected parts confirmed the presence of malignant tissue.The distribution of the carcinoma was reconstructed from the sections. RESULTS AND DISCUSSIONThe NMR images ( Fig. 1A and B) reflect the general outline of the specimen. This correlates well with the outline of the fixed specimen (Fig. 2). In...