A novel Bayesian functional spatial partitioning method with application to prostate cancer lesion detection using MRI

Abstract

Spatial partitioning methods correct for nonstationarity in spatially related data by partitioning the space into regions of local stationarity. Existing spatial partitioning methods can only estimate linear partitioning boundaries. This is inadequate for detecting an arbitrarily shaped anomalous spatial region within a larger area. We propose a novel Bayesian functional spatial partitioning (BFSP) algorithm, which estimates closed curves that act as partitioning boundaries around anomalous regions of data with a distinct distribution or spatial process. Our method utilizes transitions between a fixed Cartesian and moving polar coordinate system to model the smooth boundary curves using functional estimation tools. Using adaptive Metropolis-Hastings, the BFSP algorithm simultaneously estimates the partitioning boundary and the parameters of the spatial distributions within each region. Through simulation we show that our method is robust to shape of the target zone and region-specific spatial processes. We illustrate our method through the detection of prostate cancer lesions using magnetic resonance imaging.

Keywords: biomedical imaging; functional estimation; spatial partitioning; spatial statistics.

Figure 1.

An example of how spatial locations are assigned given a boundary. A boundary B is shown in black. s = (x, y) is a spatial location, c = (x_c, y_c) is the centroid at the current iteration, θ is the angle between s and c, and f(θ) is the magnitude of the boundary at θ In this example, s is categorized as outside the target region: the distance from s to c is greater than f(θ).

Figure 2.

Distributions of sensitivity, specificity, and Dice coefficient of BFSP, BFSP Spline, CART, KM, BayesBD, and BayesImageS over all 600 simulated data sets.

Figure 3.

From top to bottom: average partitioning results of the BFSP, BFSP Spline, KM, CART, BayesBD, and BayesImageS methods in the simulation study. Each image represents 50 simulations of the same setting. The color represents the proportion of time the method classified each spatial location as within the target region. The black outline shows the true boundary.

Figure 4.

From left to right: Map of voxelwise predicted cancer probabilities from four sample slices (Jin et al., 2018), true cancer status, partitioning results from BFSP, CART, KM, BayesBD, BayesImageS, and Thresholding.