A Bayesian analysis of dual autoradiographic images

Abstract

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 angiogenesis, the other measuring hypoxia) were simultaneously injected into the animals, the tumors were 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 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 unknown 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. Via simulation studies, we show that it is necessary to account for both the spatial correlation and the correlation between the two tracers.

Introduction

This work is motivated by a study investigating the relationship between hypoxia and angiogenesis in an animal tumor model. Oxygen deficiency (hypoxia) is a common feature of malignant tumors and has the well known effect of decreasing the sensitivity of tumors to ionizing radiation. It has been identified as a factor for tumor progression and for resistance to cancer treatment. The proliferation of new vessels from pre-existing capillaries (angiogenesis) is a key in the pathological development of solid tumors and for their ability to metastasize. It has long been conjectured that, as solid tumors grow, the central core of the tumor becomes necrotic due to hypoxia and that this core of hypoxic tissue is surrounded by an annulus of tissue with high angiogenic activity. With the recent development of in vivo tracers for hypoxia and angiogenesis, researchers are now able objectively study the relationship between hypoxia and angiogenesis within tumors.

The study consisted of eleven Swiss nude mice. Each mouse received two xenografts of EMT6 tumors approximately 1 mm³ to both sides of the thoracic back. Two weeks later, the mice were injected with two radio-labeled tracers: [¹⁸F]FAZA and [¹²⁵I]gluco-RGD. [¹⁸F]FAZA ($t_{1/2}=109.7min$.) measures hypoxia (Piert et al., 2005) and [¹²⁵I]gluco-RGD ($t_{1/2}=59.4\mathrm{days}$) measures angiogenesis (Haubner et al., 1999). Three hours later the animals were euthanized, their tumors dissected, frozen and cut into 20 μm sections. Immediately after sectioning, digital autoradiography was performed for two hours. After complete decay of ¹⁸F, digital autoradiography was again performed for two hours. The resulting image–representing ¹²⁵I activity–was subtracted from the first image resulting in the “true” ¹⁸F distribution. Digital autoradiography was again performed for 24 h to capture the ¹²⁵I distribution. To account for the disparate exposure times, tracer activity in the tumor was normalized by dividing it by the mean tracer activity in healthy muscle tissue; after subtracting mean background activity from both tumor and muscle. Henceforth the [¹⁸F]FAZA to muscle ratio will be denote “FAZA” and the [¹²⁵I]gluco-RGD to muscle ratio will be denoted “RGD”. See Picchio et al. (2008) for full details of the study. The purpose of this article is to present, in detail, the imaging model used to study the spatial relationship between hypoxia and angiogenesis.

We present a Bayesian model that accounts for both the spatial correlation (the spatial distribution of the tracers) as well as the correlation between tracers. To our knowledge, this is the first, fully Bayesian, imaging model that has been applied to dual autoradiographic images. We model the correlation between tracers with a mixture of bivariate normal distributions. The spatial correlation is modeled by a hidden Markov random field. The number of classes in the mixture is assumed to be unknown and is estimated via reversible jump Markov chain Monte Carlo (RJMCMC) simulation (Green, 1995) with a novel reallocation scheme utilizing the Swendsen–Wang algorithm (Swendsen and Wang, 1987). Our main interest is not in directly estimating the number of classes, nor in segmentation, but rather in the distributional relationship between the two tracers. Thus, we marginalize over the number of classes in our posterior inference. Results are sensitive to the regularization parameter used in our prior model, thus this parameter is estimated from the data (however, it is commonly assumed known).

Image analysis crosses many fields of study including statistics, engineering and computer science, to name a few. As a result, a variety of terminology has been used in the literature. To clarify this exposition and make it accessible to a wide audience, we present the model and estimation procedure using set theory ideas and notation, which should be familiar to most researchers across the various fields of study. This notation will also aid in describing our novel method for reallocating pixels to classes: which is necessary for the RJMCMC proposals.

We begin in Section 2 by introducing the model and notation. Section 3 is devoted to detailing our estimation of the posterior, including a description of the Swendsen–Wang algorithm and the RJMCMC algorithm. Results from the motivating example are given in Section 4. A simulation study and sensitivity analysis are presented in the penultimate section. We wrap up the paper with a short discussion.