We present a Bayesian bivariate image model and apply it to a study that was designed to investigate the relationship between hypoxia and angiogenesis in an animal tumor model. Two radiolabeled tracers (one measuring angio- genesis, the other measuring hypoxia) were simultaneously injected into the animals, the tumors removed and autoradiographic images of the tracer concentrations were obtained. We model correlation between tracers with a mixture of bivariate normal distributions and the spatial correlation inherent in the images by means of the celebrated Potts model. Although the Potts model is typically used for image segmentation, we use it solely as a device to account for spatial correlation. The number of classes in the model is assumed un- known and is estimated via reversible jump MCMC, marginalizing over the number of classes for posterior inference. We present the model and estimation method using set theory notation which will assist us in introducing a novel reallocation scheme used in the reversible jump proposals. We also estimate the spatial regularization parameter in the Potts model prior. Through a simulation study, we show that it is necessary to account for both the spatial correlation and the correlation between the two tracers.


Statistical Models