Ultrasound blood flow waveforms may be used in the diagnosis of arterial occlusive disease in human legs. We develop a statistical model to predict disease severity, conditional on the ultrasound data and some training data. It belongs to the class of models known as seemingly unrelated regressions, for which the Bayesian predictive density function cannot be evaluated analytically. Allowing for missing components of response vectors in the training data, we describe a first-order approximation to the predictive density, based on a Bayes estimate of the precision matrix. This approximation is then used to generate cross-validated predictions of disease severity in a set of 31 patients. We conclude with a discussion of the results.