Flow and diffusion images from Bayesian spectral analysis of motion-encoded NMR data

E.J. Fordham
Schlumberger Cambridge Research,
P.O. Box 153, Cambridge CB3 0HG

D. Xing, J.A. Derbyshire, S.J. Gibbs, T.A. Carpenter and L.D. Hall
Herchel Smith Laboratory for Medicinal Chemistry,
Robinson Way, Cambridge CB2 2PZ


Quantitative imaging of steady laminar flow fields in up to three dimensions is achieved by NMR imaging with the addition of motion-encoding field gradient pulses; coherent flow is encoded as a phase shift, diffusive or dispersive processes as an attenuation. A sequence of images with incremented motion-encoding gradient pulse areas displays, at any given image pixel, a damped sinusoidal oscillation of signal intensity with period inversely proportional to a convective flow velocity, and a Gaussian envelope of inverse width proportional to the square root of the local effective diffusivity. The required parameters of velocity and diffusivity are to be obtained from a spectral analysis of such oscillations, pixel by pixel over the image grid. Traditional Fourier analysis has been used with a large number of images in such a sequence; Callaghan and Xia (1991) used 18 images, with extensive zero-filling to 256 sampling points. Such an approach is not economical with data acquistion time; nor are error estimates available.

Rapid imaging protocols such as the Echo Planar (EPI) method minimise the data acquisition time required for each image in the sequence, but a time-efficient protocol must nevertheless minimise the number of images to be acquired. The Bayesian spectral analysis of Bretthorst (1988, 1991), although currently applied mainly to spectroscopic data containing several thousand data points, permits also the routine analysis, for period and attenuation parameters, of noisy, heavily truncated, non-uniformly and sparsely sampled data. Bayesian error intervals are also available.

We demonstrate (Xing et al., 1994) a non-uniform sampling strategy that requires only four images (four complex data points per pixel) to obtain velocity and diffusion images for laminar liquid flow in a cylindrical pipe. The Bayesian error intervals demonstrate the significance of small departures, arising from flow tube imperfections and other non-ideal flow conditions, from the ideal Poiseuille profile of flow velocity. Using both EPI and the Bayesian analysis of motion-encoding, total experimental time is reduced from several hours to a few seconds, the new data analysis being responsible for a factor of 4.5.

We also demonstrate these techniques on steady non-Newtonian flow (of a 1% solution of Xanthan gum) in a straight cylindrical pipe; and finally (Derbyshire et al. 1994) on a 3-dimensional flow of water in an object of complex geometry.


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G.L. Bretthorst (1988) Bayesian Spectrum Analysis and Parameter Estimation,Lecture Notes in Statistics, 48, Springer-Verlag, New York.

G.L. Bretthorst (1991) J. Magn. Reson. 93, 369.

D. Xing, S.J. Gibbs, J.A. Derbyshire, E.J. Fordham, T.A. Carpenter & L.D. Hall (1994) J. Magn. Reson., in press.

J.A. Derbyshire, S.J. Gibbs, T.A. Carpenter & L.D. Hall (1994) A.I.Ch.E. Jnl., in press.

MaxEnt 94 Abstracts / mas@mrao.cam.ac.uk