A Bayesian hidden Potts mixture model for analyzing lung cancer pathology images

Abstract

Digital pathology imaging of tumor tissues, which captures histological details in high resolution, is fast becoming a routine clinical procedure. Recent developments in deep-learning methods have enabled the identification, characterization, and classification of individual cells from pathology images analysis at a large scale. This creates new opportunities to study the spatial patterns of and interactions among different types of cells. Reliable statistical approaches to modeling such spatial patterns and interactions can provide insight into tumor progression and shed light on the biological mechanisms of cancer. In this article, we consider the problem of modeling a pathology image with irregular locations of three different types of cells: lymphocyte, stromal, and tumor cells. We propose a novel Bayesian hierarchical model, which incorporates a hidden Potts model to project the irregularly distributed cells to a square lattice and a Markov random field prior model to identify regions in a heterogeneous pathology image. The model allows us to quantify the interactions between different types of cells, some of which are clinically meaningful. We use Markov chain Monte Carlo sampling techniques, combined with a double Metropolis-Hastings algorithm, in order to simulate samples approximately from a distribution with an intractable normalizing constant. The proposed model was applied to the pathology images of $205$ lung cancer patients from the National Lung Screening trial, and the results show that the interaction strength between tumor and stromal cells predicts patient prognosis (P = $0.005$). This statistical methodology provides a new perspective for understanding the role of cell-cell interactions in cancer progression.

Keywords: Double Metropolis–Hastings; Hidden Potts model; Lung cancer; Markov random field; Mixture model; Pathology image; Potts model; Spatial point pattern.

Fig. 1.

(a and d) The observed cell distribution maps for two sample images from different patients, where lymphocyte, stromal, and tumor cells are marked in black, red, and green, respectively. (b and e) The estimated hidden spins in the -by- lattice. (c and f) The estimated AOI indicators by choosing (median model), where the blue indicates . For the first example, of which as shown in (c), , , , , , ; for the second example, of which as shown in (f), , , , , , . Note that in the bottom-left of (d), (e), and (f), the empty region is the alveolus.

Fig. 2.

Illustration of a -by- auxiliary lattice and the point emission process by different choices of . The empty points represent the observed cells and the filled points represent the hidden spins in the lattice. The square, circle, and triangle shapes stand for class , 2, and 3, respectively.

Fig. 3.

(a and d) The true maps of the -by- binary matrix for scenarios 1 and 2, respectively. Each vertex in the lattice is represented by either an empty square if or a filled square if . (b and e) The true maps of the -by- hidden spins from one dataset generated from scenarios 1 and 2, respectively. The black, red, and green colors stand for class , , and , respectively. (c and f) The observed cell distribution maps generated from the log Gaussian Cox process conditional on the hidden spins, as shown in (b and e), respectively.

Fig. 4.

The ROC curves on the posterior probabilities of inclusion on , in terms of the boxplots of TPRs under different FPRs, over datasets generated from each point process and each setting of . (a) Homogeneous Poisson process - scenario 1; (b) Homogeneous Poisson process - scenario 2; (c) Log Gaussian Cox process - scenario 1; (d) Log Gaussian Cox process - scenario 2.