# 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

### Abstract

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.

### References

P.T. Callaghan & Y. Xia (1991) *J. Magn. Reson.*
**91**, 326.
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